Full-Field Spectroscopy at Megahertz-Frame-Rates: Application of Coherent Time-Stretch Transform

UNIVERSITY OF CALIFORNIA Los Angeles

Full-Field Spectroscopy at Megahertz-frame-rates: Application of Coherent Time-Stretch Transform

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics

Peter Thomas Setsuda DeVore

2014

ABSTRACT OF THE DISSERTATION

Full-Field Spectroscopy at Megahertz-frame-rates: Application of Coherent Time-Stretch Transform

Peter Thomas Setsuda DeVore Doctor of Philosophy in Physics University of California, Los Angeles, 2014 Professor Bahram Jalali, Co-chair Professor Seth Putterman, Co-chair

Outliers or rogue events are found extensively in our world and have incredible effects. Also called rare events, they arise in the distribution of wealth (e.g., Pareto index), finance, network traffic, ocean waves, and e-commerce (selling less of more). Interest in rare optical events exploded after the sighting of optical rogue waves in laboratory experiments at UCLA. Detecting such tail events in fast streams of information necessitates real-time measurements. The

Coherent Time-Stretch Transform chirps a pulsed source of radiation so that its temporal envelope matches its spectral profile (analogous to the far field regime of spatial diffraction), and the mapped spectral electric field is slow enough to be captured by a real-time digitizer.

Combining this technique with spectral encoding, the time stretch technique has enabled a new class of ultra-high performance spectrometers and cameras (30+ MHz), and analog-to-digital converters that have led to the discovery of optical rogue waves and detection of cancer cells in blood with one in a million sensitivity.

Conventionally, the Coherent Time-Stretch Transform maps the spectrum into the temporal electric field, but the time-dilation process along with inherent fiber losses results in reduction of peak power and loss of sensitivity, a problem exacerbated by extremely narrow molecular linewidths. The loss issue notwithstanding, in many cases the requisite dispersive optical device is not available. By extending the Coherent Time-Stretch Transform to the temporal near field, I have demonstrated, for the first time, phase-sensitive absorption spectroscopy of a gaseous sample at millions of frames per second. As the Coherent Time-Stretch Transform may capture both near and far field optical waves, it is a complete spectro-temporal optical characterization tool. This is manifested as an amplitude-dependent chirp, which implies the ability to measure the complex refractive index dispersion at megahertz frame rates. This technique is not only four orders of magnitude faster than even the fastest (kHz) spectrometers, but will also enable capture of real-time complex dielectric function dynamics of plasmas and chemical reactions (e.g. combustion). It also has applications in high-energy physics, biology, and monitoring fast high- throughput industrial processes.

Adding an electro-optic modulator to the Time-Stretch Transform yields time-to-time mapping of electrical waveforms. Known as TiSER, it is an analog slow-motion processor that uses light to reduce the bandwidth of broadband RF signals for capture by high-sensitivity analog-to-digital converters (ADC). However, the electro-optic modulator limits the electrical bandwidth of TiSER. To solve this, I introduced Optical Sideband-only Amplification, wherein electro-optically generated modulation (containing the RF information) is amplified at the expense of the carrier, addressing the two most important problems plaguing electro-optic modulators: (1) low RF bandwidth and (2) high required RF drive voltages. I demonstrated drive voltage reductions of 5x at 10 GHz and 10x at 50 GHz, while simultaneously increasing the RF bandwidth.

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The dissertation of Peter Thomas Setsuda DeVore is approved.

Bahram Jalali, Committee Co-chair Seth Putterman, Committee Co-chair George Morales George Valley

University of California, Los Angeles 2014

To my parents, wife and mother-in-law who all supported me

Table of Contents

Chapter 1: Introduction ...... 1 Detecting Outliers in Optics ...... 1 Impact of Outliers ...... 1 Outliers in Optics and Limitations of Conventional Detection Methods ...... 2 Detection of Outliers Enabled by High-Throughput Real-Time Instruments ...... 8 Coherent Time-Stretch Transform ...... 9 Coherent Time-Stretch Transform in the Near Field ...... 11 The Electro-optic Conversion Bottleneck ...... 11 Time-Stretch Dispersive Fourier Transform to Detect Electrical Outliers ...... 11 Electro-optic Modulators in Optical Links ...... 14 Optical Sideband-only Amplification: Addressing the Electro-optic Conversion Bottleneck ...... 15 Chapter 2: Coherent Time-Stretch Transform (CTST) ...... 16 Introduction ...... 16 Principle ...... 24 Stage 1 of Coherent Time-Stretch Transform ...... 24 Stage 2 of Coherent Time-Stretch Transform: Complex-field Recovery ...... 26 Chapter 3: Coherent Spectroscopy in the Temporal Near Field using the Coherent Time-Stretch Transform with a Local Oscillator ...... 32 Introduction ...... 33 Principle ...... 35 Experiments ...... 37 Simulations ...... 39 Conclusion ...... 41 Chapter 4: Optical Sideband-only Amplification: Overcoming the Electro-optic Conversion Bottleneck43 Introduction ...... 44 Principle ...... 46 Electro-optic Modulation and Distortions ...... 46 Noise in Analog Optical Links and Time-Stretch Digitizer ...... 49 Optical Sideband-only Amplification ...... 51 Simulations ...... 55 Experiments ...... 58 Conclusion ...... 64 vi

Chapter 5: Future Direction: Measuring Spectro-Temporal Outliers and their Statistics using CTST ...... 66 Chapter 6: Conclusion ...... 69 References ...... 72

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Acknowledgements

I wanted to thank in particular my advisor Prof. Bahram Jalali, as well as two postdoctoral scholars who have been critical in my tutelage at UCLA: Dr. Daniel R. Solli, and Dr.

Mohammad H. Asghari, and finally Dr. George Valley of Aerospace Corporation who gave me the chance to experience the inner workings of science on the other side of academia (i.e.

FFRDCs / national laboratories). I also thank in general all members of the UCLA Photonics

Laboratory who have taught me so much that it amazes even me. I am very grateful for this very awesome opportunity that I have had.

Large portions of this work are based on research authored by me and performed under my advisor, Prof. Bahram Jalali (see “Publications and Presentations”). In particular:

 Chapter 1 borrows heavily from “Enhancing electro-optic modulators using

modulation instability,” published in the Journal of Optics in 2012.

 Chapter 3 borrows heavily from “Coherent Time-Stretch Transform for Near-Field

Spectroscopy,” published in the Photonics Journal in 2014.

 Chapter 4 borrows heavily from “Enhancing electro-optic modulators using

modulation instability,” published in Physica Status Solidi: Rapid Research Letters in

2013, and “Demonstration of Vpi Reduction in Electrooptic Modulators using

Modulation Instability,” submitted.

 Chapter 6 borrows from “Measurement of spectro-temporal outliers,” submitted.

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Vita B.A. Physics 2008 University of California, Berkeley Berkeley, California, USA M.S. Physics 2010 University of California, Los Angeles Los Angeles, California, USA First Japan Society 2011 for the Promotion of Science Silicon Photonics School: Full Scholarship 1st UCLA IEEE Photonics Society 2014 Presentation Competition: 3rd place

Publications and Presentations Journal

P. T. S. DeVore, B. W. Buckley, M. H. Asghari, D. R. Sollli, and B. Jalali, “Coherent Time- Stretch Transform for Near-Field Spectroscopy,” IEEE Photon. J. 6, 3300107 (2014). http://dx.doi.org/10.1109/JPHOT.2014.2312949 P. DeVore, et. al., “Light-weight ﬂexible magnetic shields for large-aperture photomultiplier tubes,” Nucl. Instrumen. Meth. A 737, 222 (2014). http://dx.doi.org/10.1016/j.nima.2013.11.024 P. T. S. DeVore, D. Borlaug, and B. Jalali, “Enhancing electro-optic modulators using modulation instability,” Phys. Status Solidi RRL 7, 566 (2013). http://dx.doi.org/10.1002/pssr.201307174 P. T. S. DeVore, D. R. Solli, D. Borlaug, C. Ropers, and B. Jalali, “Rogue events and noise shaping in nonlinear silicon photonics,” J. Opt. 15, 064001 (2013). Invited. Review. http://dx.doi.org/10.1088/2040-8978/15/6/064001 P. T. S. DeVore, D. R. Solli, P. Koonath, C. Ropers, and B. Jalali, “Stimulated supercontinuum generation extends broadening limits in silicon,” App. Phys. Lett. 100, 101111 (2012). http://dx.doi.org/10.1063/1.3692103 C. M. Gee, G. Sefler, P. T. S. DeVore, and G. C. Valley, “Spurious-Free dynamic range of a high-resolution photonic time-stretch analog-to-digital converter system,” Microwave and Opt. Technol. Lett. 54, 2558 (2012). http://dx.doi.org/10.1002/mop.27114 A. M. Fard, P. T. S. DeVore, D. R. Solli, and B. Jalali, “Impact of Optical Nonlinearity on Performance of Photonic Time-Stretch Analog-to-Digital Converter,” J. Lightwave Technol. 29, 2025-2030 (2011). http://dx.doi.org/10.1109/JLT.2011.2157304

Conference

B. Jalali, D. Borlaug, P. DeVore, O. Boyraz, and A. Rostrami, “Enhancing the Electro-optic Effect using Optical Sideband-only Amplification,” in Frontiers in Optics (OSA 2014), accepted. Invited. D. Borlaug, P. T. S. DeVore, and B. Jalali, “Demonstration of Vpi Reduction in Electrooptic Modulators using Modulation Instability,” in IEEE Summer Topicals (IEEE 2014), paper WD2.3. Best Student Poster. http://programme.exordo.com/sum2014/delegates/presentation/69/ D. Borlaug, P. T. S. DeVore, and B. Jalali, “Experimental Demonstration of Vpi Reduction in EO Modulators using Modulation Instability,” in Conference on Laser & Electro-Optics (OSA 2014), paper SM1G.4. http://dx.doi.org/10.1364/CLEO_SI.2014.SM1G.4 P. T. S. DeVore, B. W. Buckley, M. H. Asghari, D. R. Solli, and B. Jalali, “Near-field and complex-field time-stretch transform,” in Proc. SPIE 9141, Optical Sensing and Detection III, (SPIE, 2014), poster 91411P. http://dx.doi.org/10.1117/12.2050979 P. T. S. DeVore, D. Lam, C. Kim, and B. Jalali, “Boosting Electro-optic Modulators for Optical Communications,” in Frontiers in Optics (OSA, 2013), paper FW4B.5. http://dx.doi.org/10.1364/FIO.2013.FW4B.5 P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Energy-efficient coherent supercontinuum generation in silicon,” in 9th International Conference Group IV Photonics (GFP) (IEEE, 2012), pp. 186-188. http://dx.doi.org/10.1109/GROUP4.2012.6324128 P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Energy efficient and Stable Supercontinuum Generation in Silicon,” in CMOS Emerging Technologies (2012). http://books.google.com/books/p/pub-2963883207600822?id=yZpLPIwpjMsC&pg=PA108 P. T. S. DeVore, D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Stimulated Modulation Instability in Silicon for Energy Efficient Supercontinuum Generation,” in Nonlinear Photonics (OSA, 2012), paper NM2C.4. http://dx.doi.org/10.1364/NP.2012.NM2C.4 C. Gee, G. Sefler, P. DeVore, and G. Valley, “Spurious-Free Dynamic Range of a High- Speed Photonic Time-Stretch A/D-Converter System,” in Signal Processing in Photonics Communications (OSA, 2012), paper SpTu4A.2. http://dx.doi.org/10.1364/SPPCOM.2012.SpTu4A.2

Chapter 1: Introduction

Detecting Outliers in Optics

Impact of Outliers

Rare events, otherwise known as tail events, rogue events, or outliers arise in a wide variety of complex systems and often have significant ramifications. These events are known to appear in finance [1], network traffic [2], ocean waves [3-6], and elsewhere [7-10]. In popular contexts, such distributions have been noted to arise, for example, in the distribution of wealth [11] (e.g.,

Pareto index) and e-commerce [12] (selling less of more). Anomalous events are often dubbed outliers based on an implicit expectation of the familiar Gaussian distribution, or in other words, because we intuitively expect observables to follow Gaussian statistics, which predict an exponentially decreasing probability for extreme occurrences. Non-Gaussian statistics, ubiquitous in science and technology, are a signature of complex dynamics and hallmark of stochastic effects in these systems [13-15]. Rogue phenomena follow probability distributions with extended tails, often termed heavy-tailed or L-shaped statistics. Heavy-tailed statistics appear due to a nonlinear dependence on initial conditions; nonlinearity can reshape the Normal

Distribution into a highly skewed, heavy-tailed output distribution. The form of the distribution can be very important, indicating fundamental, even useful properties of the dynamics that may be otherwise hidden or overlooked [13, 15, 16].

