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University of Colorado, Boulder CU Scholar Undergraduate Honors eses Honors Program Spring 2017 A Pound-Drever-Hall based Repetition Rate Stabilization Technique for Mode-locked Lasers Liangyu Chen [email protected] Follow this and additional works at: hps://scholar.colorado.edu/honr_theses Part of the Atomic, Molecular and Optical Physics Commons , and the Optics Commons is esis is brought to you for free and open access by Honors Program at CU Scholar. It has been accepted for inclusion in Undergraduate Honors eses by an authorized administrator of CU Scholar. For more information, please contact [email protected]. Recommended Citation Chen, Liangyu, "A Pound-Drever-Hall based Repetition Rate Stabilization Technique for Mode-locked Lasers" (2017). Undergraduate Honors eses. 1310. hps://scholar.colorado.edu/honr_theses/1310
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  • University of Colorado, BoulderCU Scholar

    Undergraduate Honors Theses Honors Program

    Spring 2017

    A Pound-Drever-Hall based Repetition RateStabilization Technique for Mode-locked LasersLiangyu [email protected]

    Follow this and additional works at: https://scholar.colorado.edu/honr_theses

    Part of the Atomic, Molecular and Optical Physics Commons, and the Optics Commons

    This Thesis is brought to you for free and open access by Honors Program at CU Scholar. It has been accepted for inclusion in Undergraduate HonorsTheses by an authorized administrator of CU Scholar. For more information, please contact [email protected]

    Recommended CitationChen, Liangyu, "A Pound-Drever-Hall based Repetition Rate Stabilization Technique for Mode-locked Lasers" (2017). UndergraduateHonors Theses. 1310.https://scholar.colorado.edu/honr_theses/1310

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  • A Pound-Drever-Hall based Repetition Rate Stabilization

    Technique for Mode-locked Lasers

    by

    Liangyu Chen

    University of Colorado, 2017

    A thesis submitted to the

    Faculty of the University of Colorado in partial fulfilment

    of the requirements for the degree of

    Bachelor of Arts in Physics

    Department of Physics

    2017

  • This thesis entitled:A Pound-Drever-Hall based Repetition Rate Stabilization Technique for Mode-locked Lasers

    written by Liangyu Chenhas been approved for the Department of Physics

    Prof. Thomas Schibli

    Prof. Tobin Munsat

    Prof. Juliet Gopinath

    Date

    The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above

    mentioned discipline.

  • iii

    Chen, Liangyu (B.A., Physics)

    A Pound-Drever-Hall based Repetition Rate Stabilization Technique for Mode-locked Lasers

    Thesis directed by Prof. Thomas Schibli

    Femtosecond mode-locked lasers have become one of the indispensable tools for spec-

    troscopy and microwave generation for its capability to generate high-quality ultra-short

    pulses, and thus optical frequency combs with a wide spectrum. Particularly, many applica-

    tions of the optical frequency combs require the high spectral purity and long-term stability

    of the combs, which are limited by the phase noise, or timing jitter, of the mode-locked lasers.

    Therefore, several techniques have been successfully developed recently for stabilizing the

    mode-locked lasers, although they suffer many drawbacks in the sense of applicability because

    of their relative complex designs. In this thesis, an attempt to apply the Pound-Drever-Hall

    technique, a powerful frequency stabilization technique for continuous wave laser, to the

    mode-locked lasers is demonstrated with success. Our results indicate that with this simplified

    and robust system, the repetition frequency of the mode-locked laser has been stabilized

    and there is significant phase noise suppression at low frequency, and it is capable to reduce

    the integrated timing jitter of the mode-locked laser by nearly four times without the need

    of previously stabilized transfer lasers or RF signals as references. The success from this

    stabilization system reveals the great potential of utilizing Pound-Derver-Hall technique

    for future inspirations of a cost-effective and field-deployable system that could provide

    state-of-the-art stabilization for the mode-locked lasers.

  • Dedication

    To my dear friends and family, for their incredible support in the endeavour.

  • v

    Acknowledgements

    I would like to express my sincere gratitude to all the members of the Schibli’s Lab.

    First and foremost, to Professor Thomas Schibli for his incredible mentoring that have guided

    me to become a better researcher. I would like to thank Luke Charbonneau, a dear friend

    and former graduate student from the lab, without whom this project wouldn not have been

    possible. I would also like to thank Dr. Mamoru Endo for his incredible help during the final

    stages of the project, and Tyko Shoji for his constant support as friend and colleague since

    my first day working in the lab. Alas, there are many other who helped and supported me

    along the path, I would like to thank everyone who has made this experience a wonderful

    journey.

  • Contents

    Chapter

    1 Introduction 1

    1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1.1 Microwave Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.1.2 Long Distance Laser Sensing . . . . . . . . . . . . . . . . . . . . . . . 3

    1.1.3 Dual Comb Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.2 The Basics of Mode-locked Lasers . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.3 Mode-locking Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.3.1 Active Mode-locking . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.3.2 Passive Mode-locking . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    2 Noise of the Mode-locked Laser 10

    2.1 Intensity Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    2.2 Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    2.3 Timing Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    2.4 Timing Jitter Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2.4.1 Comparing to Microwave Reference . . . . . . . . . . . . . . . . . . . 15

    2.4.2 Using Intensity Electro-optical Modulation . . . . . . . . . . . . . . . 16

    2.4.3 Optical Heterodyne Cross-correlation . . . . . . . . . . . . . . . . . . 19

  • vii

    3 Laser Stabilization Technique 24

    3.1 Passive Reference Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    3.2 The Pound-Drever-Hall Stabilization Technique . . . . . . . . . . . . . . . . 26

    3.2.1 Conceptual Model of Pound-Drever-Hall . . . . . . . . . . . . . . . . 26

    3.2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    3.2.3 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    3.2.4 Noise Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    3.3 Mode-locked Laser Stabilization Technique . . . . . . . . . . . . . . . . . . . 37

    4 Experimental setup and results 41

    4.1 500 MHz Modelocked Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

    4.2 Passive Reference Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

    4.3 Electro-optic Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    4.4 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    4.5 Environmental Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    4.6 Lock Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

    4.6.1 Noise Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    4.7 Phase Noise Measurement and Results . . . . . . . . . . . . . . . . . . . . . 54

    5 Conclusion 59

    5.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    References 61

    Appendix

    A Kigre Inc. Erbium-doped Ytterbium Glass, QX/Er Datasheet 65

  • viii

    B Fiber-Based Lithium Niobate EOM-PM Datasheet 66

    C Layertech Mirror Dielectric Coating Datasheet 67

  • Figures

    Figure

    1.1 Dual-comb spectrometer diagram . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.2 A schematic setup of an actively mode-locked laser . . . . . . . . . . . . . . 6

    1.3 Progression of optical power and losses in an actively mode-locked laser . . . 7

    1.4 A schematic setup of a passive mode-locked laser . . . . . . . . . . . . . . . 8

    1.5 Progression of optical power and losses in a passively mode-locked laser . . . 9

    2.1 Simulated intensity noise spectrum . . . . . . . . . . . . . . . . . . . . . . . 12

    2.2 A visual representation of timing jitter . . . . . . . . . . . . . . . . . . . . . 15

    2.3 A block diagram for the intensity electro-optical modulation measurement

    scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    2.4 Optical heterodyne experimental setup . . . . . . . . . . . . . . . . . . . . . 20

    2.5 Optical heterodyne cross-correlation experimental setup . . . . . . . . . . . . 21

    2.6 Optical heterodyne cross-correlation discriminator signal . . . . . . . . . . . 22

    3.1 The transmitted signal from a Fabry-Perot cavity . . . . . . . . . . . . . . . 27

    3.2 The reflected signal from a Fabry-Perot cavity . . . . . . . . . . . . . . . . . 28

    3.3 experimental schematics of a Pound-Drever-Hall stabilization technique . . . 29

    3.4 The amplitude and phase of the reflection coefficient of the caivty . . . . . . 30

    3.5 Reflection coefficient in the complex plane . . . . . . . . . . . . . . . . . . . 32

    3.6 The PDH error signal with F = 447 cavity . . . . . . . . . . . . . . . . . . . 35

  • x

    3.7 The PDH error signal with F = 3140 cavity . . . . . . . . . . . . . . . . . . 36

    3.8 Concept for full frequency comb stabilization with two CW laser . . . . . . . 38

    3.9 Experimental setup for full frequency comb stabilization . . . . . . . . . . . 39

    4.1 A schematic of the experimental setup . . . . . . . . . . . . . . . . . . . . . 42

    4.2 The circuit diagram for the PI servo loop filter . . . . . . . . . . . . . . . . . 46

    4.3 The experimental environmental isolation setup . . . . . . . . . . . . . . . . 47

    4.4 Transmitted signal without phase modulation . . . . . . . . . . . . . . . . . 48

    4.5 Reflected signal without phase modulation . . . . . . . . . . . . . . . . . . . 49

    4.6 Transmitted signal with phase modulation . . . . . . . . . . . . . . . . . . . 50

    4.7 Reflected signal with phase modulation . . . . . . . . . . . . . . . . . . . . . 51

    4.8 The experimental PDH error signal . . . . . . . . . . . . . . . . . . . . . . . 52

    4.9 The transmitted signal through the reference cavity with lock engaged . . . . 53

    4.10 The phase noise measurement results using the intensity electro-optical modu-

    lation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    4.11 The phase noise measurement results using the optical heterodyne cross-

    correlation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

  • Chapter 1

    Introduction

    The goal of this project is to suppress phase noise and timing jitter of a mode-locked

    laser with a PDH based setup. The frequency lock should be sensitive, robust and have long

    term stability, and the setup does not require building other lasers or mode-locked lasers as

    reference. The thesis will outline the motivation and the background of this project, including

    a introduction to the properties of mode-locked lasers. In chapter 2, the different sources of

    noise will be introduced and analysed, including a focused discussion on the timing jitter. In

    chapter 3, the Pound-Drever-Hall stabilization technique will be introduced and explained

    with conceptual and quantitative models. In chapter 4, the experimental setup of this project

    will be detailed as well as an analysis of the result. In chapter 5, an outlook will be provided

    for possible improvements.

