Non-Stationary Photodetection Shot Noise in Frequency Combs: a Signal Processing Perspective

Non-stationary photodetection shot noise in frequency combs: a signal processing perspective

Thèse

Jean-Daniel Deschênes

Doctorat en génie électrique Philosophiae Doctor (Ph.D.)

Québec, Canada

Résumé Cette thèse examine le bruit de photon lors de la détection d’impulsions provenant d’un peigne de fréquences. En premier lieu, nous faisons abstraction du mécanisme physique produisant le bruit de photon, réduisant son effet à celui d’une source de bruit additif non-stationnaire (avec des statistiques variables dans le temps). Ce modèle de traitement de signal est ensuite utilisé dans l’analyse de deux expériences importantes pour l’utilisation d’un peigne de fréquence comme mécanisme de compteur de fréquence dans une horloge optique : la conversion du train d’impulsions optiques en un train d’impulsions électriques, et le battement hétérodyne entre un peigne de fréquences et un laser à onde continue.

Nous démontrons en premier lieu que le bruit de photon lié à la photodétection produit principalement du bruit d’amplitude, et une quantité presque négligeable de jigue aléatoire de temps sur le signal électrique détecté. Des résultats expérimentaux viennent confirmer nos prédictions théoriques. Nous explorons ensuite les limites de ce mécanisme en considérant la physique de la photodétection, ce qui révèle un étalement du temps de transit qui peut affecter la jigue aléatoire produite par cette conversion.

Dans un deuxième temps, nous démontrons que la nature pulsée du peigne de fréquences peut être utilisée pour donner un rapport signal-sur-bruit plus élevé que celui qui est prédit en considérant seulement le battement d’un seul mode du peigne avec le laser à onde continue. La première technique développée, le GATOR, rejette une grande partie du bruit de photon produit par le laser à onde continue afin d’améliorer le rapport signal-sur-bruit lorsque la puissance du peigne est faible. Avec cette technique, nous démontrons un rapport signal-sur- bruit 100 fois plus élevé que la limite en admettant l’utilisation d’un seul mode. Nous démontrons ensuite un raffinement de cette technique qui utilise le glissement de fréquence de l’impulsion optique afin d’utiliser efficacement tous les photons du peigne dans une bande passante déterminée. Cette technique nous a permis de produire un battement avec le plus grand rapport signal-sur- bruit parmi les résultats dans la littérature, 68.3 dB, obtenu en normalisant dans une bande passante commune de 100 kHz.

iii

Abstract This thesis is a study of shot noise in the photodetection of pulses from a frequency comb. We first make abstraction of the physical mechanism of shot noise to reduce its effects to that of an additive, non-stationary (meaning with time-varying statistics) noise source. This signal processing model is then used to analyze two experiments of importance for the operation of optical clockwork based on frequency combs: the conversion of the optical pulse train into an electrical pulse train by a photodetector, and the heterodyne (or beating) experiment between a frequency comb and a continuous wave laser.

For the detection of the optical pulse train, we show that photodetection shot noise yields mostly amplitude noise and vanishingly low timing jitter on the electrical signal. Experimental results confirm our theoretical predictions. We then explore the limits of this jitter when considering practical photodetection physics. This reveals a transit time spread parameter that can affect the jitter produced by this conversion.

Next, we turn our attention to the heterodyne experiment. We show that the pulsed nature of the frequency comb can be exploited in different schemes to yield higher signal-to-noise ratio (SNR) that is predicted by the use of the beating of a single comb mode with the continuous wave laser. The first technique that we develop, the GATOR, gates out shot noise from the continuous wave, and improves the SNR in the case of low comb power. Using this technique, we have demonstrated a factor of 100 higher SNR than the single-mode limit. We then show a further refinement of the technique which uses chirping of the optical pulse to effectively use all the available photons from the comb in a given bandwidth. This technique enabled us to produce the beating with the highest SNR reported in the literature of 68.3 dB, when normalizing to the common detection bandwidth of 100 kHz.

Table of Contents Résumé ...... iii Abstract ...... v Table of Contents ...... vii List of Tables ...... ix List of Figures ...... xi Abbreviations and acronyms ...... xvii Remerciements ...... xix 1 Introduction ...... 1 1.1 Motivation ...... 1 1.2 Methodology and scope of this thesis ...... 3 1.2.1 We deal with photodetection shot noise ...... 3 1.2.2 Distinction from noise from the laser and photodetection shot noise . 4 1.2.3 Semi-classical model ...... 4 1.2.4 Two main thrusts: Shot noise in electrical clock output, shot noise in comb-CW beat ...... 6 1.3 What is shot noise, what is noise stationarity and why should we care? .. 7 1.3.1 Shot noise definition ...... 7 1.3.2 Noise stationarity ...... 12 2 Shot noise in the detection of the repetition rate: operation as output of an optical clock ...... 15 2.1.1 Timing jitter on a single pulse due to photodetection shot noise ...... 15 2.1.2 Phase noise and timing jitter on a sine wave ...... 17 2.1.3 Quadrature splitting ratios in the literature ...... 26 2.1.4 Experimental results ...... 29 2.1.5 Additional factors influencing the I-Q noise split ...... 35 2.1.6 Conclusion and outlook ...... 48 3 The effect of shot noise in a comb-CW heterodyne experiment: gated optical noise reduction and chirped pulse heterodyne ...... 49 3.1 SNR limit for a single comb mode beating with a CW laser ...... 50 3.2 Gated optical noise reduction (GATOR) ...... 53 3.2.1 The gated optical noise reduction concept in the time domain ...... 53 3.2.2 The GATOR concept in the frequency domain ...... 57 3.2.3 Experimental results ...... 59 3.3 Optimal detection bandwidth for GATOR ...... 69 3.3.1 Interpretation of the theoretical result ...... 78 3.4 Chirped pulse heterodyne...... 79 3.4.1 Case of a single chirped pulse beating with a CW laser ...... 81 3.4.2 Case of a chirped pulse train beating with a CW laser ...... 86 3.4.3 Interpretation of the results and observations ...... 87 3.4.4 Signal detection strategies ...... 89 3.4.5 Experimental results ...... 98 3.5 Comparison of beat SNRs in the literature ...... 107 3.6 Alternative filter topology for pulse stretching ...... 110

vii

Conclusion ...... 115 Bibliography ...... 117 Annex A - Deriving the SNR given in [REI99] ...... 125 Annex B - Using an adjustable beam splitter at the optimum point yields the same SNR as using a 50-50 beam splitter with a balanced photodetector. 127

viii

List of Tables Table 3-1 Definition of parameters for the derivation of the snr...... 70 Table 3-2 definition of Parameters for the derivation of the snr in the chirped pulse case ...... 82 Table 3-3 Comparison of the comb-CW beat SNRs reported in the literature. ... 108

List of Figures Figure 1-1 - High-level block diagram of one type of optical clock. The output of this particular clock is the electrical pulse train at the repetition rate of the comb. The long-term accuracy is given from the transition frequency of a cold atom, first transferred to a CW laser, then to the frequency comb. D1 is a photodetector which detects a heterodyne beat between the CW laser and the comb, while D2 detects the repetition rate signal, which in this example would be the useable output of the clock...... 2 Figure 1-2 - Noisy photocurrent due to a source of constant optical power. The straight, orange line is the expected photocurrent, the vertical arrows indicate the photon absorption events, and the wavy line shows the shape of the current as filtered by detection electronics, which follows the local density of photodetection events...... 10 Figure 1-3 - Photon arrivals produced by a pulsed optical source. Note that when there is no expected photon, the expected noise is also zero because we detect no photon. The noise is thus tightly concentrated around the pulse of optical power...... 11 Figure 1-4 - Example of stationary vs non-stationary white noise. A) is the time series of a realization of white, stationary noise, while C) is the time series of non-stationery, pulsed noise. B) and D) show the respective power spectrum. The average PSD (thick, horizontal black lines) of the two noises and thus their time-averaged variance are the same even though they have drastically different time-domain statistics. Also revealed in D) is the 1/T periodicity of the spectrum of the pulsed noise source...... 12 Figure 1-5 - Example of a slowly varying signal corrupted by the two noise sources of Figure 1-4. Blue: constant signal and the two noise sources, green: weighted version of the blue signal. Using the fact that the noise is partly non-stationary allows us to essentially remove the impact of one of the noise sources, increasing our SNR by almost a factor of two (2.78 dB), as shown in the spectrum...... 14 Figure 2-1 - Computing the in-phase and quadrature noise contributions from additive noise. On the left, white, stationary noise is shown to split equally into nI and nQ since the statistics are completely unrelated to the reference cosine and sine templates. For short pulse photodetection shot noise on the other hand, the noise statistics are time-varying and related to the reference template. Most of the noise ends up in the in-phase noise (amplitude noise) function, and very little in the quadrature noise (phase noise)...... 20 Figure 2-2 - To illustrate the correlation between the shot noise at different harmonics of the repetition rate. On the left, the shot noise is multiplied by the first harmonic of the repetition rate as part of the process of decomposing the noise into its different constituents, while on the right the same noise realization is multiplied by the second harmonic. It can be seen that as long as the shot noise burst is short compared to the signal template, the noise at every harmonic is multiplied by the same value for the in-phase noise, while it is multiplied by a linear ramp for the quadrature noise. The waveform of xi

the noise is thus the same for all harmonics: in-phase noise is the same for all harmonics, while quadrature noise has a simple gain linking the noise realization in each harmonic...... 30 Figure 2-3 - Experimental setup used to measure the shot noise level in both signal quadratures. PC: Polarization controller, PBS: Polarization beam splitter, VOA: Variable optical attenuator, DL: optical delay line, BPF: Electrical bandpass filter. The experiment entails measuring the noise level while varying the optical power on the balanced photodiode D1, once without and once with the dispersive fiber spool used to stretch the pulses...... 31 Figure 2-4 - Experimental results showing the noise level as a function of the optical power on the balanced photodiode. The linear increase in noise spectral density validates that the noise is due to shot noise. The shot noise appears completely in the in-phase (amplitude noise) quadrature of the 100 MHz repetition rate signal, when the pulses are kept short, while when detecting long pulses (almost constant power), the shot noise splits almost equally in both quadratures. In the bottom graph, the symmetric vertical displacement about the center line is due to errors in the reference phase used for demodulation because of the lower SNR in the phase reference due to the longer pulses...... 33 Figure 2-5 - Two sidebands, oscillating with a positive and negative frequency offset, have to combine in a precise amplitude and phase relationship to yield purely A) amplitude noise and B) phase noise...... 36 Figure 2-6 - The current generated by an electron-hole pair depends on the depth of absorption of the photon. Most importantly, the center of mass of each current impulse is different for each depth. Based on [KAS01, chapter 5]. In this example, the holes drift speed is 0.7e5 m/s, while the electrons drift speed is 1e5 m/s, corresponding to the saturation velocities for silicon...... 44 Figure 2-7 - Average transit time and transit time spread for a P-I-N photodetector using parameters for Si (electrons speed = 1e5 m/s, holes speed = 0.7e5 m/s). The length of the detector was chosen to be equal to the penetration depth for 800 nm to yield reasonable quantum efficiency (63%). The absorption of Si at 1060 nm corresponds to the left limit of this figure at 1e3 m-1, while the absorption at 800 nm is 1e5 m-1 [ADA12], giving a transit time spread of around 10 ps for this range of wavelength. Lines: analytical result, dots: simulation results...... 47 Figure 3-1 - Illustrating the many modes of the comb's spectrum. The spacing between the modes is given by the repetition rate of the comb. The spacing between the mode and the comb bandwidth is not to scale in this figure, as usually there are tens to hundreds of thousands of modes in a typical comb’s spectrum...... 51 Figure 3-2 - The three experimental setups used in this section. Optical fibers are solid lines and dashed lines are electrical coaxial cable. BPF: Optical bandpass filter, PC: polarization controller, DL: Adjustable optical delay line, D: wideband photodetector, LPF: low-pass filter, LNA: Low-noise amplifier...... 54 Figure 3-3 - The GATOR processing chain in the all-software approach. Trace a) and b) are raw measurements recorded by the oscilloscope. xii

The vertical scale of each signal has been adjusted to clearly show all relevant features. The noise seen between the pulses on trace b) is shot noise from the CW laser. Trace c) is generated in software by multiplying a) and b) after adjusting the gate and the beat signal so that they overlap perfectly. Intuitively, from the b) trace, we would expect that the SNR of this beat should be very good as even visually, we can easily follow the phase of the beat signal. While the GATOR does indeed provide a clean sine wave from this signal, Figure 3-5 shows that simply filtering the beat signal as is conventionally done yields very poor SNR as all the inter-pulse noise is integrated into the resulting signal...... 57 Figure 3-4 - The spectrum of the wideband beat photocurrent contains many spectral copies of the beat signal and these can be averaged together to reduce the impact of shot noise from the CW laser. The bottom trace is simply a zoomed in version of the top trace, to clearly show the two spectral copies of the beat per repetition rate (100 MHz) span...... 58 Figure 3-5 - Comparison of the results using the all-software approach for the GATOR. The time-domain traces are both measured with a 50 MHz low-pass filter while the spectrum is computed using 173 kHz resolution bandwidth. The GATOR signal shows 20 dB improvement over the shot-noise limited conventional heterodyne beat...... 61 Figure 3-6 - Hardware real-time implementation of the GATOR using a mixer. Both time domain signals are shown using a 50 MHz low-pass filter, while the spectra are computed in 1.7 MHz of bandwidth. The narrowband beat (blue, upper trace) was measured by bypassing the mixer from the setup, and yields a 17 dB SNR, which is within 1 dB of the conventional shot noise limit. The GATOR signal (red, lower trace) was measured using the same photodiode, low-pass filter, RF amplifier and optical power levels and shows 35 dB of SNR, an improvement of 18 dB, or 63 on a linear scale. The beat frequency difference is due to CW laser drift between taking the two measurements...... 63 Figure 3-7 - Comparison results between a conventional heterodyne setup and a GATOR measuring the same frequency comb and CW laser beating together. The time-domain trace shows that the phase of the two beats agree within 100 mrad peak-to-peak over the full measurement duration, while the spectrum shows that the relative beat has better stability than 1 part in 1017 (1 mHz width over a carrier frequency of 194 THz). The green (dashed line) trace in the spectrum show the expected transform-limited shape of the window function used to compute the spectrum; highlighting the fact that the oscillating structure in the spectrum is due to the window function...... 66 Figure 3-8 - Modified Allan deviation of the GATOR signal (blue), the signal of the narrowband conventional heterodyne (green), the difference of the two (red). The red vertical lines indicate the +/- 1 sigma confidence interval for the red trace. The teal trace is the modified Allan deviation of the difference signal at longer timescales, while the purple trace shows this same signal after correcting for the phase mismatch of the low-pass filters. The black diagonal lines indicate theoretical t −1.5 behavior for white phase noise...... 67 xiii

Figure 3-9 - By varying the integration time T, the minimum amount of total shot noise is obtained when the amount from the comb and the CW laser are matched...... 76 Figure 3-10 - SNR improvement from GATOR vs slow detection. The required bandwidth axis shows the necessary optical and electrical bandwidth to achieve the SNR improvement in terms of the noise- equivalent bandwidth...... 79 Figure 3-11 - Illustrating the scaling between the chirped and unchirped pulses...... 83 Figure 3-12 - Illustrating the time-frequency representation of the chirped optical pulses from the comb as the blue diagonal lines indicating linear chirp, and the orange horizontal line depicting the CW laser signal...... 91 Figure 3-13 - Time-frequency representation of the chirped beat signal. Each chirp (blue diagonal line) is produced by the beating of the CW laser and a single pulse of the comb. The frequency axis has the same span as for the optical signal, but is now centered around zero...... 92 Figure 3-14 - Filtering the chirped beat signal (blue diagonal lines) with a chirp compensating filter yields a transform-limited pulse every Tr period (orange vertical lines) since the group delay of each frequency is now adjusted to be the same. This signal can then be gated to remove noise while keeping signal unaffected...... 93 Figure 3-15 - Multiplying the chirped beat signal (blue diagonal lines) with a repeating template of the complex conjugate of the chirp collapses the chirp to a single frequency (orange horizontal lines). Most of the energy in the signal is now concentrated into a single, constant frequency component. All that is left is the amplitude fluctuation of the beat signal, which can be averaged out in the usual manner by filtering...... 94 Figure 3-16 - Three different implementations of chirped pulse heterodyne. In A), the comb pulses are recorded on the oscilloscope to use as a gate. Optical fibers are solid lines and dashed lines are electrical coaxial cable. BPF: Optical bandpass filter, PC: polarization controller, DL: Adjustable optical delay line, D: wideband photodetector, LPF: low- pass filter, LNA: Low-noise amplifier. cFBG: Chirped fiber Bragg grating. h*(-t): Matched filter. B) and C) show two possible hardware implementations of the technique and the signal flow...... 95 Figure 3-17 - Fiber Bragg grating impulse response characterisation, shown in two different aspects ratio to enable easy visual comparison with the chirped pulse heterodyne results...... 100 Figure 3-18 - Raw chirped pulse heterodyne signal. On top: time-frequency representation and bottom: time-domain signal...... 101 Figure 3-19 - Chirped pulse heterodyne results after processing by the correlation approach. The bottom blue trace shows the signal after multiplication with the chirp template, while the green trace is the output result, low-pass filtered in the Nyquist range of the comb, 50 MHz...... 102 Figure 3-20 - Chirped pulse heterodyne results after processing by the matched filter approach. The bottom blue trace shows the signal after dechirping, the red dots show the gated/sampled points, while the green signal is the gated and low-pass filtered signal...... 103 xiv

Figure 3-21 - Spectrum of the chirped pulse heterodyne signal, before and after processing using the correlation approach. The processed spectrum reaches 56 dB in 1.7 MHz resolution bandwidth, a 9.5 dB of improvement over the single comb mode SNR limit. The raw beat spectrum shows indication of excess noise, most likely being ASE from the comb itself...... 103 Figure 3-22 - Left: phase noise extracted from both the chirped pulse heterodyne processed signal (green, offset for clarity) and a single beat harmonic without any other processing (blue), and their difference. The width of the blue trace is due to white additive noise and as we can see in the spectrum (right figure), the difference signal is completely dominated by the white noise level of the conventional heterodyne signal, while the chirped pulse heterodyne signal shows no sign of a white measurement noise floor up to the highest possible frequency of this measurement, limited by the repetition rate...... 105 Figure 3-23 - Modified Allan deviation of the chirped pulse heterodyne signal (green), conventional heterodyne signal (blue) and difference (red). The vertical red lines represent the estimated ± 1 sigma confidence interval for the red trace. The black diagonal line represents expected t −1.5 for additive white noise...... 106 Figure 3-24 - Comparison of beat SNR reported in the literature. A) Shows the SNR normalized in 100 kHz BW and thus also contains variation because of the repetition rate of the combs used. B) Normalizing the SNR by the bandwidth and the repetition rate allows a comparison which should only contain differences due to the experimental conditions (power levels, detection noise floor and technique)...... 109 Figure 3-25 - Experimental setup used to stretch the comb pulses with cascaded Mach-Zender interferometers. BPF: Optical bandpass filter, PC: Polarization controler...... 111 Figure 3-26 - Time-domain signal for a CMZI generated beat. Using a GATOR on the 8 pulse sets yield the 8 detected beats, overlaid as pale lines...... 111 Figure 3-27 - Spectrum of all 8 beat signals, and the coherent sum obtained after removing the phase difference compared to the first beat...... 112

Abbreviations and acronyms

AC Alternating current

ADC Analog-to-digital converter

AM Amplitude modulation

ASE Amplified spontaneous emission

CMZI Cascaded Mach-Zender interferometers

CW Continous wave dB decibel

DBM Double-balanced mixer

EHP Electron-hole pair

FBG Fiber Bragg grating

FPGA Field-programable gate array

FWHM Full-width at half maximum

GATOR Gated optical noise reduction

InGaAs Indium-Gallium-Arsenide

LNA Low-noise amplifier

MZI Mach-Zender interferometer

PM Phase modulation

PSD Power spectral density

RBW Resolution bandwidth

SNR Signal-to-noise ratio

STFT Short-time Fourier transform

xvii

Remerciements La conclusion de cette thèse représente pour moi la fin de la meilleure période de ma vie jusqu’à maintenant. Tout ce parcours n’aurait été possible sans la contribution de plusieurs personnes qui me sont chères.

En tout premier lieu, j’aimerais remercier mon directeur Jérôme Genest pour sa contribution inestimable à mon développement en tant que chercheur, mais aussi pour avoir étendu mes horizons par les multiples occasions de stages à l’étranger et conférences.

Je souhaite remercier Philippe Giaccari, qui a agi comme co-directeur pour ma maîtrise et lors mon premier stage. Il a été mon premier contact avec le domaine de la recherche, et par cette expérience positive qu’il m’a convaincu de poursuivre des études graduées.

Il me faut mentionner la contribution de mes collègues Sylvain Boudreau, Julien Roy et Simon Potvin pour les discussions stimulantes, autant sur des sujets techniques qu’humoristiques, qui a fait que ces trois années ont toujours été agréables.

J’aimerais aussi remercier mes parents, Lise et Claude Deschênes, qui m’ont soutenu tout au long de mes études, et particulièrement mon père, qui m’a donné la piqûre de la science et de la technologie dès mon plus jeune âge.

Finalement, j’aimerais remercier l’amour de ma vie, Olyvia Labbé, qui m’a accompagné au travers de tout ce processus, et m’a soutenu dans les moments plus difficiles, surtout dans les derniers mois. Elle est le pilier qui amène l’équilibre et rend surmontables même les pires épreuves. Finalement, je lui dois un grand merci pour la correction de la langue dans cette thèse.

Ces travaux ont été réalisés grâce au soutien financier du COPL, du FQRNT ainsi que du CRSNG, auxquels je suis infiniment reconnaissant.

xix

1 Introduction

1.1 Motivation

The measurement of time is such a fundamental operation that it is unsurprising that precise timing information is necessary for so many applications. Today, atomic clocks enable the global positioning system to provide location information for anyone equipped with a suitable receiver and provide safer long distance travel. This system operates using the measurement of propagation delays of radio signals emitted at precisely synchronized instants from satellites having known locations. High-bandwidth cellular networks need accurate clocks for multi-cell synchronization and efficient re-use of the bandwidth resource and use high accuracy Quartz oscillators slaved to the same radio signals emitted from the GPS system. The Internet network with its multi-gigabit backbone links needs low phase noise oscillators to provide the service which we now take for granted and keeps the world connected. These are just three of the ways that high precision oscillators are used in everyday life, but the fact is that high precision oscillators are necessary for many of the technologies that characterize the interconnected modern life, where distances are no barrier.

The next generation of clocks will be based on optical clocks [TAK05], because of two main advantages that they possess over radio frequency clocks. First, the high frequency of optical oscillation provides finer subdivisions of time, facilitating precision, and second, because cold isolated atoms with oscillation frequencies in the optical region provide the most reproducible systems available, facilitating accuracy.

Such an optical clock can be separated in two distinct parts, as shown in Figure 1-1: an optical oscillator which generates the accurate time markers in the oscillation of the electric field, and clockwork which produces a much slower electronic output signal that can be interfaced to other systems. While the development of the optical oscillator poses formidable challenges, we will limit our study to the clockwork part, which will be based on frequency comb technology.

Optical oscillator Optical clockwork

fcw control nfr control fceo control

Cold atom Frequency 1f-2f CW laser standard comb Interferometer

Optical clock output

Electrical clock output

Figure 1-1 - High-level block diagram of one type of optical clock. The output of this particular clock is the electrical pulse train at the repetition rate of the comb. The long-term accuracy is given from the transition frequency of a cold atom, first transferred to a CW laser, then to the frequency comb. D1 is a photodetector which detects a heterodyne beat between the CW laser and the comb, while D2 detects the repetition rate signal, which in this example would be the useable output of the clock.

Frequency combs are now a ubiquitous tool in optical frequency metrology, either as part of an optical clock or as a high-precision optical frequency counter. At the most general level, a comb is a broadband optical pulse train with precise timing characteristics. This fact makes it a versatile tool for optical measurements, the same way that an electrical pulse train is a ubiquitous tool in electronics. We will focus on combs as part of optical clockworks, which is in our view the most impressive and promising of their applications. Indeed, these lasers and the techniques required to perform measurements of astonishingly high precision with them were the subject of the 2005 Nobel Prize in physics, awarded in part to John L. Hall and Theodor W. Hänsch. The advent of the frequency comb has rightfully been called a revolution in frequency metrology [HAN06], because it dramatically simplifies the task of counting optical frequencies and consequently the implementation of an optical clock. Previous schemes for counting optical frequencies involved complex frequency chains, taking up many rooms full of equipment, to coherently link the optical and electrical frequencies in discrete steps using harmonic generation, frequency mixing and phase-locked loops [HAL00]. With a suitable frequency comb, one can now directly compare an optical frequency to an electrical one in a single step

2 of much reduced complexity, a critical milestone in bringing the full potential of optical clocks to more applications. The simplification is to the point where it can be expected that optical clocks will one day be common place equipment, rather than an epic endeavor that only national metrology labs can maintain.

Figure 1-1 shows a high level diagram of one type of optical clock using a comb to generate an electrical output, such as the sort depicted in [HOL01]. The optical oscillator consists of a very narrow-linewidth continuous wave (CW) laser, locked onto a transition frequency of a trapped cold atom. The clockwork consists of a comb which has one of its mode stabilized to the CW laser fcw based on the signal at detector D1, and its carrier-to-envelope offset frequency fceo independently cancelled [JON00], yielding the relation fcw = nfr between the n-th comb’s mode, its repetition rate fr and the CW laser. In this way, the output of detector D2 labeled electrical clock output contains a pulse train at the repetition frequency of the comb, which is coherently linked to the cold atom’s transition frequency. Such a scheme is often called an optical frequency divider [QUI11], since the electrical output is effectively the divided optical frequency fr = fcw/n and it is this very large division ratio (n ≈106) between the optical and electrical frequencies which facilitates the precision of such a clock. This thesis will focus on the photodetection process in the comb-heterodyne experiment at detector D1 and the conversion of the optical pulse train into an electrical output at detector D2.

1.2 Methodology and scope of this thesis

1.2.1 We deal with photodetection shot noise

While many types of noise and imperfections can affect the operation of optical clockworks based on frequency combs, some are more fundamental than others, in terms of being intrinsic to the physical system used, and not some technological limitation. Although some technical noises can have very severe impact on combs applications, we will study the effect of shot noise, as it is arguably the most fundamental type of noise encountered in many combs applications, including optical clockwork. We also believe that there is much to be done to improve frequency comb systems to make the most of each situation in the presence of shot noise.

1.2.2 Distinction from noise from the laser and photodetection shot noise

More specifically, we will focus on the effect of shot noise in the photodetection process, and neglect any type of noise that comes from the actual pulses and would be present even if the photodetection process were continuous. Note that the noise imprinted on the optical pulses also contains quantum noise which arises from the generation and amplification of optical radiation inside or outside the laser cavity, but there is a wealth of research that has been put into the effect of such noise and how this relates to the noise parameters on the resulting pulse train [AME03, PAS04, ABL06, WAH08, PRO09]. Thus, we will develop techniques which mitigate the impact of the fact that optical radiation is detected in quanta and models which capture the effect of such quantization of detection.

1.2.3 Semi-classical model

This thesis will work with a semi-classical model of shot noise, positioning itself as close as possible to the boundary between a quantum and a classical description of noise, while keeping with a completely phenomenological explanation of the phenomena. The emphasis will be on making the most out of frequency comb experiments using signal processing techniques, knowing that photon number fluctuations affect the photodetection process. As we will show, there is plenty that can be done to improve techniques and models at the shot noise limit even when considering such a simple model. This model makes only two assumptions: that light is detected in units of fundamental quanta called photons, each carrying an energy Ephoton = hf, where f is the frequency of the optical radiation and h is Planck’s constant, and that the arrival of each photon is independent [CAM10]. The second assumption rules out any so-called non- classical light or squeezed states where correlations between individual photons can appear [CAV81, XIA87, LI99, GLO04], but the first assumption still allows for time-variations and correlations in the arrival rate, which is related to the classical power in a light beam by Rate(t) = P(t)/Ephoton.

