High Repetition Rate Ti:sapphire Modelocked
Laser
by
Oktay Onur Kuzucu
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2003
@
Massachusetts Institute of Technology 2003. All rights reserved.
Author ..
...... / ..
Department of E ctrical
..........................
ering and Computer Science
May 8, 2003
Certified by ......................
Franz X. Kaertner
Associate Professor
Thesis Supervisor
Accepted by .........
Arthur C. Smith
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUL 0 7 2003
LIBRARIES
I.
High Repetition Rate Ti:sapphire Modelocked Laser
by
Oktay Onur Kuzucu
Submitted to the Department of Electrical Engineering and Computer Science
on May 8, 2003, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
Recent advancements in frequency metrology and optical coherence tomography require broadband optical sources with ultra-stable spectra. Maintaining the stability
of the optical comb lines in the frequency domain is possible, if the output spectrum of
the modelocked laser spans a full octave. In this theses, a novel cavity configuration is
proposed, where the traditional asymmetric laser cavity structure is designed for high
repetition rate operation, which permits the generation of a full octave directly out
of a modelocked Ti:Sapphire laser through enhanced Kerr Lens Modelocking (KLM)
and dispersion compensation. Numerical analysis of KLM indicates the importance
of the cavity geometry in laser operation.
The most critical elements in the generation process of a coherent octave spanning spectrum are the strength of KLM and the dispersion compensation in the
cavity. While numerous methods are available for compensating the dispersion, it is
preferable to have fixed optical elements rather than using movable components like
prisms. This is also critical in the sense that the system becomes less vulnerable to
the mechanical vibrations and air turbulence. Therefore, the development of so called
Double-Chirped Mirrors (DCM) has played a ground-breaking role in the quest for
an octave-spanning spectrum out of Ti:sapphire lasers. The properties of DCMs will
also be discussed along with the design of output couplers to utilize the intracavity
spectrum to a full extent for further experiments.
The advantages of a compact cavity are numerous. The frequency comb generation with a smaller cavity structure results in a higher repetition rate. Furthermore,
higher repetition rate provides more power per cavity mode. Also, the setup becomes more stable, which is very important for applications. The thesis concludes
by demonstrating that the developed laser results in an octave spanning spectra such
that the carrier-envelope frequency comb is detected with a large signal to noise ratio.
The current laser can be used as the clockwork for future optical atomic clocks and
high-resolution laser spectroscopy.
Thesis Supervisor: Franz X. Kaertner
Title: Associate Professor
3
4
Acknowledgments
I would like to thank Franz X. Kdrtner, for the support and guidance he provided
during these two years. His motivation for achieving the best accomplishments was
always remarkable and stimulating for his graduate students. I am also grateful to the
group members, Dr. Thomas R. Schibli, Jung-Won Kim, Christian Jirauschek, Felix
J. Grawert, Lia Matos, Alexander W. Killi, Dr. Lingze Duan and Christian Koos.
The discussions and our informal gatherings were always fruitful. Also, I am thankful
to my friends at MIT; Emre Koksal, Baris Dolek, Cagri Etemoglu, Bora "the fridgemeister" Tokyay, Volkan Muslu , Gokhan Dogan, Murat Seyhan and Cagri Savran,
for sharing all the fun and excitement in various occasions. I will not forget my longtime friends in Turkey; Ozgur Onur, Medeni Soysal, Yagiz Yasaroglu, Ekin Dino and
all the rest of Enkare members, the life would not be bearable without them.
Last, but not the least, I gratefully acknowledge all the help and support I received
from my family; Sahika, Sermet, Aysegul and Eren. It would not be possible, without
their never-ending motivation. I am especially thankful to my mother, for her care
and support, even though we were thousands of miles apart.
5
6
Contents
1
2
3
Introduction
17
1.1
High Repetition Rate Laser
. . . . . . . . . . . . . . . . . . . . . . .
18
1.2
Dispersion Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.3
Numerical Simulations
. . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.4
O verview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Kerr-Lens Modelocking in Ti:sapphire lasers
25
2.1
Theoretical Model.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2
Experimental Aspects of KLM . . . . . . . . . . . . . . . . . . . . . .
28
2.2.1
Dispersion Compensation
. . . . . . . . . . . . . . . . . . . .
29
2.2.2
Output Coupler Design . . . . . . . . . . . . . . . . . . . . . .
35
2.2.3
The spatial considerations for KLM . . . . . . . . . . . . . . .
39
Spatial model for analysis of KLM
43
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2
Gaussian Beam Propagation . . . . . . . . . . . . . . . . . . . . . . .
44
3.3
Astigmatism Compensation
. . . . . . . . . . . . . . . . . . . . . . .
48
3.4
Kerr-Lensing Effects
. . . . . . . . . . . . . . . . . . . . . . . . . . .
49
3.4.1
Tangential Plane nonlinear parameter . . . . . . . . . . . . . .
50
3.4.2
Sagittal Plane nonlinear parameter
. . . . . . . . . . . . . . .
51
3.4.3
Further improvements in the model . . . . . . . . . . . . . . .
51
3.5
Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.6
Definition of Figure-of-Merit . . . . . . . . . . . . . . . . . . . . . . .
54
7
4
5
3.6.1
Hard aperture modelocking
. . . . . . . . . . . . . . . . . . .
57
3.6.2
Soft aperture modelocking . . . . . . . . . . . . . . . . . . . .
59
Experimental Results
67
4.1
Pump beam characterization . . . . . . . . . . . . . . . . . . . . . . .
69
4.2
Optimization of the resonator
. . . . . . . . . . . . . . . . . . . . . .
70
4.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Future applications and conclusion
75
5.1
Frequency metrology applications . . . . . . . . . . . . . . . . . . . .
75
5.2
C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
A Kramers-Kr5nig Relations for Reflection and Transmission
81
B Sellmeier Coefficients for Optical Elements
83
8
List of Figures
1-1
The geometry of the alternative ultrafast laser with linear configuration. Dispersion Compensating Mirrors and Barium Fluoride wedges
are shown. This laser generally has a lower repetition rate, ranging
from 70 MHz to 100 MHz. . . . . . . . . . . . . . . . . . . . . . . . .
1-2
20
Standard Bragg-stack high-reflecting output coupler. The plots on the
left hand side are reflectivities on different scales. Group Delay and
group delay dispersion introduced by the coupler are on right hand side. 21
2-1
2-2
Parametric plots for (a)pulsewidth, (b)chirp parameter, (c)net gain,
(d) phase shift per pass.
Different normalized values for SPM and
dispersion are considered.
. . . . . . . . . . . . . . . . . . . . . . . .
The recent history in modelocked lasers in terms of pulse durations
achieved with different gain media . . . . . . . . . . . . . . . . . . . .
2-3
28
30
The comparison of three mirror structures. (a) Standard Bragg Mirror,
(b) Chirped mirror, (c) Double-Chirped Mirror with matching sections
to reduce spurious oscillations in group delay.
2-4
. . . . . . . . . . . . .
31
The pre- and post-pulses after being reflected from DCM pairs acquire
opposite phases, therefore they cancel each other after two bounces
from a pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
32
2-5
The white-light interferometry setup for the measurement of group
delay and group delay dispersion. The phase information gathered by
the photodetector is interfaced to a PC. The HeNe laser is used both for
alignment and triggering purposes. As a white-light source, a standard
tungsten filament lamp is used.
2-6
. . . . . . . . . . . . . . . . . . . . .
33
The reflectivity of the mirror with the pump window as thick solid
line with scale to the left. The group delay design goal for perfect
dispersion compensation of a Ti:sapphire laser is shown as thick dashdotted line with scale to the right. The individual group delay of the
designed mirrors is shown with rippled plots and its average as a solid
line, which is almost identical with the design goal over the 650-1200
nm range.
2-7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
The experimental result and the prediction by Sellmeier's coefficients
are shown and lower curve gives the difference between the experiment
and the predicted value. Once again, the overall agreement is excellent
in the sensitivity range of the Silicon detector used in the experiment.
The discrepancy in the lower wavelengths is caused by the RG 690
filter to eliminate the HeNe laser beam for trigger signal. . . . . . . .
2-8
35
The plots on the left hand side are reflectivities on different scales.
Group Delay and group delay dispersion introduced by the coupler are
on right hand side.
2-9
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
The crossed line indicate the target values for the reflectivity. The upper solid contour is the achieved reflectivity and the lower solid contour
is the transmission for the range of interest 550 nm - 1250 nm
. . . .
38
2-10 The crossed line indicate the target values for the the group delay in
reflection and in transmission. The slanted solid contour is the achieved
group delay in reflection and the flatter contour is the achieved group
delay in transmission for the range of interest 550 nm - 1250 nm . . .
10
39
2-11 The stability diagram of a z-fold resonator. The plot shows the spot
size variation at the focus of the resonator as a function of the curved
mirror separation.
The arm lengths are L1=33.6 cm, L2=62.3 cm.
The radii of curvature of both folding mirrors are 7.5 cm. The center
wavelength is 800nm . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
40
(a) Four-mirror folded- cavity linear resonator with Ti:sapphire crystal
as gain medium. (b) Schematic of beam propagation from input plane
i to output plano o. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-2
The stability curve for the asymmetric resonator with asymmetry param eter,
3-3
45
( = 1.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
The stability curve for the asymmetric resonator with asymmetry parameter, ( = 3. It is seen that, the stability region is narrower, whereas
the focussing is better. The spot size variation with respect to the previous case is 25% smaller.
3-4
. . . . . . . . . . . . . . . . . . . . . . . .
48
Beam Diameter on end mirrors, M1(blue) and M4(violet), and on
curved mirrors, M2(red) and M3(green), for sagittal incidence. Asymmetry parameter, ( = 1.2. The variables are the curved mirror separation and the crystal position. The laser parameters are given as:
L=1.7m,
112
03 = 9.71'.
= 313 4 , lc=7.5mm, Ppeak=200kW,
L is resonator length, l
123
= 106.6mm,
02 =
is the path length of the gain
m edium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-5
58
Beam Diameter on end mirror, M1, for sagittal incidence. Asymmetry
parameter, ( = 1.2 in hard-aperture modelocking. The variables are
the curved mirror separation and the crystal position. . . . . . . . . .
3-6
Asymmetry parameter is changed to (
11
=
59
3 for hard-aperture modelocking 60
3-7
The laser mode (solid lines) and pump beam (dashed lines) spot size
changes inside the crystal by split-step parabolic index profile algorithm, [12]. The intracavity power is 200kW, and the crystal thickness
is 7.5mm. Asymmetry parameter is ( = 3, the plots are given for different overlapping conditions between pump and laser mode; (a) w0 ,L
= wo,L = 18pm ;
3-8
(b)
wo,L =
05WO,L = 9pm ; (c) wo,L = 2 wo,L = 36pm
61
3 parameter for the soft-apertured KLM calculation on sagittal plane.
Asymmetry parameter, ( = 1.2. The variables are the curved mirror
separation and the crystal position. Discrimination between CW and
3-9
pulsed regimes is prominent towards the edge of the stability region.
62
6 parameter when asymmetry parameter is changed to ( = 3. ....
62
3-10 Second order dispersion profile for two cases. Dispersion is given in
f s 2 , wavelength in microns. (a) No prisms, 5 DCM bounces and 2 mm
crystal (b) Prisms, 6 DCM bounces and 2 mm crytal. The dispersion
of air is also taken into account. . . . . . . . . . . . . . . . . . . . . .
64
3-11 The spectrum of the ring resonator in the presence of a -1000mm radius
of curvature convex mirror. The effect of the convex mirror is distorting
the symmetry and enhancing the KLM action in the cavity . . . . . .
65
4-1
The total group delay in the cavity in the presence of all optical elements. 68
4-2
The high-repetition rate laser with DCM pairs and dispersion adjusting
BaF 2 plates and wedges. . . . . . . . . . . . . . . . . . . . . . . . . .
