COHERENT DETECTION OF ULTRA-WEAK ELECTROMAGNETIC … · J.J. Thomson [3] physicists have been eagerly searching for more pieces to the cosmic puzzle. In a little over 100 years since

COHERENT DETECTION OF ULTRA-WEAK ELECTROMAGNETIC FIELDS USINGOPTICAL HETERODYNE INTERFEROMETRY

By

ZACHARY RONALD DYLAN THOMAS BUSH

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2018

© 2018 Zachary Ronald Dylan Thomas Bush

To my parents, who guided and nurtured me to success

ACKNOWLEDGMENTS

First, I would like to thank my friends and colleagues for their tremendous help and

support. Our useful discussions, often over coffee and cookies, provided the much needed

insight to get me to where I am today. In particular I would like to express my thanks to Simon

Barke, Johannes Eichholz, Harold Hollis, Aaron Spector, Ayman Hallal, Giuseppe Messineo,

Mauricio Diaz-Ortiz, Ryan Goetz, and Paul Fulda. I would also like to thank my advisors Dr.

Guido Mueller and Dr. David Tanner for their support and understanding throughout my

graduate career. Finally, I would like to thank my parents and family for always being there for

me with a guiding hand.

4

TABLE OF CONTENTSpage

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1 Beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 CP Violation and the Strong CP Problem . . . . . . . . . . . . . . . . . . . . 12

1.2.1 CP Symmetry Breaking in the QCD Lagrangian . . . . . . . . . . . . 131.2.2 The Peccei-Quinn Solution . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2.1 The QCD Axion . . . . . . . . . . . . . . . . . . . . . . . . 151.2.2.2 Axion-like Particles . . . . . . . . . . . . . . . . . . . . . . 151.2.2.3 Constraints on the Axion . . . . . . . . . . . . . . . . . . . 15

1.3 Axions/ALPs: More than a Strong CP Solution . . . . . . . . . . . . . . . . 171.3.1 TeV Transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.2 Anamolous White Dwarf Cooling . . . . . . . . . . . . . . . . . . . . 181.3.3 Axions as Dark Matter Candidates . . . . . . . . . . . . . . . . . . . 18

1.4 Axion/ALP Search Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.1 Helioscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.2 Haloscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4.3 Light Shining through Walls . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 Structure of this Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 THE ANY LIGHT PARTICLE SEARCH . . . . . . . . . . . . . . . . . . . . . . . 22

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Overview of the ALPS Experiment . . . . . . . . . . . . . . . . . . . . . . . 222.3 Resonance Enhancement Techniques in ALPS . . . . . . . . . . . . . . . . . 232.4 Past and Future Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Other LSW Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 252.4.2 ALPS I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.3 ALPS IIa and ALPS IIc . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.4 ALPS Experimental Sensitivities . . . . . . . . . . . . . . . . . . . . . 27

2.5 Detection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 OPTICAL HETERODYNE INTERFEROMETRY THEORY . . . . . . . . . . . . . 30

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Mathematical Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5

3.2.1 Signal Present Unknown Phase . . . . . . . . . . . . . . . . . . . . . 323.2.1.1 Demodulation at the signal frequency . . . . . . . . . . . . 323.2.1.2 Demodulation away from the signal frequency . . . . . . . . 34

3.2.2 Z(N) and the Discrete Fourier Transform . . . . . . . . . . . . . . . . 353.2.3 Noise Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.4 Confident Detection Threshold . . . . . . . . . . . . . . . . . . . . . 383.2.5 Signal and Noise Combined . . . . . . . . . . . . . . . . . . . . . . . 403.2.6 Summary of Output Behaviors . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Noise Sources in Optical Heterodyne Detection . . . . . . . . . . . . . . . . . 433.3.1 Relative Intensity Noise . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.2 Dark Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.3 ADC Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.4 Quantization Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.5 Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Fundamental Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.5 Double Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.1 Influence on Signal Behavior . . . . . . . . . . . . . . . . . . . . . . . 503.5.2 Influence on Noise Behavior and Confident Detection . . . . . . . . . . 52

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 HETERODYNE DETECTION EXPERIMENTAL SETUP . . . . . . . . . . . . . . 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Optical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Phase Lock Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.2 Electro-Optic Modulation . . . . . . . . . . . . . . . . . . . . . . . . 604.2.3 Polarization Considerations . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Analog Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.1 Notch Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2 Bandpass Filters and Analog Voltage Amplification . . . . . . . . . . . 634.3.3 Measurement of the Combined Analog Gain . . . . . . . . . . . . . . 664.3.4 Measurement of the Photodetector Gain and Quantum Efficiency . . . 674.3.5 Ensuring a Shot-Noise Limited System . . . . . . . . . . . . . . . . . 69

4.4 Digital Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.4.3 Digital PLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2 Simulated Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2.1 Simulated Noise Behavior . . . . . . . . . . . . . . . . . . . . . . . . 795.2.2 Simulated Signal Behavior and Confident Detection . . . . . . . . . . 80

5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6

5.3.1 Noise Behavior and Device Sensitivity . . . . . . . . . . . . . . . . . . 825.3.2 Optical Signal Genearation and Detection . . . . . . . . . . . . . . . . 84

5.3.2.1 Calculating the Expected Signal Photon Rate . . . . . . . . 845.3.2.2 Calibration Using Stronger Signal Fields . . . . . . . . . . . 855.3.2.3 Detection of an Ultra-Weak Signal Field . . . . . . . . . . . 87

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 IMPROVEMENTS, IMPLEMENTATION IN ALPS, AND FURTHER RESEARCH . 92

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.2 Improvements to Optical Heterodyne Detection . . . . . . . . . . . . . . . . 93

6.2.1 Elimination of Added Noise Due to Multiple Demodulation Stages . . . 936.2.2 Demodulation with a Known Signal Phase . . . . . . . . . . . . . . . 986.2.3 Implementation of Both Noise Reduction Techniques . . . . . . . . . . 1026.2.4 Future Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.2.5 Hardware Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.3 Heterodyne Detection in ALPS IIc . . . . . . . . . . . . . . . . . . . . . . . 1076.4 Fractional Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.5 Future Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5.1 Two-Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.5.2 Three-Laser Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

APPENDIX

A MATLAB SCRIPTS FOR SECOND DEMODULATION AND POST-PROCESSING . 120

B MATLAB SCRIPTS FOR SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . 128

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7

LIST OF TABLESTable page

4-1 Linear Spectral Density Measurement at the Photodetector Output . . . . . . . . . 71

4-2 Linear Spectral Density Measurement After Analog Components . . . . . . . . . . . 72

5-1 Parameters for a Simulated Signal Measurement. . . . . . . . . . . . . . . . . . . . 80

5-2 Parameters for a Strong Signal Calibration Measurement. . . . . . . . . . . . . . . 86

5-3 Multiple Calibration Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5-4 Parameters for an Ultra-weak Signal Field Measurement. . . . . . . . . . . . . . . . 88

6-1 Test Parameters and Results for Double Demodulation Noise Suppression. . . . . . 97

6-2 Test Parameters and Results for Phase Search Techniques . . . . . . . . . . . . . . 101

6-3 Measured Photon Rates for Various Improvement Techniques . . . . . . . . . . . . 105

8

LIST OF FIGURESFigure page

1-1 Region of Parameter Space for the Axion Mass, ma, vs. its Two Photon Coupling,gaγγ, with Various Experimental Sensitivities. Figure from Ref. [1] . . . . . . . . . . 16

1-2 Axions and TeV Transparency in the Universe . . . . . . . . . . . . . . . . . . . . . 17

1-3 General Light Shining Through a Wall Experiment . . . . . . . . . . . . . . . . . . 20

2-1 Simplified Design of the ALPS II Experiment . . . . . . . . . . . . . . . . . . . . . 22

2-2 Exclusion Limits Set by ALPS I (In Vacuum) and Projected Design Sensitivity ofALPS IIc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2-3 Voltage Pulse from a Transition Edge Sensor. Figure from Ref. [2] . . . . . . . . . 29

3-1 Basic Concept of Heterodyne Interferometry . . . . . . . . . . . . . . . . . . . . . 30

3-2 Example Single-sided PSD Detailing a Locally Flat Region around fd. . . . . . . . . 39

3-3 Expectation Behaviors and 5-Sigma Detection Threshold . . . . . . . . . . . . . . . 42

3-4 Single-Sided FFT of the Time Series Out of the FPGA Showing Experimental DCBias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3-5 Single-Sided FFT of the Time Series Out of the FPGA with a Beat Note at Frequencyfδ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4-1 Optical Design to Test Heterodyne Interferometry for Ultra-weak Signal Fields. . . . 58

4-2 Frequency Space Describing Beat Note Generation between First-order Sidebandsand the Local Oscillator Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4-3 Measured Transfer Function of a Notch Filter Showing Attenuation at 30 MHz . . . 63

4-4 Noise LSD vs. Frequency Directly After Measurement Photodiode . . . . . . . . . . 64

4-5 Result of Bandpass Filtering to Attenuate Relative Intensity Noise . . . . . . . . . . 65

4-6 Time-series After Bandpass Filters and Amplification Stages . . . . . . . . . . . . . 66

4-7 Chain of Analog Components After the Measurement Photodetector PD2 . . . . . . 66

4-8 Measurement of a Function Generator Signal Before Analog Amplification . . . . . 67

4-9 Measurement of a Function Generator Signal After Analog Amplification . . . . . . 68

4-10 Measurement Photodetector Circuit Diagram . . . . . . . . . . . . . . . . . . . . . 68

4-11 Measurement of Carrier-Carrier Beat Note to Determine Photodetector Gain . . . . 69

9

4-12 Noise LSD Measurements Before Analog Components . . . . . . . . . . . . . . . . 70

4-13 Noise LSD Measurements After Analog Components . . . . . . . . . . . . . . . . . 72

4-14 Digital Design of the Heterodyne Detector . . . . . . . . . . . . . . . . . . . . . . 73

4-15 Simulink Block Diagram Interface for FPGA Configuration . . . . . . . . . . . . . . 75

5-1 Simulated Noise Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5-2 Simulated Signal Measurement of 100 Photons/Second . . . . . . . . . . . . . . . 81

5-3 Experimental 19 Day Shot-Noise Limited Noise Measurement . . . . . . . . . . . . 83

5-4 Real Optical Signal Measurement of 8.3× 106 Photons/Second . . . . . . . . . . . 87

5-5 Real Optical Signal Measurement of 3.3 ×10−2 Photons/Second . . . . . . . . . . 89

6-1 FFT of the Time Series out of the FPGA Demonstrating Additional Noise PickupDue to Double Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6-2 Digital Design To Eliminate Noise Pickup Due to Second Demodulation . . . . . . . 95

6-3 Measurement Demonstrating Suppression of Noise Pickup from Double Demodulation 97

6-4 Signal Measurement when Suppressing Noise Pickup from Double Demodulation . . 98

6-5 Measurement Demonstrating Noise Suppression Using Phase Search Techniques . . 100

6-6 Signal Measurement Using Phase Search Techniques to Suppress Noise Pickup . . . 102

6-7 Noise Measurement Implementing Both Double Demodulation Suppression and PhaseSearch Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6-8 Signal Measurement Demonstrating Noise Suppression Using Both Techniques . . . 105

6-9 Heterodyne Implementation in ALPS IIc . . . . . . . . . . . . . . . . . . . . . . . 108

6-10 Time-Series of the Data Stream when Disconnecting/Reconnecting the Drive Signalto the EOM Every 100 Seconds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6-11 Fractional Photon Measurement with Signal Field at 4.2× 104 Photons/Second . . 111

6-12 Fractional Photon Measurement with Signal Field at 0.1 Photons/Second . . . . . . 112

6-13 Two-Laser Heterodyne Interferometry Experiment without Modulation . . . . . . . 113

6-14 Three-Laser Heterodyne Interferometry Experiment without Modulation . . . . . . . 116

10

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

COHERENT DETECTION OF ULTRA-WEAK ELECTROMAGNETIC FIELDS USINGOPTICAL HETERODYNE INTERFEROMETRY

By

Zachary Ronald Dylan Thomas Bush

December 2018

Chair: Guido MuellerMajor: Physics

The Any Light Particle Search (ALPS) is a type of “Light Shining through Walls”

experiment designed to generate and detect axions/axion-like particles in the laboratory.

Current axion search experiments utilize the coherent conversion between an electromagnetic

field and the axion/axion-like particle field. The probability of conversion between an

electromagnetic field and axions/axion-like particles is enabled under the presence of an

external magnetic field. ALPS II, the future iteration of this experiment, will use two optically

resonant cavities to further enhance this conversion process. The design sensitivity of ALPS

II is set by various experimental parameters. Consequently, this also sets the sensitivity

requirement of the detection system. The current design requires a detection scheme sensitive

to weak electromagnetic laser fields with strengths on the order of 10−5 photons/second.

Heterodyne interferometry offers a solution by utilizing the coherent nature of the regenerated

signal field to make detection possible. This dissertation details the design of a stand-alone

testbed built in order to assess the viability of heterodyne interferometry as a detection

method for the ALPS II experiment. Results presented demonstrate a dark count rate

from a single trial better than 10−5 photons/second with a specified level of confidence,

surpassing the ALPS II requirement. Additional results detail the successful generation and

detection of an ultra-weak signal laser field with an equivalent photon rate on the order of

10−2 photons/second. Improvements to this detection method and plans for implementation

into ALPS II are also discussed.

11

CHAPTER 1INTRODUCTION

1.1 Beyond the Standard Model

It is the ultimate goal of physicists to understand the universe by means of its behavior.

Presently, this concept manifests itself into the Standard Model. The Standard Model (SM)

combines all of our current understanding of the known fundamental particles and their

interactions into a single theory. Since the first discovery of an elemental particle in 1897 by

J.J. Thomson [3] physicists have been eagerly searching for more pieces to the cosmic puzzle.

In a little over 100 years since Thomson’s cathode ray experiment the SM has grown to include

numerous fundamental particles as well as the strong, weak, and electromagnetic forces of

nature.

Two of these particles have only just recently been directly detected. Technological

advances since the turn of the century have confirmed the existence of the tau neutrino in

2000 by the Direct Observation of the Nu Tau (DONUT) collaboration [4] and the well-known

Higgs boson in 2012 by the Large Hadron Collider [5]. Confirmed SM particles are the

result of decades of work involving theoretical predictions ultimately leading to experimental

observations. Yet even with these recent significant discoveries, the SM is far from complete.

While the SM incorporates three of the fundamental forces, it does not include gravity as

described by general relativity. Many physical phenomena also remain unexplained. This

includes neutrino oscillations in matter [6] and the acceleration of the expansion of the

Universe [7, 8]. Notable for this thesis is the issue of baryon asymmetry. Specifically, we discuss

the puzzle of Charge-Parity (CP) violation, or lack thereof, within quantum chromodynamics

(QCD), dubbed the Strong CP problem.

1.2 CP Violation and the Strong CP Problem

Charge-Parity (CP) symmetry states that the laws of physics are invariant when

interchanging a particle with its antiparticle (C symmetry) and when inverting its spatial

coordinates (P symmetry). If this is true for all interactions then immediately following the

12

Big Bang the universe should have been comprised of equal parts of matter and antimatter

[9]. However, it is clearly evident that the universe is now dominated by ordinary matter. If CP

symmetry is broken in certain physical interactions it may explain why there are more particles

than antiparticles.

In 1964, James Cronin and Val Fitch showed the first indirect experimental evidence of

CP violation in electroweak interactions, specifically in the decay process of neutral kaons

[10]. This significant discovery won Cronin and Fitch the Nobel Prize in 1980 and motivated

the search for direct detection. Direct detection of CP violation was first shown in the same

neutral kaon decay process through experiments at CERN in 1986 and 1997 [11]. These results

were later verified by FermiLab in 1999 [12]. Since then, direct CP violation has been shown

in the decay process of B mesons by the BaBar collaboration in 2001 as well as violation in

strange B mesons by LHCb in 2013 [13, 14]. Unfortunately, all of these results revealed that

the contribution from the weak interaction does not fully explain the particle/anti-particle

asymmetry [15]. Therefore, if CP violation is the reason baryonic matter is prominent in the

universe today it must arise from another source, namely, another type of interaction.

In principle, there is no reason for CP symmetry to be conserved for strong interactions.

As we will see in the next section, the QCD Lagrangian naturally includes terms that are able

to break CP symmetry. Interestingly, however, no experimental evidence of CP violation in

QCD has been observed. Why then does the strong force appear to conserve CP symmetry

when there is no fundamental reason to exclude these violating terms? To begin to answer

this question, we first look at the specific term in the QCD Lagrangian that does not obey CP

symmetry. We then move on to the most well known proposed solution introduced by Peccei

and Quinn.

1.2.1 CP Symmetry Breaking in the QCD Lagrangian

The standard QCD Lagrangian details the interactions between gluons and quarks. The

sole term that is not invariant under CP transformation arises from a non-trivial topology in

13

the QCD vacuum [16]. This so-called θ term is given by,

Lθ = θg2s

32π2Ga

µνGaµν . (1–1)

In this equation, gs is the QCD gauge coupling and Gaµν is the gluon field strength tensor with

Gaµν as its dual.

As the quantity GµνGµν is CP odd, this entire term, Lθ, is not CP invariant for θ = 0.

Recent measurements on the neutron electric dipole moment leads to an upper limit on

θ = θ − arg(det mq) of |θ| < 9 × 10−11 [17]. This is therefore the issue of CP violation

in QCD. There is no fundamental reason that this angle should be as small as experiments

demonstrate. In the late 1970s Peccei and Quinn proposed an elegant solution to this Strong

CP problem.

1.2.2 The Peccei-Quinn Solution

Rather than setting the theta term to a constant value, Peccei and Quinn promoted this

parameter to a dynamic field by introducing a new global U(1) symmetry into the theory. This

symmetry, often called the Peccei-Quinn symmetry, U(1)PQ, is spontaneously broken due to a

non-zero vacuum expectation value. The result of this spontaneous symmetry breaking yields

a new pseudo-Nambu-Goldstone boson with a non-zero mass, ma [18, 19]. This particle has

since been named the axion as it “cleans up” the issue of CP violation in the strong force.

The total QCD Lagrangian now takes the form,

LQCD,tot = LQCD + θg2s

32π2Ga

µνGaµν +

a

fa

g2s32π2

GaµνG

aµν − 1

2∂µa∂

µa . (1–2)

where a is the axion field. fa is the axion decay constant and the energy scale at which

the Peccei-Quinn symmetry is broken. Terms that do not contribute to our interest of CP

symmetry are ignored. The addition of the axion field in the QCD Lagrangian causes the

vacuum energy to shift to a minimum in which θ + a/fa → 0. The θ term therefore vanishes in

the QCD Lagrangian and CP symmetry is conserved.

14

1.2.2.1 The QCD Axion

The non-zero mass of the axion arises from its interactions with neutral pions [20–22]. Its

mass is therefore related to the pion mass, mπ, and pion decay constant, fπ.

ma ∼mπfπfa

≈ 6 eV(106 GeV

fa

)(1–3)

Notably for experimental purposes, axion mixing with neutral pions leads to a characteristic

two-photon coupling, gaγγ. The strength of this coupling is calculated to be,

gaγγ =α|gγ|πfa

(1–4)

where α = 1137

is the fine structure constant in electrodynamics [23, 24]. The quantity

gγ is a dimensionless model dependent constant on the order of unity. For example, in the

Dine-Fischler-Srednicki-Zhitnitskii model gγ ≈ 0.36 [25, 26] while in the Kim-Shifman-Vainshtein-

Zakharov (KSVZ) model gγ ≈ −0.97 [27, 28].

Following the previous equations, it is clear that the axion has an inherent dependency

between its two photon coupling parameter and its mass. While this relationship confines the

axion to a specific band of parameter space, a more general set of solutions may also exist.

1.2.2.2 Axion-like Particles

Similar symmetry-breaking mechanisms in other extensions to the SM lead to a general

set of particles, called axion-like particles (ALPs). Unfortunately, ALPs do not solve the strong

CP problem. Differing from axions, ALPs do not have a dependency between their mass and

corresponding coupling parameter [29]. Therefore, experimental searches should not limit

themselves to the QCD axion band and instead explore a broader parameter space spanned by

gaγγ and ma.

1.2.2.3 Constraints on the Axion

Astrophysical processes are used to set upper bounds on ma. Stellar evolution models

place constrains on the couplings of axions and axion-like particles [20, 30]. For axions, but not

ALPs, this translates to an upper bound on the axion mass. The lifetimes of horizontal branch

15

stars further constraint on the couplings of axions and ALPs to gaγγ < 10−10 GeV−1 [31].

Upper limits on the axion mass have also been imposed through measurements on neutrino

emission during core collapse from SN1987A [32]. These measurements confine the axion

mass to ma < 10−2 eV. This upper bound is not relevant to ALPs as they do not necessarily

couple with nucleons. A lower bound on the axion mass on the order of 10−5 eV comes from

the requirement that axions produced through the realignment mechanism do not overclose the

universe [20].

Figure 1-1. Region of Parameter Space for the Axion Mass, ma, vs. its Two Photon Coupling,gaγγ, with Various Experimental Sensitivities. Figure from Ref. [1]

Figure 1-1 shows a region of parameter space of the axion/ALP-photon coupling gaγγ

vs. its mass ma. The yellow band denotes the QCD axion in which these two parameters are

directly related. The width of this band comes from the model dependent constant gγ. For

reference, the KSVZ model parameter space (gγ ≈ −0.97) is shown in green. Sensitivity curves

for various axion search experiments have also been included.

16

1.3 Axions/ALPs: More than a Strong CP Solution

1.3.1 TeV Transparency

Not only do axions/ALPs potentially solve the Strong CP problem, their properties and

interactions with other Standard Model particles may also provide explanations for some

unanswered astrophysical phenomena. For example, high energy (E > 100 GeV) cosmic

photons should be partially absorbed as they propagate through space via electron-positron pair

production through interactions with extragalactic background light [33, 34]. This is shown

in part A of Figure 1-2. However experimentally we observe a larger number of these high

energy photons reaching Earth than theoretically expected [35]. Therefore, some other process

must be present in order to make the universe appear more transparent to TeV photons than

originally thought. Conversion of TeV photons into ALPs may possibly explain this phenomena.

A B

Figure 1-2. A. Expected propagation of TeV photons depleting through electron/positron pairproduction B. Conversion of such high energy photons into axions/ALPs as apossible solution to this transparency anomaly. Images by Manuel Meyer.

Cosmic TeV photons have some non-zero probability of converting into ALPs as they

propagate through space and pass through regions with a non-zero magnetic field due to the

coupling parameter gaγγ. As ALPs only weakly interact with matter, the resulting particles

continue to propagate unimpeded therefore reducing the attenuation factor due to pair

production. These ALPs then have a similar probability of reconverting back into observable

17

photons of the same TeV energy before reaching Earth. Conversion into ALPs is therefore one

possible explanation for the overabundance of measurable TeV photons.

1.3.2 Anamolous White Dwarf Cooling

Another astronomical effect that may potentially be solved with the introduction of

axions/ALPs involves the cooling process of white dwarfs. Stellar cooling of a white dwarf

leads to a change in its rotational period per unit time. Observations made on the period of

two white dwarfs, R548 and G117-B15A, indicate an over-efficient cooling process [36]. This

corresponds to an unexpected loss of energy in simple evolutionary cooling models. Axion/ALP

production within the cores of white dwarfs may account for this discrepancy as it allows a

new channel for energy loss to occur. In fact, including axion/ALP emission in the stellar

cooling models significantly improves the fit to the observed measurements of both R548 and

G117-B15A [37, 38].

1.3.3 Axions as Dark Matter Candidates

Axions are also considered to be good candidates for dark matter as they are weakly

interacting and because a population of cold axions is naturally produced in the early universe

through the vacuum realignment mechanism [39, 40]. The mass range for cold dark-matter

axions depends on conditions of the early universe. If inflation occurs before the Peccei-Quinn

symmetry is spontaneously broken, the most plausible mass range for which axions are the

dark matter is 10−5 to 10−4 eV. However, if inflation occurs after the Peccei-Quinn symmetry

is spontaneously broken, the axion field homogenized by inflation takes the same value

everywhere [41]. In this case, it may accidentally be close to the CP conserving value. The

most likely value is of order 10−5 eV, but there is chance, p, that the value is of order p2×10−5

eV, e.g. a 1% chance that it is of order 10−9 eV. There can also be a cold population of ALPs,

which may be the dark matter, provided that the ALPs are massive. The preferred value of the

ALP mass depends on its decay constant, fALP , but the relationship is not known a-priori [42].

