ModeLocking of Fiber Lasers at High Repetition Rates

1
Mode-Locking of Fiber Lasers at High Repetition
Rates
Nicholas G. Usechak
Supervised by G. P. Agrawal J. D. Zuegel
The Institute of Optics, University of Rochester,
Rochester, New York 14627
and Laboratory for Laser Energetics, Rochester,
New York 14623
Thesis Defense, August 9, 2006
2
Talk Outline
Motivation

• A unique application for a high-repetition rate
  mode-locked Yb laser which served as the
  motivation for this work will be introduced.

Experiment

• Results from an experimental investigation of a
  passively mode-locked laser, focusing on multiple
  pulses simultaneously circulating in a laser
  cavity, will be discussed.
• Experimental findings from a high-harmonic
  actively mode-locked laser will be presented.

Numerical Simulations

• An overview of AM and FM mode locking from noise
  will be given.
• Computer simulations of the actively mode-locked
  laser will be used to predict its performance and
  explain our experimental results.

Analytic Theory

• A new analytic approach was developed to
  analytically investigate mode-locked lasers.

3
Laser-driven Fusion systems require target
preparation, which has a limited efficiency, due
to nonlinear conversion
Split to 15 lines
Split to 60 lines
Frequency conversion
Target chamber
Split to 3 lines
Source
4
A high repetition-rate mode-locked fiber laser
operating at 1053 nm would be useful in fusion
experiments
Continuous main pulse has higher intensity and
lower bandwidth (2-5 Å)
foot composed of picket pulses with Dl 11Å
Picket intensityPower on target
main
foot
t
Dtmain 1-4 ns
Dtfoot 2-5 ns
Dtfoot 3-7 ns
5
A high repetition-rate mode-locked fiber laser
operating at 1053 nm would be useful in fusion
experiments
Continuous main pulse has higher intensity and
lower bandwidth (2-5 Å)
10 GHz pulse train
foot composed of picket pulses with Dl 11Å
Picket intensityPower on target
main
foot
t
Dtmain 1-4 ns
Dtfoot 2-5 ns
Dtfoot 3-7 ns
Since the repetition-rate of most Yb lasers is lt
100 MHz, a mode-locked pulse train 1000X those
currently available is required.
6
Talk Outline
Motivation

• A unique application for a high-repetition rate
  mode-locked Yb laser which served as the
  motivation for this work will be introduced.

Experiment

• Results from an experimental investigation of a
  passively mode-locked laser, focusing on multiple
  pulses simultaneously circulating in a laser
  cavity, will be discussed.
• Experimental findings from a high-harmonic
  actively mode-locked laser will be presented.

Numerical Simulations

• An overview of AM and FM mode locking from noise
  will be given.
• Computer simulations of the actively mode-locked
  laser will be used to predict its performance and
  explain our experimental results.

Analytic Theory

• A new analytic approach was developed to
  analytically investigate mode-locked lasers.

7
To increase the repetition rate of a mode-locked
laser, one can either decrease the round-trip
time or increase the of pulses in the cavity

• Decreasing the cavity round-trip time

• Increasing the number of pulses inside a laser
  cavity

8
To increase the repetition rate of a mode-locked
laser, we can either decrease the round-trip time
or increase the of pulses in the cavity

• Decreasing the cavity round-trip time

Can be achieved by making shorter cavities

• For non-semiconductor lasers, this limits the
  mode-locking mechanisms that can be used (NLPR is
  no longer practical)

• Increasing the number of pulses inside a laser
  cavity

9
To increase the repetition rate of a mode-locked
laser, we can either decrease the round-trip time
or increase the of pulses in the cavity

• Decreasing the cavity round-trip time

Can be achieved by making shorter cavities

• For non-semiconductor lasers, this limits the
  mode-locking mechanisms that can be used (NLPR is
  no longer practical)

• Increasing the number of pulses inside a laser
  cavity

Can be achieved through passive mode locking by
causing the pulses to split

• Done by increasing the pump power or changing the
  soliton quantization conditions

Occurs when a coupled-cavity architecture is
employed

• Unfortunately, this approach requires
  interferometric stabilization of the two
  sub-cavity lengths

Can easily be achieved in actively mode-locked
lasers by exciting the laser with a higher
harmonic (integer multiple) of the cavitys
fundamental repetition rate
10
Using nonlinear polarization rotation, a
mode-locked Erbium-doped fiber ring laser was
constructed

• Pump power ? 70 mW _at_ 976 nm
• Mode-locking threshold ? 40 mW
• Fundamental repetition rate 12.7 MHz

11
By adjusting the polarization controller,
fundamental mode locking can be achieved

• Although autocorrelation was not performed on
  this laser, a 17.3 nm bandwidth indicates 150
  fs pulses.

