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DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING
MSc in Optical Communications and Signal Processing
Active & Passive Mode-Locked Fibre Ring Lasers and
Multi-wavelength Generation
By Botonakis George
21st
September, 2012
Supervisor: Professor Mark Thomson
A dissertation submitted to the University of Bristol in accordance with the
requirements of the degree of MSc in Optical Communications and Signal
Processing in the Faculty of Engineering
Department of Electrical and Electronic Engineering

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ABSTRACT
The aim of this thesis is to explore the active and passive mode-locking in fibre ring lasers. Mode
locking is a phenomenon where the circulating modes exhibit a fixed-phase relationship, so that a
strong and ultra-short pulse is generated. As we know, strong and ultra-short pulses are two key
features in long-haul optical links due to the high data rate and attenuation tolerance the
communication system can provide.
Optimization techniques will also be presented, covering a wide range of pulse-shaping factors
such as pump rate, modulating frequency and the material used in the amplifying and absorbing
fibres. A direct comparison between these two mode-locking techniques, and the benefits of
using either passive or active mode locking techniques will be discussed.
Ultimately, a passive mode-locked fibre ring laser will be modified so it can support
multiwavelength generation through Four-Wave mixing. The reason for this modification is to
explore whether the mode-locking performance can be enhanced if pulses of different
wavelength circulate in the ring laser.

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ACKNOWLEDGMENTS
I would like to express sincere thanks to my supervisor Dr. Mark Thomson for his invaluable
guidance throughout the course of this thesis. His ideas, guidance kindness and support played a
key role inspired me to conduct a far more deep research than my original expectations.
Special Acknowledgment to Dr. Siyuan Yu for his valuable suggestions and revision of this project.
I would also like to thank Dr. John Rarity and Dr. Martin Cryan for their exceptional teaching skills
since their transferred knowledge has been used throughout this thesis.
Finally, I would like to thank my parents for giving me the chance to pursue and achieve my goals.

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BIBLIOGRAPHY
The following bibliography has been used as a source in this thesis and are strongly recommended
books for anyone whose interests include nonlinear phenomena in optical fibres, short pulse
generation, mode-locking and fibre ring lasers. The number inside the brackets correspond to the
reference number used in the thesis.
Ultrafast Optics by ANDREW M. WEINER [2]
Ultra-Fast Fiber Lasers: Principles and Applications with MATLAB® Models by
Le Nguyen Binh and Nam Quoc Ngo [3]

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Nonlinear Fiber Optics Third Edition by
GOVIND P. AGRAWAL [17]

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Table of Contents
Section 1 Introduction to Mode Locked Lasers ......................................................................... 8
Mode Locking Theory.................................................................................................................................... 9
Bandwidth Limitations................................................................................................................................. 10
Gain & Absorbing medium Effect on the pulse ........................................................................................... 10
Section 2 Active Mode-Locked Lasers..................................................................................... 13
Active Mode-locking techniques................................................................................................................. 13
Active Mode-locked fibre ring laser Simulation .......................................................................................... 15
Acousto-Optic Modulator............................................................................................................................ 15
Tuning the amplitude of the RF modulating frequency & its consequences .............................................. 16
Tuning the RF modulating frequency & its consequences .......................................................................... 17
Saturation Power of the gain medium ........................................................................................................ 18
Comparison Between Slow & fast Saturation ............................................................................................. 19
Optimizing the response of the saturable amplifier ................................................................................... 19
Test Pulse Distribution................................................................................................................................. 19
Active Mode Locked fibre ring laser model.................................................................................................. 18
Section 3 Passive Mode-Locked Fibre Ring Lasers................................................................... 28
Passive Mode-Locking using Saturable Absorbers...................................................................................... 28
Saturable Absorber model.......................................................................................................................... 28
Simulating the response of the saturable absorber .................................................................................... 31
Passive Mode-Locked Fibre Ring Laser Simulations .................................................................................... 32
Bandwidth Effect ......................................................................................................................................... 35
Mode Locking Optimization Techniques ...................................................................................................... 37
Section 4 Four Wave Mixing .................................................................................................. 38
Origin of Four Wave Mixing.......................................................................................................................... 38
Different Types of Four Wave Mixing.......................................................................................................... 39
Approximate and Analytic Four Wave Mixing solution ............................................................................... 40
Mode-locking enhanced performance via FWM......................................................................................... 42
Four Wave Mixing considering energy conservation ................................................................................... 44
Simulation Parameters ................................................................................................................................. 47
Laser Gain Bandwidth................................................................................................................................... 47
Four Wave Mixing Simulations without considering energy conservation ................................................. 48
Four Wave Mixing Simulations considering energy conservation ............................................................... 50
Four Wave Mixing Simulation model ........................................................................................................... 52

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Section 4 Conclusions............................................................................................................ 53
References............................................................................................................................ 54

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Section 1:
Introduction to Mode-locked Lasers
Mode locked semiconductor, or fibre lasers are photon emitting devices which for many years
have been used in a large range of photonic applications. The reason for this is due to their
inherent ability to produce ultra-short pulses on the order of picoseconds ( ) or
femtoseconds ( ), their low timing jitter, high repetition rate and their cost effective
manufacturing
The basis of mode-locking is to force a fixed relationship between the propagating modes of the
laser mode cavity. In this case, the laser is said to be mode-locked or phase-locked. The result of
this fixed phase relationship, is the formation of a periodic ultra-short pulse.
Due to the characteristics mentioned above, there are a large number of various applications and
scientific fields where a mode-locked techniques are applied, such as: Nonlinear optics (second-
harmonic generation, parametric down-conversion, optical parametric oscillators, and generation
of Terahertz radiation), Optical Data Storage, Nuclear fusion (inertial confinement fusion),
Photonic Sampling, using the high accuracy of lasers over electronic clocks to decrease the
sampling error in electronic ADCs, Corneal Surgery, optical clock distribution and radio over fiber
signal generation [4]-[8].
The difference between a simple laser a mode-locked laser is that in a simple laser, the circulating
longitudinal modes exhibit no fixed phase relationship between them. The phase of each mode is
not constant, but instead it varies in a random fashion due to thermal instabilities of the
integrated laser material. So, there are cases where neighbouring frequencies have a temporary
phase relationship, leading to superposition of their individual energies and hence an
instantaneous short-term increase in the laser intensity occurs, causing beating effects in the laser
output. A Fabry-perot laser for example, emits a large number of longitudinal modes and because
this phenomenon average out the output intensity, the laser is said to exhibit a continuous wave
operation.
A mode locked laser on the other hand, operates quite differently. In the steady-state operation,
a considerate large number of the longitudinal modes manage to lock their relevant phase and
Instead of a constant intensity emission, a periodic burst of light pulse is emitted due to the
periodic constructively interference. Such a laser is said to be phase or mode-locked. The period
of these pulses is 2 /T L c , where T is the cavity round-trip time and the mode spacing of the
laser is then 1/  .
Parameters of the optical pulse such as pulse duration, pulse peak power and techniques to
modify these parameters, will be described later in this thesis.

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Mode locking Theory
A mode-locked laser is consisted of either an active element (an Acousto-optical modulator for
example) or a nonlinear passive absorbing element, a saturable gain medium and an optical
coupler to exit the produced pulses. Both the amplifier and the absorbing element cause the
formation of an ultra-short pulse which circulates in the laser resonator [1].
Figure 1.1: Model of the passive or active mode-locked ring laser that has been used in this thesis
There are many factors that affect the formation of the short pulse as it propagates through the
cavity and can be of great importance in mode-locked lasers:
Gain. As in all lasers, a gain medium is mandatory to compensate the loss that is introduced by
the cavity. In the case of mode-locked lasers, some (not all) mode locked lasers (MLL) the gain
should saturate (ideally, dynamic and not linear saturation can be proved to be more desirable –
more on this matter, later in this thesis).
Linear loss. Except the dynamic loss caused from the saturable absorber, or from the modulator,
the linear loss is present in every optical fibre.
Non-linear loss: The nonlinear loss is the key to mode-locking and can be:
 Active modulation: This refers to an externally driven optical modulator that modulates
either the phase or the amplitude of the circulating pulse. The modulating frequency (RF
frequency) must be equal to the cavity round-trip time [2] or a harmonic of the cavity
fundamental frequency. In the case of the active modulation, we are referring to an
Active mode-locked laser.
Frequency Domain Filtering
(Simulates the Bandwidth
of the ring laser)
Saturable Gain
Medium
White Gaussian
Noise Source
Saturable absorber (Passive mode-locking) or
Acousto-optic modulator (Active mode-locking)
Optical
Coupler (10%)
ISOLATOR
Spectrum
Analyzer

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 Saturable absorber: (self amplitude modulation). In contrast to the active modulator that
was mentioned above, a different technique that contributes to the pulse-shaping is the
implementation in the cavity of a saturable passive modulator. The loss in this material is
a function of the pulse intensity or the pulse energy. What happens is that the loss
changes dynamically in response to the pulse, which is its self modulated by the
dynamically changing loss. In the case of the active modulation, we are referring to a
Passive mode-locked laser.
Bandwidth limitations.
It is known that the bandwidth of an optical pulse is inversely proportional to its duration. So, in
order to obtain the shortest possible pulse, the bandwidth must be as large as possible. Any BW
limitations arise due to frequency-dependent loss elements or from the finite bandwidth of the
laser gain medium [2].
(a) (b)
Figure 1.2: Due to the fact that the bandwidth of a photonic device is defined by the density
of electrons in each allowed transition figure 1.2 a, the bandwidth is simulated with a Gaussian
filter in the frequency domain figure 1.3 b.
Dispersion: As the pulse propagates through an optical fibre, it can be broadened or
contracted due to passive, frequency-dependent phase variations encountered from laser cavity
[2]. Thus, dispersion can be a strong limitation factor for ultra-short pulse generation.
Effect of the gain and loss medium in the optical pulse:
As the pulse propagates through the gain medium the peak but also the tails of the pulse are
amplified. In the time domain, this means that the pulse is spread whereas in the frequency
domain is compressed. As the expanded (from the gain medium) pulse enters the loss modulator
(either passive or active), begins to compress again in the time domain and expand in the
frequency domain. Ideally, a stable mode locked laser will produce an optical pulse (as it exits the
loss modulator) exactly identical as the pulse that entered the gain medium. During this procedure
of expansion and compression (both in the time and frequency domain) the various modes that
travel though the cavity manage eventually to lock together and form an ultra short stronger
pulse. But why stronger and compared to which pulse is stronger?