18

1.4 Axion/ALP Search Experiments

Experimental searches for axions/ALPs are clearly well motivated. Modern day

experiments utilize the two-photon coupling for detection. In 1983, Pierre Sikivie showed

that detection is in fact possible via photon conversion by modifying Maxwell’s Equations to

include the axion. The classical equations that result from this calculation are given by:

L = −1

4FµνF

µν +e2N

12π2

a

νFµνF

µν +1

2∂µa∂

µa− 1

2m2

aa2[1 +O(a2/ν2)] (1–5)

∇ · E =e2N

3π2νB · ∇a, ∇× B − ∂E

∂t=

e2N

3π2ν

[E ×∇a− B

∂a

∂t

], 2a =

e2N

3π2νE · B −m2

aa

(1–6)

From these equations we find that the presence of a static magnetic field enables the axion

conversion process through what is called the “Sikivie effect” [43]. Modern experiments rely

on this process in order to enable the conversion of axions/ALPs into detectable photons. In

his original paper Sikivie discussed the possibility of experimentally testing for axion emission

from sources such as the Sun (helioscopes) and the cosmic halo (haloscopes). Notable for

this dissertation, “Light Shining through Walls” (LSW) experiments designed to generate

axions/ALPs in the laboratory and reconvert them back into detectable photons have since

emerged as well.

1.4.1 Helioscopes

Thermal photons within the cores of stars should lead to the production of axions/ALPs

due to the Primakoff process. The Sun, being our closest star, is therefore the largest source

of axion/ALP emission. External magnetic fields can be used to boost the reconversion of

solar axions into detectable X-rays. The CERN Axion Solar Telescope (CAST) was one such

experiment designed to search for axions originating from the Sun. Mesasurements from

the CAST collaboration set an exclusion limit (95% confidence level) on the axion-photon

coupling strength of gaγγ / 0.66 × 10−10 GeV−1 for ma ≤ 0.2 eV [44]. The International

Axion Observatory (IAXO) is an upcoming experiment deigned to search for solar axions and is

currently in its development phase.

19

1.4.2 Haloscopes

Other experiments have been designed to search for axions originating from the Milky

Way’s cold dark-matter halo. Sikivie’s contributions have played a pivotal role in the formation

of one such collaborative search, the Axion Dark Matter eXperiment (ADMX). The energy

level of axions from the galactic halo lead to microwave photons after conversion. ADMX

therefore utilizes a tunable microwave resonant cavity within a large superconducting magnet

for detection of such axions.

1.4.3 Light Shining through Walls

While haloscope and helioscope experiments both look for axion emission from

cosmological sources, Light Shining through Walls (LSW) experiments are designed to

generate axions in the laboratory. In this case, the resulting axions do not depend on any

astrophysical models. By generating axions in the lab, LSW experiments have the added

benefit of being able to set the axion energy. The most notable LSW experiment today is the

Any Light Particle Search (ALPS). While this specific experiment will be discussed in greater

detail in Chapter 2, the general concept of all LSW experiments is described in Figure 1-3.

Figure 1-3. Light Shining Through a Wall (LSW) experimental concept. Axions generated onthe left-hand side of the wall are used as the source for detection

Light at a known energy is incident from the left-hand side into a region with a static

magnetic field. The injected photons convert into axions/ALPs with the same initial energy

via the Primakoff process. While photons are blocked by an absorbing barrier, the weakly

20

interacting axions/ALPs pass through to the right-hand side. Another region with a similar

static magnetic field exists to the right of the barrier. The generated axions/ALPs then convert

back into detectable photons with the same energy as the incident light.

1.5 Structure of this Dissertation

The first Chapter of this dissertation briefly discussed the Strong CP problem of QCD,

a puzzle left unsolved by the Standard Model. The most well-known solution proposed by

Peccei and Quinn involves the introduction of a new global U(1) symmetry. This symmetry

is spontaneously broken yielding a new particle dubbed the axion. Similar extensions to the

Standard Model lead to a class of particles, called axion-like particles (ALPs). Other unsolved

astronomical observations may be explained by the existence of axions/ALPs. This includes

the issue of TeV transparency of the Universe and anomalous white dwarf cooling. In addition,

these particles are prime candidates for cold dark matter. This chapter explored experiments

designed to search for axions/ALPs originating from the Sun (helioscopes) and the cold

dark-matter galactic halo (haloscopes) as well as those generated in the laboratory (LSW). One

such LSW experiment directly related to the research detailed in this dissertation is the Any

Light Particle Search (ALPS).

Chapter 2 will discuss the ALPS experiment and its iterations in further detail. Calculations

within this chapter demonstrate the need for a detection system capable of measuring

extremely weak electromagnetic fields. Optical heterodyne interferometry offers one

solution and is the subject of this dissertation. Chapter 3 will then overview the theoretical

concepts behind optical heterodyne detection. This will include the various noise sources to be

considered, expected output behaviors, and a calculation for confident detection.

In Chapter 4, I detail the optical and electrical design of a stand-alone tabletop experiment

used to test the viability of this detection method. Simulated and experimental results from

this stand-alone experiment are shown and discussed in Chapter 5. Finally, Chapter 6 focuses

on improvements to the current design, future related experiments, and a discussion on the

implementation of a heterodyne detection scheme for ALPS IIc.

21

CHAPTER 2THE ANY LIGHT PARTICLE SEARCH

2.1 Introduction

The Any Light Particle Search (ALPS) is a type of Light Shining through Walls

experiment designed to generate and detect axions/ALPs in the laboratory. Similar to other

axion search experiments, ALPS uses an external magnetic field in order to enable the

axion-to-photon conversion process. Because these axions/ALPs do not arise from astronomical

sources, ALPS and other LSW experiments do not depend on cosmological models.

In LSW experiments photons are injected into a region with a strong external magnetic

field. Some of these photons convert into axions/ALPs via the Primakoff process. As the

generated particles only weakly interact with ordinary matter, they pass through a light-tight

barrier unimpeded. On the other side of the barrier exists a similar strong magnetic field used

to enable the conversion process from axions/ALPs back into detectable photons via the Sikivie

process. In order to enhance the axion/photon conversion process the ALPS experiment makes

use of two resonant Fabry-Perot cavities.

2.2 Overview of the ALPS Experiment

Wall

Axion field

HERA dipole magnetsB = 5.3 T

1064 nm laserP = 30 W

Detector

HERA dipole magnetsB = 5.3 T

Figure 2-1. Simplified design of the ALPS II Experiment.

A simplified layout of the ALPS II experiment is shown in Figure 2-1. Infrared laser light

(λ = 1064 nm) is injected from the left-hand side (LHS) into a Fabry-Perot cavity immersed

in a 5.3 T magnetic field. Photons within this cavity have some probability of converting into

axions/ALPs. The LHS cavity is therefore referred to as the production cavity (PC). The

length of this cavity is tuned to be resonant with the injected laser light via Pound-Drever-Hall

22

locking techniques [45]. Power buildup of the production cavity creates a higher circulating

power. More photons in the production cavity generate a larger number of axions/ALPs.

Axions/ALPs generated in the production cavity have the same energy as the injected

photons due to energy conservation. The particles then traverse a light-tight barrier and

enter another Fabry-Perot cavity on the right-hand side (RHS) also immersed in a 5.3 T

magnetic field. Axions/ALPs within this cavity have some probability of reconverting back into

detectable photons. The RHS cavity is therefore referred to as the regeneration cavity (RC).

The length of this cavity is also locked to the resonance of the injected laser field using error

feedback and similar Pound-Drever-Hall techniques. Because the regenerated photons have the

same energy, and therefore frequency, as the initial beam they are resonant in the RC. Similar

to before, this resonance causes power buildup increasing the strength of the signal field we

wish to measure.

The ALPS experiment poses some interesting challenges for successful operation. The

lengths of both the production and the regeneration cavities must remain locked to the

resonance frequency of the injected laser over the course of the measurement time. The

spatial modes of the two cavities must also be aligned (and remain aligned) in order for the

regenerated photons to be resonant in the RC. Angular and lateral shifts due to temperature

fluctuations must be taken into account. The experiment is also extremely sensitive to stray

light leakage into the RC. Because the regenerated photon field has the same frequency as

the initial beam, any light from the injection laser transmitted into the RC will appear as a

coherent signal.

2.3 Resonance Enhancement Techniques in ALPS

In order to determine the design sensitivity of the ALPS experiment we must calculate the

probability of axion-to-photon conversion (and reconversion) under the presence of a magnetic

field. Using Sikivie’s result, we can rewrite Equations 1–5 and 1–6 in terms of gaγγ, ma, and

23

the axion field a. In natural units (~ = c = 1) this is given by [43, 46, 47]

L = −1

4FµνF

µν +1

2(∂µa ∂µa−m2

aϕ2p)−

1

4gaγγaFµνF

µν (2–1)

∂µFµν = gaγγ∂µ(aF

µν), (∂µ∂µ +m2

a)a = gaγγE · B. (2–2)

Solving the set of equations in Eq. 2–2 for a gives,

a(±)(r, t) = e−iωt

∫d3r′

1

4π

e±ika|r−r′|

|r − r′| gaγγE(r′) · B(r′) (2–3)

where ω is the energy of the axion and the plus/minus indicates boundary conditions on a [47].

In the ALPS experiment, the magnetic field strength in both the PC and RC is set to the

same value, B0. The magnetic field on each side of the barrier spans across length, L. The

superconducting magnets are arranged such that the produced magnetic field is transverse

to the direction of propagation of the injected photons. In this case, the problem becomes

one dimensional. For the given rectangular shape and form factor of the magnetic field in the

ALPS experiment, this yields

a(+)(r, t) = iE0(gaγγB0L/2ka)2

qLsin

qL

2ei(kax−ωt) (2–4)

where q is the momentum transfer (q = ω − ka). One can then solve for the probability of

photon to axion conversion [43, 47–49]

P =1

4

1

βa

√ϵ(gaγγB0L)

2

(2

qLsin

qL

2

)2

(2–5)

in terms of the speed of the axion, βa, and the dielectric function, ϵ (assumed constant). The

probability conversion from axions to photons is also equal to P .

Because this probability goes as a sinc function in qL/2, it has a maximum when q ≈ 0.

Let us assume that we are in vacuum such that ϵ = 1. In the case of light axions (ma/ω ≪ 1)

we find that the axion speed is approximately the speed of light (βa ≈ c = 1). Therefore,

24

1βa

√ϵ≈ 1 and q = m2

a/2ω ≪ 1. The probability can be simplified to

Pa↔γ ≈ 1

4(gaγγB0L)

2 (2–6)

Referring back to Figure 2-1, laser light with electric field strength E0 is incident into the

PC from the left. The number of incident photons is given by N0. Ignoring effects of cavity

resonances, the axion field has an amplitude equal to E0

√P [48]. After transmission through

the barrier these particles have the same probability to convert back into photons. The electric

field amplitude of the regenerated photons is ES = E0P . Therefore, without considering the

optical cavities the number of regenerated photons is NS = P 2N0.

We now include the effects of the PC and the RC into this calculation. One can show

(see Ref. [48]) that the introduction of these two cavities causes the number of regenerated

photons to go as

NS =

(4TPC

(TPC + VPC)2

)(4TRC

(TRC + VRC)2

)η2P 2N0 . (2–7)

The quantity η is the mode matching efficiency between the eigenmodes of the two cavities

describing how well they are aligned. TPC is the transmissivity of the input mirror in the PC

while VPC is its internal losses. Similarly, TRC is the transmissivity of the RC input mirror and

VRC denotes the losses in the RC.

By designing the two cavities such that they are impedance matched (T = V ) we can

write NS in terms of the finesse of each cavity, F . Replacing the probability of conversion with

Equation 2–6, we find

NS = η2FPC

π

FRC

π

1

16(gaγγB0L)

4N0 . (2–8)

The design sensitivity is determined using experimental specifications and Equation 2–8.

2.4 Past and Future Iterations

2.4.1 Other LSW Experiments

ALPS is not the only axion/ALP search experiment using the “Light Shining through a

Wall” technique. The BFRT collaboration pioneered some of the first LSW experiments in

25

the early 1990s [50]. Since then other LSW experiments have been constructed. The LIPSS

[51], GammeV [52], and BMV [53] experiments have all concluded while the OSQAR [54]

experiment plans to continue its operation in the near future. Results from these experiments

set important exclusionary limits to rule out regions of the axion/ALP parameter space.

2.4.2 ALPS I

The first generation of the ALPS experiment ran from 2007 to 2010. Unlike the design

shown in Figure 2-1, ALPS I used a single 8.8 meter long production cavity. A regeneration

cavity was not implemented. The external magnetic field was produced by a single HERA

superconducting dipole magnet generating a 5.3 T magnetic field on both sides of the

light-tight barrier.

Instead of using infrared light ALPS I injected green (λ = 532 nm) light into the

production cavity. Power buildup in this cavity resulted in a total circulating power of 1.2

kW. Due to energy conservation, regenerated photons would also have a wavelength of 532

nm. It was therefore possible to use a PIXIS CCD camera as the primary detection scheme.

Experimental results from ALPS I, published in 2009 and 2010, set the most sensitive limits

of its time for the existence of axion-like particles [55]. The 95% confidence level exclusion

limits from a 31 hour exemplary run measured in vacuum are shown by the green region in

Figure 2-2.

2.4.3 ALPS IIa and ALPS IIc

The second iteration of the ALPS experiment is currently under development and will

be split into two stages ALPS IIa and ALPS IIc. Unlike the first generation experiment, both

stages of ALPS II will inject infrared (λ =1064 nm) laser light into the PC. Additionally, both

stages plan to include a RC. However, specific experimental parameters vary between the two

versions.

For ALPS IIa, the length of the two cavities will be 10 meters. ALPS IIa will also be

performed without the superconducting magnets in place. The second stage, ALPSIIc, will

extend both cavity lengths to 100 m. Ten superconducting HERA dipole magnets will be

26

placed outside the cavities in order to produce the same 5.3 T magnetic field used in ALPS I.

Within each cavity the magnetic field spans a length of 88 m. Longer effective magnetic field

lengths lead to longer interaction times between the injected photons and the magnetic field in

the PC and axions/ALPs and the magnetic field in the RC.

2.4.4 ALPS Experimental Sensitivities

The exclusion limits (95% C.L.) measured by ALPS I in vacuum are shown by the green

region of Figure 2-2. Improvements in the optical design show the projected 2000-fold increase

in sensitivity on the coupling parameter from ALPS I to ALPS IIc. The projected design

sensitivity of ALPS IIc, shown in blue, is determined using experimental specifications and the

equations described above. For reference, the projected design sensitivity of ALPS II compared

Mass ma in eV

Cou

plin

g co

nsta

nt g

aγγ

in G

eV−1

10–5 10–4

10–11

10–10

10–9

10–8

10–7

10–6

10–5

ALPS–IIc

ALPS–I

10–3

Figure 2-2. Exclusion Limits Set by ALPS I (In Vacuum) and Projected Design Sensitivity ofALPS IIc

to other axion search experiments such as ADMX, IAXO, and CAST is shown in Figure 1-1.

2.5 Detection Methods

ALPS II plans to inject infrared (λ =1064 nm) light into the PC for axion/ALP

generation. This corresponds to half of the photon energy compared to the ALPS I experiment

27

(1.165 eV in ALPS II vs. 2.33 eV in ALPS I). The same PIXIS CCD-based detector cannot be

used in ALPS II due to a reduced quantum efficiency at 1064 nm. We must therefore design

and implement a new detection scheme. However, to achieve the projected design sensitivity

we find that this is a non-trivial task. Let us first write Equation 2–8 in terms of the circulating

power in the PC

NS = η2FRC

16π(gaγγB0L)

4NPC . (2–9)

The design sensitivity of ALPS IIc makes it able to detect regenerated photons if the

coupling parameter gaγγ is larger than 2 × 10−11 GeV. The sensitivity requirement of the

photon detector is determined by this minimum coupling, Equation 2–9, and the projected

experimental parameters. Recall that for ALPS IIc, the magnetic field strength will be 5.3 T

and each cavity will be 100 meters in length. However, the effective length of the magnetic

field region is L = 88 m. The circulating power in the PC is expected to be approximately 150

kW. The finesse of the RC will be approximately 120,000. Finally, we require an efficiency of

η ≥ 95%. Therefore, ALPS II requires a detector sensitive to electromagnetic fields with power

levels on the order of 10−24 W. For λ = 1064 nm, this corresponds to a rate of approximately

2× 10−5 photons/second.

One possible detection scheme for the ALPS IIc experiment is a Transition Edge Sensor

(TES). This technology uses a superconducting material operating near its phase transition

temperature. The regenerated photon field is absorbed by the sensor causing the material

to become non-superconducting. In this case the resistance becomes non-zero resulting in a

voltage pulse as shown in Figure 2-3. Unfortunately, this detection method is sensitive to

black-body photon pileup. Any absorbed photons with an energy above the phase transition

energy of the material level will result in a voltage pulse and appear as a signal regardless of

the source. The intrinsic dark count rate of current TES measurements in a dark environment

is limited to 10−4 seconds−1 [2]. The energy resolution was measured to be ∆E/E < 8% for

four different wavelengths.

28

Figure 2-3. Voltage Pulse from a Transition Edge Sensor. Figure from Ref. [2]

Optical heterodyne detection offers an alternative approach that utilizes the coherent

nature of the signal field. Demonstration of the viability of this detection scheme for the ALPS

II experiment forms the core research within this dissertation.

29

CHAPTER 3OPTICAL HETERODYNE INTERFEROMETRY THEORY

3.1 Introduction

The principle of heterodyne interferometry involves overlapping the signal field with

a second reference field, called a local oscillator (LO), at a non-zero offset frequency. The

combined beam is then incident onto a photodetector. This is shown visually in Figure 3-1.

Figure 3-1. Basic Concept of Heterodyne Interferometry. The signal field is interfered with alocal oscillator field at a non-zero difference frequency.

Let the signal have electric field Esignal, frequency f , and arbitrary phase ϕ1. The LO has

electric field ELO, frequency f + f0, and phase ϕ2. The spatial overlap integral between the

eigenmodes of the two beams is given by κ. Let us assume that the polarizations of the two

beams are parallel. In this case, Esignal · ELO = EsignalELO. Classically mixing these two fields

yields [56]∣∣∣ Esignalei2πft+ϕ1 + ELOe

i2π(f+f0)t+ϕ2

∣∣∣2 = E2signal + E2

LO + κEsignalELO cos (2πf0t+ ϕ) (3–1)

where we let ϕ = ϕ2 − ϕ1.

For now, let us also assume complete spatial overlap of the eigenmodes such that κ = 1.

Because the average laser power is a more easily measured quantity we rewrite this equation in

terms of Psignal and PLO .∣∣∣∣√Psignalei2πft+ϕ1 +

√PLOe

i2π(f+f0)t+ϕ2

∣∣∣∣2 = Psignal + PLO + 2√PLOPsignal cos (2πf0t+ ϕ)

(3–2)

30

When the combined beam is incident onto a photodetector the first two terms on the RHS

simply lead to DC offsets. The third term results in an AC signal at the difference frequency,

f0. Time varying terms arising from this type of mixing process are called beat notes. As seen

in Equation 3–2, beat notes carry both phase and amplitude information of the interfered

fields. Regardless of the strength of the two fields, overlapping the signal field with the LO

field generates a measurable AC quantity.

In our implementation of the heterodyne readout the photodetector output is digitized via

an analog-to-digital converter (ADC) on-board a Field Programmable Gate Array (FPGA) card

with a 1 V reference voltage. We ensure that the digitization rate of the ADC, fs, satisfies the

Nyquist criterion for sampling signals at f0. The band-limited signal is then separately mixed

into two quadratures, denoted I and Q, shifted in phase by 90 degrees. This is done through

multiplication with a cosine/sine waveform at a specified demodulation frequency, fd. For a

given input of n discrete samples, x[n], I/Q demodulation gives

I[x[n]] = x[n]× cos (2πfdfsn) ,

Q[x[n]] = x[n]× sin (2πfdfsn) .

(3–3)

Each quadrature is then individually summed over the total number of samples, N . These

terms are used to compute

Z(N) =[∑N

n I]2 + [∑N

n Q]2

N2(3–4)

We find that Z(N) is proportional to the photon rate of the signal field, our quantity of

interest.

3.2 Mathematical Expectations

In this section we calculate the expected outcome of the process described in Section 3.1

using various input conditions. We first examine the behavior when only a signal is present

at the ADC (absence of noise). We separately consider two cases (1) when the demodulation

frequency is equal to the signal frequency (fd = fsig) and (2) when the demodulation frequency

is not equal to the signal frequency (fd = fsig).

31

Additionally, we show that Z(N) is related to the magnitude of the discrete Fourier

transform (DFT) of the input. When considering the effects of the noise within our system we

utilize this relationship to write Z(N) in terms of the analog power spectral density (PSD). We

then examine the case in which noise and signal are linearly combined at the input. Finally, this

section concludes with a calculation of a confidence threshold used to distinguish between the

random nature of the noise and detection of a coherent signal.

3.2.1 Signal Present Unknown Phase

Consider the simple case in which only a coherent beat note between the signal field and

the LO field is present at a photodetector with gain, G, in V/W. Noise effects and DC offsets

are ignored. We denote the frequency of the beat note as fsig. Let the signal phase, ϕ, be

an unknown quantity constant with time. The photodetector output is digitized into discrete

samples via the ADC using a 1 V reference voltage. Digitization is performed at a sampling

rate of fs. Following Equation 3–2, the digitized input is given by

xsig[n] = 2G√PLOPsignal cos (2π

fsig

fsn+ ϕ) , (3–5)

3.2.1.1 Demodulation at the signal frequency

We first look at the case in which the demodulation frequency is equal to the signal

frequency (fd = fsig). Performing I/Q demodulation on the digitized input given by

Equation 3–5 yields

I[xsig[n]] = xsig[n]× cos (2πfsig

fsn) ,

Q[xsig[n]] = xsig[n]× sin (2πfsig

fsn) .

(3–6)

32

To simplify the following calculation, let the amplitude of the beat note A = 2G√PLOPsignal.

The I quadrature then becomes

I[xsig[n]] = A cos (2πfsig

fsn+ ϕ)× cos (2π

fsig

fsn)

=A

2[cos (ϕ) +

:cos (2π

2fsig

fsn+ ϕ)]

=A

2cos (ϕ)

(3–7)

where the term cos (2π2fsigfs

n) is removed via filtering.

Similarly for Q,

Q[xsig[n]] = A cos (2πfsig

fsn+ ϕ)× sin (2π

fsig

fsn)

=A

2[:sin (2π

2fsig

fsn+ ϕ)− sin (ϕ)]

= −A

2sin (ϕ)

(3–8)

Each quadrature is then summed over the total number of samples, N . Because we assume A

and ϕ are constants, this simplifies to

N∑n

I[xsig[n]] =N∑n

A

2cos (ϕ)

=AN

2cos (ϕ)

(3–9)

andN∑n

Q[xsig[n]] =N∑n

−A

2sin (ϕ)

= −AN

2sin (ϕ)

(3–10)

These terms are then used to compute the quantity Z(N) given by Equation 3–4.

Z(N)sig =A2N2

4cos2 (ϕ) + A2N2

4sin2 (ϕ)

N2

=A2

4

(3–11)

33

Substituting A = 2G√PLOPsignal gives

Z(N)sig = G2PLOPsignal (for fd = fsig) . (3–12)

Setting the demodulation frequency equal to the signal frequency causes Z(N) to be constant

with integration time. The photon rate of the signal field is given by Psignal/(hν) where h is

the Planck constant and ν is the laser frequency so that hν is the photon energy. In this case,

we find that Z(N) is therefore directly proportional to the photon rate of the signal field.

3.2.1.2 Demodulation away from the signal frequency

It is important to understand why the signal frequency must be known and fixed. Suppose

fsig is constant but not equal to the demodulation frequency fd. Using the same input given by

Equation 3–5 we find

I[xsig[n]] = A cos (2πfsig

fsn+ ϕ)× cos (2π

fdfsn)

=A

2[cos (2π

fsig − fdfs

n+ ϕ) +:

cos (2πfsig + fd

fsn+ ϕ)]

(3–13)

and

Q[xsig[n]] = A cos (2πfsig

fsn+ ϕ)× sin (2π

fdfsn)

=A

2[:

sin (2πfsig + fd

fsn+ ϕ)− sin (2π

fsig − fdfs

n+ ϕ)]

(3–14)

While the higher frequency terms can again be ignored due to filtering, the mixing process

does not yield any DC terms when fd = fsig. In this case the summations, and thus the square

of the sums, are bounded. Computing Z(N), we find that the numerator is a bounded quantity

while the denominator increases as N2.

