Phase Stabilized Modelocked Lasers: from opticalfrequency metrology to waveform synthesis ofultrashort pulses
byTara Michele Fortier
B.A., Concordia University, Montréal 1998

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2004

This thesis entitled:
Phase Stabilized Modelocked Lasers: from optical frequency metrology to waveform
synthesis of ultrashort pulses
written by Tara Michele Fortierhas been approved for the Department of Physics

Dr. Steven T. Cundiff

Dr. Jun Ye

Date

The final copy of this thesis has been examined by the signatories, and we find thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii
Michele Fortier, Tara (Ph.D., Laser Physics)
Phase Stabilized Modelocked Lasers: from optical frequency metrology to waveform
synthesis of ultrashort pulses
Thesis directed by Prof. Dr. Steven T. Cundiff

By drawing on the techniques of single-frequency laser stabilization and on improvements of ultrafast lasers, stabilization of Ti:sapphire lasers has led to great advances in both the fields of optical frequency metrology and ultrafast science. This
thesis presents the development of the technology and application of modelocked laser
stabilization to both domains. Towards the latter I will discuss the use of extremely
phase-coherent pulse trains for exploration of quantum interference in semiconductors.
As applications to the former, we present synchronization between a Ti:sapphire andErbium doped fiber laser to produce an absolute comb at 1.5 µm. I also demonstrate
an octave spanning Ti:sapphire laser that is a candidate as a transfer oscillator for an
all-optical clock.

Dedication

To my grandmother, Martha (Moonbeam) Fortier. In life she has served a strong rolemodel and in death as a continuing source of courage for me.

v

Acknowledgements

I have had the very good fortune of sauntering into the budding field of stabilization of modelocked lasers. The time that I have spent here and the work that I havebeen a part of has been nothing short of an eye opening experience for me, very much
fuelled by the dynamic energy of the people and the quality of the science at JILA.
My mentor in this adventure has been my advisor Steve Cundiff, whose knowledge
in ultrafast lasers and semiconductors has been an invaluable part of my experimental
education. The patience and independence that he has allowed me has permitted muchof my personal scientific growth, for this I am fully indebted.
Much thanks is also due to my surrogate advisor Jun Ye. Although not an official
member of his group, he has always welcomed me into his lab and made me feel likeone of his own. Jun’s constant personal and scientific support will never be forgotten.
There are a number of people who have directly contributed to the work in thisthesis. One person in particular is David Jones who, aside from passing on to me
his amazing breadth of experimental knowledge, has also taught me the joys of rockclimbing! David has helped me through thick and thin and has been much like a big
brother to me during my adventures with physics. The work that is presented in thisthesis could now have proceeded as smoothly without his constant support. I have also
had the pleasure of working with Pete Roos, who aside from providing much needed
fun and laughter, has provided great help in some of my concluding work at JILA. A
source of quiet inspiration for me has been John (Jan) Hall. His everlasting enthusiasm

vi
for science and his remarkable electronic designs, without which none of this work could
have been done, continue to baffle and amazed me.
Of course, none of the work done here in the last years could have been possible without the amazing skill of the JILA technical staff. In particular I would liketo acknowledge Terry Brown and James Fung-A-Fat for their invaluable guidance inelectronics and Hans Green for being a friend and phenomenal machinist.
I have also had the opportunity to work with and become friends with somewonderful people. I owe my gratitude to my colleagues and friends for their constantsupport, especially during the writing of this thesis. Specifically, Martin Griebel has
gone above and beyond his duty as a post doc and friend by reading my thesis cover
to cover, while at the same time providing amazing editorial help. I would also like tothank my dear friends Emily Gibson, Quentin Diot, Qudsia Quraishi and Paul Haljan
for being there during the rough spots.
Much thanks is also due to some of my mentors along the way: Chris Greene andMisha Ivanov for making me appreciate the beauty of theoretical physics, Greg Lopinski
for providing guidance during my first true experiences in experimental physics, andPaul Corkum for his past and continuing support in my pursuit of knowledge.
Finally, thank you to my committee members, Steve Cundiff, Jun Ye, Jan Hall,
Leo Hollberg, Henry Kapteyn and David Jonas for being kind inquisitioners and thorough readers.

vii

Contents

Chapter
1 Introduction

1

1.1

Optical frequency metrology . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Carrier-envelope phase stabilization . . . . . . . . . . . . . . . . . . . . .

52 Phase stabilization of modelocked Ti:sapphire lasers
2.1

8Time and frequency domain aspects of modelocked lasers . . . . . . . .

9

2.1.1

Measurement of the laser comb parameters . . . . . . . . . . . .

12

Self-referencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2.1

. . . . . . . . . . . . . . . . . . . . . . . .

15

2.3

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.4

Phase stabilization of the Ti:sapphire laser . . . . . . . . . . . . . . . . .

18

2.4.1

Stabilization of the laser offset frequency . . . . . . . . . . . . . .

18

2.4.2

Stabilization of the laser repetition rate . . . . . . . . . . . . . .

20

2.4.3

Passive stability of the Ti:sapphire laser . . . . . . . . . . . . . .

21

The stabilized comb . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.2

2.5

Microstructured fiber

3 Phase noise characterization of microstructured fiber
3.1

3.2

25

Amplitude-to-phase noise conversion in microstructured fiber . . . . . .

26

3.1.1

Transfer of fiber noise onto the output of the laser . . . . . . . .

31

Birefringence and nonlinear polarization rotation in microstructure fiber

33

viii
4 Carrier-envelope phase coherence and the “absolute” phase
4.1

The“absolute” phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Measurement of the carrier-envelope phase coherence time and its importance to phase-sensitive measurement

4.3

4.4

5.2

39

. . . . . . . . . . . . . . . . . . .

40

4.2.1

Noise analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.2.2

Experimental set up . . . . . . . . . . . . . . . . . . . . . . . . .

42Characterization of extra-cavity sources of phase noise . . . . . . . . . .

46

4.3.1

MS fiber noise . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.3.2

Interferometer noise . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.3.3

Long-term phase drifts: coherence measurement using a lock-in
amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50“Absolute” phase measurement . . . . . . . . . . . . . . . . . . . . . . .

51

5 Octave spanning lasers
5.1

38

54

The Ti:sapphire laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

5.1.1

Cavity geometry . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

5.1.2

Pulsed operation of the Ti:sapphire laser . . . . . . . . . . . . . .

56

5.1.3

Modelocking and the Kerr effect . . . . . . . . . . . . . . . . . .

57

Characteristics of an octave spanning Ti:sapphire laser . . . . . . . . . .

60

5.2.1

Phase stabilization of an octave spanning laser . . . . . . . . . .

66

5.2.2

Broadband Ti:sapphire oscillators as transfer oscillators for alloptical clocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Applications of the stabilized Ti:sapphire laser

68
70

6.1

Quantum interference control in semiconductors: the matter interferometer 70

6.2

Synchronization between a fiber laser and a stabilized Ti:sapphire laser:
An absolute comb at 1.5 µm . . . . . . . . . . . . . . . . . . . . . . . . .

78

ix
7 Conclusions and outlook

85

Bibliography

89

Appendix
A Phase stabilization of Ti:sapphire lasers: the details

95

A.1 Frequency difference measurement of the optical comb . . . . . . . . . .

95

A.2 Measurement of the laser repetition rate . . . . . . . . . . . . . . . . . .

97

A.3 Measurement of the laser offset frequency, f0 . . . . . . . . . . . . . . .

98

A.4 Fun with microstructure fiber . . . . . . . . . . . . . . . . . . . . . . . .

99

A.5 Self-referencing: the interferometer . . . . . . . . . . . . . . . . . . . . . 100
A.5.1 The Mach-Zender interferometer . . . . . . . . . . . . . . . . . . 100
A.5.2 The prism interferometer . . . . . . . . . . . . . . . . . . . . . . 104
A.6 The laser stabilization loop . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.6.1 Stabilization of the laser offset frequency . . . . . . . . . . . . . . 107
A.6.2 Negative feedback . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.6.3 Stabilization of the laser repetition rate . . . . . . . . . . . . . . 112
A.7 Measurement of the offset frequency stability . . . . . . . . . . . . . . . 113
A.7.1 Phase noise vs. amplitude noise . . . . . . . . . . . . . . . . . . . 114
A.7.2 Measurement of the phase noise spectrum . . . . . . . . . . . . . 115

xFiguresFigure1.1Time and frequency domain aspects of the optical field produced by amodelocked laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

2

A map of the electromagnetic spectrum showing selected measured secondary standards and their corresponding frequencies. . . . . . . . . . .

4

2.1

Pulse formation in a modelocked laser. . . . . . . . . . . . . . . . . . . .

9

2.2

Summary of the time and frequency representations of the Ti:sapphirelight field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

Homodyne detection of the laser repetition rate. . . . . . . . . . . . . .

13

2.4

Nonlinear heterodyne of the laser offset frequency. . . . . . . . . . . . .

14

2.5

Continuum generation in microstructure fiber. . . . . . . . . . . . . . . .

15

2.6

Experimental realization of self-referencing for measurement of the laser
offset frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.7

Experimental apparatus of a carrier-envelope phase-stabilized fs Ti:S laser. 19

2.8

Ti:sapphire laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.1

Extracavity noise sources in the laser stabilization loop. . . . . . . . . .

27

3.2

Experimental setup used to determine the amplitude-to-phase conversion
coefficient for MS fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

xi
3.3

Experimental setup used to determine whether amplitude-to-phase conversion in MS fiber is transferred onto the laser output by the action of
the servo loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

32

Extinction of the primary axis of MS fiber at both low and high coupled
input power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.5

Linear and nonlinear birefringence in microstructure fiber. . . . . . . . .

35

3.6

Nonlinear polarization rotation in MS fiber. . . . . . . . . . . . . . . . .

37

4.1

Schematic for measurement of the coherence in the pulse train carrierenvelope phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Phase noise power spectral density of the phase fluctuations on the laser
offset frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Calculated microstructure fiber phase noise generated from fluctuations
in laser output power. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

45

Estimated microstructure fiber phase noise generated from power coupling fluctuations.

4.4

43

48

Schematic for the measurement of the extra-cavity noise caused by interferometer drift.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.6

Long term phase drifts in the carrier-envelope phase. . . . . . . . . . . .

51

4.7

Relationship between pulse train and interferometer output. . . . . . . .

52

5.1

The basic Ti:sapphire laser cavity. . . . . . . . . . . . . . . . . . . . . .

55

5.2

Nonlinear effects in the Ti:sapphire crystal (self-focussing and self-phasemodulation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

Experimental schematic for stabilization of an octave spanning Ti:sapphirelaser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

59

60

Beam profiles of the optical modes of an octave spanning Ti:sapphire laser. 62

xii
5.5

M 2 measurement of beam propagation of light from an octave spanning
Ti:sapphire laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

5.6

Far field beam profiles for CW and modelocked light at 780 nm.

65

5.7

Power spectral density of the phase noise fluctuations of the offset fre-. . . .

quency measured using the direct, unbroadened output of an octave spanning Ti:sapphire laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.8

Schematic of an all optical clock based on molecular iodine. . . . . . . .

69

6.1

Depiction of quantum interference of one- and two- photon pathways
between the valence and conduction bands in LT-GaAs. . . . . . . . . .

6.2

Experimental setup used for quantum interference control of injected photocurrents in LT-GaAs. . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3

75

Measurement of the laser offset frequency using quantum interference ina semiconductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4

71

76

Solid state detection of static shifts in the pulse train carrier-envelope
phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

6.5

Schematic of the setup for synchronizing a fiber laser to a Ti:sapphire laser. 80

6.6

Fiber laser offset frequency noise power spectral density. . . . . . . . . .

82

7.1

PTB harmonic frequency chain to the Ca intercombination line. . . . . .

87

7.2

Current and future setup for stabilization of a Ti:S laser . . . . . . . . .

88

A.1 Beating between two optical carriers. . . . . . . . . . . . . . . . . . . . .

96

A.2 Optical interference between multiple coherent optical fields.

. . . . . .

97

A.3 High-stability mounting of MS fiber. . . . . . . . . . . . . . . . . . . . .

99

A.4 The Mach-Zender ν–to–2ν interferometer. . . . . . . . . . . . . . . . . . 101
A.5 Configurations for using an acousto-optic modulator as a frequency shifterand a noise eater. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

xiii
A.6 The prism ν–to–2ν interferometer. . . . . . . . . . . . . . . . . . . . . . 105
A.7 The electronic diagram for phase lock of the offset frequency and therepetition rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.8 Block diagram of the fs synchronizer (f0 PLL). . . . . . . . . . . . . . . 108
A.9 Schematic of error detection using a digital phase detector. . . . . . . . 109
A.10 Block diagram of a tracking filter.

. . . . . . . . . . . . . . . . . . . . . 110

A.11 Piezo-electric actuator for stabilization of the laser offset frequency. . . . 111
A.12 Piezo-electric actuator for stabilization of the laser repetition rate. . . . 112
A.13 Phase noise versus amplitude noise. . . . . . . . . . . . . . . . . . . . . . 114
A.14 Block diagram of a frequency-to-voltage converter. . . . . . . . . . . . . 116

Chapter 1

Introduction

Laser stabilization is a powerful technique that is used in many aspects of atomic
physics and optical spectroscopy. Possibly more powerful, however, is the application of
single frequency laser stabilization techniques to modelocked lasers, which is the work
horse for studying ultrashort time scale physical processes. The marriage of the two
has brought the techniques and challenges of what were previously two disparate fields
into a union that has dramatically impacted both optical frequency metrology as wellas ultrafast science.
To most, upon first reflection it isn’t immediately obvious that one can use single
frequency techniques to stabilize the seemingly complicated optical spectrum of a modelocked laser. This is because modelocked lasers are pulsed lasers, where the laser pulses
are formed from millions of optical frequencies. As shown in Fig. 1.1, a modelocked
laser periodically emits nearly delta function time duration pulses, separated in timeby one cavity round trip, trt . A natural consequence is that the Fourier spectrum ofa train of delta function pulses in time is delta function comb in frequency, with eachfrequency element separated by frep = 1/trt . The amazing stability in the periodicity(trt ) of the emitted pulses yields in a very frequency stable comb. What makes it possible to simply characterize the frequencies in the comb is the forced phase coherence
between the optical modes of the laser necessary for pulse formation. The strong modelocking condition, in particular for the modelocked Ti:sapphire (Ti:S) laser (Kerr-lens2a) Time domain

∆φCE

E(t)

2 ∆φCEt
1/ frep = trtb) Frequency domain
I(ν)

0

f0
frep

νn = n frep + f0

νFigure 1.1: The time a) and frequency b) aspect of the modelocked laser. The phaseshift between the adjacent pulse in the pulse train, ∆φCE results is proportional to the0.
ratio of the offset frequency, f0 to the laser repetition rate, frep , ∆φCE = 2π ffrep

modelocking), produces a very intrinsically narrow, and equally spaced comb lines. As
a result, each comb element in the modelocked spectrum may be characterized by only
two radio frequencies, the laser repetition rate, frep and an offset frequency, f0 , that isνn = nfrep + f0 ,(1.1)

where the integer n ∼ 106 takes frep into the optical frequencies [see Fig. 1.1 b]. For a
more detailed discussion of this please consult Chap. 2.
The fact that the seemingly complicated optical spectrum of a modelocked laser
may be characterized by only two rf frequencies is an incredible result. The maininterest, however, as seen via eqn. 1.1, is that the optical comb spacing provides a direct optical-to-microwave link. The fundamental existence of the repetition rate, which
provides 105 division from optical to microwave permits absolute stabilization of thelaser spectrum using a single optical oscillator and a microwave reference [see Chap. 2].