12
By adjusting the polarization controller, the
mode-locked pulse will bifurcate and the laser
will operate with packets of pulses
The Fundamental pulse train (red) was
superimposed on our results for quick comparison
on the figures in this section

• Pulse packets could be made to remain constant
  through adjustment of the polarization
  controllers.

13
By adjusting the polarization controller, the
mode-locked pulse will bifurcate and the laser
will operate with packets of pulses

• Pulse packets could be made to remain constant
  through adjustment of the polarization
  controllers.

14
By inserting a 5 m section of SMF-28, the
spectrum acquired Kelly sidebands due to the
increase in dispersion map strength

• The spectrum shown here is consistent with a
  soliton, and the expected 490 fs pulse width is
  a reasonable value.

15
By adjusting the polarization controller, the
mode-locked pulse splits and the laser operates
with packets of pulses

• The extra fiber increased the anomalous
  dispersion in the laser cavity changing the
  soliton quantization conditions. As a result,
  the packets now consist of more pulses than
  before.

16
Both harmonic and sub-harmonic mode locking can
also be achieved using this laser

• By careful adjustment, the repetition rate of the
  laser can either be increased or even decreased.
• Although interesting, none of these regimes of
  operation are particularly attractive from a
  stability standpoint.

17
An FM fiber laser, mode locked at 10.3 GHz, was
experimentally demonstrated using a bulk phase
modulator

• Pump power 150 mW
• Optical Optical conversion efficiency 25
  (30 total)
• Slope efficiency 32 (40 total)
• Mode-locking threshold 30 mW
• Fundamental repetition rate 37 MHz (280 pulses
  _at_ 10 GHz)

Fiber Laser N. Usechak, et. al., Opt. Lett.
29, 1360 (2004).
FM Modulator J. D. Zuegel and D. W.
Jacobs-Perkins, App. Opt. 43, 1946-1950 (2004).
18
This was the first FM mode-locked fiber laser to
report a hyperbolic secant optical spectrum, and
it was tunable over 60 nm
19
This was the first FM mode-locked fiber laser to
report a hyperbolic secant optical spectrum, and
it was tunable over 60 nm
Tunable over 58 nm
20
Autocorrelation results indicate a 0.43
time-bandwidth product
Pulse characteristics Pulse energy 4 pJ Peak
power ? 2 W
Autocorrelation results Pulse width 2
ps (Transform limit 1.5 ps) Time-bandwidth
product 0.43
N. Usechak, et. al., Opt. Lett. 29, 1360 (2004).
21
Analysis of the microwave pulse-train power
spectrum yields sub-picosecond timing jitter
performance
gt72 dB

• Side-mode suppression gt 72 dB
• RMS jitter (10 Hz 12 kHz) 383 fs (total)
  282 fs (signal generator)
• RMS energy fluctuations 17 fJ ? 0.85

D. von der Linde, Appl. Phys. B, 39 201 (1986).
N. Usechak, et. al., Opt. Lett. 29, 1360 (2004).
22
A review of our experimental findings indicates
that the FM mode-locked fiber laser is a
candidate for the LLE

• Although high-repetition rates may be achieved
  passively, through pulse splitting, they
  generally result in unreliable pulse trains
  subject to large amplitude fluctuations and
  timing jitter.
• High-repetition rates may be reliably achieved
  through harmonic mode locking using FM modulation
  with good jitter and amplitude performance.
  Moreover, they can be easily incorporated into
  turnkey systems.
• The FM laser demonstrated in this work
  essentially meet all of the LLE specs.
• The FM laser demonstrated here is still the
  highest repetition rate Yb fiber laser to-date.

23
Talk Outline
Motivation

• A unique application for a high-repetition rate
  mode-locked Yb laser which served as the
  motivation for this work will be introduced.

Experiment

• Results from an experimental investigation of a
  passively mode-locked laser, focusing on multiple
  pulses simultaneously circulating in a laser
  cavity, will be discussed.
• Experimental findings from a high-harmonic
  actively mode-locked laser will be presented.

Numerical Simulations

• An overview of AM and FM mode locking from noise
  will be given.
• Computer simulations of the actively mode-locked
  laser will be used to predict its performance and
  explain our experimental results.

Analytic Theory

• A new analytic approach was developed to
  analytically investigate mode-locked lasers.

