Abstract Quantum Noise in Optical Parametric

ABSTRACT

Title of dissertation: QUANTUM NOISE IN OPTICAL PARAMETRIC AMPLIFIERS BASED ON A LOSSY NONLINEAR INTERFEROMETER

Pape Maguette Sylla, Doctor of Philosophy, 2009

Dissertation directed by: Professor Julius Goldhar Department of Electrical and Computer Engineering

Optical Parametric Amplifiers (OPA) have been of wide interest for the past decades due to their potential for low noise amplification and generation of squeezed light. How- ever, the existing OPAs for fiber applications are based on Kerr effect and require from few centimeters to kilometers of fiber for significant gain.

In this thesis, I review the principles of phase sensitive ampliﬁcation and derive the expression for gain of a lossless Kerr medium based nonlinear Mach-Zehnder Interferom- eter (NMZI OPA) using a classical physics model . Using quantum optics, I calculate the noise of a lossless Kerr medium based OPA and show that the noise ﬁgure can be close to zero.

Since in real life a Kerr medium is lossy, using quantum electrodynamics, I derive equations for the evolution of a wave propagating in a lossy Kerr medium such as an optical fiber. I integrate these equations in order to obtain the parametric gain, the noise and the noise figure. I demonstrate that the noise figure has a simple expression as a function of loss coefficient and length of the Kerr medium and that the previously published results by a another research group are incorrect. I also develop a simple expression for the noise

ﬁgure for high gain parametric ampliﬁers with distributed loss or gain.

In order to enable construction of compact parametric ampliﬁers I consider using different nonlinear media, in particular a Saturable Absorber (SA) and a Semiconductor

Optical Amplifier (SOA). Using published results on the noise from SOA I conclude that that such device would be prohibitively noisy. Therefore, I perform a detailed analysis of noise properties of a SA based parametric amplifier. Using a quantum mechanical model of an atomic 3 level system and the Heisenberg’s equations, I analyze the evolution in time of a single mode coherent optical wave interacting with a saturable absorber. I solve the simultaneous differential equations and find the expression for the noise figure of the SA based NMZI OPA. The results show that noise figure is still undesirably high. The source of the noise is identified. A new approach for low noise parametric amplifier operating with short pulses is proposed. QUANTUM NOISE IN OPTICAL PARAMETRIC AMPLIFIER BASED ON A LOSSY NONLINEAR INTERFEROMETER

Pape Maguette Sylla

Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy 2009

Advisory Committee: Professor Julius Goldhar, Chair/Advisor Professor Wendell T. Hill III Professor Christopher C. Davis Professor Thomas E. Murphy Doctor Marshall Saylor c Copyright by

Pape Maguette Sylla 2009 Dedication

To my family

ii Acknowledgments

I owe my gratitude to all the people who have made this thesis possible. I want to thank with my father and mother for supporting me to seek higher education. I want to also thank my two brothers, my two sisters and my two brothers in law for helping me when school was requiring too much of my time. I want to thank all my friends, my aunt Rokhaya and her husband, my grandparents, and an uncle who refuses to be named for their encouragement and advices. I want to extend a special thanks to Symantec

Corporation for its ﬂexibility and for allowing me to work full time while going to school.

I also want to express my gratitude to Professor Julius Goldhar for his dedication to science and engineering but also for the many weekends he sacriﬁced going over with me derivations. I thank him for his patience in general and his patience whenever ”I fell off the cliff”, but also for taking the time to listen to me. I thank him for his countless suggestions and ideas. I pray to God that he achieves his goals and that he seriously positively impacts science. I also thank Dr. Marshall Saylors for spending a lot of time proofreading and helping me reorganize this thesis. It is impossible to remember all, and

I apologize to those I’ve inadvertently left out.

Lastly, thank you all and thank God!

iii Table of Contents

List of Tables viii

List of Figures ix

List of Abbreviations xi

1 Introduction 1 1.1 Overview ...... 1 1.2 TheBalancedHomodyneReceiver ...... 4 1.2.1 TheLosslessOpticalBeamSplitter ...... 4 1.2.2 Classical Analysis of the Balanced Homodyne Detector ..... 7 1.3 TheLinearMZI...... 8 1.4 TheNonlinearMZIOpticalParametricAmplifier ...... 10 1.5 BriefIntroductiontoQuantumMechanics ...... 11 1.5.1 SingleModeFieldOperators...... 13 1.5.2 NumberState...... 14 1.5.3 NumberOperator...... 15 1.5.4 CoherentStates...... 16 1.6 Quantum Noise Added During Amplification and Attenuation ...... 17 1.6.1 QuantumNoiseAddedDuringAttenuation ...... 17 1.6.2 NoiseinPhaseInsensitiveAmplification ...... 21 1.7 DefinitionofSignaltoNoiseRatioandNoiseFigure ...... 22 1.7.1 Definition of the Photon Number Fluctuation Noise Figure.... 25 1.7.2 Definition of the Field Amplitude Squared Noise Figure ..... 28 1.7.3 NFFAS ofanPhaseInsensitiveAmplifier...... 28 1.7.4 Definition of the Quadrature Field Squared Noise Figure..... 29 1.7.5 DefinitionoftheQualityFactor ...... 30 1.8 BalancedHomodyneDetectionofaCoherentState ...... 31 1.9 NonlinearMach-Zehnder Interferometer OPA Noise ...... 33

2 Classical Treatment of the Kerr Medium Based NMZI 36 2.1 KerrEffect ...... 36 2.2 NonlinearMZIwithKerrMedia ...... 38 2.3 FrequencyResponseofaNMZI ...... 41 2.3.1 PhaseInsensitiveGain ...... 42 2.3.2 PhaseSensitiveGain ...... 43

3 Quantum Mechanical Analysis of the Lossless Kerr Medium BasedNMZI 45 3.1 CalculationofAverageQuantitiesUsingQM ...... 45 3.2 Quantum Mechanical Noise With Loss in Linear Elements ...... 48 3.3 QuantumMechanicalNoiseofNMZIwithLumpedLoss ...... 50 3.3.1 Quantum Mechanical Noiseof KerrBased NMZI withall the loss attheInput ...... 50

iv 3.3.2 Quantum Mechanical Noise of Lossy Kerr Based NMZI with all thelossattheOutput...... 51 3.4 Simple Expression of Noise Figure for High Gain Parametric Ampliﬁer ...... 54

4 QuantumMechanicalNoiseofaNMZIwithLossyKerrMedia 59 4.1 Overview ...... 59 4.2 NoiseFigureofaLossyKerrBasedNMZIOPA ...... 60 4.2.1 Differential Equation of Field Operator in Lossy Kerr Medium . . 60 4.2.2 UsingFirstOrderPerturbationTheory ...... 61 4.2.3 Solving the Differential Equation for the Average Value ..... 62 4.2.4 Solving the Differential Equation for the FluctuatingTerms . . . 62 4.3 Detailed Comparison with Imajuku’s Calculations ...... 67 4.4 Quantum Mechanical Noise of a NMZI with Gain in the Kerr Medium . . 69

5 Discussions of the Results from the Lossy Kerr Based NMZI OPA 73

6 Semiclassical Treatment of Optical Parametric Amplifier Based on Saturable Ab- sorber and Amplifier 77 6.1 Saturableabsorber/AmplifierOverview ...... 77 6.1.1 LossCharacteristics ...... 77 6.1.2 PhaseVariation...... 79 6.2 ParametricGainofSAbasedNMZI-OPA ...... 79 6.2.1 Approximate Expression for the Differential Absorption ..... 82 6.2.2 ParametricGainCalculation ...... 83 6.3 BandwidthoftheSABasedNMZI ...... 84 6.3.1 ResponseofaSAtoaModulatedSignal...... 85 6.3.2 OverallSystemOutput ...... 89

7 Quantum Mechanical Model for Interaction of Light with a Saturable Absorber 91 7.1 Overview ...... 91 7.2 QuantumMechanicalModel ...... 91

8 Solving Differential Equation of SA 104 8.1 Overview ...... 104 8.2 Deriving Simultaneous Differential Equations of the Inphase and Quadra- turePhasecomponentofNoise ...... 104 8.3 Solving the Simultaneous Differential Equations ...... 109 8.4 NoiseFigure...... 116

9 Discussions 119

A Detailed Derivation of Noise Field Operator in Lossy Kerr Medium 122 A.1 Solving the Differential Equation for the Average Terms ...... 122 A.2 DerivationoftheOutputNoiseFieldOperator ...... 123 A.3 ParametricGainDerivation ...... 123

v A.4 Correlation of the Vacuum Noise in the two Arms of the NMZI ...... 125

B Detailed Derivation of NF for the Lossy Kerr Medium Based NMZIOPA 127 2 ˆ ˆ B.1 Expression for ∆ Eout† Eout ...... 127 ˆ ˆ ˆ ˆ B.2 Calculation of δEout† δEoutδEout† δEout ...... 128

B.2.1 CalculationD of Nˆ (w)Nˆ †(y)Nˆ (Ez)Nˆ †(x) ...... 128

B.2.2 Calculation of DNˆ (w)Nˆ (z)Nˆ †(x)Nˆ †(y)E ...... 129 2 D ˆ ˆ E B.3 Simpliﬁed Expression for ∆ Eout† Eout ...... 130

B.4 Computing Nˆ †Nˆ Nˆ †Nˆ ...... 132 D ˆ ˆ E ˆ ˆ B.5 Computing δEout† δEoutδEout† δEout ...... 138 2 D ˆ ˆ E B.6 Computing ∆ Eout† Eout ...... 139 B.7 NoiseFigure...... 139 C DerivationofNoiseforNMZIBasedKerrMediumwithGain 140 C.1 ASEofKerrMediumwithGain ...... 140 C.2 NoiseFigure...... 142

D Derivation of SA based NMZI-OPA Parametric Gain 143 D.1 Approximation of Γ2 Γ1 ...... 143 D.2 DerivativeoftheSaturatedLoss− ...... 143 D.3 Approximation of β β ...... 145 2 − 1 E BandwidthoftheSaturableAbsorberBasedNMZI 146 E.1 ResponseofaSaturableabsorbertononCWsignal ...... 146 E.2 OverallSystemOutput ...... 148 E.3 ParametricGain...... 149

F Quantum Mechanical Model for Interaction of Light with SaturableAbsorber 151 F.1 Solving for σˆ+ ...... 151 ˆ F.1.1 Properties of Noise Source Fab(t) ...... 152 F.1.2 PhotonNumberEquation...... 154 F.2 NoiseFromaSingleAtom ...... 155 F.3 NoiseFromaCollectionofAtom ...... 158

G Solving Differential Equation of SA 160 G.1 SolutionfortheFluctuatingTerms ...... 160 ˆ ˆ G.1.1 Finding an Expression for Fβ(t)Fβ(t′) ...... 162 Dˆ ˆ E G.1.2 Finding an Expression for Fab(t)Fβ(t′) ...... 163 G.2 Calculating ˆx(T )ˆx(T ) ...... 165D E G.2.1 AmpliﬁedInitialNoiseh i ...... 166

vi G.2.2 NoiseduetoAbsorption ...... 166 G.2.3 NoiseduetoRelaxation ...... 167 G.2.4 BeatNoise ...... 167 G.2.5 TotalAmplitudeNoise ...... 168 G.3 Calculating ˆy(T )ˆy(T ) ...... 168 h i Bibliography 172

vii List of Tables

5.1 Different ﬁber that could be used in a Kerr based NMZI based OPA and theiroptimumlengthandtheirmaximumgain...... 75

viii List of Figures

1.1 BalancedHomodyneReceiver ...... 5

1.2 BeamSplitter ...... 6

1.3 Schematic Diagram Linear Mach-Zehnder Interferometer: MZI in linear regime, with proper phase adjustment does not mix the signal andthepump. 9

1.4 Nonlinear Interferometer. It is important to note that Φ iscomplex . . . . 11

1.5 Example of Attenuator: Beam coupler. Additional noise from a reservoir channel ...... 18

1.6 Schematicofanidealreceiver ...... 23

1.7 NoiseFigureindBasafunctionofattenuation ...... 26

1.8 Noise Figure in dB as a function of gain (equation (1.67)) ...... 27

1.9 Schematic of NMZI OPA. Appropriate operators as detailed in the text areshown...... 34

2.1 Plots of gain as a function of input phase difference for 1 km of Kerr medium with γ = 10 andpumppowerof1W ...... 40

2.2 SchematicofaSagnac ...... 42

3.1 Representation of the electric-ﬁeld properties of the coherent before and afterparametricampliﬁcation...... 46

4.1 Plots of NF vs loss for a LossyKerr medium based NMZI-OPA . .... 68

4.2 PlotsofNFNPS vs Length for a Lossy Kerr medium based NMZI-OPA as a comparison to Imajuku’s et al. ﬁgure6[8]...... 70

4.3 PlotsofNFNPS vs Loss for a Lossy Kerr medium based NMZI-OPA as a comparison to Imajuku’s et al...... 71

4.4 PlotsofNFNPS vs Gain for Kerr Medium with gain based NMZI-OPA . . 72

5.1 PlotofgainversuslengthforaHNLF(γ = 15.8)/km/W, β = 0.7dB/km )...... − 76

ix 6.1 Minimum and maximum parametric gain vs p/Psat for a SA based NMZI OPA with αH = 25, β0L =4...... 85

6.2 Minimum and maximum parametric gain vs. p/Psat for a SOA based NMZI OPA with α =5, β L = 4...... 86 H 0 − 6.3 Parametric gain in dB vs Ω in 1/τ for a SA based NMZI OPA with p/Psat = 25, αH =5and β0L =4...... 90

7.1 SingleModeQuantumMechanicalModel ...... 92

7.2 3LevelSystem ...... 95

ˆ ˆ ˜ 8.1 Illustration of spectrum of Fβ(t) relative to the spectrum of B(t). Fβ(Ω) ˆ ˜ is the he Fourier Transform of Fβ(t) and B(Ω) is the Fourier transform ˆ ˆ ˆ of B(t). The spectrum of Fβ(t) looks ﬂat relative to the spectrum of B(t) 111

8.2 Noise Figure as a function of n(0) for αH =5, β0 =2 and nsat =1 . . . . 117

8.3 Parametric gain (blue) and noise ﬁgure (red) as a function of n(0) for αH = 25, β0 =2 and nsat =1 ...... 118

9.1 Parametric gain (blue) and noise ﬁgure (red) as a function of n(0) for αH = 25, β0 =2 and nsat =1 ...... 121

x List of Abbreviations

α alpha β beta

NF NoiseFigure SA Saturable Absorber SOA Semiconductor Optical Ampliﬁer MZI Mach-Zehnder Interferometer NMZI Nonlinear Mach-Zehnder Interferometer OSNR Optical Signal to Noise Ratio SNR (Electrical) Signal to Noise Ratio OPA Optical Parametric Ampliﬁer PNF PhotonNumberSquared FAS Field AmplitudeSquared QFS Quadrature Field Squared

xi Chapter 1

Introduction

1.1 Overview

Optical communication plays an important role in the information age we are in.

One fiber optic link allows us to reliably send data over thousands of kilometers at very high bandwidth (hundreds of Gb/s). At this time, no other medium can do better. As we are pushing the fiber optic bit rate closer to its channel capacity, it is important to keep the noise generated by the fiber optic devices and the medium low. One of the devices that produces the most noise is the optical amplifier [1]. In chapter one, after a brief introduction to quantum mechanics, I give two established definitions of the Signal to

Noise Ratio (SNR), a measure of the quality of a signal, and show why those definitions are not best suited for our problem. I introduce a new definition of the SNR which will be more appropriate. I show that any phase insensitive optical amplifier, such as the commonly used EDFA, degrades Signal to Noise Ratio (SNR) by at least 3 dB. This is particularly important at the optical receiver when the input signals are very weak. There exists a class of amplifier, which is phase sensitive, that does not degrade SNR. The goal of this thesis is to provide a correct quantum mechanical analysis of phase sensitive amplifiers, which include lossy elements. Phase sensitive amplifiers operate on a principle very similar to balanced homodyne detector, which will also be briefly discussed in the introduction.

1 In chapter 2, I discuss in detail a well known phase sensitive ampliﬁer: The lossless

Kerr medium based Nonlinear Mach-Zehnder Interferometer Optical Parametric Ampli-

ﬁer (NMZI-OPA). There are numerous publications which demonstrate the feasibility and applications of this device, which uses nonlinear ﬁber as the Kerr medium [4, 5, 6, 7, 8,

15, 16]. Phase sensitive gain of this device is calculated using a simple classical model.

In chapter 3, I use a quantum mechanical model to calculate the noise properties of a lossless NMZI OPA and I show that it does not degrade SNR. A possible definition of the Noise Figure (NF) of an amplifier is the ratio of its input to its output SNR. With this definition, the noise figure is zero (0) dB for this ideal NMZI OPA. Unfortunately, there is always loss in a real Kerr medium, which results in a higher noise figure.

This problem was analyzed in detail by Imajuku et al. [8]. However, I found that some of their results are incorrect. In order to bracket the noise ﬁgure for a lossy NMZI OPA, I consider the effect of lumped loss in front of the Kerr medium and after the Kerr Medium.

The true answer for parametric ampliﬁer with distributed loss must fall within those limits.

These results, unlike other results which will be discussed later, agree with the calculations of Imajuku et al.. In general, calculation of noise for this type of device is tedious[8].

Therefore, I derived a general simple expression for noise ﬁgure for an OPA in the limit of large parametric gain, which drastically reduces the complexity of the calculations.

In chapter 4, I calculate the noise figure of a NMZI-OPA with uniformly distributed loss in the medium. This problem was also addressed by Imajuku et al. However, they made some wrong assumptions and performed unexplainable mathematical steps, which produced results which I believe to be incorrect. My approach gives a simple and elegant expression for noise figure. I also derive the noise figure for a NMZI-OPA with distributed

2 laser gain instead of loss. I show that even small amount of gain produces 3dB increase in noise figure. In chapter 5, I discuss the implication of the quantum mechanical and the classical noises in the NMZI-OPA. I also find the optimum length of an optic fiber used as a Kerr medium based on its nonlinear property and its loss coefficient.

In order to be able to build a compact OPA, another nonlinear medium besides optical ﬁber is required. There has been extensive research in optical signal processing, which utilized nonlinear optical properties of semiconductors ampliﬁers and saturable absorbers (TOAD, wavelength converter, fast optical switch, fast pulse generator ....[9,

10, 11, 12, 13]) . These media are very attractive because of the nonlinearities are very high and the devices are compact. In chapter 6, I use a classical model to show that phase sensitive ampliﬁcation can be achieved with semiconductor optical ampliﬁers and saturable absorbers. Recently, there has been some interest in this class of devices [17].

Based on the resultin chapter 4, I conclude thatan OPA based on SOAs will produce unacceptable noise figure. The calculation of noise figure for an OPA with saturable absorber is non-trivial. Chapter 6 and 7 are dedicated to this problem. In chapter 7, I use full quantum mechanical model to derive basic differential equations for light interaction with saturable absorber. In chapter 8, I solve these equations and evaluate the noise figure for this type of parametric amplifier based on saturable absorber.

In chapter 9, I discuss practical implication of results obtained in the previous chapter.

3 1.2 The Balanced Homodyne Receiver

The balanced homodyne receiver model has some common characteristics with an optical parametric amplifier. They both have gain sensitive amplification. Therefore, the analysis of an balanced homodyne receiver can give some insight into phase sensitive amplification. Homodyne receivers use interference of a weak signal with a strong local oscillator of the same frequency and with proper phase to produce gain in the receiver

[14]. They are the best available receiver for coherent optical signals [3]. However, the local oscillator introduces its own noise, which can be comparable to or greater than the incoming signal [48]. The balanced homodyne receiver overcomes the problem of part of the local oscillator noise, namely its Relative Intensity Noise (RIN), but not its phase noise as we will see in equation (1.8). It works as follows. The message signal and the local oscillator are split in two and mixed using semi-transparent mirror, as shown in

ﬁgure 1.1. The two signals must have perfect spatial mode overlap, must have the same polarization state and must have the same carrier frequency. The two optical signal are then detected, turned into an electrical signal and subtracted. These operations produce a larger electrical signal than direct detection and cancel RIN from the oscillator. In order to explain the operations of a balance homodyne detector and later the NMZI OPA, the analysis of the optical beam splitter is looked at.

1.2.1 The Lossless Optical Beam Splitter

The optical beam splitter is shown in ﬁgure 1.2. It has two inputs, waves with complex amplitude (E1 and E2), and two outputs (E3 and E4) as shown in the ﬁgure.

4 Figure 1.1: Balanced Homodyne Receiver

There is a linear relationship between the input ﬁelds and the output ﬁelds which are summarized by equation 1.1 and 1.2

E3 = r31E1 + t32E2 (1.1) and

E4 = t41E1 + r42E2 (1.2)

where rx and tx are respectively complex reﬂection and transmission coefﬁcient. The above relations can be written in matrix format:

E3 r31 t32 E1   =     . (1.3) E4 t41 r42 E2            

5 Figure 1.2: Beam Splitter

In order to satisfy conservation of energy and the phase requirement, we can choose a convention where for a 50/50 splitter the matrix is [53]

1 i 1   . (1.4) √2 1 i     It is important to note that this solution is not unique. For example, the following matrix could have been chosen

1 i 1 −  . (1.5) √2  1 i  −   

6 1.2.2 Classical Analysis of the Balanced Homodyne Detector

If ELO and Es are respectively the complex amplitudes of the electric ﬁeld of the local oscillator and the signal, at the detectors the amplitude of the electric ﬁelds are

1 E1 = (iELO + Es) (1.6) √2 and 1 E2 = (ELO + iEs) (1.7) √2 The currents of the two detectors are subtracted, the net output current is

η 2 2 I = q E1 E2 ~ω0 | | −| | η (1.8) = iq (ELOEs∗ ELO∗ Es) ~ω0 − η =2q ELO Es sin (φ + δφ) , ~ω0 | || | where η is the quantum efﬁciency, q is the electric charge, φ is the average phase of the complex number ELO∗ Es, δφ is the phase ﬂuctuation of the complex number ELO∗ Es and

ω0 it the optical frequency. We can see that the currents due to the local oscillator cancel, and therefore, their amplitude ﬂuctuations, their RIN, do not appear in the current output.

We can also see that there is a power gain compared to direct detection of the signal, which is I 2 G |η | 2 ≡ q ~ Es | ω0 | (1.9) =4 E 2 sin2 (φ + δφ) . | LO| The gain is phase sensitive; it is dependent upon the relative phase between the local oscillator and the signal. However, it is also dependent on phase noise, which leads to

RIN noise in the output. I will calculate the noise of the detected signal later in this chapter.

7 1.3 The Linear MZI

The Mach Zehnder Interferometer (MZI) is a commonly used component in many optical devices [18, 19, 20, 21]. It is often used as a building block of more complex optical devices and functionalities such as optical ﬁlters, wavelength demultiplexers, channel interleavers, intensity modulators, switches and optical gates [23]. A schematic diagram of typical MZI based on guided waves in shown on ﬁgure 1.3. It consist of two 50:50 optical couplers, which are analogous to beam splitters discussed in the previous section.

The two outputs of the ﬁrst coupler become inputs to the second one. It is assumed that the two outputs of the ﬁrst coupler travel through similar medium, with same length before arriving at the second coupler. It is straightforward to perform classical calculations of the transmission of signal through this interferometer [24]. The input of one arm of the

first coupler is a signal described by its complex electric field amplitude Es. The input to the second arm is Ep, which is the field of the local oscillator, which I will also call the pump field.

After the ﬁrst beam coupler, I have

1 Eout12 = (Ep + iEs) (1.10) √2 at the output of the ﬁrst arm and

1 Eout22 = (Es + iEp) (1.11) √2 at the output of the second.

The signals go directly to the second coupler with equal delays and amplitude. I

8 Figure 1.3: Schematic Diagram Linear Mach-Zehnder Interferometer: MZI in linear regime, with proper phase adjustment does not mix the signal and the pump.

obtain at the output of one the arm of the MZI

1 Eout1 = (Eout22 + iEout12) √2 1 1 1 = (Es + iEp)+ i (Ep + iEs) √2 √2 √2 (1.12) 1 1 i i = E E + E + E . 2 s − 2 s 2 p 2 p

= iEp

9 Similarly, at the output of the second arm

1 Eout2 = (Eout12 + iEout22) √2 1 1 1 = (Ep + iEs)+ i (Es + iEp) √2 √2 √2 (1.13) 1 1 i i = E E + E + E 2 p − 2 p 2 s 2 s

= iEs

Therefore, it can be seen from the balanced MZI that the two signals’ amplitude and their phase difference are preserved.

1.4 The Nonlinear MZI Optical Parametric Ampliﬁer

In a NMZI, a nonlinear medium is placed between the two optical couplers as shown in ﬁgure 1.4. It is assumed that the nonlinear media have similar properties and the same length. In general, because of interference between the signal and the pump, similar to the balanced homodyne detector, the total intensity is different in the upper and lower arm. In a nonlinear medium, this results in different attenuations, gains and/or phase shift of propagating waves. Unbalancing of MZI redirects some pump power into the signal output port, resulting in effective ampliﬁcation of the signal. This is a conventional implementation of the OPA [25].

In this thesis, I will ﬁrst consider a nonlinear optical Kerr effect medium, which produces a phase shift as the function of power intensity in the medium. I will also consider a saturable absorber (SA) as a nonlinear medium in which there is a phase shift and amplitude variation of the ﬁeld as a function of the intensity. The average properties at the output can be calculated either classically or using a quantum mechanical model. How-

10 Figure 1.4: Nonlinear Interferometer. It is important to note that Φ is complex

ever, noise properties must be calculated correctly using quantum mechanics. Calculating the noise properties is the main objective of this thesis.

1.5 Brief Introduction to Quantum Mechanics

In quantum mechanics, for any system, there is a state vector v , containing every- | i thing there is to know about that system at a given instant [26]. Any physical quantity is described by an operator Oˆ , which is Hermitian if the physical quantity is observable. The average value (the expected value) of the physical quantity can be calculated

v Oˆ v = O for the system represented by v . If the physical quantity is observable | i D E then O is real. We are interested in time evolution of physical quantities. There are two

11 approaches for calculation of time evolution of physical quantities. In the Schr¨odinger picture, the time evolution of a state vector is given by the Schr¨odinger s equation [56]

d v(t) i~ | i = Hˆ v(t) (1.14) dt | i where Hˆ is the Hamiltonian of the system, which is the energy operator. The average of a physical quantity is

O(t)= v(t) Oˆ v(t) . (1.15) D E and Oˆ can be considered constant as a function of time. In Heisenberg picture, state vectors remain constant in time and the operators evolve according to the Heisenberg equation [27] dOˆ (t) 1 = Oˆ (t), Hˆ . (1.16) dt i~ h i The average of a physical quantity is

O(t)= v Oˆ (t) v . (1.17) D E

There is also the interaction picture, which is a combination of the Schr¨odinger and the

Heisenberg picture. In this thesis, I exclusively work in Heisenberg picture because the equations for operators are analogous to classical system [28] and I avoid complications associated with entangled states present in the Schr¨odinger picture and hidden in the

Heisenberg picture[29].

The variance of a physical quantity can be also be calculated. Itis

2 (∆O)2 = v Oˆ 2 v v Oˆ v . (1.18) − D E D E

Using the deﬁnition of the variance above, the uncertainty involved in the simultaneous

12 measurement of two observable can be calculate using the Heisenberg uncertainty principle, which states [30] 1 ∆A∆B Aˆ , Bˆ , (1.19) ≥ 4 Dh iE where Aˆ , Bˆ is the commutator of Aˆ and Bˆ and is deﬁned Aˆ , Bˆ = Aˆ Bˆ BˆAˆ . In the − h i h i following sections (1.5.1-1.5.4) are some examples of operator and states I will be using.

1.5.1 Single Mode Field Operators

I will closely follow the formalism of Loudon [31]. Since the thesis is mainly dealing with electric field propagation, it is important to look at the quantum mechanical operator that describes it. Consider an electromagnetic field that has excited a single travelling-wave mode along the z direction with wavevector k . The electric field is expressed as follows ∂A˜ E˜ = , (1.20) − ∂t where A˜ is the vector potential. In the Heisenberg picture, the scalar electric field operator for a given linear polarization is written

+ Eˆ(χ)= Eˆ (χ)+ Eˆ −(χ) (1.21) ~ ω iχ iχ = Aˆ e− + Aˆ †e , 2ǫ0V r h i where the positive and negative frequency of the ﬁeld operator correspond respectively to the two terms of the right hand side and V is the volume of the cavity in which the electric

field is contained. Aˆ is the destruction or the annihilation operator and Aˆ † is the raising operator. They are the coefficient of the amplitude of the vector potential. χ is defined as

13 follows π χ = ωt kz . (1.22) − − 2

It is convenient to remove the square root factor from equation (1.21). Therefore, by convention the electric ﬁeld is measured in units of 2 ~ω . The operator reduces to 2ǫ0V q 1 iχ 1 iχ Eˆ(χ)= Aˆ e− + Aˆ †e 2 2 (1.23) = Xˆ cos(χ)+ Yˆ sin(χ), where Xˆ and Yˆ are quadrature operators and are deﬁned as follows

Aˆ + Aˆ Xˆ = † (1.24) 2 and

Aˆ Aˆ † Yˆ = − . (1.25) 2i

1.5.2 Number State

The photon number states or Fock state are the eigenstates of the quantum theory of light. They form a complete set for the states of a single mode. They are denoted n | i where n is the number of photon. The action of the destruction operator on the number state is as follows

Aˆ n = n n 1 (1.26) | i | − i and the of the creation operator is

Aˆ † n =(n + 1) n +1 . (1.27) | i | i

14 From this and using equation (1.23) I can calculate the average value of the electric ﬁeld in a number state n , which is | i

n Eˆ(χ) n =0. (1.28) D E

We can also calculate the variance of the electric ﬁeld for that number state which is

2 2 (∆E(χ))2 = n Eˆ(χ) n n Eˆ(χ) n − D E 2 = n Eˆ(χ) n (1.29) 1 1 = n + . 2 2 Therefore, for a vacuum state 0 the variance of the electric ﬁeld is 1 . This is also referred | i 4 to as the vacuum ﬂuctuations.