Z(N)sig =A2

4

1

N2

[ N∑n

cos (2πfsig − fd

fsn+ ϕ)

]2+

[−

N∑n

sin (2πfsig − fd

fsn+ ϕ)

]2(3–15)

We find that the result falls off as a sinc function. In the large N limit this simplifies to

limN→∞

Z(N)sig = 0 (for fd = fsig) . (3–16)

34

If the demodulation frequency and beat note frequency are not equal we do not expect any

pickup of the coherent signal. The remainder of the equations within this dissertation assume

fd = fsig when a beat note signal is present unless otherwise stated.

3.2.2 Z(N) and the Discrete Fourier Transform

The quantity Z(N) as it is defined in Equation 3–4 is related to the magnitude of the

DFT of the sampled input. To show this, consider a set of discrete samples x[n] where

n = 1, ..., N . The Fourier transform of this discrete set is given by

X [f ] =N∑

n=1

x[n]ei2πfn (3–17)

Because x[n] includes a total of N discrete samples it follows that there are N discrete

frequencies, f = jN

where j = 1, ..., N . Evaluating the DFT at the specific discrete frequency,

f = fdfs

and taking the square of its magnitude gives

∣∣∣∣X [fdfs]∣∣∣∣2 = X

[fdfs

]·X∗

[fdfs

]=

(N∑

n=1

x[n] cos (2πfdfsn)

)2

+

(N∑

n=1

x[n] sin (2πfdfsn)

)2

(3–18)

Recall

I[x[n]] = x[n]× cos (2πfdfsn)

Q[x[n]] = x[n]× sin (2πfdfsn) .

(3–19)

The magnitude squared of the DFT then becomes∣∣∣∣X [fdfs]∣∣∣∣2 =

[N∑

n=1

I[x[n]]

]2+

[N∑

n=1

Q[x[n]]

]2(3–20)

Relating this result to Z(N) using Equation 3–4 we find

Z(N) =

∣∣∣X [fdfs ]∣∣∣2N2

. (3–21)

There is a small caveat to Equation 3–21. For this equation to hold true, the total number

of samples needs to be N = 2πlfd/fs, where l is an integer. Without this requirement, this

35

windowing process results in spectral leakage [57]. However, this error is bounded. Because

Z(N) goes as 1/N2, in the large N limit the error due to spectral leakage becomes relatively

insignificant.

3.2.3 Noise Considerations

We must also consider the effect of noise present at the photodetector output. Types of

noise that must be considered within our experiment include optical shot noise, laser intensity

noise, and electronic noise. We calculate the expected behavior of Z(N) in the absence of

a beat note signal (Psignal = 0) in order to understand the effects of such noise. For this

calculation we utilize the relationship with the DFT to write Z(N) in terms of the analog

PSD. We only assume that the noise arises from a random process and is both ergodic and

wide-sense stationary so that its mean and autocorrelation function are time invariant [58].

Let x(t) be a continuous time sequence from t = −∞ to ∞. Suppose this sequence is

digitized into a continuous set of samples x(n) where

x(n) = x(t/fs) (3–22)

The discrete time Fourier transform (DTFT) of this series is given by,

Fx(n) = X

(fdfs

)=

∞∑n=−∞

x(n)e−i2πfdfs

n (3–23)

The single sided PSD in the digital domain (DPSD) is given by the DTFT of the autocorrelation

function, rx(k).

DPSD(fdfs

)= Frx(k) =

∑k

rx(k)e−i2

fdfs

k (3–24)

where

rx(k) = E x(n) · x∗(n− k) (3–25)

and E denotes the expectation value. Since we assume the random process is ergodic we can

write the autocorrelation function as,

rx(k) = limN→∞

1

2N + 1

N∑n=−N

x(n) · x∗(n− k) (3–26)

36

Realitistically, only a finite set of data can be sampled. This windowing process therefore limits

the result to an estimate of the autocorrelation function

rx[k] =1

N

N∑n=0

x[n] · x∗[n− k] (3–27)

where x[n] is the finite discrete set of samples of x(t) that runs from n = 0 to N . Using the

convolution theorem we can express rx[k] as

rx[k] =1

N(x[n] ∗ x∗[n]) (3–28)

In the finite case, the expectation value of the DFT of rx[k] in the large N limit is equal to the

analog PSD evaluated at fd.

DPSD(fdfs

)= lim

N→∞E [Frx[k]]

= limN→∞

E[F

1

Nx[n] ∗ x∗[n]

] (3–29)

Using properties of the Fourier transform we can rewrite this equation as

DPSD(fdfs

)= lim

N→∞E[1

NX

[fdfs

]X∗[fdfs

]]= lim

N→∞E

[1

N

∣∣∣∣X [fdfs]∣∣∣∣2] (3–30)

Relating this to Znoise(N) using Equation 3–21 we find

limN→∞

E [Znoise(N)×N ] = DPSD(fdfs

)(3–31)

Solving for Znoise(N) yields

limN→∞

E [Znoise(N)] =DPSD

(fdfs

)N

(3–32)

Finally, we wish to relate the DPSD in V2/(sample)−1 to the analog PSD in V2/Hz. This is

done by using the sampling frequency as a scaling factor.

PSD(fd) =1

fsDPSD

(fdfs

)(3–33)

37

We now write Znoise(N) in terms of the analog PSD

limN→∞

E [Znoise(N)] =fs PSD(fd)

N(3–34)

Using N = τfs this yields

limN→∞

E [Znoise(N)] =PSD(fd)

τ(3–35)

It is important to note from Equation 3–35 that E [Znoise(N)] depends on the analog PSD only

at the demodulation frequency and not across the entire spectrum.

While Equation 3–35 relates the expectation of Znoise(N) to the analog PSD we are

primarily interested in the outcome of a single run. Due to the windowing process of

digitization the result of an individual run of Znoise(N) provides only an estimate of the

analog PSD. Because the noise is assumed to be stationary, the PSD is by definition constant

with time. Therefore individual runs of Znoise(N) will also tend to fall off as 1/τ . The set of

final values of Znoise(N) for multiple runs over the same integration time has some non-zero

variance

σ2Z =

(PSD(fd)

τ

)2

(3–36)

Due to the behavior of such noise, there is a non-zero probability that the output can appear

as if a coherent signal is present. In order to distinguish between the random nature of noise

and pickup of a coherent signal, we must understand the statistical behavior of a single

run of Znoise(N) in order to create a confidence threshold. When Z(N) has a value above

this limit for a predefined number of samples, N , we can claim with a specified confidence

that a coherent signal is present at the demodulation frequency. From this point forward we

assume N to be sufficiently large such that Equation 3–35 and its derivatives provide good

approximations to real world applications.

3.2.4 Confident Detection Threshold

In order to simplify this calculation, let us assume that the analog PSD is locally flat

around fd. Suppose we appropriately band-pass filter the input around fd and downsample

38

such that the resulting frequency spectrum is also locally flat. This concept is visualized in

Figure 3-2.

Frequency (Hz)

PS

D (

W/H

z)

d

Figure 3-2. Example Single-sided PSD Detailing a Locally Flat Region around fd.

From Equation 3–35 we showed that E [Znoise(N)] depends only on the analog PSD

at the demodulation frequency and the total integration time. Because it is independent of

the sampling rate, the computation of Znoise(N) for this downsampled band will yield the

same result. Due to the central limit theorem, X[fdfs

]tends to a white Gaussian variable

independent of other X[

ffs

]in the large N limit [59, 60]. Because Znoise(N) goes as∣∣∣X [ f

fs

]∣∣∣2 it behaves as an exponential distribution.

The cumulative distribution function defines a probability P of measuring the final value

of Znoise(N) between 0 and an upper limit u for a given integration time.

P(u) = 1− e−u/σZ (3–37)

From the inverse of Equation 3–37 we can define a probability range for individual outcomes

of Znoise(N) to fall between 0 and an upper limit u for a given probability. For 5-sigma

39

confidence, P5s = 0.9999994. In this case,

u(P5s)[Znoise(N)] = −ln(1− P5s) σZ (3–38)

Using Equation 3–36 we find,

u(P5s)[Znoise(N)] = −ln(6× 10−7)PSD(fd)

τ(3–39)

When the outcome of Z(N) has a value above the threshold given by Equation 3–39 we can

claim with 99.99994% confidence that it is due to the presence of a coherent signal at the

demodulation frequency.

3.2.5 Signal and Noise Combined

We now consider the realistic scenario in which both noise and a beat note signal

are present at the photodetector output. Because the noise sources in this experiment are

independent of the beat note signal the two terms combine linearly. The total input of a

discrete set of N samples is given by

xtotal[n] = xsig[n] + xnoise[n] (3–40)

Using the DFT to evaluate Ztotal(N) yields

Ztotal(N) =

∣∣∣Xtotal

[fdfs

]∣∣∣2N2

(3–41)

Where,

Xtotal

[fdfs

]= F xtotal[n] = F xsig[n] + xnoise[n]

= F xsig[n]+ F xnoise[n]

= Xsig

[fdfs

]+Xnoise

[fdfs

] (3–42)

40

Therefore,

Ztotal =1

N2

∣∣∣∣Xsig

[fdfs

]+Xnoise

[fdfs

]∣∣∣∣2=

1

N2

(Xsig

[fdfs

]+Xnoise

[fdfs

])(X∗

sig

[fdfs

]+X∗

noise

[fdfs

]) (3–43)

However, because the signal and noise terms are uncorrelated this leads to,(Xsig

[fdfs

]X∗

noise

[fdfs

])=

(X∗

sig

[fdfs

]Xnoise

[fdfs

])= 0 (3–44)

So that we are left with

Ztotal =1

N2

(Xsig

[fdfs

]X∗

sig

[fdfs

]+Xnoise

[fdfs

]X∗

noise

[fdfs

])

=

∣∣∣Xsig

[fdfs

]∣∣∣2N2

+

∣∣∣Xnoise

[fdfs

]∣∣∣2N2

(3–45)

Substituting in Equation 3–12 and Equation 3–35

E [Ztotal(N)] = G2PLOPsignal +PSD(fd)

τ(3–46)

We find that a linear combination of noise and a beat note signal yields a linear combination of

Equation 3–12 and Equation 3–35. For a relatively weak signal field and short integration times

the noise term dominates E [Ztotal(N)] causing the resultant curve to fall off as 1/τ . However,

after enough integration the signal term takes over causing E [Ztotal(N)] to remain constant

with time.

3.2.6 Summary of Output Behaviors

So far we have derived the expected output behavior of Z(N) for various input criteria.

With only a coherent beat note signal given by Equation 3–5 present at the demodulation

frequency we find that Z(N) remains constant with integration time.

Z(N)sig = G2PLOPsignal (for fd = fsig) .

41

Noise (expected value) Coherent signal

5 sigma confidence level

– loge(6 × 10–7) PSD(fd) / τPSD(fd) / τPLO × Psignal

signa

l

τ

τ5sτx

Integration time

Figure 3-3. Expectation Behaviors and 5-Sigma Detection Threshold. A coherent signal at theinput causes Z(N) to remain constant with time, shown in yellow. On the otherhand, noise falls off as 1/τ . A 5-sigma confidence threshold is shown in red.

An example of this signal behavior is shown by the yellow curve in Figure 3-3 where we let

G = 1. Additionally, in this case Z(N)sig is directly proportional to the photon rate of the

signal field, our quantity of interest.

We then determined the expected behavior when only noise is present at the input. In

this case we define the input, xnoise[n], to be a finite set of discrete samples arising from a

wide-sense stationary random process. The expectation of the DFT of this sampled input was

found to be related to the analog PSD at the demodulation frequency.

E [Znoise(N)] =PSD(fd)

τ

An example expectation value when only noise is present is shown by the green line in Figure

3-3. Because we assume the noise is stationary, individual runs of Z(N)noise also tend to fall

42

off as 1/τ . However, a single run only provides an estimate of the analog PSD at fd. A set

of final values for multiple runs of Z(N)noise for the same integration time therefore has a

non-zero variance.

In order to distinguish between the random nature of noise and pickup of a coherent

signal, we calculated a 5-sigma confidence threshold. For this calculation we assume that the

analog PSD is locally flat around fd. We determined the probability range for Z(N) between 0

and an upper limit

u(P5s)[Z(N)] = −ln(6× 10−7)× PSD(fd)

τ.

The red line in Figure 3-3 shows an example of the behavior of the 5-sigma confidence

threshold. If Z(N) crosses this threshold then we can state with 99.99994% confidence that a

coherent signal is present at the demodulation frequency. The level at which Z(N) flattens out

to can then be used with Equation 3–12 to compute the photon rate of the signal field.

Finally, we considered the realistic case where a beat note signal and noise are both

present. We showed that a linear combination of a beat note signal and noise yields

E [Ztotal(N)] = G2PLOPsignal +PSD(fd)

τ

In Figure 3-3, τx denotes the integration time at which the noise term and the signal term in

Equation 3–46 are equal. The integration time required for E [Ztotal(N)] to cross the 5-sigma

threshold is given by τ5s.

3.3 Noise Sources in Optical Heterodyne Detection

Various types of noise must be considered for optical heterodyne detection. Within this

section we examine the behavior of relative intensity noise (RIN), dark noise, ADC noise,

and shot noise. We characterize our various noise sources by measuring their linear spectral

densities (LSDs). The LSD has units of V/√

Hz and is simply the square root of the PSD.

3.3.1 Relative Intensity Noise

As its name implies, relative intensity noise arises from fluctuations in a laser’s intensity

relative to a normalized absolute power level. For a laser with average power P , the optical

43

power as a function of time is P (t) = P + δP (t) where δP (t) denotes the fluctuations with a

non-zero mean. The relative intensity noise is given by

RIN =δP (t)

P(3–47)

RIN is typically larger at lower frequencies. Measurements of the RIN for the lasers used in our

optical setup are presented and discussed in detail within Chapter 4.

3.3.2 Dark Noise

Dark noise refers to fluctuations in the output voltage of a photodetector when no light

is incident on the device. This type of noise arises from random excitation of electrons in the

absorptive material of a photodiode. The measure of these fluctuations is often called the

noise equivalent power (NEP) of a photodetector. The NEP is an intrinsic property of the

device that depends on photodetector parameters such as the bandwidth, detector size, doping

levels, and bias voltage. It is defined as the incident power required to give a signal-to-noise

ratio (SNR) of 1 in a 1 Hz bandwidth [61]. Dark noise measurements of our custom-built

photodetector are shown in Chapter 4.

3.3.3 ADC Noise

Real ADCs introduce noise into the system arising from both resistor noise and “kT/C”

noise from internal capacitors. The level of ADC noise can be determined by terminating the

input with a 50-Ohm terminator and measuring the output. For the Xilinx FPGA used in this

experiment the LSD of the ADC noise is measured to be ≈ 1µV/√

Hz at the frequency of

interest.

3.3.4 Quantization Noise

Quantization noise arises when a time-varying analog signal is encoded into its digitized

version. ADCs have quantized digital values for which they can represent real analog signal

levels. When an analog signal is sampled its value is converted to the nearest representable

digital value. The resolution depends on the number of bits in the ADC, M , as well as the full

44

scale voltage range, EFSR.

∆ =EFSR

2M(3–48)

For an analog waveform that varies with time there exist multiple analog values that can only

be represented by the same digital value. This rounding error is known as quantization noise.

The LSD of quantization noise (qn) is determined by the resolution and bandwidth, BW, of the

ADC [62, 63].

LSDqn =∆

BW√12

(3–49)

For the 14-bit ADC (1 signed bit) on-board the FPGA card the bandwidth is given to be 100

MHz. We therefore calculate the LSD of quantization noise to be 3.5 nV/√

Hz. This is far

below the measured ADC noise of the FPGA card used in this experiment.

3.3.5 Shot Noise

Shot noise arises due to fluctuations in the number of photons detected per unit time

[64]. Shot noise follows Poisson statistics [65] and is a property of the field itself [66]. For the

purposes of this detection method, it is important to determine an expression for the analog

shot-noise PSD. The single-sided PSD in A2/Hz is given by [67]

PSDsn(f) = 2qIDC

[A2

Hz

](3–50)

where q is the electron charge and IDC is the average DC photocurrent. We use the

transimpedance, T , in V/A in order to write the PSD in V2/Hz .

PSDsn(f) = 2T 2qIDC

[V2

Hz

](3–51)

The average DC photocurrent depends on the photodetector responsivity, R in A/W and the

total incident power. Because PLO >> Psignal we can write IDC as

IDC = R× PLO (3–52)

The PSD becomes

PSDsn(f) = 2qRPLO (3–53)

45

Because R is a property of the photodetector and the average LO power is assumed to be

constant with time we find that the PSD of shot noise is also constant. Equation 3–53 is

therefore independent of frequency and the shot-noise PSD is flat in the frequency domain.

When calculating the confidence threshold we assume that the analog PSD is locally flat

around fd. We therefore design our system so that shot noise is the limiting source at the

beat note signal frequency, fsig. Additionally, the ALPS II experiment will also be limited by

shot-noise at the demodulation frequency.

We can now determine the behavior of Z(N) for a system dominated by shot noise at fd.

However, let us first rewrite the shot-noise PSD in terms of the photodetector gain, G. The

responsivity of the photodetector is related to its quantum efficiency, η.

R =q

hνη (3–54)

where hν is the photon energy. We can then write the shot-noise PSD as,

PSDsn = 2T 2PLOq

hνη (3–55)

From Equation 3–12 we found that Zsig(N) is proportional G2. We also solve the analog

shot-noise PSD in terms of G2. Using the equation for the responsivity, R we find

PSDsn = 2T 2R2hνPLO1

η(3–56)

The analog shot-noise PSD written in terms of the photodetector gain G = T × R is thus

given by

PSDsn = 2G2hνPLO1

η(3–57)

3.4 Fundamental Limits

From this point forward we scale Zsignal(N) to the photon rate of the signal field,

Psignal/(hν). A scaling factor of 1/(G2hνPLO) is applied to Equation 3–12 to yield

Zsig(N)

G2hνPLO=

Psignal

hν. (3–58)

46

We must similarly apply this scaling factor to Znoise(N) in Equation 3–35.

E [Znoise(N)]

G2hνPLO=

PSD(fd)

G2hνPLO × τ. (3–59)

Shot noise (sn) is the fundamental source of noise at fd in our stand-alone experiment as well

as in ALPS II. Substituting the analog PSD from Equation 3–57 into Equation 3–35 gives the

behavior of Znoise(N) in a shot-noise limited system.

E [Zsn(N)]

G2hνPLO=

2

ητ(3–60)

The left-hand side of Equation 3–60 is equivalent to the photon rate of the signal field

when it is present.2

ητx, sn=

Psignal

hν(3–61)

Using Equation 3–61 we can predict the integration time required for the signal to cross the

expected value of this fundamental noise limit.

τx, sn = 2hν

ηPsignal(3–62)

We apply the same scaling factor of 1/(G2hνPLO) to the 5-sigma confidence threshold in

Equation 3–39.u(P5s)[Z(N)]

G2hνPLO=

−ln(6× 10−7) PSD(fd)

G2hνPLO × τ. (3–63)

When the system is dominated by shot noise this threshold goes as

u(P5s, sn)

G2hνPLO=

−2 ln(6× 10−7)

ητ. (3–64)

Again, because the LHS is equivalent to the photon rate of the signal field, we can set the

RHS equal to Psignal/(hν).−2 ln(6× 10−7)

ητ5s, sn=

hν

Psignal(3–65)

47

Using Equation 3–65 we can predict the integration time required for Z(N) to cross this

detection threshold.

τ5s, sn = −2 ln(6× 10−7)hν

ηPsignal≈ 29

hν

ηPsignal(3–66)

As an example, suppose a signal field with a strength equivalent to 1 photon per second

is present and the system is shot noise limited at fd with η = 1. Using Equation 3–62 we find

that an integration time of 2 seconds is required for the shot noise level to equal the signal

level. However, it takes ≈ 29 seconds for Z(N) to cross the 5-sigma threshold in order to

claim a detection of this signal with 99.99994% confidence.

We can generalize Equations 3–62 and 3–66 for any PSD(fd).

τx =PSD(fd)

G2× 1

PLOPsignal(3–67)

τ5s =PSD(fd)

G2× −ln (6× 10−7)

PLOPsignal. (3–68)

The ratio between these two quantities is independent of the PSD, the average laser powers,

and the sampling frequency fs.

τ5s

τx= −ln

(6× 10−7

)≈ 14 (3–69)

Looking at Equations 3–59 and 3–67 we can see the importance of a higher power LO

when the system is not shot noise limited. In this case, larger LO powers yield a lower value for

E [Znoise(N)]/(G2hνPLO) and require less integration time for the signal to cross the expected

noise limit. Therefore a higher LO power improves the SNR when the system is not shot noise

limited.

Once the LO power is large enough so that the system is shot-noise limited, increasing

PLO provides no additional benefit for detection. This is shown in Equations 3–60 and 3–62.

In this case, τx and the SNR do not depend on the LO power. While larger LO powers will

48

result in larger beat note amplitudes (see Equation 3–2) the SNR will not be improved once

the system is shot noise limited.

3.5 Double Demodulation

We have mathematically shown that a coherent beat note signal can be decoupled from

the noise within our system. Regardless of the strength of the signal field, the resulting beat

note is observable using optical heterodyne interferometry provided enough integration time.

Theoretically, it is possible to directly mix down to DC during the first demodulation stage.

However, tests performed with this configuration found spurious DC signals generated within

the FPGA card no matter the demodulation frequency. A fast Fourier transform (FFT) of the

time-series out of the FPGA with only shot-noise at the input is shown in Figure 3-4.

Amplitude

Figure 3-4. Single-sided FFT of the digitized time series out of the FPGA referenced to a 1 Vsource. The large DC bias overshadows any weak beat note measurements.

Data are written to file at ∼ 20 Hz such that the Nyquist frequency is ∼ 10 Hz. The

strength of this DC bias is orders of magnitude larger than the beat notes of interest thus

preventing any useful measurements. This issue is solved by first mixing the beat note signal

down to an intermediate frequency, fδ. The first demodulation frequency on the processing

card is thus set to f1 = fsig + fδ. Data are then written to file and a second demodulation

49

stage is performed on a desktop PC. This double demodulation shifts the unwanted spurious

signal to a non-zero frequency where it integrates away. Figure 3-5 shows the result of mixing

the signal down to an intermediate frequency of fδ = 2.4 Hz. Using this configuration the beat

note can be accurately measured.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2x 10

−6

Frequency (Hz)

Am

plitu

de

Figure 3-5. Single-sided FFT of the time series out of the FPGA referenced to a 1 V sourcewith a Beat Note at Frequency fδ = 2.4 Hz. While the DC bias is still present, itno longer dominates the beat note signal.

After data are written to file the DFT is evaluated at the second demodulation frequency

f2 = fδ. The DC bias integrates away whereas the beat note of interest sums coherently.

Double demodulation solves this issue experimentally, however, we must update our theoretical

predictions to include this second mixing stage.

3.5.1 Influence on Signal Behavior

Because we have shown that Z(N)total is simply a linear combination of Z(N)signal and

Z(N)noise we may consider the effects of double demodulation on each case separately. We

first examine the case in which only a beat note signal is present at the photodetector. The

digitized input again takes the form of xsig[n] from Equation 3–5.

50

The first mixing stage on the FPGA multiplies the input by a cosine at frequency

f1 = fsig + fδ. The result of this mixing process is denoted by ysig[n].

ysig[n] = xsig[n]× cos (2πf1

fsn)

= A cos (2πfsig

fsn+ ϕ)× cos (2π

fsig + fδfs

n)

=A

2

[cos (2π

fδfsn+ ϕ) + cos (2π

2fsig + fδfs

n+ ϕ)

] (3–70)

The output of this mixing process is then downsampled using moving average filters so that

data are written to file at a rate f ′s. Individual samples are denoted by n′ and the total number

of samples written to file is N ′ = τf ′s. Higher frequency components arising from this mixing

process are removed during filtering. The filtered version of ysig[n] takes the form

ysig, filtered[n′] =

A

2cos (2π

fδf ′s

n′ + ϕ) (3–71)

I/Q demodulation is then performed on this recorded data set.

