Stable Delay of Arbitrary Radio-Frequency Waveforms over

Stable Delay of Arbitrary

Radio-Frequency

Waveforms over Optical

Fibers

Assaf Ben-Amram

Submitted in partial fulfillment of the requirements for the Master's

Degree in the Faculty of Engineering, Bar-Ilan University

Ramat-Gan, Israel 2015

This work was carried out under the supervision

of Prof. Avi Zadok from the Faculty of

Engineering, Bar-Ilan University

ACKNOWLEDGEMENTS

Now that the work is done, I can gladly say that the last two years were worth it. The

knowledge and experience I gained throughout this research are highly valuable and

useful. However, it could only happen with the help and guidance of many

individuals whom I mention next.

First and foremost, my full gratitude goes to my advisor, Prof. Avi Zadok. He warmly

welcomed me to his research group and gave me the opportunity to work with him.

Besides sharing his great knowledge and experience with me, he also patiently

guided me during this research and always treated me as equals, which is absolutely

not to be taken for granted. The writing of this thesis could not be completed

without his constant and endless support. His friendliness to his group members

induced a comfort environment that made the research process much less stressful.

For these and more, I am truly grateful.

Special thanks are given to Yoni Stern, who taught and guided me from the very

beginning of my way through the Master's degree. Most of my experience in working

with lab equipment is the result of his patient and close guidance. Yoni also helped

me understand the theoretical background of this research with his great

knowledge, and always answered kindly to any of my questions. I would like to thank

him for all his help and patience.

I would also like to thank all of Avi's former and current group members for sharing

their experience, their wise advises and support: Dr. Arkady Rudnitsky, Dvir Munk,

Eyal Preter, Idan Bakish, Nadav Arbel, Orel Shlomi, Raphi Cohen, Ran Califa, Shahar

Levi, Yair Antman and Yosef London.

I would like to express my great appreciation to the Faculty of Engineering at Bar-Ilan

University, and especially to Mrs. Dina Yeminy, for all her help and support from the

beginning of my Bachelor's degree till now.

Last but not least, I'm grateful for my loving and supporting parents and family that

encouraged me during the past two years, and express my gratitude to my lovely

wife Tzofia and my adorable daughter Noam. Thank you all!

TABLE OF CONTENTS

LIST OF FIGURES

GLOSSARY

NOTATIONS

LIST OF PUBLICATIONS

ABSTRACT ................................................................................................................................................. I

1. INTRODUCTION .............................................................................................................................. 1

1.1. MICROWAVE PHOTONICS ................................................................................................................ 1

1.1.1. BACKGROUND .......................................................................................................................... 1

1.1.2. RADIO-OVER-FIBER................................................................................................................... 3

1.1.3. VARIABLE OPTICAL DELAY LINES................................................................................................. 5

1.1.3.1. MOTIVATION .................................................................................................................................. 5

1.1.3.2. IMPLEMENTATION EXAMPLES OF VARIABLE ODLS ......................................................................... 6

1.2. STABLE RADIO-OVER-FIBER DISTRIBUTION ........................................................................................ 8

1.2.1. ENVIRONMENTAL EFFECTS ON OPTICAL FIBERS ............................................................................ 8

1.2.2. THE NEED FOR STABILIZED OUTPUT PHASE .................................................................................. 9

1.2.3. CURRENT SOLUTIONS FOR RADIO-OVER-FIBER DISTRIBUTION ...................................................... 10

1.3. HIGH-RESOLUTION BRILLOUIN SENSING ......................................................................................... 12

1.3.1. STIMULATED BRILLOUIN SCATTERING (SBS) .............................................................................. 12

1.3.2. HIGH-RESOLUTION DISTRIBUTED FIBER-OPTIC SENSORS BASED ON SBS ANALYSIS ......................... 14

1.3.2.1. BRILLOUIN OPTICAL TIME DOMAIN ANALYSIS .............................................................................. 14

1.3.2.2. BRILLOUIN OPTICAL CORRELATION DOMAIN ANALYSIS ................................................................ 15

1.4. LINEAR FREQUENCY MODULATED WAVEFORMS .............................................................................. 17

1.5. OBJECTIVES OF RESEARCH ............................................................................................................. 19

2. RESEARCH METHODOLOGY ......................................................................................................... 21

2.1. PRINCIPLE OF OPERATION ............................................................................................................. 21

2.2. STABILIZATION USING TWO LASER SOURCES ................................................................................... 24

2.3. SCHEMATIC ILLUSTRATION OF THE EXPERIMENTAL SETUP ................................................................. 25

2.4. THE CORRECTION FACTOR ............................................................................................................. 28

3. EXPERIMENTAL SETUPS ............................................................................................................... 31

3.1. OPEN LOOP MEASUREMENTS WITH A SINGLE LASER SOURCE ............................................................ 31

3.1.1. STAGE 1: OBSERVATION OF THE CHROMATIC DISPERSION EFFECTS ............................................... 31

3.1.2. STAGE 2: CHARACTERIZATION OF THE RELATIONS BETWEEN WAVELENGTH, RF PHASE SHIFTS AND DC

VOLTAGE ...………………………………………………………………………………………………………………………………………… 35

3.2. CLOSED-LOOP, STABLE DISTRIBUTION OF A RF SINE WAVE .............................................................. 37

3.3. CLOSED-LOOP, STABLE DISTRIBUTION OF AN ARBITRARY RF WAVEFORM ........................................... 39

4. EXPERIMENTAL RESULTS ............................................................................................................. 46

4.1. OPEN LOOP MEASUREMENTS WITH A SINGLE LASER SOURCE ............................................................ 46

4.1.1. STAGE 1: OBSERVATIONS OF THE CHROMATIC DISPERSION EFFECTS ............................................. 46

4.1.2. STAGE 2: CHARACTERIZATION OF THE RELATIONS BETWEEN THE WAVELENGTH, PHASE SHIFTS AND DC

VOLTAGE …………………………………………………………………………………………………………………………………………… 47

4.2. CLOSED-LOOP, STABLE DISTRIBUTION OF A RF SINE WAVE .............................................................. 49

4.3. CLOSED-LOOP, STABLE DISTRIBUTION OF AN ARBITRARY RF WAVEFORM ........................................... 53

5. STABILIZED DELAY WITHIN HIGH-RESOLUTION BRILLOUIN ANALYSIS .................................. 59

5.1. B-OCDA EXPERIMENTAL SETUP .................................................................................................... 59

5.2. EXPERIMENTAL RESULTS ............................................................................................................... 62

6. DISCUSSION AND SUMMARY ...................................................................................................... 66

6.1. SUMMARY OF RESULTS ................................................................................................................. 66

6.2. LIMITATIONS AND FUTURE WORKS................................................................................................. 67

REFERENCES .......................................................................................................................................... 69

בעברית תקציר א .…….…………………….……………………………….…………………………………………………

LIST OF FIGURES

Figure 1: The basic concept of a MWP system [1] ................................................. 2

Figure 2: General RoF system structure [27]. The optical carrier and sidelobes are

transmitted over optical fibers from the central station to base stations, and after

detection to the air by microwave antennas. ODN: Optical Distribution Network, CS:

Central Station, BS: Base Station .......................................................................... 4

Figure 3: A phased array antennas structure scheme. The steering of the beam is

achieved by differential phase shifts between the array elements [29]. ................... 6

Figure 4: A fiber-optic prism feeding phased array antenna elements for the steering

of the scanning beam [1]. EOM: Electro-optic Modulator ....................................... 7

Figure 5: The phase stabilized downlink transmission scheme [74]. The reference

signal is mixed with its replica after round-trip travel and the outcome signal changes

the wavelength of the tunable laser. MZM: Mach-Zehnder modulator, PolM:

polarization modulator, BPF: bandpass filter ....................................................... 11

Figure 6: An example for a phase-coded B-OCDA setup. Note the 25 km-long fiber

delay line that is placed prior to the fiber under test [63]. .................................... 17

Figure 7: Illustration of the PSLR, ISLR and resolution of the impulse response of an

LFM waveform [29]........................................................................................... 19

Figure 8: Experimental setup for the realization of a long, stabilized fiber-optic delay

of arbitrary RF waveforms. BPF: bandpass filter, PC: polarization controller ........... 26

Figure 9: An illustration of the setup used in the observation of the chromatic

dispersion effects. LPF: low-pass filter, EDFA: erbium doped fiber amplifier ........... 31

Figure 10: Radio frequency spectrum of the waveform generated by the microwave

generator without the LPF. ................................................................................ 32

Figure 11: Radio frequency spectrum of the waveform generated by the microwave

generator, with use of the LPF. .......................................................................... 33

Figure 12: Optical power spectral density of an optical carrier modulated by a 3 GHz

sine wave......................................................................................................... 34

Figure 13: An illustration of the setup used in the characterization of the relations

between wavelength, RF phase shift and mixer DC voltage. LPF: low-pass filter, EDFA:

erbium doped fiber amplifier, DVM: digital voltmeter .......................................... 35

Figure 14: An illustration of the setup used in the closed feedback, phase stabilized

delay of an RF sine wave ................................................................................... 38

Figure 15: An illustration of the setup for the stabilized delay of an arbitrary RF

waveform. LPF: low-pass filter, EDFA: erbium doped fiber amplifier, DVM: digital

voltmeter, BPF: band-pass fiter .......................................................................... 40

Figure 16: RF power spectral density of a LFM waveform at the output of the delay

line. ................................................................................................................. 43

Figure 17: Measured optical power spectral density of the delayed waveform. An

optical bandpass filter was used to retain one optical carrier that was modulated by

the control tone (right-hand peak), and reject a second optical carrier which was

modulated by the user's RF waveform. ............................................................... 44

Figure 18: RF power spectral density of the control sine wave following two-way

propagation along the fiber delay line. ............................................................... 44

Figure 19: A 3 GHz control sine wave, detected following two-way propagation in a

8.8 km-long dispersive fiber delay line. Each trace corresponds to a different

wavelength of the tunable optical carrier, in 0.1 nm increments. The change in group

delay between successive acquisitions is about 30 ps. .......................................... 46

Figure 20: (left) Output phase of a 3 GHz control sine wave following two-way

propagation in a 8.8 km long delay fiber, as a function of tunable laser wavelength,

(right) DC reading of the mixing between the delayed control sine wave and its

reference replica, as a function of the tunable laser source wavelength. ................ 48

Figure 21: The DC voltage at the output of the mixer as a function of time, in open

feedback loop operation. .................................................................................. 49

Figure 22: Closed loop, phase stabilized two-way delay of a 3 GHz sine wave over 8.8

km of fiber. (top): DC voltage obtained by the mixing the delayed sine wave with its

reference replica, as a function of time. (bottom): wavelength setting commands to

the tunable laser source, as a function of time. ................................................... 50

Figure 23: Persistence traces of the output 3 GHz control sine wave, following two-

way delay in 8.8 km of fiber, accumulated over 10 minutes of operation at 0.5

second intervals, with (left) and without (right) closed loop feedback. ................... 51

Figure 24: Persistance traces of the 3 GHz control sine wave foolowing delay over 8.8

km of fiber: feedback loop open and no external heating of the delay fiber (top left);

feedback loop open during the heating of the fiber by a halogen lamp (top right);

feedback loop closed during ongoing heating (bottom left); and feedback loop closed

following the removal of the heating lamp and during the cooling down of the delay

fiber (bottom right). .......................................................................................... 52

Figure 25: The DC voltage at the output of the mixer as a function of time, during the

four stages of the experiment described in Figure 24. .......................................... 53

Figure 26: The DC voltage at the output of the mixer as a function of time, during 2

minutes of open feedback loop operation. .......................................................... 54

Figure 27: Wavelength of the tunable laser source of the RF control sine wave

(green) and of the LFM signal (blue), as a function of time during two minutes of

closed-loop operation. ...................................................................................... 55

Figure 28: The DC voltage at the output of the mixer as a function of time during

closed-loop operation. ...................................................................................... 55

Figure 29: Persistence traces of the compressed shapes of LFM waveforms, sampled

at the output of the fiber delay over 2 minutes, without (top) and with (bottom)

closed-loop wavelength control. ........................................................................ 57

Figure 30: Schematic illustration of a phase-coded Brillouin optical correlation-

domain analysis (phase-coded B-OCDA) setup, incorporating a stabilized fiber-optic

delay line in the path of the Brillouin signal wave. Solid lines denote fiber paths,

dashed lines represent RF cable paths, and dashed-dotted, black lines correspond to

DC control signals. Blue color represents paths and components related to the

sensing functionality, whereas green paths and components are part of the delay

stabilization module. The 'Wavelength control' module is described in detail in Fig.

15. Details of the 'Pump wave processing' module are not shown, for better clarity

(see [64]). ........................................................................................................ 60

Figure 31: Top – Wavelength of the control channel (left-hand axis), and the implied

thermal drift in the path length of the 25 km-long delay line (right-hand axis), as a

function of time. Thermal drift on the order of 10 cm is observed. Bottom –

Measured Brillouin gain as a function of frequency offset between pump and signal,

and of time. The phase-coded B-OCDA setup was adjusted to monitor the Brillouin

gain spectrum in a 2 cm-wide hot-spot. The delay of the Brillouin signal was free-

running, without closed-loop feedback to the wavelength of the sensor laser. The

Brillouin gain spectrum changed after 10 minutes, from that of the hot-spot to that

of the fiber under test at room temperature. ...................................................... 63

Figure 32: Top – Wavelength of the Brillouin sensor laser source (left-hand axis), and

the implied thermal drift in the path length of the 25 km-long delay line (right-hand

axis), as a function of time. Thermal drift on the order of 9 cm is observed. Bottom –

Measured Brillouin gain as a function of frequency offset between pump and signal,

and of time. The phase-coded B-OCDA setup was adjusted to monitor the Brillouin

gain spectrum in a 2 cm-wide hot-spot. Unlike Fig. 31, the delay of the Brillouin signal

was stabilized through closed-loop feedback to the wavelength of the sensor laser.

The Brillouin gain spectrum remained that of the hot-spot throughout the

measurements. ................................................................................................ 64

Figure 33: Measured Brillouin frequency shifts as a function of time. The phase-

coded B-OCDA setup was adjusted to monitor the location of a 2 cm-wide hot-spot.