In many systems, attempts to find the specific causes and probabilities of extreme occurrences are hindered observationally by their intrinsic rarity and experimentally by difficulties in performing controlled tests. As a key example, determining the origin and frequency of

anomalously large ocean waves has presented significant scientific challenges [4]. Deviant outcomes may signify either some form of measurement anomaly or that the observable in fact follows a skewed distribution, i.e., one with high kurtosis. In many situations, it can be difficult to distinguish these two scenarios because too few events are available to adequately sample the makeup of the distribution, particularly its tails, where rare events reside. Even in situations where the dynamics and critical variables are well understood, predictive power is often limited by sensitivity to initial conditions, which cannot be determined with arbitrary precision. Yet, it is crucial to know how frequently the proverbial “needle in the haystack” appears in a given situation.

Outliers in Optics and Limitations of Conventional Detection Methods

The generation of rare events in optical systems has become a dynamic field of research catalyzed by the recent discovery of optical rogue waves in table-top experiments [14]. These rare broadband flashes of light arise spontaneously in supercontinuum generation, and are mathematically and physically analogous to freak ocean waves. Supercontinuum generation involves large spectral broadening of radiation in an optical medium due to a host of nonlinear effects [17]. This high-brightness white light source has found use in fields as far reaching as spectroscopy [18, 19], frequency metrology [20], biology [21, 22], and telecommunications [23-

25]. Knowledge of the seed conditions underlying rogue waves has been harnessed to actively control the instability [26-29] and generate a new ultra-stable white light known as stimulated supercontinuum radiation. This research thus has importance for both fundamental science and applications, and continuing strides are being made in both of these directions. Their study has already generated significant new findings, including emergent behavior [30]. Experiments have

also shown that reverse heavy-tailed distributions [31] and a wide variety of statistical shapes can be obtained [32].

Figure 1. Optical rogue waves, outliers analogous to freak oceanic waves, are discovered in laboratory experiments. High-throughput real-time techniques collect sufficient statistics to reveal the underlying non-Gaussian, heavy-tailed distributions. (a) Experimental setup. A mode-locked laser and Yb-doped (Ytterbium) fiber amplifier generate high peak power pulses, which travel through a nonlinear fiber (low group velocity dispersion and small modal area) and experience large optical nonlinearities, generating supercontinuum. The pulses are filtered to reveal the long-wavelength energy statistics, and recorded in real-time using a high-speed photodetector and oscilloscope. (b,c,d) Above, intensity vs. time. Rare spikes appear far above the expected on an otherwise quiet background. Below, number of events vs. intensity bins. The gray line denotes the noise floor of the detection process. Most of the events are

Rogue waves were first detected in high-speed pulse trains by pulse-resolved filtering of long wavelengths [14] (cf. Figure 1). When driven by high repetition rate laser sources, it becomes possible to acquire voluminous data sets in short time periods, which may include large numbers of statistically rare events. This yielded important yet limited information about the nonlinear dynamics. Spectrometers, capable of measuring the spectral power vs. optical wavelength over extremely broad bandwidths, have been key in optically distinguishing the chemical makeup of samples, and so are of vital importance in chemistry and biology [33-36]. Though they can capture significantly more information about optical rogue waves and other optical outliers, spectrometers are hindered by low refresh rates. Spectrally-resolving optical fields is conventionally based on spatially-separated wavelengths with a diffraction grating, and either filtering a wavelength with a slit onto a single pixel detector, detecting all colors in parallel with a detector array. Although the latter is significantly faster (at the cost of resolution), its speed is still limited to kHz frame rates [33-36], unsuitable for capture of fast dynamic systems like optical rogue waves at their native repetition rate (10’s of MHz).

Optical rogue waves and other so-called “ultrafast” pulses vary faster than state-of-the-art electronic bandwidths can acquire (multi-GHz), so capture of their temporal information as well requires more sophisticated approaches. Frequency-resolved optical gating (FROG) [37, 38] and spectral phase interferometry for direct electric-field reconstruction (SPIDER) [39, 40] are able to capture the full electric field of ultrafast pulses, and have been critical in the study of many areas of optical science, including characterizing few-cycle pulses [41] and revealing fine- structure in supercontinuum [42]. As these techniques are also based on spectrometers, they are subject to the same refresh-rate limitations and hence cannot operate at high throughputs. 4

Figure 2. Histogram of on-off gain in stimulated Raman scattering for a noisy pump. The nonlinear gain dependence transforms the pump’s thin-tailed statistics to a fat tail (here, a power law), exhibiting a non-negligible chance to see extreme events [43] © 2011 IEEE.

Anomalous signals could significantly degrade information fidelity, so the investigation of outliers at high speeds is paramount in optical communications. Stimulated Raman amplification, a nonlinear optical process whereby a Stokes wave is amplified at the expense of a shorter- wavelength pump wave, provides significant performance enhancement in telecommunications

[44] and high-throughput real-time measurement techniques [45]. Outliers in and statistical properties of Raman scattering in silicon were experimentally explored at mid-infrared wavelengths [43], where two-photon absorption and free-carrier effects are absent [46]. These experiments were able to determine the cause of the extreme behavior: in-band pump fluctuations [43, 47]. Experiments show a pump that exhibits large shot-to-shot energy fluctuations whose statistics are well described by the Rician distribution [48], a thin-tailed distribution similar to the Gaussian. The nonlinear dependence of Raman amplification on the pump imparts in-band pump fluctuations to the Stokes sideband, resulting in a power-law gain distribution (cf. Figure 2). As the pump pulse energy follows a symmetric, mean-centric

distribution [43], the observed heavy-tail must result from the Raman process and confirms that the input distribution is transformed.

Figure 3. Histogram of bandpass filtered energy showing extreme statistics and outliers in stimulated supercontinuum generation in silicon. The supercontinuum generation process is sensitive to noise. Without seeding at low pump intensity (U (10) GW/cm2), the power statistics exhibit a strong rightward skew and several outliers. At higher pump intensities (U (500) GW/cm2), the process stabilizes ever so slightly, but now exhibits a more Gaussian-like dependence. Stimulating the process with a weak seed at low pump powers (S (10) GW/cm2), however, strongly stabilizes the process. Not only are a wide variety of statistical shapes seen, but the noise characteristics may be shaped and engineered via the pump and seed powers [29] © 2012 American Institute of Physics.

Yet another area of nonlinear optics in which outliers appear is supercontinuum generation in silicon. Studies revealed that seeding supercontinuum outliers offers the possbility of overcoming silicon’s free carrier loss, the most fundamental problem in nonlinear silicon photonics [29]. While stable seeding leads to controlled supercontinuum generation, stochastic seeding can yield signficant output fluctuations, including extreme events [46]. In the latter case, one-photon-per-mode (i.e., shot) noise has been used as the source of broadband noise to initiate stochastic behavior. It was found that stimulating the process with a stronger seed overcomes the limit on the supercontinuum bandwidth set by nonlinear losses from two-photon absorption and free-carrier absorption [49]. Furthermore, stimulating modulation instability (MI) in silicon 6

waveguides with anomalous dispersion has improved supercontinuum stability, bandwidth, and energy efficiency [29], and resulted in coherent radiation [50]. Spontaneous (noise seeded) supercontinuum generation at a moderate pump intensities exhibited highly skewed long tail statistics, arising from out-of-band noise [46] (cf. Figure 3). At extremely high pump intensities

(near the breakdown intensity of silicon for these picosecond pulses [51]), the tail is no longer highly skewed, likely due to pump depletion and much larger bandwidth of the self-phase- modulated central peak. In the presence of free carrier losses, the generation of extreme distributions in supercontinuum results from moderate pumping and broadband noise, while higher pump powers or controlled seeding tend to reduce fluctuations significantly [46]. As with optical rogue waves, study of other optical outliers would benefit from identification methods faster than conventional spectrometer-based techniques.

Rare cells play significant roles in our well-being and in pathology, and optics has been crucial in their discovery and identification. In 1 milliliter of blood, red blood cells constitute the overwhelming majority – more than one billion in number. In that same milliliter of blood of a cancer patient may be as few as 10 circulating tumor cells (CTCs) [52]. But these CTCs may have a huge impact, as they could leave cancer sites to spread in the body. Hematopoietic stem cells [53], antigen specific T-cells [54], and circulating fetal cells in pregnant women [55] are examples of other rare but important cells.

Optical techniques, such as microscopy and fluorescence tagging, are instrumental in detecting and identifying these anomalous cells. The dominant techniques for outlier cell detection are optical microscopy and flow cytometry [56]. However, there exists a trade-off between throughput and discriminating power. Conventional flow cytometry boasts high throughputs of 100,000 cell/s [57], but offers few measurement points as it detects forward

and/or side scatter with a number of different fluorophores [58]. Automated microscopy, on the other hand, exchanges higher resolving power for limited throughput (1,000 cells/s) [59]. The speed in the latter case is limited by digital imaging technology, typically charge-coupled device

(CCD) and complementary metal-oxide-semiconductor (CMOS) cameras. Although these cameras are able to approach 1 million frames/s continuously with great difficulty [60], these rates are only achieved at low pixel numbers or for a few frames [61], trading information content for speed. A more fundamental problem is that collecting enough light in such photon- starved situations requires long shutter and exposure times (> 1 μs), blurring fast changes. As a result, automated microscopy is unable to detect and classify such cells in feasible time periods

[56].

In the cases described earlier, conventional optical measurement instruments find limited utility in detecting and classifying outliers, as they place stringent requirements that the system vary slowly (millisecond for spectrometers, microsecond for imagers [14, 56, 62]), be repetitive, or lack discerning power. Beyond such cases, it might be possible to reduce the rate of information by e.g. capturing one out of every thousand events. However, this vastly reduces the throughput and may make such measurements infeasible, and in cases of extreme importance one cannot afford to miss even a single event, e.g. in sorting rare cells [56, 63], or in destructive evaluation [64, 65].

Detection of Outliers Enabled by High-Throughput Real-Time Instruments

Observation of rare events in fast streams of information necessitates high-throughput real- time measurements. In the search for the Higgs boson, the Large Hadron Collider, the world’s largest and most powerful particle collider, has had to detect and classify 1 billion particle interactions per second [66]. In this flood of data, the system needs a rejection ratio of >106:1 to 8

eliminate most background events. To make matters more difficult, a mere 4% of those captured have had the possibility to result in new physics, epitomizing the search for the proverbial

“needle-in-a-haystack”. Elsewhere in high-energy physics, precise timing accuracy of single- neutron detection time-of-flight enabled observation of Deuterium-Deuterium fusion reactions in plasma [67]. Highly sensitive, fast, single-molecule single-photon fluorescence detection has been used to temporally resolve the dynamics of fluorescence lifetimes, revealing conformational dynamics that were otherwise hidden in ensemble measurements [68]. Recently, a new real time technique to measure the sequencing of a single strand of deoxyribonucleic acid (DNA) has the opened the potential to study large genomes, a limitation of previous averaging techniques [69].

Coherent Time-Stretch Transform

High-throughput real-time measurements require high-speed over large record lengths.

The Time-Stretch Dispersive Fourier Transform (DFT) [70-72] uses group delay dispersion to reduce the envelope (or modulation) bandwidth of broadband optical pulses and map the spectrum into time, and capture the spectral amplitude at megahertz repetition rates with real- time oscilloscopes (cf. Figure 4). By projecting various physical quantities onto the spectrum, this technique has yielded unprecedented 30+ MHz continuous frame-rates for spectrometers

[19, 73] and imagers [56, 65] and extremely high performance analog-to-digital converters [45,

74, 75], by granting access to the large pixel-bandwidth product of photodiodes. It has been pivotal in capturing the statistics of fast events and finding extreme events therein (see e.g.

Figure 5). In Chapter 2, we discuss the Coherent Time-Stretch Transform (CTST), which is the full Time-Stretch Dispersive Fourier Transform, as it detects the entire complex field rather than just the amplitude [76-78].

Figure 4. General schematic of operation of Time-Stretch Dispersive Fourier Transform (TS-DFT). A broadband optical pulse contains data to be digitized, but is too fast to be directly detected. Dispersion imparts a wavelength dependent delay which stretches the pulse, reducing the envelope bandwidth and simultaneously performing an analog Fourier transform, so that a single pixel detector and real-time oscilloscope may rapidly acquire single-shot spectra. Unknown optical pulses can be the signal; alternatively, known optical pulses may be encoded with, e.g., an image, an electronic waveform, or chemical absorption lines.

Figure 5. Outliers and a variety of spectral distributions found via high-throughput real-time measurements, enabled by the Time-Stretch Dispersive Fourier Transform. Entire spectra are captured in real-time, on a shot-by-shot basis, at the laser’s repetition rate (25 MHz). After digitization, one may 10

perform a wide variety of digital signal processing to extract anomalies from large volumes of data. Here, spectral bands are digitally filtered and their statistics examined. Each row denotes a different experimental condition (peak optical power of the pulse before supercontinuum generation), and each column denotes a different digitally-filtered wavelength. By varying either the condition or the wavelength band, widely differing statistics are observed, including Gaussian, top-hat, and fat tails, and are a result of the nonlinear processes involved [32]. © IOP Publishing Ltd & London Mathematical Society. Reproduced by permission of IOP Publishing. All rights reserved. Coherent Time-Stretch Transform in the Near Field

Although the Coherent Time-Stretch Transform has the ability to capture previously unseen rare events, it is limited by finite resolution inherent to the Dispersive Fourier Transform process.

