sub-femtosecond optical timing distribution for next-generation light sources

Sub-Femtosecond Optical Timing Distribution

for Next-Generation Light Sources

By

Michael Y. Peng

B.S., University of California, Berkeley (2009)

S.M., Massachusetts Institute of Technology (2011)

Submitted to the Department of Electrical Engineering and Computer Science

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2015

© Massachusetts Institute of Technology 2015. All rights reserved.

Author………………………………………………………………………………………………………..

Department of Electrical Engineering and Computer Science

August 7, 2015

Certified by…………………………………………………………………………………………………..

Franz X. Kärtner

Adjunct Professor, Electrical Engineering

Thesis Supervisor

Certified by…………………………………………………………………………………………………..

Erich P. Ippen

Elihu Thomson Professor of Electrical Engineering

and Professor of Physics, Emeritus

Thesis Supervisor

Accepted by…………………………………………………………………………………………………..

Leslie A. Kolodziejski

Professor of Electrical Engineering

Chairman, Department Committee on Graduate Theses

3

Sub-Femtosecond Optical Timing Distribution

for Next-Generation Light Sources

By

Michael Y. Peng

Submitted to the Department of Electrical Engineering and Computer Science

on August 7, 2015, in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Abstract

Precise timing distribution is critical for realizing a new regime of light control in next-generation

X-ray free-electron lasers. These facilities aim to generate sub-femtosecond (fs) X-ray pulses with

unprecedented brightness to realize the long-standing scientific dream to capture chemical and physical

reactions with atomic-level spatiotemporal resolution. To achieve this, a high-precision timing system is

required to synchronize dozens of radio frequency (RF) and optical sources across kilometer distances

with sub-fs precision. Since conventional RF timing systems have already reached a practical limit of

50 fs, next-generation systems are adopting optical technology to achieve superior performance. In this

thesis, an optical timing distribution system (TDS) is developed using ultrafast mode-locked laser

technology to deliver sub-fs timing stability.

Optical domain components of the TDS are first presented. The timing jitter of commercial mode-

locked lasers is characterized to confirm their viability as optical master oscillators for timing distribution.

Stabilization of a 1.2-km dispersion-compensated polarization-maintaining fiber link is demonstrated as a

proof-of-concept for eliminating polarization-induced timing drifts. The link is then enhanced to achieve

state-of-the-art timing distribution across a 4.7-km fiber network with 0.58 fs RMS residual drift for over

52 hours. For a complete end-to-end TDS, a remote laser is stabilized at the output of a 3.5 km fiber link

with 0.2 fs RMS residual drift. All demonstrations depend critically on the balanced optical cross-

correlator for high-precision optical timing measurements.

Second, the coverage of the TDS is extended into the RF domain using balanced optical-microwave

phase detectors (BOMPD). Two generations of BOMPDs are developed to achieve sub-fs noise

performance with MHz-level bandwidth capabilities and robust AM-PM suppression ratios (>50 dB).

Optical-to-RF synchronization is demonstrated with 0.98 fs RMS drift for over 24 hours, while RF-to-

optical synchronization is demonstrated with 0.5 fs RMS.

Lastly, an Erbium Silicon Photonics Integrated OscillatoR (ESPIOR) based on optical frequency

division (OFD) is developed for ultralow-noise microwave generation. Since f-2f interferometry is

unavailable on-chip, an alternative fCEO control scheme called quasi-OFD is proposed to improve

stabilization of an integrated frequency comb. The ESPIOR concept is demonstrated in a discrete testbed

to achieve low-noise RF generation with -63 dBc/Hz phase noise at 10 Hz offset for a 6-GHz carrier

frequency. This corresponds to an OFD ratio of 85 dB, which is close to the ideal OFD ratio of 90 dB.

Thesis Supervisor: Franz X. Kärtner

Title: Adjunct Professor of Electrical Engineering

Thesis Supervisor: Erich P. Ippen

Title: Elihu Thomson Professor of Electrical Engineering

and Professor of Physics, Emeritus

5

Acknowledgements

With the completion of my PhD, I am particularly grateful to the inspiring faculty and colleagues that I

have met during my years at MIT. I would like to extend my deepest thanks…

To my research advisor and mentor, Professor Franz X. Kärtner, who has provided me the

opportunity to pursue fascinating research at the edge of optics and electronics on ultrafast timescales. I

deeply value his guidance and encouragement over the years in helping me develop as an independent

researcher. While I am often in the details of engineering, I am always inspired by his ability to

conceptualize technological innovations from a high-level perspective while remaining firmly rooted in

fundamental physics and practical design principles. He is a brilliant teacher whose creativity I hope to

have learned well. I would like to thank him for the opportunity to develop my thesis at MIT as well as

DESY in Hamburg, Germany.

To my research co-advisor, Professor Erich P. Ippen, who never ceases to amaze me with his

ability to simplify complex physical phenomena into intuitive building blocks. He was instrumental in

developing the noise model for the ESPIOR. It was a privilege working with one of the fathers of “mode-

locking”. I have learned from him the art of balancing spoken words against deep thoughts to be an

effective communicator in the scientific community.

To my thesis committee members, Professor Rajeev J. Ram and Professor Michael R. Watts, for

overseeing the development of my thesis and cultivating my philosophy of engineering to new heights.

To my academic advisor, Professor Tomas Palacios, for his invaluable advice on career

development and the greater things that are yet to come in life.

To all the funding agencies for their financial support: Department of Energy, Center for Free-

Electron Laser, Air Force of Scientific Research, and Defense Advanced Research Projects Agency.

To the many research collaborators I have had the privilege to work with over the years: Ming

Xin and Kemal Safak for their experimental and theoretical work on the 3.5-km link at CFEL DESY.

Jungwon Kim and Jonathan Cox for their preceding work on the first-generation TDS. John Fini at OFS

Laboratories for designing and fabricating the PM fiber links. Stefano Valente for the 1.2-km link

simulations. Patrick Callahan for the integrated BOC. Aram Kalaydzhyan and Amir Nejadmalayeri for

help with BOMPD improvements. Duo Li for his preceding work on the ESPIOR testbed. Jinkang Lim

and Noah Chang for experimental and theoretical work on the ESPIOR project.

To all past and present members of the OQE and PMG group at MIT as well as CFEL group at

DESY for stimulating discussions and a fun working environment.

To Dorothy Fleischer and Donna Gale for keeping the group running seamlessly, especially with

the large volume of purchase orders, travel arrangements, and budget proposals over the years.

To all my friends for the many happy memories. They have always managed to keep my spirits

high after my countless hours in a windowless laser laboratory. I owe my sanity to them.

And most importantly, to my family – my parents, Chih-Kang and Mee-Yu, and my sister,

Jennifer – for their constant love and support. I would not have made it this far without their guidance.

This thesis is dedicated to them.

7

Table of Contents

Acknowledgements ....................................................................................................................... 5

Table of Contents .......................................................................................................................... 7

List of Figures .............................................................................................................................. 11

List of Abbreviations .................................................................................................................. 15

1 Introduction ......................................................................................................................... 17

1.1 Seeded X-ray FEL ..................................................................................................................... 17

1.2 System Requirements ............................................................................................................... 19

1.3 Optical and RF Comparison .................................................................................................... 20

1.4 Optical Timing Distribution System ....................................................................................... 24

1.5 Thesis Outline ............................................................................................................................ 26

2 Optical Timing Distribution and Synchronization ........................................................... 27

2.1 Introduction ............................................................................................................................... 27

2.2 Principle of Operation .............................................................................................................. 27

2.2.1 Balanced Optical Cross-Correlator ..................................................................................... 27

2.2.2 Optical-Optical Synchronization ......................................................................................... 29

2.2.3 Link Stabilization (Timing Distribution) ............................................................................ 30

2.3 Experimental Demonstration ................................................................................................... 32

2.3.1 Laser Jitter Characterization ............................................................................................... 32

2.3.2 Single 1.2-km Link Stabilization ........................................................................................ 35

2.3.3 Single 3.5-km Link Stabilization ........................................................................................ 39

2.3.4 4.7-km Fiber Network Stabilization .................................................................................... 42

2.3.5 Remote-Laser Synchronization ........................................................................................... 44

2.4 Design Considerations .............................................................................................................. 47

2.4.1 Link Construction ............................................................................................................... 47

2.4.2 Pulse Duration ..................................................................................................................... 48

2.4.3 Temperature and Humidity ................................................................................................. 48

2.4.4 Fiber Nonlinearity ............................................................................................................... 50

2.4.5 Link-Enhanced Timing Jitter .............................................................................................. 53

8

2.4.6 Balanced Optical Cross-correlator ...................................................................................... 53

2.4.7 Noise Model ........................................................................................................................ 55

2.5 Conclusion and Future Work .................................................................................................. 56

3 Optical-RF Synchronization ............................................................................................... 57

3.1 Introduction ............................................................................................................................... 57


3.2.1 Optoelectronic Phase-Locked Loop .................................................................................... 58

3.2.2 Balanced Optical-Microwave Phase Detector .................................................................... 59

3.3 Experimental Demonstration ................................................................................................... 60

3.3.1 Second-Generation BOMPD ............................................................................................... 61

3.3.2 Third-Generation BOMPD ................................................................................................. 67

3.4 Design Considerations .............................................................................................................. 71

3.4.1 AM-PM Suppression........................................................................................................... 71

3.4.2 Temperature / AM-PM sensitivity ...................................................................................... 73

3.4.3 Electronics Optimization ..................................................................................................... 77

3.4.4 High-Frequency Modulation / Low-Frequency Detection .................................................. 78

3.4.5 Electro-Optic Sampling ....................................................................................................... 79

3.4.6 Phase Margin ...................................................................................................................... 81

3.4.7 Noise Model ........................................................................................................................ 82

3.5 Alternative Implementations ................................................................................................... 82

3.5.1 Integrated BOMPD ............................................................................................................. 82

3.5.2 Mach-Zehnder BOMPD ...................................................................................................... 86

3.6 Conclusion and Future Work .................................................................................................. 89

4 Ultra-Low Noise Microwave Oscillator ............................................................................. 91


4.1.1 Optical Frequency Division ................................................................................................ 91

4.1.2 Erbium Silicon Photonics Integrated OscillatoR ................................................................ 93

4.2 Noise Characterization ............................................................................................................. 94

4.2.1 Cross-Modulation Dynamics .............................................................................................. 94

4.2.2 Fixed Point Modulation ...................................................................................................... 96

4.2.3 Suppression Ratio ............................................................................................................... 98

4.2.4 Quasi OFD ........................................................................................................................ 100

9

4.2.5 Noise Verification ............................................................................................................. 103

4.2.6 Discussion ......................................................................................................................... 106

4.3 MIMO Phase Noise Model ..................................................................................................... 108

4.4 System Demonstrations .......................................................................................................... 109

4.4.1 OFD Demonstration .......................................................................................................... 109

4.4.2 ESPIOR RF Readout ......................................................................................................... 113

5 Conclusion .......................................................................................................................... 121

Appendix A Phase-Locked Loop Basics ............................................................................. 123

Appendix B Noise Model for Optical-Optical Synchronization ....................................... 127

Appendix C Noise Model for Timing Link Stabilization .................................................. 133

Appendix D Noise Model for Optical-RF Synchronization .............................................. 139

Appendix E Derivation for Fixed Point Modulation Theory ........................................... 145

Bibliography .............................................................................................................................. 149

11

List of Figures

Figure 1-1 Operating principle for an externally-seeded X-ray FEL .......................................................... 18

Figure 1-2 RF approach to pulse timing error measurement ...................................................................... 21

Figure 1-3 Schematic of a pulsed optical TDS for a seeded X-ray FEL facility. ....................................... 25

Figure 2-1 Operating principle for a BOC. ................................................................................................. 28

Figure 2-2 Operating principle for optical-optical synchronization ........................................................... 29

Figure 2-3 Operating principle for timing-stabilization of a fiber link. ...................................................... 31

Figure 2-4 Experimental set-up for optical-optical synchronization. ......................................................... 33

Figure 2-5 Measurement results for laser timing jitter characterization. .................................................... 34

Figure 2-6 Experimental set-up for 1.2-km link stabilization. .................................................................... 36

Figure 2-7 Measurement results for 1.2-km link stabilization. ................................................................... 38

Figure 2-8 Experimental set-up for 3.5-km link stabilization. .................................................................... 40

Figure 2-9 Measurement results for 3.5-km link stabilization. ................................................................... 41

Figure 2-10 Experimental set-up for 4.7-km fiber network stabilization. .................................................. 42

Figure 2-11 Measurements results for 4.7-km fiber network stabilization. ................................................ 43

Figure 2-12 Experimental set-up for remote laser-laser synchronization ................................................... 44

Figure 2-13 Measurement results for remote laser-laser synchronization. ................................................. 45

Figure 2-14 Time deviation estimator for long-term stability. ................................................................... 46

Figure 2-15 Construction of the 1.2-km PM fiber link ............................................................................... 47

Figure 2-16 Simulation results for nonlinear pulse propagation in the 1.2-km PM link. ........................... 51

Figure 2-17 Improving polarization extinction ratio in the BOC ............................................................... 54

Figure 3-1 Operating principle for a BOMPD in an optoelectronic PLL ................................................... 58

Figure 3-2 Operating principle for electro-optic sampling in the BOMPD ................................................ 59

Figure 3-3 Experimental set-up for remote-station locking with 2nd

-generation BOMPDs. ...................... 62

Figure 3-4 Measurement results for short-term remote-station locking with 2nd

-generation BOMPDs. .... 63

12

Figure 3-5 Measurement results for long-term remote-station locking with 2nd

-generation BOMPDs ...... 65

Figure 3-6 Measurement results for short-term base-station locking wtih 2nd

-generation BOMPDs. ........ 66

Figure 3-7 Experimental set-up for remote-station locking with 3rd

-generation BOMPDs. ....................... 67

Figure 3-8 Improved noise floor in 3rd

-generation BOMPDs ..................................................................... 68

Figure 3-9 Measurement results for short-term remote-station locking with 3rd

-generation BOMPDs. .... 69

Figure 3-10 Measurement results for long-term remote-station locking with 3rd

-generation BOMPDs. ... 70

Figure 3-11 Measurement results for short-term base-station locking with 3rd

-generation BOMPDs. ...... 71

Figure 3-12 Measuremed AM-PM suppression ratios ................................................................................ 72

Figure 3-13 Measured detector saturation under high-power pulse train detection. ................................... 79

Figure 3-14 Simulation results for BOMPD phase sensivity KPD versus sampling pulse width. ............... 80

Figure 3-15 Experimental set-up for remote-station locking with the integrated SGI-BOMPD. ............... 83

Figure 3-16 Measurement results for remote-station locking with the integrated BOMPD ....................... 85

Figure 3-17 Operating principle for the MZI-BOMPD. ............................................................................. 86

Figure 3-18 Experimental set-up for remote-station locking with the MZI-BOMPD. ............................... 87

Figure 3-19 Measurement results for remote-station locking with the MZI-BOMPD ............................... 88

Figure 4-1 Operating principle for optical frequency division. .................................................................. 92

Figure 4-2 Operating principle for the ESPIOR. ........................................................................................ 93

Figure 4-3 Measured Kij transfer coefficients for a stretched-pulse Er-fiber laser. .................................... 95

Figure 4-4 Illustration for comb “breathing” due to fixed point modulation. ............................................. 97

Figure 4-5 Measured fixed points for cavity length, ML, and pump power, MP, modulation ..................... 98

Figure 4-6 Illustrations of suppression ratios for different fixed point relationships. ................................. 99

Figure 4-7 Operating principle for QOFD with synchronous feedback modulation. ............................... 100

Figure 4-8 Simulated fixed point for synchronous modulation ................................................................ 102

Figure 4-9 Simulated fixed point for synchronous modulation at 100 Hz offset ...................................... 102

Figure 4-10 Simulated suppression ratio for synchronous modulation..................................................... 103

13

Figure 4-11 Experimental set-up for evaluating QOFD. .......................................................................... 103

Figure 4-12 Measurement results for QOFD. ........................................................................................... 104

Figure 4-13 Measured suppression ratio for QOFD ................................................................................. 105

Figure 4-14 Simulated suppression ratio for QOFD with three noise sources. ......................................... 107

Figure 4-15 MIMO phase noise model for frequency comb dynamics .................................................... 108

Figure 4-16 Experimental set-up for measuring OFD ratio ...................................................................... 110

Figure 4-17 Experiment versus simulation for a stretched-pulse Er-fiber laser ....................................... 111

Figure 4-18 Experiment versus simulation for a soliton Er-fiber laser ..................................................... 112

Figure 4-19 Experimental set-up for the inital ESPIOR testbed demonstration. ...................................... 114

Figure 4-20 Measurement results for a 5-GHz ESPIOR. .......................................................................... 115

Figure 4-21 Experimental set-up for the final ESPIOR testbed demonstration ........................................ 116

Figure 4-22 Measurement results for a 6-GHz ESPIOR with a fully-stabilized frequency divider. ........ 118

Figure 4-23 Measurement results for a 6-GHz ESPIOR for various comb stabilitzation schemes. ......... 119

Figure A-1 Operating principle for a RF PLL .......................................................................................... 123

Figure A-2 Phase noise model for a RF PLL ............................................................................................ 124

Figure B-1 Feedback noise model for optical-optical synchronization. ................................................... 127

Figure B-2 Example simulations for optical-optical synchronization. ..................................................... 130

Figure B-3 Simulation for laser jitter characterization under loose-locking. ............................................ 130

Figure B-4 Simulation for optical-optical synchronization under tight locking. ..................................... 131

Figure C-1 Feedback noise model for timing link stabilization ................................................................ 133

Figure C-2 Open-loop gain response for link stabilization. ...................................................................... 134

Figure C-3 Closed-loop transfer coefficients for link stabilization........................................................... 135

Figure C-4 Noise model for link stabilization demonstrations. ................................................................ 136

Figure C-5 Experiment versus simulation for 1.2-km link stabilization ................................................... 137

14

Figure C-6 Experiment versus simulation for 3.5-km link stabilization. .................................................. 138

Figure D-1. Feedback noise model for optical-RF synchronization with a BOMPD. .............................. 139

Figure D-2 Open-loop gain response for optical-RF synchronization. ..................................................... 140

Figure D-3 Closed-loop transfer coefficients for optical-RF synchronization ......................................... 141

Figure D-4 Noise model for experimental BOMPD demonstrations. ....................................................... 141

Figure D-5 Simulation for remote-station locking with 2nd

-generation BOMPDs. .................................. 142

Figure D-6 Simulation for remote-station locking with 3rd

-generation BOMPDs. ................................... 143

Figure D-7 Simulation for base-station locking with 3rd

-generation BOMPDs. ....................................... 144

15

List of Abbreviations

AM Amplitude Modulation

ASE Amplified Spontaneous Emission

BOC Balanced Optical Cross-correlator

BOMPD Balanced Optical-Microwave Phase Detector

BPD Balanced Photodetector

BPF Bandpass Filter

CEO Carrier Envelope Offset

CW Continuous-wave

DAQ Data Acquisition Card

DCF Dispersion-Compensating Fiber

DIV Frequency Divider

DPD Digital Phase Detector

EDFA Erbium-doped fiber Amplifier

EDFL Erbium-doped Fiber Laser

EPHI Electronic-Photonic Heterogeneous Integration

FEL Free-Electron Laser

Er Erbium

FH First (fundamental) Harmonic

FWHM Full-Width at Half Maximum

LNA Low Noise Amplifier

LPF Lowpass Filter

LPNA Low Phase Noise Amplifier

MLL Mode-Locked Laser

MZI Mach-Zehnder Interferometer

OC Output Coupler

ODL Optical Delay Line

OFD Optical Frequency Division

OPLL Optical Phase-Locked Loop

PBS Polarization Beam Splitter

PD Photodetector

PI Proportional-Integral

PI2 Proportional-Double-Integral

16

PLL Phase-Locked Loop

PM Phase Modulation

PM Polarization-Maintaining

PMD Polarization Mode Dispersion

PPKTP Periodically-Poled KTiOPO4

PSD Power Spectral Density

PZT Piezo-electric Transducer

QOFD Quasi-Optical Frequency Division

RF Radio Frequency

RIN Relative Intensity Noise

RMS Root-Mean-Square

SGI Sagnac Interferometer

SH Second Harmonic

SMF Single-Mode Fiber

SNR Signal-to-Noise Ratio

SOD Second-Order Dispersion

SSA Signal Source Analyzer

SSB Single Side-Band

TDS Timing Distribution System

TOD Third-Order Dispersion

TJSD Timing Jitter Spectral Density

Chapter 1. Introduction

17

1 Introduction

High-precision long-distance transfer of optical frequency and timing standards has witnessed

extraordinary progress in the last decade [1]–[5] and are highly desired in large-scale scientific facilities to

achieve unprecedented performance levels. It will open new frontiers of scientific discovery in various

fields including geodesy [6], very-long-baseline interferometry [7], high precision navigation [8],

telesurgery [9], and advanced light source generation [10]–[13]. Next-generation X-ray free-electron

lasers (FEL) in particular, such as the European XFEL [14] and the Linac Coherent Light Source II [15],

aim to generate sub-fs X-ray pulses with unprecedented brightness to realize the long-standing scientific

dream to capture chemical and physical reactions with atomic-level spatiotemporal resolution. To

achieve this, it is necessary to develop a high-precision timing distribution system (TDS) to tightly

synchronize dozens of radio frequency (RF) and optical sources across the km-scale facilities. Ideally,

the timing precision of the TDS must be commensurate to, if not better than, the X-ray pulse width itself.

Since conventional RF timing systems based on electronic phase-locking techniques have already reached

a practical limit of 50-100 fs RMS timing precision [16]–[19], next-generation timing systems are

adopting optical technology to achieve superior performance. Optical techniques are envisioned to

surpass this stability barrier by using the ultralow noise properties of optical pulse trains from mode-

locked lasers, efficient signal transport from mature fiber-optic technology, and ultimately higher timing

measurement sensitivities in the optical domain.

1.1 Seeded X-ray FEL

The target application for the optical TDS developed in this thesis is for next-generation seeded

X-ray FELs, such as the Linac Coherent Light Source II. The basic operating principle of the FEL is

shown in Figure 1-1. Starting at the photoinjector, a photocathode is illuminated with a laser pulse from

the injector laser to emit an electron bunch. A strong RF acceleration field then launches the electron

bunch through a series of linear accelerators (linac) to increase the electron bunch energy up to relativistic

energy levels in the multi-GeV range. The linacs are interleaved with magnetic chicanes (compressors) to

compress the bunch in intermediate steps to maintain good beam profile. Due to shot-to-shot variations in

the energy and momentum distribution of the initial electron bunches as well as instabilities in the RF

accelerating fields, the electron bunch emerging from the linacs can accumulate excess timing jitter. To


18

reduce this, the bunch arrival times at the end of each compressor are measured against a stable optical

reference. The detected timing error is actively fed back to control the amplitude and phase of the RF

accelerating fields in each linac to stabilize the beam. Once the electron bunch is at its final relativistic

energy, it propagates through the undulator to generate self-amplified spontaneous emission (SASE) X-

ray emission. The generated X-ray FEL pulses continue onward to the experiment hall for pump-probe

measurements.

Pump-probe measurements are important for studying ultrafast phenomena on the femtosecond

timescale or faster. In these measurements, the target is first excited with a laser (pump) pulse to invoke

some reaction. After an adjustable time delay, the target is illuminated with the FEL (probe) pulse to

measure the target’s response. By repeating these measurements for a range of time delays, the time-

evolution of the target’s response on ultrafast timescales can be recorded. The sequence of images is

compiled to reveal how atoms and chemical bonds rearrange with atomic-level spatiotemporal resolution.

This unprecedented resolution will open a new frontier of scientific discovery.

Seeded FEL operation is envisioned to replace SASE to achieve sub-fs X-ray pulses in next-

generation FELs. In current SASE FELs, reducing the FEL pulse duration towards the sub-fs regime is

increasingly difficult due to synchronization instabilities in the RF accelerating fields as well as the

quality of the final bunch compression stage. In seeded operation, a seed laser pulse is injected with the

driving electron bunch in the undulator to exert additional control over the temporal and spectral

properties of the FEL pulse formation. There are two approaches for seeded FEL operation. Internal self-

seeding is achieved by feeding back a fixed spectral slice of the spontaneous X-ray emission as the seed

pulse itself. External seeding is achieved by using an external independent laser pulse generated from

high-harmonic generation or directly seeding at the desired wavelength. In the last case, the FEL pulse is

Figure 1-1 Operating principle for an externally-seeded X-ray FEL based on FLASH at

DESY.


19

fully coherent in the time and frequency domain with the seed laser pulse. This level of control will

enable sub-fs X-ray pulse generation.

1.2 System Requirements

Several independent optical and RF subsystems need to be tightly synchronized for the FEL to

operate at its fullest potential. The timing requirements for critical system paths are as follows:

For high-resolution pump-probe measurements, the relative time delay between the pump and FEL

probe pulse as they arrive in the experimental hall needs to be controlled with a resolution better than

the FEL pulse width. While the pump laser is placed in the experimental hall to control the arrival

time of its pulse with high precision, the arrival time of the FEL pulse is highly dependent on the FEL

pulse generation process, which begins several kilometers away at the photoinjector. The TDS must

therefore span the entire length of the FEL and deliver sub-fs stability.

For good beam quality, the linacs must employ feedback to stabilize the amplitude and phase of its

RF accelerating field to control the energy and momentum distribution of the electron bunch, as this

will directly affect the FEL pulse width. Conventional RF methods for detecting and stabilizing the

accelerating field directly are limited to the 100 fs range [20], while optical methods for measuring

the bunch arrival times in the optical domain have higher timing sensitivities and have so far achieved

a timing precision of 5 fs RMS [21]. These optical methods require the distribution of a stable optical

reference with sub-fs stability to multiple linacs spaced across a couple kilometers.

For seeded FEL operation, the seed pulse must be synchronized with the arrival of the electron bunch

at the input of the undulator (similar to the pump-probe timing requirement).

In addition to the timing requirements, practical design requirements must be met to realize a

robust and long-term stable TDS in a real facility. As stated from the European XFEL Technical Design

Report, the TDS must meet the following requirements [14]:

it should serve as a timing reference to the XFEL, providing femtosecond stability between all

significant points throughout the facility with small or negligible drifts over days and weeks. This

reference system must be self-contained, without the need for recalibrations;


20

it must provide RF signals or the possibility to lock ultra-low-noise RF local oscillators to the timing

reference at different frequencies;

it must provide a mechanism to lock various laser systems, as in those used for electron beam

generation, beam diagnostics, pump-probe experiments, seeding, and other applications;

the system stability, robustness, and maintenance should not limit machine availability or delay

commissioning;

the failure modes should be transparent and allow for rapid repair and start-up;

expenditure should be moderate and cost-effective.

These performance and technical requirements of the TDS can only be achieved using an optical

approach for the following reasons. First and foremost, higher timing measurement sensitivities can be

achieved in the optical domain compared to the RF domain to enable sub-fs synchronization. Second, all

components are commercial and can be easily operated and repaired; e.g. commercial, turn-key mode-

locked lasers can already achieve ultralow timing jitter and maintain good mode-locking for extended

operation times. Moreover, mature fiber-optic technology allows low-loss, robust timing distribution

over kilometer distances. While initial timing systems employ some free-space optics, future

implementations can all be fiber-coupled to reduce the system size, complexity and cost as well as

improve system robustness. Lastly, the pulsed optical timing signals contain more useful timing

information than a single RF or continuous-wave (CW) optical timing signal; the harmonics of the

repetition rate can be extracted simultaneously to synchronize multiple instruments operating over a wide

range of RF frequencies.

1.3 Optical and RF Comparison

Timing and Phase Measurement Sensitivity

Optical phase detection offers fundamentally higher timing sensitivity than RF phase detection

due to scaling of the carrier signal frequency. Assuming sinusoidal waveforms, the simplest form of

phase detection in the optical and RF domains are interferometry and frequency mixing, respectively.

While both techniques offer the same phase detection range of [0, 2π], the carrier frequency for RF

signals are in the 10-GHz regime while that for optical signals are in the 100-THz regime. Using the


21

basic relation Δθ=ω∙Δt, the timing resolution scales inversely proportional with the carrier frequency,

resulting in orders-of-magnitude higher timing sensitivity in the optical domain than in the RF domain.

As relevant to this thesis, this argument can be extended to pulsed waveforms. Additional

bandwidth around the carrier frequency can be utilized to generate pulsed waveforms with sharper rise

times and short pulse durations. RF pulses are limited to 10-ps-level pulse durations (~10 GHz

bandwidth), while optical pulses generated from modern mode-locked laser can easily reach the sub-ps

level (~1 THz bandwidth) and beyond into the fs regime. Envelope detection schemes can be

implemented in both domains to measure the pulse timing. Due to significantly shorter pulse durations in

the optical domain, higher timing measurement sensitivity by orders of magnitude is still achieved.

This fundamental scaling directly impacts the level of precision by which optical and RF sources

can be synchronized. This is best explained with the following example where the timing between two

mode-locked lasers is measured using conventional RF phase detection. The optical pulse trains from two

lasers are first detected with photodiodes to convert the pulses into the RF domain. Ideally, the detector

should preserve the pulse properties; however, due to bandwidth limitations and detector nonlinearities,

the RF pulses are longer in duration than the optical pulses and may contain distortions. Since the

distorted pulse shapes may not be reliable for timing measurements, a harmonic of the repetition rate is

filtered to obtain a clean sinusoidal RF signal that is in-phase with the original pulse train. The phase of

this signal is then measured using a RF phase detector such as a double-balanced mixer. This is a proven

technique that can achieve ps-level timing resolution; however, it faces serious limitations when scaling

towards the fs-level and beyond due to signal-to-noise ratio (SNR) degradations in the electronics.

Figure 1-2 RF approach to pulse timing error measurement via direct detection and RF

phase detection. (a) System schematic (b) Typical voltage response of a double-balanced

mixer as a function of pulse timing error.