Thermal drift in a 25 km-long delay line incorporated in the setup leads to incorrect

interrogation after 10 minutes of free-running operation (blue). Delay drift is

overcome with closed-loop, stabilized delay (red). ............................................... 65

GLOSSARY

AM Amplitude Modulator

AWG Arbitrary Waveform Generator

B-OCDA Brillouin Optical Time Domain Analysis

B-OTDA Brillouin Optical Correlation Domain Analysis

BPF Band Pass Filter

BS Base Station

CFBG Chirped Fiber Bragg Grating

CS Central Station

CW Continuous Wave

DBG Dynamic Brillouin Grating

DC Direct Current

DSP Digital Signal Processing

DVM Digital Voltmeter

EDFA Erbium-Doped Fiber Amplifier

EMI Electro-Magnetic Interference

EOM Electro-Optic Modulator

FBG Fiber Bragg Grating

FIR Finite Impulse Response

FWHM Full Width Half Maximum

IF Intermediate Frequency

IIR Infinite Impulse Response

I/Q In-phase and Quadrature

ISLR Integrated Side-Lobe Ratio

LCFBG Linearly-Chirped Fiber Bragg Grating

LFM Linear Frequency-Modulated

LO Local Oscillator

LPF Low Pass Filter

LTE Long Term Evolution

MWP Microwave Photonics

MZM Mach-Zehnder Modulator

NLFM Nonlinear Frequency-Modulated

ODL Optical Delay Line

ODN Optical Distribution Network

OFDM Orthogonal Frequency-Domain Multiplexing

PAA Phased-Array Antenna

PC Polarization Controller

PolM Polarization Modulator

PSD Power Spectral Density

PSLR Peak-to-Side-Lobe Ratio

RF Radio Frequency

RoF Radio-over-Fiber

SBS Stimulated Brillouin Scattering

SMF Single Mode Fiber

SNR Signal to Noise Ratio

SOA Semiconductor Optical Amplifier

TLS Tunable Laser Source

TTD True Time Delay

WDM Wavelength Division Multiplexing

NOTATIONS

A Complex envelop of a wave z Distance

0,g g SBS gain parameters mZ Distance between correlation

, Angular frequency peaks

B Brillouin gain line SN Total number of symbols in

P Optical power phase sequence

0, ,f f Frequency sT Phase symbol duration

Lifetime

T Duration of LFM signal

Temperature

B Bandwidth of a signal

,t Time

Instantaneous phase

L Length

n Refractive index

c Speed of light

L Linear thermal expansion

Coefficient

D Chromatic dispersion

Coefficient

Optical wavelength

0,V V Voltage parameters

Correction factor

S Dispersion slope parameter

LIST OF PUBLICATIONS

Journal papers:

1. A. Ben-Amram, Y. Stern, Y. London, Y. Antman and A. Zadok, "Stable closed-loop

fiber-optic delay of arbitrary radio-frequency waveforms," submitted to Optics

Express, 2015 (in review).

Conference papers:

1. A. Ben-Amram, Y. Stern and A. Zadok, " Fiber-optic distribution of arbitrary

radio-frequency waveforms with stabilized group delay," in Proc. of Conference

on Lasers and Electro-Optics (CLEO) 2015, (OSA, 2015), paper JTh2A.54.

2. A. Ben-Amram, Y. Stern, Y. Antman, Y. London and A. Zadok, "Stabilized Fiber-

Optic Delay of Arbitrary Waveforms with Application in Distributed Sensors,"

accepted for presentation in IEEE Microwave Photonics Topical Meeting,

Paphos, Cyprus, Oct. 2015.

I

ABSTRACT

Microwave photonics (MWP) is an area of research that merges between the

optical and radio-frequency (RF) domains. MWP techniques provide many potential

advantages over the processing of signals in the electrical domain, such as broad

bandwidth, long reach with low attenuation, and immunity against electro-magnetic

interference. One important application of MWP is the fiber-optic implementation of

variable RF delay lines, which are critical components in beam steering within

phased-array radar systems. Another significant application is the transmission of RF

signals from cellular and wireless networks over long distances using optical fibers,

or radio-over-fiber (RoF). RoF extends the reach and coverage of such networks.

Various precision applications require that the time delay of RF waveforms

over long fibers remains stable. Examples include the distribution of local oscillators

in large antenna arrays, and the testing and calibration of radar systems. Recently

another application emerged for long stable delay, as part of distributed sensors of

strain and temperature based on Brillouin scattering. Such sensors address cm-scale

segments along many km of fiber. Unfortunately, delay along optical fibers changes

when exposed to variations in the surrounding temperature. A change of 1 ºC

modifies the optical path length by 0.75 cm per 1 km of standard optical fiber.

Presently available solutions are either too complicated for implementation in most

radar and sensor systems, or provide insufficient stability.

The method employed in this thesis relies on chromatic dispersion in order to

compensate for thermal delay drifts. This basic principle was independently

II

proposed by our group and in parallel by a group from the Beijing University of Posts

and Telecommunication. A single tunable laser source is modulated by both a control

RF sine wave and the input RF waveform to be delayed. Following distribution along

a fiber, the RF phase of the output control tone is tracked and used for identifying

delay drifts. The wavelength of the tunable laser source is then adjusted in a closed

feedback loop, so that delay variations due to chromatic dispersion and

environmental changes cancel out. However, the initial embodiment relied the

separation between input and control waveforms in the RF domain, hence their

spectra were required to be non-overlapping. This restriction is difficult to

accommodate in systems that support broadband waveforms, such as high-

resolution fiber sensing protocols.

In this work, I introduce and demonstrate a significant extension of the

stabilization technique. Rather than use a single source, the control RF tone and the

input waveform modulate two separate tunable laser sources with different

wavelengths. Feedback provided by the output phase of the delayed control tone is

used to adjust the wavelengths of both sources, to obtain a stable delay of both

waveforms. The two waveforms were separated in the optical domain, therefore no

restrictions were imposed in their RF spectra. The attainable performance of the

stabilized delay line is analyzed, and tradeoffs between the range of temperature

variations that can be accommodated and residual delay uncertainty are identified. A

figure of merit is proposed, and an upper bound on performance is established in

terms of the specifications of the laser sources used.

III

Three main experiments are demonstrated in as part of this work. First, the

stable distribution of the control sine wave over 9 km of fiber is demonstrated with

residual delay variations of ±4 ps. Second, the stable distribution of broadband,

linearly frequency-modulated (LFM) waveforms, which are employed in many radar

systems, is demonstrated over the same distance. Residual delay variations are

reduced from 200 ps in open-loop operation to 20 ps in closed-loop operation,

limited by the timing jitter of our sampling oscilloscope. Last, the stable interrogation

of a 2 cm-wide local hot spot in high-resolution distributed Brillouin analysis is

achieved, in the presence of thermal delay drifts that are several times larger.

Correct interrogation could not be performed under free-running conditions.

The experimental results demonstrate the applicability of the proposed

stabilization technique to the processing of broadband microwave signals as part of

more complicated systems. The method could be further extended to the active

mitigation of acoustic and mechanical vibrations, with broader-bandwidth feedback.

Introduction 1

1. INTRODUCTION

1.1. MICROWAVE PHOTONICS

1.1.1. BACKGROUND

Over the past thirty years, the research field that merges between analog

radio-frequency (RF) and microwave signals with photonic means, known as RF-

photonics or microwave-photonics (MWP) [1], has been a very productive one [1, 2].

In MWP systems, an optical carrier wave is modulated by the electrical signal of

interest. After modulation, the optical signal propagates through optical fibers and

devices, and finally the electrical signal is reconstructed by photo-detection. MWP

processes have several potential advantages over their all-RF equivalents [1]: optical

fibers provide a low propagation loss regardless of the microwave frequency; optical

fibers have an ultra-broad transmission bandwidth of several THz; electro-magnetic

interference (EMI) has little effect on optical fibers; use of several optical carriers can

provide for parallel processing, etc. When taking into account all these advantages,

MWP setups could prove as improved optical versions of current RF systems, or

perform tasks that currently cannot be carried out in the RF domain. The basic

concept of a MWP system is illustrated in Figure 1.

Amongst the many potential applications of MWP available in the research

literature, a relatively simple one, which is already being used in analog links for the

defense sector [1], is that of antenna remoting. In order to connect between, for

example, an end unit of a radar system and a central office, an optical fiber can be

used as a transmission link. Fibers provide an attractive alternative for electrical

Introduction 2

cable paths: waveforms can propagate over tens of km without amplifiers along the

way, and with group velocity dispersion that is much smaller than that of RF cables.

Another commonly discussed application is that of MWP filters, implementing both

finite-impulse-response (FIR) and infinite-impulse-response (IIR) RF transfer

functions in optical media [3]. Other applications include the photonic generation of

arbitrary and ultra-wideband RF signals [4, 5]; the photonic implementation of

advanced RF modulation formats [6]; and analog-to-digital conversion using

photonic devices [7].

Figure 1: The basic concept of a MWP system [1].

The field of MWP continues to produce exciting advances. Examples of the

last 2-3 years include, among many others: in-phase and quadrature (I/Q) detectors

for radars [8]; analog distribution networks [9]; generation of arbitrary waveforms

and arbitrary filter transfer functions [10, 11, 12, 13, 14, 15, 16, 17]; and integrated

MWP transmitters and receivers on a single chip [10, 17, 18, 19, 20].

Introduction 3

In this work I focus on two main applications of MWP: distribution of radio

waveforms over optical fibers, and long stabilized optical delay lines. Both are

introduced next.

1.1.2. RADIO-OVER-FIBER

In many scenarios, where cellular wireless reception cannot be provided or

when long distances must be covered, data of wireless networks is distributed over

cables instead. As the demand for large data transmission bandwidth over wireless

access networks increases continuously, RF coaxial cables can no longer support the

necessary transmission. One important solution, introduced several years ago, is the

distribution of RF signals from wireless networks over long optical fibers, or radio-

over-fiber (RoF) [1, 21, 22]. The basic concept of RoF networks is that RF signals

modulate the optical carrier in the central station. The optical carrier and sidebands

are then transmitted over optical fibers to remote base stations. After detection at

the base station, the reconstructed RF signals are radiated by microwave antennas. A

general RoF system structure is illustrated in Figure 2. RoF networks benefit from the

low propagation loss and broadband bandwidth of the optical fibers and extend the

reach and coverage of wireless communication networks.

Introduction 4

Figure 2: General RoF system structure [27]. The optical carrier and sidelobes are transmitted over optical fibers from the central station to base stations, and after

detection to the air by microwave antennas. ODN: Optical Distribution Network, CS: Central Station, BS: Base Station, PD: Photo-Detector, RN: Remote Node, MT: Mobile

Terminal.

RoF networks are incorporated in satellite communication, mobile radio

communication, and other wireless network systems [23]. For example, the long

term evolution (LTE) technology standard for wireless communication systems,

published in 2011, relies on optical fibers to connect between the base stations and

the central stations [24]. The research in RoF in recent years focuses on increasing

the carrying capacity using technologies such as wavelength division multiplexing

(WDM), orthogonal frequency-domain multiplexing (OFDM) [23, 25], or a

combination of both [26]. A full-duplex RoF link for optical/wireless integration

based on analog RoF for the uplink transmission and digital RoF for the downlink was

implemented in [27]. Objectives of ongoing research include cost reduction,

increased bandwidth, better simplicity and higher transmission power [28]. In one

example, the modulation depth in RoF links was enhanced by a ring resonator notch

filter that removed the optical carrier [29].

Introduction 5

1.1.3. VARIABLE OPTICAL DELAY LINES

1.1.3.1. MOTIVATION

The delay of RF waveforms is required as part of the testing and calibration of

many radar systems, and in the emulation of remote targets within testing facilities.

One way to accomplish the necessary delay relies on high rate digital signal

processing (DSP) for generating replicas of an input waveform. The capabilities of

DSP advance continuously. However, even today, state-of-the-art analog to digital

converters and DSP elements cannot reach tens-of-GHz rates with sufficient vertical

resolution, that is separation between adjacent quantization levels. Therefore, the

delay of waveforms based on propagation over long optical fibers, referred to as

optical delay lines (ODLs), represents an attractive alternative. Many principles and

technologies are common to ODLs and RoF networks. The separation between them

is somewhat arbitrary, and it is made based on context, system considerations and

the types of waveforms being transmitted.

In the applications above the delays are typically fixed. However, much

attention was also given over the last thirty years to the optical implementation of

variable RF delay lines [4]. Variable RF delay lines play a major role in radar systems

based on phased-array antennas (PAAs), in which beam steering relies on the proper

interference between the radiation patterns from individual antenna elements, as

illustrated in a Figure 3 [30]. Each individual antenna within the array radiates

equally to all directions. When the signals feeding the antennas are in-phase, the

intensity peak of the interference pattern is formed perpendicular to the aperture.

When the signals are phase-shifted from one another, the interference pattern is

Introduction 6

reconstructed at a different angle governed by the incrementing phases, causing an

effective tilt of the beam. PAAs are faster and more reliable than mechanical beam

steering elements [31, 32].

Figure 3: A phased array antennas structure scheme. The steering of the beam is achieved by differential phase shifts between the array elements [30].

The steering of beams using incremental phase delays is adequate for single-

frequency or narrow-bandwidth transmission. When broadband radar signals are

used, incremental phase delays are not enough to avoid spatial dispersion of the

beam. Variable group delay elements, often referred to as true time delays (TTDs),

must be used instead [33]. Variable optical delay lines can provide, in principle, the

necessary TTDs.

1.1.3.2. IMPLEMENTATION EXAMPLES OF VARIABLE ODLS

A switching matrix between fibers of different lengths provides a simple form

of an ODL. In a series of works by Tur and coauthors [34, 35, 36], discrete ODLs were

realized through wavelength-selective switching among different paths. The quality

Introduction 7

of the delayed waveforms, in terms of distortion, was excellent. However, it is

difficult to vary the delay values continuously in such networks. The principle was

extended in networks of many fibers, each connected to a series of fiber Bragg

gratings (FBGs) set for different reflectivity wavelengths [37, 38]. Continuous delay

tuning was achieved by linearly-chirped FBGs (LCFBGs) in conjunction with tunable

laser sources (TLSs) [39]. Other approaches include so-called fiber prisms [40, 41],

consisted of an array of both highly-dispersive and non-dispersive segments, as

illustrated in Figure 4, and slow light-based techniques using stimulated Brillouin

scattering (SBS) in fibers [42, 43, 44, 45, 46, 47], or propagation in semiconductor

optical amplifiers (SOAs) [48, 49]. A large number of delay resolution points was

achieved by a combination of a switching matrix together with FBGs and tunable

laser sources [50].

Figure 4: A fiber-optic prism feeding phased array antenna elements for the steering of the

scanning beam [1]. EOM: Electro-optic Modulator, PD: Photo-Detector.

In a recent series of works by Ofir Klinger et al. of our group [51, 52, 53, 54],

ODL setups were adjusted to specifically accommodate linear frequency -modulated

Introduction 8

(LFM) and nonlinear frequency-modulated (NLFM) waveforms, which are common in

many radar systems. These types of waveforms are introduced in detail later in this

chapter. Long delay variations of over 100 ns were obtained. However, the setup

cannot support most other types of RF signals.

The key metric of variable MWP TTD elements is the product of the

waveform bandwidth that can be supported, multiplied by the range of delay

variations, subject to the constraints of sufficient signal quality (known as the 'delay-

bandwidth product'). Many ODL implementations struggle to reach delay-bandwidth

products above the order of unity. Recent state-of-the-art examples of ODLs include

a 7-bit compact silicon-based reconfigurable optical TTD providing a delay of 1.27 ns

with a 10 ps resolution [55]; a continuously tunable ODL based on micro-ring

resonators providing a delay of 100 ps with 168 GHz bandwidth [56]; moveable

mirrors based on dynamic Brillouin gratings (DBGs) providing the delay of 1 Gbit/s

data by as much as 10 ns [57]; and lastly, systems based on nonlinear wavelength

conversion and dispersive propagation, that achieved delay variations of micro-

seconds and supported 10 Gbit/s data [58].

1.2. STABLE RADIO-OVER-FIBER DISTRIBUTION

1.2.1. ENVIRONMENTAL EFFECTS ON OPTICAL FIBERS

Transmission over optical fibers provides many potential advantages as noted

above. One drawback, however, is the sensitivity of fibers to numerous

environmental parameters, first and foremost to temperature [59]. Instabilities in

the phase of an RF signal reconstructed at the output of a long fiber-optic delay line

Introduction 9

stem from the thermo-optic properties of silica. The group delay in silica fibers

changes by 7.5 ppm per C. Therefore, a change of 1 C in the surrounding

temperature is analogous to the lengthening or shortening of every 1 km of fiber by

an extra 0.75 cm. These delay variations may add up to many cm over long fibers. On

the other hand, an RF signal of 3 GHz, for example, corresponds to a wavelength of

10 cm. Thus, the phases of RF waveforms at the output of a long free-running delay

line might become completely arbitrary. The problem worsens with higher RF, and

with increase in the extent of delay (length of fiber).