3
The“simple” stabilization scheme, described in detail in Chap. 2, has far reaching consequences in precision optical frequency metrology and has all but made obsolete theuse of the very technically challenging harmonic frequency chains [60, 15, 14].

1.1

Optical frequency metrology
High precision optical frequency measurements began about 40 years ago with the

inception of the first laser. Since then, the development of more stable and spectrally
narrow lasers have brought the laser to the forefront of high precision measurement. A
laser with a potential dynamic resolution of 1017 (for a mHz linewidth centered in the
optical) may be used to determine the value of some fundamental constants [75, 68] with
unprecedented precision. The high frequency resolution available using stabilized lasers
has allowed for experimental investigation into various aspects of fundamental physics[35] such as gravitational wave detection [29], time variation of fundamental constants[60, 15, 14] and quantum electrodynamics [51], to name a few. Aside from providing ameans for determination of fundamental physics, the techniques pioneered in precision
measurement have also led to advances in optical communications [41] and precision
length measurement.
The full potential for the high resolution of the laser remains untapped, however,unless its frequency can be determined absolutely against the current time/frequency
standard, the Cs clock, established as the international time standard in 1967. The Cs
clock provides our current definition of the second; 1 second equaling 9 192 631 770
atomic cycles between the F = 4, mf = 0 → F = 3, mf = 0 transition in the hyperfine
ground state of 133 Cs [61].
The challenge in optical frequency metrology is to coherently link optical frequencies (several hundreds of THz) to the Cs microwave standard. The first high precision
measurements of optical frequencies were achieved by connecting a high-harmonic of
the Cs standard to optical frequencies using a series of single frequency intermediate

4

Wavelength (µm)
0.657

Cs HCOOH CH3OH

10.8 3.39

1.55

1.06

0.778

CO2 CH4

C2H2

I2/2

Rb

0.532

0.563

0.486

Hg+ I2

H

Ca

LasersVisible

1010

1011

1013

1014

1015

88.4

Frequency (Hz)

200

266

384 456

Frequency (THz)

532
616
563

Figure 1.2: A map of the electromagnetic spectrum showing selected measured secondary standards and their corresponding frequencies.

phase-locked oscillators [36, 18, 44]. Aside from being used to provide high precision
measurement of optical atomic transitions, these harmonic frequency chains allowed formeasurement of secondary optical standards [see Fig. 1.2], which then serve as portableabsolute optical frequency references. Using the few available secondary optical standards, several bridging schemes, namely frequency interval dividers [73, 75] and comb
generators [21, 50, 42] had to be devised to span the remaining frequency gap between
the secondary optical standard and an optical transition of choice.
In the late 1970’s the modelocked laser was introduced to optical frequency metrology since its optical spectrum could be used to span several tens to hundreds of THz
in the optical [21]. Once one line of the frequency comb [see eqn. 1.1] is referenced to asecondary optical standard, the comb can be used as an optical ruler to measure large
optical difference frequencies [76, 65, 64].
With a potential optical bandwidth of 300 nm, modelocked Ti:S lasers set thescene for broadband comb generators. However, this potential was greatly increased in2000 when it was shown that using microstructured (MS) fiber [62] it was possible tocoherently extend the Ti:S comb over the entire visible spectrum. The increased optical
bandwidth makes it possible to use nonlinear self-comparison for direct measurement of

5the laser offset frequency, without comparison to a secondary standard. The simplified
referencing scheme using MS fiber enables absolute stabilization of the optical comb
using a single microwave reference [45, 11, 59]. Stabilization of the Ti:S optical comb
in this manner results in the production of greater than 106 coherently linked CW linesthat may be used for high precision absolute optical frequency measurement [40].

1.2Carrier-envelope phase stabilization
The above stabilization scheme for the Ti:S laser, aside from providing an enor-

mous simplification to optical metrology, also allows for control of the electric field of
the laser pulses [74, 45]. As seen in Fig. 1.1 (a), the phase evolution of the laser pulse
train is governed by the characteristic frequencies of the comb. This further feature of
the phase stabilized Ti:S laser now permits waveform synthesis of ultrashort pulses, inthat full characterization can be attained by controlling the laser pulse parameters, thespectral amplitude and phase via pulse shaping and now the phase of the optical carrier.
With the current opportunities for generating pulses with temporal durations of ∼ 2
optical cycles [17], the strong contrast of the field strength across a phase controlled
pulse can be used to influence the outcome of extremely nonlinear processes [58, 6, 3].
The applications of the stabilized modelocked laser to physics in general appear endless.
The introduction of the phase stabilized Ti:S laser to ultrafast science heralds a new ageof phase-control possibilities from the low to the high field regimes [58, 6, 3] and fromatomic to semiconductor [27] systems. The onslaught of absolute optical frequency andphase sensitive measurements, since the first realization of phase stabilization of Ti:S
lasers is a testament to the power of the technique.
Naturally, the above brief historical perspective skips much of the painstaking
work that has taken us from the development of the first laser, to high precision measurement, to phase control of physical processes. This thesis attempts to fill out a small

6slice of the work necessary for the generation of highly phase-stable modelocked lasers
for application to both optical frequency metrology and utltrafast science.
Much of the work that I have performed during the course of my graduate carrier has focussed on the development of the techniques necessary for the productionof a highly phase stable Ti:S modelocked laser, primarily within the context of thecarrier-envelope phase control of the laser pulse train [see Chap. 2]. This has involved
development of the laser itself and of the feedback system for stabilization. As in any
field of physics, pushing the envelope requires an impeccable tool for exploration ofscience. This naturally involves the necessary first task, when developing a scientificinstrument, of determining its effectiveness. With this consideration in mind I havespent a considerable amount of time on characterization of the laser and stabilization
loop. Given, the relative newness of the field, there have been a number of questions
concerning the basic stabilization techniques that have required some consideration.
One question that arose, when stabilization of Ti:S lasers was first demonstrated, waswhether external broadening in MS fiber, given its extreme nonlinear nature, would preserve the optical coherence of the comb. This question is addressed in Chap. 3, which
explores the nature of phase noise in MS fiber and determines its effect on the stability of
the carrier-envelope phase of the laser pulse train [28]. More generally, Chap. 4 discusses
the noise contribution from some of the additional elements comprising the stabilizationloop, as well as presenting measurements indicating the stability that has been attainedusing the stabilized laser system [26]. The present system allows for generation of lightpulses with a carrier-envelope phase that is coherent over tens of minutes! That means
that, given a laser repetition rate of 100 MHz, we can produce > 10 10 pulses with nearly
identical phase [25]. This ability to produce very highly phase stable pulse trains has
opened the possibility for exploration of phase sensitive experiments. To this end, we
have exploited the phase stability of the pulse train to investigate quantum interference
effects in semiconductors. In particular, using the stabilized output of the laser we7demonstrate directional control of injected photocurrents in GaAs [27]. The details of
this experiment are described in Sect. 6.1.
An application of the phase stable Ti:S laser, more toward metrology applications,is to use Ti:S lasers as a reference laser to which other lasers (single frequency ormodelocked) may by locked. Specifically, using of the Ti:S laser as a transfer oscillator,
for other modelocked lasers can permit coherent extension of the optical comb to the
UV or IR, or alternatively for generation of super spectra with the ultimate aim ofgenerating single optical cycle pulses. Here, we have demonstrated synchronization ofa modelocked Erbium-doped fiber laser. This experiment, described in Sect. 6.2, aside
from exploring the feedback mechanisms with which to control the comb parameters of
the fiber laser, also demonstrates an absolute comb at 1.5 µm [63].
One of the ultimate goals for phase control of the Ti:S laser is the possible realization of all-optical clocks [12, 80], which are hoped to one day rival the resolution of
the current primary atomic Cs microwave reference. All-optical clocks, using the Ti:S
laser as a clockwork to connect the optical and microwave regimes has been previously
demonstrated by Ye et al. and Diddams et al.. Final replacement of atomic Cs by aprimary optical standard, however, requires further scientific and technological maturity of the both the optical standard as well its possible transfer oscillator. To this end,
Chap. 5 discusses the construction of an octave spanning Ti:S laser that produces thenecessary bandwidth for self-referenced stabilization within the laser oscillator alone,
without the use of MS fiber [24]. A laser that may be stabilized without MS fiber brings
to the fore the possibility of a more reproducible, simpler system with a potential longer
term stability than stabilization systems based on MS fiber.

Chapter 2

Phase stabilization of modelocked Ti:sapphire lasers

The work horse for the experiments presented in this thesis is the Kerr-lens modelocked Ti:Sapphire (Ti:S) laser. It is a broadband optical oscillator whose bandwidth
may span up to several hundred nm’s in the near infrared. The large gain bandwidth ofthe laser (700 - 1000 nm) makes it ideal for optical frequency metrology and ultrafast
phenomena, whereby the former takes advantage of the large coherent bandwidth tospan several hundred THz in the the near infrared, and the latter for the formation of
femtosecond pulses [4]. The challenge in metrology is overcoming the frequency gap between the optical and microwave regimes. Ti:S lasers, because of the incredible stability
in the periodicity of the emitted pulses, result in a discretization of the laser spectrum
into a stable comb of equally spaced, coherently related CW frequency components.
This comb of frequencies may be used to provide a direct link between microwave andoptical frequencies, thus making the modelocked laser a powerful tool in optical metrology. Additionally, absolute stabilization of the laser spectrum allows for deterministic
control of the pulse electric field.
This Chapter is intended as an overview toward the concepts of phase stabilization of a modelocked Ti:S laser. The experimental details, optical and electronic, arecontained within App. A, which is written more as manual for reconstruction of theexperiment.

9a)Coherent addition ofmodes results in theformation of a pulse

λ λ/2

λ/3 λ/4b)c)Spectrum of lasing light

Spectrum of lasing lightννFigure 2.1: Schematic of pulse formation in the Ti:S laser cavity. a) The resonant modesof the cavity represent integer multiples of the fundamental cavity mode, frep = c/λ(optical cavity modes are 106 times higher in frequency than those depicted here.)If the field of longitudinal modes are added with the same phase, they result in theformation of a pulse b). c) Neglecting dispersion, the larger the number of coherent
modes comprising the light pulse, the shorter the temporal duration and the larger thecontrast in field strength across the pulse.

2.1

Time and frequency domain aspects of modelocked lasers
Formation of pulses in the cavity result from the coherent addition of extremely

narrow CW wavelengths that satisfy the resonant condition of the cavity [see Fig. 2.1].
This condition forces the optical spectrum to be composed of a comb of equally spaced
optical elements, νn , that are separated in frequency by,frep =

c
c
=
,
λ
2L

(2.1)

the fundamental resonant mode of the cavity. Here c is the speed of light and 2L isthe round trip cavity length. The mechanism behind pulse formation in the Ti:S cavity
is Kerr-lens modelocking, which is discussed in more detail in Sect. 5.1.3. Just briefly,
modelocking, as the term implies, forces a fixed phase relationship between the resonant
longitudinal modes of the cavity, νn . This coherent addition of the field of the optical

10frequencies, νn , results in the formation of a train of light pulses,
E(z, t) =n
XEν exp[ i {n ωn t + φCE (t)}].

(2.2)

Modelocking of the laser spectrum requires coherence between the longitudinal modesof the cavity, forcing each mode to have a common phase, φCE , defining the phase of
optical carrier. Essentially, each light pulse is a carrier-frequency that is modulated bya strong envelope function, A(z, t). Using this representation we can rewrite the above
equation in the form,E(z, t) =

n
X

Ã

z
− n trt
A t−
vg

!

"

cos ωc

Ã

z
t−
vp

!#

.

(2.3)

Here, ωc = N (2 π frep ), where N is a large integer multiplying frep up to the optical.
As shown in Fig. 2.1, the greater the number of coherently added modes comprising
the pulse, the narrower the pulse envelope in time, and consequently the stronger thecontrast in field strength across the pulse.
Neglecting cavity dispersion, vg = vp and consequently dφdtCE = 0. For this condition the laser pulse train is a series of identical pulses separated in time by one cavity
round trip, trt = 1/frep . The inclusion of cavity dispersion causes a difference between
the group and phase velocities, vg = c/ng and vp = c/np , of the light pulses in thedn. This results in an intra-cavity time-rate ofcavity, where np = n and ng = n + ω dωchange in the relative phase between the pulse envelope and the carrier wave,
dφCE= 2π ωc
dt

Ã

1
1
−
vg
vp

!

c = 2π ωc (ng − np ) = 2π ωc2dn(ωc )
.
dω

(2.4)

This phase, φCE , common to each optical mode [see eqn. 2.2] results in a rigid
spectral shift, f0 , of the entire optical comb [16, 78, 74]. As a result, each optical comb
element, νn , is expressed as
νn = f0 + n frep ,

(2.5)

11
νn = n frep + fo

E(ν n ) = Ene

f0
f0 =
0

frep=1/trt

νi[ 2π nfrep t +φ CE (t )]1 dφCE
2π dt

f 
φCE (n t rt ) = n 2π 0  + φ 0
f rep 

Fourier Transform∆φCE = 2π fo /frep

E(t)

φ0

∆φCE+φ02∆φCE+φ0

Phase accumulated in one
cavity round tript
1/ frep = τFigure 2.2: Summary of the time and frequency representations of the Ti:S light field.
φ0 is a phase offset whose origins and importance are explained in detail in Sect. 4.1.

where,
f0 =1 dφCE
.
2π dt

(2.6)

To summarize, the modelocked spectrum of a Ti:S laser is a comb of frequencies whose
spacing is governed by the laser repetition rate, frep , and whose absolute position isdetermined by the laser offset frequency, f0 . Given the fundamental connection between
the pulse train and laser spectral components, f0 and frep govern the evolution of thephase of the pulse electric field, φCE . Figure 2.2 summarizes this discussion.
Given that both frep and f0 are stabilized absolutely the frequency spectrumof the Ti:S laser is an absolute optical comb with approximately 106 coherent lines
spanning the visible to near IR. As a result the stabilized Ti:S laser is a convenienttransfer oscillator to which other laser, single frequency or pulsed, may be synchronized.
Additionally, phase stabilization of the laser results in a train of pulses with a phaseevolution determined by eqn. 2.6.

12
Environmental perturbations, however, result in instability in the comb components. Specifically, perturbations within the laser cavity are responsible for fluctuations
in f0 due to pointing and other instabilities. Such instabilities yield a change in the
net cavity dispersion and also in amplitude-to-phase conversion in the laser crystal due
to extreme nonlinearity resulting from high intra-cavity powers [38, 39]. Once a pulse
exits the laser, however, its phase evolution is governed by dispersion outside the cavity.
Linear dispersion imparts a common phase shift to the entire pulse train, not a shift in
the frequency of the comb. Nonlinear dispersion, in contrast, can result in extra-cavity
phase noise due to fluctuations in the intensity of the laser output [see Chap. 3].
Because of these difficulties, the free running Ti:S laser does not serve as a phasestable source of short pulses, nor does it provide a frequency-stable comb. As a result,active stabilization of the comb parameters is necessary.