The subtle distinction between correlations in the arrival of individual photons (which are not modeled by this semi-classical model) and correlations in the arrival rate (which are completely modeled by a semi-classical model) can be the source of confusion, and it is sometimes hard to distinguish whether one

4 particular experiment deals with one or the other because of imprecise nomenclature, particularly in the older literature [see for example HAN56, which deals exclusively with correlations in the arrival rate]. This distinction is however important because as we will show here, frequency combs techniques have plenty of room for improvement simply by exploiting the correlations in the classically- computed rates.

1.2.3.1 The shot noise limit is already very low; we look towards the future.

The conventional shot noise limit is usually technically challenging to reach in an experiment, and the system used usually needs to be very clean and healthy before shot noise starts to become an issue. Regardless, we look towards the future needs by developing signal processing models and techniques which achieve better performance than is commonly accepted, even within the framework of the semi-classical model. Thus, most of the situations and techniques presented in this thesis deal with even lower levels of noise than the conventional shot-noise limit and as such, might be unnecessary for many applications, but we believe that there is intrinsic value in developing models and techniques to push the theoretical limits of what can be accomplished, at least in the interest of future-proofing the technology of frequency combs. Indeed, in the field of frequency transfer, Foreman states [FOR07]:

“The need for highly linear, high-current photodetectors is clear, since current performance levels for transferring frequency references over ﬁber networks are rapidly approaching the fundamental shot noise limit associated with milliwatt-level optical powers. When future optical clocks deliver the promised instability of < 1x10-16 at 1 s, we will urgently need such advances to enable accurate transfer.”

This provides at least one example of a field where performance which is considered state-of-the-art at one point quickly gets superseded to satisfy the growing demands of applications. Frequency combs are championing precision measurements in many domains such as frequency metrology and spectroscopy, where every noise source is a potential problem and will benefit from lowering their fundamental limitations.

1.2.4 Two main thrusts: Shot noise in electrical clock output, shot noise in comb-CW beat

We will thus give a description of the photodetection shot noise produced by a frequency comb, and relate this to its effects on two important parts of optical clockworks. The first part involves the extraction of the electric clock signal represented in the repetition rate of the comb, such as by D2 in our example optical clock of Figure 1-1, while the second concerns the heterodyne beating between a comb and a continuous wave laser, which is used to lock a comb mode to the optical oscillator as in D1, or simply to count an optical frequency. The relative depth of the two sections should underline the fact that significantly more of our research time was spent on the comb-CW heterodyne experiment than on the detection of repetition rate. Indeed, the emphasis of our studies was placed on the heterodyne experiment, but we still thought worthwhile to include our results on the detection of the repetition rate as it ties nicely into the theme and while our results are not fully original, they validate cutting edge results from another group1.

We will first show that the electrical pulse train produced by a frequency comb in a photodetector can yield much better jitter performance than would be naively assumed from stationary shot noise statistics. There is significant confusion in the literature as to the effect of photodetection shot noise on the phase noise of the photodetected comb repetition rate and only recently has it been demonstrated [QUI12, QUI13a, QUI13b] that this shot noise produces mostly amplitude noise and orders of magnitude less phase noise. Interestingly, this effect was theoretically predicted by Paschotta [PAS04] and the mechanics for the conversion of shot noise into only amplitude noise was understood in tangentially related domains well before [BRU68, NIE91, GAR06], but only in 2012 was the link between the two made by Quinlan and the effect clearly demonstrated for the case of detecting the repetition rate of a comb. Here, we will present a temporal model which explains the same phenomena and show experimental validation of

1 The initial goal of this work was not merely to replicate previously published results. However, we were unaware of Quinlan’s 2012 conference proceedings on the subject and obtained our results before seeing their paper in Nature Photonics in 2013. Since we used different techniques to validate the same phenomena, we think it worthwhile to report here, at the very least as an independent validation of their results. 6 our results. We will then give a theoretical explanation and simulation of the limits to the correlation of this shot noise due to different factors, most importantly due to the photodetection physics.

The second part and the main result of this thesis is a family of techniques for overcoming the single-mode shot noise signal to noise limit in a heterodyne experiment between a comb and a CW laser. Optical heterodyning is a very mature field, particularly in the case of CW-CW beats. Indeed, the experiment is almost as old as the laser itself [FOR61]; it was almost immediately apparent after the invention of the laser that the RF technique of the superheterodyne receiver could be applied to measure optical fields [JAV62]. At a fundamental level, the very high photon flux from the laser can be identified as the reason for their success in the most precise physical measurements [HAL00]. While comb technology already allows for very high precision, mostly at longer time scales, they still offer significant challenges before arriving at the optimal solutions for the new problems that they pose. Indeed, the current comb-CW heterodyne technology makes inefficient use of the available photons from the comb: oftentimes only 1 photon in 105-106 is used. The techniques presented in chapter 3 aim to improve the efficiency of comb-CW heterodyne technology to enable higher precision measurements and/or shorter time scales.

1.3 What is shot noise, what is noise stationarity and why should we care?

In this section, we will introduce the simple model of shot noise that we will use throughout this thesis and provide motivation for studying more in detail the characteristic of non-stationary noise arising from photodetection shot noise produced by a comb.

1.3.1 Shot noise definition

Photodetection shot noise is the deviation in the detected photocurrent from the expected value due to the random arrival of photons. As stated previously, the absorption of optical power from the photodetector material is known to happen in quanta called photons. Although the optical power can be assumed to follow laws of classical electromagnetism (Maxwell’s equations), photodetection has to 7 involve elements of quantum mechanics. For our purposes, we will only consider the quantized detection events, each carrying Ephoton = hf of energy, and assume that these events are independent, that is, detection of one photon does not affect the probability of detection in the next time interval. These two assumptions lead to the description of shot noise in terms of a stochastic process known as a Poisson points process [PAP02]. This process models any situation in which discrete events are known to happen at random at a certain rate, with no correlation between each event. Under these statistics, the probability that we detect n events in a time interval ∆t is equal to:

n (λ∆t) Pr{ n events in ∆ t} = eλ∆t , (1.1) n! where λ is the expected event rate during this interval. By construction, the expected number of events in a time interval is equal to the rate of events times the duration of the interval, that is:

En{ } =λ ∆= t N, (1.2) where N is the expected number of events. Less trivially, the variance of the detected number is also equal to the expected number of events:

σ2=En{ 22} − µλ = ∆= t N, (1.3) which highlights the important fact that the standard deviation of the expected arrival rate scales with the square root of the average rate. That is, if we expect N events, we should also expect fluctuations about this value of ± N . To relate this to photodetection, we only need to relate the number of detected photons to the energy transferred by a source of optical power Pexpected. From the definition of the optical power Pexpected as the rate of flow of energy E as a function of time, we know that:

∆E = Pexpected ∆= t NE ⋅ photon (1.4) N= Pexpected ∆ tEphoton .

As we define shot noise as the deviation in detected power from the expected value, we will subtract the expected value of the number of photons to isolate the deviations. The resulting process is what we call shot noise, and is now for each

8 time interval a zero-mean random variable with variance N. Furthermore, for values of N above approximately 10, which is the desired regime of operation of most techniques presented here, the probability density function is well approximated by a Gaussian with the same variance [PAP02] and we no longer need to think in terms of a Poisson distribution. So in essence, shot noise can be reduced to an additive noise source with similar characteristics to thermal noise in electronics for example, but with a variance that is controlled by the optical power that we are detecting; or equivalently a sort of multiplicative noise with respect to the optical power.

Finally, practical photodetectors do not detect each incoming photons but instead are characterized by a certain probability h of detecting each photon called the quantum efficiency, reducing both the expected number of photons and variance to hN instead of N . This efficiency includes many effects of differing nature, all leading to the fact that the actual detection rate is lower than the maximum rate due to the transfer of energy from the source.

From these basic facts about shot noise, we can make one deceptively simple statement: If you are not expecting any photon, then you also do not expect any fluctuation. This is the basic fact about shot noise which enables all the material presented in this thesis. The rest merely deals with ensuring that the desired signal and shot noise get into each other’s way as little as possible.

1.3.1.1 Continuous expected power

If we are detecting power coming from a source that we assume to be constant as a function of time, the expected number of photons in each interval is constant, and we can simply apply the probability density function of equation (1.1) without any regard to time. Thus the variance is the same at each time, and the shot noise in each time interval is uncorrelated to any other time interval due to the assumption of independence. This process is called a stationary process because all its describing statistics (the joint probability distribution) are unchanged by a shift in the time origin. Another property of this noise source is that the autocorrelation function is composed of a Dirac-like pulse around zero lag, because of the assumption of independence between photodetection events. Any time-domain correlation of the power and thus spectral non-uniformity is due to 9 the detection by a physical detector and subsequent filtering of the resulting photocurrent. The photocurrent produced by such a source would look somewhat like shown in Figure 1-2, which shows the independent photodetection events as Dirac pulses and the resulting noisy photocurrent after filtering by a detector. We have thus produced a source of white noise, which is described by a single-sided power spectral density (PSD) equal to [GEN00, section 6.3.2]:

Sf() =⋅⋅2 λ A2 , (1.5) where λ is the average event rate, and A is the area under the curve of each event. Depending on the chosen units, we have either in terms of optical power:

λh= PEphoton

AE= photon (1.6)

Sp () f= 2,h PEphoton or in terms of the electrical current produced by this power:

λ = Iq Aq= (1.7)

SI () f= 2, qI where q is the charge of an electron and I is the average current produced by the detected power.

Photocurrent(t)

Time

Figure 1-2 - Noisy photocurrent due to a source of constant optical power. The straight, orange line is the expected photocurrent, the vertical arrows indicate the photon absorption events, and the wavy line shows the shape of the current as filtered by detection electronics, which follows the local density of photodetection events.

1.3.1.2 Time-varying expected optical power

If the optical power varies as a function of time, we can still apply equation (1.1) [PAP02, WIN97], however we now need to include the time dependence of the expected number of photons. This time dependence can be found in the same way as in equation (1.4), by considering time intervals short enough so that the power is constant, or using the average of the power over that interval, leading to:

Nt()()= Pexpected t ∆ tEphoton . (1.8)

If the optical power is composed of a very short pulse, such as shown in Figure 1-3, the photon arrivals will be tightly concentrated around this pulse, with a time-varying average rate and variance:

Ent{}() =λ ()() t ∆= t Nt (1.9)

σ2()()t= En{} 22 t − µλ()()() t = t ∆= t Nt. (1.10)

This noise now has statistics which vary with time and is thus called non- stationary. When detecting a femtosecond pulse train from a frequency comb, the duty cycle of the optical radiation will be rather extreme, around 10-5 to 10-6, yielding very high concentration of the photons arrivals, and practically no shot noise in between the pulses. The various consequences of this extreme concentration will be exploited in this thesis to produce systems specifically tailored to the detection of optical radiation from a comb.

Photocurrent(t)

Time

Figure 1-3 - Photon arrivals produced by a pulsed optical source. Note that when there is no expected photon, the expected noise is also zero because we detect no photon. The noise is thus tightly concentrated around the pulse of optical power.

An interesting observation that we can make is that equations (1.6) and (1.7) still give the correct average power spectral densities for the time-averaged photodetection rate, even if the power is pulsed [WIN97, IVA03b]. The PSD is indeed still white and shaped only by the detection electronics’ transfer function. This however can lead to a deceptive picture of the situation by giving a false equivalence between such a pulsed noise source and one with constant variance.

1.3.2 Noise stationarity

The fact that the PSD for a pulsed noise source as produced by the shot noise from a comb is the same as the PSD for a source of constant power gives us a hint that the PSD captures only a subset of the characteristics of a noise source. Here, we will show two examples of noise with the same PSD, but different time- domain statistics, and show how a signal estimation system can perform better if it is designed with non-stationarity in mind. Two realizations of noise with the same average PSD are shown in Figure 1-4. As the level of the PSD is same, the variance of both noises when averaged over long time scales is the same; the time series for both noises reveal that the noise is simply concentrated into short bursts of noise for the non-stationary noise. This information is simply not contained in the average PSD, but can be seen in the spectrum of each particular realization of the noise. The time-domain repeating characteristics at every T period reveal themselves in the correlation between different spectral regions, with characteristic span 1/T [GAR86].

Figure 1-4 - Example of stationary vs non-stationary white noise. A) is the time series of a realization of white, stationary noise, while C) is the time series of non-stationery, pulsed noise. B) and D) show the respective power spectrum. The average PSD (thick, horizontal black lines) of the two noises and thus their time-averaged variance are the same even though they have drastically different time-domain statistics. Also revealed in D) is the 1/T periodicity of the spectrum of the pulsed noise source.

As an example of a case where we can obtain different performance when faced with non-stationary noise, we will consider that the two noises shown in Figure 1-4 are superimposed together onto a desired slowly varying signal which has a lower bandwidth than 1/T, that is, changes slowly compared to the noise modulation period T. A small portion of such a signal is shown in Figure 1-5, where the two noises sit on an average value around 10, which we want to estimate. If we neglect that fact that one of our noise sources is non-stationary, we could look at the spectrum of our signal, and design a filter with the desired bandwidth to capture all the signal information, and simply accept that we have a certain noise level that corrupts our signal, shown here at 0 dB. However, looking at our time-domain signal, we could decide to first apply a weighing function to our points before applying the filter; putting less weigh on the points that contain a lot more variance than the others. In fact, in a situation like this where the signal is slowly varying and the expected signal amplitude is the same for all points, the weighing which optimizes the signal-to-noise ratio is called the inverse-variance weighing, where each point gets a weigh that is inversely proportional to its variance [HAR11]. Applying this weighing to the signal in blue yields the signal in green, where the points with high variance have been reduced to almost zero and thus have very little impact on our estimate of the signal amplitude. We have effectively suppressed one of the noise sources, while only slightly reducing our average signal amplitude. The spectrum shows the Fourier transform of both time-domain signals, with the signal level normalized to the same value. In the spectrum we can see that we have reduced the noise level and thus increased our SNR by 2.78 dB. Since the two noise sources were of equal PSD, the removal of the non-stationary noise in the time-domain signal gave a reduction in the noise floor of a factor of 2 (3 dB), while reducing the signal by 0.22 dB. We can now apply the same filtering that we would have if we had not applied the weighing to our signal. The peaks that can be seen around every harmonic of 1/T are caused by the signal mixing with the weighing function, but have no effect on the outcome provided that we can filter the result with a filter with less than half this bandwidth, as given by the sampling theorem [GEN00].

Figure 1-5 - Example of a slowly varying signal corrupted by the two noise sources of Figure 1-4. Blue: constant signal and the two noise sources, green: weighted version of the blue signal. Using the fact that the noise is partly non-stationary allows us to essentially remove the impact of one of the noise sources, increasing our SNR by almost a factor of two (2.78 dB), as shown in the spectrum.

This situation is similar to the one addressed by the GATOR in section 3.2, but is not exactly the same because in this case the signal itself is composed of pulses that sit on an offending noise pedestal. This example only goes to show that designing our systems with non-stationary noise in mind can sometimes offer ways to lessen its impact on performance. One fortunate case of non-stationary noise is the impact of photodetection shot noise on jitter in the detection of the repetition rate of a comb. In this case, no re-design of the experiment is necessary to reap the benefits afforded by non-stationary noise, and we show in the next section that if we incorrectly assume that shot noise from comb is stationary, we overestimate the timing jitter of pulse train by orders of magnitudes [QUI13a].

2 Shot noise in the detection of the repetition rate: operation as output of an optical clock

In this chapter, we will examine the timing jitter that can be expected because of the quantized nature of the photodetection process when detecting the pulse train from a short pulse laser such as a frequency comb. Since we are interested only in the jitter added by this process, we will consider the timing fluctuations from an ideal center of each optical pulse and neglect any fluctuation that the center of these pulses can contain before photodetection. We will first explore a very simple situation involving a single pulse, and then we will extend the process to a more practical situation where a system outputs a sine wave which oscillates at the repetition rate of the comb or one of its harmonics. For this situation, we will consider two very important types of noise, the first being additive, white stationary noise, and the second being photodetection shot noise. We will then explain how the timing jitter from the idealized situation relates to the non- stationary character of shot noise in an actual experiment, yielding mostly amplitude noise and much less phase noise than stationary noise of the same variance. An experimental validation of this unequal effect of shot noise on amplitude and phase noise will be presented and finally, we will explore how photodetection physics in a semi-conductor can preserve or affect these properties.

2.1.1 Timing jitter on a single pulse due to photodetection shot noise

2.1.1.1 Single photon received, square pulse

Let us first consider the simple situation where an optical pulse with a square shape in the time domain with width Tp is incident on a photodetector. Let’s assume that the energy in this pulse is such that this pulse transfers approximately one photon per pulse. Taking this one step further, let’s examine only the situation where we detect exactly one photon from this pulse. By construction, the optical pulse cannot transfer any energy outside of this Tp period, but the absorption event could have happened anywhere within this time period. We could thus say that the span of the timing jitter from this situation is

15 equal to Tp, which would give a lower bound on the jitter from this situation, completely neglecting any jitter that could be added by our detection apparatus. This jitter is solely due to the discrete nature of our detection, because if somehow we could detect the exact power envelope, we could instead imagine finding the center of the pulse to better than Tp accuracy, insofar as the signal to noise ratio would permit. If we repeat this experiment, we cannot guarantee that for each pulse, one and only photon will be detected and thus the timing jitter obtained from a practical experiment would be slightly different than that, but this simple situation offers very important insight into the photodetection of a pulse train from a short pulse laser.

2.1.1.2 N photons received, square pulse

If instead N photons are detected for the same pulse, assuming that the detection of the individual photons is independent, we could instead conceive of averaging over the detection time of the N different photons to yield a timing jitter that is a factor of N smaller2. Once again, as we cannot guarantee that exactly N photons will be detected for each pulse, the exact number of detected photons in each pulse will vary according to Poisson statistics, but if N is large enough, the number of photons will be relatively close to N, and the standard deviation of the timing jitter expected from each pulse should follow very closely this N scaling.

These simplistic arguments allow us to assert that the timing jitter that is solely due to the quantization of the photodetection process should support timing jitter of the order TNp , with the exact value depending on the shape of the optical

pulse, equal to a standard deviation of TNp 12 for a square temporal distribution of the pulse. Interestingly, a very similar argumentation was put forward by Paschotta [PAS04], which computed the variance of the timing jitter for a hyperbolic secant squared pulse shape such as produced by a soliton laser.

2 To make this argument more precise, we could instead consider the standard deviation of the time of 0.5 arrival of the photons, which for a square probability density function of the time of detection would be Tp/12 , averaging to Tp/(12N)0.5. 16

2.1.2 Phase noise and timing jitter on a sine wave

In a practical situation where an optical pulsed source such as a comb is used to generate an electrical clock, the output is very often in the shape of a sine wave oscillating at the repetition rate of the comb or one of its harmonics, and the relevant timing jitter is often described in terms of the phase noise on this sine wave.

Time can be measured by the timing of a feature of the sine wave, and the location of the zero crossings is the most sensible choice, as the signal has more variation around those. If a sine wave at nominal frequency f0 and amplitude A0 is measured around its zero crossing with an additive perturbation nt( ) , the

measurement can be written as A0 sin( 2π f0 t) + n(t) . Assuming that nt( ) is almost constant around the true location of the zero crossing, in this case t = 0 , the observed location of the zero crossing will approximately be:

0= An0 sin( 2π ft0 ) + ( 0)

02≈+π An0 ft0 ( 0) (2.1) n(0) t ≈ . 2π A0 f0

That is to say, a small perturbation on the “y” value of the sine wave around the zero crossing is undistinguishable from a small perturbation on the time value t, leading to jitter on the detected time. The conversion factor is equal to the inverse of the slope of the sine wave. We can now examine more closely how such a perturbation can occur as the result of additive noise.

For small perturbations compared to the amplitude of the sine wave, and assuming that we place a bandpass filter around the desired frequency, we can refer to the well-known model of bandpass noise [TRE89] for describing the fluctuations caused by additive noise. Since we are concerned with the effect of additive noise on the signal (recall that our model of shot noise reduces to additive noise with a time-varying variance) we consider the signal to be a perfect sine wave at a certain frequency with no perturbation other than additive noise. Since the noise is bandpass filtered around our frequency, the noise is also oscillatory with approximately the same frequency, and it is instructive to

17 consider the decomposition of the noise into two functions, one oscillating exactly in phase with the signal, and another oscillating in quadrature. Assuming a perfect oscillating signal perturbed by additive noise, the model can be written as:

st( ) = A0 cos( 2π ft0 ) + nt( ) (2.2) st( ) =++ A0 cos( 2πft0) (nIQ( t)cos( 2 ππft 00) n( t)sin( 2ft)) , where A0 is the amplitude of the signal, f0 is the frequency of the oscillation, nt( )

is the additive noise, while ntI ( ) is called the baseband (slowly varying) in-phase noise and ntQ ( ) is the baseband quadrature noise. In this model, the in-phase can be seen to perturb the amplitude of the signal only:

nt( ) ππ+=+I π A00cos( 2 ft00) ntI ( )cos( 2 ft) A1 cos( 2 ft0 ) , (2.3) A0 while the quadrature noise can be seen to perturb the location of the zero crossings of the signal, or equivalently, the phase. This second fact is somewhat less trivial, but in essence it is because the sine function is at a maximum and almost constant around the location of the zero crossings of the cosine function, fulfilling the assumptions of the intuitive argument above. This can also be seen mathematically:

A nn Ancosφφ− sin =0  exp j φ + exp −− j φQQexp j φ + exp − j φ 0 ( ) Q ( ) ( ) ( ) ( ) ( ) 2 jA00jA A nn  =0 expjjφφ 1 +QQ +− exp jj  1 − ( )( )  2 AA00  A nn   ≈0 expjjφφ expQQ+− exp j exp  − j  (2.4) ( ) ( )   2 AA00   A  n n  ≈ 0 expj φ + Q +−expj φ + Q 2  A  A   0  0  n ≈ A cosφ + Q , 0   A0  where we have used the first order Taylor expansion exp( jx) ≈+ 1 jx . The phase in

units of radians is thus perturbed by a signal equal to nAQ 0 . Putting (2.3) and

(2.4) together while with neglecting the second-order effect that the amplitude fluctuations have on (2.4), we can rewrite (2.2) as:

nt( ) nt( ) st≈+ A 1I cos 2π ft+ j Q ( ) 0 0 . (2.5) A0 A0

The missing piece is now how to convert from any type of additive noise, say either white stationary noise, or photodetection shot noise, into the in-phase and quadrature noise ntI ( ) and ntQ ( ) . Fortunately, this conversion is very easy to apply, as it can be found by virtue of the orthogonality of the sine and cosine, simply by multiplying the noise function with both a cosine and a sine template, and temporally filtering to remove the oscillating terms at 2f0:

A0 ntI ( ) st( )cos( 2π ft0 ) =++oscillating terms around 2 f0 22 (2.6) ntQ ( ) s(tf)sin( 2π ft0 ) = + oscillating terms around 20 . 2

Figure 2-1 illustrates this process graphically with two important types of additive noise: white stationary noise and photodetection shot noise for a short pulse. These two types of noises are multiplied by both the cosine and sine template to separate the two noise quadratures. Clearly as all the characteristics of the stationary noise are by definition independent of time, the choice of a particular reference phase for our templates does not matter, and the statistics of the noise in both quadratures should be independent of our choice of phase. In other words, for white stationary noise, the in-phase and in-quadratures noise functions have the same statistics; the total additive noise is split equally in both quadratures and each get half the variance of the additive noise.

Figure 2-1 - Computing the in-phase and quadrature noise contributions from additive noise. On the left, white, stationary noise is shown to split equally into nI and nQ since the statistics are completely unrelated to the reference cosine and sine templates. For short pulse photodetection shot noise on the other hand, the noise statistics are time-varying and related to the reference template. Most of the noise ends up in the in-phase noise (amplitude noise) function, and very little in the quadrature noise (phase noise).

For the photodetection shot noise for a short pulse on the other hand, all the variance of the noise is concentrated around the maximum of the cosine template and most importantly, around the zero crossing of the sine template. For a very short pulse compared to the reference frequency, the cosine template is almost constant during the optical pulse and almost all the noise variance gets transferred to the in-phase noise function. On the other hand, the sine function being at a zero crossing around the noise burst produced by the signal, very little noise gets transferred from the additive noise to the quadrature noise function. From the equivalence of the in-phase noise with amplitude noise (also called amplitude modulation noise, or AM noise) and the quadrature noise with phase noise (similarly, sometimes called phase modulation noise, or PM noise), we arrive at the central result of this section: photodetection shot noise for a short pulse yields mostly amplitude noise and very little phase noise. This reconciles the view of shot noise as additive noise with the small predicted timing jitter from the time-of-arrival of the photons of section 2.1.1 and most importantly, provides a simple mechanic to explain that we can indeed expect low timing jitter from a practical experiment which uses a sine wave output from a comb as time markers.

To compute the transfer ratios of additive noise to the in-phase and quadrature noises for the case of shot noise from a short pulse laser, first assume that a single pulse produces a non-stationary noise source with time-varying variance

2 σ N (t) centered on the arrival time of the pulse at t = 0 . As explained in section

1.3.1.2, this variance will have the same shape as the optical power detected as a function of time due to randomness in photon arrivals. We want to compute the ratio of variance that exist in the in-phase quadrature to the ratio of the variance that exists in both quadratures, as observed through a bandpass filter which integrates over a much longer time than the pulse, centered around our signal harmonic at Mfr , where fr is the repetition rate of the comb. Since we know that we can compute the variance of the sum of many uncorrelated noise sources as the sum of the variances, we can apply this argument to the noise at different times, which for a continuous-time process means that the variance is proportional to the integral of the variance over time. Since we are concerned

21 only with the ratio of the variances, we can write without special attention to the normalization:

s 2 α ≡ I I 2 s T

Tr 2 s22π ∫ Nr(t)cos( 2 Mf t) dt − = Tr 2 (2.7) TTrr22 s2 2π+ s22π ∫∫N(t)cos( 2 Mf rN t) dt( t)sin( 2 Mf r t) dt −−TTrr22

Tr 2 s22π ∫ Nr(t)cos( 2 Mf t) dt − = Tr 2 , Tr 2 s 2 ∫ N (t) dt −Tr 2

2 where σ I is the variance of the in-phase noise observed if we repeat this

2 experiment many times, while σT is the variance observed in both quadratures.

Now considering the repetitive pulse train, since all the pulses have similar statistics, the variances ratio that we have computed apply to all pulses equally and furthermore, there is no longer any fast variation of the statistics due to the smoothing action of the bandpass filter. Equation (2.7) thus applies to all times equally and thus also describes the conversion ratio between the total observed

PSD ST and the PSD of the in-phase noise SI . A similar argument can be made for the quadrature noise, replacing the cosine template with a sine template:

Tr 2 s22(t)sin 2π Mf t dt 2 ∫ Nr( ) s − α ≡=Q Tr 2 Q 2 . (2.8) s Tr 2 T s 2 ∫ N (t) dt −Tr 2

These parameters are experimentally accessible as they simply describe the relative split of the PSD of additive noise into both quadratures, that is SI = α I ST and SQ = αQ ST . An interesting observation that can be made from these equations is that:

Tr 2 s22π ∫ Nr(t)sin( 2 Mf t) dt −Tr 2 αQ = Tr 2 s 2 ∫ N (t) dt −Tr 2

Tr 2 s22− π ∫ Nr(t)(1 cos( 2 Mf t)) dt − = Tr 2 Tr 2 (2.9) s 2 ∫ N (t) dt −Tr 2

Tr 2 s22π ∫ Nr(t)cos( 2 Mf t) dt − =1 − Tr 2 Tr 2 s 2 ∫ N (t) dt −Tr 2

=1, −αI which is to be expected, as the I-Q decomposition that we have applied conserves the power of the noise: any noise that does not end up in the in-phase noise will inevitably end up in the quadrature noise.