4-3
69
The pump waist measurement as a function of longitudinal displacement from the laser output.
The measured data is then fit to the
Gaussian beam with M 2 parameter 1. The dashed line is for the vertical axis and the solid line is for the horizontal axis.
4-4
. . . . . . . . . .
70
The output spectrum of the high repetition rate Ti:sapphire laser in
logarithmic (red) and linear (blue) scales. The octave spanning spectrum is obtained approximately 20 dB lower than the peak level. . . .
12
72
4-5
The effects of relative pump and laser offset on the differential of the
gain.
The top curve in this figure corresponds to the fully aligned
beam. The lower curves are for an increasing offset of w,/2 and w,
respectively [44] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-1
73
In the top figure, the slipping of the carrier wave is shown. If one
extends the spectral lines towards DC, there is a certain offset which
is not equal to zero but smaller than frep [8].
5-2
76
78 Mhz Ti:sapphire laser generating an octave-wide spectrum. fceo is
measured with an RF analyzer.
5-3
. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
77
(a) Measured output spectrum on a linear and logarithmic scale for 78
MHz repetition rate (dashed line) and for high repetition rate (solid
line). (b) Measured carrier-envelope beat signal from the 78 MHz laser
[48].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
78
14
List of Tables
3.1
The ABCD Matrices for various optical elements in the laser resonator
47
3.2
Optical element parameters for numerical simulations . . . . . . . . .
53
15
16
Chapter 1
Introduction
Over the last decades, the investigation of ultrafast phenomena led to ground-breaking
achievements in various disciplines[2, 3]. Time-resolved spectroscopy, frequency metrology [7], optical coherence tomography (OCT) [18] are among these fields. Instantaneous intensities reaching 10 GW/cm 2 are possible with 5 fs pulse duration, directly
from the laser. Generation of continuum from visible to near-IR wavelengths constitute the recent challenge in ultrafast optics.
The main limitation comes from
the chromatic dispersion induced by the optical elements in the cavity. Methods of
compensation for dispersion and KLM optimization have paved the way leading to
octave-spanning spectra.
Since the spectrum of an ultrafast laser analogously acts as a "ruler" in frequency
domain, it has become more interesting for frequency metrology [7, 8, 9, 10]. It has
been shown that precise measurements of the narrow optical resonances are possible. There had not been a mechanism to utilize these stable transitions as an optical
clock, in other words, it was practically not possible to measure terahertz frequencies
with same precision. However, the development of ultrafast lasers has changed this
picture. The frequency comb generated by a modelocked laser can function as an
optical clockwork with a potential to outperform today's cesium clocks for frequency
standards with the ability to count 10" cycles per second [8].
Mechanical oscilla-
tions, temperature fluctuations, variable humidity in the ambient medium can affect
the stability of the frequency comb structure. In frequency domain, the environmen17
tal perturbations manifest themselves in two free parameters of the frequency comb
structure. These are pulse repetition rate and carrier-envelope slip rate, denoted by
frep
and fceo respectively. In order to make precise measurements with high resolution,
a stabilized spectrum is necessary. The laser dynamics influence these parameters in
a different manner, so it is possible to obtain a quasi-orthogonal control on frep and
fceo. Maintaining long-term carrier-envelope phase coherence plays a crucial role in
construction of an all-optical atomic clock [9, 10]. Through the following sections, a
brief outline of these methods will be introduced.
In this thesis, a novel ultrafast laser configuration will be discussed. The ultimate aim of this experiment is to generate an octave-spanning spectrum from 600
nm to 1200 nm by optimizing the dispersion compensation and the cavity geometry
to enhance Kerr-Lens Modelocking (KLM) which is the passive modelocking technique to provide saturable loss necessary for pulsed operation and currently gives the
shortest pulse durations along with the broadest spectrum from a laser. Theoretical
justification of the model will be presented together with the numerical simulations.
Finally, the construction of the laser will be discussed and experimental results will
be presented.
1.1
High Repetition Rate Laser
Conventional Ti:sapphire cavity designs have a repetition rate between 50 and 100
MHz. Higher repetition rates are more desirable for several applications, because the
comb separation would be larger, so that it would be possible to resolve individual
spectral lines with a wavemeter. Furthermore, a higher repetition rate ensures that
spectral comb lines would have more power individually for the same average output
power. Physical advantages can be understood in a simple manner, a shorter cavity
configuration provides a robust and compact laser.
However the disadvantages arise in terms of high pump power requirement and less
pulse energy. Furthermore, if one includes a prism pair as the dispersion compensating
mechanism, the overall geometry of the cavity will not permit the required short
18
round-trip path.
If the dispersion compensation is maintained by double-chirped mirrors (DCM)
[36, 37], then as long as the positive group delay introduced by Ti:sapphire and
air is compensated, it is possible to alter geometry of the cavity to optimize KLM.
Ring lasers are advantageous in this case as compared to the linear cavities.
For
the dimensions of comparable magnitude, the ring laser will have half of the roundtrip propagation path in comparison to linear cavity. This will effectively double
the repetition rate. As an example, 30 cm cavity length would correspond to 1 GHz
repetition for a ring laser, whereas for this length a linear configuration corresponding
rate would be 500 MHz, because the same path length should be traversed back for
completing a round trip.
We investigate a linear resonator configuration around 150 MHz repetition rate.
This configuration actually constitutes the first step of upscaling of the repetition
rates for laser resonators. The critical aspect of this experiment is to test the strength
of KLM for a lower pulse energy. With increasing repetition rate, the pulse energy
decreases and this degradation can limit the spectral broadening, because KLM action
becomes weaker. If significant degradation occurs in the broadening of the spectrum,
that would in turn, determine the upper limit in the cavity length for the ultrafast
lasers in the absence of external broadening. For external broadening, one should
bear in mind that using a microstructure fiber can broaden a narrow spectral output
beyond an octave, however this has issues with stability and reproducibility along
with the deterioration of the spectrum [10].
Furthermore, an analysis on the optimization of the cavity geometry will be presented. This issue is critical, because the layout of the linear resonator along with
issues on cavity misalignment can determine the strength of KLM directly. A figureof-merit will be introduced for characterizing the strength of KLM. This parameter
will allow the experimenter to decide on the cavity asymmetry.
The configuration shown in Figure 1-1 is intended for repetition rates lower than
100 MHz. The pulse energies will be higher in this case, if the average output power
in both situations are equal. The longer cavity length requires better alignment.
19
1mm BaF2
Laser crysal:
OC 1
2mm Ti:AI203
BaF2 - wedges
Base Length = 30cm for 82 MHz Laser
Figure 1-1: The geometry of the alternative ultrafast laser with linear configuration.
Dispersion Compensating Mirrors and Barium Fluoride wedges are shown. This laser
generally has a lower repetition rate, ranging from 70 MHz to 100 MHz.
Another exceptional situation occurring in this case is balancing of the dispersion
on two sides of the crystal. One can refer to literature about the investigation of
dispersion balancing [61.
The results show that shortest pulses produced by the
cavity are maintained through dispersion balanced scheme. Intuitively, this is easy
to understand, if one can imagine that different frequency components entering the
gain medium as a tight lump or loose one. The Kerr-Lens effect, which is responsible
for the saturable gain will not favor the delayed frequency components and only the
frequencies which are close to the peak intensity point in the pulse will benefit from
the gain, which is also analogous to a shutter action. Therefore, the best approach
would be trying to keep the temporal symmetry of the pulse throughout one roundtrip.
Thus, the BaF 2 wedges in both arms, will provide this balancing together with the
arm-length asymmetry. Remarks will be made on the arm-length asymmetry issue
in the following sections. The fine tuning in dispersion is again provided by movable
BaF 2 wedges in one of the arms. The cavity has the same number of bounces on
DCMs to flatten average dispersion around 0 fs 2 . The dispersive characteristics of
BaF 2 and the other cavity elements will be presented in the following chapters.
20
MgF2/ZnSe LHLHLHLHLHSub 5 pairs Center 800nm
80
0
.98-
60
.96
-40
020
.94-
0
0.92
500
-20
600
700
800
900 1000 1100
lambda
T r ..-I--------
1200 13 00
Mg F2: 145 nm
700
800
900 1000 1100
lambda
1200 131
700
800
900 1000 1100
lambda
1200
ZnSe: 79.nm
50-
0.8
0
0.6
'04
-50
0.2
50
600
700
800
900 1000 1100
lambda
_1001
1200 1300
500
1. 1
600
300
Figure 1-2: Standard Bragg-stack high-reflecting output coupler. The plots on the
left hand side are reflectivities on different scales. Group Delay and group delay
dispersion introduced by the coupler are on right hand side.
The output is transmitted through a few percent output couplers. In conventional
ultrafast lasers, the output coupler structure is a standard high-reflecting Bragg-stack,
composed of high and low-index materials.
The bandwidth of the coupler has a
direct relation to the index contrast between the layers constituting the Bragg-stack.
Generally, the group delay dispersion of the coupler is designed to be flat around 0
fs2 . A sample output coupler design can be viewed in Figure 1-2.
1.2
Dispersion Profile
Since, dispersion is the limiting factor for a broadband laser, attention will be paid to
each element's contribution. The dispersive elements in the cavity are the Ti:sapphire
crystal (gain medium), BaF 2 wedges and air. The compensation for the overall group
delay is provided by Double Chirped Mirrors. The group delay of each element has
been found experimentally with a white-light interferometer in order to ensure that
21
numerical values from handbooks are correct.
A direct comparison to the group
delay predicted by Sellmeier coefficients is made especially for BaF 2 wedges because
the manufacturing process could easily alter the material characteristics.
Further
consideration on this issue will be given in the following chapters.
1.3
Numerical Simulations
It's a non-trivial task to obtain an intuitive picture for KLM. One can obtain an analytic formulation only under certain approximations such as paraxial approximation
without any aberration [19],[20]. In subsequent chapters, a brief description of KLM
will be given. Numerical study of KLM gives a quantitative treatment of the strength
of modelocking after the definition of a related figure-of-merit. The code numerically
iterates the q-parameter inside the resonator until it reaches a self-consistent solution. This computation can be time consuming, because the real steady-state which
is reached within milliseconds in a laser, but in a computer this can take much longer
time. The main reason behind this is that the self-consistent solution is reached after thousands of round trips and this computation is carried out for many different
conditions.
1.4
Overview
The gain medium used in the cavity exploited a nonlinear phenomenon known as selffocusing effect. This is due to the refractive index being a function of the intensity.
Thus, the Gaussian beam is subject to a nonhomogeneous refractive index, as it
propagates through the gain medium. If n 2 , the nonlinear coefficient of the refractive
index is positive and large enough to make significant changes in n, the refraction
will be stronger in the center of the transverse intensity profile. So, the Kerr medium
behaves like a converging lens and focusses the beam just like a lens (Kerr Lens)
[3]. Under certain conditions, a perturbation in the cavity can experience more gain
while suffering less from diffraction losses with self-focusing. Thus, at every following
22
round trip, this growing perturbation is favored and is amplified to a giant pulse
in the cavity while getting shorter in duration. The limitation to this shortening is
imposed by group-delay dispersion.
The pulsed regime of operation of this setup is named as Kerr-Lens Modelocking,
whose aspects are investigated widely in last decades. Modes are locked without any
need for an external modulation (active locking) or for a saturable absorbing medium
(passive locking). A good understanding of the KLM operation of the practical laser is
essential for the intended experiments and applications, since KLM has the capability
of generating the shortest pulses out of a laser cavity.
In the numerical simulations, the self-focussing effect for a standard z-folded cavity
is investigated with Brewster-cut Ti:sapphire crystal as a gain medium. A numerical
code, which utilizes the ABCD matrix method for Gaussian wave propagation and
the split-step parabolic refractive index model is used to simulate the operation in the
cavity. Another consideration for the resonator is compensation of astigmatic effects
[13, 14], which optimizes the focussing conditions and cavity stability.