1.5.3 Number Operator

The number operator is the observable that counts the number of photons. It will be very useful in this thesis as measurements of ampliﬁed signals will be needed. It is deﬁned as follows

ˆn Aˆ †Aˆ . (1.30) ≡

One way to look at this operator is that it counts the photons by removing and replacing them. It acts as follows on a number state

ˆn n = n n . (1.31) | i | i

It is important to note that for a number state, while the number of photon can be accu- rately calculated, its phase is undeﬁned. This is consistent with an uncertainty principle,

15 1 which states that ∆n∆φ = 2 [32]. Conversely, if there is a state for which the phase is well defined, the pure number would be undefined. Because its phase is undefined, the number state cannot be used as a quantum mechanical model for coherent light.

1.5.4 Coherent States

The coherent state or Glauber state, typically denoted α is deﬁned as the following | i superposition of number states

∞ n 1 α 2 α α = e− 2 | | n . (1.32) | i √ | i n=0 n! X where α is any complex number. It can be observed from the above equation that the coherent state has a Poissonian number distribution. In other words, the probability of detecting n photons while measuring a coherent state is Poisson distributed and its distribution is n α 2 α p(n)= e−| | | | . (1.33) n!

The action of the destruction operator on a coherent state is as follows

Aˆ α = α α . (1.34) | i | i

Therefore α is an eigenstate of the destruction operator and α its eigenvalue. Although, | i α is not an eigenstate of the creation operator, the creation operator does satisfy the | i left-eigenvalue relation conjugate to equation (1.34)

α Aˆ † = α α∗. (1.35) h | h |

We can calculate the average number of photon in a coherent state

2 α Aˆ †Aˆ α = α . (1.36) | | D E

16 The ﬂuctuation in photon number for a coherent state can also be calculated

(∆n)2 = α ˆn2 α ( α ˆn α )2 − h | | i

= α 4 + α 2 α 4 (1.37) | | | | −| | = n

The variance of a coherent state is equal to its average number of photons. The coherent state is very often used to model a single mode laser. While it is not the only model used for single mode laser, it is the one for which δnδφ is minimum, making a model of choice in this thesis as I will be looking for the minimum noise ﬁgure in different OPAs.

1.6 Quantum Noise Added During Ampliﬁcation and Attenuation

In this section, I go over the process of attenuation and ampliﬁcation and look at the noises added and some of their properties. These noises will play a very important role when OPA will be analyzed.

1.6.1 Quantum Noise Added During Attenuation

I will first look at the quantum noise added during attenuation. Consider a signal represented at the input by a coherent state α (see figure 1.5). Consider also the electric | i field represented by the lowering operator Aˆ . The operator Aˆ satisfies the following commutator relation [59]:

Aˆ , Aˆ † =1. (1.38) h i Following the formalism of Haus [57], the operator Aˆ evolves to Bˆ at the output of the attenuator. If L is the fraction of power transmitted through the attenuator, one would

17 Figure 1.5: Example of Attenuator: Beam coupler. Additional noise from a reservoir channel

expect

Bˆ = √LAˆ . (1.39)

However, the evolution of the electric ﬁeld inside the attenuator is described by the

Heisenberg equation. The Heisenberg equation preserves the commutator relation. There- fore, Bˆ must obey the same commutator relation as Aˆ . Therefore, equation (1.39) is incorrect. What is physically happening is that energy is dissipated. By the

ﬂuctuation-dissipation theorem [44], this results in an addition of noise. Therefore, a

ˆ ˆ Langevin noise operator NL must be added to the expression of B. A specific example of an attenuator is a beam splitter (see figure 1.5), where the field from the second input con- tributes to the output. The second input of the beam splitter is represented by a vacuum

18 state 0 and its ﬁeld operator is represented by Aˆ . Therefore, at the output, one would | i 1 get

ˆ ˆ ˆ B = tA + rA1, (1.40) where t is the transmission coefficient and r the reflection coefficient of the beam splitter.

I choose t = √L. Therefore, r = i√1 L. I calculate the commutator of Bˆ, which is −

Bˆ, Bˆ † = L Aˆ , Aˆ † + (1 L) Aˆ , Aˆ † =1. (1.41) − 1 1 h i h i h i Bˆ obey the same commutator relation as Aˆ , which is what is expected. It can also be noted that the term i √1 L Aˆ is an additional noise term proportional to a lowering − 1 operator. In general, this noise term is needed to preserve the commutation relation of the evolving ﬁeld operator. Thus, in general, for the output of an attenuator, it is

ˆ ˆ ˆ B = √LA + NL, (1.42)

ˆ where NL is proportional to some annihilation operator[57] operating on the vacuum state

0 . Let us ﬁnd the commutation relation of Nˆ by computing | i L

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ B, B† = L A, A† + NL, NL† = L + NL, NL† =1. (1.43) h i h i h i h i

Therefore, Nˆ , Nˆ † =1 L. L L − h i In order to characterize the output, let us compute the output number of photons and the photon number ﬂuctuations The number of photons after attenuation is:

n = Bˆ †Bˆ h bi D E ˆ ˆ ˆ ˆ = L α A†A α + √L α A† α 0 NL 0 (1.44) D E D ED E √ ˆ ˆ ˆ ˆ + L α A α 0 NL† 0 + 0 N L† NL 0 . D ED E D E

19 ˆ ˆ ˆ ˆ The terms 0 NL 0 and 0 NL† 0 are zero. The term 0 NL† NL 0 is zero since D E D E D E ˆ NL is a lowering operator. Therefore,

2 nb = L α h i | | (1.45) = L n h ai

2 as expected, where n = α Aˆ †Aˆ α . Now, n is calculated h ai h b i D E

2 n = Bˆ †BˆBˆ †Bˆ . (1.46) h b i D E The operators need to be arranged in normal order, which means having all the lowering operator on the right of the terms and all the raising operatoron theleft, as I willshow. For

2 this, the commutator relation BˆBˆ † =1+ Bˆ †Bˆ is used. The expression of n becomes h b i

2 ˆ ˆ ˆ ˆ nb = B† B†B +1 B h i (1.47) D E = Bˆ †Bˆ †BˆBˆ + Bˆ †Bˆ D E D E ˆ Now, the operators are in normal order. They can easily be evaluated by noting that NL operates on the state 0 and therefore does not make any contribution. Therefore, | i

2 2 ˆ ˆ ˆ ˆ ˆ ˆ nb = L A†A†AA + L A†A h i (1.48) D E D E = L2 α 4 + L α 2. | | | | Therefore,

(∆n)2 = L n . (1.49) h ai

The output variance is equal to the output average number of photons. This is due to the

ˆ fact that the output signal is a coherent state. The added noise NL guaranties that the output ﬁeld satisfy the Heisenberg uncertainty principle.

20 1.6.2 Noise in Phase Insensitive Ampliﬁcation

Closely following Haus [58], an input coherent state α is assumed, whose electric | i field I will represent by a lowering operator Aˆ . It goes through a phase insensitive amplifier with field gain √G. I will denote the output Bˆ. Just as in the previous section, in

ˆ ˆ ˆ order to have B, B† =1, an extra noise term NG needs to be added: h i ˆ ˆ ˆ B = √GA + NG. (1.50)

This noise term Nˆ satisfies Nˆ , Nˆ † = 1 G. Since G > 1, Nˆ , Nˆ † is negative. G G G − G G h i h i ˆ Therefore, NG is proportional to a raising operator. Physically, this noise comes from the Heisenberg uncertainty principle. Phase insensitive amplification can be seen as a simultaneous measurement and amplification of the inphase and quadrature phase component, which are two noncommutative observables. Therefore, an associated uncertainty is added. The output number of photons is

ˆ ˆ nb = B†B h i (1.51) D E ˆ ˆ √ ˆ ˆ ˆ ˆ ˆ ˆ = GA†A + G A†NG + ANG† + NG† NG D E since Nˆ † Nˆ = Nˆ Nˆ † + G 1, the expression becomes G G G G −

n = G n + G 1 (1.52) h bi h ai − where n = α Aˆ †Aˆ α and G 1 is extra noise added by the ampliﬁer. Now, let us h ai − D E calculate n2 h b i

2 n = Bˆ †BˆBˆ †Bˆ h b i D E = BˆBˆ † 1 BˆBˆ † 1 (1.53) − − D E = BˆBˆBˆ †Bˆ † 3BˆBˆ † +1 . − D E 21 With n2 expressed in anti normal ordering, the argument can be made that the noise h b i ˆ operator NG does not make any contribution. Therefore,

2 2 n = G Aˆ Aˆ Aˆ †Aˆ † 3GAˆ Aˆ † +1 . (1.54) h b i − D E I can now express the relation in normal ordering to evaluate it, which is

2 2 2 n = G Aˆ †Aˆ +1 Aˆ †Aˆ +1 + G 3G Aˆ †Aˆ +1 +1 h b i − D E 2 2 = G Aˆ †Aˆ †Aˆ Aˆ +3Aˆ †Aˆ +1 + G 3G Aˆ †Aˆ +1 +1 (1.55) − D E = G2 n 2 +4G2 n 3G n +2G2 3G 1 h ai h ai − h ai − − In this fashion, I can compute the variance

2 2 2 ∆nb = nb nb h i−h i (1.56) = G n +2G(G 1) n + G(G 1). h ai − h ai − It can be veriﬁed that the output signal variance is not equal to the output average number of photons. Therefore, the output signal is not a coherent state.

1.7 Deﬁnition of Signal to Noise Ratio and Noise Figure

In this thesis, I will use the Signal to Noise Ratio (SNR) as a measure of quality of an optical signal. There are more than one way to deﬁne SNR. In general, for an electrical signal the deﬁnition used is [45]

Average Power of Electrical Signal SNR = . (1.57) Average Power of Electric Noise

My problems will involve the characterization of optical signals. This can be done by characterizing the signal and the noise of an ideal receiver illuminated by an optical

22 Figure 1.6: Schematic of an ideal receiver

signal (see ﬁgure 1.6). In an ideal receiver, for each photon received, the detector emits one electron in a circuit. To characterize the electric current generated by the receiver, I consider a repeated experiment where the statistical average number of photons received

n is deﬁned as the signal. The generated current ﬂows for a time T equal to the duration h i of the optical signal in a circuit with a resistance R, where R is large enough not to allow any oscillation in the circuit. The average current in this circuit is

q n i = h i. (1.58) h i T

Therefore, the average electric signal power is

q2 n 2 P = h i . (1.59) S τ 2R

23 The average total electric power is

i2 q2 n2 P = h i = h i. (1.60) Tot R τ 2R

I can deﬁne the noise power as

q2 n2 q2 n 2 q2 P = P P = h i h i = n2 n 2 , (1.61) N Tot − S T 2R − τ 2R T 2R h i−h i which is proportional to the variance of the number of photons. Therefore, the SNR of the detected optical signal is

P n 2 n 2 SNR S (1.62) PNF = = 2 h i 2 = h i 2 . PN n n (∆n) h i−h i

This deﬁnition of SNR is called the photon number ﬂuctuations SNR [41]. This SNR is in principle measurable in an experiment and can be readily calculated for some common types of signals. For example, for a coherent state α with n = α 2 and (∆n)2 = α 2 | i h i | | | | it is

α 4 SNR = | | = α 2. (1.63) PNF α 2 | | | | When an optical signal goes through a device, its SNR changes. To compare the noise performance of different devices, a quantity called Noise Figure (NF) is introduced. There are more than one deﬁnition of NF based on which SNR is used. In general, the noise

ﬁgure is [43] SNR NF = 10 log out . (1.64) − SNR in where SNRin is the Signal to Noise Ratio of the signal before the device and SNRout is the Signal to Noise Ratio of the signal after the device.

24 1.7.1 Deﬁnition of the Photon Number Fluctuation Noise Figure

I can use the deﬁnition in equation (1.64) to calculate the Photon-Number-Fluctuation

NF (NFPNF) of an attenuator with ﬁeld loss √L. It has been seen in section 1.6.1 that for an input coherent state α , the output average number of photons is L α 2 and the photon | i | | number ﬂuctuations is L α 2. Therefore, the output SNR is | |

SNR = L n (1.65) PNF h ai

Since the input SNR = α 2, the noise figure based on the photon number fluctuation PNF | | defition is

SNR NF = 10 log out PNF − SNR in (1.66) = 10 log(L) − (see figure 1.7). Let us now calculate the noise figure for a phase insensitive amplifier with field gain √G. It has been seen in section 1.6.2 that for an input coherent state α , | i the output average number of photons is G α 2+G 1 and the photon number fluctuations | | − is G α 2 +2G(G 1) α 2 + G(G 1). Therefore, the output SNR is | | − | | −

G2 α 2 NF = 10 log | | , (1.68) PNF − G α 2 +2G(G 1) α 2 + G(G 1) | | − | | − which is plotted in figure 1.8. With higher amplification, there is a higher deterioration of the SNR with the NFPNF converging to 3 dB. It can also be seen that the NFPNF for phase insensitive amplifiers is signal dependent for weak signal, which can be a problem.

25 Figure 1.7: Noise Figure in dB as a function of attenuation

In section 5, I will show that this signal dependence of the NFPNF is also true for NFPNF of Kerr based NMZI OPA.

However, if a large signal is assumed (n 1 ), then the NF for phase insensitive ≫ 2 PNF ampliﬁers becomes

G2 α 2 NF 10 log | | PNF ≈ − G α 2 +2G(G 1) α 2 | | − | | (1.69) G2 = 10 log . − G +2G(G 1) − The noise ﬁgure is no longer signal dependent. I will show in section 3.4, assuming a large signal, that the NFPNF is signal independent.

26 For large values of G,

G α 2 +2G(G 1) α 2 + G(G 1) 2G2 α 2 (1.70) | | − | | − ≈ | |

Therefore,

G2 n 2 SNR h ai PNF ≈ 2G2 n h ai (1.71) n = h ai. 2 By equation (1.68) NF = 10log(2) 3 dB. That 3 dB is the minimum in conse- PNF ≈ quence of the uncertainty principle. A simultaneous measurement of two noncommuting variables must double the Heisenberg uncertainty [41]. For other reasons, a practical ampliﬁer will have a larger noise ﬁgure.

Figure 1.8: Noise Figure in dB as a function of gain (equation (1.67))

27 1.7.2 Deﬁnition of the Field Amplitude Squared Noise Figure

Haus proposed another definition of SNR called the field amplitude squared SNR whose noise figure is not signal dependent. It is defined as follows

E 2 SNRFAS = h i . (1.72) (∆E)2

For a coherent state,

2 α 2 SNRFAS = | 1| =2 α =2 n . (1.73) 2 | | h i

For an attenuator with ﬁeld loss √L with an input coherent state α , it was said in | i section 1.6.1 that the output is also a coherent state with average number of photon L n . h i Therefore, its output SNR is

2 α 2 SNRFAS = | 1| =2 α =2L n . (1.74) 2 | | h i

Therefore, its noise ﬁgure is

NF = 10 log(L)= NF . (1.75) FAS − PNF

1.7.3 NFFAS of an Phase Insensitive Ampliﬁer

Let us calculate the field amplitude squared noise figure of the phase insensitive amplifier with field gain √G and with a coherent state α as an input. Using equation | i (1.50), I calculate the variance of the output field, which is

∆Bˆ 2 = ∆Xˆ 2 + ∆Yˆ 2 (1.76) 1 = G . − 2 28 Since the average number of photons at the output is G α2 , the output SNR is | |

G α 2 SNR = | | . (1.77) FASout G 1 − 2

Since the input SNR =2 α 2, the noise ﬁgure is FASin | |

2G NF = 10log . (1.78) FAS G 1 − 2 It can be seen that this noise ﬁgure in independent of the signal without any assumption on the value of the signal. Also, for large values of G the ﬁeld amplitude squared noise

figure goes to 3 dB, which is roughly equal to the photon number fluctuations noise figure.

However, this definition also has its problems. The electric field is not a measurable quantity. Therefore, for the calculations of noise figure, the signal to noise ratio has to be estimated. Moreover, this definition of the SNR assumes that the inphase component of the noise and the quadrature phase component of the noise are have the same power, which makes it unsuitable for phase sensitive optical parametric amplifiers as I will show.

1.7.4 Deﬁnition of the Quadrature Field Squared Noise Figure

A new deﬁnition of SNR and its associated noise ﬁgure that is more suitable to

OPAs is introduced. It is the quadrature ﬁeld squared SNR and is deﬁned as follows

n SNRQFS = h i . (1.79) ∆Xˆ 2 D E If this deﬁnition is applied to a coherent state α , its signal to noise ratio is | i

SNR =4 α 2. (1.80) QFS | |

29 For an attenuator with ﬁeld loss √L with an input coherent state α , the output SNR is | i

SNR =4L α 2. (1.81) QFSout | |

Since SNR =4 α 2, the attenuator quadrature ﬁeld squared noise ﬁgure is QFSin | |

NF = 10 log(L)= NF = NF . (1.82) QFS − PNF FAS

To calculate the quadrature phase squared noise figure for a phase insensitive amplifier with field gain √G, I first calculate the quadrature phase noise, which is

1 2 G δXˆ = − 2 . (1.83) 2 D E Since the average number of photons at the output is G α2 , the output SNR is | | 2G α 2 SNR = | | . (1.84) QFSout G 1 − 2 Since the input SNR =4 α 2, the noise ﬁgure is QFSin | | 2G NF = 10 log = NF . (1.85) QFS − G 1 FAS − 2

For large values of G, it can be veriﬁed that NFQFS for the phase insensitive ampliﬁer goes to 3 dB, which is roughly equal to the NFPNS. The added advantage of the NFQFS over the NFFAS is that it is in principle measurable.

1.7.5 Deﬁnition of the Quality Factor

Another measure of a signal quality is called Quality Factor or Q. It is to similar

SNR and is very applicable to On-Off Keying (OOK). It is deﬁned as follows [42]

n n Q = h 1i−h 0i , (1.86) ∆n1 + ∆n0

30 where n and n are respectively the average number of photons when bit one (1) is h 1i h 0i sent and bit zero (0) is sent, and ∆n1 and ∆n0 are respectively the standard deviation of the signal when bit on (1) and bit zero (0) is sent. The Q factor of an Pulse-Amplitude-

Modulated (PAM) OOK coherent state signal can be calculated. α = 0 is chosen for the bit zero (0) and α = 0 is chosen for the bit one (1). Since for a coherent state α the 6 | i variance ∆n = α 2, ∆n = α 2 and ∆n =0. Therefore, | | 1 | | 0

Q = α = n . (1.87) | | h i p The Q factor is very convenient for calculating the Bit Error Rate (BER) [42], which is.

1 Q BER = erfc . (1.88) 2 √2 In this thesis, I will be analyzing the output signal of OPA and using the different noise ﬁgures deﬁned above.

1.8 Balanced Homodyne Detection of a Coherent State

Because of similarities to Phase-Sensitive-Ampliﬁed detection, I will ﬁrst analyzed the Balanced Homodyne Detector (BHD), which is more widely known. For this, I assume that all dark and thermal noise is negligible in comparison to the photon noise.

Balanced Homodyne Detectors can be readily described in Quantum Mechanics using

ˆ ˆ ˆ ˆ the Heisenberg picture [2]. I denote ALO, As, B1 and B2 respectively, the operators of the complex amplitude of the electric ﬁeld of the local oscillator, the incoming signal, and the signal incident on the photodetectors after an ideal splitter.

ˆ 1 ˆ ˆ B1 = ALO + iAs , (1.89) √2 31 ˆ 1 ˆ ˆ B2 = As iALO . (1.90) √2 − The difference between the charges collected by the two detectors is

ˆ ˆ ˆ ˆ δˆq = q B1†B1 B2†B2 − (1.91) = iq Aˆ † Aˆ Aˆ †Aˆ . − LO s − s LO I assume that the frequency of the local oscillator is the same as the incoming signal frequency. Using the coherent states α and α respectively as the wave vectors of | LOi | si the local oscillator signal and the incoming signal, I have

ˆ ˆ ˆ ˆ δˆq = αs αLO iq ALO† As As†ALO αLO αs h i − − (1.92) D D E E

=2q αLOα s sin( φ), | | where φ is the phase of αLO∗ αs Under optimum phase,

δˆq =2q α α . (1.93) h i | LO s|

The mean square ﬂuctuation is

∆(δˆq)2 = (δˆq)2 δˆq 2 −h i

= q2 α 2 + α 2 (1.94) | LO| | s| = q2 ( n + n ) , h LOi h si where

n = Aˆ † Aˆ (1.95) h LOi LO LO D E and

n = Aˆ †Aˆ . (1.96) h si s s D E For large relative values of the local oscillator signal, n can be neglected h si

∆ˆq2 = q2 n . (1.97) h LOi 32 Therefore,

SNR =4 n , (1.98) PNS h si which is equal to twice ﬁeld amplitude squared SNR of the optical signal and four times its photon number square SNR. However, the SNR is not the whole story when it comes to determining sensitivity of a receiver in digital communications. Let us calculate its quality factor. To calculate Q for this receiver, αs = 0 is chosen for the bit zero (0) and

α =0 is chosen for the bit one (1). It is s 6

2q αLOαs Q = | | = ns . (1.99) 2 2 q nLO + q nLO h i h i h i p p p The quality factor of a ideal detector and a balanced homodyne detector are the same.

1.9 Nonlinear Mach-Zehnder Interferometer OPA Noise

In this section, I derive the expression for the noise at the output of a NMZI OPA without input signal using quantum mechanics, and I compare the results with the one from classical analysis. Assuming that a ﬁeld operator Aˆ can be expanded as a perturbation which is small around the average value, the ﬁeld can be represented as

Aˆ = A + δAˆ (1.100) where A Aˆ . This is known as Quazi Linearization. This assumes that A δAˆ ≡ ≫ D E .Also, the ﬂuctuations can be broken up into a inphase and a quadrature phase as follows

Aˆ = A + δXˆ + iδYˆ . (1.101)

33 Figure 1.9: Schematic of NMZI OPA. Appropriate operators as detailed in the text are shown.

Now, let us consider a NMZI OPA as shown in ﬁgure 1.9, with an input pump ﬁeld

ˆ ˆ denoted with the operator Ap and a signal ﬁeld denoted with the operator As. In the following chapters, I will show how the evolution of the ﬁeld operator is calculated in a nonlinear medium. Below is the computation of the NMZI output noise assuming that

ﬁeld operators at the output of the two nonlinear medium are known.

ˆ ˆ I denote respectively A1 and A2 the ﬁeld operators at the inputs to the output coupler. Under normal conditions, the signal comes out in one arm and the pump in the other.

On the signal arm, I have

ˆ 1 ˆ ˆ Aout = A1 + iA2 √2 (1.102) 1 ˆ ˆ ˆ ˆ = A1 + iA2 + δX1 + iδY1 + i δX2 + iδY2 . √2 h i h i 34 Assuming that the input signal power is zero, the average ﬁeld at the input of the nonlinear medium a the upper arm of the NMZI is Ap , while the one at the lower arm of the √2 nonlinear medium is iAp . Since the power of the ﬁeld going through the two nonlinear √2 medium are the same, their amplitude and phase response will be the same. Therefore, at the output iA1 = A2, so that equation (1.102) becomes

ˆ 1 ˆ ˆ ˆ ˆ Aout = δX1 δY2 + i δY1 + δX2 . (1.103) √2 − h i ˆ ˆ The noise in the upper arm of the NMZI OPA described by δX1 and δY1 is uncorrelated

ˆ ˆ to the noise in the lower arm described by δX2 and δY2)(see appendix A.4). Therefore, the Ampliﬁed Spontaneous Emission (ASE) is

ˆ ˆ ASE = Aout† Aout D E 1 ˆ 2 ˆ 2 ˆ ˆ ˆ 2 = δX1 + δY1 + i δX1, δY1 + δX1 (1.104) 2" D E D E Dh iE D E ˆ 2 ˆ ˆ + δY2 + i δX2, δY2 . # D E Dh iE Since the noise at the output of the nonlinear medium have the same statistics

ˆ 2 ˆ 2 ˆ ˆ (1.105) ASE = δX1 + δY1 + i δX, δY . hD E D E Dh iEi It is well known that δXˆ and δYˆ do not commute. Using equations (1.24) and (1.25) I can calculate their commutator, which is[55]

i δXˆ , δYˆ = , (1.106) 2 h i and obtain

2 2 1 ASE = δYˆ + δXˆ . (1.107) 1 1 − 2 hD E D Ei

35 Chapter 2

Classical Treatment of the Kerr Medium Based NMZI

2.1 Kerr Effect

Discovered in 1875 by John Kerr, the DC Kerr effect or the quadratic electro-optic effect (QEO effect)is a change in the refractive index of a material in response to the power of an electric field. The optical Kerr effect or the AC Kerr effect is the case in which the change is due to light. The relationship for Optical Kerr effect is described as follows [52, 33]. Let us assume an electric vector field E = Es cos(ωt). For a nonlinear material, in general the electric polarization field P will depend on the electric field E such that

3 3 3 3 3 3 (1) (2) (3) Pi = ǫ0 χij Ej + ǫ0 χijkEjEk + ǫ0 χijklEjEkEl j=1 j=1 k j=1 k l X X X=1 X X=1 X=1 (2.1) + + HOT, ···

(n) where ǫ0 is the vacuum permittivity and χ is the n-th order component of the electric susceptibility of the medium, and where i = 1, 2, 3. It is assumed that 1 represents x,

2 y and 3 z. For materials exhibiting a non-negligible Kerr effect, the third, χ(3) term is signiﬁcant, with the even-order terms typically cancelling due to inversion symmetry of the nonlinear medium (a change in sign in E means a changing sign in P). From that, the following equation can be obtained [33]:

(1) 3 (3) 2 P ǫ χ + χ Es Es cos(ωt). (2.2) ≈ 0 4 | | 36 This equation looks like the polarization equation for linear material

P = ǫ0χEs cos(ωt). (2.3)

The difference is that in (2.2), the linear susceptibility has an extra nonlinear term:

χ = χLin + χNL (2.4) (1) 3 (3) 2 = χ + χ Es . 4 | |

1 1 Since the index of refraction n = (1+ χ) 2 , we deﬁne n0 = (1+ χLin) 2 . The Kerr medium’s index of refraction can be written

1 n = (1+ χLin + χNL) 2

1 1 1 = (1+ χLin) 2 (1 + χNL) 2 1+ χLin 1 1 2 = n 1+ χ 0 n2 NL 0 (2.5) 1 n 1+ χ ≈ 0 2n2 NL 0 3 (3) 2 = n 1+ χ Es 0 4n2 | | 0

= n0 + n2I.

There are many phenomena such as four wave mixing, polarization rotation and cross phase modulation built into the nonlinear index n2. However, the principal phe- nomenon is a phase shift due to the retardation of propagation in proportion to the intensity, called self phase modulation. The Kerr medium of choice for many applications is

Highly Nonlinear Fiber (HNLF)[4].

37 2.2 Nonlinear MZI with Kerr Media

I will assume a noiseless pump and a negligeable thermal noise. Let us consider an experimental setup as in ﬁgure 1.4. At one input, a strong pump signal is assumed represented by the complex phasor Ep, which I will pick to be real without loss of generality.

At the second input of the NMZI, a weak signal Es is assumed. After the ﬁrst coupler, the expression of the two outputs are

1 E¯out11 = (Ep + iEs) (2.6) √2 and 1 E¯out21 = (Es + iEp) . (2.7) √2

As discussed earlier, when those two signals go through the Kerr media in both arms of the NMZI, they self phase modulate and pick up a phase shift. It is assumed that the Kerr media in both arms have the same length. The expression of the electric ﬁeld at the output of the two Kerr media can be expressed as follows

eiΦ1 E¯out12 = (Ep + iEs) (2.8) √2 and eiΦ2 E¯out22 = (Es + iEp) . (2.9) √2 the phase of shifts are calculated as follows [52].

2π (E iE )(E∗ + iE ) 2π Φ = n L s − p s p + n L 1 λ 2 2 λ 0 (2.10) π 2 2π n L E + iE (E E∗) + n L ≈ λ 2 p p s − s λ 0

38 and

2π (E iE )(iE∗ + E ) 2π Φ = n L p − s s p + n L 2 λ 2 2 λ 0 (2.11) π 2 2π n L E iE (E E∗) + n L. ≈ λ 2 p − p s − s λ 0 The approximations are valid because Es is weak compared to Ep.