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The output electric field of the generated light-wave in the temporal domain is the summation of
all the oscillating modes given as [3]
   0
(1)R nj n t
n
n
e t E e
    
 
Where:
 0 is the referenced centre oscillating frequency
 nE and n are the amplitude and phase of the nth mode, respectively
When the laser is in the free oscillating state, En and n can take any value without any bound
leading to the generation of a continuous wave (CW) source [3] (figure 1.3).
Figure 1.3 When the modes are out of phase, the power of the generated signal is
distribution in a random fashion [3]
Figure 1.4: Temporal Intensity of an active mode locked laser after a single run. Observe that
the mode-locking haven’t yet been achieved and hence the optical signal looks like noise.
On the other hand, when 0n  and 1nE  then the equation (1) becomes [3]
 
 
 0
sin / 2
cos
sin / 2
R
R
N t
e t t
t



 (2)

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Equation (2) describe an oscillation at frequency 0 modulated with the sinc function [3]
 
 
 
sin / 2
sin / 2
R
R
N t
f t
t


 (3)
The average power is then:
 
 
 
2
2
sin / 2
sin / 2
R
R
N t
P t
t


 (4)
From the last expression, we can make the following observations [3]:
1. The pulse period is
2. Because the various modes have been “locked” together, the peak power is N times the
average power.
3. The peak field amplitude is N times the amplitude of a single mode.
4. As the number of contributing modes increase, the pulse width (time from the peak to
the first zero) decrease as N increases with the relationship /T N  .
An example of a laser in which the various modes have been locked together can be seen in figure
1.6.
Figure 1.5 Electrical field amplitudes of five in-phase individual modes and (b) the total
power of a periodic pulse train.

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Section 2: Active Mode-Locking
Mode-locking Techniques
The techniques used to achieve mode-locking are classified as either passive or active. Passive
methods require a non-linear absorber integrated in the laser cavity, capable to cause self-
modulation of the light. On the other hand, active mode-locking involve an external RF signal
which drive a modulator (an acousto-optic modulator for example), so that a periodic loss is
applied to the optical signal.
The most common active mode-locking technique require the implementation of a standing wave
acousto-optic modulator into the laser cavity which apply a sinusoidal amplitude modulation of
the light in the laser cavity
Variations of Active Mode-locking: Although the most commonly used modulating technique
is via synchronous modulation, there are several other variations of active mode-locking, each
one serve a different purpose [2]:
 Synchronous phase modulation [9],[10]: In this case the modulation depth m
takes an imaginary value. This mean that in the frequency domain a time derivative of the
phase give a frequency shift to the pulse and due to the finite gain bandwidth, the pulse is
attenuated.
 Regenerative mode-locking: With the help of a photodiode, the beat note between the
pulses is detected, amplified and then it is used to drive an intracavity modulator.
Therefore, the modulating frequency is automatically adjusted to the cavity mode
spacing. In some sense, regenerative mode-locking is a combination between active and
passive moode locking due to this “automated” function;
 Harmonic mode locking: This technique is used to achieve shorter pulses and the
generation of pulse trains spaced more closely than the cavity round-trip time. Instead of
driving the modulator with the cavity fundamental frequency, an integer multiple of this
frequency is used. This is especially useful in fibre lasers where the round-trip time is too
long (often in the scale of hundreds of nanoseconds) and the desired bit rate may be in
the range of gigahertz [11]. One flaw of this technique is that there is no guarantee that
the gain peak (transmission peak )of the amplifying medium will coincide with the pulse
peak and possible fluctuation of the pulse’s intensity may occur.
Figure 1.6: Example of a harmonic mode-locked laser using x4 times the ring
fundamental frequency
 Synchronous pumping: Most appropriate for optically pumped systems, the laser is
synchronously pumped by the pulses of another mode-locked laser [12], [13].
 Hybrid mode-locking: In this case the active mode-locking meets the passive mode-
locking using a satutable absorber to shorten the pulses, where an electrically [14] or
optically driven modulator amplify the pulse

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Figure 1.7: (a) Actively mode-locked laser arrangement, with an intracavity modulator driven
at the cavity round-trip period; (b) periodic modulator transmission and resulting mode-locked
pulses.
Considering the principles of communication systems, when we modulate a signal of (optical)
frequency ‘ν’ with a signal at frequency mf the resulting modulated signal has sidebands at
(optical) frequencies &m mv f v f  . Because we desire the maximum transition to occur at
the cavity mode spacing (figure 1.7, (b)) so that the optical signal have the minimum attenuation,
the modulating frequency must be set at the cavity-mode spacing v , so that the generated
energy from the sidebands can “feed” the two neighbouring optical pulses. Due to the fact that
the generated side-bands are driven in-phase with the central pulse and because all the past and
future optical pulses have been modulated with the same principle, the optical side-bands and
the main pulses are co-phased. Because they are co-phased, their amplitudes can be algebraically
superimposed and the new pulse (main pulse plus the sideband) will increase its energy (energy
transfer).
Figure 1.8: Figure source: [3]. (a) energy transfer to two sidebands, (b) oscillating modes in the
cavity with fundamental frequency of Rf , (c) energy distribution when modulating the oscillating
modes with frequency m Rf f and (d) energy transfer between the modes when m Rf f .

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Active mode-locked Fibre Ring Laser Simulation
With the basic principles of an active mode-locked laser have been presented, the next step is to
model the devices of the ring laser.
Acousto-Optic modulator
An acousto-optic modulator is a device which allows the modulation of the frequency, intensity
and direction of a laser beam. Modulation of the laser beam can be achieved by varying the
amplitude and frequency of the acoustic waves travelling through the crystal.
Assume that the laser has length L with an acousto-optic modulator of length mL , with mL L .
The pulse after the transmission through the modulator becomes [31]:
       exp 1 cos 5m m mA t A t a L t    
Where m is the modulating frequency, ma is the attenuation factor.
(a) (b)
Figure 1.9: In figure (a) you can see an acousto-optic modulator and its difraction effect on
the incident front wave. Altering parameters such as the modulating frequency, the absorption
coefficient and the length of the modulator, a different loss is introduced to the optical pulse.
Observing figure 1.9 (a), we notice that the front wave splits into multiple front waves with the
intensity getting weaker as the angle θ (with reference non-diffracted wave) increases. Recalling
the formula of Snell’s critical angle 1
sin air
critical
fibre
n
n
 
 , as the angle of a front-wave gets
critical  then portion of the pulse-energy is lost.

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Figure 1.10: Configuring the parameters of the acousto-optic modulator, the energy loss of the
pulse is also altered. Higher diffraction will force larger portion of the pulse energy to follow higher
diffraction angle and if this angle exceeds the critical angle, then the optical pulse is essentially
attenuated.
Tuning the amplitude of the RF modulating frequency and its consequences
Tuning the amplitude of the modulating RF frequency, different loss curves are obtained as a
bigger portion of the pulse energy follow diffraction angles above the critical angle critical (see
figures 1.9 and 1.10). This modification essentially allows to configure the evolution of the pulse
(convergence speed, pulse amplitude and pulse width).
(a) (b)
Figure 1.11: Loss curves for different attenuation factor. Figure 1.11 (a) used alpha=80db and
Figure 1.11 (b) used alpha=50db. Observe the peak of the loss curve has a value equal to ‘one’ and
the tails have a value less than one. This essentially attenuates the tails of the optical pulse with a
factor proportional to the attenuation factor. It should also be mentioned that a large attenuation
factor helps the formation of narrower pulses.

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Tuning the RF frequency and its consequences
It has already been mentioned that the period of the modulating RF frequency must be
equal to the round-trip time, so that the minimum loss of the modulator coincide with the peak of
the optical pulse in order to preserve the amplitude of the pulse and “clear” the tails of the pulse,
creating the shortest possible pulse. Another technique that can be used to compress the pulse
even further, is to use a harmonic of the fundamental modulating frequency (harmonic mode-
locking). By using this technique, the loss curve gets narrower, attenuating the tails of the pulse
even stronger.
(a) (b)
Figure 1.12: Increasing the modulation frequency, the attenuation applied to the tails of the
pulse is stronger (compare for example figure 1.12 (a) and figure 1.12 (b)) and hence narrower
pulses can be obtained.
Gain Medium
The next key element to maintain a stable function of the mode-locked laser is the gain medium.
The assumptions that were made for the amplifier are the following [2]:
 The laser spectrum is centred at the peak of the gain spectrum
 The gain can medium can be fast or slow saturable, depended by factors such as the
spontaneous emission rate, the pump rate and pulse power. In this thesis we will
investigate the advantages and disadvantages of either case
In order to analyze the gain medium we will use a four-level model:
Figure 1.13: Energy-level structure for a four-level atom [2].