2.1.1

Measurement of the laser comb parameters

Measurement of the comb parameters for stabilization is obtained using frequency
difference measurement, whose technical maturity and corresponding convenience is theresults of many year of development. A huge advantage of the optical comb provided
by the modelocked laser is that, given suitable bandwith, the comb parameters may bemeasured absolutely without the cumbersome use of a secondary optical standard. For
instance, the laser repetition rate is directly measurable using a fast photodetector (PD).
This is possible because adjacent comb lines in the laser spectrum optically heterodyne
yielding an rf spectrum of harmonics of the repetition rate [see Fig. 2.3],|νm − νn | = |(nfrep + f0 ) − (mfrep + f0 )| = (m − n)frep .

(2.7)

Given a cavity length of ∼ 3 m, frep = 1/trt is an rf frequency close to 100 MHz,
which allows for direct comparison to a microwave standard. Because f 0 is bounded13
Optical Comba)b)rf Comb

0frep

n frepf
2 frep 5 frep

14 frep

Figure 2.3: Depiction of homodyne detection of frep . The component in the rf spectrum
at frep results from beating between all the adjacent components in the optical comb.
The components at n frep are generated by beating between optical comb components
separated in frequency by n frep . Since all optical comb components contribute to
the beat signal, the generated rf spectrum exhibits a very strong signal-to-noise ratio(SNR) for each rf comb component. The rf signal that results from this interference, incontrast to that shown here, typically exhibits a gain roll off determined by the speedof the photodetector.

in frequency by frep , it is also an rf frequency. However, because the offset frequency
is common to each comb line, the linear optical heterodyne observed when measuring
frep is insensitive to f0 [see eqn. 2.7]. As a result, f0 is determined by nonlinear selfcomparison, or also possibly via comparison against a secondary optical standard.

2.2

Self-referencing
The major advance in 2000 was measurement of f0 independent of a secondary

optical standard. The measurement scheme was first conceived of in the frequency
metrology community and proposed by Telle et al. in 1999 [74]. The idea was to use theTi:S laser spectrum for self-comparison. This is achieved by cross-referencing harmonics
of the spectral extremes of the optical comb, |N (nfrep + f0 ) − M (mfrep + f0 )| [45]. The

14a) Optical Combf0b) rf Combfrep

2 frep3 frep4 frep5 frep

n

n+1
frep

f
f0

(n+1) frep- f0

n frep+ f0

Figure 2.4: a) Depiction of the measurement of the laser offset frequency, f 0 . The rf
spectrum generated via self-referencing of the laser offset frequency can be thought of asheterodyning two optical combs offset in frequency by f0 . b) There are two components
to the rf spectrum, the signal produced by each optical spectrum separately that resultsin the rf spectrum of harmonics of frep . The second is the cross-interference of the optical
spectra that generate f0 sidebands on the repetition rate harmonics. The sidebands at
the offset frequency are enhanced by the collective addition due interference of themany contributing comb components. The rf signal that results from this interference,in contrast to that shown here, typically exhibits a gain roll off determined by the speedof the measurement device.

harmonics are chosen such that spectral overlap is achieved, that is N n = M m. If thiscriteria is satisfied, optical interference of the harmonics results in a beating at the rf
difference frequency (N − M )f0 [see Fig. 2.4].
Although the implementation of the self-referencing scheme may appear trivial,
the major drawback is the requirement of an extremely large bandwidth. To perform thesimplest optical comparison, where N = 1 and M = 2, the bandwidth requirement is an
optical octave (∼500 nm). The typical bandwidth available from a Ti:S laser, however,is on the order of 100 nm. In 2000, microstructure (MS) fiber technology provided asimple solution to the bandwidth problem as it enabled continuum generation using the15b)

cladding

20 dB/div

a)

ν2n

νn
generated
continuum

initial
pulse

10 µm
core (2 µm)

600c) four-wave mixing:

1000

1200

2ν2-ν12ν1-ν2

ν1 ν2

800

wavelength (nm)

2ν1

ν

ν1 ν2

2ν2

ν

Figure 2.5: a) Cross-section of microstructure fiber developed by Lucent Technologies
to work at Ti:S wavelengths. b) The broadening of a Ti:S input pulse to an opticaloctave with the octave points marked. The ν and 2ν point of the continuum used for
self-referencing are marked. c) Schematic depicting the generation of new wavelengths
via four-wave mixing of light at ν1 and ν2 . The dashed line represents the square of the
field at frequency νi that is optically heterodyned with the field at νj to produce lightwith frequency 2νi − νj .

pulse energy available directly from the unamplified laser output [62]. More recently,
however, ultra-broadband laser have been demonstrated that allow for self-referenced
measurement of f0 using the laser spectrum directly [see Sect. 5.2].

2.2.1

Microstructured fiber

One method used to increase the spectrum of a pulsed laser is external broadening
in optical fiber. This technique takes advantage of the high intensity generated andmaintained by confining a light pulse within a small mode field diameter defined by thefiber core. Broadening is obtained in optical fiber via the third-order nonlinear process
of four-wave mixing, which results in photo-mixing of the incoming light pulse withitself. This process results in the generation new frequencies at (2nfrep − mfrep ), wherem and n define different comb lines of the frequency spectrum.

16
Continuum generation of modelocked nJ laser pulses from a Ti:S oscillator alone
requires a long nonlinear interaction length in optical fiber. The effect of material
dispersion on the light pulses launched into standard telecom fiber, however, limits thenonlinear interaction length to only a mm or so. This is because the zero dispersion
point for telecom fiber is at ∼1.31 µm whereas Ti:S lasers are centered ∼ 800 nm. The
difficulty posed by dispersion may be overcome by using microstructured (MS) fiber,
which is engineered such that the geometrical waveguide dispersion cancels the material
dispersion within the Ti:S gain bandwidth.
The cladding of MS fiber is composed of a honeycomb structure of air holes [seeFig. 2.5 a]. By changing the filling fraction of the air-glass cladding and the core size thezero dispersion point of the fiber may be shifted by several hundred nanometers, making
MS fiber a convenient method for continuum generation for different modelocked laser
sources in the visible and IR [53]. Additionally, the large index step between the coreand cladding results in strong confinement of light within the fiber core, and in single
mode propagation of light over a bandwidth of at least an optical octave. This is because
the large index step causes very weak coupling to higher order modes.
For MS fiber designed to work at Ti:S wavelengths, the high confinement coupledwith the zero dispersion point at 780 nm allow for high intensities to be maintained
over a much longer interaction length, thereby leading to the incredible broadening
observed in Fig. 2.5 b). Broadening in the fiber allows for generation of coherent lightfrom 400 nm to greater than 1300 nm, more than sufficient to perform a self-referenced
measurement of the laser offset frequency.

2.3

Experimental setup
Using the continuum generated in MS fiber it is possible to perform a measure-ment of the laser offset frequency using the self-referencing technique described above.
Self-referencing is implemented by using a ν–to–2ν interferometer [see Fig. 2.6 b]. This17b)a) Self-referencing

ν-to-2ν interferometer
530nm
AOM

1060nmeiφ nφn = 2π (n frep + fo) t + φ






PBS

e iφ 2nx2

φ2n = 2π (2n frep + fo) t + φ






νn



ν 2n







E(t) ∝ Cos(2π fo t + φ (t))

IF

×2



PD
P

PBS – polarizing beamsplitter
AOM – accousto-optic modulator
P – polarizer
IF – interference filter (530 nm)
×2 – frequency doubling crystal
PD - photodetector

Figure 2.6: a) Depiction of self-referencing. Optical heterodyne between harmonics atthe edges of the spectrum results in amplitude modulation of the light at frequency 2ν nwith a modulation frequency, f0 . b) Experimental realization of self-referencing using a
ν–to–2ν interferometer. For further experimental details consult Sect. A.5.1.

interferometer performs the simplest N = 1, M = 2 harmonic comparison [Fig. 2.6][45], requiring the least number of nonlinear stages but the most optical bandwidth.
The ν–to–2ν interferometer uses a Mach-Zender geometry. Light entering theinterferometer is split by a dichroic beamsplitter that reflects the low frequency endof the MS fiber broadened spectrum (for simplicity 1060 nm), and transmits the highfrequency end of the spectrum (for simplicity 530 nm.) Light in the 1060 nm arm of
the interferometer is then frequency doubled in a nonlinear crystal (for the experiments
here either 1 cm LiB3 O5 (LBO) or 4 mm β Barium Borate (BBO) crystals cut for
type I phase matching were used.) Light from the arms of the interferometer are thenrecombined at a beamsplitter. Measurement of an interference signal requires temporal,
polarization, spectral, and spatial overlap between the two beams of the interferometer.
The former requires adjustment of the relative path length of the two interferometerarms. The latter is achieved by matching the mode sizes, the wavefront curvature of thebeams and by obtaining good spatial and angular overlap between the two beams. Oncethe beams are recombined, an optical filter (center = 530 nm with 10 nm bandpass)and polarizer are used to satisfy the second and third conditions. Aside from ensuring

18
spectral overlap, an interference filter or grating selects out only that light contributing
to the beat signal. This helps to increase the signal-to-noise ratio (SNR) of the beatsignal and makes sure that the photodiode (PD) does not saturate.
The comb-offset frequency is measured as amplitude modulation of the filteredpulse train between the green portion (∼530 nm) and the doubled near IR portion(∼1060 nm) of the spectrum. This modulation is measured using a fast PD and appears
as amplitude sidebands on the repetition rate rf spectrum. Once the heterodyne beatsignal is measured, it is electronically filtered, amplified and used in a feedback loop forcomparison against a known rf reference.

2.4

Phase stabilization of the Ti:sapphire laser
Absolute stabilization of the laser spectrum is obtained by phase locking both

frep and f0 to synthesized reference frequencies. A phase lock is obtained when thefrequency and phase of a slave signal tracks that of a reference signal by the action of aphase locked loop (PLL) [see Fig. 2.7]. For electronic comparison, the PLL uses a mixer
that measures the difference frequency between the slave and reference signals. If thetwo signals have the same frequency, the output of the mixer is a phase comparison thatcan serve as an error signal to actively correct for errors in the slave signal [see Fig. 2.7].
A loop filter (LF) is used to electronically shape (integrate, filter and amplify) theerror signal to match the transfer function of the negative feedback actuator. Negative
feedback to the laser acts as a voltage-to-frequency converter that uses the electronic
feedback signal to correct for instabilities in frequency/phase.

2.4.1

Stabilization of the laser offset frequency

In the case of the offset frequency, stabilization is actuated by either tilting thelaser end mirror after an intra-cavity prism sequence, or via amplitude modulationof the laser pump power [see Fig. 2.7]. The latter is obtained by placing an AOM

19a)
Ti:sapphire laser

MS

AOM

EM

PUMP

fAOM
530nm
AOM

1060nmfAOM + fo

BBO

fAOM
BP

+ 60 dB
LF

PBS

ν-to-2ν

Locking electronicsb)
BBO – β Barium Borate

fref

fref
LF

f0

V

F

AOM – acousto-optic modulator

stabilized
outputMS – microstructure fiber
EM – end mirror
PBS - polarizing beamsplitter
LF – loop filter
BP – electronic bandpass filter

Figure 2.7: Experimental apparatus of a carrier-envelope phase-stabilized fs Ti:S laser.
The box in the lower left shows the ν–to–2ν interferometer used to measure f 0 . An
AOM is included in the ∼530 nm arm of the interferometer to shift the frequency of one
arm relative to the other such that f0 may be set to zero [see Sect. A.5.1.2]. For further
details concerning the ν–to–2ν interferometer and the stabilization electronics please
consult Sect. A.5.1 and Sect. A.6, respectively. b) Block diagram of a phase locked loop
(PLL). An input signal to be phase locked, f0 , is compared against a reference frequency,
fref , using a mixer. The phase difference between the two serves as an error signal thatis shaped (gain and filtering) to match the characteristics of the feedback actuator. The
shaped voltage signal is then fed back to a voltage-to-frequency converter, here the Ti:S
laser, to correct for the phase disparity.

in the path of the laser pump beam [59]. This method takes advantage of nonlinear
amplitude-to-phase conversion in the Ti:S crystal to control the carrier-envelope phase(the speed of the feedback of this method is limited by the upper state lifetime of the
Ti:S crystal, ∼ 10 µs.) The former method allows for control of f0 by adjusting the phasedelay between successive longitudinal cavity frequencies. After a compressor prism pair,intra-cavity light is linearly dispersed on the laser end mirror (EM). Tilting of the EM
in the horizontal plane results in the addition of a linear time delay vs frequency, whichleads to a constant shift in frequency between each pair of comb components. Tilting is20
actuated by placing the laser EM on a piezo-electric actuator (PZT), whose speed (< 30
kHz) is determined by the engineering of the mirror + PZT system [see Fig. A.11].
If stabilization of the Ti:S laser is to be used only to produce a CE phase stabletrain of pulses, it is not necessary to stabilize the frequency comb spacing. Given
eqn. 2.6, the accumulated pulse-to-pulse carrier envelope phase between adjacent pulses
is linearly related to the ratio of the comb offset frequency to the laser repetition rate,
by1∆φCE =

Z trt(2πf0 )dt = 2π

f0
frep

(2.8)

As a result, stabilization of f0 to a reference signal derived from the laser repetition
rate [see Sect. A.6.1.1] is sufficient for producing a phase stable train of pulses, evenif frep is free running. Thus the relative stability of the offset frequency is linked tothat of the repetition rate (in this case one establishes stability in the pulse phase, notcoherence of the optical comb elements.) This significantly reduces the complexity of thestabilization system since neither the offset frequency, nor the comb spacing need to bereferenced absolutely. It is the stability in the ratio between the two which determines
the coherence of the pulse train [see Chap. 4].

2.4.2

Stabilization of the laser repetition rate

Stabilization of the comb spacing uses a high harmonic of the laser repetition rate
for stabilization. This is necessary given that errors in frep are multiplied up by 106 toreach optical frequencies [see eqn. 2.5]. Because phase noise power on the repetition rate
harmonics in the rf spectrum scales as the square of the harmonic number, in principle,
the higher the harmonic used for stabilization the better the phase sensitivity obtained. 2
The highest sensitivity measurement of the comb spacing is obtained using an optical
1

This equation holds only for the simple case that f0 is a constant. In reality, of course, f0 is subject
to phase noise, whereby ∆φCE picks up a noise term that takes in account noise. This issue is exploredin detail in Chap. 4
2
The quality of the microwave reference source may be poorer, however, at higher frequency.

21
as opposed to rf reference [see Fig. 5.8] and is the subject of of all-optical clocks [80, 12].
Stabilization of frep is similar to that used for the offset frequency. The rf spectrum of the repetition rate harmonics are measured on a fast PD. A high harmonic fromthe rf spectrum is then filtered and amplified such that it may be compared against a
synthesized reference frequency. Frequency comparison, which is again obtained using
a mixer, results in an error signal that is shaped by a loop filter and used for negative
feedback to the laser. Corrections to frep are controlled by adjusting the laser cavity
length by either placing the laser end mirror or output coupler on a PZT.
Instabilities in frep correlate directly to time changes in optical cavity length thatare caused by time changes in the intracavity indices of refraction. Such changes also
naturally influences dispersion within the laser cavity thus affecting f0 [34].