With the numerical model [12, 19, 22]we have the capability of comparing different
cavity geometries and locating the best KLM operation regions. The model will be
developed in future, by the inclusion of the gain guiding, gain saturation effects.
These theoretical findings will be verified with the experimental setup, in the high
repetition rate laser in order to gain insight for optimum cavity geometry.
23
24
Chapter 2
Kerr-Lens Modelocking in
Ti:sapphire lasers
2.1
Theoretical Model
The artificial saturable absorber action necessary for the passive modelocking can be
generated with an isotropic Kerr-medium. Previous ultrashort pulse generation was
accomplished by dye lasers until it was realized that Kerr-effect observed in solidstate lasers was causes a significant self phase-modulation. The intensity dependent
focussing by Kerr-effect causes an artificial fast saturable absorber action by imposing
an intensity dependent phase profile upon the beam propagating through the solidstate gain medium. The discovery of this mechanism dates back to 1991 and this
method of generation of ultrashort pulses is called Kerr Lens Modelocking. [28]
The significance of the group velocity dispersion and the self-phase modulation
inside the cavity can be understood through studying the modified master equation
for fast saturable absorber modelocking. [29]
1
8/
T16Ta = (g-f)a +
1
*2
I+jD
2a
+ (- - j)Ia a
(2.1)
In this equation, D represents GVD parameter and the gain filtering with homogeneous broadening is given by (1/Qf 2 )a. The GVD parameter is related to the
25
dispersive medium with propagation constant 3 and length L as D = /Y"L/2. The
SPM coefficient is given by 6 = (27r/A)n 2 L/Aeff, where A is the carrier wavelength,
n 2 is the intensity dependent refractive index and Aeff is the effective mode crosssectional area. The gain parameter is assumed to be constant, since in general, gain
relaxation time is much larger than the pulse duration and the gain saturation energy
is much larger than the pulse energy. The steady-state solution is found as: [30]
-
a(t) = Aosecho-i-3
(2.2)
T
The concept of the offset between carrier and the pulse envelope can be understood
through group velocity dispersion, or in general unequal group and phase velocities.
This additional effect can be introduced in the master equation by considering the
carrier phase slip with respect to envelope in one round-trip, j'Oa. After modifying
the right-hand side of (2.1), one can obtain two complex equations after separating
terms with same dependence. [29]
jb=g (2 + 3jo -
02)
+
/1
jD) (1+ j1)
2
21
+
jD2D
where, W is the pulse energy given by W = 2A 0
(I
2
T.
2
(2.3)
j6)W = 0
(2.4)
One can understand the dynam-
ics of important parameters like pulse duration, chirp parameter, net gain and the
nonlinear phase shift as dispersion and SPM coefficient are varied. All these parameters can be written in terms of D and &, by separating the real and imaginary parts
of (2.3) and (2.4). After straightforward simplifications, the explicit dependence in
terms of the following normalized parameters can be obtained:
Dn= DQf 2 1
n = WQf
2
0
y/6,
6n =
WQf
2
o/6
(2.5)
where ro is the normalizing pulsewidth for 6 = D =0. First, one can express pulsewidth
26
in a simplified fashion:
TO
-=
-(3/2)(s 2 y4
--
sDn) + v/[(3/2)(s27.y
-
sDs)j2 + 2s 2
(2.6)
with,
1 +Dn
s
2
(2.7)
D Yn + 6
Furthermore, the chirp parameter, net gain and the nonlinear phase shift terms
can be more explicitly written as follows, after skipping the algebraic simplification
details.
13 - D
+ 8YnT
1 + Da'2o
(g
-
2
2)Q0 2 = (/32 _
1 + 2D,3)(T02/T 2 )
+ D (1-23 2 )
=
Qf2TO
2
(2.8)
(2.9)
(2.10)
T
where, Eq. (2.10) indicates the phase change per round trip. The finite value of
Vb induces a change in phase velocity of the pulse. For few cycle pulses, the phase
between carrier and envelope becomes important.
The following figure depicts the relative change of all four parameters with respect
to the normalized dispersion and SPM coefficient. It is possible to make a comparison
to the case when SPM and dispersive effects are not present. All parametric plots
picture the characteristics of the system for vanishing 6n and Dn.Thus one can make
an important observation from the plots that in the presence of self-phase modulation,
slight amount of negative dispersion reduces the pulsewidth. Whereas, with positive
dispersion, the pulse always broadens. On the other hand, for the excess values of the
negative dispersion, the pulsewidth converges to the linearized variation expressed by
T
=
(41D)/(3W). At this range the solution represents the fundamental soliton from
the solution of Nonlinear Schrbdinger Equation [31]. This case also overlaps with the
chirp-free hyperbolic secant solution, when the normalized SPM coefficient (positive)
27
and the normalized dispersion (negative) have the same magnitude. [291
a(T, t) = Aosech
k
e-
(2.11)
o2T/2TR
D1--+
8,
8
=0
-2
Sn=0
-3
Sn =2
=2
)n
0
4
(b)
(a)
=0
-.
8-4
=
2
2
t
,4gif
f 0
,=
Dn
2
5-44
n6,0
-
(d)
(C)
Figure 2-1: Parametric plots for (a)pulsewidth, (b)chirp parameter, (c)net gain, (d)
phase shift per pass. Different normalized values for SPM and dispersion are considered.
Finally, one should note that the net gain parameter also determines the stability
condition. For a stable solution it is required that
S
=
1-
2 -
20D, > 0
(2.12)
This is why, negative dispersion is required in the presence of strong self-phase modulation to stabilize the pulse. This can be seen in Figure (2.1) part (c).
2.2
Experimental Aspects of KLM
The self-locking of modes was first observed accidentally in a laser whose amplifying
medium consisted of a Ti:sapphire crystal pumped by an Ar+ laser, which operated
28
in a CW regime in a cavity without a saturable observer. Scottish scientists [1] then
noticed that the laser went into a pulsed regime when they leaned on the table on
which the laser was mounted.
The recent history in ultrafast optics has reached its climax by the generation of
two and sub-two cycle pulses [32], [33]. Several developments generated this result
such as broadband laser materials and broadband dispersion compensation. Today's
solid-state laser materials suitable for the generation of shortest pulses are Ti:sapphire,
Cr:LiSAF/LiCAF for visible spectra, and Cr:YAG, Cr:forsterite for near-infrared
spectra.
As soon as one obtains the spectral broadening by passive modelocking,
the main limitation comes from the dispersion. A solution to this problem is the
utilization of prism pairs [30], chirped mirrors [34], [35], and double-chirped mirrors
[36]. The latter structure has a potential to support spectra spanning an entire octave
utilizing the broad gain bandwidth of Ti:sapphire. Figure 2.2 depicts the development
of the ultrafast lasers in last decades. Even though the pulse duration has reached
its limits recently, the spectral width obtained has drawn more attention due to its
interesting applications in frequency domain.
2.2.1
Dispersion Compensation
Group delay dispersion inside a laser resonator can be compensated in various ways.
The DCM structures mentioned in the previous section can satisfy the two most
critical conditions, namely broadband reflectivity and broadband negative group delay
which effectively causes blue to travel faster than red. It is useful to understand the
limitations imposed by the standard optical coatings. The well-known Bragg-mirror
reflectivity is determined by the index contrast between high-index and low-index
layers, denoted by
nH
and nL respectively.
r
=
/.Xf
-
Ao
=
nH -
L
L
nH -+nL
H
(2.13)
where, fo is the center frequency, Lf is the high-reflectivity bandwidth. Inserting the refractive indices for Silicon Dioxide and Titanium Dioxide and 800 nm for
29
-11
10
Nd:glass
Nd:YAG
Nd:YLF
S-P Dye
Dye
1-12_
1
-Diode
CW Dye
Color
Center
Cr:YAG
Cr:U*S(C)AF
Er:fibe~r
Cr:forsterite
_
1-13
CPM
Nd:fiber
20 fs
14 fs
-14
10
Compression
Tsapphire
9fs
5fs
1965
I
I
1970
1975
I
I
I
1985
1990
1995
I
1980
I
I
2000 2002
Figure 2-2: The recent history in modelocked lasers in terms of pulse durations
achieved with different gain media
center wavelength yield a reflectivity bandwidth of approximately 180 nm. For this
calculation, the relative bandwidth factor (Af/fo)
=
0.23. This falls short of sup-
porting an octave spectrum. Furthermore, one can see that group delay is only fairly
flat for the central part of the high-reflectivity bandwidth which cannot impose a
counter-balancing action to positive dispersion.
An alternative design is chirping the Bragg-wavelength (AL) for the coating, i.e. AL
increases through the coating to reflect red later than blue. This structure can achieve
larger reflectivity bandwidth along with negative group delay characteristics. However, the limitations come from the matching problem between the incident medium
and between the layers. Therefore large oscillations around an average group delay
are observed. This problem has been solved to extent by the double-chirped mirrors
which introduce a broadband AR-coating to overcome the mismatch between the
ambient medium and the coating, and provide superior impedance matching between
the layers [37]. The layout of these structures are given in Figure 2-3.
30
M
M
(a) Bragg-Mirror:
TiO 2/ SiO 2
-
Si2_
Substrate
(b) Chirped Mirror:
U
-
4
- Layers
E~i
1
Only Bragg-Wavelength, B Chirped
*
Sio2 2-__Dispersion:
u
11
Substrate
]
Negative
xU"
X2 >
2
W
kl
(c) Double-Chirped Mirror: Bragg-Wavelength and Coupling Chirped
d h'; XB/
SiO2-
AR-
Substrate
Coating
4
Impedance-Matching Sections
Figure 2-3: The comparison of three mirror structures. (a) Standard Bragg Mirror,
(b) Chirped mirror, (c) Double-Chirped Mirror with matching sections to reduce
spurious oscillations in group delay.
Although the group delay oscillations in DCMs are much smaller in amplitude
than in chirped mirrors, they still impose a limit on the spectral broadening. The
reason for this limitation is that there are generally multiple bounces in a cavity and
if there is a 1 fs deviation from the average group delay of one mirror, the overall
deviation will be multiplied by the number of bounces. Therefore, in one round-trip
some spectral components may be delayed or advanced further than the main pulse
and this creates so-called pre- and post-pulses. In extreme deviations, this could cause
a spectral filter and inhibit further broadening in frequency domain and shortening
in time domain.
One alternative to overcome this problem is to use the DCMs in pairs. These pairs
are designed to have almost identical broadband reflectivity. However, the group delay
31
Mirror-Pairs Covering One Octave
Si02 Substrate
ARCoating
AR-
SiO -
Subtrateoating
Quarter Wave
Layer
o
U
Figure 2-4: The pre- and post-pulses after being reflected from DCM pairs acquire
opposite phases, therefore they cancel each other after two bounces from a pair.
and group delay dispersion oscillations are complementary. The oscillations around
the average group delay have the opposite sign. This results in the cancellation of
pre- and post pulses which would have the opposite phase terms after a bounce on a
pair, in other words a coherent subtraction between the pre- and post pulses can be
obtained. This situation is shown in the Figure 2-4. The complementary characteristics in group delay are obtained by the insertion of a quarter-wave layer after the
AR-coating. The dispersive characteristics of the mirror pairs are measured by white
light interferometry [38]. The measurement setup involves a standard Michelson interferometer, along with an electronic box which is capable of retrieving the phase
information and transferring to a a PC for Fast-Fourier Transform. Precise group
delay and group delay dispersion information can be gathered from this setup.
Measured group delay shows a good agreement with the design from 600 nm to
1100 nm. After 1100 nm, the sensitivity of the detector deteriorates significantly.
Maximum deviation from the design curve occurs around 900 nm with a magnitude
of 1.5 fs. The consequences of the deviation will be discussed in the following chapters,
regarding the spectral effects.