22

The first set of SNR degradations occurs during optical-to-RF conversion. To maintain the high

SNR of the optical signal at photodetection, the incident optical power on the photodiode must be

maximized. The maximum electrical SNR is determined by the thermal or shot noise floor at detection

and the maximum extractable signal power at detector saturation. This is typically much smaller than the

optical SNR. The SNR can be further degraded by detector nonlinearities. With high peak pulse power,

nonlinearities such as space-charge effects will dominate (and at much lower average optical power levels

compared to CW detection) [22]. Select nonlinearities will couple amplitude modulation (AM) noise into

temporal distortions in the pulse envelopes, which will translate to phase modulation (PM) noise of the

RF harmonics. This process is called AM-PM conversion and may severely degrade the timing stability

of the detector. Typical AM-PM conversion coefficients are on the order of ~ps/mW for 1 mW incident

average power (0.1 rad/RIN) [23]. Moreover, the thermal drift of the detector may result in ~350 fs/K

drift [24] and degrade the long-term phase stability of the RF signal.

The second set of SNR degradations is at RF phase detection. Double-balanced mixers, which

are commonly used as RF phase detectors when operated at quadrature, typically require a +10 dBm

signal at the local oscillator (LO) port to drive the diode-switching circuitry. To avoid mixer

nonlinearities, the input signal power to the RF port should be 10 dB below that of the LO power to

prevent the RF signal from overpowering the LO signal. Furthermore, additional active RF components

such as amplifiers may further degrade the SNR with additive and parametric noise.

Accounting for all these factors, the timing resolution of this conventional RF method is

practically limited to the 10-fs-level. For an input optical pulse train with 0 dBm average optical power

and a 200-MHz repetition rate, the regenerated 10-GHz RF signal after photodetection using a standard

PIN diode and bandpass filtering is shot-noise-limited at -170 dBm/Hz with -30 dBm signal power. The

signal is amplified further with negligible excess noise to 0 dBm to yield a maximum phase discriminant

of 0.22 V/rad in the mixer, which is equivalent to about 10 μV/fs. The amplified thermal noise floor

filtered by a 10-MHz low-pass filter (LPF) at the mixer output already yields 71 μV of background noise.

The timing precision of the mixer is therefore only 7 fs, which is unsuitable for sub-fs timing applications.

This shortcoming calls for an alternative method that can simultaneously provide higher timing

sensitivities in the mV/fs regime while minimizing background detection noise to achieve sub-fs

resolution. To achieve this, it is advantageous to perform timing measurements directly in the optical

domain where the optical SNR remains the highest. For example, a common practice in ultrafast optics is

to use two time events of comparable timescales to evaluate each other, i.e. use one pulse to evaluate


23

another in a technique known as optical cross-correlation. Cross-correlation can preserve the timing

precision of the pulses and yield timing resolutions significantly shorter than the pulse width itself,

therefore enabling sub-fs timing resolution or better.

In this thesis, two timing measurement devices operating directly in the optical domain are used

to achieve sub-fs synchronization. The first device is a balanced optical cross-correlator (BOC) [25],

which measures relative pulse arrival times to achieve link stabilization for timing distribution as well as

optical-optical synchronization. The second device is a balanced optical-microwave phase detector

(BOMPD), which measures the pulse arrival times relative to the RF signal zero-crossings to achieve

optical-RF synchronization. Both devices circumvent the limitations of conventional RF phase detection

by extracting timing information directly in the optical domain.

Timing Jitter and Phase Noise

In addition to timing measurement sensitivity, the noise performance of signal sources in the

optical domain can be better than that in the RF domain. The timing jitter of a mode-locked laser due to

amplified spontaneous emission [26] and shot noise is [27]:

𝑆∆𝑡,𝑜𝑝𝑡𝑖𝑐𝑎𝑙 = 𝑆∆𝑡,𝐴𝑆𝐸 + 𝑆∆𝑡,𝑆𝐻𝑂𝑇 ≈ [0.5𝜏𝐹𝑊𝐻𝑀

2

𝐸𝑝

𝑔

𝑇𝑟𝑡

ℎ𝜈]1

𝑓2+ [0.26

ℎ𝜈

𝑃𝑎𝑣𝑔

𝜏𝐹𝑊𝐻𝑀2 ]

where τFWHM is the pulse width at full-width half-maximum for a sech2 pulse, Ep is the pulse energy, Trt is

the repetition period, g is the incremental gain per round trip, hν is the photon energy, Pavg is the average

output power. The Leeson phase noise model for an RF oscillator converted to timing jitter is [28]:

𝑆∆𝑡,𝑅𝐹 = 𝑆∆𝑡,𝐿𝐸𝐸𝑆𝑂𝑁 + 𝑆∆𝑡,𝑇𝐻𝐸𝑅𝑀 ≈𝑇0

2

(2𝜋)2[

1

𝑓2(

𝜈0

2𝑄)

2

+ 1]2𝐹𝐾𝑇

𝑃𝑠

where To is the oscillator signal period, ν0 is the oscillator angular frequency, Q is the loaded cavity

quality factor, F is the gain noise figure (assume unity for simplicity), KT is the thermal energy, and Ps is

the average signal power.

Using these expressions, the noise components can be directly compared to reveal the scaling

advantages of optical sources. Comparing the white noise and f –2

noise components yield:

𝑆∆𝑡,𝑂𝑃𝑇,𝑆𝐻𝑂𝑇

𝑆∆𝑡,𝑅𝐹,𝑇𝐻𝐸𝑅𝑀

≈ (𝜏𝐹𝑊𝐻𝑀

𝑇0

)2

(ℎ𝜈

𝐾𝑇) (

𝑃𝑎𝑣𝑔

𝑃𝑠

)−1


24

𝑆∆𝑡,𝑂𝑃𝑇,𝐴𝑆𝐸

𝑆∆𝑡,𝑅𝐹,𝐿𝐸𝐸𝑆𝑂𝑁

≈

𝜏𝐹𝑊𝐻𝑀2

𝐸𝑝

ℎ𝜈

𝜏𝑝ℎ

𝑇02

𝐸𝑚𝑜𝑑𝑒

𝐹𝐾𝑇

𝜏𝑟𝑓

≈ (𝜏𝐹𝑊𝐻𝑀

𝑇0

)2

(ℎ𝜈

𝐾𝑇) (

𝐸𝑝𝜏𝑝ℎ

𝐸𝑚𝑜𝑑𝑒𝜏𝑟𝑓

)

−1

The stored energy in the resonator (Ep and Emode) and its storage lifetime (τph and τrf) are related to the

cavity quality factor Q. These can vary widely for RF oscillators as well as optical oscillators. For the

sake of comparison, assume that these quantities are comparable. This leaves two fundamental terms.

Although the photon energy hν is a factor of 30 higher than thermal noise KT, the quadratic scaling of the

pulse width τFWHM relative to the time period T0 of the RF signal yields a 104

– 106 noise reduction for

mode-locked lasers. This applies for both white and f –2

noise components. This is a sizeable advantage

for using mode-locked laser as timing sources. In practice, it is necessary to further evaluate real

oscillator design parameters (e.g. resonator quality factor Q, cavity power, noise figure, operating

frequency) to verify that this performance advantage can be fully realized. Environmental perturbations

(e.g. temperature) and excess timing jitter (e.g. pump-RIN-induced jitter) may degrade the timing jitter of

the mode-locked laser to levels much higher than that predicted with this simple quantum-limited

approximation. Additional noise suppression techniques such as optical frequency division can be used to

greatly suppress technical noise in mode-locked lasers. Ultra-stable optical reference cavities can be used

to stabilize a comb line of a mode-locked laser in the optical domain with mHz precision. Using the laser

frequency comb as an optical divider, the stability of the optical line can be divided down into the

microwave regime to generate ultralow phase noise microwaves [29], [30].

1.4 Optical Timing Distribution System

A pulsed optical TDS has been under development for the past decade to address the strict timing

requirements for FEL facilities (see Figure 1-3) [3]. The TDS begins with a mode-locked laser (optical

master oscillator) that is locked to a local high-quality microwave standard. The timing signal, which is

the optical pulse train from the master laser, is distributed through timing-stabilized optical fiber links to

remote locations throughout the FEL facility in a star network topology. The distributed timing signals

are used for three main purposes. First, the timing signal is used to synchronize remote laser sources,

such as the photo-injector laser, seed laser, and the pump laser. Optical-to-optical synchronization is

performed using the BOC in an optical phase-locked loop (PLL). Second, it is used directly as a high-

precision optical reference for beam diagnostics and controls such as bunch arrival monitors. Lastly, it is


25

used to synchronize various RF systems. Optical-to-RF synchronization is performed using a BOMPD in

an opto-electronic PLL to maintain the high timing precision of the optical signal into the RF domain. All

components of this timing distribution and synchronization system will enable next-generation FEL

facilities to achieve system-wide stability at the sub-fs level for sub-fs X-ray pulse synthesis and pump-

probe experiments.

The first-generation optical TDS was previously demonstrated by Kim et al. in [3]. Previous

laboratory demonstrations with a 300-m stabilized fiber link using standard single-mode fiber (SMF)

showed that polarization mode dispersion (PMD) limited the link stability to ~10 fs over few days of

operation [3] and caused delay jumps as much as 100 fs when the fiber was significantly perturbed [31].

This first-generation system has since been improved and adopted at FLASH in Germany [21] and

FERMI at ELETTRA in Italy [32] to realize the world’s first femtosecond optically-synchronized X-ray

FELs. For optical-RF synchronization, first-generation BOMPDs achieved 40 fs pk-pk drift over 10

hours of operation and a total integrated RMS jitter of 6.8 fs over a 28 μHz – 1 MHz range. The BOMPD

noise floor was approximately -135 dBc/Hz, which corresponds to a 3-fs RMS detection noise floor

integrated over a 1 MHz bandwidth. This inherent noise in the BOMPD is too large for sub-fs

synchronization in next-generation light sources.

Figure 1-3 Schematic of a pulsed optical TDS for a seeded X-ray FEL facility.


26

1.5 Thesis Outline

The focus of this thesis is on the development of a second-generation TDS that improves the

timing stability and distance coverage by an order of magnitude to achieve sub-fs stability across 3.5 km.

This thesis is divided into three chapters. The first chapter covers the optical domain components of the

optical TDS. This includes link stabilization for timing distribution and optical-optical synchronization.

Both depend critically on the BOC as the high-precision timing measurement device. The second chapter

covers the synchronization of signal sources across the optical and RF domains to extend the coverage of

the optical TDS. The BOMPD is the critical timing measurement device that enables optical-RF

synchronization. The last chapter covers the development of a photonic-based microwave oscillator for

ultralow noise microwave generation.

Chapter 2. Optical Timing Distribution and Synchronization

27

2 Optical Timing Distribution and Synchronization

2.1 Introduction

Since current FELs can already produce sub-10-fs X-ray pulses [33] and concepts for sub-fs pulse

generation are in place, significant system developments are in order to improve the TDS towards the

sub-fs stability level. The BOC, which is the primary pulse timing measurement device, is already

capable of attosecond timing resolution and should theoretically set the minimum stability of the TDS.

However, the first-generation TDS was only able to achieve ~10 fs stability, which is several orders-of-

magnitude larger. In this thesis, the system limitations in the first-generation TDS are addressed and

solved to demonstrate a second-generation TDS that can achieve long-term sub-fs stability. In summary,

this is achieved by systematically suppressing technical noise from the environment (i.e. temperature and

humidity changes), polarization-induced drifts, and AM-PM conversion noise. Progress into the sub-fs

regime and beyond also warrants a deeper understanding of pulse shaping and system noise dynamics at a

fundamental level.

This chapter is organized as follows. First, the principles of operation for key components in the

TDS are described from a phenomenological perspective. Experimental demonstrations of the second-

generation TDS are then presented, starting with timing jitter characterization of the optical master

oscillator, followed by timing stabilization of the 1.2-km and 3.5-km fiber links, and ending with remote

laser-laser synchronization across the 3.5-km link. Lastly, design considerations and technical issues for

achieving sub-fs stability are discussed.

2.2 Principle of Operation

2.2.1 Balanced Optical Cross-Correlator

The critical component in the optical timing distribution system that enables sub-femtosecond

stabilization is the balanced optical cross-correlator (BOC). It is a timing measurement device that can

measure the relative timing error between two optical pulses with extremely high timing precision in the


28

attosecond (as) regime. The BOC is based on balanced detection of double-pass, type-II phase-matched

second harmonic generation (SHG) in a periodically-poled KTiOPO4 (PPKTP) crystal [34].

The principle of operation for the BOC is shown in Figure 2-1(a). Two input pulses that are

orthogonally polarized and initially offset by a relative time delay ΔT are sent into the PPKTP crystal

aligned to its principle axes. The birefringence between the axes causes one pulse to travel faster and

walk through the other pulse. The difference in propagation time between the two axes is ΔT0 = TY – TX =

L/vg,y – L/vg,x, where L is the effective crystal length, and vg,y and vg,x are the group velocities of the two

principle axes. The crystal length (4 mm) is long enough to achieve appreciable pulse walk-off on the

order of the pulse width. During the walk-off, the PPKTP crystal generates a second harmonic (SH)

signal whose energy corresponds to the pulse area overlap. At the far end of the crystal, a dichroic filter

transmits the SH and reflects the fundamental harmonic (FH). The FH is not significantly depleted since

the SH conversion efficiency is low near η = 8×10-3

[25]; typically, the FH is at the 10-mW level while

the SH is at the 10-μW level. On the reverse pass, the faster pulse continues to walk through the other to

generate a second SH signal. Another dichroic filter separates out the SH on the reverse pass. The two

SH signals are detected on a balanced photodetector (BPD) to measure the energy difference. The output

voltage signal at baseband represents the relative pulse timing error.

An example voltage response of the BOC is shown in Figure 2-1(b). The x-axis is the pulse

timing error after the forward pass through the crystal, i.e. Δt = ΔT – ΔT0. The zero-crossing point occurs

when the SH generated on the forward and reverse pass are equal; that is, the position where the two FH

pulses cross each other is exactly at the end of the forward pass (ΔT=ΔT0). If the faster pulse arrives

Figure 2-1 Operating principle for a BOC. (a) Schematic of a double-pass, thick-crystal

BOC (b) Example voltage response as a function of pulse timing error. Note that Δt is the

pulse timing error at the end of the forward pass


29

earlier at the BOC input (ΔT<ΔT0), then this crossing position is earlier during the forward pass. More

SH is generated on the forward pass, resulting in a positive voltage at the detector output. The opposite

situation occurs when the faster pulse arrives later (ΔT>ΔT0) at the BOC input to generate a negative

output voltage. To first-order approximation, the voltage signal near the zero-crossing of the BOC curve

is directly proportional to the timing error V = KS∙Δt, where KS is the BOC timing (slope) sensitivity.

Beyond the linear regime, the BOC curve decays back towards zero because the initial pulse delay is too

large and out-of-range for proper balanced detection.

The BOC offers a number of advantages over conventional RF phase detection. First, timing

error information is converted into amplitude modulation in the optical domain, thus circumventing AM-

PM conversion issues at photodetection. Second, the balanced detection suppresses amplitude noise to

first-order approximation since the input pulses generates its own pair of forward and reverse SH signals,

which can perfectly cancel at the zero-crossing. Third, baseband detection with low-noise balanced

detectors with high transimpedance gain can be used to minimize background noise and achieve high

SNR. Further discussion of the BOC can be found in Section 2.4.6.

2.2.2 Optical-Optical Synchronization

Using the BOC as a high-resolution timing detector, an optical PLL (OPLL) can be implemented

to synchronize two optical oscillators. This synchronization scheme is based on the conventional RF PLL

Figure 2-2 Operating principle for optical-optical synchronization with a BOC. Pulse

timing error is measured with a BOC and fed back to the slave laser to achieve a timing

(phase) lock.


30

(see Appendix A). The principle of operation for optical-optical synchronization is shown in Figure 2-2.

Pulse trains from two independent mode-locked lasers with an initially small repetition rate difference Δfr

are sent to the BOC to measure their relative timing errors. Due to Δfr, the input pulse timing error to the

BOC will increase linearly with time as Δte(t) = (fr /Δfr)t, where fr is the fundamental repetition rate. This

will generate a train of BOC voltage response curves with a repetition period 1/Δfr at the BOC output.

The voltage error signal is filtered by the PI controller and fed back to the cavity length PZT (or pump

power) of the slave laser to control its repetition rate. Once the PLL is closed, the transient response for

lock acquisition can be understood in two steps. First, the feedback loop will pull the slave laser

repetition rate closer to that of the master laser. The decrease in Δfr effectively stretches the train of BOC

curves in time. Once Δfr is sufficiently small, the PLL response is fast enough to lock to the linear region

of a single BOC curve and minimize the error voltage to the zero-crossing to achieve lock.

2.2.3 Link Stabilization (Timing Distribution)

The optical PLL is not restricted for synchronizing two independent oscillators. It can also

synchronize two pulse streams generated from the same laser. For example, if a laser pulse train is split

into two paths and one path is subjected to noise, the two pulse trains can be recombined and measured

with a BOC to detect the noise introduced in the noisy path. The detected error can be used to remove the

accumulated noise in the noisy pulse train to recover its original stability. This is the fundamental

principle behind the timing-stabilized links for timing distribution.

The principle of operation of the timing-stabilized link is shown in Figure 2-3. The timing signal

is a low-noise optical pulse train from a mode-locked laser (master laser) that is distributed through a

dispersion-slope-compensated polarization-maintaining (PM) fiber link to a remote location. The master

laser is locked to a local microwave standard to improve its low-frequency noise. When the link is not

stabilized, slow fiber fluctuations due to temperature, humidity, acoustics, and mechanical disturbances

will induce timing errors in the output pulse arrival times. For example, ambient temperature fluctuations

of 1 K for a 1.2-km fiber link with a thermal expansion coefficient of 5×10-7

cm/cm∙K will induce length

changes equivalent to picoseconds of timing error. This is four orders-of-magnitude larger than the target

goal of sub-fs performance.


31

The goal of link stabilization is to maintain a constant time-of-flight for pulses from the link input

to the link output. Link stabilization begins with detecting the round-trip timing error. The output

coupler at the end of the link partially reflects the output pulse back to the input. A BOC then measures

the difference in arrival times, Δt, between the round-trip pulse and a new laser pulse and outputs a

corresponding error voltage, V = KS∙Δt, where KS is the BOC timing sensitivity. The error voltage is fed

back to a variable delay in the link to correct for the detected timing errors. The feedback forces the error

voltage to zero to suppress link fluctuations within the locking bandwidth to achieve long-term timing

stabilization. Once the link is stabilized, the pulse train at the link output can be used for optical-to-

optical and optical-to-RF synchronization as well as direct timing measurements.

There are a few restrictions to this link stabilization scheme. First, since the variable delay is

bidirectional, the timing errors acquired by the link pulse during the forward- and reverse-pass must be

equal; otherwise, eliminating the round-trip timing error may still result in a residual timing error at the

link output. Second, timing fluctuations occurring on timescales faster than the round-trip propagation

time cannot be compensated. The high-frequency timing jitter of the master laser must therefore be in the

sub-fs regime because it will cause timing fluctuations in the BOC on a shot-by-shot basis and set the

fundamental timing stability for link stabilization. Third, no noise should penetrate into the reference

path to preserve the original laser noise. The reference path can be made significantly short to improve its

robustness against environmental noise.

Figure 2-3 Operating principle for timing-stabilization of a fiber link. Pulse timing error,

Δt, acquired during round-trip propagation in the fiber link is measured with a BOC and

corrected by a variable delay in a negative feedback loop.


32

2.3 Experimental Demonstration

This section presents a series of experimental demonstrations that culminate in a complete end-to-

end optical TDS operating with sub-fs stability. First, the timing jitter of a commercial mode-locked laser

is characterized to confirm its viability as an optical master oscillator for sub-fs timing distribution.

Second, a 1.2-km dispersion-compensated PM fiber link is stabilized as a proof-of-concept demonstration

for long-term sub-fs stability. Third, the link is improved upon to demonstrate state-of-the-art timing

distribution across a 3.5 km link for over 96 hours of operation. An additional out-of-loop measurement

between simultaneous operation of the 1.2-km and 3.5-km links is presented to confirm sub-fs

performance for a 4.7-km fiber network. Lastly, a remote laser is stabilized to the output of the 3.5 km

fiber link with sub-fs residual jitter to demonstrate an end-to-end optical TDS.

2.3.1 Laser Jitter Characterization

The optical TDS requires a low-noise optical master oscillator to provide a clean timing reference

for the entire fiber network. While the low-frequency drift of the laser can be stabilized to a high-quality

RF reference, the high-frequency jitter cannot be suppressed and will fundamentally limit the timing

precision of the TDS. It is therefore critical to first confirm that commercial mode-locked lasers can

deliver the minimum jitter necessary for sub-fs timing distribution.

In literature, it is well-documented that solid-state lasers such as Ti:Saph lasers, which have high

pulse energies and low resonator losses, can exhibit extremely low jitter at high frequencies [35], [36] and

may rival the best commercial RF oscillators. Fiber lasers are also approaching superior noise

performance levels as well [37], [38], especially when stabilized to ultra-stable reference cavities [29],

[30], [39]. Although these state-of-the-art lasers may already satisfy the minimum requirements for sub-

fs timing jitter, their main drawback for a TDS is that they require constant attention and maintenance for

long-term performance. These lasers are operated in a laboratory setting and employ bulk optics, which

are sensitive to beam misalignment issues and environmental fluctuations. The basic requirements of the

TDS for the European XFEL is that the master laser should operate with negligible drift over days and

weeks and not limit machine availability due to maintenance [14]. Therefore, the focus of this thesis is on

using robust commercial turn-key lasers to demonstrate a high-precision TDS for a FEL facility.


33

Experimental Set-up

The timing jitter of a commercial mode-locked laser is measured with a dual-oscillator

measurement set-up, which is a standard RF technique for measuring the absolute phase noise of two

identical oscillators [40]. The timing jitter is measured by loosely locking two identical lasers using the

BOC in an optical PLL and characterizing the free-running absolute timing jitter at frequencies beyond

the locking bandwidth. The mode-locked lasers (Origami-15, OneFive) operate with a 216.667-MHz

repetition rate, 150-fs pulse width, 1554.7-nm center wavelength, and +22.4-dBm average power.

The jitter measurement set-up is shown in Figure 2-4. The input beams are combined at

orthogonal polarizations using a polarization beam splitter. A half-wave plate (λ/2) aligns the beams with

the principle axes of the PPKTP crystal. The BOC consists of a single 4-mm PPKTP crystal operated in a

double-pass configuration with appropriate dichroic elements [25]. The crystal is phase-matched for a

fundamental wavelength of 1550 nm (second harmonic at 775 nm) with a 100-nm bandwidth, which is

large enough to support 200-fs pulses. A dichroic mirror (DM) coating is applied on the rear facet of the

crystal to filter out the forward-pass SH. A focusing lens is used to tightly focus the input beam on the

rear facet of the crystal to achieve high SH conversion as well as symmetry between the forward and

reverse pass. The reverse-pass SH is filtered with a dichroic beamsplitter (DBS). The SH signals are

detected using a low-noise BPD (Newport; 2307) with variable transimpedance gain and 1-MHz

maximum bandwidth. Additional bandpass filters centered at the SH are added to each detection port to

decrease ambient noise. The error signal is filtered by a PI controller (New Focus; LB1005) and applied

to the repetition rate tuning port of the slave laser. The output voltage spectral density from the BPD is

Figure 2-4 Experimental set-up for optical-optical synchronization. A loose-lock is

implemented between the two mode-locked lasers to characterize the laser timing jitter

beyond the PLL bandwidth.


34

measured with a signal source analyzer (SSA) (Agilent 5052B) and divided by the BOC sensitivity to

obtain the timing jitter spectral density in the optical domain.

From a simple feedback loop analysis (see Appendix B), the PLL locking bandwidth with only

proportional gain for the loop filter can be easily calculated as f3dB = KBOCKPIKPZT/(2πfr) where KBOC (V/fs)

is the BOC sensitivity, KPI (V/V) is the proportional gain of the PI controller, KPZT (Hz/V) is the repetition

rate tuning sensitivity of the slave laser, and fr is the laser repetition rate. Using typical values observed

from experiment, e.g. KBOC = 1 mV/fs, KPZT = 17.4 Hz/V, fr = 216 MHz, and KPI = [-40, 0] dB, the locking

bandwidth can be tuned through the 100 Hz – 10 kHz range.

Measurement Results

The timing jitter measurements are performed in collaboration with Kemal Safak and Ming Xin at

CFEL DESY. The jitter spectra for select proportional gain values, as shown in Figure 2-5(a), agree with

simulation (see Appendix B). Decreasing the loop gain decreases the locking bandwidth accordingly.

The curves overlap beyond the locking bandwidth because the jitter is free-running and therefore add

incoherently to produce the absolute timing jitter spectrum of both lasers combined. Assuming identical

lasers, the measured jitter is divided by two to obtain the absolute timing jitter for a single laser. The jitter

spectral density above 30 kHz is buried by the BOC detection noise floor (black), which is set by the

electronic noise of the BPD. The SH power is insufficient to observe the shot noise limit.

Figure 2-5 Measurement results for laser timing jitter characterization. (a) Timing jitter

spectral density for various proportional gain values. (b) Timing jitter measurement for

0 dB gain and its integrated RMS jitter in the frequency range [f, 1 MHz].


35

The jitter spectral density curve corresponding to 0 dB proportional gain is redrawn in Figure

2-5(b). The laser jitter is directly observed since the locking bandwidth is below 1 kHz. The jitter

spectral density is integrated from 1 MHz to lower frequencies to yield the integrated RMS jitter curve

(red). The total jitter over the 1 kHz – 1 MHz range is about 0.4 fs. The jitter increases without bound for

frequencies below 1 kHz. This low-frequency jitter can be suppressed by stabilizing the laser to a RF

standard with superior low-frequency noise. Above 30 kHz, the BPD noise floor yields 170 as, which

indicate that the laser jitter may be much lower. If the laser is quantum-limited by amplified spontaneous

emission, the laser jitter spectrum will follow a -20 dB/decade behavior until the shot noise floor is

reached at high offset frequencies. This yields only 60 as RMS jitter over the 30 kHz – 1 MHz range.

The main jitter requirement for the optical master oscillator is that its high-frequency jitter for

timescales faster than round-trip propagation time in the fiber link must be in the sub-fs regime. For a

3.5-km link in the European X-FEL, the round-trip time is 35 μs, which corresponds to a cut-off

frequency of 29 kHz for the high frequency jitter. The lasers characterized here clearly meet this

criterion. Furthermore, sub-fs timing jitter down to 1 kHz indicates that the laser can support sub-fs

timing distribution over significantly longer distances (up to 100 km).

2.3.2 Single 1.2-km Link Stabilization

Timing-stabilized fiber links are critical for distributing the timing signal from the maser laser to

various locations throughout a FEL facility. Previous demonstrations of a timing-stabilized 300-m link

based on SMF were limited by PMD-induced timing drifts at the 10-fs level [3], [41]. To surpass this

limit, a PM fiber link is developed. A proof-of-concept demonstration is first performed to assess the

viability of PM fiber to eliminate PMD-induced errors.

Experimental Set-up

The timing stabilization set-up for the 1.2-km PM link is shown in Figure 2-6. The master laser is

a free-running Er-doped fiber mode-locked laser (Menlo Systems, M-Comb LH082) that generates an

optical pulse train with 170-fs pulse duration, 200-MHz fundamental repetition rate, 1557.66-nm center

wavelength, and +20-dBm average power. The pulse width and repetition rate are selected to reduce

higher-order dispersion and nonlinear effects in the fiber link. The laser timing jitter is near 1 fs in the


36

83 kHz – 10 MHz range [34], which is sufficient to test the PM fiber link for PMD suppression since

PMD-induced drifts are larger at the 10-fs level. For subsequent link demonstrations, this laser is

replaced with that characterized in Section 2.3.1 for sub-fs timing jitter performance.

The set-up is divided into two sections: the in-loop section which performs link stabilization and

the out-of-loop section which evaluates the link stability. In the in-loop section, the laser pulses are

divided into reference and link pulses. The reference pulses serve as a timing reference for the in-loop

BOC. The link pulses are transmitted through the link path, which consists of a voltage-controlled optical

delay line (ODL), 45° Faraday rotator, half-wave plate, 1.2-km PM link, and a 90/10 reflection/

transmission output coupler. The link stabilization scheme is optimal when the timing errors accumulated

during the forward and reverse transmissions are equal. This restricts the round-trip pulse propagation

along a single polarization axis in the PM link. To achieve this, the half-wave plate aligns the input

Figure 2-6 Experimental set-up for 1.2-km link stabilization. (L1, collimator; FR, 45°

Faraday rotator; FC, fiber collimator; PMF, standard PM fiber; PM-DCF, dispersion-

compensating PM fiber; OC, output coupler; BOC, balanced optical cross-correlator;

DBS, dichroic beam-splitter; L2, focusing lens; PPKTP, periodically-poled KTiOPO4;

DM, dichroic mirror; BPD, balanced photodetector; PI, proportional-integral controller;

HVA, high voltage amplifier; LPF, low-pass filter; DAQ, data acquisition.)


37

polarization with the slow axis of the PM link. The 45° Faraday rotator induces a 90° round-trip

polarization rotation external to the fiber link so that the returning link pulses can reach the in-loop BOC.

The dispersion-compensated 1.2-km PM link is fabricated by OFS (see Section 2.4.1 for details).

Residual link dispersion is compensated by adding 2.4 m of standard PM fiber to achieve a minimum

pulse width of 250 fs at the link output.

The power coupled into the link input is +11 dBm, which is near the onset of fiber nonlinearities.

With a 10% output coupler, the power reflected back to the link input is enough to achieve a high BOC

timing sensitivity of 20 mV/fs for tight in-loop locking. The lower output power yields a lower timing

sensitivity of 3.5 mV/fs for the out-of-loop BOC. This is sufficient for measuring long-term drifts but not

high-frequency jitter because the latter is buried by the BOC detection noise floor. Although this is not

ideal, it is acceptable for this demonstration because PMD-induced timing drifts occur only on very slow

timescales (hours) and can be measured above the detection noise floor. Moreover, without the use of an

link EDFA, the link path consists only of passive elements, which minimizes additive noise. Subsequent

link demonstrations include an EDFA to increase the SNR to measure high-frequency noise.

The feedback stabilization begins with the in-loop BOC, which measures the round-trip timing

error by comparing the arrival times between the round-trip link pulses and the reference pulses. The

locking electronics consists of a PI controller (Menlo Systems, PIC210), high voltage amplifier (Menlo

Systems, HVA150) and a voltage-controlled ODL built from mounting a retroreflector on a stacked

combination of a 40-μm PZT (Thorlabs, PAS009) and a 25-mm motorized translation stage (PI, M-

112.12S) for short- and long-term stabilization, respectively. The feedback bandwidth is around 20 Hz.