2.4.3

Passive stability of the Ti:sapphire laser

In theory, if the electronics and feedback are infinitely fast, feedback to the laser
will take out any intrinsic laser instabilities. Limits to the speed of the PLL, however,make it that passive stability is important for producing a stable laser. As a result of thisconsideration, the laser used here is engineered to reduce the effects of environmental
perturbations to stability [see Fig. 2.8].
The three primary factors affecting phase stability are mechanical vibrations, airpressure changes, and temperature changes. Vibrations are addressed by using low profile and solid mirror mounts throughout the laser cavity, which is not typical practicefor ultrafast lasers. Of course, high quality optical tables with vibration isolation legs
are required. Enclosing the laser in a rigid box helps to reduce vibrations due to acoustics and also addresses the problem of air pressure changes. Air pressure fluctuations
are detrimental because they change the density of the air in the cavity, which in turnchanges the effective optical path length. Although this is a small effect, it can easily change the cavity length a wavelength or more, which in turn, changes the comb

22

M3

DI

M2

VE
R

xtal
M1

EM

OC

PD

Figure 2.8: The Ti:S laser used in our experiments. Construction details are in the text.

frequencies by frep or more. Although the long term drift of f0 and frep caused by temperature changes can easily be compensated by the feedback loop, due to the limited
actuator dynamic range, the duration of locking is limited.
To obtain long term locking, the laser is constructed on a 5 cm thick cast aluminum plate. Typically, the thermal mass of such a plate is sufficient so that active
temperature control is not required, although that depends to some extent on the ambient temperature fluctuations. A cavity mirror mounted on a PZT with one wavelength
worth of travel, together with thermal stabilization via a gentle water flow throughthe laser base plate, can usually maintain feedback lock without interruption for many
hours. A secondary benefit of the cast plate is a greater immunity to higher frequency

23(airborne) vibrations. By carefully isolating the cast plate (typically with rubber stoppers/springs having a resonant frequency of 400 Hz) from the table and arranging therigid enclosure to entirely span the cast plate, the airborne vibrations are not strongly
coupled to the laser. Instead, for the most part, they are directly shorted to the optical
table. Additionally, beam tubes are also used to cut down on pointing and pressure
instabilities of the pump beam as it is directed to the Ti:S laser. The same generalprecautions are taken into account in the ν–to–2ν interferometer design.

2.5

The stabilized comb
Once both the comb spacing and the laser offset frequency are stabilized, using

the techniques described above, the stabilized Ti:S laser may provide both a source ofphase-stable pulses as well as an absolute optical comb in the IR. The stability of thetwo characteristic frequencies, however, determines the quality of the optical comb andthe pulse train.
In metrology, the width of the comb lines and uncertainty in their absolute position present the primary limits on precision and accuracy. Because the accuracy andstability of the comb components of the Ti:S laser are fundamentally limited by thesources to which they are stabilized, much effort must be placed into development of
the primary reference [37]. In the microwave regime, the primary standard is the Cs
clock transition, although possibilities exist in the optical. Stabilization of the Ti:S frequency comb, for example to neutral atomic Ca [57] and Hg+ [71] or molecular Sr [79],for instance, represent the first generation of optical clocks that in time should rival thecurrent Cs standard (as an application modelocked lasers to all-optical clocks see ref.
[12, 80] and also Sect. 5.2.2.)
Phase stabilization of a Ti:S laser for use in waveform synthesis of ultrashortpulses has different requirements from those of optical frequency metrology. In ultrafast
experiments that exhibit sensitivity to the electric field of the light pulses, the primary

24
concern is the carrier-envelope coherence of the laser pulse train, not the stability of theindividual comb components. Unlike optical frequency measurement, ultrafast applications are concerned with the relative stability of f0 to frep [see eqn. 2.4.1]. The stability
in the ratio of f0 to frep is paramount to the coherence of the pulse train since instabilities result in accumulated phase error with time [see eqn. 2.4.1]. The accumulationof a π phase error results in loss of coherence of the pulse train and thereby determines
the time duration over which a phase sensitive measurement may be performed. Thisis significantly different from optical frequency metrology, where long time averaging
yields an increase in measurement stability of an optical line. Additionally, stabilization of the laser pulse train does not require the use of an absolute reference. In absolute
optical frequency measurement, as discussed in the previous paragraph, the reference
serves as the primary limit to precision and accuracy. In contrast, carrier-envelope
phase stabilization using an rf synthesizer, in practice, is fundamentally limited by thecharacteristics of the PLL, not the frequency reference.

Chapter 3

Phase noise characterization of microstructured fiber

One of the first concerns plaguing the self-referencing technique was whether
external broadening of Ti:S pulses conserved the coherence of the frequency comb. This
was an issue considering the extremely nonlinear process that would be necessary. Before
the development of MS fiber, it was common to employ broadening of amplified pulses
by focusing the pulses into a short piece of glass [48]. The high intensities provided
by the amplified pulses allowed for self phase modulation (SPM) of the laser light toproduce a continuum of wavelengths. Although a common practice, it was only provenin 2000 by Bellini and Hänsch that this broadening technique preserved the coherence ofthe laser pulse train [8]. The coherence of the continuum was demonstrated using a quasi
double slit experiment where amplified pulses were focused to two points, closely spaced,in a piece of glass. The diverging continuum from both points were then observed toyield interference fringes demonstrating spatial and phase coherence of the continuum.
This helped to validate the possible notion of self-referencing, although it was not first
demonstrated using this broadening scheme.
In 1999 Lucent Technologies demonstrated extreme spectral broadening of unamplified Ti:S pulses using specially designed microstructured (MS) fiber [62]. The
combinations of MS fiber, Jan Hall and the Cundiff group proved a potent combination,
whose net result was the first demonstration by Jones and Diddams of self-referenced
stabilization of a Ti:S laser [45]. One of the main concerns was whether ref. [45] demon-

26
strated true coherence of the pulse train. This claim was questioned by the scientificcommunity whose concern was whether the use of MS fiber for broadening resulted inphase noise on the laser pulse train. Clearly, it had been demonstrated by Jones et al.
that the broadened comb was well behaved enough to produce a clean f 0 beat signal andthat the resultant beat was possible to stabilize. Out-of-loop coherence of the resulting
pulse train, however, was demonstrated using a cross-correlator that allowed for thegeneration of a fringe resolved correlation trace between one pulse and a pulse emitted
two cavity round trips later. Due to the complexity (length considerations) of the correlator, the coherence measurement was limited to 20 ns, definitely not long enough toclaim that the use of MS fiber allowed for long term phase coherence in the laser pulse
train [see Chap. 4]. It remained to be demonstrated, as it had been for broadening of
amplified pulses in a glass plate, that the continuum generated by MS fiber preserved
the coherence of the laser pulse train.

3.1

Amplitude-to-phase noise conversion in microstructured fiber
The concern of using MS fiber for external broadening of Ti:S pulses, was the sameas it had been for SPM generated light of amplified pulses in glass. That is because
the highly nonlinear process used to broaden the laser spectrum might corrupt thestability of the offset frequency, thereby overwhelming the true value of the CE phase.
From the results by Jones et al. it could be assumed that this generated noise was notstrong enough to make it impossible to stabilize the broadened spectrum. However,even if one could perfectly stabilize the output of the fiber, a problem still remains.
When the stabilization loop for the laser is closed, the PLL can only use the laser tocompensate for errors in f0 . The stabilization loop, however, contains many possible
noise contributions apart from the laser, for instance the ν–to–2ν interferometer [seeSect. 4.3.2], PD’s, electronics, and the MS fiber. If any of these components adds noise
to the offset frequency, the PLL will write the noise onto the laser output as it corrects

27
a)fo + εL

εE
ν-to-2νεLb)fo –εELASER

εE

fo + εL+ εE

ν-to-2ν

εL
LASER-εL- εEfo

in loop

fo + εLlocking
electronics
in loopfo - εE

out-of-loop

out-of-loop

Figure 3.1: a) Schematic of the noise sources within the laser stabilization loop. A
ν–to–2ν interferometer measures the laser offset frequency, f 0 , plus noise from the laser²L . Measurement of f0 adds extracavity noise, ²E , yielding the net sum of the noise on
f0 of (²L + ²E ). When the stabilization loop is closed, the locking electronics correct for
the error on f0 via negative feedback to the laser b). The feedback signal suppresses
the laser noise completely (given it has an infinite bandwidth) but in the process writes
the extracavity noise onto the laser output, −²E . Subsequent determination of the laser
offset frequency using an in-loop interferometer results in perfect cancellation between−²E written onto the laser and ²E resulting during measurement.

for the phase error [see Fig. 3.1].
A particular concern with MS fiber is that small amounts of amplitude noise on
the input pulses will be converted to phase noise in the fiber. This occurs because of theindex of refraction’s dependence on intensity, n(I) = n0 + n2 I, where n0 is the linear
index, n2 is the nonlinear index and I is the intensity of the light in the fiber core (i.e.,
including the effective area). Propagation of light with frequency ω over the length of
the fiber, l0 , results in phase shift in the carrier-envelope phase,∆φ = ω l0

Ã

1
1
−
vg
vp

!

=

ω 2 l0 d(n0 + I n2 )
,
c
dω

(3.1)

resulting from the group and phase velocity differences, vp = c/np and vg = c/ng , ofdn
the fiber core, where np = n and ng = n + ω dω.

Linear dispersion in eqn. 3.1 only adds a constant phase shift to the pulse train,
but does not result in added instability to f0 . The nonlinear contribution of the Kerreffect to the carrier-envelope phase, φN L , results in phase noise. This same process isalso responsible for intracavity generation of phase noise in the Ti:S laser crystal due to28
fluctuations in laser pump power [59]. The time-dependent shift is directly proportional
to the dispersion of n2 and the change in intensity (∆I), that is,
φN L =

ω 2 lo dn2∆I = CAP ∆P.
c dω

(3.2)

This equation defines the amplitude-to-phase conversion coefficient for microstructure
fiber, CAP , which relates the nonlinear shift in the carrier-envelope phase, φN L , to achange in power, ∆P , on the input beam. This definition for CAP is valid, however,
only for a very narrow parameter space, determined by pulse chirp, fiber length, opticalfrequency.
A ν–to–2ν interferometer is used to measure amplitude-to-phase (AM-to-PM)
conversion in MS fiber [Fig. 3.2 a]. This is done by imposing a known amplitude
modulation on the laser output, P (t) = Po + ∆P sin(ωmod t), before the fiber. The
applied amplitude modulation of the laser intensity is converted to nonlinear phase
1 dφN Lmodulation, φN L (t) = CAP P (t), or in turn, frequency modulation on f0 (t) = 2π
dt =
ωmod
2π

CAP ∆P cos(ωmod t). As a result, the measured frequency deviation of f0 , ∆f ,should be directly proportional to the deviation in power, ∆P , that is,∆f = fmod CAP ∆P.

(3.3)

The rms frequency deviations of f0 resulting from AM-to-PM in MS fiber are
determined using a frequency-to-voltage (F-to-V) converter. The F-to-V converter measures the FM fluctuations on a high frequency carrier, and converts these to a voltage signal, whose amplitude is directly proportional to frequency deviation [see Sect. A.7.2.2].
Unfortunately, because the F-to-V converter only has a dynamic range of 1 MHz, its
use requires stabilization of f0 . The offset frequency, measured in the ν–to–2ν interferometer, is stabilized by tilting the laser end mirror with a net servo bandwidth < 1kHz. To ensure that the phase noise generated by the fiber is not transferred to thelaser, the laser power is amplitude modulated at a modulation frequency, fmod , significantly outside the servo bandwidth (> 10 kHz). This also helps to reduce contamination

29

a)

b)
100

PUMP
LCM

MS

ν-to-2ν

f0

F V

FFT

Std. Dev (rad.)Ti:sapphire laser

polarizer

CAP= 5.93 rad/mW
80
60
40
experiment
20

linear fit

LCM - liquid crystal modulator
FFT - signal analyzer
F-V - frequency to voltage converter
MS – microstructure fiber

0.5

c)

1.5

2.0

20

30

2.5

d)
20

250

15

200

10
fmod = 10 kHz
fmod = 20 kHz
fmod = 30 kHz
fmod = 40 kHz

5
0∆f (kHZ)∆f (kHZ)

1.0

Modulation depth ∆P (mW)

150
100
50
0

0.0

0.5

1.0

1.5

∆P (mW)

2.0

2.5

0

10

40

fmod(kHz)

Figure 3.2: a) Experimental setup used to determine the amplitude-to-phase conversion
coefficient, CAP , for MS fiber. Amplitude modulation of the laser power into the fiber isobtained using a liquid crystal modulator and a thin polarizer. A frequency-to-voltage
converter (F-V) is used to distinguish between amplitude and phase noise on the beatsignal and an FFT is used after the F-to-V converter to measured the frequency deviations at ωmod . b) The measured deviations in f0 , ∆f , as a function of modulation depth,∆P performed at a modulation frequency fmod = 10 kHz. A linear fit to the experimentally measured deviations results in an amplitude-to-phase conversion coefficient, C AP
for a 4.5 cm piece of MS fiber of 5.93 rad/mW at a 100 MHz repetition rate with anaverage input pulse power of 50 mW for a minimum temporal pulse width of 15 fs at the
input to the fiber. c) The observed frequency deviations in f0 due amplitude-to-phaseconversion in MS fiber as a function of different modulation frequencies. d) As expected
given eqn. 3.3, the frequency deviation, ∆f , is linearly proportional to the modulation
frequency, fmod .

of the measurement due to imperfect stabilization of f0 that results in an increase inbroadband noise below 5 kHz. This contamination is further reduced by narrowing thedetection bandwidth of the signal analyzer (FFT) that is used to analyze the signal
from the F-to-V converter.

30
By varying the amplitude modulation depth, ∆Prms , and comparing this to theresulting change in frequency deviation, ∆frms , it is possible to determine CAP =∆frms /(fmod ∆Prms ) [see Fig. 3.2 b]. The strength of the rms deviation in frequency,∆frms , is detected at the amplitude modulation frequency, 10 kHz, using a signal analyzer, [FFT in Fig. 3.2 a]. Each data point in Fig. 3.2 b) measures the rms frequency
deviation versus amplitude modulation depth. For a refractive index with only first orderdependence on intensity, the strength of the phase modulation should increases linearlywith power modulation. The experiment was repeated for four different modulation
frequencies. By fitting the data obtained from these different modulation frequencies
to a straight line we determine an average value of CAP = 5.93 rad/mW. This value
for CAP was determined using a 4.5 cm piece of fiber with an average coupled power of
50 mW at a 100 MHz repetition rate. The pulses at the input of the fiber were chirp
compensated to yield a ∼ 15 fs pulse duration. This was confirmed by using an autocorrelator containing the same optical elements. Consequently, this translates to C AP
= 593 rad/ nJ. Knowledge of CAP for the fiber then makes it possible to determine thecontribution of fiber phase noise to the carrier-envelope phase by measuring the powerfluctuations on the laser output [see Sect. 4.3.1 for details].
The validity of this measurement, however, is limited to a very narrow parameter
space, but one that is typical for our system. A more thorough examination of MS
fiber noise would require systematically studying the affect on CAP due to changes indifferent parameters such as pulse chirp, coupled power, and fiber length. On the subjectof fiber length, we believe that the conversion coefficient, given compressed pulses, may
be relatively insensitive to propagation length. This is possible since, for short pulses(∼ 20 fs), most of the broadening appears to occur within the first few mm of fiber.
Short pulses incident to the fiber exhibit high enough intensity to allow for broadeningof incident ∼ 40 nm bandwidth light to an optical octave within 3-4 mm of MS fiber,
sufficient for measurement of f0 . The rapid broadening in the fiber results in heavy

31
chirp of the wavelength components not at 770 nm (zero dispersion point of the fiber)
which results in a rapid decrease in intensity within only a few mm’s of fiber. Therefore,
the amplitude-to-phase conversion coefficient should only exhibit a strong dependence
on fiber length within the first few mm’s of fiber.

