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A Pound-Drever-Hall based Repetition RateStabilization Technique for Mode-locked LasersLiangyu [email protected]
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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
A Pound-Drever-Hall based Repetition Rate Stabilization
Technique for Mode-locked Lasers
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
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
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
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.
To my dear friends and family, for their incredible support in the endeavour.
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
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
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
A Kigre Inc. Erbium-doped Ytterbium Glass, QX/Er Datasheet 65
B Fiber-Based Lithium Niobate EOM-PM Datasheet 66
C Layertech Mirror Dielectric Coating Datasheet 67
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
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
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.
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 . 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
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 . 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
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 . 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 . 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,
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
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 . 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:
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
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 . This thesis will continue to examine this type of noise
and the implement a stabilization technique based on the Pound-Drever-Hall approach.
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 .
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 . Additionally, this pulse shortening effect is even less
effective for shorter pulse durations, thus essentially limits the pulse duration on the order of
When compared with the later passive mode-locking in the field of ultra-fast optics,
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 .
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, essentially being
”saturated” with intense light.
Figure 1.4: A schematic setup of a passive mode-locked laser using a saturable absorber
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 . 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
. 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 . 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 .
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 .
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.
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 . 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 , which is calculated with:
δPrms =√〈(P (t)− Pavg)2〉 (2.1)
where Pavg is the average power.
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
From the power spectral density measurement, one can calculate the r.m.s. relative
intensity noise with:
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 .
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 . 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 .
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 . 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
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 .
noise pump source and constructing a mechanically stable laser system. 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. 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. 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 .
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 .
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 .
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 . It is derived from the fact that phase noise and frequency noise are
really the creation of one source, just a matter of perspective:
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 .
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. 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 .
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 . Timing jitter can also couple to the
intensity noise of the mode-locked lasers through the saturable absorber depending on its
recovery rate .
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.
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 .
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
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
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.
Interleaved 500 MHz MLL(1 GHz carrier output)
Ultra-stable Monolithic MLL(1 GHz carrier output)
(Fiber Couple) (Fiber Couple)
PI Servo to lock carrier
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. 
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 . This technique is conceptually simple and
extremely sensitive, capable of achieving less than 1 zs/√Hz resolution .
Figure 2.4: Experimental setup for the optical heterodyne technique . 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.
1GHz Monolithic Mode-Locked Laser
500MHz Mode-Locked Laser
RF Signal Analyzer
Polarization Maintaining Fiber
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
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.
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
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.
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
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
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 .
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 . 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
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
And then the Q-factor can be found with:
Q = ν0Trt2π
where ν0 is the optical resonance frequency and Trt is the round trip time, l is the fractional
optical power loss per round trip .
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
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 . 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
frequency has a much larger effect on power transmission than intensity fluctuation .
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 .
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 .
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 . 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 . 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
small variation and monitor the amplitude response, essentially measuring the slope with a
small df . This conceptual idea only really works when the frequency detuning is slow
enough for the cavity to completely respond . Otherwise, the output will not follow the
curve shown in figure 3.2 . Although, the technique will still work at higher modulation
frequencies, and both the noise suppression and loop bandwidth will be improved .
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 .
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 . 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
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 . 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 .
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 .
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 .
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
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 . However, their
relative phase depends greatly on the the frequency of the laser with respect to the resonance
 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
. 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.
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 .
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:
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 .
Then, the amplitude reflection coefficient, F (ω), can be found for a lossless symmetric
F (ω) =ErefEinc
1− r2exp(i ω∆νFSR
where ∆νFSR =C2L
is the free spectral range 1.1, and r is the reflectivity of each cavity
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 . 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
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.
the phase reference to enable the phase measurement of the reflected verses the leaked beam
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 :
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 . 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
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  :
Pref = |Eref |2 =Pc|F (ω)|2 + Ps[|F (ω + Ω)|2 + |F (ω − Ω)|2
[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)]
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 . 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
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
is applied with Ω >> ∆ν, the linewidth of the cavity, and therefore it is safe to assume
F (ω±Ω) ≈ −1 . 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 . 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
Thus, the final error signal becomes :
� = −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 :
� ≈ −8√PcPs
∆νδf = Dδf (3.13)
where ∆ν ≡ ∆νFSRF is the linewidth of the cavity and D ≡ −8√PcPs∆ν
is called the frequency
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
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
. 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
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 . 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 .
However, the quantum shot noise is the fundamental limit of how quiet the error signal
can be. 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,
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 :
Dividing this equation by the frequency discriminant D will give the apparent frequency
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
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.
, 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 .
Figure 3.8: The concept for microwave generation directly from a fiber mode-locked laser
. 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.
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. . 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
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 .
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 . 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 . 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
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.
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
The experimental setup is shown in figure 4.1.
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 .
4.1 500 MHz Modelocked Laser
This part was mainly built by Luke Charbonneau and it is well detailed in his thesis
. 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
design was calculated with ABCD matrices for Gaussian beams  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
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  and Prasankumar
. 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:
λc = 1551 nm
∆λ = 13 nm
fr = 495.67 MHz
Pout ≈ 52.3 mW
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) . The finesse of the cavity can be found with
F = π(r1r2)14
≈ 447.23 (4.2)
and the optical linewidth ∆νcav of the cavity was:
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
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) . 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.
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
of this servo loop is provided in figure 4.2.
Figure 4.2: The circuit diagram for the PI servo loop filter .
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
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
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.
-0.10 -0.05 0.00 0.05
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.
-0.10 -0.05 0.00 0.05
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
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
187 mV= .545 ≈ 55% (4.4)
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
-0.10 -0.05 0.00 0.05
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
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
-0.10 -0.05 0.00 0.05
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.
-4x10-3 -2 0 2 4
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
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.
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:
(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
where Pc is the measured reflected power with the correction of the percentage of coupling
efficiency and the percentage of po