1.2.2. THE NEED FOR STABILIZED OUTPUT PHASE

The importance of a stabilized output phase is apparent, for instance, in the

case of PAAs that was mentioned above. When the relative phases between

neighboring elements fluctuate, the transmitted beam becomes spatially distorted.

RF phase stability is also necessary in the testing and calibration of radars; in the

emulation of distant targets; in the distribution of local oscillator microwave tones

among multiple sites, such as in large radio-astronomical arrays [60]; in microwave

generation within electro-optic oscillators [61]; in optical coherence tomography

based on CFBGs and ODLs [62]; and in certain types of lidars [63].

Recently, the significance of stable delay of modulated sequences, carried

over tens of km of fiber, has been also highlighted in the context of distributed fiber-

optic sensors that are based on Brillouin scattering analysis [64, 65]. This particular

application is pursued in our group as well, hence a stabilized delay represented a

pressing need of specific setups. High-resolution Brillouin sensors, and the effects of

delay propagation instability in such setups, are addressed later in this chapter.

Introduction 10

1.2.3. CURRENT SOLUTIONS FOR RADIO-OVER-FIBER DISTRIBUTION

Early attempts to overcome the instability of the output RF phase relied, for

example, on the passive thermal isolation of a 32 km long fiber inside a container

[66]. However the stability that can be achieved using passive means is restricted. As

discussed in subsequent chapters, the extent of thermal stability that is needed for

supporting GHz-frequency signals over many km is on the order of 1e-3 C. Some

approaches rely on all-optical phase-locked loops to provide stabilization on the

scale of the optical wavelength. These ultra-precision delay lines are used in particle

accelerators and colliders [67, 68]. Other solutions rely on a heterodyne

optoelectronic delay-locked loop [69]; adjusting the phase of a laser by pump

modulation and cavity length control [70]; and DSP compensation [71]. Most of

these techniques deal with phase drifts within the optical domain, which makes the

compensation systems extremely complicated, and are not suitable for incorporation

in radar systems.

The solution path proposed in this research relies on chromatic dispersion in

the delay fiber to compensate for the thermo-optic variations in path length. The

approach employs a closed feedback loop that offsets the wavelength of the tunable

laser source, onto which the RF signal is modulated, so that path length variations

due to the thermo-optic effect and chromatic dispersion cancel out. The approach

was suggested in parallel by our group and, initially unknown to us, by the group of

Prof. Kun Xu at the Beijing University of Posts and Telecommunication [72, 73, 74,

75]. In their setup, which is illustrated in Figure 5, the desired RF waveform of the

user and an internal control tone jointly modulate the output of a single tunable

Introduction 11

laser. Following propagation along the fiber, the two waveforms are detected and

separated by RF filters. The output phase of the control tone is then monitored and

used to drive the feedback loop and adjust the source wavelength to obtain a stable

path length [75].

Figure 5: The phase stabilized downlink transmission scheme [75]. The reference signal is mixed with its replica after round-trip travel and the outcome signal changes the

wavelength of the tunable laser. MZM: Mach-Zehnder modulator, PolM: polarization modulator, BPF: bandpass filter, WTL: wavelength tunable laser, PD: photo-detector,

BC: bias controller, EDFA: erbium doped fiber amplifier, CIR: circulator, OSC:

oscilloscope, SA: spectrum analyzer.

The system reported in [75] supports a broad range of user waveforms,

however it requires that the RF spectra of the user waveform and of the control tone

do not overlap. This restriction is not easily met in high-resolution Brillouin analysis

setups, which involve broadband sequence modulation. The setup therefore is not

directly applicable to Brillouin sensing, and significant modifications are necessary.

Introduction 12

1.3. HIGH-RESOLUTION BRILLOUIN SENSING

1.3.1. STIMULATED BRILLOUIN SCATTERING (SBS)

Stimulated Brillouin scattering (SBS) is a nonlinear optical effect that can

couple between two optical waves along a standard fiber [76]. In SBS, a relatively

intense pump wave interacts with a counter-propagating signal wave, which is

typically weaker and detuned in frequency. The combined intensity to the two waves

stimulates, through electrostriction, an acoustic wave. The frequency of the acoustic

wave equals the difference between the optical frequencies of the pump and signal

waves, and its wavenumber is the sum of their wavenumbers. Through photo-

elasticity, the acoustic wave leads to a travelling index grating, which scatters the

light waves. The travelling grating can couple optical power between the counter-

propagating pump and signal waves. Effective coupling requires that the difference

between the central frequencies of the two waves should closely match the Brillouin

frequency shift of the fiber 11B GHz (for standard single-mode fibers at 1550 nm

wavelength) [76, 77].

SBS requires the lowest power threshold among all nonlinear mechanisms in

standard silica fibers [78], which makes it very attractive for all-optical signal

processing [15, 43, 79, 80], and sensing [81, 82, 83] applications. In most of the

applications and throughout this work, the power of the pump is strong enough so

that it is barely affected by the amplification of the probe, a regime known as that of

an undepleted pump [76]. For this case, the complex envelop of the pump wave

Introduction 13

pumpA is considered as constant, and the complex envelop of the probe wave probeA

is exponentially amplified along the fiber:

( ) (0)exp ( )2probe probe probe

zA z A g

(1)

In equation (1) above, 1( ) mprobeg defines the complex gain function,

probe is the frequency of the probe wave and z is the distance the probe wave

travelled along the fiber. The complex gain function, for a continuous-wave (CW)

pump laser, is of Lorentzian shape [76, 84]:

2

0( )

1 2

pump

probe

pump probe B B

g Ag

j

(2)

Here, the frequency of the pump is pump , 2 30 Mrad/sB denotes the

narrow inherent linewidth of the SBS process, and 1

0 W mg

is the gain coefficient

at the peak of the Brillouin gain line. In case the pump wave is modulated to obtain a

broadened power spectral density (PSD) pump pumpP , the complex gain function is

given by a convolution of the pump PSD with the SBS line shape [43]:

0

( )1 2

pump

probe pump

pump probe B B

g Pg d

j

(3)

The value of B varies with both temperature and mechanical strain. Hence,

a mapping of the local Brillouin gain spectrum along standard fibers is being used in

distributed sensing of both quantities for 25 years [85, 86, 87].

Introduction 14

1.3.2. HIGH-RESOLUTION DISTRIBUTED FIBER-OPTIC SENSORS BASED ON

SBS ANALYSIS

1.3.2.1. BRILLOUIN OPTICAL TIME DOMAIN ANALYSIS

One of the major challenges facing Brillouin sensing technology is reaching

cm-scale spatial resolution. The most widely employed measurement protocol is that

of Brillouin optical time domain analysis, or B-OTDA [85], proposed initially by

Horiguchi and coworkers 25 years ago. B-OTDA relies on SBS amplification of a CW

signal by counter-propagating pump pulses. Since the pump pulses are limited to a

certain time frame, the SBS effect only occurs when and where the pulses exist along

the fiber. Therefore, the SBS gain of the signal wave is time dependent, and spatial

distinction is achieved through temporal analysis. At every section of the fiber, if the

frequency difference between the pump pulse and the signal wave equals to the

local value Brillouin frequency shift B , SBS amplification takes place. When strain is

applied, for example, in some sections, it changes the Brillouin frequency shift so

that B , which in turn leads to a decrease in the SBS amplification.

The spatial resolution of this fundamental scheme is limited to the duration

of the pump pulses, which is in turn restricted to the acoustic lifetime 1 B ~ 5

ns, or longer. Consequently, currently available commercial equipment provides a

spatial resolution on the order of 1 m, with 50 km range. Much effort is being

dedicated to resolution enhancement in B-OTDA, using elaborate schemes for the

shaping and control of pump pulses. A detailed account of these efforts is beyond

the scope of this research, which is focused on microwave photonics. State of the art

Introduction 15

B-OTDA setups reported in the literature reach 2 km range with 2 cm resolution [88],

or 5 km range with 5 cm resolution [89].

1.3.2.2. BRILLOUIN OPTICAL CORRELATION DOMAIN ANALYSIS

In the late 90's, Hotate and coworkers proposed a different sensing technique

known as Brillouin optical correlation domain analysis, or B-OCDA [64, 90]. B-OCDA

reaches higher spatial resolution than that of B-OTDA, on the scale of few mm [91].

B-OCDA relies on the broadband modulation of the Brillouin pump and signal waves,

so that their complex envelopes are in correlation only within discrete and narrow

peaks along the fiber. Hotate's initial method was to frequency-modulate both the

pump and signal waves by a common sine wave function over a broad frequency

range. By that, the difference between their instantaneous frequencies remained

fixed at B in specific, narrow segments only. Only in those positions could the SBS

effect be built.

The first demonstrations provided mm-scale resolution. However, multiple

periodic correlation peaks were generated, effectively limiting the number of

resolution points that could be unambiguously interrogated to only a few hundreds.

Similarly to B-OTDA, the unambiguous measurement range of B-OCDA was

continuously extended over the last 15 years, using more elaborate frequency

modulation protocols. A full account of these efforts is outside the present

discussion.

Recently, our research group and coworkers proposed to jointly modulate the

pump and signal waves with a phase code in order to increase the number of

Introduction 16

resolution points [92]. The phase codes employed are binary sequences, designed to

exhibit low correlation sidelobes. Using this method, the resolution is determined by

the duration of an individual bit, whereas the separation between neighboring,

periodic peaks is set by the length of the code, which could be chosen arbitrarily. By

slightly changing the bit duration, the positions of all the peaks (except for the zero-

order one) are effectively scanned along the fiber under test. Modulation rates can

be as high as 12 Gbit/s, yielding a resolution of 9 mm [65]. The phase-coded B-OCDA

principle was recently extended by our group to the analysis of a 2.2 km-long fiber

with a spatial resolution of 2 cm [64].

One difficulty that is associated with the setup is the need for a deliberate

delay imbalance between the paths that lead the pump and signal waves towards

the opposite ends of the fiber under test. This imbalance is necessary to guarantee

that high-order correlation peaks are in overlap with the test fiber. The delay must

be several times longer than the fiber under test itself, reaching 60 km in some

experiments. An example for a B-OCDA setup is illustrated in Figure 6.

Introduction 17

Figure 6: An example for a phase-coded B-OCDA setup. Note the 25 km-long fiber delay line that is placed prior to the fiber under test [63]. PC: polarization controller, EDFA: erbium doped fiber

amplifier.

The proper function of B-OCDA requires that the path lengths leading the two

waves towards the correlation peaks remain stable to within 1 cm over 1-2 hours.

The requirement is the same as that of a fiber-optic delay lines with a stabilized

phase of the output RF waveform. Solutions developed towards stabilize microwave-

photonic delay lines could be highly instrumental in high-resolution Brillouin sensor

setups.

1.4. LINEAR FREQUENCY MODULATED WAVEFORMS

A group of signals that is widely employed in radar systems is that of linear

frequency-modulated (LFM) waveforms. A LFM signal of duration T , bandwidth B

and central radio frequency 0f can be expressed as:

2

0 0( ) cos 2 rectLFM

B tA t A f t t

T T

(4)

Introduction 18

Here rect( ) equals 1 for 1 2 , and zero elsewhere. The instantaneous

frequency ( )f t of the LFM waveform is obtained from the derivative of its

instantaneous phase ( )t , which is the argument of the cosine in equation (4):

2

0 0

1( ) 2

2

d B Bf t f t t f t

dt T T

(5)

It can be deduced from equation (5) that the instantaneous frequency of LFM

signals is time dependent, sweeping across a broad bandwidth over relatively long

durations. The broad bandwidth of the LFM waveform provides high ranging

resolution, whereas their long duration leads to a high overall energy which helps

improve the signal-to-noise ratios (SNRs) of the measurements.

Equation (5) reveals another advantage of LFM signals: the equivalence

between temporal delays and frequency offsets. This quality is used in ranging

measurements: the delayed echoes that are reflected from a target are mixed with a

replica of the transmitted waveform. The beating between the two is centered at an

intermediate frequency that is directly proportional to the distance to the target.

The same property was used in MWP TTDs of these waveforms [51].

LFM signals are of constant amplitude, and are simply generated and

processed. The post-detection processing of LFM signals is carried out by a cross-

correlation of a LFM reference signal and the detected waveform. The result of the

cross-correlation is referred to as the impulse response function. It is characterized

by a strong and narrow main-lobe peak and low side-lobes.

Introduction 19

Three figures of merit define the quality of the impulse response function, as

illustrated in Figure 7: the peak-to-side-lobe ratio (PSLR) is the ratio between the

power of the main-lobe peak and the power of the highest side-lobe peak; the

integrated side-lobe ratio (ISLR) is the ratio between the integrated energy within

the full-width-half-maximum (FWHM) of the main-lobe and the integrated energy

elsewhere; and the resolution simply defined by the FWHM of the main-lobe. The

first two are measured in dB, and the last is measured in seconds.

Figure 7: Illustration of the PSLR, ISLR and resolution of the impulse response of an LFM waveform [29]. PSLR: pick to side-lobe ratio, ISLR: integrated side-lobe ratio.

In this work, we use the stabilized, long fiber-optic delay of LFM signals in

proof-of-concept experiments.

1.5. OBJECTIVES OF RESEARCH

The main goals of this work are: 1) to stabilize the output timing and phase of

arbitrary RF signals, following their MWP delay over long optical fibers; and 2) to

employ the solution in high-resolution Brillouin analysis of fixed fiber positions over

an extended period of time. The proposed solution relies on the cancellation of

thermo-optic path length drift by chromatic dispersion as noted above. Unlike [75],

Introduction 20

however, the user waveform and the control tone will be modulated onto separate

tunable laser sources. The separation between the two waveforms will be carried

out in the optical domain, so that all restrictions on the RF spectra of the two signals

are removed. The research program consisted of the following tasks:

1) Preliminary tests to examine the relations between the RF phase shift and the

wavelength of the tunable laser. The design of a proper control feedback

loop. The generation of an error signal, and its application in controlling the

transmission wavelength of a tunable laser diode.

2) The demonstration of a stabilized output phase of a single-tone RF sine wave

at the output of closed-loop fiber-optic delay line.

3) The extension of the setup to support a user waveform and a control tone,

modulated onto two separate lasers. The stabilization of the delay of the user

waveform based on a feedback provided by a control sine wave that is

carried over a separate laser source.

4) The stabilization of the output phase of arbitrary RF signals, such as

broadband LFM waveforms.

5) The incorporation of the stabilized delay line within a high-resolution

Brillouin analysis setup, and the demonstration of its added value.

Experimental Setups 21

2. RESEARCH METHODOLOGY

As was discussed in the previous chapter, the temperature of the

surroundings of an optical fiber changes its optical path length. Due to the thermo-

optic properties of the silica, the group delay along the fiber changes by 7.5 ppm per

C. These changes lead to phase variations in delayed RF tones. On the other hand, a

travelling signal in an optical fiber experiences delay variations that stem from

chromatic dispersion. In this work, a method for cancelation of the temperature-

induced phase variations by dispersion-induced delay variations is proposed.

2.1. PRINCIPLE OF OPERATION

Let us assume a delay line of length L which is exposed to a temperature

change T . The 'nominal' delay of the signal in the optical fiber is nL

c , where

1.45n is the group index of light in the fiber, and c is the speed of light. The

variations of the delay as a result of T are:

1 1 1 1T

n L n n LL T T T

n T L T c n T L T

(6)

In equation (6) above 1

7.5 /n

nppm C

n T

is the thermo-optic

coefficient of silica fibers, 1

0.5 /L

Lppm C

L T

is their linear thermal expansion

coefficient, and 1LT

.