    1.1 Motivation

    Lasers have been essential tools in science and technology since the successful demon-

    stration in 1960. Many exciting developments followed shortly after as researchers poured in

    with numerous ideas on its improvements and possible applications. The mode-locked laser,

    or frequency comb, was one such amazing product developed by Hargrove in 1964 [1]. It

    is capable to produce pulse trains consisted with ultra-short pulses with incredible spatial

    coherence. The magnificent invention enabled novel investigations in the field of spectroscopy,

    metrology and microwave generation that requires a very quiet and stable source that normal

  • 2

    lasers simply could not provide. Therefore, it is crucial to continuously refine and improve

    upon the outstanding capability of the mode-locked laser. In practice, the effort is focused

    on stabilization and noise suppression of the mode-locked laser, especially focusing on the

    timing jitter that could disrupt a perfectly periodic pulse train that many applications depend

    greatly on. Currently, there are many successful stabilization techniques developed for the

    mode-locked laser involving full frequency comb lock. Although these stabilization techniques

    can achieve significant noise suppression, they are usually very complex and difficult to

    construct, therefore limiting the options for possible applications. Examples of effort to

    reduce the complexity and improve the applicability of the system will be presented later, as

    a recurring theme in either continuous wave laser or mode-locked laser stabilization technique.

    As a commonality, they usually required previously frequency stabilized continuous wave

    lasers, using a powerful technique called the Pound-Drever-Hall technique, as references

    for the frequency comb. Therefore, this project will attempt a different path from these

    well-established methods and employ the Pound-Drever-Hall technique to the mode-locked

    laser without any intermediate steps, stabilizing its repetition frequency directly. This method

    would significantly reduce the complexity, and in turn, enable more on-field applications for

    mode-locked laser with appreciable timing jitter and phase noise suppression.

    1.1.1 Microwave Generation

    Low noise microwave signals are highly desired to improve metrology, radar and

    telecommunication. Traditional microwave generation method such as cryogenic sapphire

    oscillator that requires a vacuum chamber and cooling to a few kelvin is very costly to

    construct and maintain [2]. As a better alternative, mode-locked lasers are used in microwave

    generation by converting their repetition rate to a RF microwave signal via a photodiode.

    Through this way, essentially the relatively high stability of the mode-locked lasers are

    transferred down to microwave without the space and cost to maintain a vacuum chamber.

    Therefore, as the stability of the pulse train directly determines the quality of the microwave

  • 3

    signal, it is essential to suppress the timing jitter and stabilize the repetition rate of the

    frequency comb as good as possible and ensure long term stability. Also, as the applications

    for microwave generation are plenty, a portable and efficient system is mostly desired, further

    motivate the attempt of this project.

    1.1.2 Long Distance Laser Sensing

    Because of its long coherence length, mode-locked lasers have been used in ultra long

    distance metrology. For example, intra-satellite ranging, which are of great relevance for

    space missions [3]. It have a very specific requirements and constrains on accuracy, sampling

    rate, flexibility, complexity and reliability, which greatly limit the options of designs and level

    of stabilization and mode-locked laser is among the best candidates for its superior spatial

    coherence from ultra pulses. This serves as another neat example for the need of a robust

    and reliable stabilization system that is capable of significant phase noise suppression.

    1.1.3 Dual Comb Spectroscopy

    Spectroscopy is a powerful technique in understanding the mechanics of elements and

    chemicals. In particular, their vibrational, rotational and low frequency modes. It is also one

    of the most important applications of femtosecond frequency comb. Traditional spectroscopy

    method requires mechanical components, such as a scanning mirror mounted on a movable

    stage, which could be a limiting source of noise. The latest the method in spectroscopy

    utilizes the frequency combs of two mode-locked laser, eliminating any mechanical part in the

    system [4]. The two frequency combs have a slight mismatched repetition rate depending on

    the sample. The setup essentially is an interferometer of these two lasers, and the combined

    light is sent through the sample. Any absorption from the sample will reflect in heterodyne

    spectrum. This method provide much faster sampling rate, higher resolution and less noise

    than any other mechanical method, and relying on the best stabilization technique that could

    suppressing the timing jitter in the repetition rate to function at best performance. However,

  • 4

    employ a state-of-the-art stabilization technique on two sets of the mode-locked laser would

    be an enormous effort, therefore, a simplified and effective noise suppression system will be

    very appreciable.

    Figure 1.1: A dual-comb spectrometer uses another comb rather than scanning mirrors as

    the reference arm for the first comb. The scan rate and resolution is much higher than other

    method in spectroscopy.

    1.2 The Basics of Mode-locked Lasers

    Mode-locked lasers are capable of producing trains of ultra-short pulses on the order

    of picoseconds or femtoseconds in time domain [5]. This is achieved by maintaining a fixed

    phase relationship between the longitudinal modes of the cavity. In contrast, a multi-mode

    continuous wave laser would have a random phase relationship between the modes, therefore

    the output would be a random interference pattern between the modes repeated every round

    trip time in the cavity. In a mode-locked laser, ideally all the different frequencies oscillating

    in the cavity will constructively interfere at one point and destructively interfere at everywhere

    else. This will create a pulse circulating around the cavity rather than a continuous wave.

    Thus repetition rate of this pulse train is essentially the free spectral range (FSR) of the laser

    cavity, is simply found with:

    frep =1

    τ=

    c

    2L(1.1)

  • 5

    where τ is the cavity round trip time, c is the speed of light in the cavity medium and

    L is the geometric length of the cavity.

    Similar to the result of a Fourier transformation from time domain to frequency domain,

    each pulse will contain a wide range of frequencies that are allowed by the cavity modes,

    essentially creating a frequency comb consisting of the resonant frequencies of the cavity.

    Thus each frequency component, or ”comb tooth”, in the frequency comb can be found with:

    f(n) = f0 + nfrep (1.2)

    where f0 is the offset frequency of the carrier envelope, and n is an integer that represents

    a specific ”comb tooth”.

    f0 of the pulse describes the phase difference between the carrier wave and the envelope.

    There is a certain change in the envelope phase each round trip in the cavity and thus can be

    related to the repetition frequency as:

    f0 =∆φCEO mod 2π

    2πfr (1.3)

    where the mod 2π term means the fact that only the modulus 2π phase shift between

    the carrier envelope and carrier phase, per round-trip, is relevant. The f0 stabilization is

    important in full frequency comb lock, and will be explained in chapter 3, section 3.3.

    Ideally, the repetition rate will remain constant during laser operation. However,

    in practise, there are many possible sources of noise that could disrupt the perfect pulse

    production. The focus of this project is to reduce the the timing jitter of the mode-locked

    laser, which, simply put, describes the deviations of the temporal pulse position from those

    in a perfectly periodic pulse train [6]. This thesis will continue to examine this type of noise

    and the implement a stabilization technique based on the Pound-Drever-Hall approach.

  • 6

    1.3 Mode-locking Techniques

    There are two major categories of mode-locking techniques: active and passive mode-

    locking. At the start they are both in normal laser operation, producing a continuous wave

    laser with random phase between different cavity mode. These techniques create a locked

    phase relation from there through different means.

    1.3.1 Active Mode-locking

    In active mode-locking, an amplitude modulator is utilized, which amplified the pulse at

    frequency at the free spectral range of the cavity all the cavity modes are linked to the phase

    of this modulation, after a few thousands operations, pulses from the interference between

    cavity modes are created inside the cavity. Then the selection process begins: the pulse pass

    through at the correct times would experience minimum losses, and vice versa for the other

    undesired pulses, creating a steady pulse oscillating around the cavity with the phase set by

    the initial modulation signal.

    Figure 1.2: A schematic setup of an actively mode-locked laser [7].

    This method provides pulse shortening only to a limited extent, as there is a little

    attenuation to the tails, but would be eventually offset by other effects such as gain narrowing

    that could broaden the pulse [8]. Additionally, this pulse shortening effect is even less

    effective for shorter pulse durations, thus essentially limits the pulse duration on the order of

    picoseconds[8].

    When compared with the later passive mode-locking in the field of ultra-fast optics,

  • 7

    active mode locking is at a disadvantage for the requirement for an optical modulator and

    only capable to generate longer pulses. Therefore, it is mostly used when there is a need to

    synchronize the pulse with a electronic signal or with other lasers, such as the cases in optical

    fibre communication or laser array sensors.