3.1.1

Transfer of fiber noise onto the output of the laser

Next it is interesting to determine whether fiber generated phase noise is transferred onto the laser output as a result of the stabilization process. To gain insight
into the transfer of extra-cavity sources of phase noise, a measurement of the noisemust be performed independent of the stabilization loop. Such a measurement is often termed out-of-loop, and may be obtained by running two ν–to–2ν interferometerssimultaneously, each with its own piece of MS fiber.
To measure the transfer of fiber noise to the laser via the action of the servo loop,
sinusoidal amplitude modulation is imposed only on the light into the fiber in the firstinterferometer [Fig.

3.3 a]. This will result in contamination of f0 that is then used

in a feedback loop for locking the laser. The action of the servo loop is such that thephase error is compensated using the laser as an actuator. The error, then written onto
the output of the laser, is measured using an independent ν–to–2ν interferometer [28].
Figure 3.3 a gives a sample set of data, where the sinusoidal deviations in f0 due tophase modulation from the fiber are measured using a frequency counter. To make surethat the frequency noise from the fiber was transferred to the laser, modulation of thelaser power was at 0.1 Hz, well within the servo loop bandwidth.
Aside from demonstrating transfer of phase noise onto the laser output, this outof-loop measurement may also be used to compare with the previous measurement of
CAP in Sect. 3.1. By varying the modulation depth of the laser power coupled into the
MS fiber in the first ν–to–2ν interferometer and measuring the rms frequency deviationsof f0 on the second we once again determine CAP = 2∆frms /(fmod ∆Prms ) [see Fig. 3.3

32a)% modulation depthb)

Ti:sapphire laser

0

1

2

3

PUM
P
LCM

MS

δin

ν-to-2ν

ELECTRONIC
SMS

ν-to-2ν

LOCKING

δout

300

3

200

2

100

1

0
COUNTERstd. dev. (radians)std. dev (mHz)

4

0
0.0

0.4

0.8

1.2

1.6

mod. depth (mW)δ (Hz)

c)

1
0.5
0
0.5
-1
0

50

100

150

200

record length (s)Figure 3.3: a) Experimental setup used to determine whether amplitude-to-phase conversion in MS fiber is transferred onto the laser output by the action of the servo loop.
The experiment uses a dual interferometer setup, where AM noise on the laser light toone fiber (with the stabilization loop) induces phase noise on f0 . This noise is transferredby the action of the servo loop onto the output of the laser, whereby the deviations induced in f0 are measured using a second, independent (out-of-loop) interferometer using
a frequency counter. A sample data set is included that shows the sinusoidal deviations
in f0 due to sinusoidal modulation of the laser power into the locking interferometer.
This demonstrates the transfer of fiber noise onto the laser by the action of the servo
loop. b) The measured deviations in f0 , ∆f , as a function of modulation depth, ∆P . A
fit to the measured data CAP =6.02 rad/ mW for a 5 cm piece of MS fiber at a 100MHz
repetition rate with an average input pulse power of 40 mW.

b].
Each data point in Fig. 3.3 b) is determined from a time record at a 1.0 s gate
time over 200 s [see Fig. 3.3 c]. Intrinsic phase noise, η, due to imperfect stabilizationof f0 , measured at zero modulation depth, adds in quadrature with those due to AMto-PM modulation of MS fiber, yielding a total fluctuation given bypη 2 + (CAP ∆P )2 .

Fitting the data to this yields CAP = 602 rad/nJ, confirming the result obtained in theprevious experiment [CAP = 593 rad/nJ, Sect. 3.1].
Side-by-side ν–to–2ν interferometers also yield insight into long-term phase stability and allow verification that accumulated extra-cavity noise does not corrupt this

33
b)

a)

power (arb. units)

40

250

low power

high power

200

30

150
20
100
10

50

0

0
760

800

840

880

500

600

wavelength (nm)

700

800

900

1000

1100

wavelength (nm)Figure 3.4: Extinction of the primary axis of MS fiber for low (244 µW) a) and high b)
intensity light. The dotted (solid) line indicates light launched along the fiber primary
axis and analyzed with a polarizer co- (cross) polarized with the input polarization.
Imperfect extinction may results from twist in the core along the length of the fiber.

stability. This may be done by analyzing the phase noise spectra of the stabilized offset
frequency, determined from each interferometer individually. This measurement determines the noise spectra of the offset frequencies both in- and out-of loop. Ideally, thein-loop measurement is limited by the feedback loop only. Fluctuations in the out-ofloop measurement are due to extra-cavity noise sources as well as residual laser noise
not removed by the feedback loop. From eqn. 2.6 it may be seen that an out-of-loop
measurement of the total phase noise on f0 will give insight into the coherence of thecarrier-envelope phase, which is the subject of Chap. 4.

3.2

Birefringence and nonlinear polarization rotation in microstructure fiber
The geometry of MS fiber cladding consists of a honeycomb structure of air holes.

Asymmetry of the cladding that results during the pulling of the MS fiber, however,can produce a slight transverse birefringence. The effect of the resulting retardance isparticularly obvious in these fibers, whereby optimal broadening is obtained along apreferred polarization axis.
Light that is injected along the primary axes of the fiber maintains its input

34
polarization [see Fig. 3.4]. Light incident to the fiber, with a polarization off of the
primary axes, exits with a polarization that is rotated with respect to the input, indicating fiber birefringence. The strength of the linear retardance of the fiber may beeasily determined by launching low intensity, broadband laser light from a Ti:S laser
into the fiber at 45 degrees to the primary axes [see Fig. 3.5 a]. By analyzing the output
using a polarizer, set with its polarization axis either co- or cross-polarized with the
input light, one obtains a fringe pattern indicating the strength of polarization rotation
due to the fiber birefringence. Figure 3.5 b) shows the fringe pattern in wavelength
across the transmitted spectrum. This retardance measurement was performed for a 10cm long piece of MS fiber, given to us courtesy of Lucent Technologies, with a coupledaverage power of 0.574 mW at a 100 MHz repetition rate. The Ti:S laser used in themeasurement operates at a center wavelength of 800 nm, with 15 fs pulses at the input
to the fiber. Analysis of the fringe pattern determines a value of the linear retardance
for the fiber to be Γλ = 2π(19914)/λ. At 800 nm the 10 cm fiber yields a net retardanceof, Γ800 = 49.78 π.
Increasing the light intensity from the laser, coupled into the fiber, we observea shift in the fringe pattern, or a nonlinear change in the fiber birefringence, Fig. 3.5c), and in more detail in Fig. 3.6. For light incident with a polarization that has anunequal distribution of power along the two polarization eigenaxes, a change in fieldstrength will result in a nonlinear differential index between the slow and fast fiber
axes, thereby resulting in a nonlinear contribution to the fiber birefringence. Thismeans that amplitude fluctuations on the input beam will be converted to fluctuationsin polarization of the broadened output. Nonlinear polarization change in fiber can beunderstood by considering the full vector model of the propagation of light along two
orthogonal polarization axes of the fiber. Propagation in the fiber can be describedby the coupled nonlinear Schrödinger equations (CNSE) in the slowly varying envelope

35a)input power
adjust

P2

P1

P P

OSA

LASER

b)

c)
coupled power = 0.574 mW

coupled power = 3.74 mW

power (arb. units)

8
60
6
40
4
20

2

0

0
720

760

800

Wavelength (nm)

840

880

720

760

800

840

880

Wavelength (nm)Figure 3.5: a) Experimental schematic to measure the linear and nonlinear birefringenceof microstructured (MS) fiber. The two polarizers, P , are used to adjust the input powerof the pulses. The pulse duration was measured using interferometric autocorrelation
with similar amounts of glass as present for the optical components before the MS fiber.
b) and c) show the results of the retardance measurement for MS fiber for low (0.574mW) and high (3.7 mW) coupled power, respectively. Input light was launched at 45
degrees with respect to the MS fiber principle axis. The dotted (solid) line show thetransmitted output analyzed co- (cross-) polarized with the input.

approximation,
i

∂u
∂u
1 ∂2u
+i
+ γu ++ (|u|2 + A|v|2 )u + Bv 2 u∗ = 0
∂z
∂t
2 ∂t2

(3.4)

∂v
1 ∂2v
∂v
−i
− γv ++ (|v|2 + A|u|2 )v + Bu2 v ∗ = 0
∂z
∂t
2 ∂t2

(3.5)

i

Here, u and v are the complex envelope components along the slow and fast axes,
respectively, 2δ and 2γ are the normalized group and phase velocity differences, z andt are the normalized length and time coordinates, and A and B are coefficients that
express the relative strength of the cross phase modulation and coherent energy exchange
coefficients, respectively. The direction of energy flow due to coherent energy exchange
is determined by the relative phase between the two polarization eigenaxes.
In the case we consider of an input polarization of 45 degrees, only the last term

36(the energy exchange term) in eqn. 3.5 yields a nonlinear dependence of the polarizationon the light intensity. For the previous two terms in eqn. 3.5, SPM and cross phasemodulation, a change in field strength results in an common imparted phase to boththe slow and fast axes because the energy is distributed equally between them. Thisis also the case since χ3 is nominally the same for both the slow and fast axes, that is
nslow = n0slow + In2 and nf ast = n0f ast + In2 , where n2 = n2self + n2cross . Changes
in field strength, as induced by the energy exchange term result in transfer of energy
between the fast and slow axes, thereby causing a non-zero propagation phase between
the two. As a result, the light energy is not equally partitioned between the two axes,
yielding a nonlinear change to the fiber birefringence even if light is incident at a 45
degree angle to the slow an fast axes of the fiber.
The issue of nonlinear polarization is problematic for the applications of interest
here since it can result in an addition of nonlinear phase noise to the broadened pulse
train. This occurs since changes in the pulse intensity at the input to the fiber will
result in a time-dependent change in polarization of light at the output of the fiber.
Aside from fluctuations in the phase of the broadened output, nonlinear polarization
rotation results in amplitude modulation of the beat since it is polarization sensitive.
The effect nonlinear polarization rotation in optical fiber has already been laidout analytically by Winful et al. [19]. The analytical result, however, was derived for
CW light with a light intensity that is constant as a function of propagation distance.
Given the complexity of the propagating spectral components and third order dispersion
in the fiber results in a complicated pulse chirp and decrease in pulse intensity that isdetermined by the interaction between the laser pulse parameters and the characteristicsof the MS fiber. Also, because of the complicated interplay between nonlinearity anddispersion, a richer understanding of the effect of nonlinear polarization requires a more
complicated simulation of the experiment.
A theoretical study of the above system was conducted by our collaborators

37
coupled power(mW)

normalized power

100

5.7

80

4.22
60

3.16

40

2.10
1.20

20

0.30
0
760

800

840

880

wavelength (nm)Figure 3.6: More detailed measurement of nonlinear polarization rotation in MS fiber.
Light with indicated coupled power is input and analyzed at 45 degrees to the fiber
primary axes. As the light power is increased a shift in the fringe pattern is observed.

I. T. Lima Jr., B. S. Marks and C. R. Menyuk at the University of Maryland. Tosummarize their results, using the nonlinear Schrödinger equation and a split-step algorithm they determined that the factor that allows for an accumulated effect of nonlinear
polarization rotation in the fiber are dispersive effects on the propagating light pulse.
Without dispersion, the coherence time of the pulses is a fraction of the differential
group delay produced by the fiber birefringence. Therefore, without pulse chirp thenonlinear interaction length over which energy transfer between the two polarization
axes occurs is limited [23].

Chapter 4

Carrier-envelope phase coherence and the “absolute” phase

Having determined the extent of phase noise generation in MS fiber one starts toquestion the general stability of the stabilization loop. This is an important question
to answer for both metrologists and those interested in exploring phase-sensitive experiments. Phase coherence of the stabilized laser, however, is prerequisite for the latteras discussed in Sect. 2.5.
The first demonstration of the coherence of a stabilized laser pulse train was byJones et al.. The authors demonstrated that by choosing the ratio between f 0 and frep ,one had control over the pulse-to-pulse CE phase of the unbroadened pulse train [seeeqn. 2.4.1]. The measurement was performed using an unwieldly cross-correlator [45]that allowed for the generation of a fringe resolved correlation trace between one pulse
and a pulse emitted two cavity round trips later. Although, this demonstrated out-ofloop control over ∆φCE , it only proved that the process was coherent over 20 ns, thetime between pulse i and i + 2. Phase errors due to instabilities in f0 result from an
integration of the frequency error in time [see Sect. 4.2]. This is the natural consequenceof frequency being the time derivative of phase. Integration times of 20 ns (50 MHz)are insignificant when it comes to accumulation of phase error. This is particularly truesince the main source of phase error results from mechanical vibrations in the (100 Hz- 10 kHz) range, producing noise that is measurable only at much longer time scales
20 ns. The complexity of the cross-correlation process represents a severe limitation

39
for extension of measurement time given the huge path length difference between thecorrelator arms. Consequently, a different method had to be devised to determine thecoherence of the laser pulse train.
The importance of a long term coherence time is predicated on the ability tocontrol the“absolute” phase of the laser pulse train. This quantity, to some, represents
the “Holy grail” for ultrafast phase-sensitive measurement. The idea behind wanting
to measure and control the “absolute” phase is that, given that one is interested indetermining the phase dependence of some physical process, one needs to know what
the absolute phase is to make the necessary correlation.

4.1

The“absolute” phase
So what is the “absolute” phase? The main interest in the absolute phase liesin the extreme nonlinear optics community, whereby the individual shot-to-shot pulse
phase can influence the outcome of a high field experiment. In the frame of the solitary
pulse, the “absolute phase” is defined relative to the peak pulse envelope. However, thisdefinition for the pulse phase is in no way absolute in the context of the pulse train.
Life also becomes complicated when there is an evolution of the CE phase of the pulse
train, f0 6= 0.
To understand the origin of the absolute phase, let us look for a moment at
eqn. 2.6. If we integrate eqn. 2.6 and evaluate it at intervals of trt we obtain a linear
relationship between the carrier-envelope phase of the nth pulse in the train and f0 .
φCE (n trt ) = n (2 π

f0
) + φ0 = n ∆φCE + φ0 (x0 , t0 )
frep

(4.1)

This is only mathematically valid, however, if f0 is constant, or noiseless (issues of instabilities in f0 are considered in Sect. 4.2). Within this limiting case, in the above0
equation ∆φCE = 2 πf0 trt = 2 π ffrep
is the CE phase shift accumulated between ad-

jacent pulses. The one quantity common, however, to each pulse is φ0 . This phase40
is often defined as the “absolute” phase of the pulse train. Mathematically φ 0 is theintegration constant determined from the initial conditions of the laser. The question
remains, however, which initial condition (x0 and t0 ) do we choose? The fact that φ0 is
so dependent on the choice of initial condition indicates how un-“absolute” it is.
What makes the absolute phase so important is the fact that phase stabilizationof the Ti:S laser makes it possible to produce a pulse train such that every pulse in the
train has exactly the same carrier-envelope phase, φ0 . By examining eqn. 4.1, this isaccomplished by making f0 = 0, which thus forces ∆φCE to also be zero, leaving uswith only φ0 . The current interest in the absolute phase is that it is hard to measure(Sect. 6.1), but relatively easy to control (control over the carrier-envelope phase is
obtained via stabilization of the offset frequency, given of course, that one can obtain aphase lock). If one could both control and measure the absolute phase, one could boast
the ability to determine the phase-dependence of coherent control-type and extreme
nonlinear optics experiments [13, 58].