I [xsig[n]] = ysig, filtered[n′]× cos (2π

fδf ′s

n′)

Q [xsig[n]] = ysig, filtered[n′]× sin (2π

fδf ′s

n′)

(3–72)

Looking at each quadrature individually,

I [xsig[n]] =A

2cos (2π

fδf ′s

n′ + ϕ)× cos (2πfδf ′s

n′)

=A

4

[cos (ϕ) + cos (2π

2fδf ′s

n′ + ϕ)

] (3–73)

Q [xsig[n]] =A

2cos (2π

fδf ′s

n′ + ϕ)× sin (2πfδf ′s

n′)

=A

4

[− sin (ϕ) + sin (2π

2fδf ′s

n′ + ϕ)

] (3–74)

51

We then take the sum of each quadrature from n′ = 1 to N ′ and square the result. The higher

frequency terms thereby integrate away yielding[N ′∑

n′=1

I[xsig[n]]

]2=

[N∑

n′=1

A

4cos (ϕ)

]2=

A2N ′2

16cos2 (ϕ)

[N ′∑

n′=1

Q[xsig[n]]

]2=

[N∑

n′=1

A

4− sin (ϕ)

]2=

A2N ′2

16sin2 (ϕ)

(3–75)

Finally we compute Z2,sig, where the subscript “2” denotes double demodulation. Substituing

in the beat note amplitude yields

Z2,sig(N′) =

G2

4PLOPsignal (3–76)

Solving for the photon rate of the signal field we find

4 Z2sig(N′)

G2hνPLO=

Psignal

hν. (3–77)

Using this new scaling factor of 4/(G2hνPLO) we obtain a quantity equal to the photon rate of

the signal field after two demodulation stages.

3.5.2 Influence on Noise Behavior and Confident Detection

Next we consider the effects of double demodulation when only noise is present at

the input. For this calculation we must take into account the appropriate scaling factor of

4/(G2hνPLO). We again assume that the noise arises from an ergodic, wide-sense stationary

process. We also assume that the PSD is locally flat around fd = f1 + f2. The DPSD when

data is recorded to file, DPSD′, is related to the DPSD immediately after the ADC. Following

the flow of the signal, the input noise is mixed with a cosine as part of the first demodulation

stage. This reduces the DPSD by a factor of 2. The output of the mixer is then downsampled

via moving average filters. Because the downsampling stages involve averaging they do not

affect the level of the DPSD. The DPSD of the recorded data can be written in terms of the

analog PSD.

DPSD′(f

f ′s

)=

1

2DPSD

(f

fs

)=

fs2

PSD(fd) (3–78)

52

The DFT of the recorded data evaluated at f2 is related to DPSD′.

DPSD′(f2f ′s

)= E

∣∣∣X [f2f ′

s

]∣∣∣2N ′

= EZ2, noise(N′)×N ′ (3–79)

Solving E [Z2,noise(N′)] in terms of the analog PSD yields

E [Z2,noise(N′)] =

PSD(fd)

2τ(3–80)

We have found that applying a scaling factor of 4/(G2hνPLO) sets the LHS equal to the

photon rate of the signal field using double demodulation. Applying this scaling factor to

Equation 3–80 yields4 E [Z2,noise(N

′)]

G2hνPLO=

2 PSD(fd)

G2hνPLO × τ. (3–81)

In this case, the set of final values of Z2, noise(N′) for multiple runs over the same integration

time now has a variance of

σ22,Z =

(PSD(fd)

2τ

)2

(3–82)

Replacing PSD(fd) with Equation 3–57 gives the result when shot noise is the limiting source

of noise at the demodulation frequency.

4 E [Z2,sn(N′)]

G2hνPLO=

4

ητ. (3–83)

Comparing the RHS of Equation 3–83 to the RHS of Equation 3–60 it is clear that double

demodulation increases the noise pickup by a factor of 2. In both cases the level the curve

flattens out to is equal to the photon rate of the signal field. Therefore, the addition of a

second mixing stage causes the SNR to decrease by a factor of 2. Consequently, this also

means that τx is a factor of 2 larger as well.

τ2, x, sn = 4hν

ηPsignal(3–84)

53

Additionally, double demodulation requires twice as long of an integration time in order for

EZ2,total to cross the 5-sigma threshold.

4 u(P2, 5s)[Z(N′)]

G2hνPLO=

−4 ln(6× 10−7) PSD(fd)

G2hνPLO × τ. (3–85)

When shot noise is the limiting source at fd this becomes

4 u(P2, 5s, sn)

G2hνPLO=

−4 ln(6× 10−7)

ητ. (3–86)

The amount of integration time required Z2,total to cross the threshold is given by

τ2, 5s, sn = −4 ln(6× 10−7)hν

ηPsignal≈ 57

hν

ηPsignal(3–87)

It has been shown that E [Ztotal(N)] is simply a linear combination of Zsig(N) and

E [Znoise(N)]. Therefore we can state

4 E [Z2total(N′)]

G2hνPLO=

Psignal

hν+

4

ητ. (3–88)

For short integration times and a low photon rate, 4/(ητ) is the dominating term. After long

enough integration the signal term takes over causing the curve to remain constant with

time. These equations now reflect the expected output behaviors when implementing double

demodulation.

3.6 Summary

Within this chapter, we define a quantity Z(N) and determine its expected output

behavior under a variety of input conditions. When only a beat note between a signal field and

the LO is present at the photodetector we find that Zsig(N) remains constant with integration

time. More importantly, we found that Zsig(N) is directly proportional to the photon rate of

the signal field, our quantity of interest. When only noise is present we find that E [Znoise(N)]

falls as 1/τ . When a beat note and noise are both present we found that the combined Z(N)

is simply a linear combination of Zsig(N) and E [Znoise(N)].

54

While the expectation value of [Znoise(N)] tends as 1/τ , there is a non-zero probability for

individual runs to appear as if a coherent signal is present. We therefore determined a 5-sigma

confidence threshold to distinguish between the statistical nature of our noise and pickup of a

coherent signal. When Z(N) crosses this threshold we can state with 99.99994% confidence

that a coherent signal is present at the demodulation frequency. An example of the expected

output behaviors and this confidence threshold using a single demodulation stage is shown in

Figure 3-3.

In order to avoid an intrinsic DC bias of the FPGA card, we modify our theoretical

predictions to include a second demodulation stage. The beat note signal is mixed down to

an intermediate frequency before being written to file. Compared to single demodulation,

including a second mixing stage results in an additional pickup in noise by a factor of 2. This

causes the SNR to decrease by the same factor. With both a beat note and noise present at

the photodetector this results in

4 E [Z2 total(N′)]

G2hνPLO=

Psignal

hν+

4

ητ. (3–89)

For a weak signal field and short integration times this quantity is dominated by the noise term

and therefore falls off as 1/τ . After enough integration time the signal term takes over causing

E [Z2 total(N′)] to remain constant. The integration time at which the signal term equals the

noise term is given by

τ2, x, sn = 4hν

ηPsignal. (3–90)

We also calculated the integration time required for E [Z2 total(N′)] to cross the 5-sigma

confidence threshold if a signal is present.

τ5s, sn = −4 ln(6× 10−7)hν

ηPsignal≈ 57

hν

ηPsignal. (3–91)

In chapter 6 we discuss and implement a scheme proposed by Aaron Spector designed

to eliminate the factor of 2 noise pickup due to double demodulation. With his design we are

able to recover the original sensitivities determined when using a single demodulation stage.

55

We also discuss a technique designed to calculate and utilize the signal phase to reduce the

noise pickup by an additional factor of 2. Using these two techniques in parallel results in a

suppression of the noise pickup by a factor of 4. The amount of integration time required to

claim a detection of a coherent signal with 99.99994% confidence is also significantly reduced.

56

CHAPTER 4HETERODYNE DETECTION EXPERIMENTAL SETUP

4.1 Introduction

Within this chapter we detail the design of a stand-alone testbed constructed in order to

experimentally verify the concepts described in Chapter 3. Measurements with this testbed are

used to demonstrate that heterodyne detection is a viable option for ALPS IIc. While the basic

concept of heterodyne interferometry is shown in Figure 3-1, in practice we must also take

experimental considerations into account. In all of the calculations in Chapter 3, we assume

that the beat note frequency is known and constant with time. We additionally assume that

each field is linearly polarized and that the directions of their corresponding electric field are

parallel to one another. We also assume the spatial eigenmodes of the two fields are perfectly

matched so that κ = 1. Realistically, heterodyne interferometry requires more effort than

simply overlapping two laser fields.

The first section of this chapter details the optical design of our stand-alone experiment.

We discuss the process of experimentally generating an optical beat note between a LO field

and a weak signal field at a known photon rate. We also describe an error feedback system

required to keep the beat note at a fixed frequency, fsig. We then describe the various analog

components used in our stand-alone experiment including notch and band-pass filters, voltage

amplifiers, and a custom built photodetector. With these components we ensure that shot

noise from the LO is the dominant source of noise within our system.

We conclude this chapter with an overview of our digital design. We discuss the FPGA

architecture used for digitization, first demodulation, filtering, and downsampling. The FPGA

card is also used to implement the error feedback loop to fix the beat note frequency at fsig.

Second demodulation and post-processing are both performed in MatLab. The custom MatLab

scripts are included in Appendix A.

57

4.2 Optical Design

We construct the optical setup shown in Figure 4-1. Using this design we can generate

an optical beat note between an ultra-weak signal field and a LO field that takes the form of

Equation 3–5 after digitization. We can therefore test the theory discussed in Chapter 3.

BS

PM Fiber

Mirror

Servo Loop

Laser 1 Laser 2

PI

λ/2λ/2

λ/2

λ/2

BS

PolBS

PolBS

EOM

to DataAcquisition

PD1

PD2

sin(2π fcc t)

sin(2π fEOM t)

ND

Mixer

Figure 4-1. Optical Design to Test Heterodyne Interferometry for Ultra-weak Signal Fields.

In our setup, Laser 1 (L1) acts as our LO field while Laser 2 (L2) is used for generation

of the weak signal field. A half-wave plate (HWP) and polarizing beam splitter (PolBS) pair

is placed at the start of each beam path. The combination of a HWP and PolBS is used for

power control purposes and to ensure that the outgoing light is linearly polarized.

L2 is then sent through another HWP before entering an electro-optic modulator (EOM).

We use this HWP to align the field polarization to the optical axis of the EOM crystal. The

EOM is used to generate sidebands on L2. The purpose of the EOM will be discussed later

in this section. Laser 2 then passes through two neutral density (ND) filters with a combined

attenuation factor on the order of 105. The filters are used to reduce the power of the signal

field down to the appropriate level.

Laser 1 and Laser 2 are both incident into the same 50/50 power beam splitter (BS). Half

of the average light power of each beam is transmitted while the other half is reflected. The

two laser fields are overlapped at this BS. The combined beam is then sent into a single-mode

58

polarization-maintaining optical fiber. The optical fiber is designed to transmit only the TEM00

mode of each beam. By sending the combined beam into the optical fiber we ensure complete

overlap of the spatial eigenmodes at the output coupler so that κ = 1.

After the fiber, the combined beam is incident into another 50/50 power BS. Each path is

then individually focused into two separate photodetectors, PD1 and PD2. The output of PD1

is used for an error feedback loop to Laser 1. This feedback loop locks the difference frequency

between Laser 1 and Laser 2 to a constant value, fCC. PD2 is a custom-built photodetector

used for our signal measurements. The average power of each laser measured at the input of

photodetector PD2 is given by PLO and PL2.

4.2.1 Phase Lock Loop

Interfering the two laser fields results in a beat note signal at both PD1 and PD2. The

beat note between the two main laser fields is called the carrier-carrier (CC) beat note. The

frequency of the CC beat note, fCC must remain constant in order to use the theory described

in Chapter 3. We lock the CC beat note frequency to a fixed value using error feedback to the

control box of Laser 2. This type of error feedback is often called a phase lock loop (PLL).

The digitized CC beat note has the form

xCC[n] = A cos (2πfCC

fsn+ ϕCC) (4–1)

where ϕCC is the phase and A = 2G√PLOPL2. The digitized beat note is multiplied with a

sine wave with phase ϕPLL at the desired difference frequency, fCC. This yields

xCC[n]×sin (2πfCC

fsn+ ϕPLL) =

A

2:

sin (2π2fCC

fsn+ ϕCC + ϕPLL)+

A

2sin (ϕPLL − ϕCC) (4–2)

where the first term is low pass filtered away. The output of this mixing process thus only

depends on the difference in phase. This serves as the error signal for our feedback loop.

The error signal is then sent into a proportional-integral (PI) controller. A digital-to-analog

converter (DAC) is used to turn this output into a voltage which is then sent to the control

box for Laser 2. The control box then drives a piezo actuator to adjust the frequency of

59

Laser 2 in the appropriate direction to keep the difference in phase constant. When this phase

difference is constant the CC beat note frequency is fixed and equal to fCC .

The error signal depends on both the servo gain as well as the beat note amplitude,

A. While the servo gain is used to amplify the voltage output after mixing it also amplifies

any noise present in the system. Setting the servo gain too high can cause instability in the

feedback loop. The LO power must be set to a high enough level to provide adequate error

feedback and maintain a stable PLL.

Experimentally, we have shown that the CC beat note amplitude must have a minimum

power of 1 µW in order to keep the PLL stable over the required integration times. The

largest LO power at the input without saturating either photodetector is 5 mW. This leads to

a minimum average power of Laser 2 of P2 ≥ 60 pW to maintain stability of the PLL. This

is equivalent to a minimum rate of 3 × 108 photons/second. Because we wish to reduce our

signal field strength to a level below 1 photon/second we cannot use the CC beat note as our

measurement source. We make use of sideband generation from electro-optic modulation to

produce ultra-weak signal fields at fixed frequencies.

4.2.2 Electro-Optic Modulation

Laser 2 is sent through a broadband EOM before being interfered with the LO field.

Our EOM consists of a magnesium oxide doped lithium niobate crystal (MgO:LiNbO3). An

electric drive signal at frequency fEOM applied to the crystal changes its refractive index via

the electro-optic effect. This phase modulates the beam as it passes through the crystal.

Phase modulation generates sidebands both above and below the laser frequency, fL2. These

sidebands occur at k integer multiples of the drive frequency, fEOM. The amount of light

power in the kth order sideband is [68]

PSB, k = [Jk(m)]2PL2 . (4–3)

where Jk(m) is the kth order Bessel function and m is the modulation depth. The modulation

depth is dependent on the amplitude of the drive signal to the EOM. Thus PSB,k can be fine

60

tuned to a specified level. When the ND filters are placed in the beam path the optical power

of Laser 2 and all of the resultant sidebands are attenuated by a factor of 105.

After the EOM, the beam is sent into the 50/50 BS where it is overlapped with the LO

field. Interfering these two beams generates the CC beat note. All of the sidebands also beat

with the LO to produce AC signals with amplitudes given by

Ak = 2√

PSB, k PLO (4–4)

This is visualized for the k = ±1 sidebands in Figure 4-2. While higher order sidebands

are present experimentally they are not shown in the figure.

Figure 4-2. Frequency Space Describing Beat Note Generation between First-order Sidebandsand the Local Oscillator Field

Using this configuration, the average power of Laser 2 is set to maintain a stable PLL.

Phase modulation generates ultra-weak sidebands with power levels comparable to the

projected sensitivity of ALPS IIc. Interference between these sidebands and the LO generate

beat notes at known, fixed frequencies. These sideband-LO beat note signals are measurable

using the theory described in Chapter 3. We simply set the demodulation frequency equal to

the frequency of the sideband-LO beat note of interest. When performing measurements we

use the 2nd order sideband (k = +2). We set the CC beat note frequency to fCC = 30 MHz

and the EOM drive frequency to 23 MHz + 1.2 Hz. This sets the 2nd order sideband-LO

61

beat note to be at a frequency of 16 MHz + 2.4 Hz. The first demodulation frequency is

therefore set to f1 = 16MHz. In order to measure the signal amplitude we then perform I/Q

demodulation in MatLab at a frequency of f2 = 2.4Hz.

4.2.3 Polarization Considerations

Throughout chapter 3 we assume that the polarization of the signal field is parallel to

the polarization of the LO. In this case the dot product Esignal · ELO yields a maximum value

of EsignalELO. We experimentally align the two field polarizations to be parallel using HWPs

placed at specific positions in each beam path.

Following the path of Laser 2, after the EOM the laser light is linearly polarized. We place

a HWP in the beam path in order to align the field polarization to match the optical axis of

the single-mode polarization-maintaining fiber. The path of Laser 1 is much simpler. Recall

that after Laser 1 passes through the initial HWP/PolBS pair the light is linearly polarized. We

thus simply place a HWP in this beam path to align the polarization of Laser 1 to the optical

axis of the fiber.

Because the polarization of each beam is aligned to the optical axis of the same

single-mode fiber they are therefore parallel to each other. We use a polarization-maintaining

fiber so that the polarizations of the two fields remain parallel at the output coupler and thus

at PD1 and PD2.

4.3 Analog Components

Because PD1 and PD2 are used for different purposes they require different analog

components at each output. PD1 is used in the error feedback loop. We require the peak

voltage of the CC beat note to be large enough to provide an adequate error signal and

maintain the stability of the PLL. A voltage gain amplifier is placed directly after the output of

PD1 to increase the voltage by a factor of 10. The remainder of the PLL is performed digitally.

The AC-coupled output of PD2 is used to measure the beat note between the 2nd order

sideband and the LO. We must add two voltage amplification stages in order to ensure that

the shot noise at fsig is greater than the ADC noise of the FPGA. However, we must be careful

62

to prevent saturation of the ADC. In the remainder of this section we discuss the various

analog components placed at the output of PD2 to satisfy these criteria.

4.3.1 Notch Filter

Recall that the CC beat note is also present at PD2. If we simply add a voltage amplifier

at the photodetector output the CC beat note will saturate the ADC. We thus use a notch

filter to attenuate the CC beat note before implementing any amplification stages. The

25 30 35

−100

−80

−60

−40

−20

0

Frequency (MHz)

dB

Notch Filter Transfer Function

Figure 4-3. Measured transfer function of a notch filter showing attenuation at 30 MHz. Wemeasure a Q-factor of 86 with this filter.

measured transfer function of the notch filter used in this experiment is shown in Figure 4-3.

Our notch filter is centered at 30 MHz with a bandwidth of 3 MHz and a measured Q-factor of

86. This filter attenuates the CC beat note by approximately 100 dB.

4.3.2 Bandpass Filters and Analog Voltage Amplification

We must also consider the level of the relative intensity noise (RIN) at the photodetector

output. Figure 4-4 shows the measured LSD vs. frequency at the output of PD2. For this

63

measurement only the LO is incident onto the photodetector with PLO = 5.0 mW. The peak at

lower frequencies is due to RIN.

Figure 4-4. Noise LSD vs. Frequency Directly After Measurement Photodiode. The peak atlower frequencies is due to RIN. We therefore choose our signal frequency to be16MHz + 2.4Hz where RIN is not the dominant source of noise.

We plan to implement two voltage amplifiers with a combined amplification factor of

100. However, we must be careful that the RIN does not saturate the ADC after amplification.

The blue curve in Figure 4-5 shows a time-series of the voltage output directly after the notch

filter. This measurement yields a peak-to-peak noise voltage of ≈ 30 mV. If nothing is done

to reduce the level of RIN then after amplification we expect a peak-to-peak noise voltage of

≈ 3.0V, which will saturate the ADC.

We thus add two bandpass filters with a center frequency of 16 MHz and a bandwidth of

6 MHz before amplification. Our choice of signal frequency comes from the center frequency of

these bandpass filters. The red curve in Figure 4-5 shows the time-series of the noise after the

notch filter and two bandpass filters. We measure a peak-to-peak noise voltage of ≈ 1.3mV.

After amplification by a factor of 100, we expect a peak-to-peak noise voltage of 130 mV.

Thus, we can now add our two voltage amplifiers without saturating the ADC.

64

0 0.05 0.1 0.15 0.2 0.25−20

−10

0

10

20

Time (ms)

Vol

ts (

mV

)

Effect of Bandpass Filters Before Amplification

Without Bandpass FiltersWith Bandpass Filters

Figure 4-5. Effect of band pass filters on peak-to-peak noise. The blue curve shows the noisereadout on an oscilloscope without the use of bandpass filters. In this case the RINcauses a peak-to-peak voltage of ≈ 30 mV. Amplification by a factor of 100 wouldsaturate the ADC. The red curve shows the result when two bandpass filterscentered at 16 MHz are used. The resulting peak-to-peak voltage is ≈ 1.3mV.

We add the two bandpass filters and two voltage amplifiers after the notch filter. We

measure the peak-to-peak noise voltage after amplification to verify that the bandpass

filters prevent saturation of the ADC. The blue curve of Figure 4-6 shows the result after

amplification if we remove the bandpass filters. As expected we measure a peak-to-peak noise

voltage of ≈ 3.0 V. The red curve in Figure 4-6 shows the result after amplification when the

two bandpass filters are included. In this case we measure a peak-to-peak noise voltage of

≈ 120 mV, in agreement with expectations. We now have a set of analog components that

amplifies the noise level at the signal frequency without saturating the ADC input.

65

0 0.05 0.1 0.15 0.2 0.25

−1

0

1

2

3

Time (ms)

Vol

ts (

V)

Effect of Bandpass Filters After Amplification

Without Bandpass FiltersWith Bandpass Filters

Figure 4-6. Time-series After Bandpass Filters and Amplification Stages.

4.3.3 Measurement of the Combined Analog Gain

The combined analog component chain from the output of PD2 to the input ADC is

shown in Figure 4-7.

Figure 4-7. Chain of Analog Components After the Measurement Photodetector PD2

Because the bandpass filters are not ideal they slightly attenuate the output at 16MHz +

2.4Hz. The combined gain factor of the analog components is therefore not exactly 100. We

thus experimentally measure the total analog gain using a function generator as our input. We

set our input to be a sine wave at 16 MHz + 2.4 Hz with a root-mean-square (RMS) voltage

of 3.8 mV. Figure 4-8 shows a measurement of this input signal using a spectrum analyzer.

We send the function generator signal through the chain of analog components described in

66

15.995 16 16.005

1

2

3

4

Frequency (MHz)

Vol

tage

(m

V)

Figure 4-8. Measurement of a function generator signal before analog amplification. The RMSvoltage is measured to be 3.8 mV

Figure 4-7. The resulting output is shown in Figure 4-9. We measure an RMS voltage after

the analog components of 250 mV. The combined gain factor of the analog components is

calculated by dividing the output RMS voltage with the input RMS voltage. This yields a

total gain factor of ≈ 66. The total analog component gain must be accounted for during

calibration of the heterodyne detector.

4.3.4 Measurement of the Photodetector Gain and Quantum Efficiency

The measurement photodetector, PD2, was designed and constructed by Ayman Hallal

using a 300 µm diameter InGaAs Hamamatsu G12180-003A photodiode. The circuit

diagram for this detector is shown in Figure 4-10. We use the AC only output port for our

measurements.

Recall that the equations derived in Chapter 3 to determine the photon rate of the

signal field depend on the photodetector gain, G, in V/W. We therefore must measure the

AC gain of the photodetector at the signal frequency of 16 MHz + 2.4 Hz. Average laser

powers are measured using an Ophir NovaP/N7Z01500 power meter. This power meter has

67

15.995 16 16.005

50

100

150

200

250

Frequency (MHz)

Vol

tage

(m

V)

Figure 4-9. Measurement of a function generator signal after analog amplification. The RMSvoltage after the notch filters, bandpass filters, and voltage amplifiers is measuredto be 250 mV

R3

1.2k

I1

AC 1 R7

400 M

C2

6p

C3

2.2p

R6

49.9

R10

0

C7

100p

R11

10k

R4

49.9

R14

0

U3

THS4031

U2

THS4031

R5

390

R8

390

R1

1.2k

U1

THS4031

VC

CVD

D

VC

CVD

D

AC

VC

CVD

D

DC_AC

160k

xHamamatsu_G12180-003A

Figure 4-10. Circuit diagram of the measurement photodetector, PD2. The AC output port isused for measurements. The transimpedance is T = 2.4 kΩ. This photodetectorwas designed and built by Ayman Hallal.

an accuracy of ± 7% when using the PD300-IR head with the filter installed [69]. For this

measurement we set the average local oscillator power at PD2 to PLO = 5.0 mW and Laser 2

to PL2 = 3.2µW and generate the CC beat note. The peak CC beat note amplitude in power

68

is given by A = 2√

PLOPL2. In this case this yields A ≈ 25 mW. We set the CC beat note to

16 MHz + 2.4 Hz and measure the voltage output using a spectrum analyzer. The result of

this measurement is shown in Figure 4-11.

12 14 16 18 200

50

100

150

200

250

300

Frequency (MHz)

Vol

tage

(m

V)

Carrier−Carrier Beat Note After Measurement PD

Figure 4-11. Measurement of Carrier-Carrier Beat Note to Determine Photodetector Gain

We measure an RMS voltage of the CC beat note of 260 mV. Converting this to a peak

voltage yields VCC, peak ≈ 360 mV. From this measurement we calculate the AC gain of PD2 at

16 MHz + 2.4 Hz to be G ≈ 1.4× 103 V/W.

The circuit diagram of the photodetector gives a transimpedance of T = 2.4 kΩ.