On the other hand, the change in delay due to an offset of the optical carrier

wavelength by a certain is given by:

( ) ( ) ( )D

cD L D

n (7)

In equation (7) above, ps nm kmD is the chromatic dispersion coefficient

of the optical fiber. It is assumed for the moment that the central wavelengths of the

two laser sources, that of the user waveform and that of the control tone, are

sufficiently close so that the same value of chromatic dispersion coefficient applies

to both. Subject to this condition, the same wavelength correction can be applied to

both sources. Even if this condition is not met, knowledge of the fiber dispersion

slope parameter could provide for the necessary correction in wavelength increment

between the two sources (this issue is addressed in subsequent sections).

Comparison between (6) and (7) reveals the range of temperatures variations

that can be compensated by a tunable laser of scanning range :

( )cDT

n

(8)

For a standard SMF-28 fiber, whose chromatic dispersion parameter is

17 ps nm kmD , and for a tunable laser source with a wavelengths scanning

range of 100 nm, the supported temperature span would be 50 C .

The residual phase uncertainty of the compensated delay line is related

to the radio frequency of the input waveform f , the fiber dispersion, and the

residual uncertainty of the laser source wavelength :


1

2DL

f

(9)

Using equation (9), we can set a performance bound on the extent of the

nominal delay and the maximal radio frequency that can be supported by a given

fiber of known dispersion, and a given laser source of known wavelength instability,

subject to a constraint of maximum tolerable residual phase error:

1

2 2 2

n nf

fDL fDc Dc

(10)

Equation (10) suggests that a longer delay and a higher radio frequency

would require better wavelength stability. The relation also suggests a tradeoff

between the delay-bandwidth product and the range of temperature variations that

can be supported: The former requires an optical fiber with low chromatic

dispersion, whereas the latter requires an optical fiber that is highly dispersive (see

equation (8)). For instance, the use of a tunable source with wavelength instability of

1 pm and a standard SMF-28 fiber, would allow for the delay of signals with phase

stability of one angular degree only when the delay-bandwidth product f does not

exceed 610 . The tradeoff can be expressed in terms of a figure of merit, which sets a

bound on the nominal delay of the fiber, the range of temperature variations, and

the delay variations that are attainable (in units of C):

maxmax

1 1FoM T N

(11)


Here max is the tuning range of the laser source. Equation (11) highlights

the significance of the tunable laser quality, in term of its "effective number of

wavelength resolution points" maxN

, in setting upper bound on the

performance of the stabilized fiber delay line. Tunable lasers with a scanning range

of 100 nm and wavelength resolution of 1 pm, or about 100,000 wavelength

resolution points, are readily available. Such sources could support a stabilized fiber-

optic delay line with a figure of merit of about 1.5e10 C.

The RF bandwidth of the input waveform is restricted by dispersion-induced

fading. Assuming double-sideband, small-signal modulation, one can show that the 3

dB RF bandwidth of delayed distribution is given by [93, 94]:

3

2

1 1 2 1

2 3 2 3 2dB

nf

L D D

(12)

Here 2

2 2D

c , in units of ps2/km. The bandwidth limitation for a 25

km-long, standard single-mode fiber delay line is about 7 GHz.

2.2. STABILIZATION USING TWO LASER SOURCES

The stabilization principle relies on the transmission of a single-tone wave,

which is referred to as the control signal, along with the user's waveform. After a

round-trip, the control signal is mixed with its local replica. The DC level, which is

filtered out of the mixer's output, represents the variations of the output RF phase of

the control tone. The phase stabilization is carried out by changing the wavelength of

the tunable laser source in increments that are determined by the DC level and a


correction factor. This correction factor is linearly proportional to the dispersion

parameter (more about the correction factor is provided later in this chapter).

In previous arrangements both RF signals, the control tone and that of the

user, modulated a single tunable laser source. When using this configuration, it is

possible to stabilize the output phase of the user's waveform only when the two

signals can be separated in the RF domain with no overlap between them.

Consequently, this configuration might be unsuitable for broadband waveforms.

In this work we propose a method that uses two tunable laser sources in two

different wavelengths. Each RF signal, the control tone and that of the user,

modulates a different laser source. If the separation between the wavelengths of the

two laser sources is large, the dispersion parameters of the fiber may not be the

same for both, and each laser source would require a different correction factor. The

two correction factors would be related by a simple linear relation, which is

governed by the dispersion slope parameter of the fiber. In using this scheme, the

phase of an arbitrary RF signal can be stabilized based on the feedback provided by

tracking the phase of an independent RF control tone, without any restriction on its

spectrum.

2.3. SCHEMATIC ILLUSTRATION OF THE EXPERIMENTAL SETUP

A schematic illustration of experimental setup for the stabilized fiber-optic

delay of arbitrary RF waveforms is shown in Figure 8. The generic description is given

here to help clarify the stabilization principle, whereas a more detailed account will

be provided in the next chapter. Solid black lines denote optical fiber paths, green


lines indicate radio frequency signal paths that are part of the control circuitry, red

lines correspond to the path of the user's signal to be delayed, and dashed black

lines indicate DC signals of the control feedback loop. The setup is modular: the

feedback loop can be disconnected, and either of the two RF inputs, that of the

intended user and the control sine wave, may be switched on and off independently.

A control sine wave modulates the upper tunable laser source whose

wavelength is denoted as 𝜆1. A replica of the RF control sine wave is retained at the

near end of the fiber, to be used as reference. The input RF waveform to be delayed

modulates the lower tunable laser source whose wavelength is denoted as 𝜆2. The

nominal wavelengths of the two lasers must be sufficiently distinct to allow for

convenient optical filtering. The two signals are combined at the input of the fiber

delay line.

Amplitude

Modulator

User RF

RF splitter

RF control

(sine wave)

RF mixer

Feedback

Tunable laser 𝝀𝟏

Wavelength

control

Control

sine wave

Reference voltage

Amplitude

Modulator Tunable laser 𝝀𝟐

PC

PC

1 2

3

Det. 2

BPF 𝝀𝟏

Fiber

mirror

Det. 1

User RF

BPF 𝝀𝟐

2:1 1:2

Figure 8: Experimental setup for the realization of a long, stabilized fiber-optic delay of arbitrary RF waveforms. BPF: bandpass filter, PC: polarization controller.


At the far end of the fiber delay line the optical signal is split in two. One copy

of the signal is filtered by an optical bandpass filter (BPF) to retain only the

waveform of the user and eliminate the control sine wave, and is then detected by a

photo-receiver (Det. 1). The filtered signal serves as the stabilized RF output of the

setup, to be used by a larger system such as a Brillouin fiber sensor, if necessary. The

output of the second coupler branch is retransmitted back along the fiber delay line

by a fiber mirror.

Back at the transmitter end of the fiber link, the returning optical waveform

is filtered by a second BPF whose central wavelength is set to 𝜆1 to retain the control

sine wave only, and detected by a second photo-receiver (Det. 2). The sine wave,

having propagated through a two-way delay, is mixed with its reference replica in a

RF mixer, generating a DC voltage. The voltage depends on the RF phase differential

between the two sine wave copies, ranging from a maximal positive value when the

two are in-phase, through zero value for a 90 phase differential, and maximum

negative value for a 180 phase difference. Therefore, the stabilization of the DC

output of the mixer at any given value guarantees the phase stability of the control

sine wave.

The DC reading is used as the input of a negative feedback loop mechanism,

which modifies the operating wavelength of both laser diodes in attempt to retain

the DC value of choice. For convenience, our experiments aim to maintain zero

voltage, since this working point provides a linear relation between phase drift and

correction voltage (see also below). The phase stabilization of the control sine wave

is reached through a compensation of thermal drifts by chromatic dispersion. It


invariably guarantees the phase stabilization of the user waveform as well at the

output of Det. 1, irrespective of any particular properties of that waveform.

In most applications, the waveform of the user must be transmitted to a

remote location. Therefore it propagates only one-way along the fiber delay line. The

control sine wave, on the other hand, propagates back and forth since its reference

replica is retained at the transmitter end.

2.4. THE CORRECTION FACTOR

The DC voltage V is sampled by a computerized setup controlling the

tunable laser sources, and a command for setting their wavelengths to new values is

sent at pre-determined time intervals based on their present operating wavelength,

the sampled voltage and the correction ratio:

next prev V (13)

The correction factor α between the DC voltage reading and the necessary

wavelength increment correction can be deduced from equation (13):

V V

(14)

The first term on the right-hand side is related to the length of the fiber, its

dispersion parameter, and the radio-frequency of the control sine wave. According

to (7) and (9):

1 nm

2 radRFf DL

(15)


The second term on the right-hand side of equation (14) refers to the change

in the phase per DC voltage. The output voltage follows a sine wave dependence of

the phase difference between the delayed and local replicas of the RF tones. As

mentioned earlier, we look to stabilize a phase difference of 90 between the two RF

inputs of the mixer. Denoting the residual phase error with respect to 90 as , we

may write:

0( ) sin( )V V (16)

Here 0V is the maximum reading of the mixer output voltage. Hence, in the

vicinity of zero residual phase error:

0 0( ) cos( ) ( ) ( )dV V d V d (17)

We therefore find that the second term on the right-hand side of equation

(14) is simply given by the magnitude of the sinusoidal variations of the mixer

output:

0

1 rad

mVV V

(18)

The numerical values of both terms are calibrated using simple experimental

procedures. Bringing (15) and (18) together:

0

1 1( )

2 ( )RFV f LV D

(19)

Two correction factors are necessary in our stabilization protocol, one for

each tunable laser source. Since the wavelengths of the two are typically separated


by about 10 nm, the different dispersion coefficient of the fiber at each wavelength

must be taken into consideration:

2 1 2 1( ) ( ) ( )D D S (20)

Here S is the dispersion slope parameter of the fiber. For standard SMF-28

fiber it is given by 20.092[ps nm km]S . For instance, if the separation between

the wavelengths of the laser sources is 10 nm, a difference between dispersion

coefficients of 0.92 ps nm kmD must be taken in consideration when setting

the two correction factors 1 2( , ) in equation (18).

Residual delay variations stem from uncertainties in both the setting of

wavelengths and in the measurement of voltage. The wavelength setting error in

1 was ±30 pm, corresponding to of about ±5 ps over 9 km. The standard

deviation of the noise V in voltage measurements was on the order of ±5 mV,

corresponding to delay variations 02V fV on the order of ±2.5 ps.


3. EXPERIMENTAL SETUPS

3.1. OPEN LOOP MEASUREMENTS WITH A SINGLE LASER

SOURCE

The preliminary experiments we conducted are divided into two stages. In

the first we observed the effect of chromatic dispersion on the output phase of a RF

sine wave, and in the second we examined the relations between the DC level at the

mixer output, the phase shift of the RF output waveform and the wavelength of the

tunable laser source.

3.1.1. STAGE 1: OBSERVATION OF THE CHROMATIC DISPERSION EFFECTS

Real-time

Scope

Signal

Generator

LPF -

𝑓𝑐 = 3.3𝐺𝐻𝑧

Tunable laser

𝜆1 = 1550 𝑛𝑚 Amplitude

Modulator

Fiber

mirror

Amplifier +40dB

Det. 1

EDFA

Figure 9: An illustration of the setup used in the observation of the chromatic dispersion effects. LPF: low-pass filter, EDFA: erbium doped fiber amplifier.


The purpose of the first experiment was to modulate a tunable laser source

by a RF sine wave, and measure the RF phase shifts at the output of a long fiber-

optic delay line due to chromatic dispersion. The setup for this experiment is shown

in detail in Figure 9. Black lines denote optical fiber paths and green lines indicate

radio frequency signal paths. A sine wave of 3 GHz frequency and 27 dBm power was

generated by a high frequency microwave generator. The frequency of the sine wave

was chosen according to the bandwidth limitations of the RF amplifiers and the

photo-detector that are part of the setup. At the output of the signal generator a

low-pass filter (LPF) with cut-off frequency 3.3cf GHz was placed, in order to

prevent aliasing. This LPF also blocked off harmonic distortions, as shown in Figures

10 and in Figure 11.

Figure 10: Radio frequency spectrum of the waveform generated by the microwave generator without the LPF.


Figure 11: Radio frequency spectrum of the waveform generated by the microwave generator, with use of the LPF.

After the LPF, the RF sine wave was split in two paths: one was used to drive

an electro-optic amplitude modulator (AM), which was placed at the output of a

tunable laser diode source (see next). The other replica of the sine wave was

sampled by a 6 GHz bandwidth real-time oscilloscope and used as a timing

reference.

The light source used in the experiment was a continuous wave (CW) tunable

laser diode at the 1550 nm wavelength range, with 8 dBm output power. The output

of the laser passed through a polarization controller (PC) and connected to the AM.

The PC was needed for matching the state of polarization of the laser light with that

of the modulator. The optical carrier and the two modulation side-bands propagated

through a magneto-optical fiber circulator and into an 8.8 km-long optical fiber delay

line that was wrapped around a spool. Figure 12 shows the optical carrier and

modulation side-lobes as measured by an optical spectrum analyzer (OSA). The fiber


delay line was terminated by a fiber mirror, which in this particular experiment was a

bare cleaved facet. The Fresnel relative power reflectivity of facet is about -14 dB

(4%).

Figure 12: Optical power spectral density of an optical carrier modulated by a 3 GHz sine wave.

Following a round-trip in the fiber delay line and back through the circulator,

the optical signal was amplified by a programmable erbium doped fiber amplifier

(EDFA) from an optical power of -29 dBm to 0 dBm. It is important to keep the

output power of the EDFA below 0 dBm in order to prevent damage of the photo-

receiver. The output of the EDFA was then detected by a fast photo-receiver with a

rise-time of 35 ps, denoted as Det. 1, and amplified by a RF amplifier that provided a

total gain of 40 dB. The output of the amplifier was sampled by a second channel of

the real-time digitizing oscilloscope. The oscilloscope was connected by the internal

communication network of the laboratory to a personal computer.


The timing of the reference signal channel was held steady by the internal

trigger of the oscilloscope, whereas the timing of the delayed waveform in the other

channel could vary across the screen. The wavelength of the tunable laser source

was tuned by a matlab code within the range of 1550-1551 nm, in increments of 0.1

nm, every 2 seconds. The time interval between measurements was defined by the

minimal time required to save data on the oscilloscope. The RF phase difference

between the reference sine wave and the delayed replica, for each wavelength step,

was measured on the oscilloscope.

3.1.2. STAGE 2: CHARACTERIZATION OF THE RELATIONS BETWEEN

WAVELENGTH, RF PHASE SHIFTS AND DC VOLTAGE

DVM

Signal

Generator

LPF - 𝑓𝑐 = 3.3𝐺𝐻𝑧

Tunable laser


Modulator

Fiber

mirror

Amplifier +40dB

Det. 1

EDFA

Attenuator -16dB

Mixer

IF

LO RF



Figure 13: An illustration of the setup used in the characterization of the relations between wavelength, RF phase shift and mixer DC voltage. LPF: low-pass filter, EDFA:

erbium doped fiber amplifier, DVM: digital voltmeter.


The purpose of this experiment was to characterize the relations between

the wavelength of the tunable laser source and the phase shift of a delayed RF sine

wave, and the corresponding DC voltage reading that is obtained by mixing the

delayed RF sine wave and the reference tone. That DC reading is later used as

feedback for the tunable laser wavelength adjustment.

The setup for this experiment is illustrated in Figure 13. It is very similar to

the previous one of Figure 9, except for a few additions. The cleaved facet at the end

of the delay line was replaced by a fiber mirror with 99% power reflectivity.