    Figure 1.3: Progression of optical power and losses in an actively mode-locked laser. The

    modulator causes some losses for the pulse tails, effectively shortening the pulses [7].

    1.3.2 Passive Mode-locking

    As an improvement upon the active mode-locking, passive mode-locking incorporates

    a saturable absorber into the laser resonator. The saturable absorber has a certain optical

    loss at low optical intensities, which is then reduced at high intensities[9], essentially being

    ”saturated” with intense light.

  • 8

    Figure 1.4: A schematic setup of a passive mode-locked laser using a saturable absorber

    mirror [7].

    This saturation effect can occur in semiconductors, which will reduce the absorption for

    photons above the bandgap energy when most of its electrons are excited into the conduction

    band [9]. In passive mode-locking, a semiconductor saturable absorber mirror, or SESAM, is

    commonly used for this purpose. The SESAM will attenuate low intensity light, creating a net

    losses for the continuous wave light, and when a high intensity pulse is formed, it will saturate

    the SESAM and experience a low loss. Thus, a short window of positve net-gain is formed

    [5]. Meanwhile, as the tails of the pulse will have lower intensity, it has a pulse-shortening

    effect as long as it can relax to the ground state quickly enough so that it could absorb

    again before the arrival of the next pulse. Comparing to active mode-locking, utilizing the

    saturable absorber allows the generation of much shorter pulses, on the order of femtoseconds,

    as the saturable absorber can provide a much faster loss modulation than any electronic

    modulator could achieve [5]. However, if the saturable absorber does not have the appropriate

    properties, there can be many possible issues relating to passive Q-switching, which can incur

    massive energy fluctuations in the laser cavity or other unstable mode of operation. Careful

    consideration must be made in the construction of a passive mode-locked laser [5].

  • 9

    Figure 1.5: Progression of optical power and losses in a passively mode-locked laser with a

    saturable absorber. The shorter the pulse, the faster the loss modulation will become. The

    gain stays approximately constant, as the saturation of the gain medium is less noticable [7].

    Passive mode-locked laser has been at the frontier of ultra-fast optics ever since its

    invention. It is another key ”enabling technology” as scientists and engineers have given to

    the invention of laser, providing us a new powerful tool to the understanding of our universe.

  • Chapter 2

    Noise of the Mode-locked Laser

    In this chapter, the types and sources of noise in a mode-locked laser system will be

    explored and analyzed in order to understand what issues a stabilizing system needs to

    address. Noises can generally be categorized into two major categories: quantum noise, which

    is closely related to the spontaneous emission of photons in the gain medium, and technical

    noise, which originates from the components and laser system construction, including the

    noise from the pump source, from mechanical vibrations of the mirrors or from thermal

    fluctuation [10]. At the end, three measuring methods for the timing jitter in mode-locked

    laser are introduced with emphasis on the optical heterodyne cross-correlation technique,

    which is the most sensitive of the three.

    2.1 Intensity Noise

    Intensity noise describes fluctuations in the laser output power of the laser. To measure

    intensity noise, one would simply monitor the fluctuation of the output power over a desired

    time interval using a photodiode. In experiments, the intensity noise is usually quantified by

    the root-mean-squared value of the output power fluctuations [11], which is calculated with:

    δPrms =√〈(P (t)− Pavg)2〉 (2.1)

    where Pavg is the average power[11].

  • 11

    As a laser can exhibit different rate of power fluctuations, the bandwidth of the

    measurement devices could underestimate or distinct the noise at different frequencies.

    Therefore, a power spectral density (PSD) is most useful in this case by measuring noise at

    different frequency intervals in order to display the spectral distribution of the total noise

    power.

    From the power spectral density measurement, one can calculate the r.m.s. relative

    intensity noise with:

    dP

    Pavg

    ∣∣∣∣rms

    =

    √∫f2f1SI(f)dt (2.2)

    where f1 and f2 are the lower and upper noise frequency bonds for the interval, respectively,

    and SI(f) is the power spectral density over that frequency interval [11].

    In solid-state and in most diode lasers, there is a characteristic peak in the relative

    intensity noise called the relaxation oscillations, which result from the dynamic interaction

    between the energy in the light-field an the stored energy in the gain, if the excited-state

    lifetime is much longer than the cavity damping time [12]. These under-damped oscillations

    are a fine example of quantum intensity noise. In the frequency domain around fro, the

    laser is highly sensitive to external perturbation. At frequencies well above fro, external

    perturbations are strongly suppressed and follow a second-order low-pass-filter characteristic.

    The impact of this quantum noise depends on many parameters. It can be minimized with

    high intractivity powers, low resonator losses and a long round-trip time inside the cavity, all

    of which had to do with the damping of the relaxation oscillations [11].

    Another common noise source is the mode beating in the resonator when the laser is in

    multi-mode operation. The frequency of the beat is strongly dependent on the laser type,

    ranging from several Ghz for diode lasers to a few kHz for long fiber lasers [10]. However,

    this does not play a role in mode-locked lasers.

    At last, to minimize the intensity noise of the laser, there are many possible methods.

    The first approach is to minimize any external influences on noise, such as using a low

  • 12

    Figure 2.1: Simulated intensity noise spectrum of a diode-pumped solid-state laser. Graycurve is plotted with a quantum-limited pump source. Red curve has a 30 dB excess noisefrom the pump source. Figure used with permission from Dr. Rdiger Paschotta [11].

    noise pump source and constructing a mechanically stable laser system[10]. Secondly, one

    can optimize the laser parameters to mitigate the effect of quantum noise by adjusting

    the relaxation oscillation frequency into a less noisy domain or using a long low-loss laser

    cavity[10]. Finally, the noise can be actively countered with a feedback system stabilizing the

    output power. The potential of these methods greatly depends on the circumstances.

    In the end, the ultimate limit of the intensity noise is the quantum shot noise, which is

    basically the random occurrence of photons as discrete packets of energy[10]. In common

    scenarios, the measured noise floor will be well above the quantum shot noise. However,

    in ultra-fast optics experiments, it is quite often that we are trying to measure the noise

    much below the quantum noise floor. or at least hope to achieve. As quantum shot

    noise is proportional to the square-root of the average output power that falls onto the

    photodector, essentially it is the signal-to-noise ratio of a Possion distribution, SNR =√N .

  • 13

    The photodetector often sets a piratical upper limit for the output power one can detect, so

    that shot noise would often show up as the measurement noise floor.

    Therefore, it is prominent at low optical intensity, and thus must be taken into consid-

    eration when taking measurements.

    2.2 Phase Noise

    Phase noise describes the phase fluctuations of the electromagnetic waves of the laser

    output. The present of the phase noise results in a finite linewidth in the frequency domain

    for each frequency component in the laser output, and leads to a finite temporal coherence of

    the laser. To have a numerical perspective, the coherence length could be merely centimetres

    for a humble laser pointer, but could be of the order of millions of kilometers for single-mode

    operation continuous wave lasers or mode-locked lasers depending on specific designs [13].

    Fundamentally, the origin of the phase noise is again quantum noise that arises from the

    spontaneous emission of the gain medium. The spontaneous emission essentially generates

    photons with random phase and polarization, unlike the desired stimulated emission which

    generates photons with the same polarization as the previous ones that are already oscillating

    in the laser cavity. Additionally, it can be affected by other technical noises like the intensity

    noise, which may lead to the coupling of those two noise [6].

    In practice, phase noise is generally measured and quantified using a phase noise power

    spectral density (PSD), Sφ(f). It is also common to use a frequency PSD to better understand

    the phase noise. These two are directly related by:

    Sν(f) = f2Sφ(f) (2.3)

    where Sν(f) is the frequency PSD and f is the respective offset frequency from the

    carrier frequency [13]. It is derived from the fact that phase noise and frequency noise are

    really the creation of one source, just a matter of perspective:

  • 14

    φ(t) =

    ∫f(t)dt (2.4)

    f(t) =dφ(t)

    dt(2.5)

    In order to acquire data on phase noise one usually has to compare the desired source

    with a known oscillator, preferably one with less noise than the source, and measure the beat

    signal between them. This reference oscillator could also be another laser with the same

    parameters or part of the same laser output with a known delay [6].

    2.3 Timing Jitter

    Timing jitter is another classification of noise in mode-locked lasers, which is commonly

    discussed in the context of ultra-fast optics or microwave oscillators. As the output of a

    mode-locked laser is a pulse train with repetition frequency, frep, the timing jitter describes

    the deviations of the pulse position in time domain from a perfectly periodic pulse train,

    as shown in figure 2.2[14]. It is mostly related to the phase noise corresponding to each

    frequency component of the pulse train with a factor of the quantity square of the oscillator

    frequency, f 20 . To be more precise, the timing jitter does not depend on f20 , but the phase

    noise does. Great effort has been made by Haus et al. in 1991 to tackle the timming jitter

    with mathematical model in order to understand this problem more thoroughly [15].

    Many other effects and noises can contribute to timing jitter in a mode-locked laser. It

    can be affected by center frequency shifts as the same in a continuous wave laser, due to the

    spontaneous emission form the gain medium. This will lead to a corresponding random shift

    in the pulse position and changes in group velocity [15]. Timing jitter can also couple to the

    intensity noise of the mode-locked lasers through the saturable absorber depending on its

    recovery rate [15].