For example, for a pulse with rectangular shape in time of width Tp, equation (2.7) becomes:

Tp 2 2 p ∫ cos( 2 Mfr t) dt −Tp 2 α I = Tp 2 ∫ dt −Tp 2

T 2 1p  11 = + cos( 4p Mfr t) dt T ∫ 22 p −Tp 2

11Tp = + sin( 2p Mfrp T ) (2.10) p Tpr2 4 Mf 11 = + sin( 2p Mfrp T ) p 2 4 Mfrp T 1 =(1 + sinc( 2Mfrp T )) , 2 where we have used the normalized sinc function, defined as sinc( x) ≡ ( sin(ππxx)) ( ) .

The quadrature noise becomes:

ααQI=1 − 1 (2.11) =(1 − sinc( 2Mfrp T )) . 2

These results are exactly equal to that derived by Quinlan [QUI13b] using a fairly different frequency domain approach and accounting for the spectral correlation of the noise due to the non-stationarity [GAR86]. Of course, this spectral domain approach is equivalent to the temporal approach used here in terms of the predictions even though the path taken is quite different. As is often the case in science, these results were unbeknownst to us derived and published a few months before ours. In the limit of a long pulse compared to the harmonic considered, both conversion ratios converge to 12, which is the well-known result for a stationary noise source. For a short pulse compared to the period of the harmonic considered, that is 2Mfrp<< T , the sinc factor in (2.10) and (2.11) is almost equal to 1, and these can be approximated from their second order Taylor series expansion:

1 2 αQ≈ (2pMfrp T ) 12 (2.12)

ααIQ=1. −

As the qualitative explanation above predicted, these equations indicate a very uneven split between both quadratures; most noise will end up in-phase with the signal, producing amplitude noise, while only a small fraction of the noise will end up in quadrature with the signal, producing phase noise.

We will now show that this model of bandpass noise yields the same predictions as simply considering the random time-of-arrival of the photons such as done in section 2.1.1. Evaluating the timing jitter in this bandpass additive noise model is essentially an exercise in cascading the required conversion ratios. Collecting here the required conversions, we first have the conversion between phase noise to timing jitter for a given harmonic:

1 σσ22= t 2 ϕ . (2.13) (2π Mfr )

To get the phase fluctuations, we can base ourselves on equation (2.5), which provides the conversion ratio between quadrature noise and phase noise. For an

24 average signal amplitude I0 , the phase fluctuations are related to the quadrature noise fluctuations by:

2 22 σσϕ = Q I0 . (2.14)

The variance of the quadrature fluctuations is related to the PSD of the

2 quadrature noise and to the measurement bandwidth R, by σ QQ= SR. The PSD of the quadrature noise is related to the power spectral density of the noise by

SSQ= α QT, yielding:

2 σαQ= QTSR. (2.15)

The PSD of the shot noise is related to the average signal level by equation (1.7):

2 σαQQ= 2qI0 R . (2.16)

Putting (2.13), (2.14) and (2.16) together yields the timing jitter:

11 σα2 = 2qI R tQ22I 0 (2π Mfr ) 0 (2.17) 12qR = α . 2 Q I (2π Mfr ) 0

Equation (2.17) can be further simplified by (2.12):

2 112 2qR σt= (2pMfrp T ) 2 12 I (2p Mfr ) 0 (2.18) 122 qR = Tp . 12 I0

Recognizing that I0 2 qR can be expanded in the number of electrons integrated in the response time of the filter of bandwidth R , (2.18) is exactly equal to the result obtained in section 2.1.1.2:

2 2 1 Tp σ t = , (2.19) 12 Ne with the only differences being that the number of electrons is used instead of the number of photons, which are related by the quantum efficiency of the photodetector.

Another interesting fact that can be gathered from this equation is that the timing jitter variance due to shot noise is independent of the harmonic chosen to demodulate the pulse train. This is because the higher sensitivity of higher harmonics to timing jitter is exactly compensated by the increase in the splitting ratio of shot noise into quadrature noise. Of course, the higher harmonics retain their higher immunity to stationary additive noise and in this respect do offer an improvement over lower harmonics as a clock signal.

2.1.3 Quadrature splitting ratios in the literature

As the types of additive noise encountered in practice are usually stationary, it is customary to assume that this noise splits equally in both quadratures when trying to estimate the effect of additive noise on the amplitude and the phase of a signal. Indeed, the literature is rife with papers which simply assume this equal split in both quadratures to estimate the amount of phase noise or timing jitter to expect from a given level of photodetection shot noise [IVA01, HOL01, IVA02, IVA03a, IVA03b, IVA07, FOR07, NAK10, FOR11, FOR12, HAB11, HEI11, MCF05, OZD01, QUI09, QUI11, SCH11, STU10, ZHA12]. Only a minority of these papers even discuss the assumption of equal split in both quadratures.

To understand the magnitude of the error committed when assuming an equal split of the shot noise in both quadratures when using a frequency comb, let us consider an example with reasonable parameters: an average photocurrent of 1 mA produced by a pulse train of 100 fs pulses at 100 MHz repetition rate. The shot noise PSD is:

S= 2 qI T (2.20) =3.2 × 10−22 A Hz , while the PSD of the quadrature noise is:

SQQ=α 2 qI 1 2 = (22p Mfrp T) qI (2.21) 12 =M 2 ⋅×1.1 10−31 A Hz .

The ratio of the noise PSD predicted by the equal split model to the actual noise PSD in quadrature with the signal is:

11S⋅ 2 qI 22T = SQQα 2 qI 1 = (2.22) 2αQ =M 29 ⋅×1.5 10 .

Note that this ratio is independent of the signal level, but the individual PSDs are. Assuming that we detect the first harmonic of the repetition rate, that is a misprediction by almost 9 orders of magnitude. In practice, clock setups usually measure a high harmonic of the repetition rate, say M = 100 , so that the error is somewhat smaller, but still many orders of magnitude.

An interesting fact is that Paschotta [PAS04] had correctly predicted the detection quantum noise-limited timing jitter on a short pulse train back in 2004, but it seems to have been somewhat ignored by the scientific community, perhaps because the link between this very low timing jitter and the much higher shot noise PSD in the form of the splitting ratio between the in-phase and quadrature noise was left unexplored.

There are probably several reasons that contribute to the fact that so many papers use an incorrect formula for the phase noise prediction. First of all, working at or beyond the shot noise limit is a taxing requirement for any experiment and this makes it non-trivial to measure precisely; a noise that is predicted to be 70 to 90 dB below the standard shot noise limit will undeniably be very hard to study. Indeed, for many years papers have dealt with more pressing issues which yielded noise floors much higher than the shot noise, such as beam- pointing noise [IVA02] and non-linear AM-to-PM conversion in photodiodes [DID09]. A second factor which probably contributed to the spread of this wrong assumption is that the nice and smooth average power spectrum of noise as usually envisioned lacks any mechanism for accounting for non-stationary noise and this can make the effects of this nonstationarity mysterious. Indeed, to account for such time-varying noise sources, the less well-known cross-spectrum needs to be considered when making predictions based on the spectral domain [GAR86]. Interestingly, for a cyclo-stationary noise source (periodically varying statistics like an optical pulse train producing shot noise), the spectrum of a realization of the noise can give an accurate picture to understand the spectral

27 correlations, but this quantity is very seldom looked at, partly because for conventional (stationary) noise sources this spectrum contains no additional information compared to an average power spectrum, and partly because usual laboratory tools used to measure spectra (electrical or optical spectrum analyzers) go out of their way to show their users nice and smooth, time-averaged power spectra. Fortunately for our case, the time domain yields very intuitive insight into the nature of the effects of non-stationarity, chiefly because unsurprisingly the time domain is the natural domain in which time-varying phenomena is described. This will become a recurring theme in this thesis, where explanations of different phenomena will take on a very simple form in the time domain, and the explanation in the frequency domain will be more contrived. Of course, it remains best to understand a single situation from as many angles as possible in order to have a more complete picture and as such, we will try to give explanations in both domains as much as possible.

This non-equal splitting ratio was then explored thoroughly starting from 2012 by Quinlan and co-workers at NIST, when they published their first measurement of timing jitter below the level of equal shot noise split, proving the theoretical effect [QUI12]. In 2013, they revisited this subject with a thorough theoretical paper [QUI13b] which explained the effect in the frequency domain using the cross- spectrum concept, accounting for spectral correlation in the noise. This model predicts the same result as our temporal model presented above and spectral correlation between noise sidebands can be shown using simple modulation theory to imply unequal split between the in-phase and quadrature noise and vice versa. Quinlan and co-workers have also published a seminal paper in Nature Photonics in 2013 showing compelling results with actual clock phase noise measured to be 5 dB below the commonly accepted shot noise level, assuming equal split of the noise in both quadratures [QUI13a].

2.1.3.1 Shot noise is highly correlated in all harmonics

As a side result of examining the conversion of the shot noise into the two quadratures at different harmonics, one can notice that the noise in each harmonics will be highly correlated. Indeed, for a very short pulse compared to the repetition period, the conversion of nt( ) into ntI ( ) involves multiplication by

28 cos( 2π Mfr t) ≈ 1, which is mostly the same value for each harmonic. This means that all the amplitude noises detected in the different harmonics will be equal.

For the quadrature noise, the template is instead of the form sin( 2ππMfrr t) ≈ 2 Mf t , which is a simple linear ramp for all the harmonics. Indeed, the only difference between each harmonic is a gain equal to the harmonic number, but the baseband noise realization will have the same shape. This process is also shown graphically in Figure 2-2. As a consequence of this, we can state that trying to average out shot noise by simultaneously using the signal from many harmonics is useless in this situation. Of course, as higher harmonics are more sensitive to timing noise, the corresponding effect of stationary additive noise such as thermal noise will be lesser, and the measurement of the timing at a higher harmonic is more sensitive than at lower ones with respect to additive noise from the electronics. Thus averaging the information from many harmonics together will give same variance if shot noise dominates the measurement, but the signal to noise performance will improve if thermal noise is the dominating process. Note that this is a different observation than the one made at the end of section 2.1.2, because here we assert that the realization of the noise in each harmonic will be the same (they are highly correlated), while previously we simply computed that the variance of the noise is the same for each.

2.1.4 Experimental results

We have designed an experiment specifically to isolate photodetection shot noise from other noise sources, and show that this type of noise is in phase with the repetition rate signal and thus produces almost only amplitude noise and vanishingly low phase noise.

Figure 2-2 - To illustrate the correlation between the shot noise at different harmonics of the repetition rate. On the left, the shot noise is multiplied by the first harmonic of the repetition rate as part of the process of decomposing the noise into its different constituents, while on the right the same noise realization is multiplied by the second harmonic. It can be seen that as long as the shot noise burst is short compared to the signal template, the noise at every harmonic is multiplied by the same value for the in-phase noise, while it is multiplied by a linear ramp for the quadrature noise. The waveform of the noise is thus the same for all harmonics: in-phase noise is the same for all harmonics, while quadrature noise has a simple gain linking the noise realization in each harmonic.

It is very challenging to get a shot-noise limited measurement of the repetition rate signal: the required dynamic range is huge and this translates to very stringent requirements on the linearity and noise properties of the photodetector and associated electronics. It is even much harder to have electronics which have noise much below the shot noise limit as we intend to measure; some groups have used custom photodetectors specifically designed for their high dynamic range for this reason [QUI12, FOR13]. Rather than measuring the full repetition rate signal at the same time as the noise, we instead opted to use differential optical detection to get rid of the repetition rate signal, leaving only the shot noise to be handled by the electronic amplifiers. The setup is depicted in Figure 2-3.

Figure 2-3 - Experimental setup used to measure the shot noise level in both signal quadratures. PC: Polarization controller, PBS: Polarization beam splitter, VOA: Variable optical attenuator, DL: optical delay line, BPF: Electrical bandpass filter. The experiment entails measuring the noise level while varying the optical power on the balanced photodiode D1, once without and once with the dispersive fiber spool used to stretch the pulses.

The signal from a frequency comb producing 100 fs pulses at 100 MHz was sent to a 50-50 fiber coupler to generate two identical copies of the pulses, including any amplitude and phase noise already present on the pulses. The polarization controller and polarizing beam splitter were used as a precaution to remove any potential sensitivity of the experiment to polarization noise. The variable attenuator VOA2 and the variable delay line DL are used to very carefully match the amplitude and the phase of the detected signals to maximize rejection. These two pulses are detected on two separate photodetectors and the resulting photocurrents are subtracted. Since both photocurrents measure the same optical power as a function of time, the subtraction (ideally) removes the mean photodetection rate, and leaves the same shot noise variations as if the entire signal were detected by a single photodetector, since subtracting two uncorrelated types of noise has the same effect on the statistics as summing them.

Note that this technique supresses (by a factor equal to the common mode rejection ratio of the photodetector pair) all the noise sources other than shot noise, which should be the only uncorrelated noise source between the two detectors. Any signal or noise which is present in the expected photon generation rate will be common, yielding only the rate deviations between the two detectors, effectively isolating the detection shot noise process. Whether or not the actual pulse train from the comb has such low jitter characteristics as that can be supported by the shot noise is a separate issue than photodetection shot noise and has been treated extensively elsewhere [PAS04, QUI12, QUI13a, QUI13b].

The ADC was used in a subsampling configuration at 125 Megasamples/s, aliasing down to 25 MHz the bandwidth around the 100 MHz signal carrier. The noise was prefiltered using 10 MHz bandwidth bandpass filters to avoid any overlapping aliasing. Detector D2 was used as the phase reference to generate the reference template of the repetition rate. In a first calibration step, the optical signal was removed from the inverting photodetector, to yield the phase difference between the signal at D1 and at D2. This allowed us to generate the in-phase and quadrature templates for the signal at D1 while monitoring the phase at D2. This calibration step was done at very low power, to avoid any saturation of the higher gain detector D1. After calibration, the setup was reconnected into balanced mode, and the noise level in the I and Q quadratures in a fixed bandwidth (1 MHz around the carrier) was measured by software demodulation.

To further prove that the noise that we measure is indeed shot noise from the comb, we have measured the PSD level for varying optical powers by varying the attenuation at VOA1, before the fiber coupler. These results are shown in Figure 2-4. The noise level sits on a baseline due to the noise in the system when there is no optical power incident on the photodetectors; this dark noise is denoted

SDark . The linear increase of the noise variance with average detected photocurrent confirms the fact that we are indeed measuring photodetection shot noise rather than any type of noise which would be present in the expected photodetection rate, which would show a variance which scales as the square of the detected power. The experiment was run once with ultrashort pulses of less than 1 ps. The model of section 2.1.2 predicts in this case that all the shot noise

Figure 2-4 - Experimental results showing the noise level as a function of the optical power on the balanced photodiode. The linear increase in noise spectral density validates that the noise is due to shot noise. The shot noise appears completely in the in-phase (amplitude noise) quadrature of the 100 MHz repetition rate signal, when the pulses are kept short, while when detecting long pulses (almost constant power), the shot noise splits almost equally in both quadratures. In the bottom graph, the symmetric vertical displacement about the center line is due to errors in the reference phase used for demodulation because of the lower SNR in the phase reference due to the longer pulses.

ends up in phase with the signal (α I ≈ 1), and almost none of the noise ends up in the quadrature phase (αQ ≈ 0 ). The measurements agree very well with those predictions. The small discrepancy at very low powers is due to the diminishing signal-to-noise ratio on the reference phase. In a second run of the experiment, a

33 fiber spool was added to stretch the pulses to almost the full 10 ns repetition period, making the signal very close to stationary noise. In this case our model predicts roughly equal split of the shot noise between the two quadratures, which once again agrees very well with the results. The residual imbalance between the two quadratures is due to remaining modulation of the signal as a function of time; the noise was not strictly stationary. In this configuration, the reference signal at D2 which produced the phase reference had less power in the signal harmonic because of the stretching and thus the phase reference was slightly noisier. This explains the symmetric vertical displacement about the center line of the bottom figure.

This setup allowed us to measure shot noise at very low power, ensuring linear operation. We monitored for any sign of non-linearity by checking the phase of the subtraction residual, which should be constant as we change power equally on both photodetectors. This detection technique has all the advantages of supressed-carrier techniques [IVA03a, IVA07] for the measurement of very small noise on a large carrier, but the carrier suppression is done by separately detecting the (ideally) same optical signal twice and subtracting the result, leaving only the much smaller differential noise to be handled by the subsequent electronics, instead of using a microwave interferometer. This avoids driving the RF amplifier with a strong carrier, enabling very accurate noise measurements.

Such balanced detection is very widely used in many coherent detection schemes [YUE83] where a strong LO and a weak signal field are combined onto a beam splitter and are differentially detected to remove all noise coming from the local oscillator, except for the shot noise. In our case here, this LO shot noise is precisely the desired quantity, and thus we use no second field at the other input of the beam splitter. Although we have independently developed this technique for our particular goal of measuring the two quadratures of shot noise from our comb, it turns out that this technique has been frequently used in experiments with squeezed light to isolate the shot noise contribution from technical noises as far back as 1987 [XIA87, FRE93, RAL95, LI99].

2.1.5 Additional factors influencing the I-Q noise split

The effects discussed in the previous sections have shown that in the ideal case of photodetection of an ultrashort pulse, there can be a widely unbalanced split between the shot noise that gets in the I and Q quadratures, leaving vanishing low jitter. While the principal factor controlling the magnitude of this inequality is the optical pulse length, controlling the time spread of photons arrival, several other factors can affect this in an experiment. We will quickly go over some of these factors, some as well known results already, and some as direction for areas of future research that needs to be accomplished before fully understanding the photodetection jitter problem.

2.1.5.1 AM-to-PM conversion by deterministic, linear time-invariant filtering

Even if we assume that we have a noise source which gives us only amplitude noise or noise in phase with the signal only, even a fully deterministic, linear time-invariant system can convert some of this noise to phase noise. This effect occurs when using a filter with an asymmetric frequency response around the desired carrier frequency. This case is of practical importance to experiments using detection of the repetition rate of a comb, as one is easily tempted to push the detected harmonic as close as possible to the photodetector cut-off (or even beyond), as higher harmonics provide better sensitivity with respect to thermal noise. This mechanism has been very well understood since at least 1937 [ROD37], as it has implications for communications systems which use phase or amplitude modulation of a carrier wave. We will thus only give an intuitive explanation without delving into too much detail.

The mechanism for linear AM-to-PM conversion is best understood in the frequency domain, using the phasor representation for our signals, where only the complex exponential of our signal is considered, allowing a simple representation of the signal and noise in the complex plane. In this representation, our signal is a large vector oscillating (rotating in the complex plane) at the desired frequency. It is thus customary to remove this rotation of the signal and the noise, a sort of coordinate transformation, to make our signal a single large vector pointing along the origin without any rotation as a function of time (Figure 2-5).

A) Amplitude noise B) Phase noise

Imaginary Imaginary

+2π∆ft -2π∆ft +2π∆ft

Real Real

-2π∆ft

Figure 2-5 - Two sidebands, oscillating with a positive and negative frequency offset, have to combine in a precise amplitude and phase relationship to yield purely A) amplitude noise and B) phase noise.

In this view, amplitude noise can be seen as the addition of a vector that is exactly in line - in phase - with our signal, while phase noise is the addition of a vector that is 90 degrees offset - in quadrature - with our signal. The further step is to consider that noise at a frequency offset +∆f from our signal simply adds a rotating complex exponential to our signal. To yield pure amplitude noise for example, a second noise vector with the same amplitude must be added which oscillates at −∆f so that the vector sum of these two noise sidebands stays in a given direction in the complex plane. The initial phase offset between these two sidebands thus controls whether these two sidebands contribute only amplitude noise (Figure 2-5 A), phase noise (Figure 2-5 B), or a combination of both (if the vector sum neither points exactly along the Real or Imaginary axis).

Thus, for a pair of noise sidebands to give only noise in one quadrature of the signal I or Q, two conditions must be realized: First, the amplitude of the sidebands must be equal, and second, the initial phase offset between the sidebands and the signal must be precisely either 0 degrees (for amplitude noise), or 90 degrees (for phase noise). Any filter which alters this amplitude and phase relationship between those three frequency components of a signal will produce linear AM-to-PM and PM-to-AM conversion. It can be shown that the AM-to-PM

36 and PM-to-AM conversion transfer functions are the same, and for a filter Gf( ) are given by [TRE85]:

2 1 Gff( +−00) Gf*( f) Hfiq ( ) = − , (2.23) 4 Gf( 00) G*( f)

where Hfiq ( ) is the modulus squared of the conversion filter, f is the modulation frequency of the noise considered (or its offset relative to the carrier) and f0 is the carrier frequency of the signal.

The simplest example of a linear filter which implements this conversion is a brick wall filter which passes all frequencies up to and including the carrier frequency, and then blocks all frequencies above the carrier. Then, all the noise sidebands are complex exponentials at a negative frequency offset, with no correlated sideband at a positive frequency. The noise in this case is called Single Sideband noise (SSB) and produces equal amplitude and phase noise, regardless of the initial split of the noise before the filter.

Another less trivial example of a filter with AM-to-PM that is important practically is a first order filter. This filter is important to photodetection because photodetector frequently form a first-order filter due to the interaction of their capacitance with the input resistance of the measuring circuit. When measuring a signal either close or beyond the cut-off frequency of the filter, AM-to-PM conversion can become significant as the sidebands on one side will be attenuated less than the signal and the corresponding sideband on the other side. This effect is thus the smallest close to the carrier and increases for higher offsets from the carrier. Indeed, for a signal much beyond cut-off and noise modulation frequencies well below the cut-off frequency of the filter, equation (2.23) can be expanded into:

2 1 Gff( +−00) Gf*( f) Hfiq ( ) = − 4 Gf( 00) G*( f) 1 Gf( ) = 1+ jf fc f Gf( ) ≈− jc beyond cutoff f 2 ff − jjcc 1 ff+− f f Hf( ) ≈−00 iq 4 ff − jjcc ff00 2 1 ff ≈−00 4 ff+−00 f f 2 f 2 11 ≈−0 4 ff+−00 f f 2 2 f f00−− f( ff + ) ≈ 0 4 ( ffff+−00)( )

2 f 2 −2 f ≈ 0 22 4 ff− 0 ff22 (2.24) ≈ 0 222 ( ff− 0 )

2 f 2 2 ≈  for ff<< 0 . f0

The conversion filter in this condition is thus equal to a first-order high-pass filter at modulation frequencies below the carrier frequency. Such AM-to-PM conversion only becomes significant if we expect to be measuring noise in one quadrature well below the other quadrature, or when measuring at modulation frequencies close to the cut-off. In this case, the approximation of (2.24) is invalid and (2.23) must be used to evaluate the conversion. As shown in [TRE85], this conversion filter can even be higher than unity, providing even more phase noise at the output than amplitude noise at the input.

While this effect should not significantly affect most noise measurements when the experiment is designed with the possibility of AM-to-PM in mind, some early papers examine the PSD level of shot noise while keeping only one sideband [IVA03b], preventing the measurement of any correlation between the two; effectively forcing the equal splitting of the shot noise in amplitude and phase

38 noise (SSB noise). Other papers use a photodetector which is close to or beyond its cut off frequency [IVA03b]. In this case, only the PSD level of the total noise was considered and then assumed to split equally into AM and PM. However, it might also give the wrong idea that there is nothing wrong with using a photodetector past its cut-off frequency to measure phase noise levels.

2.1.5.2 Non-linear AM-to-PM conversion

As mentioned earlier, one significant hurdle that had to be solved in the development of precision time signals from frequency combs was photodiode saturation which effectively converts amplitude fluctuations to phase fluctuations. Diddams explains this effect very clearly in [DID09]: essentially, P-I- N photodetectors suffer from a type of saturation where the impulse response distorts at higher powers, way before any saturation effect can be noticed on the average photocurrent. This impulse response change implies a change in the group delay that the optically generated carriers experience. Any modulation of the pulse power thus has an impact on the timing of the detected pulse, causing AM-to-PM conversion. Physically, the effect is caused by the fact that the large numbers of mobile charge carriers that are generated by the optical pulse create an electric field that counteracts the bias field that is present in the photodiode [WIL94]. The reduction in the electric field impacts the velocity of the carriers, increasing the transit time and thus the response time.

2.1.5.3 Repetition rate multiplication using a passive optical filter outside of the laser cavity

An interesting fact is that a number of papers in the literature, in the interest of circumventing the non-linear AM-to-PM conversion problem in photodiodes, have taken an approach with can affect the I-Q imbalance of the shot noise. The idea is that since only a single harmonic of the repetition rate will be detected, the signal contained in the other harmonics only contributes to reduce the available dynamic range from the photodiode and produces broadband shot noise. At least two optical filter geometries have been proposed to prevent the unused harmonics from being created on the detector, the first being to use an optical Fabry-Perot cavity to keep only one mode from the comb at every N, in order to produce a pulse train with a repetition rate equal to Nfr , with fr being the repetition rate of 39 the comb [DID09]. This approach has certain disadvantages in that the total power is also reduced by the same factor N, although the high-speed photodiodes usually saturate at much lower power than is available from the comb and thus there is still a net gain to be had in the noise floor. The second type of optical filter that was proposed to generate a pulse train with a N times higher repetition rate was a cascade of Mach-Zender interferometers (MZI) [HAB11]. This filter geometry is attractive for this application since it provides a 2M increase in the repetition rate, where M is the number of cascaded sections, for a fixed power loss of 50% for an ideal structure, regardless of the number of sections.

To see why this scheme can influence the I-Q split of photodetection shot noise, let us consider a simple two-times repetition rate multiplier: a single Mach- Zender interferometer. We further assume that the pulses are infinitely small such that ideally no phase noise is expected from photodetection shot noise: only amplitude noise will be produced. For each pulse that is sent to the input, two pulses will be detected at the output, with a delay equal to the fiber delay that was added internally to the MZI. If this delay is exactly equal to half the repetition period of the input pulse train, the photodetection rates will look exactly like that of a comb with twice the repetition rate and thus no increase in phase noise is expected. However, if the delay is incorrectly set, say by ∆T , the resulting pulse train will be composed of the sum of two pulse trains, each with pure amplitude noise. This pulse train at twice the repetition rate cannot be detected by a simple bandpass filter3 without being affected both by AM and PM noise. The reason for this is that no single phase reference can be found that will exactly align the zero crossings of the signal template with the photon arrivals due to both set of pulses. The best that can be done in this case is to choose the reference phase as a compromise between the two ideal reference phases, with the understanding that part of the amplitude noise will couple into the quadrature noise. For infinitely short pulses producing shot noise, the amount of coupling that can be expected will be equivalent to:

3 One could theoretically envision a complex, time-variant system that could compensate for this non-ideal interleaving but that would be hardly practical. 40

∆∆TTT ddt+ +− t r + sin2 2π 2Mf t dt ∫( r ) 2 22 αQ = ∆∆TTT dd+ +−r + ∫t t dt 2 22 1 ∆∆TTT =−+sin22 2ππ 2Mf sin 2 2Mf r + rr 22 22

1 22∆T  ∆T = sin 2π 2Mfr + sin 22π Mfr  (2.25) 222

2 ∆T = sin 2π 2Mfr 2

1  2 ∆T =1 − cos 2π 2Mfr , 22 where d (t) is the Dirac-delta function and we have used its sifting property:

∫d (thtdth) ( ) = (0) . Assuming a small timing error compared to the signal period, we have:

2 11∆T απ≈1 −− 1 22Mf Qr 22 2 2 1 ∆T ≈ 22π Mfr (2.26) 42 2 ≈∆(π Mfr T ) .

Comparing with (2.12), we can say that this is equivalent to the splitting ratio obtained using an optical pulse with effective length ≈∆T 12 . Thus, very special care needs to be applied when constructing the MZI cascade if one wants to achieve the low timing jitter afforded by very short pulse lengths.

In the case of [DID09], where a Fabry-Perot cavity is used to increase the repetition rate, a feedback loop is used to keep the free spectral range exactly equal to a multiple of the repetition rate and thus each output pulse is expected to overlap coherently without increasing the effective pulse duration.

These papers still demonstrate that an improvement in the phase noise floor was reached; highlighting the fact that the shot-noise limited timing jitter is very low and technically challenging to reach.