DCM pairs have already been used successfully [33]. Yet, use of DCM pairs only
32
uonzrouter<Reference
Photo-
detectors(Mtl
HeNe
Lens
Motorized
Translation Stage
Beamsplitter
Lens
Single Mode
Fiber
Dielectric
Mirror
(under test)
Figure 2-5: The white-light interferometry setup for the measurement of group delay
and group delay dispersion. The phase information gathered by the photodetector is
interfaced to a PC. The HeNe laser is used both for alignment and triggering purposes.
As a white-light source, a standard tungsten filament lamp is used.
would require a perfect manufacturing process and a single cavity configuration. The
sensitivity to ambient conditions that affect the overall group delay in the cavity
can change the performance of the laser drastically.
Therefore, it is desirable to
incorporate a fine-tuning element for the dispersion. The prism pairs seem to be the
first solution. However, the positional sensitivity of the prism pairs become more
critical for an octave-spanning spectrum. Since, the prisms also impose a significant
spatial chirp on the pulse. Due to the finite beam size, it's also quite challenging to
provide the negative dispersion uniformly in spatial domain. Thus, a more compact
way of fine-tuning is necessary.
For this experiment, this fine-tuning ability is provided by BaF 2 wedges. BaF 2 has
the lowest ratio between third and second order dispersion between 600 nm and 1200
nm. The processing of BaF 2 is difficult, because, one requires quite thin wedges and
plates in order to have the required dispersion balancing. The material dispersion is
also dependent on the process, thus, with the same white light interferometry tech-
33
-
1.0000
0.9990
0.9980
0.9970
1.0
-
0.8
-
80 a
60
-
S 0.6
S
100
0.4 -
-40
D
0.2
-20
--
0.0
'-0
600
1000
800
Wavelength, nm
1200
Figure 2-6: The reflectivity of the mirror with the pump window as thick solid line
with scale to the left. The group delay design goal for perfect dispersion compensation
of a Ti:sapphire laser is shown as thick dash-dotted line with scale to the right. The
individual group delay of the designed mirrors is shown with rippled plots and its
average as a solid line, which is almost identical with the design goal over the 6501200 nm range.
nique, the BaF 2 samples are measured and compared with the Sellmeier coefficients
[39], also stated in Appendix B. Figure 2-7 shows this comparison. The available
plates for the experiment have the thicknesses: 0.7, 1 and 1.3 mm. The wedges are 1"
long and the thickness at the thin end is 1.65 mm. This thickness linearly increases
towards the other edge by 0.75 . Thus, a continuous tuning range is obtained for the
material insertion from 1.65 mm to 1.98 mm. This insertion range corresponds to a
path length variation of approximately 0.8 mm/wedge.
Using wedges and plates instead of prism pairs makes the resonator quite robust
and stable. It is known from practical experience that, the starting behavior and
long term operation of a prism-less laser is better than the configuration with the
prisms. The BaF 2 wedges provide the ability to maintain a constant laser output
power without any alignment over a long period of time.
34
Experimental
Seilmeren
60
50
40
30
20
10
Difference
0
0.7
0.9
lambda
0.8
1
1.1
Figure 2-7: The experimental result and the prediction by Sellmeier's coefficients are
shown and lower curve gives the difference between the experiment and the predicted
value. Once again, the overall agreement is excellent in the sensitivity range of the
Silicon detector used in the experiment. The discrepancy in the lower wavelengths is
caused by the RG 690 filter to eliminate the HeNe laser beam for trigger signal.
2.2.2
Output Coupler Design
Another element which plays a crucial role in the resonator is the output coupler.
Generally, for a standard laser one uses a high-reflecting Bragg-stack, which provides
a constant output coupling over a 150 nm range and a flat group delay both in
reflection and transmission. However, outside this region the reflectivity sharply rolls
off and the group delay also deviates appreciably. A sample output coupler profile
is shown below. The figure indicates a Bragg-stack made from MgF 2 (n=1.45) and
ZnSe (n=2.35).
For a broadband Ti:sapphire laser, a simple Bragg-stack will not function satisfactorily. The reflectivity profile is narrower as compared to the DCM pairs. Also
the overshoots in the group delay will delay the spectral components far more than
KLM can support, hence it will act as a spectral filter. The most convenient output
coupler that is designed as a Bragg-stack was ZnSe/MgF 2 , however ZnSe additionally
introduces insertion loss, which is estimated as 2 percent. There are also manufac-
35
MgF2/ZnSe LHLHLHLHLHSub 5 pairs Center 800nm
1 fill
80
0.98
60
.2.96
-40
0
(G 20
2b.94
0
0.92 -
-20
500600
700
800
900 1000 1100
lambda
-4U
500
1200 1300
Mg F2: 145 nm
600
700
800
900 1000 1100 1200 1300
lambda
700
800
900 1000 1100 1200
lambda
ZnSe: 79.nm
OU
0.8 F
0.
20.6
-0.
-50
0.2 -
0500
600
700
800
-100
500
900 1000 1100 1200 1300
lambda
600
1300
Figure 2-8: The plots on the left hand side are reflectivities on different scales. Group
Delay and group delay dispersion introduced by the coupler are on right hand side.
36
turing difficulties associated with this material. It is difficult to handle ZnSe with
conventional ion-beam sputtering machines.
Inspired from the DCM structure, one can design a dielectric coating for the
output coupler with a reasonable transmission and group delay characteristics. The
desired features for the coupler can be listed as:
9 2 percent output coupling over a range of 350 nm, centered around 800 nm.
9 Smooth roll-off characteristics towards the band edges, 600 nm - 1200 nm.
* The group delay variation in reflection, which is either constant or can be compensated by the BaF 2 wedges.
9 The group delay variation in transmission should be as flat as possible. The allowed
maximum deviation should be much less than the expected pulse duration.
Thus, an optical coating design software (OptiLayerTM) is used to find a suitable
dielectric design with the specified criteria above. The inherent impedance matching
problem reflects itself once again with this design attempt. Although a smooth reflectivity profile can be obtained without any problem, the group delay variation is
more problematic. After specifying several target designs, it has been observed that
the flat group delay target in reflection over an octave bandwidth, can distort the
reflectivity profile significantly. In order to eliminate this effect, the target for group
delay in reflection has been modified from flat profile to a negative dispersive profile.
Since BaF 2 wedges and plates for fine tuning the dispersion are already available, the
negative group delay introduced by the output coupler can be directly compensated
by additional BaF 2 insertion. The amount of negative dispersion by the output coupler is found by the plate thickness increments. The available plate thicknesses differ
by 0.3 mm, which corresponds to 0.72mm path length in one round trip. The overall
dispersion characteristics can be preserved if a BaF 2 plate is replaced by a thicker
plate and a flat group delay output coupler is replaced by the dielectric coupler which
should compensate for the additional insertion. The following figures, (2-9, 2-10), depict the target points for reflectivity and group delay along with the achieved profile
with the 38-layer SiO 2 and TiO 2 design.
From the figure, one can see that there are similar ripples as seen in DCM design,
37
0
10 - --
RP
- --
--- - - -
- -- - - - - - -
--
- ------ - --
- - ----- --
---
-- --
-
Tp
....
40
0
9Sa
Angle(s):
1100
1200
Th - 3671.9
N - 38
Meuit
1000
Wavelwgth. rvn
aoo. o.O0'
Function - 5952E4d01
Figure 2-9: The crossed line indicate the target values for the reflectivity. The upper
solid contour is the achieved reflectivity and the lower solid contour is the transmission
for the range of interest 550 nm - 1250 nm
around the average group delay target profile. The maximum deviation is ± 4 fs,
but one should bear in mind that there's only single bounce on the coupler, and
the deviations are likely to be cleaned by double passage through Kerr-Lens. The
consequences will be seen in the experimental results. The group delay in transmission
is fairly flat over a range from 650 nm to 1150 nm. Preserving the flatness is important
because if the pulse is to be compressed externally, the minimum pulse duration can be
achieved only if the transmission group delay has a profile which can be compensated
by other elements. Thus, arbitrary group delay variations which cannot be balanced
externally are not allowed. Another interesting point that one should bear in mind
is that the group delay in transmission and reflectivity are connected to each other.
The group delay in transmission and the transmission are connected to each other
via Kramers-Kr6nig relations, see Appendix A. Furthermore, reflectivity is expressed
as r = 1-t.
Thus, the resonance features seen in the reflectivity show up in the
transmission group delay.
38
GD
fn
ODrp.
..............
--------------
40
-------
-------
---------
Xx
.I.
30-
/\!
A
Xx/'I
......
......
..
. ...
X
XXX
20-
'XXXX_:
10-
0-
700
80
900
1000
1100
1200
Wanelength in
Angle(s): O0.00
Th- 3671 .3
Th - 3671.9
N - 38
Meih unctin - 5975241E+001
Figure 2-10: The crossed line indicate the target values for the the group delay in
reflection and in transmission. The slanted solid contour is the achieved group delay
in reflection and the flatter contour is the achieved group delay in transmission for
the range of interest 550 nm - 1250 nm
The design for the broadband output coupler has been manufactured by ion-beam
sputtering technique, which is also utilized for the manufacturing of the DCMs. The
final design had 36 layers with elimination of one pair without any significant variation
from the design target. The coating is deposited on a 1mm thick fused silica substrate,
whose material dispersion can be compensated externally. The substrate also has a
30' wedge angle in order to eliminate etalon effects.
2.2.3
The spatial considerations for KLM
After carefully accounting for the dispersive effects, one should be concerned about
the spatial layout of the resonator. The geometry of the resonator can make a strong
impact on the strength of KLM as it will be elaborated in the next chapter. However,
one of the primary issues while setting the resonator are the stability regions, mode
size change within stable region and the interaction area with the pump beam.
39
The conventional z-fold cavity which was shown before has 2 stability regions.
This results from the Gaussian beam analysis under paraxial approximation which is a
valid approximation for most practical cases. Every resonator geometry can be scaled
down to a 2-mirror resonator configuration
[14, 3]. The resonator mode characteristics
can be investigated with ABCD matrix analysis which will be introduced in the next
chapter. A sample stability diagram is shown below in figure (2-11).
18
16
C/)-Cu
Cu
N
32
C) 72
.47.7..8..2
.4.......
0
7.2
7.4
7.6
7.8
8
8.2
8.4
8.6
Mirror separation (cm)
Figure 2-11: The stability diagram of a z-fold resonator. The plot shows the spot size
variation at the focus of the resonator as a function of the curved mirror separation.
The arm lengths are L1=33.6 cm, L2=62.3 cm. The radii of curvature of both folding
mirrors are 7.5 cm. The center wavelength is 800nm.
The discrimination between (continuous wave) CW and pulsed operation can be
obtained towards the edges of the stability windows. Generally best CW operation is
obtained at the center region of both stability windows, however it is not possible to
see KLM action. In practice, the best operation for KLM is the inner edge of the outer
stability region. Usually, one aligns the laser for the best CW operation in second
stability window and later brings folding mirrors closer. Meanwhile, the crystal should
40
also be translated to be centered around focus of the laser mode. However, for the
optimum KLM case, the focus of the laser mode should not be at the center but on
the crystal facet [18]. The position of the crystal has a direct effect in determining
the favoring of KLM over CW.
41
42
Chapter 3
Spatial model for analysis of KLM
3.1
Introduction
The gain medium used in the cavity exploited a nonlinear phenomenon known as
the self-focusing effect. This is due to the refractive index being a function of the
intensity (Optical Kerr Effect): n
=
no + n 2 1. Thus, the Gaussian beam is subject to
a nonhomogeneous refractive index, as it propagates through the gain medium. If n 2 ,
the nonlinear coefficient of the refractive index is positive and large enough to make
significant changes in n, the refraction will be stronger in the center of the transverse
intensity profile. So, the Kerr medium behaves like a converging lens and focusses
the beam just like a lens (Kerr Lens).