Following two-way delay, photo-detection and amplification of the RF sine wave, the

output of the RF amplifier was connected to the RF input port of a RF mixer. The

reference replica of the same RF tone was attenuated by a 16 dB RF attenuator and

connected to the local oscillator (LO) input port of the same mixer. The purpose of

the attenuator was to make sure that the amplitudes of both sine waves are equal at

the input ports of the mixer. The output of the RF mixer is the product between the

two sine waves, and can be written as:

1 2

1 2 1 2 1 2 1 2( ) cos ( ) cos ( )2

mix

A AV t C t t (21)

Here, 1,2A , 1,2 and 1,2 are the amplitudes, frequencies and instantaneous

phases of the two sine waves, respectively. C denotes a proportionality constant

that is determined by the responsivity of the detector, RF amplification and

efficiency of the mixer. In this particular case, the frequencies of both sine waves

were equal, hence the right-hand cosine term in (20) becomes a DC voltage. Two

cascaded LPFs with a cut-off frequency 0 2.25f GHz were placed at the output


port of the mixer, in order to block-off the signal at the doubled frequency.

Consequently, the signal measured by a digital voltmeter (DVM) was:

1 2( ) cos ( )

2

A AV t C t (22)

In (21), ( )t denotes the difference between the instantaneous phases of

the two sine waves.

Similarly to the previous experiment, we used a matlab code to change the

wavelength of the tunable laser source within the range of 1550-1551 nm, in

increments of 0.1 nm and every 0.5 seconds. For each step we observed the RF

phase shift between the two sine waves on the oscilloscope, and simultaneously the

DC voltage. The results provided the necessary correction factor that was used later

in the conversion of residual phase errors to wavelength offsets of tunable lasers

(see Chapter 2).

3.2. CLOSED-LOOP, STABLE DISTRIBUTION OF A RF SINE WAVE

The purpose of this stage in the experiment was to distribute a single-tone RF

sine wave over a fiber-optic delay line, measure the DC voltage that represents

phase shifts as above, and use this voltage in a closed feedback loop that changes

the wavelength of the laser source and maintains a stable output RF phase of the

sine wave.

The setup for this experiment is illustrated in Figure 14. The dashed black

lines indicate DC signals of the control feedback loop. The setup was that of the

previous stage, except for the feedback loop that was added after the DVM. The


conversion factor between measured voltage and necessary wavelength correction

was extracted in the previous experimental phase. Both the DVM and the tunable

laser source were connected to a computer.

Using a LabVIEW code, we implemented a closed feedback loop according to

equation (12), which is repeated here for convenience:

next prev V (23)

Every half a second, the LabVIEW code issued a set of commands to measure

the DC voltage and set the tunable laser source to a new wavelength. The feedback

loop was designed to maintain zero DC voltage, which corresponds to a stabilized

DVM

Signal

Generator


Tunable laser


Modulator

Fiber

mirror

Amplifier +40dB

Det. 1

EDFA

Attenuator -16dB

Mixer

IF

LO RF



Figure 14: An illustration of the setup used in the closed feedback, phase stabilized delay of an RF sine wave. EDFA: erbium doped fiber amplifier, LPF: low-pass filter, DVM: digital

voltmeter.


relative phase of 90º. This feedback loop repeated itself automatically, until

terminated manually or until the wavelength of the tunable laser source drifted by

more than 5 nm.

3.3. CLOSED-LOOP, STABLE DISTRIBUTION OF AN ARBITRARY

RF WAVEFORM

The purpose of this experiment was to transmit a RF sine wave as the control

signal over one optical wavelength, and an arbitrary RF waveform, referred to as the

user's waveform, on a different optical carrier. The tracking of the output RF phase

of the control signal provided the necessary feedback for adjusting the wavelengths

of both optical carriers, and stabilizing the delays and output phases of both.

The setup for this experiment is illustrated in Figure 15. Red lines correspond

to the path of the user's signal to be delayed. On the upper branch, a CW tunable

laser source was initiated to a wavelength of 1527.5 nm, denoted here as 1 , and to

an output power of 5 dBm. The output of the laser source was connected to a PC

and then to an electro-optic AM. A RF LFM signal was generated using an arbitrary

waveform generator (AWG). The LFM signal was defined by a pulse duration of

5T s , a central frequency of 0 1.75f GHz and a bandwidth of 500B MHz .

The central frequency of the LFM signal was chosen according to the bandwidth of

the photo-receiver we have in our disposal.


Real-time

Scope

Mixer

LO

IF

RF

DVM

Signal

Generator


Amplitude

Modulator

Tunable laser

𝜆2 = 1544 𝑛𝑚

Arbitrary

Waveform

Generator


Attenuator -12dB

Amplifier +40dB


Tunable laser

𝜆1 = 1527.5 𝑛𝑚

2:1 Amplitude

Modulator 1:2

Fiber

mirror


Amplifier +30dB

Optical BPF

𝜆1 = 1527.5 𝑛𝑚

Det. 1

EDFA

Amplifier +40dB

Optical BPF

𝜆2 = 1544 𝑛𝑚

Det. 2

EDFA



Correlation

processing

Figure 15: An illustration of the setup for the stabilized delay of an arbitrary RF waveform. LPF: low-pass filter, EDFA: erbium doped fiber amplifier, DVM: digital

voltmeter, BPF: band-pass fiter.

Attenuator -16dB


The AWG used for this setup has a maximal sampling rate of 5 Gsample/s.

The output of the AWG was split into two paths. The signal in one path was used as a

reference signal for the post-detection correlation processing of the delayed

waveform, and was therefore connected directly to the real-time oscilloscope.

The RF LFM waveform in the other path was filtered by a LPF with a cut-off

frequency 2.25cf GHz in order to prevent aliasing. The filtered waveform was

attenuated by a -12 dB RF attenuator, and then amplified by a fixed-gain 40 dB

broadband RF amplifier. The attenuator was required to prevent saturation of the

amplifier. Following amplification, the waveform passed through another LPF with a

cut-off frequency of 2.25cf GHz to ensure that only the LFM signal at the correct

central frequency is used in modulation of the laser source, with no copies at other

frequencies. At that point, the power of the LFM signal was 7 dBm. Lastly, the

processed LFM waveform was used to modulate the light at the output of the laser

source via the electro-optic AM.

Going down to the lower branch of the setup, a second CW tunable laser

source was initiated to a wavelength of 1544 nm, denoted as 2 , and an output

power of 8 dBm. The wavelengths of both laser sources were chosen arbitrarily;

however, they must be a separated but at least 4 nm to allow for their proper

separation by the optical band-pass filters (BPFs) available to us.

Similarly to previous setups, a sine wave of 3 GHz frequency and 27 dBm

power was generated by a high frequency microwave generator. This sine wave was

split into two paths: one copy was attenuated by a -16 dB RF attenuator and then


connected to the LO input port of a RF mixer, as the reference signal for phase

stabilization, and the other copy was filtered by a LPF with a cut-off frequency of

3.3cf GHz and used to modulate the second tunable laser source through a

second electro-optic AM.

Next, both optical carriers and their side-lobes were combined by a 50/50

optical coupler, and directed through a magneto-optical fiber circulator into an 8.8

km-long fiber delay line. At the far end of the delay line, the combined signals were

split by a 50/50 optical splitter. One copy of the both modulated carriers was

reflected back by a 99% reflectivity fiber mirror.

The other copy of the signal, with an optical power of -15 dBm, was amplified

by a programmable EDFA to an output power of 8 dBm and filtered by a 4 nm-wide

tunable optical band-pass filter (BPF) with 1 as its central wavelength, to retain only

the optical carrier with the user's waveform, which in our particular case was the

LFM signal. The filtered waveform was detected by a fast photo-receiver with a rise-

time of 35 ps, denoted as Det. 1. The output power of the EDFA was chosen to

maximize the signal power, without reaching the upper limit of the photo-receiver

dynamic range.

The reconstructed, delayed LFM signal was amplified by a 30 dB RF amplifier,

since the electrical output power of the detector was too low for further processing.

The transfer function of the optical BPFs in our disposal provided a rejection ratio of

about 20 dB, hence the amplified RF waveform was filtered yet again with a LPF with

a cut-off frequency of 2.25cf GHz , to remove residual cross-talk of the control


sine wave. Lastly, the output LFM waveform was sampled by the real-time

oscilloscope for further off-line processing. Figure 16 shows the LFM waveform at

the output of LPF prior to the oscilloscope.

Figure 16: RF power spectral density of a LFM waveform at the output of the delay line.

The reflected copy of the optical waveforms traveled back through the fiber

delay line and then through the circulator. The overall optical power of the

waveforms at the circulator output was -23 dBm. The waveforms were then

amplified by another programmable EDFA to an output power of 5 dBm, and filtered

by a second 4 nm-wide tunable optical BPF with a central wavelength of 2 to retain

only the control sine wave. The output of the BPF as measured by the OSA is shown

in Figure 17. The filtered signal was detected by another fast photo-receiver with a

rise-time of 12 ps, denoted as Det. 2, and was then amplified by a 40 dB RF amplifier.

Following the amplifier, the signal was connected to the RF input port of the RF

mixer. Figure 18 shows the filtered RF sine wave prior to the mixer.


Figure 17: Measured optical power spectral density of the delayed waveform. An optical bandpass filter was used to retain one optical carrier that was modulated by the control tone (right-hand peak), and reject a second optical carrier which

was modulated by the user's RF waveform.

The RF mixer multiplied the delayed sine wave with its reference replica. The

output of the mixer was filtered twice by LPFs with cut-off frequencies of

Figure 18: RF power spectral density of the control sine wave following two-way propagation along the fiber delay line.


2.25cf GHz , and 3.3cf GHz , in order to retain only the DC voltage component,

as explained in detail in the previous sub-section.

A LabVIEW code was used to interrogate the DVM readings of the DC voltage.

The correction factor between DC voltage and the necessary wavelength offsets to

the laser diode of the control waveform was known from previous experiments. The

corresponding correction factor for the tunable laser of the LFM waveform was

extrapolated based on the dispersion slope parameter of the delay fiber, using

equation (19). New wavelength settings were sent to both tunable laser sources by

the LabVIEW code, and the process repeated itself until termination.

In parallel to the tracking of the control sine wave and the setting of

wavelengths, the timing of the delayed LFM waveform at the far end of the delay

line was monitored by its cross-correlation with its reference replica (see in Chapter

1 for compression of LFM waveforms). The timing of the main lobe peak was used as

a measure of the effective delay of the waveform. A properly stabilized delay line

manifests in reduced timing jitter of the compressed main lobe. The smallest jitter

we could observe corresponds to the timing inaccuracy of the sampling oscilloscope,

which is on the order of ±1 ppm of the acquisition trace duration.

Experimental Results 46

4. EXPERIMENTAL RESULTS

4.1. OPEN LOOP MEASUREMENTS WITH A SINGLE LASER

SOURCE

4.1.1. STAGE 1: OBSERVATIONS OF THE CHROMATIC DISPERSION EFFECTS

Figure 19 shows the delayed control sine wave following propagation along

the fiber delay line, for 10 wavelengths of the tunable laser diode source. The

wavelength was changed by 0.1 nm increments between successive acquisitions. The

wavelength variations translate into group delay variations and changes to the

output RF phase, according to the chromatic dispersion of the delay fiber. Each 0.1

nm wavelength increment corresponds to a delay variation of about 30 ps, in good

agreement with the predictions of equation (7).

Figure 19: A 3 GHz control sine wave, detected following two-way propagation in a 8.8 km-long dispersive fiber delay line. Each trace

corresponds to a different wavelength of the tunable optical carrier, in 0.1 nm increments. The change in group delay between

successive acquisitions is about 30 ps.


The overall duration of the experiment was about 20 seconds. Variations in

the temperature of the air-conditioned laboratory are limited to 1-2 C over an hour.

Therefore, temperature drifts within the experimental durations were unlikely to

exceed 0.01 C. The corresponding delay changes are below 0.1 ppm, or few ppm at

most. These possible thermal drifts are two orders of magnitude smaller than the

dispersion-induced variations observed in the measurements.

4.1.2. STAGE 2: CHARACTERIZATION OF THE RELATIONS BETWEEN THE

WAVELENGTH, PHASE SHIFTS AND DC VOLTAGE

Figure 20 shows the output phase of a delayed, 3 GHz control sine wave

(left), and the DC voltage obtained at the output of the mixer (right), as a function of

the tunable laser diode wavelength. The phase varies linearly with wavelength

whereas the DC voltage follows a sinusoidal pattern, in agreement with equation

(21). Given the dispersion parameter of the delay fiber and the frequency of the RF

sine wave, the expected delay variation is approximately 300 ps/nm. The change in

delay corresponds to a RF phase variation of about 324 angular degrees per nm. The

observed phase variations were 303 angular degrees per nm. The experimental

duration was sufficiently short to avoid thermal delay drift in between successive

measurements.


Figure 20: (left) Output phase of a 3 GHz control sine wave following two-way propagation in a 8.8

km long delay fiber, as a function of tunable laser wavelength, (right) DC reading of the mixing between the delayed control sine wave and its reference replica, as a function of the tunable laser

source wavelength.

The correction factor α between the DC voltage reading and the necessary

wavelength increment can be deduced from the results of Figure 20. The

term of equation (14) can be calculated based on experimental parameters, but can

also be extracted from the left-hand panel of Figure 20. The slope of the curve

suggests that:

1 1 nm

2 0.84 2 radRFf DL

(24)

The V term (see equation (17)) is simply given by the magnitude of the

DC voltage variations with wavelength:

0

1 1 rad

110 mVV V

(25)

Multiplying the two together, we find:

1550

1 1 pm( ) 1.6

0.84 2 110 mVV

(26)


4.2. CLOSED-LOOP, STABLE DISTRIBUTION OF A RF SINE WAVE

This section describes the experimental, closed-loop stable distribution of a

RF sine wave. First, Figure 21 shows the voltage at the mixer output that was

monitored over 45 minutes with the feedback loop open. Two periods of the DC

voltage were observed over 40 minutes, corresponding to delay variations of 660 ps

or about 7.5 ppm of the nominal delay of the fiber. These delay variations therefore

suggest a temperature change of about 1 ºC. The measurements suggest that

temperature is drifting linearly in time. One can reasonably expect that such slow

drifts in the temperature of an air-conditioned laboratory would be monotonous and

nearly linear. The magnitude 0V of voltage changes in this particular experiment is

200 mV. The results suggest a correction factor of pm

0.8mV

.

Figure 21: The DC voltage at the output of the mixer as a function of time, in open feedback loop operation.

Figure 22 shows the DC voltage at the mixer output (top) and the wavelength

setting commands sent to the tunable laser source (bottom), as a function of time


during over 2 minutes of closed-loop operation. Following the first 5 seconds of

transient operation, the mixer output reaches zero voltage and is held stable with

small-scale oscillations around that value. At the same time, the laser diode

wavelength continues to adjust and compensate for ongoing thermal drifts. As

further, direct illustration of phase stability, Figure 23 shows persistence trace

recordings of the control sine wave at the output of Det. 1, accumulated over 10

minutes following the initial stabilization. The output phase is entirely random for

open-loop operation (right), and is nicely stabilized with the loop closed (left). The

results provide a proof of the proposed stabilization concept.

Figure 22: Closed loop, phase stabilized two-way delay of a 3 GHz sine wave over 8.8 km of fiber. (top): DC voltage obtained by the mixing the delayed sine wave with its reference replica, as a

function of time. (bottom): wavelength setting commands to the tunable laser source, as a function of time.


Figure 23: Persistence traces of the output 3 GHz control sine wave, following two-way delay in 8.8 km of fiber, accumulated over 10 minutes of operation at 0.5 second intervals, with (left) and

without (right) closed loop feedback.