Remarkably, the Coherent Time-Stretch Transform may perform a complex temporal Fourier transform, in mathematical analogy to Fresnel spatial diffraction theory [72]. A minimum dispersion is required to enter the so-called temporal far field, in deference to the Fraunhofer condition in Fourier optics. Fine spectral features make sufficient dispersion difficult or undesirable, entailing large loss and reduction of peak power and repetition rate. In Chapter 3 we introduce the use of Coherent Time-Stretch Transform using local-oscillator based complex field detection in the temporal near field to ameliorate these issues [79]. There, enabled by this new method, we perform phase-sensitive chemical spectroscopy at multi-megahertz rates for the first time. This technique is four orders of magnitude faster than even the fastest (kHz) spectrometers, and will enable capture of real-time spectral dynamics of chemical reactions (e.g. combustion), high-energy physics and living cells, and monitoring fast high-throughput industrial processes.

The Electro-optic Conversion Bottleneck

Time-Stretch Dispersive Fourier Transform to Detect Electrical Outliers

Electro-optic modulators (EOM) impose changes to optical waves by electrical signals. They are versatile instruments, performing a variety of operations in different areas. The most

important use in industry is in telecommunications [80, 81] to transmit voice, video and Internet traffic, by encoding electrical data on low-loss high-bandwidth optical links. In science, they provide several other applications, including pulse picking to reduce the repetition rate of pulsed lasers [82], optical signal processing of ultrafast pulses [83, 84], and regenerative amplification

(for very large pulse energy) [85], but in most applications their use has been constrained by low

RF bandwidths and high drive voltages (half-wave voltage, pronounced “vee-pie”).

Figure 6. Schematic of the photonic time-stretch (TiSER), an optical pre-processor that retards high- bandwidth RF waveforms for digitization by lower bandwidth but higher sensitivity analog-to-digital converters. TiSER builds on TS-DFT by encoding RF signals onto a time-stretched pulse with an electro- optic modulator, then subsequently slowing down the modulation with a second dispersion. Large modulator bandwidth is imperative as the modulator is exposed to the fast RF signals.

The photonic time-stretcher (TiSER) [71, 74, 86] combines an electro-optic modulator with the Time-Stretch Dispersive Fourier Transform to create a photonic pre-processor that reduces the bandwidth of broadband RF signals for capture by highly-sensitive analog-to-digital converters (ADC) (see Figure 6). TiSER’s high bandwidth and sensitivity make it the perfect platform for catching electrical outliers [87]. TiSER modulates the RF signal of interest onto a chirped optical pulse in the far field with a high-speed electro-optic modulator (EOM). In this

stage, the RF signal temporal waveform is stamped on the spectrum. The pulse is then further dispersed, spreading the colors (and hence modulation) in time, slowing the RF signal. Upon detection, this decelerated RF waveform is captured by precise ADCs. As the time is effectively slowed down by a stretch factor, this photonic pre-processor effectively multiplies the sampling rate and bandwidth of any backend ADC. TiSER’s capability to dramatically slow microwave waveforms has enabled demonstrations of up to 10 trillion samples per second [45], and 8 effective number of bits up to 10 GHz [75], orders of magnitude beyond the state-of-the-art. As the power consumed by electronics scales superlinearly with speed, it has been shown that

TiSER is much more efficient than purely electronic ADCs [88]. In addition, it has demonstrated the potential to replace costly and power hungry sampling oscilloscopes for real-time in-service eye diagram (bit-error-rate) generation and analysis in telecommunications [89].

There is no fundamental limitation to the fastest signal that TiSER can slow down.

Practically, though, the electro-optic modulator limits the bandwidth, as the optical components are inherently broadband, and high-speed photodiodes (which typically boast speeds far beyond modulators) are barely taxed by already stretched signals. The highest performance robust modulators available are integrated Lithium Niobate (LiNbO3) electro-optic Mach-Zehnder modulators. A material’s electro-optic (Pockels) effect induces a change in its refractive index proportional to a strong applied electric field. The Mach-Zehnder interferometer takes a single beam, splits it into two copies, and interferes the two. By adding a different phase onto the two arms with the Pockels effect, this interferometric configuration maps the electric field into an optical amplitude modulation. To achieve the highest speeds and circumvent RC time constants, the electrodes are placed in a traveling-wave configuration [90], such that the RF voltage wave and optical wave travel together to keep their phases identical, necessary as the electro-optic

effect is phase-sensitive. Longer interaction lengths accrue larger optical phase changes, reducing the required voltage, but high frequency effects such as mismatch between optical and

RF phase velocities, and electrode loss, become more severe [90]. Thus, traveling-wave modulators may be designed for low or large bandwidth, but not both. Commercial traveling- wave modulators are in the 10’s of GHz range with on the order of 5 V. Though 100 GHz modulators have been demonstrated in controlled laboratory experiments [91, 92], these are done so at the cost of higher drive voltages or using less well-tested materials. This trade-off limits

TiSER’s bandwidth and sensitivity.

Electro-optic Modulators in Optical Links

The optical link, the most extensive of electro-optic modulator applications, carries many orders of magnitude more information over longer distances than electrical cables [81, 90].

Sending information via coaxial cables over kilometer distances incurs a large loss due to the skin effect in metals [93], an effect which is exacerbated at higher frequencies. Loss in communication links is typically compensated with repeaters, but high-speed coaxial cables require repeater spacings of ~ 1 km [81], which is costly to operate. The low loss in optical fibers

(0.2 dB/km in standard Single Mode Fiber at 1550 nm), the high carrier frequency (~200 THz), and consequently large bandwidths make optical links a feasible alternative at high speeds [81].

Optical links contain a light source (typically electrically pumped diode lasers for low cost and high power efficiency) as the carrier, a modulator to encode the electronic signal onto the optical carrier, a glass fiber to guide the optical wave, and a photodetector to convert the optical information back to electronic.

Analog optical links have more stringent requirements than their digital counterparts.

Whereas receivers in digital links need only distinguish between discrete physical quantities, 14

analog links need very low noise and distortion to reproduce precisely the original signal.

Sources of distortion in an optical link include optical nonlinearities and modulator nonlinearities

[80, 81]. Thus, the signal-to-noise-and-distortion ratio ( , sometimes )

( 1 )

where the three terms on the right hand side are the detected RF powers of each, and spurious- free dynamic range ( ) (range of detected signal powers above the noise floor for which the distortion is below the noise floor) are standard measures of analog links. Poor electro-optic modulator bandwidth and degrade these measures [90]. TiSER may be viewed as an optical link where the RF signal is modulated on a chirped pulse [71, 86], so fast and sensitive electro- optic modulators are vital for both applications.

Optical Sideband-only Amplification: Addressing the Electro-optic Conversion Bottleneck

In Chapter 4, we introduce Optical Sideband-only Amplification, which decreases the half- wave voltage and increases the RF bandwidth simultaneously by utilizing modulation instability.

Inspired by stimulating optical rogue waves to tame their properties, we stimulate modulation instability with electro-optically generated sidebands of an optical carrier which acts as the pump. The gain increases with carrier-sideband frequency separation, which is used to solve two of the most critical issues with high-speed electro-optic modulators: limited RF bandwidth and high voltages required to drive the modulator. We test its capabilities in the context of the analog optical link, a prevalent and demanding application, and measure its effect on the and

as well its ability to reduce and increase the RF bandwidth.

Chapter 2: Coherent Time-Stretch Transform (CTST)

This chapter begins by laying out the important features of the Time-Stretch Dispersive

Fourier Transform and its coherent version, the Coherent Time-Stretch Transform. It then briefly goes over the theory behind the Coherent Time-Stretch Transform, and concludes with two specific implementations.

Introduction

The Time-Stretch Dispersive Fourier Transform (TS-DFT) enables capture of fast dynamics, and collection of sufficient data to generate meaningful statistics and have a realistic chance of finding outliers. TS-DFT has achieved sub-GHz spectral resolution spectroscopy at 1.25 MHz frame rate [73], and was instrumental in the discovery of optical rogue waves [14] and properties of time-constrained modulation that were hidden in ensemble-averaged measurements [30]. The

ADC version of TS-DFT (TiSER) achieved 8 effective number of bits [75] and 52 dB SFDR [94] at 10 GHz, considerably better than the best purely electronic ADCs. The stretching process reduced ADC jitter down to 8 fs, and has performed 1.2 Tb/s real-time optical communications performance monitoring [89]. It has also enabled a new class of imagers, which have imaged

cells at a flow rate of 100,000/s (100 times conventional imaging methods) and achieved 1 in a million sensitivity of detecting rare cells (100 times conventional flow cytometry), significantly relaxing the trade-off between speed and sensitivity [56].

TS-DFT stretches a short optical pulse with group delay dispersion – typically by propagating in a long single mode optical fiber. The delay serves two purposes: chirping the signal enough so that the spectral power is mapped into the temporal waveform, and stretching the time-domain envelope sufficiently for capture with electronics. Combined together, they reduce the envelope bandwidth sufficiently so that the spectrum may be captured with a photodetector and oscilloscope. By using a real-time oscilloscope, single-shot spectra may be acquired. Real-time oscilloscopes boast an array of strengths for a wide variety of experiments, including large memory, wide analog bandwidths, and high sampling rates.

TS-DFT inherently measures the spectra of optical pulses. This has been used to characterize the properties of the light itself. Alternatively, the signal of interest may be placed on top of an optical pulse that serves as the background. The simplest configuration sends the pulse through a chemical sample to probe its absorption spectrum. In addition to the time-stretching element, employing a pre-chirped optical source and modulator slows RF waveforms to effectively enhance the bandwidth and sampling speed of analog-to-digital converters (TiSER). High- throughput imaging (>90 MHz demonstrated [95]) may be performed by using a diffraction grating to map changes in spatial reflectivity into the wavelength domain, while dispersion maps this into time, achieving a space-to-time mapping.

A key metric for continuous operation with high-throughput is the total channel capacity

, which is defined here to be the Shannon single-channel capacity [96] multiplied by the number of detector pixels :

( ) ( 2 ) where is the bandwidth of a single detector element, and is the signal-to-noise ratio.

Hypothetically, one can improve the throughput by adding more pixels in parallel, but practically one is limited by the frame rate of such a sensor array, which is far less than the frame of TS-

DFT. Hence, an important observation is that the detector-bandwidth product, defined as , measures the throughput. TS-DFT exploits the large continuously-operating detector-bandwidth product of photodiodes and real-time oscilloscopes (> 10 GHz) to capture enormous quantities of information quickly, and the sizable memory depth of real-time oscilloscopes to store this information over long record lengths. The high-throughput obtained by this method allows one to capture fast transients [65], calculate statistics and capture outliers [14, 30, 32], and even calculate statistics and capture outliers of fast transients which is introduced in Chapter 5.

The primary method for stretching has been optical fibers, taking advantage of the enormous investment in long-haul communications. Silica fiber boasts the lowest loss of any cable known to man, ~0.2 dB/km at 1550 nm, resulting in the most important telecommunications wavelength

[97, 98]. Combined with a group velocity dispersion of 17 ps/nm/km (= -21 ps2/km, both at 1550 nm) arising from intrinsic material properties and change of mode geometry vs. wavelength [97], the net dispersion per loss is very high, 80 ps/nm/dB (-100 ps2/dB). Dispersion compensating fiber has slightly worse loss (0.5 dB/km) but much higher dispersion (100 ps/nm/km = -125 ps2/km), so that the dispersion to loss ratio is even superior (200 ps/nm/dB = -250 ps2/km or even higher). To illustrate the ease with which optical fiber stretches ultrashort pulses, a 100 fs wide,

10 THz bandwidth pulse sent through 4 km would be stretched to ~12 ns (cf. Eqn. 5), sufficiently slow to capture many points with today’s modern multi-GHz digitizers, and only lose 37% of the original energy.

In cases where extreme group delay dispersion is necessary (> 10 ns/nm) or when stretching at wavelengths far outside the telecommunications band, alternative stretching techniques are necessary. Chirped fiber Bragg gratings provide an extremely large dispersion over very short distances [99]. Physically, axial variations of the refractive index tune the Bragg reflection condition vs. distance along a fiber. In other words, different colors are reflected at different points along a fiber, changing the geometrical distance and hence delay. However, the fabrication process introduces flaws which cause a ripple of the group velocity dispersion profile and impacts the time-stretching process. In the context of TiSER, where especially large dispersion is desired, there has been significant work in correcting these group delay ripples

[100, 101]. In many biological and chemical applications, shorter wavelengths are needed [102].

Although TS-DFT has been demonstrated at these wavelengths with single mode fiber [102], the fiber loss increases quickly due to the dependence of Rayleigh scattering [81]. An alternative method employs mapping colors into modes of a multi-mode fiber, using large modal dispersion to reduce the fiber length by several orders of magnitude [103].