  • 15

    Figure 2.2: A visual representation of timing jitter. Case A shows a perfectly periodic pulsetrain. Case B shows a pulse train with timing jitter. Small pulse to pulse variations can resultin a large timing drifts on the pulse train. This is the characteristic of a random walk[14].

    2.4 Timing Jitter Measurements

    As a focus of this project, it is essential to have suitable methods to measure the timing

    jitter of a laser. In this section three measuring methods will be examined. The later two

    will be utilized in this project for the final measurement.

    2.4.1 Comparing to Microwave Reference

    This is the traditional method of measuring timming jitter developed in the early

    days of active mode-locking, and the resolution is quite limiting. The pulse train from the

    mode-locked laser is converted to RF signal at the frequency of the repetition rate of the laser

    via a photodector. This signal is mixed with an electronically generated microwave signal

    ideally at exactly the same frequency as the repetition frequency of the mode-locked laser [14].

  • 16

    The mixer essentially calculates the product between those two signals and the output will

    be measured using a signal analyzer to get the power spectral density of the phase noise of

    the device under tested. The sensitivity of this right straightforward method mostly depends

    on the quality of the microwave reference. The sensitivity and resolution also depends on

    the measurement bandwidth of the specific methods, but the carrier frequency is the most

    important and limiting factor. Therefore, the resolution of the microwave reference method

    with carriers on the order of Mhz, is pale in comparison to the optical methods comparing to

    other lasers, utilizing the optical frequency as the carrier which is on the order of hundreds

    of Thz.

    2.4.2 Using Intensity Electro-optical Modulation

    This method was developed in Schibli’s Lab, utilizing an ultra-stable low-noise 1 GHz

    monolithic mode-locked laser, which is the only one of its kind in the world, also developed

    in Schibli’s Lab. It is a rigid, one-piece, mode-locked laser with no mechanical parts. The

    cavity is made out of the calcium fluoride CaF2 crystal which has low thermal expansion and

    ultra high transparency, which length can only be controlled via thermal expansion. Thus,

    the monolithic laser has ultra-low noise and ultra stable.

    As a specific example with schematic shown in figure 2.3, our stabilized 500 MHz

    mode-locked laser will first need to double its repetition frequency to have a main carrier

    of1 GHz by using an optical interleaver, essentially overlapping one half of the pulse train

    onto the other one delayed by a half of the round trip time in the laser cavity, which is 2 ns

    for the 500 MHz laser.

    Then, the light from the 500 MHz laser, now doubled to 1 GHz , is converted into a

    RF signal via a photodiode. At the other end, the light from the 1 GHz monolithic laser is

    sent into an intensity electro-optical modulator (EOM) with dual output. At the modulator,

    due to its operation method, there is an output only if when a light pulse from the 1 GHz

    monolithic laser is present, providing a regular sampling rate of the RF signal. After the

  • 17

    modulator, the two beams are focused onto a pair of balanced photodetectors. Ideally, if the

    two photodiodes are well balanced and there are no RF signals applied to the modulator,

    the signal from the balanced detectors should be exactly zero. Then, the RF signal from

    the 500 MHzlaser is applied onto one arm of the electro-optical modulator. There will be

    a sinusoidal discrimination signal from the balanced detectors, and the two lasers will be

    loosely locked so that a 90 degrees relative phase difference between the two is maintained.

  • 18

    Interleaved 500 MHz MLL(1 GHz carrier output)

    Ultra-stable Monolithic MLL(1 GHz carrier output)

    (Fiber Couple) (Fiber Couple)

    Dual-outputIntensity

    EOM

    (RF Input)

    (Slow PZT)

    Spectrum Analyzer

    PI Servo to lock carrier

    RF Amp

    Figure 2.3: block diagram for the intensity electro-optical modulation measurement scheme.

    The red solid lines represent optical beam (in fiber or wave guide) paths and the dashed black

    lines represent electrical connections. [16]

  • 19

    For the noise measurement, at the output of the balanced detectors, if the two lasers

    are perfectly quiet, the output will remain exactly zero. Therefore, any phase noise in the

    RF signal will create a relative phase shift between the carrier frequencies of the two lasers

    causing they to overlap, thus this frequency noise will be converted to intensity modulation

    which would offset the balance. This offset then will be measured and recorded by the RF

    signal analyzer. As EOM the essentially mix the signal down to DC with no carrier frequency

    presented, it requires less dynamic range on the signal analyzer, allowing it to make more

    accurate measurements comparing to measuring the noise at the 1 GHz carrier frequency.

    Improvements and rigorous tests based on this method is still being conducted in

    Schibli’s lab. Currently, the effort on suppressing the quantum shot noise and other technical

    noise like the thermal noise has been successfully employed via creating multiple channels

    of uncorrelated light paths and using two pairs of balanced detectors rather than a single

    pair. After these improvements, the current limiting factor on noise measurement is the 1/f

    flicking noise in the electro-optical modulator, which could only be improved by designing

    better modulators or by using the cross-correlation techniques which will be introduced next.

    2.4.3 Optical Heterodyne Cross-correlation

    This is another method developed in Schibli’s lab, improved upon the optical heterodyne

    technique. The optical heterodyne technique essentially compares the frequency components

    from two uncorrelated mode-locked lasers [17]. This technique is conceptually simple and

    extremely sensitive, capable of achieving less than 1 zs/√Hz resolution [17].

  • 20

    Figure 2.4: Experimental setup for the optical heterodyne technique [17]. The two lasers are

    two 500 MHz mode-locked lasers with the same parameters. HWP, half-wave plate; QWP,

    quarter-wave plate; BS, beamsptlitter; PBS, polarizing beamsplitter; LFP, low-pass filter;

    LN-AMP, low-noise amplifier.

    As stated, the current phase noise measurement setup used in this project is a improved

    version of the one shown above, displayed in the schematic 2.5, introducing two more uncor-

    related light path and have cross correlation between two pairs of balanced photodetectors

    instead of one pair.

  • 21

    1GHz Monolithic Mode-Locked Laser

    500MHz Mode-Locked Laser

    ∆2

    QWPHWP

    LPF

    LPF

    Mixer

    PBS Lens

    Photodiode

    Grating

    BoardbandMirror

    LN-AMP

    Collimator

    LPF

    Piezo Driver

    Loop Filter

    RF Signal Analyzer

    Polarization Maintaining Fiber

    PBS

    PBS

    Grating

    Collimator

    Figure 2.5: The curent experiemental setup for the optical heterodyne technique. HWP,

    half-wave plate; QWP, quarter-wave plate; PBS, polarizing beamsplitter; LFP, low-pass filter;

    LN-AMP, low-noise amplifier.

    In this setup, the 1 GHz monolithic mode-locked laser in our lab is used as a reference.

    The beams from the two lasers are first combined via a polarizing beam splitter with one

    beam horizontally polarized while the other one is vertically polarized. The combined beam

    is shone onto a diffraction grating to acquire the board spectrum. Another grating is used

    here to recollimate the spectrum to parallel beam in order to avoid astigmatism at the cost of

    around 43% of the total optical power. Then, the light passes through a half wave plate with

    optical axes at 22.5 degrees with respect to the horizontal, essentially totalling polarization

    by 45 degree. Afterwards, the spectrum is cut in halves using a high-reflectivity boardband

  • 22

    gold mirror. Each half of the spectrum is separated in two arms using another polarizing

    beam splitter; one of the halves have an extra quarter wave plate in place to ensure there

    is a 90 degree phase difference between the two arm so that the final discrimination signal

    is sinusoidal and only have one zero-crossing which can be used as a locking point for the

    feedback circuit. Then, the two beams from each half after the polarizing beam splitter are

    focused onto a pair of balanced photodetectors respectively.

    -0.10

    -0.05

    0.00

    0.05

    0.10

    Vol

    tag

    e (V

    )

    400x10-62000-200

    Time (S)

    Figure 2.6: A example of the measured discriminator signal that is used as feedback to theloop filter.

    When power on each photodiode is balanced so that the the DC output of each pair

    of photodiode is zero, the signal is without signal from the carrier frequency and are left

    with phase noise and offset frequency from each laser. A 120 MHz low-pass-filter is put

  • 23

    place between each pair of balanced detector and the mixer to block the beats between the

    offset frequency which is above the Nyquist frequency. When the repetition frequencies of

    these two laser are really close, the 500 MHz mode-locked laser is locked to the zero-crossing

    in discriminator signal from the mixer, as shown in figure 2.6. The in-loop residual noise,

    which is essentially the timing jitter between the two lasers, will be measured by a RF signal

    analyzer. Comparing with the previous electro-optical modulation method, this method

    directly compares the lasers at optical frequencies instead of converting to a RF carrier

    frequency, thus have a greatly improved sensitivity. The benefit of having two pairs of

    balanced detector is that: first, through optimization of optical compoenents, more light

    can be collected in this method than in the original optical heterodyne method, increasing

    signal to noise contrast; secondly, as there are now four light paths instead of two paths, the

    polarizing beam splitter uncorrelated the quantum noise between each path, so the average of

    the four can reduce the quantum noise; thirdly, similar to the last point, the thermal noise is

    also uncorrelated between them, so the thermal noise will also decrease. Therefore, by using

    crossing correlation between two pair of balanced detectors, much of the the technical noise

    and quantum noise from the measuring devices. There are also many optimization along the

    light path so more percentage of the light can be measure, further increase the signal to noise

    ratio. Therefore, this method is capable of higher sensitivity and less systematic noise than

    the other methods presented here.