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According to figure 1.13, electrons are excited from the lower level (state 1) to level 2 with the
help of a pump. The electrons in level 2 are assumed to exhibit a rapid relaxation to level 3, so the
population of electrons in level 2 remain close to zero. The transition e to 4 is the lasing transition
(stimulated emission), contributing in the energy of the laser action (rate S in figure 1.13), or they
can also relax spontaneously down to level 4 with rate 1
G 
in which case their energy is not
available to laser field [2] due to the random phase of the spontaneous emission.
The gain medium equation which was used in this thesis is the following [2]:
 
  2
3 3
3 3
| |
G
G G G
a tN N
W N N N
t P 

   
 (6)
0 G
G
G G
P

 

 (7)
Where: GN is the total population density of ions (etc.) responsible for gain, and 3N is the
population density in level 3, the upper laser level, G is the beam cross-sectional area in the gain
medium, G is the 3 4 gain cross section, 0 is the photon energy and GP is the amplifier
saturation power.
Solving this first order differential equation with the span t to be equal for the time needed to
reach the pulse its peak (i.e. half the time window of the simulation), we get different steepness of
the gain curve depended by the pump rate, the time span t , the power of the pulse   2
| |a t and
the relaxation time G .
Saturation power of the gain medium
Setting   2
| |GP a t , the equation 6 becomes:
 
 
 
   
2
3 3 3 3 3 3
3 3 3 32
| |
| |
G G G
G G G G
a tN N N N N N
W N N N W N N W N N
t a t t t   
  
           
  
Which is a simple integral over the time span t and denotes that since the saturation point has
been ‘reached’ the amplification is linear and proportional to the pump rate. In other words,
when the power of the pulse   2
| |a t is equal to the saturation power GP , the amplification is not
dynamic but linear.
Having obtained the density of the electrons over the span t , we got a range of values for 3N ,
each for every sample point of the pulse. This essentially means that the degree of amplification
wont be uniform across the span of the pulse. The gain is given by the equation:
 g t
e , where
   3 / 2G gg t N t l [2]. So, a large 3N value means stronger amplification.

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Figure 1.14: The basic requirement for stable laser operation is the net gain (gain-loss) to be
greater than zero in the region around the pulse’s peak and negative in the tails of the pulse. We
also need to consider the coupled output power from the laser cavity. Therefore, the net gain
become: gain minus absorber/modulator loss minus coupler loss.
Comparison between Fast and Slow Saturation
Before we proceed to simulations, consider the basic principle in maintaining a pulse of stable
amplitude: The net gain must be greater than zero around the pulse’s peak and smaller than zero
at the tails of the pulse. This technique allows only the peak of the pulse to grow and the tails to
be attenuated. In order to achieve this “selective amplification”, we should modify the gain
medium so that it (ideally) saturates exactly at the peak of the pulse. This slow-grow of the gain
curve can be achieved with a slow saturable medium G pt  . In order to “control” the
simulation of the saturable medium response to different values of pulse power, pulse width,
pump rate, relaxation time and number of electrons, a Gaussian and a hyperbolic secant pulse has
been used as a test pulse.
Optimizing the response of the saturable absorber
The following simulations show how the response of the gain medium change by changing
various parameters of the gain medium such as the pump rate, number of electrons and
relaxation time. Also we will see how the characteristics of the input pulse such as pulse width,
pulse power or even the pulse’s distribution, affect the amplifier’s response. In any case, we will
see that in order to maintain the optimal performance of the mode-locking laser, the pulse’s
characteristics will need to change given a non-tunable amplifier, and vice-versa.
Test Pulse Distribution:
In order to see if the distribution of the input pulse affects the response of the gain medium, the
default source (white Gaussian nose source) was de-activated and instead the input pulse was
either Gaussian, or Hyperbolic-secant. The reason why these two distributions were tested, is
because the propagating optical pulse is better described with one of these distributions.
 The simulation results showed that both distributions caused almost the same response of
the saturable amplifier. For this reason, the test pulse was chosen to follow a Gaussian
distribution.

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(a) (b)
Figure 1.15: Simulation Parameters: Relaxation time: 14
90 10 secG 
  (a),
12
90 10 secG 
  (b), pump rate
11
50 10 / secW electrons  . The red line is the upper value is
the total number of electrons in the gain medium. This is the also the highest possible
amplification according to the formula:    3 / 2G gg t N t l
(a) (b)
Figure 1.16: Simulation Parameters: Relaxation time: 12
90 10 secG 
  (a), pump rate
11
50 10 / secW electrons  , pulse half width: 300 femto-second (a), 1500 femto-second (b). As
the pulse duration increase the degree of saturation also increases. Observe that the gain has
achieved its maximum value approximately 0.5 μsec away from the peak pulse (the peak pulse
will occur at 1.5 μsec). This is not desirable since the tails of the pulse will also be amplified and
the pulse will broaden. On the other hand when the pulse half width (zero to peak amplitude) is 5
times shorted (300 femto-second) the amplification may be lower, but only the peak of the pulse
will receive the highest amplification.

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(a) (b)
Figure 1.17: Simulation Parameters: Relaxation time: 12
90 10 secG 
  (a), pump rate
11
50 10 / secW electrons  , pulse half width: 300 femto-second, peak pulse power 5 mWatt (a),
peak pulse power 1 kWatt (b). These figures show the saturation degree of the gain medium and
how this is changed with different pulse power. Observe that when the pulse power is increased,
the saturation degree lowers. This happens because higher power, force a larger number of
electrons to be “used” for amplification in the very beginning of the pulse and after
15
100 10 sec

any amplification will be defined by the relaxation time G .
(a) (b)
Figure 1.18: Simulation Parameters: Relaxation time: 12
90 10 secG 
  (a), pump rate
11
50 10 / secW electrons  , pulse half width: 1500 femto-second. peak pulse power 5 mWatt
(a), peak pulse power 1 kWatt (b). Increasing the pulse duration the saturation effect occur even
further away from the pulse peak. In the extreme case (b), the early saturation make the gain
medium exhibit a flat gain-something not desirable in mode-locking.

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(a) (b)
Figure 1.19: Simulation Parameters: Relaxation time: 12
90 10 secG 
  (a), pulse half width:
300 femto-second. Peak pulse power:5 mWatt, pump rate
10
50 10 / secW electrons  (a)
12
50 10 / secW electrons  (b). With the help of these figures the importance of the pump rate
becomes more apparent. In the first case (a) the pump rate is too low and most possibly the net
gain (gain from amplifier minus loss from modulator/absorber minus coupled ratio) will be smaller
than zero around the pulse peak, leading in a steady decrease of the pulse’s power. On the other
hand when the pump rate is too high (b) the pulse will receive the highest possible amplification,
but the tails of the pulse will also be amplified (early saturation) and the pulse will broaden, unless
a strong modulator/absorber is used, “clearing” the tails of the pulse. We see that even in the
case of a non-optimized gain medium, the performance of mode-locking can be balanced-out
with an optimal configured loss medium, and of course vice-versa.
Optimization techniques for the saturable amplifier.
The tails of the pulse should have the minimum amplification and the maximum attenuation so
they get suppressed. As we see the loss figures from the acoustic-optic modulator (figure 1.11 &
1.12), the maximum attenuation occur at the tails of the pulse. The technique that was followed
in this thesis is to periodically pump the ring laser with a frequency equal to the pulse repetition
rate. This may require a precise synchronization circuit and will increase the price of the
experiment, but the advantages overweight the higher expense:
 If a continuous pump is used then, if the pulse repetition rate is lower than the relaxation
rate 1 1
p Gt  
 then the amplified spontaneous emission noise will increase. This is not
desirable because the phase of this type of transition is random and hence non-coherent
with the phase of the optical pulse, leading to a possible distortion of the pulse in the
time domain and disturbance in the frequency domain.
 In the case of erbium-doped fibres, the electrons relax in (let’s say energy state 3 for
simplification reasons). When the pulse enters the amplifier, a large number of these
electrons will be used for the amplification of the pulse and more specifically, the tails of
the pulse. This is not desirable because:
o Possible violation of the basic mode-locking principle “The net gain must be
greater than zero around the peak of the pulse”.