In order to accelerate the thermal drift of the delay, a halogen lamp was

placed in close proximity to the fiber spool. Heat radiated from the lamp introduced

strong temperature gradients over many minutes. Figure 24 shows several

persistence traces of the delayed, control sine wave. First, the feedback loop was

disconnected (top left), and the output was recorded over 7.5 minutes without

deliberate heating. Next, the halogen lamp was brought near the fiber for 2.5

minutes, and the feedback remained disconnected (top right). Then, feedback was

applied, and recording were taken over another 2.5 minutes (bottom left). Last, the

lamp was removed, and the output was recorded for additional 7.5 minutes during

the cooling down of the delay fiber (bottom right). In all panels, traces were acquired

once every 10 seconds. The result show drifts of the output RF phase when the

feedback loop was opened, both with and without the heating of the lamp. The

closed feedback was able to stabilize the output phase, even during the cool-down

transient of the delay fiber.


Figure 24: Persistance traces of the 3 GHz control sine wave following delay over 8.8 km of fiber: feedback loop open and no external heating of the delay fiber (top left); feedback loop open during the heating of the fiber by a halogen lamp (top right); feedback loop closed during ongoing heating (bottom left); and feedback loop closed following the removal of the heating lamp and during the

cooling down of the delay fiber (bottom right).

Figure 25 shows the DC reading as a function of time along the entire

experiment, with the instances of transition between the four stages of the

experiment as described above indicated on the curve. The DC voltage varied

gradually during open loop operation. The rate of the voltage sweep depended on

the rate of the temperature changes, hence sharper voltage transitions were

observed with heating installed. The voltage remained fixed at zero value following

the closing of the loop 10 minutes into the measurements, both with and without

the presence of the lamp.


Figure 25: The DC voltage at the output of the mixer as a function of time, during the four stages of the experiment described in

Figure 24.

4.3. CLOSED-LOOP, STABLE DISTRIBUTION OF AN ARBITRARY

RF WAVEFORM

This section describes the experimental, closed-loop stable distribution of a

LFM waveform as an example for the processing of arbitrary user signals. Similarly to

the previous experiment, we started with the feedback loop open in order to extract

the magnitude of the DC voltage at the output of the mixer for the calibration of the

correction factor. Once again we used the halogen lamp to accelerate the thermal

drift of the delay. Figure 26 shows the DC voltage at the output of the mixer over 2

minutes of open feedback loop operation, indicating a phase change of about 3π. In

this particular experiment the magnitude 0V was 350 mV, hence the correction

factor for the tunable laser source of the RF control sine wave was set to

pm0.6

mV

. Based on the dispersion slope of the delay fiber, the correction


factor for the other tunable laser, that of the user waveform, was adjusted to

1

pm0.54

mV

.

Figure 26: The DC voltage at the output of the mixer as a function of time, during 2 minutes of open feedback loop operation.

Figure 27 shows the wavelengths of the tunable laser source of the RF control

sine wave (green) and of tunable laser source of the LFM signal (blue), as a function

of time during 2 minutes of closed-loop operation in which the halogen lamp was

placed near the delay fiber. Both wavelengths were swept in order to maintain a

stable delay, according to the feedback provided by the control channel. Figure 28

shows the DC voltage reading at the output of the mixer as a function of time.

Following an initial transient of a few seconds, the control voltage remained fixed at

a constant value while the delay fiber was being heated.


Figure 27: Wavelength of the tunable laser source of the RF control sine wave (green) and of the LFM signal (blue), as a function of time during two minutes of closed-loop operation.

Figure 28: The DC voltage at the output of the mixer as a function of time during closed-loop operation.

Unlike previous experiments, the DC voltage reached a value of 11 mV which,

although small, was nevertheless different from zero. This behavior stems from a

combination of comparatively rapid heating, alongside rather slow and simplistic

feedback. As can be seen in Figure 27, the feedback loop changed the wavelengths

of the tunable lasers by approximately 1 nm within 100 seconds. Considering the

dispersion parameter of the fiber, the 1 nm wavelength offset compensated for

thermal drifts of the one-way delay in the 8.8 km-long fiber by approximately 150 ps.


This delay variation represents 3.3 ppm of the 45 µs-long nominal one-way delay, or

an increase of 0.3 ºC in the average temperature of the fiber. In other words, the

thermal drift of the delay during each 0.5 second-long interval was associated with a

wavelength correction of 5 pm, which is in turn equivalent to a voltage drift of about

10 mV.

The feedback procedure relies on a simple proportionality correction, and

does not include any extrapolation of the wavelength correction term based on

previous samples. In addition, the effective loop bandwidth of 2 Hz was very narrow.

Hence the feedback loop could not reconstruct the overall trend of the

instantaneous temperature, or correct for drifts that take place within a sampling

interval. The results demonstrate the limitations of this basic feedback approach. In

this particular experiment, the heating rate of the delay fiber was nearly constant.

Therefore, the shortcoming of the feedback mechanism manifested only in a nearly-

fixed, small bias of the mixer voltage, which leads to the stabilization of the output

waveform to a slightly different delay. This difference in itself is usually insignificant,

as long as no additional delay jitter is induced. However, this drawback might have

more severe implications in cases of rapid temperature changes of non-constant

rates. The problem may be solved by using faster and more careful feedback

protocols.

Lastly, Figure 29 shows persistence traces of the compressed shapes of the

delayed LFM waveforms, sampled every 5 seconds following their one-way

propagation over the fiber delay line. The delayed waveforms were compressed

using off-line matched-filter post-processing. The resolution of the 500 MHz-wide


LFM waveforms following their compression is on the order of 2 ns. The top and

bottom panels show the results of open-loop and closed-loop operation,

respectively. The timing drift of the main lobe peak position is reduced from 200 ps

during open-loop operation to 20 ps with the feedback loop closed. The latter value

is limited by the timing inaccuracy of the digitizing oscilloscope.

Figure 29: Persistence traces of the compressed shapes of LFM waveforms, sampled at the output of the fiber delay over 2 minutes, without (top) and with (bottom) closed-

loop wavelength control.

One of the tunable lasers available to us could only perform a continuous

wavelength sweep over approximately 2 nm. Any discontinuity in the lasers

wavelengths immediately disrupts the closed-loop operation. Therefore, the largest


timing jitter that could be compensated for in the particular experiment was on the

order of 300 ps.

Stabilized Delay Within High-Resolution Brillouin Analysis 59

5. STABILIZED DELAY WITHIN HIGH-RESOLUTION BRILLOUIN ANALYSIS

The purpose of this experiment is to demonstrate the applicability of the

proposed stabilization technique to the processing of broadband microwave signals

as part of more complicated systems. As was mentioned in Chapter 1, high-

resolution B-OCDA systems require a fiber optic delay line of many km in order to

achieve cm resolution measurements. However, thermal delay drifts on the long

fiber might lead to the incorrect interpretation of data and to loss of resolution. The

principles of B-OCDA were addressed in the introduction. In this chapter, I first

describe the phase-coded B-OCDA experimental setup and then proceed to show the

significance of stabilized delay for its proper functions.

5.1. B-OCDA EXPERIMENTAL SETUP

In this section I provide more detail on the resolution and spatial scanning

protocols in phase-coded B-OCDA systems. Both pump and signal optical waves are

coded by a repeating, high-rate binary phase sequence of symbol duration sT and a

period of sN bits. A large number of correlation peaks is typically introduced along

the fiber under test, with a spatial extent 12 sz c n T (which also signifies

resolution), and separation between neighboring peaks of sN z [65, 92, 95]. Post-

detection data analysis protocols are able to simultaneously and unambiguously

interrogate Brillouin interactions taking place at thousands of correlation peaks [65,

95]. One of the main challenges in the realization of phase-coded B-OCDA is the


spatial scanning of correlation peaks positions. The favored solution path to-date

relies on long fiber delay lines.

Figure 30 shows a schematic illustration of a phase-coded B-OCDA setup,

which includes a stabilized delay line based on the principles of this work. A tunable

laser source at 2 of 1551 nm is used as the common source for Brillouin pump and

signal waves. Light from the laser output is phase-modulated by a repeating binary

sequence ( sN = 211 bits), chosen for its favorable correlation properties [96]. The

symbol duration sT of 200 ps corresponds to a spatial resolution of 2 cm.

Figure 30: Schematic illustration of a phase-coded Brillouin optical correlation-domain analysis (phase-coded B-OCDA) setup, incorporating a stabilized fiber-optic

delay line in the path of the Brillouin signal wave. Solid lines denote fiber paths, dashed lines represent RF cable paths, and dashed-dotted, black lines correspond to

DC control signals. Blue color represents paths and components related to the sensing functionality, whereas green paths and components are part of the delay

stabilization module. The 'Wavelength control' module is described in detail in Fig. 15. Details of the 'Pump wave processing' module are not shown, for better clarity

(see [65]).

The modulated light is split into pump and signal branches, that are linked

through a 200 m long fiber under test to form a fiber loop. The value of B in the


fiber under test at room temperature is 10.85 GHz. Light in the pump branch is offset

in frequency by an adjustable using an electro-optic single-sideband modulator,

restricted to pulses by an electro-optic amplitude modulator, and amplified by an

EDFA. These components are not shown in Fig. 30, for better clarity and focus on the

current work (see [65] for full detail). Light in the signal branch passes through a 25

km-long delay fiber and enters the fiber under test from the opposite direction. The

signal wave at the output of the fiber under test is filtered by an optical band-pass

filter that retains 2 , detected by a photo-receiver, and sampled for off-line

processing.

The role of the long fiber delay is as follows: The middle of the fiber loop is

defined as the location of equal distances from the splitting point at the phase

modulator output (see Fig. 30), going in clockwise and counter-clockwise directions.

Correlation peaks are introduced at distances 12m s s sZ m N z m N c n T from

the middle of the loop, where m is an integer. The locations mZ of all peaks, with

the exception of the zero-order one, may be moved with slight changes sT to the

symbol duration: m m s sZ Z T T .

Let us denote the highest (lowest) order of correlation peaks that is in

overlap with the fiber under test as maxm ( minm ). A sufficiently long delay imbalance

in the signal path guarantees that max min minm m m . Consequently, the positions

of all correlation peaks within measurement range can be scanned with practically

equal increments, which are set to match the resolution step: mZ z for all


maxminm m m . The long fiber delay allows for the rapid, all-electrical scanning of

correlation peak positions, with no moving parts and over long ranges [92].

The proper function of the sensing setup requires that the long fiber delay

remains stable to within 1

2sT throughout the duration of the analysis, which might

require 1-2 hours. Stabilization was achieved using the protocol proposed in this

work. Light from a laser diode source of wavelength 1 of 1530 nm was modulated

by a 3 GHz control sine wave and coupled into the signal path. The control channel

was filtered and detected at the output of the fiber under test, and used to adjust

both 1 and 2 as discussed above. In order to test the stability of the setup, a 2 cm-

wide hot-spot was introduced at an arbitrary location along the fiber under test, and

sT was fine-tuned so that one of the correlation peaks was in overlap with the hot-

spot.

In a first experiment, closed-loop feedback was provided only to the

wavelength of the tunable laser of the control signal, whereas the delay of the B-

OCDA signal wave was free-running. In a second experiment, feedback was provided

to the wavelengths of both tunable lasers. In both experiments, the Brillouin gain

spectrum at the position of the hot spot was acquired every 30 s.

5.2. EXPERIMENTAL RESULTS

Figure 31(top) shows the instantaneous setting of 1 during the first

experiment. Intentional heating of the delay line by a halogen lamp began six

minutes into the measurements. Starting at that stage, the wavelength was adjusted


over a 1.2 nm range (left-hand axis), indicating a thermal drift in path length of over

10 cm (right-hand axis). Figure 31(bottom) shows the Brillouin gain spectra as a

function of time. Ten minutes into the experiment, the measured gain spectrum was

shifted from that of the hot-spot to that of the fiber under test at room temperature.

Thermal delay drift therefore led to an incorrect interpretation of the measurement

data.

In a second experiment, closed-loop feedback was applied to the wavelength

of the sensor laser 2 as well. Figure 32(top) shows the settings of that wavelength

Figure 31: Top – Wavelength of the control channel (left-hand axis), and the implied thermal drift in the path length of the 25 km-long delay line (right-hand

axis), as a function of time. Thermal drift on the order of 10 cm is observed. Bottom – Measured Brillouin gain as a function of frequency offset between pump and signal, and of time. The phase-coded B-OCDA setup was adjusted to monitor

the Brillouin gain spectrum in a 2 cm-wide hot-spot. The delay of the Brillouin signal was free-running, without closed-loop feedback to the wavelength of the sensor laser. The Brillouin gain spectrum changed after 10 minutes, from that of

the hot-spot to that of the fiber under test at room temperature.


as a function of time. Here too, heating of the delay line started after six minutes,

and the wavelength changes indicate a thermal drift of about 9 cm, or several times

larger than both z and the extent to the hot-spot. This time, however, the

measured Brillouin gain spectrum remained that of the hot-spot.

Figure 32: Top – Wavelength of the Brillouin sensor laser source (left-hand axis), and the implied thermal drift in the path length of the 25 km-long delay line (right-hand axis), as a function of time. Thermal drift on the order of 9 cm is observed. Bottom – Measured Brillouin gain as a function of frequency offset between pump and signal, and of time. The phase-coded B-OCDA setup was adjusted to monitor the Brillouin

gain spectrum in a 2 cm-wide hot-spot. Unlike Fig. 31, the delay of the Brillouin signal was stabilized through closed-loop feedback to the wavelength of the sensor

laser. The Brillouin gain spectrum remained that of the hot-spot throughout the measurements.

Lastly, Fig. 33 shows the experimental estimated B at the position of the

hot-spot as a function of time, for both experiments. The instantaneous adjustments

to 2 were accounted for in the data analysis of the second experiment, as B is


inversely proportional to the optical wavelength [76]. The stabilization of the delay

line allowed for correct interrogation of the narrow hot-spot in the presence of

thermal drift. Note that the temperature of the hot-spot was not the same in the

two experiments.

Figure 33: Measured Brillouin frequency shifts as a function of time. The phase-coded B-OCDA setup was adjusted to monitor the location of a 2 cm-wide hot-spot. Thermal drift in a 25 km-long delay line incorporated in the setup leads to incorrect

interrogation after 10 minutes of free-running operation (blue). Delay drift is overcome with closed-loop, stabilized delay (red).

Discussion and Summary 66

6. DISCUSSION and SUMMARY

6.1. SUMMARY OF RESULTS

A technique for the distribution of arbitrary RF waveforms over long optical

fiber with a stabilized delay was proposed and demonstrated in this work. The

method is based on the compensation of thermal drifts in the long fiber-optic delay

line using chromatic dispersion. This basic concept was previously reported and

demonstrated in [72, 75]. However, in this work we extended this principle for the

distribution of broadband input RF waveforms alongside the control RF sine wave,

with no spectral restrictions. To that end, we used two tunable laser sources with

different wavelengths: one was modulated by the control RF sine wave and the

other by the arbitrary input waveform. By monitoring the RF phase of the control

sine wave, reconstructed after two-way propagation, the wavelengths of both

tunable laser sources were adjusted in a closed feedback loop to maintain a stable

delay of both waveforms. Since the signals were separated in the optical domain, no

restrictions were imposed on the RF spectrum of the input waveform.