For TS-DFT, a constant group delay dispersion arising from quadratic phase is desirable so that the recorded time may be easily converted to optical frequency. No dispersive element imparts perfectly quadratic phase, and for larger bandwidths the linear mapping assumption becomes less accurate. As long as the dispersion is smooth enough though, the frequency is still mapped to time, and one may simply shift the times to correct these so-called time warps [94].

TS-DFT has even been demonstrated for an octave of bandwidth, taking into account the cubic phase [104]. Fine spectral variations in the group delay occur for chirped fiber Bragg gratings and require more effort to correct [100]. Astonishingly, certain nonlinear group delay profiles bring added flexibility over their linear counterparts. This anamorphic stretch transform (AST),

inspired by anamorphic stretching in the arts, may be tuned, for example, to reduce the required

RF bandwidth for a given stretched duration, resulting in a compression of the time-bandwidth product that is absent with a linear frequency-to-time mapping [105].

The physics of chromatic dispersion is mathematically analogous to spatial diffraction [72].

Free-space propagation in the Fresnel approximation involves evolution of the transverse spatial of the electric field as

( 3 )

where is the axial distance, is a transverse distance, and is the wavenumber. For single- mode waves undergoing constant group velocity dispersion, the electric field changes in time as

( 4 )

where is the group velocity dispersion and is time. Thus, all results in Fourier optics hold for dispersion as well. In particular, sufficient propagation for a given shape sends the field into the temporal version of the far field, as assumed for TS-DFT (cf. Figure 7). The more general time- stretch transform (TST) does not necessarily assume far-field operation, but still assumes constant (or near constant) group delay dispersion vs. wavelength. The most general anamorphic stretch transform (AST) makes no assumption on the shape of the dispersion and though the analogy between diffraction and dispersion breaks down, new possibilities to process the signal arise [106]. Thus, when operating in the near-field or using AST, the temporal waveform no longer reflects the spectrum, so one must detect the entire electric field to backpropagate before stretching, necessitating complex-field detection (see Figure 8 for the relation between these transforms).

Figure 7. Schematic illustration of the difference between temporal near field and far field, named as per the analogy to spatial diffraction. In the near field, the pulse is relatively short and exhibits sharp temporal features, whose shape evolves as the pulse continues to propagate. The long dispersion limit results in the pulse outline converging towards the spectrum, while the temporal duration continues to increase.

Figure 8. Relationship between Time-Stretch Dispersive Fourier Transform (TS-DFT), Coherent Dispersive Fourier Transform (cDFT), Time-Stretch Transform (TST), and Anamorphic Stretch Transform (AST), after Fig. 1 in Ref. [106]. The Anamorphic Stretch Transform uses dispersion with a general profile to manipulate the envelope bandwidth and temporal duration independently. More specifically, the Time- Stretch Transform involves a quadratic phase profile (or nearly so) to reduce the envelope bandwidth for capture by electronics, and leave the time-bandwidth product intact. The Time-Stretch Dispersive Fourier Transform furthermore stretches the pulse into the far field, creating a frequency-time mapping, and when combined with phase recovery is dubbed the Coherent Dispersive Fourier Transform. Both AST and TST require knowledge of the complex field, underpinning the great need for coherent detection, and even TS-DFT benefits from phase information and becomes cDFT.

Researchers have taken the space-time analogy further by introducing the concept of a time lens

[83]. A time lens imparts a quadratic temporal phase that performs the job of adding or removing quadratic phase in the spatial (time) domain. Such an object enables temporal imaging [83, 84,

107], which may magnify or demagnify optical pulses. Although dispersion is used in all cases, the stretching mechanism is fundamentally different from TS-DFT and TiSER. In particular,

TiSER also involves a quadratic phase imparted in the time domain (from the chirped optical

pulse at the modulator), but the signal pulse is not initially dispersed, and thus does not need satisfy a lens equation [71].

The TS-DFT stretching process involves an inherent reduction in peak power and loss, trading speed for sensitivity. Optical amplification has been used to counter this complication, and even provide as much as 30 dB net gain [56]. Optical signals may be amplified after detection in the electronic domain, but poor sensitivity results from inevitable electronic noise added during optoelectronic conversion, as well as from electronics’ gain-bandwidth trade-off [108]. In contrast, optical amplification avoids amplification of electronic noise, permitting much higher signal fidelity and providing multi-THz gain bandwidths [109]. There are multiple methods for optical amplification. Rare earth doped fiber amplifiers provide high gain in short lengths of optical fiber in several wavelength bands [109]. As TS-DFT optical pulses are usually centered at 1550 nm, discrete erbium-doped fiber amplifiers are commonly employed. However, discrete amplification compromises either high or low optical nonlinearity as the signal must start very weak or end very powerful, respectively [45]. By performing distributed amplification in the propagation fiber, distributed Raman amplification circumvents this limitation by maintaining the signal at a moderate power, and has proven immensely successful in telecommunications [44]. The stretching fiber provides a convenient medium for distributed

Raman amplification in CTST [110], and has crucially advanced both CTST speed and sensitivity [73]. Fortuitously, the time-stretch fiber of choice at 1550 nm, dispersion compensating fiber, has a relatively large Raman amplification coefficient [44]. In addition, as

Raman scattering involves a virtual transition, it may be employed at a much wider range of wavelengths. At shorter wavelengths where attenuation in the dispersive fiber is a much higher

concern, Raman amplification has recently been demonstrated [111] and can improve the sensitivity over direct detection by almost 20 dB [112].

The large carrier frequency in optics is too fast for electronic digitization. The common method for capturing optical information, photodetection, is sensitive to the carrier averaged intensity of the field, recording the optical intensity as an electronic voltage. The Coherent Time-

Stretch Transform contains a photonic processing stage that encodes the complex electric field into multiple intensity measurements [76, 77, 79, 113]. TS-DFT, in contrast, does not capture the optical phase. Knowledge of the full complex field provides significantly more information and experimental freedom than the spectral amplitude alone. CTST may act as a FROG or SPIDER but with over 4 orders of magnitude higher throughput. Knowledge of the complex electric field permits one to digitally Fourier transform, reconstructing the optical pulse profile in the time domain. The full optical field is necessary to digitally backpropagate through well-characterized optical elements, letting one “look back in time” and thoroughly analyze the optical evolution.

Backpropagation also eliminates the need for frequency-to-time mapping, permitting operation in the near field [76, 79]. Even when employing frequency-to-time mapping, the phase is found to provide valuable information about the system of interest, including depth measurement in optical coherence tomography [114], higher specificity classification of cells [115] and eliminating the interference-induced bandwidth limitation of TiSER [116].

Implementing the Coherent Time-Stretch Transform requires a phase retrieval technique.

Phase recovery techniques are used in a vast array of fields, including communications, image recovery, hologram generation, ultrashort pulse reconstruction, electron microscopy, adaptive optics, and X-ray imaging [38, 117-121]. They can be iterative or direct. The high-throughput real-time nature of CTST places several demands on the technique used. While increasing the

numbers of measurements can improve reconstruction accuracy and bring other benefits, such as consistency checks, doing so may strain both hardware and software. The duty cycle limits the number of measurements a single pixel may capture, beyond which the acquisition hardware requires multiple channels. With the exception of pure averaging, more components are required to manipulate the signal to introduce measurement diversity. At the same time, numerical processing cannot increase the information available based on the set of measurements, but instead generally introduces distortion and noise. This degradation is aggravated by the limited sensitivity of fast ADCs which are needed to capture the time-stretched signals. Thus, the technique must minimize signal deterioration. Furthermore, real-time processing must be employed for e.g. feedback or data reduction, creating a compelling need for an efficient algorithm to handle the increased data rate, and in some cases a field programmable gate array

(FPGA) may also be required to speed up the computation. Although a number of algorithms for

CTST have been demonstrated, they have yet to be optimized against such stringent requirements.

Principle

Stage 1 of Coherent Time-Stretch Transform

In this section, we describe the theory behind the two stages of the Coherent Time-Stretch

Transform: (1) the time-stretching process and (2) complex-field recovery. The first step of the

Coherent Time-Stretch Transform is the Time-Stretch Transform, i.e., stretching the pulse.

Sufficient stretching is required to reduce the modulation bandwidth as well as perform the

Dispersive Fourier Transform, but with complex-field recovery the analog Fourier transform is

no longer required. Nevertheless, it is important to be aware of whether a signal is in the far field, as knowledge of frequency-to-time mapping influences the back-end processing.

To determine the far field requirement, as well as find out its significance, it is instructive to highlight the main points of the derivation of the Time-Stretch Dispersive Fourier Transform from Refs. [122, 123]. Let ( ) be the slowly varying electric field envelope of a single transverse mode in time such that one may ignore the second derivative term for the axial variation of the electric field (see e.g. [122-124]). Then upon propagation a distance through a dispersive medium characterized by a group velocity dispersion ,

( ) [ ] ∫ ̃ ( ) [ ( ) ]

where is the angular frequency of the electric field envelope and ̃ ( ) is the Fourier transform of ( ). Here, we are interested in finding the conditions under which one can take the ̃ term outside the integral. If ̃ ( ) varies slowly enough (which we quantify soon), then the integral only involves the exponential term. By invoking the stationary phase approximation

[123],

( ) √ [ ] ̃ ( )

which also yields the time-frequency mapping

( ) ( 5 )

Except for the exponential phase factor and some constants, the spectrum has been mapped into time. Here, we emphasize that the electric temporal field matches the spectral field (not just amplitude). Let the finest spectral feature in ̃ ( ) be of width . The stationary phase approximation requires that ̃ ( ) not vary much in a range over which the exponential term 25

starts to become larger than (the first zeros of the cosine and sine components). This is commonly expressed as

√

where denotes the approximate crossover frequency for a spectral feature to leave the near- field [123]. We emphasize that is not to be interpreted as the spectral resolution of the Time-

Stretch Transform. Rather, if any feature is narrower than that value, then that feature remains in the near field. Simulations [76, 123] and experiments [76] show that these near-field features are realized as complex interferograms with sharp features rather than just a blurred version of a narrow feature (cf. Figure 7). In fact, the information in these interferograms has been used for one method to recover the spectrum with intensity only measurements [76].

Stage 2 of Coherent Time-Stretch Transform: Complex-field Recovery

There are multiple methods to recover the phase in the Coherent Time-Stretch Transform [76-

78, 113, 125]. Stereo Time-Stretch is one such technique, and I introduce a new near-field capable local-oscillator-based CTST in Chapter 3, so these are described here in more detail.

Stereo Time-Stretch, also known as Stereopsis-inspired Time-stretched Amplified Real-time

Spectrometer (STARS), is based on the diversity of a filtered and unfiltered measurement [78,

125]. Stereo Time-Stretch is analogous to depth perception in eyes, or locating the source of sound with two ears, wherein the two detectors provide different versions of the same information. The schematic setup is shown in Figure 9. A signal under test is dispersed in a dispersive element to slow down the envelope bandwidth and map the frequency to time. The stretched pulse is then split: one copy is directly detected yielding the spectrum, and the other copy is optically filtered before detection. The STARS algorithm recovers the spectral phase 26

from these two measurements. The filter may be engineered for the specific signal of interest, and must only have a nonzero instantaneous response and satisfy certain phase requirements

(described later).

Figure 9. Stereo Time-Stretch, or Stereopsis-inspired Time-stretch Amplified Real-time Spectrometer (STARS). The signal is stretched in a dispersive element, performing the time-stretch transform. One copy is directly detected as the spectrum, and the other is optically filtered. The spectral phase is recovered using these two measurements as specified in Equation 9.

The electric field after stretching | | is optically filtered by a filter with temporal response ( ) | ( )| ( ). As the signals will be sampled with period , we follow the

discrete time signal analysis in Ref. [125] to calculate the filtered field | |

( 6 ) ( ) ∑ ( ) (( ) )

where the sum starts from as long as the filter is causal. Eqn. 6 may be written as the sum

of the instantaneous response ( ) ( ) and the delayed response | |

( 7 ) ( ) ( ) ( ) ∑ ( ) (( ) ) ( ) ( )

Photodetectors measure power, so the unfiltered power is measured as | ( )| , and the filtered power is measured as

| ( )| | ( ) ( )| | | | ( )|| ( )|| | ( ) ( ) ( 8 ) [ ]

At this point, the full filter response ( ) is known, and | | and | | have been measured, assuming knowledge of from time to time ( ) . Then, the delayed response is calculated from all previously reconstructed electric field samples (due to the causality of ( )), and all terms in Eqn. 8 are known except ( ), so one simply inverts the equation to find the instantaneous phase:

| ( )| | ( ) ( )| | | ( 9 ) ( ) ( ) [ ] | ( )|| ( )|| |

Induction on the temporal step yields knowledge of the past phase from time to time

( ) . Once the phase is known, frequency-to-time mapping or backpropagation is used to find the initial input. Arccosine exhibits an ambiguity, so the interference phase must be limited to a range of [ ] or [ ]. It has been discussed that, for signals in the far field, it is sufficient for the spectral phase response to be bounded within one of the previously mentioned ranges [125].