23. Page | 23
o there is a possibility of electrons depletion before the peak of the pulse enters the
amplifying medium if:
 The number of available electrons is small.
 Pulse width is large.
 The tails of the pulse are strong.
o If any of the above cases occur, then the peak of the pulse might be amplified
only from the electrons provided from the pump.
 This periodic pumping dictated to pump with a frequency equal to the pulse rate (If
harmonic active mode-locking technique is used, then the pump frequency must be
increased at the same harmonic since the pulse period is shorter)
 The proposed optimization will not work unless the pump duration is also taken into
account: If the pump is turned on for a time greater than the pulse width (null to null
width and not Full Width Half Maximum) then, the mode-locking performance will get
lower since the amplified spontaneous emission (ASE) is out of phase with the mode-
locked pulse.
o In the case of erbium-doped fibres, the relaxation rate can be very large, in the
order of micro-soconds. In this case, we must make sure that after the passage of
the pulse, no electrons remain in energy state 3, because either they will increase
the ASE (if the fibre length is very large, or they will amplify the front-tails of the
pulse after one circulation). Therefore, we must make sure that the pump duration
is shorter than the pulse width, so that every electron in state 3 is ‘used’.
o A practical pump calibration technique for minimum ASE, is to set the “on” state
of the pump a time greater than the maximum expected pulse width. Next, using
a spectrum analyzer and lowering with small steps the “on” time, a “cleaner”
spectrum will be observed. This happens because no electrons are pumped to
higher energy states unnecessarily.
o This technique could also be used for the estimation of the “lag” time the
electrons experience until they spontaneously return to the lower energy state.
Active mode-locked fibre Ring laser Simulation
What characterize a mode-locked laser is its capability to lock the phase of the circulating modes,
so that their amplitudes can contribute constructively in forming of a strong, and ultra-short
optical pulse. The methodology that has been used in this thesis is the following:
 In order to show that modes with random phase can lock together, the Matlab function
white Gaussian noise “wgn(.)” was used as an input pulse (in time-domain).
Techniques for strong output pulses: (a) increase the amplifying medium length, (b) decrease the
absorption medium length, (c) change the amplifying medium with another medium with greater
number of electrons, (d) replace the absorbing medium with another with lower number of
electrons (passive mode-locking case), (e) lower the modulating frequency (active mode-locking),
and (f) change the attenuation factor of the acousto-optic modulator (active mode-locking). Later
in this thesis, we will investigate how four-wave mixing can affect the power of the optical pulses.

24. Page | 24
 In order to simulate the ring laser finite Bandwidth, the pulse was filtered with a Gaussian
filter (the pulse was converted from time domain to frequency domain, multiplied with the
Gaussian transfer function and then it was re-converted to the time-domain).
 Next the (not yet “formed” pulse) was inserted into the saturable amplifier. The gain was
given from the equation   3( ) exp / 2G ggain t N t l . To find the value of 3N for
every t of the pulse, a time-window was defined with length as the approximate time
needed from the pulse to pass through the amplifier. Having obtained the gain for every t,
an element-by-element multiplication between the optical pulse and the gain gives the
amplitude of the pulse, after the passage of the pulse through the amplifier.
 The next step is to enter the amplified pulse in the acousto-optic modulator to shape the
pulse according to:      exp 1 cosm m mA t A t a L t     .
 Finally a percentage of the pulse energy is coupled out of the ring laser (a typical 10%
couple ratio is assumed), and the pulse re-enters the amplifier in a circle.
Figure 1.20: Active mode-locked fibre ring laser model with an optimized pump.
Simulation Parameters:
Left Figure Right Figure
Modulating Frequency Ring Fundamental Frequency Ring Fundamental Frequency
Amplifier length 7 m. 7 m.
Acousto-optic modulator length 0.1m. 0.1m.
Pump rate 10
50 10 secelectrons 10
50 10 secelectrons
3 4N N Relaxation coefficient 13
9 10 sec
 13
9 10 sec

Acousto-optic modulator attenuation
coefficient
80 db 50 db

25. Page | 25
High Attenuation Low Attenuation
Figure 1.21: Tuning the diffraction coefficient of the acousto-optic modulator, the shape and
the temporal intensity of the pulse change. It is interesting to notice that an increased
attenuation/diffraction factor (left figure), the pulse is compressed in the time domain, but also
gets stronger! This happens for this reason: The electron population increase in energy state 3
initiates about 20 μsec before the peak of the pulse. Because the tails are already attenuated from
the modulator, fewer electrons will return to energy state ‘1’ during the period  20 ~ 0 sec .
Because a greater number of electrons will be ‘stored’ in state 3, the stimulated emission will also
increase-amplifying the pulse!
Figure 1.22: As we can see, a higher attenuation factor (left figure) forces the fibre ring laser
to converge faster. In this case, a 30 db increase the convergence time x16.

26. Page | 26
Figure 1.23: From the above figures we can see that as the pulse width contracts (left figure:
700 femto-seconds due to increased attenuatio ), the spectrum expands. On the other hand, the
spectrum of a wider pulse (right figure: 3 picosecond) is narrower and the spectrum tails are
weaker
Simulation Parameters:
Left Figure Right Figure
Modulating Frequency Ring Fundamental Frequency Ring Fundamental Frequency
Amplifier length 7 m. 7 m.
Acousto-optic modulator length 0.1m. 0.1m.
Pump rate 10
30 10 secelectrons 11
30 10 secelectrons
3 4N N Relaxation coefficient 13
9 10 sec
 13
9 10 sec

Acousto-optic modulator attenuation
coefficient
80 db 80 db
Low Pump Rate High Pump Rate
Figure 1.24: The increased pump rate (right figure) widens the positive net gain window and
hence the pulse is spread in the time domain. Also because the pulse’s amplitude is higher, the
convergence time is increased.

27. Page | 27
Figure 1.25: It is very interesting to notice how must the spectrum of the pulse changes with
the increase of the pump rate by one order of magnitude.
Figure 1.26: As the pump rate is increased (right figure), the convergence becomes slower.
Active mode-locked fibre ring laser conclusions:
 The RF modulating frequency must be either equal or a multiple of the ring’s fundamental
frequency to avoid pulse distortion.
 Increasing the amplitude of the RF frequency the attenuation is larger.
 The saturation of the amplifier should occur close to the peak of the pulse so that the tails
of the pulse remain as weak as possible.
 The net gain must be positive only close to the peak of the pulse and negative everywhere
else.
 In case the net gain is negative also at the peak of the pulse, the simplest solution is to
increase the pump rate and/or decrease the amplitude of the RF modulating frequency
 The criterion which defines which modes are allowed to propagate within the ring laser, is
this: in ring laser length  , where n=integer and i is the wavelength for the i-
mode. In other words, each wavelength must perform an integer number of cycles inside
the ring laser unless it will fade away.

28. Page | 28
Section 3: Passive Mode-Locking
PASSIVE MODE-LOCKING USING SATURABLE ABSORBERS
In this section, we introduce a technique to achieve passive mode-locking, using saturable
absorbers instead of an externally driven modulator. The greatest benefit of this technique is that
the problem of the modulation frequency which must be matched precisely to the cavity mode
spacing, doesn’t exist (However, this can be an advantage for applications where synchronization
to an external clock is required [2]).
In the case of passive mode-locking, the saturable absorber is a non-linear device in which the
absorption is depended on the laser pulse intensity. The ultra-short pulse which circulates in the
laser, will modulate the intracavity loss as a function of the intensity of the optical pulse. This
“automated” modulation is in perfect synchronization with the laser pulses.
Figure 2.1: Model of a passively mode-locked laser
Saturable Absorber Model
The most common material which have been used for the non-linear absorber, are organic dye
solutions and semiconductors [2]. These can be modelled as a four level system as seen in figure
2.2
Figure 2.2: Model of a four-level saturable absorber

29. Page | 29
In order to describe the effect of absorption, imagine a pulse of a certain power   2
| |a t which
“enters” and absorbing medium. Because the material act as an absorber, the density of the
electrons in energy state 1, is (ideally) many orders of magnitude larger than the corresponding
density in state 2 so the absorption strength is proportional to the population densities 1 2N N
(where jN is the density in units of
3
m
of absorbers in level j) [2]. It is assumed that the
transition 3 4 does not amplify the laser pulse because amplified pulse would be red-shifted
compared to the transition 1 2 . Finally, the transitions 2 3 and 4 1 are assumed
instantaneous.
These assumptions lead to a simpler model of absorption which claims that the absorption
spectrum is homogeneously broadened [2]. Due to the duality between the time domain and the
frequency domain, this means that in the pulse is compressed in the time domain.
The rate equation which describes the absorber is [2]:
 
 
2
31
1 2
0
| |
A A
a tNN
N N
t A

 

  

(1)
And
1 3 2 4& 0AN N N N N    (2)
Where: 0 is the photon energy,  is the 1 2 absorption cross section, and AA is the
beam cross-sectional area in the absorber.
Analysis of the equation 1: The 1st
term is relaxation from the state 3. As the time A
rise, less electrons move to state 1 and the absorber can become fast saturated (occurs when the
pulse duration is smaller than the relaxation time, p At  ). This happens because when all the
electrons “stored” in state 1 have been depleted, any absorption will be proportional to the
steady flow rate 3 AN  from state 3 and the absorber becomes saturated. The time needed from
the absorber to become saturated, is not solely depended from the relaxation time A . Observe
that also for large   2
| |a t values, the rate 1N t  drops due to the increase of the 2nd
term in
equation 1. Therefore, because the peak of the optical pulse is the point with the largest   2
| |a t ,
the corresponding loss for that ‘t’ is minimum and the tails of the pulse (where the absorber is not
yet saturated), will have a bigger absorbing factor.
Simplification of the equation 1: Using eq. (2) we can re-write the eq.(1) as:
  2
1 1
1
| |A
A A A
a tN N N
N
t P 
 
 

(3)
Where:
0 A
A
A
A
P

 
 (4)
The equation 4 define the saturation power (the power needed so that the absorber introduces a
steady loss to the pulse). Setting   2
| |AP a t , the equation 3 becomes:

30. Page | 30
 
 
2
1 1 1 1 1 1
12
| |
| |
A A A
A A A A A
a tN N N N N N N N N
N
t a t t t    
    
      
  
Which is a simple integral over the pulse duration t and defines that after the absorber has
been saturated, the loss is no longer dynamic, but linear and inversely proportional to the
relaxation time a
The solution of the equation 3 is of the form  1N t . This mean that the available number of
electrons is different for every t, and hence the loss vary across the duration of the pulse.
Because the rate 1N t  becomes minimum for   2
| |AP a t (saturation point), the loss
   12 al t N t l
 is also minimum for any value   2
| |AP a t . What needs to be considered is
that the time-depended net gain       0Tg t g t l t l   , should be positive around the peak of
the pulse and negative at the tails, so they get suppressed ( 0l is the cavity loss and is assumed to
be zero)
Figure 2.3: the net gain must be positive close to the peak of the pulse and negative
everywhere else.
What happens though is slight more complex. The initial pulse might be a simple noise source so
the shape of the optical pulse will not take its final shape in the first runs (the number of runs
needed for a stable function varies with the cavity configuration). This means that the minimum
loss minl will not coincide (initially) with the (final) pulse peak. Assume that there is a ‘t’ such that
amplifying medium gets saturated   2
| |AP a t . Because the differential equation that describe
the amplifier follow the same principle with the equation 2 there is a time difference
missmatch pulse widthdt t which describes the time difference between the desired saturation
point-pulse peak (time needed until pulse peak: pulse widtht ) and the actual saturation point for
that run ( missmatchdt ). In other words, the saturation may occur before or after the (final) peak of
the pulse. Ideally, in the limit Number of runs  , 0missmatch pulse widthdt t  . This mean that
as the pulse circulates in the ring laser, the saturation point occurs closer and closer to the peak of
the pulse, until they (ideally) coincide. Imagine the gain and loss curves as a “cage” with
dimensions missmatch pulse widthdt t where within this cage the pulse is “allowed” to grow (this time
difference can also introduce a time-jitter of the pulse). As the pulse circulate the ring, these cage
dimensions contract, leaving a smaller positive net gain area, and hence a narrower pulse is
obtained with the minimal jitter.

31. Page | 31
Simulating the response of the saturable absorber: The focus of this section is to show the
depletion of the electrons in energy state 1, for various values of the relaxation coefficient a , pulse power,
pulse width and number of electrons AN . The test pulse followed the same distribution as the simulations
conducted for the saturable amplifier (Gaussian).
Figure 2.4: Simulation Parameters: Relaxation coefficient 14
10 10 seca 
 , pulse power: 1
mWatt (Left Figure), pulse power: 500Watt (Right Figure), pulse half width:
15
300 10 sec
 ,
electrons population 15
70 10 .AN   As you can see in the figures above, in the case of a weak
pulse (1mWatt), the electrons close to the peak of the pulse are fewer than the case of a stronger
pulse where the attenuation is almost even across the duration of the pulse. This ‘behaviour’ allow
weaker pulses to grow faster (left figure)
Figure 2.5: Simulation Parameters: Relaxation coefficient 14
10 10 seca 
 , pulse power: 1
mWatt, pulse half width:
15
30 10 sec
 (Left figure), pulse half width:
12
3 10 sec
 (Right figure),
electrons population 15
70 10 .AN   In this case, because the width of the pulse is very small (30
femtoseconds-left figure) the electron decrease in energy state 1 is small. On the other hand, a 3
picosecond pulse will manage to almost deplete the electron population in the very early stages of

32. Page | 32
the pulse. From this point and until the pulse leave the absorber, any attenuation will be provided
from the electron flow at a rate 1
.a 
Figure 2.6: Simulation Parameters: Relaxation coefficient 13
10 10 seca 
 (Right),
15
10 10 seca 
 (Left), pulse power: 1 mWatt, pulse half width:
15
300 10 sec
 , electrons
population 15
70 10 .AN   In this case we can see how the relaxation time a affects the electron
population in energy state 1: Initially, a number of electrons rise from state 1 and end up in state
3, where they relax for a seconds. If this time is large compared to the pulse width, then no
electrons will complete the energy state cycle more than once. On the other hand, a low a value
will allow
Serious problems may occur for the following parameter combination: Strong pulse, low AN ,
large a and large pulse width. For this combination, a large number of electrons will rise to state
3 and after a seconds this large number will provide a sudden increase of electrons in state 1.
Before this happens, the pulse will pass with the minimum attenuation through the absorber (since
the electrons have risen before, at state 3). When this massive electron number return to state 1,
the pulse will have a strong attenuation until the electrons are depleted again. This procedure may
split the original pulse at two separate pulses (of the same wavelength)
Passive Mode-Locked Fibre Ring Laser Simulations
Simulation Parameters Left Figure Right Figure
Fibre Ring Laser Bandwidth 10 nm 10 nm
Number of electrons in the absorber 15
300 10AN   15
300 10AN  
Number of electrons in the amplifier 15
70 10GN   15
70 10GN  
Relaxation time (Amplifier) 14
65 10 secG 
  14
65 10 secG 
 
Relaxation time (Absorber) 14
10 10 secA 
  14
90 10 secA 
 

33. Page | 33
Figure 2.7: A decreased relaxation coefficient A (left figure) cause the spectrum to widen
due to the compression of the pulse in the time domain
Figure 2.8: As we can see in the above figures, decreasing the relaxation time A (left figure)
cause the more electrons to return to energy state 1, after their initial excitement to higher energy
states. This mean that the attenuation is higher in the tails of the pulse and so due to energy
conservation, this energy will cause the pulse to increase it amplitude.
Figure 2.9: As we can see, a decreased A (left figure), shortened the Full width Half
Maximum by 2.5 times

34. Page | 34
Simulation Parameters
Left Figure Right Figure
Fibre Ring Laser Bandwidth 10 nm 10 nm
Number of electrons in the absorber 15
50 10AN   15
900 10AN  
Number of electrons in the amplifier 15
70 10GN   15
70 10GN  
Relaxation time (Amplifier) 14
65 10 secG 
  14
65 10 secG 
 
Relaxation time (Absorber) 14
10 10 secA 
  14
10 10 secA 
 
Figure 2.10: In this case, the saturable absorber has been replaced with a heavier doped one
(right figure). Because the electron pupulation is larger, the saturation occur closer to the peak of
the pulse and hence the tails are heavier attenuated. This has as a concequence a compression
ratio of 27.77 (check figure 2.12)
Figure 2.11: As we can see, the spectrum is more expended in the doped absorber case (right
figure). This was expected due to the duality between time and spectrum domain- Expansion in
the time domain (left figure) means compression in the spectrum domain.

35. Page | 35
Figure 2.12: As we can see, replacing the absorber with a heavier doped one, except the
increased compression effect, the convergence speed is also enhanced. For example, a light doped
absorber (left figure) the pulse need almost 600 round-trips, where in the doped case the pulse
need about 240 round trips.
Bandwidth Effect
The following simulation intent to show the importance of the Bandwidth of the laser in obtaining
ultra-short pulses. What we expect to see is a compression in the time domain when the
bandwidth is increased. An increased BW, essentially allow more modes to propagate in the fibre
ring laser. The main factors that limit the BW, is the 3db BW of the gain medium, and because our
model is consisted with various fibre types, the BW limit is also limited from the fibre with the
smallest Numerical Aperture since  2V a   is the normalized frequency and the
number of supported modes is given by:   2
0.5 2V g g   . Where g is the profile parameter.
Simulation Parameters Left Figure Right Figure
Fibre Ring Laser Bandwidth 5 nm 60 nm
Number of electrons in the absorber 15
500 10AN   15
500 10AN  
Number of electrons in the amplifier 15
70 10GN   15
70 10GN  
Relaxation time (Amplifier) 14
65 10 secG 
  14
65 10 secG 
 
Relaxation time (Absorber) 14
10 10 secA 
  14
10 10 secA 
 

36. Page | 36
Figure 2.13: As we can see in the above figures, our assumptions have been validated. Because
the number of propagating modes is larger (right figure), and under the assumption that the
performance of mode-locking is good, then the larger number of modes will lock their phases and
through constructive interference, an ultra-short, stronger pulse will e formed.
Figure 2.14: It is very interesting to see that in the case of the increased BW (right figure), the
spectrum of the pulse show that the wavelength peak (1550 nm) is much stronger when the pulse
in time domain is also strong. We must say though that this is a log-scaled figure and every value
is compared with the peak value. This is why the spectrum ‘tails’ appear greener in the right
figure. In linear scale, we would see lower tail-values in the lower BW case.
Figure 2.15: As we can see, a 12x bandwidth increase (5nm to 60nm), the pulse compression
rate is almost 5x (1 pico-second to 200 femto-second)