4.2

Measurement of the carrier-envelope phase coherence time andits importance to phase-sensitive measurement
The interest in determining the strength of phase noise generation in MS fiber in

Sect. 4.3.1 was to determine its effect on the CE phase of the laser pulse train. The
time scale over which pulses emitted from the laser maintain their coherence determines
the amount of time one has for performing a phase sensitive experiment. The numberof pulses emitted from the laser with this phase (within some standard deviation) is
determined by the quality of the stabilization process. Even for a perfectly stabilized
laser, extra-cavity sources of noise can contaminate the output of the laser and thus
limit the time-scale over which the pulses maintain their coherence [see Fig. 3.1].
In the derivations of the CE phase for the pulse train above, one big assumption
was made: that f0 is constant. This would be the case if f0 were stabilized perfectly.

41
More realistically, the limits of the servo system and extra-cavity noise sources cause f 0
to fluctuate about its lock point. To understand the effect this has on the coherence time,
we return once again to eqn. 4.1. The time dependence of the fluctuations on f 0 may
be written by separating the error in f0 , δf , from the carrier, that is f0 (t) = f0 + δf (t).
If we include the time dependence of f0 (t) into eqn. 4.1 we see that fluctuations of f0
result in phase noise, δφ, on φ0 ,φCE (n trt ) = n ∆φCE +µZ n trt¶

2π δf dt + φ0 = n ∆φCE + (δφ + φ0 )(4.2)

From this equation we see that just as f0 is related to φCE , noise on f0 is related tonoise on φCE . In particular, if f0 = 0, its fluctuations are transferred to fluctuations in
φ0 by
δf =

1 d(δφ)2 π dt(4.3)

Therefore, for ultrafast experiments relying on the stability of the CE phase [13, 58],
knowledge of f0 ’s stability is paramount as dephasing of φ0 determines the maximum
possible duration of a phase sensitive measurement.

4.2.1

Noise analysis

The stability of the offset frequency is determined via analysis of f0 ’s lineshapeonce it is locked. This is a frequency domain analysis of the fluctuations in f0 about
its carrier. If f0 were a constant, its lineshape would be a delta function in frequency.
Frequency deviations on f0 , however, result in the addition of noise sidebands to theperfect carrier. These sidebands, depending upon the detection scheme for f 0 may yield¯¯¯ F {δφ(t)} ¯2¯¯ (rad2 /Hz) , or a frequency noise spectrum,a phase noise spectrum, Sφ (f ) = ¯ √∆fRB ¯¯¯¯ F {δf (t)} ¯2¯ (Hz2 /Hz) [see Sect. A.7]. These two quantities are called the power¯√
Sf (f ) = ¯∆fRB ¯spectral density (PSD) of the noise measured with a resolution bandwidth, ∆f RB . The
connection between the noise PSD in frequency and phase is given by,Sf (f ) = f 2 Sφ (f ).

(4.4)

42
Either Sf (f ) or Sφ (f ) may be used to determine the linewidth of f0 , ∆ν, or the coherencetime of φ0 , τcoh . Both quantities are determined by integration of the PSD, which yields
the total accumulated rms error, in phases2sZ −1/(tobs )∆φrms |tobs =Z −1/(tobs )−∞Sφ (f ) df =s2

Z −1/(tobs )−∞Sf (f )
df ,
f2

(4.5)

or in frequency,∆frms |tobs =2−∞

f 2 Sφ (f ) df =s2

Z −1/(tobs )−∞Sf (f ) df .

(4.6)

What these equations tell us is that over an observation time tobs we can expect an rms error in phase or frequency given by ∆φrms or ∆frms , respectively. For
shorter observation times, one would typically expect these deviations to be less thanat longer observation times. Additionally, these equations provide a general definition of the carrier-linewidth, ∆ν, or equivalently, the coherence time, τcoh . In themetrology community it is common practice to define the coherence time of a carrierin terms of the observation time at which ∆φrms accumulates to 1 radian, that is,
tobs
τcoh = (2π)= 1/(π ∆ν).

4.2.2

Experimental set up

To obtain a true measurement of the coherence time for the stabilized laser requires performing an out-of-loop measurement. The measurement scheme is the sameas that used to determine the transfer of MS fiber noise onto the laser output, andfollows the same basic principles [see Sect. 3.1.1]. The measurement is performed using
two ν–to–2ν interferometers [see Fig. 4.1] [26]. One interferometer stabilizes the laser,
while the second determines the carrier-envelope phase noise.
The carrier-envelope phase coherence time is determined for the stabilized laser
system described in Chap. 2. Negative feedback to the laser is obtained with a bandwidth of ∼ 18 kHz via tilting the laser end mirror using a piezo-electric actuator (PZT).

43

Ti:sapphire laser

AOM

fAOM
MS

PUMP

532nm
AOM

fAOM + f0 inBBO

1064nmν-to-2ν IN-loop

LOCKING
ELECTRONICS

fref

ν-to-2ν OUT ofMS

MS- microstructure fiber

loop

lock-in
amp

BBO

FFT

AOM- accousto-optic
modulator
BBO – Beta Barium Borate
FFT - Fast Fourier
Transform signal analyzer

f0 outFigure 4.1: Experimental setup showing how the coherence of φCE is measured. One
interferometer is used to stabilize the laser while the second ν–to–2ν interferometer
determines the phase coherence. The noise of the second interferometer is minimized
by making the ν–to–2ν comparison as common mode as possible by using prisms forspectral dispersion [see Sect. A.5.2].

Phase noise produced within the stabilization loop, but external from the laser, is transferred onto the laser output [see Fig. 3.1]. The strength of this transfer is governed by
the gain of the PLL as a function of frequency.
The offset frequency measured at the output of the stabilization interferometer
gives an in-loop measurement of f0 ’s phase noise. This noise is only the residual noise nottaken out by the PLL, which may be used for monitoring the effectiveness of the feedback
loop. The offset frequency, measured at the output of the second interferometer, incontrast, measures the external noise transferred from the stabilization loop as well as
the residual noise not taken out by the locking electronics. This out-of-loop noise alone

44
would present a true measure of the out-of-loop phase noise on f0 . Unfortunately, thesecond interferometer is also a source of noise. Thus, noise measurement of f 0 using thesecond interferometer itself results in an overestimation of the total phase noise on thelaser.

4.2.2.1

Phase noise or frequency noise?

As mentioned previously, the measurement of a frequency or phase noise spectrumof f0 , depends on the measurement technique. There are two devices that may be used
to determine the noise spectrum of a signal. Both act to remove the noise from thecarrier, such that noise spectrum begins at DC. Such a baseband measurement allows
for generation of high resolution spectrum using a signal analyzer (FFT) [see Fig. 4.1].
A double-balanced mixer may be used to produce a phase noise spectrum of a quiet
signal (with phase noise sidebands << 1 rad) [see Sect. A.7.2.1]. A frequency-to-voltage
converter (F-to-V) is used to obtain a frequency noise spectrum of a signal with strong
amplitude or phase fluctuations [see Sect. A.7.2.2]. In the noise measurements presented
here, noise spectra of the stabilized offset frequency were obtained using a mixer. A
frequency spectrum of the noise of the unstabilized f0 was obtained using a F-to-V
converter, and was converted to a phase noise spectrum using eqn. 4.4. Note that inanalyzing the single sideband noise spectrum produced using the mixer and F–to–V
converter, the factor of two in eqn. 4.6 and eqn. 4.5 is not necessary.
Figure 4.2 presents the results of the coherence measurement. A measurementof the unstabilized offset frequency is used to indicate the available noise suppression.
From this, we see a many order magnitude suppression in phase noise at low frequency.
The difference between the out–of–loop and in–loop spectra yields the extra-cavity phasenoise present within the stabilization loop (e.g., feedback electronics, the ν–to–2ν interferometer, microstructure fiber, etc.) and the differential noise between the two loops.
As expected, the out-of-loop noise is significantly higher than the in-loop noise.

45
observation time (s)

10

3

10

2

10

1

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

0.8

5

out of loop

unlocked
0.6

10

0

in loop
10

0.4

-5

0.2
10

-10

0.0
10

-3

-2

10

10

-1

10

0

10

1

10

2

10

3

10

4

10

accumulated phase noise (rad)phase noise (rad 2/Hz)

105frequency (Hz)Figure 4.2: Phase power spectral density (left axis), Sφ (f ), for the in-loop, black line,
and out-of-loop spectra, grey line, of f0 versus frequency offset from the carrier. An
unlocked spectrum is included to indicate the strength of the phase noise suppression
resulting from negative feedback to the laser. The dashed lines show the integrated
phase error (right axis) of Sφ (f ) as a function of observation time. Integration of theunlocked spectrum is not present.

Integration of the phase noise PSD out–of–loop (in–loop), in Fig. 4.2 results inan accumulated phase error of 0.8 rad (0.109 rad) over the interval 102 kHz down to
488 mHz (resolution limited). Given that the out-of loop accumulated phase error isless than 1 rad, the lower frequency integration bound determines the coherence time,τcoh = 1/(2π 488µHz) = 326 s. This indicates that the phase of the pulses in the pulse
train evolve in a coherent manner for > 30 billion pulses.
Comparison between the unlocked and the in-loop phase noise PSD densities
indicates a servo bandwidth ∼ 20 kHz. The in-loop PSD shows that additional phasenoise results from the action of the servo loop at frequencies higher than 20 kHz, andthat there is insufficient gain in the acoustic range (∼100 Hz - 5 kHz). This frequency
range is also responsible for the majority of the out–of–loop phase noise contribution.

46

4.3

Characterization of extra-cavity sources of phase noise

4.3.1

MS fiber noise

A major source of out–of–loop noise present in the stabilization loop results from
amplitude to phase modulation in MS fiber, where amplitude noise on the laser output[see Fig. 4.4], or amplitude noise resulting from coupling fluctuations [see Fig. 4.3] isconverted to phase noise via the fiber nonlinear index of refraction [see Sect. 3.1]. This is
detrimental when the direct output of the laser is to be used in an experiment, however
it does not affect an experiment that uses the output of the MS fiber.
To measure the contribution of fiber noise that results from power fluctuations due
to coupling, we couple a HeNe laser into the fiber and measure the power fluctuations
at the fiber output. From this measured noise spectrum we subtract in quadrature theintrinsic power fluctuations due to the HeNe itself. The net result of the fractional
power fluctuations due to coupling are shown in Fig. 4.3 as well as the calculated fiber
phase noise that would result. The calculated fiber phase noise was determined using
the amplitude-to-phase noise coefficient measured in Sect. 3.1 for 40 mW of coupled
Ti:s light at a 100 MHz repetition rate.
A shortcoming of the dual interferometer method is that MS fiber is used inboth interferometers. As a result, the dual-interferometer may not fully reflect MS
generated phase noise resulting from common mode power fluctuations. To estimate thecontribution of these fluctuations to the out-of-loop measured phase noise, we measure
the laser amplitude fluctuations and use them in conjunction with the amplitude-tophase conversion factor for MS fiber, CAP = 5.93 rad/ mW, as measured on f0 [seeSect. 3.1]. This measurement is also used to compare the fiber noise induced by the
PZT stabilization scheme described above to that obtained via modulation of the laser
pump power [26]. Stabilization using the latter method is actuated by placing an AOM
in the path of the pump beam for the Ti:S.

47
observation time (s)100phase noise (rad2 /Hz)

10

0.01

0.001

1.0

-2

0.8

-4

0.6

-6

0.4
0.2

-8

accumulated phase (rad)

10

0.1

0

10

10

1

1.2

10

10

10

2

0.0

0.01

0.1

1

10

100

1000

frequency (Hz)Figure 4.3: Phase power spectral density, Sφ (f ) in rad2 /Hz, of the calculated fiber
induced phase noise on f0 resulting from power fluctuations resulting from fluctuationsin the power coupled into MS fiber, for a coupling power of 40 mW. Integration of thenoise spectrum yields the accumulated phase error as a function of observation time.

One drawback of modulating the pump power as a means for feedback to the Ti:S
oscillator is the possibility of inducing fluctuations on the output power [34]. Figure 4.4
shows the amplitude noise PSD as well as the calculated accumulated fiber phase noiseof the PZT versus AOM stabilized systems. A spectrum of the amplitude noise for theunstabilized laser is shown for comparison. As can be seen in Fig. 4.4, the PZT system
contributes little additional amplitude noise during stabilization, whereas the opposite
is true for the AOM stabilized laser.1 Integration of the AOM and PZT stabilized
noise spectra from 8 Hz to 3.2 kHz yields the rms fractional laser power fluctuation,(∆P/P0 )rms , to be 0.00473 and 8.34 × 10−5 , respectively. For a coupled laser power of
50 mW at 100 MHz and an amplitude-to-phase conversion coefficient for MS fiber of 593
rad/nJ [28], the amplitude noise would result in 1.40 rad of phase jitter on f 0 for the
AOM stabilized system. To connect back to the coherence measurement, the additional
fiber noise contributed by the PZT stabilized system to the accumulated out-of-loop
1Note that the opposite is true if the dominant source of noise in the laser cavity is due to fluctuations
in pump power, as has been shown to be the case for the prismless laser [59].

48

observation time (s)
0.1

0.01

0.001

(∆P/Po)2/ Hz

10

1.2
1.0

AOM lock
-9

10

0.8
0.6

-11

10

0.4
PZT lock
0.2

unlocked

-13

10

0.0
10

100

accumulated phase noise (rad)

1.4
-7

1000

frequency (Hz)Figure 4.4: Phase noise spectra (left axis) for the unlocked laser, locked by modulating
the 532 nm pump power (AOM) and locked using the fast PZT versus frequency. Calculated accumulated phase jitter resulting from amplitude-to-phase conversion in MS
fiber (right axis) versus observation time. Both the unlocked and the PZT lock spectra’s
accumulated phase noise are close to zero.

phase noise presented in Fig. 4.2 is determined to be 0.155 rad rms. Assuming that theout-of loop measurement does not reflect amplitude-to-phase noise conversion in MS
due to common mode power fluctuations then the calculated fiber noise brings the total
accumulated phase noise (fiber generated noise + out-of-loop phase noise in Fig. 4.2)
to 0.955 rad, bringing the observed coherence time determined in Fig. 4.2 dangerously
close to 1 rad after tobs = 125 ms.
MS fiber contributes more than 50% of the measured out-of-loop phase noise, anddependent on the laser power fluctuations may overwhelm the coherence entirely. Asidefrom the noise measurements presented here, additional mechanisms in the MS fiber havebeen demonstrated that result in production of broadband noise, which may limit thestability of the fiber based stabilization scheme in optical frequency measurement [56, 1].
Therefore, elimination of the MS fiber would prove an invaluable step in increasing thesimplicity of the phase stabilized Ti:S system [for example see Sect. 5.2].