With the feedback engaged, any timing error that remains uncompensated will cause a timing

drift at the link output. The out-of-loop BOC monitors this drift by measuring the timing error between

the link output pulses and new laser pulses. For long-term drift measurements, the out-of-loop BOC

voltage is filtered with a 0.5-Hz anti-aliasing low-pass filter (LPF) and recorded with a data acquisition

card (DAQ) at a 1-Hz sampling rate. The motor stage delay is recorded simultaneously to monitor the

timing error compensated by the in-loop stabilization.

Temperature stabilization and vibration isolation of the free-space optics are critical for sub-fs

stability. Separate enclosures are built for the free-space optics and fiber link. Each enclosure consists of

an external 2" layer of extruded polystyrene insulation foam and an internal Aluminum enclosure that is

temperature controlled with a resistive heater pad and PI controller. Lead foam is placed beneath the set-


38

up to dampen table vibrations. Thermal fluctuations in free-space optics are reduced further by

minimizing differential path lengths and the usage of kinematic mounts. The enclosures have a

temperature set point at 27°C. Moreover, the link temperature is modulated by 0.05°C with a period of 20

min to measure the timing errors induced by thermal expansion in the fiber link.

Experimental Results

Sub-fs timing stabilization of the 1.2-km PM link is achieved for 16 days without interruption.

The recorded data for the residual link drift and compensated link delay are shown in Figure 2-7(a).

Although the data log for the motorized ODL faulted in day 13, the link stabilization remained unaffected.

The feedback corrected for over 65 ps of timing fluctuations, while the residual drift at the link output

showed only 0.6 fs RMS with 2.5 fs pk-pk deviation. This indicates that timing fluctuations are

suppressed by a factor of more than 20,000 over 16 days to achieve an average timing instability of

1.8×10-21

. PM fiber is concluded to be effective at eliminating large 100-fs delay jumps previously

caused by PMD and overcoming the 10-fs stability limit for multi-day operation.

Figure 2-7 Measurement results for 1.2-km link stabilization. (a) Link drift measured by

the out-of-loop BOC and link delay compensated by the ODL; (b) Relative temperature

changes for the PM link and free-space enclosures and ambient temperature.


39

The link drift is limited by environmental fluctuations in the free-space optics. Recorded data for

relative temperature changes within the enclosures for the PM link and free-space optics as well as that of

the ambient temperature are plotted in Figure 2-7(b). A strong correlation exists between the link drift,

free-space enclosure temperature, and ambient temperature. This is reasonable because the free-space

enclosure is large in volume, making it difficult to isolate the enclosed optics from the environment. In

free-space, 1 fs of timing error is equivalent to a length change of only 0.3 μm. Length changes on this

scale in uncompensated path lengths are suspected of introducing false timing drift in the measurements.

Link drift is also sensitive to mechanical stress in the optical breadboard; slight pressure along one edge

of the breadboard due to tension in power cables can result in a slow linear drift greater than 3 fs over a

few days (not shown here). Resolving these mechanical stability issues in the free-space optics is

necessary to achieve sub-fs stability. This is evident from the 0.13 fs drift during days 11-14 when

ambient fluctuations are minimal. Direct modulation of the link temperature by 0.05°C induced less than

1 ps of timing error; this corresponds to a temperature coefficient of delay of 17 fs/m/K, which is within

range of coefficients reported for comparable fiber types [42]. Scaling towards ±0.5°C temperature

instability in FEL facilities [32], the expected 10-ps-level timing errors can still be easily compensated by

using a longer ODL.

2.3.3 Single 3.5-km Link Stabilization

The successful demonstration of a 1.2-km PM link with sub-fs stability over multi-day operation

is a key milestone for optical timing distribution. Based on this demonstration, a second testbed is built at

Deutsches Elektronen-Synchrotron (DESY) in Hamburg, Germany to achieve state-of-the-art timing

distribution across a 3.5-km PM link. Initial work is developed in collaboration with fellow colleagues,

Kemal Safak and Ming Xin, at CFEL DESY. The final results are summarized in the following sections.

Full system details are covered in Safak’s dissertation[43].

Experimental Set-up

Key improvements for the 3.5-km PM link demonstration are summarized as follows.

The link is extended from 1.2 km to 3.5 km to cover the length of the European X-FEL.


40

The laser from Section 2.3.1 is used to achieve sub-fs high-frequency timing jitter performance. The

laser repetition rate is chosen to match operating frequencies at the European XFEL [14], and is

locked to a local RF synthesizer to reduce timing drifts below 10 Hz.

A bi-directional PM-EDFA is inserted near the end of the link to boost the link output power to allow

measurement of the high-frequency link noise.

The free-space PZT and motorized delay stage are replaced with a fiber stretcher and fiber-coupled

motorized delay stage, respectively, to eliminate free-space beam misalignment issues.

The breadboard is replaced with a temperature-stabilized breadboard surfaced with a super invar sheet

to improve the long-term temperature stability of the free-space optics.

The electronic feedback loop is improved for tighter locking, e.g. the PI controller is replaced with a

proportional-double-integral-differential (PI2D) controller (Vescent D2-125).

The link is placed external to the temperature enclosure and subjected to ambient conditions.

Experimental Result

Link stabilization is achieved for 96 hours. The residual link drift and compensated link delay are

shown in Figure 2-9(a). The ODL corrected over 8 ps of link fluctuations. The residual drift at the link

output is 0.57 fs RMS with 3 fs pk-pk deviation. This is comparable to that achieved with the 1.2-km PM

Figure 2-8 Experimental set-up for 3.5-km link stabilization.


41

link, but over three times the distance. Temperature fluctuations are suppressed down to ±0.1K. Relative

humidity changes are uncontrolled and fluctuated by 4%. The ODL is found to be correlated with

humidity because humidity affects the refractive index of fiber and introduces group delay (see Section

2.4.3). The out-of-loop signal is measured directly with a SSA (Agilent E5052B) to obtain the high-

frequency link noise above 1 Hz in Figure 2-9(b). The Fourier transform of the long-term drift covers the

range below 1 Hz.

To calculate the RMS timing jitter, the jitter spectral density is integrated from the lowest

frequency. This direction of integration is better for assessing the in-loop locking performance since the

noise beyond the locking bandwidth can be ignored. The integrated jitter from 3 μHz to the 2-kHz

locking bandwidth is 0.55 fs RMS. The total integrated jitter is 0.62 fs RMS. The main jitter contribution

is from noise components below 20 μHz, which corresponds to slow drift on the 10-hour scale. This drift

is suspected to be from link power fluctuations due to beam misalignments in the ODL (see Section 2.4.4).

This is supported by the high correlation between the out-of-loop drift and ODL. Negligible jitter is

contributed in the 20 μHz – 1 kHz range due to tight in-loop noise suppression. The increase in residual

Figure 2-9 Measurement results for 3.5-km link stabilization. (a) Link drift over 96

hours and the corresponding link delay compensated by the ODL; (b) Complete jitter

spectral density from 3 μHz to 1 MHz and its corresponding integrated timing jitter.


42

jitter in the 1 kHz – 10 kHz range is the laser jitter filtered by delayed self-homodyne detection after

round-trip propagation [34]. The noise beyond 50 kHz is solely from the BPD noise floor and bandwidth

roll-off. These experimental results are verified with phase noise simulations in Appendix C.

2.3.4 4.7-km Fiber Network Stabilization

Experimental Set-up

Both 1.2-km and 3.5-km timing links are then operated in parallel to demonstrate a 4.7-km

timing-stabilized fiber network. The experimental set-up is summarized in Figure 2-10. The fiber link

spools are placed external to the enclosure to subject them to environmental fluctuations in the laboratory

and on opposing sides of the optical table to minimize common mode noise. The SNR in the BOC is

improved further by using a beta barium borate (BBO) crystal with large birefringence to increase the

polarization extinction ratio (see Section 2.4.6). Since the feedback PZT resonances are near 10 kHz, the

feedback loop bandwidth is set lower at 1 kHz for sufficient phase and gain margin. The stabilized link

outputs are compared in the out-of-loop BOC to evaluate the timing stability of the fiber network.

Figure 2-10 Experimental set-up for 4.7-km fiber network stabilization.

The dominant source of power fluctuation in the system is from fiber coupling losses at the link

input. The long travel range of the motorized ODL (both free-space and fiber-coupled versions)

introduces beam steering errors, which causes fluctuations in the power coupled into the link. These

fluctuations within the fiber link cause nonlinear pulse distortions, which introduce timing errors at the fs-


43

level (see Section 2.4.4). To reduce these effects, feedback on the EDFA pump current is used to stabilize

the link power. The feedback begins with monitoring the link power drifts with the BPD within the BOC.

Power fluctuations are sampled by a DAQ and fed back to the EDFA through a TCP/IP network to

control its pump current. The 100-ms network latency in the feedback path is low enough to achieve

proper feedback control since large power fluctuations (>1%) occur on timescales slower than 1 Hz.


The relative timing stability between the link outputs is monitored for 52 hours of continuous

operation. The long-term drift below 1 Hz is only 0.2 fs RMS, as shown in Figure 2-11(a). The complete

jitter spectral density from 7 μHz to 1 MHz is shown in Figure 2-11(b). The total integrated jitter is only

0.58 fs, which yields an overall average timing stability of 3.1×10-21

. The ODLs compensate for about

8.6 ps and 35 ps link fluctuations in the 1.2-km and 3.5-km links, respectively. Link noise is therefore

suppressed by over four orders of magnitude. Link power fluctuations reduced to ±0.2% with active

power stabilization in the 1.2-km link. The 3.5 km link is only stabilized to ±1% due to the coarse

resolution of EDFA pump current and is the main drift contribution in the 300 μHz – 1 Hz range.

Figure 2-11 Measurements results for 4.7-km fiber network stabilization. (a) Out-of-loop

drift between the link outputs below 1 Hz; (b) Complete jitter spectral density and its

corresponding integrated jitter.


44

2.3.5 Remote-Laser Synchronization

After timing distribution through the fiber network, the final step is to synchronize a remote laser

at the link output for remote optical-optical synchronization. This is important in a FEL facility to

synchronize the seed pulse with the driving electron bunch for X-ray FEL pulse generation or the pump

laser pulses with the FEL pulses for pump-probe measurements.

Experimental Set-up

The remote optical-to-optical synchronization setup is an extension of the 3.5-km link

stabilization set-up (see Figure 2-12). The setup consists of three sections: link stabilization, remote laser

locking, and out-of-loop measurement. The remote laser is identical to the master laser and is the same

laser as that from Section 2.3.1. The remote locking section begins with combining the remote laser and

link output in a BOC to generate the timing error signal. The error voltage is filtered by a PI controller

and separated into two paths to optimize feedback for slow and fast noise independently. This is

necessary because the limited frequency response of the high voltage amplifier prevents the PI output

from being directly amplified and used for feedback control. The first path is sampled by a DAQ card

and used to generate a DC offset voltage to compensate slow timing drifts. The second path remains

Figure 2-12 Experimental set-up for remote laser-laser synchronization


45

unamplified to maintain good phase margin for high locking bandwidths to compensate fast noise. The

voltage adder recombines the two paths to drive the PZT of the remote laser. With the link stabilization

and remote locking engaged, the master and slave laser output are combined in the out-of-loop BOC for a

residual timing jitter measurement.


Remote optical-to-optical synchronization across the 3.5-km timing link is achieved for over 44

hours of continuous operation. The out-of-loop residual timing drift below 1 Hz is 94 as RMS as shown

in Figure 2-13(a). The complete jitter spectral density is plotted in Figure 2-13(b). Within the 1-kHz

locking bandwidth of the link stabilization set-up, the integrated jitter is less than 0.2 fs RMS. The total

integrated jitter is 0.68 fs RMS. The bump in the 1 kHz – 10 kHz range is suspected to be from residual

link-locking jitter induced by the master laser as well as electronic noise amplified by the remote laser-

locking feedback loop. Further analysis is needed to suppress this noise.

Figure 2-13 Measurement results for remote laser-laser synchronization. (a) Long-term

laser-laser timing drift; (b) Complete residual jitter spectral density from 6 μHz to 1 MHz

and its corresponding integrated timing jitter.


46

Allan Deviation for Time Series

The measured data can also be represented using a different estimator known as the time

deviation (TDEV), which is derived from the modified Allan deviation [44]. The Allan deviation is a

standard figure of merit used in the time/frequency community to assess the long-term performance of

oscillators in terms of average fluctuations over a sampling period. The TDEV is calculated for the out-

of-loop drift as well as the link delay, temperature, and humidity for the remote laser synchronization

demonstration. Typical temperature and humidity coefficients of group delay are used to convert their

drifts into expected timing fluctuations in the link. As the averaging time increases, the average out-of-

loop drift decreases as τ-1

and approaches the ~10-21

stability limit. The temperature and humidity,

however, remains above ~10-16

, which is 4-5 orders of magnitude higher. Since the link delay

compensates for these drifts, it is similarly limited above ~10-16

. While averaging may improve oscillator

performance for conventional clock applications, the timing requirement for the TDS in FEL facilities is

very strict and does not allow for averaging since sub-fs stability is required from pulse-to-pulse.

Therefore, the jitter spectral density and integrated jitter is a more important figure of merit than the Allan

deviation.

Figure 2-14 Time deviation estimator for long-term stability. Based on remote laser-laser

synchronization results.


47

2.4 Design Considerations

Extensive analysis of system limitations is required to achieve long-term sub-fs stability in the

optical TDS. Critical design choices and issues encountered during the development of the system are

discussed in the following sections.

2.4.1 Link Construction

Dispersion-compensated PM fiber links are developed and fabricated in collaboration with OFS

Laboratories to eliminate PMD-induced timing drifts. The first PM link demonstration is based on a

1.2-km PM fiber link, which is constructed from 1088 m of standard TruePhase™ PM panda-style fiber

matched to 190 m of custom PM dispersion-compensating fiber (DCF). The PM-DCF has a PANDA-like

geometry containing Boron stress rods with a 35-μm diameter and core index profile similar to

conventional DCF. It has a measured birefringence of 2.9×10-4

, loss of 0.42 dB/km, and wavelength cut-

off at 1520 nm. The dispersion and dispersion slope are -104.1 ps/nm∙km and -0.34 ps/nm2∙km,

respectively, for the slow axis at 1550 nm. Due to the small mode size of the PM-DCF (Aeff = 22 μm2), an

intermediate bridge fiber (PM Raman fiber) is used to minimize the splicing loss between the PM-DCF

and standard PM fiber. The single-pass link transmission loss is -2.7 dB and the polarization extinction

ratio is >20 dB. For the 3.5-km PM link demonstration, the section lengths are extended to 2946 m of

standard PM fiber and 511 m of PM-DCF. The transmission loss is increased to -8 dB, and the

polarization extinction ratio is degraded to 16.7 dB.

Figure 2-15 Construction of the 1.2-km PM fiber link


48

2.4.2 Pulse Duration

Short pulse durations in the TDS are desired for two reasons. First, laser timing jitter scales

quadratically with the pulse duration. Since the high-frequency timing jitter of the master laser sets the

fundamental stability limit in the TDS, it is highly desirable to use a laser that can generate short pulses.

Second, shorter pulses have higher peak powers (assuming fixed average power). Since SHG requires

high power for SH conversion, it is desirable to use short pulses to achieve high SNR in the BOC for tight

synchronization. For example, if the pulse duration is increased by a factor of 10 (assuming constant

pulse energy) from 400 fs to 4 ps, the SH conversion would decrease quadratically to degrade the BOC

precision from 2 as to 200 as.

During link transmission, however, long pulse durations are favored. First, the larger optical

bandwidths associated with shorter pulses increase the difficulty for dispersion compensation in the fiber

link due to higher-order dispersion. Currently, the dispersion compensation is sufficient for second- and

third-order dispersion for pulses with about 20 nm of optical bandwidth. Decreasing the pulse width by a

factor of 2 will increase the optical bandwidth to 40 nm, which will greatly increase the effect of residual

higher-order dispersion. These effects can be introduced in the form of deterministic pulse distortions

(see Section 2.4.4) as well as link-enhanced timing jitter (see Section 2.4.5), both of which would degrade

the timing stability of the TDS.

These practical constraints suggest that the pulse duration should be limited to the 100 fs – 1 ps

range, assuming transform-limited pulses. Although the working principle of the timing distribution

scheme is still valid for pulse durations outside of this range, order-of-magnitude changes in either

direction will greatly degrade the timing stability of the system. It may be possible to correct for these

timing instabilities with additional compensation techniques, e.g. external pulse compression, but this is at

the expense of increased system complexity. In a real FEL facility, the ideal timing distribution system

should be simple to install and robust over long operation times. The pulse duration should be optimized

to achieve a low-maintenance, robust TDS.

2.4.3 Temperature and Humidity

Uncompensated path lengths, such as the reference path in the link stabilization set-up as well as

paths in the out-of-loop measurement set-up, are highly susceptible to timing instabilities induced by the


49

environment. Temperature fluctuation is a major source of long-term timing instability. In free-space

propagation, 1 fs is equivalent to 0.3 μm. Length fluctuations on this scale are easily introduced into free-

space path lengths through thermal expansion of the optical breadboard. For example, daily laboratory

temperature fluctuations of ±1 K for an Aluminum breadboard with a thermal expansion coefficient of

α = 23.1 μm/m∙K will cause timing instabilities of ±0.75 fs/cm of path length. For the 4-cm reference

path length in the link stabilization set-up, its timing instability is ±3 fs, which is unacceptable. For the

out-of-loop measurement set-up, uncompensated path lengths are typically longer and suffer higher

timing instabilities. To improve the overall system stability, the Al breadboard is replaced by a water-

cooled breadboard surfaced with a Super Invar sheet (α = 0.64 μm/m∙K) and stabilized to ±0.1K to yield a

timing instability of ±2 as/cm. This allows for uncompensated path lengths up to 5 meters without

incurring timing instabilities exceeding the sub-fs level.

In addition to uncompensated free-space path lengths, extra care must be taken to identify all

uncompensated lengths in fiber. For example, for remote laser-laser synchronization in a real facility, it

may be necessary to use fiber patch cords to transport the timing signal from the link output and the

remote laser signal to a convenient location on an optical table to perform locking. These patch cords,

which may be as short as 1 meter, are not timing-stabilized and can introduce serious timing errors under

typical ambient conditions. The temperature and humidity coefficients of fiber group delay for various

fiber types have been reported in literature [42]. Depending on the optical fiber construction, the

temperature coefficient may vary from 3.2 to 41 fs/m/K. The upper value represent fibers that have

multiple protective coating layers with a high degree of thermal expansion rate mismatch, such as normal

SMF with a 250-μm buffer. PM fiber is arguably similar to SMF in performance but may have a higher

coefficient due to added thermal effects from the stress rods. The lower value represents fibers that have

minimal coatings and matched thermal expansion rates, such as Furukawa’s phase-stabilized optical

fibers. The humidity coefficient ranges from 0.41 to 2.46 fs/m/% for similar reasoning. For 1-m patch

cords in typical laboratory conditions with ±1K temperature and ±2% relative humidity changes, the

expected timing fluctuations are already at the fs-level. These patch cords should therefore be timing-

stabilized, if not minimized in length, to achieve sub-fs stability.

The next-generation TDS is expected to take on an all-fiber approach with the advent of a fiber-

coupled, integrated PPKTP BOC [45]. An all-fiber implementation is advantageous in a real FEL facility

for many practical reasons such as ease-of-implementation, increased system robustness, and elimination


50

of free-space alignment issues. The main advantage relevant here is its small form factor. The total

system size can be significantly reduced to fit within a small volume for better temperature and humidity

control. Small commercial temperature-humidity chambers can easily provide ±0.01K temperature

stability and ±1% humidity stability. Even with this improvement, the temperature and humidity timing

drifts are still non-negligible. Due to splicing constraints, the minimum reference path fiber length is

practically limited to about 30 cm. The temperature and humidity drifts in the chamber would be 0.24 fs

and 1.5 fs respectively, assuming normal SMF. Furthermore, the temperature dependence of the

birefringence of PM fiber has not yet been accounted for and may degrade the system stability further.

To suppress this instability down below the sub-100-as regime, the ideal solution is to combine optical

elements in a compact fiber (or integrated) module to minimize critical path lengths. For example, the

reference path would require packaging a Faraday rotation mirror (equivalent to the quarter-wave plate

and mirror) with the polarization beam splitter to achieve path lengths below 1 cm for sub-fs stability.

2.4.4 Fiber Nonlinearity

Numerical simulations for nonlinear pulse propagation through the 1.2-km fiber link are

performed to calculate the deterministic timing errors caused by fiber nonlinearities. The split-step

Fourier method is used to simulate pulse propagation in the presence of real losses, second-order

dispersion (SOD), third-order dispersion (TOD), self-phase modulation (SPM), stimulated Raman

scattering (SRS), and self-steepening (SS) [46]. The link input pulses are noiseless transform-limited

sech2 pulses with 170-fs pulse durations and 1557.66-nm center wavelength and are linearly polarized

along the slow axis of the PM link. Birefringence and PMD are ignored. After single-pass and round-trip

link transmission, the distorted pulses are cross-correlated with clean reference pulses using the thin-

crystal BOC approximation [41] and normalized to generate the output voltage curves for the out-of-loop

and in-loop BOCs, respectively, as shown in Figure 2-16(a-b). The simulation is repeated for pulse

energies from 1 fJ to 110 pJ. The BOC simulations in Figure 2-16(a-b) are performed by Valente [46].

Increasing the pulse energy generally steepens the zero-crossing slope (y-axis) for a higher BOC

timing sensitivity until severe pulse distortions dominate. The impact on link timing accuracy (x-axis) is

less obvious and requires careful analysis. Asymmetric pulse distortions due to dispersion and fiber

nonlinearities will shift the center of gravity of the link pulse and introduce a timing offset in the zero-

crossing of the BOC curve with respect to the origin. This offset is considered timing error and can be


51

extracted from the in-loop and out-of-loop BOCs curves in Figure 2-16(a-b) for the round-trip and single-

pass pulses, respectively, as shown in Figure 2-16(c).

The total timing error at the output of a stabilized link can be calculated as follows. Consider first

the round-trip timing error. The link stabilization scheme feeds back to a free-space ODL to compensate

for voltage fluctuations about the zero-crossing in the in-loop BOC. The feedback however cannot

compensate for time-shifts in the time-axis of the BOC curve. If the round-trip timing error time-shifts

the BOC zero-crossing by tRT, the feedback loop will unknowingly track this change and falsely introduce

-½tRT into the link path, which will propagate to the link output as timing error. Similarly, single-pass

propagation will introduce its own timing error tSP. The total timing error at the link output is the

summation of both contributions ΔtLINK = tSP − ½tRT.

Figure 2-16 Simulation results for nonlinear pulse propagation in the 1.2-km PM link.

(a-b) Normalized output voltage curves for the out-of-loop and in-loop BOC for select

pulse energies; (c) Single-pass and half round-trip timing error, as extracted from the

zero-crossings in the out-of-loop and in-loop BOC output curves, respectively, and the

resulting link timing error; (d) Link timing drift, as extracted from the link timing error

curve, corresponding to 0.1%, 1%, and 10% pulse energy fluctuations.


52

For low pulse energies below 1 pJ, the link error is due to residual TOD only. Above 1 pJ,

intensity-dependent nonlinearities such as SPM and SRS begin to take effect. The round-trip error is

typically larger than the single-pass error because the round-trip pulse travels twice the link length

compared to the single-pass pulse and experiences more nonlinear distortions. While the single-pass error

increases monotonically, the round-trip error initially decreases due to residual link dispersion and SPM

counteracting each other to reduce the pulse asymmetry. These opposing trends combine to lower the

link error below 3 fs for pulse energies up to 70 pJ. Beyond 90 pJ, complete pulse-splitting and severe

distortions due to SPM and SRS cause the link error to increase sharply above tens of fs. These

distortions can be identified in the in-loop BOC curves as nonlinear behavior near the zero-crossing (for

100 pJ) or multiple zero-crossings (for 105 pJ). The 1.2-km link demonstration is operated with 63 pJ

pulse energy to maximize the BOC timing sensitivity while avoiding the nonlinear threshold at 70 pJ.

The timing distribution systems for FELs only require relative timing stability rather than an

absolute clock system [32]. When the timing system is initially calibrated for a specific pulse energy

operating point, the corresponding link timing error in Figure 2-16(c) is set as the zero-error reference

point. After this calibration, fluctuation about this operating point is considered timing drift. Using the

link error curve in Figure 2-16(c), the link timing drift due to 0.1%, 1%, and 10% power fluctuations are

calculated and plotted in Figure 2-16(d). For 63-pJ pulse energy and typical ±5% power fluctuations due

to beam misalignment, the link drift is expected to be in the sub-fs level, which agrees with experiment

where the timing drift is 0.6 fs. Since these nonlinearity-induced drifts are power-dependent, the link

power can be lowered to reduce link drift. Operating at much lower power levels will require EDFAs or

integrated BOCs to maintain high SNR for tight locking.

Additional simulations for the 3.5-km link are performed by Xin et al. [47] to calculate

deterministic timing drifts due to residual link dispersion and average link power. Residual TOD

equivalent to 10 m of uncompensated fiber can cause 5 fs of timing drift for ±5% power fluctuations. An

average link power of +10 dBm can now cause fs-level drift for 1% power fluctuations. This is a

significant increase from the simulation results for the 1.2-km link where 1% power fluctuations only

yielded 0.05 fs (at 46 pJ pulse energy). This is reasonable because the link length increased three-fold,

allowing significant pulse evolution and distortions to occur. In Figure 2-16 (a-b), it is clear how quickly

distortions can accumulate when scaling the propagation length from 1.2 km to 2.4 km. For a round-trip


53

propagation length of 7 km, the link drift increases significantly. Therefore, it is necessary to implement

link power stabilization to minimize power fluctuations below 1% to achieve sub-fs stability.

2.4.5 Link-Enhanced Timing Jitter

In addition to the amplitude-to-timing conversion, the link also enhances the timing jitter of the

master laser through timing-to-timing conversion. A more advanced numerical model is developed by

Xin et al. [47] to simulate nonlinear pulse propagation in the presence of laser timing jitter. The

simulation results are summarized here. The inherent laser jitter can be enhanced through two processes.

First, pulse center frequency fluctuations are coupled to timing jitter via residual link dispersion,

particularly second order dispersion (SOD) and third order dispersion (TOD). This jitter contribution is

considered Gordon-Haus jitter and can amount to 0.1 fs for SOD equivalent to 2 m of standard PM fiber

and 0.3 fs for 3 m of fiber. Second, timing and amplitude noise due to spontaneous emission noise are

coupled and enhanced by the link nonlinearities. This jitter contribution is limited to 0.13 fs for input

power below +12 dBm but escalates exponentially to 1.4 fs at +14 dBm. For optimal system

performance, the input power should be increased for high BOC timing sensitivity while avoiding the

+12-dBm threshold for significant nonlinearity-induced jitter.

2.4.6 Balanced Optical Cross-correlator

Polarization Extinction Ratio

Ideally, the input pulses to the BOC are aligned along the two principle axes of the PPKTP crystal

for maximum SHG. The link pulse ELy and reference pulse ERx are the desired signal pulses. Due to finite

polarization extinction ratios in the optical elements upstream from the BOCs, the pulses are projected

along the undesired polarization axes. These pulse components (ELx and ERy) are considered noise pulses.

These noise pulses may interfere with the signal pulses during SHG (ELy with ERy and ERx with ELx) and

generate excess amplitude noise in the BOC. This noise cannot be eliminated by balanced detection. To

eliminate this effect, a highly birefringent material, such as a beta barium borate (BBO) crystal, is placed

before the BOC. The large birefringence adds a significant delay between the axes such that the noise

pulses do not overlap and interfere in the SHG process (see Figure 2-17).


54

Figure 2-17 Improving polarization extinction ratio in the BOC with a birefringent crystal.

Timing Sensitivity Calibration

The timing sensitivity of the BOC is measured using two methods. The first method is for the

link stabilization set-up where the two input pulse trains to BOCs have the exact same repetition rate.

The relative delay between the pulse trains is swept with a motorized ODL while the response voltage of

the BOC is recorded. With the voltage response plotted as a function of the relative delay, the BOC

timing sensitivity is calculated as the slope at the zero-crossing in units of V/fs.

The second method is for the laser-to-laser synchronization set-up where the two lasers have

different repetition rates. The free-running master and slave laser pulse trains are combined in the BOC

to generate a train of BOC curves. The repetition rate difference is coarsely tuned to a small value such

that the rise time of the BOC curve is not limited by the BPD bandwidth. The instantaneous repetition

rate difference and BOC curve is recorded on an oscilloscope simultaneously. The rate difference can

alternatively be measured using photodetectors and a mixer to measure the beat frequency between the

fundamental repetition rate tones. The real time scale of the BOC curve is calibrated by multiplying the

oscilloscope time scale tosc with the ratio of the recorded frequency difference to the laser center repetition

frequency Δt = (Δfr/fr) tosc. The slope at the zero-crossing is the BOC timing sensitivity.

Beam Alignment

Due to the symmetry of balanced cross-correlation, precise alignment is necessary to achieve

optimal BOC performance. The PPKTP crystal has a reflective dichroic coating on its rear facet. The

input beams must be focused exactly on this rear facet of the PPKTP crystal to ensure symmetry for the

forward and reverse pass. Mounting the focusing lens on a translation stage will help to fine tune the

focus position to achieve a balanced (antisymmetric) BOC curve. Also, DC voltage offset tuning in the


55

PI controller is critical after the PLL is locked because the operating point for AM noise cancellation may

not be exactly at 0V. Experimentally, it was necessary to tune the offset voltage by tens of mV to achieve

the best AM noise cancellation. Lastly, the input beams should be as collinear as possible to achieve high

SH efficiency. To aid in alignment, it is helpful to first couple the input beams into the same fiber

collimator placed equidistant to the PPKTP crystal and maximize the coupling efficiency for each beam.