The proposed method was successfully demonstrated in three main

experiments: First, the stable distribution of the control sine wave in closed-loop

operation was reported, with residual delay variations of less than 10 ps. Second, the

stable distribution of broadband LFM waveforms, which are prevalent in many radar

systems as they provide both high SNR and high resolution, had been achieved. In

this experiment the timing drifts in free-running operation were reduced by an order

of magnitude in closed-loop operation. Last, the stable interrogation of a local hot


spot in high-resolution distributed Brillouin analysis was demonstrated, in the

presence of thermal delay drifts that were several times larger than its extent. This

demonstrated the applicability of the proposed stabilization technique to the

processing of broadband microwave signals as part of more complicated systems.

6.2. LIMITATIONS AND FUTURE WORKS

The results obtained in this work were limited by several factors. Some of the

limitations are inherent to the stabilization principle. The method relies on chromatic

dispersion, hence the fiber-optic delay line must be sufficiently long so that the delay

variations are appreciable. The same dispersion, however, introduces delay

uncertainties. In extreme conditions, propagation over long dispersive fibers might

distort the temporal envelope profile of broadband waveforms.

Other limitations are related to the specific implementation chosen in our

experiments. One such issue is the choice of the radio-frequency of the control sine

wave. The sensitivity of the feedback mechanism to small-scale delay variations

would improve with higher control tone frequencies. A major restriction is imposed

by the low bandwidth of the feedback loop. The update rate of the tunable lasers

wavelengths was limited by the protocols of their computer-controlled interfaces to

only twice per second. The fundamental limitation on the rate of wavelength

adjustments, on the other hand, is set by the two-way delay along the fiber and may

be as high as tens of KHz. A broader feedback loop may also compensate and

stabilize the effects of structural and acoustic vibrations.


Yet another practical restriction has to do with the tuning range of the laser

sources and the bandwidth of optical filters used in the separation between the two

waveforms. As noted earlier, the tuning range of the source restricts the system's

figure of merit. One of the laser sources available to us could only be tuned

continuously over 2-3 nm. Optical bandpass filters impose a similar limitation: when

the laser wavelength is swept beyond their transmission bandwidth, they too must

be adjusted. Our optical filters had a bandwidth of 4 nm, and they could only be

tuned manually. Use of electrically-controlled filters would resolve this problem.

The introduction of the proposed technique into real-world systems can

provide many opportunities. Unlike already existing solutions, the proposed method

provides a stable delay with variations of only few ps, over long fiber-optic delay

many km of fiber and wide range of temperatures. Applications such as beam

forming in phased-array antenna, that rely on the precise phase difference between

neighboring elements, may find the proposed method crucial for maintaining proper

function. The dispersion of the delay fiber can be chosen to emphasize long delay,

high precision or broad temperature range, as expressed in the figure of merit of the

proposed system.

Several topics remain for further academic inquiry. For instance, the

development of a broadband feedback loop that is able to compensate for rapid

changes and sharp gradients in the surrounding temperature, as well as for

structural and acoustic vibrations. Another direction would be the incorporation of

the stabilized delay as a module within a radar test and measurement setup, or

phased array antennas.

References 69

REFERENCES

[1] J. Capmany and D. Novak, "Microwave photonics combines two worlds," Nat. Phot. 1,

319-330 (2007).

[2] P. Candelas, J. M. Fuster, J. Marti and J. C. Roig, "Optically generated electrical-

modulation formats in digital-microwave link applications," Jour. Light. Tech. 21(2),

496-499 (2003).

[3] C. H. Cox and E. I. Ackerman, "Microwave photonics: past, present and future," in

International topical meeting on Microwave photonics, 9-11, Gold Coast, Qld, Sept.

2008.

[4] J. Capmany, B. Ortega and D. Paster, "A tutorial on microwave photonic filters," Jour.

Light. Tech. 24(1), 201-229 (2006).

[5] A. Zadok, D. Grodensky, D. Kravitz, Y. Peled, M. Tur, X. Wu and A. E. Willner, "Ultra-

Wideband Waveform Generation Using Nonlinear Propagation in Optical Fibers," in

Ultra Wideband Communications: Novel Trends - Antennas and Propagation, M. Martin

ed. (InTech 2011).

[6] I. S. Lin, J. D. McKinney and A. M. Weiner, "Photonic synthesis of broadband microwave

arbitrary waveforms applicable to ultra-wideband communication," Micro. Wire. Comp.

Lett. IEEE 15(4), 226-228 (2005).

[7] R. A. Becker, C. E. Woodward, F. J. Leonberger and R. C. Williamson, "Wide-band

electrooptic guided-wave analog-to-digital converters," Proceedings of the IEEE, 72(7),

802-819 (1984).

[8] H. Emami and N. Sarkhosh, "Reconfigurable microwave photonic in-phase and

quadrature detector for frequency agile radar," Jour. Opt. Soci. Am. A 31(6), 1320-1325

(2014).

[9] D. Novak and R. Waterhouse, "Commercial and defense applications of microwave

photonics," Avionics, Fiber-Optics and Photonics Conference, IEEE, San Diego, CA, Oct.

2013.

[10] J. Capmany, G. Li, C. Lim and J. Yao, "Microwave Photonics: Current challenges towards

widespread application," Opt. Exp. 21(19), 22862-22867 (2013).

[11] J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret and S. Sales, "Microwave Photonic

Signal Processing," Jour. Light. Tech. 31(4), 571-586 (2013).

[12] J. Yao, "A Tutorial on Microwave Photonics, Part 1," IEEE Phot. Soc. New. 26, 5-12

References 70

(2012).

[13] L. Li, X. Yi, T. X. H. Huang and R. Minasian, "High-Resolution Single Bandpass Microwave

Photonic Filter With Shape-Invariant Tunability," IEEE Phot. Tech. Lett. 26(1), 82-85

(2014).

[14] C. Wang and J. Yao, "Fiber Bragg gratings for microwave photonics subsystems," Opt.

Exp. 21(19), 22868-22884 (2013).

[15] Y. Stern, K. Zhong, T. Schneider, R. Zhang, Y. Ben-Ezra, M. Tur and A. Zadok, "Tunable

sharp and highly selective microwave-photonic band-pass filters based on stimulated

Brillouin scattering," Phot. Res. 2(4), B18-B25 (2014).

[16] H. Wang, J. Y. Zheng, W. Li, L. X. Wang, M. Li, L. Xie and N. H. Zhu, "Widely tunable

single-bandpass microwave photonic filter based on polarization processing of a

nonsliced broadband optical source," Opt. Lett. 38(22), 4857-4860 (2013).

[17] M. Burla, L. R. Cortés, M. Li, X. Wang, L. Chrostowski and J. Azaña, "Integrated

waveguide Bragg gratings for microwave photonics signal processing," Opt. Exp. 21(12),

25120-25147 (2013).

[18] J. Capmany, S. Sales, I. Gasulla, J. Mora, J. Lloret and J. Sancho, "Innovative concepts in

microwave photonics," Waves 4, 43-58 (2012).

[19] D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales and J. Capmany,

"Integrated microwave photonics," Las. Phot. Rev. 7(4), 506-538 (2013).

[20] P. Munoz, J. Capmany, D. Perez, J. Fandino, J. S. Fandino and J. D. Domenech,

"Integrated microwave photonics: state of the art and future trends," 16th

International Conference on Transparent Optical Networks, IEEE, Graz, Austria, Jul.

2014.

[21] J. Yao, "A Tutorial on Microwave Photonics, Part 2," IEEE Phot. Soc. New. 26, 4-12

(2012).

[22] A. M. J. Koonen and M. G. Larrode, "Perspectives of Radio over Fiber Technologies,"

paper OThP3 in Optical Fiber Communication Conference, (OSA), San Diego, CA, Feb.

2008.

[23] R. Karthikeyan and S. Prakasam, "A Survey on Radio over Fiber (RoF) for Wireless

Broadband Access Technologies," Inter. Jour. Com. Appl. 64(12), 15-19 (2013).

[24] ETSI TS 136 410 V9.1.1, Euro. Telecom. Stan. Inst. May 2011,

http://www.etsi.org/deliver/etsi_ts/136400_136499/136410/09.01.01_60/ts_136410v

090101p.pdf

http://www.etsi.org/deliver/etsi_ts/136400_136499/136410/09.01.01_60/ts_136410v090101p.pdf

http://www.etsi.org/deliver/etsi_ts/136400_136499/136410/09.01.01_60/ts_136410v090101p.pdf

References 71

[25] A. K. Vyas and N. Agrawal, "Radio over Fiber: Future Technology of Communication,"

Inter. Jour. Emer. Tre. Tech. Com. Sci. 1(2), 233-237 (2012).

[26] M. Morant, J. Pérez and R. Llorente, "Polarization Division Multiplexing of OFDM Radio-

over-Fiber Signals in Passive Optical Networks," Adv. Opt. Tech. 2014, 1-9 (2014).

[27] V. A. Thomas, S. Ghafoor, M. El-Hajjar and L. Hanzo, "A Full-Duplex Diversity-Assisted

Hybrid Analogue/Digitized Radio Over Fibre for Optical/Wireless Integration," IEEE

Comm. Lett. 17(2), 409-412 (2013).

[28] J. Beas, G. Castanon, I. Aldaya, A. Aragón-Zavala and G. Campuzano, "Millimeter-Wave

Frequency Radio over Fiber Systams: A Survey," IEEE Comm. Sur. Tut. 15(4), 1593-1619

(2013).

[29] A. Perentos, F. Cuesta-Soto, A. Canciamilla, B. Vidal, L. Pierno, N. S. Losilla, F. Lopez-

Royo, A. Melloni and S. Iezekiel, "Using a Si3N4 Ring Resonator Notch Filter for Optical

Carrier Reduction and Modulation Depth," IEEE Phot. Jour. 5(1), (2013).

[30] Y. Stern, "All-Optical Signal Processing and Analysis of Radio Frequency Waveforms,"

M.S. thesis, Dept. Elect. Eng., Bar-Ilan Univ., Ramat-Gan, Israel.

[31] S. Yegananarayanan and B. Jalali, "Wavelength-selective true time delay for optical

control of phased-array antenna," IEEE Phot. Tech.Lett. 12, 1049-1051 (2000).

[32] W. V. Aulock, "Properties of phase arrays," Proceedings of the IRE 48, 1715-1727

(1960).

[33] B. H. Kolner and D. W. Dolfi, "Intermodulation distortion and compression in an

integrated electrooptic modulator," App. Opt. 26, 3676-3680 (1987).

[34] O. Raz, S. Barzilay, R. Rotman and M. Tur, "Submicrosecond scan-angle switching

photonic beamformer with flat RF response in the C and X bands," Jour. Light. Tech. 26,

2774-2781 (2008).

[35] O. Raz, R. Rotman and M. Tur, "Wavelength-controlled photonic true time delay for

wide-band applications," IEEE Phot. Tech. Lett. 17, 1076-1078 (2005).

[36] R. Rotman, O. Raz and M. Tur, "Analysis of a true time delay photonic beamformer for

transmission of a linear frequency-modulated waveform," Jour. Light. Tech. 23, 4026

(2005).

[37] A. Molony, C. Edge and I. Bennion, "Fibre grating time delay element for phased array

antennas," Elec. Lett. 31, 1485-1486 (1995).

[38] H. Zmuda, R. A. Soref, P. Payson, S. Johns and E. N. Toughlian, "Photonic beamformer

for phased array antennas using a fiber grating prism," IEEE Phot. Tech. Lett. 9, 241-243

References 72

(1997).

[39] J. Cruz, B. Ortega, M. Andres, B. Gimeno, D. Pastor, J. Capmany and L. Dong, "Chirped

fibre Bragg gratings for phased-array antennas," Elec. Lett. 33, 545-546 (1997).

[40] B. Vidal, T. Mengual, C. Ibanez-Lopez and J. Marti, "Optical beamforming network

based on fiber-optical delay lines and spatial light modulators for large antenna arrays,"

IEEE Phot. Tech. Lett. 18, 2590-2592 (2006).

[41] R. D. Esman, M. Frankel, J. Dexter, L. Goldberg, M. Parent, D. Stilwell and D. Cooper,

"Fiber-optic prism true time-delay antenna feed," IEEE Phot. Tech. Lett. 5, 1347-1349

(1993).

[42] A. Zadok, A. Eyal and M. Tur, "Extended delay of broadband signals in stimulated

Brillouin scattering slow light using synthesized pump chirp," Opt. Exp. 14(19), 8498-

8505 (2006).

[43] A. Zadok, A. Eyal and M. Tur, "Stimulated Brillouin scattering slow light in optical

fibers," App. Opt. 50(25), E38-E49 (2011).

[44] K. Y. Song, M. G. Herráez and L. Thévenaz, "Gain-assisted pulse advancement using

single and double Brillouin gain peaks in optical fibers," Opt. Exp. 13, 9758-9765 (2005).

[45] E. Shumakher, N. Orbach, A. Nevet and G. Eisenstein, "On the balance between delay,

bandwidth and signal distortion in slow light systems based on stimulated Brillouin

scattering in optical fibers," Opt. Exp. 14, 5877-5884 (2006).

[46] A. Zadok, S. Chin, L. Thévenaz, E. Zilka, A. Eyal and M. Tur, "Polarization-induced

distortion in stimulated Brillouin scattering slow-light systems," Opt. Lett. 34, 2530-

2532 (2009).

[47] M. D. Stenner, M. A. Neifeld, Z. Zhu, A. Dawes and D. J. Gauthier, "Distortion

management in slow-light pulse delay," Opt. Exp. 13(25), 9995-10002 (2005).

[48] S. Sales, W. Xue, J. Mork and I. Gasulla, "Slow and fast light effects and their

applications to microwave photonics using semiconductor optical amplifiers," IEEE

Tran. Micro. Theo. Tech. 58, 3022-3038 (2010).

[49] F. Öhman, K. Yvind and J. Mørk, "Slow light in a semiconductor waveguide for true-time

delay applications in microwave photonics," IEEE Phot. Tech. Lett. 19, 1145-1147

(2007).

[50] N. A. Riza, M. A. Arain and S. A. Khan, "Hybrid analog-digital variable fiber-optic delay

line," Jour. Light. Tech. 22, 619 (2004).

[51] O. Klinger, "Long Microwave-Photonic Variable Delay of Chirped Waveforms,"

References 73

M.S.thesis, Dept. Elect. Eng., Bar-Ilan Univ., Ramat-Gan, Israe, (2012).

[52] O. Klinger, Y. Stern, F. Pederiva, K. Jamshidi, T. Schneider and A. Zadok, "Continuously

variable long microwave-photonic delay of arbitrary frequency-chirped signals," Opt.

Exp. 37, 3939-3941 (2012).

[53] O. Klinger, Y. Stern, T. Schneider, K. Jamshidi and A. Zadok, "Long Microwave-Photonic

Variable Delay of Linear Frequency Modulated Waveforms," IEEE Phot. Tech. Lett.

24(3), 200-202 (2012).

[54] Y. Stern, O. Klinger, T. Schneider, K. Jamshidi, A. Peer and A. Zadok, "Low-distortion

long variable delay of linear frequency modulated waveforms," IEEE Phot. 4(2), 499-503

(2012).

[55] J. Xie, L. Zhou, Z. Li, J. Wang and J. Chen, "Seven-bit reconfigurable optical true time

delay line based on silicon integration," Opt. Exp. 22(19), 22707-22715 (2014).

[56] J. Xie, L. Zhou, Z. Zou, J. Wang, X. Li and J. Chen, "Continuously tunable reflective-type

optical delay lines using microring resonators," Opt. Exp. 22(1), 817-823 (2014).

[57] Y. Antman, L. Yaron, T. Langer, M. Tur, N. Levanon and A. Zadok, "Experimental

demonstration of localized Brillouin gratings with low off-peak reflectivity established

by perfect Golomb codes," Opt. Exp. 38(22), 4701-4704 (2013).