Recall that the photodetector gain is given by G = T × R where R is the responsivity.

The responsivity is related to the quantum efficiency in Equation 3–54. We thus calculate a

quantum efficiency of η = 0.7 for this photodetector at the desired frequency.

4.3.5 Ensuring a Shot-Noise Limited System

We must make sure that for the given LO power our system is limited by shot noise at the

signal frequency, fsig = 16 MHz + 2.4 Hz. If shot noise is the dominant source of noise at this

frequency then we can use Equation 3–57 to calculate an expected value of the analog PSD.

69

Because the spectrum analyzer used in our laboratory measures the noise LSD we write

LSDsn =√

PSDsn =

√2G2hνPLO

η(4–5)

From Equation 4–5 we find that if our system is shot-noise limited at fsig then increasing the

LO power by a factor of 2 should increase the measured LSD by a factor of√2.

Let us first test that the measurement photodetector itself is shot-noise limited for

PLO > 2.0 mW. For this measurement we do not include the notch filter, bandpass filters,

or voltage amplifiers. The laser power is measured using an Ophir NovaP/N7Z01500 power

meter. Only the LO is incident onto PD2. We initially set the average LO power to PLO =

2.7mW. The measured LSD for this optical LO power is given by the red curve in Figure 4-12.

This figure also demonstrates that the LSD (and thus PSD) is locally flat around the desired

Figure 4-12. Noise LSD measurements before analog components. We find that the LSDincreases by a factor of

√2 when we double the incident optical light power. Thus

we confirm that the photodetector is shot-noise limited for PLO = 5.3 mW.

signal frequency. Using Equation 4–5 with η = 0.7 we expect the LSD to be 54 nV/√

Hz.

At 16 MHz + 2.4 Hz we measure a LSD of 66 nV/√

Hz. We then double the LO power to

70

PLO = 5.3 mW. We expect the LSD to increase by a factor of√2 to 76 nV/

√Hz. The blue

curve in Figure 4-12 shows the result of this measurement. We obtain a value for the LSD of

93 nV/√

Hz. A measurement of the dark noise, shown in green in Figure 4-12, yields an LSD

of 18 nV/√

Hz. For reference, the spectrum analyzer noise is measured to be ≈ 7.6 nV/√

Hz

at the desired frequency.

In both instances the measured noise level was slightly higher than our expectations.

This error arises from the uncertainty in the power meter measurements. Regardless, the ratio

between the two LSD measurement yields 93/66 ≈√2. When we double the optical power

we see an increase in the LSD by a factor of√2. This confirms that the photodetector is

limited by shot noise when PLO = 5.3 mW at 16 MHz + 2.4 Hz. These results are compiled in

Table 4-1.

Table 4-1. Linear Spectral Density Measurement at Photodetector Output.PLO (mW) Expected LSD (nV/

√Hz) Measured LSD (nV/

√Hz)

2.7 54 655.3 76 93

Note: The expected LSD values are derived from powermeasurements performed with the Ophir power meter. Uncertaintiesin the calibration of the power meter as well as the quantumefficiency and gain factors of the photodetector circuit are likelyresponsible for the discrepancy between the expected and measuredresults.

Next we check that shot noise is the dominant source of noise after all of the analog

components. In this case, we measure the LSD of dark noise to be 0.96µV/√

Hz. We again

perform measurements at PLO = 2.7 mW and 5.3 mW and measure the resulting LSD after

the notch filter, two bandpass filters, and voltage amplifiers. Using the calculated analog gain

we expect the LSD to be 3.6 µV/√

Hz when the LO power is set to 2.7 mW. The result of this

measurement is shown by the red curve in Figure 4-13. The effects of the two bandpass filters

are apparent in this figure. With this incident LO power we measure a LSD of 4.4 µV/√

Hz.

The difference in measured vs. expected LSD values again arises from the uncertainty in laser

power measurements using the Ophir power meter.

71

Figure 4-13. Noise LSD measurements after analog components. Again when we double theLO power the LSD increases by a factor of

√2. Thus we confirm that shot noise

is the dominant source of noise after the analog components when PLO = 5.0mW.

We then increase the LO power to PLO = 5.3 mW. We expect the LSD to increase

to 5.1 µV/√

Hz. In this case, we obtain a measured value of the LSD of 6.3µV/√

Hz at

16MHz + 2.4Hz. This is shown by the blue curve in Figure 4-13. By doubling the LO power

we find that the LSD increases by a factor of 6.3/4.4 ≈√2. Thus, shot noise is still the

dominant source of noise at the desired frequency after all of the analog components. These

measurements are summarized in Table 4-2.

Table 4-2. Linear Spectral Density Measurement After Analog Components.PLO (mW) Expected LSD (µV/

√Hz) Measured LSD (µV/

√Hz)

2.7 3.6 4.45.3 5.1 6.3

Note: The expected LSD values are derived from powermeasurements performed with the Ophir power meter. Uncertaintiesin the calibration of the power meter as well as the quantum efficiencyand gain factors of the photodetector circuit are likely responsible forthe discrepancy between the expected and measured results.

72

Finally, we compare the level of shot noise to the noise level of the ADC. We measure

the LSD of the ADC noise at the desired signal frequency to be ≈ 1.0 µV/√

Hz. Referring to

Figure 4-13, when the LO power is set to 5.3 mW the measured LSD is 6.3 µV/√

Hz at 16

MHz + 2.4 Hz. The shot-noise level is approximately a factor of 6 above the ADC noise level

at this frequency. Our measurements therefore confirm that shot noise is in fact the dominant

source of noise after digitization for the provided average LO power.

4.4 Digital Design

After the signal from PD2 passes through all of the analog components it is digitized via

an ADC at a rate of fs = 64 MHz on-board a Field Programmable Gate Array card. This

versatile card can be configured using Very High Speed Integrated Circuit Hardware Description

Language (VHDL) to perform various digital processing tasks as outlined by the user. A

simplified digital design detailing the path of the photodetector signal is shown in Figure 4-14.

FPGA

Data Processing (20 Hz)Data Acquisition (64 MHz)

fromOptical Setup

PD2

∑

∑

ADC

CIC Filter

1 х cos(2π f1 / fs n)

A х sin(2π fsig / fs n)

1 х sin(2π f2/ fs' n')

1 х cos(2π f2 / fs' n')

FPGA

Figure 4-14. Digital Design of the Heterodyne Detector

The first demodulation stage takes place on the FPGA card. The input signal is mixed

with a waveform at frequency f1 = fsig − fδ generated by a numerically controlled oscillator

(NCO) using a look-up table (LUT). After the mixing process the signal passes through a

73

cascaded integrated comb (CIC) filter to remove the higher frequency terms. The CIC filter is

a type of moving average filter [70]. Data are downsampled by a factor of 2048 at this stage to

a reduced rate of 31.25 kHz and then written to a buffer, not shown in the figure.

A direct memory access (DMA) transfer is initiated using LabView to stream data from

this buffer to a desktop computer. A second moving average filter downsamples the data by

a factor of 1562. Data are then written to file at a rate of f ′s ≈ 20 Hz. Our choice of the

intermediate frequency, fδ = 2.4 Hz, comes from the Nyquist frequency of the recorded data of

approximately 10 Hz.

Data are then imported into MatLab for second demodulation and post-processing. The

signal at fδ is decomposed into its in-phase (I) and quadrature (Q) components via separate

mixing with a cosine and sine NCO at f2 = fδ, respectively. We then compute Z2(N) and

apply the appropriate scaling factor in order to obtain an equivalent photon rate of the signal

field. The result is then plotted vs. integration time, τ .

4.4.1 Hardware

The FPGA card used in this experiment is a Xilinx model PMC-AX3065 [71]. It uses

14-bit ADCs to sample data at a rate of fs = 64 MHz. VHDL is used to program the first

demodulation stage and CIC filtering described above. We use a Stanford model DS345

function generator to produce the drive signal to the EOM. The frequency, amplitude, and

phase of this drive signal can be easily modified. We adjust the frequency of the function

generator output to 23 MHz + 1.2 Hz. With the CC beat note at 30 MHz this sets the beat

note between the 2nd order sideband and the LO to 16 MHz + 2.4 Hz. In order to prevent

cycle slips, the FPGA card is synchronized to a master clock operating at a frequency of 64

MHz. The function generator driving the EOM is also synchronized to the master clock via a

10 MHz timebase reference signal.

4.4.2 Software

In order to write the necessary VHDL code to program the FPGA card we use a library

within MATLAB called Simulink. Simulink uses blocks diagrams connected by wires to

74

represent mathematical operations and the flow of data. Figure 4-15 shows the Simulink block

diagram of the first demodulation stage. The complete Simulink design includes a phase meter

channel and the digital PLL and is much more complex. The Simulink code for the phase

meter channels and digital PLL was written by Johannes Eichholz. Within the digital PLL,

Figure 4-15. Simulink Block Diagram Interface for FPGA Configuration

a phase meter is used to actively track the frequency value of the CC beat note. Recall that

the frequencies of the sideband-LO beat notes depend on the frequency of the CC beat note.

While the CC beat note is locked to 30 MHz, this value can vary slightly with time due to

laser frequency fluctuations in L1 and L2. We denote the measured CC beat note frequency

by “Phase Meter Frequency.” Based on the drive frequency to the EOM we manually set

the “EOM Freq” value to 46 MHz. Looking at Figure 4-15 we first calculate the difference

frequency between these two quantities. This is equal to the first demodulation frequency,

f1 = 16 MHz. We send this frequency value into an accumulator in order to obtain a phase

value. This phase is used to generate both sine and cosine waveforms at frequency f1 with an

internal LUT. Looking at the digital design in Figure 4-14 we only use a cosine waveform at

frequency f1 during the first demodulation stage. In Chapter 6 we find that it is beneficial to

include the sine channel as well in order to suppress noise pickup due to double demodulation.

We then individually multiply each waveform out of the LUT with the ADC input channel. The

result of each multiplication stage is sent through separate CIC filters that downsample the

data by a factor of 2048. After the CIC filters we scale the data appropriately and output each

channel to the buffer.

75

We use LabView to transfer data from the buffer to a local desktop computer. Similar

to Simulink, LabView also represents mathematical operations with blocks and wires. Timing

and data transfer management are both handled by LabView through communication with

the FPGA card at a rate of 31.25 kHz. We implement custom-built moving average filters to

further downsample the data by a factor of 1562 to a rate of ≈ 20 Hz. We then save the data

to multiple text files on a desktop computer. Second demodulation and post-processing are

performed in MatLab using the custom scripts presented in Appendix A.

4.4.3 Digital PLL

The digital PLL is also performed using FPGA card. The output of PD1 is sent through a

voltage amplifier and is digitized by another ADC. We implement a phase meter design in order

to determine the phase of digitized CC beat note. We use this phase and a LUT to generate a

reference waveform at frequency fCC. We multiply the digitized PD1 signal with this reference

waveform and the result is sent into a digital PI controller. This creates the error signal used

in the feedback loop. A 16-bit digital to analog converter (DAC) on-board the FPGA card is

used to convert this error signal into a real voltage. We send this error voltage into the piezo

actuator of Laser 2’s control box in order to keep the beat note locked at fCC.

The measured frequency value of the CC beat note is also written to file at a rate of ≈ 20

Hz. In order to characterize the PLL we set PLO = 5.0 mW and PL2 = 17 µW. We lock the

two lasers at a difference frequency of 30 MHz. The value of the CC frequency is measured

using a phase meter and data are written to file. After 30 minutes of measurement time we

observe a standard deviation in the CC beat note frequency of 9 × 10−4 MHz. The maximum

variation in the measured CC beat note frequency was determined to be 3.9× 10−2 MHz.

4.5 Summary

This chapter focused on the design of our stand-alone experiment built in order to test

the concept of optical heterodyne detection of an ultra-weak signal field. We first detailed the

optical setup in which we overlap two laser fields to produce a beat note signal. One of the

main experimental considerations is the requirement that the difference frequency between the

76

two lasers remains constant with time. We implement a phase lock loop to generate an error

signal that is fed back to the control box of Laser 2. For this feedback loop to remain stable

over the necessary integration times we cannot lower the signal laser power down to the desired

level. We therefore use an electro-optic modulator to generate sidebands that also beat with

the LO field at known, fixed frequencies. The power in each sideband is easily adjustable and

can be calculated using Bessel functions. We choose the 2nd order sideband as our signal field

and measure the resulting beat note at frequency fsig.

Within this chapter, we demonstrated that our system is limited by shot noise at the

desired signal frequency through measurements of the noise linear spectral density at various

LO power levels. We additionally showed the need for a notch filter and two bandpass filters in

order to prevent saturation of the ADC input. The photodetector gain, quantum efficiency, and

analog amplification factor were also measured at the desired signal frequency.

We then described the digital design of the experiment detailing the path of the

signal through the two demodulation stages. The FPGA card is used to perform the first

demodulation stage along with the digital PLL. After data are written to file we use MatLab to

perform the second demodulation stage. We then compute Z2(N) and scale it to an equivalent

photon rate of the signal field. We plot the result vs. integration time.

In the end, our testbed allows us to produce a measurable beat note at a fixed frequency,

fsig, between an ultra-weak signal field and the LO. The strength of the signal field is easily

adjustable by changing the amplitude of a function generator. We are now ready to test our

design and determine if heterodyne detection if a viable option for the ALPS IIc experiment.

77

CHAPTER 5RESULTS

5.1 Introduction

With the theory and design of a heterodyne detection system in place, we proceeded to

perform simulated and real optical measurements. We concerned ourselves with two cases:

(1) when only noise is present and (2) the linear combination of noise and a beat note signal.

Simulations were performed entirely in MatLab using the scripts presented in Appendices A

and B. We then constructed the setup described in Figure 4-1 in order to measure real optical

signals in the laboratory. We first investigated the output behavior of Z2(N) when only the

LO field is incident onto PD2. From this measurement we verify the expected 1/τ behavior of

Znoise(N) and calculate an equivalent device sensitivity for the given integration time.

We then interfere our two laser fields to generate observable optical beat notes. For

calibration purposes, we initially set the power of the 2nd order sideband relatively high

(> 103 per second). In this case the beat note between the 2nd order sideband and the LO

is measurable using a spectrum analyzer. We can therefore compare our result to the readout

from the spectrum analyzer. We then adjust the power of the 2nd order sideband to an

equivalent rate on the order of 10−2 photons/second. In this case, the resultant beat note is

below the noise floor of the spectrum analyzer. We perform a 3-day measurement using the

design described in Chapter 4 and present results demonstrating successful detection of this

signal field.

5.2 Simulated Results

MatLab is used to perform simulations designed to test the heterodyne detection scheme

presented in Chapter 4. Beat note signals and shot noise are both artificially generated and

linearly combined at a sampling rate of 64 MHz in order to represent the photodetector output

after digitization, x[n]. We perform first demodulation and CIC filtering in MatLab at the

same sampling rate in order to simulate the processes within the FPGA card. The CIC filter

downsamples the data by a factor of 2048 to a rate of 31.25 kHz. Due to memory constraints

78

of the PC used for these simulations we are only able to simulate data for a total integration

time of 10 seconds. In order to obtain a larger number of samples, we decide to skip the

second downsampling stage discussed in Chapter 4. We write our simulated data to file at

a rate of 31.25 kHz. The MatLab script used to generate these data files is presented in

Appendix B. We then perform the second demodulation stage on the saved data using the

appropriate sampling rate. We calculate Z2(N) and scale it to an equivalent photon rate. The

result is then plotted vs. integration time.

5.2.1 Simulated Noise Behavior

We first simulate the case in which only the LO field is incident onto PD2. Looking at

Appendix B we simply set the photon rate of the signal field to zero. We generate an array

of white Gaussian noise (WGN) with the appropriate PSD to represent shot noise at the

photodetector output arising from PLO = 5.0 mW with a gain of G = 1.4 × 103 V/W. After

10-4 10-3 10-2 10-1 100 101

Integration Time (s)

10-210-1100101102103104105106107

Phot

ons/

seco

nd

Simulated ResultShot-Noise Expectation Value5-sigma Threshold

Figure 5-1. Simulated result for the case when only noise is present at the demodulationfrequency (blue). The curve follows Equation 3–83 (shown in red) and falls off as1/τ as expected. The 5-sigma threshold for detection is also shown in purple.

two demodulation stages we expect Z2(N) to fall off as 1/τ . The result of this simulation

79

is shown in blue in Figure 5-1. The simulated result follows the expectation value (shown in

red) for a system limited by shot noise at the demodulation frequency given by Equation 3–83.

The purple curve shows the 5-sigma confidence threshold for detection given by Equation 3–

86 using a quantum efficiency of η = 0.7. The result of this simulation agrees with our

expectations in the case where shot noise is dominant at the demodulation frequency and no

beat signal is present.

5.2.2 Simulated Signal Behavior and Confident Detection

We next simulate the case in which both shot noise and a beat note signal are present at

the photodetector output. The simulated beat note is generated using a LUT and is set to a

signal frequency of fsig = (16 MHz + 5 kHz). For this measurement, we suppose the power in

the simulated 2nd order sideband is PSB, 2 = 1.9 × 10−17 W. For λ =1064 nm laser light this

is equivalent to 100 photons/second. We again suppose the LO power is 5.0 mW and generate

a similar WGN array to represent shot noise. The noise array and beat note signal array are

then linearly combined. The first demodulation waveform is also generated using a LUT and is

set to a frequency of f1 = 16 MHz. The various parameters for this simulation are shown in

Table 5-1.

Table 5-1. Parameters for a Simulated Signal Measurement.Quantity ValuePLO 5.0 mWPsignal 1.9× 10−17 W

Expected Photon Rate 100 photons/sHeterodyne Measured Photon Rate 102 photons/s

After data are written to file we perform the second demodulation stage at a frequency

f2. We then calculate the equivalent photon rate of the signal field and plot the result vs.

integration time. The result of this simulation is shown in Figure 5-2.

When we set the second demodulation frequency equal to the signal frequency (f2 =

5 kHz) the resulting curve, shown in blue, flattens out after some integration time. The curve

eventually crosses the 5-sigma threshold, shown in purple, after ≈ 0.8 seconds signifying

80

10-4 10-3 10-2 10-1 100 101

Integration Time (s)

10-210-1100101102103104105106107

Pho

tons

/sec

ond

Simulated ResultShot-Noise Expectation Value5-sigma ThresholdDemod. Off Signal Frequency

Figure 5-2. Simulated results with an expected signal rate equivalent to 100 photon persecond. Demodulation at the signal frequency (blue) yields a measured rate of 102photons per second. Demodulation away from the signal frequency (yellow) yieldsthe 1/τ behavior of noise. The expected value for this level of noise is shown in redand the 5-sigma confidence threshold is shown in purple.

confident detection. The level that this curve flattens out to yields the photon rate of the

signal field. For this trial, we measure a rate of 102 photons/second, in agreement with our

expectations. One possible source of error is spectral leakage as discussed in Chapter 3.

Additionally, the ability to integrate for a longer amount of time should yield more accurate

results.

The yellow curve shows the result when we set the second demodulation frequency not

equal to the signal frequency (f2 = 5 kHz). In this case only shot noise is present in the

measurement frequency bin. The curve therefore follows the 1/τ behavior of Equation 3–83

as expected. The constant level of the measured signal rate crosses the expected value curve,

shown in red, at ≈ 6 × 10−2 seconds. For η = 0.7 and PSB, 2/(hν) = 100 photons/second we

expect τ2, x, sn ≈ 6 × 10−2 seconds. Similarly, we can also calculate the time we expect Z2(N)

81

to cross the 5-sigma threshold, τ5s, sn ≈ 0.8 seconds. Both of these intersection points agree

with the result of our simulation.

The simulations shown above successfully demonstrate that the digital design of our

heterodyne detection system works as intended. Although these simulations are limited to

a total integration time of 10 seconds due to computer memory constraints we are able to

accurately measure a signal with a field strength of 100 photons/second. There is no reason to

believe that weaker fields should not be detectable as well. We therefore proceed to generating

real optical beat notes in the laboratory.

5.3 Experimental Results

We construct the optical setup as described in Chapter 4. The output of PD2 is sent into

an ADC on-board the FPGA processing card. We investigate the behavior of Z2(N) for two

separate cases: (1) when only the LO field is incident onto PD2 and (2) when both lasers are

incident and a beat note between the LO field and a 2nd order sideband is present at the signal

frequency, fsig = 16 MHz + 2.4 Hz.

5.3.1 Noise Behavior and Device Sensitivity

We first perform a measurement with no signal field present to study the behavior of

the noise in our system. Only the LO beam with power PLO = 5.0 mW is incident onto

PD2. In Chapter 4 we have shown that our custom photodetector is limited by shot noise

at a demodulation frequency of 16 MHz + 2.4 Hz for this incident light power. We expect

the resulting curve to fall off as 1/τ in agreement with Equation 3–83. Data were collected

continuously for a total integration time of 19 days. We compute Z2(N′) and the result is

scaled to an equivalent photon rate. The result of this measurement plotted against integration

time, τ , is shown in Figure 5-3.

The yellow curve shows the result when setting the second demodulation frequency equal

to f2 = 2.5 Hz. The curve follows the expected value for the given LO power shown in red.

The 5-sigma threshold is calculated using Equation 3–86 and is shown in purple. Using the

same data run we compute Z2(N′) 50 additional times with different demodulation frequencies

82

τIntegration time

5 sigma confidence levelExpected value

Measurement data (2.5 Hz)50 run average (2.5-3.0 Hz)Double demodulation limit

Shot noise limit

Figure 5-3. 19 day shot-noise limited measurement with only the LO is incident withPLO = 5.0mW. The result of a single trial scaled to photons/second is shown inyellow. For this single measurement, because the curve does not cross the 5-sigmathreshold, shown in purple, we find that no spurious signals appeared over theentire integration time. We additionally compute Z2(N) for 50 separatedemodulation frequencies near 2.5 Hz. These data are then averaged to producethe dark blue line. This average follows the expected value line, shown in red. Thefundamental shot-noise limit if only one demodulation stage was required is drawnin light blue for comparison. The second demodulation stage increases theshot-noise limit by a factor of 2, shown by the dashed green line. Because theexpected value sits on top of this theoretical limit we show that shot noise is thedominant noise source in our setup.

near 2.5 Hz. The results are then scaled to a photon rate and averaged together. This average

is shown by the dark blue line in Figure 5-3 and is similar to the red expectation value curve in

both amplitude and behavior.

The light blue line shows the expected fundamental shot-noise limit if only one

demodulation stage was used. Because we require a second demodulation stage, the

amount of shot noise, scaled to photons/second, increases by a factor of 2. The theoretical

83

shot-noise limit after two demodulation stages is shown by the dashed green line. Because

the expectation value of our data lies on top of the double demodulation shot-noise limit we

confirm that shot noise is, in fact, the dominant noise source in our setup.

For the single measurement shown in yellow, because the curve does not cross the 5-sigma

threshold no spurious signals are picked up over the entire 19-day integration time when

Laser 2 was turned off. In this trial, the curve drops down to a level on the order of 10−6

photons/second. This level is an order of magnitude lower than the sensitivity requirement set

by ALPS II. However, if we perform multiple measurements over the same integration time the

results will yield different levels due to the statistical nature of the noise. Additionally, if we

perform a large number of measurements with no beat note signal present there is a non-zero

probability that some of the trials will cross the 5-sigma threshold. We therefore must use the

results from our statistical analysis in order to make claims with a certain level of confidence.

5.3.2 Optical Signal Genearation and Detection

Our previous result showed that for a single measurement no spurious signals appeared

after 19 days of consecutive integration time. However, we must also demonstrate that beat

notes at the demodulation frequency are detectable using this method. Furthermore, we

must ensure that our measurements yield photon rates that agree with expectations. We

calibrate our device by setting the power of the signal field to a high enough level so that we

can independently compare our results with measurements using a spectrum analyzer. After

calibration, we then lower the power of the signal field to sub-photon/second levels.

5.3.2.1 Calculating the Expected Signal Photon Rate

Referring to Figure 4-1 we first remove the ND filters in order for the 2nd order

sideband-LO beat note amplitude to be measurable using a spectrum analyzer. We set the

drive amplitude to the EOM to a given value and measure the voltage amplitude of both the

CC beat note and the sideband-LO beat note with the spectrum analyzer. The corresponding

modulation depth is calculated by

VSB, k = Jk(m)VCC (5–1)

84

where Jk(m) is the kth order Bessel Function. For small modulation depths we can use the

approximation

Jk(m) ≈ mk

2kk!(5–2)

We then determine the power in the 2nd order sideband (k = +2) using Equation 4–3.

PSB, 2 = [J2(m)]2PL2 .

Dividing by the photon energy, hν, yields the expected photon rate of the 2nd order sideband.