Fiber-Coupled Integrated BOC

An integrated waveguide version of the PPKTP crystal can achieve higher timing sensitivities at

lower power levels compared to bulk-crystal BOCs. This is because the waveguide offers tighter mode

confinement for higher SH conversion efficiency. The integrated BOC offers many benefits for the

timing system at large. First, the integrated device can be fiber-coupled to enable an all-fiber

implementation of the TDS, which is easier to implement and maintain compared to free-space optics.

System failure modes can be easily diagnosed and rapidly repaired for quick system start-up times.

Second, since the BOCs no longer require high input power, the link operation power can be lowered to

support more links for a fixed output power from the master laser. Furthermore, the low link power will

reduce the nonlinearity-induced timing errors.

A fiber-coupled, integrated PPKTP waveguide was developed by Callahan et al. [48] in

collaboration with AdVR and used to realize an all-fiber implementation of the timing system [49]. The

fiber-coupled BOC shows an improvement by a factor of 20 over a bulk-optic BOC. However, due to

excessive uncompensated fiber (between the PPKTP and the BPD), temperature and humidity effects

limited the timing drift of the system to the 1 fs – 10 fs level for over 40 hours of operation. Further

development is needed to integrate the WDM coupler and detectors on-chip with the PPKTP waveguide

to realize a true fully-integrated BOC with negligible uncompensated path lengths.

2.4.7 Noise Model

PLL noise models are critical for understanding and debugging the link stabilization and optical-

optical synchronization demonstrations. The models developed in this thesis are presented in Appendix B

and Appendix C.


56

2.5 Conclusion and Future Work

Long-term stable optical timing distribution and synchronization has been demonstrated at the

sub-fs level, which is an order-of-magnitude improvement over the first-generation TDS. In the 4.7-km

fiber network stabilization demonstration, the relative timing stability between the link outputs for over

52 hours of continuous operation is only 0.58 fs RMS, compared to the 6.4 fs RMS over 72 hours for the

first-generation TDS based on the 300-m SMF link [3]. In the remote laser-laser demonstration, the

remote laser is synchronized to the optical master oscillator across the 3.5 km link with less than 0.2 fs

RMS residual jitter for over 44 hours of continuous operation, compared to the 8 fs pk-pk over 3.5 hours

across the 300-m SMF link [41]. Key developments that enable sub-fs stability are the dispersion-

compensated PM fiber links, improved SNR in the BOC (via EDFA and improved polarization extinction

ratio), active stabilization of the link power, and noise simulations in the fiber link.

Progress towards the sub-100-as performance level will involve solving a range of technical

challenges. A deeper understanding of pulse propagation effects at a fundamental level in the fiber link

need to be developed and verified. Feed-forward compensation schemes, e.g. Smith predictor, should be

explored to mitigate the time-delay latency in the feedback loop to increase the locking bandwidth. An

all-fiber implementation with tight temperature and humidity control should be explored. With decreased

link operating powers from using integrated BOCs, the nonlinearity-induced timing errors may become

negligible, but remains to be confirmed. Temperature-dependent birefringence effects in PM fiber and its

impact on timing need to be studied. The PLL noise model for the feedback control loops need to include

higher-order effects to accurately predict the system noise under tight locking, especially in the 1 - 10 kHz

range. Lastly, in the sub-100-as regime, every physical aspect of the TDS must be re-considered for

robustness against environmental perturbations since 0.1 fs corresponds to 0.03 μm.

Chapter 3. Optical-RF Synchronization

57

3 Optical-RF Synchronization

3.1 Introduction

In addition to optical-to-optical synchronization, it remains equally important to develop

techniques to preserve the ultralow-noise optical timing signals into the RF domain for optical-to-RF

synchronization. Over the past few years, ultralow-noise microwave generation from optical frequency

combs [29], [50]–[53] have achieved such a high level of stability that it has become imperative to

re-evaluate the methods for optical-to-RF conversion. Conventional methods using direct detection in

standard photodiodes cannot achieve sub-fs performance due to nonlinearities at photodetection. Under

high power illumination, the detector is subject to, for example, space-charge effect, which limits the

output RF power and increases the amplitude-to-phase (AM-PM) conversion noise [22], [54]. Advanced

detector designs can mitigate these effects to a certain extent. For example, unitraveling-carrier diodes

can achieve high linearity at output powers approaching the 1-W level [55] and low AM-PM conversion

[56]. However, the conditions for the latter are highly sensitive to the average photocurrent and bias

voltage [23], [56], [57], thus imposing difficulty for reliable operation in real systems where significant

power drifts may occur. Various compensation techniques [58], [59] can improve direct detection, but the

sensitivity of the AM-PM suppression remains too large for practical applications.

To circumvent these issues, an alternative approach for optical-RF conversion is pursued here. A

balanced optical-microwave phase detector (BOMPD) is developed for use within an optoelectronic PLL

[60], [61]. This PLL approach has two advantages. First, the BOMPD can convert the phase error

between an optical pulse train and RF signal into an amplitude-modulated (AM) signal directly in the

optical domain. This AM signal can be detected via direct detection without concern for AM-PM

conversion. Second, the BOMPD derives its high phase sensitivity from the optical modulation depth

more so than the absolute optical power at detection. Therefore, the BOMPD approach does not require a

dynamic range as large as that for the direct detection approach to achieve the same level of

synchronization.

Few groups have modified the basic BOMPD scheme with nonreciprocal biasing and balanced

detection to improve the BOMPD noise floor down to -154 dBc/Hz and performed optical-RF extraction

with sub-fs absolute timing jitter [62] as well as verified up to 60 dB of AM-PM suppression [63].


58

However, this modified scheme requires additional components that break the inherent symmetry of the

Sagnac loop, thus degrading its long-term stability.

For this reason, the basic BOMPD scheme is developed further in this thesis. While the first-

generation BOMPDs were limited to stability levels in the sub-10-fs regime [3], the improved BOMPDs

can achieve optical-to-RF (and RF-to-optical) synchronization with long-term-stable residual timing jitter

in the sub-fs regime as well as a robust AM-PM suppression ratio greater than 50 dB. This confirms the

feasibility of BOMPDs for sub-fs synchronization in a large-scale TDS for next-generation light sources.


Optical-RF synchronization is performed using an optoelectronic PLL with a BOMPD as its

phase detector – see Figure 3-1. Its principle of operation is best understood first at the system level as a

PLL, then at the component level for the BOMPD.

3.2.1 Optoelectronic Phase-Locked Loop

The PLL begins with two input signal sources: a mode-locked laser (MLL), which generates an

optical pulse train with repetition rate fr, and a voltage-controlled oscillator (VCO), which generates a

single RF frequency near a multiple of the repetition rate ~Nfr. As a black box model, the BOMPD

accepts the two input signals and generates a voltage error signal ΔV that corresponds to the phase error

Δθ between the input RF zero-crossings and pulse positions. When Δθ is small, the output error signal is

directly proportional to the input phase error according to ΔV = KPD∙Δθ, where KPD is the BOMPD phase

Figure 3-1 Operating principle for a BOMPD in an optoelectronic PLL to synchronize a

VCO to a MLL. Assume N=1 for simplicity.


59

sensitivity in units of V/rad. The error signal is sent to a PI loop filter and fed back to the VCO frequency

tuning port to complete the PLL. While this feedback configuration locks the VCO to the laser, the error

signal can alternatively be fed back to the laser repetition rate to lock the laser to the VCO. In both cases,

the PLL minimizes the phase error signal to zero. The loop filter is optimized to maximize the loop

bandwidth and in-loop noise suppression for tight synchronization.

3.2.2 Balanced Optical-Microwave Phase Detector

Internal to the BOMPD is a fiber Sagnac interferometer (SGI), which consists of a 50:50 coupler,

fiber loop, and a unidirectional electro-optic phase modulator. Its operating principle is based on electro-

optic sampling in the time domain – see Figure 3-2:

Figure 3-2 Operating principle for electro-optic sampling in the BOMPD under various

modulator biasing conditions. (a) No RF signal applied; (b) Reference bias signal applied for

alternating quadrature bias; (c) VCO signal applied in addition to quadrature bias. Yellow

dots indicate electro-optic sampling points.

Consider an input pulse, which is first split into two counter-propagating sub-pulses in the fiber

loop. When no RF signal is applied to the modulator, the sub-pulses will accumulate a net zero

differential phase δϕ after one round-trip due to the loop symmetry. The sub-pulses will interfere upon

exiting the loop and, according to the SGI transmission function POUT = PIN sin2(δϕ/2), result in zero

transmission of the input pulse – see Figure 3-2(a). This loop symmetry is crucial because it passively

biases the SGI at a fixed point in its transmission function, providing the BOMPD its inherent robustness

against long-term drifts.

A reference signal is then applied to bias the SGI at quadrature for phase detection. Specifically,

the SGI is biased at alternating quadrature points with opposite slope polarity – see Figure 3-2(b). While

the quadrature bias maximizes the BOMPD phase sensitivity, the alternating polarity generates a bipolar


60

voltage zero-crossing in the error signal necessary for phase-locking. The reference signal is derived

from the input optical signal: the pulse train is detected, filtered at the Mth odd harmonic, and frequency-

divided to a half-repetition-rate frequency Mfr/2. The signal is then time-delayed and amplified so that

the clockwise sub-pulses, when passing through the modulator, will align with and electro-optically

sample the peak signal voltages to acquire alternating ±π/2 phase shifts. The counterclockwise sub-pulses

will acquire zero phase shifts in the modulator due to unidirectional phase modulation. The resulting net

δϕ = ±π/2 phase shifts will therefore achieve alternating quadrature bias. The output pulse train contains

zero amplitude modulation and serves as the reference level for the subsequent phase error detection.

The VCO signal is then added to detect its phase error relative to the pulse train. The pulses will

sample the VCO signal at a rate equal to δf = mod(~Nfr, fr) due to aliasing – see Figure 3-2(c). Contrary

to quadrature biasing, the phase shifts acquired here have the same polarity for adjacent pulses; e.g. a

slow positive (or negative) phase error drift will perturb the bias points to the right (or left), resulting in

amplitude modulation of the output pulse train with the respective polarity. Upon detection, the

modulated pulse train will contain error sidebands centered at the half-repetition rate frequencies. One set

of sidebands is filtered and down-mixed in-phase with another reference signal to generate the baseband

voltage error signal.

3.3 Experimental Demonstration

The second and third generation BOMPDs are developed and presented in this thesis. The second

generation BOMPD is developed for compatibility with an integrated BOMPD (see Section 3.5.1).

Although technical limitations prevented low noise performance with the integrated BOMPD, the fiber-

based BOMPD is able to achieve 1-fs-level noise performance up to 1 MHz locking bandwidths. The

third generation BOMPD is a hybrid optimization of the first and second generation BOMPDs. Signal

paths are optimized for high- and low-frequency operation as well as AM/PM sensitivity to deliver sub-fs

noise performance up to multi-MHz bandwidth capabilities. These BOMPDs are used to demonstrate

both scenarios where optical-RF synchronization is needed in a FEL facility: 1) base-station locking,

where the master laser is synchronized to a local stable RF standard, and 2) remote-station locking, where

remote RF oscillators are synchronized to the distributed timing signal.


61

3.3.1 Second-Generation BOMPD

Key design considerations and improvements over the first-generation BOMPDs are summarized

here (see Section 3.4 for details). First, the reference signal for quadrature bias is increased into the multi-

GHz (6.3 GHz) regime to achieve unidirectional phase modulation in the phase modulator. The SGI

becomes repetition-rate independent since the arrival time of counterclockwise sub-pulse in the modulator

is no longer important, rendering the SGI more robust against imperfect loop length as well as

environmental noise. Second, the BOMPD is designed such that the only PM-sensitive path is the VCO

signal path. All other signal paths, i.e. reference signal generation, quadrature biasing, and error signal

detection, are AM-sensitive. The electronics for these signal paths are optimized accordingly. Third, the

VCO signal is amplified beyond Vπ of the phase modulator to increase its zero-crossing slope to

maximize KPD. Although this generates higher-order nonlinearities in the error signal, they can be ignored

because the PLL locks only to the zero-crossing where linearity is preserved. Moreover, the resultant null

error signal when the PLL is locked poses no risk for saturation or nonlinearities in the detection

electronics. This is fundamentally different from conventional RF phase detection where the signal power

at the RF mixer is always present and competes with the LO power, resulting in possible mixer

nonlinearities.

Experimental Set-up

The experimental set-up for remote-station locking using the second-generation BOMPDs is

shown in Figure 3-3. The input pulse train is generated from the same Er-doped fiber mode-locked laser

used in the 1.2-km link demonstration (see Section 2.3.2). The pulses are split into the reference path and

main SGI path. The reference path contains a high-speed 14-GHz photodiode (Discovery; DSC40S),

bandpass filter (TTE) for the 63rd

harmonic of the pulse train, and divide-by-2 digital frequency divider

(Hittite; HMC492LP3) to generate a reference signal at 6.3 GHz. The optical power to the reference path

is limited to 0 dBm to avoid detector saturation. Low noise amplifiers, variable RF attenuators, and phase

shifters are employed as necessary to achieve quadrature bias. The main SGI path is constructed using

PM fiber components, namely a 50:50 coupler (AFW) and 10-GHz phase modulator (EOSpace). DCF is

essential for minimizing the pulse width at the modulator input for high-accuracy electro-optic sampling.

With the BOMPD biased at quadrature, the VCO (PSI; DRO-10.225-FR), which is to be phase-

locked to the 51st harmonic (10.220 GHz) of the pulse train, is then added to the phase modulator using a


62

RF diplexer. To maximize the BOMPD phase sensitivity, the VCO signal is amplified with a low phase

noise amplifier (Microsemi; AML1011PNB3001) to drive the modulator as hard as possible at +24 dBm.

An additional noise filter is necessary after amplification to suppress the amplified broadband noise, since

this noise will alias down to baseband during electro-optic sampling and degrade the BOMPD noise floor.

At the SGI output, the modulated pulse train is detected with -3 dBm of power on a high-speed

photodiode (Discovery Semi; DSC50S), filtered at 6.3 GHz (K&L; 6C60-6360/T100-O/O), amplified and

down-converted in-phase with the reference signal in a double-balanced mixer to generate the baseband

error signal. The optical power and RF gain in the signal detection path is set to maximize the SNR and

linearity. A PI2D controller (Vescent; D2-125) feeds the error signal back to the VCO tuning port to close

the PLL with a 100-kHz locking bandwidth.

To characterize the PLL performance, a second nearly-identical BOMPD is implemented for an

out-of-loop measurement. The out-of-loop BOMPD is configured with a lower noise floor than the in-

loop BOMPD. Low-noise DC preamps with 36-dB gain are used to amplify the error signals above the

measurement noise floor. The in-loop and out-of-loop BOMPD sensitivities measured after the DC

preamps are 4.4 mV/fs and 4.8 mV/fs, respectively, referred to 10.220 GHz. For long-term drift

Figure 3-3 Experimental set-up for remote-station locking with 2nd

-generation BOMPDs.

Second BOMPD is for an out-of-loop residual phase error measurement. (BPF, bandpass

filter; LNA, low noise amplifier; DIV, frequency divider; LPF, lowpass filter, LPNA, low

phase noise amplifier; DCF, dispersion-compensation fiber; ϕM, phase modulator; DC,

DC preamplifier)


63

measurements, the out-of-loop error signal is low-pass filtered at 0.5 Hz and sampled at 1 Hz with a DAQ

card. Temperature changes at various locations are also recorded. For short-term jitter measurements, the

voltage PSD of the error signals are measured with a spectrum analyzer in the 1 Hz – 1 MHz range and

divided by KPD2 to convert to the single sideband (SSB) phase noise spectra.

Remote-Station Locking

For remote-station locking, the VCO is locked to a mode-locked laser that has lower noise

performance. The timing link is omitted here since it can be achieved in principle, as demonstrated by the

remote laser-laser synchronization in Section 2.3.5. Short-term stability measurements are shown in

Figure 3-4. The free-running absolute phase noise spectra of the signal sources are shown in black for

reference. The VCO phase noise is obtained via loose-locking with a second identical VCO in a dual-

oscillator phase noise measurement set-up. The laser phase noise is obtained via direct detection and

measurement of the repetition rate harmonic at 10.220 GHz with a SSA. The residual phase error

Figure 3-4 Measurement results for short-term remote-station locking with 2nd

-

generation BOMPDs. (a) Residual phase error referred to 10.220 GHz and (b) its

corresponding timing jitter integrated from 1 Hz. Out-of-loop jitter is <2 fs for a

100-kHz locking bandwidth.


64

measured by the in-loop and out-of-loop BOMPDs while the PLL is locked are shown in blue and red,

respectively. The BOMPD noise floor is characterized to assess the BOMPD’s limitations for phase

detection independently from the PLL’s limitations for the overall optical-RF synchronization. The latter

is highly dependent on the signal sources and PLL locking parameters, which are external to the BOMPD.

With the improved BOMPDs, optical-to-RF synchronization is for the first time clearly limited by

the detection noise floor in the in-loop BOMPD (see green curve). The detection noise floor contains two

dominant power-law noise processes. The white noise floor above 100 Hz represents thermal noise at

detection, while the f -2

behavior below 100 Hz represents colored noise from the electronics. Exceptions

to these trends are noise spurs from ground loops (multiples of 60 Hz), specific RF components (200-

600 Hz) and vibrations (near 6 Hz). The corresponding integrated timing jitter is shown at the bottom of

Figure 3-4. The noise floor shows capability for sub-fs synchronization up to a 300-kHz locking

bandwidth. Even including the measured noise spurs from the in-loop curve (red), which total 0.6 fs

RMS between 1 Hz and 1 kHz, the sub-fs range for the BOMPD noise floor is degraded to only 200 kHz.

This bandwidth is sufficient for synchronizing most high-end commercial oscillators.

For this demonstration, the VCO is selected with relatively high phase noise to evaluate the

BOMPD performance over large locking bandwidths. The out-of-loop timing jitter integrated from 1 Hz

to the 100-kHz locking bandwidth is just below 2 fs RMS and is limited by the VCO noise in the 30 kHz

– 300 kHz range rather than the BOMPD noise floor. The noise performance can easily be scaled into the

sub-fs regime by using a lower-noise VCO (see Section 3.3.2). Another option is to optimize the PLL

phase margin to extend the locking bandwidth; however, this is challenging due to the presence of higher-

order zeroes and poles in the detection electronics that generate an instable resonance near 1 MHz.

The thermal noise floor at detection is not yet a fundamental limit for the BOMPD. A figure of

merit for the BOMPD is the SNR at detection rather than the absolute noise floor. In phase noise

measurements, the effective noise floor can be suppressed by increasing the signal power, or equivalently

KPD. This is demonstrated with the third-generation BOMPDs in Section 3.3.2.

Long-term stability is achieved with 1.0 fs RMS and < 7 fs pk-pk drift for over 10 hours of

continuous operation and 0.8 fs RMS over the first 6 hours – see Figure 3-5(a). The out-of-loop drift

exhibits a correlation of 0.5 with the ambient temperature fluctuations – see Figure 3-5(b). Despite these

fluctuations, this is still a factor of 5 improvement compared to the previous stability limit of 5.18 fs RMS

for the first-generation BOMPDs [61]. These results are comparable to that of FLOM-PDs recently

reported in [62]; however, their long-term drift is caused in the optical domain by coupling ratio drifts,


65

beam drifts, and imbalanced attenuation in the modified SGI. The simplified SGI demonstrated here is

insensitive to these types of drifts. Instead, drift is caused by environmental perturbations to the

mechanical stability of the electronics; vibrations from heat sink fans, mechanical stress, and thermally-

induced phase drifts in the VCO input signal path (see Section 3.4.2). Localized heating of specific signal

paths is performed to confirm the high correlation between timing drift, temperature and the VCO path.

Preventative measures against temperature- and vibration-induced drifts are taken, such as replacing RF

cables with rigid connectors, securing components to the breadboard, and placing lead foam beneath the

set-up. However, due to the large set-up and overheating issues with the RF power amplifiers, no isolated

temperature enclosure is used. A strong noise component is recorded near 4.5 mHz with an RMS value of

0.5 fs – see inset of see Figure 3-5(a). Ground loop noise and electro-static discharge spikes are suspected

of leaking past the 1-Hz LPF through the grounding and aliasing down to 4.5 mHz by the DAQ card; i.e.

the sampling rate and power line noise are not exact integer multiples of each other. This noise can be

eliminated with proper galvanic isolation of the electronics (see Section 3.3.2).

Figure 3-5 Measurement results for long-term remote-station locking with 2nd

-generation

BOMPDs. (a) Out-of-loop drift referred to 10.220 GHz and (b) relative temperature changes.

Out-of-loop measurement shows 1 fs RMS drift (< 7 fs pk-pk) over 10 hours of operation

and 0.8 fs RMS drift over the first 6 hours.


66

Base-Station Locking

For base-station locking, the signal sources are exchanged for a lower noise mode-locked laser

(OneFive; Origami) and Sapphire-loaded cavity oscillator (SLCO) (Raytheon; SLCO-10.833-NCS)

whose 10.833-GHz center frequency is a multiple of the 216.667-MHz laser repetition rate. The feedback

is reconfigured to lock the laser to the SLCO. Moreover, the RF power amplifier in the reference path is

upgraded to one with superior AM noise performance (CiaoWireless; CA67-345-LP-HB). The remaining

set-up is left unmodified. The in-loop and out-of-loop BOMPD sensitivities are 6.0 mV/fs and 5.2 mV/fs,

respectively, referred to 10.833 GHz. Under tight synchronization, the residual phase error spectra in

Figure 3-6 shows a loop bandwidth limited to 20 kHz as well as a moderate servo resonance. These arise

from the laser PZT resonance close in at 30 kHz, which degrades the PLL phase margin. Despite this, the

out-of-loop integrated jitter is only 0.5 fs from 1 Hz to 20 kHz, which is in the sub-fs regime. Due to

higher signal power available from this laser, the improved noise floor extends sub-fs synchronization

capabilities beyond 1 MHz. The PLL performance is again limited by the in-loop detection noise floor.

The new low-frequency behavior obeys the f -1

flicker noise of the replacement amplifier.

Figure 3-6 Measurement results for short-term base-station locking wtih 2nd

-generation

BOMPDs. (a) Residual phase error referred to 10.833 GHz and (b) its corresponding

timing jitter integrated from 1 Hz. Out-of-loop jitter is 0.5 fs for a 20-kHz locking

bandwidth. Noise floor data below 10 Hz is discarded due to data acquisition error.


67

3.3.2 Third-Generation BOMPD

The BOMPD noise floor can be suppressed further by increasing the SNR at photodetection.

While a high-frequency reference signal is advantageous for repetition-rate independence in the SGI, it is

disadvantageous for error signal detection since the detector responsivity and saturation power are lower

at high frequencies (See Section 3.4.4). The 6.3-GHz detection electronics are replaced to perform down-

conversion at 108.333 MHz, the lowest half-repetition-rate possible. To maximize the RF signal power,

the power incident on the detector is increased to +5 dBm, which is near saturation at 108.333 MHz. The

detection electronics begins with a 50Ω terminated photodiode (EOTech, 3500F), followed by a DC block

and LPF serving as a large-bandwidth BPF, and a cascade of LNAs. The reference signal generation path

employs similar electronics in addition to a frequency divider. Path length matching is attempted to

minimize thermal expansion effects in the VCO signal path (see Section 3.4.2).

Figure 3-7 Experimental set-up for remote-station locking with 3rd

-generation BOMPDs.

Additional reference signal generation path is introduced to perform error signal detection

at 108 MHz, the lowest possible frequency.

Noise Floor Characterization

The experimental set-up is first optimized to characterize the minimum noise floor achievable for

a single BOMPD. For this measurement, the VCO signal is removed from the RF input port. The

detection noise floor, which is set by the optical input and additive noise of the electronics, is measured at


68

the BOMPD output with a spectrum analyzer. The VCO signal is temporarily applied to the RF input to

obtain KPD to convert the measured noise floor in units of V2/Hz to SSB phase noise in units of dBc/Hz.

The expected improvement in SNR from switching to low-frequency detection is verified in

Figure 3-8, which reveals a measured -161 dBc/Hz detection noise floor referred to 10.833 GHz. This is

a 25 dB improvement over that achieved with the first-generation BOMPD [3]. The integrated jitter up to

1 MHz is just below 0.2 fs, which shows great potential for sub-fs synchronization over large locking

bandwidths. Few noise spurs exist, such as 60-Hz power line noise, a 500-Hz spur due to particular

electronics and a drifting tone near 700-800 Hz due to parasitic pick-up of the SLCO signal transmitted

through air. Their contribution to the overall jitter is, however, negligible. The white noise floor

represents shot noise from the higher photocurrent. It can be improved further by using a high-power

linear diode [55] to increase the RF signal power at photodetection by +20 dB, which would yield a noise

floor below -180 dBc/Hz. Long-term stability is also expected to improve with lower detection frequency

since it decreases the BOMPD’s sensitivity to temperature-induced phase shifts (see Section 3.4.2). The

f -3

noise represents noise from the low-frequency detection electronics.

Figure 3-8 Improved noise floor in 3rd

-generation BOMPDs using low-frequency detection

electronics for increased signal power. The -161 dBc/Hz noise floor yields a sub-fs noise floor

for bandwidths beyond 1 MHz.

Remote-Station Locking

For remote-station locking with the third-generation BOMPDs, the SLCO is used as the remote

oscillator. Since the remote VCO has lower noise than the optical timing signal, optical-RF

synchronization will lock the VCO up to the mode-locked laser noise. Although the VCO noise will

increase, it remains tightly synchronized to the laser with sub-fs precision.


69

Short-term stability measurements are shown in Figure 3-9. The free-running absolute phase

noise spectra of the signal sources are shown in black for reference. The SLCO phase noise is obtained

via a loose phase-lock with a second identical SLCO in a dual-oscillator phase noise measurement set-up

with interferometric carrier suppression [64]. The laser phase noise is obtained via direct detection and

measurement of the 10.833 GHz harmonic with a SSA. The residual phase error measured by the in-loop

and out-of-loop BOMPDs while the PLL is locked are shown in blue and red, respectively. Below the

loop bandwidth, the in-loop noise suppression shows that the SLCO is tightly synchronized to the laser.

The integrated jitter from 1 Hz to the 20-kHz loop bandwidth is about 0.14 fs RMS, which is in the sub-fs

regime. The loop response near the locking bandwidth is well-behaved due to good phase margin. At

higher frequencies, the free-running phase noise of the laser is observed until the BOMPD noise floor is

reached. The BOMPD noise floor is limited by shot noise at photodetection and by colored noise from

the detection electronics below 30 Hz. The noise floor is higher at -152 dBc/Hz because the limited

power from the signal sources is shared to operate two BOMPDs rather than one, which decreases KPD.

Figure 3-9 Measurement results for short-term remote-station locking with 3rd

-

generation BOMPDs. (a) Residual phase error referred to 10.833 GHz and (b) its

corresponding timing jitter integrated from 1 Hz. Out-of-loop jitter is 0.14 fs for a 20-

kHz locking bandwidth.


70

Long-term stability is maintained for over 24 hours of continuous operation with 1 fs RMS drift

and 6 fs pk-pk deviation. During a 10-hour subset (between Hours 7.5 and 17.5) the drift is only 0.9 fs

RMS. These are conservative estimates because the digitizing measurement noise of the digitizer

contributes up to 0.25 fs RMS. Combining this long-term performance with the short-term stability and

accounting for the digitization noise, the total integrated RMS jitter from 12 μHz to the 20-kHz locking

bandwidth is 0.98 fs RMS, which confirms long-term stable sub-fs optical-RF synchronization. Galvanic

electrical isolation of the BOMPD is critical for eliminating the electronic noise seen previously in Figure

3-5(a) to achieve sub-fs stability.

Figure 3-10 Measurement results for long-term remote-station locking with 3rd

-generation

BOMPDs. Out-of-loop drift is < 1 fs RMS drift over 24 hours.

Base-Station Locking

Similarly, the RF-to-optical synchronization results are shown in Figure 3-11. Due to laser PZT

resonance, the locking bandwidth is optimized near 20 kHz to maximize the locking bandwidth while

limiting the servo resonance to achieve sub-fs synchronization. The integrated jitter from 1 Hz to the

20-kHz locking bandwidth is about 0.5 fs RMS.


71

Figure 3-11 Measurement results for short-term base-station locking with 3rd

-generation

BOMPDs. (a) Short-term residual phase error referred to 10.833 GHz, (b) corresponding

integrated timing jitter starting from 1 Hz. Out-of-loop jitter is 0.5 fs for a 20-kHz locking

bandwidth.

3.4 Design Considerations

3.4.1 AM-PM Suppression

The AM-PM suppression ratio is a figure of merit for the BOMPD. It is defined as the

suppression of input amplitude noise to the output phase noise. To measure this, the optical input to the

BOMPD is amplitude-modulated with an electro-optic modulator while the resultant phase modulation at

the BOMPD output is monitored. The amplitude modulator is driven by a RF synthesizer swept from

100 Hz to 1 MHz. The input RIN spectrum in units of dBc/Hz is subtracted from the output phase noise

spectrum in units of dBrad2/Hz to calculate the AM-PM suppression ratio. This ratio can also be reported

as α in linear units of rad/(ΔI/I). Contrary to traditional AM-PM conversion processes, the AM-PM

conversion in the BOMPD is an AM-AM conversion process; i.e. the AM-AM leakage of input AM noise

to the output AM noise is measured and converted to the AM-PM conversion coefficient using KPD.


72

Measurements are performed with the PLL in both the on and off states. In the “PLL off” state,

the VCO signal is removed during measurement but is applied temporarily to measure KPD for a given

input VCO power. The measured AM-PM suppression is shown in Figure 3-12. It exhibits a wideband

flat response of -50 dB, which is equivalent to α = 0.003 rad. The measurement is repeated with the PLL

on, where the VCO is applied and phase-locked to the laser with a 10-kHz locking bandwidth (see blue

curves). This technique is similar to that conventionally used for characterizing the closed-loop response

of a PLL. For frequencies beyond the locking bandwidth (>20kHz), the PLL response behaves as a unity-

gain high-pass filter. With fine adjustment of the DC offset on the order of a few mV in the loop filter,

the AM-PM suppression ratio can be improved beyond 50 dB. This is because the DC offset

compensates for amplitude imperfections in the SGI, which may result from AM noise in the quadrature

biasing and finite polarization extinction ratio.