I can also write Φ = Φ + Φ and Φ = Φ Φ , where the common phase is 1 10 11 2 10 − 11 π 2π Φ = n LE2 + n L, (2.12) 10 λ 2 p λ 0 and the phase difference is,

π Φ = 2 n LE (E ), (2.13) 11 − λ 2 pℑ s where ( ) means imaginary part. After the two signals propagate through the second ℑ · coupler, I get at one of the outputs of the NMZI

Ep iEs E = eiΦ2 eiΦ1 eiΦ2 + eiΦ1 . (2.14) out 2 − − 2 I can rewrite eiΦ1 eiΦ10 (1 + iΦ ) and eiΦ2 eiΦ10 (1 iΦ ) since Φ is small. This ≈ 11 ≈ − 11 11 allows a further reduction of the electric ﬁeld output equation to.

E = iE eiΦ10 Φ ieiΦ10 out − p 11 −

= ieiΦ10 (E Φ + E ) (2.15) − p 11 s

iΦ10 = ie (iΦ (E E∗)+ E ) . − 10 s − s s If I rewrite E = E eiφ, then s | s|

E = ieiΦ10 E 2Φ sin(φ)+ eiφ . (2.16) out − | s| 10 Therefore, the gain of the NMZI is E 2 G = out E s (2.17) 2 2π 2 iφ = n2 LE sin(φ)+ e , λ p

39 which is a phase sensitive gain.

A common constant used to describe the characteristics of a Kerr medium is γ in

1 1 units of W− km− [34] which is deﬁned as follows:

2πn γ = 2 , (2.18) λAeﬀ

where Aeff is the effective area. In this case Aeff = 1 For 1 km of nonlinear fiber with

γ = 10 and a pump signal of 1 W and a signal of 1 mW, a phase sensitive gain is shown in ﬁgure 2.1

Figure 2.1: Plots of gain as a function of input phase difference for 1 km of Kerr medium with γ = 10 and pump power of 1 W

40 2.3 Frequency Response of a NMZI

Even for linear MZI, unbalancing the interferometer by making one arm shorter than the other can introduce bandwidth limitations. For example, let us consider a MZI with the same parameters as the one in section 2.2 but without nonlinear effects and with one arm longer by ∆L. It can be shown following the same procedure as the one in that section that the output ﬁeld is

i 2π n L+ ∆L π π E = ie λ 0( 2 ) E sin n ∆L E cos n ∆L . (2.19) out p λ 0 − s λ 0 h i Looking only at the ﬁeld signal output, I assume that its maximum is reached at frequency

ω0. Therefore, ω cos 0 n ∆L =1. (2.20) 2c 0

The minimum is reached at frequency ω = ω0 + ∆ω, where

πc ∆ω = . (2.21) n0∆L

∆ω is the estimated bandwidth. For ∆L = 1mm, the bandwidth is about 100 Ghz. To avoid this bandwidth limitation, an interferometer called Sagnac is used in practice (see

ﬁgure 2.2). The Sagnac is a MZI folded on itself so that one arm is used instead of two.

It guaranties that the interferometer is always balanced, while its behavior is similar to a regular MZI. Therefore, for an easier analysis, the MZI is used. But in practice, the

Sagnac is used.

41 Figure 2.2: Schematic of a Sagnac

2.3.1 Phase Insensitive Gain

Here I am considering a case where the signal and the pump have a different carrier frequency and there is no dispersion in the Kerr medium. If Ω is the difference between

i[Ωt+φ] their carrier frequencies, then Es = As(t)e is the input signal and Ep = Ap is the pump. Based on equation (2.15), the total output electric ﬁeld of the NMZI is:

T iω0t iΦ10 i[Ωt+φ] Eout = ie Ase Φ10(2 sin(Ωt + φ)) + e (2.22) iω0t iΦ10 (Ωt+φ) i(Ωt+φ) i[Ωt+φ] = ie A e Φ (4i e e− )+ e . s 10 −

42 i[(Ω)t+φ] Note that the output electric ﬁeld includes the idler signal As(t)e− , which is not part of the input signal. To look at the ampliﬁcation of the signal, we keep only the terms that includes the input signal and obtain:

ES = ieiω0tA eiΦ10 iΦ ei(Ωt+φ) + ei[Ωt+φ] . (2.23) out s − 10 From this I get an expression for the gain, which is:

ES 2 G = | out| E 2 | s| = iΦ +1 2 (2.24) |− 10 |

2 = Φ10 +1, with Φ = π n L A 2. It can be seen that a phase insensitive ampliﬁcation is obtained 10 λ 2 | p| and that it is independent of frequency detuning Ω. In the real world, the bandwidth is determined by the ﬁnite response time of χ(3), which can be almost instantaneous.

2.3.2 Phase Sensitive Gain

Consider the signal and the pump to be at the same carrier frequency. However, the

iφ signal is amplitude modulated. Therefore, I can write Es = As(t)e and Ep = Ap where

As(t) real. Then, based on equation (2.15), the electric ﬁeld output of the NMZI is:

iΦ10 iω0t iφ Eout = iAs(t)e e Φ10(2 sin(φ)) + e . (2.25) I can then get an expression for the gain, which is:

E 2 G = | out| E 2 | s| (2.26) 2 2 = 4Φ10 sin (φ)+2Φ10 sin(2φ)+1 , 43 where Φ = π n L A 2. It can be seen that the gainis sensitiveto phase but is bandwidth 10 λ 2 | p| independent.

In conclusion, we can say that the Kerr medium is an important ampliﬁer with large gain and wide bandwidth.

44 Chapter 3

Quantum Mechanical Analysis of the Lossless Kerr Medium Based NMZI

3.1 Calculation of Average Quantities Using QM

Using notation from section 1.6.2 I can write the output ﬁeld operator Bˆ in terms of

ˆ the input ﬁeld operator A by replacing in equation (2.15). Further, by replacing Es with

ˆ ˆ ˆ A, Es∗ with A†, and Eout with B. I obtain:

ˆ iΦ10 ˆ ˆ ˆ B = ie iΦ10 A A† + A − − (3.1) = µAˆ + νAˆ † where µ ieiΦ10 (iΦ + 1) and ν eiΦ10 Φ . As a cross check, using equation ≡ − 10 ≡ − 10 (3.1), I compute the commutator of Bˆ and obtain

2 2 Bˆ, Bˆ † = µ ν =1. (3.2) | | −| | h i This time, unlike in section 1.6.2, the commutator has been preserved by the device without the need of the addition of an extra noise term operator in the expression of Bˆ. This is interesting because there is gain. To explain what happens physically consider a phasor representing the electric field of a coherent state as shown in figure 3.1 and a disk representing uncertainties in amplitude and phase of the electric field. After amplification, the inphase component of the noise with the gain gets amplified while the out of phase component of the noise gets attenuated, in a such way that the product of photon number

1 and phase uncertainties is kept ∆n∆φ = 2 . This noise is called squeezed noise.

45 Figure 3.1: Representation of the electric-ﬁeld properties of the coherent before and after parametric ampliﬁcation

For an input signal modeled by a coherent state α , I can compute the number of | i photons at the output:

α Bˆ †Bˆ α = α µ∗Aˆ † + ν∗Aˆ µAˆ + νAˆ † α (3.3) D E D E 2 2 2 2 2 = µ α +2 ν∗µα + ν α +1 , | | | | ℜ | | | | where ( ) means real part. From the above equation, it can be seen that ( µ 2+ ν 2) α 2+ ℜ · | | | | | | 2 2 2 (ν∗µα ) is the ampliﬁed input signal and ν is the ASE. The maximum signal output ℜ 2 2 happens when (ν∗µα ) is maximum, which happens when all the term ν∗µα is real. ℜ

46 Therefore, the maximum signal output relative to signal input phase occurs when

2 2 ν∗µα = µ ν α . (3.4) ℜ | || || | The output is then

2 2 2 2 2 α Bˆ †Bˆ α = µ α +2 µ ν α + ν α +1 max | | | | | || || | | | | | D E 2 = ( µ + ν ) α 2 + ν 2 (3.5) | | | | | | | | = G α 2 + ν 2, | | | | where

G =( µ + ν )2 (3.6) | | | | and is the parametric gain. Some useful equations involving the parametric gain are:

1 G + G− 2 ν 2 = − (3.7) | | 4 and G + G 1 +2 µ 2 = − . (3.8) | | 4

Using these equation, the maximum signal output relative to signal input phase, which occurs when α is real can be obtained. It is

1 G + G− 2 n = ( µ + ν )2 α 2 + − h bimax | | | | | | 4 1 G + G− 2 = G n + − (3.9) h ai 4 G G n + , ≈ h ai 4

2 where n α Bˆ †Bˆ α and n α . Since G is assumed to be large, in the order h bi ≡ h ai≡| | D E of 20 dB, the following approximation can be made

G n G n + . (3.10) h bimax ≈ h ai 4 47 Similarly, the minimum output, which occurs when α is purely imaginary, is:

3.2 Quantum Mechanical Noise With Loss in Linear Elements

To compute the noise, I calculate the variance of power signal ∆n. In order to do so, I compute Bˆ †BˆBˆ †Bˆ using (3.1). D E 4 2 2 2 2 2 Bˆ †BˆBˆ †Bˆ = µ α (1 + α ) +2 µ µ ν∗α (2 + α ) | | | | | | ℜ | | | | D E 2 2 2 2 2 4 2 +2 µ µ ν∗α α + µ ν 4 α +8 α +2 ℜ | | | | | | | | | | | | (3.12) 2 2 4 2 2 2 +2 µ ν∗ α +2 ν∗µ ν α 4+2 α ℜ ℜ | | | | + ν 4 1+3 α 2 + α 4 . | | | | | | I then ﬁnd n 2 using (3.3) , which is h i

2 4 4 2 2 4 4 2 4 n = µ α +2 ν∗ µ α + ν 1+2 α + α h i | | | | ℜ | | | | | |

2 4 4 2 2 2 2 2 (∆n) = µ + ν α +4 µ µ ν∗α +4 ν∗µ ν α | | | | | | ℜ | | ℜ | | (3.14) + µ 2 ν 2 6 α 2 +2 . | | | | | | when the relative phase between the signal and the pump is adjusted to give maximum gain, I use (3.4)

(∆n)2 = µ 4 + ν 4 α 2 +4 µ 2 + ν 2 µ ν α 2 + µ 2 ν 2 6 α 2 +2 . (3.15) | | | | | | | | | | || || | | | | | | | 48 Substituting equation (3.7) and (3.8) in the above equation and using the approximation

µ 2 ν 2, I get the variance for the maximum output to be | | ≈| |

1 (∆n)2 G2 n + . (3.16) ≈ h ai 8 This implies that the output photon number squared SNR is

2 2 G na SNRPNS = h i (3.17) out G2 n + 1 h ai 8 Since SNR = n , the photon number squared NF is PNSin h ai n NF = 10log h ai . (3.18) PNS n + 1 " h ai 8 # Even for high parametric gain, the photon number squared NF of this OPA is signal dependent for weak signal. For large signal ( n 1 ), the photon number squared NF h ai ≫ 8 approaches 0 dB. Also, I have

G + G 1 (∆E)2 = − . (3.19) 4

Therefore, the output ﬁeld amplitude squared SNR is

4G n SNR a (3.20) FASout = h i1 . G + G−

Since SNR =2 n , the ﬁeld amplitude squared NF is FASin h ai

2G NF = 10 log . (3.21) FAS − G + G 1 −

For large parametric gain, NFFAS approaches -3 dB, which shows that NFFAS is not appropriate for this OPA. The quadrature ﬁeld noise variance is

1 G + G− 1 ∆Xˆ 2 = − . (3.22) 4 D E 49 Therefore, the output quadrature ﬁeld squared SNR is

4G n SNR = h ai . (3.23) FASout G + G 1 1 − − Since SNR =4 n , the ﬁeld amplitude squared NF is QFSin h ai

G NF = 10 log . (3.24) QFS − G + G 1 1 − −

For a large parametric gain G, NFQFS converges to 0 dB.

3.3 Quantum Mechanical Noise of NMZI with Lumped Loss

Real couplers are usually the largest practical loss and contribute heavily to total noise. To get an estimate of their impact in the NMZI OPA’s noise ﬁgure, I will consider a lumped loss before the Kerr medium and one after.

3.3.1 Quantum Mechanical Noise of Kerr Based NMZI with all the loss

at the Input

If there is a loss that is placed before each Kerr Medium, it is equivalent to placing the losses at the inputs of the NMZI’s. I have shown that for an attenuator with loss L in section (1.6.1) NF = NF = NF = 10 log(L). The lossless parametric QFS FAS PNS − amplifier has a noise figure of NF = NF = 0 dB and NF = 3 dB for large QFS PNS FAS − parametric gain. Therefore, for a NMZI OPA with lumped loss placed before the Kerr medium, the noise figures are

NF = NF = 10 log(L) (3.25) PNS QFS − 50 and

NF = 10 log(L) 3 (3.26) FAS − −

Let us now look for noise ﬁgures expressions with the lumped loss after the Kerr medium.

3.3.2 Quantum Mechanical Noise of Lossy Kerr Based NMZI with all

the loss at the Output

If the losses are placed after the Kerr Medium, it is equivalent to placing them after the NMZI’s. The output ﬁeld operator of the OPA is as given by equation (3.1). The output after the loss is

Cˆ = √LBˆ + Nˆ , (3.27) where Nˆ is Langevin noise operator proportional to the annihilation operator and that

Nˆ †, Nˆ = 1 L. In order to characterize this output signal, I need to calculate its − h i average and its variance. The average signal output is

2 2 2 2 2 Cˆ †Cˆ = L µ α +2 ν∗µα + ν α +1 (3.28) | | | | ℜ | | | | D E To calculate its variance, I ﬁrst calculate Cˆ †CˆCˆ †Cˆ D E 2 4 2 2 2 2 2 Cˆ †CˆCˆ †Cˆ = L µ α (1 + α ) +2 µ µ ν∗α (2 + α ) | | | | | | ℜ | | | | D E n 2 2 2 2 2 4 2 +2 µ µ ν∗α α + µ ν 4 α +8 α +2 ℜ | | | | | | | | | | | |

2 2 4 2 2 2 +2 µ ν∗ α +2 ν∗µ ν α 4+2 α (3.29) ℜ ℜ | | | |

2 2 4 4 2 2 2 2 2 (∆n) = L µ + ν α +4 µ µ ν∗α +4 ν∗µ ν α | | | | | | ℜ | | ℜ | | n + µ 2 ν 2 6 α 2 +2 (3.31) | | | | | | o 2 2 2 2 2 + L 2 ν∗µα + µ 1+ α + ν α . ℜ | | | | | | | | h i When the relative phase between the signal and the pump is adjusted to give maximum gain, I use equation (3.4) to get

(∆n)2 = L2 µ 4 + ν 4 α 2 +4 µ 2 + ν 2 µ ν α 2 + µ 2 ν 2 6 α 2 +2 | | | | | | | | | | || || | | | | | | | n o + L 2 ν µ α 2 + µ 2 1+ α 2 + ν 2 α 2 . | || || | | | | | | | | | h i (3.32)

For high parametric gain, using equations (3.7) and (3.8), the approximation µ 2 ν 2 | | ≈ | | can be made. Substituting equation (3.7) and (3.8) in the above equation and using the approximation that µ 2 ν 2, I get the variance for the maximum output to be | | ≈| |

1 1 (∆n)2 L2G2 n + + LG n + . (3.33) ≈ h ai 8 h ai 4 For high parametric gain G much larger than the loss L, I have

1 (∆n)2 L2G2 n + . (3.34) ≈ h ai 8 52 With the output signal being GL n , the output photon number squared SNR is h ai G2L2 n 2 SNR = h ai PNSout L2G2 n + 1 h ai 8 (3.35) 1 n + . ≈h ai 8 Since SNR = n , PNSin h ai

n NF = 10 log h ai . (3.36) PNS − n + 1 h ai 8

For large signals n 1 , NF is roughly 0 dB. h ai≫ 8 PNS To calculate the ﬁeld amplitude squared NF, I compute (∆E)2, which is

1 G + G− 2 1 (∆E)2 = L − + . (3.37) 4 2

From this, it is then easy to show, using equations (1.64) and (1.72) , that

2GL NF = 10 log . (3.38) FAS − LG + LG 1 2L +2 − −

For a large parametric gain G, NFFAS converges to 0 dB.

2 To calculate the quadrature ﬁeld squared NF, I compute ∆Xˆ , which is 1 2 G + G− 2 1 ∆Xˆ = L − + . (3.39) 4 4 It is then easy to show that

2GL NF = 10 log . (3.40) QFS − LG + LG 1 2L +1 − −

For a large parametric gain G, NFQFS converges to 0 dB.

Let us ﬁnd an expression that will reduce the amount of necessary calculations for the NF.

53 3.4 Simple Expression of Noise Figure for High Gain Parametric

Ampliﬁer

In this section, I derive an expressionfor the noise for a general high gain parametric ampliﬁer, which will simplify the necessary calculations in the following chapters. For important applications, the ampliﬁer gain will be at least 10 dB. Unlike in the section 1.9,

I assume that there is an input signal that is much greater than the noise. I calculate the

ˆ output photon number variance of the OPA. I denote Aout, the ﬁeld operator at the output of the parametric ampliﬁer. Assuming that perturbation is valid, i.e. Aˆ δAˆ , I out ≫ out D E can write

ˆ ˆ ˆ Aout = Aout + δAout, (3.41) D E ˆ ˆ where Aout is the amplified signal field and δAout is the noise field operator. From this D E equation above, I find the variance of photon number, which is (see appendix B.1)

2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ∆ Aout† Aout = Aout δAout† δAout + Aout δAout† δAout†

D E D2 E D E D2 E ˆ ˆ ˆ ˆ ∗ ˆ ˆ (3.42) + Aout δAoutδAout† + Aout δAoutδAout

D E D E D E D 2 E ˆ ˆ ˆ ˆ ˆ ˆ + δA† δA outδA† δAout δA† δAout . out out − out D E D E I assume that the signal power is much greater than the power of the noise. Therefore, I only keep the beat terms between noise and signal since these are the source of the noise in the electrical domain. Therefore, I have

2 2 2 ∆ Aˆ † Aˆ Aˆ δAˆ † δAˆ + Aˆ δAˆ † δAˆ † out out ≈ out out out out out out D E D2 E D E D2 E ˆ ˆ ˆ ˆ ∗ ˆ ˆ (3.43) + Aout δAoutδAout† + Aout δAoutδAout

D E D E D 2 E D E ˆ ˆ ˆ ∗ ˆ = Aout δA out† + Aout δAout . D E D E

54 2 ˆ The average number of photon at the output is Aout . At the input, I denote the D E average input number of photon α 2. I denote by G , the parametric gain of the OPA. | in| p Therefore,

Aˆ = G α eiφ, (3.44) out p| in| D E p where φ is some phase. | |

2 2 2 iφ iφ ∆ Aˆ † Aˆ = G α δAˆ † e + δAˆ e− (3.45) out out p| in| out out

I deﬁne δXˆ and δYˆ , such that

iφ iφ δAˆ † e + δAˆ e δXˆ out out − (3.46) ≡ 2 and

iφ iφ δAˆ † e δAˆ e− δYˆ out − out . (3.47) ≡ 2i

This is equivalent to the following expression

ˆ ˆ ˆ iφ δAout = δX + iδY e . (3.48) Substituting this expression and equation (3.44) into equation (3.43), I get

2 2 2 ∆ Aˆ † Aˆ 4G α δXˆ . (3.49) out out ≈ p| in| D E In equation (1.107), we saw that

2 2 1 ASE = δYˆ + δXˆ . (3.50) − 2 hD E D Ei 2 2 Under the condition that δXˆ δYˆ ≫ D E D E 2 ASE = δXˆ . (3.51) D E 55 This condition is expected since the component of the noise that is in phase with the signal is expected to be considerably ampliﬁed compared to the component of the noise in the quadrature phase. Therefore,

2 2 ∆ Aˆ † Aˆ =4G α ASE. (3.52) out out | in| I can compute using equation (1.62) the output photon number squared SNR, using equation (3.44) for the signal and (3.52) for the noise. I get

G α 2 SNR = | in| . (3.53) PNSout 4ASE

For a coherent state input, the input SNR is α 2. Therefore, the photon number squared | in| noise ﬁgure is

G NF = 10 log . (3.54) PNS − 4ASE

It is important to note that this result is only valid for large parametric gain (greater than

10dB) and for large signal power (greater than 10 photons).

To compute the quadrature ﬁeld squared noise ﬁgure, I assume an input coherent state α for an OPA. The output quadrature phase squared SNR is | ini

2 G αin SNRQASout | 2| . (3.55) ≈ δXˆ D E For large gain, assuming the out of phase component of the noise is negligible relative to the inphase component, the SNR becomes

G α 2 SNR | in| . (3.56) QASout ≈ ASE

56 The input quadrature field squared SNR of an coherent state α is 4 α 2, the field | ini | in| amplitude squared noise figure is

G NF = 10 log . (3.57) QFS − 4ASE

This means that for large signals and large parametric gain,

NFQFS = NFPNS. (3.58)

Now, I compute NFFAS. I ﬁrst calculate the ﬁeld amplitude squared output SNR for an OPA with input coherent state α , which is | ini 2 G αin SNRFAS = | | out (∆E)2 2 G αin = 2 | | 2 (3.59) δXˆ + δYˆ

D E2 D E G αin = | | 1 . ASE + 2 For large gain, I have

G α 2 SNR | in| . (3.60) FASout ≈ ASE

Since the input field amplitude squared SNR of an coherent state α is 2 α 2, the field | ini | in| amplitude squared noise figure is

G NF = 10 log , (3.61) FAS − 2ASE which shows that for large signal and large parametric gain,

NF = NF 3= NF 3. (3.62) FAS PNSout − QFS −

57 This result shows one more time that the NFFAS is inappropriate for OPAs. Therefore,

I will stop using it. The expression for NFQFS and NFNPS will be used to signiﬁcantly reduce the necessary calculations in the following chapters of this thesis.

58 Chapter 4

Quantum Mechanical Noise of a NMZI with Lossy Kerr Media

4.1 Overview

In the previous chapter, we have looked at the NMZI noise in which loss is either at the input or at the output of the Kerr medium . In this chapter, we look at another case in which the loss is uniformly distributed throughout the Kerr medium. In section 3.3,

I have shown that the signal is least degraded when all the loss is located after the Kerr medium, in which case NF NF 0. I have also shown that the degradation is QFS ≈ PNS ≈ the strongest when all the loss is placed before the Kerr medium, in which case NF QFS ≈ NF 10 log(L). Therefore, I can expect the noise ﬁgure for the distributed loss PNS ≈ − in the Kerr medium to be bracketed by the previous two extreme cases, which is to be between 10 log(L) and 0 dB. − While, this problem has been considered before by Imajuku et al. [8], their derivation seems to have errors. In this chapter I show a proper method for solving the problem.

In order to solve this problem, I derive the differential equation involving the quantum operators in the lossy Kerr Medium. I then use ﬁrst order perturbation theory to linearize and solve the differential equation.

59 4.2 Noise Figure of a Lossy Kerr Based NMZI OPA

4.2.1 Differential Equation of Field Operator in Lossy Kerr Medium

For this analysis, I chose a small segment on the Kerr medium. I can choose a model for that segment from a variety of possibilities. For example, it can be a lossless propagation followed by loss, or loss followed by lossless propagation, or half of loss followed by lossless propagation and then followed by half of loss. These different models for the small segments adds to the same Kerr medium. Therefore, they are all expected to yield the same results. I chose the model in which there is lossless propagation followed by loss because I expect the calculations to be simpler. It is important to note that the results in the previous chapter in which it was said that loss in front of a lossless Kerr medium yield different results from loss after the Kerr medium, assumes a high gain. For inﬁnitesimal length, the gain produced by the Kerr medium is inﬁnitesimal. Therefore, those results do not apply.

During the lossless propagation, the ﬁeld goes through self phase modulation which can be described by the following equation [54]

dAˆ (z) = iγAˆ †(z)Aˆ (z)Aˆ (z). (4.1) dz

Since Aˆ †(z)Aˆ (z) is invariant of motion within the lossless Kerr medium after propagation dz, I have

Aˆ ′(z) = exp iγAˆ †(z)Aˆ (z)dz Aˆ (z) (4.2) 1+ iγAˆ †(z)Aˆ (z)dz Aˆ (z). ≈

60 Then the signal goes through loss. The signal ﬁeld operator is expressed as follows:

βdz Aˆ (z + dz)= Aˆ ′(z)e− + dˆf (4.3)

= Aˆ (z)+ iγAˆ †(z)Aˆ (z) β Aˆ (z)dz + dˆf, − where dˆf is Langevinian noise such that

βdz 2 dˆf, dˆf † =1 e− − (4.4) h i 2βdz. ≈ Therefore, I have dAˆ (z) = iγAˆ †(z)Aˆ (z) β Aˆ (z)+ Nˆ dz − (4.5) , where

Nˆ (z), Nˆ †(z′) =2βδ(z z′), (4.6) − h i where δ( ) is a Dirac delta function. ·

4.2.2 Using First Order Perturbation Theory

To solve equation (4.5), I introduce δAˆ (z) such that

δAˆ (z) Aˆ (z) Aˆ (z) . (4.7) ≡ − D E I substitute this in the differential equation and separate ﬂuctuating terms from steady state terms, keep the linear terms and throw away the higher order terms, I get:

2 d ∗ Aˆ (z) + δAˆ (z) = iγ Aˆ (z) + Aˆ (z) δAˆ †(z)+ Aˆ (z) δAˆ (z) dz " D E D E D E D E

β Aˆ (z) + δAˆ (z) + Nˆ . − # D E (4.8)

61 Therefore dδAˆ (z) 2 2 2 = iγ Aˆ (z) δAˆ (z)+ Aˆ (z) δAˆ †(z)+ Aˆ (z) δAˆ (z) dz " D E D E D E (4.9)

βδAˆ (z) + Nˆ . − # For the average value equation, I get:

ˆ d A(z) 2 = iγ Aˆ (z) β Aˆ (z) . (4.10) D dz E − D E D E

4.2.3 Solving the Differential Equation for the Average Value

To solve (4.10), I take an integral on both side of the equation as follows ˆ x d A(z) x 2 = iγ Aˆ (z) β dz. (4.11) D E 0 Aˆ (z) 0 − Z Z D E

D E Since input state is α , Aˆ (0) α. Since this is a propagation of a ﬁeld through a | i ≡ D E lossy medium with loss coefﬁcient β,I have

βz Aˆ (z) = αe− . (4.12) D E I substitute this result in the equation then evaluate the integral to get an expression for

Aˆ (z) (see appendix A.1) D E βz iγ 2 2βz Aˆ (z) = αe− exp α 1 e− . (4.13) 2β | | − D E

4.2.4 Solving the Differential Equation for the Fluctuating Terms

Finding a solution for equation (4.9) is a bit more involved. I start by simplifying the equation with a series of substitution. I begin with the following for short hand notation:

γ 2 2βz φ(z) α 1 e− + θ . (4.14) ≡ 2β | | − in 62 These equation are then substituted in equation (4.9)

ˆ dδA(z) 2 2βz = i2γ α e− β δAˆ (z) dz | | − (4.15) 2 2βz 2iφ(z) + iγ α e− e δAˆ †(z)+ Nˆ . | | I make a series of change of variable starting with δAˆ (z) δBˆeiφ(z), which gives ≡

ˆ dδB(z) 2 2βz 2 2βz iφ(z) = iγ α e− β δBˆ(z)+ iγ α e− δBˆ †(z)+ Nˆ e− . (4.16) dz | | − | | βz Then, substituting δBˆ(z)= δCˆ(z)e− in equation (4.16) gives

ˆ dδC(z) 2 2βz iφ(z) βz = iγ α e− δCˆ(z)+ δCˆ †(z) + Nˆ e− e . (4.17) dz | | Then, I introduce the variables

δCˆ(z)+ δCˆ (z) δXˆ † (4.18) ≡ 2 and

δCˆ(z) δCˆ †(z) δYˆ − . (4.19) ≡ 2i

δXˆ and δYˆ are Hermitian and

δCˆ(z)= δXˆ + iδYˆ (4.20)

Therefore, if I substitute this relation in equation (4.17),I get

d 2 2βz iφ(z) βz δXˆ + iδYˆ = iγ α e− δCˆ(z)+ δCˆ †(z) + Nˆ e− e . (4.21) dz | | The Hermitian conjugate of the equation is

d 2 2βz iφ(z) βz δXˆ iδYˆ = iγ α e− δCˆ(z)+ δCˆ †(z) + Nˆ †e e . (4.22) dz − − | | 63 Summing equations (4.21) and (4.22), I get

βz d e iφ(z) iφ(z) δXˆ = Nˆ †e + Nˆ e− . (4.23) dz 2 Subtracting equations (4.21) and (4.22), I get