37. Page | 37
Mode-locking optimization techniques
Several optimizations techniques have been proposed for semiconductor and for fibre mode-
locked lasers.
 The most profound optimization regards the saturable absorber, the key device for
effective mode-locking. What happens is that when we force the ring laser to produce
pulses at a higher rate, there is a possibility that the absorbing medium will not manage
to recover completely from the passage of the previous pulse. This occur when the
relaxation time A is larger than the pulse interval
1
pulse AR 
 and so, some electrons
may have left in state 3 instead of having the ideal condition 1 total electronsN N at the
beginning of each pulse. One way to achieve this condition is after the passage of each
pulse to apply a strong inverse field and force the remaining electrons in state 3 to return
to state 1.
 The use of a modified Density of state (DOS) function offer the following privileges:
o Due to the nature of the modified material to provide wavelength-selective
amplification (variety of energy transitions), a broader emission spectra [16] and
a larger portion of longitudinal modes of the mode-locked pulse will have their
amplitude amplified. So the coherent superposition of these modes will produce a
stronger mode-locked pulse.
o In order to minimize the jitter between the pulses, ideally the saturation
absorption and amplification must ideally occur at the centre of the pulse. The
modified DOS function essentially force the absorption and gain dynamics to be
more “symmetrical” due to the smaller number of stored electrons in each state.
o Smaller amplified spontaneous emission noise (ASE). What the modified DOS
function essentially tries to achieve, is to “distribute” the electrons in each state
(amplifier case), so that every electron has been used for the amplification of the
pulse and the minimum number of electrons return to state 1 increasing the ASE.
o In the case of semiconductor laser devises,
Conclusions/Sum up
We have seen that passive mode-locking is more ‘robust’ to modulating frequency mismatch
(active mode-locking). On the other hand because it is not easy to change the saturable absorber
if we notice that the provided absorption is too weak or to strong (where a simple amplitude
tuning of the RF modulating frequency could easily solve this problem), extra attention must be
given during the design of the passive mode-locked fibre ring laser. Once any optimizations are
done, its performance is exceptionally good and can give ultra-short pulses with minimum jitter.

38. Page | 38
Section 4: Four Wave Mixing
Introduction
There is a class of nonlinear processes referred as parametric processes because they involve
modulation of the medium parameter such as the refractive index. Extensive research have been
conducted because these processes find potential applications such as wavelength conversion
[18], [19] new frequency generation [20][21], parametric oscillator [22], and quantum information
processing [23], [24].
Origin of Four-Wave Mixing
The response of any dielectric material to light becomes nonlinear for electromagnetic fields of
high intensity and optical fibres is no exception. On an atomic level, the origin of this nonlinear
response is related from the an-harmonic motion of bound electrons under the influence of an
applied field [17]. As a result, the total polarization induced by electron dipoles is also non-linear
and is expressed as [30]-[33]:
     
 1 2 3
0 ...P        (1)
Where 0 is the vacuum permittivity and  
 1,2,3,...
j
j  is the jth order susceptibility.
The origin of parametric processes lies in the nonlinear response of bound electrons of a material
to an applied optical field [17].More precisely, the polarization the optical pulse induces to the
medium is not linear in the applied field and hence higher order non-linear terms are generated.
The magnitude of these higher order terms is defined by the non-linear susceptibilities [25]-[29].
In this expansion, the second order susceptibility  2
 which is responsible for non-linear effects
such as second harmonic generation and sum-frequency-generation [31], vanishes for silica
glasses and hence optical fibres do not exhibit second-order non-linear effects. Therefore, the
lowest order non-linearity with the largest energy contribution to the total polarization, is the
third order susceptibility,  3
 .
The third order susceptibility is responsible for phenomena such as four wave-mixing, third
harmonic generation and nonlinear refraction [31]. In order to point out the effects of the third-
order susceptibility, we consider the third-order polarization in Eq. 1 given as:
 3
0NLP    (2)
Where  is the electric field, NLP is the induced nonlinear polarization and 0 is the vacuum
permittivity.
Consider now four optical waves oscillating at frequencies 1 2 3 4, , &    , all linearly polarized
along the axis x. In that case, the total electric field can be written as [17]:
 
4
1
1
ˆ exp . .
2
j j j
j
E x E i k z t c c

     (3)
From the last equation, we have four propagating fields of different frequency and with different
propagation constant /j j jk n c [17].
Substituting Eq. 3 into Eq. and expressing the nonlinear polarization in the same form as Eq.3, we
have [17]:
 
4
1
1
ˆ exp . .
2
NL j j j
j
P x P i k z t c c

     (4)

39. Page | 39
From the above expression, we find that jP  1 4j to is consisted of a large number of terms
involving the products of three electric fields. For example, 4P is expressed as [17]:
 
 
   
3 2 2 2 20
4 4 4 1 2 3 4
*
1 2 3 1 2 3
3
[| | 2 | | | | | |
4
2 exp 2 exp ...]
P E E E E E E
E E E i E E E i



  
   
  
(5)
Where and   are defined as 17:
   
   
1 2 3 4 1 2 3 4
1 2 3 4 1 2 3 4
k k k k z t
k k k k z t
    
    


       
       
(6)
In equation 5, the first four terms which include the field 4E are responsible for the self phase
modulation (SPM) and cross phase modulation (CPM). Our point of interest is on the last terms
which describe the four wave mixing (FWM). Due to phase mismatch, the number of these terms
which are in effect contribute in the parametric coupling is limited and so, the requirement for
significant FWM is referred as phase matching which requires a specific set of contributing
frequencies and refractive indices to occur.
Different Types of Four-Wave Mixing
Observing the equation (5) we can see two types of FWM: The case  which refers to the case
where three photons of different frequencies, transfer their energy to a fourth photon at the
frequency 4 1 2 3      . This case refers to phenomena called “third-harmonic generation”
when 1 2 3    , or “frequency conversion” when 1 2 3    . There is a difficulty though in
satisfying the phase matching condition in optical fibres with high efficiencies and hence the
dominant FWM effect is the case  . In this case, two photons at frequencies 1 2and  are
annihilated with the simultaneous creation of two photons at frequencies 3 4and  , so that
1 2 3 4      respecting the law of energy conservation. The phase-matching condition for
the  case is [17]:
 
3 4 1 2
3 3 4 4 1 1 2 2= / 0
k k k k k
n n n n c   
    
   
(7)
 From equation 7, we observe that when 1 2  the phase matching is satisfied due to
the fact that the  case is demands the equality 1 2 3 4      . This case is similar
to Stimulated Raman Scattering (SRS),which creates two sidebands called “stokes” and
“anti-stokes” waves, placed symmetrically at frequencies 3 4&  with a frequency shift
[17]:
1 3 4 1s        

40. Page | 40
Approximate and Analytic FWM Solution
In order to fully describe the FWM, a numerical approach is needed to solve the following coupled
mode equations [17]:
(8)
(9)
(10)
(11)
Where:
* *
2 2 2 2 1 2
| | | | | | | | | |
i j k l
ijkl
i j k l
F F F F
f
F F F F
 (12)
is the overlap integral.
The above coupled equations include the effects of self-phase modulation, cross-phase
modulation and pump depletion.
We call as pump wave a strong pulse which is launched into the fibre along with a (usually)
weaker signal at frequency 3 . With this technique the weak signal is amplified, a new signal at
4 is instantaneous generated and the pump depletes a portion of its energy respecting the law
of energy conservation. This type of amplification is called “parametric amplification” and is
also used for the amplification of optical signals along a long-distance fibre.
In this thesis, two approaches are used to show the four-wave mixing effect: The first approach
assumes the pump waves are assumed to be so strong compared with the generated waves, so
that the energy conservation has no effect on the pump waves. This is an approach a lot simpler
than the case where the energy conservation is considered. With this simplified approach, the
propagation of every single wavelength must be treated explicitly as it circulated inside the ring
laser.
In order to get a better understanding of the FWM phenomenon, the simple approach is first
presented. In this approach a number of simplifications took place
The overlap integrals ijklf are considered equal for every combination of ijkl [17]:
 1/ , 1,2,3,4ijkl ij efff f A i j   (13)
Where effA is the effective core area. Next, we introduce the nonlinear parameter j using the
definition [17]:

41. Page | 41
 '
2 (14)j j effn cA   
Next, because this approach treat the pump fields orders of magnitude stronger than the
generated sidebands, they acquire only a phase shift occurring as a result of self-phase
modulation SPM and cross-phase modulation CPM [17]:
   
   
1 1 1 2
2 2 2 1
exp 2 (15)
exp 2 (16)
A z P i P P z
A z P i P P z


   
   
Substituting the equations (xxx and xxx) to equations xxx10 and xxx11, we obtain two linear
coupled equations for the pump fields and the generated sidebands [17]:
 
 
*3
1 2 3 1 2 4
*
*4
1 2 4 1 2 3
2 (17)
2 (18)
i
i
dA
i P P A PP e A
dz
dA
i P P A PP e A
dz






  
  
Where:  1 23 (19)k P P z      
To solve these equations, we introduce [17]:
   1 2exp 2 , 3,4 (20)j jB A i P P z j     
Using Eqs. (xxx15)-(xxx19), the side-bands are obtained [17]:
*3
1 2 4
*
4
1 2 3
2 (21)
2 (22)
ikz
ikz
dB
i PP e B
dz
dB
i PP e B
dz




 
Where the net mismatch is given by [17]:
 1 2 (23)k P P    
This simplified approach is valid only when the conversion efficiency of the Four-Wave Mixing
process is relatively small, so that the pump waves remain largely undepleted. In order to
include pump depletion, it is mandatory to solve the complete set of four equations, Eqs. (8)-(11)

42. Page | 42
Mode-Locked Modes
Ideal FWM
Wavelength
Spacing.
The envelope contracts
while it propagates
through the amplifier.
A
B C
Mode-locking Enhanced performance with Four-Wave Mixing
The purpose of this section is to show if Four-Wave mixing can improve the mode-locking (ML)
performance. How do we define the performance of a mode-locked laser?
 It has already been mentioned that a laser can support a number of circulating modes and
through mode-locking techniques, these modes can lock their phase to form a strong,
ultra-short pulse. The main question needs to be answered is, what is the percentage of
these modes that finally achieved to lock their phase with the dominant mode. How this
percentage can be improved, and how the varying dispersion along the fibre ring laser
affect the mode-locking performance, are some questions that need to be answered.
Hypothetically speaking, a mode-locked fibre ring laser (MLFRL) with close to zero dispersion all
along its length, the ML performance will be increased becase the propagation constant will be
the same for each mode.
Figure 3.1: Blue line: a wave packet, red line: the envelope of the wave packet (the envelope
moves at the group velocity)
With the help of figure 3.1 (above) the ML performance can be defined as the ratio of the mode-
locked modes, to the total number of modes that circulate in the Ring laser. Lets see now how this
ratio is affected from the laser absorber and amplifier: As we know, when the pulse expands
propagates through the amplifier and contracts in the absorber. In the frequency domain this
mean that the envelope of the wave packet expands in the absorber and contracts in the
Total Number of Modes in the Ring Laser
The envelope expands while it
propagates through the absorber.