2.1.5.4 Photodetector transit time spread

Up until now, we have always assumed that the noise source produced by photodetection shot noise could be accurately modeled as a single noise source with time-varying statistics controlled by the instantaneous optical power. Here, we consider the spatial extent of the photodetector along with the physics of the conversion of photon absorption events into a measurable current to examine whether the low phase noise predicted by the simpler model can be realized in practice. This effort is in no way comprehensive, mostly because of time constraints linked with the end of our studies and thus much more work will be needed in this area to cover its whole depth.

The methodology that we follow in this section is to consider the photodetection events associated with the promotion of an electron-hole pair (EHP) in the conduction band of the semiconductor material of the photodetector. We examine how this generation translates into an electrical current in the measuring circuit. The photodetector model that is used is presented in Figure 2-6 and mostly follows Kasap’s derivation [KAS05], extending the result to the relevant effects at hand. This model follows a mostly classical approach to the generation of a current in a semiconductor, abstracting the underlying quantum physics. A P-I-N (doped with acceptors, intrinsic and doped with donors) photodiode is considered under side illumination, with a strong reverse bias field in the intrinsic region. We assume that all photons are absorbed in the intrinsic region, neglecting any absorption in the doped regions. EHPs are randomly generated at different depths in the material, such as illustrated in the figure where an EHP is generated at 1, 8 and 17 μm . Once these EHP are promoted to the conduction band of the semiconductor, they start drifting under the deterministic action of the bias electric field E . The holes drift in the same direction of the electric field vector until they reach the p-doped region, where they stop drifting as the field is negligible and they eventually recombine. The electrons instead drift in the direction opposite of the bias field, ending up in the n-doped region with low field. This drifting action is what causes a current to be measured in the outside circuit. Indeed, the bias source acts to generate a force eE on the moving charges and this requires mechanical work, as the carriers are moving. Equating the electrical work done by the bias source to the mechanical 42 work done on the electron, knowing that to move an electron a distance dx in time dt , the work required is [KAS05]:

W= eEdx = Vie ( t) dt . (2.27)

Assuming a constant bias field E= VL throughout the intrinsic region, L being the length over which the potential V drops, and denoting the velocity of electrons ve ( t) = dx dt , the current generated by the electron is given by:

V e dx= Vi( t) dt L e e dx = it( ) (2.28) L dt e ev( t) e = it( ). L e

This simplistic argument allows to intuitively understand that the electron current produced by the photon absorption event will have a certain time spread, even considering this event to be infinitesimally short. This current will indeed have the same shape as the velocity of the electron as a function of time. To get a feel for the timescales involved, we can assume that the velocity of the electron is equal to the saturation velocity in the semiconductor material, as long as the electron is under the action of a strong bias field. With this assumption, the current waveform will be equal to a rectangular pulse in time, with a spread equal to the transit time of the electron in the intrinsic region. For an electron promoted to the conduction band at depth l in the material and intrinsic region length L , we have tee=( Llv − ) . A similar argument can be made for the hole, yielding a

hole current that lasts thh= lv , where vh is the average hole velocity in the material. The total current seen at the output will then be the linear sum of the hole and electron current and the shape will depend on the absorption depth l . For an EHP generated close to the p region, most of the current will be due to the electron as the hole current will last for a very short time. For an EHP generated at the furthest depth of the photodetector, the current will instead be produced mostly by the hole. In the middle, both currents will contribute but for a shorter amount of time overall. In all cases, the total integrated charge can be shown to be exactly equal to the charge of the electron.

Figure 2-6 - The current generated by an electron-hole pair depends on the depth of absorption of the photon. Most importantly, the center of mass of each current impulse is different for each depth. Based on [KAS01, chapter 5]. In this example, the holes drift speed is 0.7e5 m/s, while the electrons drift speed is 1e5 m/s, corresponding to the saturation velocities for silicon.

Based on this model, we can see that such a photodetector has an impulse response that depends on l , the depth at which each photon gets absorbed. Since this absorption depth is random for each photon, the model violates our assumption that the shot noise can be represented by a single noise source followed by a single impulse response. The model indeed has to be amended to include many different noise sources, corresponding to shot noise produced in different sub-regions of the photodetector, each followed by an impulse response. Note that only in the limit of a photodetector which has the same impulse response for each of its sub-regions can the model be reduced to a single noise source with a single impulse response that filters the current.

To quantify the amount of jitter produced by the existence of different transfer functions, we reduce the effect of the different impulse responses to a simple delay. For example, Figure 2-6 shows the center of mass4 of the impulse response corresponding to an EHP generation at the three depths. That is, we reduce the photodetector impulse response model to a single function tl( ) which deterministically gives the input to output delay as a function of spatial position of the absorption. For example, for the p-i-n detector with the impulse response as a function of position described above, the time delay as a function of position can be found to be:

+ ∫(iteh( ) ittdt( )) tl( ) = + ∫(iteh( ) itdt( )) tt 1 ehev ev =ehtdt + tdt eL∫∫ L 00 (2.29) vt22 vt =ee + hh LL22 2 1 (Ll− ) l 2 = + . 2  L vveh

Using a similar argument to that of section 2.1.1.1, we will first consider the jitter seen when detecting a single photon from a pulse, this time considering no time spread of the optical pulse. What introduces randomness in this model is the absorption depth of the photon, as characterized by the absorption probability pl( ) as a function of depth. Knowing that a single photon is absorbed at time 0, this photon could still have been absorbed at any depth in the semiconductor, and each of those depths will yield a certain delay. Since the spatial position of the photon absorption is a random variable and there exists a deterministic mapping from the spatial position to the delay, the delay can also be seen as a random variable itself. This new random variable can be completely characterized by knowledge of the probability of absorption at every spatial position in the detector, and the mapping of position to delay. The average and variance of this delay will be given by:

4 It can be shown that as can be expected intuitively, the center of mass of an impulse response is equal to the group delay calculated at low frequencies. 45

µ = = t Etl{ ( )} ∫ tl( ) pldl( ) 2 22 σ t =Et{ ( l)} − ( Etl{ ( )}) (2.30)

2 σ 22= − t ∫t( l) p( l) dl( E{ t( l)}) ,

where µt is the well-known average transit time of the photodetector and we have introduced σ t , the transit time spread of the photodetector. This model can easily be expanded to include three dimensional positions, if the photodetector is known to have delay and absorption probability characteristics which vary with other spatial dimensions than depth, as is expected of any real photodetector.

The time spread computed this way corresponds to the time spread of a single detected photon, and thus produces an effect analogous to the length of the optical pulse. When detecting a pulse train with N expected photons per pulse, the detected timing jitter variance will be N times lower as the time spread effect will tend to average out.

One way of interpreting this result is to consider that we sum over many different shot noise processes, each giving amplitude noise only, but with a different central time, and that prevents us from finding a single correct phase center to use for the demodulation, in the same manner as explained in section 2.1.5.3. We thus get both amplitude and phase noise.

We have computed both analytically and integrated numerically equation (2.30) for a simple photodetector, assuming that the probability of absorption as a function of depth is given by:

pl( ) = kexp(α l) , (2.31) where α is the absorption coefficient of the material and k is a normalization constant. The absorption coefficient was chosen as that of silicon for the 800 nm wavelength. The length of the intrinsic region was chosen as L = 10 μm to yield reasonable quantum efficiency (h = 0.63 due to incomplete absorption only) at the design wavelength. Figure 2-7 shows the results of both the average transit time and the transit time spread. The transit time spread can be seen to vary significantly depending on the absorption coefficient. This is because for a high

46 absorption coefficient compared to the depth of the material, most of the photons are absorbed in a very narrow region and thus experience very similar delay.

Figure 2-7 - Average transit time and transit time spread for a P-I-N photodetector using parameters for Si (electrons speed = 1e5 m/s, holes speed = 0.7e5 m/s). The length of the detector was chosen to be equal to the penetration depth for 800 nm to yield reasonable quantum efficiency (63%). The absorption of Si at 1060 nm corresponds to the left limit of this figure at 1e3 m-1, while the absorption at 800 nm is 1e5 m-1 [ADA12], giving a transit time spread of around 10 ps for this range of wavelength. Lines: analytical result, dots: simulation results.

For the spectral regions with lower absorption coefficient, the transit time spread for this photodetector is shown to be around 10 ps. In general, the transit time spread will be much worse for lower absorption coefficients, as the spatial extent of the photogenerated EHP will be greater. Another effect that can be seen in this figure is that the average transit time varies from approximately 4 to 5 ps over the range of absorption coefficients. If an optical pulse with a wide range of wavelength if detected by such a detector, different wavelengths will see a different absorption coefficient and thus different delays and some extra transit time variation can be expected, although this effect is likely to be dwarfed by the spread due to the absorption depth.

2.1.6 Conclusion and outlook

We have shown that using a simple model considering the photon arrivals as dictated by the optical power as a function of time, the fundamental timing jitter due to photodetection shot noise is vanishingly low for short pulses such as those from a femtosecond laser. This effect can be seen in experiments by very unequal split between the I and Q noise quadratures, showing that the very common assumption of equal split in both quadratures does not apply to shot noise from frequency combs. In I-Q view, the shot noise indeed yields 2qI PSD, but this appears almost completely in I (amplitude noise) quadrature. Our experimental results validate our claims of very unequal split in the case of short pulses, and these results are supported by Quinlan’s [QUI12, QUI13a, QUI13b].

We have explored some mechanisms that can affect the noise splitting and might come into play in an actual experiment. Some of these mechanisms are well- known and have been studied thoroughly (linear and non-linear AM-to-PM conversion), while the limits that we predict due to photodetection physics and passive repetition rate multiplication are merely theoretical predictions. For the limit due to the photodetector, the effect is much higher when the absorption coefficient of the semiconductor at the wavelength of operation is low, meaning that InGaAs detectors should show very low effects as the absorption cuts off very abruptly. On the other hand, indirect bandgap materials such as Silicon and Germanium have much smoother roll-off of the absorption coefficients and care will have to be applied when low jitter is expected in regions of low absorption. These predictions will have to be carefully evaluated for geometries closer to actual photodetectors used in an experiment, and the jitter will have to be measured experimentally and compared to the theoretical predictions to fully validate the model.

3 The effect of shot noise in a comb-CW heterodyne experiment: gated optical noise reduction and chirped pulse heterodyne

In this section, we will concentrate on shot noise in the comb-CW laser heterodyne experiment5. Such a beating is at the heart of the frequency comb revolution, since it is the tool that allows measuring an optical frequency relative to the comb repetition rate (or equivalently, the comb modes). This measurement is either used as the useful output itself, as an optical frequency counter, or it can also be used in a feedback loop to lock one of those frequencies to the other, transferring the accuracy of one reference to the other.

The SNR of this beat signal is an important parameter in many applications, either because the experiment has a particular threshold under which it is impossible or very difficult to accomplish, or because the noise of the beat signal limits the precision of the measurement or its bandwidth. For example, in most frequency counting applications, the SNR requirement is simply that the beating is sufficiently low-noise to be counted without cycle slips, although higher SNR means that the frequency can be counted more accurately with better sub-cycle precision. Other applications fall in the second category, for example the generation of low-noise microwave frequencies [DID03], servo-control of the frequency comb parameters [BAU09], or monitoring the relative fluctuations of two combs [DES10], require the lowest amount of noise possible from the beating.

Obtaining a low-noise beating between a frequency comb and a CW laser is not trivial. As there are a very large number of modes in the comb's spectrum, the power in each of those is usually very small, and this can lead to poor beat SNR depending on the amount of power from the CW laser, the comb, and the dark noise from the detection setup. Stated another way, the field from a frequency comb is comprised of a very intense, very low duty cycle pulse train, and this is very poorly matched with the continuous wave field from the CW laser; yielding very little interference between the two.

5 Various parts of this section are taken from our paper [DES13a] published in 2013 in Physical Review A. 49

In the past, all comb-CW beating experiments have considered the comb in the frequency domain, reasoning that its spectrum is composed of discrete lines, and computing the SNR predictions for a single comb mode beating with the CW line, assuming without mentioning that the shot noise is stationary. Starting with this analysis in section 3.1, we will then switch to a temporal explanation of the interference, which will lead us to derive different techniques that can achieve SNR above the single-mode limit. The first technique that we will present in section 3.2, gated optical noise reduction (GATOR), is most useful when the shot noise can be considered stationary, which is the case when the CW laser produces much more shot noise than the comb. In section 3.3, we will then derive the limit to this technique in the general case of non-stationary noise, when both the shot noise from the comb and the CW laser have to be considered, and provide an optimal detection bandwidth which maximizes the SNR. In section 3.4, we will describe in detail theoretically and experimentally a slightly more involved technique which makes even better use of the available optical power and works even in the case of high comb power. Section 3.5 will present a review of the SNRs reported in the literature for comb-CW beats and finally, section 3.6 will introduce a variation of the technique and compare it to that of section 3.4.

3.1 SNR limit for a single comb mode beating with a CW laser

In [REI99], some guidelines to produce a low noise beating are given, namely to use an optical bandpass filter to remove as much power as possible from the frequency comb. This is to produce a beating limited by the shot-noise from the CW laser, which is implied in this paper and even sometimes directly stated as being the optimal SNR that can be obtained [TEL99]. This corresponds to shot- noise limited detection of a single mode from the comb, but as we shall demonstrate starting from section 3.2, the achievable SNR can be higher than this in certain conditions. First, we will derive this limit that we will call the single-mode shot noise SNR limit.

Comb spectrum

Optical Frequency

Figure 3-1 - Illustrating the many modes of the comb's spectrum. The spacing between the modes is given by the repetition rate of the comb. The spacing between the mode and the comb bandwidth is not to scale in this figure, as usually there are tens to hundreds of thousands of modes in a typical comb’s spectrum.

The comb spectrum is composed of many modes as shown in Figure 3-1. We start by assuming that the comb passes through an optical bandpass filter and that N modes of the comb reach the detector, each with power Pn, while the CW laser provides power PCW. Thus, we detect a signal which can be written as:

N Pdetected ()φ=++h PPCW h∑ n 2 hPPCW n cos()φ+ , (3.1) n=1 where h is the quantum efficiency of the detector and φ is the phase difference between the CW and the comb mode of interest6, and where we have explicitly written only the interference term which will be detected (by using an appropriate bandpass filter for example). To compute the electrical SNR of this signal, we need the electrical power in the signal in which we are interested in, and the noise variance. Since we are computing a signal ratio, the units in which we do our calculations do not matter and we will keep the detected signal in units of optical power, Watts. The electrical power in the beat signal will be, up to a constant:

2 = ηφ Pelectrical,signal (2PCWP n cos())

22 = 4ηPPCW n cos φ (3.2) 2 = η PPCW n ,2

6 We did not put a frequency difference between the two beams, but this would not change the result in any way, because we will average the results over all possible phase differences. 51 where the angled brackets denote averaging over all possible phases. We now need the variance of the noise, which we assume to be shot noise produced by the optical power arriving on the detector. From equation (1.5), we know that the shot noise variance will be given by:

22 sshot =⋅⋅2 λAR ⋅ , (3.3) where λ is the average temporal rate of arrival of photons, and A is the area under the signal produced by each photon, and R is the detection bandwidth. In our case, since the signal is scaled in units of Watts, the area under each photon detection event is equal to the energy transferred by the photon, Ephoton, while the rate of arrival is equal to the average detected optical power divided by the energy in each photon:

N hhPPCW + ∑ n s 22= ⋅ n=1 ⋅⋅ shot 2 ERphoton (3.4) Ephoton

=2.hEphoton ⋅ R(PNCW+ P n )

The SNR is thus given by combining (3.2) and(3.4):

P = electrical,signal SNR 2 s shot 2h 2 PP = CW n (3.5) 2hERphoton ⋅ (PCW+ NP n ) h PP = CW n . ⋅ ERphoton (PCW + NPn )

Assuming that we have removed enough comb modes so that PCW >> NPn, we have:

h SNR = Pn . (3.6) ERphoton ⋅

The SNR is thus a function of the desired bandwidth, and the optical power present in a single mode of the comb, yielding very inefficient use of the power available from the hundreds of thousands of modes from the comb. Immediately, one could think of detecting N such beats, and somehow averaging all the information to improve the SNR, but looking at the frequency domain picture, this does not seem like an easy task. Fortunately, as we will show in the next

52 section, the time domain signal has a very simple structure: it is composed of narrow pulses and this makes it simple to exploit to improve the SNR.

3.2 Gated optical noise reduction (GATOR)

If we take the power available from the CW laser and the power available from the comb as given conditions of the experiment, as is usually the case in practice, we can identify four types of conditions, where the power from the CW laser and the comb are independently either high or low, with high and low being defined as arbitrary soft limits that depend on the required SNR and the dark noise of the detection setup. Here we will concentrate on the case where very little power is available from the comb, while plenty of power is available from the CW laser. This case is particularly troublesome because the three other cases either produce a beat signal with sufficient SNR, as in the case of high CW and high comb power, or one could use a different setup to achieve the required SNR. For example, if low CW power but high comb power is available, one could use a second, potentially less stable, higher power CW laser as an intermediate oscillator to produce two intermediate beats each with higher SNR, and electrically beat them together to get the desired beating at sufficient SNR. If both the comb and the CW laser are low power, then a combination of using an intermediate oscillator and the technique presented in this section could be used to attain sufficient SNR.

This section is organized as follows: section 3.2.1 presents the technique in the time domain, while section 3.2.2 provides a frequency domain interpretation. Section 3.2.3.1 and 3.2.3.2 contain two experimental implementations of the technique and section 3.2.3.3 presents an experimental validation of the long term accuracy compared to a conventional heterodyne setup.

3.2.1 The gated optical noise reduction concept in the time domain

First, let us describe the experimental setup to be discussed, and the assumptions that we will make. The setup which is considered is depicted in Figure 3-2 a). The optical field from the comb is band-pass filtered to remove unneeded modes from the spectrum, keeping only those around the CW laser's frequency and is then superimposed onto the CW electric field inside a fiber

Figure 3-2 - The three experimental setups used in this section. Optical fibers are solid lines and dashed lines are electrical coaxial cable. BPF: Optical bandpass filter, PC: polarization controller, DL: Adjustable optical delay line, D: wideband photodetector, LPF: low-pass filter, LNA: Low-noise amplifier. coupler. The purpose of the filtering operation is to send only the light which is needed on the photodetector, as any excess light will produce more shot noise than is necessary, as is explained in [REI99]. However, we assume that the filter has wider finite bandwidth and lets a certain bandwidth through. We will assume that this bandwidth is at least equal to or higher than the highest photodetection bandwidth to be considered, namely tens of GHz (approximately 0.5 nm around 1550 nm). We will assume that the CW laser has much higher

54 power than the filtered comb light, and thus the shot noise seen on the detector is predominantly caused by the CW light. The difference between this setup and a conventional heterodyne setup is two-fold. First, it requires an additional photodetector which serves to give a clock signal whose purpose will soon be apparent, and second, the photodetectors used should have the highest bandwidth possible (preferably tens of GHz) instead of only the desired final measurement bandwidth or half the comb repetition rate (usually tens of MHz).

The photocurrents in both photodiodes are amplified by very fast transimpedance amplifiers and are recorded by a fast AC-coupled oscilloscope. Figure 3-3 a) and b) shows actual experimental signals recorded by this setup, with a 100 MHz frequency comb and a 10 GHz oscilloscope. The noise seen on the beat signal is in fact shot noise from the CW laser, while the modulated pulses carry the beating information. The observation central to this technique is that the beating information is present only when there is temporal overlap between the comb pulses and the CW field, while the dominant noise process (shot noise from the CW laser) is spread out temporally.

In a conventional heterodyne experiment, the beat information would be extracted from this signal using either a slower photodetector (slower than half the pulse repetition rate to respect the Nyquist criterion) so as to temporally spread the signal pulses to produce a continuous signal, or using either an electrical low- pass or bandpass filter, which would accomplish the same result. The filtering would reduce the noise variance through temporal averaging (or equivalently, bandwidth narrowing). However, we argue that this averaging will also unnecessarily integrate the inter-pulse noise into the measurement, which bears no useful information. It should thus be possible to reduce the noise level from the measurement while keeping most of the beat signal amplitude by gating out the inter-pulse noise. While practical electronics might not be fast enough to limit the gate time to the filtered optical pulse duration, as long as we can find electronics that are faster than the repetition rate, we can still gate out a sizeable fraction of the CW shot noise. We call this particular improvement of the conventional heterodyne setup: GATed Optical noise Reduction (GATOR).

At this point, the purpose of the clock signal that is recorded along with the beat signal should be apparent; it serves to generate a gate signal with low jitter relative to the beat signal. In fact, one could either use this signal to regenerate a gate signal of arbitrary shape, or even simply use the clock signal directly as a gate. Indeed, provided that there is no delay between the pulse arrival time in both channels, multiplying the gate signal with the beat signal will accomplish the desired gating operation, as shown in Figure 3-3 c). This operation does not affect the beat signal information in any way, but it removes most of the measurement noise, effectively increasing the SNR. Assuming that the SNR was shot-noise limited by the noise from the much stronger CW laser, one can see that the noise variance (averaged over one pulse repetition period or more) was reduced by a factor equal to the duty cycle of the gate signal. Note that the bandpass filtering mentioned previously will improve the SNR of both the conventional and the GATOR signal in the same ratio. Indeed, bandpass filtering amounts to averaging the signal over a number of temporal pulses. The noise improvement of this filtering comes from the fact that the noise in different pulses is mostly uncorrelated, and since the GATOR does not introduce any additional correlation in the remaining noise, it can benefit from the same filtering. Additionally, if the dominant noise process of the experiment is photodetector dark noise or transimpedance amplifier noise, the gating operation will still provide the same SNR gain as in the shot-noise limited case, enabling the use of less sensitive detectors than might otherwise be required.

A potential limitation to this technique is that if the beat signals seen by the photodetector are chirped, then gating this signal will remove a significant portion of the desired signal. Fortunately, the amount of chirp that can actually be tolerated is very large. Indeed, the requirement is simply that the pulses be small compared to the response time of the electronics, over the bandwidth of the comb that is actually used (in the tens of GHz range). Assuming a response time of 65 ps, the optical pulses could be chirped to 6 ps duration without significant stretching of the beat signal. For 0.5 nm of bandwidth used (60 GHz at 1550 nm), this corresponds to 6 ps/(0.5 nm) = 12 ps/nm. Assuming SMF-28 fiber with 13 ps/(nm km) of group delay dispersion, this means that the technique works even with pulses stretched by almost 1 kilometer of fiber.

Figure 3-3 - The GATOR processing chain in the all-software approach. Trace a) and b) are raw measurements recorded by the oscilloscope. The vertical scale of each signal has been adjusted to clearly show all relevant features. The noise seen between the pulses on trace b) is shot noise from the CW laser. Trace c) is generated in software by multiplying a) and b) after adjusting the gate and the beat signal so that they overlap perfectly. Intuitively, from the b) trace, we would expect that the SNR of this beat should be very good as even visually, we can easily follow the phase of the beat signal. While the GATOR does indeed provide a clean sine wave from this signal, Figure 3-5 shows that simply filtering the beat signal as is conventionally done yields very poor SNR as all the inter-pulse noise is integrated into the resulting signal.

3.2.2 The GATOR concept in the frequency domain

This scheme also has a simple explanation in the frequency domain. Figure 3-4 shows the spectrum of the beat photocurrent as seen by a 10 GHz photodetector, which contains many spectral copies of the beating, with a white noise floor coming from the shot noise from the CW laser. Intuitively, one could imagine that recording the N spectral beats at the same time, and averaging over the temporal phase or frequency information given by each would give an N-fold improvement in terms of signal power to noise power ratio. It can be shown that this is exactly how the proposed technique achieves its SNR improvement. Indeed, by the convolution theorem, we know that the time-domain multiplication of the gate and the beat signal implies that to compute the spectrum of the result, the spectra of the two signals should be convolved. Since the spectrum of the gate signal is comprised of strong peaks at harmonics of the repetition rate, the

Figure 3-4 - The spectrum of the wideband beat photocurrent contains many spectral copies of the beat signal and these can be averaged together to reduce the impact of shot noise from the CW laser. The bottom trace is simply a zoomed in version of the top trace, to clearly show the two spectral copies of the beat per repetition rate (100 MHz) span. convolution operation amounts precisely to carrying out the sought-after averaging over many different spectral copies of the beating signal; in other words, time-domain gating and averaging over many spectral copies of this signal are the same operation. This averaging operation over the correlated beat notes will only slightly change the amplitude and add a constant phase offset to the beat signal. In other words, the averaging operation leaves the amplitude of each beat harmonic essentially the same as before the averaging. On the other hand, the shot noise from the CW or thermal noise is stationary and is thus uncorrelated in the many harmonics, and so its power spectral density will average down linearly with the number of spectral copies averaged [PAP02]. Note that this also implies that if the dominant noise process is non-stationary such as the shot noise produced by the comb, the noise in the harmonics will be correlated and averaging over them will not provide an improvement. In this case, a different technique such as the one presented in section 3.4 should be used to improve the SNR.

Intuitively, the noise power spectral density reduction can be approximated as the number of spectral copies, N, that are averaged. For a comb repetition rate of

Fr and a flat measurement bandwidth of BW, we have N = 2B/Fr, which accounts for the fact that there are two copies of the beating signal in each Fr harmonic. Assuming that the signal power is mostly unaffected by the gating/spectral averaging operation, the beat SNR improvement over the shot noise limited detection of the beat of a single mode of the comb with the CW laser can also be approximated as N. However, since each spectral copy of the beating will be multiplied by the relative amplitude and phase of the corresponding clock signal harmonic, the exact transformation that the noise and signal amplitude receive depends on the spectral shape of the signal, the noise, and the gate.

Note that averaging out over many spectral copies like that might seem like a very difficult operation, in terms of maintaining phase coherency between the different copies, however it is very easy to understand in the time domain that since the gate signal is derived directly from the comb pulses, it is virtually impossible for the two signals to walk off each other. The only way this could happen would be to have a varying delay between the splitting of the comb light and the multiplication of the two signals. In practice, this implies that this delay must be constant up to a small fraction of the electrical pulse length, meaning over a scale of tens of picosecond in our case with 10 GHz detectors, otherwise the actual beating signal will be gated out of the measurement. This alignment is straightforward to maintain in practice, but even in the case where the delay does drift or jitter somewhat, this type of error would mostly produce an amplitude variation in the gated beating signal, and this would only apply a delay on the detected beat signal, not a frequency shift. In any case, we have performed an experimental validation on the agreement between the gated beat signal and the un-gated beat signal in section 3.2.3.3.

3.2.3 Experimental results

In order to show that the GATOR is realizable in practice and that the SNR is indeed improved without adversely affecting the frequency and phase of the beat signal, we have carried out three demonstrations. We will show in this section two different practical implementations of the GATOR, and we will validate that

59 the GATOR beat and the conventional narrowband beat generate the same information over long time scales. All these demonstrations use the same frequency comb (Menlo C-Comb, Fr = 100 MHz) with its power attenuated to simulate a low comb power situation, and a PLANEX narrow linewidth external- cavity DFB CW laser from Redfern Integrated Optics, with 8 dBm of output power and 10 kHz of Lorentzian linewidth. The bandpass filter is a JDS Uniphase TB9 grating filter with 32 GHz of bandwidth.

3.2.3.1 First approach: all-software with a fast oscilloscope

The first approach uses the optical setup in Figure 3-2 A). The beat and clock signals are simply recorded by a very fast oscilloscope. This is the simplest way of implementing the GATOR since all the operations are done in software, but it requires a very fast oscilloscope or acquisition card with much higher bandwidth than the repetition rate to achieve significant SNR gain. It is also unable to generate beat signal information for all time as it is limited by the oscilloscope memory, and it would require very fast digital electronics to implement in real- time. It is nonetheless very instructive to implement as it allows visualizing the signals at different intermediate steps, as shown in Figure 3-3. Traces a) and b) correspond to the raw beat and gate signals as recorded by the oscilloscope. The oscilloscope (Agilent Infiniium DSO81004A) and photodetector (Bookham PT10G) both have 10 GHz bandwidth and the resulting beat pulses are 65 ps of full-width at half maximum (FWHM). Trace c) is the result of the multiplication operation, after carefully removing any residual delay between the two channels. The CW power on the detector is 2.8 mW, and the comb power is 19 pW/mode (6 nW total). Figure 3-5 depicts the time-domain result of low-pass filtering in the same bandwidth both the raw beat signal, as would be done in a conventional heterodyne experiment, and the GATOR signal. The spectra of both the raw beat signal and the GATOR signal are shown in the same figure, computed in 170 kHz resolution bandwidth. At this CW power level, the dominant noise on the beat signal is shot noise, 6 dB above the dark noise of the scope and detector setup, which implies that using this particular setup, the narrowband setup achieves very closely the shot-noise limited SNR of a single comb mode beating against the CW laser, which we can verify using equation (3.6).