βz d 2 2βz e iφ(z) iφ(z) δYˆ =2γ α e− δXˆ + Nˆ e− Nˆ †e . (4.24) dz | | 2i − From equation (4.23), I get

L βz e iφ(z) iφ(z) δXˆ (L)= δXˆ (0) + Nˆ †e + Nˆ e− dz. (4.25) 0 2 Z From equation (4.24), I get

L βz 2 2βz e iφ(z) iφ(z) δYˆ (L)= 2γ α e− δXˆ (z)+ Nˆ e− Nˆ †e dz 0 | | 2i − Z γ 2 2βL = δYˆ (0) + α 1 e− δXˆ (0) β | | − (4.26) L z βx 2 2βz e iφ(x) iφ(x) + 2γ α e− Nˆ †e + Nˆ e− dxdz | | 2 Z0 Z0 L βz e iφ(z) iφ(z) + Nˆ e− Nˆ †e dz, 0 2i − Z where x is a dummy variable and not a physical dimension. Therefore,

2 2 δXˆ (L) = δXˆ (0) D E D LE L 1 β(z+z′) iφ(z) iφ(z) iφ(z′) iφ(z′) + e Nˆ †e + Nˆ e− Nˆ †e + Nˆ e− dzdz′ 4 Z0 Z0 L L 2 1 β(z+z′) = δXˆ (0) + e Nˆ (z)Nˆ †(z′) dzdz′ 4 Z0 Z0 D E 2βL D E 2 e 1 = δXˆ (0) + − 4 De2βL E = , 4 (4.27)

2 2 2 γ 4 2βL 2 2 δYˆ (L) = δYˆ (0) + α 1 e− δXˆ (0) 4β2 | p| − D E D LE L D E 1 β(z+z′) iφ(z) iφ(z) iφ(z′) iφ(z′) + e Nˆ †e Nˆ e− Nˆ e− Nˆ †e dzdz′ 4 − − Z0 Z0 2 L L z z′ γ 4 β(x 2z+y 2z′)+i(φ(x) φ(y)) + α e − − − Nˆ Nˆ † dxdydz′dz 4 | p| Z0 Z0 Z0 Z0 2βL 2 2 D E 2βL e γ 4 2βL 2 1 γ 4 2βL 1 e− = + α 1 e− + α 1 e− L − 4 16β2 | p| − 8 β | p| − − 2β 2βL 2 e 1 γ 4 2βL = + α 1 e− L 4 2 4β | p| − (4.28)

It is conventional to deﬁne

2βL 1 e− L − . (4.29) eﬀ ≡ 2β

Therefore,

2βL 2 2 e γ δYˆ (L) = + α 4L L. (4.30) 4 4 | p| eﬀ D E Substituting back the changes of variables, it is easy to show that

iφ(L) βL δAˆ (L)= δXˆ (L)+ iδYˆ (L) e − . (4.31) 1 (iφ(L) βL) 1 (iφ(L) βL) I deﬁne δXˆ δXˆ e 2 − and δYˆ δYˆ e 2 − . Therefore, B ≡ B ≡

2 1 δXˆ (L) = (4.32) B 4 D E and

2 2 1 γ 4 2βL δYˆ (L) = + α L Le− (4.33) B 4 4 | p| eﬀ D E 65 Using equation (1.107), I get the output ASE of the parametric ampliﬁer, which is

ˆ 2 ˆ 2 1 ASE = δXB(L) + δYB(L) − 2 (4.34) D2 E D E γ 4 2βL = α L Le− 4 | p| eﬀ The large gain expression of the parametric ampliﬁer is derived in appendix A.3 and is

2 2βL 4 2 G = γ e− α L . (4.35) | p| eﬀ

The observation of equation (A.24) shows that the ﬁeld gain phase is ieiφ(L) relative to that of the input ﬁeld, while that of the noise is eiφ(L) (see equation (4.31)). Therefore, at the

π output, the noise ﬁeld is 2 out of phase relative to our phase frame of reference. Therefore, ˆ the in phase component of noise with the output signal is δYB and the quadrature phase

ˆ ˆ 2 ˆ 2 component is δXB. δYB(L) is much greater than δXB(L) . Therefore, I can use D E D E equation (3.54) to get the noise ﬁgure since it requires that the in phase component of the noise be much larger than its quadrature phase. I calculate the noise ﬁgure and get

2βL 1 e− NF = NF = 10 log − . (4.36) QFS PNS − 2βL See figure 4.1. This expression was obtained with the approximate expression (3.54) valid only for high gain and large signal. In Appendix B, I derive the photon number squared noise figure of a lossy Kerr Medium based NMZI assuming only that the parametric gain is high. In other words, the input signal can be weak, the expression of the noise figure is as follows

2 αs “ NFPNS = 10 log . (4.37) −  L 2 1 L2 L  L αs + 8 L2 L +1 eff eff − eff   The noise figure shows that Haus [41] is right since it is dependent on the input signal, which is a problem for this choice of definition of the noise figure.

66 If I assume that the input signal is large, I can ignore in the noise ﬁgure expression any term not containing the input signal power. I get

2 αs NFPNS 10 log . (4.38) ≈ − L α2 " Leﬀ s #

Therefore, 2βL 1 e− NF = 10 log − , (4.39) PNS − 2βL which is in aggreement with equation (4.36).

This result is different from the result given by Imajuku et al. [8], which is

2 1 NF = 10 log 1+ e2βL 1 2βL (2βL)2 (4.40) PNS − 2 − − − 2 (2βL)

4.3 Detailed Comparison with Imajuku’s Calculations

In this section, I want to show the steps in Imajuku et al.’s calculations that led to the differences in our results. Imajuku et al. [8] write that the output ﬁeld operator of the

NMZI OPA is

βL 2βL Aˆ (L)= Aˆ (L)e− + 1 e Γˆ (L), (4.41) 0 − − s p where

2 2 2 i(ψ+θ′) γ αout L iθ′ γ αout L iθ′ Aˆ (L)= e 1+ | | Aˆ (0)e− + | | Aˆ †(0)e , (4.42) 0  2 2  s   2βL 1 e− α 2 = α 2 − , (4.43) | out| | | 2βL

α 2 is the averaged photon number of the input pump, | |

π 2 φ0 θ′ = φ + − , (4.44) b 2 67 Figure 4.1: Plots of NF vs loss for a Lossy Kerr medium based NMZI-OPA

γ α 2L φ = arctan | out| , (4.45) 0 2 γ α 2L ψ = φ + | out| , (4.46) 0 2

φb is the pump ﬁeld phase at the output of the NMZI,

2βL L ′ i(ψ+θ ) 2βe− iφ0 βz Γˆ − ˆc d s(L)= e 2βL e e 0(z) z r1 e− 0 − Z (4.47) 2 L z γ αout iφ0 βz βz + | | ie− e ˆc0(z′)+ e ˆc0†(z′) dz′dz , 2 0 0 ! Z Z h i 2 iθ′ i γ|αout| z ˆc0(z)= e− − 2 ˆc(z), (4.48)

ˆc (z)+ ˆc (z) ˆc(z)= m,1 m,2 , (4.49) √2

68 ˆcm,1(z) and ˆcm,2(z) are the vacuum operator in each nonlinear medium of the NMZI.

They obey the following commutation relation

ˆc (z)+ ˆc† (z′) = δ(z z′), (4.50) m,i m,i − h i where i =1, 2. At this point, this result is similar to the one I obtained in equation (A.8).

Without any explanation, they use the following mathematical identity (possibly it is an approximation) z ˆc (z′)dz′ =(L z)ˆc (z) (4.51) 0 − 0 Z0 Consequently, in equation (53) of their publication, they write

2βL L ′ i(ψ+θ ) 2βe− iφ0 βz Γˆ − ˆc d s(L)= e 2βL e e 0(z) z r1 e− 0 − Z (4.52) 2 L γ αout iφ0 βz βz + | | (L z) ie− e ˆc0(z)+ e ˆc0†(z) dz . 2 0 − ! Z h i From then on, our results diverge. See ﬁgure 4.2 and 4.3

4.4 Quantum Mechanical Noise of a NMZI with Gain in the Kerr Medium

It has been shown that the loss in the nonlinear medium of NMZI OPA deteriorates its noise ﬁgure. This gives me an idea on what to expect if the nonlinear medium is a saturable absorber. However, it does not give much insight of what to expect in the case where the nonlinear medium is an SOA. For this, I am going to consider a NMZI OPA with gain in the Kerr medium instead of loss.

With a process similar to the previous section (see appendix C), I get the noise

ﬁgure of a NMZI with gain in the Kerr medium instead of loss. If I choose

e2g0L 1 Leﬀ − , (4.53) ≡ 2g0

69 Figure 4.2: Plots of NFNPS vs Length for a Lossy Kerr medium based NMZI- OPA as a comparison to Imajuku’s et al. ﬁgure 6 [8]

then I have

2 2g L 4 2 1 Lg0 Eˆ † Eˆ 0 (4.54) δ outδ out γ e αpLeﬀ 2g L . ≈ "2 − 2(e 0 1)# D E − Under those condition, the expression of the parametric gain is

G γ2eg0Lα4L2 . (4.55) ≈ p eﬀ

I use the noise ﬁgure formula I derived earlier

G NF 10 log NPS ≈ − 4ASE (4.56) L = 10log 2 . − L eﬀ 70 Figure 4.3: Plots of NFNPS vs Loss for a Lossy Kerr medium based NMZI- OPA as a comparison to Imajuku’s et al.

It can be seen that the noise ﬁgure goes very quickly to 3 dB( See ﬁgure 4.4).

71 Figure 4.4: Plots of NFNPS vs Gain for Kerr Medium with gain based NMZI-OPA

72 Chapter 5

Discussions of the Results from the Lossy Kerr Based NMZI OPA

In the previous chapters, I have looked the affects of the loss in Kerr based NMZI

OPA on its noise figure. In order to calculate its noise, I calculated quantum mechanically the fluctuations of the field after the nonlinear medium. From that result, I calculated the output ASE of the OPA. From those results interesting things can be noticed.

First, in classical physics, when a noisy signal goes through a lossy medium, both the signal and the noise decay the same way. Therefore, the SNR does not degrade. We have seen that in section 4 that in quantum mechanics this is not true. It was seen that the in phase component of the noise remaining constant (see equation (4.32)) while the quadrature phase component grew (see equation (4.33)). Quantum mechanics guaranteed that the noise did not go below the minimum threshold imposed by the Heisenberg uncertainty principle. Since the signal decays, the SNR degrades. Furthermore, it was seen that the inphase component of the noise, while remaining constant, made the quadrature phase component grow (see equation (4.26)). This difference between classical physics and quantum mechanics justiﬁes the use of quantum mechanics in that section. Moreover,

The ASE at the output is the ampliﬁed vacuum ﬂuctuations. Let us get a typical value of the ASE for a gain of 30 dB. From equation (4.34) and (4.35), I have

G 2βL ASE = . (5.1) 4 1 e 2βL − −

It can be seen in the absence of gain, there is no ASE. Let us choose 2βL = ln(2) (this

73 choice will be justiﬁed later in this chapter). Then the expression of the ASE becomes

G ASE = ln(2). (5.2) 2

Therefore, for a 30 dB gain, the ASE is about 347 photons. Note that in the absence of loss, it is 250 photons. It is also important to note that classical OPA has classical noise, for example from pump ﬂuctuation (RIN and phase noise) and thermal noise. However, in the absence of photon quantization there is no quantum noise due to the uncertainty of n h i and no vacuum input giving rise to ASE. In actuality, for both the classical and quantum

OPA, the pump noise is the dominant noise. Loss in classical OPA does however lead to noise. Thermal heat causes the index of refraction to have ﬂuctuations, introducing phase noise, and the thermal Rayleigh scattering causes RIN. These are thoroughly classical noises.

Second, in section 1.6.2, I calculated an expression for the ASE of a NMZI OPA using quantum mechanics. From equation (1.107), the expression of the ASE is

2 2 1 ASE = δYˆ + δXˆ . (5.3) 1 1 − 2 hD E D Ei This expression tells us that if the output from the nonlinear medium are coherent beams,

ˆ 2 ˆ 2 1 for which δY1 = δX1 = 4 there will be no ASE. We have also seen that the out- D E D E put of a NMZI OPA amplifying a coherent state always contains ASE. Therefore, in the absence of ASE, there is no gain. ASE as derived here is in number of photons. Typi- cally for high gain parametric ampliﬁcation, ASE is in the order of 10 to a 100 photons.

Therefore, the 1 in the expression of the ASE is not important and can be neglected. − 2 The expression becomes the same as the one given by classical physics. Therefore, in that scenario quantum mechanics is not needed.

74 Table 5.1: Different ﬁber that could be used in a Kerr based NMZI based OPA and their optimum length and their maximum gain.

Fiber Type γ (/km/W) β (dB) Opt. Length (km) Max. Gain (dB)

SF 2.2 0.1 15 20.2 − HNL Fiber 15.8 0.35 4.3 31.9 − 3 3 BOBF 460 1.9 10 0.8 10− 23.8 − · · 3 3 LSF 640 1.3 10 1.2 10− 28.4 − · ·

Now, for practical devices Kerr based NMZI OPA, let us calculate the best performance we can expect if different types of lossy ﬁber were used as Kerr medium. In table

5.1, I show the loss coefﬁcient and the nonlinear coefﬁcient of Standard Fiber (SF)[8],

Highly Nonlinear Fiber (HNLF)[8], Bismuth-Oxide Based Fiber (BOBF) [39], Lead Sil- icate Fiber (LSF) [40]. Let us calculate their optimum gain. The expression of the parametric gain from (4.35) is

2βL 2 4 2βL 1 e− G = γ α e− − . (5.4) | p| 2β

The parametric gain is length dependent (See ﬁgure 5.1) . Its maximum as a function of

L is reached when ln(2) L = . (5.5) opt 2β

The above expression justiﬁes the earlier choice of 2βL = ln(2). The maximum gain is

γ2 G = α 4 (5.6) max 8β | p|

75 Figure 5.1: Plot of gain versus length for a HNLF (γ = 15.8)/km/W, β = 0.7dB/km ) −

At maximum gain, the noise ﬁgure is

1 NF = NF = 10 log 1.4186. (5.7) QFS PNS − 2 ln(2) ≈

To illustrate this, I compare the maximum gain for OPAs based on the ﬁbers in table 5.1 for an input pump of 2 Watt. It shows that HNLF ﬁber would yield the highest gain.

In the next chapter, I consider a different nonlinear medium.

76 Chapter 6

Semiclassical Treatment of Optical Parametric Ampliﬁer Based on

Saturable Absorber and Ampliﬁer

In this chapter, I show that a Saturable Absorber (SA) or Semiconductor Optical

Amplifier (SOA) can be used as a non linear medium in the NMZI OPA [65]. The steady state parametric amplifier gain is calculated from a simple classical model. I assume that variation of gain or loss is usually accompanied by large phase modulation as generally described by the Kramers-Kronig relation, which is typical of semiconductor media. The calculations are done for saturable absorber. The results are identical for an amplifier except for the sign of the absorption coefficient.

6.1 Saturable absorber/Ampliﬁer Overview

6.1.1 Loss Characteristics

For a cross section σ , the absorption or gain of a travelling wave is [51]

dP (z) = σN(z)P (z), (6.1) dz − where P (z) is the ﬁeld amplitude squared, z is the axis of propagation and N(z) is the number of atoms per unit length. For a SA/SOA, the absorption/ stimulated emission of a photon changes N. Assuming one atom is removed for each photon and also assuming that atoms relaxes and return with some time constant τ, the equation of the number of

77 atoms is dN(z) N(z)P (z) N N(z) = σ + 0 − , (6.2) dz − ~ω τ where N0 is the total number of atoms per unit length. In steady state

N N(z)= 0 , (6.3) 1+ P (z) Psat

~ω where Psat = στ . Psat is the saturation power. Substituting the equation above in (6.1), I get 1 dI(z) σN = 0 , (6.4) P (z) dz −1+ P (z) Psat The absorption/gain coefﬁcient is β(z)= σN(z). Multiplying equation (6.2) by σ I get

dβ(z) β(z)P (z) β β(z) = + 0 − , (6.5) dz τPsat τ where β0 = σN0. β0 is the loss/gain per unit length in the material. Also,[49]

1 dP (z) β = − 0 . (6.6) P (z) dz 1+ P (z) Psat

The same equation may represent the behavior of a SOA if the sign of β is inverted. β0 is positive if the device is a SA and is negative if it is an SOA. The solution of this equation in implicit algebraic form for a length L of the device is

P (L) P (L) P (0) ln + − = β L. (6.7) P (0) P − 0 sat The solution to this equation is

P (0) P (0) β0L P (L)= P W e Psat − , (6.8) sat · P sat where W (y) is the Lambert W function deﬁned as the inverse of the function.

y = W eW . (6.9)

78 The loss ratio is then [65] P (L) I Γ= = out (6.10) P (0) Iin

I deﬁne

β = ln(Γ). (6.11) −

Therefore,

P (0) β(z) Psat P (0) β0z e = W e Psat − , (6.12) P (0) · P sat

6.1.2 Phase Variation

The SA also shifts the phase of the electric ﬁeld as it propagates through it. This change in phase depends on the Henry-alpha factor, αH , of the medium which is the ratio between the real and imaginary parts of a complex loss. Typical values for the Henry- alpha factor ranges between three and ﬁve. For an input E(0) in an SA, the output electric

ﬁeld after the SA is [65]

β (iαH +1) E(L)= E(0)e− 2 . (6.13)

Therefore, β(z) θ = α + θ , (6.14) out − 2 H in where θin is the phase at the input to the medium.

6.2 Parametric Gain of SA based NMZI-OPA

Here I consider an NMZI as shown in ﬁgure 1.4 in which the nonlinear medium is a saturable absorber. At the input of one arm of the NMZI, I have the electric ﬁeld of the

79 pump Ep, which I will chose to be real, and on the other arm, I have the CW input signal

iθin Ese , where θin is the relative phase between Es and Ep. As previously, after the ﬁrst beam coupler, I have

1 iθin Eout11 = Ep + iEse , (6.15) √2 at the output of the ﬁrst arm and

1 iθin Eout11 = Ese + iEp (6.16) √2 in the other arm. I denote by Pa(z) and Pb(z) respectively the power in the ﬁrst SA and second SA. From equations (6.15), I obtain

1 P (0) = E2 + E2 2E E sin(θ ) (6.17) 1 2 s p − p s in and from equation (6.16)

1 P (0) = E2 + E2 +2E E sin(θ ) . (6.18) 2 2 s p p s in Assuming that Es is weak, then to the ﬁrst order I have

1 P (0) E2 2E E sin(θ ) (6.19) 1 ≈ 2 p − p s in and

1 P (0) E2 +2E E sin(θ ) . (6.20) 2 ≈ 2 p p s in I will deﬁne the total relative loss of each SA using (6.8)

P (L) Γ 1 1 ≡ P (0) 1 (6.21) P1(0) Psat P1(0) β0L = W e Psat − P (0) P 1 sat 80 and P (L) Γ 2 2 ≡ P (0) 2 (6.22) P2(0) Psat P2(0) β0L = W e Psat − . P (0) P 2 sat I write the exponential loss β ln(Γ ) and β ln(Γ ). Since at the input of each 1 ≡ − 1 2 ≡ − 2 SA I had

1 iθin Eout11(0) = Ep + ie Es (6.23) √2

1 iθin Eout12(0) = e Es + iEp . (6.24) √2 At the output of each SA I have

1 (iαH +1)β1L e− 2 Eout11(L)= (Ep + iEs) (6.25) √2 and

1 (iαH +1)β2L e− 2 Eout12(L)= (Es + iEp) . (6.26) √2 The output ﬁelds of the NMZI are

1 1 1 (iαH +1)β1L (iαH +1)β2L Eout1 = e− 2 e− 2 Ep 2 − (6.27) h 1 1 iθin (iαH +1)β1L (iαH +1)β2L + ie e− 2 + e− 2 Es and i

1 1 1 (iαH +1)β1L (iαH +1)β2L Eout2 = i e− 2 + e− 2 Ep 2 (6.28) h 1 1 iθ (iαH +1)β2L (iαH +1)β1L + e in e− 2 e− 2 E . − s I am only interested in output ﬁeld one since it is the signal output, whichi can be rewritten as 1 (iαH +1)(β1+β2)L 4 1 1 e− (iαH +1)(β1 β2)L (iαH +1)(β1 β2)L Eout1 = e− 4 − e− 4 − Ep 2 − (6.29) 1 h 1 iθin (iαH +1)(β1 β2)L (iαH +1)(β1 β2)L + ie e− 4 − + e− 4 − Es . i 81 Next, I simplify the above expression by deﬁning the differential absorption β β 1 − 2 and calculating its approximate expression.

6.2.1 Approximate Expression for the Differential Absorption

I substitute (6.19) and (6.20) in (6.21) and (6.22) and I obtain

P Γ = sat 1 E2 2E E sin(θ ) p − p s in 2 (6.30) 2 (Ep−2EpEs sin(θin) Ep 2EpEs sin(θin β0 W − e Psat − × Psat ! and

Psat Γ2 = 2 Ep +2EpEs sin(θin) 2 (6.31) 2 (Ep+2EpEs sin(θin) Ep +2EpEs sin(θin β0 W e Psat − . × Psat ! For any differentiable function f(x), to the ﬁrst order, I have

f(x + ∆) f(x ∆) 2f ′(x)∆. (6.32) − − ≈

I choose ∆=2EpEs sin(θin). Therefore (see appendix D.1),

2 Γ Γ = Γ(∆) Γ( ∆) 2∆Γ′(E ), (6.33) 2 − 1 − − ≈ p where

x Psat x β0 Γ(x)= W e Psat − . (6.34) x · P sat It can be shown (see appendix D.2)

Γ(Γ 1) 2 (6.35) Γ′(Ep )= − 2 . − Psat +ΓEp

82 Therefore,

(Γ 1) β2 β1 =2EpEs − sin(θin), (6.36) − ≈ Psat +Γp where p 1 E2 (see appendix D.3). ≡ 2 p

6.2.2 Parametric Gain Calculation

Since ∆ is small, to the ﬁrst order β + β 2β where β ln(Γ(E2)). Therefore, 2 1 ≈ ≡ p equation (6.29) can be written

β (iαH +1) 2 1 1 e− (iαH +1)(β1 β2) (iαH +1)(β2 β1) Eout1 e− 4 − e− 4 − Ep ≈ 2 − (6.37) h 1 1 iθin (iαH +1)(β1 β2) (iαH +1)(β2 β1) + ie e− 4 − + e− 4 − Es . i Substituting for the approximation of β β from equation (6.36) and introducing p 1 − 2 ≡ 1 2 2 Ep , I get

β (iαH +1) iθ (1 Γ) sin(θin) E e− 2 ie in (iα + 1) − p E . (6.38) out1 ≈ − H Γp + P s sat I deﬁne

iθin Ein = Ese . (6.39)

Substituting for Ein, I get

β (iαH +1) (iαH + 1) (1 Γ) E = e− 2 iE − p (E E∗ ) . (6.40) out1 in − 2i E2Γ+ P in − in p sat I deﬁne µ and ν such that

Eout1 = µEin + νEin∗ , (6.41) then I have

β (iαH +1) i (1 Γ) µ e− 2 i + (iα + 1) − p (6.42) ≡ 2 H Γp + P sat 83 and

β (iαH +1) i (1 Γ) ν e− 2 (iα + 1) − p . (6.43) ≡ 2 H Γp + P sat Therefore, the parametric gain is

2 iθin iθin Gpar = µe + νe− . (6.44)

The maximum parametric gain that can be attained by adjusting the relative phase between the pump and the signal is

G = µ + ν 2 . (6.45) par | |

The minimum parametric gain that can be attained by adjusting the relative phase between the pump and the signal is

G = µ ν 2 . (6.46) par | − |

See ﬁgures 6.1 and 6.2

It can be seen that a phase sensitive ampliﬁer is obtained with a possibility of more than 10 dB gain difference between the minimum and the maximum depending on the relative phase.

6.3 Bandwidth of the SA Based NMZI

So far, I have considered a CW signal. In this section, I will look for the bandwidth and noise of the SA based NMZI. As before, I assume that the arms are normally balanced so that that the low power bandwidth is very large.

84 Figure 6.1: Minimum and maximum parametric gain vs p/Psat for a SA based NMZI OPA with αH = 25, β0L =4.

6.3.1 Response of a SA to a Modulated Signal

To ﬁnd the response of a SA to a modulated signal, I will assume a weak modulated

iθin signal at the input and a strong CW pump. I will write for our signal Es = As(t)e where As(t) a real signal and and Ep = √2Ap for the pump. From equation (6.5), the expression of the absoption coefﬁcient is

∂β(z) β β(z) β(z) E(z) 2 = 0 − | | (6.47) ∂t τ − Psatτ with

E(z) 2 A2(z) 2A (z)A (z, t) sin(θ ) (6.48) | | ≈ p − p s in

85 Figure 6.2: Minimum and maximum parametric gain vs. p/Psat for a SOA based NMZI OPA with α =5, β L = 4. H 0 −

and τ is the carrier lifetime. To simplify the expressions, I will set Psat = 1, which is equivalent to saying that powers are normalized to Psat. I will reintroduce Psat at the end by dividing all the powers by Psat. Using a perturbation technique , I write

β(z) = βs(z)+ δβ(z), where βs(z) is the steady state loss and δβ(z) is the perturbation in loss. 2A (z)A (z, t) sin(θ ) is the driving term of the perturbation. I substitute these − p s in relations in equation (6.47) and separate the constant terms from the pertubation terms and obtain the two equations:

∂δβ(z) τ =2β (z)A (z)A (z, t) sin(θ ) δβ(z)(1 + A2(z)) (6.49) ∂t s p s in − p

β β βA2(z) 0= 0 − p . (6.50) τ − τ

86 Equation (6.50) gives me the steady state solution, which is

β β (z)= 0 , (6.51) s 1+ P (z)

2 where P (z)= Ap(z). Taking the Fourier transform of equation (6.49), I get

iΩτδβ˜ (z)=2β (z)A (z)A˜ (z, Ω) sin(θ ) δβ˜ (z)(1 + A2(z)), (6.52) Ω s p s in − Ω p

˜ where Ω is the signal envelope frequency, δβΩ(z) is the Fourier transform of δβ(z) and

A˜s(z, Ω) is the Fourier transform of As(z, t). I shall here after represent the Fourier transform of any variable by a over the variable symbol. Solving for δβ˜Ω, I get

β (z)∆P (z, Ω) δβ˜ (z)= − s s (6.53) Ω 1+ P (z)+ iΩτ

The equation of the power inside each SA is governed by [52]

dP (z) t = β(z)P (z). (6.54) dz − t

Taking the Fourier transform of the equation, I get

dP˜ (z, Ω) t = β˜ (z, Ω)P˜ (z, Ω). (6.55) dz − Ω t

Using a perturbation technique , I will write P˜t(z, Ω) = P (z) + ∆P˜s(z, Ω) where P (z) is the constant portion and ∆P˜s(z, Ω) is the perturbation of the power. Obviously,

P (z) A2(z) (6.56) ≡ p and

∆P˜ (z, Ω) 2A (z)A˜ (z, Ω) sin(θ ). (6.57) s ≡ − p s in

I substitute this perturbation equation in equation (6.55) and obtain

dP (z) d∆P˜ (z, Ω) + s = β (z)P (z) δg˜ (z)P (z) β (z)∆P˜ (z, Ω). (6.58) dz dz − s − Ω − s s 87 I separate this equation into a steady state equation and a perturbation equation. For the steady state equation, I have

dP (z) = β (z)P (z), (6.59) dz − s

where βs(z) is given by equation (6.51). Therefore,

L 1 L +1 dP (z)= β dz. (6.60) P (z) − 0 Z0 Z0

It has been already seen that the solution to this equation is

P (0) β0z P (z)= P (0)W P (0)e − . (6.61) For the perturbation equation, I have

d∆P˜ (z, Ω) s = δβ˜ (z)P (z) β (z)∆P˜ (z, Ω). (6.62) dz − Ω − s s

P (L) If I deﬁne the total steady state gain as Γs = P (0) , then I can show that the solution of this equation is (see appendix E.1)

1+ P (0) + iΩτ ∆P (L, Ω) = ∆P (0, Ω)Γ . (6.63) s s s 1+ P (L)+ iΩτ

I will deﬁne L ∆β(Ω) = δβ˜Ω(z)dz. (6.64) Z0 Therefore, I can prove that (see appendix E.1)

(1 Γs) ∆β(Ω) = − − ∆Ps(0, Ω). (6.65) 1+ΓsP (0) + iΩτ

This results is consistent with the previous results since for Ω=0, I obtain equation

(6.36).