The pulsed regime of operation of this setup is named Kerr-Lens Modelocking,
whose aspects have been investigated widely over the last decade. Modes are locked
without any need for an external modulation (active locking) or for a saturable absorbing medium (passive locking). A good understanding of the operation of this laser
is therefore essential for experiments and applications, since KLM has the capability
of generating the shortest pulses directly from a laser. The popular applications of
KLM lasers include optical coherence tomography(OCT) [4], [5] and frequency metrology [7].
In this chapter, the self-focussing effect for a standard z-folded cavity with Brewstercut Ti:sapphire crystal as a gain medium is studied. A numerical code, which utilizes
43
the ABCD matrix method for Gaussian wave propagation and the split-step parabolic
refractive index model is used to simulate the operation of the cavity. Another consideration for the resonator is compensation of astigmatic effects, which optimizes
the focussing conditions and cavity stability. Finally, a comparison of cavities with
different arm-length ratios and their advantages and disadvantages are discussed.
3.2
Gaussian Beam Propagation
The theory of Gaussian beams and their transformation by optical systems are wellstudied in various sources [2], [3]. The formation of an image by an optical system
is described in ray optics in terms of the transformation of the rays emerging from
a point on the object so as to intersect a point of the image. This description is
consistent with paraxial Fresnel diffraction theory. The beam propagation in a laser
cavity is described in terms of the ABCD matrix concept. Each optical element or free
propagation in the cavity has a corresponding matrix element. The ABCD matrices
for various resonator elements are given in Table 3.1.
The ABCD matrix in Eq. 3.1 relates the transversal position and slope of the
input and output beams, see Figure 3.2. The whole optical system can be described
in terms of cascaded matrices for each element.
Mtota
=
t
(3.1)
M
=t
Ct Dt
Ci
Di
For the resonator, a self-consistent solution is sought. This means that the initial
beam and the beam after one round-trip must have the same q-parameters.
The
q-parameter is a complex quantity which completely describes a Gaussian beam. The
TEMOO mode is characterized by its q-parameter as:
1
q(z)
_
=
-1
I
R(z)
.A A L
irw2(z)
(3.2)
where, AL is the wavelength of operation, R(z) is the radius of curvature and w(z) is
44
M
12
M2
BA
(a)
y3--
M3
4
M4
z
(b)
r7
10
Figure 3-1: (a) Four-mirror folded- cavity linear resonator with Ti:sapphire crystal
as gain medium. (b) Schematic of beam propagation from input plane i to output
plano o.
the transverse beam radius at z.
After passing through an optical element with known ABCD parameters, the qparameter of the input and output beams are related by:
Aqin + B
ot-Cqin + D
=q~
-on
(3.3)
For the self-consistent solution, qin and q%,u are set to be equal and the solution for the
Gaussian Beam parameter and the stability analysis is trivial. However, considering
Kerr-Lensing, the ABCD matrix components themselves become dependent on the
q-parameter. In this case, the calculation is started with an arbitrary q-parameter
and the ray tracing in the resonator is repeated until the relative change of beam
radius on one of the end mirrors,
namely
[Wm+1(Zend)
-
(R(Zend) =
Wm(Zend)/Wm(Zend)
<
45
oc), satisfies the convergence condition,
,
where c is the desired accuracy [12].
Here,
Wm+1(Zend)
is related to
Wm(Zend)
from Eq.(3.3)
1 (Zend)
(rw2)
At(wm)qm(zend) + Bt(Wm)
SZ=
Ct(wn)qm(zend) + Ct(Wm)
(3.4)
On the other hand, the stability of the solution is an important concern. Initially,
one should investigate the stability of the linear cavity where the ABCD parameters
are independent of the applied laser power (P=O). The stability region for the laser
is considered to be independent of the crystal position between the curved mirrors,
for the sake of simplicity, as long as it stays close to the focus. Therefore, we basically limit our calculations to the curved mirror separations, where the discriminant,
[(Dt - At) 2 + 4BtCt]1/ 2 is an imaginary number [2].
The stability curves for two different configurations are shown in Fig.(3-2) and
(3-3). The spot size in the cavity is plotted against the curved mirror separation. In
the first case, the resonator has a certain ratio of arm length , 112 =1.2134; the second
case refers to a different asymmetric configuration, where, 112
0.22
=
3l 3 4 .
109
110
-
0.20.18
E
0.16-
2 0.14
0 0.12
.)
(>
N
0.1
0.0.08
0.06
0.04
0.0f02
103
104
105
106
107
108
Curved Mirror Separation (mm)
Figure 3-2: The stability curve for the asymmetric resonator with asymmetry parameter, ( = 1.2.
46
Table 3.1: The ABCD Matrices for various optical elements in the laser resonator
Optical Element
ABCD Matrix
Free space
-/R
Mirror (Normal Incidence)
(
1
1
Brewster Plate (Normal Incidence)
Brewster Plate (Sagittal Incidence)
(1
Brewster Plate (Tangential Incidence)
(
0)/R
1
(
/no
lB/n)
(1/I5
(-hcs(-yl
Csi(Ys )
-coBy, sin(ylB
)
T
1/no
1B
cos(-)
Kerr-Lens (Normal Incidence)
)
(-nLnB-yt sin(-YtelB)
47
0
1
c
-2/(R cos 0) 1
Mirror (Tilted Incidence, tangential plane)
Kerr-Lens (Tangential Incidence)
1)
(-2 cos
Mirror (Tilted Incidence, sagittal plane)
Kerr-Lens (Sagittal Incidence)
)
1 ) sin(yl)
(f' )os(YlB)
8
y.) sin(-YlB)
(1/
COS(.,1B)
(1/LK)sin(tlB)
COS(-YtlB)
0.18
0.16-
0.14-
E
E
C0.12
C.,
0
00.08
-
W 0.06-
0.04
0.04
T02
104
106
108
110
Curved Mirror Separation (mm)
112
Figure 3-3: The stability curve for the asymmetric resonator with asymmetry parameter, ( = 3. It is seen that, the stability region is narrower, whereas the focussing is
better. The spot size variation with respect to the previous case is 25% smaller.
3.3
Astigmatism Compensation
The astigmatism in the four mirror z-folded cavity is compensated by balancing the
astigmatic distortion of the curved mirrors and the astigmatism introduced by the
Brewster plate. Reducing the astigmatic effects in the cavity leads to the best focussing conditions and cavity stability [13], [14]. The compensation requires that the
focal lengths in the sagittal and tangential planes are identical, as far as two curved
mirrors and a Brewster-cut crystal are concerned. The curved mirror incidence and
the Brewster plate incidence in tangential plane result in the following focal length:
ft=-
R cos 0
l'
+ -
(3.5)
2 L
Same calculation can be carried out for the sagittal plane as well (i.e. sagittal incidence on curved mirror and the Brewster plate [12]).
f
R
-
2cosO
48
+
l
nL
(3.6)
The mirror tilting angle can be found in terms of the crystal length after equating
two focal lengths,
f, =
ft. Notice that to compensate the astigmatism of two mirrors,
we require the real crystal thickness IB to satisfy, IB
6 = 7r - arccos (21B
(n4
[n' -1-
-
=
21B-
2n2 +I 1+ 4R2 n2B/1)])
(3.7)
For a 2 mm crystal, the resulting compensation angle in Eq.(3.7) is 5.5 degrees,
whereas for a 7.5 mm crystal, 9.71 degrees.
3.4
Kerr-Lensing Effects
The optical Kerr Effect describes the intensity dependence of refractive index of the
material.
To observe this phenomenon, high intensities are necessary because the
intensity dependent part of n is rather small.
n = nL + n21
(3.8)
The essential mechanism in KLM is self-focussing, which is due to the Optical Kerr
Effect. If a Gaussian beam with intensity I is incident on a medium with positive
n2, this will essentially cause the medium to act as a lens. The intensity-dependent
transverse index-modulation is called Kerr-lens since the radially varying retardation
of the phase focuses the beam and if this effect can compensate the diffraction spreading of the beam, an inward focussing will be observed. This will, in turn, increase the
intensity at the center of the beam and eventually the focussing will become stronger.
This is called whole-beam self-focussing [3], [17].
To investigate the spatial profile of the refractive index, consider a Gaussian
beam in TEMOO mode. The spatial intensity distribution of the beam can be wellapproximated by the first two terms of its Taylor expansion. This indicates the the
refractive index has a parabolic profile [12], [15], [17]. However, a better approximation can be obtained through minimization of the mean-square error [40], [41]. This
will be presented in the improvements section.
49
The intensity profile and the refractive index variation are approximated to be
parabolic for initial analysis:
I(r) = Ioexp[-2(r/w)2 ] ~ Io[l - 2(r/w)2 ]
n
=
no + n2IO[I
-
(3.9)
2(r/w) 2 ]
Thus, the overall refractive index with the radial dependence can be written as,
n = (nL + n21 0 )
o
r2)
(no + n210~w
(
I -
= h(1 -
1
2
2 2
(3.10)
(3.11)
r )
where the constant part of the refractive index can be expressed simply as,
S= nrL + n2Io
(3.12)
and finally the nonlinear parameter is extracted:
[4n 2IO-
here P
=
1/2 =
[8n2P
1/2
1
(3.13)
-rw 2 Io/2 is the laser power. -y term signifies the strength of nonlinearity.
For zero-power case, it vanishes. This -y value is valid for the normal incidence case,
because the constant portion of the refractive index in sagittal and tangential planes
appear in different powers as indicated in Table 3.1.
3.4.1
Tangential Plane nonlinear parameter
The tangential plane parameters are obtained after making the necessary manipulations on beam radius and the intensity terms
50
[12]. Then, the nonlinear parameter ge,
which is also seen in ABCD matrix elements for Kerr-lens, is written as:
421
1
n 3inB
W
8n P )1/21
W2
n 3 B7
(3.14)
where
hB =nL +
3.4.2
(3.15)
n0
Sagittal Plane nonlinear parameter
The nonlinear parameter for the sagittal plane is found in a similar way:
4n2Io 1/2 1
y=--
nonhB )
=
W
8n2P
13.16
nohBXr
W
(3.16)
Comparing (3.14) and (3.16), one can see the the nonlinearity induced on the
sagittal plane is larger than the one in tangential plane. This is due to the additional
no factor in the denominator of (3.14).
Therefore, it is in general the case that
focussing inside the crystal is stronger in the sagittal plane.
3.4.3
Further improvements in the model
The parabolic approximation for the refractive index variation is not the best approach.
Assuming that, there is a Gaussian elliptic intensity profile, a quadratic
approximation can be made by the minimization of mean-square error [40], [41]. The
intensity dependent part of the refractive index and the quadratic approximation can
be written as:
An = n21 =
An
In Eq.
n2
2P
7rwXWY
-
exp (-2X
T2
I-b - 2
2
22
axwx
2
2
2
)
(3.17)
(3.18)
aw2
(3.18), parameters ax, a. and b are found by the minimization of the
mean-square error by squaring the difference between the Gaussian and quadratic
51
functions, weighting the result by the Gaussian function and then integrating over
the x-y plane [20]. So, the parabolic expansion which is inherently a,=aY=1, becomes
ax=ay=4 and b=3/4. The parabolic approximation is only valid around the beam
center. This gives an effective intracavity power that is too large by a factor of 4.
Thus, the critical power, or the power which a Gaussian beam eventually undergoes
catastrophic self-focusing is rewritten as:
,,it = aPi = a 8
(3.19)
Within the quadratic approximation Pcrit= 4 P1, since a=4. Accurate numerical calculations made without the quadratic approximation show the critical power to be
3.77 P 1 . For Ti:sapphire this value is 1.82 MW.
3.5
Numerical Analysis
As mentioned before, the accurate analysis of the resonator eigenmode requires numerical techniques to model the beam propagation inside the crystal.