Figure 3-5 - Comparison of the results using the all-software approach for the GATOR. The time-domain traces are both measured with a 50 MHz low-pass filter while the spectrum is computed using 173 kHz resolution bandwidth. The GATOR signal shows 20 dB improvement over the shot-noise limited conventional heterodyne beat.

Note that this equation is only valid in the limit of high CW power compared to the power of all comb modes reaching the detector. Evaluating (3.6) using the parameters used to generate Figure 3-5 (h = 0.76 , R = 1.7 MHz ), we get a shot- noise limited SNR of 28 dB versus an experimentally measured 26.9 dB, approaching the shot noise limit to within 1.1 dB, while the GATOR spectrum shows 46.9 dB, a 20 dB or a 100-fold SNR improvement over the mostly shot- noise limited beat. Note that the GATOR signal is still limited by the shot-noise, but with a much lower limit. While it would be tempting to use the approximation 2B/Fr as the SNR gain, it can be seen from Figure 3-4 that the signal spectrum is not flat over the specified 10 GHz bandwidth, and thus the effective bandwidth is lower. Note that at this bandwidth and comb power, the raw beat signal is unsatisfactory to use even in a frequency counting experiment, and would be disastrous to use in an experiment which requires the phase information at high bandwidth. This signal could be cleaned up by bandpass filtering, however this imposes additional constraints on the experiment, namely that the phase/frequency information is not needed over short timescales, and 61 that the beat frequency can be set to the center of the bandpass filter, the filter tuned, or that a tracking filter is used. Moreover, the GATOR beat signal would also benefit from the SNR increase of this filtering, and the SNR improvement between the two signals would be the same, yielding a signal with very high accuracy for the GATOR.

3.2.3.2 Real-time hardware gating with a double-balanced mixer

The second approach that we have implemented is depicted in Figure 3-2 b) and uses a double-balanced mixer (DBM) to accomplish real-time hardware gating of the beat signal. After gating by the DBM, the signal is low-pass filtered to remove the now redundant spectral copies of the beat signal and the noise, and is amplified by a low-noise amplifier (LNA). The DBM is driven by the short pulses from D2, operating as the gate for the beat signal.

This particular implementation of the GATOR has the advantage of producing a continuous beat signal, but it also has more constraints than the all-software approach. The gate signal has to be already aligned with the beat signal, otherwise the beat will be gated out instead of passing through the gate unaffected. This implies either that the delay alignment be done once by splicing optical fibers or coaxial cables of the correct length, or that some form of variable delay line be used. We used the delay line approach for this demonstration. The photodiode which generates the gate signal needs only -19 dBm of comb light, which can be taken from a different wavelength than where the beat is produced, rather than using the fiber coupler as shown in the experimental setup. Additionally, the beat signal must be much smaller than the gate signal at the input of the mixer so as to stay in the linear range of the mixer. Finally, the amplifier used after the filter has to be ultra-low noise, as most of the shot-noise and dark noise has been gated out of the signal at this point, and we must make sure that the amplifier's noise does not significantly affect the noise level of the measurement. Note that after the filter and low-noise amplifier, the beat signal is an analog signal, which is completely equivalent to the signal that would be obtained with a lower bandwidth detector, but with lower noise, and can be used in the same manner without further modifying existing setups.

For the DBM, we used an inexpensive SYM-63LH+ mixer from Mini-Circuits, rated at 1-6000 MHz of bandwidth. In fact, in terms of implementation costs, this type of mixer is so inexpensive that its cost is completely negligible compared to the cost of the wideband photodetectors. Care should be used however to use this bandwidth figure in the equation predicting the SNR gain, as the frequency response is hardly flat over the whole bandwidth. Indeed, the experimentally observed SNR improvement is 18 dB over the SNR of the narrowband setup as is shown in Figure 3-6, and thus is 17 dB over the theoretical shot-noise limit of narrowband heterodyne for these conditions.

Figure 3-6 - Hardware real-time implementation of the GATOR using a mixer. Both time domain signals are shown using a 50 MHz low-pass filter, while the spectra are computed in 1.7 MHz of bandwidth. The narrowband beat (blue, upper trace) was measured by bypassing the mixer from the setup, and yields a 17 dB SNR, which is within 1 dB of the conventional shot noise limit. The GATOR signal (red, lower trace) was measured using the same photodiode, low-pass filter, RF amplifier and optical power levels and shows 35 dB of SNR, an improvement of 18 dB, or 63 on a linear scale. The beat frequency difference is due to CW laser drift between taking the two measurements.

This improvement would mean an effective bandwidth of 3 GHz, assuming that the noise at the output of the LNA is comprised of shot noise only. In fact, the particular LNA that we used (MITEQ AU-4A-010, 60 dB gain, 1.6 dB Noise Figure), while very low-noise, contributes to 25% of the output noise power, and thus reduces the obtained SNR gain by the same amount. Note that this amplifier only needs Fr/2 of bandwidth (or the desired final bandwidth of the beat signal), and LNAs with even lower noise figure could conceivably be used to lower the penalty associated with its noise.

The linearity limit of this GATOR setup has been measured experimentally and can be given in terms of its third order intercept point (IP3). IP3 is usually used to specify the behavior of weakly nonlinear RF systems and can be understood as the power level in the fundamental tone that would give the same power in the third harmonic. Based on this definition, we find that IP3 for our mixer-based implementation of the GATOR is when the comb input power is 60 nW/mode. This means that if a maximum power in the third harmonic of -30 dBc is desired, the comb input power would have to be at least 15 dB below IP3, meaning below 2 nW/mode.

3.2.3.3 Long-term comparison against conventional heterodyne setup

In order to validate that all our assumptions are satisfied in practice and that there are no mechanisms that we have failed to include in our analysis that would degrade the accuracy of a GATOR setup, we have set out to experimentally demonstrate that the GATOR setup supports comparison between a CW laser and a comb at mHz-level accuracy with no observable degradation. As we did not have access to a mHz-level linewidth comb and CW laser, we opted to validate the GATOR against the conventional, proven technique of narrowband heterodyne. To accomplish this, we have ran a GATOR setup continuously for 1400 seconds, in parallel with a conventional heterodyne setup, which measured the beat signal between the same comb and CW laser (Figure 3-2 c)). The goal was to verify that the beat signal given by the GATOR and the narrowband setup would stay at the same relative phase for long times, validating the GATOR's long term accuracy. We trust that by proving that the GATOR provides a beat signal with the same

64 phase as the narrowband heterodyne technique, the GATOR does not introduce unknown frequency shifts or drifts.

We used essentially the same setup as explained in section 3.2.3.2 using the DBM as a hardware gate. Both resulting signals were recorded at 250 MS/s by an oscilloscope, which would acquire data for 200 us every second, for 1400 seconds, yielding 1400 data windows. Once all the data was recorded, the resulting beat signals were bandpass filtered in software, then mixed together to measure their relative phase, averaged over 50 us. The decision was made to implement the phase comparison in software rather than hardware, as the transfer function of the software filters could be exactly matched for both measurement channels, while the mismatch of hardware analog bandpass filters could have introduced an extra phase term between the two beats as the beat signal drifts in frequency over the course of the measurement. This process was repeated for all 1400 data windows, and the relative phase of the two beat signals is displayed after removing a constant phase offset in Figure 3-7. The resulting 100 mrad peak-to-peak phase excursion are mostly due to the limited SNR of the narrowband setup, but could also have been improved somewhat by better matching the analog low-pass filters used in both measurement channels. Most importantly, the phase difference stays well below 1 rad over the complete measurement time which implies that the two beats are fully coherent over the same time scale. Taking the Fourier transform of the resulting beat signal with a Blackman window yields a transform-limited beat signal of 1 mHz of full-width at half maximum (FWHM), as can be seen in the same figure. At the carrier frequency of the CW laser of 194 THz, this implies better than 1 part in 1017 agreement between the two techniques. The oscillations in the spectrum are due to the spectral shape of the Blackman window used. Another measure of stability that we can use to characterize the agreement between the conventional heterodyne and the GATOR is the Allan variance [RIL08]. There are many flavors of such a metric but all of those can be explained as being a measure of the variance of a signal (usually an instantaneous frequency as a function of time) as filtered by a filter whose duration can be adjusted. The exact choice of the type of measurement filter leads to the many flavors of this variance with somewhat different characteristics.

Figure 3-7 - Comparison results between a conventional heterodyne setup and a GATOR measuring the same frequency comb and CW laser beating together. The time-domain trace shows that the phase of the two beats agree within 100 mrad peak-to-peak over the full measurement duration, while the spectrum shows that the relative beat has better stability than 1 part in 1017 (1 mHz width over a carrier frequency of 194 THz). The green (dashed line) trace in the spectrum show the expected transform-limited shape of the window function used to compute the spectrum; highlighting the fact that the oscillating structure in the spectrum is due to the window function.

In Figure 3-8, we have chosen the modified Allan variance for its ability to properly discriminate between white phase noise and other types of similar noise. Indeed, we expect that the difference between the GATOR and the conventional heterodyne beat signal will be composed only of white additive noise, which is converted to white additive phase noise after extracting the phase of the beat signal. Any other type of noise indicates an unforeseen instability or inaccuracy of the setup. Computing the Allan variance of both the GATOR and the conventional heterodyne should give the same result for the timescales where the laser phase noise (really, the phase difference between the comb and the CW laser) dominates, and should give a lower result for the GATOR at timescales

Figure 3-8 - Modified Allan deviation of the GATOR signal (blue), the signal of the narrowband conventional heterodyne (green), the difference of the two (red). The red vertical lines indicate the +/- 1 sigma confidence interval for the red trace. The teal trace is the modified Allan deviation of the difference signal at longer timescales, while the purple trace shows this same signal after correcting for the phase mismatch of the −1.5 low-pass filters. The black diagonal lines indicate theoretical t behavior for white phase noise. where the measurement noise dominates. These two features can be seen in the lower part of the figure which gives a detailed view of the short timescales, where the GATOR and narrowband heterodyne modified Allan deviations can be seen to converge for longer timescales where the laser phase noise dominates, and diverge for shorter timescales where the narrowband heterodyne becomes

67 dominated by additive noise. Another interesting feature shown by this figure is that the narrowband trace stops much earlier than the GATOR trace, because the lower SNR of the narrowband trace meant that phase unwrapping could not be applied to the beat signal, while the higher SNR of the GATOR signal meant that shorter timescales could be observed without any unwrapping problems. Note that in a frequency counting experiment, the same problem would have been encountered where a narrower filter would have to be used to properly count the beat frequency without any cycle skips. The red trace shows the modified Allan deviation of the difference of the GATOR and narrowband phase signals. The good agreement with a t −1.5 slope is consistent with white phase noise caused by additive noise on the narrowband heterodyne signal, revealing no significant deviation which would indicate that the GATOR might degrade the long-term stability of the beat. We can also see from this plot that the white, additive phase noise of the GATOR setup was in this case one order of magnitude lower in standard deviation than the narrowband setup (comparing the two black t −1.5 slopes). At longer timescales, we have used the phase difference measured at each data window taken every second, yielding the teal trace. This trace has been divided by the deadtime correction factor of 50000.5 = 70 which accounts for the sparse sampling of the phase data (200 us every 1 second). Some slight deviations from the expected extrapolated t −1.5 behavior can be observed (which can also be observed on Figure 3-7, where the residual noise can be seen to have some slower structure), but this is attributed to the phase mismatch between the two analog low-pass filters used in this validation experiment. Indeed, each filter has a deterministic phase response which is a function of frequency, and since the beat frequency was not exactly constant during this experiment, the two filters add a small phase modulation to the two signals. Fortunately, we can retrieve this phase difference as a function of frequency by fitting a low order polynomial (we used a second order) to the phase difference signal as a function of frequency and removing this fit from the phase data. This process effectively cancels for the deterministic phase difference which was added by the phase mismatch. The corrected phase yields the modified Allan deviation shown in purple, which now follows closely the extrapolated t −1.5 slope, supporting our claim that the phase difference between the GATOR and narrowband heterodyne

68 is composed of additive noise, even at very long timescales. Note that at long timescales, the value of fractional frequency stability reaches the surprisingly low value of 5.3x10-23 (3.7x10-21 without the deadtime correction factor) at 250 seconds averaging time, but that this number only indicates the level of agreement between two separate measurements of the same quantity. These values do not reflect the absolute stability of the frequency measurement by our technique; it simply means that both the GATOR and the conventional heterodyne technique are affected by the same instabilities and in the same manner.

3.3 Optimal detection bandwidth for GATOR

In this section, we will derive the SNR for the beating of the comb and CW laser considering both the shot noise from the comb and that from the CW. This will allow us to compute the bandwidth which optimizes the SNR, knowing that comb shot noise increases with detection bandwidth, while CW shot noise decreases with the bandwidth. To do this, we will compute the SNR for the interference of a single pulse from the comb with the field of the CW laser, knowing that the SNR will be the same for all the pulses, and assuming that the shot-noise of the optical fields is the dominating noise process. This will correspond to an upper- bound for the SNR as other noise processes can only degrade this SNR, or hopefully, can be mitigated to below this level. With the analytic expression for this SNR, it will be possible to compute the optimal bandwidth as a function of the fixed parameters. We start by defining the parameters of the two sources in Table 3-1. Let us now define the experimental setup. Both fields are combined in a fiber coupler. We assume that the polarization of both fields is the same. The power quantities were defined as measured at the output of the coupler when there is no interference; that is to say that we do not include the factor of 2 because of the splitting of the power in two different ports of the coupler. We write all electric fields in units of root power to simplify notation. We also use the phasor notation to simplify the trigonometric computations. In front of the photodetector is an adjustable optical bandpass filter centered on the frequency of the CW laser that we can use to vary the comb bandwidth that is used in the experiment.

TABLE 3-1 DEFINITION OF PARAMETERS FOR THE DERIVATION OF THE SNR

Comb average power Pcomb

CW laser power Pcw Ratio between the average power of

the CW laser and the average power α = PPCW comb of the comb

Comb repetition period Tr

Energy in a single pulse from the Epulse = PcombTr comb before filtering

Length of a single unfiltered pulse Tp from the comb

Comb bandwidth BWoptical = TBWP/Tp

Time-bandwidth product of the pulses TBWP = Tp BWoptical Number of modes in the comb’s M=TrBWoptical=TBWP Tr/Tp unfiltered spectrum Ratio of transmitted comb power by 1 = = the optical filter optical 푇푝 훽 푇푓퐵푊 푇푓푇퐵푊푃 To keep the equations simple, we assume that this filter has a rectangular impulse response which is much longer than the unfiltered length of the pulse; the operation of this filter is to increase the duration of the pulse to a new length

Tf by selectively transmitting only a portion of the comb’s spectrum. When giving bandwidth or time duration values, we will use the noise-equivalent bandwidth: NEB = | ( )| max{| ( )| } as this will simplify the equations compared to the 2 2 traditional∫ 퐻 푓full⁄-width퐻 at푓 half푑푓 maximum (FWHM) used in ultrafast optics. With this normalization, the bandwidth of the filter is 1 f and it can be verified that ( ) the filter will transmit a ratio p f of the energy⁄푇 of each pulse. To ensure the fact that each optical pulse푇 ⁄ 푇is푇퐵푊푃 detected independently on the detector, we require , which in practice is a hard to break assumption for moderately low-repetition푇푓 ≤ 푇푟 rate combs (~100MHz).

We can now write the expression for the instantaneous power as seen by the detector:

2 Pt( ) =η ECW ( t) + Epf ( t) , (3.7)

70 where ECW et Ep are respectively the electric fields for the CW laser and that of the filtered comb, and h is the quantum efficiency of the detector. We now write the field of the filtered pulse as being a rectangular pulse in time:

  Ppf exp( j 2p ftc ) tT≤ f 2 Epf (t) =  (3.8) >  0 t Tf 2.

To compute the peak power of the filtered pulse, Ppf, we will start with the energy of the transmitted pulse:

Tf /2 T PdtP= T = PT p (3.9) ∫ pf pf f comb r T TBWP −Tf /2 f

P TT = comb r p Ppf 2 . (3.10) Tf TBWP

By combining (3.10) and (3.8) we obtain the expression for the electric field for the filtered pulse:

 1 PcombTT r p  expj 2p f t tT≤ 2 = ( c ) f Etpf ( ) Tf TB WP (3.11)  >  0 t Tf 2.

The electric field for the CW laser is simply:

Etcw ( ) = Pcw exp( j( 2pjftc + )) , (3.12) where is the phase difference between the interfering fields and is essentially the variable휑 that we want to measure. In practice, we will satisfy ourselves with measuring the cosine of this phase. We can now express the detected power as:

 1 Pcombrp TT 1 PPcw combTTrp hhP++2c h os(ϕ) tT≤2 = cw 2 f Pt( )  Tf TBWP Tf TBWP (3.13)  η >  Ptcw Tf 2.

Let’s include the effect of the response time of the electronic acquisition chain. To do this, we will consider that the value that will be retained is the integral of the power over a certain time, which amounts to assuming that the impulse response is rectangular and is sampled at a certain time. The units of this quantity will thus be those of optical energy (Joules). We will assume that the electronics

71 integrate over a time longer or equal to the length of the optical filtered pulse, that is . We will also separate the beating signal from the two other terms of the detect푇푓 ≤ 푇ed푒 energy, which are proportional to the power of the CW laser and the filtered pulse:

EEtotal= sources + E beat . (3.14)

Esources contains no information on the desired phase, while Ebeat is the desired signal:

Tf /2 1 PPcw combTTrp Ed= 2ηcos(ϕ) t (3.15) beat ∫ T TBWP −Tf /2 f

PP TT E = 2ηcw comb rpcos(ϕ) . (3.16) beat TBWP

It is interesting to note that this value is independent on the integration time of the electronics, which is expected because the interference of the fields is only present for the duration of the optical pulse, and integrating for a longer time will not change the result. Another less trivial point that we can make from this equation is that the amplitude of the signal is also independent of Tf, and thus the optical bandwidth selected by the optical filter, since the amplitude of the electric field of the filtered pulse decreases at the same rate as its length increases and thus the integrated beating energy stays the same. This fact is very interesting for the optimization of the SNR, because increasing Tf lowers the amount of shot noise produced by the pulse, but conserves the amplitude of the beating signal. This is a well-known fact that is frequently used to optimize comb-CW beats. The amplitude of the term which is independent on the beating is:

Te /2 1 Tp E = P( t) dt=ηη P T + E . (3.17) sources ∫ sources cw e TBWP T pulse −Te /2 f

We can see that this signal is made of two terms, one proportional to the integration time of the electronics, and a second which is only related to the energy in the pulse. This will have an impact on the behaviour of the SNR. In fact, the variance due to shot noise on the measured value will be proportional to the detected energy:

2 s shot = EEphoton total , (3.18) where Ephoton is the energy transferred by a single photon for this optical frequency. We will simplify somewhat this equation by using Esources to evaluate the variance of the shot noise instead of Etotal, since Esources corresponds to the average value of the total detected energy:

1 Tp s2 =hE TP + E shot photon e cw pulse . (3.19) TBWP Tf

In accordance with the convention of defining the SNR in terms of an electrical power ratio, the corresponding SNR is equal to the mean-square beating signal7 divided by the noise variance:

2 TTrp 2 4h PP E cw comb SNR =beat = TBWP . (3.20) shot s 2 sho t 1 Tp h + 2 ETphoton eP CW Epuls e TBWP Tf

A striking fact revealed by this equation is that the shot noise from the comb can be lowered in a seemingly arbitrary manner, by increasing Tf, the duration of the filtered pulse, in the limit that Te>Tf>Tp. Thus, independently of all other parameters, it seems desirable to increase Tf until the shot-noise term from the comb becomes negligible compared to the shot-noise from the CW laser. This implies that very little of the detected energy comes from the comb, while most comes from the CW laser. Once this is accomplished, we can neglect the second term in the denominator to obtain:

TT TT 42hh2 PP rp P r p cw comb TBWPcomb TBWP SNRshot = = (3.21) 2hETPphoton e CWe ETphoton

ETpulse pp11 T SNRshot = 22hh= Nphotons , (3.22) ETphoton eeTBWP T TBWP where Nphotons is the number of photons in the unfiltered pulse. In appendix A, the derivation is continued to show that this result is equal to the one given in

7 The averaging is done over all the possible values of the phase difference between the fields and reduces the beating power by an additional factor of 2 compared to the peak beating power. 73

[REI99], where the SNR of the beating of a single mode of the comb with a CW laser is given8. We call this case “slow detection”, as it corresponds to a case where the electronics response time is much longer than the duration of the filtered pulse. As we have shown experimentally in section 3.2, it is possible to improve the SNR to a better value by using a short integration time, essentially by gating out the shot noise from the CW laser that does not interact with the comb. Here, we will theoretically derive the SNR including the effect of the comb shot noise to show that there is an optimum integration time which optimizes the SNR.

In equation (3.22), we can see that the SNR corresponds to the number of photons in the pulse, multiplied by a factor p ( e ). Since the SNR scales as

e , it deteriorates when the integration time푇 ⁄of 푇the푇퐵푊푃 electronics is increased. This −1 푇decline comes from the shot noise from the CW power which is temporally separated from the pulse, but is still integrated in the measurement of the interference. From this observation, it is clear that to improve the SNR, the integration time must be reduced until the integration is performed over the shortest temporal window, which is during the pulse only, i.e. e = :

푇 푇푓 Tp SNRshot = 2h Nphotons . (3.23) Tf TBWP

This time, we can see that it is desirable to keep a small Tf in order to optimize the SNR. This being in conflict with the previous optimization goal that we used to neglect the shot noise from the comb, we have to backtrack to a more accurate equation in order to compute the optimal value. Starting with equation (3.20), we will also simplify the notation by expressing Ebeat and shot in terms of the pulse energy and two new parameters, namely the ratio of the휎 average power of the CW laser to the average power of the unfiltered pulses from the comb: = CW comb, ( ) and = p f , the ratio of the bandwidth of the comb that is훼 transmitted푃 ⁄푃 by the훽 optical푇 ⁄ 푇 푇퐵푊푃filter:

8 Appendix B pushes this one step further to show that the adjustable beam splitter technique presented in [REI99] yields the exact same improvement as using balanced detection and thus we do not pursue this approach any further. 74

E 2 = beat SNRshot 2 (3.24) s shot

42η2PP TTPETEPTEηa2 22ηa2 ηa22 E 2 = cw comb r p = combpulse p = pulse cr= pulse (3.25) beat 2TBWP TBWP M M

2 s shot = EEsources photon (3.26)

Tp Esources =+=+ηηη PTCW f Epulse hα PTEcomb f ηb pulse (3.27) Tf TBWP

αTf α EEsources=h pulse  += bh Epulse  + b (3.28)  β Tr M

2 α sshot =hEEphoton pulse + β. (3.29) M β

From (3.27), we can also separate the integrated shot-noise variance contribution from the CW laser and that of the filtered pulse:

α s2 =hEE (3.30) shot,cw M β photon pulse

2 s shot,comb =hb EEphoton pulse . (3.31)

Combining (3.24), (3.25) and (3.29) allows to compute the SNR:

Epulse α 1 SNRshot = 2h . (3.32) Ephoton M α + β M β

From this equation, we notice that filtering influences the amount of noise on the measurement in a non-trivial way. In effect, the integrated shot-noise from the CW laser increases when is reduced, while the integrated shot-noise from the pulse lowers with decreasing훽 . There is thus an optimal value for which will minimize the amount of noise,훽 and at the same time optimize the SNR,훽 since as we mentioned earlier, the signal amplitude is independent of the optical filtering or the integration time. Since the only free parameter for a given comb and CW laser is the optical filtering, we will minimize the variance with respect to the fractional bandwidth :

훽

dd2 αα− sshot =hEEphoton pulse +==β01 + (3.33) dβ dMββM β 2

α β = . (3.34) optimum M

By substituting the optimal value of beta in (3.30) and (3.31), the amount of shot noise which comes from both fields is:

α PP ss22= =hEE= hcmo b cw ET. (3.35) shot,cw,optimum shot,comb,optimum MMphoton pulse photon r

This highlights the fact that in order to minimize the sum of both noise terms when they have this particular functional relationship, one has to match both quantities. Indeed, this can be seen graphically on a log-log graph such as Figure 3-9, where one type of noise scales as the integration time, while the other scales inversely with the integration time and the sum reaches a minimum at the intersection point.

Figure 3-9 - By varying the integration time T, the minimum amount of total shot noise is obtained when the amount from the comb and the CW laser are matched.

In this case, this result also implies that the energy of the filtered pulse is the same as the integrated energy of the CW during the pulse. As the integration duration is the same for both signals, this implies that in order to optimize the

SNR, one has to filter the comb until the peak power of the pulses is the same as the average power of the CW laser, a result which closely resembles a similar situation involving the beating of two CW lasers. Then, the resulting optical pulse on the detector must be integrated over a time exactly equal to the pulse duration, or the noise has to somehow be gated out of the integration.

From this result, we can compute the SNR in the case of optimal filtering and short integration time by substituting (3.34) in (3.32):

Epulse a 1 SNRgator = 2h (3.36) Ephoton M aa + M M

Epulse a SNRgator =h . (3.37) Ephoton M

This result can further be simplified by acknowledging the fact that Epulse/Ephoton is the number of photons in a single unfiltered pulse from the comb:

a SNR=h N =hβ N . (3.38) gator photons M optimum photons

For the sake of completeness, we will state here the different assumptions that we have made during this derivation; this equation is subject to the following constraints:

TTTefp= = β= T pM α

M α ≤ TTpr (3.39) α M ≤ 1.

The first constraint is the optimal filtering condition, the second is to ensure that each pulse can be detected independently ( ), and the last one simply states that one cannot reduce the length of the pulse푇푓 ≤ 푇 푟by filtering ( > 1). The last two can be further simplified and combined to yield a single훽 criterion for the possibility of achieving the SNR in (3.38):

PP maxcw , comb < M , (3.40) PPcomb cw that is, the ratio between the larger power between the comb or the CW laser to the lower power has the be lower than the number of modes in the spectrum of

77 the unfiltered comb. Since the number of modes is usually a very large number (~38e3 for our 100 MHz system), this means that a very wide range of CW power can be detected with optimal performance before the optical filter needs to be as long as the comb repetition period, or that no optical filtering is necessary.

3.3.1 Interpretation of the theoretical result

Equation (3.38) shows that the maximum achievable SNR for this situation is equal to the SNR of shot-noise limited detection of the complete power of the comb transmitted by the optical filter. This is in sharp contrast to the equation given in [REI99], where the SNR is limited by the power in a single mode of the comb. The SNR improvement is thus equal to the number of modes in the transmitted portion of the comb. The number of modes of the comb that contribute to the beating and incidentally the SNR improvement factor is related to the transmitted fraction of the comb bandwidth and the number of modes in the unfiltered comb:

SNR Improvement Factor = M used = βaMM= . (3.41)

As explained in section 3.2.2, this improvement can also be understood in the frequency domain, where the integration and sampling of the photocurrent, synchronous with the repetition rate, can be seen as folding many aliases of the beating signal on the exact same frequency, effectively using many harmonics of the comb simultaneously to measure the beating. For the special case where the same amount of power is available from the comb and the CW laser ( = 1), which is close to many practical scenarios, the improvement is simply used훼 = . For our 100 MHz system, the possible improvement in this case is 푀 200. √ Anoth푀 er interesting comment that we can make based on equation √(푀3.41≈ ) is that the maximum improvement for a given comb system will be limited by the maximal bandwidth that can be achieved by the sampler. This improvement occurs when is at its maximum value constrained by the electrical system; for a given value of훽 M, we have:

BWelectrical,max BW electrical,max Max Improvement Factor = M = . (3.42) BWoptical,max Fr

To evaluate if the SNR gain from the GATOR is worth the increase in complexity of the faster detection, we plot in Figure 3-10 the SNR improvement for different cases of CW and comb power for our 100 MHz comb system. It can be seen that for a maximum electrical bandwidth of approximately 40 GHz, a maximum SNR improvement of 26 dB can be expected from this system, irrespective of the comb or CW power. It is worthwhile to remember that the bandwidth values given here are noise-equivalent bandwidths and thus for a first-order roll-off, a -3dB bandwidth of 40 GHz corresponds to ~63 GHz of noise-equivalent bandwidth.