For calibration purposes, we do not place the ND filters back into the beam path so that we

may compare the readout from the spectrum analyzer to our own measurement. We place the

ND filters back into the beam path in order to attenuate the sideband power to sub-photon

per second levels. We use the known modulation depth to calculate the sideband power

before attenuation. Dividing by the attenuation factors of the ND filters we thus calculate the

expected photon rate of an ultra-weak signal field.

5.3.2.2 Calibration Using Stronger Signal Fields

For calibration purposes, we generate a 2nd order sideband with a relatively high photon

rate (>103 photons/second) so that the resulting beat note with the LO is observable on

a spectrum analyzer (SA). The RMS voltage amplitude of this beat note can be used to

determine the photon rate of the signal field as measured by the spectrum analyzer. We use

the RMS sideband voltage amplitude along with the photodetector gain G and the LO power

to calculate the power in the 2nd order sideband.

PSB, 2 =(VSB, RMS)

2

2G2PLO(5–3)

We then divide by hν in order to obtain an equivalent photon rate. We compare this result to

the photon rate measured using our heterodyne detection system. We set up a measurement

with the parameters described in Table 5-2. For these calibration measurements because the

2nd order sideband power is relatively large we reduce the LO power to PLO = 2.7 mW.

85

Table 5-2. Parameters for a Strong Signal Calibration Measurement.Quantity ValuePLO 2.7 mWPL2 4.7µW

Drive Amplitude 9 VppVCC, RMS 240 mVVSB, RMS 140 µV

m 6.7× 10−2

Expected Photon Rate 8.1× 106 ph/sSA Measured Photon Rate 9.2× 106 ph/s

Heterodyne Detection Measured Photon Rate 8.3× 106 ph/s

For this measurement, we calculate an expected photon rate of approximately 8.1 × 106

photons/second. Using the spectrum analyzer we measure a rate of 9.2× 106 photons/second.

We then measure the photon rate using the digital setup from Figure 4-14. Because the

signal field strength is relatively large we set the integration time to a few minutes. The

measurement yielded a rate of 8.3 × 106 photons/second and is shown in Figure 5-4. We

repeated this process for various EOM drive amplitudes and sideband signal powers. The

results of these measurements are compiled in Table 5-3. The maximum percent difference

between the spectrum analyzer reading and the heterodyne system for the measured data is

16%. This error is attributed to uncertainties in power measurements using the power meter,

photodetector gain and modulation depth measurements, windowing processes of the spectrum

analyzer, and spectral leakage as discussed in Section 3.2.2. We therefore find that our

Table 5-3. Multiple Calibration Measurements.Expected Photon Rate (ph/s) Heterodyne Measured Rate (ph/s) SA Measured Rate (ph/s)

5.2× 103 5.0× 103 5.9× 103

2.2× 104 2.4× 104 2.5× 104

3.0× 105 3.0× 105 3.4× 105

2.7× 106 2.7× 106 3.0× 106

8.1× 106 8.3× 106 9.2× 106

system independently yields results in agreement with spectrum analyzer measurements within

an error we find acceptable. We proceed to attenuate the power 2nd order sideband using

86

10−2

10−1

100

101

10210

5

106

107

108

Integration Time (s)

Pho

tons

/sec

ond

Expected Photon RateHeterodyne MeasurementSpectrum Analyzer Measurement

Figure 5-4. Measurement of an optically generated signal arising from the interference of twolasers. The signal photon rate is set to approximately 8.1× 106 photons/secondthrough measurements using a spectrum analyzer (red). The resulting plot ofZ2(N) scaled to photons/second is shown in blue. The curve flattens out yieldinga measured photon rate of 8.3× 106 photons/second, in agreement withexpectations.

the ND filters so that the resulting beat note is not observable on the spectrum analyzer and

perform measurements using our design.

5.3.2.3 Detection of an Ultra-Weak Signal Field

We first determine the modulation depth by removing the ND filters as described in

Section 5.3.2.1. We set the drive amplitude to the EOM to yield a modulation depth of

m = 1.1× 10−2. The two laser powers are measured at the input of PD2 to be PLO = 5.0mW

and PL2 = 5.7µW. The two ND filters are then placed back into the beam path. We

previously measure the combined attenuation factor of the ND filters to be approximately

2.0 × 105 at 1064 nm. We therefore calculate a 2nd order sideband power after attenuation of

6.3 × 10−21 W. For 1064 nm light this is equivalent to an expected photon rate of 3.4 × 10−2

87

photons/second. While weaker signals should also be observable, reducing the photon rate in

Table 5-4. Parameters for an Ultra-weak Signal Field Measurement.Quantity ValuePLO 5.0 mWPL2 5.7µWm 1.1× 10−2

fsig 16 MHz + 2.4 HzExpected Photon Rate 3.4 ×10−2 photons/s

Heterodyne Measured Photon Rate 3.3 ×10−2 photons/s

our setup was not possible due to the appearance of spurious electronic signals on the order

of 10−4 photons/second. However, we stress that these spurious signals disappear when the

function generator driving the EOM is turned off. Therefore these signals are not artifacts

of the weaker laser field but instead are due to the modulation process itself and will not be

a concern for ALPS IIc. This issue has been investigated and currently is thought to arise

from cross-talk between the drive signal to the EOM and the FPGA card. Additional work is

required to eliminate these spurious signals in order to lower the photon rate even further.

We phase lock the two lasers and measure the 2nd order sideband-LO beat note at

16 MHz + 2.4 Hz over a total integration time of approximately 3 days. The result of this

measurement is shown in Figure 5-5. Setting the demodulation frequency not equal to the

signal frequency (f2 = fsig) is shown in yellow. The resulting curve falls off as 1/τ and follows

the expected value line (red) for the given LO power.

The blue curve shows the result when we set the demodulation frequency equal to the

signal frequency (f2 = fsig = 2.4 Hz). The curve initially follows the 1/τ behavior of noise in

agreement with Equation 3–88 for an ultra-weak signal field and short integration times. The

noise dominance continues until the signal begins to take over, causing the curve to flatten out

and subsequently cross the 5-sigma threshold, shown in purple. The level at which this curve

flattens out yields a rate for the sideband of 3.3× 10−2 photons/second.

The time required for the noise to drop down to the signal level, τ2,x, and for the curve

to cross the 5-sigma threshold, τ2, 5s, also agree with expectations. For a photon rate of 3.4

88

τIntegration time

Signal present at 2.4 Hz5 sigma detection limit

3.33 x 10-2 photonsper second

Demodulation at exactly 2.4 HzDemodulation at 2.4003 Hz

Demodulation 2.5 HzExpected value (no signal)

Figure 5-5. Measurement of a 2nd order sideband with a photon rate on the order of 10−2

photons/second. The blue line shows the result when demodulating at the signalfrequency of 2.4 Hz. While noise is dominant at the beginning of themeasurement, eventually the beat note signal coherently adds, causing the curve tocross the 5-sigma threshold signifying confident detection. The level this curveflattens to yields a measured photon rate of 3.3 ×10−2 photons/second.Demodulating away from the signal frequency (f2 = 2.5Hz) causes the curve to falloff as 1/τ , shown in yellow. This follows the expected value line of noise, shown inred. The result when demodulating 300 µHz away from the signal (green)demonstrates the energy resolution of this system.

×10−2 photons/second the blue signal curve crosses the red noise expected value line after

≈ 170 seconds in agreement with Equation 3–84. Confident detection is made after ≈ 2400

seconds, in agreement with Equation 3–91. From this measurement, we confirm that our

system is capable of both generating and detecting weak signal fields on the order of 10−2

photons/second.

89

The importance of maintaining phase coherence throughout the entire measurement is

demonstrated by the green curve in Figure 5-5. In this case, the demodulation frequency is

set to be 300 µHz away from the signal frequency. For shorter integration times, the size of

the measurement frequency bin is large enough to pick up the signal. Eventually, the device

resolution increases past the point at which signals 300 µHz apart are distinguishable. When

this happens the integrated I and Q values begin to oscillate at |fsig − f2|. This causes the

curve to fall off as a sinc function, preventing it from crossing the 5-sigma threshold. We

must maintain phase coherence throughout the full integration time to ensure that signals are

detectable using heterodyne interferometry.

5.4 Summary

Within this chapter, we presented and discussed our results from simulations and real

optical measurements. All of our simulations are performed entirely in MatLab using the scripts

shown in Appendices A and B. Due to computer memory constraints our total integration time

for these simulations is limited to 10 seconds. We first investigated the case in which the input

consists solely of shot noise with an analog PSD equivalent to when PLO = 5.0mW. Results

from this simulation agree with the expected 1/τ behavior from Equation 3–83. No spurious

signals appeared during the measurement and the resultant curve did not cross the 5-sigma

threshold.

We then artificially generated an AC waveform using a LUT in MatLab to represent a beat

note between the LO field and a signal field with a strength equivalent to 100 photons/second.

We then performed a measurement in which the input is given by the linear combination of

simulated noise and this artificial beat note. When we set the demodulation frequency equal

to the signal frequency the resulting curve initially behaves as noise. Eventually, the signal

term in Equation 3–88 takes over causing the curve to flatten out and cross the 5-sigma

threshold. The level at which this curve flattens out yields a rate of 102 photons/second. We

attribute the error in the measured photon rate to limitations in the total integration time due

to computer memory constraints.

90

With simulations yielding successful results we proceeded to real optical measurements

in the laboratory with the setup described in Figure 4-1. We first investigated the output

behavior when only the LO field was incident onto PD2 with PLO = 5.0 mW. The result of a

measurement performed with Laser 2 off did not reveal any spurious signals that would degrade

the sensitivity of our setup after 19 days of integration. In this trial, the curve dropped down

to a level on the order of 10−6 photons/second. Recall that the sensitivity requirement for

ALPS IIc is on the order of 10−5 photons/second. However, due to the nature of the noise,

multiple trials will yield varying final values with some possibly above the 5-sigma threshold

even when a beat note is not present. We therefore use the statistical analysis presented in

Chapter 3 in order to make any claims with a given amount of confidence.

Our results also demonstrate successful generation and detection of signals at various field

strengths. We first calibrate our device by removing the ND filters and setting the 2nd order

sideband power to a level so that the resulting beat note with the LO is visible on a spectrum

analyzer. We thus independently compare our measurement with the calculated result using

the RMS voltage amplitude readout from a spectrum analyzer. We additionally compare these

results to the expected photon rates obtained through modulation depth measurements.

We then placed the ND filters back into the beam path and performed a measurement

with an expected field strength of the 2nd order sideband signal equivalent to 3.4 ×10−2

photons/second. After a 3-day integration time the resultant curve crosses the 5-sigma

threshold and yields a rate of 3.3 ×10−2 photons/second, in agreement with expectations. Our

system is therefore capable of both generating and measuring sub-photon/second signals.

Longer integration times and improvements in the generation of ultra-weak laser fields are

required to achieve low power levels which are comparable to the projected sensitivity of ALPS

IIc. Work on the generation, implementation, and detection of weaker signal fields is currently

ongoing. Our results also highlight the importance of maintaining phase coherence and stability

throughout the measurement. These limitations to heterodyne detection must be taken into

account during implementation into the ALPS II experiment.

91

CHAPTER 6IMPROVEMENTS, IMPLEMENTATION IN ALPS, AND FURTHER RESEARCH

6.1 Introduction

Results from Chapter 5 showed that, for a single measurement, no spurious signals

appeared after a 19-day integration time. Continuously keeping the ALPS II experiment

operational for months at a time is impractical when taking into account experimental

considerations. Such considerations include maintaining phase stability throughout the entire

measurement time and the ability to keep both cavities locked to the same resonant frequency.

Fortunately, we found ways to improve the digital design of our heterodyne detection system in

order to reduce the total noise pickup and lower the integration time required to claim 5-sigma

confident detection.

In this chapter, we discuss and implement two methods designed to reduce the total noise

pickup within our system. We first discuss a design proposed by Aaron Spector to eliminate

the noise pickup arising from the introduction of the second demodulation stage. The second

concept involves measuring and using the phase of the beat note during demodulation. When

this phase is known we find that the Q quadrature contains only noise and can be ignored.

Results from our stand-alone experiment demonstrate that heterodyne interferometry

can be applied as a single photon detector. We must therefore consider how we plan to

implement this detection technique into the ALPS IIc design. As we have seen, maintaining

phase coherence between the signal field and the demodulation waveform is crucial for the

operation of this system. A simplified layout of the current design is discussed in detail. We

discuss the two PLLs used to transfer phase information to the measurement hardware as well

as the stability requirements of the optical components.

While the optical design presented in Chapter 4 utilizes an EOM to generate the

ultra-weak signal field at sub-photon/second levels, some critics may argue that we simply

measure modulation rather than actual photons. We therefore discuss two future stand-alone

experiments designed to test optical heterodyne detection without the use of phase modulation.

92

6.2 Improvements to Optical Heterodyne Detection

In order to decrease the integration time required to claim 5-sigma confident detection

of coherent signals we must reduce the amount of noise pickup within our system. Less noise

pickup causes the noise term in Equation 3–89 to be lower so that the signal term dominates

after a shorter integration time. We investigate two methods to reduce the total noise pickup

within our original design.

6.2.1 Elimination of Added Noise Due to Multiple Demodulation Stages

In Chapter 3 we showed that the addition of the second demodulation stage causes an

increase in noise pickup by a factor of 2. The effect of this additional noise contribution is

evident in our measurement in which only the LO was incident onto PD2, shown in Figure 5-3.

Suppose a beat note is present at fsig = f1 + f2 as shown by Figure 6-1. With our original

design the total noise pickup includes contributions at frequency f1 + f2 as well as those at

frequency f1 − f2. Pickup of the noise at frequency f1 + f2 is unavoidable as it is in the same

frequency bin as the beat note. However, using a design proposed by Aaron Spector we are

able to eliminate the noise contributions at frequency f1 − f2.

Figure 6-1. FFT of the time series out of the FPGA card with a signal at frequency f1 + f2.Double demodulation picks up noise contributions from both the signal frequency,f1 + f2, as well as at frequency f1 − f2.

93

Referring to Figure 4-14, the first demodulation process involves multiplying the digitized

photodetector signal with a cosine waveform at frequency f1 generated by a LUT. Let us

denote the result of this mixing process as channel I out of the FPGA card. Now consider the

addition of a second channel on the FPGA card in which the digitized photodetector signal is

separately multiplied by a sine waveform at the same frequency, f1, and phase, ϕ. We denote

the result of this mixing process by channel Q out of the FPGA card. Data from each channel

are separately written to file. We then perform second demodulation on each individual channel

to yield a total of four terms: II’, IQ’, QI’, and QQ’. The prime indicates second demodulation.

Figure 6-2 shows an updated version of the digital design with both channels out of the FPGA

card and the resulting four terms. By taking a specific linear combination of these terms we are

able to reduce the noise pickup due to double demodulation without affecting the signal level.

We therefore recover our original sensitivities as if only a single demodulation stage is present.

Let us express these four terms mathematically. We again denote the photodetector output

by x[n]. We set the first demodulation frequency equal to f1 and the second demodulation

frequency equal to f2. The resulting four terms after both demodulation stages are given by

II ′ = x[n] cos (2πf1fsn)× cos (2π

f2fsn) ,

IQ′ = x[n] cos (2πf1fsn)× sin (2π

f2fsn)

QI ′ = x[n] sin (2πf1fsn)× cos (2π

f2fsn)

QQ′ = x[n] sin (2πf1fsn)× sin (2π

f2fsn) .

(6–1)

We then compute the following linear combinations

II ′ −QQ′ =x[n]

2cos (2π

f1 + f2fs

n) ,

IQ′ +QI ′ =x[n]

2sin (2π

f1 + f2fs

n) .

(6–2)

94

Figure 6-2. Digital Design To Eliminate Noise Pickup Due to Second Demodulation

This result is equivalent to a single demodulation stage design with fd = f1 + f2. We compare

this result to the original design in Chapter 4 in which we only use IQ’ and II’.

II ′ =x[n]

2

[cos (2π

f1 + f2fs

n) + cos (2πf1 − f2

fsn)

],

IQ′ =x[n]

2

[sin (2π

f1 + f2fs

n)− sin (2πf1 − f2

fsn)

].

(6–3)

From these equations we clearly see the additional pickup at frequency f1 − f2. The pickup at

f1−f2 is canceled out by including the second output channel on the FPGA and computing the

appropriate linear combination of terms. We then use the terms in Equation 6–2 to compute

an equivalent Z2(N) given by

Z2(N) =[∑N

n=1(II′ −QQ′)]2 + [

∑Nn=1(IQ

′ +QI ′)]2

N2(6–4)

95

We first consider the case in which only a beat note is present at frequency fsig = f1 + f2 with

peak amplitude A = 2G√

PLOPsignal. Solving for Z2, sig(N) in terms of the photon rate of the

signal field we findZ2,sig(N)

G2PLOhν=

Psignal

hν(6–5)

In this case the scaling factor is the same as when only one demodulation stage was used.

Next, we compute Z2, noise(N) in which only shot noise is present at the demodulation

frequency. After applying the appropriate scaling factor of 1/ G2PLOhν we find

Z2, noise(N)

G2PLOhν=

2

ητ(6–6)

Comparing the RHS of this result to the RHS of Equation 3–83 we find that this process

reduces the noise pickup by a factor of 2. We therefore regain the sensitivity of a design with

only one demodulation stage.

We proceed to test this concept using real optical signals. We reconfigure the FPGA card

to output both channels as described in Figure 6-2. After both channels are written to file we

use MatLab to generate the four terms and compute the appropriate linear combinations. We

then calculate Z2(N) using this method and scale the result to photons/second.

First we investigate the case in which only the LO is incident onto PD2. We set PLO =

6.0 mW. We compute Z2(N) for 50 different demodulation frequencies of a single run and

determine the average in order to obtain a better estimate of the expectation value. Results

from this measurement are shown in Figure 6-3.

The red curve shows the result when using only IQ’ and II’. This follows the behavior of

Equation 3–83 as expected. The blue curve shows the result when implementing this design to

suppress noise due to double demodulation. Multiplying our result after noise suppression by a

factor of 2 yields the green curve. Because the red and green curve overlap we verify that this

technique does in fact decrease the noise pickup by a factor of 2.

We must also confirm that the signal level remains the same. We therefore turn on Laser

2 and generate a beat note signal at 16 MHz + 2.4 Hz. We set the laser powers to PLO = 6.0

96

10−1

100

101

102

103

10−2

100

102

Integration Time (s)

Pho

tons

/sec

ond

Implementing Double Demod SuppressionWithout SuppressionImplementing Suppression Times 2

Figure 6-3. Measurement Demonstrating Suppression of Noise Pickup from DoubleDemodulation

mW and PL2 = 18µW measured at PD2 without the ND filters in place. We drive the EOM

so that the modulation depth is equal to m = 1.1 × 10−2. We thus compute an expected

photon rate for the 2nd order sideband of ≈ 0.11 photons/second after the ND filters are

placed back into the beam path. The properties of this run are compiled in Table 6-1.

Table 6-1. Test Parameters and Results for Double Demodulation Noise Suppression.Quantity ValuePLO 6.0 mWPL2 18µWm 1.1× 10−2

Expected Photon Rate 0.11 ph/sHeterodyne Measured Photon Rate-No Suppression 0.13 ph/s

Heterodyne Measured Photon Rate-With Suppression 0.14 ph/s

Results of this measurement are shown in Figure 6-4. Part A of the figure shows the

result when only II’ and IQ’ are used to compute Z(N). In this case, shown in blue, the curve

crosses the 5-sigma threshold after approximately 750 seconds. This measurement yields a

photon rate in the signal field of 0.13 photons/second. Part B of Figure 6-4 shows the result

97

when we use both channels out of the FPGA and suppress the noise pickup due to double

demodulation. When demodulating at the signal frequency, shown in blue, the curve now

crosses the 5-sigma threshold at approximately 370 seconds. This measurement yields a rate of

0.14 photons/second in the signal field. Both curves therefore tend to a similar photon rate.

Errors arise from uncertainties in the power meter measurements as well as modulation depth

measurements and spectral leakage. We therefore confirm an improvement in the SNR by a

factor of 2 when using this technique. This also improves our device sensitivity by a factor of

2 over the same integration time. The amount of integration time required to claim 5-sigma

confident detection of coherent signals is therefore reduced back to τ5s, sn ≈ 29hν/(ηPsignal).

10−1

100

101

102

103

10−4

10−3

10−2

10−1

100

101

102

103

104

Integration Time (s)

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/sec

ond

No Suppression Techniques ImplementedNoise Expectation Value5−sigma ThresholdDemod off Signal Freq.

A

10−1

100

101

102

103

10−4

10−3

10−2

10−1

100

101

102

103

104

Integration Time (s)

Pho

tons

/sec

ond

Double Demod. Noise Suppression ImplementedNoise Expectation Value5−sigma ThresholdDemod off Signal Freq.

B

Figure 6-4. A. Without either of the discussed techniques implemented. For this setup, itshould take approximately 750 seconds to cross the 5-sigma threshold.B. Implementing double demodulation noise suppression. By suppressing the noisepickup due to double demodulation the integration time required to cross the5-sigma threshold is reduced to approximately 370 seconds.

6.2.2 Demodulation with a Known Signal Phase

Another way to reduce the amount of noise pickup in our experiment involves measuring

the phase of the beat note between the signal field and the LO. When this phase is unknown

we must measure both I and Q quadratures in order to fully recover the beat note amplitude.

However, if the phase is known then we can use it during demodulation so that only the I

quadrature is needed. The Q quadrature, and thus the noise contributions in this quadrature,

can therefore be ignored.

98

Let the photodetector output contain a linear combination of noise and a beat note signal,

xtotal[n] = xsignal[n] + xnoise[n]. Suppose the constant signal phase, ϕ is now a known quantity.

We use this phase when generating the demodulation waveform. For now let us only consider a

single demodulation stage design. When the phase is known I/Q demodulation yields

I = xtotal[n]× cos (2πfd

fsn+ ϕ) ,

Q = xtotal[n]× sin (2πfd

fsn+ ϕ) .

(6–7)

The digitized beat note is given by xsignal[n] = A cos (2πfsigfsn+ ϕ) where A = 2G

√PLOPsignal.

In this case

I =

[A cos (2π

fsig

fsn+ ϕ) + xnoise[n]

]× cos (2π

fd

fsn+ ϕ) ,

Q =

[A cos (2π

fsig

fsn+ ϕ) + xnoise[n]

]× sin (2π

fd

fsn+ ϕ) .

(6–8)

When we set the demodulation frequency equal to the signal frequency we find

I =A

2+ xnoise[n] cos (2π

fd

fsn+ ϕ) ,

Q = xnoise[n] sin (2πfd

fsn+ ϕ) .

(6–9)

While the noise contribution in the Q quadrature is normally picked up when the phase is

unknown in this case we can discard Q entirely. We therefore reduce the total noise pickup

by a factor of 2 using this method. We then only need to use the I quadrature to compute

Z2(N).

Z2(N) =(∑N

n=1 I)2

N2(6–10)

When only a beat note is present at the demodulation frequency we find

Zsig(N)

G2PLOhν=

Psignal

hν(6–11)

Again note the scaling factor of 1/ G2PLOhν.

Next consider the case in which only shot noise is present at the demodulation frequency.

Because Equation 6–10 no longer computes the DFT, solving for Znoise(N) is slightly more

99

complicated. By using the known signal phase and ignoring the Q quadrature it can be shown

thatE [Z2 noise(N)]

G2PLOhν=

1

ητ(single demodulation) (6–12)

Experimentally we must still use two demodulation stages to avoid the DC bias of the FPGA

card. We use the known phase during the second demodulation stage performed in MatLab.

Without suppressing the noise due to double demodulation, the scaling factor becomes

4/(G2hνPLO). This yields

4 E [Z2,noise(N)]

G2PLOhν=

2

ητ(double demodulation) (6–13)

Comparing this result to Equation 3–83 we reduce our noise pickup by a factor of 2.

In order to verify this concept experimentally we first perform a measurement with only

the LO incident onto PD2 with PLO = 6.0 mW. In this case we are only concerned with

the decrease in noise pickup when Q is ignored. The result of this measurement is shown in

Figure 6-5. We again compute Z2(N) at 50 separate demodulation frequencies in order to

10−1

100

101

102

103

10−2

100

102

Integration Time (s)

Pho

tons

/sec

ond

Implementing Phase SearchWithout Phase SearchImplementing Phase Search Times 2

Figure 6-5. Measurement Demonstrating Noise Suppression Using Phase Search Techniques

100

obtain an estimate of the expected value lines. The red curve shows the result when using both

I and Q quadratures. The result when Q is ignored and only I is used to compute Z2(N) is

shown by the blue curve. The green curve is equivalent to the blue curve multiplied by a factor

of 2. Because the red curve lies on top of the green curve we demonstrate that by ignoring Q

we decrease our noise pickup by a factor of 2, as expected.