24
Mode-locking theory predicts that FM mode-locked
lasers produce chirped Gaussian pulses this is
at odds with our results
25
An AM mode-locked fiber laser can be constructed
by inserting an amplitude modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

26
An AM mode-locked fiber laser can be constructed
by inserting an amplitude modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

27
An AM mode-locked fiber laser can be constructed
by inserting an amplitude modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

28
An AM mode-locked fiber laser can be constructed
by inserting an amplitude modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

29
An AM mode-locked fiber laser can be constructed
by inserting an amplitude modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

30
An AM mode-locked fiber laser can be constructed
by inserting an amplitude modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

31
An AM mode-locked fiber laser can be constructed
by inserting an amplitude modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

32
An AM mode-locked fiber laser can be constructed
by inserting an amplitude modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

33
An FM mode-locked fiber laser can be constructed
by inserting a frequency modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

34
An FM mode-locked fiber laser can be constructed
by inserting a frequency modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

35
An FM mode-locked fiber laser can be constructed
by inserting a frequency modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

36
An FM mode-locked fiber laser can be constructed
by inserting a frequency modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

37
An FM mode-locked fiber laser can be constructed
by inserting a frequency modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

38
An FM mode-locked fiber laser can be constructed
by inserting a frequency modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

39
An FM mode-locked fiber laser can be constructed
by inserting a frequency modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

40
An FM mode-locked fiber laser can be constructed
by inserting a frequency modulator in the basic
cw laser
Experimental Setup
Initiation from noise
Physical Mechanism

• The modulator should be driven at a frequency
  identical to that of the cavity or at harmonic of
  the cavitys fundamental frequency.
• Produces 100 ps 500 fs pulses

41
An extensive model considered the effect of
propagation on a slowly varying envelope as it
passed through each cavity element
2nd and 3rd order fiber dispersion
Self-phase modulation
loss
Finite-bandwidth saturable gain (in passive fiber
g0 0)
propagation
Loss
Loss
Loss
N. Usechak, et. al., IEEE JQE 41, 753 (2005).
42
This detailed model requires many parameters
which can not simply be looked up in the
literature or data sheets
Active Fiber Parameters

• second-order dispersion
• nonlinearity
• Gain Saturation

• third-order dispersion
• finite gain bandwidth
• small-signal gain

Passive Fiber Parameters

• second-order dispersion
• nonlinearity

• third-order dispersion

Grating Parameters

• second-order dispersion

• third-order dispersion

Bulk Parameters

• loss

• polarization issues?

Modulator Parameters

• modulation depth

43
This detailed model requires many parameters
which cannot simply be looked up in the
literature or data sheets
Active Fiber Parameters

• second-order dispersion
• nonlinearity
• Gain Saturation

• third-order dispersion
• finite gain bandwidth
• small-signal gain

Passive Fiber Parameters

• second-order dispersion
• nonlinearity

• third-order dispersion

Grating Parameters

• second-order dispersion

• third-order dispersion

Bulk Parameters

• loss

• polarization issues?