88 6.3.2 Overall System Output

In this section, I will follow a procedure identical to the one used in section 6.2.2

(see appendix E.2). I deﬁne

E˜ (Ω) A˜ (Ω)eiθin . (6.66) in ≡ s

The output ﬁeld is then

˜ ˜ ˜ Eout = µEin(Ω) + νEin∗ (Ω), (6.67) where

βs (iαH +1) 1 (Γ 1) µ ie− 2 1 (iα + 1) − p (6.68) ≡ − 2 H Γ p + P + iΩτP s sat sat and

βs (iαH +1) 1 (Γ 1) ν ie− 2 (iα + 1) − p , (6.69) ≡ 2 H Γp + P + iΩτP sat sat 2 where P is the power of the input pump on each arm of the NMZI: p = Ap. Reintroducing

Psat, I see that for Ω=0, equation (6.68) and (6.69) are equivalent to equation (6.42) and

(6.43). I can now ﬁnd the parametric gain, which is

E˜ 2 G(Ω) = | out| A˜ (Ω) 2 | s | (6.70) iθ iθ 2 = µ(Ω)e in + ν(Ω) e− in | | | See ﬁgure 6.3.

To calculate the noise ﬁgure, I need to use quantum mechanics to calculate the noise generated by this device.

89 Figure 6.3: Parametric gain in dB vs Ω in 1/τ for a SA based NMZI OPA with p/Psat = 25, αH =5and β0L =4.

90 Chapter 7

Quantum Mechanical Model for Interaction of Light with a Saturable

Absorber

7.1 Overview

I considered two possible candidates for the new type of phase sensitive ampliﬁer, based on either SOA or on SA. Based on results from Shtaif et al. [66], I concluded that the SOA produces too much ASE noise to be useful. Therefore, I decided to investigate the noise properties of non linear media based saturable absorbers. The question which

I am trying to answer in the next several chapters is what is the minimum noise in a SA based NMZI OPA. In this chapter, I will follow closely the formalism of Professor Rana

[61] to derive the equations for time evolution of operators describing the ﬁeld and the medium.

7.2 Quantum Mechanical Model

Figure 7.1 shows schematically a saturable absorber interacting with a single quan- tized mode of electromagnetic field. I will use Jaynes-Cummings formalism to describe the electromagnetic field interacting with atoms [46, 62]. I will start with the Hamiltonian, which gives the energy of the atom, the energy of the electric field and their coupling.

ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1 H = E1N1 + E2N2 + ~ω0A†A + κAˆσ+ + κ∗A†σˆ + , (7.1) − 2

91 Figure 7.1: Single Mode Quantum Mechanical Model

where κ is a complex constant that I will discuss later,E1 and E2 are the energy levels of

ˆ ˆ the atom σˆ , σˆ+, N1 and N2 are atomic state operators. For two atomic energy levels e1 − | i and e2 , the operator σˆ+ raises the atomic state from level 1 to level 2, while σˆ lowers | i − it. Therefore,

σˆ = e e , (7.2) + | 2ih 1|

σˆ = e1 e2 , (7.3) − | ih | thus

σˆ = σˆ+† . (7.4) −

Further,

Nˆ = e e (7.5) 1 | 1ih 1|

92 and

Nˆ = e e . (7.6) 2 | 2ih 2|

From this, it follows that

ˆ N1 = σˆ σˆ+ (7.7) − and

ˆ N2 = σˆ+σˆ . (7.8) −

The Hamiltonian for the electric ﬁeld interacting with N atoms becomes

N ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1 H = E1N1 k + E2N2 k + κAˆσ+ k + κ∗A†σˆ k + ~ω0A†A + , (7.9) − 2 k X=1 where Nˆ and Nˆ , k 1, 2, .., N are the raising operator and lowering operator of 1 k 2 k ∈ { } the kth atom. I deﬁne N Nˆ Nˆ , (7.10) 1 ≡ 1 k k X=1 N Nˆ Nˆ , (7.11) 2 ≡ 2 k k X=1 N

σˆ σˆ k (7.12) − ≡ − k X=1 and N σˆ σˆ . (7.13) + ≡ + k k X=1 The Heisenberg equation for a general operator Oˆ (t) is given by

dOˆ (t) 1 = Oˆ (t), Hˆ . (7.14) dt i~ h i Applying this equation to each of the operators, and using the relations between the operators, I get ˆ dN2(t) 1 ˆ ˆ = κσˆ+(t)A(t) κ∗σˆ (t)A†(t) , (7.15) dt i~ − − h i 93 dNˆ (t) dNˆ (t) 1 = 2 , (7.16) dt − dt ˆ dN1(t) 1 ˆ ˆ = κσˆ+(t)A(t) κ∗σˆ (t)A†(t) , (7.17) dt −i~ − − ˆ h i dA(t) ˆ iκ∗ = iω0A(t) σˆ , (7.18) dt − ~ −

dσˆ+(t) iκ∗ = iω σˆ (t) Aˆ † Nˆ (t) Nˆ (t) . (7.19) dt 21 + − ~ 2 − 1 h i where E E ω 2 − 1 . (7.20) 21 ≡ ~

Any population in level 2 causes spontaneous emission, which is very detrimental.

Because, I am trying to consider only saturable absorbers introducing minimum noise into the parametric ampliﬁer, I will consider only a system in which the upper level is quickly depopulated by fast transitions to other levels lumped as level 3. As illustrated in ﬁgure

7.2 this level then slowly relaxes back to ground state. This is typical of what occurs in many semiconductor absorbers.

Therefore, I am going to add fast relaxation and dephasing to equation (7.19) [50]

dσˆ+(t) iκ∗ =(iω γ) σˆ (t)+ Aˆ †Nˆ (t)+ Fˆ (t), (7.21) dt 21 − + ~ 1 +

ˆ where γ is the relaxation and dephasing rate and F+(t) is the noise associated with it. I also add an empirical term for repopulation of level 1 to equation (7.17), which results in

ˆ ˆ dN1(t) N N1(t) i ˆ ˆ ˆ = − + κσˆ+(t)A(t) κ∗σˆ (t)A†(t) + FN (t), (7.22) dt τ ~ − − h i ˆ where τ is the rate of relaxation and FN (t) is the noise due to relaxation. I will see later

ˆ ˆ in this chapter why F+(t) and FN (t) had to be added and I will evaluate their properties.

Now I have a complete system of differential equations. To solve them, I will ﬁrst ﬁnd an

ˆ expression for σˆ+, which I will later substitute in the differential equation in A(t).

94 Figure 7.2: 3 Level System

From equation (7.21), it can be shown that (see appendix F.1),

t iκ∗ ′ ˆ ˆ (iω21 γ)t (iω21 γ)t Aˆ Nˆ (iω21 γ)t σ+(t)= σ+(0)e − + ~ e − †(t′) 1(t′)e− − dt′ Z0 (7.23) ~ iω t + Fˆ† (t)e 0 , iκ ab where

t ′ iκ iω0t (iω21 γ)(t t ) Fˆ† e− Fˆ (t′)e− − − dt′. (7.24) ab ≡ ~ + Z0 For the semiconductor I am considering, γ is large. As a result, the ﬁrst term of the above equation can be neglected and the initial conditions are quickly forgotten by the system.

95 Therefore,

t ′ ~ iκ∗ (iω21 γ)t (iω21 γ)t iω0t σˆ (t)= e − Aˆ †(t′)Nˆ (t′)e− − dt′ + Fˆ† (t)e . (7.25) + ~ 1 iκ ab Z0

iω0t (iω21 γ)t Let Aˆ (t) Bˆ(t)e− . Nˆ (t) and Bˆ(t) are varying much slower than the term e − . ≡ 1 Therefore, they can be moved in front of the integral

t ′ ~ iκ∗ (iω21 γ)t (i(ω21 ω0) γ)t iω0t σˆ (t)= e − Bˆ †(t)Nˆ (t) e− − − dt′ + Fˆ† (t)e . (7.26) + ~ 1 iκ ab Z0 After evaluating the integral, I have

Bˆ Nˆ ~ κ∗ †(t) 1(t) iω0t (iω21 γ)t iω0t σˆ (t)= e e − + Fˆ† (t)e . (7.27) + i~ (i(ω ω ) γ) − iκ ab 21 − 0 − (iω21 γ)t When γ is very large, I can again neglect e − . I then obtain the expression for σˆ+, which is

ˆ ˆ ~ κ∗ A†(t)N1(t) iω t σˆ (t)= + Fˆ† (t)e 0 . (7.28) + ~ ((ω ω ) iγ) iκ ab 0 − 21 − ˆ Now, I will substitute this expression of σˆ+ into the differential equation of A(t).

Equation (7.18) is a differential equation for A(t) in terms of σˆ − ˆ dA(t) ˆ iκ∗ = iω0A(t) σˆ (7.29) dt − ~ −

I can use the expression for σˆ+ from equation (7.28) to get an expression of σˆ , since −

σˆ+ = σˆ† , which can be substituted in the above equation. I get − dAˆ (t) κ 2 γ (ω ω ) = | | Aˆ (t)Nˆ (t) i 0 − 21 dt ~2 1 −((ω ω )2 + γ2) − ((ω ω )2 + γ2) 0 − 21 0 − 21 (7.30) ˆ iω0t ˆ + Fab(t)e− + iω0A(t),

I deﬁne ℧ such that

κ 2 γ (ω ω ) ℧ | | i 0 − 21 (7.31) ≡ ~2 −((ω ω )2 + γ2) − ((ω ω )2 + γ2) 0 − 21 0 − 21 96 The first term corresponds to absorption of the field with coefficient

κ 2 γ ℧ | | , (7.32) r ≡ − ~2 ((ω ω )2 + γ2) 0 − 21 and the second corresponds to index of refraction change

κ 2 (ω ω ) ℧ | | 0 − 21 . (7.33) i ≡ ~2 ((ω ω )2 + γ2) 0 − 21 ˆ Now, I will calculate the properties of Fab(t), required when I solve the differential equation in Aˆ (t).

In equation (7.21), I added an empirical term for the noise associated with relaxation

ˆ and dephasing. This noise lead to the expression for Fab(t) in equation (7.24). The noise

ˆ ˆ ˆ term is necessary so that so that A(t), A†(t) = 1 for all values of t. Without Fab(t), I h i would have

dAˆ (t) = ℧Aˆ (t)Nˆ (t)+ iω Aˆ (t). (7.34) dt − 1 0

I deﬁne Cˆ ℧Nˆ (t)+ iω ≡ 1 0

dAˆ (t) = CˆAˆ (t). (7.35) dt −

I then have

Cˆt Aˆ (t)= Aˆ (0)e− . (7.36)

Therefore

(Cˆ +Cˆ †)t Aˆ (t), Aˆ †(t) = Aˆ (0), Aˆ †(0) e− (7.37) h i h i (Cˆ +Cˆ †)t = e− ,

97 which is obviously wrong as the answer should be one (1) . To ﬁx this, I must add the following noise term

dAˆ (t) iω0t = CˆAˆ (t)+ Fˆ (t)e− (7.38) dt − ab

I can assume that Fˆ (t), Fˆ† (t′) = Cˆ δ(t t′), since the noise ﬂuctuates much faster ab ab 0 − h i that the other variables(on a time scale 1/γ) [63]. The solution of this equation is

t Cˆ Cˆ ′ ′ ˆ ˆ t (t t ) iω0t ˆ A(t)= A(0)e− + e− − − Fab(t′)dt′. (7.39) Z0 Computing the commutator of Aˆ (t), I can show that (see appendix F.1.1)

ˆ ˆ ˆ C0 = C + C† (7.40) ˆ =2℧rN1(t).

Therefore,

Fˆ (t), Fˆ† (t′) =2℧ Nˆ (t)δ(t t′). (7.41) ab ab r 1 − h i In general, starting with equation (7.21), I would obtain

Fˆ (t), Fˆ† (t′) = 2℧ Nˆ (t) Nˆ (t) δ(t t′), (7.42) ab ab − r 2 − 1 − h i and

Fˆ† (t)Fˆ (t′) =2℧ Nˆ (t) δ(t t′) (7.43) ab ab r 2 − D E D E and [64]

Fˆ (t)Fˆ† (t′) =2℧ Nˆ (t) δ(t t′). (7.44) ab ab r 1 − D E D E 98 ˆ Because I am neglecting N2(t), I can set

ˆ ˆ (7.45) Fab† (t)Fab(t′) =0. D E ˆ N1(t) cannot be neglected due to its importance. Now, I will derive a differential equation for ˆn(t), the number of photons in the mode operator, which will later help us solve the system of differential equations. By deﬁnition ˆn(t) = Aˆ †(t)Aˆ (t). Taking the derivative of this equation, I get

dˆn(t) dAˆ (t) dAˆ (t) = Aˆ †(t) + Aˆ (t). (7.46) dt dt dt

After substituting equation (7.38) I get

dˆn (t) iω0t iω0t = 2℧ Nˆ (t)ˆn(t)+ Aˆ †(t)Fˆ (t)e− + Fˆ† Aˆ (t)e . (7.47) dt − r 1 ab ab

The term 2℧ Nˆ (t)ˆn(t) represent absorption. Using equation (7.39), I can compute − r 1

ˆ ˆ iω0t ˆ ˆ iω0t A†(t)Fab(t)e− and Fab† (t)A(t)e . I can obtain (see appendix F.1.2) D E D E

ˆ ˆ iω0t (7.48) A†(t)Fab(t)e− =0. D E Similarly

ˆ ˆ iω0t (7.49) Fab† (t)A(t)e− =0. D E Using perturbation theory in equation (7.47), it can easily be shown that

dn(t) = 2℧ N n(t), (7.50) dt − r 1 where N Nˆ (t) and is assumed to be constant, and n(t) ˆn(t) . Solving this 1 ≡ 1 ≡ h i D E differential equation gives us

n(T ) ln = 2℧ N T. (7.51) n(0) − r 1 99 κ Referring back to equation (7.32) for the deﬁnition of ℧r, we can see that ~ is the Rabi frequency [37]. It can also be expressed as follows [38]

κ 2 ω | | = 0 ~e D~ 2 ~2 ~ 12 2ǫ0 V | · | (7.52) , where V is the volume, D12 is the dipole moment. Therefore,

κ 2 ω | | = 0 ~e D~ 2, (7.53) ~2 2ǫ~SL| · 12| where S is the cross sectional area of the optical beam and L the length of the cavity.

L Since T = c , I get

n(T ) ω γ ln = 0 ~e D~ 2 N . (7.54) n(0) − ǫ c~S | · 12| ((ω ω )2 + γ2) 1 0 0 − 21 Since this expression is the absorption coefﬁcient, I deﬁne

ω γ βˆ(t) 0 ~e D~ 2 Nˆ (t). (7.55) ≡ ǫ c~S | · 12| ((ω ω )2 + γ2) 1 0 0 − 21 Returning to the differential equation of Aˆ (t), I simplify equation (7.30) by substi-

iω0t tuting for Aˆ (t)= Bˆ(t)e− . Iget

dBˆ(t) = ℧Nˆ (t)Bˆ(t)+ Fˆ (t). (7.56) dt − 1 ab

Equation (7.56) can be rewritten

dBˆ(t) ℧ = ℧ 1+ i i Nˆ (t)Bˆ(t)+ Fˆ (t). (7.57) dt − r ℧ 1 ab r Then using equation (7.55), I have

dBˆ(t) βˆ(t) ℧ = 1+ i i Bˆ(t)+ Fˆ (t). (7.58) dt − 2T ℧ ab r 100 Let

℧i αH , (7.59) ≡ ℧r the Henry alpha factor. Finally, the equation becomes

dBˆ(t) βˆ(t) = (1 + iα ) Bˆ(t)+ Fˆ (t). (7.60) dt − 2T H ab As a check on this equation, I consider the case of unsaturable loss. For unsaturable loss βˆ(t)= β is constant. After the interaction, I have

β ˆ ˆ (1+iαH ) ˆ Bout = Bine− 2 + Floss, (7.61) where Bˆ Bˆ(0) is the input wave, before interaction, Bˆ Bˆ(T ) is the output wave in ≡ out ≡ and

T β cβ (1+iαH ) (1+iαH )t Fˆ e− 2 Fˆ (t)e 2L dt. (7.62) loss ≡ ab Z0 I use equation (7.45) to get

ˆ ˆ (7.63) Floss† Floss =0. D E I then use equation (7.44)to get

β Fˆ Fˆ† =1 e− , (7.64) loss loss − D E which is consistent with what I said in section 1.6.1 and αH has no effect on the loss induced noise.

ˆ Now, I will obtain a differential equation for N1(t). I substitute the expression of

σˆ and σˆ+ from equation (7.28) in equation (7.22), I get − ˆ ˆ dN1(t) ˆ ˆ N N1(t) = 2℧rˆn(t)N1(t)+ FN (t)+ − dt − τ (7.65)

iω0t iω0t Aˆ †(t)Fˆ (t)e− Fˆ† Aˆ (t)e . − ab − ab 101 ˆ To solve the differential equation above, I will need to ﬁnd the properties of FN (t). Iuse

ˆ an argument similar to the one I used for Fab(t) to justify its addition in equation (7.17) and to prove that for a single atom (see appendix F.2)

1 Nˆ (t) 1j (7.66) Fˆ (t)Fˆ (t′) = − δ(t t′). Nj Nj D τ E − D E For a collection of N atoms, it is (see appendix F.3)

N Nˆ (t) 1 (7.67) Fˆ (t)Fˆ (t′) = − δ(t t′). N N Dτ E − D E I can now simplify equation (7.65) by substituting Aˆ (t)= Bˆ(t)eiω0t. Iget ˆ ˆ dN1(t) ˆ ˆ N N1(t) = 2℧rˆn(t)N1(t)+ FN (t)+ − dt − τ (7.68)

Bˆ †(t)Fˆ (t) Fˆ† Bˆ(t). − ab − ab I substitute in the above relation equation (7.55) and after some algebraic manipulations,

I get

dβˆ(t) β βˆ(t) = 0 − 2℧ ˆn(t)βˆ(t)+ Fˆ (t), (7.69) dt τ − r L where

L β 2 ℧ N (7.70) 0 ≡ c r and

L L Fˆ (t) 2 ℧ Bˆ †(t)Fˆ (t)+ Fˆ† Bˆ(t) +2 ℧ Fˆ (t). (7.71) L ≡ − c r ab ab c r N ˆ ˆ ˆ FL(t) is Hermitian. I can compute its properties by computing FL(t)FL(t) . For this, I D E ﬁrst substitute equation (7.55) in equation (7.44) to get

1 Fˆ (t)Fˆ† (t′) = βˆ(t) δ(t t′). (7.72) ab ab T − D E D E 102 I then use this result to get

ˆ β0 + (2τ℧r ˆn(t) 1) β(t) Fˆ (t)Fˆ (t′) =2℧ T h i − δ(t t′). (7.73) L L r  τ D E − D E   ˆ I multiply equation (7.71) by Fab(t) and take the average to get

Fˆ (t)Fˆ† (t′) = 2℧ Bˆ †(t) βˆ(t) δ(t t′). (7.74) L ab − r − D E D ED E ˆ I multiply equation (7.71) by Fab† (t) and take the average to get

Fˆ (t)Fˆ (t′) = 2℧ Bˆ(t) βˆ(t) δ(t t′). (7.75) ab L − r − D E D ED E In conclusion, I obtained from this chapter the following results

dβˆ(t) β βˆ(t) = 0 − 2℧ ˆn(t)βˆ(t)+ Fˆ (t), (7.76) dt τ − r L

dBˆ(t) 1 = βˆ(t)(1+ iα ) Bˆ(t)+ Fˆ (t), (7.77) dt − 2T H ab and I can rewrite equation (7.47) as follows

dˆn(t) 1 = βˆ(t)ˆn(t)+ Bˆ †(t)Fˆ (t)+ Fˆ† Bˆ(t). (7.78) dt − T ab ab

103 Chapter 8

Solving Differential Equation of SA

8.1 Overview

In the previous chapter, I derived equation (7.76) and equation (7.77), which are the simultaneous differential equation describing the interaction between the field operator and a saturable absorber. In this chapter, I will find a solution to those differential equation. In these solutions, I assumed that the noise terms are small compared to the average values of the parameters. Therefore, I will use first order perturbation theory.

8.2 Deriving Simultaneous Differential Equations of the Inphase and Quadra-

ture Phase component of Noise

It is important to note that the average is a statistical average, or ensemble aver- h·i age and not a time average. I use perturbation theory to solve the differential equation of

ˆn(t) and βˆ(t). For that, I make the following substitution

Bˆ(t)= δBˆ(t)+ Bˆ(t) . (8.1) D E To simplify notation, I will use B(t) Bˆ(t) . Similarly ≡ D E βˆ(t)= δβˆ(t)+ βˆ(t) . (8.2) D E

104 To simplify notation, I will use β(t) βˆ(t) . ≡ D E ˆn(t)= δˆn(t)+ ˆn(t) . (8.3) h i

To simplify notation, I will use n(t) ˆn(t) . Ihad ≡h i

dˆn(t) 1 = βˆ(t)ˆn(t)+ Bˆ †(t)Fˆ (t)+ Fˆ† (t)Bˆ(t). (8.4) dt − T ab ab

I substitute equation (8.3) in equation (7.78) and looking at the averaged equation, I get

dn(t) 1 = β(t)n(t). (8.5) dt − T

Substituting equation (8.2) in equation (7.76) gives

dδβˆ(t) dβ(t) β β(t) δβˆ(t) + = 0 − − + Fˆ (t) d d L t t τ (8.6) 2℧ n(t)+ δˆn(t) δβˆ(t)+ β(t) . − r Separating the averaged terms to the ﬂuctuating terms, I get the following equation for the ﬂuctuating terms

ˆ dδβ(t) ˆ ˆ ˆ τ = δβ(t)+ τFL(t) 2℧ τ δβ(t)n(t)+ β(t)δˆn(t) (8.7) dt − − r and the following equation for the averaged terms

dβ(t) β β(t) = 0 − 2℧ β(t)n(t). (8.8) dt τ − r n(t) is the driving term in the above equation. I will assume that n(t) changes slowly enough for the atom to follow without oscillation. In other words that dβ(t) 0. There- dt ≈ fore,

β β(t)= 0 . (8.9) 1+2τ℧rn(t)

105 Substituting this result in equation (8.5), I get

1 dn(t) 1 β = 0 . (8.10) n(t) dt − T 1+2τ℧rn(t)

I deﬁne

1 nsat , (8.11) ≡ 2τ℧r and use this deﬁnition in equation (8.10) to get

1 1 dn(t) β0 = T . (8.12) n(t) dt − 1+ n(t) nsat which is a differential equation which agrees with the semiclassical result I had in equation (6.6), where the differential equation was

1 dP (z) β = − 1 , (8.13) P (z) dz 1+ P (z) Psat where β1 is the absorption coefﬁcient per unit length. The solution of equation (8.12) is

n(0 1 n(0) β0T n(t)= n(0) W e nsat − T (8.14) · n sat n(0) is the input number of photons and n(T ) is the output number of photons, I have

nout nin β0 n = n W e nsat − . (8.15) out in · n sat Again, this agrees with semiclassical results, where the solution to equation (6.6) is

Pin Pin β1l P = P W e Psat − . (8.16) out in · P sat I use (8.11) to get

1 τ = . (8.17) 2℧rnsat

106 Using the above equation, equation (7.76) can be rewritten

ˆ ˆ ˆ dβ(t) β0 β(t) ˆn(t)β(t) ˆ = − + FL(t), (8.18) dt τ − nsatτ equation (8.7) can be written

dδβˆ(t) 1 τ = δβˆ(t)+ τFˆ (t) δβˆ(t)n(t)+ β(t)δˆn(t) , (8.19) dt − L − n sat equation (7.73) can be written

n(t) ˆ β0 + 1 β(t) T nsat Fˆ Fˆ − (8.20) L(t) L(t′) = 2 δ(t t′), nsat  τ D E − D E   equation (8.9) can be written

β β(t)= 0 , (8.21) 1+ n(t) nsat and equation (7.74) and (7.75) can be written consecutively

ˆ ˆ 1 ˆ ˆ (8.22) FL(t)Fab† (t′) = B†(t) β(t) δ(t t′), − τnsat − D E D ED E and

ˆ ˆ 1 ˆ ˆ Fab(t)FL(t′) = B(t) β(t) δ(t t′). (8.23) − τnsat − D E D ED E It can be veriﬁed that if n , I am back to the non saturable case. In that case sat →∞ equation (8.20) becomes

ˆ ˆ FL(t)FL(t′) =0. (8.24) D E ˆ ˆ Since FL(t) is Hermitian, in that case FL(t)=0.

107 Bˆ(t) can be written

Bˆ(t)= B(t)+ δXˆ (t)+ iδYˆ (t), (8.25) where δXˆ (t) and δYˆ (t) are the in phase and the quadrature phase component of δBˆ(t)(t).

Bˆ(t) can also be written δXˆ (t) δYˆ (t) Bˆ(t)= B(t) 1+ + i B(t) B(y) ! (8.26) = B(t)(1+ ˆx(t)+ iˆy(t)) , where δXˆ (t) ˆx(t) , (8.27) ≡ B(t) and δYˆ (t) ˆy(t) . (8.28) ≡ B(t)

Substituting equation (8.26) in equation (7.77) and after some algebraic manipulation (see appendix G.1) I get the following equations

1 1 dˆy(t) B− (t)Fˆ B∗− (t)Fˆ† α = ab − ab H δβˆ(t), (8.29) dt 2i − 2T and 1 1 ∂ˆx(t) 1 B (t)Fˆ + B (t)Fˆ† = δβˆ(t)+ − ab ∗− ab . (8.30) dt −2T 2

As a check, I assume that n , which is the case for unsaturable loss. In that sat →∞ case equations (8.30)and (8.19) can be easily solved and gives

eβ0 1 ˆx(T )ˆx(T ) = ˆx(0)ˆx(0) + − . (8.31) h i h i 4n(0)

β0 Multiplying the whole equation by n(T )= n(0)e− , I get

β 2 2 0 β0 1 e− δXˆ (T ) = e− δXˆ (0) + − , (8.32) 4 D E D E 108 which is what is expected.

8.3 Solving the Simultaneous Differential Equations

The solution of (8.19) is

t 2n(t′)β(t′) 1 δβˆ(t)= H (t′) Fˆ (t′) ˆx(t′) dt′ H− (t), (8.33) 1 L − τn 1 Z0 sat since δβˆ(0) = 0 and where

t nsat + n(t′) H (t) exp dt′ . (8.34) 1 ≡ τn Z0 sat It is important to note from equation (8.19) that for time long compared to τ δβˆ(t) can be considered memoryless and can be determined by event that occur at time very close to t.

Therefore,

n(t) t H (t) exp 1+ . (8.35) 1 ≈ n τ sat This approximation can be used because δβˆ(t) is memoryless. The equation can further be reduced as follows t t 2n(t)β(t) 1 δβˆ(t) H (t′)Fˆ (t′)dt′ ˆx(t) H (t′)dt′ H− (t) ≈ 1 L − τn 1 1 Z0 sat Z0 t t ′ 2n(t)β(t) 1+ n(t) t ˆ n τ 1 H1(t′)FL(t′)dt′ ˆx(t) e sat dt′ H− (t) (8.36) ≈ − τn 1 Z0 sat Z0 t 2 n(t) β(t) Fˆ d nsat ˆx 1 H1(t′) L(t′) t′ n(t) (t)H1(t) H1− (t), ≈ " 0 − 1+ # Z nsat where the lower values of the integral has been ignored, since δβˆ(t) forgets and only remembers what happened at time t and ˆx(t), n(t) and β(t) are considered slowly varying and can be taken outside of the integral. The equation gives

t 2 n(t) β(t) ˆ 1 Fˆ d nsat ˆx (8.37) δβ(t) H1− (t) H1(t′) L(t′) t′ n(t) (t), ≈ 0 − 1+ Z nsat 109 ˆ since everything is slowly varying except for FL(t). Substituting this result in equation

(8.30) gives us.

n(t) ∂ˆx(t) 1 β(t)ˆx(t) nsat ˆ = + Nx, (8.38) dt T n(t) +1 nsat where

1 1 B (t)Fˆ + B (t)Fˆ† Nˆ − ab ∗− ab + Fˆ (t), (8.39) x ≡ 2 β where

t ′ 1 1+ n(t) t −t ˆ n τ ˆ Fβ(t) e sat FL(t′)dt′. (8.40) ≡ −2T Z0 The solution of equation (8.38) is then

t 1 ˆ ˆx(t)= H2− (t) H2(0)ˆx(0) + H2(t′)Nxdt′ , (8.41) Z0 where

′ T n(t ) 1 β(t′) nsat d H2(t) = exp n(t′) t′ T t +1 ! Z nsat (8.42) n(t)+ n = sat . n(T )+ n sat I use this result to get the expression of the power of the inphase noise

ˆx(T )ˆx(T ) = ˆx(0)ˆx(0) H (0)2 h i h i 2 T T (8.43) ˆ ˆ + H2(t′)H2(t′′) Nx(t′)Nx(t′′) dt′dt′′, 0 0 Z Z D E ˆ ˆ where ˆx(0)Nx(t′) =0 and Nx(t′)ˆx(0) =0. D E D E It is clear that in order to get an expression for ˆx(T )ˆx(T ) I will need an expression h i ˆ ˆ ˆ ˆ ˆ ˆ for Fβ(t)Fβ(t′) , Fab(t)Fβ(t′) and Fβ(t)Fab(t′) . Let us first find an expression D E D E D E ˆ ˆ for Fβ(t)Fβ(t′) . D E 110 ˆ ˆ Figure 8.1: Illustration of spectrum of Fβ(t) relative to the spectrum of B(t). ˜ ˆ ˜ Fβ(Ω) is the he Fourier Transform of Fβ(t) and B(Ω) is the Fourier transform ˆ ˆ ˆ of B(t). The spectrum of Fβ(t) looks flat relative to the spectrum of B(t)

ˆ On the scale at which I look at this problem, Fβ(t) is roughly a delta correlated

ˆ noise. This can be understood by observing that Fβ(t) oscillates much faster than the signal and can be approximated by a delta correlated function. Yet, another way to look

ˆ at it (see ﬁgure 8.1) is to see that the spectrum of Fβ(t) relative to the spectrum of the signal looks constant, similar to the one of a delta correlated function.