CTST with a local oscillator (or reference) is inspired by coherent detection in optical communications [77, 116]. Coherent detection involves interfering the incoming signal with a local oscillator [81, 116, 117], resulting in a cross intensity term proportional to the signal and reference electric fields. It provides all the benefits of complex-field detection plus signal gain by beating with a strong reference. Interference is achieved by combining the signal and local oscillator with e.g. a beam splitter or fiber coupler, providing complementary outputs. The two complementary outputs are modeled as

( ) [| ( )| | ( )| | ( )|| ( )| ( ( ) ( ))] ( 10 )

where and are the local oscillator (reference) and signal electric fields, respectively, and and are their phases, as functions of time . As the carrier in the Time-Stretch Transform is wideband, beating with a narrowband local oscillator would result in a THz-range bandwidth interference term, too broad for electronics to capture. Instead, by interfering the time-stretched signal with an equally chirped reference, the beat chirp is cancelled [116] (c.f. Figure 10).

Figure 10. Schematic of Local-Oscillator based CTST, which uses coherent detection to recover the signal phase. Both the signal under test and the local oscillator are chirped in dispersive elements. The signal is delayed with respect to the local oscillator. Upon interfering in the combiner, the frequency-time mapping translates the delay into an intermediate frequency , performing heterodyne detection. The two complementary outputs are digitally processed which, along with a well- characterized local oscillator, reconstructs the full optical field.

Heterodyne detection is a form of coherent detection where the reference carrier frequency differs from the signal’s, typically in the GHz range. This results in shifting the center frequency of the beat frequency away from , so the phase information is contained in RF sidebands. As the Coherent Time-Stretch Transform introduces a chirp, a time delay between the reference and signal results in the desired upshifted beat frequency [116] (cf. Figure 11a)

( ) ( 11 ) where is the group velocity dispersion and is the length of the dispersing element. Balanced detection subtracts the complementary outputs to yield the beat term

| ( )|| ( )| [( ( ) ( ))] ( 12 )

Sufficient upshift is required to separate the sidebands from DC so that the Hilbert transform may recover the analytic representation of the beating term

| ( )|| ( )| [ ( ( ) ( ))] ( 13 )

Summing the complementary outputs eliminates the beat term, leaving the sum of the two independent intensities

| ( )| | ( )| ( 14 )

Knowledge of the reference electric field is then sufficient to reconstruct the signal via

( 15 ) ( ) √( ( ) | ( )| ) ( ( ) ( )) where denotes the argument of a complex number. The key assumption for recovering the analytic representation via the Hilbert transform is that phase information contained in the beat frequency modulation mixed down to does not extend into negative frequencies, which would create an undesired image interference, a well-known problem in electrical heterodyne detectors. Mathematically, we require that

( ) ( 16 )

Near field features make this requirement difficult, as they are narrow in frequency content, but broad in time, so a time delay no longer translates into a constant frequency shift. Upon beating with a time-stretched local oscillator, the beat frequency modulation is chirped and broadband

(cf. Figure 11b). Nevertheless, a frequency-to-time mapped pulse with near field features may still be recovered with this algorithm, provided that the beat sidebands do not cross zero so that there is no undesired image. Hence, the top feature in Figure 11b will be accurately reproduced, whereas, the bottom feature will not.

Figure 11. Schematic of optical time-frequency representation of local oscillator (LO) based Coherent Time-Stretch Transform. (a) The signal and local oscillator are in the far field, and a time delay between them results in a constant beat frequency for heterodyne detection [116]. (b) The signal and local oscillator backgrounds are in the far field, but the signal exhibits some near field features. The top near field feature will cause a chirp in the beat frequency, but remains upshifted from DC, so its phase is reproduced accurately. In contrast, the bottom near field feature crosses the local oscillator time- frequency curve, so that the beat frequency modulation extends into negative frequencies. This does not allow the Hilbert transform to reproduce the analytic representation, so here a larger time delay would be required.

Chapter 3: Coherent Spectroscopy in the Temporal Near

Field using the Coherent Time-Stretch Transform with a

Local Oscillator

The Time-Stretch Dispersive Fourier Transform (TS-DFT) has enabled high-throughput real- time nanosecond-resolved spectroscopy of non-repetitive phenomena. However, molecular and other chemical spectra exhibit linewidths so narrow that a key TS-DFT assumption breaks down: that of frequency-to-time mapping. In this “temporal near-field,” the intensity waveform contains insufficient information to reconstruct the spectrum. The Coherent Time-Stretch Transform is a form of TS-DFT that captures the complex optical spectrum. In this chapter, I apply a new form of the Coherent Time-Stretch Transform which I introduced in Chapter 2 which recovers these spectra from temporal interference patterns with a reference laser (local oscillator) in the temporal near field [79]. I also experimentally demonstrate phase-sensitive absorption spectroscopy at megahertz repetition rates for the first time. Fortuitously, the spectral phase, absent from conventional spectrometer measurements, is found to improve the absorption 32

strength dynamic range. Temporal near field spectroscopy is expected to be used for monitoring fast chemical dynamics such as combustion or high-energy physics.

Introduction

Many natural phenomena, such as impairments in data networks, phase transitions, protein dynamics in living cells and chemical reactions, occur on fast time scales and are non-repetitive, creating a need for high-throughput real-time measurements. The Time-Stretch Dispersive

Fourier Transform (TS-DFT) [19, 70, 72, 113, 126, 127] has proven invaluable for high- throughput, real-time detection of fluctuations in nonlinear processes, including optical rogue waves and pulse-to-pulse fluctuations in modulation instability and supercontinuum generation

[14, 30, 32, 62, 104]. It has also enabled a new class of imaging systems able to identify cancer cells with a one-in-a-million sensitivity [56, 65, 128], as well as high-resolution, fast analog-to- digital-converters [71, 74, 75, 116]. The time-dilation process along with the inherent fiber losses results in reduction in peak power and loss of sensitivity; a key innovation has been the introduction of distributed Raman amplification (with a concomitant lower noise figure and distortion than discrete amplification), resulting in single-shot and high-sensitivity performance

[45, 65, 110]. Additionally, the TS-DFT has been applied to real-time spectroscopy in a variety of systems [19, 110, 129-132]. In short, the TS-DFT chirps a pulsed source of radiation so that its temporal envelope matches its spectral profile (frequency-time mapping), and the waveform is slow enough to be captured by a real-time digitizer. The former result has been termed the far- field regime in analogy with the corresponding situation in spatial diffraction theory – the

Fraunhofer limit.

We may also consider the temporal near-field regime in analogy with the corresponding case in spatial diffraction, i.e., the Fresnel limit. Specifically, a waveform may be stretched with a reduced amount of dispersion, resulting in a temporal near-field signal [76]. In this limit, the time-stretch transform may permit capture of the waveform in the time domain, but the intensity envelope does not correspond to the spectral profile: in other words, a one-to-one correspondence between frequency and time is not achieved. However, it has been demonstrated that an iterative procedure can be used to recover both the spectral amplitude and the phase from dual near-field measurements, eliminating the need to work exclusively in the far-field regime

[76]. This recovery is possible because the constituent spectral components are not fully separated in the near-field waveform, resulting in an interferogram that contains amplitude and phase information. Thus, the dispersive element generates a near-field time-stretch transform of the waveform in this limit. Alternatively, the result can be considered a Coherent Dispersive

Fourier Transform as both spectral amplitude and phase information are available, comprising a vector measurement. In many systems, the spectral phase can be computed from the amplitude spectrum based on the causal requirements formulated in the Kramers-Kronig relations.

Nevertheless, phase measurements provide a complementary source of information, which can supplement or enhance spectral amplitude data impaired by, e.g., noise, finite frequency resolution, and limited frequency span. For example, proper phase reconstruction via the

Kramers-Kronig relations may require measurement of spectral features outside the measurement bandwidth. Indeed, simultaneous measurements of spectral amplitude and phase have proven useful in the investigation of a variety of physical processes [133]. Moreover, in the general case, phase and amplitude are not uniquely connected [134]; thus, amplitude and phase channels can

contain distinct information, and dramatically different phase profiles may even be measured for identical amplitude spectra [135].

Principle

Here, we introduce a form of the time-stretch transform wherein near-field measurements are enabled by dispersive stretching with simultaneous optical amplification followed by complex- field detection and reconstruction. In this Coherent Time-Stretch Transform (CTST), interfering the signal with a reference waveform and stretching the interferogram in time allows recovery of the spectral amplitude and phase in the near-field limit (cf. Figure 12). Other detection schemes may also be possible (e.g., self-referenced using optical filters [78, 125]). Complex-field detection offers another route to near-field operation, which avoids the usual far-field requirement of the TS-DFT. Near-field acquisition reduces the dispersion requirement, decreasing loss and allowing measurements at an increased repetition rate (i.e., avoiding pulse overlap). The bandwidth limitations of the photodetector and digitizer are then the remaining constraints on smallest permissible dispersion [76]. The coherent DFT, which is similar to the

CTST but assumes the far-field regime, has enabled real-time coherent digitizers [116], phase- sensitive imagers [115], and vector optical spectrum analyzers [77, 136].

Figure 12. The unknown optical field input is stretched with group-velocity dispersion (GVD), slowing down the signal in time and reducing its modulation bandwidth so that it may be captured by electronic instruments. The complex optical field is detected (using an independent reference signal or self- referenced), and the desired input is reconstructed digitally [79].

Figure 13. Experimental block diagram of the Coherent Time-Stretch Transform (CTST) applied to spectroscopy. Pulses of 350 GHz full-width at half maximum bandwidth around 1530 nm are stretched 12 using group-velocity dispersion (GVD). One copy traverses the sample (80 cm C2H2 acetylene gas cell), and another passes through the reference path, delayed at to create the upshift frequency. The complimentary outputs of the interferometer are interleaved at half the pulse period and detected by a photodetector and 16 GHz oscilloscope. The time-stretch segment can be applied at any point in the sequence prior to the photodetection, and does not necessarily have to be applied in the particular order shown here. MLL: mode-locked laser [79].

Use of a well-characterized reference signal (local oscillator – LO) provides the benefits of coherent detection, including heterodyne gain and reduced noise. We demonstrate real-time near- field spectroscopy of time-stretched narrow absorption lines of acetylene gas. Figure 13 shows an experimental implementation of the CTST technique. An optical source provides both the signal and reference radiation, both of which are chirped in a Raman-amplified dispersive element. The signal travels through an absorption cell and beats with a time-delayed reference that upshifts the beating term into an intermediate frequency. At the photodetector, the recorded signal is:

( ) [| ( )| | ( )| { ( ) ( )}] ( 17 )

Where ( ) is the recorded intensity/current, ( ) and ( ) are the reference and signal optical fields, and denotes the real part. By performing the Hilbert transform on the beating term, the phase difference is recovered, and knowledge of the reference allows reconstruction of the full optical field. This works as long as the envelope is in the far field, whereas the spectral features themselves may be in the near field. In contrast to algorithms that make the far-field

assumption [137-139], here the interferometry and phase recovery is performed in the time domain. Once the time-domain waveform is reconstructed, a discrete Fourier transform with back-propagation results in the complex spectrum; thus, the instrument operates as a complex- field spectrum analyzer. The phase accuracy depends on a sufficiently large upshift frequency so that the interference term separates into two sidebands with no frequency overlap and the proper analytic representation is obtained. On the other hand, the upshift frequency must be within the bandwidth of the receiver for proper detection.

Experiments

Figure 14 and Figure 15 show results from experimental measurements of an 80 cm Triad

12 Technology C2H2 acetylene gas cell at a pulse repetition rate of 36.7 MHz. The CTST measurements are able to recover the narrow gas lines with only -1910 ps/nm of dispersion.

Figure 14a shows an unprocessed digitizer waveform which contains a 15 GHz beat frequency from the unbalanced interferometer. Although the dominant absorption features in this range have very narrow linewidths (~450 MHz FWHM native linewidth), the large absorption of this gas cell broadens the perceived widths of the features in a normal spectral measurement: ~3 GHz and ~5 GHz for the odd and even-numbered lines, respectively, given the dynamic range of the experiment. These features are dispersed into near-field waveforms with the present dispersion, evidenced in the reconstructed time domain shown in Figure 14b. (A discussion of dispersion requirements separating the near and far field is presented in Ref. [76].) Thus, the unprocessed trace is not the actual spectrum, although acetylene features can be discerned as dips. Dispersing the lines into the far field would require significantly more dispersion; as noted, large amounts of dispersion introduce loss and limit the maximum repetition rate. 37

Figure 14. (a) Unprocessed experimental trace, voltage vs. time. The beat (or intermediate) frequency at 15 GHz between the reference and signal is observed, created by a time delay between the two waveforms. This interference allows the spectral amplitude and phase to be recovered via the Hilbert transform for complex field spectroscopy. Refresh rate is 36.7 MHz set by the mode-locked laser. (b) Reconstructed time-domain power (blue, left axis) and phase (green, right axis) vs. time, as measured by the oscilloscope. The power trace alone is equivalent to the Time-Stretch Dispersive Fourier Transform, but the Coherent Time-Stretch Transform also captures the phase information, allowing reconstruction of the spectrum without a frequency-time mapping. The overshoots following the absorption features are a signature of the temporal near field [76, 79].