  • Chapter 3

    Laser Stabilization Technique

    In this chapter, the idea of using an Fabry-Perot cavity as a passive reference is

    introduced and explored. More will be dedicated on the Pound-Drever-Hall technique to

    stabilize continuous wave lasers, including the conceptual understanding, experimental setup

    and mathematical explanation. At the end, other current mode-locked laser stabilization

    technique will be examined and compared.

    3.1 Passive Reference Cavity

    The fundamental idea behind most laser stabilization techniques is to have a much

    quieter reference, so that when comparing to the reference, the measured noise is ideally

    equal to the absolute noise from the laser. Therefore, the idea of using a passive reference

    cavity as reference becomes fairly straightforward, which is essentially a much more stable

    copy of the laser resonator, with no amplified spontaneous emission, no heat induced and

    very little optical loss. This type of optical cavity made of two parallel mirrors is also called

    Fabry-Perot cavity for its first appearance in the field of optics as Fabry-Perot interferometer

    in 1899.

    Today a Fabry-Perot optical cavity is usually made of two parallel planoconcave high-

    reflectivity mirrors, in order to eliminate reflection loss and degenerate modes. Only when

    the laser frequency is the same as or integer multiple of the resonance of the cavity, the light

    could be coupled into the cavity and form a standing wave. Light that is not on resonant

  • 25

    would simply be reflected by the first mirror before it even enters in the cavity. The cavity

    spaces needs to have low thermal expansion to prevent long term dirft. Mathematically, one

    can find the change in length due to thermal expansion of the material using:

    ∆L ≈ αL∆TL (3.1)

    where αL is the linear thermal expansion coefficient of the specific material, ∆T is the

    temperature difference in kelvin. L is the distance between the mirrors [18].

    This effect would become noticeable if the cavity is significantly shorter or having a very

    high finesse. In these cases, the thermal expansion and the Brownian motion of the mirrors

    becomes the limiting noise floor of the cavity. For most cases, the mechanical vibration of

    the mirrors and the thermal fluctuation of the medium in the cavity contributes the most to

    its noise, and this greatly depends on the quality of environment isolation of the cavity.

    Another important property of the cavity is the Q-factor, which is commonly used

    to describe the ”sharpness” of the resonance of a resonator. The higher the Q-factor, the

    narrower the linewidth around the resonant frequency, and the higher the first order resonance

    will be in amplitude [19]. In the case of an optical cavity, it is related to the finesse of the

    cavity, which describes the optical losses for the light oscillating within the cavity, and found

    by using:

    F =c

    2nL

    ∆ν 12

    =π(r1r2)

    14

    1− (r1r2)12

    (3.2)

    where F is the cavity finesse, c2nL

    is the free spectral range, ∆ν 12

    are the full spectral width at

    half-maximum, n is the index of refraction of the medium in the cavity, L is the length of the

    cavity and r1 and r2 are the power reflectivity of the two mirrors in the cavity, respectively

    [19].

  • 26

    And then the Q-factor can be found with:

    Q = ν0Trt2π

    l=

    ν0FSR

    F (3.3)

    where ν0 is the optical resonance frequency and Trt is the round trip time, l is the fractional

    optical power loss per round trip [19].

    A high finesse cavity in laser stabilization is always appreciable in Pound-Drever-Hall

    technique that will be discussed later, as this will increase the sensitivity to frequency shift

    in the discrimination signal and thus can provide a better feedback.

    3.2 The Pound-Drever-Hall Stabilization Technique

    In this section, the Pound-Drever-Hall technique to stabilize continuous wave lasers

    will be explained in details, including the conceptual understanding, experimental setup and

    mathematical explanation.

    3.2.1 Conceptual Model of Pound-Drever-Hall

    Since the early days of laser technology, there have been many attempts to stabilize

    lasers with a passive reference cavity. These early techniques mostly relied on the transmitted

    signal from the cavity to provide the feedback to the laser [20]. The basic idea was that the

    transmitted signal would be maximized if the laser frequency is on resonance. The major

    problem is the coupling of intensity noise and frequency noise, as the deviation from the

    maximum power could be caused by either the fluctuation in intensity or a drift in center

    frequency. Therefore, such an error signal would not be robust. One approach to this problem

    is to stabilize the laser intensity separately, which has been done with some success in the

    seventies. As shown in figure 3.1, this signal is symmetrical around the resonant frequencies

    that are integer multiple of the free spectral range. Thus, one can also employ the side-fringe

    locking technique, which use the slope on either side of the transmission peak, where changing

  • 27

    frequency has a much larger effect on power transmission than intensity fluctuation [20].

    However, besides the fact that this error signal still couples to the intensity noise, it also

    suffers from the narrow locking range. It will easily break if the frequency drifts from one

    side of the center frequency to the other side [20].

    Figure 3.1: The transmitted signal from a Fabry-Perot cavity as a function of frequency inthe unit of free spectral range. The cavity has a very low finesse to display the features moreeasily [21].

    A improved stabilization technique was demonstrated by R. Pound, R. Drever, and

    John L. Hall in 1983, utilizing the reflected signal from the cavity instead of the transmitted

    signal, and it is able to decouple the intensity and phase noise of the laser [22]. As shown in

    figure 3.2, the reflected signal is zero when the laser is on resonance, regardless of the laser

    intensity. Although the signal is still symmetrical around the resonance, thus, one can not

    simply tell which side of resonance by measuring the amplitude [21]. However, the derivative

    of this signal is asymmetrical, being positive above resonance and negative just below the

    resonance, thus it can be utilized as an error signal used to lock the frequency at resonance.

    Also, it is can be realized in practice as well: one can just dither the laser frequency with a

  • 28

    small variation and monitor the amplitude response, essentially measuring the slope with a

    small df [21]. This conceptual idea only really works when the frequency detuning is slow

    enough for the cavity to completely respond [21]. Otherwise, the output will not follow the

    curve shown in figure 3.2 [21]. Although, the technique will still work at higher modulation

    frequencies, and both the noise suppression and loop bandwidth will be improved [21].

    Figure 3.2: The reflected signal from a Fabry-Perot cavity as a function frequency. If thefrequency is modulated with a small shift, one can tell from the slope that which side ofresonance the frequency is on [21].

    3.2.2 Experimental Setup

    The Pound-Drever-Hall technique aims to achieve this purpose experimentally. As

    shown in figure 3.3, the light from the stabilization target laser is modulated with a small

    frequency variation via an electro-optical modulator driven by a local oscillator [22]. The

    modulation frequency needs to be well outside the linewidth of the cavity, which is in term

    determined by its finesse, so it would completed reflected from the cavity. Then, the beam

    enters the cavity: the frequency component that is on resonance will form a standing wave in

    the cavity, transmitting a leakage signal out of the cavity. The transmitted light on the far

  • 29

    side will be monitored, and the one on the entry side will destructively interfere with the

    light reflected from the first mirror of the cavity, whose amplitude depends on the frequency

    detuning of the laser [22]. The optical isolator, usually a polarizing beamsplitter and a quarter

    wave plate, picks up the reflected beam and sends it to be collected by a photodetector. The

    output from the photodetector is compared with the local oscillator signal via a mixer [22].

    The output of the mixer is essentially the product of the reflected signal and the modulation

    signal, so the output contains a low frequency signal, which has the information about the

    derivative of the intensity, and twice the modulation frequency. Then, there is a low-pass

    filter to isolate the low frequency signal, and send it to the feedback loop to tune the laser,

    locking it to the reference cavity [22].

    The optical isolator, together with a Faraday isolator at the input, can prevent any

    reflected beam to re-enter the laser cavity, which could destabilize the laser. The phase shifter

    between the local oscillate and mixer provides a 90 degrees phase difference between signal

    sent to the modulator and the mixer.

    Figure 3.3: A basic outline of the Pound, Drever, Hall (PDH) stabilization setup. Solid linesrepresent the beam’s path, whereas dotted lines represent electronic connections [22].

    The reflected beam, as briefly mention above, is actually a result of interference between

    two different beams: one is the beam reflected from the first cavity mirror, which never

  • 30

    entered the cavity. The other is the leakage light from the standing wave oscillation in the

    cavity. For a lossless symmetric cavity, they will have the same frequency and their intensities

    are almost the same as well, depending on the choice of cavity mirror [21]. However, their

    relative phase depends greatly on the the frequency of the laser with respect to the resonance

    [21] as shown in figure 3.4. In the case when the laser frequency is perfectly the same as the

    cavity resonance, the beams have the same amplitude and exactly 180 degrees out of phase,

    and thus they completely destructively interfere and the reflected signal will be exactly zero

    [22]. If the laser frequency is off resonance or integer multiple of the resonance, but near

    enough than some light could oscillating in the cavity, the beams will not have exact 180

    degree phase difference, so they could not cancel each other out. The phase of this resulted

    reflect beam will indicate which side of resonance the laser frequency is on, and thus will be

    explored mathematically in the next section for better understanding[22].