[58] Y. Wang, C. Yu, L. Yan, A. E. Willner, R. Roussev, C. Langrock, M. M. Fejer, J. E. Sharping

and A. L. Gaeta, "44-ns Continuously Tunable Dispersionless Optical Delay Element

Using a PPLN Waveguide With Two-Pump Configuration, DCF, and a Dispersion

Compensator," IEEE Phot. Tech. Lett. 19(11), 861-863 (2007).

[59] M. Johnson, Optical fibers, cables and systems, 2009.

[60] J. M. Payne and W. P. Shillue, "Photonic techniques for local oscillator generation and

distribution in millimeter-wave radio astronomy," Int.Topical Meeting on Microwave

Photonics (IEEE MWP) 1, 9-12, Nov. 2002.

[61] K. Volyanskiy, J. Cussey, H. Tavernier, P. Salzenstein, G. Sauvage, L. Larger and E.

Rubiola, "Applications of the optical fiber to the generation and to the measurement of

low-phase-noise microwave signals," Jour. Opt. Soci. Am. B 25(12), 2140-2150 (2008).

[62] E. Choi, J. Na, S. Y. Ryu, G. Mudhana and B. H. Lee, "All-fiber variable optical delay line

for applications in optical coherence tomography: feasibility study for a novel delay

line," Opt. Exp. 13(4), 1334-1345 (2005).

[63] G. N. Pearson, K. D. Ridley and D. V. Willetts, "Chirp-pulse-compression three-

dimensional lidar imager with fiber optics," App. Opt. 44(2), 257-265 (2005).

References 74

[64] Y. London, Y. Antman, R. Cohen, N. Kimelfeld, N. Levanon and A. Zadok, "High-

resolution long-range distributed Brillouin analysis using dual-layer phase and

amplitude coding," Opt. Exp. 22, 27144-27158 (2014).

[65] D. Elooz, Y. Antman, N. Levanon and A. Zadok, "High-resolution long-reach distributed

Brillouin sensing based on combined time-domain and correlation-domain analysis,"

Opt. Exp. 22, 6453-6463 (2014).

[66] I. L. Newberg, C. M. Gee, G. D. Thurmond and H. W. Yen, "Long microwave delay fiber

optic link for radar testing," Micro. Sym. Dig., IEEE MTT-S International, 2, 693-696

(1989).

[67] J. M. Byrd, L. Doolittle, A. Ratti, J. W. Staples and R. Wilcox, "Timing distribution in

accelerators via stabilized optical fiber links," Proceeding of LINAC, Knoxville, 577-579

(2006).

[68] V. A. Lebedev, J. Musson and M. G. Tiefenback, "High-precision beam-based rf phase

stabilization at jefferson lab," Proceedings of the 1999 Particle Accelerator Conference,

IEEE, 2, 1183-1185 (1999).

[69] L. Zhang, L. Chang, Y. Dong, W. Xie, H. He and W. Hu, "Phase drift cancellation of

remote radio frequency transfer using an optoelectronic delay-locked loop," Opt. Lett.

36(6), 873-875 (2011).

[70] B. Ning, P. Du, D. Hou and J. Zhao, "Phase fluctuation compensation for long-term

transfer of stable radio frequency over fiber link," Opt. Exp. 20(27), 28447-28454

(2012).

[71] D. Hou, P. Li, C. Liu, J. Zhao and Z. Zhang, "Long-term stable frequency transfer over an

urban fiber link using microwave phase stabilization," Opt. Exp. 19(2), 506-511 (2011).

[72] A. Zhang, Y. Dai, F. Yin, T. Ren, K. Xu, J. Li, Y. Ji, J. Lin and G. Tang, "Stable radio-

frequency delivery by λ dispersion-induced optical tunable delay," Opt. Lett. 38, 2419-

2421 (2013).

[73] Z. Wu, Y. Dai, A. Zhang, K. Xu, J. Li and J. Lin, "Broadband downlink stable radio

frequency phase delivery exploiting fiber chromatic dispersion," Opt. Comm. Net.

(ICOCN), 12th International Conference, IEEE, (2013).

[74] Z. Wu, Y. Dai, F. Yin, K. Xu, J. Li and J. Lin, "Stable radio frequency phase delivery by

rapid and endless post error cancellation," Opt. Lett. 38(7), 1098-1100 (2013).

[75] A. Zhang, Y. Dai, F. Yin, J. Li, T. Ren, K. Xu and G. Tang, "Phase stabilized downlink

transmission for wideband radio frequency signal via optical fiber link," Opt. Exp. 22,

21560-21566 (2014).

References 75

[76] R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

[77] G. P. Agrawal, Nonlinear fiber optics, Springer, 2000.

[78] D. Cotter, "Observation of stimulated Brillouin scattering in low-loss silica fibre at 1.3

μm," Elec. Lett. 18, 495-496 (1982).

[79] A. Zadok, A. Eyal and M. Tur, "Gigahertz-wide optically reconfigurable filters using

stimulated Brillouin scattering," Jour. Light. Tech. 25, 2168-2174 (2007).

[80] Y. Antman, N. Primerov, J. Sancho, L. Thévenaz and A. Zadok, "Localized and stationary

dynamic gratings via stimulated Brillouin scattering with phase modulated pumps,"

Opt. Exp. 20, 7807-7821 (2012).

[81] K. Hotate and M. Tanaka, "Distributed fiber Brillouin strain sensing with 1-cm spatial

resolution by correlation-based continuous-wave technique," IEEE Phot. Tech. Lett. 14,

179-181 (2002).

[82] T. Sperber, A. Eyal, M. Tur and L. Thévenaz, "High spatial resolution distributed sensing

in optical fibers by Brillouin gain-profile tracing," Opt. Exp. 18(8), 8671-8679 (2010).

[83] J. C. Beugnot, M. Tur, S. F. Mafang and L. Thévenaz, "Distributed Brillouin sensing with

sub-meter spatial resolution: modeling and processing," Opt. Exp. 19, 7381-7397

(2011).

[84] Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd and A. E.

Willner, "Numerical study of all-optical slow-light delays via stimulated Brillouin

scattering in an optical fiber," Jour. Opti. Soc. Am. B 22, 2378-2384 (2005).

[85] T. Horiguchi, T. Kurashima and M. Tateda, "A technique to measure distributed strain in

optical fibers," IEEE Phot. Tech. Lett. 2, 352-354 (1990).

[86] M. Niklès, L. Thévenaz and P. A. Robert, "Simple distributed fiber sensor based on

Brillouin gain spectrum analysis," Opt. Lett. 21, 758-760 (1996).

[87] X. Bao and L. A. Chen, "Recent progress in Brillouin scattering based fiber sensors,"

Sensors-Basel, 11, 4152-4187 (2011).

[88] Y. Dong, H. Zhang, L. Chen and X. Bao, "2 cm spatial-resolution and 2 km range Brillouin

optical fiber sensor using a transient differential pulse pair," App. Opt. 51(9), 1229-1235

(2012).

[89] S. M. Foaleng, M. Tur, J. C. Beugnot and L. Thévenaz, "High spatial and spectral

resolution long-range sensing using Brillouin echoes," Jour. Light. Tech. 28(20), 2993-

3003 (2010).

References 76

[90] K. Hotate and T. Hasegawa, "Measurement of Brillouin gain spectrum distribution along

an optical fiber using a correlation-based technique -proposal, experiment and

simulation," IEICE T. Electorn, E83-C(3), 405-412 (2000).

[91] K. Song, Z. He and K. Hotate, "Distributed strain measurement with millimeter-order

spatial resolution based on Brillouin optical correlation domain analysis," Opt. Lett.

31(17), 2526-2528 (2006).

[92] A. Zadok, Y. Antman, N. Primov, A. Denisov, J. Sancho and L. Thévenaz, "Random-access

distributed fiber sensing," Laser Phot. Rev. 6(5), L1-L5 (2012).

[93] J. Wang and K. Petermann, "Small signal analysis for dispersive optical fiber

communication systems," Jour. Light. Tech. 10(1), 96-100 (1992).

[94] R. Rotman, O. Raz and M. Tur, "Small signal analysis for analogue optical links with

arbitrary optical transfer function," Elec. Lett. 40(8), 504-505 (2004).

[95] Y. London, Y. Antman, N. Levanon and A. Zadok, "Brillouin analysis with 8.8 km range

and 2 cm resolution," accepted for presentation in Optical Fiber Sensors (OFS-24)

Conference, Curitiba, Brazil, 2015.

[96] Y. Antman, N. Levanon and A. Zadok, "Low-noise delays from dynamic Brillouin gratings

based on perfect Golomb coding of pump waves," Opt. Lett. 37(24), 5259-5261 (2012).

‏א‏

תקציר

-רדיומגנטיים בתדרי רדיו ומיקרוגל באמצעים פוטוניים, או -עיבוד אותות אלקטרו

. לעיבוד אותות באמצעות תחומי ההתמחותשני מחקר המשלב בין תחום הינו ,פוטוניקה

פוטוניקה יתרונות רבים על פני עיבוד ישיר בטכניקות מבוססות התקנים -טכניקות רדיו

וחסינות בפני ,סרט גדול, טווח שידור ארוך עם ניחות נמוך-אלקטרוניים, כדוגמת רוחב

מוש של קווי מיפוטוניקה הוא -ת רדיוומגנטיות. יישום חשוב של טכניק-הפרעות אלקטרו

לאותות בתדרי רדיו באמצעים אופטיים, המשמשים כרכיבים הכרחיים מתכוונניםהשהייה

יישום משמעותי נוסף הוא שידור אותות להטיית אלומות במערכות מכ"ם מרובות אלמנטים.

: ארוכים על גבי סיבים אופטיים טווחיםת לסלולרית ואלחוטי שמקורם בתקשורתבתדרי רדיו

אלו. רשתות תקשורתשל והכיסוי סיב. יישום זה מרחיב את טווחי השידור-יגב-על-רדיו

ההשהייה הזמנית של אותות מחייבים כי יישומים שונים בעלי רמת דיוק גבוהה

מתנדים מקומיים הפצה של דוגמאות לכך כוללות רדיו על גבי סיבים ארוכים תשאר יציבה. ה

רכות מכ"ם. לאחרונה הופיע יישום נוסף וכיול של מע ותבמערכי אנטנה גדולים, ובדיק

וטמפרטורה מחיישנים מפולגים למעוות מכאני , כחלק ארוכה ומיוצבתהשהייה המחייב

של סנטימטרים נדרשים לאבחנה מרחביתאלו חיישנים המבוססים על פיזור ברילואן.

הסיבים הדרך האופטית לאורךהמזל, לרועשל סיב. רבים לאורך קילומטרים בודדים,

לשינויים בטמפרטורת הסביבה. שינוי של מעלת פיםחשו אשר הםהאופטיים משתנה כ

סיב אופטי סטנדרטי. של ס"מ לכל קילומטר 0.75משנה את הדרך האופטית ב אחת צלסיוס

כים מדי למימוש עבור רוב מערכות מסוב לתיקון סחיפות כגון אלו יום כהפיתרונות הקיימים

מספקים את היציבות הנידרשת. או שאינם חיישנים,מכ"ם והה

תיזמון הנובעים השינויי קיזוזזו מבוססת על עבודהב תיהשיטה שבה השתמש

בסיסי הוצע במקביל העיקרון ה. דיספרסיה כרומטיתבסיס על משינויי טמפרטורת הסביבה

קבוצה מאוניברסיטת בייג'ינג ידי-עלבאופן בלתי תלוי על ידי קבוצת המחקר שלנו וו

על ידי כןו ,מקור לייזר מתכוונן בודד מאופנן על ידי אות בקרה סינוסי בתדר רדיו .תלתקשור

‏ב‏

התפשטות לאורך הסיב, פאזתבתדר רדיו אותו נרצה להשהות. לאחר שרירותי אות כניסה

תזמון. אורך הגל של ב סחיפות יהוילז ומשמשת המוצא בתדר הרדיו של אות הבקרה נמדדת

תיזמון כתוצאה מדיספרסיה הכך ששינויי ,מעודכן בחוג משוב סגורמקור הלייזר המתכוונן

אולם, התיכנון הראשוני התבסס על ומשינויים סביבתיים. אלו הנובעיםאת יםקזזמכרומטית

הפרדה בין אות הכניסה לאות הבקרה במרחב תדרי הרדיו, ולכן נדרש שלא תהיה חפיפה

סרט, כגון -רכות התומכות באותות רחביבמעלהשגה קשה הזאילוץ . במרחב התדר ביניהם

סיב ברזולוציה גבוהה. נישיחי

בעבודה זו אני מציג ומדגים שיפור משמעותי של טכניקת הייצוב. במקום להשתמש

במקור לייזר בודד, אות הבקרה בתדר הרדיו ואות הכניסה מאפננים שני מקורות לייזר

קבל על ידי פאזת המוצא של אות מתכוונים נפרדים בעלי אורכי גל שונים. המשוב המת

את אורכי הגל של שני מקורות הלייזר, בכדי לקבל ייצוב השהייה משנההבקרה המושהה

התדרים האופטי, ולכן אין מגבלה על האותות מופרדים במרחבעבור שני האותות. שני

ניתוח מפורט של ביצועי השיטה המוצעת מצביע על החפיפה שלהם במרחב תדרי הרדיו.

שרות המתחייבות בין טווח שינויי הטמפרטורה הניתן לקיזוז לבין שגיאות תזמון שיוריות. הפ

הניתוח מוביל להצעת מדד איכות עבור ביצועי מערכת ההשהייה המיוצבת, ולחסם עליון על

הביצועים הניתנים להשגה במגבלות המפרט הטכני של מקורות הלייזר.

מודגמת השהיה ניסוי הראשוןבמעבודה זו. שלושה ניסויים עיקריים מודגמים כחלק

של חוסר יציבות שיוריעם ,ק"מ 9של אות הבקרה הסינוסי על גבי סיב באורך של מיוצבת

סרט, -של אותות בעלי אפנון תדר לינארי רחבפצה מודגמת הניסוי השני בשניות. -פיקו 4±

הסחיפה השיורית אורך. אותו העל גבי סיב בהמצויים בשימוש נרחב במערכות מכ"ם,

שניות בחוג סגור, -פיקו 20 -שניות בחוג פתוח ל-פיקו 200 -מ ההופחת ת האותהשהייב

מדידה הושגה . לבסוף, מכשור המעבדהשגיאת התיזמון של ידי -מוגבלת עלכאשר המדידה

ברילואן מפולג ברזולוציה חיישן ס"מ על ידי 2יציבה של נקודה חמה מקומית ברוחב של

. כדי סחיפה סביבתית מכוונת של השהיה לאורך הסיב בשיעור גבוה פי כמהתוך גבוהה,

.שכזו אינה ניתנת להשגה בחוג פתוחמדידה

‏ג‏

עבור עיבוד הייצוב המוצעתשיטת את הישימות של מדגימות תוצאות הניסויים

ניתנת להרחבה עבור שיטה הסרט כחלק ממערכות מורכבות יותר. -אותות מיקרוגל רחבי

סרט גדול -חוג משוב בעל רוחב באמצעות, ות לאורך הסיבואקוסטי ותמכאני ותהפרע ביטול

יותר.

עבודה זו נעשתה בהדרכתו של פרופ' אבי צדוק,

אילן-מהפקולטה להנדסה של אוניברסיטת בר

אותות כלליים יה מיוצבת של יהשה

בתדרי רדיו על גבי סיבים אופטיים

עמרם-בן אסף מוסמך תואר קבלת לשם מהדרישות כחלק מוגשת זו עבודה

אילן-בר אוניברסיטת של להנדסה בפקולטה

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