ˆ ˆ I will compute the integral of Fβ(t)Fβ(t′) and rewrite it as a delta correlated D E function (see appendix G.1.1). I can rewrite equation (8.20) as such

Fˆ (t)Fˆ (t′) = Kδ(t t′), (8.44) L L − D E

111 where n(t) ˆ β0 + 1 β(t) T nsat − (8.45) K 2 . ≡ nsat  τ D E

Equation (8.40) can be rewritten  

t Fˆ (t)= G(τ )Fˆ (t τ )dτ , (8.46) β 1 L − 1 1 Z0 where

τ 1 1+ n(t) 1 nsat τ (8.47) G(τ1)= e− . 2T

Therefore,

1 τ τK n(t) − 1+ n(t) 1 Fˆ Fˆ − nsat τ (8.48) β(t) β(t τ1) 2 1+ e . − ≈ 8T nsat D E I now compute the integral

2 2 ∞ τ K n(t) − Fˆ Fˆ d (8.49) β(t) β(t τ1) τ1 2 1+ . − ≈ 4T nsat Z−∞D E Therefore,

2 τ 2K n(t) − Fˆ Fˆ ′ ′ (8.50) β(t) β(t ) 2 1+ δ(t t ). ≈ 4T nsat − D E ˆ ˆ I now look for an expression for Fab(t)Fβ(t′) . Equation (8.23) can be rewritten D E

Fˆ (t)Fˆ (t′) = K δ(t t′), (8.51) ab L 1 − D E where B(t) K1 β(t). (8.52) ≡ −nsatτ

112 Following a procedure identical to the one in the previous section (see appendix G.1.2), I get

1 ˆ ˆ τK1 n(t) − Fab(t)Fβ(t′) 1+ δ(t t′). (8.53) ≈ T nsat − D E Similarly

1 ˆ ˆ τK1 n(t) − Fβ(t′)Fab(t) 1+ δ(t t′). (8.54) ≈ T nsat − D E ˆ ˆ ˆ ˆ ˆ ˆ I can now use the expressionfor Fβ(t)Fβ(t′) , Fab(t)Fβ(t′) and Fβ(t)Fab(t′) D E D E D E to get an expressionfor the power of the inphase noise, ˆx(T )ˆx(T ) . I take equation (8.43) h i ˆ and use equation (8.39) to substitute the expression of Nx(t), with the results in equations

(8.53),(8.54), (8.50) and (7.72), I get (see appendix G.2)

ˆx(T )ˆx(T ) = ˆx2(0) H2(0) h i h i n(t′) T β0 + 1 β(t) 1 β(t ) nsat 2 ′ − d + H (t′)  + 2  t′ 4T 0 n(t′) n(t′) (8.55) Z nsat n +1  sat  T   1 2 β(t′) + H (t′) dt′. n(t′) T 0 n +1  Z sat nsat The inphase noise can be broken into four noises: the amplified initial noise, the relaxation noise, the absorption noise and the beat noise between the the absorption noise and the relaxation noise. The amplified initial noise is noise due to the amplification of the incoming signal’s noise ˆx2(0) . The relaxation noise is the noise due to h i ˆ ˆ ˆ ˆ Fβ(t)Fβ(t′) . The absorption noise is noise due to Fab(t)Fab(t′). The beat noise is D E ˆ ˆ ˆ ˆ noise due to Fab(t)Fβ(t′) and Fβ(t)Fab(t′) . D E D E The amplified initial amplitude noise is

Init =H2(0) ˆx2(0) . (8.56) x 2 h i 113 Using equation (8.42), I have

1 n(0) + n 2 1 Init = sat . (8.57) x 4 n(T )+ n n(0) sat

I compute the ﬁrst integral of equation (8.55)(see appendix G.2.2), which is noise due to absorption

n2 n2 2n ln n(0) + sat sat + n(0) n(T ) 1 sat n(T ) n(T ) − n(0) − (8.58) Noiseabsx = 2 , 4 (n(T )+ nsat) which, as n converges to sat →∞

1 eβ0 1 Noise = − , (8.59) absx 4 n(0) which is what was expected. I compute the second integral of equation (8.55) (see appendix G.2.3), which is noise due to relaxation

1 n(0) n(T ) (8.60) Noiserelx = − 2 2 (n(T )+ nsat)

I compute the third integral of equation (8.55)(see appendix G.2.4), which is the beat noise between the relaxation noise and the absorption noise

n(0) n(T )+ n ln n(0) − sat n(T ) (8.61) Noisebeatx = 2 . (n(T )+ nsat)

The power of the amplitude noise is the sum of all the noises and is

2 2 n(0) nsat nsat 2 6nsat ln + 1 n(0) + nsat 1 1 n(T ) n(T ) − n(0) ˆx(T )ˆx(T ) = + 2 h i 4 n(T )+ nsat n(0) 4 (n(T )+ n ) sat (8.62) 3 (n(0) n(T )) + − 2 . 2 (n(T )+ nsat)

114 Now, I calculate the quadrature phase noise ˆy2(T ) Using equation (8.29) and h i equation (8.30), It can be shown that (see appendix (G.3))

(1 + α2 ) 1 α2 1 n(0) + n ˆy2(T ) = H + α2 ˆx2(T ) H sat 4 n(T ) H − 2 n(0) n(T )+ n sat n(0) n n (8.63) 2 3 ln + sat sat αH n(T ) n(T ) − n(0) − 2  n(T)+ nsat    For veriﬁcation, I set n . Then using equation (8.31), I can verify that sat →∞

1 1 ˆy2(T ) = , (8.64) 4 n(T ) which is what was expected. The incoming signal’s noise is ampliﬁed. At the output, it becomes

1 n(0) + n 2 n(0) + n 1 Init = (1 + α2 )+ α2 sat 2α2 sat . (8.65) y 4 H H n(T )+ n − H n(T )+ n n(0) " sat sat #

From equation (8.63), the expression of the absorption noise is

(1 + α2 ) 1 1 Noise = H + α2 Noise absy 4 n(T ) − n(0) H absx n(0) n n (8.66) 2 ln + sat sat αH n(T ) n(T ) − n(0) − 2  n(T )+ nsat    From equation (8.63), the expression of the relaxation noise is

2 (8.67) Noiserely = αH Noiserelx .

From equation (8.63), the expression of the beat noise is

ln n(0) 2 2 n(T ) Noisebeaty = αH Noisebeatx αH . (8.68) − n(T)+ nsat    Now that I have the inphase and quadrature phase noise of the saturable absorber, I use it to get the noise ﬁgure of the NMZI OPA.

115 8.4 Noise Figure

From equation (3.54), assuming a large parametric gain and a large signal, a good estimate of the noise ﬁgure is

G NF = NF +3= NF 10 log par , (8.69) QFS FAS PNS ≈ − 4ASE with 1 ASE = n(T ) ˆy2(T ) + ˆx2(T ) . (8.70) − 2 It is easy to prove that the parametric gain is

2 iθin iθin Gpar = µe + νe− . (8.71)

where

β (iαH +1) i (1 Γ) µ e− 2 i + (iα + 1) − n(0) (8.72) ≡ 2 H Γn(0) + n sat and

β (iαH +1) i (1 Γ) ν e− 2 (iα + 1) − n(0) . (8.73) ≡ 2 H Γn(0) + n sat I calculated the noise figure for this device for a large range of parameters and found that it is always above3 dB. I plot thenoise figure of a SA based NMZI OPA with a Henry alpha factor 5, β0 = 2 and nsat = 1 (see figure 8.2). It can be seen that the noise figure remains high, more than 8 dB. I plot the gain and the noise figure for a saturable absorber of αH = 25, β0 =2 and nsat =1 on the same graph (see figure 8.3).

116 Figure 8.2: Noise Figure as a function of n(0) for αH =5, β0 =2 and nsat =1

117 Figure 8.3: Parametric gain (blue) and noise ﬁgure (red) as a function of n(0) for αH = 25, β0 =2 and nsat =1

118 Chapter 9

Discussions

In this thesis, I have done the following:

I went over two deﬁnitions of noise ﬁgure (NF and NF ) and shown that the • NPS FAS

NFNPS is signal dependent for weak signals in OPAs. I have also shown that the

NFFAS is inappropriate for OPAs. Therefore, a new deﬁnition of the noise ﬁgure

was introduced, namely the quadrature phase noise ﬁgure (NFQFS ), which works

very well for OPAs and phase insensitive ampliﬁers. In other words, it is not signal

dependent and is much easier to calculate.

I have a derived a simple expression to get the noise ﬁgure for OPA in high gain • regime and large signal based on their parametric gain and ASE.

I have derived the proper expression of the noise ﬁgure of a lossy Kerr based NMZI • OPA. I have also found the expression for the optimum length for the length of a

ﬁber used as a Kerr medium based on its nonlinear properties and its loss coefﬁcient.

I have shown that Kerr based NMZI OPA with gain instead of loss under high gain • have a minimum 3dB noise ﬁgure.

I have demonstrated the feasibility of a SOA based and SA based NMZI OPA as an • alternative to the Kerr based NMZI OPA. I have

– I have calculated their steady state parametric gain.

119 – I have calculated their steady state bandwidth.

– I have used Quantum Mechanics to show that the noise ﬁgure of SA Based

NMZI OPA is very high, in the order of 9 dB.

While at the beginning, the unexpectedly high noise ﬁgure of the SA based NMZI OPA came as a little surprise, because the expected noise ﬁgure was in the order of 1 dB, after a closer look at the expression something interesting became apparent. I expected most of

ˆ the noise to come from the absorption noise Fab , the noise due to the signal absorption.

ˆ However, most of the noise comes from relaxation noise Fβ(t), due to the relaxation of the electrons. This noise is well known and was by Yamamoto et al. [67]. The magnitude of the relaxation noise could be due to an overestimate because of the way that noise was calculated. Indeed, the noise average value was calculated in appendix F.2 based on the assumption that

ˆ ˆ ˆ N3N3 = N3 (9.1)

This is actually only true if the atom is isolated, which is not really the case. While I could get a better estimate of this noise, it is better to completely cancel that noise or make it not interfere with the signal. There are two ways this could be achieved.

The ﬁrst way is to not let the electrons relax back to level one. Instead, the electrons and their associated holes can be removed by applying a very-low-noise current. A possible technique to produce very low ﬂuctuations current is discussed by Yamamoto et al.

[67]. Yamamoto et al. explain that if such current is used to pump a laser oscillator, it is no longer subject to the standard quantum limit. It can then produce amplitude squeezed states.

120 The second way, is to send very short pulses as the signal and the pump, much shorter that the relaxation time τ but still much longer than the decoherence time 1/γ. In this fashion, by the time the relaxation occur, the experiment is over.

ˆ Assuming these techniques are applied, then the noise due to relaxation Fβ(t) can be neglected. This significantly improves the noise figure and places it in the order of 1 to 0.5 dB (see figure 9.1). However, the application of these techniques are left to future research.

Figure 9.1: Parametric gain (blue) and noise ﬁgure (red) as a function of n(0) for αH = 25, β0 =2 and nsat =1

121 Appendix A

Detailed Derivation of Noise Field Operator in Lossy Kerr Medium

A.1 Solving the Differential Equation for the Average Terms

To solve (4.10), I take an integral on both side of the equation as follows ˆ x d A(z) x 2 = iγ Aˆ (z) β dz. (A.1) D E 0 Aˆ (z) 0 − Z Z D E

D E To solve it, I define Aˆ (0) α. Since this is a propagation of a field through a lossy ≡ D E medium with loss coefficient β, I have

βz Aˆ (z) = αe− . (A.2) D E I substitute this result in our equation. I get

x d Aˆ (z) x 2 2βz = iγ α e− β dz. (A.3) Dˆ E | | − Z0 A(z) Z0 I evaluate the right left handD side ofE the equation. I get

Aˆ (x) x 2 2βz ln = iγ α e− β dz. (A.4) D ˆ E | | − A(0) Z0 After some more algebraic D manipulation,E  I get

Aˆ (x) iγ 2 2βx ln = α 1 e− βx . (A.5) D α E −2β | | − − I set x = z and obtain the following solution of the differential equation

iγ 2 2βz Aˆ (z) = α exp α 1 e− βz 2β | | − − (A.6) D E βz iγ 2 2βz = αe− exp α 1 e− . 2β | | − 122 A.2 Derivation of the Output Noise Field Operator

The deﬁnition of δAˆ (L) is

iφ(L) βL δAˆ (L)= δXˆ (L)+ iδYˆ (L) e − . (A.7) Substituting for the values of δXˆ (L) and δYˆ (L) from equation (4.25) and (4.26), I get

iθ iθ δAˆ (L)= µ(α)δAˆ (0)e in + ν(α)δAˆ †(0)e− in + Nˆ , (A.8) where

iφ(L) βL γ 2 2βL µ(α) e − 1+ i α 1 e− , (A.9) ≡ 2β | | − iφ(L) βL γ 2 2βL ν(α) e − i α 1 e− (A.10) ≡ 2β | | − and

L z 2 β(x 2z) iφ(x) iφ(x) Nˆ γ α e − Nˆ †(x)e + Nˆ (x)e− dxdz ≡ | | Z0 Z0 (A.11) L βz iφ(z) + e Nˆ (z)e− dz. Z0 A.3 Parametric Gain Derivation

I consider Kerr based NMZI with a strong pump in the first input with average field intensity αp. Similarly, at the second input of the NMZI I assume a weak signal with average field intensity αs. After the first coupler, I have the two outputs

1 Eout11 = (αp + iαs) (A.12) √2 at the output of the ﬁrst arm and

1 Eout21 = (αs + iαp) . (A.13) √2

123 To get an expression for the output of the ﬁrst Kerr medium, I use the previous results, namely equation (A.6) and (A.8), which gives

βL iγ 2 2βL E = α e− exp α 1 e− (A.14) out12 1 2β | 1| − where 1 α1 = (αp + iαs) (A.15) √2

Similarly, the output of the second Kerr medium is

βL iγ 2 2βL E = α e− exp α 1 e− (A.16) out22 2 2β | 2| − where 1 α2 = (αs + iαp) (A.17) √2

At the output of the NMZI, I have

βL βL αpe− iαse− E = eiΦ2 eiΦ1 eiΦ2 + eiΦ1 , (A.18) out 2 − − 2 where

γ 2 2βL Φ = α 1 e− (A.19) 1 2β | 1| − and

γ 2 2βL Φ = α 1 e− . (A.20) 2 2β | 2| − I deﬁne Φ Φ + Φ , Φ Φ Φ , 1 ≡ 10 11 2 ≡ 10 − 11

γ 2 2βL Φ α 1 e− , (A.21) 10 ≡ 4β | p| − and

γ 2βL Φ 1 e− (α α∗ α∗α ). (A.22) 11 ≡ 4β − s p − s p 124 Using the above deﬁnitions, we can simplify equation (A.18) further and get

βL iΦ10 E = ie− e (α Φ + α ) . (A.23) out − p 11 s

Substituting Φ11 into equation (A.23), we get

βL iΦ10 γ 2βL iθ E = ie− e 1 e− sin(θ )+ e in α . (A.24) out − 2β − in | s| This equation can be rewritten as follows

iθ iθ E = µ(α) α e in + ν(α) α e− in , (A.25) out | s| | s| where

iφ(L) βL γ 2 2βL µ(α) e − 1+ i α 1 e− , (A.26) ≡ 2β | | − iφ(L) βL γ 2 2βL ν(α) e − i α 1 e− . (A.27) ≡ 2β | | − Therefore, the maximum parametric gain is

2 2βL 4 2 G = γ e− α L . (A.28) | p| eﬀ

A.4 Correlation of the Vacuum Noise in the two Arms of the NMZI

ˆ ˆ It is most important to note that δA2(0) and δA1(0) are standard vacuum ﬂuctua- tions that are uncorrelated. The proof is as follows:

1 † δAˆ (0)Aˆ †(0) = δAˆ + iδAˆ iδAˆ + δAˆ 1 2 2 lo lo D E 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = iδAδA† + δAδAlo† + δAloδA† + iδAloδAlo† 2 − (A.29) i D i E = − 2 =0

125 and

1 † δAˆ †(0)Aˆ (0) = δAˆ + iδAˆ iδAˆ + δAˆ 1 2 2 lo lo D E (A.30) =0.

ˆ ˆ ˆ ˆ It is very easy to see that δA1(0)A2(0) = δA1†(0)A2†(0) =0. D E D E

126 Appendix B

Detailed Derivation of NF for the Lossy Kerr Medium Based NMZI OPA

2 B.1 Expression for ∆ Eˆout† Eˆ out ˆ ˆ To calculate the variance of Eout† Eout, I use the deﬁnition

Eˆ Eˆ + δEˆ . (B.1) out ≡ out out D E ˆ ˆ I can then compute Eout† Eout, which is

2 ˆ ˆ ˆ ˆ ˆ ˆ ∗ ˆ ˆ ˆ (B.2) Eout† Eout = Eout + Eout δEout† + Eout δEout + δEout† δEout D E D E D E 2 ˆ ˆ I can then compute Eout† Eout , which is 2 4 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ∗ ˆ Eout† Eout = Eout +2 Eout Eout δEout† +2 Eout Eout δEout

D E 2 D E D E 2 D E D E ˆ ˆ ˆ ˆ ˆ ˆ +3 Eout δ Eout† δEout + Eout δEout† δ Eout†

D E D E 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ + E out δE out† δEout† δEout + Eout δEoutδEout†

D E 2 D E ˆ ∗ ˆ ˆ ˆ ∗ ˆ ˆ ˆ + Eout δEoutδEout + Eout δEout δEout† δEout D E D E ˆ ˆ ˆ ˆ ˆ ∗ ˆ ˆ ˆ + Eout δEout† δEoutδEout† + Eout δEout† δEoutδEout D E D E ˆ ˆ ˆ ˆ + δEout† δEoutδEout† δEout

(B.3)

Taking the average of equation (B.2), I get

2 ˆ ˆ ˆ ˆ ˆ (B.4) Eout† Eout = Eout + δEout† δEout . D E D E D E

127 From the above equation, I get

2 4 2 ˆ ˆ ˆ ˆ ˆ Eout† Eout = Eout + δEout† δEout (B.5) D E D E 2D E ˆ ˆ ˆ +2 Eout δEout† δEout D E D E Taking the average of equation (B.3), I get

4 ˆ ˆ ˆ ˆ ˆ Eout† EoutEout† Eout = Eout

D E D E 2 2 ˆ ˆ ˆ ˆ ˆ ˆ +3 Eout δEout† δEout + Eout δEout† δEout† (B.6) D E2 D E D E2 D E ˆ ˆ ˆ ˆ ∗ ˆ ˆ + Eout δEoutδEout† + Eout δEoutδEout D E D E D E D E ˆ ˆ ˆ ˆ + δEout† δE outδEout† δEout . D E Therefore,

2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ∆ Eout† Eout = Eout δEout† δEout + Eout δEout† δEout†

D E D2 E D E D2 E ˆ ˆ ˆ ˆ ∗ ˆ ˆ (B.7) + Eout δEoutδEout† + Eout δEoutδEout

D E D E D E D 2 E ˆ ˆ ˆ ˆ ˆ ˆ + δE† δ EoutδE† δEout δE† δEout . out out − out D E D E

B.2 Calculation of δEôut† δEôutδEôut† δEôut D E To calculate the noise commutator, I need to compute the following averages:

ˆ ˆ ˆ ˆ B.2.1 Calculation of N(w)N†(y)N(z)N†(x) D E From equation (4.6)

Nˆ (z), Nˆ †(z′) =2βδ(z z′). (B.8) − h i where it is a Dirac delta function. Therefore,

Nˆ (z)Nˆ †(x) Nˆ †(x)Nˆ (z)=2βδ(z x). (B.9) − − 128 Therefore,

Nˆ (z)Nˆ †(x) =2βδ(z x). (B.10) − D E Therefore,

Nˆ (w)Nˆ †(y)Nˆ (z)Nˆ †(x) Nˆ (w)Nˆ †(y)Nˆ †(x)Nˆ (z)=2βδ(z x)Nˆ (w)Nˆ †(y). (B.11) − −

Since

Nˆ (w)Nˆ †(y)Nˆ †(x)Nˆ (z) =0, (B.12) D E I have

Nˆ (w)Nˆ †(y)Nˆ (z)Nˆ †(x) Nˆ (w)Nˆ †(y)Nˆ †(x)Nˆ (z) =2βδ(z x) Nˆ (w)Nˆ †(y) . − − D E D E (B.13)

Therefore,

Nˆ (w)Nˆ †(y)Nˆ (z)Nˆ †(x) =2βδ(z x) Nˆ (w)Nˆ †(y) − (B.14) D E D E =4β2δ(z x)δ(w y). − −

B.2.2 Calculation of Nˆ (w)Nˆ (z)Nˆ †(x)Nˆ †(y) D E From the commutator relationship, I have

Nˆ (z)Nˆ †(x) Nˆ †(x)Nˆ (z)=2βδ(z x). (B.15) − −

Therefore,

Nˆ (z)Nˆ †(x)Nˆ †(y) Nˆ †(x)Nˆ (z)Nˆ †(y)=2βδ(z x)Nˆ †(y). (B.16) − −

Therefore,

Nˆ (w)Nˆ (z)Nˆ †(x)Nˆ †(y) Nˆ (w)Nˆ †(x)Nˆ (z)Nˆ †(y)=2βδ(z x)Nˆ (w)Nˆ †(y). (B.17) − − 129 Therefore,

2 Nˆ (w)Nˆ (z)Nˆ †(x)Nˆ †(y) Nˆ (w)Nˆ †(x)Nˆ (z)Nˆ †(y) =4β δ(z x)δ(w y). (B.18) − − − D E Using the result of the previous section, I have

2 2 Nˆ (w)Nˆ (z)Nˆ †(x)Nˆ †(y) 4β δ(w x)δ(z y)=4β δ(z x)δ(w y). (B.19) − − − − − D E Therefore,

2 Nˆ (w)Nˆ (z)Nˆ †(x)Nˆ †(y) =4β δ(z x)δ(w y)+ δ(w x)δ(z y) . (B.20) − − − − D E h i

2 B.3 Simpliﬁed Expression for ∆ Eˆout† Eˆ out It was seen that the noise from the noise from the upper arm of the NMZI and the lower arm are uncorrelated. Therefore, the noise at the output of the NMZI is of the same form as the ones in each of the arm of the NMZI given in equation (A.8), which is

ˆ ˆ iθin ˆ iθin ˆ (B.21) δEout = µ(αp)δA(0)e + ν(αp)δA†(0)e− + N.

I make the following approximation

L z 2 β(x 2z) iφ(x) iφ(x) Nˆ = γ α e − Nˆ †(x)e + Nˆ (x)e− dxdz | p| Z0 Z0 L βz iφ(z) + e Nˆ (z)e− dz (B.22) Z0 L z 2 β(x 2z) ˆ iφ(x) ˆ iφ(x) γ αp e − N†(x)e + N(x)e− dxdz ≈ | | 0 0 Z Z The term is neglected since it will small relative to the pump power α 2. In general, I | p| am only going to keep terms with factor of α 8 in variance of the output . Also, I will | p| use the high gain approximation

µ 2 ν 2. (B.23) | | ≈| | 130 ˆ ˆ ˆ ˆ I compute δEout† δEoutδEout† δEout , which is D E 4 δEˆ † δEˆ δEˆ † δEˆ µ δAˆ δAˆ †δAˆ δAˆ † out out out out ≈| | D E D E 4 + µ δAˆ δAˆ δAˆ †δAˆ † | | D E 2 2 + µ Nˆ †δAˆ δAˆ †Nˆ + µ Nˆ †Nˆ δAˆ δAˆ † (B.24) | | | | D E D E 2 2 + µ δAˆ δAˆ †Nˆ †Nˆ + µ δAˆ Nˆ Nˆ †δAˆ † | | | | D E D E + Nˆ †Nˆ Nˆ †Nˆ . D E I use the following expressions

δAˆ δAˆ †δAˆ δAˆ † =1, (B.25) D E

δAˆ δAˆ δAˆ †δAˆ † =2, (B.26) D E Nˆ †δAˆ δAˆ †Nˆ = Nˆ †Nˆ , (B.27) D E D E Nˆ †Nˆ δAˆ δAˆ † = Nˆ †Nˆ , (B.28) D E D E δAˆ δAˆ †Nˆ †Nˆ = Nˆ †Nˆ , (B.29) D E D E δAˆ Nˆ Nˆ †δAˆ † = Nˆ Nˆ † , (B.30) D E D E and the fact that

Nˆ †Nˆ = Nˆ Nˆ † . (B.31) D E D E I substitute these expressions into the equation, I get

4 2 δEˆ † δEˆ δEˆ † δEˆ =3 µ +4 µ Nˆ †Nˆ + Nˆ †Nˆ Nˆ †Nˆ . (B.32) out out out out | | | | D E D E D E

131 Also, it can easily be shown that

ˆ ˆ ˆ ˆ δEout† δEout = δEoutδEout† D E D E ˆ ˆ = δEoutδEout (B.33) D E ˆ ˆ = δEout† δEout† . D E Therefore,

2 2 2 ∆ Eˆ † Eˆ = δEˆ † δEˆ 2 Eˆ +2 Eˆ out out out out out ℜ out D E D E D E (B.34) 2 + δEˆ † δEˆ δEˆ † δEˆ δEˆ † δEˆ out out out out − out out D E D E ˆ ˆ ˆ ˆ B.4 Computing N†NN†N D E

4βL 8 L x′ 4 e− αp β(x 2x′) iφ(x) iφ(x) Nˆ †Nˆ Nˆ †Nˆ = γ | | e − Nˆ †(x)e + Nˆ (x)e− dxdx′ 16 * 0 0 D E Z Z L w′ β(w 2w′) iφ(w) iφ(w) e − Nˆ †(w)e + Nˆ (w)e− dwdw′ × Z0 Z0 L y′ β(y 2y′) iφ(y) iφ(y) e − Nˆ †(y)e + Nˆ (y)e− dydy′ × Z0 Z0 L z′ β(z 2z′) iφ(z) iφ(z) e − Nˆ †(z)e + Nˆ (z)e− dzdz′ . × 0 0 + Z Z (B.35)

132 Keeping only the terms that are not going to average to zero, I have

4βL 8 L x′ L w′ L y′ L z′ 4 e− αp Nˆ †Nˆ Nˆ †Nˆ = γ | | 16 0 0 0 0 0 0 0 0 D E Z Z Z Z Z Z Z Z β(x+w+y+z 2(x′+w′+y′+z′)) e − × ˆ ˆ ˆ ˆ N(x)N†(w)N(y)N†(z) (B.36) × " D E

+ Nˆ (x)Nˆ (w)Nˆ †(y)Nˆ †(z) # D E

dxdx′dwdw′dydy′dzdz′. × I now use the results in section B.2, I get

4βL 8 L x′ L w′ L y′ L z′ 2 4 e− αp Nˆ †Nˆ Nˆ †Nˆ = β γ | | 4 0 0 0 0 0 0 0 0 D E Z Z Z Z Z Z Z Z β(x+w+y+z 2(x′+w′+y′+z′)) e − ×

δ(x w)δ(y z)+ δ(w y)δ(x z) (B.37) × " − − − −

+ δ(x y)δ(w z) − − #

dxdx′dwdw′dydy′dzdz′. × From this, I get

4βL 8 L L w′ L y′ L z′ 2 4 e− αp Nˆ †Nˆ Nˆ †Nˆ = β γ | | 4 0 0 0 0 0 0 0 D E Z Z Z Z Z Z Z β(2w+y+z 2(x′+w′+y′+z′)) e − δ(y z) × " −

β(w+y+2z 2(x′+w′+y′+z′)) (B.38) + e − δ(w y) −

β(w+2y+z 2(x′+w′+y′+z′)) + e − δ(w z) − #

dx′dwdw′dydy′dzdz′. ×

133 From this, I get

4βL 8 L L L L w′ y′ 2 4 e− αp Nˆ †Nˆ Nˆ †Nˆ = β γ | | 4 ( 0 0 0 0 0 0 D E Z Z Z Z Z Z β(2w+2y 2(x′+w′+y′+z′)) e − dydwdx′dw′dy′dz′ × (B.39) h L L L L L y′ z′i β(2y+2z 2(x′+w′+y′+z′)) + e − 0 0 0 0 0 0 0 Z Z Z Z Z Z Z h β(2y+2z 2(x′+w′+y′+z′)) + e − dzdydx′dw′dy′dz′ . ) i From this, I get