Figure 15. Spectral power (blue, left axis) and phase (green, right axis) recovered from the Coherent Time-Stretch Transform (CTST) applied to an acetylene gas cell. Time-averaged spectrum obtained with an optical spectrum analyzer (OSA) shown for reference [79].

The reconstructed spectrum is shown in Figure 15. Neither averaging nor smoothing have been used. The overshoots in the near-field features in Figure 14b are absent upon spectral reconstruction, demonstrating that the Coherent Time-Stretch Transform can reconstruct the

spectrum from near-field measurements. The present absorption feature widths provide a conservative measure of the spectral resolution (~4 GHz); given the bandwidth of ~500 GHz, the time-bandwidth product (TBP) is at least 125. Much larger bandwidths (and hence time- bandwidth products) should be possible with system improvements. A measurement by a conventional optical spectrum analyzer (OSA) is also shown for reference. Small discrepancies between the traces may arise from uncompensated optical nonlinearities, incomplete knowledge of the reference phase, and limited oscilloscope bandwidth; refinement of the recovery algorithm is also likely to bring improvement. For perspective, it may be noted that the CTST spectra are collected ~108 times faster than those from the OSA. Also, as the CTST data are obtained in a single shot, interferometric stability is not as critical as it would be in time-averaged interferometry, where interference fringes must remain stable over a very large number of pulse pairs.

Simulations

Simulations of the full CTST system are shown in Figure 16 and include the full complex transfer function, where the phase has been determined from the HITRAN absorption data [140] by applying causal requirements to a Lorentzian approximation of the lineshape. They support the experimental measurements and demonstrate that residual artifacts in the reconstruction can be reduced by employing a digitizer with larger electronic bandwidth. A larger digitizer bandwidth will also better resolve narrower linewidths, further improving the time-bandwidth product. Figure 16a shows a simulation of the spectral amplitude under the experimental conditions (16 GHz digitizer bandwidth and 15 GHz intermediate frequency). In the case of unlimited electronic bandwidth, the ripples in the spectral amplitude are eliminated (cf. Figure 39

16b). With a 32 GHz bandwidth, double that used in the experiment, some amplitude ripples remain, but fidelity is improved (cf. Figure 16c,d). A simulated phase trace is included in Figure

16d, showing the characteristic dispersive shape seen in the experimental measurement.

Figure 16. Simulations of spectral power vs. frequency of the input spectrum (red) and the recovered spectrum (blue) using the Coherent Time-Stretch Transform. (a) The reference is delayed to create a 15 GHz beat (intermediate) frequency ( , and is detected with a 16 GHz bandwidth (BW) receiver. (b) Infinite beat frequency and bandwidth. (c) 30 GHz beat frequency and 32 GHz bandwidth receiver. (d) Simulations of Coherent Time-Stretch Transform power (blue) and phase (green) with a 30 GHz beat frequency and 32 GHz bandwidth receiver [79].

As previously mentioned, it should also be noted that the absorption lines are actually much stronger than observed in the present amplitude spectra and that the even numbered lines are approximately 3x stronger than the odd lines on a logarithmic scale, as calculated from information in the HITRAN database [140]. The absorption strengths observed with the OSA and time stretch are limited by both spectral resolution and noise in the former case and primarily by the noise in the latter case. Although these limitations mask the difference in the absorption strengths of the even and odd lines, the phase measurements record a much larger phase shift for 40

the even lines. Thus, even though amplitude measurements frequently have difficulty distinguishing differences in very strong resonances, the phase shifts can be indicative of activity beyond the dynamic range of the amplitude measurements. The absolute phase shifts observed in the present measurements have been reduced by the finite electronic bandwidth of the digitizer; a simulation assuming detection with double the electronic bandwidth shows larger phase shifts

(cf. Figure 16d). Nevertheless, a comparison of the phase shifts of the even and odd lines from the experimental data still indicates that the lines have substantially different absorption strengths, even though the digitizer bandwidth is limited to 16 GHz.

Conclusion

In conclusion, the Coherent Time-Stretch Transform retrieves spectral amplitude and phase information from near-field interference measurements at extremely high throughput. It builds upon the Time-Stretch Dispersive Fourier Transform by adding complex field detection and operating in the near field, and the ability to capture phase data offers an additional source of information, which may enhance both dynamic range and sensitivity. We have demonstrated for the first time that this method can be used to perform phase-sensitive absorption spectroscopy of a gaseous sample at millions of frames per second. This technique can also be applied to spectral measurements of other samples and scattering processes. It may also find application in the study of the statistical and dynamical properties of laser sources and supercontinuum generation. Both the TS-DFT and CTST utilize linear or near-linear dispersive profiles, conserving the time- bandwidth product of the intensity envelope. Yet, in some cases, measurement efficiency can be enhanced by altering the waveform’s native time-bandwidth product. The anamorphic stretch transform (AST), which also measures the complex-field in real-time, aims to engineer the time- 41

bandwidth product of the waveform by harnessing specific dispersion functions [105]. This type of manipulation has great potential to facilitate real-time measurements over long record intervals by compressing the data volume.

Chapter 4: Optical Sideband-only Amplification:

Overcoming the Electro-optic Conversion Bottleneck

Conventional high-performance electro-optic modulators, which place electronic information on optical waves, are the bottleneck of optical communication links and photonic time-stretch analog to digital converters, suffering a trade-off between low-voltage operation and large microwave bandwidths. Modulation instability, a nonlinear optical phenomenon that is the precursor to supercontinuum generation (large optical spectral broadening), amplifies optical sidebands at the expense of the carrier, and whose effect increases with frequency separation. In this chapter, I introduce the concept of Optical Sideband-only Amplification, wherein I harness modulation instability’s properties to amplify electro-optically generated sidebands to simultaneously reduce the required voltage and increase the RF bandwidth, and demonstrate large gains both numerically and experimentally [141, 142]. This has the potential to enable modulators to catch up to electronic processing that has progressed beyond 100’s of GHz.

Introduction

Modulation instability (MI) is a nonlinear physical process whereby tiny disturbances spontaneously grow on an otherwise quiet background [97]. First discovered in hydrodynamics where it is known as the Benjamin-Feir instability [143], MI arises in diverse contexts such as sand-dune formation [144], free electron lasers [145], optics [146], and matter-waves [147], and has been described as “one of the most ubiquitous types of instabilities in nature” [148]. This process has been proposed as the mechanism to give rise to oceanic rogue or freak waves [3], giant “walls of water” [149] that appear suddenly even out of calm seas [150]. Discovery of their optical counterpart, optical rogue waves [14], confirmed that specific noise stimulates their generation via modulation instability. Purposefully stimulating the creation of optical rogue waves has led to enhanced bandwidth, stability and brightness [26-29] of the nonlinearly broadened light known as supercontinuum [17]. Here, we show how this phenomenon can be exploited to boost the weak electro-optic effect, one of the most fundamental and pressing predicaments in optical communication. In our approach, modulation sidebands stimulate MI in a

3rd order nonlinear optical material placed after the electro-optic device. This effect enhances

( ) of the electro-optic material and also compensates for the device related high frequency roll- off. Boosting ( ) in one material with the ( )-induced MI in another is an intriguing concept that enables low voltage electro-optic modulation at ultrahigh frequencies.

Electro-optic (EO) modulation is the critical bridge that connects electronic computing to optical communication [90, 151]. While today’s communication, sensing, and computing systems are predominantly electronic, optics has become indispensable for high-speed communication. Unfortunately, the development of EO modulators has not kept up with the tremendous pace of electronics over the past decades. To be sure, access to broadband optical 44

channels is curtailed by the limited bandwidth of EO modulators that is currently in the 50-

100 GHz range [151]. These EO modulator speeds are achieved at the cost of increase in modulation voltage brought about by the decline of electro-optic interaction due to the increased loss of metallic electrodes at high frequencies. As the electronic frontier continues to progress toward the terahertz regime, the modulator bottleneck will become more pronounced. Existing approaches for improving EO modulators are primarily focused on overcoming the RC time constant and increasing the interaction length between optical and electrical waves using traveling wave electrodes [90]. While these have achieved improved frequency responses, the progress has come at the cost of increased electrical drive voltage due to the shorter interaction length required for these bandwidths [91]. As a result, devices can be designed for either low voltage or high frequency, but not both. A different technique achieved EO gain over a multi-

GHz bandwidth by interfering the two outputs of a Mach-Zehnder modulator after undergoing self-phase modulation [152]. Another approach for improving EO modulation has been utilizing exotic materials with a strong electroabsorption effect such as transparent conductive oxides

[153] and graphene [154, 155].

A new concept is proposed as a path to broadband low-voltage electro-optic modulation

[141]. Our method enhances any intensity modulator including those based on electrorefraction as well as electroabsorption effects. Optical Sideband-only Amplification is numerically demonstrated to reduce a Mach-Zehnder modulator's half-wave voltage, known as , by 17 dB, as well as nearly doubling its bandwidth by exploiting the intrinsic frequency dependence of the

MI gain spectrum.

Principle

Electro-optic Modulation and Distortions

There are many physical mechanisms by which electronics impose information onto optical waves [124], but the most proven high-speed instrument is the electro-optic Mach-Zehnder modulator [90]. The electro-optic effect [124] involves the direct change of the refractive index of a material in response to an electric field. The refractive index may be expanded in a Taylor series as a function of electric field strength to find the linear and quadratic electro-optic coefficients, respectively called the Pockels ( ( )) and Kerr ( ( )) coefficients (N.B. The Pockels and Kerr coefficients are often expressed as and , and their relation to the more fundamental ( ) and ( ) is given in Ref. [124]). In non-centrosymmetric materials, the Pockels coefficient is nonzero and its effect is generally stronger than the Kerr effect. Placing electrodes on a Pockels material and applying a voltage linearly alters the refractive index, causing a proportional increase in the phase retardance of an optical wave that passes through it as given by

( 18 ) where

( ) ( ) ( 19 ) with electrode separation , optical refractive index , wavelength , electro-optic overlap factor , and interaction length [124]. This directly shows why is called such: when

, the phase is changed by . Varying the phase difference between two beams impingent on the same spot changes their interference pattern, so a time-dependent phase produces a time- dependent amplitude. A Mach-Zehnder interferometer involves splitting a single beam into two

which accrue independent phase delays, and subsequently combining the beams. The Mach-

Zehnder modulator electrically changes the phase shift in one or both paths via the Pockels effect, modulating the optical field. This geometry is amenable to fabrication as an integrated optical device to be compact, reliable, low cost, and fairly insensitive to fabrication defects [90].

An ideal Mach-Zehnder modulator may be modeled with

( ) { [ ]} ( 20 ) where and are the optical powers before and after the modulator, respectively [90].

Intuitively, the cosine arises from the interference between optical waves with different phases,

and mathematically it comes from | | . Real Mach-Zehnder modulators have losses and uneven splitters and combiners. As long as these non-idealities are voltage-independent, their only effect is to reduce the extinction ratio [90]

( ) {( ) ( ) [ ]} ( 21 ) where is the extinction (dark state) and is the loss (light state). Fortunately,

Equation 21 has the same form as Equation 20, so even imperfect Mach-Zehnder modulators may function in the same way. As these imperfections will not affect our analysis, we use

Equation 21 and split the voltage into a bias voltage and a time-varying voltage , such that

( ) { [ ( ) ]} ( 22 )

Here we analyze a sinusoidal modulation by choosing (“quadrature” bias), and

( ) ( ),

( ){ [ ( )]}

where is the modulation depth. Using the Jacobi-Anger expansion and assuming small modulation depth , we find that the lowest order terms are

( ) { ( ) ( )} ( 23 )

The term carries the information of interest, and the term is a 3rd harmonic distortion. A smaller for the same peak is hence equivalent to a larger modulation depth . When two or more different frequencies are present, they mix to create third order nonlinear distortions of similar strength to the term. Although third order distortions are dominant over other orders in a quadrature biased analog optical link [90], deviations from quadrature create second order distortions. In TiSER, dispersion induced phase may also generate second order spurs, which are theoretically eliminated by using a differential detection method [71, 86]. However, care must be taken to ensure that TiSER is properly calibrated or the second orders may not cancel and instead significantly degrade the signal [156].