49
a)

b)

observation time (s)

HeNe

AOM

PD

BBO
PBS

High V

LF

1

0.1

0.01

0.001

no box

10

0.20

-5

0.15

enclosed

10

-10

0.10
stabilized

10

0.05

-15

0.00
0.1

1

FFT

10

100

1000

accumulated phase noise (rad)ν-to-2ν

phase noise (rad2/ Hz)10frequency (Hz)Figure 4.5: a) Schematic for the measurement of the extra-cavity noise caused by interferometer drift. The interferometer is enclosed in a plexiglass box. b) The power
spectral density of the phase noise measured using light at 632 nm for three different
conditions: interferometer with no box, enclosed in a box and stabilized. Integrationof the noise spectrum yields the accumulated phase noise as a function of observation
time.

4.3.2

Interferometer noise

Much of the low frequency contributions to noise of the carrier-envelope phase are
due to long time scale drift in the ν–to–2ν interferometers. These drifts are particularly
villainous because noise at long time scales results in a greater accumulated phase noise
than noise at fast time scales. Phase noise in the ν–to–2ν interferometers results fromfluctuations in the relative path length between the interferometer arms, l(t).
∆φint =

2π
∆l
λ

(4.7)

Phase noise resulting from path length differences are measured by threading a HeNelaser through the ν–to–2ν interferometer. Changes in relative path length between
the arms of the interferometer result in a time-dependent fluctuation in fringe contrast
that can be measured using a PD [Fig. 4.5 a]. The resulting fluctuations in voltage
measured on the PD are calibrated by determining the voltage equivalent to a π wrap
in phase. This calibration, used in tandem with the noise spectrum from the PD is
used to determine the phase noise PSD of the interferometer. As described in detail50
in Sect. 4.2.1, integration of this noise yields the accumulated phase noise due to theinterferometer of 0.3 rad over tobs = 20 s [see Fig. 4.5 b].
As seen in Fig. 4.5 b), drift in the interferometer only contributes noise overlong time scales, with an accumulated phase noise of 0.24 rad (60 mHz - 3.2 kHz).
By enclosing the interferometer, this noise may be cut down by more than an order ofmagnitude to 0.011 rad. A further two orders of magnitude noise suppression may beobtained via active stabilization of the relative path length (8.5 × 10−5 rad). This isdone by placing two glass wedges in one arm of the interferometer, where one wedge is
attached to a shearing PZT [see Fig. 4.5 a]. The voltage signal obtained from the PD
that measures the fringe contrast is used as an error signal to stabilize the interferometer.

4.3.3

Long-term phase drifts: coherence measurement using a lock-in
amplifier

Given a pulse train with a highly stable φCE , an additional method for determining longer term phase drifts may be obtained by looking at the time fluctuations in φ CE
directly. The only difference in the experiment is that f0 is measured using a lock-in
amplifier instead of an FFT [see Fig. 4.1]. With a dual-phase lock-in, it is possible todirectly measure the phase difference between the reference channel and signal channel.
This allows for a more visual representation of the phase fluctuations on the pulse train.
To be able to use a standard lock-in, we lock f0 ∼ 100 kHz. This reference
signal used for stabilization of f0 is also provided to the reference channel of the lock-in.
Figure 4.6 shows a 400 sec time record of the lock-in phase. The standard deviationof the phase fluctuation, at a 1.0 s time constant, is 3.8 deg (0.066 rad) rms, with amaximum deviation of 20.8 deg (0.363 rad) rms. Using a lock-in amplifier, set witha long time constant (1 s), allows for observation of the CE phase noise only within
a limited bandwidth at very low frequencies, masking the higher frequency noise. To
obtain the long term stability shown in Fig. 4.6 required placing all interferometers in51boxes and enclosing the beams between interferometers in beam tubes. Turbulent air
currents can easily result in 2π phase wrap in a matter of seconds.

Figure 4.6: Time record of the lock-in phase (lower trace, left axis) and amplitude(upper trace, right axis). The inset shows an enlargement to show that the amplitude
and phase fluctuations are uncorrelated.

4.4“Absolute” phase measurement
In an ideal case, the phase of the f0 signal, measured using a ν versus 2ν com-

parison, does not add additional phase offsets to φ0 during measurement. Since aphase-locked loop establishes a fixed phase, it seems that φCE is fully determined by
the reference signal to which f0 is locked, or that a phase sensitive measurement of the
f0 signal gives full knowledge of φCE . One limiting case is f0 = 0 in which case φCEdoes not evolve pulse–to–pulse and thus is just given by φ0 , see eqn. 4.1. However, any
ν–to–2ν interferometer is non-ideal and introduces an arbitrary phase shift that prevent this direct connection from being correct. This is illustrated in Fig. 4.7. The phaseshifts in the ν–to–2ν interferometer arise from differences in the path length of the twoarms and dispersion in any optical elements. (In principle an explicit interferometer is
not needed if a chirp free pulse is used, however there is still unavoidable dispersion in
the second harmonic crystal.)

52

Figure 4.7: Schematic showing relationship between pulse train phase and the phaseat the ν–to–2ν interferometer output. a) Pulse train showing pulse-to-pulse changein ∆φCE = π/4. b) Output of ν–to–2ν interferometer, solid line is the output of an
ideal interferometer where zero signal is coincident with the pulse that has φ CE = 0.
However, an actual signal from an interferometer has an arbitrary phase shift relative
to the ideal signal.

A number of interferometric schemes have been demonstrated that would result
in determination of φ0 while minimizing added offsets during measurement. In 2002
there was a demonstration by Mücke et al. that by using the 5 fs unamplified output
of a broadband (400 nm) Ti:S laser, one could generate both an octave continuum
and the necessary second harmonic for nonlinear comparison, all in a 1 µm piece ofZnO [54]. This is wonderful since it eliminates both the need for broadening in MS
fiber and the doubling crystal that are necessary in the ν–to–2ν interferometer. Morerecently, Apolonski et al. showed that interaction of unamplified 5 fs pulses with agold cathode result in a photo-emitted current that observed a dependence on φ 0 [3].
Additionally, as will be discussed in Sect. 6.1, using octave spanning, chirp free pulses
the injected photocurrent generated in a direct gap semiconductor can be used for solidstate detection of φ0 .
Although the above experiments seems very promising for measurement of the“absolute” phase, each of these techniques still suffer phase offsets during measurement.
The offsets are particularly bad for measurement of φ0 using semiconductors since it isdifficult to determine how far the light penetrates into the material. Even though theestimated phase offset may be small one has to keep in mind the fact that, in the worst

53
case scenario of using the common ν–to–2ν interferometer, the maximum error that canbe measured on φ0 is 2π (even if the actual error is thousands of radians, the detected
phase error is always modulo 2π).
Aside from the measurement phase error, there also exist serious problems if one
chooses the above methods for“absolute” phase stabilization of the laser. Let us assume
first that the system used to determine the absolute phase does not suffer unnecessary
phase offsets. On top of the errors that result during measurement (let’s not forget the
Guoy phase ambiguity), the absolute phase can accrue errors during stabilization or
simply by encountering dispersive elements outside the laser cavity. The former resultsin time changes in φ0 but may be minimized by careful design of the feedback loop
(path length stabilization of the interferometers, minimizing MS fiber noise, taking out
phase drifts due to temperature changes in BNC cables, etc. [see Sect. 4.2]). Material
dispersion outside the cavity only adds a constant phase offset to φ0 . The problem
with an added offset is if the beam going to the absolute phase measurement and thatgoing to the phase sensitive experiment travel through differing amounts of dispersivematerial there will exist a phase error between the two pulse trains.
Only the final scientific results that show a direct phase-dependence without ambiguity will really count. For the moment, at best we can maintain the coherence φ 0
and shift it in a systematic manner....

Chapter 5

Octave spanning lasers

Aside from the problems posed by fiber induced phase fluctuations [see Sect. 4.3.1],
stabilization of the Ti:S laser using MS fiber presents challenges to short pulse experiments because of the large cubic fiber dispersion [62]. Additionally, complexities in thefiber alignment often lead to loss of fiber coupling and degradation in the f 0 signal overtime. This is a considerable hindrance in optical metrology since long term averaging isnecessary to increase the measurement precision. To overcome these problems requires
finding a means for continuum generation that exhibits better stability than is achievable in MS fiber. One possibility is external broadening of laser pulses via SPM in athin glass plate [8]. This, however, requires very high intensities that may be achievedby amplification of the laser pulses. One other possibility, which is the focus of thischapter, is the generation of continuum directly within the laser cavity.

5.1

The Ti:sapphire laser

5.1.1

Cavity geometry

The Ti:S laser presented here is a standing wave cavity [see Fig. 5.1] boundedby two flat mirrors: the laser end mirror (EM) and the output coupler (OC). The
OC is used to couple a portion of the intra-cavity light out of the laser, with a loss
determined by the mirror reflectivity. The Ti:S crystal is optically pumped at 532 nm. 1

55

a)

b)

EM

OC
OC
EM

M3
L

L

PUMP

PUMP
M1

M1

M2

M2

dispersion compensation:
Dispersion due to the laser crystal
severely reduces the number of modes
that can be resonant in the laser cavity

prism compressor
chirped mirror stack

Figure 5.1: a)X-folded Ti:S laser. Intracavity dispersion compensation is used to maximize the lasing bandwidth of the modelocked laser spectrum. b) Kerr-lens modelocked
Ti:S laser cavity. Intracavity prisms and chirped mirrors are used to provide negative
second order dispersion compensation. A prism compressor results in negative second
order dispersion compensation through a combination of light refraction and geometry.
Light that is incident on the first prism is refracted such that red light is refracted morestrongly than blue. The diverging rays from the first prism are then incident on thesecond prism in the geometry shown. Light exiting the second prism is horizontallydispersed but not diverging. Because of the orientation of the second prism, red light
is forced to travel through more glass than blue thus causing the optical path lengthof the red light to be greater than that of the blue. Negative disersion in a chirped
mirror is obtained, in the simplest case, by a dielectric stack with indicies, n1 and n2 .
By varying the thickness of the layers as shown the half wave condition for reflection of
red light occurs deeper in the dielectric stack of the mirror than it does for blue light.

The pump beam is focussed with a 10 cm anti-refection coated lens, L, coupled into
the cavity through the first curved mirror, M1. The pump light is focussed to obtain
efficient population inversion in the Ti:S crystal. The two curved mirrors (f = 5 cm),
M1 and M2 , are used for focusing of the Ti:S laser light into the crystal (x-tal). For
optimal single frequency lasing (CW), M1, M2 and L are aligned such that the Ti:Slasing light and the pump beam are spatially overlapped. Stable CW operation of thelaser is obtained for M 1 − xtal = 5.0 cm and M 2 − xtal = 5.1 cm for OC − xtal = 0.62
m and EM − xtal = 1.173 m.2
1

We use a 6 W Verdi laser from Coherent Lasers. This pump laser is preferable since it is single
mode and that it has low amplitude fluctuation.
2
The stability of the laser for a given geometry is obtained using ABCD matrix cavity stability
calculations, which assumes a Gaussian TEM00 mode. The optimal mirror positions are determined by
the cavity length of each arm since the cavity optics determine the spot size and divergence of the beam

56
To reduce reflections of the lasing beam from the crystal surfaces, the crystal iscut at Brewster’s angle. This makes it such that light in the horizontal, x-direction, andthe vertical, y-direction, are refracted differently in the laser crystal. The astigmatism
caused by focussing the lasing beam at an angle, θ with respect to the crystal axis
[fx = f cos(θ) and fy = f /cos(θ)] may be used to compensate the astigmatism due tothe cut of the crystal [49]. The astigmatism is balanced for,sin2 (θ)
t=cos(θ)
f√√n4 − 1 n2 − 1
.
n4

(5.1)

Here n is the index of Sapphire and t is the thickness of the crystal.

5.1.2

Pulsed operation of the Ti:sapphire laser

Pulsed operation of the Ti:S laser is achieved by passive modelocking of thelaser spectrum. Just as the term implies, modelocking is achieved by establishing afixed phase relationship between resonant longitudinal modes in the laser cavity, which
results in the formation of a light pulse [see Fig. 2.1].
Formation of the pulse train requires that the longitudinal modes of the cavity
be equally spaced in frequency. This is naturally the case for a dispersionless cavity. In
reality, however, the bandwidth of the resonant modes is limited by dispersion of thecavity optical elements that yield a wavelength dependence of the index of refraction,n(λ). This causes the different wavelengths of light to travel different optical path
lengths in the cavity, making the mode spacing uneven in frequency.3
in the laser crystal.
3
Outside the laser cavity, dispersion results in dephasing of the different wavelengths of light, where
each wavelength picks up a phase φλ = k(λ) d = k0 d n(λ), as they travel the length of the medium, d.
This dephasing causes the carrier frequency of the pulse to be chirped (frequency modulated) such that,for normal dispersion, the leading edge of the pulse has a longer wavelength than that of the trailingedge of the pulse. Because n(λ) is not linear in frequency, Taylor expansion of φλ about the center
wavelength of the pulse, λ0 is performed to give the order of the dispersion:·

φλ = φ0 + k0 (λ − λ0 )d2 ndn
+ (λ − λ0 )2 2 + ......
dλ
dλ

¸

(5.2)

typically, this equation is expressed in terms of derivatives in ω. Although pulse chirp does not change
the frequency components of the pulse it does act to broaden the pulse envelope. This is because thetemporal profile of the pulse is determined by the Fourier transform of the spectral components including

57
The limiting effect of dispersion on the resonant bandwidth of the laser also hasstrong consequences on the nonlinear processes in the laser cavity that are responsible
for pulsed formation. Therefore construction of the modelocked Ti:S laser requires intracavity dispersion compensation, which is typically achieved using a prism compressor[22] [see Fig. 5.1]. Alternatively, chirped mirrors (CM) [52, 47] may be used, which inprinciple can compensate any order dispersion [see Fig. 5.1], limited by the complexityof the coating design. For the laser presented here, dispersion compensation is obtainedusing a combination of both chirped mirrors and intra-cavity prisms. Because the prism
compressor is in one arm of the laser only, the dispersion map of the laser is asymmetric
about the crystal center. As a result, nonlinear processes in the crystal primarily occur
only on one pass through the crystal, on the return trip from the prism compressor.

5.1.3

Modelocking and the Kerr effectPulse formation and the generation of ultra-broadband spectra in a passively
modelocked laser is a complicated process, which for the most part is still not fully wellunderstood. The basic idea for the typical Kerr-lens modelocked laser, however, is asfollows: if a perturbation is introduced into the laser cavity, this will cause a certain
number of longitudinal modes to add coherently, forming a weak pulse. As this pulse
travels through the laser crystal it feels the intensity-dependent nonlinear refractive
index of the crystal,n(x, t) = n0 + I(x, t)n2 ,(5.3)

where n2 = 2 n2 /(²0 c n0 ). Here n0 is the linear refractive index, I(x, t) is the instantaneous pulse intensity and n2 is the nonlinear index of refraction and indicates thecoupling strength between n and the electric field, E(x, t). Nonlinearity in the laser
crystal satisfies two purposes. The first is the generation of new wavelengths by a protheir phase. The narrowest temporal profile for a light pulse is that for which each component in thefrequency spectrum has the exact same phase, this is termed a transform limited pulse.