4βL 8 L L L L w′ 4 e− αp Nˆ †Nˆ Nˆ †Nˆ = βγ | | 8 ( 0 0 0 0 0 D E Z Z Z Z Z β(2w 2(x′+w′+z′)) β(2w 2(x′+w′+y′+z′)) e − e − × − h L L L L y′ i dwdx′dw′dy′dz′ + (B.40) × Z0 Z0 Z0 Z0 Z0 β(2y 2(x′+w′+y′)) β(2y 2(x′+w′+y′+z′)) 2 e − e − × − h i dydx′dw′dy′dz′ . × ) From this, I get

4βL 8 L L L L 4 e− αp Nˆ †Nˆ Nˆ †Nˆ = γ | | 16 ( 0 0 0 0 D E Z Z Z Z 2β(x′+z′) 2β(x′+w′+z′) e− e− × − h 2β(x′+w′+y′+z′) 2β(x′+y′+z′) + e− e− − L L L L i dx′dw′dy′dz′ + (B.41) × Z0 Z0 Z0 Z0 2β(x′+w′) 2β(x′+w′+y′) 2 e− e− × − h 2β(x′+w′+y′+z′) 2β(x′+w′+z′) + e− e− − i dx′dw′dy′dz′ . × )

134 Rewriting the integrals, I have

4βL 8 L L L L 4 e− αp Nˆ †Nˆ Nˆ †Nˆ = γ | | 16 ( 0 0 0 0 D E Z Z Z Z 2β(x+z) 2β(x+w+z) 2β(x+w+y+z) 2β(x+y+z) e− e− +3e− 3e− × − − h i 2β(x+w) 2β(x+w+y) +2 e− e− dxdwdydz . − ) h i (B.42)

From this, I get

4βL 8 L L L 4 e− αp Nˆ †Nˆ Nˆ †Nˆ = γ | | 32β ( 0 0 0 D E Z Z Z 2βz 2β(L+z) 2β(L+w+z) 2β(w+z) e− e− + e− e− × − − h 2β(w+y+z) 2β(L+w+y+z) +3e− 3e− (B.43) −

2β(L+y+z) 2β(y+z) 2βw 2β(L+w) +3e− 3e− +2 e− e− − − h i 2β(L+w+y) 2β(w+y) +2 e− e− dwdydz . − ) h i From this, I get

4βL 8 L L 4 e− αp 2βz 2β(L+z) Nˆ †Nˆ Nˆ †Nˆ = γ | | Le− Le− 32β ( 0 0 − D E Z Z h 2β(L+y+z) 2β(y+z) 1 2β(L+z) +3Le− 3Le− + e− − 2β

2β(L+z) 2βz 2β(y+z) 2β(L+y+z) + e− e− +3e− 3e− − − (B.44)

2β(2L+z) 2β(2L+y+z) 2β(L+y+z) e− +3e− 3e− − −

2βL 2β(2L) 2βL 2β(L+y) +2 1 e− +2 e− e− +2 e− − − h i h i h 2β(2L+y) 2β(L+y) 2βy e− +2 e− e− dydz . − − ! ) i h i

135 Rewriting the equation, I have

4βL 8 L L 4 e− αp 2βz 2β(L+z) Nˆ †Nˆ Nˆ †Nˆ = γ | | Le− Le− 32β ( 0 0 − D E Z Z h 2β(L+y+z) 2β(y+z) 1 2β(L+z) +3Le− 3Le− + 2e− − 2β

2β(2L+z) 2βz 2β(y+z) 2β(L+y+z) (B.45) e− e− +3e− 6e− − − −

2β(2L+y+z) 2βL 2β(2L) +3e− +2 1 2e− + e− − h i 2β(L+y) 2β(2L+y) 2βy +2 2e− e− e− dydz . − − ! ) h i From this, I get

4βL 8 L 4 e− αp 2 2βz 2 2β(L+z) Nˆ †Nˆ Nˆ †Nˆ = γ | | L e− L e− 32β ( 0 − D E Z h 3L 2β(L+z) 2β(2L+z) 2β(L+z) 2βz + e− e− + e− e− 2β − − h i 1 2β(L+z) 2β(2L+z) 2βz + 2Le− Le− Le− 2β − −

3 2βz 2β(L+z) 2β(2L+z) 2β(L+z) (B.46) + e− e− +2e− 2e− 2β − − h 2β(2L+z) 2β(3L+z) 2βL 2β(2L) + e− e− +2L 1 2e− + e− − − i h i 1 2βL 4βL + 2e− 2e− β − h 6βL 4βL 2βL + e− e− + e− 1 dz . − − ! ) i

136 Rewritting the equation, I have

4βL 8 2 4 e− αp L 2βL 4βL Nˆ †Nˆ Nˆ †Nˆ = γ | | 1 2e− + e− 32β ("2β − D E h i 3L 2βL 4βL 6βL + 3e− 3e− + e− 1 4β2 − − h i 1 L 2βL 4βL 6βL + 3e− 3e− + e− 1 2β 2β − − h i 3 2βL 4βL 2βL (B.47) + 1 e− +3e− 3e− 4β2 − − h 4βL 6βL 8βL 6βL +3e− 3e− + e− e− − − i 2 2βL 2β(2L) +2L 1 2e− + e− − h i L 2βL 4βL 6βL + 3e− 3e− + e− 1 . β − − !#) h i From this, I get

4βL 8 4 e− αp L 2βL 4βL 6βL Nˆ †Nˆ Nˆ †Nˆ − − − =3γ | 2 | 3e 3e + e 1 64β ( β − − D E h i 1 2βL 4βL 8βL 6βL + 1 4e− +6e− + e− 4e− (B.48) 4β2 − − h i 2 2βL 4βL + L 1 2e− + e− . − ) h i Which reduces to

4βL 8 e α L 3 Nˆ Nˆ Nˆ Nˆ 4 − p 2βL † † =3γ | 2 | − 1 e− 64β ( β − D E (B.49) 1 2βL 4 2 2βL 2 + 2 1 e− + L 1 e− . 4β − − ) I deﬁne 2βL 1 e− L − . (B.50) eﬀ ≡ 2β

Substituting Leﬀ in the equation, I get

4βL 8 ˆ ˆ ˆ ˆ 4 e− αp 1 4 1 2 2 3 N†NN†N =3γ | | Leff + L Leff LLeff (B.51) 8 (2 2 − ) D E

137 B.5 Computing δEôut† δEôutδEôut† δEôut D E

If I rewrite equation (A.10) using Leﬀ , I get

4 4 4βL γ 8 4 µ e− αp Leﬀ | | ≈ 16| | (B.52) ν 4. ≈| |

Using equation (B.22), I find an expression for Nˆ †Nˆ , which is D E 2βL 2 e− 4 Nˆ †Nˆ = γ α (L L ) L . (B.53) 4 p − eff eff D E ˆ ˆ Using equation (B.21), I find an expression for δEout† δEout , which is D E 2 2βL γ e− 4 δEˆ † δEˆ = α LL . (B.54) out out 4 p eff D E I use these results in equation (B.32) to get

4 2 δEˆ † δEˆ δEˆ † δEˆ 3 µ +4 µ Nˆ †Nˆ + Nˆ †Nˆ Nˆ †Nˆ out out out out ≈ | | | | D E D 4 E D 3E 4 4βL 8 3Leff (L Leff ) Leff γ e− αp + − ≈ | | ( 16 4 (B.55) 3 1 1 + L4 + L2L2 LL3 8 2 eff 2 eff − eff ) 4 γ 4βL 8 2 2 3 4 = e− α 3L L 2LL +2L . 16 | p| eff − eff eff Putting everything together, I have

2 4 ˆ ˆ γ 4βL 8 2 2 3 4 ∆ δEout† δEout = e− αp 2L Leff 2LLeff +2Leff 16 | | − (B.56) 4 γ 4βL 8 2 2 3 4 = e− α L L LL + L 8 | p| eff − eff eff

138 2 B.6 Computing ∆ Eˆ out† Eˆout Since

2 2 2 ∆ Eˆ † Eˆ = δEˆ † δEˆ 2 Eˆ +2 Eˆ out out out out out ℜ out D E D E D E (B.57) 2 + δEˆ † δEˆ δEˆ † δEˆ δEˆ † δEˆ , out out out out − out out D E D E I have

2 2 2βL ˆ ˆ γ e− 4 2 2 ∆ Eout† Eout = αpLLeﬀ 2Gαs +2Gαs cos(2Φ10) 4 (B.58) 2 ˆ ˆ + ∆ δEout† δEout , where the expression of Φ10 is given by (A.21). The maximum noise in then when cos(2Φ10)=1. After some algebraic manipulation, I get

2 2 2 L 2 1 L L ˆ † ˆ (B.59) ∆ EoutEout = G αs + 2 +1 . Leff 8 Leff − Leff It can be verified that if β 0 −→

2 2 2 1 ∆ Eˆ † Eˆ = G α + , (B.60) out out s 8 which is the result I obtain in equation (3.16). Therefore, the output SNR is

4 αs SNRout = . (B.61) L 2 1 L2 L L αs + 8 L2 L +1 eff eff − eff B.7 Noise Figure

2 With SNRin = αs, I get the following expression of the noise ﬁgure

α2 NF = 10 log s . (B.62) −  L 2 1 L2 L  L αs + 8 L2 L +1 eff eff − eff  

139 Appendix C

Derivation of Noise for NMZI Based Kerr Medium with Gain

C.1 ASE of Kerr Medium with Gain

To get the ﬁeld output of Kerr Medium with gain instead of loss, I follow the same procedure as in section 4.2 The differences in the equation are that instead of β, I have

g , g being the gain coefﬁcient, and instead of Nˆ being a lowering operator, it is a − 0 0 raising operator. I get

iγ Aˆ (z) = αeg0z exp α 2 e2g0z 1 , (C.1) 2g | | − 0 D E L g0z e− iφ(z) iφ(z) δXˆ (L)= δXˆ (0) + Nˆ †e + Nˆ e− dz (C.2) 0 2 Z and γ δYˆ (L)= δYˆ (0) + α 2 e2g0L 1 δXˆ (0) g0 | | −

L z g0x 2 2g0z e− iφ(x) iφ(x) + 2γ α e Nˆ †e + Nˆ e− dxdz (C.3) 0 | | 0 2 Z Z L g0z e− iφ(z) iφ(z) + Nˆ e− Nˆ †e dz. 0 2i − Z Therefore,

L L 2 2 ′ 1 g0(z+z ) δXˆ (L) = δXˆ (0) + e− Nˆ †(z)Nˆ (z′) dzdz′ 4 0 0 D E D E Z Z D E (C.4) 2g0L 1 1 e− = + − , 4 4

ˆ 2 1 where I used δX (0) = 4 since our input signal is a coherent state. To calculate

2 D E δYˆ (L) , I ﬁrst make the following approximation based on the fact that α 2, the pump | | D E 140 power is very large

2 2 γ 2 2 ˆ 4 2g0L ˆ δY (L) = 2 α e 1 δX (0) 4g0 | | − D E 2 L L z Dz′ E ′ γ 4 g0(2z x+2z y)+i(φ(x) φ(y)) + α e − − − Nˆ †Nˆ dxdydz′dz 4 | | 0 0 0 0 Z Z Z Z D E 2 2 2g0L γ 4 2g0L 2 1 γ 4 2g0L e 1 = α e 1 + α 1 e− − L . 16g2 | | − 8 g | | − 2g − 0 0 0

γ2e2g0L e2g0L 1 L = α 4 e2g0L 1 − 4g0 | | − " 2g0 − 2 # 2 2g0L γ e 2 1 Lg0 = α 4 e2g0L 1 . 4g2 | | − 2 − 2(e2g0L 1) 0 " − # (C.6)

I deﬁne e2g0L 1 Leﬀ − . (C.7) ≡ 2g0

Therefore, I have

2 1 Lg0 Yˆ 2 2g0L 4 2 (C.8) δ (L) = γ e α Leﬀ 2g L . | | "2 − 2(e 0 1)# D E −

141 C.2 Noise Figure

The gain of this OPA is calculated using the same steps as in appendix A.3, but with

g instead of β. I obtain − 0

G = γ2eg0L α 4L2 . (C.9) | | eﬀ using the same argument as in section 4.2, it can be shown that δYˆ is the component of noise in phase with the output signal. Therefore, I can use the equation (3.54) to geet the noise ﬁgure, which is

G NF 10 log ≈ − 4ASE 2Lg = 10log 2 0 (C.10) − (e2g0L 1) − L = 10log 2 . − L eﬀ

142 Appendix D

Derivation of SA based NMZI-OPA Parametric Gain

D.1 Approximation of Γ2 Γ1 −

I substitute (6.19) and (6.20) in (6.21) and (6.22) and I obtain

P Γ = sat 1 E2 2E E sin(θ ) p − p s in 2 (D.1) 2 (Ep−2EpEs sin(θin) Ep 2EpEs sin(θin β0 W − e Psat − × Psat ! and

Psat Γ2 = 2 Ep +2EpEs sin(θin) 2 (D.2) 2 (Ep+2EpEs sin(θin) Ep +2EpEs sin(θin β0 W e Psat − . × Psat ! For any differentiable function f at x, to the ﬁrst order I have

f(x + ∆) f(x ∆) 2f ′(x)∆. (D.3) − − ≈

I choose ∆=2EpEs sin(θin). Therefore,

2 Γ Γ = Γ(∆) Γ( ∆) 2∆Γ′(E ), (D.4) 2 − 1 − − ≈ p

D.2 Derivative of the Saturated Loss

I set p 1 E2. Therefore, ≡ 2 p

p Psat p β0L Γ(p)= W e Psat − . (D.5) p P sat 143 Taking the derivative with respect to p

p p Psat p β0L Psat d p β0L Γ′(p)= W e Psat − + W e Psat − (D.6) − p2 P p dp P sat sat

p p p β0L p β0L For short hands, I will write W for W e Psat − and W ′ for W ′ e Psat − . Psat Psat Then the derivative of the gain can be written

p p Psat 1 p β0L β0L Γ′(p)= W + e Psat − + e Psat − W ′. (D.7) − p2 p P sat

Substituting for the derivative of the Lambert W function which is

dW (x) W (x) = (D.8) dx x (1 + W (x))

I get

p p Psat 1 p β0L β0L W Psat Psat Γ′(p)= W + e − + e − p 2 p β0L − p p Psat (1 + W ) e Psat − Psat P 1 p W sat (D.9) = 2 W + +1 − p p Psat (1 + W ) p Psat P 1 P W = sat W + sat +1 . − p2 p p (1 + W ) Using Γ as short for Γ(p) and substituting for

p Psat p β0L Γ(p)= W e Psat − , (D.10) p P sat

I get

Γ Γ 1 p Γ′(p)= + Γp +1 − p p 1+ Psat Psat (D.11) Γ(Γ 1) = − − Psat +Γp

144 D.3 Approximation of β2 β1 −

I denote β(p) ln(Γ(p)). Therefore, ≡ −

β(p + ∆) ln(Γ(p)) ∆Γ′(p) ≈ − − (D.12) Γ = ln(Γ(p)) ∆ ′ − − Γ Therefore,

Γ′ β′(p)= − Γ (D.13) (Γ 1) = − . Psat +Γp Since β = β(p + ∆) and β = β(p ∆), 2 1 − (Γ 1) β β 2∆ − 2 − 1 ≈ P +Γp sat (D.14) (Γ 1) =2EpEs − sin(θin). Psat +Γp

145 Appendix E

Bandwidth of the Saturable Absorber Based NMZI

E.1 Response of a Saturable absorber to non CW signal

I substitute for δβΩ(z) using (6.53) in (6.62) and obtain

d∆P˜ (z, Ω) β (z)∆P˜ (z, Ω) s = s s P (z) β (z)∆P˜ (z, Ω). (E.1) dz 1+ P (z)+ iΩτ − s s

I then factor for βs(z) and ∆P˜s(z, Ω), I get

1 d∆P˜ (z, Ω) P (z) s = β (z) 1 . (E.2) ˜ dz s 1+ P (z)+ iΩτ − ∆Ps(z, Ω)

I take the integrals. I get

L d∆P˜ (z, Ω) L P (z) s = β (z) 1 dz. (E.3) ∆P (z, Ω) s 1+ P (z)+ iΩτ − Z0 s Z0

We introduce a change of variable in right hand side of the equation X = P (z). From equation (6.59), we have dX = β (z)P (z). (E.4) dz − s

Therefore,

dX = β (z)P (z)dz. (E.5) − s

After substitution into the integral, I obtain

L d∆P˜ (z, Ω) P (L) 1 1 s = dX. (E.6) ∆P˜ (z, Ω) P X − 1+ X + iΩτ Z0 s Z (0)

146 The evaluation of the integrals leads to

L X P (L) ln ∆P˜s(z, Ω) = ln , (E.7) 0 1+ X + iΩτ P (0) with

P (0) β0z P (z)= P (0)W P (0)e − . (E.8) I can then solve for ∆P˜s(L, Ω) as follows

∆P (L, Ω) P (L)(1 + P (0) + iΩτ) ln s = ln . (E.9) ∆P (0, Ω) P (0)(1 + P (L)+ iΩτ) s

P (L) If I deﬁne the total steady state gain as Γs = P (0) , I then have

1+ P (0) + iΩτ ∆P (L, Ω) = ∆P (0, Ω)Γ . (E.10) s s s 1+ P (L)+ iΩτ

I deﬁne the total loss of an SA as

L L βs(z)dz δβ (z)dz Γ(Ω) = e− 0 − 0 Ω . (E.11) R R

I know that

L βs(z)dz Γs = e− 0 . (E.12) R

I deﬁne L ∆β(Ω) = δβΩ(z)dz. (E.13) Z0 Therefore,

Γ(Ω) = Γ (1 ∆β(Ω)). (E.14) s −

I then have

Γ (1 ∆β(Ω))(P (0)+∆P (0, Ω)) = P (L) + ∆P (L, Ω). (E.15) s − s s

147 I expand the equation, dropping the higher order terms. I get

Γ P (0)+Γ ∆P (0, Ω) Γ ∆β(Ω)P (0) = P (L) + ∆P (L, Ω). (E.16) s s s − s s

Since ΓsP (0) = P (L), I have

Γ ∆P (0, Ω) Γ ∆β(Ω)P (0) = ∆P (L, Ω). (E.17) s s − s s

Substituting for ∆Ps(L, Ω), I have

1+ P (0) + iΩτ Γ ∆P (0, Ω) Γ ∆β(Ω)P (0) = ∆P (0, Ω)Γ . (E.18) s s − s s s 1+ P (L)+ iΩτ

Solving for ∆β(Ω), I get

∆P (0, Ω) 1+ P (0) + iΩτ ∆β(Ω) = s 1 . (E.19) P (0) − 1+ P (L)+ iΩτ

Therefore, ∆P (0, Ω) P (L) P (0) ∆β(Ω) = s − , (E.20) P (0) 1+ P (L)+ iΩτ which can be rewritten

(1 Γs) ∆β(Ω) = − − ∆Ps(0, Ω). (E.21) 1+ΓsP (0) + iΩτ

This results is consistent with our previous results since for Ω=0 I obtain equation

(D.14).

E.2 Overall System Output

At the output of each SA, I have

βs+∆β(Ω) 1 (iαH +1) E1(L, Ω) = E1(0, Ω)e− 2 (E.22) √2 148 and

βs−∆β(Ω) 1 (iαH +1) E2(L, Ω) = E2(0, Ω)e− 2 , (E.23) √2 where β = ln(Γ ), s − s

1 iθin E1(0, Ω) = iAp A˜s(Ω)e (E.24) √2 − and

1 iθin E2(0, Ω) = Ap + iA˜s(Ω)e . (E.25) √2 − Therefore, the NMZI ﬁeld output at one of its arms is:

βs+∆β(Ω) 1 iθin (iαH +1) Eout = − i( iAp + A˜s(Ω)e )e− 2 2 − −

βs−∆β(Ω) iθin (iαH +1) +(Ap iA˜s(Ω)e )e− 2 − ! (E.26) βs−∆β(Ω) βs+∆β(Ω) 1 (iαH +1) (iαH +1) = − Ap e− 2 e− 2 2 − βs−∆β(Ω) βs+∆β(Ω) iθin (iαH +1) (iαH +1) iA˜s(Ω)e e− 2 + e− 2 . − ! βs ∆β(Ω) (iαH +1) (iαH +1) ∆β(Ω) factoring e− 2 and expanding e− 2 1 (iα + 1), I get ≈ − 2 H βs (iαH +1) e− 2 E − ∆g (iα + 1)A 2i(A˜ (Ω)eiθin ) out ≈ 2 Ω H p − s βs (E.27) (iαH +1) 2 e− 2 2(1 Γs)Ap sin(θin) = − − (iα + 1) 2ieiθin A˜ (Ω). 2 − 1+Γ A2 + iΩτ H − s s p E.3 Parametric Gain

I deﬁne

E˜ (Ω) A˜ (Ω)eiθin . (E.28) in ≡ s

Eout1 can then be rewritten

βs (iαH +1) (1 Γs)p sin(θin) iφ E = e− 2 − (iα +1)+ ie A˜ (Ω), (E.29) out1 1+Γ p + iΩτP H s s sat 149 2 where P is the power of the input pump on each arm of the NMZI: p = Ap. I reintroduce

Psat. Substituting for E˜in, I get

(1 Γ )p E˜ (Ω) E˜ (Ω) βs s in in∗ (iαH +1) − − Eout1 = e− 2 (iαH +1)+ iE˜in(Ω) . (E.30)  Psat +Γ sp + iΩτPsat    I deﬁne µ and ν such that

˜ ˜ Eout1 = µEin(Ω) + νEin∗ (Ω). (E.31)

Therefore, I have

βs (iαH +1) 1 (Γ 1) µ ie− 2 1 (iα + 1) − p (E.32) ≡ − 2 H Γ p + P + iΩτP s sat sat and

βs (iαH +1) 1 (Γ 1) ν ie− 2 (iα + 1) − p . (E.33) ≡ 2 H Γp + P + iΩτP sat sat I can be seen that for Ω=0, equation (E.32) and (E.33) are equivalent to equation (6.42) and (6.43) I can now ﬁnd the parametric gain, which is

E 2 Γ(Ω) = | out| A˜ (Ω) 2 | s | (E.34) iθ iθ 2 = µ(Ω)e in + ν(Ω) e− in . | | |

150 Appendix F

Quantum Mechanical Model for Interaction of Light with Saturable

Absorber

F.1 Solving for σˆ+

(iω21 γ)t I multiply equation (7.21) by e− −

dσˆ (t) iκ∗ (iω21 γ)t + (iω21 γ)t ˆ Aˆ Nˆ (iω21 γ)t e− − = e− − (iω21 γ) σ+(t)+ ~ † 1(t)e− − dt − (F.1) ˆ (iω21 γ)t + F+(t)e− − , which is

d (iω21 γ)t iκ∗ (iω21 γ)t (iω21 γ)t σˆ (t)e− − = Aˆ †Nˆ (t)e− − + Fˆ (t)e− − . (F.2) dt + ~ 1 +

Therefore,

t iκ∗ ′ ˆ ˆ (iω21 γ)t (iω21 γ)t Aˆ Nˆ (iω21 γ)t σ+(t)= σ+(0)e − + ~ e − †(t′) 1(t′)e− − dt′ Z0 (F.3) t ′ (iω21 γ)t ˆ (iω21 γ)t + e − F+(t′)e− − dt′. Z0 I deﬁne

t ′ iκ iω0t (iω21 γ)(t t ) Fˆ† e− Fˆ (t′)e− − − dt′. (F.4) ab ≡ ~ + Z0

I use this deﬁnition in the expression of σˆ+(t), I get

t iκ∗ ′ ˆ ˆ (iω21 γ)t (iω21 γ)t Aˆ Nˆ (iω21 γ)t σ+(t)= σ+(0)e − + ~ e − †(t′) 1(t′)e− − dt′ Z0 (F.5) ~ iω t + Fˆ† (t)e 0 . iκ ab

151 I assume sufﬁcient time lapse so that the initial conditions are forgotten by the system. I obtain

t ′ ~ iκ∗ (iω21 γ)t (iω21 γ)t iω0t σˆ (t)= e − Aˆ †(t′)Nˆ (t′)e− − dt′ + Fˆ† (t)e . (F.6) + ~ 1 iκ ab Z0

iω0t let Aˆ (t) Bˆ(t)e− . Nˆ (t) and the envelope Bˆ(t) are pretty much constant relative to ≡ 1

(iω21 γ)t the oscillation e − . The equation becomes

t ′ ~ iκ∗ (iω21 γ)t (i(ω21 ω0) γ)t iω0t σˆ (t)= e − Bˆ †(t)Nˆ (t) e− − − dt′ + Fˆ† (t)e . (F.7) + ~ 1 iκ ab Z0 After evaluating the integral, I get

Bˆ Nˆ ~ κ∗ †(t) 1(t) iω0t (iω21 γ)t iω0t σˆ (t)= e e − + Fˆ† (t)e . (F.8) + i~ (i(ω ω ) γ) − iκ ab 21 − 0 − Assuming that γ far more signiﬁcant than the oscillation ω21, I get

ˆ ˆ ~ κ∗ A†(t)N1(t) iω t σˆ (t)= + Fˆ† (t)e 0 . (F.9) + ~ ((ω ω ) iγ) iκ ab 0 − 21 −

F.1.1 Properties of Noise Source Fˆab(t)

ˆ ˆ ˆ ˆ Fab(t) is chosen so that A(t), A†(t) = 1 for all values of t. Without Fab(t), I h i have

dAˆ (t) = ℧Aˆ (t)Nˆ (t)+ iω Aˆ (t), (F.10) dt − 1 0 where ℧ ℧ + i℧ . Iset Cˆ ℧Nˆ (t)+ iω . This gives ≡ r i ≡ 1 0

dAˆ (t) = CˆAˆ (t). (F.11) dt −

I solve this equation and get

Cˆt Aˆ (t)= Aˆ (0)e− . (F.12)

152 Therefore,

(Cˆ +Cˆ †)t Aˆ (t), Aˆ †(t) = Aˆ (0), Aˆ †(0) e− (F.13) h i h i Cˆ +Cˆ † t = e−( ) , which is obviously wrong as it has to be one (1) regardless of t To ﬁx this, I add the following the noise term back

dAˆ (t) iω0t = CˆAˆ (t)+ Fˆ (t)e− (F.14) dt − ab and I require that that Fˆ (t), Fˆ† (t′) = Aδ(t t′). The solution is ab ab − h i t Cˆ Cˆ ′ ′ ˆ ˆ t (t t ) iω0t ˆ A(t)= A(0)e− + e− − − Fab(t′)dt′. (F.15) Z0 I compute the commutator

(Cˆ +Cˆ †)t Aˆ (t), Aˆ †(t) = Aˆ (0), Aˆ †(0) e− (F.16) h i h t t i ˆ ′ ′ ˆ † ′′ ′′ C(t t ) iω0t C (t t )+iω0t + A e− − − e− − δ(t′ t′′)dt′dt′′. − Z0 Z0 After some algebra, I get

t ˆ ˆ (C+C†)t (Cˆ +Cˆ †)(t t′) 1= e− + A e− − dt′. (F.17) Z0 I evaluate the integral

ˆ ˆ 1 (C+C†)t (Cˆ +Cˆ †)t − 1= e− + A 1 e− Cˆ + Cˆ † . (F.18) − Therefore,

A = Cˆ + Cˆ †

=2 (℧)Nˆ (t) (F.19) ℜ 1 ˆ =2℧rN1(t).

153 Therefore,

Fˆ (t), Fˆ† (t′) =2℧ Nˆ (t)δ(t t′). (F.20) ab ab r 1 − h i Remembering that I wrote Nˆ (t) because Nˆ (t) 0. In reality it was Nˆ (t) Nˆ (t). 1 2 ≈ 2 − 1 Therefore,

Fˆ (t), Fˆ† (t′) = 2℧ Nˆ (t) Nˆ (t) δ(t t′). (F.21) ab ab − r 2 − 1 − h i Therefore,

ˆ ˆ ℧ ˆ Fab† (t)Fab(t′) =2 r N2(t) δ(t t′) − (F.22) D E D E =0 and

Fˆ (t)Fˆ† (t′) =2℧ Nˆ (t) δ(t t′). (F.23) ab ab r 1 − D E D E ˆ Fab(t) is a lowering operator.