To investigate the source of AM-PM conversion, the reference and main SGI paths are modulated

independently. Modulation of the reference path yields an AM-PM suppression that is limited by the

measurement noise floor, which is greater than 90 dB. This is expected because the digital frequency

divider used for reference signal generation is insensitive to AM noise at its input and is able to generate a

low AM noise waveform at its output. These effects compound favorably to decouple the input AM noise

from the output PM noise. Modulation of the SGI path, however, reproduced the 50-dB suppression.

This confirms that the AM-PM conversion is limited by optical AM-AM leakage in the SGI.

The AM-PM suppression ratio is robust with respect to large changes in optical power. The AM-

AM leakage is fixed because it originates from fixed optical power attenuation from the imperfect

Figure 3-12 Measuremed AM-PM suppression ratios with the PLL unlocked (red) and

locked (blue). BOMPD exhibits a nominal, wideband AM-PM suppression of -50 dB with

further improvement via DC offset adjustment.


73

construction of the SGI. For example, finite polarization extinction ratio from fiber splices within the SGI

may degrade the overall extinction ratio of the SGI transmission function and allow input AM noise to

leak through to the SGI output. The effective AM-PM suppression ratio therefore changes as a linear

function of optical power. This was verified experimentally by varying the optical power over several

dBs and observing commensurate changes in the AM-PM suppression ratio from its nominal 50-dB value.

This is in contrast to direct detection techniques, such as in [56], [65], where the minimum AM-PM

suppression exists only at localized points for select bias voltages and average photocurrents. Local

minima values for the suppression ratio may vary widely from -45 to +6 dB and can only be maintained if

power fluctuations remain within a small percentage. The BOMPD is a more robust method for low AM-

PM conversion in FEL facilities where the optical power may vary greatly throughout the TDS.

Future work will be to investigate the fundamental limit for this leakage and improve the

suppression ratio. Although the nominal 50-dB suppression ratio is currently 10 dB lower than that in

[63], the immediate benefit of the simplified BOMPD is the improvement in long-term stability due to the

increased robustness of the simplified SGI. Both BOMPDs still offer a suppression ratio that is

significantly higher and more robust than that of direct detection.

3.4.2 Temperature / AM-PM sensitivity

Temperature is a dominant noise source that degrades long-term stability. Signal paths in the

optical and RF domains are subject to different thermal expansion effects. Practical design constraints for

critical system paths due to thermal expansion are analyzed here.

The thermal expansion rate between optical fiber and RF coaxial cable may be orders-of-

magnitude apart. Optical fiber has a relatively low thermal coefficient of expansion at α = 0.55 μm/m∙K

assuming a SiO2 core. It is more useful to divide this value by the propagation velocity v to obtain the

coefficient of group delay αg= α/v= 2.75 fs/m∙K. Measured delay coefficients of various fiber types are

higher in the 3.2 to 41 fs/m∙K range [42] due to additional thermal mismatch between protective coating

layers in the fiber construction. RF cables exhibit orders-of-magnitude larger thermal sensitivities. RF

coaxial cables are typically constructed with a copper conductor core and Teflon (PTFE) dielectric

insulator. While the delay coefficients for the individual materials are 82.5 fs/m∙K and 675 fs/m∙K,

respectively, the overall delay coefficient for the cable is highly variable depending on the cable material,

geometry, and operating temperature. The coefficients of various coaxial cables, as measured in [66],


74

[67] show that a typical delay coefficient of 350 fs/m∙K near room temperature can be expected, but may

range as high as 700 fs/m∙K and as low as 0 fs/m∙K as a function of temperature due to molecular

transitions in PTFE.

Long-term stability in the BOMPD against temperature, acoustics, and mechanical stress is best

maintained by using rigid RF connectors rather than flexible coaxial cables to adjust path lengths.

Rugged variable phase shifters may work for adjusting the high-frequency quadrature bias (6.3GHz)

signal since a 180° tunable phase delay range can be achieved easily with a short length adjustment. For

longer signal wavelengths (108 MHz), rigid RF connectors should be used. The minimum path

adjustment in this case is therefore the length a 2-cm RF connector. Assuming a delay coefficient of

350 fs/m∙K near room temperature with 1K fluctuation, the absolute timing instability of one connector is

already at 7 fs. Fortunately, however, the BOMPD performance is limited by the relative timing between

pairs of signal paths rather than their absolute timing, which relaxes the temperature requirements. In the

following, the phase sensitivities of three primary signal pairs are analyzed.

VCO Input vs Optical Input

The only signal pair that is phase-sensitive in the BOMPD is the signal paths from the input ports

of the BOMPD to the input ports of the phase modulator where electro-optic sampling occurs. This is

because the optical pulses sample the zero-crossing of the RF signal, where PM sensitivity is at its

maximum. The minimum practical length for the RF path is about 10 cm due to RF cabling and the

diplexer size. Assuming αg,RF = 350 fs/m∙K, a 1K fluctuation will change the delay of this RF path by

about 35 fs. The optical path length for the clockwise propagating pulses is about 1 m due to splicing

constraints and DCF for pulse compression. Note that the optical path length for the counter-propagating

pulses is not included because they do not participate in electro-optic sampling due to unidirectional

phase modulation. Assuming αg,fiber = 41 fs/m∙K, the same 1K fluctuation will change the optical delay by

only 2.8 fs. The relative timing between the optical and RF path is about 32 fs, which is unacceptable for

sub-fs stability.

To minimize this differential temperature drift, the RF and optical path lengths must be carefully

designed to match their thermal expansion rates. Since the thermal coefficient for the RF path is

significantly higher, the absolute length of the RF path length should be minimized (about 10 cm). The

fiber length should then be increased to match the thermal expansion rates such that the timing instability

remains below 1 fs:


75

|𝛼𝑔,𝑟𝑓𝐿𝑟𝑓 + 𝛼𝑔,𝑓𝑖𝑏𝑒𝑟𝐿𝑓𝑖𝑏𝑒𝑟|𝛥𝑇 < 1 𝑓𝑠

A 10-cm RF path with 1K fluctuations will require a fiber length between 1.28m ± 24mm which is an

unrealistically tight tolerance.

To achieve a practical system, additional temperature compensation techniques must be employed

to increase this tolerance. First, cables with Teflon (PTFE) dielectric material should be avoided since

they exhibit large temperature coefficient swings around room temperature due to the molecular transition

behavior of PTFE. Specialty cables with near-zero delay coefficients exist commercially. However, if

PTFE cables must be used due to their low-cost and low-loss, its operating temperature can be raised

above the molecular transition of PTFE (>24°C) to approach a stable, low thermal coefficient in the 0 to

50 fs/m∙K range and stabilized within ±0.1K to reduce temperature drifts below 1 fs. Using these

improved values, the tolerance range on fiber length is relaxed to ±24 cm, which is reasonable. In the

experiments, temperature stabilization is achieved as a by-product of heat dissipation from the RF power

amplifier operating near 40°C. Although a temperature gradient exists in the RF cable, the stability of the

gradient is maintained during the long-term measurements to achieve sub-fs stability.

Quadrature Bias vs Optical Input

Another signal pair is the quadrature bias signal (from the optical input to the RF port of the

phase modulator) and optical input (from the optical input to the optical port of the phase modulator). For

ideal quadrature biasing, the extrema of the bias signal are aligned in-phase (Δϕquad = 0) with the incoming

pulse positions in the phase modulator and adjusted in amplitude (Vrf = Vπ/2) to achieve alternating

quadrature bias. Consider a single pulse that is aligned with the maximum of the quadrature bias signal.

𝛷𝑃𝑀 = 𝜋 (𝑉𝑟𝑓

𝑉𝜋

) 𝑐𝑜𝑠(𝛥𝜙𝑞𝑢𝑎𝑑) = +𝜋

2

Thermal expansion between these paths will introduce an additional relative phase delay

𝛥𝜙𝑡ℎ𝑒𝑟𝑚 = 2𝜋𝑓𝑟𝑓[(𝛼𝑔,𝑟𝑓𝐿𝑟𝑓 + 𝛼𝑔,𝑓𝑖𝑏𝑒𝑟𝐿𝑓𝑖𝑏𝑒𝑟2) − (𝛼𝑔,𝑓𝑖𝑏𝑒𝑟𝐿𝑓𝑖𝑏𝑒𝑟1)]𝛥𝑇

that degrades the quadrature bias

𝛷𝑃𝑀 = +𝜋

2𝑐𝑜𝑠(𝛥𝜙𝑞𝑢𝑎𝑑 + 𝛥𝜙𝑡ℎ𝑒𝑟𝑚)


76

Due to the cosine function’s insensitivity to phase at its maximum as well as the extremely small phase

delay expected from thermal expansion, the quadrature bias signal path is insensitive to PM noise to first-

order approximation. Phase fluctuations due to thermal effects at quadrature can be approximated to

second-order as

𝛷𝑃𝑀 = 𝛷𝑃𝑀,0 − 𝛿𝛷𝑃𝑀 = +𝜋

2(1 −

1

2𝛥𝜙𝑡ℎ𝑒𝑟𝑚

2) =𝜋

2−

𝜋


2

The expected average output power fluctuations from the SGI is therefore

𝑃𝑜𝑢𝑡 =1

2𝑃𝑎𝑣𝑔[1 − 𝑐𝑜𝑠(𝛷𝑃𝑀)]

𝑃𝑜𝑢𝑡 =1

2𝑃𝑎𝑣𝑔[1 − 𝑠𝑖𝑛(𝛿𝛷𝑃𝑀)]

𝑃𝑜𝑢𝑡 = 𝑃𝑜𝑢𝑡,0 + 𝛿𝑃𝑜𝑢𝑡 ≈1

2𝑃𝑎𝑣𝑔 −

1

2𝑃𝑎𝑣𝑔𝛿𝛷𝑃𝑀

The SGI output power is directly proportional to KPD to yield

𝛿𝐾𝑃𝐷

𝐾𝑃𝐷

=𝛿𝑃𝑜𝑢𝑡

𝑃𝑜𝑢𝑡

= 𝛿𝛷𝑃𝑀 =𝜋


2

Therefore, thermal fluctuations in the quadrature bias signal path will cause second-order fluctuations in

the BOMPD phase sensitivity.

In the experiments, the maximum KPD is calibrated only once, so any fluctuations thereafter may

cause further degradation and lead to an underestimation of the actual phase error. The magnitude of this

calibration error is fortunately negligible because of the second-order dependency. For example, the

current BOMPD noise floor is around 0.1 fs. The phase sensitivity must degrade by 32% to raise the

noise floor to 1 fs. This implies that unmatched RF path lengths of up to 6 m and temperature

fluctuations up to 10 K can be tolerated, which is an order-of-magnitude larger than what exists currently

in the BOMPDs. Furthermore, assuming that such fluctuations are possible, the absolute timing position

of the zero-crossing in the BOMPD response remains unaffected for zero timing error. Thermal

fluctuations only affect the scaling factor for non-zero voltages away from the zero-crossing. This is not

important for the in-loop BOMPD since it is locked to the zero-crossing, but it may be an issue in the out-

of-loop BOMPD measurements when the error signal drifts away from the zero-crossing. This issue can


77

be remedied with power monitoring for post-processing correction or active power stabilization for real-

time stabilization of the BOMPD sensitivity, if needed.

Error Signal Downconversion

The last pair of signals to consider is the reference signal and error signal at down-conversion.

Using similar arguments as the previous case, this signal pair is also PM insensitive to first-order

approximation and can tolerate large mismatched paths and temperature swings. Any thermal

fluctuations that do exist between these two signal paths will degrade the KPD through a second-order

effect. Moreover, it is also beneficial to operate the detection electronics at lower frequencies (longer

wavelengths) to increase temperature stability. The effective phase delay for a given temperature-induced

length fluctuation is decreased for lower RF frequencies according to Δϕtherm = 2πfRFαg,RFLRFΔT. By

switching from 6.3 GHz to 108 MHz, the phase change would decrease by a factor of nearly 60. While

working with longer wavelength may increase the difficulty for initial phase alignment for in-phase

down-conversion, it is worth the SNR improvement of several dBs at photodetection due to higher

responsivity and lower saturation effects at lower detection frequencies.

3.4.3 Electronics Optimization

The PM-insensitive paths are instead AM-sensitive and require optimization of the electronics for

low residual AM noise. In the detection electronics, the best SNR can be obtained by using a highly

linear, low-noise PIN-TIA. The high transimpedance gain will reduce the effective thermal noise floor at

detection while amplifying the signal with low noise figure for high SNR. If a low-noise TIA is

unavailable, it is acceptable to use a 50Ω terminated photodiode followed by a LNA with low noise figure

to maximize the SNR for the ensuing amplifier chain. This latter approach is used in the experiments to

achieve sub-fs noise performance. Moreover, extra care must be taken when selecting LNAs because the

noise figure typically quoted by manufacturers do not include the close-to-carrier noise performance [68].

Noise figures are typically defined in the absence of an input signal and averaged over a large offset

frequency range. In the presence of an input signal, parametric up-conversion of DC noise may generate

close-to-carrier noise components. This is detrimental for the BOMPD because it easily degrades the

BOMPD noise floor with noise spurs well above the sub-fs-level. It is safer practice to employ low


78

phase-noise amplifiers in every RF signal path. In addition, galvanic electrical isolation is implemented

to avoid ground loops, which may disturb both long-term and short-term stability.

For amplifying the VCO signal, a LPNA must be used since this signal path is phase-sensitive.

An anti-alias bandpass filter is necessary to suppress the amplified white noise at large offset frequencies

(>10 MHz) from the VCO frequency because electro-optical sampling will alias this noise down to

baseband and degrade the BOMPD noise floor. The intra-loop modulator is currently a Lithium Niobate

modulator, which has a highly linear electro-optic coefficient but requires high drive voltage. Future

work will involve finding a modulator with lower Vπ as well as higher damage threshold to further

increase KPD. Moreover, decreasing the operating power will improve the thermal control and power

consumption; currently, overheating problems with the RF power amplifier is preventing the use of a

temperature enclosure to stabilize the BOMPD temperature.

3.4.4 High-Frequency Modulation / Low-Frequency Detection

Unidirectional phase modulation in the SGI is ideal to increase its robustness to noise. It is

achieved with a high-frequency travelling-wave electrode design for the electro-optic phase modulator. A

rule of thumb for travelling-wave modulation is that the crystal length exceeds the wavelength of the

driving RF waveform by factor of 10. For example, the frequency cut-off for a 50-mm crystal is around

400 MHz. There are a number of advantages with unidirectional modulation. First, the SGI becomes

repetition rate independent since counter-propagating pulses are not impacted by the phase modulation.

In the first-generation BOMPDs, the Sagnac loop length had to be tailored to an exact length such that the

counter-propagating pulses experience equal but opposite phase shifts from the quadrature bias signal.

With unidirectional modulation, the length restriction on the loop length is removed. Second,

temperature-dependent group delays in the SGI no longer impact its long-term stability. Third, pulse

alignment with the quadrature bias signal is easier since phase adjustments for high frequencies can be

achieved with short tunable delays.

Low-frequency detection, on the other hand, is desired to improve the SNR in the error detection

signal path. The SGI can currently handle at least +10 dBm of average power before significant pulse

distortions degrade the BOMPD performance; however, the SGI output is currently attenuated to 0 dBm

to avoid saturation and nonlinear effects in the photodetector. Photodiodes generally have higher

responsivity and saturation levels at low frequencies when detecting optical pulse trains. Characterization


79

of a high-speed PIN diode (Discovery; DSC50s) shows that switching the detection frequency from

6.3 GHz to 108 MHz can increase SNR by about 8 dB for the same incident power of 0 dBm – see Figure

3-13. Furthermore, increasing the power to +5 dBm can achieve an extra boost of >10 dB due to higher

saturation power at 108 MHz. This is implemented in the third-generation BOMPDs to achieve sub-fs

noise performance. Future improvements to the BOMPD may be to use high-power MUTC photodiodes

to fully utilize the +10 dBm optical power available at the SGI output.

Figure 3-13 Measured detector saturation under high-power pulse train detection. A high-

speed PIN diode (Discovery DSC50s) detects a 200-MHz pulse train at various average

optical power levels. Curves are interpolations between peak values of the repetition rate

harmonics. Higher linearity and saturation levels are observed at lower frequencies.

3.4.5 Electro-Optic Sampling

Optical Pulse Duration

Dispersion management is critical for achieving short pulses for high-resolution electro-optic

sampling. If 1 fs timing resolution is desired, the Nyquist criterion strictly requires that the sampling

pulse have a maximum pulse duration of 1 fs. Fortunately, since the RF signal is a well-behaved sinusoid

and not an arbitrary waveform, it is acceptable if the pulse duration is wider since it will sample and

average a linear range of values near the zero-crossing of the VCO signal. The sampled information is


80

still preserved. If the VCO power is increased to drive the phase modulator as hard as possible to increase

KPD, there is an additional constraint on the upper limit for pulse duration. Higher VCO power means that

the slope of the VCO signal at its zero-crossing is very steep and that the fixed pulse width will sample a

larger voltage range. If the corresponding range of electro-optic phase shifts exceed the linear range of

the SGI transmission function (at quadrature), then the pulse duration is effectively averaging out the

BOMPD response curve and degrading the BOMPD timing sensitivity.

This effect in the BOMPD voltage response is simulated for varying pulse duration in the 200 fs

to 10 ps range to match experiments. Dispersion in SMF28 fiber is about 18 ps/nm∙km. For 1, 10, and

20 m of SMF 28, the duration for a transform-limited pulse with 30-nm optical bandwidth increases to

0.54, 5.4 and 10.8 ps due to SOD. These lengths are realistic in the experiments because several fiber-

coupled devices, each with 0.5-m fiber pigtails, are used between the laser output port and the phase

modulator in the SGI where electro-optic sampling occurs. Figure 3-14(a) shows the voltage response

curve for the BOMPD when driving with high VCO power (beyond multiple Vπ). The time axis

corresponds to one period of the VCO signal at 10.833 GHz, which is 92 ps. The BOMPD phase

sensitivity KPD is the zero-crossing slope at time zero. Ideally, the optical pulse is as short as possible (see

Figure 3-14 Simulation results for BOMPD phase sensivity KPD versus sampling pulse

width. (a) BOMPD voltage responses for high VCO drive voltage (multiple Vpi) vs time

delay between the pulse train and VCO signal (Time span equals 1 period of the VCO

signal at 10.833 GHz); (b) Corresponding KPD degradations as a function of pulse width.


81

blue curve) to accurately sample the VCO signal to achieve maximum phase sensitivity. However, as the

pulse duration broadens, the pulse duration effectively samples a wider time window of the BOMPD

response curve and averages out the information, resulting in degraded phase sensitivity. The change in

phase sensitivity relative to the maximum phase sensitivity is plotted in Figure 3-14(b). The units in dB

directly represent the expected degradation to the BOMPD noise floor. For example, a degradation of

10 dB for 9-ps pulse durations would increase the BOMPD noise floor from 0.1fs to the 1-fs level. Note

that for lower RF drive power, the phase sensitivity degradation is not as serious as that depicted here. As

safe practice, it is best to keep the pulse duration below 1 ps by using DCF or minimizing excess fiber

length below 2 m.

Laser Timing Jitter

The laser timing jitter sets the fundamental limit for electro-optic sampling in the SGI. For sub-fs

synchronization, the laser timing jitter must be strictly below 1 fs. This limitation is also known as

aperture jitter. Unlike the timing-stabilized link, the BOMPD does not suffer from fiber-nonlinearity-

induced or link-enhanced timing error. The former is because the optical power in the BOMPD is limited

to 0 dBm due to detector saturation, which is too low to induce significant pulse timing errors. As for the

latter, the fiber length in the BOMPD is short enough that residual dispersion can be well-compensated to

minimize fiber-enhanced timing jitter.

3.4.6 Phase Margin

If high locking bandwidths is desired, the BOMPD phase response must not add considerable

phase at high offset frequencies to achieve good phase margin. Typically, the phase margin is limited by

the feedback tuning mechanism of the oscillators themselves, such as the PZT resonance of the slave laser

in the 10 kHz range. For RF oscillators, however, their feedback bandwidth can be much larger in the

MHz range. To support this larger bandwidth, the electronics in the BOMPD must have minimal phase

response up to the MHz range as well. These include the photodetector, LNAs, divider mixer, and

especially the bandpass filters.

Bandpass filters are usually the limiting components since they exhibit narrow passbands and

large phase excursions near the passband edges. This is especially problematic for filtering high

frequency signals in the multi-GHz regime. For example, to filter out the 12.7 GHz repetition rate tone


82

for quadrature biasing, the filter must reject the neighboring repetition rate harmonics with >60 dBc

suppression, while maintaining a flat phase response for a ±10 MHz passband. This demands a very high

Q cavity filter. For low-frequency detection, these design requirements are relaxed because a simple DC

block and lowpass filter combination can serve as a bandpass filter with flat phase response over a large

bandwidth.

It is important to make a clear distinction between PM insensitivity and phase response.

Although signal paths may be PM insensitive, it does not mean it is insensitive to phase response. PM

insensitivity means that small phase fluctuations can be ignored, whereas the phase response is

responsible for large-signal phase changes that will limit the PLL locking performance.

3.4.7 Noise Model

See Appendix D for the PLL noise model used to simulate and verify the optical-RF synchronization

demonstrations with the BOMPDs.

3.5 Alternative Implementations

Two additional variations of the BOMPD are developed. The first is an integrated version of the

BOMPD with the SGI integrated on a silicon photonics platform. The second is a discrete BOMPD based

on a Mach-Zehnder Interferometer (MZI). Full functionality of both devices are demonstrated

experimentally. Long-term and short-term stability issues that limit their timing precision at the 10-fs

level will be discussed. Thee fiber-based SGI-BOMPD is ultimately favored for long-term and short-term

stability.

3.5.1 Integrated BOMPD

The first variation of the BOMPD is an attempt towards full-integration of the fiber-based

BOMPD. An integrated BOMPD offers many benefits such as substantially smaller form factor, lower

cost, large volume production, high isolation against environmental perturbations, and integration with

high-speed CMOS electronics. As a proof-of-concept demonstration, the SGI is first fabricated on a

silicon photonics platform while the remaining BOMPD components remain in discrete form.


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Experimental Set-up

The integrated SGI is designed and fabricated in collaboration with Sandia National Laboratories

on a silicon photonics platform using silicon nanowire waveguides [69]. The chip is comprised of on-

chip inverse-tapered couplers, a 50:50 coupler, and two high-speed phase modulators within the Sagnac

loop. The design and fabrication of the modulators are explained in [70]. For simplicity, the modulators

are designed as lumped element devices without travelling-wave electrodes. Since phase modulation is

bidirectional, the modulators are positioned at specific locations within the Sagnac loop to achieve

optimal differential phase shifts for quadrature bias at 6.36 GHz and maximum phase error detection of

the VCO signal at 10.176 GHz. High-frequency operation in the GHz regime is necessary to

accommodate the reduced Sagnac loop length. The modulators are 1000 µm long each with a high Vπ of

about 10 V. Bias tees are used to reverse bias the modulators to operate them in depletion mode. On-off

chip coupling is performed using silicon nitride waveguide inverse tapers matched to cleaved SMF-28

fibers. Index matching liquid is used to facilitate beam size matching to minimize coupling losses.

The BOMPD architecture is similar to that for the fiber-based SGI-BOMPD. The laser source is

a home-built Er-doped fiber soliton laser with a 508.8-MHz fundamental repetition rate [71]. The VCO

(Poseidon; DRO-10.225) is a dielectric resonant oscillator operating at 10.176 GHz, which is near the 20th

Figure 3-15 Experimental set-up for remote-station locking with the integrated SGI-

BOMPD. Fiber-based BOMPD used for out-of-loop measurement. (DC, DC voltage

offset; BT, bias-tee)


84

harmonic of the laser. The quadrature bias signal is extracted from the 25th harmonic of the repetition rate

at 12.72 GHz, divided to 6.36 GHz, and amplitude- and phase-adjusted to drive the first modulator ϕ1.

Due to the high Vπ (10V) of the modulator, the quadrature bias signal needs +18 dBm RF power (5Vpp)

to achieve differential ±π/2 phase shifts for quadrature bias as well as a >2.5V DC reverse bias to avoid

forward biasing the pn junction. This bias signal also serves as the reference signal for error signal down-

conversion at the SGI output. The VCO signal is applied to the second modulator ϕ2 to detect its phase

error with respect to the optical pulse train. An EDFA is used to boost the optical power at the SGI input

to 15 mW to increase the SNR at detection. The PLL is closed with a 300-kHz loop bandwidth. In-loop

locking is performed with the integrated BOMPD while the out-of-loop measurement is performed with

the fiber-based BOMPD.


The in-loop noise floor is at -110 dBc/Hz due to thermal noise at detection. Under tight PLL

locking, the in-loop noise is suppressed below the thermal noise floor to -140 dBc/Hz; this does not truly

represent the synchronization between the VCO and MLL because the PLL is also synchronizing the

VCO to the thermal noise. The out-of-loop measurement captures this phenomenon and is limited by the

in-loop detection noise floor below the locking bandwidth. The residual timing error in the 10 Hz –

300 kHz range is about 28 fs RMS.

Sub-fs stability is not possible in this current experimental set-up because optical losses in the

integrated SGI severely limit the SNR at error detection. First, on-off chip coupling losses introduce a

fixed loss of a few dBs. Second, limited RF power delivered to the modulator (due to limited RF gain

and parasitic cable losses) as well as the high Vπ of the modulator degrade the SGI bias point below

quadrature. These factors attenuate the SGI output power to 60 μW, which is significantly lower

compared to the mW-level signals achieved in the fiber-based BOMPD, and ultimately degrades KPD. All

these factors effectively degrade the BOMPD noise floor to -110 dBc/Hz. Note that these performance

limitations are from technical origins and can be corrected in next-generation devices. Future

improvements will be to further decrease Vπ of the phase modulators by increasing the modulator length,

minimize the coupling losses, and optimize the driving RF electronics to approach a noise floor level that

is comparable to that in the fiber-based BOMPD (-150 dBc/Hz).


85

Two immediate attempts to increase the SNR at detection were made with no success. The first

attempt was to use an EDFA to amplify the SGI output; however the SGI output power is too low to seed

proper amplification. The second attempt was to use the reflection port of the SGI for error detection

since the reflected power is higher than the transmitted power due to reciprocity of the SGI. The reflected

signal can be detected using an optical circulator at the SGI input. In practice, however, background

noise generated from back-reflection at the fiber-to-chip coupling interface overpowers the weaker signal

reflected from the SGI and degrades the BOMPD noise floor considerably. Both attempts were unable to

improve the SNR in the BOMPD.

Long-term sub-fs stability is also not possible due to fiber-to-chip coupling issues. Air currents

and table vibrations can quickly change the on-chip coupling ratio to degrade long-term stability and

cause the system to lose lock after a couple hours of operation. Ideally, all optical components should be

securely packaged into the same fiber-coupled, integrated device to achieve long-term stability. Although

thermal expansion is high in silicon waveguides, the compact size in a fully integrated device can be

easily temperature stabilized.

Figure 3-16 Measurement results for remote-station locking with the integrated BOMPD

for in-loop locking and the fiber-based BOMPD for out-of-loop measurement.


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In addition to stability issues, the non-idealities in the silicon phase modulator will limit the

maximum BOMPD sensitivity achievable. The phase modulators are based on pn junctions, which

exhibit typical diode I-V characteristic with regions of reverse breakdown, depletion, and forward bias.

While Lithium Niobate modulators are highly linear over large voltage ranges, silicon phase modulators

are limited to the depletion-mode voltage range and exhibit a nonlinear electro-optic phase response.

Since high KPD requires large input voltage swings, this may problematic for a number of reasons. First,

the DC reverse bias must be sufficient to avoid forward biasing as well as reverse bias breakdown. Also,

the baseband noise must be minimized as much as possible to maintain the inherent stability of the SGI.

Second, limited slew rate and hysteresis will cause asymmetry in the phase shifts and result in residual

amplitude modulation at the SGI output. Additional AM may also result directly from residual AM noise

in the phase modulator itself. All these factors will degrade the timing precision of the BOMPD.

3.5.2 Mach-Zehnder BOMPD

The second variation of the BOMPD is based on a Mach-Zehnder interferometer (MZI) [72]. The

MZI serves the same purpose as the SGI in converting time-varying phase changes into amplitude

modulation. Its principle of operation is shown in Figure 3-17. The MZI is similarly biased at quadrature

for maximum phase sensitivity. There are notable differences between this approach and the SGI

approach. Balanced detection is used to detect the dual complementary outputs of the MZI. The phase-

encoded amplitude modulation is extracted from the average power of the pulse train rather than

modulation at a half-repetition rate, which offers higher SNR at error detection. However, the MZI is not

inherently biased at a fixed point in its transmission function and is highly susceptible to environmental

drifts. For long-term stability, the MZI requires an additional quadrature bias stabilization scheme, which

has yet to be implemented.

Figure 3-17 Operating principle for the MZI-BOMPD.


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Experimental Set-up

The experimental set-up used to test the MZI-BOMPD is shown in Figure 3-18. The laser is an

Er-fiber mode-locked laser with a 508.8-MHz repetition rate and the VCO is a dielectric resonant

oscillator with 10.176 GHz center frequency, which is at the 20th harmonic of the fundamental repetition

rate. The quadrature bias is coarsely tuned and maintained with a manual DC voltage supply. This is

suitable for short-term jitter measurements, but insufficient for long-term drift measurements. The dual

MZI outputs are length- and amplitude-adjusted with variable ODL and attenuators to achieve perfect

cancellation at balanced detection for quadrature bias. The reference signal for error signal down-

conversion is derived from the 2nd

harmonic of the repetition rate (1.02GHz).

Figure 3-18 Experimental set-up for remote-station locking with the MZI-BOMPD.


The preliminary in-loop measurement is shown in Figure 3-19. A PLL locking bandwidth of

about 500 kHz is achieved with about 60 fs RMS jitter. While improvements to the optics and electronics

can improve the high-frequency noise, these do not solve the long-term stability issues due to quadrature

bias drifts in the MZI.