Modulator Parameters

• modulation depth

14 different parameter values are required!
44
Total cavity dispersion was measured by mode
locking the laser with a SESAM and using an in
situ technique
By inserting a razorblade in front of HR2, we can
adjust the carrier wavelength.
A change in carrier frequency changes the lasers
repetition rate through dispersion. Therefore,
we can determine the dispersion of the laser
cavity by monitoring these two frequencies.
W. H. Knox, Opt. Lett. 17 514 (1992).
N. Usechak, et. al., IEEE JQE 41, 753 (2005).
45
Fiber dispersion can also be measured using
white-light interferometry
When interfered with itself, an incoherent
broad-band source will produce fringes determined
by the (wavelength dependant) propagation
constant.
46
The modulation depth of a modulator driven gt 5
GHz may be determined using a cw narrow-band
source and an OSA
FM modulation creates multiple sidebands at
locations related to the driving frequency and
with strengths related to the modulation depth.
N. Usechak, et. al., IEEE JQE 41, 753 (2005).
47
Gain is perhaps the most difficult parameter to
determine however, it can be measured by using a
gain fiber in an amplifier
This figure indicates a small-signal gain of
3.3 1/m and a gain-bandwidth of 25 nm.
N. Usechak, et. al., IEEE JQE 41, 753 (2005).
48
The computer model predicts hyperbolic-secant
pulses with a temporal FWHM of 1.5 ps and a
spectral FWHM of 0.8 nm
The numerical results are in excellent agreement
with our experimental findings, which allows us
to explore our laser numerically.
N. Usechak, et. al., IEEE JQE 41, 753 (2005).
49
By shifting the phase of the driving electronics
by a half-cycle, we can investigate the stability
of the mode-locked pulses
In this case, the pulse switching is initiated
via the wings of the secondary pulse. However,
this type of switching can also be seeded through
noise.
N. Usechak, et. al., IEEE JQE 41, 753 (2005).
50
By viewing the phase shifts effect on the
temporal and spectral widths, two distinct states
(and a selection mechanism) are found
This figure reveals the laser mechanism that
favors the dominant pulsespectral filtering.
Note the dominant pulse spectrum is 1/4 that of
the secondary pulse spectrum.
N. Usechak, et. al., IEEE JQE 41, 753 (2005).
51
If the pulse energy is increased, the pulse can
survive the phase shift and will relocate under
the influence of TOD
Here, pulse switching is initiated via TOD. The
pulse remains intact while it resynchronizes with
the modulator extrema satisfying ?FM?2 lt 0.
N. Usechak, et. al., IEEE JQE 41, 753 (2005).
52
These results allowed us to observe two different
pulse-shifting mechanisms in FM mode locked
lasers
N. Usechak, et. al., IEEE JQE 41, 753 (2005).
53
Using our computer model, we can also investigate
the effect of repetition-rate detuning between
the cavity and the modulator
This figure shows the effect of a small
pulse-modulator detuning on a mode-locked pulse.
In this figure, the laser remains mode locked,
although the carrier frequency of the pulse
shifts.
54
Large values of detuning prohibit stable pulse
formation and result in FM oscillation or chaotic
behavior in the time domain
This figure shows the effect of a 94 kHz
detuning on our laser and the results predicted
by our numerical simulations are in agreement.
55
Our numerical simulations indicated that the FM
instability was not an issue in this laser

• Using the experimentally measured values computer
  simulations of the FM mode-locked laser agree
  with our experimental findings.
• The FM instability was investigated and we found
  that it is suppressed in the presence of residual
  second-order dispersion.
• The role of third-order dispersion and
  nonlinearity on pulse instability was
  investigated.
• The effect of both small and larger repetition
  rate detunings was investigated.

56
Talk Outline
Motivation

• A unique application for a high-repetition rate
  mode-locked Yb laser which served as the
  motivation for this work will be introduced.

Experiment

• Results from an experimental investigation of a
  passively mode-locked laser, focusing on multiple
  pulses simultaneously circulating in a laser
  cavity, will be discussed.
• Experimental findings from a high-harmonic
  actively mode-locked laser will be presented.

Numerical Simulations

• An overview of AM and FM mode locking from noise
  will be given.
• Computer simulations of the actively mode-locked
  laser will be used to predict its performance and
  explain our experimental results.

Analytic Theory

• A new analytic approach was developed to
  analytically investigate mode-locked lasers.

57
Analytic theories are not able to accurately
predict the pulse parameters from actively
mode-locked lasers
By comparing this approach with full simulations,
good agreement is found in both dispersion
regimes. The results from prior theories are
shown for comparison.
Anomalous dispersion regime
Normal dispersion regime
58
We developed a quasi-analytic approach for
investigating pulses in mode-locked lasers by
introducing rate equations

• Mode-locked lasers are time consuming to model -
  parametric studies are not convenient.
• By applying the moment method1 to the master
  equation of mode locking, one obtains a system of
  coupled rate equations for the mode-locked pulse
  parameters.
• Using these equations the pulse energy, timing,
  frequency shift, chirp, and width may all be
  investigated.
• This approach allows accurate parametric studies
  to be performed over large parameter spaces in
  fractions of a second.
• This approach also allows us to visualize pulse
  stability in FM mode-locked fiber lasers.