43. Page | 43
amplifier. During this cycle-procedure, the modes that experience the minimum change in their
propagation constant value in reference to the dominant mode, so that they manage to still be
‘members’ of this wave packet, will eventually lock their phases. Imagine the envelope of the
wave packet as a cage that violently expands and contracts. All the modes that manage be still in
the cage after this cage ‘oscillation’, are capable for mode-locking.
How dispersion affect the size of the wave packet:
 When the fibre’s dispersion is low, the propagation constants of the circulating modes are
simmilar with each other. This mean that when the pulse enters either the amplifier, or
the absorber, the group velocity change is less than the case of higher dispersion fibres.
This mean that more modes manage to remain ‘members’ of the wave packet. In the
extreme case of a zero-dispersion (something not possible) amplifying and absorbing
fibre, the wave packet will include all the propagating modes and the mode-locking
performance aproach the 100%.
How the amplifier and the absorber affects the dispersion:
 If we relate the dispersion as the degree of expansion or contraction the pulse experience
as it propagates through the absorber or amplifier, then the dispersion rate can be
increased either by increasing the pump rate, or with a heavier doped fibre, or by using a
heavier doped absorber.
Slow versus Fast convergence speed of the MLFRL, their connection with dispersion and
ultimately, the MLFRL performance.
 It is reasonable to claim that as the number of modes that eventually will lock their phase
with the dominant mode increase, then the convergence time will increase. But to have a
large number of mode-locked modes, then the dispersion difference (amplifier’s
dispersion minus absorber dispersion) must be small (remember that a zero dispersion
will allow the performance to reach 100%). If this dispersion difference increase, then the
wave packet will contract, the laser will converge faster, and the performance will drop.
How the performance of a MLFRL can be increased via Four-wave mixing
 Observe that figure 3.1 includes one big wave packet (peak A) and four other wave
packets (peaks B & C) with lower amplitude. Lets say that these packets include all the
modes that are able to lock their phases with the dominant mode in each wave packet.
o Suppose that the horizontal axis in figure 3.1, is the frequency axis.
 Therefore, the number of modes that circulate in the MLFRL can form groups where each
group include these modes that can lock their individual phases (that’s because each
group has simmilar propagation constants), with the dominant mode in each group.
 The next step is to force each group to lock their phases with the dominant mode of each
group. This is where four wave mixing shows its importance: If we don’t feed the ring
laser with more than one pulse (of different wavelength), then the MLFRL will
automatically force only the modes of the dominant group (group A from figure 3.1) to
mode-lock (otherwise we would have multiwavelength generation).

44. Page | 44
 To support FWM, a non-linear fibre of zero dispercion (for maximum energy transfer)
must be integrated to the ring laser. To initiate the FWM process, two (or more) pulses
must be used as an input source.
 The wavelength spacing between these pulses need to be very carefully chosen for this
reason: The nature of every ring laser (without FWM) is to ‘automatically search’ and
form a group of the modes that can mode-lock (say group A from figure 3.1). Therefore,
any pulse of wavelength diferent than the centre wavelength from each group, is not
optimal and should be avoided. The wavelength spacing must be such that the second
wavelength will coincide with the dominant mode of the adjacent wave packet (group B
or C). This technique will force all the modes from the adjacent group, to mode-lock with
the pump wavelength.
o A methodology that can be followed in the laboratory to find both the dominant
mode and the optimal wavelength spacing could be like this: Set the Laser into
free-oscillating mode (without using a pump pulse) and with a spectrum analyzer
find the peak of the output pulse. Next, drag the cursor until you reach the first
null and measure the wavelength distance from the peak of the pulse. Assuming
that each wave packet is of equaly (frequency) length, then the sub-optimal
wavelength spacing would be twice the distance peak to null.
o Any amplifier of absorber re-configuration might lead to different performance
because the wave packet size might change. This occur because the applied
dispersion is altered with a stronger pump for example (the amplifier’s dispersion
is increased in this case) This dictates the repetition of the above methodology to
find the new wavelength spacing.
 Each time these pulses enter the nonlinear fibre, the FWM process dictates that
additional pulses, equally spaced in the spectral domain, will be generated. These pulses
will be presicely positioned right in the centre of every wave packet (considering an
optimum wavelength spacing). Therefore, these groups will mode-lock with the pulse that
just appeared amongst them, and the performance of the mode-locked fibre ring laser
will maximize.
Four wave mixing considering energy conservation
The four-wave mixing process that has already been presented, does not consider energy
conservation because for every generated wave, the pulses that were involved in the mixing
process must loose a portion of their energy, such that the summation of the ‘substracted’ energy
is equal to the energy of the generated pulse.
The simulation in this case is quite simmilar to the non-conserving case, since every pulse (of
different wavelength) is propagated independedy. The main difference is that three instead of
two pulses are used as a source with a wavelength spacing 1nm (for the reasons presented
above-simulation speed). This approach is sub-optimal because the wavelength spacing must be
different for every mode locked laser configuration.

45. Page | 45
Figure 3.2: As we can see, figure (a) shows a FWM process with two pump waves (no-energy
conserving case) and the frequencies 112 221&f f are the generated pulses of frequency
112 1 1 2f f f f   and 221 2 2 1f f f f   . In figure (b) we see a FWM process with three pump
waves (energy conserving case) and the generated frequencies.
As you might have noticed the energy-transfer processes increase as  3 2
2N N , where N is
the number of pulses. This number rises very fast as we can see from the figure below:
Figure 3.3: Four-wave mixing products for increased number of wavelengths [26],[27]
Consequence of the number of FWM products in the performance of the MLFRL:
 As the number of FWM products rise very fast (new products will be generated
every time these pulses propagate through the zero dispersion fibre of the ring
laser), so is the performance of the MLFRL, because more pulses will appear
inside each wave packet.
The ammount of transferred energy is given from the formula [25]:
     
2
6
2
11114 2 2
1024
exp (24)
eff
ijk i j k
eff
L
P L DX PP P L
n c A

 

  
         
Where: ‘η’ is the FWM efficiency, ‘n’ is the refractive index of the core, ‘λ’ is the wavelength, ‘c’ is
the speed of light in vacuum, ‘D’ is the degenerancy factor whose values equal 1,3 and 6
respectivaly for the cases i j kf f f  , i j kf f f  and i j kf f f  respectively, 1111X is the
third-order nonlinear susceptibility, effL is the effective fiber length given as [25]

46. Page | 46
 1 expeffL aL a     , where α is the fiber attenuation coefficient, effA is the effective area
of the fiber core and , &i j kP P P are the input pump powers.
Figure 3.4: As we can see from the figure above [26], [27] the use of a zero dispersion fibre
will provide the maximum conversion efficiency  5 0.3162db  or 31.62% up to 1nm channel
spacing. This is the value of mixing efficiency  0.3162n  that has been used in the simulations
since the wavelength spacing was 1 nm and the fibre’s dispersion is 0 ps/nm/km
According to the law of energy conservation, the power of the old waves decreases with the
increasing power of the new ones, and the decrements can be obtained as [26],[28]:
     
     
     
(25)
(26)
(27)
i ijk j k i k j k i j
j ijk i k i k j k i j
k ijk i j i k j k i j
P L P L
P L P L
P L P L
       
       
       
    
    
    
Phase matching condition [26]:
 
 
 
3 3 4 4 1 1 2 2
3 3 4 4 1 1 2 2
1 2
0M W NL
M
W
NL
k k k
k n n n n c
k n n n n c
k P P

   
   

      
    
        
  
 Where , &M W NLk k k   denote the phase mismatch induced by material dispersion,
waveguide dispersion and nonlinear effects.
 The second reason for the choise of a zero-dispersion fibre is that the phase matching
condition is simplified due to the equalization of the propagation constants of the
propagating modes and hence 0Wk  . To simplify the phase matching condition even
more, it is assumed that both the amplifier’s and absorber’s dispersion is such that the
phase matching is not possible and hence the FWM process is isolated only in the zero-
dispersion fibre (any energy transfer in the amplifier and absorber is negligible).
 In addition, the nonlinear phase-mismatch can be ignored if the input pump power is not
very large [26]