    Figure 3.4: The amplitude and phase of the reflective coefficient of the caivty F (ω). Thediscontinuity in phase is caused by the reflected sign vannishing at resonance [21].

    3.2.3 Mathematical Model

    With the conceptual model established, a quantitative model is necessary to further

    understand the technique. To describe the incident and reflected beam at one point outside

    the cavity, their equations can be written as following, respectively:

  • 31

    Einc = E0eiωt (3.4)

    Eref = E1eiωt (3.5)

    where E0 and E1 is complex to account for the relative phase difference between them,

    ω is the angular frequency of the laser [21].

    Then, the amplitude reflection coefficient, F (ω), can be found for a lossless symmetric

    cavity:

    F (ω) =ErefEinc

    =r(exp(i ω

    ∆νFSR)− 1)

    1− r2exp(i ω∆νFSR

    )(3.6)

    where ∆νFSR =C2L

    is the free spectral range 1.1, and r is the reflectivity of each cavity

    mirror [22].

    To better understand this complex equation, it is useful to plot it out and look at its

    evolution. As shown in figure 3.5, F (ω) will always fall on the edge of the dash-lined circle

    center along the real axis. The laser frequency ω will determine where exactly on the circle

    the F (ω) lies. The right edge of the circle where it crosses the real axis is the resonance. It

    crosses the origin because of the lossless symmetric cavity we assumed. As laser frequency ω

    increases it will trace out this circle counter-clockwise [21]. As one can see, the intensity of

    the reflected beam, |F (ω)|2, is symmetrical around the resonance, however, its phase, the

    imaginary part, is different. Very near the resonance, F (ω) is almost only on imaginary axis,

    being in the lower half when below resonance and upper half when above resonance.

    This motivate the use of a phase modulator is necessary in the Pound-Drever-Hall

    technique. The electro-optical phase modulator will create two sidebands with definite

    phase relationship with the incident and reflected beam without entering the cavity, as their

    modulation frequency is chosen to guarantee their reflection at the first mirror of the cavity.

    As there is no way to measure the optical phase directly, these sidebands essentially serve as

  • 32

    Figure 3.5: A plot of the reflection coefficient, F (ω), in the complex plane. As the laserfrequency increases, the imaginary part will trace out a counter-clockwise circle. The rightedge of the circle where F (ω) cross the real axis is when the laser frequency is on resonance[21].

    the phase reference to enable the phase measurement of the reflected verses the leaked beam

    [21].

    Phase modulation will have the similar effect as frequency modulation, so there will

    be two sidebands around each frequency component with addition and subtraction of the

    modulation frequency, respectively. To it show mathematically, the E field of the incident

    light after phase modulation is:

    Einc = E0ei(ωt+β sin(Ωt)) (3.7)

    Then, it can be expanded using Bessel functions [22]:

  • 33

    Einc ≈ [J0(β) + 2iJ1(β) sin(Ωt)]eiωt (3.8)

    = E0[J0(β)eiωt + J1(β)e

    i(ω+Ω)t − J1(β)ei(ω−Ω)t] (3.9)

    where Ω is the phase modulation frequency, β is a constant known as the modulation

    depth, and J0 and J1 are Bessel functions of the first kind [22]. This equation essentially

    means there are three components in the incident beam: a main carrier with frequency ω

    and two sidebands with frequency ω ± Ω.

    The power in the sidebands are small if the modulation depth is small (β < 1) . Then,

    we can treat these components as individual beams and apply the reflection coefficient

    calculated before to each part in order to find the total electric field of the reflected beam

    [22]:

    Eref = E0[F (ω)J0(β)eiωt + F (ω + Ω)J1(β)e

    i(ω+Ω)t − F (ω − Ω)J1(β)ei(ω−Ω)t] (3.10)

    As photodetectors can only measure optical power, we need to find Pref = |Eref |2 [22] :

    Pref = |Eref |2 =Pc|F (ω)|2 + Ps[|F (ω + Ω)|2 + |F (ω − Ω)|2

    ]+ 2√PcPs

    [Re[F (ω)F ∗(ω + Ω)− F ∗(ω)F (ω − Ω)] cos(Ωt)

    + Im[F (ω)F ∗(ω + Ω)− F ∗(ω)F (ω − Ω)] sin(Ωt)]

    + (2Ω-terms)

    (3.11)

    where the main carrier power, Pc = J20P0 , and the sideband power, Ps = J

    21P0, with P0 being

    the total reflected power.

    The resulting power is a wave with an envelope showing the beat pattern between the

    two frequencies [21]. The Ω term originates from the interference between the main carrier

    and the sidebands, and the 2Ω term is the cross term of the two sidebands interfering with

    each other.

    As it is better to ensure the sidebands are completely reflected of the first cavity mirror

    in order to avoid transferring cavity power onto the sidebans, usually a fast modulation

  • 34

    is applied with Ω >> ∆ν, the linewidth of the cavity, and therefore it is safe to assume

    F (ω±Ω) ≈ −1 [21]. With that assumption, the expression F (ω)F ∗(ω+ Ω)−F ∗(ω)F (ω−Ω)

    is purely imaginary, and the cosine term in equation 3.11 become negligible [21]. Then,

    the reflected signal is mixed with the modulation signal. To ensure the DC signal is zero

    after the mixer, a 90 degree phase shift is introduced by the phase shifter mentioned in the

    experimental setup.

    Thus, the final error signal becomes [22]:

    � = −2√PcPsIm[F (ω)F

    ∗(ω + Ω)− F ∗(ω)F (ω − Ω)] (3.12)

    Near resonance, the error signal is nearly linear and asymmetric, and can be approxi-

    mated with [22]:

    � ≈ −8√PcPs

    ∆νδf = Dδf (3.13)

    where ∆ν ≡ ∆νFSRF is the linewidth of the cavity and D ≡ −8√PcPs∆ν

    is called the frequency

    discriminant.

    This linear relation is very useful as feedback via a servo loop, usually a PI or PID

    controller, to the laser cavity or current control, locking the laser frequency to the resonance

    of the reference cavity.

    Furthermore, two graphs of the error signal from equation 3.12 are plotted to show the

    effect of cavity finesse on the error signal with the same modulation frequency of 15 MHz

    and free-spectral range of 500 MHz :

    The most important features in these graphs are the slopes of their center fringe, or

    frequency discriminant D in equation 3.13. As the feedback servo will lock to the zero crossing

    of that center fringe, which is where the resonance is at, a small change in frequency will

    change the voltage by Dδf amount. Essentially, it means, the frequency discriminant D,

    determines the sensitivity of the servo lock. Comparing to earlier method, the fluctuation

  • 35

    Figure 3.6: A plot of the normalized PDH error signal as a function of frequency in the unitsof modulation frequency from resonance. The finesse 447.23 is calculated from having themirror reflectivity of 99.3%. The two intermediate curves are from the interaction of the tailsof the main carrier with the sidebands.

    in intensity will not affect the location of the zero-crossing as the mixer is not sensitive to

    amplitude modulation, although since the slope is depended on the carrier power, it does

    have some effect on the sensitivity of the lock.

    3.2.4 Noise Limitations

    As any noise in the error signal itself is indistinguishable from the frequency noise in

    the laser, it is important to understand what factors contribute to the noise in the error

    signal. From the error signal equation 3.12, it can be derived that the deviation of the cavity

    length from the resonance has the same order of impact as frequency detuning near resonance

    [21]. In fact, among many possible parameters, including the intensity noise of laser, the

    modulation frequency Ω, the modulation depth β, sensitivity of the photodiode, and the

  • 36

    Figure 3.7: A plot of the normalized PDH error signal as a function of frequency in the unitsof modulation frequency from resonance. The finesse 3140.02 is calculated from having themirror reflectivity of 99.9%. Note that its center slope is considerably sharper than that inthe previous graph.

    relative phase difference between the signals sent to the mixer; none of these contributes to

    the error signal to the first order [21]. The error signal is only first-order sensitive to the

    sideband power fluctuations, which can be reduced by using a higher modulation frequency

    Ω as most noises fall off at higher frequencies [21].

    However, the quantum shot noise is the fundamental limit of how quiet the error signal

    can be[21]. The quantum shot noise arose from the random occurrence of photon absorption

    event in a photodector, and related to the quantum discreteness of photons. It has a flat

    spectrum in PSD, constant along all frequencies, therefore it becomes prominent at very high

    frequencies. Mathematically it is proportional to the square-root of the average intensity

    the falls onto the photodector, essentially the signal-to-noise ratio of a Poisson distribution,

  • 37

    SNR =√N as mentioned in section 2.1. In this case, the reflected power falling on the

    photodetector when it is on resonance is the average power in the two sidebands. Therefore,

    mathematically, the quantum shot noise can be calculated as [21]:

    Se =

    √2hc

    λ(2Ps) (3.14)

    Dividing this equation by the frequency discriminant D will give the apparent frequency

    noise [21]:

    Sf =

    √hc3

    8

    1

    FL√λPc

    (3.15)

    where F is the finesse of the cavity, L is the length of the cavity, λ is the wavelength of the

    laser and Pc is the carrier power.