1S.N.Vlasov, et al., Radiophys.Quantum Electron.
14 1062 (1971).
59
All numerical simulations and analytic
investigations in this work are based on the
master equation of mode-locking
2nd order dispersion and gain filtering
3rd order dispersion
SPM
propagation
Loss and saturated gain
FM Mode-Locker
Where
and
The Master equation of mode-locking is written in
terms of two different time scales one with
respect to the local pulse time t and another
with respect to the round trip time T. The
parameters are also averaged over the cavity.
60
The moment method allows a pulses moments to be
monitored as it propagates under the influence
of a governing equation
We focus on pulse energy, timing, frequency
shift, chirp, and width
Energy
Timing
Frequency shift
Chirp
Width
Autosoliton
Gaussian
61
The moments are differentiated with respect to
the propagation variable, T, to give the rate
equations for FM mode locking
In these equations constants Cn 1 and the
functions ?n depend on the pulse shape assumed
when applying the moment method
62
Solving this system of coupled equations using
realistic parameters for a FM mode-locked fiber
laser gives
Anomalous dispersion regime
Normal dispersion regime
This figure shows pulses converge faster in the
anomalous dispersion regime. In the normal
dispersion regime the modulator pulse forming
effect must overcome both dispersion and
nonlinearity.
63
The real power of this method, however, is its
ability to accurately predict the steady-state
pulse parameters
By comparing this approach with full simulations,
good agreement is found in both dispersion
regimes. The results from prior theories are
shown for comparison.
Anomalous dispersion regime
Normal dispersion regime
Our work reproduces the other results shown above
in the appropriate limits.
64
Parametric studies may be performed very quickly
(ltlt 1 s)while yielding accurate values for the
steady-state pulses
For example, the effect of varying the residual
cavity dispersion on the pulse width, chirp,
timing, and frequency shift is easily
investigated.
Anomalous dispersion regime
Normal dispersion regime
65
Of course parametric studies may be performed
with multiple parameters, this is not feasible to
do with full simulations
The black spot in the center of this movie
represents the parameters used to perform all of
the prior results in this talk.
Pulse Width
Anomalous dispersion regime
66
We have developed a quasi-analytic approach for
investigating pulses in mode-locked lasers by
introducing rate equations

• We have also applied this treatment to AM mode
  locking and passive mode locking with similar
  results.
• By applying the moment method1 to the master
  equation of mode locking, one obtains a system of
  coupled rate equations for the mode-locked pulse
  parameters.
• Using these equations the pulse energy, timing,
  frequency shift, chirp, and width may all be
  investigated.
• This approach allows accurate parametric studies
  to be performed over large parameter spaces in
  fractions of a second.
• The rate-equation approach represents a general
  treatment of this problem its results collapse
  to those of previous theories.

67
I would like to acknowledge the following people
for their support and/or assistance during my
time as a graduate student

• My Advisors

Professor Agrawal, Jonathan Zuegel, and the
Horton fellowship

• Committee Members

Govind Agrawal, Nicholas Bigelow, Wayne Knox,
Chunlie Guo, Jonathan Zuegel

• Group Members

Qiang Lin, Fatih Yaman, Jayanthi Santhanam,
Lianghong Yin, Weihua Guan

• LLE Scientists and Machinists

John Marciante, Jake Bromage, Wade Bittle, Robert
Keck, Vincent Bagnoud, Ildar Begishev, Christope
Dorrer, Joseph Henderson, and Richard Fellows

• Optics Students (in no particular order)

John Heebner, Vincent Wong, Giovanni Perieda,
Colin OSullivan-Hale, George Gehring, Fei Lu,
Yujun Deng, Rolf Sagar, Greg Brady, Li Ding,
Alberto Marino, Jason Neiser, Alex Radunsky,
Aaron Schweinsburg, Michael Storm, and Petros
Zerom

• Optics Staff (in no particular order)

Per Adamson, Brian McIntyre, Joan Christian,
Betsy Benedict, Gina Kern, Gayle Thompson, and
Noelene Votens
68
Thank You
69
Since closed form solutions do not exist, prior
theories ignored many important effects
Previous analytic efforts investigated AM
mode-locked lasers using

• Taylor expansion of the modulation cycle and ?2
  ?3 ? 02
• Autosoliton theory3 (assumes pulse is fixed by
  cavity elements)
• Soliton perturbation theory4 (seeks to include
  modulator)

These results all fail in the normal dispersion
regime in the presence of nonlinearity.
They also do not provide a good physical picture
for pulse stability under FM Mode-locking.
Using the moment method we gain

• A better understanding of pulse stability in FM
  mode-locked fiber lasers
• The inclusion of a chirp parameter sheds light on
  how to produce nearly transform-limited pulses
• Fast results

2D. Kuizenga and A. Siegman, IEEE J. Quantum
Electron. QE-6, 694 (1970).
3H. Haus and Y. Silberberg, IEEE J. Quantum
Electron. QE-22, 325 (1986).
4F. Kärtner, D. Kopf, and U. Keller, J. Opt. Soc.
Am. B 12, 486 (1995).
70
Parametric studies may be performed with respect
to any laser parameter
This figure shows that the pulse chirp in an FM
mode-locked laser is independent of nonlinearity
Anomalous dispersion regime
Normal dispersion regime
71
By focusing on the pulse timing and
frequency-shift equations, we are able to map out
the stable/unstable operating locations
FM mode-locked lasers are prone to a switching
instability where the pulses may align themselves
with either modulator extrema.
Anomalous dispersion regime