47. Page | 47
Simulation Methodology
Two pulses of equal amplitude but of different wavelength have been used as pump pulses. The
wavelength spacing was chosen to be 1 nm for two reasons:
a) As the channel spacing becomes smaller, the sampling rate must be increased in order
for these two pulses to be distinguishable in the frequency domain. The minimum
number for 1nm spacing, has been found (through simulations) to be 13
2 samples per
pulse. Increasing the sample number to 14
2 , the spectrum resolution is increased in the
expense of simulation speed.
b) The four wave mixing conversion efficiency is about 100% in a zero-dispersion fibre for
wavelength spacing up to 1nm. This is convenient since 13
2 samples per pulse allow this
resolution with reasonable simulation speed
Because energy conservation wasn’t considered in the simplified approach, the energy transfer
is initiated only if the adjacent pulse power exceeds a certain threshold.
Laser Gain Bandwidth:
 The simulations for the active and passive mode-locked lasers used a four-energy state
system to simplify the simulations. A flaw with this approach is that the electron
population does not follow a uniform distribution across the states (equal amplitude for
every mode) but instead it follow a lorentzian pdf:
Due to the limited gain bandwidth, a simple approach to limit the maximum amplitude of the
generated modes, is to limit the amplifier’s electron population for every mode, regarding the
distance of this mode from the dominant mode (where the dominant mode uses the maximum
number of electrons). This technique is easy to be implemented into the code, since the matrix
which contains the pulses is populated such that the middle element is the dominant mode.
(a) (b)

48. Page | 48
Figure 3.5: Electron population curves, created from a gaussian distribution. In both cases,
the 3db BW allow the propagation of 40 modes. Observe that the middle mode will use the
maximum number of electrons in the amplifier and hence its amplitude saturation level is larger
than any other mode. It is very interesting to mention that the curve shape, can change by
changing the standard deviation parameter. This modification essentially set the 3db BW, since
a low standard deviation, will cause the net gain to be less than zero even at the peak of the
pulse, and hence this pulse will not ‘survive’ because it will be attenuated from the absorber
(case (a)).
Four-Wave Mixing Simulation without considering energy conservation
In this section, we will see the FWM evolution for various configurations of our ring laser model
Zero-dispersion Fibre length Effect
(1a) (1b)
(2a) (2b)
Figure 3.6: As we can see from the above figures, each time the pulses propagate through the
zero-dispersion fibre new wavelengths are generated. The number of generated pulses is
essentially limited from the 3db BW.

49. Page | 49
(1a) (1b)
(2a) (2b)
Figure 3.7: Left figures: Log scaled spectral Intensity, right figures: Linear scaled spectral
Intensity,As you can see, decreasing the length of the zero-dispersion fibre, the transferred energy
and ultimately the number of the generated pulses, is lower.
Laser Gain Bandwidth Effect:
Figure 3.8: Log Scaled spectral Intensity. As you can see, increasing the standard deviation
(right figure) which defines the spread of the gaussian distribution and hence the 3db BW, the
number of propagating modes is increased.

50. Page | 50
Four-Wave Mixing Simulation with energy conservation
In this case the simulation algorithm has considered the energy conservation.
(1a) (1b)
(2a) (2b)
Figure 3.9: In these figures we can see the FWM process regardng energy concervation. As
you may have noticed, the generated pulses in figures 2a & 2b do not have a large difference from
the pump pulses. This happens because: 1) The pump waves loose portion of their power due to
energy conservation and 2) the BW is larger than the simulations with no energy conservation. The
difference between figures 1x) and 2x) is that in the second case the third order nonlinearity is
larger. (10-15
instead of 10-16
). Observe that a high nonlinearity promote a higher energy transfer.

51. Page | 51
Figure 3.10: in this case the 3rd
order nonlinearity was 10-14
. It is interesting to observe that the
power of the different wavelength pulses fluctuate violently as the pulses enter the sero-dispersion
fibre. For example, in the next run, the spectral profile might look very different as the amount of
transferred energy is large for a large nonlinear coefficient. Therefore, a fibre with a high
nonliearity should be avoided because the output will not be stable and problems may arise in
any optical systems that the ring laser feeds
Figure 3.11: This figure shows the temporal intensity evolution when the 3rd
order nonlinearity
is very large (10-13
). The pulse distortion in this figure confirms the claim that a fibre with high
nonlinearity should not be integrated in a fibre ring laser. On the other hand a fibre with
nonlinearity 10-16
, caused every generated pulse to have minimum fluctuations (the figures in this
case are ommited since they are simmilar to the temporal evolution figures, presented in Section
3)
How to overcome the distortion problem caused from a high non-linear fibre: In case
you need to choose a zero dispersion fibre but its nonlinearity is high, then choosing the one with
the larger effective area will lower the nonlinear refractive index and any unwanted phenomena
simmilar to the one in figure xxx, will be less severe. Also by lowering the pump power will help in
this case.

52. Page | 52
Figure 3.12: Flow chart which describes the model used for the non-conserving FWM process.
The model used for the energy conserving case, does not include a threshold but limits the number
of propagating modes respecting the 3db BW due to the long simulation time.
START
FEED THE FIBRE RING LASER WITH TWO
PULSES, WITH WAVELENGTH SPACING 1nm
HAS THE PULSE
POSITIONED AT THE
EDGE OF THE
SPECTRUM REACHED
THE FWM
THRESHOLD?
NO YES
INITIATE THE FWM PROCESS:
 Call the function to solve the coupled
first order differential Equations.
 Add the generated pulses in the
matrix which contains all the
generated pulses (No information
about wavelength is contained in this
matrix since it is known that every
pulse is 1nm away from the adjacent
pulse)
 Since each pulse is treated
independently, the fibre I ring laser is
feeded sequentially from the elements
of the above matrix.
Passive Mode-locked Fibre Ring Laser
 Propagate every from the matrix which
contains the pulses generated through
the FWM process.
 The POSITION of each pulse in the
matrix, defines its wavelength (the
middle element for example, is the
dominant mode). This position defines
the maximum amplitude the pulse can
take in the fibre ring (different number
of electrons for every wavelength –
gain BW simulation)
HAS THE PULSE
POSITIONED AT THE
EDGE OF THE
SPECTRUM REACHED
THE FWM
THRESHOLD?
Optical Coupler
Output Coupled Ratio: 10%
Output
 Spectrum Analyzer
 Possible Source to a single-mode mode-locked fibre ring laser

53. Page | 53
Section 5: Conclusions
After a number of simulations, there comes a point where critical questions such as: ‘Is it better to
choose passive or active mode-locking’, or ‘Does four wave mixing enhance the performance of a
mode-locked fiber ring laser’ are needed to be answered:
The question passive versus active mode locking cannot be answered, since both techniques have
pros and cons and their application use is different.
 Active mode-locking should only be used with the combination of an accurate
synchronization circuit with the minimum possible jitter or drift in its modulating RF
frequency. Any deviation from the required frequency will cause the ring laser to produce
either distorted pulses or nothing at all due to the violation of the positive net gain
requirement. If there such a problem doesn’t exist, then active mode-locking provide the
flexibility to swap from short to ultra-short pulses, with a simple increase of the
modulator;s absorption coefficient. Also, if the pulse rate requirement is high but the
large length of the fibre ring laser, does not permit such high pulse rates to be obtained,
then modulating at a harmonic of the ring fundamental frequency, the pulse rate is
increased proportionally to the order of the used harmonic.
 Passive mode-locking on the other hand is more robust. Synchronization problems
doesn’t exist, any pulse can feed the ring laser and it will automatically drop, or increase,
the power of the pulse, until it ‘match’ the ring’s default pulse amplitude. On the other
hand, passive mode locking may suffer from jitter if the amplifier and absorber are not
optimally configured.
o An optimal configured passive mode-locked fibre ring laser has a narrow
positive net gain region , otherwise the peak of the pulse may occur anywhere
within this region and so it is said that the pulse is jittered.
 Adding a zero dispersion fibre so that four-wave mixing can be supported, can enhance
the mode-locking performance. Great attention should be paid in the wavelength
spacing of the pump pulses which can be different in every Ring laser, as a function of
the amplifier’s and absorber’s strength. Any sub-optimal wavelength spacing value may
lead to lower (in comparison to no-FWM supporting laser, or even worse, the distortion
of the pulses due to spectral smearing). It must be noted though, that the pulse
repetition rate in such a laser is much lower due to the extra distance the pulses need to
travel. On the other hand, lower pulse rate will decrease the spectrum expansion (do not
forger that high pulse rates inevitably cause wider spectrum).
 High-nonlinear fibres should be avoided in a FWM-supporting ring laser due to intense
instabilities of the output pulses. The use of a fibre with lower 3rd
order nonlinearity may
exhibit slower convergence time due to lower energy transfer but the stable pulses
overcome this problem.
 This multi wavelength generation can be used as a source which feeds an array of
passive mode-locked rings, each optimized for a specific wavelength. This technique will
equalize the uneven amplitude, caused from the 3db gain BW and any fluctuation from
the energy transfer. Next modulating the output from each secondary mode-locked ring,
a Wavelength Division Multiplexed System (WDM) can be formed

54. Page | 54
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[1] http://www.rp-photonics.com/mode_locked_lasers.html

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