By knowing the signal phase and implementing it into the second demodulation stage

we can reduce the amount of integration time required for 5-sigma detection. Recall that

the second demodulation waveforms are generated LUTs in MatLab. We create a loop that

sweeps the phase of the demodulation waveform from 0 to 2π. We then compute Z2(N) from

Equation 6–10 for each phase and scale the results to photons/second. By maximizing the I

quadrature, and consequently the measured photon rate, we are able to determine the beat

note phase. Instead of setting up a completely new measurement, we are able to test this

Table 6-2. Test Parameters and Results for Phase Search TechniquesQuantity ValuePLO 6.0 mWPL2 18 µWm 1.1× 10−2

Expected Photon Rate 0.11 ph/sHeterodyne Measured Photon Rate-Unknown Phase 0.13 ph/sHeterodyne Measured Photon Rate-Known Phase 0.13 ph/s

concept using the same parameters from the double demodulation suppression measurement

in Section 6.2.1. We again expect a photon rate of 0.11 photons/second in the 2nd order

sideband. Data are taken using the original FPGA design with a single output channel. After

data are written to file we use MatLab to calculate the signal phase and implement it during

second demodulation. Results of this test compared to older methods with an unknown phase

are shown in Figure 6-6.

The result using the design detailed in Chapters 3 and 4 is shown in Part A of Figure 6-6.

Part B shows the result when the phase is measured and used during second demodulation.

Demodulating at the signal frequency, shown in blue, we obtain a rate of 0.13 photons/second

101

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100

101

102

103

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10−3

10−2

10−1

100

101

102

103

104

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tons

/sec

ond

No Suppression Techniques ImplementedNoise Expectation Value5−sigma ThresholdDemod off Signal Freq.

A

10−1

100

101

102

103

10−4

10−3

10−2

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100

101

102

103

104

Integration Time (s)

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tons

/sec

ond

Phase Search ImplementedNoise Expectation Value5−sigma ThresholdDemod off Signal Freq.

B

Figure 6-6. A. Without either of the techniques discussed implemented. Again, for this setupτ2, 5s ≈ 750 s.B. Determining the phase of the beat note we can ignore the Q quadraturecompletely reducing the noise pickup by a factor of 2. This reduces the integrationtime requires to cross the 5-sigma threshold.

in the signal field. Because the signal level remains the same we therefore improve our SNR by

a factor of 2. This, in turn, reduces the amount of integration time required to claim 5-sigma

confident detection. The factor of 2 in reduction of the 5-sigma threshold still needs to be

evaluated by a statistical analysis which is planned for the future.

6.2.3 Implementation of Both Noise Reduction Techniques

Each of the improvement techniques mentioned above have individually been shown

to reduce our noise pickup by a factor of 2. By measuring the signal phase and using both

channels out of the FPGA card it is in fact possible to combine these methods to reduce the

total noise pickup by a factor of 4. Suppose our input is given by

xtotal[n] =

[A cos (2π

fsig

fsn+ ϕ) + xnoise[n]

](6–14)

102

We again calculate the four terms II’, IQ’, QI’, and QQ’, however, we now suppose the phase is

known and use it during second demodulation. We find

II ′ =

[A cos (2π

fsig

fsn+ ϕ) + xnoise[n]

]cos (2π

f1fsn)× cos (2π

f2fsn+ ϕ) ,

IQ′ =

[A cos (2π

fsig

fsn+ ϕ) + xnoise[n]

]cos (2π

f1fsn)× sin (2π

f2fsn+ ϕ)

QI ′ =

[A cos (2π

fsig

fsn+ ϕ) + xnoise[n]

]sin (2π

f1fsn)× cos (2π

f2fsn+ ϕ)

QQ′ =

[A cos (2π

fsig

fsn+ ϕ) + xnoise[n]

]sin (2π

f1fsn)× sin (2π

f2fsn+ ϕ) .

(6–15)

Demodulating at the signal frequency, simplifying, and ignoring higher order terms yields

II ′ =A

4+ xnoise[n] cos (2π

f1fsn) cos (2π

f2fsn+ ϕ) ,

IQ′ = xnoise[n] cos (2πf1fsn) sin (2π

f2fsn+ ϕ)

QI ′ = xnoise[n] sin (2πf1fsn) cos (2π

f2fsn+ ϕ)

QQ′ = −A

4+ xnoise[n] sin (2π

f1fsn) sin (2π

f2fsn+ ϕ) .

(6–16)

In this case when the phase is known the IQ’ and QI’ terms contain only noise and can both be

ignored. We then compute Z2(N) as

Z2(N) =[∑N

n=1(II′ −QQ′)]2

N2(6–17)

Looking first at the signal term we find

Z2,sig(N)

G2PLOhν=

Psignal

hν(6–18)

Calculation of the noise term is again non-trivial as we no longer use the DFT to compute

Z2(N). It can be shown thatE [Z2,noise(N)]

G2PLOhν=

1

ητ(6–19)

By implementing both techniques described in this chapter we are able to increase our device

sensitivity by a combined factor of 4 when compared to Equation 3–83.

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We now test this combination of our two improvement techniques using real optical

measurements. We first turn off Laser 2 so that only the LO is incident onto PD2 with

6.0 mW of optical light power. While we cannot determine a phase without a beat note

present, this measurement demonstrates a reduction in noise pickup when IQ’ and QI’ are

ignored in conjunction with eliminating the noise pickup due to double demodulation. We

again measure Z2(N) at 50 separate demodulation frequencies and compute the average. The

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Figure 6-7. Noise Measurement Implementing Both Double Demodulation Suppression andPhase Search Techniques

output when no improvement techniques are implemented is shown by the red curve. The

blue curve shows the result when we use both channels out of the FPGA card and use the

beat note phase in the second demodulation stage. The green curve is equal to the blue curve

multiplied by a factor of 4. Because the red curve lies on top of the green curve we confirm

that implementing both improvements decreases our noise pickup by a combined factor of 4.

We again must verify that the signal level remains the same. In order to easily compare

the result with previous tests we use the same recorded data from the measurement described

in Table 6-1. The expected rate of the 2nd order sideband is again 0.11 photons/second.

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Figure 6-8. A. Result without implementing any of the improvements described.B. Implementing improvement techniques in parallel reduces the noise pickup by afactor of 4. The signal level remains the same as before. This reduces the amountof time required to cross the 5-sigma threshold. The factor of 4 reduction in the5-sigma threshold still needs to be confirmed via statistical analysis planned for thefuture.

Part A of Figure 6-8 again shows the result when neither technique is implemented.

The result when using both improvement techniques in parallel is shown in Part B of the

figure. In this case, demodulating at the signal frequency, shown in blue, yields a rate of 0.14

photons/second in the signal field. By using both channels out of the FPGA card and knowing

the phase of the beat note the integration time required to cross the 5-sigma threshold is

greatly reduced. Numerical results of all of the measurements in this section are shown in

Table 6-3.

Table 6-3. Measured Photon Rates for Various Improvement Techniques.Type of Improvement Measured Photon Rate (ph/s)No Suppression Used 0.13

Double Demodulation Suppression 0.14Phase Search Suppression (Known Signal Phase) 0.13

Both Improvement Techniques Implemented 0.14Note: The expected rate is set to be 0.11 photons/second.

Implementing both improvements we are able to increase our SNR by a combined factor

of 4. Consequently, this reduces the amount of integration time required to cross the 5-sigma

threshold for ALPS IIc to a far more feasible level of approximately 12 days.

105

6.2.4 Future Improvements

While we have so far discussed and successfully tested two methods to improve our

device sensitivity there are further upgrades that can be made to our current system. One

possible source of error in all of our results arises from measurements of the two average laser

powers. We assume both L2 and the LO remain at constant power levels throughout the entire

integration time. While the power of these lasers are mostly stable they may slightly vary over

the course of multiple day measurements. Because the LO is many orders of magnitude higher

in power than L2, fluctuations in PLO are much more influential. We can account for these

power fluctuations by constantly measuring PLO. During the calculation of Z2(N), the power

of the LO becomes an array of values rather than a single quantity. In this case, any power

fluctuations of the LO are tracked and accounted for during the calculation of the photon rate

of the signal field.

We can also improve the accuracy of our signal measurements by reducing the amount of

spectral leakage. In order Z(N) to be proportional to the magnitude squared of the DFT we

require that the total number of samples is N = 2πlfd/fs, where l is an integer. For all of

the signal measurements presented in this dissertation, the spectral leakage was assumed to be

negligible. In order to completely eliminate errors due spectral leakage we simply set the array

size of the sampled data to the nearest whole integer of 2πfd/fs.

Finally, one of the issues discussed with the current design is the appearance of spurious

signals on the order of 10−4 photons/second. While all of our signal measurements above this

level are still valid, these spurious signals prevent us from lowering the photon rate any further.

Future work is required to investigate the source of these signals in order to reduce or eliminate

them completely. We reiterate that these spurious signal have been shown to be the result of

the modulation itself and will not be an issue for ALPS IIc. This is apparent when we turn off

the drive signal to the EOM and the spurious signals disappear.

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6.2.5 Hardware Upgrades

We are currently in the process up updating the hardware used in this experiment to

more modern technology. The Xilinx PMC-AX3065 FPGA card used to perform all of the

experimental measurements in this dissertation is now obsolete. The company that originally

produced this card no longer offers technical support. Finding a replacement card in the case

of a malfunction is therefore rather unlikely. Modern FPGA cards boast higher sampling rates

and lower ADC noise. We are currently looking into using stand-alone device produced by

Liquid Instruments called a Moku:Lab for implementation into the ALPS IIc experiment. This

device has an internal FPGA card and can sample at speeds up to 500 MS/S. The stated ADC

noise of a Moku:Lab device is 30 nV/√

Hz, a factor of 33 lower than the measured ADC noise

of the Xilinx FPGA card [72].

Moku:Lab devices are easily portable and do not require integration with a desktop

computer or LabView. Data can be recorded using a tablet and saved to a standard SD card.

Additionally, an Ethernet port allows for continuous streaming of data. Simple configuration

of the FPGA card is performed via the tablet computer. More complex configurations are

programmed using either Python or MatLab code to interface with the device. Work on

transferring the digital Simulink design to a Moku:Lab system is currently being performed by

Mauricio Diaz-Ortiz. Once operational we will perform both noise and signal measurements

in order to calibrate the device and compare with the Xilinx FPGA results. Due to their ease

of use, ALPS IIc plans to use three Moku:Lab FPGA devices. Two will be used for laser phase

locking while one device is designated to signal measurements.

6.3 Heterodyne Detection in ALPS IIc

From the results of our stand-alone heterodyne interferometry experiment we are able

to meet and surpass the sensitivity requirements set by the ALPS IIc design. We must

then consider how exactly we plan to implement this detection method into the ALPS IIc

experiment. Recall the simplified ALPS IIc layout from Figure 2-1. Heterodyne interferometry

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relies on overlapping two separate laser fields. The ALPS IIc optical design must therefore be

modified in order to be compatible with this detection method.

Phase information from the source laser injected into the PC must be tracked throughout

the entire data run without any of the light leaking into the RC. We must also ensure that

the two cavities remain locked at the same resonant frequency. A proposed design using three

lasers is shown in Figure 6-9.

Figure 6-9. Heterodyne Implementation in ALPS IIc

Laser 1 acts as the injection laser into the PC. Laser 1 is frequency locked to the

production cavity via Pound-Drever-Hall (PDH) locking techniques to maintain resonance.

A reference laser is located on the central table and is phase locked to Laser 1 using a standard

PLL with the output of PD1. The NCO used in this PLL is synchronized to a master clock.

Finally, a third laser is located on the RHS end table. This laser acts as the LO field for

heterodyne detection and is injected into the RC. A second PLL locks the LO to the reference

laser using the output from PD2. The NCO used for this PLL is also synchronized to the

master clock. Lastly, the RC is length locked to the LO using PDH techniques. In the end,

the LO is phase locked to Laser 1 using the reference laser as an intermediary stage. Because

Laser 1 is locked to the PC we ensure that the two cavities are locked to the same resonant

frequency.

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Phase information from Laser 1 is transmitted to the measurement hardware in order

to track the beat note frequency between the LO field and the regenerated photon field. We

generate the demodulation waveform using an NCO set to the difference frequency between

Laser 1 and the LO with a small offset fδ in the Hz range. This NCO is also synchronized to

the master clock in order to prevent cycle slips. The current design requires a phase stability

of 0.1 cycle RMS over the entire integration time in order to recover 95% of the resulting beat

note amplitude. Work on testing various aspects of the optical design to meet this requirement

is ongoing.

The optical layout described in Figure 6-9 is designed solely with heterodyne detection

in mind. Unfortunately, this concept is not compatible when using the transition edge sensor

(TES) as a means of detection. Unlike heterodyne detection, the TES is sensitive to any

infrared light appearing in the RC. Therefore, injecting the LO field into the RC will appear

as a signal to the TES. As of now, the TES detection system and our heterodyne detection

system cannot be used concurrently. A TES design, using 532 nm light to lock the resonance

of the RC, is also currently being developed. It is still beneficial to perform individual science

runs with each detector operational. Measurements obtained using one detection method can

be confirmed by the other yielding a more confident result.

6.4 Fractional Photons

Continuously keeping the ALPS II experiment operational over months is virtually

impossible due to external distortions. We must consider the possibility of experimental

interruptions to our measurements. During these interruptions, the regenerated photon field

may not be present at the measurement photodetector. If we know the timestamps at which

these interruptions occur and can track the phase evolution during the downtimes, we can

appropriately stitch together the sections of the time series containing valid measurement data.

Using this method we can fully recover the signal amplitude. We design an experiment using

our stand-alone setup in order to test this concept in the laboratory.

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Consider a measurement using our testbed in which we disconnect/reconnect the function

generator driving the EOM at a set rate and continuously record data. When the function

generator is disconnected, no sidebands are present. By knowing the time at which the

function generator was disconnected/reconnected we can throw out these segments of the data

files. This is represented by the time-series shown in Figure 6-10. In this measurement, we

Figure 6-10. Time-Series of a Data Stream in which the Drive Signal to the EOM isDisconnected/Reconnected Every 100 Seconds. We start this measurement withthe beat note present at the photodetector for 100 seconds. Regions highlightedin red are known to not have a beat note signal present during these times. Wecan throw out data from these regions and stitch together the remainder in orderto fully recover the signal amplitude.

disconnect the function generator driving the EOM every 100 seconds. The time-series shown

in the figure spans the duration of the entire measurement. However, we know the times at

which beat note signals were not present on the measurement photodetector, highlighted

in red. In order to completely recover the signal amplitude, we are able to throw out these

highlighted regions and stitch together the remaining data in which we know a beat note is

present. By timestamping the data, leaving the function generator running while disconnected,

and synchronizing all of the electronics using a 10 MHz timebase we are able to maintain

phase coherence during demodulation. We initially perform a measurement with the 2nd

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Figure 6-11. Fractional Photon Measurement with ≈ 4.2× 104 Photons/Second Signal Field.The blue curve shows the result when the EOM is continuously driven by thefunction generator. The green curve gives the result when we disconnect drive tothe EOM for 100 seconds then reconnect it for 100 seconds. In this case, weappropriately cut the data stream to only include samples when a beat note ispresent. Cutting out samples obviously reduces the total integration time,however, the green curve yields a similar photon rate of the signal field.

order sideband at a rate of ≈ 4.2 × 104 photons/second. The result is shown in Figure 6-11.

The blue curve shows the result when the function generator driving the EOM is continuously

connected. In this case, no data stitching is necessary. This yields a measured photon rate

of the signal field of ≈ 4.7 × 104 photons/second. The green curve shows the result when

we disconnect/reconnect the function generator every 100 seconds. However, we ignore data

at times in which we know the beat note signal is not present. We then stitch together the

remaining data in order to completely recover the signal amplitude. While this obviously

reduces the total measured integration time, the result also yields a rate of ≈ 4.7 × 104

photons/second in the signal field.

We perform this measurement again but with a reduced sideband rate of ≈ 5.0

photons/second. The result is shown in Figure 6-12. Again, the blue curve shows the

result when the drive signal to the EOM is continuously present. This yields a measured

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Figure 6-12. Fractional Photon Measurement with Sub-Photon Per Second Signal Field. Wereduce the rate in the signal field to ≈ 5.0 photons/second. The blue curve showsthe result when the EOM is continuously driven and yields a rate of 5.3 photonsper second. The green curve shows the result when we disconnect/reconnect thedrive to the EOM as before and stitch together the data stream with the knowntimestamps. This results yields a rate of 5.4 photons/second in the signal field.

rate of 5.3 photons/second in the signal field. We then perform a measurement in which we

disconnect/reconnect the drive signal to the EOM but throw out data when the function

generator is disconnected. We appropriately stitch together the remaining data and plot the

result, shown in green. This measurement yields a rate of 5.4 photons/second in the signal

field.

We can therefore recover the full signal amplitude by ignoring parts of the data stream

in which we know a beat note signal is not present and stitching together the remaining data.

This will be especially useful in ALPS IIc. We can stitch together fragments of data runs as

long as we maintain phase coherence and appropriately timestamp the data stream in order to

know which segments to retain.

6.5 Future Experiments

In this dissertation we have shown successful measurements of signal fields with equivalent

photon rates on the order of 10−2 photons/second. Despite these results, some critics may

112

claim that our experiment simply measures modulation instead of actual photons. We argue

that we measure the interference between two optical laser fields and therefore there is no

difference. Nonetheless, in order to completely silence any doubts we have devised two future

experiments that do not rely on modulation for detection. The first design uses two lasers to

generate optical beat notes. Key differences, operation, and potential drawbacks of this design

are discussed. We then develop another experiment involving three separate laser sources to

avoid potential problems with the two-laser design. The three-laser design is similar to how

heterodyne detection will be implemented in ALPS IIc.

6.5.1 Two-Laser Setup

In order to measure signals at the sub-photon per second level using heterodyne detection

we must be able to track the beat note frequency over the full integration time. In our

stand-alone experiment we utilize phase modulation to generate relatively low power sidebands

and track the 2nd order sideband-LO beat note frequency using the CC beat note. Removal

of the modulation requires a redesign of the optical layout itself. Our first conceptual design

for a stand-alone heterodyne detection system without the use of modulation is shown in

Figure 6-13.

Figure 6-13. Two-Laser Heterodyne Interferometry Experiment without Modulation

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As with any heterodyne detection scheme, a minimum of two laser sources must be

present. Laser 1, at frequency fL1, acts as our source for the ultra-weak signal field. Laser 2,

at frequency fL2, represents our LO field. Light from Laser 1 is split into two separate paths

at the first beam splitter, BS1. The transmitted path is then incident onto a beam combiner,

BS2, where it is interfered with the LO field. The combined beam is then incident onto PD1.

Because Laser 1 is not attenuated along this beam path, the optical powers of each laser can

be set to sufficiently large enough levels to produce an adequate error signal and maintain a

stable phase lock loop at the difference frequency, fCC = fL1 − fL2.

The reflected path of Laser 1 is sent through multiple ND filters to attenuate the power to

desired levels. The beam is then sent into a beam combiner, BS3, where it is overlapped with

the LO field. The combined beam is then incident onto the measurement photodetector, PD2.

We therefore generate a beat note signal at frequency fCC. However, the beat note at PD2 is

between the LO and a sub-photon per second field.

In order to use two demodulation stages it is easier to reconfigure the FPGA so that

the first demodulation waveform is at frequency f1 + fδ. Because we no longer use phase

modulation it is much more difficult to lock the two beams at a signal frequency of 16MHz +

2.4Hz. Previously, we were able to offset the signal frequency by 2.4 Hz with the precision

of the function generator driving the EOM. The current phase lock loop design using the

FPGA card does not have the accuracy to lock the two lasers with such a relatively small offset

frequency. While it is possible to update the phase lock loop design, it is simpler to change

the first demodulation frequency in the FPGA to f1 + fδ. With this two-laser design we are

able to generate a beat note between an ultra-weak signal field and the LO while being able to

separately lock and track the beat note frequency without the use of an EOM.

One issue that may arise is the appearance of scattered light from Laser 1 off of the

various optical components. The beat note of interest is between the LO field and Laser

1. Unwanted scattered light present at PD2 has the potential to ruin our measurements. A

scattered light analysis should be performed in order to determine the expected power levels

114

from Laser 1 at PD2. The use of irises and other optics may be required to reduce the amount

of scattered light at the measurement photodetector. It is also possible that the equivalent

photon rate of such scattered light is at the levels desired for measurements. In this case, the

ND filters placed in the beam path of Laser 1 can simply be replaced with a wall. Additionally,

back-reflected light off of the optics leading into PD1 may also lead to larger than expected

photon rates. An analysis on such back-reflected light should also be considered.

The largest cause for concern with this two laser design is that of crosstalk between the

two photodetector outputs. Optical beat notes at the signal frequency are present at both

photodetectors. The beat note at PD1 used for error feedback is much larger in amplitude

than the beat note at PD2. The electronic signal out of PD1 therefore has the potential

to leak into the measurement channel and spoil any useful measurements. Additionally, the

waveform generated for the phase lock loop is also at the signal frequency. Crosstalk may

occur through air between various cables or through the FPGA circuit board itself. An analysis

is required in order to determine the strength of signals arising from crosstalk within this

design. Greater cable shielding may be used to combat these unwanted signals. It is also

possible to eliminate any concerns regarding crosstalk by modifying the optical design. We

have devised a solution using a third intermediary laser in order to transfer phase information.

With this approach crosstalk will hopefully no longer be an issue. This three-laser design is also

very similar to the final ALPS IIc design.

6.5.2 Three-Laser Setup

We have identified multiple causes for concern regarding the proposed two laser

experiment without the use of phase modulation as described in Figure 6-13. We seek to

improve upon this idea with the addition of a third laser source. Our updated conceptual

design is analogous to the approach of implementing heterodyne detection in ALPS IIc. The

layout of a three-laser experiment designed to test the capabilities of heterodyne detection

without the use of an EOM is shown in Figure 6-14.

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Figure 6-14. Three-Laser Heterodyne Interferometry Experiment without Modulation

Similar to the two laser design, Laser 1, at frequency fL1, acts as the source of

the ultra-weak signal field we wish to detect. Laser 2, at frequency fL2, now acts as an

intermediary reference laser. The newly added Laser 3, at frequency fL3, is our LO field.

Similar to before, the output of Laser 1 is immediately split into two paths at BS1. The

transmitted path is then overlapped with Laser 2 at BS2 to generate an optical beat note at

frequency fCC1 = fL1 − fL2. The combined beam incident onto PD1 whose output is used for

error feedback to Laser 1. With this PLL, Laser 1 is phase locked to our reference laser, Laser

2.

The reflected path of Laser 1 passes through multiple ND filters in order to attenuate

the light power to the desired level. The attenuated beam is then overlapped with Laser 2

at BS3. We then send the combined beam into another beam combiner, BS4, where it is

overlapped with Laser 3. Both the transmitted and reflected paths out of BS4 are comprised of

a combination of all three lasers. Interference at this beam combiner creates multiple optical

beat notes. Each path containing all of these beat notes is then focused onto two separate

photodetectors, PD2 and PD3.

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The beat note between Laser 2 and Laser 3 occurs at frequency fCC2 = fL2 − fL3. This

is present at the output of PD3 and is used as part of a second PLL. The optical powers of

these two lasers can be adjusted so that this loop is stable. From this feedback loop we are

able to phase lock Laser 3 to Laser 2. Recall that Laser 1 is also phase locked to Laser 2 via

the PLL at PD1. Therefore, with this design we phase lock Laser 1 to Laser 3 using Laser 2 as

a reference.

An optical beat note between Laser 3 and the attenuated beam from Laser 1 at frequency

fsig = fL1 − fL3 is also generated from the interference at BS4. This beat note signal of

interest is present at the measurement photodetector, PD2. Phase information from locking

Laser 1 to Laser 3 is sent into the FPGA card in order to track the frequency of this beat note

signal during demodulation.

We must again be careful to optimize the coupling between the spatial eigenmodes of

all three beams. We also must consider the polarization of the laser fields along each path.

Wave-plates must be placed appropriately into the design to ensure that the polarization of the

three laser fields are parallel when the beams are interfered.

With this three laser setup we generate an optical beat note between an ultra-weak signal

field and a higher power LO field at a known, fixed frequency. By adding a third laser we

ensure that crosstalk will hopefully no longer be an issue when using this updated design.