    This quantum shot noise will provide ultimate limitation on how quiet the error signal

    can be.

    3.3 Mode-locked Laser Stabilization Technique

    There are many successful attempt to stabilize the mode-locked laser to suit the purpose

    of their applications. As briefly mentioned in the motivation section 1.1, these techniques are

    capable of full frequency comb lock, stabilizing both fr and f0. As a result the experimental

    setup is usually fairly complex, thus motivating a constant desire for a simplified technique

    with comparable noise suppression level.

    In this section an example of this effort is detailed. It is presented by Swann et al.

    [23], which seeks to reduce the complexity of the previously full frequency comb stabilization

    method in the application of a portable microwave generation system with fiber based

    mode-locked lasers [24].

  • 38

    Figure 3.8: The concept for microwave generation directly from a fiber mode-locked laser

    [25]. In the frequency domain, the repetition frequency is stabilized via stabilization of the

    3.74 THz wide comb across two individual frequency components.

    The basic idea is to lock two continuous wave lasers, with center wavelength 1535 nm

    and 1565 nm to a single PDH cavity. Those two CW lasers have a stabilized center frequency

    will serve as the frequency standard for the two frequency components in a fiber based

    mode-locked. Therefore, as two components are locked to two frequency absolutely, both fo

    and frep can be stabilized simultaneously, achieving full frequency comb lock.

  • 39

    Figure 3.9: Schematic of the entire stablization system. The greeen background indicates an

    air suspensed optical table. Colored lines mean the free-space optical path, brown lines mean

    polariztion maintain fiber, black lines mean electrical signal. PDH: Pound-Drever-Hall locking

    electronics, EOM: fiber-coupled eletro-optical modulator, AOM: fiber coupled acusto-optic

    modulator BP: bandpass filter, PBS: polarizing beam splitter. [25]. In the frequency domain,

    the repetition frequency is stabilized via stabilization of the 3.74 THz wide comb across two

    individual frequency components.

    The two wavelengths 1535 nm and 1565 nm of the cw lasers are chosen to match two

    frequency components in the femtosecond fiber laser’s spectrum. The two lasers are first

    locked to an optical cavity of 10 cm with finesse of 200000 using PDH technique. As the light

    are mostly off resonance, only a few percent of the combined light is coupled into the cavity.

    One half of the fs fiber laser output is combined with the two cavity stabilized cw lasers. The

  • 40

    combined light is spectrally filtered, and the heterodyne best at 1535 nm and 1565 nm was

    measured, respectively. The two heterodyne signals are then used in a phasedlocked loop to

    stabilize the femtosecond fiber laser [23].

    This technique is already much simplified than previous attempts to lock fo and frep

    separately, which would require using a nonlinear fiber for supercontinuum generation and a

    f to 2f self-referencing scheme [24]. However, the compromise for reduction in complexity

    is that the stabilization ”moment arm” is 3.74 THz, a great difference form that of the full

    comb lock, which is ≈ 200 THz [23]. This would decrease the sensitivity of the optical locks

    to the excess phase noise by a factor of(

    3.74 THz200 THz

    )2 ≈ 12500

    comparing to full frequency comb

    lock [23].

    Considering the fact that most optical components are fiber based, this stabilizing

    system is generally better suited for portable means than other state-of-the-art frequency

    comb stabilization systems. However, this technique still require two PDH stabilized cw

    1535 nm and 1565 nm transfer lasers, which mean that they are ”transferring” the stability

    to the fs mode-locked laser, rather than stabilize the mode-locked laser directly. It is still a

    complex system by itself, and this project will try reduce this complexity even further.

  • Chapter 4

    Experimental setup and results

    In this chapter, the experimental setup to apply the Pound-Drever-Hall (PDH) technique

    on a mode-locked laser will be carefully detailed, and the results will also be presented and

    examined. This is a collaborative project between me, Liangyu Chen, and Luke Charbonneau

    M.S., a previous master student in Schibli’s lab who graduated in 2016. Therefore, in this

    chapter I will refer to work that was done as part of his master thesis. Published data from

    his thesis will be cited.

    This project applies the PDH technique to mode-locked laser to stabilize its repetition

    frequency directly. The aim is to suppress phase noise and timing jitter of the laser via

    a robust PDH setup that does not require building other transfer lasers as reference as

    required in the system described in section 3.3. This goal is achieved with a significant

    noise suppression at low frequencies on a bulk erbium-doped ytterbium (Er:Yb) 500 MHz

    mode-locked laser.

    The experimental setup is shown in figure 4.1.

  • 42

    Figure 4.1: A schematic of the experimental setup. The purple rectangle represents a acrylic

    box that provides environment isolation for the setup. The photodetector and oscilloscope

    on the left-hand side of the schematic is only used for monitoring the transmitted signal from

    the cavity [16].

    4.1 500 MHz Modelocked Laser

    This part was mainly built by Luke Charbonneau and it is well detailed in his thesis

    [16]. First, a continuous-wave, bulk, free-space 1560 nm laser was constructed with erbium-

    ytterbium gain medium (Kigre, Inc. - QX/Er - Datasheet: Appendix A) and was pumped

    by a 980 nm laser diode (Gooch and Housego - Model #: D1306077) . The laser cavity has

    a folded ”X” design and an asymmetric long and short arm, as shown in figure 4.1 . This

  • 43

    design was calculated with ABCD matrices for Gaussian beams [26] to achieve the correct

    mode size at the center of the gain medium and at the SESAM. The exact dimension of each

    arm is marked on the diagram. At the end of the long arm, a 0.5% output coupler was used

    as the output of the laser. After tuning the laser resonator to reduce the lasing threshold, the

    mirror in the short arm was replaced by a semiconductor saturable absorber mirror (SESAM)

    . The asymmetric design was to ensure the focus on the SESAM is appropriate to trigger the

    saturation behaviour.

    The SESAM was commonly used in passively mode-locked laser as described earlier in

    section 1.3.2. The SESAM used here has a combination of slow and fast saturable absorber.

    The details on SESAM is well established in the works of Ippen [27] and Prasankumar

    [28]. There is a single InGaAs quantum well absorber layer close to the top of the SESAM

    and the rest underneath is a semiconductor Brag mirror. For the designed wavelength the

    Brag mirror serves as a near-perfect reflector, as reflection from each quarter-wave layer

    always constructively interfere with the reflected light. The materials also have a larger

    bandgap energy to ensure that no absorption occurs in the mirror. The quantum well at

    the first layer create an interband relaxation transition for electrons from conduction band

    to valence band. When high intensity light is shone on the SESAM, electrons accumulate

    in the conduction band, leaving valence band depleted of electrons, therefore, reducing the

    possibility of absorption . After this saturation process, the electrons can transit at the

    interband energy level due to fast thermal relaxation, and then recombine with the carriers

    in the valence band, which is a slow relaxation. With this combination of slow and fast

    saturation behaviour, it is capable to self-starting mode-lock due to the slow behaviour and

    further pulse-shortening due to the fast behaviour.

    After aligning the SESAM and other optical components to achieve maximum output

    power, the output of this now mode-locked laser was coupled into a fiber and used an optical

    spectrum analyser and RF analyser to measure its main parameters:

  • 44

    λc = 1551 nm

    ∆λ = 13 nm

    fr = 495.67 MHz

    Pout ≈ 52.3 mW

    (4.1)

    where ∆λ is the spectral bandwidth of the pulses of the mode-locked laser.

    From the measure repetition rate, 495.67 MHz, the length of the laser resonator can be

    calculated with equation 1.1, which was 302.6 mm. A slow and a fast piezo were installed

    on the mirrors of the laser resonator to allow small variations of the cavity length, which is

    essential to lock the laser to the reference cavity. The slow piezo has a maximum displacement

    of 1.1µm when applying a 75 V voltage. The output coupler was mounted on a movable

    precision stage to allow tuning of the repetition rate within a large range.

    4.2 Passive Reference Cavity

    The passive reference cavity in this setup is composed of two concave mirrors (reflectivity ≈

    99.3% at λ = 1551 nm, LayerTech - Coating Batch #: R1009002 and F115H010, (R1009002)

    Dielectric Coating Datasheet: C) [16]. The finesse of the cavity can be found with

    F = π(r1r2)14

    1− (r1r2)12

    ≈ 447.23 (4.2)

    and the optical linewidth ∆νcav of the cavity was:

    ∆νcav =FSR

    F=

    495.67 MHz

    447.23≈ 1.11 MHz (4.3)

    The input coupling mirror has a radius of curvature of 1000 mm and the other mirror

    has a radius of curvature of 500 mm , which were chosen to reduce the undesirable higher

    order cavity modes. The length of the cavity was designed to match the length of the laser

    resonator, so it would match the free spectral range of the mode-locked laser. One of mirror

  • 45

    of the passive cavity mounted on a moveable stage so its free spectral range could be tuned

    accordingly to follow the repetition rate of the mode-locked laser has to be adjusted. A

    plano-convex lens was placed before the entrance of the cavity to improve the coupling

    efficiency into the cavity.