F.1.2 Photon Number Equation

By deﬁnition ˆn(t)= Aˆ †(t)Aˆ (t). Taking the derivative of this equation, I get

dˆn(t) dAˆ (t) dAˆ (t) = Aˆ †(t) + Aˆ (t). (F.24) dt dt dt

After substituting equation (F.14), I get

dˆn (t) iω0t iω0t = 2℧ Nˆ (t)ˆn(t)+ Aˆ †(t)Fˆ (t)e− + Fˆ† Aˆ (t)e . (F.25) dt − r 1 ab ab

154 The term 2℧ Nˆ (t)ˆn(t) represent absorption. Using equation (F.15), I can compute − r 1

ˆ ˆ iω0t ˆ ˆ iω0t A†(t)Fab(t)e− and Fab† (t)A(t)e , which contains absorbed noise. Taking the D E D E Hermitian conjugate of (F.15), I get

t Cˆ † Cˆ † ′ ′ ˆ ˆ t (t t )+iω0t ˆ (F.26) A†(t)= A†(0)e− + e− − Fab† (t′)dt′. Z0

ˆ iω0t I Multiply the equation by Fab(t)e−

t Cˆ † Cˆ † ′ ˆ ˆ iω0t ˆ ˆ ( +iω0)t ( +iω0)(t t ) ˆ ˆ A†(t)Fab(t)e− = A†(0)Fab(t)e− + e− − Fab† (t′)Fab(t)dt′. Z0 (F.27)

Since

ˆ ˆ (F.28) A†(0)Fab(t) =0 D E and

ˆ ˆ (F.29) Fab† (t′)Fab(t) =0, D E I have

ˆ ˆ iω0t (F.30) A†(t)Fab(t)e− =0. D E Similarly,

ˆ ˆ iω0t (F.31) Fab† (t)A(t)e− =0. D E

F.2 Noise From a Single Atom

ˆ I added FN (t) to equation (7.17) empirically. I remove it from (7.65). I consider separately the population decay. This is done by turning off the electric ﬁeld, setting

155 Aˆ (t)=0 in equation (7.65), and letting the system relax. For one single atom, I have:

dNˆ (t) 1 Nˆ (t) 1j = − 1j . (F.32) dt τ

I also know that

Nˆ (t)=1 Nˆ (t) Nˆ (t), (F.33) 3j − 1j − 2j because the number of carrier is conserved. Since Nˆ (t) 0, I have 2j ≈

Nˆ (t)=1 Nˆ (t). (F.34) 3j − 1j

Therefore our original equation, can be rewritten

dNˆ (t) Nˆ (t) 1j = 3j . (F.35) dt τ

Also,

dNˆ (t) dNˆ (t) 1j = 3j . (F.36) dt − dt

Therefore, the equation can be rewritten

dNˆ (t) Nˆ (t) 3j = 3j . (F.37) dt − τ

For a small ∆t, I have ˆ ˆ ˆ N3j(t) N3j(t + ∆t)= N3j(t) ∆t − τ (F.38) ∆t = Nˆ (t) 1 . 3j − τ ˆ ˆ ˆ Since N3j(t)N3j(t)= N3j(t) for any t, I have

2 2 ∆t 2 Nˆ (t + ∆t) = Nˆ (t) 1 3j 3j − τ (F.39) 2∆t Nˆ (t) 1 . ≈ 3j − τ 156 Therefore,

ˆ ˆ 2∆t N j(t + ∆t)= N j(t) 1 . (F.40) 3 3 − τ Since I started with

∆t Nˆ (t + ∆t)= Nˆ (t) 1 , (F.41) 3j 3j − τ ˆ something is wrong. Let’s add a noise term FNj(t) (phonon)

dNˆ (t) 1 Nˆ (t) 1j = − 1j + Fˆ (t), (F.42) dt τ Nj which can be rewritten

dNˆ (t) Nˆ (t) 3j = 3j + Fˆ (t). (F.43) dt − τ Nj

I require that

Fˆ (t)Fˆ (t′) = Aδ(t t′). (F.44) Nj Nj − D E Therefore,

∆t t+∆t Nˆ (t + ∆t)= Nˆ (t) 1 + Fˆ (t′)dt′. (F.45) 3j 3j − τ Nj Zt Multiplying this equation by itself, I get

2∆t 2∆t t+∆t Nˆ (t + ∆t)= Nˆ (t) 1 + Nˆ (t) 1 Fˆ (t′)dt′ 3j 3j − τ 3j − τ Nj Zt (F.46) t+∆t t+∆t ˆ ˆ + FNj(t′)FNj(t′′)dt′dt′′. Zt Zt When I take the average of the equation

2∆t Nˆ (t + ∆t) = Nˆ (t) 1 3j 3j − τ (F.47) D E D t+∆tE t+∆t ˆ ˆ + FNj(t′)FNj(t′′) dt′dt′′, t t Z Z D E 157 I get

2∆t Nˆ (t + ∆t) = Nˆ (t) 1 + A∆t. (F.48) 3j 3j − τ D E D E I choose

Nˆ (t) 3j (F.49) A = . D τ E

This implies that

∆t Nˆ (t + ∆t) = Nˆ (t) 1 , (F.50) 3j 3j − τ D E D E which works. Therefore ˆ N3j(t) Fˆ (t)Fˆ (t′) = δ(t t′) Nj Nj D E τ − (F.51) D E ˆ 1 N1j(t) = − δ(t t′). D τ E −

F.3 Noise From a Collection of Atom

I have for a single atom

dNˆ (t) 1 Nˆ (t) 1j = − 1j + Fˆ (t). (F.52) dt τ Nj

Therefore, for a collection of atoms, I have

N N dNˆ (t) 1 Nˆ (t) 1j = − 1j + Fˆ (t), (F.53) dt τ Nj j=1 j=1 X X which gives

N dNˆ (t) N Nˆ (t) 1 = − 1 + Fˆ (t). (F.54) dt τ Nj j=1 X 158 I deﬁne

N Fˆ (t) Fˆ (t). (F.55) N ≡ Nj j=1 X Then,

N N ˆ ˆ ˆ ˆ FN (t)FN (t′) = FNj(t)FNk(t′) . (F.56) j=1 k D E X X=1 D E Since Fˆ (t) and Fˆ (t) are uncorrelated for j = k, all the cross terms of the double Nj Nk 6 sum are zero. Therefore,

N ˆ ˆ ˆ ˆ FN (t)FN (t′) = FNj(t)FNj(t′) j=1 D E X D E N ˆ 1 N1j(t) = − δ(t t′) (F.57) D τ E − j=1 X ˆ N N1(t) = − δ(t t′). Dτ E −

159 Appendix G

Solving Differential Equation of SA

G.1 Solution for the Fluctuating Terms

Substituting equation (8.26) in equation (7.77), I get

d B(t) ˆ B(t)(1+ ˆx(t)+ iˆy(t)) = β(t)(iαH + 1) (1 + ˆx(t)+ iˆy(t)) dt − 2T (G.1) h i ˆ + Fab(t).

Therefore,

dB(t) d B(t) 1+ ˆx(t)+ iˆy(t) + B(t) ˆx(t)+ iˆy(t) = βˆ(t)(iα + 1) dt dt − 2T H (1 + ˆx(t)+ iˆy(t)) (G.2) × ˆ + Fab(t).

It is easy to show that by substituting Bˆ(t) = B(t)+ δBˆ(t)(t) and βˆ(t) = β(t)+ δβˆ(t) in equation (7.77) and separating the average terms from the ﬂuctuating terms, I get for the average terms

dB(t) 1 = β(t)(1+ iα ) B(t). (G.3) dt − 2T H

Substituting equation (G.3) in equation (G.2), I get

d δβˆ(t) Fˆ (t) ˆx(t)+ iˆy(t) = (iα + 1) 1+ ˆx(t)+ iˆy(t) + ab dt − 2T H B(t) (G.4) δβˆ(t) Fˆ (t) (iα + 1) + ab . ≈ − 2T H B(t)

160 where the last step is a ﬁrst order approximation. Taking the Hermitian conjugate of this equation, I get ˆ ˆ d δβ(t) Fab† (t) ˆx(t) iˆy(t) = (iαH 1) + . (G.5) dt − 2T − B∗(t) Summing the equation (G.4) and (G.5) and dividing the result by 2, I get

1 1 ∂ˆx(t) 1 B (t)Fˆ + B (t)Fˆ† = δβˆ(t)+ − ab ∗− ab (G.6) dt −2T 2

Subtracting the equation (G.4) and (G.5) and dividing the result by 2i, I get

1 1 dˆy(t) B− (t)Fˆ B∗− (t)Fˆ† α = ab − ab H δβˆ(t), (G.7) dt 2i − 2T

Check:

I assume that n , which is the case of unsaturable loss. In that case equation sat → ∞ (8.19) gives

dδβˆ(t) τ = δβˆ(t)+ τFˆ (t). (G.8) dt − L

ˆ I saw in equation (8.24) that under this condition the driving term FL(t) contains no power

ˆ ˆ and that FL(t)=0. Therefore, δβ(t)=0. Therefore,

1 1 ∂ˆx(t) B (t)Fˆ + B (t)Fˆ† = − ab ∗− ab . (G.9) dt 2

Therefore,

T 1 1 B (t)Fˆ + B (t)Fˆ† ˆx(T )= ˆx(0) + − ab ∗− ab dt. (G.10) 2 Z0 1 Tβ(t) ˆx(T )ˆx(T ) = ˆx(0)ˆx(0) + dt (G.11) h i h i 4T n(t) Z0

161 1 β0t Here, n(t)= n(0)e− T and β(t)= β0

T β0 1 β t ˆx(T )ˆx(T ) = ˆx(0)ˆx(0) + e T 0 dt h i h i 4T n(0) 0 TZ 1 β t e T 0 (G.12) = ˆx(0)ˆx(0) + 0 h i h 4n(0)i eβ0 1 = ˆx(0)ˆx(0) + − h i 4n(0)

β0 Multiplying the whole equation by n(T )= n(0)e− , I get

β 2 2 0 β0 1 e− δXˆ (T ) = e− δXˆ (0) + − , (G.13) 4 D E D E which is what is expected.

G.1.1 Finding an Expression for Fˆβ(t)Fˆβ(t′) D E ˆ At the scale at which I look at this problem, Fβ(t) is roughly a delta correlated

ˆ ˆ noise. In this section, I will compute the integral of Fβ(t)Fβ(t′) and rewrite it as a D E delta correlated function . I can rewrite equation (8.20) as such

Fˆ (t)Fˆ (t′) = Kδ(t t′), (G.14) L L − D E where n(t) ˆ β0 + 1 β(t) T nsat − (G.15) K 2 . ≡ nsat  τ D E

Also, equation (8.40) can be rewritten 

t Fˆ (t)= G(τ )Fˆ (t τ )dτ , (G.16) β 1 L − 1 1 Z0 where

τ 1 1+ n(t) 1 nsat τ (G.17) G(τ1)= e− . 2T

162 Therefore,

t t τ1 − Fˆ (t)Fˆ (t τ ) = G(τ )G(τ ) Fˆ (t τ )Fˆ (t τ τ ) dτ dτ β β − 1 2 3 L − 2 L − 1 − 3 2 3 Z0 Z0 D E t t τ1 D E − = K G(τ )G(τ )δ(t τ t + τ + τ )dτ dτ 2 3 − 2 − 1 3 2 3 Z0 Z0 t = K G(τ3 + τ1)G(τ3)dτ3, Z0 (G.18) since t′ t = τ . Substituting for G( ),I get − − 1 ·

t τ τ K 1+ n(t) 2 3+ 1 ˆ ˆ n τ Fβ(t)Fβ(t τ1) = e− sat dτ3 2 − 4T 0 D E Z 1 t τ τ τK n(t) − 1+ n(t) 2 + 1 1+ n(t) 1 = 1+ e− nsat τ e− nsat τ − 8T 2 n − sat 1 τ τK n(t) − 1+ n(t) 1 = 1+ e− nsat τ . 8T 2 n sat (G.19)

ˆ I look for the total energy of Fβ(t) as follows

1 τ ∞ ∞ τK n(t) − 1+ n(t) 1 ˆ ˆ n τ Fβ(t)Fβ(t τ1) dτ1 =2 1+ e− sat dτ1 2 − 0 (8T nsat ) Z−∞D E Z (G.20) 2 τ 2K n(t) − 1+ . ≈ 4T 2 n sat Therefore,

2 τ 2K n(t) − Fˆ Fˆ ′ ′ (G.21) β(t) β(t ) = 2 1+ δ(t t ) 4T nsat − D E

G.1.2 Finding an Expression for Fˆab(t)Fˆβ(t′) D E Equation (8.23) can be rewritten

Fˆ (t)Fˆ (t′) = K δ(t t′), (G.22) ab L 1 − D E 163 where B(t) K1 β(t). (G.23) ≡ −nsatτ ˆ I take equation (G.16) and multiply it by Fab(t) and take the average, I get

t+τ1 Fˆ (t)Fˆ (t + τ ) = G(τ ) Fˆ (t)Fˆ (t + τ τ ) dτ ab β 1 2 ab L 1 − 2 2 Z0 D E t τ1 D E − = K G(τ )δ(t t τ + τ )dτ 1 2 − − 1 2 2 Z0 (G.24)

= K1G(τ1)

τ K1 1+ n(t) 1 = e− nsat τ . 2T I take the integral of the expression, I get

τ ∞ ∞ K1 1+ n(t) 1 ˆ ˆ n τ Fab(t)Fβ(t + τ1) dτ1 =2 e− sat dτ1 0 2T Z−∞ Z (G.25) D E 1 τK n(t) − 1 1+ . ≈ T n sat Therefore,

1 ˆ ˆ τK1 n(t) − Fab(t)Fβ(t′) 1+ δ(t t′). (G.26) ≈ T nsat − D E Similarly,

1 ˆ ˆ τK1 n(t) − Fβ(t′)Fab(t) 1+ δ(t t′). (G.27) ≈ T nsat − D E

164 G.2 Calculating ˆx(T )ˆx(T ) h i

ˆ I take equation (8.43) and use equation (8.39) to substitute the expression of Nx(t),

I get

ˆx(T )ˆx(T ) = ˆx(0)ˆx(0) H (0)2 h i h i 2 T T ˆ ˆ + H2(t′)H2(t′′) Fβ(t′)Fβ(t′′) dt′dt′′ Z0 Z0 T T D 1 E B− (t) ˆ ˆ + H2(t′)H2(t′′) Fab(t′)Fβ(t′′) (G.28) 0 0 2 Z Z D E 1 B∗− (t) ˆ ˆ + Fβ(t′′)Fab† (t′) dt′dt′′ 2 ! D E T T 2 B(t′) − ˆ ˆ + H2(t′)H2(t′′)| | Fab(t′)Fab† (t′′) dt′dt′′. 0 0 4 Z Z D E Using equations (G.26),(G.27), (G.21) and (7.72) in the above expression, I obtain

ˆx(T )ˆx(T ) = ˆx2(0) H2(0) h i h i n(t′) T β0 + 1 β(t) 1 2 β(t′) nsat ′ − d ′ + H (t )  + 2  t 4T 0 n(t′) n(t′) (G.29) Z nsat n +1  sat  T   1 2 β(t′) + H (t′) dt′. n(t′) T 0 n +1  Z sat nsat   check:

I set n . Then H(t) 1 and sat →∞ →

T 2 1 β(t′) ˆx(T )ˆx(T ) = ˆx (0) + dt′. (G.30) h i h i 4T n(t ) Z0 ′ This gives us

β0 β0 1 e− δRˆ (T )δRˆ (T ) = e− δRˆ (0)δRˆ (0) + − (G.31) 4 D E D E as before.

165 G.2.1 Ampliﬁed Initial Noise

The incoming signal’s noise is ampliﬁed. At the output, it becomes the ﬁrst expression of equation (G.29), which is

Init =H2(0) ˆx2(0) . (G.32) x 2 h i

Using equation (8.42), I obtain

1 n(0) + n 2 1 Init = sat . (G.33) x 4 n(T )+ n n(0) sat

G.2.2 Noise due to Absorption

I compute the ﬁrst integral of equation (G.29), which is noise due to absorption. I get

T 1 2 β(t′) Noise = H (t′) dt′ absx 4T n(t ) Z0 ′ T 2 1 n(t′)+ nsat β(t′) = dt′ 4T n(T )+ n n(t ) Z0 sat ′ 1 n(T ) n + n 2 1 = sat dn − 4 n(T )+ n n2 (G.34) Zn(0) sat 2 n(T ) nsat 2nsat ln(n)+ n n 1 − n(0) = h 2 i − 4 (n(T )+ nsat) n2 n2 2n ln n(0) + sat sat + n(0) n(T ) 1 sat n(T ) n(T ) − n(0) − = 2 , 4 (n(T )+ nsat) As a check, I set, as n (nonsaturable loss). The above expression converges to sat →∞ 1 1 1 Noise = absx 4 n(T ) − n(0) (G.35) 1 eβ0 1 = − , 4 n(0) which is expected.

166 G.2.3 Noise due to Relaxation

I compute the second integral of equation (G.29), which is noise due to relaxation.

I do it as follows

n(t′) T β0 + 1 β(t′) 1 2 nsat Noise ′ − d ′ (G.36) relx = H (t )  2  t . 4T 0 n(t′) Z nsat n +1  sat    Since

β β(t)= 0 , (G.37) n(t) +1 nsat

I can make a change of variable and evaluate the integral as follows

n(t′) T 1 1 1 nsat Noise 2 − d relx = β(t′)H (t′)  + 2  t′ 4T 0 n(t′)+ nsat n(t′) Z nsat n +1  sat   n  n(T ) 2 1 1 1 n + n 1 nsat (G.38) sat − d =  + 2  n − 4 n(0) n n(T )+ nsat n + nsat n Z nsat n +1  sat  1 n(0) n(T )   = − 2 . 2 (n(T )+ nsat)

G.2.4 Beat Noise

I compute the third integral of equation (G.29), which is the beat noise between the relaxation noise and the absorption noise. I evaluate it as follows

T 1 2 β(t′) Noisebeat = H (t′) dt′ x n t′ T 0 n ( ) +1  Z sat nsat   n(T )1 n + n 2 1 = sat dn (G.39) − n(0) n n(T )+ nsat n n +1  Z sat nsat n(0)   n(0) n(T )+ n ln − sat n(T ) = 2 . (n(T )+ nsat)

167 G.2.5 Total Amplitude Noise

I get an expression for the total amplitude noise from the saturable absorber, which is

ˆx(T )ˆx(T ) = Init + Noise + Noise + Noise h i x absx relx beatx n n2 n2 2 (0) sat sat 1 n(0) + n 1 1 2nsat ln n(T ) + n(T ) n(0) + n(0) n(T ) = sat + − − 4 n(T )+ n n(0) 4 2 sat (n(T )+ nsat) n(0) n(T )+ n ln n(0) 1 n(0) n(T ) − sat n(T ) + − 2 + 2 4 (n(T )+ nsat) (n(T )+ nsat) n n2 n2 2 (0) sat sat 1 n(0) + n 1 1 6nsat ln n(T ) + n(T ) n(0) = sat + − 4 n(T )+ n n(0) 4 2 sat (n(T )+ nsat) 3 (n(0) n(T )) + − 2 . 2 (n(T )+ nsat) (G.40)

G.3 Calculating ˆy(T )ˆy(T ) h i

To get the phase noise, I use equation (G.7)

1 1 ∂ˆy(t) B− (t)Fˆ B∗− (t)Fˆ† α = ab − ab H δβˆ(t) (G.41) ∂t 2i − 2T and equation (G.6)

1 1 ∂ˆx(t) 1 B (t)Fˆ + B (t)Fˆ† = δβˆ(t)+ − ab ∗− ab . (G.42) ∂t −2T 2

I deﬁne

1 1 B− (t)Fˆ B∗− (t)Fˆ† Nˆ (t) ab − ab (G.43) pi ≡ 2i and 1 1 B (t)Fˆ + B (t)Fˆ† Nˆ (t) − ab ∗− ab . (G.44) pr ≡ 2 168 Therefore, equation (G.6) can be written

1 ∂ˆx(t) δβˆ(t)= Nˆ (t) (G.45) −2T ∂t − pr and equation (G.7) can be written

∂ˆy(t) ∂ˆx(t) = Nˆ (t)+ α Nˆ (t) . (G.46) ∂t pi H ∂t − pr Algebraic manipulation of equation (G.7) and equation (G.6) gives us

∂ˆy(t) ∂ˆx(t) α = Nˆ (t) α Nˆ (t). (G.47) ∂t − H ∂t pi − H pr

From this, I can write an expression of the phase noise, which is

t ˆ ˆ ˆy(t) αH ˆx(t)= ˆy(0) αH ˆx(0) + Npi(t) αH Npr(t) dt′. (G.48) − − 0 − Z The power of the phase noise is

ˆy(t)2 = ˆy2(0) + α2 ˆx2(0) + α2 ˆx(t)2 2α2 ˆx(0)ˆx(t) H H − H h i t t ˆ ˆ 2 ˆ ˆ + Npi(t′′)Npi(t′) + αH Npr(t′′)Npr(t′) 0 0 " Z Z D E D E ˆ ˆ ˆ ˆ αH Npr(t′′)Npi(t′) + Npi(t′′)Npr(t′) dt′dt′′ (G.49) − # D E D E t 2 ˆ ˆ αH Npr(t′)ˆx(t) + ˆx(t)Npr(t′) − 0 " Z D E D E ˆ ˆ + αH ˆx(t)Npi(t′) + Npi(t′)ˆx(t) dt′. # D E D E I use in the above expression the fact that ˆy2(0) = ˆx2(0) = 1 , h i h i 4n(0)

ˆ ˆ ˆ ˆ Npi(t′′)Npi(t′) = Npr(t′′)Npr(t′) (G.50) D E D E and

Nˆ (t′′)Nˆ (t′) = Nˆ (t′′)Nˆ (t′) . (G.51) pr pi − pi pr D E D E 169 I also use the following relation

T T T ˆ ˆ 1 β(t′) Npi(t′′)Npi(t′) dt′dt′′ = dt′ 0 0 4T 0 n(t′) Z Z D E Z 1 n(T ) 1 = dn (G.52) − 4 n2 Zn(0) 1 1 1 = . 4 n(T ) − n(0) Therefore, the expression of the power of the phase noise becomes

(1 + α2 ) 1 1 1 ˆy2(T ) = H + + α2 ˆx2(T ) 2α2 ˆx(0)ˆx(t) 4 n(0) n(T ) − n(0) H − H h i t 2 ˆ ˆ αH Npr(t′)ˆx(t) + ˆx(t)Npr(t′) − 0 " Z D E D E ˆ ˆ + αH ˆx(t)Npi(t′) + Npi(t′)ˆx(t) dt′. # D E D E (G.53)

I can see that

ˆx(t)Nˆ (t′) = Nˆ (t′)ˆx(t) (G.54) pi − pi D E D E and

ˆ ˆ Npr(t′)ˆx(t) = ˆx(t)Npr(t′) . (G.55) D E D E Therefore,

(1 + α2 ) 1 ˆy2(T ) = H + α2 ˆx2(T ) 2α2 ˆx(0)ˆx(t) 4 n(T ) H − H h i (G.56) T 2 ˆ 2αH Npr(t′)ˆx(t) dt′. − 0 Z D E I rewrite the integral in the expression as follows

T T ˆ 1 β(t′) 2 Npr(t′)ˆx(t) dt′ = H(t′) dt′ 0 2T 0 n(t′) Z D E Z T (G.57) 1 β(t′) + H(t′) dt′. n(t′) T 0 n +1  Z sat nsat   170 Using equation (8.42) and our usual change of variable, I have

T 3 ln(n) nsat n(0) ˆ 1 n 2 Npr(t′)ˆx(t) dt′ = − 0 2 n(T )+ nsat n(T ) Z D E n(0) nsat nsat (G.58) 1 3 ln n(T ) + n(T ) n(0) = − 2  n(T)+ nsat    Substituting this answer in the expression of the phase noise, I get

(1 + α2 ) 1 ˆy2(T ) = H + α2 ˆx2(T ) 2H (0)α2 ˆx2(0) 4 n(T ) H − 2 H n(0) n n (G.59) 2 sat sat α 3 ln n(T ) + n(T ) n(0) H − . − 2  n(T)+ nsat    Finally, using equation (8.42), I get

(1 + α2 ) 1 α2 1 n(0) + n ˆy2(T ) = H + α2 ˆx2(T ) H sat 4 n(T ) H − 2 n(0) n(T )+ n sat n(0) n n (G.60) 2 sat sat α 3 ln n(T ) + n(T ) n(0) H − . − 2  n(T)+ nsat    As a check, I set n (unsaturable loss case). Then from equation (G.12), the sat →∞ expression of the amplitude noise is

1 1 1 ˆx2(T ) = ˆx2(0) + 4 n(T ) − n(0) (G.61) 1 1 = . 4 n(T ) Therefore, equation (G.60) becomes

(1+2α2 ) 1 α2 1 α2 1 1 ˆy2(T ) = H H H 4 n(T ) − 2 n(0) − 2 n(T ) − n(0) 1 1 (α2 α2 ) 1 1 = + H − H (G.62) 4 n(T ) 2 n(T ) − n(0) 1 1 = , 4 n(T ) which is what was expected.

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[34] M Romagnoli, S Wabnitz, P Franco, M Midrio, F. Fontana, G. E. Town, ”Tunable erbium-ytterbium ﬁber sliding-frequency soliton laser”,Journal Optical Society of America B Vol. 12, No. 1, January 1995, p 72-76

[35] Caves, Carlton M. ”Quantum limits on noise in linear ampliﬁers”,Phys. Rev. D Vol. 26, No. 8, Oct. 1882, p 1817-1839

[36] Stephen B. Alexander,1997, Optical Communication Receiver Design, SPIE Optical Engineering Press, p. 68

[37] Wendell T. Hill, Chi H. Lee,2007, Light-matter Interaction: Atoms and Molecules in External Fields and Nonlinear Optics, Wiley-VCH, p. 77

[38] Fritz Riehle,2004, Frequency Standards: Basics and Applications, Wiley-VCH, p. 135

174 [39] H Ebendorff-Heidepriem,P. Petropoulos, V. Finazzi, K. Frampton,R.C. Moore, D. J. Richardson, T. M. Monro ’Highly Nonlinear Bismuth-Oxide-Based Holey Glass Fiber’, Presented at OFC 2004, Los Angeles, California, paper ThA4.

[40] P. Petropoulos, H. Ebendorff-Heidepriem, V. Finazzi, R.C. Moore, K. Frampton, D.J. Richardson, T.M. Monro, ’Highly nonlinear and anomalously dispersive lead silicate glass holey ﬁbers’. Optics Express Vol. 11, No. 26, Dec 2003

[41] H. A. Haus,The Noise Figure of Optical Ampliﬁers IEEE Photonics Technology Letters, Vol. 10, No. 11, NOV. 1998, pp 1602-1604

[42] E. Desurvire, D. Bayart, B. Desthieux, S. Bigo, 2002 Erbium Doped Fiber Ampli- ﬁersWiley pp. 184

[43] H. T. Friis, Noise ﬁgure of radio receivers, Proc. IRE, vol. 32, pp.419422, 1944.

[44] Weber J. ”Fluctuation Dissipation Theorem”. Physical Review 101 1956 p. 16201626.

[45] Govind P. Agrawal, 1998. Fiber Optic Communication Systems, Wiley-Interscience, pp. 166.

[46] E. T. Jaynes and F. W. Cummings, ”Comparison of quantum and semiclassical ra- diation theories with application to the beam maser”, Proc. IEEE Vol 51, pp 89 (1963).

[47] Hermann A. Hauss, 2000. Electromagnetic Noise and Quantum Optical Measure- ments, Springer, pp. 282.

[48] Ibid, pp. 285.

[49] Siegman, A. E., 1986, Lasers, Mill Valley, University Science Books, pp. 326

[50] Ibid, pp. 89

[51] Ibid, pp. 286

[52] Ibid, Chapter 10.2

[53] Christopher C. Gerry, Peter L. Knight, 2005, Introductory Quantum Optics, Cam- bridge University Press, pp. 138-139

175 [54] Loudon, Rodney, 2000, The Quantum Theory of light, Oxford University Press, Chapter 9, pp. 413

[55] Ibid, pp. 142

[56] Ibid, pp. 153

[57] Hermann A. Hauss, 2000. Electromagnetic Noise and Quantum Optical Measure- ments, Springer, pp. 311.

[58] Ibid, pp. 314.

[59] Loudon, Rodney, 2000, The Quantum Theory of light, Oxford University Press, pp. 134

[60] Hermann A. Hauss, 2000. Electromagnetic Noise and Quantum Optical Measure- ments, Springer, pp. 393.

[61] Farhan Rana, Professor at Cornell University. Notes can be found at http://courses.cit.cornell.edu/ece531/Lectures/Lectures.htm.

[62] Loudon, Rodney, 2000, The Quantum Theory of light, Oxford University Press, pp. 167

[63] Marlan Orvil Scully, Muhammad Suhail Zubairy, 1997, Quantum Optics, Cam- bridge University Press, pp. 273

[64] Miguel Orszag, 2000, Quantum Optics: Including Noise Reduction, Trapped Ions, Quantum Trajectories, and Decoherence, Springer, pp. 105

[65] Chad Ropp, Julius Goldhar, Nonlinear Mach-Zehnder Interferometer as a DPSK Signal Regenerator. University of Maryland ECE department MERIT research pro- gram’s report, 2006.

[66] M. Shtaif, B. Tromborg and G. Eisenstein, ”Noise Spectra of Semiconductor Opti- cal Ampliﬁers: Relation between Semiclassical and Quantum Descriptions” IEEE Journal of Quantum Electronic, Vol 34, No 5 pp. 869-878, May 1998.

[67] Y. Yamamoto, S. Machida and O. Nilson, ’Amplitude Squeezing in a Pump-Noise- Suppressed Laser Oscillator’ Physical Review A, Vol. 34 No 5, Nov. 1986

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