The major shortcoming of the MZI-BOMPD is the lack of an inherent bias point. The SGI has

inherent symmetry that biases the SGI at the minimum of its transmission function. In the MZI approach,

an additional feedback scheme is necessary to stabilize the MZI at its quadrature point with extremely

high-precision. Various quadrature stabilization schemes were attempted with limited success. The first

attempt involved direct power monitoring of the MZI outputs and feedback to the DC bias of the MZI

(ditherless operation). This failed because it is impossible to separate amplitude imbalance due to real


88

phase error between the VCO and pulse train from that cause by bias drifts within the MZI. The second

attempt involved using a commercial quadrature bias controller (YYlabs) with a dithering tone to lock to

quadrature; i.e. the even-order harmonics of the dither tone are suppressed to lock to quadrature. This

was initially promising; however, the level of stability was inadequate for sub-fs stability, let alone the

presence of a strong dither tone at a 1 kHz offset that degrades the locking capabilities.

In conclusion, the MZI BOMPD is unable to provide sub-fs stability due largely to the absence of

an effective quadrature stabilization scheme. The only foreseen benefit of the MZI-BOMPD is the

increased SNR at balanced detection. In practice, the FLOM-PD [52] combines advantages of both the

MZI-BOMPD and SGI-BOMPD. With a non-reciprocal biasing element in the SGI, the modified loop

can employ balanced detection to increase its SNR. Its long-term stability is notably higher than that of

the MZI-BOMPD because it still takes advantage of the SGI symmetry. However, with the introduction

of additional elements in the Sagnac loop, its stability is compromised compared to the basic SGI with

only an intra-loop modulator. In this thesis, the basic SGI is shown to prevail over these other BOMPD

methods to achieve both long-term and short-term stability in the sub-fs regime.

Figure 3-19 Measurement results for remote-station locking with the MZI-BOMPD (in-

loop only)


89

3.6 Conclusion and Future Work

By characterizing noise sources within the BOMPD and systematically optimizing the optics and

electronics, the third-generation BOMPDs are able to achieve sub-fs noise floor performance (-161

dBc/Hz) for up to MHz-level locking bandwidths. Optical-RF synchronization is demonstrated for over

24 hours of continuous operation with 0.98 fs RMS drift integrated from 12 μHz to a 20-kHz loop

bandwidth. RF-optical synchronization is demonstrated with 0.5 fs RMS over the 1 Hz – 20 kHz locking

bandwidth range. A nominal, wideband AM-PM suppression ratio of 50 dB is confirmed with potential

improvement via DC offset adjustment. These results confirm the feasibility of BOMPDs for sub-fs

optical-RF synchronization in a large-scale TDS for next-generation light sources.


90

Chapter 4. Ultra-Low Noise Microwave Oscillator

91

4 Ultra-Low Noise Microwave Oscillator

Photonic oscillators are attractive signal sources due to their potential for ultra-low phase noise

microwave generation. While low-noise microwaves can already be generated from commercial rack-

mounted RF oscillators (e.g. Sapphire loaded crystal oscillators), photonic oscillators offer two key

advantages. First, pulsed optical sources can offer orders-of-magnitude improvement in timing jitter due

to the fundamental scaling between optical pulse width and RF repetition period (see Section 1.3).

Second, the high-quality photonic-microwave oscillator can be potentially integrated on a silicon photonic

chip, which greatly reduces its size. These photonic oscillators may serve as timing references in both the

optical and RF domain for time/frequency metrology. While state-of-the-art photonic oscillators have

been demonstrated in laboratory settings, its size, cost, and complexity are prohibitive for widespread use.

To this effect, current research efforts are towards developing chip-scale, integrated photonic oscillators,

such as the Erbium Silicon Photonics Integrated OscillatoR (ESPIOR).

The ESPIOR is a photonic oscillator that derives its ultra-low phase noise performance from

optical frequency division (OFD) and CW-based RF generation. In this thesis, the ESPIOR concept is

demonstrated with an Er-fiber laser in a discrete fiber-optic testbed. This testbed is used to investigate

noise dynamics in the frequency comb and predict performance scaling issues from Er-fiber mode-locked

lasers to integrated lasers in the fully-integrated ESPIOR platform.


4.1.1 Optical Frequency Division

OFD is a proven technique to generate ultra-low phase noise microwaves from cavity-stabilized

CW lasers with the help of optical frequency combs [30]. The pulse train from a mode-locked laser

corresponds to a series of evenly-spaced comb lines in the frequency domain. Each comb line obeys the

basic equation vN = f0 + N fr, where vN is the comb line frequency, f0 is the carrier envelope offset (CEO)

frequency, N is the mode number, and fr is the repetition rate. Perturbations to a comb line can be

linearized as δvN = δf0 + N δfr to show that the frequency comb has two degrees of freedom, i.e. CEO

fluctuations δf0 and repetition rate fluctuations δfr. Many noise sources may perturb the frequency comb.

ASE and shot noise are the primary causes of quantum-limited timing jitter. In practice, additional noise


92

sources such as pump RIN, temperature fluctuations, acoustics, and vibrations may increase the laser

noise considerably above the quantum limit. When the optical pulse train is detected on a photodetector,

neighboring comb lines beat to generate RF harmonics of the repetition rate. The phase stability of the

each RF tone is derived from the stability of the repetition rate δfr.

OFD begins with an ultra-stable CW reference, which is implemented by stabilizing a narrow-

linewidth (~100 Hz) CW laser to an external ultra-stable passive optical cavity using the Pound-Drever-

Hall (PDH) technique to achieve sub-Hz linewidths [73]. Such a reference cavity has a higher stability

and quality factor than the mode-locked laser cavity [39]. A mode-locked laser is then referenced to this

stable CW reference by phase-locking one comb line vN to the CW line vcw with an optical PLL. This

suppresses the free-running noise of the comb line by several orders of magnitude. The remaining degree

of freedom in the frequency comb, δf0, is detected with an f-2f interferometer and stabilized to a RF

reference. The fully-stabilized frequency comb then serves as an optical frequency divider to divide the

stability of the CW reference at an optical frequency down into the microwave regime. The repetition

rate stability can be re-expressed as δfr = (δvN – δf0)/N. Typically, N is on the order of 105 to 10

6 and the

stabilized δf0 at RF frequencies is negligible compared to δvN of the reference cavity at optical

frequencies. Therefore, the CW reference stability is divided by N to achieve ultra-low repetition rate

noise in the microwave regime. The OFD ratio is defined as 1/N in units of dB. For example, the sub-Hz

Figure 4-1 Operating principle for optical frequency division. A full-stabilized frequency

comb (via combline νN stabilization to a stable CW reference and fo stabilization using f-

2f interferometry) divides the CW reference down to the microwave regime.


93

stability of a stable CW reference at an optical frequency of 192 THz can be divided down by an ideal

OFD ratio of -80 dB to yield ultra-low phase noise (-80 dBc/Hz @ 1Hz offset) at 20 GHz.

4.1.2 Erbium Silicon Photonics Integrated OscillatoR

The ESPIOR uses this OFD scheme to generate ultra-low phase noise microwaves on an

integrated platform. The stable CW laser is an integrated CW laser stabilized to a high-Q, athermal

reference cavity via PDH locking. A comb line from an integrated mode-locked laser is stabilized to the

CW reference using an optical PLL. Although low-noise microwaves can be extracted by direct detection

of the pulse train, this approach is avoided because of SNR limitations at detection. High pulse peak

powers induce detector nonlinearities (e.g. saturation and AM-PM conversion), which degrade the output

quality of the RF signal. Furthermore, the limited power per comb line limits the maximum extractable

RF power. To maximize the SNR at detection, it is desirable instead to beat two high-power CW signals

to extract a single RF beat tone at the desired RF frequency.

This CW approach is adopted for the ESPIOR RF readout scheme. A second CW signal νCW2 is

derived from the CW reference νCW1 using a SSB modulator. The VCO that drives the modulator is

initially set at the desired output RF signal frequency. A second optical PLL then stabilizes νCW2 to a

neighboring comb line νN+ΔN in the frequency comb. While νCW2 is aligned exactly with a comb line in

Figure 4-2, offset-locking can be used to lock νCW2 to non-integer multiples of fr. If the frequency comb is

fully-stabilized, then the stability of νCW1 is transferred to νCW2. The two stable CW signals νCW1 and νCW2

are then combined on a photodetector to extract a RF signal at the beat frequency ΔνCW = νCW2 − νCW1 with

Figure 4-2 Operating principle for the ESPIOR. RF readout is performed by stabilizing

a second CW frequency to the frequency comb reference and beating the two stabilized

CW signals on a photodetector.


94

phase noise that is derived from the stable reference cavity.

While this scheme works in principle, one major issue encountered for the integrated ESPIOR is

the unavailability of an on-chip f-2f interferometer. If δf0 fluctuations cannot be suppressed, then it may

overpower the reference cavity stability δvN and degrade the OFD ratio. Therefore, it is imperative to

investigate if comb line stabilization alone is sufficient to realize full OFD. If it is not, then an alternative

fo control scheme that is compatible with on-chip technology must be developed.

4.2 Noise Characterization

In this section, the stability of a partially-stabilized frequency comb (with one degree-of-freedom

secured via comb line stabilization) is modelled and characterized experimentally using an Er-fiber mode-

locked laser (Menlo; M-comb LH120). Using the fixed point modulation theory for comb dynamics, an

alternative feedback control scheme called quasi-OFD (QOFD) is proposed to improve comb

stabilization. QOFD is analyzed and verified experimentally. A comparison between QOFD and

conventional OFD with f-2f interferometry as well as its practical implications for an integrated ESPIOR

are presented last.

4.2.1 Cross-Modulation Dynamics

Two common feedback controls for a frequency comb are cavity length modulation ΔL and pump

power modulation ΔP. The 2×2 linearized transfer matrix K that maps small input voltage changes in the

cavity PZT ΔVL and pump power modulation ΔVP to output frequency changes in the repetition rate Δfr

and CEO frequency Δfo is:

[𝛥𝑓𝑟

𝛥𝑓𝑜] = [

𝐾11 𝐾12

𝐾21 𝐾22] [

𝛥𝑉𝐿

𝛥𝑉𝑃]

where Kij (i, j = 1, 2) are the transfer coefficients in units of Hz/V. Ideally, a diagonal matrix is desired so

that the feedback mechanisms independently control the two degrees-of-freedom of the frequency comb

to achieve the highest level of comb stability. This is however far from reality.

Newbury et al. have shown in their perturbative model for soliton fiber lasers that pump power

fluctuations couple strongly to both fr and fo through numerous pulse shaping mechanisms (e.g. SPM,

TOD, SS, spectral shift, resonant group velocity) and claim that a similar analysis applies to cavity length


95

fluctuations. Moreover, not only are the off-diagonal matrix elements non-zero and non-negligible, there

are internal feedback dynamics (e.g. gain saturation) that further complicate how a frequency comb can

be stabilized [74], [75]. Despite how these coefficients arise, the approach adopted in this thesis is to

accept the matrix elements as given by the internal laser dynamics and engineer an external feedback

scheme to minimize fo fluctuations without resorting to f-2f interferometry to maximize the OFD ratio.

To this effect, it is necessary to first characterize the Kij coefficients for a mode-locked laser. The

basic method is to directly modulate ΔL and ΔP with a RF function generator and record the

corresponding changes in fr and fo with a RF frequency counter. fr is observed by direct detection, while fo

is observed by f-2f interferometry. If an f-2f interferometer is unavailable, the beat note between a comb

line and a stable CW reference can be recorded instead; changes in fo are then calculated using the beat

note equation δf0 = δvN – N δfr. The modulation frequency is swept to measure the frequency response of

the coefficients. Advanced measurement set-ups involving PLLs or frequency discriminators are needed

to extract the phase response of the transfer coefficients. Fortunately, the phase responses for slow

modulation frequencies below 1 kHz are either in-phase (0°) or out-of-phase (180°) [76] and can be

directly observed with the frequency counter. This is sufficient for this thesis because the cavity PZT for

the laser under test has a limited bandwidth of 10 kHz.

Figure 4-3 Measured Kij transfer coefficients for a stretched-pulse Er-fiber laser.


96

The Kij coefficients for the laser under test are measured using the basic method (see Figure 4-3).

K11, which maps ΔVL to Δfr, is nearly constant until it approaches the PZT bandwidth near 10 kHz. A

higher bandwidth may be achieved if the cavity PZT is replaced with a shorter PZT on a lead-filled mount

to increase the mechanical resonant frequency[77]. K22, which maps ΔVP to Δfo, is also near constant.

K12, which maps ΔVP to Δfr, is approximately constant for higher frequencies but increases for frequencies

below 10 Hz due to slow thermal effects. K21, which maps ΔVL to Δfo, exhibits a strong frequency

dependence. This suggests that beam deflections from the cavity PZT are inducing fluctuations in the

intracavity power. These coefficients are in close agreement to those reported for a similar Er-fiber laser

[76]. Curve fitting in the 1 Hz – 300 Hz range is performed to simplify the subsequent analysis.

4.2.2 Fixed Point Modulation

The fixed point modulation theory is useful for visualizing perturbations to a frequency comb

[78], [79]. If the perturbation is small enough such that the comb response is approximately linear, then

the frequency comb will breathe, i.e. expand and contract, with respect to a fixed comb line position. For

a perturbation source X, its fixed-point is calculated by setting the comb line equation to zero and solving

for the mode number m:

𝜕𝑣𝑚

𝜕𝑋= 𝑚

𝜕𝑓𝑟

𝜕𝑋+

𝜕𝑓𝑜

𝜕𝑋→ 0

𝑚 = −

𝜕𝑓𝑜

𝜕𝑋𝜕𝑓𝑟

𝜕𝑋

Fixed point m is the ratio of the change in fo to the change in fr and is not restricted to integer values.

The 2×2 transfer matrix K is an example of two such perturbation sources. The fixed points for

the cavity length and pump power modulation are:

𝑚𝐿 = −𝐾21

𝐾11

, 𝑚𝑃 = −𝐾22

𝐾12

It is useful to normalize the fixed points to the mode number no of the laser center frequency.

𝑀𝐿 =𝑚𝐿

𝑛𝑜

, 𝑀𝑃 =𝑚𝑃

𝑛𝑜

A fixed point at DC is equal to 0 and a fixed point at the laser center frequency is equal to 1.


97

Typical fixed points for cavity length modulation are in the DC to few THz range because cavity

length modulation primarily affects the repetition rate. Modulation of a free-space path should ideally

yield a fixed point exactly at 0 since no mismatch between group and phase velocity is introduced.

However, secondary effects such cavity power modulation due to beam misalignment and coupling issues

may couple into fo changes, which effectively shift the ideal fixed point away from fo. Typical fixed

points for pump power modulation are much higher near the laser center frequency and vary more widely

depending on specific laser parameters. This is because intracavity gain dynamics strongly couple into fo

through group and phase velocity mismatch as well as fr through index modulation. Separate experiments

(not shown here) confirm that pump fixed points can range anywhere from tens of THz to beyond the

laser center frequency at 192 THz depending on the laser type (e.g. stretched-pulse or soliton) as well as

its mode-locking state. The beat note equation can be re-expressed in terms of the fixed point, mx:

𝜕𝑣𝑛

𝜕𝑋= (𝑛 − 𝑚𝑋)

𝜕𝑓𝑟

𝜕𝑋

This states that the modulation strength of a comb line depends linearly on its distance from the fixed

point. Figure 4-4 shows the comb breathing response for typical fixed point positions for ML and MP

relative to the laser center frequency.

Figure 4-4 Illustration for comb “breathing” due to fixed point modulation. ML and MP are

fixed points for cavity length and pump power modulation, respectively. no is the mode

number of the laser center frequency vo.

The fixed points ML and MP are calculated from the measured Kij coefficients and plotted in

Figure 4-5. ML is near zero and wideband up to the PZT bandwidth, while MP exhibits a steep decline

below 10 Hz due to thermal effects and a shallow incline at higher frequencies.


98

Figure 4-5 Measured fixed points for cavity length, ML, and pump power, MP, modulation,

for a stretched-pulse Er-fiber laser.

4.2.3 Suppression Ratio

The ESPIOR is limited to a single feedback loop to stabilize a comb line to the CW reference.

Even if this comb line stabilization is perfect, residual noise in the repetition rate fr may still exist (see

Appendix E for derivation). This is detrimental for the ESPIOR because it directly degrades the OFD

ratio. The spectral density of the residual fr noise is:

𝑆𝛥𝑓𝑟= ∑ (

𝑚𝑋𝑖− 𝑚𝑌

𝑛𝑐𝑤 − 𝑚𝑌

) 𝑆𝛥𝑓𝑟,𝑋𝑖

𝑃

𝑖=1

where SΔfr,X is the spectral density of the free-running repetition rate noise contributed by noise source X,

mX is the fixed point of the noise source X, mY is the fixed point of the feedback control Y, ncw is the mode

number of the stabilized comb line, and P is the total number of noise sources perturbing the frequency

comb. This is an important equation since it states that the fixed point relationships determine how well

each noise source is suppressed. The noise suppression ratio RX,Y for a perturbation source X due to

feedback control Y is defined as:

𝑅𝑋,𝑌 = (𝑚𝑋 − 𝑚𝑌

𝑛𝑐𝑤 − 𝑚𝑌

)


99

in terms of absolute mode numbers. In the experiments, it is common to align and stabilize the laser

center frequency νo to the CW reference νcw. This further simplifies the expression to:

𝑅𝑋,𝑌 = (𝑀𝑋 − 𝑀𝑌

1 − 𝑀𝑌

)

in terms of normalized mode numbers. This expression is simple yet effective for explaining the

suppression of repetition rate noise in a partially-stabilized frequency comb.

The suppression ratio is illustrated with the following examples (see Figure 4-6) for different

fixed point relationships. Assume that a noise source X perturbs the frequency comb with fixed point MX

(green arrow) and that feedback is controlled via cavity length modulation with fixed point ML (red

arrow). For comb line stabilization, the feedback loop minimizes fluctuations detected at the laser center

frequency νo (magenta). Any consequent stabilization of the repetition rate fr is an additional benefit.

In the first case, the feedback fixed point ML aligns perfectly with the noise fixed point MX. When

feedback is applied to stabilize νo fluctuations, its actions are exactly out-of-phase with that of the

noise source. Stabilizing the comb line coincidentally stabilizes fr fluctuations perfectly (RX,L = 0).

In the second case, MX is aligned with νo. Since vo is insensitive to perturbations by noise source X,

the feedback scheme detects no fluctuations in vo and perceives the frequency comb as stable. In

actuality, fr noise caused by X is free-running with no suppression (RX,L = 1).

In the third case, MX is set twice as far at 2νo. Comb line stabilization is achieved with a feedback

Figure 4-6 Illustrations of suppression ratios for different fixed point relationships. (left)

Simple cases for feedback on the cavity PZT, ML, and a noise source, MX; (right) General

case for a real frequency comb with two feedback controls and multiple noise soures.


100

control signal that is in-phase with the noise source; however, the repetition rate noise is doubled

(RX,L = 2). This can be reasoned by considering the comb line movement at each fixed point induced

by the opposing fixed point.

From these trends, it is intuitive that the feedback fixed point should align as close as possible with the

noise fixed point to maximize the suppression of repetition rate noise.

In the general case (see Figure 4-6), the laser has two primary feedback controls on cavity length

and pump power modulation with fixed points ML and MP (see red arrows), respectively, as well as

numerous noise sources with their own respective fixed points (see green arrows). Few noise sources are

well understood: acoustics, vibrations, and thermal expansion may introduce cavity length changes

through ML, while pump RIN and intracavity power losses may introduce gain fluctuations through MP.

For these noise sources, direct feedback to their respective feedback controls can achieve the best noise

suppression. In a real laser, additional noise sources may enter through different fixed points such as

MX,i. These noise sources cannot be fully suppressed by the feedback controls and may limit the

frequency comb stability.

4.2.4 Quasi OFD

In the case that the dominant noise source has a fixed point that does not align exactly with the

feedback fixed points, ML and MP, it is possible to operate both feedback controls simultaneously to

achieve a custom feedback fixed point to maximize noise suppression. This synchronous modulation

scheme is called quasi OFD (QOFD) and is developed in this thesis.

Figure 4-7 Operating principle for QOFD with synchronous feedback modulation.


101

Synchronous modulation begins with a feedback loop that stabilizes a comb line to the CW

reference via pump power modulation (see Figure 4-7). The feedback error signal is amplified by a

proportional gain G to drive the cavity PZT modulation port simultaneously, yielding:

[𝛥𝑓𝑟

𝛥𝑓𝑜] = [

𝐾11 𝐾12

𝐾21 𝐾22] [

𝐺𝛥𝑉𝑃

𝛥𝑉𝑃]

This set of equations can be solved for the effective synchronous modulation fixed point, MSYNC:

𝑀𝑆𝑌𝑁𝐶(𝐺) = −𝛥𝑓𝑜

𝛥𝑓𝑟

= −𝐾21𝐺 + 𝐾22

𝐾11𝐺 + 𝐾12

If G is small, MSYNC approaches the pump fixed point MP. If G is large, MSYNC approaches the cavity PZT

fixed point ML.

Gain polarity determines the behavior of MSYNC for intermediate gain values. The measured

transfer coefficients for the laser under test have the following polarities:

[𝑲𝟏𝟏 𝑲𝟏𝟐

𝑲𝟐𝟏 𝑲𝟐𝟐] = [

+ −− +

]

MSYNC can be expressed explicitly in terms of inverting (–) and non-inverting (+) gain:

𝑀𝑆𝑌𝑁𝐶,− =1 +

|𝐾21|

|𝐾22||𝐺|

1 +|𝐾11|

|𝐾12||𝐺|

𝑀𝑃

𝑀𝑆𝑌𝑁𝐶,+ =1 −

|𝐾21|

|𝐾22||𝐺|

1 −|𝐾11|

|𝐾12||𝐺|

𝑀𝑃

where G > 0. For increasing inverting gain, MSYNC,– transitions gradually from MP to ML. However, for

non-inverting gain, MSYNC,+ experiences a pole discontinuity at G=|K12|/|K11| and zero at G=|K22|/|K21|.

MSYNC is calculated from Figure 4-5 and plotted in Figure 4-8 for proportional gain values achievable with

the amplifier (New Focus; LB1005) used in experiments. The gain polarity determines the direction that

MSYNC takes to transition from MP to ML. In theory, an arbitrary fixed point can be achieved with the

proper gain. In practice, the frequency dependence can be shaped using a programmable digital filter.

Figure 4-9 shows the cross-section at 100-Hz frequency offset to better reveal MSYNC trends.


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Figure 4-8 Simulated fixed point for synchronous modulation MSYNC as a function of

increasing (a) inverting gain and (b) non-inverting gain.

Figure 4-9 Simulated fixed point for synchronous modulation at 100 Hz offset as a function

of synchronous gain.

The suppression ratio for pump and cavity PZT noise using synchronous modulation are

calculated and plotted in Figure 4-10. RP,SYNC, which is the suppression ratio for pump noise MP using

synchronous feedback MSYNC, is calculated for inverting and non-inverting gain in Figure 4-10(a). At low

gain, the highest noise suppression is achieved because MSYNC is aligned with the noise source at MP. At

high gain, a limited suppression ratio is achieved because the feedback fixed point is at ML, which is far


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from MP. For intermediate gain values, the noise suppression can take a gradual transition (inverting

gain) or experience a discontinuity (non-inverting gain). Similar reasoning can be applied to explain the

trends in Figure 4-10(b) for RL,SYNC, which is the suppression ratio for PZT noise ML using synchronous

feedback MSYNC.

4.2.5 Noise Verification

Synchronous modulation for QOFD is verified with the experimental set-up in Figure 4-11. The

OPLL stabilizes one comb line from the mode-locked laser to the CW reference. The CW laser (Orbits

Lightwave; ETH-40-1560.61) has a center wavelength of 1560.61 nm, which overlaps with the center

Figure 4-10 Simulated suppression ratio for synchronous modulation on (a) pump noise

and (b) cavity PZT noise at a 100 Hz offet frequency

Figure 4-11 Experimental set-up for evaluating QOFD.


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wavelength of the mode-locked laser. Despite this overlap, the comb line power is still insufficient to

achieve a beat note with adequate SNR for tight locking. The pulse spectrum is filtered with a narrow

0.6-nm bandpass filter and amplified with an EDFA to increase the comb line power. The amplified

comb lines and the CW reference are combined on a photodetector to generate a series of beat notes. The

CW wavelength is coarsely tuned to generate a beat note at 220 MHz. This beat note is filtered,

amplified, divided by 10, and compared against a RF reference at 22 MHz in a digital phase detector

(Menlo, DXD200). The error signal is filtered by a PI filter (New Focus, LB1005) and fed back to the

pump power modulation port to complete the OPLL. For synchronous modulation, the error signal is

further amplified by a proportional gain amplifier (New Focus, LB1005) to drive the PZT port

simultaneously. The laser output is tapped off and detected with a high-speed photodiode (EOT; 3500F)

to extract low phase noise microwaves. The phase noise of the 6-GHz harmonic (=24fr) is measured with

a SSA to directly evaluate the stability of the frequency comb.

The free-run laser noise is known to be pump-RIN limited; therefore, pump power feedback

should yield better noise suppression than cavity PZT feedback. This is verified experimentally in Figure

4-12. The free-run laser noise is shown for reference (see purple – free run). When comb line

stabilization is performed via PZT feedback, the free-run noise is suppressed by only 5 dB (see black

dashed – PZT lock). When pump feedback is used instead, the noise is suppressed by 12 dB (see black

solid – Pump lock). When synchronous modulation is applied from low gain (blue) to high gain (red),

MSYNC shifts from MP to ML, which degrades the noise suppression accordingly. For inverting gain, this

transition is gradual and upper bounded by the PZT lock curve. For non-inverting gain, the laser noise

Figure 4-12 Measurement results for QOFD. (a) inverting gain (b) non-inverting gain

from low gain (blue) to high gain (red)


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increases without bound due to the pole discontinuity in MSYNC.

These trends are clearly observed in Figure 4-13, which show cross-sections of Figure 4-12 at

specific offset frequencies. The suppression ratio R is defined relative to the free-run noise. The

measured suppression ratio RP,SYNC for pump noise at a 2-kHz offset frequency is plotted in Figure 4-13(a)

for inverting (blue) and non-inverting (red) gain. The simulated curves based on fixed point

measurements are plotted for comparison. Consider the inverting gain case first. A small suppression

ratio of -5 dB is observed at high gain (+40 dB) because the feedback fixed point MSYNC is at ML, which is

far from the pump noise at MP. As the gain is lowered, MSYNC approaches MP to improve the suppression

ratio, until it plateaus at -12 dB. This plateau indicates that another noise source with a fixed point

different from MP is present (otherwise pump feedback would suppress it). The trends for the non-

inverting gain case can be similarly reasoned. Instead of a gradual transition, there exists a pole

discontinuity from MSYNC+. Data could not be acquired for non-inverting gain higher than +5 dB because

the discontinuity prevents the PLL from locking. The measured suppression ratios for both gain polarities

are in good agreement with simulation.

Certain noise spurs exhibit opposite trends with respect to increasing gain, as seen from the inset

in Figure 4-12(a). The measured suppression ratios for these spurs are plotted in Figure 4-13(b) along

with the simulated curves for cavity PZT-induced noise. The measurements and simulations are in good

agreement, which imply that the origin of these spurs is likely from acoustics and mirror vibrations that

Figure 4-13 Measured suppression ratio for QOFD at specific offset frequencies; (a)

pump-RIN-induced noise at 2 kHz (b) various PZT-related spurs in the sub-kHz range;

green star indicates lowest measured suppression ratio using external modulation

technique to increase measurement dynamic range.


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perturb the cavity length through fixed point ML. Noise suppression is limited at high gain because the

PZT noise spurs are suppressed below the pump noise.

The measurement dynamic range in Figure 4-13 is limited because noise features in the free-run

laser noise are small in magnitude. To confirm larger suppression ratios predicted by simulation, an RF

function generator is used to directly modulate the PZT and pump power modulation ports to induce a

large modulation tone. The measurements are repeated to confirm the simulated suppression ratios down

to -30 dB, which is indicated by the green star markers.

4.2.6 Discussion

Synchronous modulation for QOFD is therefore confirmed with these measurements.

Synchronous modulation is effective for achieving a variable feedback fixed point to optimize the

suppression of a single noise source, and the suppression ratio equation is accurate in predicting the

repetition rate noise suppression based purely on fixed points relationships. These measurements,

however, also reveal a major shortcoming of QOFD. The noise suppression is limited in practice by the

presence of multiple noise sources. That is, while it may be easy to adjust the synchronous fixed point to

maximize the suppression of a single noise source, the laser is limited by the next largest noise source

operating at a different fixed point.

Multiple Noise Sources

This is illustrated in the following examples where a single feedback control (MSYNC) is used to

suppress three different noise sources (ML, MP, and MX) simultaneously.

Assume that the three noise sources contribute equally to the total laser noise. The first two noise

sources are pump noise and cavity PZT noise with fixed points MP and ML, while the third noise

source is unknown with a fixed point MX located somewhere in-between. Figure 4-14 shows the

suppression ratio of each noise source as a function of MSYNC. Since there is only a single feedback

loop in the ESPIOR, it is desirable to tune MSYNC to maximize the suppression of all noise sources

simultaneously. The best noise suppression is only -10 dB at a gain of +12 dB, which is at the

intersection between the MP and ML curves.

If noise source X dominates the other two sources by several tens of dB, then it is not necessary to

suppress all noise sources simultaneously. Instead, MSYNC is set at MX to target the most offending


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noise source and achieve the highest noise suppression possible. Once noise source X suppressed, the

laser is then limited by the next largest noise source, which in this case would be MP.

Synchronous modulation is clearly advantageous in both cases because it provides the user with

the ability to tune the feedback fixed point to maximize noise suppression for the dominant noise source.

The maximum OFD achievable, however, is then limited by the second largest noise source. The

magnitude difference between the largest and second largest noise sources vary from laser to laser and

need to be considered on a case-by-case basis. For the laser characterized in this section, the pump noise

can only be suppressed by 15 dB before being limited by the next largest noise source.

The target goal of the ESPIOR is to achieve noise suppression greater than 80 dB from the free-

running noise, which is impractical with QOFD. QOFD may suffice for moderate phase noise

performance, but for true ultra-low phase noise performance, two independent feedback controls for a

fully-stabilized frequency comb are absolutely necessary.

Comparison with Conventional OFD

In conventional OFD, the two degrees of freedom in a frequency comb are independently

stabilized to achieve maximum comb stability. The two feedback error signals are δf0 via f-2f

interferometry and δfN via beat note detection with a stable CW reference. The fixed point positions for

the cavity PZT and pump power modulation are exploited for optimal feedback control; δfN is stabilized

Figure 4-14 Simulated suppression ratio for QOFD with three noise sources.