Figure 3-10 - SNR improvement from GATOR vs slow detection. The required bandwidth axis shows the necessary optical and electrical bandwidth to achieve the SNR improvement in terms of the noise-equivalent bandwidth.

3.4 Chirped pulse heterodyne

The signal detection strategy of the GATOR achieved the highest SNR possible within the limits of the experimental setup presented. However, the GATOR technique suffers from two problems at higher comb power: first, the comb shot noise is concentrated on a single temporal bin of the detection electronics and as we will show, this produces more shot noise than is necessary, and second, the detected signal is a pulsed signal with low duty cycle and potentially high dynamic range, putting severe constraints on the linearity and noise properties of

79 the electronics. At a fundamental level, both of those problems occur because the comb pulse is very narrow in the time domain. In the optimization presented in the previous section, we had to make a compromise between the integrated shot noise from the comb and that of the CW laser because the detection bandwidth and the pulse duration were coupled. Here, we present a further improvement to this technique which removes this coupling and simultaneously addresses the two problems of the GATOR.

While the previous experimental setup assumed that the optical filter used to limit the pulse bandwidth had the minimum, transform limited duration for the chosen bandwidth, this does not need to be the case. If we allow a filter with arbitrary spectral phase, the duration of the impulse response can also be arbitrary for a given bandwidth. At the cost of increased complexity of the setup, we can now remove an important constraint which previously forced us to achieve a compromise.

For example, if we use a chirped filter with linear group delay as a function of frequency, we can spread the comb pulses over almost one full repetition period, yielding shot noise with constant variance as a function of time; essentially making the noise stationary, and at the same time dramatically reducing the dynamic range of the signal while keeping the same energy. In terms of noise and beat signal amplitude, the situation is now almost exactly like the beating of two CW lasers because the photodiode now sees almost constant power. The beating signal will now itself become chirped, but as we will show, it is still possible to extract the same information. This is the concept presented in sections 3.4.1 to 3.4.5. Other filter structures are also possible to achieve an impulse response duration longer than the transform limit and we will present a second one in section 3.6. Once we accept that the pulse duration will be stretched to approximately one repetition period regardless of the bandwidth, we are free to choose the bandwidth as we please for addressing other considerations.

To choose the bandwidth of the optical pulses, let’s first assume that we have an optical bandpass filter centered on the frequency of the CW laser which selects the spectral region of the comb that we will send to the detector. The spectral content of the comb that is offset from the CW by ±∆f will in turn produce

80 spectral content in the beat signal at frequency ∆f . Starting with a bandwidth of zero, it should be clear that increasing the bandwidth of the optical filter will increase the amount of detectable beat signal, up until the limit of the bandwidth of the electronics BWelectrical , that is to say until ∆=f BWelectrical . Indeed, increasing the optical bandwidth above the bandwidth of the electronics will produce not produce any more detectable beat signal but will still increase the total shot noise on the detector. The total optical bandwidth is now equal to the span ±BWelectrical ,

meaning a bandwidth of 2BWelectrical . The limit which defines the available power to produce a beat signal is thus that of the electronics (the photodetector and any electronic device downstream). For this reason, we call the power of the comb which lays within this bandwidth around the frequency of the CW laser the spectrally available power. This concept is important because as we will show in this section, it is possible to reduce the comb-CW heterodyne experiment to a situation that is very close to beating two CW lasers, one with the same power as the CW laser in the experiment, and one with power equal to the spectrally available power from the comb. Similarly, the spectrally available energy is the energy available in a single pulse after filtering to match the optical bandwidth to the electrical bandwidth. This type of bandwidth scaling law is different than the rule derived in section 3.3 because in the present case, the electrical bandwidth is set by what is achievable in practice; the more we can get the better, while for the GATOR, the bandwidth had an optimal value based on power and noise considerations.

3.4.1 Case of a single chirped pulse beating with a CW laser

In this section, we derive the shot-noise limited SNR for the beating of a single chirped pulse with a CW laser. We assume that the pulse has been optically bandpass filtered to match the bandwidth of the electronics, and that we can choose the pulse duration by varying the chirp. We use the definitions from Table 3-2.

TABLE 3-2 DEFINITION OF PARAMETERS FOR THE DERIVATION OF THE SNR IN THE CHIRPED PULSE CASE

Electric field of the pulse, after optical filtering to match Aunchirped(t) the electrical bandwidth, but before chirping

Electric field of the pulse, after optical filtering to match Achirped(t) the electrical bandwidth and after chirping

Electric field of the CW laser ACW(t)

Peak power of the filtered, unchirped pulse Punchirped

Length of the filtered, unchirped pulse Tunchirped

Length of the filtered, chirped pulse Tchirped Phase difference between the pulse and CW laser at t=0 φ0 Phase of the chirped pulse as a function of time φ (t) chirp

We start by writing the equation for the electric field of the unchirped pulse. We write the electric fields in units of root power to lighten the notation:

  Punchirped exp( jt 2p fc ) 0 <

Note that this pulse is already filtered in an optical bandwidth that is matched to the electrical bandwidth, but its duration is still transform-limited. If we send this pulse through a chirped optical filter, we can increase the duration of this pulse by a factor k, while conserving the initial pulse energy. Here, as illustrated in Figure 3-11, we make a simplifying assumption that the chirped pulse amplitude is still rectangular in time, as this will keep the derivation tractable. In practice, the impulse response would start smoothly and taper off gradually at the end, but by proper filter design, a well-behaved apodization window can approximate a rectangular shape to good accuracy. In any case, the results presented here will be correct up to a small correction factor which accounts for the shape; the scaling of the parameters will remain the same. Without further ado, we can write the chirped pulse at the output of the chirping filter:

 P  unchirped pf+ << = exp( j 2 ftc t j chirp ( )) 0 t kTunchirped Atchirped ( )  k (3.44)   0 elsewhere,

where φchirp (t) accounts for the extra phase of the chirped pulse.

Power

Punchirped

Punchirped/k

Tunchirped kTunchirped Time

Figure 3-11 - Illustrating the scaling between the chirped and unchirped pulses.

It can easily be verified that the chirping transformation applied here conserves the energy between the unchirped and chirped pulse. Similarly, we can write the electric field for the CW laser as:

AtCW () = PCW exp( j 2pffc tj+ 0 ) . (3.45)

The beat signal that is detected consists of the power as a function of time:

22 2 * Pt()()()=hAtACW +=chirped t hh AtCW () ++ Achirped ()t2 h Re{ Achirped ()() tACW t} . (3.46)

We assume that we have characterized a priori the chirp function and we will use it in a post-detection correlation operation to retrieve the beat information. In order to keep the SNR derivation simple, we will give all the details about the demodulation strategy in section 3.4.4. The detected interference signal will be the result of the correlation between a template of the chirp signal and the beat signal:

kTunchirped = − φ Ebeat ∫ Pt()exp()jchirp ()t dt . (3.47) 0

We will now split the interference term from the other terms, in order to independently account for the integrated energy (and thus shot noise) into the measurement, and the amplitude of the beating signal:

kTunchirped = hφ* − Ebeat ∫ 2 Re{Achirped ( tA) CW ( t)}exp( jchirp ( t))dt . (3.48) 0

Expanding, we get:

kTunchirped PP = h CW unchirped φφ−− φ Ebeat 2 ∫ cos( chirp (t) 0 )exp( j chirp (t))dt (3.49) 0 k

kTunchirped PP =hCW unchirped −+φ φφ − Ebeat ∫ exp( j0 ) exp( 2 jchirp ( tj) 0 ) dt . (3.50) 0 k

The second term in the integral will average down to zero because it is oscillating over the integral duration, yielding:

ETbeat=hφ unchirped PPCW unchirped kexp(− j 0 ) . (3.51)

As the SNR is conventionally computed as a ratio of power or energy in an electrical signal, we take the square of this signal, averaged over all initial phases:

2 2 = h ETbeat ( unchirped PPCW unchirpe d k) (3.52)

2 2 2 ETbeat =h PPCW unchirped kunchirped . (3.53)

The variance of the shot noise integrated into the result will be proportional to the total number of photons, or the optical energy integrated in the measurement. Note that multiplying the signal by the complex chirp template does not affect the variance of the result, because the noise is uncorrelated temporally and changing its phase does not introduce correlation. Averaging over all initial phases of the interference (that is, neglecting the interference terms), we get:

kTunchirped 22 = hh+ Esources ∫ AtCW ( ) Achirped ( t) dt (3.54) 0

P EP=+=+hhunchirped kT kP P T sources CW unchirped ( CW unchirped) unchirped , (3.55) k which yields a shot noise variance of:

2 sshot= EE photon Esources =hphoton (kPCW+ P unchirped) T unchirped . (3.56)

The SNR is thus:

2 2 2 Ebeat h P PkT SNR = = CW unchirped unchirped (3.57) s 2 shot hEphoton (kTPCW+ P unchirped) unchirped

2 Ebeat hPP T SNR = = CW unchirped unchirped . (3.58) s 2 1 shot Ephoton (PPCW+ k unchirped )

At this point, it is clear that we can improve the SNR of this beat signal by increasing the chirping ratio k. Here, k = 1 corresponds to the unchirped case, or the GATOR with the bandwidth set to the maximum available without any optimization. From this, we can see that the effect of adding chirp to this heterodyne signal is to lower the effect of the comb shot noise on the measurement. Indeed, in this idealized situation where we have only one pulse beating with the CW laser, we can set k arbitrarily high and totally reject the influence of the shot noise from the comb:

hPP T lim{ SNR} = CW unchirped unchirped (3.59) k →∞ Ephoton PCW

hhPEunchirpedTunchirped unchirped lim{ SNR} = = =hNunchirped , (3.60) k →∞ EEphoton photon where Eunchirped and Nunchirped are respectively the energy and the number of photons in the unchirped pulse, in other words the spectrally available energy. The interesting fact here is that all the energy in the pulse that is contained in the detection bandwidth can indeed be used to produce the beating, not just the energy contained in a Fr bandwidth (which would correspond to a single mode if we had a pulse train instead of a single pulse). The difference between this situation and the GATOR is that here we have explicitly set the measurement bandwidth to be the highest that we can achieve, without any compromise, yet we have still reached full utilization of the spectrally available energy. The fact that the number of photons in the pulse sets the SNR should not come as a surprise, as by considering a single pulse with finite energy beating with a CW laser with constant power and unlimited duration, we have implicitly assumed that our CW laser can contribute infinite energy. In essence, we have used the coherent gain of the CW laser to its maximum by stretching the pulse until only the CW shot noise dominates the measurement, to reach an SNR equal to the number of photons in the spectrally available energy. 85

3.4.2 Case of a chirped pulse train beating with a CW laser

Now that we have the result for the SNR of the beat between a single chirped pulse and a CW laser, it is very easy to derive the result for the SNR of the beat between a chirped pulse train and a CW laser. We simply have to recognize the fact that we can no longer apply unlimited chirping ratio to the pulse; we are constrained to at most make each pulse end when another one begins. Past that point, there is no more advantage to be gained from stretching even more because this will not spread the pulse photons any further. We thus use Eq. (3.58), with the additional constraint that the chirped pulse duration is equal to the pulse repetition period, Tr, which is equivalent to imposing k = Tr/Tunchirped:

hPP T SNR = CW unchirped unchirped . (3.61) Tunchirped Ephoton PPCW + unchirped ( Tr )

Note that the chirped pulses from the comb now produce a continuous flow of power; the shot noise statistics are thus stationary9. In terms of SNR, the situation should now be completely analogous to one where two CW lasers beat together as both the signal amplitude and the noise variance is constant. Indeed, the spectrally available power can be written in terms of the peak unchirped power as: PPSAP= unchirped T unchirped Tr , allowing equation ref to be written as:

hP PT SNR = CW SAP r , (3.62) Ephoton (PPCW+ SAP ) which is exactly the SNR result for two CW lasers beating together over duration

Tr, one with power equal to the CW laser, and the other with power equal to the spectrally available power from the comb [KIN78]. We can also verify that in the two limiting cases of spectrally available power much higher or much lower than the CW laser’s power, we have the same results as for two CW lasers beating:

hPTSAPr SNR = =hNSAP , (3.63) PPCW>> SAP Ephoton

hPCWTr SNR = =hNCW , (3.64) PPCW<< SAP Ephoton

9 This neglects the fact that the photon arrivals are in fact modulated by the interference, but this degree of non-stationarity is lesser than when detecting a very low duty cycle pulse. 86 where NSAP is the number of spectrally available photons per pulse, and NCW is the number of photons from the CW laser in one repetition period, the fundamental time unit used for this experiment. This fact shows that we have used all the available power from both sources to the fullest, and can be contrasted with the result of section 3.1, where only the power in a single mode of the comb could be efficiently used, and those of section 3.3, where the power in many modes was efficiently used, but we were constrained to use a specific detection bandwidth instead of the highest available as in the present case.

3.4.3 Interpretation of the results and observations

All of these schemes aim to improve the overlap between the two fields: we are trying to transform the field from the comb to match as much as possible that of the CW laser. The fundamental reason why this achieves SNR improvement is because of the scaling laws of the shot noise statistics and the heterodyne signal amplitude with respect to the pulse shape. While this statement is somewhat imprecise, we can make it more accurate by contrasting the SNR scaling of the comb-CW heterodyne experiment against two similar signal detection situations: a coherent communication system (for example, a digital radio system) operating in additive noise of a constant origin, and direct (incoherent) detection of optical pulses operating in the shot noise limit. These two situations have slightly different signal and noise scaling laws with respect to the pulse shape, and as such produce a different SNR scaling law.

In the case of a coherent communication system with constant additive white noise, the signal amplitude scaling law is the same as the chirped comb-CW heterodyne experiment (k-0.5 as a function of the chirping ratio k), while the noise scales differently: the noise variance is independent of the pulse shape. In this situation, the well-known matched filter result [SKL01] tells us that the SNR for a single bit is independent of the pulse shape and is only dependent of the pulse energy, and thus no gain can be expected from chirping.

On the other hand, the direct detection of the amplitude of an optical pulse at the shot noise limit has the same noise scaling laws as the comb-CW heterodyne experiment (the temporal variance scales as k-1), but this time the signal scaling law is different. For direct detection, the signal amplitude scales as k-1, while for

87 the heterodyne case the beat signal amplitude scales as k-0.5. Once again, this different means that the shot-noise limited for direct detection is independent of the pulse shape, and thus the chirping ratio.

From these two other cases, it shouldn’t be surprising that the SNR scaling law for the heterodyne experiment is a function of the chirping ratio. The exact functional relationship of course requires more careful attention, as shown in section 3.4.2.

Note that chirping the comb pulse is not the only possible way of achieving this good match between the two fields. We could also have worked in reverse: trying to match the CW field to the comb by chirping it using a phase modulator, then producing a narrow pulse by passing the chirped CW field through a chirped FBG with the appropriate group velocity dispersion. That however would cause practical problems at the detection, because it would worsen the dynamic range of the signal instead of lowering it like we did in the chirped pulse approach with the correlation detection. Another conceivable possibility is to use optical signal processing techniques such as a Duguay shutter [DUG71] to address a much wider detection bandwidth, although this poses formidable implementation challenges to reach low noise characteristics.

One useful characteristic of this chirped heterodyne scheme is its behavior when used with a weakly non-linear (electronic) device. Of course, the major effect of using a chirp instead of a transform-limited pulse is to dramatically reduce the amplitude of the signal for a given energy and thus excite very little any non- linearity in the system. There is however an additional non-trivial immunity of the frequency chirp to second-order non-linearity which is well known in the field of audio engineering and system identification [STA02]. To describe this advantage, let us first consider a conventional heterodyne experiment being affected by second order non-linearity. In this case, the second order non- linearity of the beat signal will produce spectral content at DC and at twice the beating frequency, which will have to be rejected by using a tighter bandpass filter, limiting the useful signal bandwidth and thus the time resolution of the heterodyne system. If however, the beat pulses are chirped, then the second- order non-linear effects will also appear at the second harmonic during the chirp.

This second harmonic signal will have twice the chirp rate and will thus not be dechirped by passing through a matched filter, or will produce very little correlation with the expected chirp template. This produces an additional rejection of this non-linearity, based on its different time-frequency characteristics instead of only its spectral characteristics.

Finally, chirped pulse heterodyne has another desirable characteristic which is absent from a conventional heterodyne between a comb and a CW laser. Indeed, when beating a transform-limited, narrow pulse from a comb with a CW laser, we are essentially sampling the field of the CW laser over a very short duration. This sampling occurs at a rate equal to the repetition rate, whose period is usually much larger than the duration of the pulse, even after optical filtering. In essence, we are grossly undersampling the CW field, and any amplitude or phase fluctuation will be aliased from the whole optical bandwidth of the comb back into the Nyquist range of the comb. On the other hand, while using chirped pulse heterodyne with the signal detection strategies presented here, the beat information is integrated over (ideally) the whole duration between each pulse, yielding much more reasonable sampling of the CW field with respect to the Nyquist frequency of the repetition rate. This means that if the CW laser contains amplitude or phase fluctuations at higher frequencies than the repetition rate, the chirped pulse heterodyne setup will be unaffected (provided that they account for less than approximately one radian of phase, which is a given if the result of even the conventional heterodyne is to make sense), while the conventional heterodyne setup will show excess phase and amplitude noise.

3.4.4 Signal detection strategies

In the derivation above, we purposely refrained from giving too many details on the detection strategy, since we wanted to keep the main focus on the SNR result. In this section, we will explain in more detail the processing strategies that one can use to retrieve from a chirped pulse heterodyne setup the same kind of beating information that is usually obtained from a conventional heterodyne setup. There are two main approaches that one can use to retrieve the beat signal, namely the matched filter approach and the correlator approach. These two signal processing techniques are widely known and use extensively in digital

89 communications [SKL01] and can be shown to give completely equivalent results, although the implementation details are quite different. Indeed, in the case of chirped pulse heterodyne, we will see that the correlator approach is to be favored in most cases.

3.4.4.1 Beat signal model

To illustrate the signal and the effect of various processing steps, it will be most useful to use the short-time Fourier transform (STFT), also sometimes called the time-frequency representation. This transform takes a one dimension signal and produces a map of spectral amplitude as a function of both time and frequency, which allows particularly simple representations for signals which contain oscillating terms with time-varying instantaneous frequencies. For example, a linear frequency chirp of the form exp( j 2π ft 2 ) will show up as a single linear region of high values. The transform is defined very simply, by computing many Fourier transforms over short temporal windows, as can be inferred from its name:

+∞ S( ft, ) =∫ stw( ) (t −− t)exp( j 2pft) d t, (3.65) −∞ where S(f, t) is the short-time spectrum, s(t) is the signal to be analyzed and w(t) is a suitable window function which tapers off after a characteristic time extent. The size of the window can be chosen at will and is a trade-off between frequency and time resolution. As can be expected, longer windows mean high frequency resolution but poor time resolution. The exact shape of the window dictates some parameters of the transform such as dynamic range (in this case, the ability to see a small signal in the presence of a larger one), sensitivity (the ability to accurately a small signal buried in noise), and the time-bandwidth product (the constant of proportionality in the time and frequency resolution trade-off). In any case, we will use this transform only for illustration purposes and the only important parameter will be to adjust the width of the window to a suitable size to capture the frequency chirp with enough time and frequency resolution. This transform will be used on the experimental results to illustrate the processing algorithm with actual data, but first we will present a model of the beat signal using pictorial representations of such transforms. 90

After filtering the pulses from the comb by the chirped filter, the optical pulses will have a time-frequency representation in the form of a linear chirp, while the CW laser will present a line of constant frequency10. For example, the representation in Figure 3-12 illustrates four consecutive pulses and the CW signal.

Optical frequency

Tr 2Tr 3Tr 4Tr Time

Figure 3-12 - Illustrating the time-frequency representation of the chirped optical pulses from the comb as the blue diagonal lines indicating linear chirp, and the orange horizontal line depicting the CW laser signal.

After detection by the photodiode, the beat signal will have a time-frequency representation that is comprised only of the pairwise difference frequency components of the CW laser and the chirped pulse, yielding the chirped beat signal shown in Figure 3-13. This signal is a direct replica of the comb signal at the optical frequency, translated to baseband by the CW signal, imprinting its phase fluctuations at the same time. As long as the chirped pulse template h(t) is known, this signal can be processed to yield a beat signal with only the relative frequency fluctuations of the comb and CW laser, making chirped pulse heterodyne completely compatible with a setup which previously used a conventional heterodyne. The SNR gain comes from the fact that the shot noise realization is set during the detection, where the noise statistics have been favorably manipulated by choosing the signal waveform appropriately. Once the

10 The frequency of the CW laser is of course not exactly constant, but for most comb-CW beats of interest, the frequency of the CW laser will be approximately constant relative to the frequency excursion of the chirp. 91

Beat frequency

Tr 2T 3Tr 4Tr Time

Figure 3-13 - Time-frequency representation of the chirped beat signal. Each chirp (blue diagonal line) is produced by the beating of the CW laser and a single pulse of the comb. The frequency axis has the same span as for the optical signal, but is now centered around zero. photon shot noise realization is set and the photocurrent has been amplified, we can safely manipulate the electrical signal as we wish without being affected by photon noise statistics.

3.4.4.2 Matched filter: De-chirping + GATOR

The first approach to process this signal is simply to use a linear, time-invariant filter which compensates for the chirp of the beating pulses to produce transform- limited pulses. This process is illustrated in Figure 3-14, where filtering the beat signal by h*(-t) introduces a frequency-dependent delay which matches the group delay characteristics of the chirped beat pulses, yielding a transform-limited beating pulse every Tr. Once this is accomplished, the signal is now completely analogous to the signal produced by a conventional heterodyne setup with no chirp11. It can thus be processed by a gate signal exactly as in the GATOR to remove all the noise that occurs at different times than the beating pulses. Finally, as in the GATOR, this gated signal can now be low-pass or bandpass filtered in the same manner as the signal from a conventional heterodyne setup,

11 Apart from the noise characteristics, of course, which have been modified by the presence of chirp at the detection. Note that the de-chirping operation does not affect the distribution of the noise, since it is uncorrelated temporally and thus has random group delay characteristics, which stay random by addition of the deterministic group delay of the filter. 92

A) Block diagram Gate pulses

Chirped beat Beat pulses h*(-t)

B) Time-frequency map Frequency

Tr 2Tr 3Tr 4Tr Time

Figure 3-14 - Filtering the chirped beat signal (blue diagonal lines) with a chirp compensating filter yields a transform-limited pulse every Tr period (orange vertical lines) since the group delay of each frequency is now adjusted to be the same. This signal can then be gated to remove noise while keeping signal unaffected. but it is now of much higher SNR. Indeed, the signal has been gathered into a single temporal bin, while the noise, because it is uncorrelated temporally, has merely been shuffled by the de-chirping operation. This is can be understood because the stationary noise has random group delay characteristics, and it will of course stay random by addition of the deterministic de-chirping group delay.

3.4.4.3 Correlator

Another possibility for processing the beat signal (Figure 3-15) is to multiply the chirped beat signal with a repeating template of the complex conjugate of the chirp signal, h*(t). This cancels the chirp to yield a continuous wave signal with constant amplitude over the duration of the beat. This beat signal can potentially

A) Block diagram Gate pulses Chirp template h*(t)

Chirped beat Baseband beat

B) Time-frequency map Frequency

Tr 2Tr 3Tr 4Tr Time

Figure 3-15 - Multiplying the chirped beat signal (blue diagonal lines) with a repeating template of the complex conjugate of the chirp collapses the chirp to a single frequency (orange horizontal lines). Most of the energy in the signal is now concentrated into a single, constant frequency component. All that is left is the amplitude fluctuation of the beat signal, which can be averaged out in the usual manner by filtering. have residual amplitude modulation that is periodic with Fr, but this is no different and even much less severe than having a sampled beat signal from a conventional heterodyne setup; it can easily be filtered out to yield a continuous signal. In this scheme, the template of the chirp is exactly matched to the chirp, but this scheme also works when the template is offset by a constant frequency from the chirp. In this case, the resulting beat signal will also be at an offset frequency different from baseband, and this can be used advantageously to enable the use bandpass RF amplifiers.

3.4.4.4 Implementations

The simplest implementation of the above techniques, shown in Figure 3-16 A), is definitely to use an ADC with wide enough bandwidth, sampling rate and

Figure 3-16 - Three different implementations of chirped pulse heterodyne. In A), the comb pulses are recorded on the oscilloscope to use as a gate. Optical fibers are solid lines and dashed lines are electrical coaxial cable. BPF: Optical bandpass filter, PC: polarization controller, DL: Adjustable optical delay line, D: wideband photodetector, LPF: low-pass filter, LNA: Low-noise amplifier. cFBG: Chirped fiber Bragg grating. h*(- t): Matched filter. B) and C) show two possible hardware implementations of the technique and the signal flow. dynamic range to properly digitize the chirped beat signal, then to implement either a correlator or a matched filter in digital programmable electronics, either as software on a processor or as hardware on an FPGA. While the dynamic range requirement should hardly be an issue for anything but situations with extremely high SNR (which means very high power from both the comb and the CW laser), the bandwidth and sampling rate requirement can pose significant challenges, especially for real-time processing. Indeed, our proof of concept presented in

95 section 3.4.5 was conveniently implemented as software post-processing using data acquired by a very high bandwidth oscilloscope (Agilent DSO81004A). Although it could not be described as state-of-the-art as it is already a discontinued model that has been superseded by much faster ones, this oscilloscope is still much faster than currently available off-the-shelf ADCs boards for FPGA processing boards, which as of 2013 range in the low GHz and GS/s range. If a fast enough ADC was available however, the implementation of the correlator algorithm in an FPGA would be a simple matter, as it requires very little hardware: it only needs one multiplier and one adder, and the algorithm can be heavily pipelined or parallelized for high throughput as there are no feedback loops.

Fortunately, both the matched filter and correlator approaches lead to a hardware, real-time implementation which does not require digital electronics. These are respectively shown in Figure 3-16 B) and C). The matched filter approach in Figure 3-16 B) is conceptually the most straightforward to implement: we simply have to design an electrical filter that has the desired group delay characteristics as a function of frequency. Such a filter can be designed and implemented straightforwardly using a structure called a “noncommensurate transmission line” [GUP10]. It is essentially a cascade of second order all-pass sections, each composed of two coupled transmission lines etched onto a dielectric substrate. This filter can approximate any group delay characteristic to fairly good accuracy and has very reasonable fabrication tolerances. The obvious downside is that each design is etched onto a printed circuit board and cannot be tuned. In Figure 3-16 C), the correlator approach is shown, once again using a transmission line all-pass filter, but this time it is used to generate the required chirp template, which is amplified and sent to the mixer to implement the multiplication operation. A fourth approach which is not shown here would be to generate the chirp template by designing another chirped fiber Bragg grating, this time covering the whole bandwidth of the comb spectrum, but with an amplitude modulation which would generate the desired chirped signal when detected.