Fast modulators use a traveling-wave geometry, aptly named as the RF wave travels with the optical wave in the same direction. By keeping the phase fronts together over a long distance, the interaction is maintained over multiple RF wavelengths. A mismatch in the phase velocities, or loss of the electrical wave, reduces the change in optical phase. The net effect is to increase the

as a function of frequency, given by [90, 157]

( 24 ) ( ) ( ) ( ) [ ] ( )( )

where ( ) is the at low frequencies, |( ) ( )| is the reflection coefficient between the microwave source and modulator with impedances and

respectively, is the RF loss of the electrodes, and is the phase drift between the RF and optical waves

( ) ( 25 )

with being the refractive indices in the modulator of the microwave, and

√ ( 26 ) where is due to conductor losses, is due to dielectric and radiation losses, and

. Equations 19 and 24 show the trade-off between low and large bandwidth: longer lengths decrease ( ) but increase ( ) at high frequencies via and . It is also a convenient model for the frequency dependence of a Mach-Zehnder modulator for simulations.

Noise in Analog Optical Links and Time-Stretch Digitizer

The noise in analog optical links and TiSER directly affects the ability to accurately capture the signal of interest. Even in absence of noise on the signal itself, the optical detection process adds several sources of noise [81]. Thermal noise arises from random electron motion in the load resistor of the photodetector circuitry, resulting in a stochastic current. Also known as Johnson noise, it may be modeled as Gaussian random process with variance

( ) ( 27 ) where is the Boltzmann constant, is the temperature, and is the detector load resistance.

The discrete nature of light and electricity results in different source of randomness, shot noise.

The number of photons arriving at the detector in a given interval is a Poissonian random process, which dictates that the variance equals the mean [81]. A semiclassical model of shot noise places the noise in the optical field by inserting one fictitious photon per mode with random phase [158, 159]. This model is particularly convenient for nonlinear simulations where 49

even weak noise may have a strong effect on field evolution [14, 29]. It creates a beat intensity between the signal and vacuum fields which is proportional to the square root of the number of photons in the signal, yielding the same variance as other shot noise models. An in-depth discussion of the quantum mechanical origin of shot noise in terms of photon statistics of spontaneous and stimulated emission processes may be found in, e.g., Ref. [160]. Regardless of the view chosen, the shot noise at the photodetector exhibits a mean square current fluctuation

( 28 ) where is the current generated by the average optical power and is the detector responsivity (number of free electrons collected per impingent photon, in A/W). Other sources of noise are not as important for typical analog optical links. Dark current arises due to thermally generated electrons in the detector, and the generation rate also obeys shot statistics.

As long as the optical power is far above the equivalent dark power (~nW [81]), this term is negligible. Relative intensity noise depends on the precise laser used and is band-limited [81], and so will be ignored here.

Optical Sideband-only Amplification

Figure 17. (a) Schematic of modulation instability (MI) gain spectrum vs. optical frequency. Sidebands modulated on a strong optical carrier grow as a result of MI. The gain increases with modulation frequency. The broadband gain spectrum spans 10’s of GHz to over 10 THz depending on the fiber dispersion, fiber nonlinearity, and optical power (cf. Eq. 2). (b) Schematic of modulation depth vs. RF frequency. Increase in MI gain with modulation frequency compensates electro-optic modulator roll-off, resulting in Optical Sideband-only Amplification or equivalently a low drive-voltage, wideband modulator. (c) Simulation of MI of a 100 GHz sideband demonstrates the strength of this effect (parameters given in text). : optical carrier frequency; : RF modulation frequency [141].

Modulation instability (MI) in optics results from the interplay between Kerr nonlinearity and anomalous group velocity dispersion. The effect can be obtained using linear stability analysis performed on analytical solutions to the nonlinear Schrödinger equation:

( ) ( ) ( 29 ) ∑ ( ) ( )| ( )|

( ) ∫ ( )| ( )| ( )

1/2 Where ( ) is the field amplitude (in Watts ), are the dispersion coefficients,

( ) ( ) is the Kerr nonlinear coefficient, is the effective modal area, is the refractive index, is the pump wavelength, is the loss per unit length, and and are 51

the Raman fractional contribution and response function respectively [97]. Over the electrically relevant pump-sideband frequency difference < 1 THz, the Raman effect and dispersion terms with can be ignored. Though the stochastic nature of temporally-confined MI [30], MI with near-zero [161], or when the sidebands grow too large [162-164] involve rather complex dynamics, the initial evolution of MI from continuous-wave radiation with sizable anomalous group velocity dispersion can be described analytically. The sideband gain per unit length under this condition without loss or pump depletion is [97]

( ) | |[( | |) ] ( 30 ) where is the pump-sideband frequency separation and is the pump optical power (cf. Figure

17a). Sidebands modulated on a strong optical carrier, acting as the pump, are thus amplified, increasing their modulation depth. The response of EO modulators is characterized by a high frequency roll-off resulting from velocity mismatch between microwave and optical waves in the traveling-wave electrodes, and from electrode microwave losses. The losses are primarily due to the skin effect, and radiative and dielectric loss (from waveguide scattering imperfections and molecular resonances, respectively) [157].

Fortuitously, the increase of Optical Sideband-only Amplification gain with pump-sideband frequency separation compensates the modulator’s high frequency roll-off, enhancing both the bandwidth and the EO sensitivity (cf. Figure 17b). Figure 17c clearly shows the growth of a 100

GHz sideband (simulation parameters described below). The enhanced modulator utilizing

Optical Sideband-only Amplification to boost modulation extinction in a communication link is shown in Figure 18. The gain spectral width depends on fiber dispersion, fiber nonlinearity, and optical power, and span tens of GHz to >10 THz [97]. Therefore this technique is applicable to

both narrowband (sub-octave) as well as broadband (multi-octave) intensity-modulated waveforms.

Figure 18. Optical Sideband-only Amplification in a communication link. The conventional link employs only the electro-optic modulator, while the boosted version shown here employs Optical Sideband-only Amplification to compensate modulation roll-off [141].

We model the effect of Optical Sideband-only Amplification on the signal by using Equation

23. The sidebands stimulate MI causing their growth at the expense of the carrier. As the

sideband powers increase by , the sideband fields increase by . As the detected RF current is a result of the carrier-sideband beating, the RF output power is proportional to the

square of , yielding:

( 31 )

where is the intrinsic Mach-Zehnder characteristic voltage, and and are the input and output RF powers. Here, by exploiting modulation instability in a fiber that follows the

Mach-Zehnder, the voltage required to achieve a given modulation depth is reduced.

From Eq. (3), the Optical Sideband-only Amplification-induced growth of by is

equivalent to reduction by , yielding the boosted modulator half-wave voltage

( 32 )

( ) ( ) ( ) 53

where is the pump (carrier) power depletion factor. Equation (4) demonstrates both key abilities of Optical Sideband-only Amplification: augmenting the modulator results in a lower

“effective ,” as well as flattening the high-frequency roll-off of the modulator. The lower

( ) effective can be interpreted as increasing the second order nonlinear susceptibility

( ) ( ). The desirable reduction of effective due to pump (carrier) depletion can be understood by recognizing that the transfer of power from the carrier to the sideband increases the ratio of AC to DC signal. For communication applications, Eq. (4) enhances system performance, as calculated in the boosted shot-noise limited RF signal-to-noise ratio ( ) [81]:

( 33 )

where is the photodetector responsivity expressed as the ratio of the photogenerated electrical current divided by the incident optical power, is the electrical impedance of the modulator, is the elementary charge, and is the RF bandwidth.

The MI effect can be tuned to optimize the Shannon-Hartley channel capacity

( ) [96], the absolute upper bound on information transmission bit-rate, with RF bandwidth . An analytical solution can be found if it is assumed that the noise is independent of Optical Sideband-only Amplification gain (valid if pump-fluctuation transfer to the sidebands and MI-amplified quantum fluctuations can be neglected [165]), and that (satisfied for any useful frequency band), as well as neglecting loss and pump depletion. The channel capacity can then be split into two terms, ( ( )) , where and are the boosted and intrinsic channel capacities, respectively. To optimize the performance, one can tune the dispersion yielding the relation for the optimal MI cut-off frequency:

( 34 ) [ ( ) ] √ { } | | [ ( ) ] where to is the electrical RF signal bandwidth. Physically, this choice maximizes the logarithmic power over this band, the quantity of importance for information capacity. It is interesting to note that the form of the pre-boosted has no effect on how the Optical

Sideband-only Amplification gain should be distributed. In the limit (single frequency), √ , as expected since this places the gain peak at the RF frequency of interest [97]. Substituting the optimal Optical Sideband-only Amplification dispersion in the channel capacity yields:

( 35 )

[( ) ( ) ] ( ( ))

where is the Optical Sideband-only Amplification length. If we consider the values below

(which optimizes a frequency band from to 191 GHz), Eq. (7) predicts a stunning 4.7 bits/s/Hz added capacity in the ideal case. Although Brillouin scattering, optical losses and added noise will reduce this in practice, the analysis does reveal the potential to strongly augment the information capacity of a communication link.

Simulations

Optical Sideband-only Amplification is simulated using the setup shown in Figure 18. A single-frequency 500 mW laser and an ideal RF signal generator with impedance feeds a high-speed Mach-Zehnder modulator. The modulator is modeled after Ref. [91] using

Equation 25 with active length , ( ) , modulator impedance ,

skin effect loss , and microwave and optical refractive indices

and with optical loss ignored. The output of the modulator feeds an EO

booster with , , , attainable in commercial highly nonlinear fibers. The radiation is detected by an ideal photodetector with impedance

and responsivity . The nonlinear Schrödinger equation is solved by the split-step Fourier method [97], demonstrated to accurately capture nonlinear optical phenomena

[28].

Figure 19. Gain ( ( ) ( )) of normal and boosted communication links vs. RF frequency. (a) Optical Sideband-only Amplification tailored for high gain at 100 GHz (parameters in text). Here the gain is improved 1000 times over the normal link. (b) Optical Sideband-only Amplification tuned for bandwidth extension ( , , ). MI’s compensation of the roll-off nearly doubles link bandwidth from 144 GHz to 271 GHz [141].

In Figure 19 we examine the normal and boosted link transfer function, ( )

( ), as an RF tone’s frequency is varied. The Optical Sideband-only Amplification gain response can be tailored via the booster parameters and laser power, here selected for large reduction (cf. Figure 19a) and bandwidth extension (cf. Figure 19b). In Figure 19a, a large negative group velocity dispersion is chosen for high-gain at 100 GHz, resulting in over 1000 times larger RF power than the normal link, vastly increasing sensitivity. Choice of smaller dispersion parameter (cf. Figure 19b) increases the Optical Sideband-only Amplification bandwidth, enabling near doubling of bandwidth from 144 GHz to 271 GHz.

The introduction of an instability into a communication link raises concerns about noise. To be sure, MI has been identified as the source of noise in generation of supercontinuum (white light) radiation [17]. However, such fluctuations arise because in conventional supercontinuum generation, MI is spontaneously triggered by pre-existing noise. Indeed, it has been shown that when MI is stimulated by a deterministic yet minute seed, the output fluctuations are dramatically reduced by as much as 30 dB [28]. In the proposed technique, the modulation sidebands that stimulate MI are phase-locked to the carrier; hence, the process will be inherently stable.

Figure 20. Signal to noise and distortion ratio ( ) (dB) vs. RF Power In (dBm) / Peak Voltage In (dBV) of normal and boosted optical links. Note how the curve of the boosted link is nearly identical to that of the normal except the input voltage is shifted by 17 dB, and exhibits little to no degradation. This logarithmic shift is consistent with a physical reduction of the characteristic voltage by 50 times. As the maximum is ( ) times the spurious-free dynamic range ( ), these curves also shows that the remains unchanged. Simulation parameters are in the text [141].

The fidelity of the output RF signal is investigated using the signal to noise and distortion ratio ( ). Two RF tones, 100 and 101 GHz, are excited in the modulator and the detected photocurrent is bandpass filtered to 95 to 105 GHz with thermal and shot-noise included (cf.

Figure 20). The stability of the Optical Sideband-only Amplification process stimulated by the deterministic sidebands is reflected in the lack of degradation for the boosted modulator in Figure 20. The observed 17 dB shift in the thus implies over 50 times reduction of effective , from 6.7 V to 114 mV.

Experiments

Upon successful simulation results, we tested Optical Sideband-only Amplification in a slightly modified analog optical link testbed, shown in Figure 21. As photodetector nonlinearity and frequency response may obscure the effect of MI on the modulator, we used a different

detection method. When modulating the amplitude of a narrowband carrier with an ideal sinusoid, the optical spectrum exhibits a tall central peak (the carrier) and two identical side peaks (the sidebands), and the sideband-to-carrier ratio gives the modulation depth, which may be read out directly with a spectrometer. A tunable laser diode at 1550 nm feeds an isolator protected erbium-doped fiber amplifier (EDFA), the output of which is polarization aligned with a phase modulator that is driven by an amplified 2.5 GHz band-limited white noise source for stimulated Brillouin scattering (SBS) suppression. The amplified noise source has a peak-to-peak voltage of 6.32 V and drives an EOSpace 10 GHz phase modulator, model number PM-0K1-10-

PFU-PFU, having a half-wave voltage of 4.0 V at 1 GHz. The SBS suppressed carrier is polarization aligned with an EOSpace 20 GHz Mach-Zehnder modulator, model number AZ-

1x2-0K1-20-PFU-SFU, having a half-wave voltage of 4.5 volts at 1 GHz. The amplitude modulator is driven by an HP 83650B synthesizer at RF frequencies from 10 to 50 GHz with 2.5

GHz granularity. The RF output power is kept constant at 1 mW. The modulated waveform is amplified by an isolator protected EDFA, variably attenuated, boosted by the highly-nonlinear fiber, and detected by an optical spectrum analyzer. At 1550 nm the HNLF is described by the

2 3 -1 - following coefficients: = 1 dB/km, = –10.9 ps /km, = 0.062 ps /km, and = 11.6 W km

1. A circulator and various power monitors are used to precisely control the power before entering the HNLF and to monitor SBS suppression.