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with ML while δf0 is stabilized with MP. The large distance between the fixed point and the source of error

detection effectively increases the feedback loop gain (or sensitivity). With these two independent

controls, any number of noise sources that enter the frequency comb can be stabilized. This is in contrast

with QOFD, where only one degree of freedom is secured via comb line stabilization. QOFD cannot

compete with the performance of conventional OFD since the feedback control can be optimized to

suppress at most one noise source.

4.3 MIMO Phase Noise Model

A multiple-input multiple-output (MIMO) phase noise model is developed to simulate the

stability of Er-fiber mode-locked lasers in the presence of cross-modulation effects. Based on the

conventional PLL noise model, the MIMO model is augmented to a two-port model to include: 1) cross-

modulation effects from pump power ΔP and cavity length ΔL modulation to frequency changes Δfr and

Δfo, and 2) simultaneous locking of dual PLLs.

The noise model for conventional OFD is depicted in Figure 4-15. The first feedback loop is for

stabilizing a comb line to a CW reference, while the second feedback loop is for stabilizing fo via f-2f

interferometry. Both loops begin with phase detection, followed by a PI controller and voltage tuning

sensitivity K for mapping voltage to cavity length and pump power changes. These matrices are diagonal.

The Hij matrix contains the cross-modulation comb dynamics, which map ΔP and ΔL into frequency

changes Δfr and Δfo. (In Section 4.2.1, the product Hij∙K is experimentally characterized as Kij). The V

Figure 4-15 MIMO phase noise model for frequency comb dynamics for conventional

OFD and quasi OFD


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matrix converts these outputs to changes in comb line frequency Δvn and Δfo. The frequency/phase block

performs time integration to convert frequency changes into phase changes. The output phases are fed

back to the phase detector to close the PLLs. While Figure 4-15 depicts a MIMO model for conventional

OFD, this model can be modified for different feedback schemes by changing the appropriate transfer

coefficients. For example, QOFD is implemented by removing the second feedback loop and changing

the appropriate matrix elements in K for synchronous modulation. Major noise sources included in the

model are ASE, pump RIN and noise from the CW laser and RF synthesizer references. For accurate

modelling, all transfer coefficients and laser noise sources are measured experimentally if possible.

Otherwise, fundamental laser parameters are used for calculations (e.g. ASE-induced timing jitter). The

MIMO model is used to predict and confirm the frequency comb stability in the following section.

4.4 System Demonstrations

The ESPIOR system is implemented in a discrete testbed using fiber-optic components and

commercial fiber lasers to benchmark its performance. The first set of system demonstrations is to

measure the maximum OFD achievable from the optical frequency divider. Two types of mode-locked

fiber lasers (i.e. stretched pulse vs soliton) are tested. Since these lasers exhibit different fixed points due

to different laser parameters, they are expected to yield different OFD ratios. The second set of system

demonstrations is to implement the complete ESPIOR system (including the RF extraction stage) to

extract low phase noise microwaves.

4.4.1 OFD Demonstration

Experimental Set-up

The maximum OFD ratio for two different mode-locked lasers is demonstrated with the

experimental set-up shown in Figure 4-16. The stable CW reference is implemented by stabilizing a

narrow 400-Hz linewidth CW laser (Orbits Lightwave; ETH-40-1560.61) to a stable high-finesse

(F>250,000) ultra-low expansion cavity using the PDH technique with an 80-kHz locking bandwidth.

The first mode-locked laser under test (Menlo; M-comb LH120) is the 250-MHz laser characterized in

this chapter. Its mode-locked state is optimized for maximum OFD. The second laser under test is a


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soliton Er-fiber laser (Menlo; M-comb LH082). In both cases, comb line vN is stabilized to the stable CW

reference via cavity PZT feedback. For the stretched-pulse laser, two feedback controls are available to

directly compare QOFD (via synchronous modulation for comb line stabilization) and conventional OFD

(via f-2f interferometry and comb line stabilization). The MIMO model is used to confirm all

experimental results.

Figure 4-16 Experimental set-up for measuring OFD ratio in a Er-fiber laser under QOFD

(via synchronous modulation) or conventional OFD (via f-2f interferometry). Both methods

use comb line (vN) stabilization to the CW reference via PZT feedback.

Stretched-Pulse Laser

The measurement results for QOFD and conventional OFD are depicted in Figure 4-17. Phase

noise spectra for the free-run and stable CW laser are shown in black. The free-run MLL noise is shown

in blue. For ideal OFD, the MLL noise should be suppressed down to the Ideal OFD curve.

For QOFD (see red curve), a comb line from the mode-locked laser is stabilized to the stable CW

reference via synchronous modulation to secure one degree of freedom in the frequency comb. The OFD

ratio achieved at a 100 Hz offset frequency is 45 dB. Synchronous modulation is able to suppress the

dominant noise source in the free run MLL by 20 dB before another noise source within the MLL

prevents further noise suppression. Independent characterization of fo noise with f-2f interferometry and

simulation with the MIMO noise model (cyan) confirms that QOFD in the 1 Hz – 5 kHz range is limited

by uncontrolled fo fluctuations. To achieve higher OFD ratios, fo must be further stabilized.


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For conventional OFD (see green curve), fo stabilization via f-2f interferometry and comb line

stabilization to the stable CW reference are implemented to achieve a fully-stabilized frequency comb for

the best OFD ratio possible. The OFD is limited by the measurement noise floor of the SSA at 60 dB.

Simulation with the MIMO noise model (cyan) predicts that the OFD should be as high as 100 dB. A

tighter feedback on the frequency comb as well as improved PDH locking for the stable CW reference

should suppress the laser noise further towards the ideal OFD limit [29], [39].

These measurements are repeated for a free-run CW reference. Interestingly, the noise

performances of QOFD and conventional OFD remain unaffected. This is an important observation

because it confirms that OFD is limited by MLL noise and not the CW reference. The OFD ratios relative

to the free-run CW noise are 100 dB and >115 dB OFD ratios for QOFD and conventional OFD,

respectively, at a 100-Hz offset frequency referred to a 1-GHz carrier.

Figure 4-17 Experiment versus simulation for a stretched-pulse Er-fiber laser under

quasi OFD and conventional OFD


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Soliton Laser

The same QOFD measurement and simulation is performed for the soliton laser; however,

conventional OFD could not be performed because an f-2f interferometer was unavailable. Moreover,

synchronous modulation could not be performed because of faulty electronics in the pump power

modulation port. Cavity PZT modulation is used for feedback only. In contrast with the stretched-pulse

laser, the soliton laser is unable to achieve any level of noise suppression in its free-run noise (blue) when

stabilized to a free-run or stable CW reference. This behavior is predicted by the MIMO noise model

(cyan). Independent characterization of the laser confirmed that the free-run MLL noise is dominated by

pump RIN and that the fixed point ML is near DC at 0.05 while MP is close to the laser center frequency at

0.9. This situation is analogous to the second case covered in Figure 4-6, where the MLL is not expected

to experience any suppression of its repetition rate noise.

Figure 4-18 Experiment versus simulation for a soliton Er-fiber laser under quasi-OFD.


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Scaling for Integrated Lasers

The MIMO noise model is accurate in predicting the noise stability of both commercial stretched-

pulse and soliton Er-fiber lasers, provided that the laser transfer coefficients and noise sources are

properly characterized. This noise model may be extended to predict the noise performance of integrated

mode-locked lasers. The Kij coefficients for an integrated laser can be characterized using the same

technique as that for the fiber lasers. A notable difference would be that the integrated laser uses thermal

modulation of the refractive index to modulate the cavity length instead of using PZT modulation in a

free-space path. Despite this, its transfer coefficients can be similarly characterized and simulated with

the MIMO model. Future work will be to develop a complete analytical model to explain the dependency

of the pump power fixed point on fundamental laser parameters (similar to [74]) as well as mode-locking

state and type (e.g. stretched-pulse versus soliton).

4.4.2 ESPIOR RF Readout

The complete ESPIOR architecture is implemented with the fiber-optic testbed to demonstrate

low-noise microwave generation. The first proof-of-concept demonstration uses a 200-MHz soliton laser

as the optical frequency divider and a free-running CW laser as the reference frequency. The final

demonstration uses a lower-noise 250-MHz stretched-pulse laser and an ultra-stable CW reference to

demonstrate the best RF extraction possible.

Experimental Set-up (200-MHz soliton laser)

The initial proof-of-concept ESPIOR demonstration is detailed below (see Figure 4-19(a) for the

system schematic):

CW Reference: The frequency reference is a free-running CW laser (Orbits; Ethernal SlowLight

ETH-40-1560.61), which generates an optical frequency at 192.1083 THz with a 400 Hz linewidth

within a 1 ms observation time.

Optical Frequency Divider: The optical frequency divider is the 200-MHz soliton fiber laser (Menlo

Systems; M-Fiber LH082). To divide the CW reference vcw1 down into the microwave regime, the

neighboring comb line vN is locked to vcw1 through two OPLLs. OPLL1 is the slow feedback loop

that locks vN to vcw1 via PZT modulation. This maximizes the suppression of cavity length noise in


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the frequency comb. The PLL bandwidth is limited to 10 kHz due to high-frequency PZT feedback

resonances. OPLL2 is the fast feedback loop that uses a fast acoustic-optic modulator (AOM) at the

laser output to extend the locking bandwidth into the 100 kHz range. In contrast to PZT modulation,

the AOM shifts the entire frequency comb. Its modulation fixed point is at infinity. Although this

feedback does not improve the laser timing jitter, it is still critical for OFD because it improves the

coherency between vcw1 and vN (and ultimately vcw2) for low-noise RF extraction.

RF Extraction: The SSB modulator is a Lithium Niobate dual-parallel Mach-Zehnder modulator

(Thorlabs; LN86S-FC). It is driven by a 5-GHz VCO (Agilent E8257D) to shift vcw1 to the second

frequency vcw2. OPLL3 locks vcw2 to its neighboring comb line vN+ΔN. With all three OPLLs engaged,

Figure 4-19 Experimental set-up for the inital ESPIOR testbed demonstration. (a) System

architecture. (b) Frequency domain illustration. (OPLL, optical phase locked loop;

AOM, acoustic-optic modulator; SSB Mod, single-sideband modulator; PD,

photodetector). vN’ is the unshifted frequency of comb number N. Offset locking in the

OPLLs is required to realize this OFD stabilization scheme.


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vcw1 and vcw2 are combined in a high speed photodetector (EOTech, 3500F) to extract a 5-GHz RF

signal. Its absolute phase noise is measured with a SSA (Agilent E5052A).


The RF spectrum of the extracted RF signal, as shown in Figure 4-20(a), includes the

fundamental and second harmonic tones. The power in the fundamental tone is limited by the CW laser

power and large insertion losses from several system components, such the SSB modulator (-5 dB).

These technical issues can be improved to increase the optical power at photodetection to increase the

signal power. The second harmonic results from imperfect biasing in the SSB modulator; the residual

lower AM sideband beats with the upper PM sideband to generate the second harmonic. It is clear that

the extracted RF signal does not contain harmonics of the laser repetition rate, which is a major advantage

of the CW approach for RF extraction. The CW approach avoids repetition rate multiplication, which

require intracavity [80], [81] or external cavity pulse interleavers [82]–[84] and are limited by detector

nonlinearities at high pulse peak powers.

The measured phase noise of the extracted 5-GHz signal (see black solid) is shown in Figure

4-20(b). The free-run CW phase noise referred to 192 THz (see green solid) and its ideal OFD

performance divided down to 5 GHz (see green dashed) are shown for reference. In an ideal oscillator,

the extracted signal should approach the ideal OFD curve. In this demonstration, however, the extracted

signal is limited by the free-running MLL noise (see blue dashed). This is consistent with the OFD

Figure 4-20 Measurement results for a 5-GHz ESPIOR. (a) RF spectrum and (b) phase

noise of the extracted 5-GHz output signal.


116

demonstration in the previous section. Since the PZT feedback fixed point is far from the pump noise

fixed point, the suppression ratio is nearly 0 dB. Therefore, the frequency comb stability does not

improve from its free-run noise.

Despite this, the ESPIOR demonstration still successfully generates a low-noise RF signal. The

phase noise of the extracted RF signal is significantly lower than that of the free-run VCO (see black

dotted). The VCO noise is suppressed by 38 dB at 10 Hz offset, and the OFD ratio relative to the free-run

CW noise is 60 dB at 10 Hz offset. A linear, time-invariant noise model is simulated (see red dashed) to

confirm the experimental result. The OPLL locking bandwidths are clearly visible from the servo

resonances in the 10 kHz – 40 kHz frequency range. To achieve lower phase noise RF extraction, the

frequency comb stability must be improved.

Experimental Set-up (250-MHz stretched-pulse laser)

With a successful proof-of-concept demonstration, the testbed is improved further to demonstrate

the best possible RF extraction. Key system improvements in this final ESPIOR demonstration are

detailed below (see Figure 4-21 for the system schematic):

CW Reference: The CW laser is stabilized to a stable optical cavity via PDH locking to reduce its

linewidth by orders of magnitude. Slow and fast feedback via laser PZT tuning and external AOM

frequency shifting, respectively, are used to achieve a locking bandwidth in the 100 kHz range and an

Figure 4-21 Experimental set-up for the final ESPIOR testbed demonstration


117

in-band suppression of the CW noise by 60 dB at 10 Hz offset.

Optical Frequency Divider: The mode-locked laser is replaced by a stretched-pulse Er-fiber laser

(Menlo systems; M-fiber LH120) with a 250-MHz fundamental repetition rate. In addition to comb

line stabilization, f-2f interferometry is implemented to stabilize f0 via pump power modulation. This

fully-stabilized frequency comb is expected to achieve the maximum OFD ratio.

RF Extraction: Phase noise measurement electronics are optimized for RF extraction at 6 GHz instead

of 5 GHz. Furthermore, the OPLL electronics are improved to increase the electronic SNR for tighter

phase-locking.


The measured phase noise of the extracted 6-GHz signal (black) is shown in Figure 4-22. The

phase noise of the stable CW reference at 192 THz (green solid) and its ideal OFD performance at 6 GHz

(green dashed) are shown for reference. Note that the stable CW* noise is measured from the in-loop

PDH error signal and remains to be confirmed with an independent out-of-loop measurement. In contrast

to the first ESPIOR demonstration, the mode-locked laser here is a fully-stabilized frequency comb.

Direct detection of the frequency comb and measurement with the SSA shows that its noise (blue) is

buried by the SSA measurement floor (gold dashed), except for the servo resonance near 20 kHz due to

the pump feedback.

With this significant improvement in the frequency comb stability, the extracted RF signal is no

longer limited by the mode-locked laser. Instead, the extracted signal is limited by the electronic noise

floor of the PLLs (see red). The PLL noise can be suppressed further by optimizing the detection

electronics with customized components for higher SNR. Furthermore, since the PLL noise floor is

currently higher than the ideal OFD curve for a free-run CW reference, additional stabilization of the CW

reference to the ULE cavity does not improve the oscillator performance.

The oscillator performance in the high frequency range above 3 kHz is limited by the PLL

locking bandwidths. The built-in pump power and PZT modulation ports of the mode-locked laser have

limited bandwidths in the 10 kHz range. Intracavity amplitude and phase modulators should be inserted

into the laser cavity for faster feedback to extend the locking bandwidth into the MHz regime and achieve

higher in-band noise suppression for lower noise RF extraction.


118

Compared to the first ESPIOR demonstration, the extracted signal is improved overall by about

20 dB. The absolute phase noise of the extracted 6-GHz signal is -63 dBc/Hz at 10 Hz offset

and -94 dBc/Hz at 1 kHz offset. The OFD ratio measured relative to the CW free-run noise is 85 dB at

10 Hz offset and 61 dB at 1 kHz offset. The ideal OFD ratio of 90 dB is presumably achieved for low

offset frequencies in the 1 Hz – 10 Hz range since the measurement is limited by the SSA noise floor.

Additional ESPIOR demonstrations are performed for a partially-stabilized frequency comb (i.e.

with comb line stabilization only). This is analogous to an integrated ESPIOR where f-2f interferometry

is unavailable. The error signal from OPLL1 is first configured for pump feedback (see Figure 4-23(a)).

Since the mode-locked laser noise is pump-RIN-limited, direct feedback on the pump power yields good

overall suppression (-15 dB) of the free-run timing jitter, except for residual PZT-induced spurs. When

OPLL1 is configured for PZT feedback (see Figure 4-23(b)), the converse is true. The PZT-induced spurs

are highly suppressed, while the pump-RIN-induced noise experiences limited suppression (-5 dB).

These trends are consistent with the suppression ratios discussed in Section 4.2.5. For completeness,

ESPIOR demonstrations for a free-running and fully-stabilized frequency comb are plotted in Figure

4-23(c) and (d), respectively. In all four cases, the RF readout faithfully reproduces the stability of the

mode-locked laser (optical frequency divider) within the locking bandwidth.

Figure 4-22 Measurement results for a 6-GHz ESPIOR with a fully-stabilized frequency

divider.


119

Figure 4-23 Measurement results for a 6-GHz ESPIOR for various comb stabilitzation

schemes. (a) Pump feedback, (b) PZT feedback, (c) free-run, (d) fully-stabilized comb.


120

Chapter 5. Conclusion

121

5 Conclusion

Long-term stable optical timing distribution and synchronization is demonstrated over 4.7 km with

sub-fs stability, which is over an order-of-magnitude improvement from the first-generation TDS in terms

of both length coverage and timing stability. This will enable next-generation seeded X-ray FELs to

generate sub-fs X-ray pulses with unprecedented brightness to realize the long-standing scientific dream

to capture chemical and physical reactions with atomic-level spatiotemporal resolution. Key

demonstrations in the optical TDS developed in this thesis are highlighted as follows:

The timing jitter of a commercial solid-state mode-locked laser is characterized with sub-fs timing

jitter for frequencies above 1 kHz. This confirms its viability as an optical master oscillator to

support sub-fs timing distribution over link lengths up to 100 km.

Proof-of-concept demonstration with the 1.2-km PM fiber link showed that polarization-induced

timing drifts, which previously limited link stability to the sub-10-fs level, is eliminated to achieve

sub-fs stability over 16 days of continuous operation. This paved the way for the development of the

second-generation optical TDS based on PM fiber.

Timing distribution across a 4.7-km PM fiber network, which is composed of two independent links

of lengths 1.2 and 3.5 km, is demonstrated with a relative timing stability of 0.58 fs RMS over 52

hours of operation.

Remote laser-laser synchronization across the stabilized 3.5-km link is demonstrated with 0.2 fs RMS

residual jitter for over 44 hours of operation.

Third-generation BOMPDs are developed to enable phase detection with sub-fs precision

(-161 dBc/Hz noise floor) up to multi-MHz bandwidths as well as robust AM-PM suppression ratios

(>50 dB)

Optical-to-RF synchronization is demonstrated for over 24 hours of continuous operation with 0.98 fs

RMS drift integrated from 12 μHz to a 20-kHz locking bandwidth, while RF-to-optical

synchronization is demonstrated with 0.5 fs RMS from 1 Hz to a 20-kHz locking bandwidth.

These results confirm the feasibility of a complete optical TDS for achieving system-wide timing stability

at the sub-fs level across both optical and RF domains in next-generation light sources.

Chapter 5. Conclusion

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Lastly, a novel integrated photonic oscillator (ESPIOR) for ultralow noise microwave generation

is demonstrated in a discrete platform. Due to the unavailability of on-chip f-2f interferometry, the

ESPIOR performance is limited by the partially-stabilized frequency comb (i.e. comb line stabilization

only). A MIMO phase noise model is developed to investigate frequency comb dynamics such as cross-

modulation effects within an Er-fiber laser. An alternative feedback control using synchronous

modulation called quasi OFD is proposed and verified to extend the feedback capabilities for comb

stabilization. While the feedback fixed point can be adjusted to maximize the noise suppression for a

dominant noise source in a laser (e.g. 20 dB for a stretched-pulse Er-fiber laser), the presence of other

noise sources will limit further noise suppression. Therefore, QOFD cannot compete with the

performance of conventional OFD for ultra-low noise microwave generation. Absolute fo control is

necessary for full OFD. RF generation from the complete ESPIOR system is verified with the discrete

testbed. Initial ESPIOR demonstrations confirm that the RF readout scheme is fully functional and is

limited by the frequency comb stability. Improved ESPIOR performance using a lower-noise laser and

stable CW reference achieved -63 dBc/Hz at 10 Hz offset and -94 dBc/Hz at a 1 kHz offset for a 6-GHz

carrier frequency. This corresponds to an OFD ratio of 85 dB at 10 Hz offset and 61 dB at 1 kHz offset,

which is close to the ideal OFD ratio of 90 dB. If the ESPIOR can be fully-realized on a silicon photonics

chip, these compact, portable photonic oscillators can then deliver high-quality microwaves to working

environments that were previously unsuitable for rack-sized RF oscillators to operate.

Appendix A. Phase-Locked Loop Basics

123

Appendix A Phase-Locked Loop Basics

The phase-locked loop (PLL) is the basic building block for timing synchronization. A review of basic

PLL theory for RF oscillators is presented here. The same principles apply for optical PLLs.

Principle of Operation

The purpose of a PLL is to synchronize the phase of a slave oscillator to that of a master

oscillator. Each free-running master and slave oscillator generates a single RF frequency at ω1 = 2πf0 and

ω2 = 2π(f0+Δf), respectively, where Δf is the initial frequency difference between the oscillators. Their

phases are θ1 = ω1t and θ2 = ω2t + θ’, where θ’ is an arbitrary phase offset since the oscillators are

independent. The phase error between the two RF signals, Δθ = θ2 ‒ θ1, is detected in a phase detector to

generate an output error signal that is directly proportional to the phase error, Ve = KPD∙Δθ, where KPD is

the phase detector sensitivity. If a double-balanced mixer is used as the phase detector, the output voltage

signal is sinusoidal, KPD∙sin(Δθ), and is directly proportional only when the phase error is small. If a

digital phase detector is used, the voltage error signal maintains linearity across a larger phase detection

Figure A-1 Operating principle for a RF PLL (a) Basic PLL schematic; (b-c) Time-

domain illustration for lock acquisition using an analog or digital phase detector.


124

range (see Figure A-1). In either case, the voltage error signal is filtered by a PI controller and fed back

to the slave oscillator to adjust its frequency. If the error voltage is non-zero, the voltage will speed up or

slow down the slave oscillator frequency. Since phase is related to frequency through the basic

relationship θ = ωt, the phase of the slave oscillator will eventually be locked to that of the master

oscillator. The PI controller is optimized to suppress the error voltage as tightly as possible to zero, where

the phase detector is linear. This type-II, 2nd

-order PLL is the most common feedback control system and

is covered in-depth in literature.

Closed-Loop Transfer Functions

Figure A-2 Phase noise model for a RF PLL

The basic block diagram for the PLL in the phase domain is shown in Figure A-2. The transfer

function of each component is determined as follows:

Phase detector: To first-order approximation, the phase detector is linear when converting input

phase error to output voltage and has a transfer function HPD(s)=KPD.

Loop filter: Tight phase-locking with high accuracy requires a PI controller, which has a transfer

function HPI(s)=KP(2πfPI+s)/s, where KP is the proportional gain and fPI is the corner frequency

between proportional and integral control.

VCO: Voltage tuning of the oscillator frequency is linear, Δf2=KVCOΔvf, to first-order approximation.

Applying the Fourier relationship for frequency-to-phase integration sθ(s)=F(s), the transfer function

for the VCO from input voltage to output phase is HVCO(s)=KVCO/s.

With the individual transfer functions defined for each component, the closed-loop transfer

functions between various nodes can be calculated to analyze the PLL noise performance. Noise sources

can be injected at various input nodes and observed at any output node using the appropriate closed-loop

transfer functions. This method is used extensively to simulate the PLL performance in this thesis. The

transfer functions are derived using Black’s formula for a negative feedback loop,


125

𝐻𝑐𝑙𝑜𝑠𝑒𝑑 =𝐺𝑓𝑜𝑟𝑤𝑎𝑟𝑑

1 + 𝐺𝑙𝑜𝑜𝑝

where Gforward is the forward gain from an input node to an output node and Gloop is the open-loop gain.

For example, to calculate how the reference phase noise, θIN, propagates through to the slave oscillator,

θOUT, one calculates the forward gain from θIN to θOUT to be:

𝐺𝑓𝑜𝑟𝑤𝑎𝑟𝑑 = 𝐾𝑉𝐶𝑂𝐾𝐿𝐹𝐾𝑃𝐷

The open-loop gain is coincidentally the same as the forward gain:

𝐺𝑙𝑜𝑜𝑝 = 𝐾𝑉𝐶𝑂𝐾𝐿𝐹𝐾𝑃𝐷

The closed-loop transfer function is therefore

𝐻𝜃𝑜𝑢𝑡,𝜃𝑖𝑛=

𝐺𝑓𝑜𝑟𝑤𝑎𝑟𝑑

1 + 𝐺𝑙𝑜𝑜𝑝

=𝐾𝑉𝐶𝑂𝐾𝐿𝐹𝐾𝑃𝐷

1 + 𝐾𝑉𝐶𝑂𝐾𝐿𝐹𝐾𝑃𝐷

Loop Stability (Phase Margin)

The upper limit for tight locking is set by the phase margin of the feedback loop. Representing

the open-loop gain as a generalized phasor |A(f)|ejθ(f)

, the closed-loop response is |A(f)|ejθ(f)

/(1+|A(f)|ejθ(f)

).

The PLL locking bandwidth occurs near the unity-gain crossover frequency, i.e. |A(f)| = 1. The phase

response is critical at this frequency. If the phase response is out-of-phase at θ = 180°, then the pole in

the denominator renders the loop unstable. Phase margin is defined as the remaining phase until 180° is

reached, i.e. PM = θ – π. As safe practice, it is best to design for a 50-80° phase margin to achieve

reasonable servo resonances near the locking bandwidths. It is possible to extend the locking bandwidth

further by optimizing the phase response of the loop with additional phase compensation techniques, such

as lag-lead compensation. In a real system, one may need to model higher-order zeros and poles in the

system since they may impact the amplitude and phase response at higher frequencies and limit loop

stability.

Locking Bandwidth

The feedback loop can be conditioned for a range of locking bandwidths with a reconfigurable PI

controller. The two ends of the locking spectrum are:


126

Loose-locking, where the slave oscillator tracks only slow phase drifts in the master oscillator and

leaves the high-frequency noise uncontrolled. This corresponds to a low locking bandwidth, which is

achieved by decreasing the loop gain and fPI.

Tight-locking, where the slave oscillator is tightly synchronized to all phase changes in the master

oscillator, including both slow and fast phase changes. This corresponds to a high locking bandwidth,

which requires high loop gain and fPI.

Both locking conditions are desired in the experiments. Loose-locking is a common practice for

characterizing the absolute noise of a pair of oscillators. The out-of-band noise, i.e. noise beyond the

locking bandwidth, is unaffected by the PLL and represents the absolute noise of the oscillators. The

voltage output of the phase detector is directly measured with a FFT spectrum analyzer and divided by the

phase detector sensitivity to yield the absolute phase noise spectrum. If the noise spectrum within the

locking bandwidth needs to be recovered, the closed-loop PLL transfer function can be characterized to

extract the absolute phase noise from the measured in-band noise, i.e. noise within the locking bandwidth.

Tight-locking is used in practice to tightly synchronize a noisy oscillator to a low-noise reference

oscillator. Intermediate locking is optimized for case-specific applications. For example, if the slave

oscillator has inherently lower noise than the reference oscillator above a certain offset frequency, then a

medium-lock is ideal to achieve a locking bandwidth equal to this cut-off to achieve low phase noise

performance across all frequencies.

Appendix B. Noise Model for Optical-Optical Synchronization

127

Appendix B Noise Model for Optical-Optical Synchronization

Feedback Noise Model

Figure B-1 Feedback noise model for optical-optical synchronization.

The feedback model for optical-optical synchronization using an optical PLL is given in Figure

B-1. The BOC is used as the timing (phase) detector. HBOC, HPI and HVCO are the transfer functions for

the BOC, PI controller and slave laser, respectively. HBOC is the timing sensitivity Kb of the BOC and is a

constant to first-order approximation. HPI can be used for proportional gain KP only or for proportional-

integral control KP(2πfPI+s)/s. HVCO is the repetition-rate tuning sensitivity Kv of the slave laser oscillator

multiplied by the frequency-to-time conversion term 1/(s∙frep), where frep is the repetition rate of the master

laser. Sv,BOC and Sv,PI are the additive voltage noise PSD (V2/Hz) of the BOC and PI controller referred to

their outputs. St,master and St,slave are the free-running TJSD (fs2/Hz) of the master and slave laser.

Closed-loop transfer functions are derived below to determine how these noise sources propagate

through the closed-loop system. Two output nodes are of interest: 1) the relative timing jitter at the BOC

input and 2) the absolute timing jitter of the slave laser at the PLL output.

Relative Timing Jitter

Relative timing jitter spectral density between the lasers referred to the BOC input is:

𝑆𝛥𝑡,𝑒𝑟𝑟𝑜𝑟 = |1

1 + 𝐻𝐵𝑂𝐶 𝐻𝑃𝐼𝐻𝑉𝐶𝑂

|2

(𝑆𝑡,𝑚𝑎𝑠𝑡𝑒𝑟 + 𝑆𝑡,𝑠𝑙𝑎𝑣𝑒) + |𝐻𝑃𝐼𝐻𝑉𝐶𝑂

1 + 𝐻𝐵𝑂𝐶 𝐻𝑃𝐼𝐻𝑉𝐶𝑂

|2

𝑆𝑣,𝐵𝑂𝐶 + |𝐻𝑉𝐶𝑂

1 + 𝐻𝐵𝑂𝐶𝐻𝑃𝐼𝐻𝑉𝐶𝑂

|2

𝑆𝑣,𝑃𝐼

The four driving noise terms can be classified as signal and background noise: St,master and St,slave are the

signal terms while Sv,BOC and Sv,PI terms are the background noise terms. SΔt,error is obtained

experimentally by measuring the voltage PSD at the BOC output, Sv,e, with a RF spectrum analyzer and

dividing by the BOC sensitivity KB to refer to the BOC input.