    4.3 Electro-optic Modulator

    This setup uses a fiber-based lithium niobate (LiNbO3) crystal electro-optic phase

    modulator from EO-Space (Model #: PM-0K5-10-PFA-PFA, Datasheet: B) [16]. In an

    earlier attempt, a free-space electro-optic phase modulator was constructed using a bulk

    (LiNbO3) crystal. However, there were parasitic amplitude modulation coupled to the phase

    modulation due to its high sensitivity to the light path in the crystal. Therefore, a fiber-based

    electro-optic modulator was used instead, providing a higher bandwidth and much less

    amplitude modulation (AM).

    The modulator was driven by a Ω = 15 MHz sinusoidal signal with 4 V peak-to-peak

    from a Rigol function generator. The modulation frequency chosen to be much larger then

    the linewidth of the cavity, ∆νcav ≈ 1.11 MHz.

    4.4 Electronics

    For this setup, a low-noise, passive photodetector with no amplification was used to

    collect the reflected signal. The final signal was obtained by mixing the reflected signal

    with the phase shifted signal from the local oscillator using a high voltage phase detector

    (Minicircuits: Model #: MPD 1+). A 10.7 MHz low pass filter (Minicircuits: Model #: BLP

    10.7+) is used to filter out 15 MHz the modulation signal. Then, the signal was sent to a

    proportional-integrator (PI) servo loop with dual output for fast and slow piezo mounted

    in the cavity. The fast loop only has a proportional-integrator, and will respond to the fast

    components in the error signal up until 188.7 kHz, which is the measured bandwidth, and the

    slow loop with an extra slow integrator will handle any long term drifts. The circuit diagram

  • 46

    of this servo loop is provided in figure 4.2.

    Figure 4.2: The circuit diagram for the PI servo loop filter [16].

    4.5 Environmental Isolation

    To provide suitable environmental isolation, the setup was constructed on top of a

    optical breadboard with damping material as substrate, and was inside a 1 inch-thick acrylic

    box to prevent air fluctuations and provide thermal isolation to some extent. The box was

    placed on a high-performance Laminar flow optical table as standard for optical experiments,

    which is able to minimize the effects of mechanical and acoustic vibrations from the lager

  • 47

    surrounding environment.

    Figure 4.3: The experimental environmental isolation setup described in section 4.5. The box

    at the left contains the mode-locked laser and the refrence cavity. The optical heterodyne

    cross-correlation setup is at the right on the air suspended optical table.

    4.6 Lock Performance

    First, the resonance of the cavity was scanned by sending a periodic triangular-sloped

    signal to the slow piezo while tuning its center voltage, and the transmitted and reflected

    signals were monitored so that the resonance could be easily identified. The transmitted and

  • 48

    reflected signal without phase modulation are shown in figure 4.4 and figure 4.5, and the

    signals with phase modulation are shown in figure 4.6 and figure 4.7, respectively.

    4

    3

    2

    1

    0

    Tra

    nsm

    itted

    Sig

    nal

    (V

    )

    -0.10 -0.05 0.00 0.05

    Time (s)

    Transmitted Signal Scanning Signal (not to scale)

    Figure 4.4: Reference cavity resonances without phase modulation. The red trace shows

    the main resonance, the peak is around 3.00 V, with around 577µW of optical power on the

    photodetector. The resoponsitivity of the photodiode is 1.04 A/[email protected] and the gain

    is 5kV/A . The blue trace represents the scanning signal, driving the slow piezo at 10 Hz

    and 500 mVpp. There are also higher order resonances presented in the cavity, but they fall

    outside of the scan range.

  • 49

    8

    6

    4

    2

    0

    Tra

    nsm

    itted

    Sig

    nal

    (V

    )

    -0.10 -0.05 0.00 0.05

    Time (s)

    0.20

    0.15

    0.10

    0.05

    0.00

    Refle

    cted Sign

    al (V)

    Transmitted Signal Reflected Signal

    Figure 4.5: The blue trace is the reflected signal from the cavity, the mean of the higher

    values is 187 mV and the mean of the lower peaks is 85 mV, with 3.62 mWoptical power on

    the photodiode at the maximum. The red trace shows the transmitted signal as reference.

    Note that the reflected signal is well above zero on resonance due to cavity loss, higher cavity

    modes.

    From the graph 4.5, the contrast between the strength of the reflected signal when the

    light was on resonance and it was off resonance would tell how much light had been coupled

    into the cavity, essentially refreshing to contrast between the lowest point and the highest

    point in the reflected intensity graph 4.5:

    DC Voltage− Reflected Peak VoltageDC Voltage

    =102 mV

    187 mV= .545 ≈ 55% (4.4)

  • 50

    Therefore, the coupling efficiency was measured to be approximately 55%. It was largely

    limited by the transverse modes in the cavity as a wider scan would show that there are

    several other transverse modes presented. The highest one being around 70% of the main

    peak.

    3.5

    3.0

    2.5

    2.0

    1.5

    1.0

    0.5

    0.0

    Tra

    nsm

    itted

    Sig

    nal

    (V

    )

    -0.10 -0.05 0.00 0.05

    Time (s)

    Transmitted Signal Scanning Signal (not to scale)

    Figure 4.6: Reference cavity resonances with phase modulation at modulation frequency

    Ω = 15 MHz with 4 V peak-to-peak. The red trace is the transmitted signal with peak value

    at 2.82 V and the blue trace shows scanning signal as the same as in the previous graph.

    Since the electro-optical phase modulator has transferred some power to the sidebands which

    are completely reflected from the cavity, therefore the maximum signal when on resonance

    is reduced. Note that the smaller peaks around the main resonance are sidebands from the

    pulse with frequencies f ± Ω and transmitted through the cavity when scanning through

    these frequencies.

    From the measurement of main carrier transmitted power before and after the modu-

    lation, approximately 12.73% of the main carrier optical power was transferred to the two

  • 51

    sidebands.

    8

    6

    4

    2

    0

    Tra

    nsm

    itted

    Sig

    nal

    (V

    )

    -0.10 -0.05 0.00 0.05

    Time (s)

    0.20

    0.15

    0.10

    0.05

    0.00

    Refle

    cted Sign

    al (V)

    Transmitted Signal Reflected Signal

    Figure 4.7: The blue trace is the reflected signal from the cavity. The red trace shows the

    transmitted signal as reference. Note the reflected signal is the result of interference between

    two beams: the two sidebands that are reflected from the first cavity mirror, and the leakage

    signal from the cavity when it is on resonance.

    The error signal can also be seen at the output of the mixer with the frequency scanning

    mentioned above in shown in figure 4.8.

  • 52

    -0.4

    -0.2

    0.0

    0.2

    0.4

    PD

    H E

    rror

    Sig

    nal (

    V)

    -4x10-3 -2 0 2 4

    Time (S)

    Figure 4.8: The experimental PDH error signal when scanning the cavity near resonance.

    The zero-crossing of the center fringe is the resonance and thus is used in the locking process.

    The discrepancy between the measured signal the simulated signal shown in figure 3.6 is

    likely due to the extra losses and higher order modes in the cavity.

    To initiate the lock to the cavity, the scanning interval was decreased step by step while

    adjusting the voltage to the slow piezo to zero-in on the resonance. When the frequency was

    really close to the resonance, the scanning was stopped and the output of the PI controller

    was connected to the slow and fast piezo instead. A manual tuning of slow piezo signal was

    usually required to let the fast lock engage on the resonance. Then, the slow integrator was

    turned on so that the slow lock was engaged as well. The combination of slow and fast piezo

    would simultaneously compensate for both fast noise up to the bandwidth of the fast loop at

    188.7 kHz , and low frequency noise and long-term drifts caused by acoustic oscillations and

    thermal fluctuations. To determine its long-term stability, the lock was kept in operation

    for around 8 hours, and the voltage drift on the slow piezo was measured to be 0.168 V per

  • 53

    hour. Given the fact that the slow loop was supplied with ±15 V, the minimum time for

    the slow piezo to run of range if the drift is constant in one direct would be ≈ 90 hours,

    therefore providing a lower bound for long-term stability of the lock without excessive external

    disturbance. The transmitted signal through the reference cavity while the lock was engaged

    is shown in figure 4.9.

    3.0

    2.5

    2.0

    1.5

    1.0

    0.5

    0.0

    Tra

    nsm

    itted

    Sig

    nal

    (V

    )

    121086420

    Time (s)

    Figure 4.9: The transmitted signal through the reference cavity with lock engaged for

    measurement period of 12 s. Note that the height of signal roughly corresponds to the

    maximum resonance peak at 2.82 V shown previously in figure 4.6.

    4.6.1 Noise Limitation

    With the parameters of these major components settled and measured, we can find the

    shot noise limitation of the error signal with equation 3.14 provided in the previous chapter:

  • 54

    Sf =

    √hc3

    8

    1

    FL√λPc

    =

    √hc3

    8

    1

    (447.23)(302.62× 10−3 m)√

    (1551.67× 10−9 m)(1− 0.13)(0.55)([3.62× 10−3 W])

    = 2.39× 10−3 Hz√Hz

    (4.5)

    where Pc is the measured reflected power with the correction of the percentage of coupling

    efficiency and the percentage of po


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