Unlike the two-laser design, there does not exist an electronic output containing a relatively

strong AC signal at frequency fsig. The un-attenuated beam from Laser 1 is never overlapped

with the LO. In this case, crosstalk from the output of PD1 will no longer create false signals

at the measurement frequency.

6.6 Summary

Within this chapter we detailed two improvements to our stand-alone experimental design

in order to decrease the total amount of noise pickup over the same integration time. The

first improvement involved adding a second channel to the FPGA card and performing I/Q

demodulation on both output channels. A linear combination of the resulting four terms

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was used to eliminate noise pickup due to the addition of the second demodulation stage.

Additionally, when a signal is present we investigated the result of measuring the phase of

the resulting beat note and using it during demodulation. We found that by knowing the

signal phase we are able to ignore the Q quadrature and “force” all of the signal into the I

quadrature. Each of these improvements individually lead to an decrease in the total noise

pickup by a factor of 2 while the signal level remains the same. These improvements were

tested using real optical signals in the laboratory.

Implementing both of these improvements in parallel when a beat note signal is present

decreases our total noise pickup by a combined factor of 4. The SNR, in turn, increases

by a factor of 4 as well. The addition of these improvements also significantly reduces the

integration time required in order to claim 5-sigma confident detection of coherent signal fields.

Future improvements to the heterodyne detection scheme were also discussed in this

chapter. This includes measuring the LO power throughout the entire integration time and

reducing the amount of spectral leakage. Also mentioned was the planned use of a newer

FPGA device called a Moku:Lab. This portable device has a lower ADC noise than the Xilinx

FPGA by a factor of 33. Work on the implementation of a Moku:Lab FPGA device into our

system is currently ongoing.

We then detailed one possible design to implement heterodyne detection into the ALPS IIc

experiment. The proposed design uses a three laser system in order to transfer the appropriate

phase information to be used during demodulation. We plan to overlap the LO field and the

regenerated photon field by injecting the LO into the regeneration cavity. Interference of

these two electromagnetic fields generates measurable beat notes observable using heterodyne

detection. Unfortunately, this proposed design is not compatible with the TES system.

Injection of the LO field into the RC will appear as a false signal using the TES.

We discussed the possibility of experimental interruprtions in ALPS II and the concept of

stitching together sections of the data stream containing valid measurement data. We tested

this method using our stand-alone setup. In this experiment, we simply disconnect/reconnect

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the function generator driving the EOM at a set rate. We leave the function generator running

and synchronized to a master clock in order to maintain phase coherence. For both relatively

large and weak signal fields, we show that by knowing when the beat note is present at the

photodetector, we can stitch together our data stream to obtain measured rates that match

with results when the drive signal is continuously connected to the EOM.

Finally, we overviewed two different stand-alone optical experiments designed to test

heterodyne detection capabilities without the use of modulation. Both a two-laser and a

three-laser setup were discussed. While the two-laser design is much simpler, crosstalk may

pose a potential problem. The three-laser design removes these concerns by using a reference

laser to transfer phase information. While still only conceptual designs, results from these

experiments will be beneficial to demonstrate the effectiveness of heterodyne detection without

the use of modulation.

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APPENDIX AMATLAB SCRIPTS FOR SECOND DEMODULATION AND POST-PROCESSING

1 function [ Time , Photon_Rate ] = Heterodyne_Demodulation (ClockTime , I , Q, Measured_CC_Frequency ,

Lock_Threshold , Lock_Frequency , Sampling_Frequency , Demodulation_Frequency ,

Local_Oscillator_Power , Photodetector_Gain , Analog_Gain , Quantum_Efficiency )

2 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

3 % This f i r s t sect ion checks that the beat note remained locked throughout the measurement .

4 % I f the system became unlocked i t outputs a message and the sample number at which th i s occurs .

5 % This s c r i p t then r e s i z e s the input arrays to ignore data when the system was unlocked .

6

7 % Counter to sweep through Measured_CC_Frequency array

8 counter = 0;

9

10 % I n i t i a l i z e 'Last_Locked ' to the end of the input array . I n i t i a l l y assume system never unlocked

11 Last_Locked = length (Measured_CC_Frequency) ;

12

13 f o r i = 1: length (Measured_CC_Frequency)

14 i f abs (Measured_CC_Frequency( i )−Lock_Frequency) > Lock_Threshold

15 i f Last_Locked > i

16 Last_Locked = i ;

17 end

18 counter = counter + 1;

19 end

20 end

21

22 i f counter == 0

23 disp ( 'Lock i s good ' ) ;

24 e l s e

25 ClockTime = ClockTime ( 1 : ( Last_Locked−1)) ;

26 I = I ( 1 : ( Last_Locked−1)) ;

27 Q = Q( 1 : ( Last_Locked−1)) ;

28 Measured_CC_Frequency = Measured_CC_Frequency ( 1 : ( Last_Locked−1)) ;

29

30 X = [ 'The PLL f i r s t became unlocked at sample number ' , num2str (Last_Locked) ] ;

31 disp (X) ;

32 end

33

34 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

35 % This next sect ion normalizes the clock sample numbers to 1 , 2 , 3 . . .

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36

37 % Determine the step s i z e between clock samples

38 Step_Size = ClockTime (2) − ClockTime (1) ;

39

40 % Subtract the minimum value from the array such that ' Clock ' s t a r t s at 0

41 Clock = ClockTime − min(ClockTime) ;

42

43 % Divide ' Clock ' by the step s i z e . Clock should now be 0 , 1 , 2 . . .

44 Clock = Clock ./ Step_Size ;

45

46 % Add '1 ' to the Clock array causes Normalized_Clock to go as 1 , 2 , 3 . . .

47 Normalized_Clock = Clock + 1;

48

49 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

50 % The clock array and the I and Q arrays are then sorted to be in time order

51 % This i s done in case of buf fer sk ips such that the clock gets unorganized (1 , 2 , 4 , 3 . . . )

52

53 [ SortedClock , SortIndex ] = sort ( Normalized_Clock ) ;

54 ysorted = I ( SortIndex ) ;

55 Sorted_I = ysorted ' ;

56 xsorted = Q( SortIndex ) ;

57 Sorted_Q = xsorted ' ;

58

59 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

60 % In th i s sect ion we generate the NCO' s used to perform second demodulation on both output

channels of the process ing card

61

62 % Set 'h ' to Planck ' s constant

63 h = 6.626e−34;

64

65 % Set 'nu ' to the frequency of 1064 nm l i g h t

66 nu = 281954887218045.06;

67

68 % Generate the time array

69 Time = SortedClock '/ ( Sampling_Frequency) ;

70

71 % Generate sample number array . Same as the Normalized_Clock

72 N = (1 : length (Time) ) ;

73

121

74 % Next we generate the s ine / cos ine waveforms for second demodulation

75 % Indiv idua l ly multiply the output channels of the process ing card by s ine / cos ine to get four

terms : II , IQ , QI , and QQ

76

77 % ' II ' i s the I output of the process ing card mult ip l ied by cos ine at the second demodulation

frequency

78 I I = Sorted_I .∗ cos (2 .∗ pi .∗Time.∗ Demodulation_Frequency) ;

79

80 % 'IQ ' i s the I output of the process ing card mult ip l ied by s ine at the second demodulation

frequency

81 IQ = Sorted_I .∗ s in (2 .∗ pi .∗Time.∗ Demodulation_Frequency) ;

82

83 % 'QI ' i s the Q output of the process ing card mult ip l ied by cos ine at the second demodulation

frequency

84 QI = Sorted_Q .∗ cos (2 .∗ pi .∗Time.∗ Demodulation_Frequency) ;

85

86 % 'QQ' i s the Q output of the process ing card mult ip l ied by s ine at the second demodulation

frequency

87 QQ = Sorted_Q .∗ s in (2 .∗ pi .∗Time.∗ Demodulation_Frequency) ;

88

89 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

90 % We then computes Z_2(N) and s c a l e s i t to an equivalent photon rate

91 % This ca l cu lat ion does not include any suppression techniques such as double demodulation

suppression or phase search techniques

92

93 % Calculated Z_2(N) . Change denominator to N to keep noise f l a t with integrat ion time

94 Z_2 = ((cumsum( I I ) .^2)+(cumsum(IQ) .^2) ) . / (N.^2) ;

95

96 % Scales Z_2(N) to an equivalent photon rate of the s igna l f i e l d

97 Photon_Rate = (4 .∗Z_2) . / ( ( Photodetector_Gain^2) ∗(Analog_Gain^2) ∗( Local_Oscillator_Power ) ∗(h∗nu) ) ;

98

99 % Compute the l a s t point of the array to y ie ld the ca lcu lated photon rate a f t e r the measured

integrat ion time

100 Last_Point = Photon_Rate(end) ;

101

102 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

103 % Compute the expectation value for shot−noise and the 5−sigma threshold using equations from the

text

104

122

105 Shot_Noise_Expectation = 4./( Quantum_Efficiency .∗Time) ;

106

107 Five_Sigma_Threshold = −1.∗Shot_Noise_Expectation .∗ log (6e−7) ;

108

109 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

110 % Plot the re su l t ing photon rate vs . integrat ion time in a log log plot along with the expectation

value fo r shot−noise and the 5−sigma threshold

111

112 l og log (Time , Photon_Rate , Time , Shot_Noise_Expectation , Time , Five_Sigma_Threshold) ;

113

114 legend ( 'Measured Result ' , ' Shot−noise Expectation Level ' , '5−Sigma Threshold ' ) ;

115 xlabe l ( ' Integrat ion Time ( s ) ' ) ;

116 ylabe l ( ' Photons/second ' ) ;

117 set ( f i n d a l l ( gcf , '−property ' , ' FontSize ' ) , ' FontSize ' ,12)

118

119 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

120 % Display use fu l ca lcu lated values such as to ta l integrat ion time , measured photon rate (

Last_Point )

121

122 % Calculate to ta l integrat ion time in hours

123 Integration_Time = Time(end) /3600;

124

125 % Display the tota l integrat ion time

126 IntTime = [ ' Integrat ion Time i s ' , num2str ( Integration_Time ) , ' hours ' ] ;

127 disp ( IntTime )

128

129 % Display the measured photon rate without using suppression techniques

130 Photon_Rate_Display = [ 'Without using suppression techniques there are ' , num2str (Photon_Rate(end

) ) , ' photons / s ' ] ;

131 disp (Photon_Rate_Display)

132

133 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

134 % Now l e t us look at the s c r i p t s designed to implement suppression techniques to decrease the

noise pickup as described in the text

135 % We f i r s t implement double demodulation suppression and phase search techniques ind iv idua l ly and

then both together

136

137 %Using e i the r technique ind iv idua l ly decreases the shot−noise expectation by a factor of 2

123

138 Shot_Noise_Expectation_One_Suppression = 2./( Quantum_Efficiency .∗Time) ;

139

140 %For double demodulation suppression th i s reduces the 5−sigma threshold by a factor of 2

141 Five_Sigma_Threshold_Double_Demod_Suppression = −1.∗Shot_Noise_Expectation .∗ log (6e−7) . / 2 ;

142

143 %Using a known beat note phase reduces the 5−sigma threshold by a factor of 2

144 Five_Sigma_Threshold_Phase_Search_Suppression = −1.∗Shot_Noise_Expectation .∗ log (6e−7) . / 2 ;

145

146 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

147 % This sect ion computes the photon rate a f t e r suppression of the noise pickup due to Double

Demodulation

148 % It does NOT include any phase search techniques and there fore must use both quadratures out of

the process ing card

149

150 % First compute the l i n e a r combination of II , IQ , QI , and QQ as described in the text

151 CombinedI = I I − QQ;

152 CombinedQ = QI + IQ ;

153

154 % Using these terms compute Z_2(N) as usual

155 Z_2_Double_Demodulation_Suppression = ((cumsum(CombinedI) .^2)+(cumsum(CombinedQ) .^2) ) . / (N.^2) ;

156

157 % Scale Z_2(N) to an equivalent photon rate

158 Photon_Rate_Double_Demodulation_Suppression = (Z_2_Double_Demodulation_Suppression) . / ( (

Photodetector_Gain^2) ∗(Analog_Gain^2) ∗( Local_Oscillator_Power ) ∗(h∗nu) ) ;

159

160 % Plot the r e s u l t vs . integrat ion time . Also plot the corresponding shot−noise expectation and 5−

sigma threshold

161 f i gu re ;

162 l og log (Time , Photon_Rate_Double_Demodulation_Suppression , Time ,

Shot_Noise_Expectation_One_Suppression , Time , Five_Sigma_Threshold_Double_Demod_Suppression) ;

163

164 legend ( 'Measured Result Using Double Demodulation Suppression ' , ' Shot−noise Expectation Level ' , '

5−Sigma Threshold ' ) ;

165 xlabe l ( ' Integrat ion Time ( s ) ' ) ;

166 ylabe l ( ' Photons/second ' ) ;

167 set ( f i n d a l l ( gcf , '−property ' , ' FontSize ' ) , ' FontSize ' ,12)

168

169 % Display a message with the r e s u l t

170 Photon_Rate_Double_Demodulation_Suppression_Display = [ ' Using only double demod suppression ' ,

num2str (Photon_Rate_Double_Demodulation_Suppression(end) ) , ' photons / s ' ] ;

124

171 disp (Photon_Rate_Double_Demodulation_Suppression_Display)

172

173 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

174 % This sect ion computes the phase of the s igna l (must be present and dominant ) in order to ignore

the Q quadrature

175 % It does NOT include noise suppression due to double demodulation

176 % Determine the phase of the s igna l f i e l d

177 % Essent ia l ly sweep the phase through 2∗ pi radians unt i l the l a s t point i s maximized when only

the I quadrature i s used .

178

179 % I n i t i a l i z e the s igna l phase to 0

180 phi = 0;

181

182 % Last_Point_Counter i s maximized when phi equals the s igna l phase . I n i t i a l i z e to 0

183 Last_Point_Counter = 0;

184

185 f o r r = 0 : ( pi /1000) :2∗ pi

186 I_Phase = Sorted_I .∗ cos (2 .∗ pi .∗Time.∗ Demodulation_Frequency + r ) ;

187 Z_2_Phase = ((cumsum(I_Phase) .^2) ) . / (N.^2) ;

188 Last_Point_Phase = Z_2_Phase(end) ;

189 i f Last_Point_Phase > Last_Point_Counter

190 Last_Point_Counter = Last_Point_Phase ;

191 phi = r ;

192 end

193 end

194

195

196 % Multiply the I channel out of the process ing card with a cos ine wave at the second demodulation

frequency using the known s igna l phase

197 II_Phase = Sorted_I .∗ cos (2 .∗ pi .∗Time.∗ Demodulation_Frequency + phi ) ;

198

199 % Compute Z_2(N) using phase search techniques . Only cos ine i s used for second demodulation .

200 Z_2_Phase_Search = ((cumsum( II_Phase ) .^2) ) . / (N.^2) ;

201

202 % Scale th i s to an equivalent photon rate

203 Photon_Rate_Phase_Search = (4 .∗Z_2_Phase_Search) . / ( ( Photodetector_Gain^2) ∗(Analog_Gain^2) ∗(

Local_Oscillator_Power ) ∗(h∗nu) ) ;

204

205 % Plot the r e s u l t vs . integrat ion time . Also plot the corresponding shot−noise expectation and 5−

sigma threshold

125

206 f i gu re ;

207 l og log (Time , Photon_Rate_Phase_Search , Time , Shot_Noise_Expectation_One_Suppression , Time ,

Five_Sigma_Threshold_Phase_Search_Suppression ) ;

208

209 legend ( 'Measured Result Using Phase Search Techniques ' , ' Shot−noise Expectation Level ' , '5−Sigma

Threshold ' ) ;

210 xlabe l ( ' Integrat ion Time ( s ) ' ) ;

211 ylabe l ( ' Photons/second ' ) ;

212 set ( f i n d a l l ( gcf , '−property ' , ' FontSize ' ) , ' FontSize ' ,12)

213

214 % Display a message stat ing the s igna l phase

215 Signal_Phase_Display = [ 'The s igna l phase i s ' , num2str ( phi ) , ' . ' ] ;

216 disp ( Signal_Phase_Display )

217

218

219 % Display a message with the r e s u l t

220 Photon_Rate_Phase_Search_Display = [ ' Using only phase search techniques ' , num2str (

Photon_Rate_Phase_Search(end) ) , ' photons / s ' ] ;

221 disp (Photon_Rate_Phase_Search_Display)

222

223 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

224 % Now l e t us use both suppression techniques together

225

226 % Using both techniques decreases the shot−noise expectation by a factor of 4

227 Shot_Noise_Expectation_Both_Techniques = 1./( Quantum_Efficiency .∗Time) ;

228

229 % This a l so decreases the 5−sigma threshold by a factor of 4

230 Five_Sigma_Threshold_Both_Techniques = −1.∗Shot_Noise_Expectation .∗ log (6e−7) ./ (4 ) ;

231

232 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

233 % To use both techniques we need to determine I I and QQ with the known s igna l phase .

234 % The terms 'IQ ' and 'QI ' only include noise in th i s case . See text fo r re f e rence .

235

236 % Multiply the Q channel out of the process ing card with a s ine wave at the second demodulation

frequency using the known s igna l phase

237 QQ_Phase = Sorted_Q .∗ s in (2 .∗ pi .∗Time.∗ Demodulation_Frequency + phi ) ;

238

239 % Double demodulation suppression used a l i n e a r combination of II , IQ , QI , and QQ.

240 % However with the phase of the s igna l known , IQ and QI only contain noise .

126

241 Combined_Phase = II_Phase − QQ_Phase;

242

243 % Compute Z_2(N) using both suppression techniques

244 Z_2_Both_Suppression_Techniques = ((cumsum(Combined_Phase) .^2) ) . / (N.^2) ;

245

246 % Scale th i s to photons per second

247 Photon_Rate_Both_Suppression_Techniques = (Z_2_Both_Suppression_Techniques ) . / ( ( Photodetector_Gain

^2) ∗(Analog_Gain^2) ∗( Local_Oscillator_Power ) ∗(h∗nu) ) ;

248

249 % Plot the r e s u l t vs . integrat ion time . Also plot the corresponding shot−noise expectation and 5−

sigma threshold when both techniques are used

250 f i gu re ;

251 l og log (Time , Photon_Rate_Both_Suppression_Techniques , Time ,

Shot_Noise_Expectation_Both_Techniques , Time , Five_Sigma_Threshold_Both_Techniques) ;

252

253 legend ( 'Measured Result Using Both Suppression Techniques ' , ' Shot−noise Expectation Level ' , '5−

Sigma Threshold ' ) ;

254 xlabe l ( ' Integrat ion Time ( s ) ' ) ;

255 ylabe l ( ' Photons/second ' ) ;

256 set ( f i n d a l l ( gcf , '−property ' , ' FontSize ' ) , ' FontSize ' ,12)

257

258 % Display a message with the r e s u l t

259 Photon_Rate_Both_Suppression_Techniques_Display = [ ' Using both suppression techniques ' , num2str (

Photon_Rate_Both_Suppression_Techniques (end) ) , ' photons / s ' ] ;

260 disp (Photon_Rate_Both_Suppression_Techniques_Display)

261

262 end

127

APPENDIX BMATLAB SCRIPTS FOR SIMULATIONS

1 function [ I_Simulated , Q_Simulated , ClockTime , Measured_CC_Frequency ] = Simulation_Test (N_pho, Tau

, CC_Frequency , P_LO, Input_Impedance , Quantum_Efficiency )

2 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

3 % IMPORTANT NOTE: These simulations sample at a rate of 31.25 kHz . This i s done due to process ing

constra ints l imi t ing the length of integrat ion time I can simulate at 64 MHz.

4 % Because I can only simulate 0.5−0.7 seconds of data I sample at a higher rate than

experimentally (31.25 kHz vs ~20 Hz)

5 % This must be taken into account when using the ' Heterodyne_Demodulation ' function

6 % The output of th i s function y i e ld s a simulated 'Measured_CC_Frequency ' array , 'ClockTime ' array

, and both ' I ' and 'Q' as they would appear out of the process ing card

7 % The user manually se t s the photon rate of the s igna l

8 % 'N_pho = 0 ' can be set fo r a noise only simulation

9

10 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

11 % N_pho i s the photon rate of the s igna l f i e l d we wish to measure

12 % Tau i s the integra i ton time

13 % CC_Frequency i s the frequency the CC beat note i s at . Not r e a l l y important here

14

15 % Define Planck ' s constant

16 h = 6.626e−34;

17

18 % Define l a s e r frequency for 1064 nm l i g h t

19 nu = 281954887218045.06;

20

21 % Determine equivalent power in s igna l f i e l d fo r 1064 nm l i g h t

22 P_signal = N_pho.∗ ( h∗nu) ;

23

24 % Speci fy photodetector gain in V/W. NOTE: Must match gain in ' Heterodyne_Demodulation ' function

25 G = 1.4 e3 ;

26

27 % Calculates s i gna l f i e l d−LO beat note Amplitude , A

28 A = 2.∗G.∗ sqrt (P_LO.∗ P_signal ) ;

29

30 % S p e c i f i e s the frequency of the s igna l

31 Signal_Frequency = 16e6 + 5e3 ;

32

33 % S p e c i f i e s f i r s t demod frequency (FPGA)

128

34 f_1 = 16e6 ;

35

36 % S p e c i f i e s sampling frequency of ADC in samples per second

37 f_s = 64e6 ;

38

39 % Calculates to ta l number of samples

40 N = f_s∗Tau;

41

42 % Determines the FPGA ClockTime/Sample number

43 Clock = ( 1 :N) ;

44

45 % Determines the time stamp of each sample number

46 Time = Clock/f_s ;

47

48 % Generates d i g i t i z e d beat note at the FPGA ADC

49 Signal = A.∗ cos (2 .∗ pi .∗Time.∗ Signal_Frequency ) ;

50

51 % Generate white Gaussian noise at the appropriate l e v e l fo r the given LO power

52 nPSD = 2.∗G^2.∗h .∗nu .∗P_LO ./ Quantum_Efficiency ;

53 n1 = wgn(N, 1 , (nPSD/2) , ' l i n e a r ' ) ;

54 eta = n1 ∗ sqrt ( f s ) ;

55

56 % Linearly combine s igna l and noise to represent photodetector output

57 PD = Signal + eta ' ;

58

59 % Mixes PD s igna l with cos ine NCO from FPGA ( I channel )

60 IHigh = PD.∗ cos (2 .∗ pi .∗tSumFPGA.∗ f_1) ;

61

62 % Mixes PD s igna l with s ine NCO from FPGA (Q channel )

63 QHigh = PD.∗ s in (2 .∗ pi .∗tSumFPGA.∗ f_1) ;

64

65 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

66 % This sect ion f i l t e r s and downsamples the data using a moving average f i l t e r

67

68 k = f l o o r ( length ( IHigh ) ./2048) ;

69 I_Simulated = zeros (1 ,k) ;

70 Q_Simulated= zeros (1 ,k) ;

71 AvgI = 0;

72 AvgQ = 0;

73

129

74 f o r i = 1: k

75 f o r j = 1:2048;

76 m = ( i −1)∗2048 + j ;

77 AvgI = IHigh (m) + AvgI ;

78 AvgQ = QHigh(m) + AvgQ;

79 end

80 I_Simulated ( i ) = AvgI/2048;

81 Q_Simulated( i ) = AvgQ/2048;

82 AvgI = 0;

83 AvgQ = 0;

84 end

85

86 % Simulated I channel from FPGA

87 I_Simulated = I_Simulated ' ;

88

89 % Simulated Q channel from FPGA

90 Q_Simulated = Q_Simulated ' ;

91

92 % Generates the clock time stamp array I expect out of FPGA

93 ClockTime = 31250.∗(1: length (SumI) ) ' ;

94

95 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

96 % Measured_CC_Frequency i s jus t used to make sure that we didn ' t break lock

97 % It isn ' t used to simulate any s i gna l s

98 % Generate a 'Measured_CC_Frequency ' fo r input into ' Heterodyne_Demodulation ' function

99

100 Measured_CC_Frequency = ClockTime ./ ClockTime ;

101 Measured_CC_Frequency = CC_Frequency .∗Measured_CC_Frequency ;

102

103 end

130

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BIOGRAPHICAL SKETCH

Zach started his journey into the field of physics at an early age where music and math

were, and still are, his passion. A high school physics instructor and mentor saw potential in

Zach’s work and challenged him to pursue physics as a career. Zach earned his Bachelor of

Science at the University of Florida (Go Gators!) and accepted an offer from the University

to attend graduate study. His research has led him into the field of laser optics where he

collaborated with a large scale experiment in Hamburg, Germany. His work has allowed him to

travel internationally and work with physicists from around the world.

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