There are
several methods proposed for the analytical modelling of the Kerr medium. These
include a simple transform method for the Gaussian beam [18] and the approximate small-signal spot size variation calculation with the aberrationless theory of
self-focussing [19], [20], [21].
However, these methods model the Kerr medium as a thin-layer element and
do not investigate the spot size variation inside the crystal. This information will
be useful for an understanding of the self-focussing mechanism. To investigate this
variation, the Kerr medium is modelled as a sequence of thin layers. Each layer acts
like a Graded-Refractive Index GRIN lens, [2]. Therefore, one needs to calculate the
beam-parameter after each layer, since the next layer's ABCD matrix is dependent
on the input parameters.
The parameters of the laser are given in Table 3.5. These are also valid for the
experimental setup. Therefore, we are able to verify the simulation results experimentally later, and observe KLM behavior.
52
Table 3.2: Optical element parameters for numerical simulations
Asymmetric config. Asymmetric config.
Parameter
((=1.2)
(( = 3)
10cm
Curved Mirror Curvatures
(R 2&R 3 )
Curved Mirror Tilting Angle
(degrees)
Resonator Length / (Rep.
Rate)
5.02
2m / 75MHz
(101.250-103.671)&
(104.591-107.022)
0.861
Stability Range(mm)
Short arm(m)
(101.250-102.991)&
(108.637-110.413)
0.473
1.419
1.036
Long arm(m)
Ti:sapphire crystal parameters
no
1.76 [25]
3.43 x 10-20 [24]
n2
lB
2mm
Varied around 123/2
12k
The numerical procedure, which was written with C++, is iterative in nature.
The eigenmode for the zero-power condition is used as an initial value. The overall q-parameter dependent ABCD matrix is calculated and the next eigenmode is
found from (3.3).
Then, the convergence condition is checked. It's trivial to com-
pare the initial and final spot sizes on one of the end mirrors.
If the desired ac-
curacy is not yet reached, the code is run again and this procedure continues until
[Wm+1(Zend)
-
< e.
Wm(Zend)]/wm(Zend)
code, a damping factor
Also, in order to enhance the speed of the
= 0.01 is applied. Physically, this corresponds to a Gaus-
sian aperture placed in front of an end-mirror, and it introduces some loss for quick
convergence. But the results are not affected considerably. The ABCD matrix for
the Gaussian aperture is given by:
1
0
.(3.20)
Mga =
17~y
53
where
Wga =
For a loss of 1 percent,
wga
W(Zed)
(3.21)
should be approximately 7 times larger than w(z) at that
point. The waist of the aperture is adaptive, it doesn't remained fixed at a certain
value to ensure that same loss is applied to the incident beam at every round-trip.
The intracavity power level used in the simulation is 200kW, which is less than
Pcrit = 420kW (which was obtained from parabolic index approximation, however
a better approximation obtained by mean-square error minimization yields almost 4
times larger critical power of self-focussing). For higher power levels, the beam collapses, because the self-focusing reaches a level where Kerr-Lensing over-compensates
for diffraction. The spatio-temporal approach, which also takes into account the temporal effects, such as dispersion, is then more accurate to use to model the beam
propagation exceeding the critical power.
The simulations in [23] show that, con-
sidering temporal effects eliminates the discrepancy occurring around Pcrit. But the
same reference indicates that for lower power levels, the spatial solution yields similar
results as compared to spatio-temporal approach. So for time efficiency, the ABCD
matrix method is more preferable for KLM analysis.
For the spatial domain on which the calculations will be carried out, previous
stability region information is used for the curved-mirror separation. Simultaneously,
the crystal position within the inner cavity is also varied. Therefore, the eigenmode
analysis is carried out as a function of two parameters: The mirror separation and
the crystal position.
3.6
Definition of Figure-of-Merit
At the final stage, one needs to define a figure-of-merit for comparing different resonator configurations. In this definition, the method of modelocking is also important.
For instance, if the laser is pushed to pulsed regime by hard-aperture modelocking,
the beam size variation at the point where the aperture is placed is important. The
54
main result that one is interested in, is the relative beam narrowing ratio, J. This
parameter is considered as a measure of the KLM performance [22], and is defined
as:
1 w(P =p, Z
P
= Zend) - w(P = 0, z = Zend)
W(P = 0, Z =Zend)
(3.22)
In this equation, p is the intra-cavity power normalized to the critical power of
self-focusing, p
P/Prit.
On the other hand, if the discrimination between CW and pulsed operation is
provided within the gain medium, then the definition of the figure-of-merit should be
altered. A fair comparison between these modes of operation can be made by the gain
change due to self-focusing. This is directly related to the spatial overlap between
the pump beam and the laser mode. Therefore, one needs to compute the overlap
integral inside the crystal.
1G(P=p)-G(P=0)
G(P = 0)
p
Here, several assumptions have to be made regarding the spatial profiles of both
beams before computing the round-trip gain. The pump and laser mode have Gaussian profiles, not necessarily circular.
fm(x, y, z)
= exp
2x22
2
WMm2 (Z)
,
-
fP(x,y, z) = exp
WMY2 (Z)
2x~
WPX22(Z)
-
22
WPY2 (Z)
(3.24)
In Eq. (3.6), the subscript m stands for the laser mode whereas, the subscript p
stands for the pump mode. For the pump mode, the axial variation of the waist is
given by:
Wpx(X, y, Z) =
w, 0 [1 + (M 2zA/7wPo2) 2 1 / 2
55
(3.25)
The model also accounts for an imperfect Gaussian beam by including the beam
quality factor, M 2 . Furthermore, according to the rate equation analysis, if I(z) is
the incident pump intensity at any plane inside the gain medium, the saturated gain
coefficient g(z) at that plane can be given as [42]:
g(z) = ?'z1(1-.t(3.26)
1 + 2Icirc/Isat
In Eq.(3.6),
system.
?7q
is the product of quantum efficiency and quantum defect of the
Isat is the saturation intensity of the gain medium,
aa
is the absorption
coefficient of the gain medium. The complete variation of the pump beam inside the
gain medium is obtained as:
JP(z) = Ap(z) fp(x, y, z) exp(-az)
(3.27)
and,
Ap(z)
fP(x, Y,z)ddy,
(3.28)
where PP is the incident pump power, fp(xyz) is the spatial profile function at a
plane z inside the gain medium and Ap(z) is the pump area at the sane plane. If
Pcirc is the total intracavity power inside the resonator, the Icirc(x,y,z) at any point
inside the gain medium can similarly be defined as:
Icirc(x, Y, Z) =
"ci" fm(X, y, Z)
Am(z)
(3.29)
where, Am(z) is the mode are at the same plane found from the numerical computation for KLM. Then, the change in circulating power in one round trip is defined
as
56
APcirc =
dI(x, y, z)dxdy = 2
ff
g(x, y, z)Icirc(X, y, z)dzdxdy
(3.30)
where 1 is the length of the crystal. Finally, for the small gain approximation, the
total saturated round trip gain G can be defined as:
2 f f f g(x, y, z)Icirc(X, y, z)dV
=
(3.31)
Pciirc
Substituting the previously defined terms, the total gain inside a resonator can be
computed as follows:
G = 2(xaPz
,
r
aP/fexp(-aaz)fp(x,
(Z)Am(Z)
3.6.1
1+
Y, Z)fm(X, Y, Z)
dv
(3.32)
A
Hard aperture modelocking
For this method of modelocking, one is especially concerned with the waist change
at the end mirror, where the aperture is placed. The beam diameter variation at the
end and curved mirrors are depicted in Figure 3-4. Here, one can see that insertion
of the aperture favors the pulsed operation when the mirror separation and crystal
position are adjusted such that, the waist on M4 reaches a dip. The CW mode suffers
from aperture losses more than the pulsed mode.
The resulting simulations are shown for the sagittal plane only, because KerrLensing is stronger in the sagittal plane as compared to the tangential plane due
to the additional no2 factor in the denominator. Thus the focussing is stronger in
sagittal domain, and in practice a vertical slit is placed in front of an end mirror to
achieve hard-aperture modelocking. Also, the plots for the cavity are given for the
first stability window, which corresponds to the left-hand lobes in Fig (3-3) and (3-2).
57
Iagittal
0.4
0.2
52
50
4
-
-- -
0.6-
--
Plane
- - -
-
3
M.
-
48
-46
44
52
50
L2K Crystal Separation (mm)
58
56
54
1.2 -
Tangential Plane
El
-M
- - -- - - M2l
N
-
-
M4
co 0.4
0.244
58
56
54
52
50
48
46
L2K Crystal Separation (mm)
An
Spot bize in Crystal
-
Z 30
-
--
- -
20 -
so
M10
0
1
2
5
3
6
7
-8
Crystal Length (mm)
Figure 3-4: Beam Diameter on end mirrors, M1(blue) and M4(violet), and on curved
mirrors, M2(red) and M3(green), for sagittal incidence. Asymmetry parameter, ( =
1.2. The variables are the curved mirror separation and the crystal position. The
laser parameters are given as: L=1.7m, 112 = 3134, lc=7.5mm, PPck=200kW, 123 =
106.6mm, 02 = 03 = 9.71'. L is resonator length, l is the path length of the gain
medium.
58
KLM Resonator, Beam diameter on M1, sagittal
E
2
1.5,
E
1060
0.5
-60
103
102.5
Cavity Separation (mm) 102
45
50
Crystal Position (mm)
101.5
101
40
Figure 3-5: Beam Diameter on end mirror, Ml, for sagittal incidence. Asymmetry
parameter, ( = 1.2 in hard-aperture modelocking. The variables are the curved mirror
separation and the crystal position.
On the other hand, using the split-step approach gives one an important advantage
over the other methods. It is possible to monitor the beam radii change inside the
crystal. To depict this one can refer to another resonator configuration, which has a
different crystal length, using the same algorithm for the spatial solution. Figure 3-7,
which is taken from [12] indicates the laser and pump mode radii variation throughout
the crystal. Although this solution is accurate enough for low power levels, it ceases
to give a valid result for the values exceeding critical intracavity power as mentioned
before. The temporal effects, such as dispersion and self-phase modulation, which are
balancing mechanisms in soliton propagation have to be considered as well. For more
advanced coverage of this issue, one can refer to the algorithms for spatio-temporal
solutions [23].
3.6.2
Soft aperture modelocking
The figure-of-merit for the soft aperture modelocking case is different as discussed
before. Computing the overlap integral for comparison of gain in CW and pulsed
59
KLM Resonator, Beam diameter on M1, sagittal
2,
E
1.5
0.6
60
103
102.5
55
255
50
102
Cavity Separation (mm)
101.5
45
101
Crystal Position (mm)
40
Figure 3-6: Asymmetry parameter is changed to (
=
3 for hard-aperture modelocking
regimes, one can create the parametric plots for mirror separation and crystal position.
It is generally seen that the asymmetric resonators perform better than the symmetric ones. Thus, a question arises about the asymmetry ratio of the arm lengths.
It will be interesting to compare different arm-length ratios for the KLM resonators.
The asymmetry parameter will be called (, and a comparison is made for (1
=
1.2
and (2 = 3. This can be viewed from following figures 3-5 and 3-6.
We notice that, the waist size on the end mirror change is not significant for
differing ( values as seen on figures 3-5 and 3-6. However, the J parameters (Figures
3-8 and 3-9) have different variations. The ( = 1.2 cavity exhibits weaker KLM when
the curved mirror separation is increased, in other words, the stability boundary is
reached. At the optimized crystal position, 3 (equivalently gain modulation depth)
sharply decays from -0.42 to -0.14. The optimum region for modelocking is maintained
when the mirror separation is adjusted to move towards the shorter stability boundary,
and the ideal crystal position in this case is slightly off the laser mode focus. In fact,
detailed investigation shows that the offset is directly related to the Rayleigh range
[18]. Hence, the beam waist is located on one surface of the crystal.