128

Absolute Timing Jitter

The absolute timing jitter of the slave laser at the PLL output is

𝑆𝑡,𝑜𝑢𝑡 = |𝐻𝐵𝑂𝐶𝐻𝑃𝐼𝐻𝑉𝐶𝑂


|2

𝑆𝑡,𝑚𝑎𝑠𝑡𝑒𝑟 + |𝐻𝑃𝐼𝐻𝑉𝐶𝑂


|2

𝑆𝑣,𝐵𝑂𝐶 + |𝐻𝑉𝐶𝑂


|2

𝑆𝑣,𝑃𝐼 + |1


|2

𝑆𝑡,𝑠𝑙𝑎𝑣𝑒

St,out can be measured by directly detecting the slave laser, filtering a RF harmonic of the repetition rate,

and measuring its absolute phase noise with a SSA. If the noise floor of the SSA is too high or AM-PM

conversion at detection is an issue, the absolute timing jitter can be calculated from the measured relative

timing jitter, if the closed-loop PLL transfer function is properly characterized.

Locking Bandwidth

The transfer function for the input master laser jitter to the output slave laser jitter can be solved for the

PLL locking bandwidth. By setting the magnitude squared of this coefficient to ½ and s=j2πf3dB, the -3dB

loop bandwidth f3dB can be calculated:

|𝐻𝐵𝑂𝐶 𝐻𝑃𝐼𝐻𝑉𝐶𝑂


|2

=1

2

For tight-locking (proportional-integral control)

The loop filter transfer function is:

𝐻𝑃𝐼 =𝐾𝑃(2𝜋𝑓𝑃𝐼 + 𝑠)

𝑠

Relative timing error between the two lasers is:

𝑆𝛥𝑡,𝑒𝑟𝑟𝑜𝑟 = |𝑠2

𝑠2 + 𝐾′(2𝜋𝑓𝑃𝐼 + 𝑠)|

2

(𝑆𝑡,𝑚𝑎𝑠𝑡𝑒𝑟 + 𝑆𝑡,𝑠𝑙𝑎𝑣𝑒) + |(𝐾𝑃𝐾𝑉/𝑓𝑟𝑒𝑝)(2𝜋𝑓𝑃𝐼 + 𝑠)

𝑠2 + 𝐾′(2𝜋𝑓𝑃𝐼 + 𝑠)|

2

𝑆𝑣,𝐵𝑂𝐶 + |𝐾𝑉/𝑓𝑟𝑒𝑝

𝑠2 + 𝐾′(2𝜋𝑓𝑃𝐼 + 𝑠)|

2

𝑆𝑣,𝑃𝐼

Absolute timing error of slave laser is:

𝑆𝑡,𝑜𝑢𝑡 = |𝐾′(2𝜋𝑓𝑃𝐼 + 𝑠)

𝑠2 + 𝐾′(2𝜋𝑓𝑃𝐼 + 𝑠)|

2

𝑆𝑡,𝑚𝑎𝑠𝑡𝑒𝑟 + |(𝐾′/𝐾𝐵)(2𝜋𝑓𝑃𝐼 + 𝑠)

𝑠2 + 𝐾′(2𝜋𝑓𝑃𝐼 + 𝑠)|

2

𝑆𝑣,𝐵𝑂𝐶 + |(𝐾′/𝐾𝐵𝐾𝑃)𝑠

𝑠2 + 𝐾′(2𝜋𝑓𝑃𝐼 + 𝑠)|

2

𝑆𝑣,𝑃𝐼 + |𝑠2

𝑠2 + 𝐾′(2𝜋𝑓𝑃𝐼 + 𝑠)|

2


The locking bandwidth is:

𝑓3𝑑𝐵 =1

2𝜋√

𝐾′

2[(𝐾′ + 2𝜛𝑃𝐼) + √(𝐾′ + 2𝜛𝑃𝐼)2 + 4𝜛𝑃𝐼

2 ]

where K’ = KBKPKV/frep.


129

For loose-locking (proportional gain only)

The loop filter transfer function is:

𝐻𝑃𝐼 = KP

Relative timing error between the two lasers is:

𝑆𝛥𝑡,𝑒𝑟𝑟𝑜𝑟 = |𝑠

𝑠 + 𝐾′|

2

(𝑆𝑡,𝑚𝑎𝑠𝑡𝑒𝑟 + 𝑆𝑡,𝑠𝑙𝑎𝑣𝑒) + |(𝐾𝑃𝐾𝑉/𝑓𝑟𝑒𝑝)

𝑠 + 𝐾′|

2

𝑆𝑣,𝐵𝑂𝐶 + |𝐾𝑉/𝑓𝑟𝑒𝑝

𝑠 + 𝐾′|

2

𝑆𝑎,𝑃𝐼

Absolute timing error of slave laser is:

𝑆𝑡,𝑜𝑢𝑡 = |𝐾′

𝑠 + 𝐾′|

2

𝑆𝑡,𝑚𝑎𝑠𝑡𝑒𝑟 + |(𝐾′/𝐾𝐵)

𝑠 + 𝐾′|

2

𝑆𝑣,𝐵𝑂𝐶 + |(𝐾′/𝐾𝐵𝐾𝑃)

𝑠 + 𝐾′|

2

𝑆𝑣,𝑃𝐼 + |𝑠

𝑠 + 𝐾′|

2


The locking bandwidth is:

𝑓3𝑑𝐵 =1

2𝜋𝐾′

Simulation for Laser Jitter Characterization

The optical PLL is simulated to verify the laser jitter measurements in Section 2.3.1. Assume that

the absolute laser timing jitter is ASE-limited with a f -2

slope and that the master laser jitter is -5 dB

below that of the slave laser (see black curves in Figure B-2). The second assumption is for pedagogical

purposes; the lasers in the experiments are identical with 0 dB noise difference. Typical values for the

transfer coefficients are KB = 1 mV/fs, KV = 17.4 Hz/V, frep = 216 MHz, and KP = -20 dB. The BOC noise

floor is based on specifications for a commercial low-noise BPD.

The simulated relative timing jitter between the two lasers as measured by the BOC is shown in

blue. Beyond the 2-kHz locking bandwidth, the measured jitter is the combined free-running jitter of the

two lasers until it is buried by the BOC noise floor. If both lasers had contributed equal jitter, the jitter of

a single laser is 3 dB lower than the measurement. In this example, however, the noisier laser clearly

dominates the measurement. Within the locking bandwidth, the f -2

laser jitter is suppressed with

a -20 dB/decade slope to yield a flat residual noise spectrum. The simulated absolute timing jitter of the

slave laser is shown in red. There are two key differences between the absolute and relative timing jitter.

Beyond the locking bandwidth, the slave laser maintains its free-running jitter spectrum. Within the

locking bandwidth, the slave laser noise approaches that of the master laser. A residual timing jitter less


130

than the absolute phase noise means that the slave laser is synchronized to the master laser noise with a

higher level of precision.

Figure B-3 shows the simulated relative timing jitter for a 40-dB range of proportional gain

values to verify the laser jitter measurements in Section 2.3.1. These results agree with those

experimentally obtained in Figure 2-5.

Figure B-3 Simulation for laser jitter characterization under loose-locking. Relative timing

error spectra shown for a range of propotional gain values (color curves)

Figure B-2 Example simulations for optical-optical synchronization. Relative and

absolute timing jitter for the slave oscillator under loose-locking.


131

Simulation for laser-to-laser synchronization

For tight synchronization, the PI controller is used with high loop gain to achieve the maximum

locking bandwidth and in-loop noise suppression. The additional integral control improves the in-loop

suppression by another -20 dB/decade. A 2nd

-order pole at 10 kHz is included to simulate the PZT

resonance of the slave laser. When the gain is too low, the phase margin decreases and causes the servo

resonance to increase. When the gain is too high, the sharp decrease in phase margin due to the PZT

resonance causes the feedback loop resonance to sharpen just below the PZT resonance frequency. This

simulation accurately explains the locking behavior observed in the experiments. Care is needed to adjust

both the PI corner frequency and proportional gain to achieve the best feedback loop response at the

locking bandwidth. Here, it occurs for KP = -20 dB.

Figure B-4 Simulation for optical-optical synchronization under tight locking. Servo

resonances caused by reduced phase margins at low and high gain values


132

Appendix C. Noise Model for Timing Link Stabilization

133

Appendix C Noise Model for Timing Link Stabilization


Figure C-1 Feedback noise model for timing link stabilization

The basic feedback model for timing link stabilization is shown in Figure C-1. The arrival time

of the round trip pulse tRT is compared against that of a new laser pulse tIN. HBOC is the timing sensitivity

of the BOC and in the simplest treatment is a constant KB in units of V/s. For high-frequency modelling,

this sensitivity is multiplied by the frequency response of the BPD, F(s), to include the gain bandwidth

roll-off and phase response. HPI is the transfer function for the PI controller and is equal to KP(2πfPI+s)/s.

HODL is the transfer function for the variable ODL within the link path, which converts input voltage to

timing adjustments of the link pulse tadj. KPZT is the response of the PZT tuning element in the ODL in

units of m/V and is divided by c, the speed of light for free-space propagation, for conversion to effective

time delay. The ODL affects the link pulse bidirectionally, i.e. tadj is injected at the beginning of the

forward pass and at the end of the reverse pass. HD represents the time delay T for single-pass

propagation through the fiber link.

Additive noise from each component is injected at its output. Sv,BOC is the voltage noise (V2/Hz)

in the BPD due to amplified thermal noise at detection. Sv,PI is the voltage noise (V2/Hz) of the PI

controller. For these electronic components, power-law noise at low offset frequencies such as 1/f flicker

noise may exist. St,laser and St,link are the free-running timing jitter spectra (fs2/Hz) of the laser and link

noise, respectively.


134

Transfer Coefficients

The open-loop gain is simulated in Figure C-2 to analyze the PLL stability. The gain magnitude

response is plotted for increasing proportional gain KP in the PI controller, where blue is low gain and red

is high gain. The PLL bandwidth occurs near the unity-gain crossover frequency, and the corresponding

phase margin determines the PLL stability (i.e. the magnitude of the servo resonance). For example, a

unity-gain crossover frequency of about 1 kHz (see orange curve) yields a phase margin of 130°, which

yields overdamped, but stable PLL operation. Ideal phase margin is around 65°.

The link stabilization feedback scheme differs from the conventional type-II, second-order PLL

(see Appendix A). The feedback on HODL directly controls the pulse timing tadj and therefore does not

have a frequency-to-time integration 1/s term. Without this, the phase margin is strictly greater than 90°,

which will theoretically yield stable locking for any PLL bandwidths. In practice, higher-order phase

responses of system components such as photodetectors (via F(s), which is not modelled here) and PZT

resonances will degrade the phase margin and limit the maximum achievable PLL bandwidth.

Furthermore, the absence of the 1/s term degrades the in-band noise suppression slope from 40 dB/decade

to 20 dB/decade only. The 40 dB/decade slope can be recovered by implementing a PI2 controller with a

second integrator, though careful design of the phase margin will be required.

Figure C-2 Open-loop gain response for link stabilization. (left axis) amplitude response for

a range of proportional gain values; (right axis) phase response in terms of phase margin.

While the open-loop gain is a useful design guideline, the actual PLL response is best understood

with the closed-loop transfer functions. The transfer functions for the four dominant noise sources in the

system are computed in Figure C-3. Noise in the optical domain (i.e. laser and link noise) are referred to

the BOC input, while noise in the RF domain (i.e. BOC and PI controller noise) are referred to the BOC

output to facilitate comparison.


135

The laser noise is suppressed via delayed self-homodyne detection after round-trip link propagation.

The round-trip pulse and reference pulse timing are subtracted in the BOC, which effectively filters

the laser noise by a sin(2πfT) function, where T is the single-pass link propagation delay. This can be

visualized as a high-pass filter with unity gain and +20 dB/dec slope suppression for low frequencies,

in addition to nulls at f = 1/T(n/2), where n is an integer. For example, the null locations for a 3.5 km

link are at multiples of 28.6 kHz. For increasing gain, the PLL bandwidth increases from below 1 Hz

to 10 kHz. The integral control improves the in-band suppression from +20 dB/dec to +40 dB/dec.

The link noise behaves differently from laser noise since the noise injected adds in-phase and is

filtered instead by a cos(2πfT) function. The null resonances occur at f = 1/T(1/4+n/2), which begin

at 14.3 kHz for n = 1. The link noise is high-pass filtered with an in-band noise suppression slope of

+20 dB/dec due to the integral control. High gain is desirable for large PLL bandwidths and high in-

band noise suppression.

BOC noise is low-pass filtered with a -20 dB/dec roll-off and is also filtered by the cos(2πfT)

function. Increasing the gain introduces more BOC noise into the system within the PLL bandwidth.

Figure C-3 Closed-loop transfer coefficients for link stabilization. (a) master laser timing

jitter and (b) link timing jitter referred to the BOC input and (c) BOC voltage noise and

(d) PI controller voltage noise referred to the BOC output.


136

PI controller noise is high-pass filtered similar to link noise. The overall noise suppression is

proportional to KBKPZT. If the PI controller is high enough, the loop gain in the denominator of the

transfer function increases to improve the noise suppression further.

Noise Model for Experiments

Figure C-4 Noise model for link stabilization demonstrations. Includes in-loop locking and

out-of-loop measurement set-ups.

The feedback model is expanded upon (see Figure C-4) to verify experimental link

demonstrations in Section 2.3.2 and 2.3.3. The basic feedback model is depicted in the in-loop section.

A more involved treatment of link noise is adopted. The link path is sub-divided and injected with

coherent link noise at all locations. This simulates laboratory conditions where the link is wound around

a common spool and experiences the same noise perturbations throughout link. This may not apply for a

real facility implementation where the link noise is injected at discrete locations along the link. (In this

latter case, the link noise will achieve null locations at different frequencies depending on its injection

location.) With the in-loop PLL engaged, the link output and reference pulse are measured in a second

BOC for an out-of-loop measurement. Higher-order frequency responses are included in this model; in

particular, the 100-kHz bandwidth response with -40 dB/dec roll-off is modelled for the balanced

detector, i.e. F(s) = (1 + jf/f-3dB)-2

, where f-3dB is the detector bandwidth.


137

Experiment vs Simulation

The noise model is simulated to verify both 1.2-km and 3.5-km link demonstrations covered in this thesis.

1.2-km Link Stabilization

Simulated and experimental results for the 1.2-km link stabilization are shown in Figure C-5.

The free-run timing jitter spectra of the laser and link noise are shown for comparison (see solid black).

The laser noise is obtained via direct detection of the laser pulses and measurement with a SSA in the

10 Hz – 50 kHz range. The link noise with f -2

slope is obtained from the long-term in-loop measurement

in Figure 2-7 in the 10-4

– 100 Hz range. The jitter spectra are extrapolated to cover the full 10

-4 – 10

6 Hz

range. Due to the delayed self-homodyne detection of the laser, the inherent laser jitter is passively

suppressed (see black dashed). With link stabilization engaged, the effective laser noise is actively

suppressed further (see black dotted). The locking bandwidth shown is in the few 100 Hz range. This

actively-suppressed laser noise serves as the noise floor for measuring link noise. Another limitation is

the BOC noise (see red dashed). The BOC noise is measured from the baseband output of the BPD with a

RF spectrum analyzer and dividing by the BOC timing sensitivity to convert to timing jitter. It exhibits

f -1

flicker noise with a corner frequency near 10 Hz. The “hump” near 100 kHz is due to electronic noise

Figure C-5 Experiment versus simulation for 1.2-km link stabilization


138

in the detector. The bandwidth response beyond 100 kHz decays with a -40 dB/dec roll-off.

The out-of-loop noise for simulation (see blue) and experiment (see green) are in good agreement.

Beyond the locking bandwidth, the measurement is limited by the inherent laser jitter and detector

bandwidth roll-off. Below the locking bandwidth, the link noise is suppressed by 20 dB/dec, which yields

a flat residual noise spectrum in the 10-2

– 102 Hz range. At lower frequencies below 10 mHz, experiment

diverges from simulation with a f -2

slope. This long-term drift is suspected to be from link power

fluctuations, which is not included in the noise model. Since the electronic noise floor of the BOC is still

many orders-of-magnitude lower, the long-term system performance can be further improved upon.

3.5-km Link Stabilization

Simulated and experimental results for 3.5-km link stabilization are shown in Figure C-6. The

laser noise is now the measured f -3

jitter spectrum of the OneFive laser. With this slope increase, the

passive suppression of the laser noise follows a -10 dB/dec slope. The f -2

link noise is improved slightly

with improved optical set-ups and laboratory conditions. The BOC noise floor is improved by two

orders-of-magnitude since the BOC timing sensitivity is larger. The same analysis as that for the 1.2-km

link is applied. The simulation (see blue) and experimental result (see green) are in good agreement. The

divergence at low frequencies is again suspected to be from link power fluctuations.

Figure C-6 Experiment versus simulation for 3.5-km link stabilization.

Appendix D. Noise Model for Optical-RF Synchronization

139

Appendix D Noise Model for Optical-RF Synchronization


Figure D-1. Feedback noise model for optical-RF synchronization with a BOMPD.

The basic feedback model for an opto-electronic PLL using a BOMPD is shown in Figure D-1.

The phase of the slave oscillator ϕOUT is synchronized to that of the master oscillator ϕIN. HPD is the phase

detection sensitivity the Sagnac interferometer KSGI multiplied by the detector responsivity R. HAMP is the

gain response of the post-detection electronics, which includes amplifiers, filters, and a down-conversion

mixer. The overall BOMPD phase sensitivity in units of V/rad is KPD = KSGIRGA. In the simplest

treatment, this sensitivity is constant and wideband. HPI and HVCO are the transfer functions for the PI

controller and slave oscillator, respectively. HPI is KP(2πfPI+s)/s for proportional-integral control and

HVCO is the frequency-to-time conversion 1/s multiplied by the frequency tuning sensitivity Kv of the slave

oscillator. The latter is multiplied by a function F(s) to include the higher-order frequency response of the

oscillator feedback control. This is important for accurately modelling the loop phase margin for PLL

stability near the locking bandwidth.

Additive noise from each component is injected at its output. Sv,PD is the voltage noise floor at

photodetection due to thermal noise or shot noise. Sv,AMP is the voltage noise of the post-detection

amplifiers and is dominated by the noise figure of the first amplifier stage. Sv,PI is the voltage noise of the

PI controller. For these electronic components, power-law noise at low offset frequencies such as 1/f

flicker noise also exist. Sϕ,master and Sϕ,slave are the free-running phase noise spectra (rad2/Hz) of the master

and slave oscillators, respectively. For laser oscillators, its timing jitter spectral density St is divided by

the RF oscillator operating frequency 2πfRF to obtain its equivalent phase noise Sϕ.


140

Transfer Coefficients

The open-loop frequency response for this noise model is simulated in Figure D-2 to analyze the

PLL stability. The open-loop gain response is plotted for increasing proportional gain in the PI controller

(blue = low gain; red = high gain). The locking bandwidth occurs at the unity gain, and the corresponding

phase margin determines the stability of the PLL servo resonance. For low and high gain, the phase

margin is below 30°, which will yield a large servo resonance. For medium gain, the phase margin is

near the optimal 45° to 60° range, which will yield stable locking.

Figure D-2 Open-loop gain response for optical-RF synchronization. (left axis) amplitude

response for varying proportional gain values and (right axis) phase response in terms of

phase margin.

These trends are best illustrated with the closed-loop transfer functions for the four dominant noise

sources. The transfer functions are referred to the BOMPD input for direct comparison.

The master and slave oscillator noise is high-pass filtered with unity gain and suppressed with a

+40 dB/dec slope within the locking bandwidth. High gain is desirable for larger locking bandwidths

and higher noise suppression. For low and high gain, the decreased phase margin causes an

appreciable servo resonance at the locking bandwidth.

Photodetection noise is similarly high-pass filtered; however, since this noise is added after phase

detection in the Sagnac interferometer, it is effectively amplified by HPD-2

. Therefore, it is desirable

to maximize HPD first before GA when optimizing the overall BOMPD sensitivity.

PI controller noise is bandpass-filtered with moderate in-band suppression. Typically, this noise

contribution is negligible because the PI controller is the last amplification stage. The PLL noise is

dominated by the photodetection noise.


141

Figure D-3 Closed-loop transfer coefficients for optical-RF synchronization referred to the

BOMPD input for (a) master oscillator phase noise, (b) slave oscilator phase noise (c)

photodetection noise and (d) PI controller noise.

Noise Model for Experiments

Figure D-4 Noise model for experimental BOMPD demonstrations. Includes in-loop locking

and out-of-loop measurement set-ups.


142

The basic feedback model is expanded upon (see Figure D-4) to verify experimental

demonstrations in Section 3.3. The basic model previously describe is depicted as the in-loop section. A

low-noise DC preamplifier with gain GPRE is used to reduce the measurement instrument noise floor. The

phase noise at the input of the BOMPD is calculated by dividing the measured voltage noise spectrum of

vm by HPDHAMPGPRE. When the in-loop PLL is engaged, the residual phase between the two oscillator

signals are measured in a second BOMPD for an out-of-loop measurement.

Experiment vs Simulation

The noise model is simulated to verify all key BOMPD demonstrations covered in this thesis.

Second-Generation BOMPD

Simulations for remote-station locking using the second-generation BOMPD are shown in Figure

D-5. The DRO is synchronized to the MLL. The absolute phase noise spectra for the DRO and MLL are

shown in black. The noise floor for the in-loop and out-of-loop BOMPDs are set by the thermal noise

floor at photodetection. The out-of-loop BOMPD is configured with a lower noise floor than the in-loop

BOMPD to measure the PLL performance. The in-loop noise consists of master oscillator noise, slave

oscillator noise, and the BOMPD noise floor. With locking engaged, the in-loop noise is suppressed with

a -40 dB/decade slope to levels much lower than the BOMPD noise floor. Physically, this means that the

DRO is being synchronized to the BOMPD noise floor in addition to the MLL noise. This is confirmed

with the out-of-loop measurement (red solid), which shows that the slave oscillator is synchronized (and

Figure D-5 Simulation for remote-station locking with 2nd

-generation BOMPDs.


143

limited) to the in-loop detection noise floor. The expected absolute phase noise spectrum of the

synchronized DRO is shown in magenta. These simulations verify the experimental demonstration in

Figure 3-4.

Third-Generation BOMPD

Simulations for remote-station and base-station locking using the third-generation BOMPDs are

shown in Figure D-6 and Figure D-7, respectively. The SLCO begins with lower noise than the laser.

The noise floor for the in-loop and out-of-loop BOMPDs is set by shot noise at photodetection due to the

increase in optical power. A PI2 controller is used to achieve tighter locking at low frequencies (e.g.

below 100Hz, the noise suppression is -60dB/dec with double integration rather than -40dB/dec with a

single integrator). For base-station locking, the reduced phase margin due to the laser PZT resonance

induces a moderate servo resonance near the locking bandwidth. These simulations verify the

experimental demonstrations in Figure 3-9 and Figure 3-11.

Figure D-6 Simulation for remote-station locking with 3rd

-generation BOMPDs.


144

Figure D-7 Simulation for base-station locking with 3rd

-generation BOMPDs.

Appendix E. Derivation for Fixed Point Modulation Theory

145

Appendix E Derivation for Fixed Point Modulation Theory

Single Perturbation Source

For a perturbation source X in a mode-locked laser, its effect on the mth comb line νm is:

𝜕𝑣𝑚

𝜕𝑋= 𝑚

𝜕𝑓𝑟

𝜕𝑋+

𝜕𝑓𝑜

𝜕𝑋

Using the concept of fixed point modulation [78], [79], this perturbation Xi will cause the frequency comb

to "breathe" (i.e. expand and contract) with respect to a fixed comb line of mode number mfix. Note that

mfix is not restricted to integer values.

𝜕𝑣𝑚𝑓𝑖𝑥,𝑋

𝜕𝑋= 𝑚𝑓𝑖𝑥,𝑋

𝜕𝑓𝑟

𝜕𝑋+

𝜕𝑓𝑜

𝜕𝑋→ 0

𝑚𝑓𝑖𝑥,𝑋 = −

𝜕𝑓𝑜

𝜕𝑋𝜕𝑓𝑟

𝜕𝑋

The mth comb line perturbation due to X can be re-expressed as:

𝜕𝑣𝑚

𝜕𝑋𝑖

= (𝑚 − 𝑚𝑓𝑖𝑥,𝑋𝑖)𝜕𝑓𝑟

𝜕𝑋𝑖

Many Perturbation Sources

In general, there may be many pathways that noise sources can perturb the repetition rate noise of the

laser. To name a few:

Pump RIN gain and pulse energy fluctuations timing jitter

Pump RIN gain and pulse energy fluctuations RIN ( timing jitter)

Mirror vibrations length fluctuations δLfree-space timing jitter

Mirror vibrations beam misalignment RIN ( timing jitter)

Thermal fluctuation length fluctuations δLfree-space and index modulation δn timing jitter

Thermal fluctuation beam misalignment RIN ( timing jitter)

ASE (from gain & loss) pulse energy fluctuations δEpulse RIN ( timing jitter)

ASE (from gain & loss) center frequency fluctuation δνcenter GVD indirect timing jitter

ASE (from gain & loss) direct timing jitter


146

Quantum (complex amplitude) noise timing jitter

Quantum (complex amplitude) noise amplitude noise ( timing jitter)

Note that many perturbations sources can contribute to timing jitter and RIN directly. Moreover, the RIN

can couple back into timing jitter through second-order effects (i.e. internal laser feedback dynamics).

For the treatment here, first-order effects on timing jitter are considered.

Let there be N number of noise sources Xi (i=1,2…,N) that contribute to the overall noise in the comb line

νm and repetition rate fr.

𝑆𝛥𝜈𝑚 ,𝑛𝑜𝑖𝑠𝑒 = ∑ 𝑆𝛥𝜈𝑚,𝑋𝑖

𝑁

𝑖=1

= ∑(𝑚 − 𝑚𝑓𝑖𝑥,𝑋𝑖)

𝑁

𝑖=1

𝜕𝑓𝑟

𝜕𝑌𝑖

𝑆𝛥𝑋𝑖

𝑆𝛥𝑓𝑟,𝑛𝑜𝑖𝑠𝑒 = ∑ 𝑆𝛥𝑓𝑟 ,𝑋𝑖

𝑁

𝑖=1

= ∑𝜕𝑓𝑟

𝜕𝑋𝑖

𝑆𝛥𝑋𝑖

𝑁

𝑖=1

Feedback stabilization of vm via cavity length modulation

Feedback controls for a frequency comb typically consist of cavity length modulation ΔL and pump

power modulation ΔP. Assume that the stabilization of νm to the CW reference is achieved with a single

feedback on ΔL. The change in comb line νm and repetition rate fr due to ΔL is

𝑆𝛥𝑓𝑟,𝑐𝑡𝑟𝑙 =𝜕𝑓𝑟

𝜕𝐿𝑆𝛥𝐿

𝑆𝛥𝜈𝑚,𝑐𝑡𝑟𝑙 = (𝑚 − 𝑚𝑓𝑖𝑥,𝛥𝐿)𝜕𝑓𝑟

𝜕𝐿𝑆𝛥𝐿

With feedback engaged, the feedback loop will minimize the error in the comb line νm

𝑚𝑖𝑛 (𝑆𝛥𝜈𝑚,𝑐𝑡𝑟𝑙 + 𝑆𝛥𝜈𝑚,𝑛𝑜𝑖𝑠𝑒) → 0

This yields

(𝑚 − 𝑚𝑓𝑖𝑥,𝛥𝐿)𝜕𝑓𝑟

𝜕𝐿𝑆𝛥𝐿 + ∑(𝑚 − 𝑚𝑓𝑖𝑥,𝑋𝑖

)𝜕𝑓𝑟

𝜕𝑋𝑖

𝑆𝛥𝑋𝑖

𝑁

𝑖=1

= 0

(𝑚 − 𝑚𝑓𝑖𝑥,𝛥𝐿)𝑆𝛥𝑓𝑟,𝐿 + ∑(𝑚 − 𝑚𝑓𝑖𝑥,𝑋𝑖)𝑆𝛥𝑓𝑟 ,𝑋𝑖

𝑁

𝑖=1

= 0


147

𝑆𝛥𝑓𝑟,𝐿 = −1

(𝑚 − 𝑚𝑓𝑖𝑥,𝛥𝐿)∑(𝑚 − 𝑚𝑓𝑖𝑥,𝑋𝑖

)𝑆𝛥𝑓𝑟 ,𝑋𝑖

𝑁

𝑖=1

Residual timing jitter frep

Assuming perfect comb line stabilization, residual repetition rate noise may still exist. The residual

timing jitter of fr can be calculated:

𝑆𝛥𝑓𝑟= 𝑆𝛥𝑓𝑟,𝐿 + ∑ 𝑆𝛥𝑓𝑟 ,𝑋𝑖

𝑁

𝑖=1

𝑆𝛥𝑓𝑟= (−

1

(𝑚 − 𝑚𝑓𝑖𝑥,𝛥𝐿)∑(𝑚 − 𝑚𝑓𝑖𝑥,𝑋𝑖

)𝑆𝛥𝑓𝑟 ,𝑋𝑖

𝑁

𝑖=1

) + ∑ 𝑆𝛥𝑓𝑟 ,𝑋𝑖

𝑁

𝑖=1

𝑆𝛥𝑓𝑟= ∑ (1 −

𝑚 − 𝑚𝑓𝑖𝑥,𝑋𝑖

𝑚 − 𝑚𝑓𝑖𝑥,𝛥𝐿

) 𝑆𝛥𝑓𝑟,𝑋𝑖

𝑁

𝑖=1

𝑆𝛥𝑓𝑟= ∑ (

𝑚𝑓𝑖𝑥,𝑋𝑖− 𝑚𝑓𝑖𝑥,𝛥𝐿


) 𝑆𝛥𝑓𝑟 ,𝑋𝑖

𝑁

𝑖=1

This is an important equation. It states that the fixed point relationships will determine how well the

noise source Xi will be suppressed. The noise suppression ratio R for perturbation source Xi due to

feedback control ΔL can be defined as:

𝑅𝑋𝑖,𝛥𝐿 = (𝑚𝑓𝑖𝑥,𝑋𝑖

− 𝑚𝑓𝑖𝑥,𝛥𝐿


)


148

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sub-femtosecond optical timing distribution for next-generation light sources

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