3.4.4.5 Comparison of the matched filter and correlator approaches.

While the matched filter and correlator approach can appear to be two completely different concepts, it can be shown that both approaches yield the same exact linear transformation from the input signal to their output. This is the aforementioned result from communication theory which can be understood by simple examination of the convolution equation implemented by the filter and the sampling operation. Indeed, by comparing the definition of both operations, the relationship is mostly trivial, apart from a time-reversal, and the fact that the convolution outputs a function of time rather than a scalar. From [SKL01, appendix A], the convolution operation of a signal x(t) by a filter a(t) can be compared to the correlation of the signals x(t) and b(t):

+∞ Convolution: yt( ) =∗≡ xt( ) at( ) ∫ x(t) at( − tt) d (3.66) −∞

+∞ Correlation: zx≡ ∫ (t)bt( ) dt . (3.67) −∞

The sampling operation merely selects the result of the convolution y(t) for a given t, for example t = 0. In this case, equation (3.66) reduces to:

+∞ y(0) =∫ xa(t) ( − tt) d, (3.68) −∞ which is exactly the same equation as the correlation (3.67), provided that a(- t)=b(t). Thus, by sampling the result of the matched filter operation at the appropriate times, the system comprised of a de-chirping filter + a gate produces the exact same results as the correlator with a time-reversal between the filter and the correlator template. However, there are significant practical differences in the two, namely in the intermediate results of the computation. Indeed, if the processing is implemented digitally, the de-chirping filter is needlessly computing the result of the correlation of the input signal and the template for every input point, but only keeping the result for one point every pulse. This is computationally very inefficient and can be optimized down to compute just the necessary results, at which time the system is now the same as a correlator. If instead the system is implemented in analog hardware, then the matched filter approach has a significant disadvantage because the intermediate signal of the

97 matched filter approach occupies much more dynamic range than the intermediate result of the correlation. Indeed, for the matched filter the signal is compressed onto a single temporal impulse, and we now have the same dynamic range problem as in the GATOR technique. This fact needlessly increases the requirements on the linearity and noise properties of the amplifier at the output of the system because after gating, most of the noise has been removed while the signal is a very narrow and high pulse. On the other hand, the correlator approach instead produces an intermediate signal which is a constant frequency sine wave and both the signal and noise have approximately the same amplitude as before the mixer, making very efficient use of the dynamic range of the output amplifier.

3.4.5 Experimental results

The setup of Figure 3-16 A) was used to produce chirped pulse heterodyne signals between our Menlo, 100 MHz, 20 mW comb and a Redfern Integrated Optics PLANEX external cavity CW laser with 8 mW of power. The power from the CW laser was reduced to 0.5 mW measured at the input of the Bookham PT10G 10 GHz photodiode to avoid non-linearity. The signal was digitized and recorded by an Agilent DSO81004A, 10 GHz, 40 GS/s oscilloscope. The comb was filtered by a chirped fiber Bragg grating from Teraxion, custom-designed to our specifications of 10 GHz of -3 dB bandwidth, approximately 6 ns of impulse response duration. This duration was chosen conservatively to avoid overlap issues between different pulses for this first proof of concept although the concept should in principle scale to impulse responses which overlap between neighboring pulses. The bandwidth was also chosen conservatively to be equal to that limited by our fastest oscilloscope at 10 GHz, on a single side of the CW laser, rather than +/- 10 GHz, once again to keep this initial demonstration simple, although we do not foresee significant problems with using a +/- 10 GHz chirp around the laser frequency. No additional attenuation was added to the comb to reach the highest SNR possible. A total of 1.74 μW spectrally available power from the comb reached the detector, yielding 14 nW per mode with 12 GHz of noise-equivalent bandwidth compared to the 10 GHz -3 dB bandwidth specification. Note that without chirping the comb pulse, the beat signal would

98 not have been measurable with this photodetector (and indeed, none of the others available in our lab) at this power level because it would have been way above the saturation of the detector. The quantum efficiency of the detector including any coupling loss between the fiber and the detector was carefully measured to be 0.7, by measuring the power with a calibrated power meter and the average photocurrent.

The complex frequency response of the FBG was characterized by the manufacturer using a Luna Technologies OVA instrument. The complex spectrum was inverse Fourier transformed to yield the impulse response in the time domain and to characterize the time-frequency representation of the grating, shown in Figure 3-17, in order to compare with the chirped pulse heterodyne results. The figure is shown in two different aspects ratios to enable easy visual comparison with both Figure 3-13 and Figure 3-18. This figure can be contrasted with Figure 3-13 to see the differences between the actual signals and their idealized representations. The first difference is that the shape of the beat signal is not quite a linear chirp. It is rather formed of two sub-pulses with two different, mostly linear chirp rates with some deviations. This is in fact simply that the shape of the impulse response of the fiber Bragg grating is not exactly as specified. The actual design was produced in house at Teraxion and is proprietary so we did not have access to the details, other than the characterization of the impulse response and spectrum. One thing that we can see is that they mostly optimized for a flat spectrum over the specified bandwidth, even though the specifications were both on the time domain and the frequency domain. This is understandable as most FBG customers specify their gratings in the frequency domain first. The good news is that this is not critical in that the technique works just the same, and the only effect is to reach a slightly lower SNR increase and slightly higher dynamic range than could have been achieved had the chirp been closer to ideal. Another unimportant difference is the fact that the beat is composed of a downchirp instead of an upchirp, which makes no difference whatsoever for the processing.

Figure 3-17 - Fiber Bragg grating impulse response characterisation, shown in two different aspects ratio to enable easy visual comparison with the chirped pulse heterodyne results.

3.4.5.1 Results and processing steps

Figure 3-18 shows the first few pulses of both the time-domain signal and its time-frequency representation. This figure can be compared to Figure 3-17. Some slight differences with the FBG characterization can be seen in the impulse response shape, and this can be explained in part by the fact that the characterization was done at Teraxion, while the FBG was still under mechanical constraint in the writing setup, while the beat signal was produced with the FBG simply resting on an optical table. Also, such a long grating is particularly sensitive to strain, which can induce local birefringence, and thus a perfect match between the two responses is not to be expected as no effort was expended to match the strain and polarization between the two setups. We also opted to display both the positive and negative frequencies of the signal, in order to display one feature of the dataset, which is some slight aliasing at low frequencies. The pulse content below and above the CW laser frequency is mapped to the same frequency, at very close times, and these two interfere when we compute the STFT. This pattern repeats with an approximate period of 4 pulses because the beat frequency for this case is around Fr/4.

100

Figure 3-18 - Raw chirped pulse heterodyne signal. On top: time-frequency representation and bottom: time-domain signal.

Note that over longer time scales, this interference only yields beat notes at two different frequencies. An interesting fact apparent from this representation is that we can clearly see signal content that does not follow the main, mostly linear chirp. This is either caused by the impulse response of the grating being different than its design specification, or because the pulses themselves had content that was not exactly coincident with the peak of the pulse, for example amplified spontaneous emission (ASE) noise. In fact, a lot of the background signal can be seen to be very well correlated with the beating at every pulse (for example around 2 ns and 42 ns, from 0 to 8 GHz) and matches with features seen in Figure 3-17 and thus in fact contributes to the beat signal and is indeed coming from the non-ideal impulse response of the grating.

Numerically processing the dataset with the correlation approach yields results shown in Figure 3-19. We can clearly see the main signal content, now concentrated around zero frequency. The oscillating content in the time domain signal is due to the negative frequencies and will be removed by the filtering operation that is performed after the multiplication. The result of this filtering operation yields the signal in green on the bottom figure, where we can see the very high SNR of the resulting beat note, as this signal was filtered in 50 MHz.

101

The blue trace shows that the beat information, after multiplication with the pulse template, is present

Figure 3-19 - Chirped pulse heterodyne results after processing by the correlation approach. The bottom blue trace shows the signal after multiplication with the chirp template, while the green trace is the output result, low-pass filtered in the Nyquist range of the comb, 50 MHz. continuously in the short-time average of the signal. For example, in the signal burst just before 35 ns, the beat going through a zero crossing which is recovered by low-pass filtering the result. Another interesting feature that is shown is that the multiplication with the template weighed the signal to almost zero in regions where the signal amplitude was already low, reflecting the fact that there is less incremental signal compared to the incremental noise to be gained from these regions.It is also interesting to compare the intermediate results of the correlation approach with that of the matched filtering approach, shown in Figure 3-20, where it is apparent that the signal is much closer to the cartoon version of Figure 3-14. This is once again because the company that did the fiber Bragg grating put more emphasis on achieving the design specifications in the frequency domain (mostly flat spectrum over 10 GHz) than in the time domain. Hence, the spectrum of the filter's response is closer to the specifications than the time-domain impulse response. On the other hand, the correlation approach reveals more easily the temporal shape of the impulse response by removing the chirp at all times.

102

Figure 3-20 - Chirped pulse heterodyne results after processing by the matched filter approach. The bottom blue trace shows the signal after dechirping, the red dots show the gated/sampled points, while the green signal is the gated and low-pass filtered signal.

3.4.5.2 SNR performance

Based on the values provided in the beginning of section 3.4.5, we can compute the expected shot-noise limited SNR for a single comb mode for the current setup at 46.5 dB and compare to the SNR achieved by the chirped pulse heterodyne setup. The spectrum for the raw signal, the processed signal, and for the measured noise floors is shown in Figure 3-21.

103

This figure shows several interesting features. First of all, we can see that at this operation point, this setup shows only 2 dB more noise when the CW shot noise is added to the detection thermal noise. This thermal noise is dominated by the amplifier embedded in the photodetector and thus a lower noise photodetector or a more linear detector (which would allow us to send more CW power and thus boost the signal compared to the thermal noise) would directly translate to an SNR increase of the setup. Another important feature is that when superimposing the comb signal to produce the beat, depending on the spectral region, we reach either mostly thermal + CW shot noise limited operation, for example around 1.45 GHz, while around 1.41 GHz, we find that the performance is almost completely dominated by additional noise of another nature. We can also rule out comb shot noise since the average power of the comb is much lower than the CW power, and this shows up even in the unprocessed beat spectrum, where the PSD level should match with the stationary noise prediction. Based on further experience that we have in using this comb, we know in fact that this noise is ASE noise that is temporally spread over all the Tr duration between each pulse, and that is partially coherent from pulse to pulse (meaning that it partially retains the particular noise realization), owing to the slight peaking that it produces in the signal spectrum. Fortunately, this signal being temporally spread, it also means that it is stationary and thus it gets rejected out like any of the other types of stationary noise from the processing steps. Another way to say this is that this noise is temporally correlated and thus colored in the frequency domain, but it is temporally equally spread out (stationary) and thus it is uncorrelated between the harmonics. This is why we can see that the achieved SNR depends on where in the spectrum we take the measurement, reaching approximately 17 dB of improvement at high offset frequencies. At closer offset frequencies, we can in fact see the actual phase information between the comb and the CW laser which contains phase noise which is the quantity being measured by this experiment, rather than additive measurement noise. Compared to the theoretical shot noise limit of 46.5 dB, we have achieved a peak SNR of 56 dB, a 9.5 dB improvement over the theoretical single-mode limit. The somewhat disappointing performance is explained by the excess thermal noise in the photodetector, as can be seen by the close thermal and thermal + CW shot

104 noise levels. Regardless, as shown in section 3.5, the beat achieved here is to our knowledge, the highest SNR reached when accounting for the bandwidth in the beating between a comb and a CW laser. Extracting the phase of both the raw beat signal and the processed beat signal allows comparing the phase noise measured by both (Figure 3-22). Ideally, with no additive noise, both phases should be the same at all timescales and the difference should be zero. One interesting fact is that we can see the improvement in the phase noise where we actually measure the real phase noise of the CW laser/comb beat up to much higher frequencies instead of simply additive noise. The decay around 26 MHz is caused by the measurement filter used to extract the phase, but the slope from 3 to 20 MHz present in the chirped pulse heterodyne signal only is the actual beat phase noise that is properly measured, while the raw heterodyne signal only allows measuring the phase properly up to approximately 4 MHz. At higher frequencies, the conventional heterodyne is measuring white phase noise, limited by the additive noise of the setup.

Figure 3-22 - Left: phase noise extracted from both the chirped pulse heterodyne processed signal (green, offset for clarity) and a single beat harmonic without any other processing (blue), and their difference. The width of the blue trace is due to white additive noise and as we can see in the spectrum (right figure), the difference signal is completely dominated by the white noise level of the conventional heterodyne signal, while the chirped pulse heterodyne signal shows no sign of a white measurement noise floor up to the highest possible frequency of this measurement, limited by the repetition rate.

We can also compute the modified Allan deviation of the two beat signals and their difference, shown here in Figure 3-23. Many interesting observations can be made on this figure. First, we can see that the blue (conventional heterodyne) and green (chirped pulse heterodyne) converge at much shorter timescales compared to Figure 3-8 which showed the modified Allan deviation for the GATOR. This is because the SNR is much higher in the situation presented here, because much more comb power was used. We can also see that both traces still

105

Figure 3-23 - Modified Allan deviation of the chirped pulse heterodyne signal (green), conventional heterodyne signal (blue) and difference (red). The vertical red lines represent the estimated ± 1 sigma −1.5 confidence interval for the red trace. The black diagonal line represents expected t for additive white noise. converge at longer timescales (here, from 20 ns) because at these timescales the measurement is dominated by actual laser phase noise, the desired quantity to be measured, which is common to both measurements. The modified Allan deviation of the difference frequency still follows the t -1.5 slope fitted to the white phase noise of the conventional heterodyne signal. We can see some structure at long timescales, but this is expected as each t computed by the modified Allan deviation is not statistically independent (as most of the data is shared between neighboring t values and combined with similar weights), and the fit is still within the uncertainty of the estimate of the deviation (vertical red lines). The SNR for the chirped pulse heterodyne is now high enough to yield phase estimates not dominated by additive noise even at the shortest t possible (widest measurement bandwidth). The bandwidth limit and shortest available timescale is now given by the Nyquist range of the beat signal (Fr/4, because we need equal bandwidth on both sides of the beat signal to measure the phase noise and the

Nyquist range is Fr/2). We can still see lower deviation for the chirped pulse heterodyne signal for almost one decade of t values, from 20 ns to 200 ns.

106

Finally, comparing Figure 3-8 to Figure 3-23, we can see that the laser phase noise is within a factor of 3 (worst case at 1 us) between the two measurements, a difference which can easily be explained by the fact that the two measurements were taken months apart, and no particular care was taken to replicate the current and temperature settings of the CW laser, and the polarization settings of the comb, all three factors which can easily account for such a difference. Furthermore, since the GATOR and narrowband setups agree for both cases (at timescales were the conventional heterodyne is not limited by additive noise), we did not pursue this matter further.

3.5 Comparison of beat SNRs in the literature

An exhaustive review of all the SNRs achieved when beating a comb and a CW laser is all but impossible to accomplish because of the sheer number of published papers that use such a beat. Indeed, a sizeable fraction of all papers using frequency combs use in some way or another a beat with a CW laser, although not all state the SNR of the beat, as usually some other quantity is of interest, and few list all the conditions of the beat, namely the comb and CW powers used, and the nature of the noise limitation. Regardless, we have compiled a sampling of the SNR of comb-CW beats given in the literature (Table 3-3) to give an idea of the spread of values achieved. This is not a completely fair review; many papers report on beats accomplished with very little comb power, for example in a distant spectral region from the center of the comb. But these are the same experiments that could have easily achieved an SNR improvement, or even an improvement in the reachable optical wavelengths using our techniques, especially considering the fact that the various flavors of GATOR techniques also reduces technical noise sources other than shot noise.

There is also a selection bias effect at play here in that clearly in the literature most if not all of reported beats will be when the experience actually worked and thus the SNR was at least sufficient to accomplish the measurement and we cannot conclude on the number of times a comb-CW beat became problematic because of low SNR since these won’t be reported in publications.

107

TABLE 3-3 COMPARISON OF THE COMB-CW BEAT SNRS REPORTED IN THE LITERATURE.

SNR SNR Reference Fr SNR RBW per pulse in 100 kHz MHz dB kHz dB dB SCH08 136 32 1 -16.3 12.0 MAL08 100 30 3 -12.2 14.8 SWA07 100 35 1 -12.0 15.0 TEL99 100 16 100 -11.0 16.0 JON01 120 35 2 -9.8 18.0 WAS04 50 38 5 1.0 25.0 VOG01 98 25 100 -1.9 25.0 NAK01 84 45 3 3.6 29.8 ADL09 136 40 10 1.3 30.0 NEW07 50 47 3 7.8 31.8 QUI11 200 30 300 4.8 34.8 SCH04 86 45 10 8.7 35.0 RUE11 152 38 100 9.2 38.0 DID00a 90 38 100 11.5 38.0 BAU09 28 65 0 18.4 39.8 DID00b 90 50 10 13.5 40.0 INA06 54 40 100 15.7 40.0 JOS02 100 45 100 18.0 45.0 YCA12 25000 53 300 6.8 57.8 OZD01 10287 50 1000 12.9 60.0 IMA99 6060 52 1000 17.2 62.0 DES13* 100 56 1700 41.3 68.3

To enable a fair comparison we have computed two relevant metrics: the reported SNR of the beat, normalized to a common resolution bandwidth (RBW) of 100 kHz. For a coherent beat in a white noise floor, the SNR should scale with the inverse of the RBW and thus we have generated our metric by computing SNR*RBW/100kHz for each, to remove the arbitrary choice of bandwidth in the comparison. This gives us a first picture of the SNRs achieved, shown in Figure 3-24 A). Our chirped pulse heterodyne result presented here (labeled DES13* in the table) shows 68.3 dB of SNR when normalized in 100 kHz, comparing favorably (6 dB higher than the second best) to the three results in our list that were achieved using multi-GHz combs. Note that although we could not account for the power used in each of these experiments, the goal of the chirped pulse

108

80,0

70,0 A) SNR in 100 kHz BW

60,0

50,0

40,0

SNR [dB] SNR 30,0

20,0

10,0

0,0

50,0 B) SNR per pulse 40,0

30,0

20,0

10,0 SNR [dB] SNR

0,0

-10,0

-20,0

Figure 3-24 - Comparison of beat SNR reported in the literature. A) Shows the SNR normalized in 100 kHz BW and thus also contains variation because of the repetition rate of the combs used. B) Normalizing the SNR by the bandwidth and the repetition rate allows a comparison which should only contain differences due to the experimental conditions (power levels, detection noise floor and technique). heterodyne technique is exactly that: to enable the use of all the spectrally available photons from our source, which a multi-GHz comb setup achieves even using a conventional heterodyne setup. When comparing with combs with different repetition rates however, one should notice that for the same average

109 power density, the fact that there are less comb modes per unit bandwidth means that the power in concentrated in fewer modes. To offset for this advantage of higher repetition combs, we can normalize by the repetition rate, in addition to the bandwidth. The equation for our comparison metric now becomes

SNR*RBW/(Fr/2). To explain the additional factor of two, first note that this has the same form as our previous normalization, but that instead the comparison bandwidth is equal to Fr/2, that is the Nyquist bandwidth for that comb. This means that this comparison basis is in fact computing the SNR per pulse of the comb. Another way to see this is that the factor RBW/(Fr/2) is roughly equal to the duration of the bandlimiting filter used for the measurement, expressed in units of repetition periods, which means that this number is equal to the number of pulses, or measurement points, that the filter has averaged. Using this normalization, the advantage of the chirped pulse repetition rate is now even more evident. Note that the vertical axis of Figure 3-24 B) is logarithmic, and thus the difference between an SNR per pulse of 41.3 dB for our result and 18.4 dB of [BAU09] is even more impressive.

3.6 Alternative filter topology for pulse stretching

One other intriguing possibility that we didn’t pursue fully is the use of a completely different optical filter topology to implement the required energy- preserving stretching of the pulses. Indeed, one way to stretch the energy of a short pulse over a longer duration is to use multiple cascaded Mach-Zender interferometers (CMZI) to produce multiple temporal copies of an input pulse [HAB11]. The optical setup to implement this would look like Figure 3-25, with potentially more MZI sections to reach the desired stretching ratio. Neglecting implementations issues which can cause more loss, this filter is almost energy- preserving, in that it produces two outputs, both with half the input power, regardless of the number of sections and thus spreading ratio. In the spectral domain, it can be shown that this filter achieves this by modifying a combination of amplitude and phase, as opposed to a pure chirp filter which affects only the phase. For an N-sections CMZI, the input pulse is split into M = 2N copies, giving a total peak power reduction of 2M=2N+1 between the input pulse and each output pulses. The heterodyne signal on the other hand is proportional to the electric

110

Figure 3-25 - Experimental setup used to stretch the comb pulses with cascaded Mach-Zender interferometers. BPF: Optical bandpass filter, PC: Polarization controler. field and thus the square root of the power, and is only reduced by a factor 2M , in the same manner as when using a chirp filter. The processing however will be different, because the exact phase difference between each of the beating copy will be interferometrically dependent on the delay in each section, and thus in the general case, the phase difference will be very hard to predict, as opposed to varying smoothly as when using a chirp filter. This means that this phase difference will have to be estimated from the same dataset that is used for combination, before generating the coherent sum of all the beats. In the digital domain, the M-1 phase differences between the first and each beat can be estimated online based on the past values of the beat phasors, and then removed from all but the first before summing the result. Figure 3-26 shows an example signal taken with the setup of Figure 3-25.

Figure 3-26 - Time-domain signal for a CMZI generated beat. Using a GATOR on the 8 pulse sets yield the 8 detected beats, overlaid as pale lines.

111

The time domain signal is complex to interpret visually because each beat has its own set of pulses separated by 10 ns, and there are 8 of those beats in each 10 ns period. The detected beats for each pulse set is overlaid in thin lines on this graph to show that there are indeed, 8 coherent beats in this signal. These 8 beats were each detected by a GATOR scheme, and the phase difference between each and the first was estimated and removed to yield the coherent sum.

Summing 8 coherent beats each with the same SNR would yield 10log10 ( 8) = 9 dB of

SNR improvement, but there are significant differences in the amplitude of each, as shown in the spectrum of Figure 3-27. Accounting for the amplitude differences yields an expected 7.35 dB of SNR improvement compared to the beat with the highest amplitude, which is indeed achieved by the current setup.

Figure 3-27 - Spectrum of all 8 beat signals, and the coherent sum obtained after removing the phase difference compared to the first beat.

This technique has some advantages over the chirped filter technique. The main advantage is that the CMZI works to spread the comb pulses over the whole comb bandwidth, whereas the chirped filter is to be tuned for a very precise spectral region and unlikely to be tunable because of the high accuracy required. Another advantage is that there is some additional information available in the form of the phase differences between each beating copy; the phase of each beat is a function of the absolute frequency of the CW laser and this mapping could conceivably be

112 inverted to retrieve this absolute frequency. Finally, there are also some disadvantages of this technique compared to the chirped filter. The fact stated previously that the phase difference between each beat is a function of the absolute frequency of the CW laser also makes it less robust. Indeed, the phase differences are also a sensitive function of the lengths of each fiber path and this can make it more challenging to coherently combine all the beats. Finally, at low beat SNR, it can be harder to calibrate the combining phases because they need to be estimated from the beat signal itself.

113

Conclusion

In this thesis, we have explored shot noise in the detection of an ultrashort pulse from a frequency comb. We have used a fairly high level abstraction of shot noise as a non-stationary noise source, and developed techniques and models to account for its impact on the performance of photodetection. We studied two experiments that are critical to the operation of a frequency comb as clockwork in an optical clock: the conversion of the optical pulse train to an electrical pulse train by a photodetector, and the heterodyne experiment between a comb and a CW laser.

We have first shown that directly detecting the ultrashort pulse train should yield vanishingly low jitter due to shot noise, as the photon arrivals themselves are very well timed by the boundaries defined by the optical pulse length. We have explained how this low jitter translates into the more common phase noise metric used to characterize the timing jitter, and how this relates to the well-known 2qI shot noise floor. Using a temporal model, we have shown that the shot noise will mostly translate into amplitude noise, in-phase with the signal and vanishingly low phase noise, in quadrature with the signal. We have demonstrated the validity of the model experimentally, and finally, we have given theoretical predictions exploring the limits imposed by practical photodetection physics. We have shown that there exists a transit time spread which is always some fraction of the average transit time and can randomly affect the detected timing of even an infinitesimally short optical pulse. This transit time spread depends on semiconductor parameters and photodetector geometry.

With the correct shot noise model, it seems that the noise floor limit at high offset frequencies for detection of the repetition rate will almost always be thermal noise in the electronics. With such a fact in mind, one can wonder whether there is anything we can do to improve the usual detection setup of a fast photodetector followed by bandpass filtering of the highest harmonic possible.

In the second main subject of our thesis, we have shown that the SNR of a heterodyne beat between a frequency comb and a CW laser can be significantly improved in situations where little power is available from the comb by using

115

GATOR. We have shown two practical implementations of this technique: the simplest one uses a fast oscilloscope but provides beat signals of duration limited by the oscilloscope memory and gives 19 dB of improvement over the shot noise- limit of a conventional setup. A real-time hardware approach was presented which is only moderately more complex, while producing beats with 17 dB SNR improvement over the shot-noise limit with no duration limitation. The long-term accuracy of the beat signal was shown to be completely indistinguishable from a conventional setup, at least at the millihertz level. Both experimental setups presented are simple and robust, and the hardware real-time approach produces a signal which is completely compatible with a conventional heterodyne setup; no further modification to the downstream experiment is necessary to take advantage of the higher SNR signals.

We have then demonstrated a further refinement of this technique, chirped pulse heterodyne, which uses chirp to spread the comb’s photons temporally and lower their impact on the SNR. This technique is more complex to implement than the GATOR, but removes the compromise that has to be made for the detection bandwidth and allows to use the highest bandwidth possible for any power levels, effectively using all the available photons in a given bandwidth. It is especially well suited to higher powers because the signals have a much lower dynamic range than required for GATOR. This technique enabled us to produce the beating with the highest SNR reported in the literature of 68.3 dB, when normalizing to the common detection bandwidth of 100 kHz.

Two applications where the GATOR technique or its variations would be especially well suited are when the frequency comb either has a low repetition rate, or when broadening a comb into a supercontinuum, as both these cases imply that the comb power is spread over a larger number of modes. Furthermore, supercontinuum generation often produces comb spectra with uneven power distribution across the whole bandwidth, with lots of valleys and troughs in the spectrum, potentially causing a situation where the CW laser falls into a region with lower power per comb mode. Finally, in the case of a very low repetition rate comb (in the low MHz, or even in the kHz range), this technique becomes unavoidable as it is accordingly easier to implement and provides even more SNR improvement potential.

116

Bibliography

ABL06 Ablowitz, M. J., Ilan, B., & Cundiff, S. T. (2006). Noise-induced linewidth in frequency combs. Optics letters, 31(12), 1875-1877. ADA12 Adachi, S. (2012). The Handbook on Optical Constants of Semiconductors: In Tables and Figures. World Scientific Publishing Company. ADL09 Adler, F., Cossel, K. C., Thorpe, M. J., Hartl, I., Fermann, M. E., & Ye, J. (2009). Phase-stabilized, 1.5 W frequency comb at 2.8–4.8 μm. Optics letters,34(9), 1330-1332. AME03 Ames, J. N., Ghosh, S., Windeler, R. S., Gaeta, A. L., & Cundiff, S. T. (2003). Excess noise generation during spectral broadening in a microstructured fiber.Applied Physics B, 77(2-3), 279-284. BAU09 Baumann, E., Giorgetta, F. R., Nicholson, J. W., Swann, W. C., Coddington, I., & Newbury, N. R. (2009). High-performance, vibration-immune, fiber-laser frequency comb. Optics letters, 34(5), 638-640. BED68 Bedrosian, E., & Rice, S. O. (1968). Distortion and crosstalk of linearly filtered, angle-modulated signals. Proceedings of the IEEE, 56(1), 2-13. BRU68 Bruyevich, A. N. (1968). Fluctuations in autooscillators for periodically nonstationary shot noise. Telecomm. Radio Eng. 23, 91–96 CAM10 Campbell, N. (1910). Discontinuities in light emission. Proc. Cambr. Phil. Soc. 15, 310–328. CAV81 Caves, C. M. (1981). Quantum-mechanical noise in an interferometer. Physical Review D, 23(8), 1693. DES10 Deschênes, J. D., Giaccarri, P., & Genest, J. (2010). Optical referencing technique with CW lasers as intermediate oscillators for continuous full delay range frequency comb interferometry. Optics Express, 18(22), 23358-23370. DES13a Deschênes, J. D., & Genest, J. (2013). Heterodyne beats between a continuous-wave laser and a frequency comb beyond the shot- noise limit of a single comb mode. Physical Review A, 87(2), 023802. DID00a Diddams, S. A., Jones, D. J., Ma, L. S., Cundiff, S. T., & Hall, J. L. (2000). Optical frequency measurement across a 104-THz gap with a femtosecond laser frequency comb. Optics letters, 25(3), 186-188. DID00b Diddams, S. A., Jones, D. J., Ma, L. S., Cundiff, S. T., & Hall, J. L. (2000, February). A phase and frequency controlled femtosecond laser for metrology and single-cycle nonlinear optics. In Advanced Solid State Lasers. Optical Society of America.