For the ( = 3 case, 3 exhibits fairly less-tilted profile. The optimum values always
60
60
001.101
40
20
0
E
--
(a)
60-
20
40 0(b)
604020-0 ~(C)
0
4
2
6
Position, z (mm)
Figure 3-7: The laser mode (solid lines) and pump beam (dashed lines) spot size
changes inside the crystal by split-step parabolic index profile algorithm, [12]. The
intracavity power is 200kW, and the crystal thickness is 7.5mm. Asymmetry parameter is ( = 3, the plots are given for different overlapping conditions between pump
and laser mode; (a) wo,L = wo,L = 1&pm ; (b) wo,L = 0. 5 wo,L = 9pm ; (c) wo,L
2
wo,L =
36pm
vary between -0.33 and -0.43, so as compared to the (
=
1.2 case, KLM is stronger
throughout the whole stability region of the cavity. Same arguments apply for the
optimum operating point for this case. However, the mirror separation is less critical
now, because of more flat characteristics.
We observe that there is a trade-off between the stability range and the effectiveness of KLM in that range. For ( = 1.2 the stability range is broader, therefore, once
modelocking is achieved, the resonator is less sensitive to the slight displacements in
the curved mirrors. On the other hand, for the ( = 3 case the observed stable region
61
0.2-
'U,
I$~ ~
0-- -0.2 -0.4-
-06
14
60
103
102
Cavity Separation (mm)
45
101
50
Crystal Position (mm)
40
Figure 3-8: 6 parameter for the soft-apertured KLM calculation on sagittal plane.
Asymmetry parameter, ( = 1.2. The variables are the curved mirror separation and
the crystal position. Discrimination between CW and pulsed regimes is prominent
towards the edge of the stability region.
0.20-0.2..
-0.4-.
_n A .
103
102.5
102
Cavity Separation (mm)
10 1 5
45
.
101
60
55
50 Crystal Position (mm)
40
Figure 3-9: 6 parameter when asymmetry parameter is changed to (= 3.
62
is narrower but a more efficient KLM profile is possible.
For an experimental verification, a laser cavity utilizing 5 bounces on Double
Chirped Mirrors (DCM) for dispersion compensation and a 1% output coupler (SiO 2
/
TiO 2 ) is used when the simulations are tested for the first time.
As a matter
of fact, the DCMs were specifically designed for CaF2 prisms to obtain a spectrum
extending from 600 nm to 1150 nm, and a 2.3 mm thick crystal. However, for the
quick experimental verification, the prisms are not used and the available crystal
thickness was 2 mm. Therefore, the bandwidth of the spectrum reduces to 100 nm
Thus, the operating conditions are not
due to sharply varying dispersion profile.
ideal, because generally a flat second order dispersion profile is sought around Ofs 2 .
The second order dispersion curves are given for the ideal and non-ideal cases below.
However, this setup was capable of verifying the theoretical expectations.
spectral curves for both cases do not differ appreciably.
The
For the same repetition
rate, we would like to observe the operation conditions for ( = 1.2 and ( = 3. Our
experiments have shown that, KLM region for the ( = 1.2 is found around the lower
end of the stability region and it performed to be very robust in operation. ( = 3
case was on contrary very fruitful for modelocking. The cavity easily modelocked
almost at every point in the stability region, but as expected, it was more sensitive
to the displacements on the curved mirrors. The operation was less reliable, thus a
more compact cavity is needed. This is another reason justifying a high repetition
rate laser, with compact layout.
The dispersion profile for this case is flat around Ofs 2 for a spectral range of
300nm. The profile is not perfectly flat because the DCMs were designed for different
crystal thickness. Another verification of the supreme KLM behavior with increasing
asymmetry parameter is shown by Thomas R. Schibli, on a Cr:Forsterite laser. It
is reported that, it was relatively much easier to modelock the laser with higher
asymmetry than the traditional values.
an easy material to work with.
Generally, it is known that Ti:sapphire is
Therefore, achieving KLM does not constitute a
big handicap. However, observing KLM for materials like Cr:Forsterite or Cr:YAG
are more difficult.
For these type of resonators, it is essential to optimize cavity
63
500
400
300
200
100
0
0.6
0.7
0.8
0.9
lambda
1
1.1
1.2
0.7
0.8
0.9
lambda
1
1.1
1.2
(A)
0-10
-20
0
04
-30
-40
-50
0.6
(B)
Figure 3-10: Second order dispersion profile for two cases. Dispersion is given in fs 2,
wavelength in microns. (a) No prisms, 5 DCM bounces and 2 mm crystal (b) Prisms,
6 DCM bounces and 2 mm crytal. The dispersion of air is also taken into account.
64
Sa
-20
o.j
C
1.0
-405
0.5
a
-60
-80
U
600
800
'
-
1000
Wavelength (nm)
46
"1200
1
1400
Figure 3-11: The spectrum of the ring resonator in the presence of a -1000mm radius
of curvature convex mirror. The effect of the convex mirror is distorting the symmetry
and enhancing the KLM action in the cavity.
conditions to start modelocking.
Therefore, the results from Cr:Forsterite suggest
that manipulating the cavity geometry, while keeping the repetition rate constant
has an influence on KLM.
Finally, another verification of the influence of the asymmetry ratio on KLM
operation comes from [26],[27].
The geometry of the resonator is a bow-tie for this
setup and the repetition rate is much higher, 1 GHz. The symmetric bow-tie ring
would contain two curved mirrors, an output coupler and a flat-mirror.
For this
case, the symmetry is disturbed by replacing the flat mirror with a slightly convex
chirped mirror (R=1000mm). The authors claim that the convex mirror is the key
element in the broadband operation of their Ti:sapphire ring laser. The presence of the
asymmetry increases self-amplitude modulation and enhances the cavity conditions
for the generation of shorter pulses. The published spectrum for this cavity is shown
below in Figure 3-11:
Elimination of the influence of non-flat dispersion on the resonator will improve
the comparison ground for the cavity geometries. Also, the results encourage one
65
to carry out the analytical aspects of the resonator geometry.
Secondary effects,
such as gain-guiding, gain-aperturing and misalignment sensitivity also have to be
included in the model. These effects can change the results in a critical way. For
instance, one of the favorable regions for KLM can be eliminated when gain-guiding
and gain-aperturing are included in the model [22].
66
Chapter 4
Experimental Results
In light of the preceding chapters, the tools for developing the cavity are now at hand.
The temporal and spatial considerations for the cavity design are well-stated and
following elements and guidelines for the construction of the resonator are available:
Temporal Elements
* The double-chirped mirror pairs spanning an octave with an improved pre- and
post pulse cancellation. A total of 12 bounces on pairs. The distribution of mirrors are
balanced on both sides of the crystal in order to provide best conditions for dispersion
managed modelocking. The optimum condition is the case where dispersion is equally
balanced in both arms individually as stated before.
* The BaF 2 plates (0.7 mm, 1 mm, 1.3 mm with 4' wedge angle)and wedges (1.65-1.98
mm thickness variation with 0.75').
* The output coupler with negative group delay variation compensating for 0.72 mm
BaF 2.
9 Overall group delay variation with optimum wedge position is shown in the following
figure, this includes the round-trip propagation through air, 2mm Ti:sapphire crystal,
BaF 2 wedges and plates, output coupler. The DCM pairs were actually designed for
a longer cavity, therefore an imperfect dispersion profile is obtained. However, the
maximum deviation from the group delay is still comparable with the expected pulse
duration. Hence, it has a potential to provide an octave spectrum.
Spatial considerations
67
3
2
0
-1
-2
0.7
0.8
0.9
lambda
1
1.1
Figure 4-1: The total group delay in the cavity in the presence of all optical elements.
o From the distribution of mirrors, wedges and plates, the configuration shown in
Figure 4-2 is realized for the repetition rate of 150 MHz, 2 m round-trip path length.
4 bounces on DCM pairs, a total of 1.5 mm BaF 2 in the short arm. 8 bounces on
DCM pairs, 2 BaF 2 wedges in the long arm.
o After the study of the cavity asymmetry issues, an arm-length ratio of (=1.85 is
established. This seems to be the optimum ratio for the placements of the elements,
whereas still providing a high contrast between arm lengths.
o The overall cavity is quite compact, has a potential to fit on a 15" x 15" board
with the pump lens (6 cm focal length, ThorLabs
TMplano-convex
lens). The layout
of the resonator is shown in Figure (4-2). In this layout OCI is the dielectric output
coupler, whereas
OC2 is replaced by a silver mirror, with approximately 2% loss.
o Due to the spatial limitations, this configuration appears to be the shortest cavity
that could be put together. Regarding the number of optical elements in the cavity,
one can see that the setup is quite dense. Therefore, next step in obtaining higher
repetition rate is the transformation to a ring structure, which would at least double
the repetition rate. Because, single path traversing in a ring-laser already corresponds
to one round-trip.
o The pump beam is obtained from a 10 W Spectra PhysicsTM MilleniaTM
68
pump
laser. The beam profile is slightly elliptic, but the collimation is quite satisfactory
with divergence angle less than 0.6 mrad.
9 The curved mirrors used for this setup have 7.5 cm radii of curvature, for which the
astigmatism compensation angle is 5.80.
1.5 mm BaF2
OC 1
I
CLaser
crystal:
2mm Ti:A12 03
I
II
-1
BaF2 - wedges
OC2
Base Length = 35cm for 150 MHz Laser
Figure 4-2: The high-repetition rate laser with DCM pairs and dispersion adjusting
BaF 2 plates and wedges.
4.1
Pump beam characterization
The focusing of the pump beam is important for the laser operation. In general, the
practice is to focus the pump mode tighter than the laser mode. The argument follows
from the computation of the overlap integral for solid state lasers [42]. A pump waist
which is 30% tighter than the laser mode waist is preferable.
Thus, initial pump
mode characterization is important in order to maintain the best focusing conditions.
The pump laser used for high-repetition rate laser experiment is Spectra PhysicsTM
TM
MilleniaTM Xs 10 W laser. The pump beam profile is measured by BeamScope
beam profiler from DataRay T M Inc. The profile is measured on several longitudinal
2
locations for the full characterization of the beam. M value of the pump beam is
very close to 1, therefore the well-known longitudinal Gaussian beam variation can
69
be fit into the measurement results in order to obtain the position and magnitude of
absolute waist. The measurement results can be observed in figure 4-3. The fitting
algorithm yields the horizontal waist as 810 Mm located at -1.09 m with respect to
laser output and the vertical waist as 588pm located at -1.9 m. It can be seen that
the laser beam has an elliptic profile.
E
1.05-
a)
-
0.95--
-.....
-.............
0 .9 0
0.6
0.8
--
1.0
1.2
1.4
Longitudinal separation [m]
Figure 4-3: The pump waist measurement as a function of longitudinal displacement
from the laser output. The measured data is then fit to the Gaussian beam with
M 2 parameter 1. The dashed line is for the vertical axis and the solid line is for the
horizontal axis.
The pump beam is focused by a plano-convex lens with f=60 mm.
After an
ABCD matrix calculation, the waist inside the crystal becomes 11.4 pum, which is
approximately 30% smaller than the laser mode around modelocking regime.
4.2
Optimization of the resonator
The laser can easily be brought into stable CW operation. The stability diagram of
this high-repetition rate laser is already shown in Figure (2-11). The observed mode
profile is very close to TEMOO. The absorption of the Ti:sapphire crystal is 73% due
to the high doping level. The lasing threshold for CW before the insertion of the
70
BaF 2 wedges and plates is at 300 mW pump power, whereas after the insertion it
rises to 470 mW. Effectively, this threshold can be made lower by retro-reflecting the
unabsorbed pump light. In order to avoid excessive losses the plates and wedges are
inserted under Brewster's angle (55.80). With correct birefringence adjustment for
the crystal, the losses from the reflections on wedges and plates can be reduced to
0.05%. This relies heavily on the quality of polishing. With high-quality processing,
scattering losses are significantly reduced. Internal losses in BaF 2 are negligible.