Figure 21. Experimental setup. A continuous-wave laser source (SOURCE), acting as the carrier and pump, is phase modulated to prevent stimulated Brillouin scattering (SBS) in the BOOSTER. The RF signal is modulated onto the carrier (SIGNAL) and is boosted in the BOOSTER. The MONITOR measures the booster input power, output power, back-reflected power, and optical spectra. The MONITOR is therefore able to verify the limited impact of stimulated Brillouin scattering on the boosting process. TLD: tunable laser diode; EDFA: erbium doped fiber amplifier; ISO: isolator; POL: polarization controller; PM: phase modulator; AWG: arbitrary waveform generator; AMP: amplifier; AM: amplitude modulator; SYNTH: synthesizer; PM [#]: power meter, numeric designator; CIR: circulator; HNLF: highly nonlinear fiber of 2 km length; OSA: optical spectrum analyzer [142].

Before discussing the core booster results it is critical to understand role of Brillouin scattering in the experiments. Figure 22 shows the transmitted and reflected power as a function of the launch power with respect to the HNLF. These power levels correspond to PM_4, PM_3, and PM_2 of Figure 21, respectively. The launch power is set manually by adjusting the variable attenuator in PM_1/ATTN. Two cases are shown: (1) the case where SBS suppression is turned off as shown using the dashed line, and (2) the case where SBS suppression is turned on as shown using a solid line.

Figure 22. Monitoring of SBS with suppression on (solid lines) and off (dashed lines). Note that the deviation from linear reflected power as a function of launch power occurs at 10 mW for the unsuppressed case and 100 mW for the suppressed case. With SBS suppression on, SBS is completely suppressed up to and including 100 mW launch powers. At 200 mW launch power 3.4 mW is reflected by SBS [142].

As seen in Figure 22, SBS suppression is successful in regions that observe a linear dependence between the reflected power and the launch power [166]. In the case of SBS suppression turned off, SBS onset begins beyond 10 mW launch power. In the case of SBS suppression turned on, SBS begins beyond 100 mW launch power. The approach achieves 10 dB of SBS suppression. SBS will play no role in experiments utilizing launch powers up to and including 100 mW with SBS suppression turned on. At 200 mW launch power with SBS suppression turned on SBS causes a 3.4 mW back reflection and a 7 mW drop in transmitted power. Suppression of SBS is critical to ensuring observed modulation enhancement is due to

MI, as desired. Past works have shown as high as 17 dB SBS suppression using similar approaches [166]. We believe our suppression capability can be improved upon by utilizing a phase modulator with an integrated polarization analyzer. Such a modulator would prohibit any un-suppressed light from making its way to the HNLF, which we believe is the cause of the reduced performance. For the following experiments SBS suppression is turned on.

Now that the SBS suppression capabilities are understood further measurements can proceed.

The power entering the HNLF was set to (1, 50, 100, 150, 200) mW and a computer was used to synchronously sweep the RF frequency and to grab OSA traces. The experimental OSA traces are shown in Figure 23. The drawn to scale insert shows the modulation envelope of the RF sideband at 1 mW carrier power and at 200 mW carrier power. Inset spectra are carrier normalized to permit comparison of the modulation depth. Notice the improved modulation depth with respect to the carrier and the flattening of the frequency response as the carrier power increases. The rise in the noise floor is due to the increase in EDFA ASE noise. The amplification of the ASE by MI gain spectrum is also evident. Use of a high power signal laser capable of providing up to 200 mW power would eliminate the need for EDFA. The satellite peaks on top of the MI amplified ASE is believed to be due to FWM, as predicted in the previous section.

Figure 23. Experimental OSA traces. Seventeen individual traces, one for each RF frequency, are taken for each power level and plotted in a single color. Drawn to scale inset compares carrier normalized modulation envelope of 1 mW RF sideband and 200 mW RF sideband. Evident are improved modulation depth and compensation of the modulator’s frequency roll off at high modulation (sideband) frequencies [142].

The half-wave voltage, or , was extracted from the experimental OSA traces via comparison to a numerically simulated OSA trace of a linear optical link. The simulated linear link consisted of an ideal laser connected to an ideal Mach–Zehnder modulator detected by an

OSA. The effective half-wave voltage was determined by numerically varying both the simulated laser power and the simulated half-wave voltage until the error between the experimental and simulated OSA traces was minimized. Figure 24 shows the effective half-wave voltage of the boosted modulator as a function of RF frequency and carrier power.

At 1 mW carrier power (blue line) the experimentally measured effective half-wave voltage closely matches the manufacturer's specification (brown line). As the carrier power is increased there is an overall reduction in the half-wave voltage as well as improved flatness over the measurement range. Note that this trend is present in all cases of increased carrier power, including 100 mW carrier power (red line) where SBS is fully suppressed (cf. Figure 22). This shows that the effect is due to MI as desired. The flatness can be measured by examining the ratio of largest to smallest half-wave voltage across the 10 to 50 GHz range. For low optical power the ratio is approximately 3.3. At 200 mW carrier power this ratio is reduced to 2, evidencing the ability of Optical Sideband-only Amplification to equalize the RF response of the modulator. Note that the required signal power requirements scale as the square of the voltage.

For carrier power of 200 mW the half-wave voltage is reduced by more than 5-fold at 10 GHz and by 10-fold at 50 GHz to 1 V and 2 V, respectively.

Figure 24. Effective half-wave voltage of boosted modulator as a function of RF frequency and carrier power. Note the EO modulator's manufacturer specification. For 200 mW the half-wave voltage is reduced by more than 5-fold at 10 GHz, at 50 GHz the reduction is 10-fold [142].

Conclusion

In summary, we introduced a new concept that exploits Optical Sideband-only Amplification to compensate for the intrinsically weak electro-optic effect, ( ), resulting in a highly desired reduction of the effective of electro-optic modulators. Simulations demonstrate more than 50 times enhancement of electro-optic effect and modulation sensitivity at frequencies in the 100’s of GHz range. Experiments resulted in a 10-fold reduction of the effective half-wave voltage at 50 GHz. Future work must consider the impact of modulation instability on the noise and dynamic range of the communication link. Stimulating modulation instability with RF modulation sidebands of an optical carrier enables high frequency low-voltage modulation, an increasingly critical capability that is needed if optical data communication is to keep pace with advances in electronics. Finally, the enhancement in ( ) may also find use in wave-mixing

applications as long as the frequency translation falls within the MI bandwidth, which can be extended to tens of THz by reduction in the dispersion, of the MI medium.

Chapter 5: Future Direction: Measuring Spectro-

Temporal Outliers and their Statistics using CTST

A compelling use for CTST with multiple applications is the capture of the electric field of ultrashort pulses at megahertz repetition rates. Knowledge of the electric field allows characterization of the spectro-temporal distribution of energy by a time-frequency representation. To characterize the variations statistically, we plot histograms of the spectro- temporal energy within different time-frequency quadrants (cf. Figure 25). Analysis of the spectro-temporal energy within different quadrants is motivated by the spectral filtering technique previously used to investigate anomalous spectra in supercontinuum. In particular, examination of energy content frequency-shifted from the input radiation has proven useful for identifying rare events and statistics generated by the nonlinear dynamics [14, 31, 32]. Likewise, here we have the chance to identify spectro-temporal outliers or skewed statistics by selecting energy content shifted in frequency and time relative to the average event. This shifted energy content may be used, for example, to identify events with anomalous chirp. Furthermore,

statistical distributions based on distinct quadrants may generally identify different sets of outliers. Figure 25 shows a hypothetical example which exhibits Gaussian-like statistics in the central area of the time-frequency space, but tends to morph into skewed forms toward the outer regions of the window, especially in the corners. It may be noted that some of the histograms have small shoulders or lobes, suggesting the possibility of different sub-populations. We note that measurements with a conventional spectrometer would produce merely the average frequency trace also shown in Figure 25, permitting no direct access to the statistics. The TS-

DFT is able to measure the pulse-resolved spectral traces and their statistics, while the full-field measurements are needed for the present analysis. Here, spectro-temporal profiles are shown in the form of the Wigner-Ville distribution, which achieves high resolution in both time and frequency; other time-frequency representations, such as the spectrogram, can also be employed.

Assembling the time-frequency representations of a large volume of pulses allows the statistical properties of the spectro-temporal dynamics to be compiled, and provides a means of identifying events with anomalous properties. This technique may have application in communications, for example, by using each spectro-temporal quadrant as an orthogonal code of an optical code- division multiple access (CDMA) scheme [167].

Figure 25. Time-frequency statistics of a synthetic data set of ~5000 pulses. a, Frequency-time representation averaged over ~5000 pulses, along with time (top) and frequency (left) projections. b, Statistics for each time-frequency quadrant in a, labeled 1-9. Statistical distributions are obtained by integrating the energy in different spectro-temporal windows and provide a measure of the spectro- temporal variability from pulse to pulse. Here we would find, for example, that skewed statistics tend to be found in regions of relatively low average energy. c, Spectral and d, temporal statistics over three frequency/time slices.

Chapter 6: Conclusion

In this thesis I used and modified the Coherent Time-Stretch Transform to enable the acquisition of optical and electrical outliers. Most significantly, I adapted the Coherent Time-

Stretch Transform to resolve molecular absorption lines by adding temporal near-field capability.

Finally, I addressed the trade-off between high-speed and low-voltage operation of the Coherent

Time-Stretch Transform digitizer, by introducing and demonstrating Optical Sideband-only

Amplification.

I have adapted the Coherent Time-Stretch Transform to measure optical pulses in the near field. For the first time, I measured the complex-field spectrum of a chemical at millions of frames per second. This technique is four orders of magnitude faster than even the fastest (kHz) spectrometers, and will enable capture of real-time spectral dynamics of chemical reactions (e.g. combustion), high-energy physics and living cells, and monitoring fast high-throughput industrial processes.

I introduced and demonstrated Optical Sideband-only Amplification, which has the potential to solve the most pressing issue in optical communication: high performance electro-optic modulation of high bandwidth, low voltage signals. With simulations, I showed that this can

drastically increase optical link gain at 100+ GHz frequencies, where traditional electronic amplification is costly, expensive and power hungry. I also demonstrated this concept experimentally, and showed both drastic reduction of (>5x across 10 – 50 GHz range), and high-frequency amplification-based equalization. As this technique is widely tunable in terms of gain and RF bandwidth, it will continue to enhance future electro-optic modulators as well. This will allow transport of wideband RF signals with more RF power over longer distances, and likely become a necessary asset for future optical links and the photonic time-stretch analog-to- digital converter.

Besides using the Coherent Time-Stretch Transform to analyze spectro-temporal outliers as presented in Chapter 5, future work involves putting the Coherent Time-Stretch Transform into the hands of more scientists and engineers, as well as widening the physical range of operation.

As result of the work outlined here and other recent technological improvements, the Coherent-

Time Stretch Transform is ready for commercialization. Already, the Coherent Time-Stretch

Transform has started to be used in a few other laboratories, but so far specialized knowledge of ultrafast optics is needed to set up such as experiment. To eliminate as many barriers to adoption as possible, we at UCLA are developing a Coherent Time-Stretch Transform prototype. This includes the full hardware necessary to transform and digitize fast phenomena in real-time with high throughput, as well as the software to control data acquisition and analyze the data.

Operation will require little knowledge of the inner workings of the Coherent Time-Stretch

Transform, which will reduce the requirements for proper use and involve the broadest possible user base. This will significantly expand the range of uses and result in the highest possible impact.

The titanium sapphire (Ti:S) laser is one of the most widely used lasers in research, with a myriad of applications including pump-probe spectroscopy, coherent control, nonlinear optics, material processing, two photon microscopy, atmospheric research, combustion research.

However, the use of the Coherent Time-Stretch Transform at its 800 nm lasing wavelength remains a challenge. This necessitates the further development of a better dispersive element at

800 nm, and chromo-modal dispersion is not only promising for characterizing the laser, but also capture of high-throughput dynamics in real-time in these applications away from telecommunications wavelengths. Chromo-modal dispersion may also pave the way for application of the Coherent Time-Stretch Transform in the visible and mid-infrared.

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