117

DID00c Diddams, S. A. (2010). The evolving optical frequency comb [Invited]. JOSA B,27(11), B51-B62. DID03 Diddams, S. A., Bartels, A., Ramond, T. M., Oates, C. W., Bize, S., Curtis, E. A., ... & Hollberg, L. (2003). Design and control of femtosecond lasers for optical clocks and the synthesis of low- noise optical and microwave signals.Selected Topics in Quantum Electronics, IEEE Journal of, 9(4), 1072-1080. DID09 Diddams, S. A., Kirchner, M., Fortier, T., Braje, D., Weiner, A. M., & Hollberg, L. (2009). Improved signal-to-noise ratio of 10 GHz microwave signals generated with a mode-filtered femtosecond laser frequency comb. Opt. Express, 17(5), 3331-3340. DUG71 Duguay, M., & Hansen, J. (1971). Ultrahigh-speed photography of picosecond light pulses. Quantum Electronics, IEEE Journal of, 7(1), 37-39. FOR07 Foreman, S. M., Holman, K. W., Hudson, D. D., Jones, D. J., & Ye, J. (2007). Remote transfer of ultrastable frequency references via fiber networks. Review of Scientific Instruments, 78(2), 021101- 021101. FOR11 Fortier, T. M., Kirchner, M. S., Quinlan, F., Taylor, J., Bergquist, J. C., Rosenband, T., ... & Diddams, S. A. (2011). Generation of ultrastable microwaves via optical frequency division. Nature Photonics, 5(7), 425-429. FOR12 Fortier, T. M., Nelson, C. W., Hati, A., Quinlan, F., Taylor, J., Jiang, H., ... & Diddams, S. A. (2012). Sub-femtosecond absolute timing jitter with a 10 GHz hybrid photonic-microwave oscillator. Applied Physics Letters, 100(23), 231111-231111. FOR13 Fortier, T. M., Quinlan, F., Hati, A., Nelson, C., Taylor, J. A., Fu, Y., ... & Diddams, S. A. (2013). Photonic microwave generation with high-power photodiodes. Optics Letters, 38(10), 1712-1714. FOR61 Forrester, A. T. (1961). Photoelectric mixing as a spectroscopic tool. JOSA,51(3), 253-256. FRE93 Freeman, M. J., Wang, H., Steel, D. G., Craig, R., & Scifres, D. R. (1993). Wavelength-tunable amplitude-squeezed light from a room-temperature quantum-well laser. Optics letters, 18(24), 2141-2143. GAR06 Gardner, W. A., Napolitano, A., & Paura, L. (2006). Cyclostationarity: Half a century of research. Signal processing, 86(4), 639-697. GAR86 Gardner, W. A. (1986). The spectral correlation theory of cyclostationary time-series. Signal processing, 11(1), 13-36. GEN00 Genest, J. and Tremblay, P. (2000) Understanding Fourier-transform spectrometers, notes de cours GEL-66010 Spectrométrie par transformée de Fourier. GLO04 Glöckl, O., Andersen, U. L., Lorenz, S., Silberhorn, C., Korolkova, N., & Leuchs, G. (2004). Sub-shot-noise phase quadrature 118

measurement of intense light beams. Optics letters, 29(16), 1936- 1938. GUP10 Gupta, S., Parsa, A., Perret, E., Snyder, R. V., Wenzel, R. J., & Caloz, C. (2010). Group-delay engineered noncommensurate transmission line all-pass network for analog signal processing. Microwave Theory and Techniques, IEEE Transactions on, 58(9), 2392-2407. HAB11 Haboucha, A., Zhang, W., Li, T., Lours, M., Luiten, A. N., Le Coq, Y., & Santarelli, G. (2011). Optical-fiber pulse rate multiplier for ultralow phase-noise signal generation. Optics letters, 36(18), 3654-3656. HAL00 Hall, J. L. (2000). Optical frequency measurement: 40 years of technology revolutions. Selected Topics in Quantum Electronics, IEEE Journal of, 6(6), 1136-1144. HAN06 Hänsch, T. W. (2006). Nobel lecture: passion for precision. Rev. Mod. Phys,78(4), 1297-1309. HAN56 Hanbury Brown, R., & Twiss, R. Q. (1956). Correlation between photons in two coherent beams of light. Nature, 177(4497), 27- 29. HAR11 Hartung, J., Knapp, G., & Sinha, B. K. (2011). Statistical meta-analysis with applications (Vol. 738). Wiley. com. HEI11 Heinecke, D. C., Bartels, A., & Diddams, S. A. (2011). Offset frequency dynamics and phase noise properties of a self-referenced 10 GHz Ti: sapphire frequency comb. arXiv preprint arXiv:1107.4562. HOL01 Hollberg, L., Oates, C. W., Curtis, E. A., Ivanov, E. N., Diddams, S. A., Udem, T., ... & Wineland, D. J. (2001). Optical frequency standards and measurements. Quantum Electronics, IEEE Journal of, 37(12), 1502-1513. HUA12 Huang, Y., Cao, J., Liu, P., Liang, K., Ou, B., Guan, H., ... & Gao, K. (2012). Hertz-level measurement of the Ca+ 4s2 S1/2–3d2 D5/2 clock transition frequency with respect to the SI second through the Global Positioning System. Physical Review A, 85(3), 030503. IMA99 Imai, K., Widiyatmoko, B., Kourogi, M., & Ohtsu, M. (1999). 12-THz frequency difference measurements and noise analysis of an optical frequency comb in optical fibers. Quantum Electronics, IEEE Journal of, 35(4), 559-564. INA06 Inaba, H., Daimon, Y., Hong, F. L., Onae, A., Minoshima, K., Schibli, T. R., ... & Nakazawa, M. (2006). Long-term measurement of optical frequencies using a simple, robust and low-noise fiber based frequency comb. Optics express,14(12), 5223-5231. IVA01 Ivanov, E. N., Diddams, S. A., & Hollberg, L. (2003). Experimental study of noise properties of a Ti: sapphire femtosecond laser. Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on, 50(4), 355-360.

119

IVA02 Ivanov, E. N., Hollberg, L., & Diddams, S. A. (2002). Analysis of noise mechanisms limiting frequency stability of microwave signals generated with a femtosecond laser. In Frequency Control Symposium and PDA Exhibition, 2002. IEEE International (pp. 435-441). IEEE. IVA03a Ivanov, E. N., Diddams, S. A., & Hollberg, L. (2003). Analysis of noise mechanisms limiting the frequency stability of microwave signals generated with a femtosecond laser. Selected Topics in Quantum Electronics, IEEE Journal of, 9(4), 1059-1065. IVA03b Ivanov, E. N., Diddams, S. A., & Hollberg, L. (2003). Experimental study of noise properties of a Ti: sapphire femtosecond laser. Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on, 50(4), 355-360. IVA07 Ivanov, E. N., McFerran, J. J., Diddams, S. A., & Hollberg, L. (2007). Noise properties of microwave signals synthesized with femtosecond lasers.Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on,54(4), 736-745. JAV62 Javan, A., & Ballik, E. A. (1962). Frequency characteristics of a continuous-wave He-Ne optical maser. JOSA, 52(1), 96-98. JON00 Jones, D. J., Diddams, S. A., Ranka, J. K., Stentz, A., Windeler, R. S., Hall, J. L., & Cundiff, S. T. (2000). Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis. Science, 288(5466), 635-639. JON01 Jones, R. J., & Diels, J. C. (2001). Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency synthesis.Physical review letters, 86(15), 3288. JOS02 Jost, J. D., Hall, J. L., & Ye, J. (2002). Continuously tunable, precise, single frequency optical signal generator. Opt. Express, 10(12), 515-520. KAS01 Kasap, S. O. (2001), Optoelectronics and Photonics: Principles and Practices. Prentice Hall. KIN78 Kingston, R. H. (1978). Detection of optical and infrared radiation. In Berlin and New York, Springer-Verlag (Springer Series in Optical Sciences. Volume 10), 1978. 150 p. (Vol. 10). LI99 Li, Y. Q., Guzun, D., & Xiao, M. (1999). Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator. Physical review letters, 82(26), 5225. LOC64 Lucovsky, G., Schwarz, R. F., & Emmons, R. B. (1964). Transit‐Time Considerations in p—i—n Diodes. Journal of applied physics, 35(3), 622-628. MAL08 Malara, P., Maddaloni, P., Gagliardi, G., & De Natale, P. (2008). Absolute frequency measurement of molecular transitions by a direct link to a comb generated around 3-µm. Opt. Express, 16, 8242-8249.

120

MCF05 McFerran, J. J., Ivanov, E. N., Bartels, A., Wilpers, G., Oates, C. W., Diddams, S. A., & Hollberg, L. (2005). Low-noise synthesis of microwave signals from an optical source. Electronics Letters, 41(11), 650-651. NAK10 Nakajima, Y., Inaba, H., Hosaka, K., Minoshima, K., Onae, A., Yasuda, M., ... & Hong, F. L. (2010). A multi-branch, fiber-based frequency comb with millihertz-level relative linewidths using an intra-cavity electro-optic modulator.Optics Express, 18(2), 1667-1676. NEW07 Newbury, N. R., & Swann, W. C. (2007). Low-noise fiber-laser frequency combs. JOSA B, 24(8), 1756-1770. NIE91 Niebauer, T. M., Schilling, R., Danzmann, K., Rüdiger, A., & Winkler, W. (1991). Nonstationary shot noise and its effect on the sensitivity of interferometers. Physical Review A, 43(9), 5022. OZD01 Ozdur, I., Akbulut, M., Hoghooghi, N., Mandridis, D., Ozharar, S., Quinlan, F., & Delfyett, P. J. (2010). A Semiconductor-Based 10- GHz Optical Comb Source With Sub 3-fs Shot-Noise-Limited Timing Jitter and 500-Hz Comb Linwidth. Photonics Technology Letters, IEEE, 22(6), 431-433. PAP02 Papoulis, A., & Pillai, S. U. (2002). Probability, random variables, and stochastic processes. Tata McGraw-Hill Education. PAS04 Paschotta, R. (2004). Noise of mode-locked lasers (Part II): timing jitter and other fluctuations. Applied Physics B, 79(2), 163-173. PRO09 Prochnow, O., Paschotta, R., Benkler, E., Morgner, U., Neumann, J., Wandt, D., & Kracht, D. (2009). Quantum-limited noise performance of a femtosecond all-fiber ytterbium laser. Optics Express, 17(18), 15525-15533. QUI09 Quinlan, F., Ozharar, S., Gee, S., & Delfyett, P. J. (2009). Harmonically mode-locked semiconductor-based lasers as high repetition rate ultralow noise pulse train and optical frequency comb sources. Journal of Optics A: Pure and Applied Optics, 11(10), 103001. QUI10 Quinlan, F., Ycas, G., Osterman, S., & Diddams, S. A. (2010). A 12.5 GHz-spaced optical frequency comb spanning > 400 nm for near- infrared astronomical spectrograph calibration. Review of Scientific Instruments, 81(6), 063105-063105. QUI11 Quinlan, Franklyn, et al. "Ultralow phase noise microwave generation with an Er: fiber-based optical frequency divider." Optics letters 36.16 (2011): 3260-3262. QUI12 Quinlan, F., Fortier, T. M., Jiang, H., Hati, A., Nelson, C., Fu, Y., ... & Diddams, S. A. (2012, September). Shot noise correlations in the detection of ultrashort optical pulses. In Photonics Conference (IPC), 2012 IEEE (pp. 14-15). IEEE. QUI13a Quinlan, F., Fortier, T. M., Jiang, H., Hati, A., Nelson, C., Fu, Y., ... & Diddams, S. A. (2013). Exploiting shot noise correlations in the

121

photodetection of ultrashort optical pulse trains. Nature Photonics. QUI13b Quinlan, F., Fortier, T. M., Jiang, H., & Diddams, S. A. (2013). Analysis of shot noise in the detection of ultrashort optical pulse trains. JOSA B, 30(6), 1775-1785. RAL95 Ralph, T. C., Taubman, M. S., White, A. G., McClelland, D. E., & Bachor, H. A. (1995). Squeezed light from second-harmonic generation: experiment versus theory. Optics letters, 20(11), 1316-1318. REI99 Reichert, J., Holzwarth, R., Udem, T., & Haensch, T. W. (1999). Measuring the frequency of light with mode-locked lasers. Optics communications, 172(1), 59-68. RIL08 Riley, W. J. (2008). Handbook of frequency stability analysis (p. 136). US Department of Commerce, National Institute of Standards and Technology. ROD37 Roder, H. (1937). Effects of tuned circuits upon a frequency modulated signal.Proceedings of the Institute of Radio Engineers, 25(12), 1617-1647. RSU95 Tsuchida, H. (1995). Generation of amplitude-squeezed light at 431 nm from a singly resonant frequency doubler. Optics letters, 20(21), 2240-2242. RUE11 Ruehl, A., Martin, M. J., Cossel, K. C., Chen, L., McKay, H., Thomas, B., ... & Ye, J. (2011). Ultrabroadband coherent supercontinuum frequency comb.Physical Review A, 84(1), 011806. RUS86 Rush, D., Burdge, G., & Ho, P. T. (1986). The linewidth of a mode- locked semiconductor laser caused by spontaneous emission: experimental comparison to single-mode operation. Quantum Electronics, IEEE Journal of,22(11), 2088-2091. SCH04 Schibli, T. R., Minoshima, K., Hong, F. L., Inaba, H., Onae, A., Matsumoto, H., ... & Fermann, M. E. (2004). Frequency metrology with a turnkey all-fiber system. Optics letters, 29(21), 2467-2469. SCH08 Schibli, T. R., Hartl, I., Yost, D. C., Martin, M. J., Marcinkevičius, A., Fermann, M. E., & Ye, J. (2008). Optical frequency comb with submillihertz linewidth and more than 10 W average power. Nature Photonics, 2(6), 355-359. SCH11 Schilt, S., Bucalovic, N., Dolgovskiy, V., Schori, C., Stumpf, M. C., Di Domenico, G., ... & Thomann, P. (2011). Fully stabilized optical frequency comb with sub-radian CEO phase noise from a SESAM- modelocked 1.5-µm solid-state laser. Optics Express, 19(24), 24171-24181. SKL01 Sklar, B. (2001). Digital communications (Vol. 2). NJ: Prentice Hall. STA02 Stan, G. B., Embrechts, J. J., & Archambeau, D. (2002). Comparison of different impulse response measurement techniques. Journal of the Audio Engineering Society, 50(4), 249-262. 122

STU10 Stumpf, M. C., Pekarek, S., Oehler, A. E. H., Südmeyer, T., Dudley, J. M., & Keller, U. (2010). Self-referencable frequency comb from a 170-fs, 1.5-μm solid-state laser oscillator. Applied Physics B, 99(3), 401-408. SWA07 Swann, W. C., Lorini, L., Bergquist, J., & Newbury, N. R. (2007, July). Narrow linewidth 1.5 μm sources and the thermal limit. In IEEE/LEOS Summer Topical Meetings, 2007 Digest of the (pp. 165-166). IEEE. TAK05 Takamoto, M., Hong, F. L., Higashi, R., & Katori, H. (2005). An optical lattice clock. Nature, 435(7040), 321-324. TEL99 Telle, H. R., Steinmeyer, G., Dunlop, A. E., Stenger, J., Sutter, D. H., & Keller, U. (1999). Carrier-envelope offset phase control: A novel concept for absolute optical frequency measurement and ultrashort pulse generation. Applied Physics B, 69(4), 327-332. TRE85 Tremblay, P., & Tetu, M. (1985). Characterization of frequency stability: effect of RF filtering. Instrumentation and Measurement, IEEE Transactions on, 34(2), 151-154. TRE89 Tremblay, P., Têtu, M., & Michaud, A. (1989). Frequency stability characterization from the filtered signal of a precision oscillator in the presence of additive noise. Instrumentation and Measurement, IEEE Transactions on,38(5), 967-973. UDE00 Udem, T., Reichert, J., Hänsch, T. W., & Kourogi, M. (2000). Absolute optical frequency measurement of the cesium D2 line. Physical Review A, 62(3), 031801. VOG01 Vogel, K. R., Diddams, S. A., Oates, C. W., Curtis, E. A., Rafac, R. J., Itano, W. M., ... & Hollberg, L. (2001). Direct comparison of two cold-atom-based optical frequency standards by using a femtosecond-laser comb. Optics Letters, 26(2), 102-104. WAH08 Wahlstrand, J. K., Willits, J. T., Menyuk, C. R., & Cundiff, S. T. (2008). The quantum-limited comb lineshape of a mode-locked laser: Fundamental limits on frequency uncertainty. arXiv preprint arXiv:0807.4172. WAS04 Washburn, B. R., Diddams, S. A., Newbury, N. R., Nicholson, J. W., Yan, M. F., & Jørgensen, C. G. (2004, May). A phase locked, fiber laser-based frequency comb: limit on optical linewidth. In Conference on Lasers and Electro-Optics. Optical Society of America. WIL94 Williams, K. J., Esman, R. D., & Dagenais, M. (1994). Effects of high space-charge fields on the response of microwave photodetectors. Photonics Technology Letters, IEEE, 6(5), 639- 641. WIN97 Winzer, P. J. (1997). Shot-noise formula for time-varying photon rates: a general derivation. JOSA B, 14(10), 2424-2429.

123

XIA87 Xiao, M., Wu, L. A., & Kimble, H. J. (1987). Precision measurement beyond the shot-noise limit. Physical review letters, 59(3), 278. YCA12 Ycas, G. G., Quinlan, F., Diddams, S. A., Osterman, S., Mahadevan, S., Redman, S., ... & Sigurdsson, S. (2012). Demonstration of on-sky calibration of astronomical spectra using a 25 GHz near-IR laser frequency comb. Optics Express, 20(6), 6631-6643. YUE83 Yuen, H. P., & Chan, V. W. (1983). Noise in homodyne and heterodyne detection. Optics Letters, 8(3), 177-179. ZHA12 Zhang, W., Li, T., Lours, M., Seidelin, S., Santarelli, G., & Le Coq, Y. (2012). Amplitude to phase conversion of InGaAs pin photodiodes for femtosecond lasers microwave signal generation. Applied Physics B, 106(2), 301-308.

124

Annex A - Deriving the SNR given in [REI99]

We start from equation (3.22), which gives the SNR for the beating in the case where the shot noise from the comb is sufficiently reduced by optical filtering:

Tp TP pp SNRslow = 2h . (A.1) TEe photon By assuming that the integration is performed over the longest range possible while still separating the pulses, we set Te = Tr and obtain the following result, where N is the number of modes in the comb’s spectrum, and En is the energy transferred by a single mode of power Pn during one repetition period of the comb:

11Epulse EPnn SNRslow = 2h= 22 hh = . (A.2) N EEEFphoton photon photon r In order to use the same conditions as in the reference, we must accept an integration time longer than the comb’s period; in terms of SNR this is equivalent to averaging the beating over a certain number of pulses, given by Fr/(2BW), where BW is the electrical bandwidth used, i.e. BW=1/(2Te) for a temporal boxcar filter. This averaging improves the SNR linearly with the number of integrated pulses (because the signal power stays the same while the variance of the noise is reduced linearly):

PPnnhhFr SNRslow, averaging = 2 = , (A.3) FEr photon 2BW Ephoton BW which is the result given in [REI99], with a small subtlety: the power is defined here as after the beamsplitter, while in the reference the power is measured before the beamsplitter. However, their experimental setup assumes an adjustable beamsplitter set to the optimum ratio, which allows to transmit practically all of the filtered comb’s power to the detector and thus the definitions are approximately equivalent.

125

Annex B - Using an adjustable beam splitter at the optimum point yields the same SNR as using a 50- 50 beam splitter with a balanced photodetector

In this section, we will show that using an adjustable beam splitter set at the optimum point yields the same SNR as using a 50-50 beam splitter with a balanced photodetector. This fact is useful because the techniques presented in this thesis cannot easily make use of the SNR improvement coming from using an adjustable beam splitter, because that entails lowering the CW laser’s signal to match the “weak” comb signal, while we instead use the pulsed nature of the comb to use more of its power, blurring the divide between the comb being the weak signal.

The technique of using an adjustable beam splitter to detect the beat between a comb and a CW laser is shown in [REI99], and can be intuitively understood this way: the idea is to use a beam splitter which transmits more of the weak signal on the photodiode, to increase the signal, while at the same time letting less of the strong beam reaching the photodiode, producing less shot noise. Of course, the actual situation is more complicated than that because we need to know the exact scaling of the signal and noise with the transmission ratio to determine the possible improvement. We will follow the derivation in section 3.1, while adding an adjustable beam splitter. The powers are now defined as measured at the input of the beam splitter rather than at the detector. We start by assuming that

N modes of the comb reach the beam splitter, each with power Pn, and are transmitted with a ratio t. Since the beam splitter is (ideally) a lossless device, the reflection coefficient for the second input which contains the CW laser will be (1-t). Thus, we detect a signal that can be as:

N Pdetected (φ) =−+(1 tP)hhCW t∑ Pn + 21 h( − t) PPCW t n cos(φ) +, (B.1) n=1

127 where h is the quantum efficiency of the detector and φ is the phase difference between the CW and the comb modes12, and where we have explicitly written only the interference term which will be detected (by using an appropriate bandpass filter for example). To compute the electrical SNR of this signal, we need the electrical power in the signal in which we are interested in, and the total noise variance. Since we are computing a signal ratio, the units in which we do our calculations do not matter and we will keep the detected signal in units of optical power, Watts. The electrical power in the beat signal will be, up to a constant:

2 =η − φ Ptelectrical,signal (2 (1 ) PCWtP n cso ( )) (B.2)

22 Ptelectrical,signal =4η(1 − ) PCWtP n cos φ (B.3)

2 Pelectrical,signal =21η ( − tt) PCWP n , (B.4) where the angled brackets denote averaging over all possible phases. We now need the variance of the noise, which we assume to be shot noise produced by the optical power arriving on the detector. From [PAP02], we know that the shot noise variance will be given by:

22 s shot =⋅⋅⋅2 rate area R , (B.5) where “rate” is the temporal rate of arrival of photons, “area” is the area under the signal produced by each photons, and R is the detection bandwidth. In our case, since the signal is scaled in units of Watts, the area under each photon detection event is equal to the energy transferred by the photon, Ephoton, while the rate of arrival is equal to the average detected optical power divided by the energy in each photon:

N (1−+tP)hhCW t∑ Pn 22n=1 s shot =2⋅⋅⋅ ERphoton , (B.6) Ephoton

2 sshot =2hERphoton ⋅ ((1−+t) PCW tNP n ) . (B.7)

The SNR is thus given by combining (B.4) and (B.7):

12 We did not put a frequency difference between the two beams, but this would not change the result in any way, because we will average the results over all possible phase differences. 128

P 21h 2 ( − tt) PP SNR =electrical,signal = CW n (B.8) s 2 shot 2hEphoton ⋅ R((1−+t) PCW tNPn )

h (1− tt) P P SNR = CW n . (B.9) ⋅ Ephoton R ((1−+tP) CnW tNP )

At this point, we can optimize the SNR by choosing an appropriate t. Setting the derivative of the SNR with respect to t to 0 yields:

∂SNR hPP((1−tP) CW +tNPn )(12 −− t) t( 1− t)(−+ PCW tNP n ) =0 = CW n (B.10) ∂t ERphoton ⋅ 

0 = ((1−t) PCW + tNP n )(12 − t) −−− t(1 t)( PCW + tNP n ) (B.11)

2 02=(PCW − NP n ) t +−( PCW) t + P CW . (B.12)

Solving this quadratic equation gives a maximum at:

2 2PCW− 44 P CW −−( P CW NP n) P CW topt = (B.13) 2(PCW− NP n )

PCW − NPn P CW topt = (B.14) (PCW− NP n )

− PCW( P CW NPn ) topt = (B.15) −+ ( PCW NPn)( P CW NPn )

PCW topt = . (B.16) + ( PCW NPn )

We can now go back to the SNR equation (B.9) with the new optimum condition (B.16):

h NP P PP SNR = n CW CW n (B.17) ⋅ 2  ERphoton + PCW NPn NP P ( ) n + CW PCW NPn PN++PNPP ( CW n ) ( CW n )

h NP P PP SNR = n CW CW n (B.18) ERphoton ⋅ + + ( PCW NPn ) (PCW NPn NPn PCW )

129

h PCW P CW NP n SNR = Pn . (B.19) ERphoton ⋅ + + + (PCWPNPPPPCW n CW NPnnnnCW NPCW NP NP )

Now we will assume that PCW >> NPn so that we can neglect all but the first term of the denominator:

h PCW P CW NP n SNR = Pn (B.20) ERphoton ⋅ (PCW PPCWnN )

h SNR = Pn , (B.21) ERphoton ⋅ which is the result given in [REI99].

If instead, we use a beam splitter with a fixed splitting ratio at 50% reflection, 50% transmission, but put a photodetector at both outputs, we now get two signals, which we can write starting from (B.1) with t = 0.5:

N + 2 Pdetected (φ) = 0.5h PPCW + 0.5 h∑ n + 2 h 0.5 PPCW n cos(φ) +, (B.22) n=1

N − 2 Pdetected (φ) = 0.5h PPCW + 0.5 h∑ n − 2 h 0.5 PPCW n cos(φ) +. (B.23) n=1

Subtracting those two signals, the beat has doubled in amplitude, and thus has quadrupled in electrical power:

2 = η2 φ Pelectrical,signal (4s0.5 PPCW n co ( )) (B.24)

2 2 Pelectrical,signal = 4ηPPCW n (cos(φ)) (B.25)

2 Pelectrical,signal = 2η PCWP n . (B.26)

In terms of noise, since we collect all the signal power onto detectors, the photon arrival rate is now simply equal to the total optical power at the input of the beam splitter divided by the energy carried by each photon:

22 s shot =⋅⋅⋅2 rate area R , (B.27)

h P+ NP 22( CW n ) s shot =⋅2 ⋅⋅ERphoton , (B.28) Ephoton

130

2 sshot = 2hREphoton ( PCW+ NP n ) . (B.29)

The SNR is simply the ratio of (B.26) and (B.29):

P 2h 2 PP SNR =electrical,signal = CW n (B.30) s 2 shot 2hREphoton ( PCW+ NP n )

P hPP SNR =electrical,signal = CW n . (B.31) s 2 shot REphoton ( PCW+ NP n )

Once again, if we assume that PCW >> NPn , we get:

hP SNR = n , (B.32) REphoton which is the same result that we get using an adjustable beam splitter set to the optimum ratio. However, in terms of practical applications, the balanced configuration has the immense advantage of rejecting the amplitude noise from the CW laser and ASE-ASE beat noise from both the comb and the CW laser, which can often dominate the measurement otherwise. It is also simpler to use compared to the adjustable beam splitter setup, where we need to optimize the ratio every time the power changes and this requires tweaking the optical setup while measuring the SNR of the beat. Finally, using the adjustable beam splitter setup implies that we get less beat signal on the photodetector than if we had used a 50-50 beam, splitter. Indeed, the beat signal scales as tt(1− ) , which is a parabola with its maximum at t = 0.5 . Using any other ratio means that the beat signal will be smaller, and if we take into account the effect of dark noise from the photodetection setup, we can easily get into a situation where we are no longer shot noise limited, and all the careful theoretical optimization of the ratio becomes useless.

131