In order to obtain the pulsed operation, the laser is operated at the inner edge of
the second stability window. As mentioned before, inner edge of the outer stability
region is the most convenient location for modelocking. Also, the numerical simulations indicated that gain for pulsed operation is higher than CW operation. The
CW power drops drastically towards the stability boundary, and if the pump focus
is not maintained well enough, higher order Gaussian modes can arise. Thus, when
the position of the crystal is changed for seeking the modelocking regime, the pump
focus should always be kept at the center of the gain medium. This can be easily
tracked with checking the CW power and maximizing it after every translation step.
Alignment is solely done by the end mirrors. The compact setup provides easy access
to both end mirrors simultaneously, as compared to the case where one only has an
access to one mirror at a time, simultaneous adjustment may not be possible for large
cavities.
After the modelocking regime is found, one optimizes the spectral broadening with
the help of an optical spectrum analyzer. The crystal position is critical, thus slow
adjustments are required. Also, the wedge insertion is brought to a level where the
spectrum is widest.
4.3
Results
Applying the methodology described above, a broad spectrum with 100 mW average
output power is reached with 3.7 W of pump power. The spatial profile is still very
close to TEMOO. At the optimum wedge insertion, the spectrum shown in Figure 4-4
71
10
0.10
-20CD
0.06
-
-30
U)
0.04
oe
-40
0.02
-50
.-
-
600
1000
800
0.00
1200
Wavelength [nm]
Figure 4-4: The output spectrum of the high repetition rate Ti:sapphire laser in
logarithmic (red) and linear (blue) scales. The octave spanning spectrum is obtained
approximately 20 dB lower than the peak level.
is obtained.
It is observed that, the spectrum extends from 560 nm to 1250 nm. The octave
is reached at 600 nm and 1200 nm with only 20 dB less power than of the peak level.
The spectral shaping of the output coupler is observed around the peak at 630 nm
due to a 4 fs deviation from the average group delay, see Figure (2-10). This spectrum
is well-capable of exploiting the lf-2f self-referencing scheme for the frequency comb
generation.
The stable modelocking operation can be maintained through long duration if the
system is isolated from environmental effects like, dust, air turbulence and mechanical vibrations. A good isolation is mandatory for the stabilization experiment and
extracting fceo information.
Another point to mention is that in order to modelock the laser a slight amount
of misalignment is necessary. The effect associated with this is the higher degree of
72
discrimination between CW and pulsed regimes with a larger misalignment [44]. One
can simulate that 6 parameter can have a larger variation with increasing misalignment. This effect is also equivalent to the offsetting the pump beam with respect to
the laser mode. The enhancement in the differential gain is plotted with respect to
the equivalent pump offset in Figure 4-5. When there is an offset between the pump
profile, and the laser beam profile, there is a greater access to gain for the beam
undergoing self-focussing relative to the CW beam. This increase in the fractional
overlap change can be crudely thought of as the reason for enhancement of J parameter under conditions of misalignment. Naturally, one has to consider the effects
of higher-order mode excitation, the precise distribution of the gain coefficient and
various other effects to gain more insight.
dG
dw
0
'8
1
14
1 6 1 8
2
Figure 4-5: The effects of relative pump and laser offset on the differential of the gain.
The top curve in this figure corresponds to the fully aligned beam. The lower curves
are for an increasing offset of w,/2 and w. respectively [44]
73
74
Chapter 5
Future applications and conclusion
5.1
Frequency metrology applications
The output of the high-repetition rate laser is ideally suited for the generation of
frequency comb. The applications of the comb are numerous for frequency metrology.
As it will be shown, one can apply the self-referencing scheme to beat one side of the
spectrum with the other side after doubling the low-frequency component with a
nonlinear crystal.
For the characterization of the frequency comb, there are two parameters that one
should control: fre, and feo are already defined in Chapter 1. The significance of feo
increases as the pulse gets shorter. This is depicted in the following figure. The slip of
the carrier with respect to the envelope is varying at every round-trip with a frequency
fceo. There exist physical phenomena where the processes are dependent directly on
the electric field rather than on the intensity. Therefore, one should stabilize the
carrier-envelope phase. The techniques used for stabilizing the carrier-envelope phase
are best understood in frequency domain [45], [46].
As each comb line is offset by the same fceo, optically heterodyning different comb
lines will not contain the fceo information. Instead, if one frequency doubles the red
end of the spectrum and beats it against the blue end of the spectrum, fc,, information
can be extracted.
Consider a single comb line on the red end, the electric field phase for a spectral
75
E(t)
A~~
r
Fourier
b
transformation
E&i
jE)
/
~
4I
fe
Figure 5-1: In the top figure, the slipping of the carrier wave is shown. If one extends
the spectral lines towards DC, there is a certain offset which is not equal to zero but
smaller than frep [8].
line (nth comb line) can be written as:
on = 27r(fn + fceo) + #on
where
#o
=
27r(nfrep + fceo) + 4Jon
(5.1)
is the optical phase constant of the n'h comb line. Similarly, an octave
away on the blue side of the spectrum, the 2nth comb line will have the phase:
02n = 2-r(f
2n
+ feco) + Oo2n = 27(2nfrep + fceo) + #o2n
(5.2)
After doubling the n'h comb line, a heterodyne beat can be obtained between the
doubled line and 2nth comb line on a photodiode. The signal will have an interference
term with phase,
76
#det - 27rfceot + 20on -
(5.3)
po2n
An fceo measurement is depicted in Figure 5-3-b. The self-referencing scheme is
carried out with a 78 MHz Ti:sapphire laser, where light at 1160 nm is doubled with
a BBO crystal and beat against 580 nm light. The fundamental and doubled light
are temporally overlapped and directed to the photo-lutiplier tube. The construction
of the experiment is identical to the one shown on Figure 5-2.
Base Length = 30cm
0
Laser crystal:
Silver 1 mm BaF2
E
Mirror
2mm Ti:A120 3
PUMF
0
0
111
1
BaF2 wedges
OC
7
~F
580nm
1mm
PMT Pol.
BBO1
11 6Onm
Figure 5-2: 78 Mhz Ti:sapphire laser generating an octave-wide spectrum.
measured with an RF analyzer.
feo is
The laser produces the required octave-wide spectrum with 50 mW average power
[48], which is quite comparable to the spectrum obtained from high repetition rate
laser. The comparison of two spectra is depicted in Figure 5-3-a. This clearly demonstrates that the fco signal can be obtained with a large signal to noise ratio. The
SNR values obtained with 10 kHz and 100 kHz resolution bandwidth are 30 dB and
25 dB respectively. The high repetition rate laser also carries the same potential to
demonstrate the fceo beat, thus this is the next step in exploring the frequency comb
generation with higher repetition rates.
77
I
0
I
I
1.2
IM7
-10
.n
v0-40
-
-
E
100 kHzRBW
-~..
0 .8
0
=3
a)-20
-6...
-...
-60
06
0.0
-30
600
a)
800
1000
Wavelength [nm]
1200
0
b)
20
40
60
Frequency [Hz]
80x10
Figure 5-3: (a) Measured output spectrum on a linear and logarithmic scale for
78 MHz repetition rate (dashed line) and for high repetition rate (solid line). (b)
Measured carrier-envelope beat signal from the 78 MHz laser [48].
After studying the comb line dynamics, one could see that the carrier-envelope
phase control can be provided by inserting an Acousto-Optic Modulator into the
pump beam.
Modulation of the pump power mostly changes fceo, but affects frep
less '. Although the exact effect is not yet well-understood, pump power modulation
changes the nonlinear phase shift in the Ti:sapphire crystal and this causes a change
in feo on the pump beam.
Thus, with an optical phase-locked loop (PLL), it is
possible to maintain control over f,, and carry out carrier-envelope phase sensitive
experiments.
As a summary, the necessity for a broadband spectrum is clear, and detection of
fceo is already carried out by several other groups [45], [46], [47]. The high-repetition
rate laser has a potential to produce a highly stabile frequency comb. The octave
is already reached 20 dB below the peak level, so it is possible to control fceo after
careful isolation of the laser and preparing feedback electronics.
5.2
Conclusion
In this thesis, the construction of a compact, high-repetition rate Ti:sapphire laser is
studied for various applications. The motivation behind this attempt is to improve
'If one desires to control frep, one of the cavity mirrors can be mounted on a piezo-electric transducer (PZT). For the absolute comb generation frep should also be locked to an optical transition,
but not an RF synthesizer.
78
the spectral output of a Ti:sapphire laser with necessary temporal and spatial adjustments. As a result, a spectrum spanning a full octave from 600 nm to 1200 nm
is reached. Generation of octave spanning lasers are broadening the field of optical
frequency metrology and make optical atomic clocks possible. Obtaining an optical
atomic clock from an octave spanning Ti:sapphire laser will constitute the future
standard for time and has a potential to outperform today's Cs clocks, with special stabilization techniques. Furthermore, the generation of the broad spectrum also
motivates the monitoring of ultrafast events with various time-domain and frequencydomain applications. The boundaries of the nonlinear optics has reached the extreme
nonlinear region with the generation of high-harmonic attosecond pulses is pursued.
Broader spectra also enhance the resolution achieved in biomedical imaging like optical coherence tomography.
Future work will focus on the upscaling of the repetition rate while preserving
the spectral output. Frequency comb stabilization will be carried out simultaneously.
Further studies of the Kerr-Lens dynamics are necessary for extracting the widest
spectra from compact resonators, utilizing both numerical and analytical approaches.
79
80
Appendix A
Kramers-Krdnig Relations for
Reflection and Transmission
The Kramers-Kr6nig relations establish a connection between the real and imaginary
parts of a complex function.
KK relations are useful in gaining insight into the
frequency dependence of the complex index of refraction. Expressing the refractive
index variation with respect to the absorption coefficient is well studied in most texts
on photonics [15, 16].
For dispersion in a traveling wave, the complex electric permittivity
E(w)
=
(A.1)
6'(w) + iE"(w)
of a dielectric as a function of angular frequency satisfies
E'(w) - 1 =
P.V.
dx
(A.2)
dx,
(A.3)
and,
(W)
2 P.V
we PV
tr
(X)
-
where PV denotes the Cauchy principal value. Using the fact that E" (x) is an odd
81
function allows Eq. A to be rewritten as:
-
1
-P.V.
7r
2
_-00
2
X
dx
(A.4)
and
=
2(w)
P.V.
2
2
dx
(A.5)
Thus, real part of the permittivity can be related to the reflectivity and transmission for a dielectric boundary, and the complex part can be related to the transmission
group delay.
82
Appendix B
Sellmeier Coefficients for Optical
Elements
In optics, the Sellmeier equation is a semi-empirical relationship between refractive
index n and wavelength A for a particular transparent medium. The usually form of
the equation is:
B1 A2
A2 - 1
B+2 A2
2
A -C
+
2
B2 A2
A2 - C2
(B.1)
where B 1 ,2 ,3 and C 1 ,2 ,3 are experimentally determined Sellmeier coefficients. These
coefficients are usually quoted for A measured in micrometers. The equation is used
to determine the dispersion of light in a refracting medium. Tabulated here are the
coefficients of some elements involved in the high-repetition rate laser [39].
BaF 2 :
B1 = 0.643356, B 2
=
0.506762, B 3 = 3.8261,
C1 = 0.057789, C 2
=
0.109680, C 3
=
46.3864
Sapphire:
B1
=
1.4313493, B 2 = 0.650547, B 3 = 5.3414,
C1
=
0.005279, C2 = 0.014238, C 3
=
325.0178
Fused silica:
B1 = 0.6961663, B 2 = 0.4079426, B3 = 0.894794,
C1 = 0.004679, C2
=
0.013512, C3
83
=
97.9340
84
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