58
cess called self-phase modulation (SPM), which is sensitive to the temporal profile ofthe laser pulse. The second is self-focussing, which is dependent on the spatial profileof the pulse. For a more thorough discussion of this please consult ref. [43].

5.1.3.1

Self-phase modulation

Let us consider a pulse with a Gaussian intensity profile. The propagation of thepulse in the laser crystal results in a time change in refractive index due to the Kerreffect, n(t) = n0 + n2 I(x, t). The leading edge of the light pulse [see Fig. 5.2 a], witha positive rising intensity results in an increase in refractive index. The time change in2n(t) results in a phase modulation, ddt2φ = k0 n2 z ∂I(x,t)∂ t , that results in a red shift ofthe pulses leading edge. Conversely, the trailing edge of the pulse has a negative slope,causing a decrease in refractive index compared to the center of the pulse. This has theeffect of blue shifting the trailing edge of the pulse with respect to the pulse center. The
net effect is the generation of new coherent wavelengths with a positive pulse chirp of thecentral portion of the spectrum. Self-phase modulation is the time-domain equivalentof four-wave mixing.

5.1.3.2

Self-focusing

The transverse effect of the light pulse on the nonlinear index of refraction of
the crystal results is the formation of a nonlinear lens with a spatial profile matchingthat of the light beam, n(r) = n0 + n2 I(r) [see Fig. 5.2 b]. This lens results in selffocussing of the laser beam.4 This nonlinear effect may be used as a passive saturable
absorber for preferential gain of modelocked over CW laser operation, and thus acts
as the laser modelocking mechanism. Using the Kerr-Lens effect as a passive saturable
absorber requires misalignment of the curved mirrors, M1 or M2, from the optimal CW
position. This is done by changing the curved mirror distance with respect to the crystal
4
For the typical KLM Ti:S oscillator, dispersion eventually reduces the pulse intensity such that
self-trapping of the beam does not occur.

59a) Self-phase modulationIb) Self focusing
n (I)
I
n(I)

dφ /dt = dn/dt

t
FT
x

I

ν

Figure 5.2: Temporal a) and spatial b) effects resulting from the Kerr- nonlinearity.
a) Depiction of self-phase modulation, (top) input pulse travels through a nonlinear∂nmedium with a Gaussian temporal profile, resulting in a phase profile, dφ
dt ∝ ∂t . (bottom) shows the incident spectrum of the pulse and the typical double Gaussian final
spectrum. b) Depiction of self-focussing.

center. The change in position causes the intracavity beam to no longer be optimallyoverlapped with the pump in the laser crystal.5 Perturbation of the laser cavity resultsin the formation of a weak circulating pulse that is in addition to the CW beam. Whenthe pulse propagates through the laser crystal, the pulse will focus more tightly thanthe CW beam because it has a higher peak intensity. If the curved mirror position
is chosen correctly, Kerr-lensing will cause the pulse to overlap better with the pumpthan the CW light, thus experiencing a higher gain. Self-phase modulation, due to thehigher peak intensity of the pulse, will act to broaden the pulse’s existing spectrum.
As the pulse then recirculates through the cavity, dispersion picked up in the crystal
is compensated by the prism compressor and the chirped mirrors. The new frequency
components and chirp compensation yield a shorter duration pulse that results in afurther increase the pulse intensity and so on and so forth. Given that dispersion is well5

Misalignment of the curved mirror position also results in the production of a slightly asymmetric
CW beam. This is because the mirror position determines the condition for asymmetry correction [seeEq. 5.1]. Often, the correct modelocking position may be determined by examining the spatial profileof the CW beam profile.

60
from locking electronics

a)

b)
20 dB

OC

EM

P2

P1


2ν

ν



PUMP
M1



M3

M2

600

800

1000

1200

Wavelength (nm)c)

BBO
filter(580 nm)

frep
30 dB

polarizer
PMT

folocking
electronics

to laser EM

Figure 5.3: a) Experimental schematic of the octave spanning Ti:S laser and the ν–
to–2ν interferometer used for measurement of the laser offset frequency. b) the octavespanning spectrum obtained from the laser that is used for the ν–to–2ν comparison. c)
The rf spectrum of the measured beat signal at a 100 kHz resolution bandwidth.

managed and that the proper curved mirror position is found, the laser will result in astable modelocking of the laser spectrum.

5.2

Characteristics of an octave spanning Ti:sapphire laser
In this section I will describe how the coherent bandwidth of the laser may be

extended such that the laser spectrum alone is broad enough to support measurement
and stabilization of the laser offset frequency, without external broadening in MS fiber[24].
Ti:S lasers have been previously demonstrated that result in intra-cavity continuum generation [17, 7]. Reference [17] describes a Kerr-Lens modelocked (KLM) Ti:S
laser using specially designed double-chirped mirror pairs for broadband dispersion compensation as well as a secondary time and spatial focus in a thin glass plate used for
additional intra-cavity spectral broadening. Reference [7] demonstrates a 1 GHz repe-61
tition rate ring laser that employs a slightly convex intra-cavity mirror to increase theeffect of SPM in the laser crystal. The laser spectrum of ref. [17] was shown to yield adirect ν–to–2ν beat signal and ref. [7] a direct 2ν–to–3ν measurement of f 0 .
Given the previous discussion of pulse formation, the less intracavity dispersion,
the higher the achievable nonlinearity in the laser crystal, and thus the greater thebroadening. As a result, all broadband lasers must, to some extent, minimize the effectof intracavity dispersion. The octave spanning laser that I will describe here is a typical
x-folded cavity that uses commercially available negatively chirped mirrors and intracavity CaF2 prisms for dispersion compensation, Fig. 5.3 a). M1, M2, M3 and EM are
all negative chirped mirrors from LayertecT M , with respective part number, 101568(02),101568(02), 100466 and 101515. The output coupler (OC) is G034-007 from Spectra
Physics and the separation between the two CaF2 prisms is 82 cm.
Before I attempt to explain how the laser works, I will first describe some of its
attributes. Figure 5.3 b) shows the spectrum of the laser for a pumping power of 6.5
W. What is remarkable about this laser is the amount of generated modelocked power.
Typically, to keep the intracavity powers high, a low transmission output coupler is used
for generation of large intracavity bandwidths. The transmission of the output coupler
used here is ∼ 5 % and results in a modelocked power > 600 mW. A major percentageof the power is simply transmitted by the output coupler since it only reflects over 200
nm bandwidth [see Fig. 5.5].
Because of the extreme breadth of the spectrum as compared to the reflectivity of
the OC, the continuum must only be generated on a single pass through the crystal. As
a result, light produced outside the bandwidth of the OC is not resonant with, and thus
is not forced to obey the spatial conditions of the cavity. For light circulating manytimes in the cavity, the cavity filters out any spatial aberrations that do not match
the permitted cavity spatial modes, thus leaving only beautiful Gaussian profiles. In
contrast, light produced on a single pass through the crystal is subject to diffraction

62

Lasing modes – TEM00
779nm

850nm

1000nm1080nm

690nm

720 nm779 nm

850 nm1000 nm1080 nm

690 nm

0

664nm

612nm

Spot Size (mm)664 nm-10
sagittal plane
tangential plane

3

-20
-30

2
-40

612 nm587nm

Laser Spectrum (dB)41100nm1100 nm

1120 nm
1151nm

1

-50
600

800

1000

1200

Wavelength (nm)1150 nm

587 nmNon-resonant modes exhibit non-ideal beam profiles (nonGaussian, higher order Gaussian)Figure 5.4: The laser beam profile is displayed for selected wavelengths. A 10/90 knifeedge fit was used to determine the spot sizes in the sagittal (filled squares) and tangential(empty squares) planes, displayed at the right of the figure. The solid and dashed lines
are the respective fits to the expected 1/λ diffraction limit divergence. Some of thediffraction rings observed for the larger beam modes may result from aperturing of thelaser mirrors.

and aperturing of the cavity optics. This results in a significantly different spatial
profiles between the resonant and non-resonant spectral modes, as shown in Fig. 5.4.
A 90/10 knife edge method was used to perform a second moment measurement of thebeam profiles in Fig. 5.4 to yield the beam spot sizes as a function of wavelength [10].
From the spot size, it is seen that the spectral extremes exhibit a significant increase inasymmetry. Additionally, it is also seen, not surprisingly, that these modes also show asignificant deviation away from the Gaussian TEM00 mode and its corresponding 1/λdiffraction limit.
The increase in spot size in the wings of the spectrum is understood by examining the beam parameters at the output of the laser, Fig. 5.5. The data in Fig. 5.5
was obtained by performing an M 2 measurement of the laser beam propagation versus

63
wavelength (given the beam profiles, Gaussian beam propagation theory may no longer
hold in the wings of the laser spectrum, and as a result the data in Fig. 5.5 is intended
only to indicate a trend in the beam characteristics.) The beam waist measurement[Fig. 5.5 a] shows a sudden decrease of the non-resonant compared to the resonantmodes. This is a little surprising but explains the greater divergence in the wings ofthe laser spectrum that leads to the larger spot sizes observed in Fig. 5.4. Figure 5.5 b)compares the measured beam divergence to that which would result from the propagation of a Gaussian TEM00 mode given the respective measured beam waists in Fig. 5.5a). The ratio of the former to the latter gives the M 2 value as a function of wavelength,
indicating the strength of the non TEM00 Gaussian beam propagation (non-Gaussianmodes and/or higher modes).
The broadband operation of this laser is speculated to result from very gooddispersion compensation, but only over the bandwidth of the output coupler (this assumption is supported by the fact that exact choice of broadband chirped mirror doesn’t
make a huge difference in determining the laser bandwidth). If the dispersion over thisbandwidth is close to zero then it is possible to form a very short pulse in the laser crystal. The resultant high intensities would cause the formation of an extremely strong
Kerr-lens that, in turn, would result in the production of the continuum. This doesn’t
seem the least bit surprising and may, in fact, be only part of the story.
Unfortunately, the role of dispersion has not, however, been experimentally confirmed. We did attempt to determine the cavity dispersion by placing a slit after thesecond intracavity prism sequence for wavelength selection, and measuring any corresponding shifts in frequency of frep (changes in round trip cavity time) as a functionof wavelength. The change in optical cavity length was obtained by measuring a highharmonic (∼ 10th ) of frep that resulted from mode beating in the cold cavity6 . The mea6

The cold cavity refers to a cavity operated CW such that there is no nonlinear effects resulting
from pulse propagation.

64
8
7

60



6
5

40

4
20

3
2

0
600

700

800

900

1000

beam divergence (mrad)b)
OC transmission (%)

waist size (102 µm)

a)

10
8
6
4
2
600

1100

700

800

900

1000

1100

wavelength (nm)wavelength (nm)c)
sagittal plane
tangential plane

2.5

M2

2.0

1.5

1.0
600

700

800

900

1000

1100

wavelength(nm)Figure 5.5: a) Measured beam waists and b) measured beam divergence at the laser
output coupler of the laser in the sagittal (filled squares) and tangential planes (empty
squares) versus wavelength. Also shown in a), the output coupler transmission (rightaxis) is included to indicate the bandwidth of the resonant versus non-resonant modes.
In b) the solid and dotted lines are the calculated TEM00 beam divergences from themeasured waist sizes shown in a) in the sagittal and tangential planes, respectively. The
ratio of the calculated beam divergence to the measured divergence indicates the M 2value for the mode, which is shown in c).

surement was unsuccessful, however, since ambient temperature drifts caused changes
in frep that overwhelmed those due to dispersion.
The laser is operated in a regime close to that used in non-octave Ti:S lasers.
Typically, KLM is obtained by displacing one curved mirror slightly away from theoptimum mirror position for CW operation [see Sect. 5.1.1]. In a non-octave spanning
Ti:S laser, the displacement of the mirror will result in a slightly elliptical CW beam.
Here, however, we take the misalignment a step further, in that the curved mirror
furthest from the pump is translated toward the crystal and to the edge of stability
for CW operation. Next the bandwidth of the modelocked spectrum is optimized bytranslating both curved mirrors, in the same direction away from the pump, just until

65
CW beam

CW beam

779nm

ML beamFigure 5.6: Far field beam profiles for CW and modelocked light at 780 nm.

the point where the laser begins Q-switching. The final curved mirror positions withrespect to the crystal yield the following results:
1) The misalignment heavily increases the discrimination between modelocked and CW
gain. This is also observed in ref. [7]. The discrimination in average power is ∼ 100 mW
CW as compared to > 600 mW modelocked. This indicates the formation of a verystrong Kerr-Lens to correct for the change in overlap with respect to the pump beam.
2) Translating M2 to the edge of CW stability yields a highly asymmetric CW beam(tangential waist/ sagittal waist = 1.56 [see Fig. 5.6]). This occurs because the misalignment of M2 no longer optimally correct for astigmatism due to the Brewster cutof the laser crystal [see Sect. 5.1.1]. Curiously enough, because of the nonlinear correction for the CW asymmetry, the Kerr-lens may in fact be astigmatic. The formationof an asymmetric Kerr-lens may explain the observed beam profile asymmetry in the
non-resonant wings of the modelocked spectrum.
3) The mirror positioning gives rise to a much more tightly focused CW beam thanthe pump beam (∼ ratio of 4:1, calculated using ABCD cavity propagation based on
the observed CW asymmetry). This, however, should result in terrible overlap with thepump beam unless either, nonlinearity during modelocking expands the waist, or thefocus of the Ti:S light is shifted in the crystal with respect to the pump.

66
There are a number of questions that remain unanswered as to what accounts forthe extremely large bandwidth and the observed mode profiles. In particular it would beinteresting to understand the physics behind the observed decrease in waist size of the
non-resonant as compared to the resonant modes. As mere speculation, the observed
factor of ∼ 2 decrease in waist size may be the simple result that, given that SPM is athird order nonlinear process, the non-resonant generated light is produced with a waist√that is 3 times smaller than the generating resonant modes. This is because, for a³2´

Gaussian beam the electric field exhibits a proportionality, E(r, z) ∝ exp − ωr 2 . Lightogenerated by SPM will have an electric field that is proportional to E(r, z) 2 E(r, z)∗ ,√thus being produced with a waist, ωo / 3. This is only possible if the generated light is
not resonant with the cavity, in which case the cavity geometry does not determine thewaist size. Perhaps the M2 data can help to uncover some of the details concerning thenonlinear effects in the Ti:S crystal during continuum generation.

5.2.1

Phase stabilization of an octave spanning laser

Although differing definitions are applied to the term “octave-spanning,” perhaps
the most appropriate for self-referenced stabilization is the bandwidth necessary toobserve a direct ν–to–2ν beat signal. This comparison requires an octave of bandwidth,
which for the results presented here, is produced directly by the laser itself. The offset
frequency is obtained via optical heterodyne between fundamental light at ∼580 nm
and doubled light at ∼1160 nm using a prism ν–to–2ν interferometer [see Fig. 5.3 a].
Because the light intensity in the wings of the spectrum is significantly lower than thatobtainable using MS fiber, a photomultiplier tube (PMT), not a photodiode, is used todetect the heterodyne beat signal. For the measurements presented here, the SNR of
the offset frequency beat note [Fig. 5.3 c] is 35 dB at 100 kHz resolution BW. This issufficient for stabilization of f0 without using a tracking filter [see Sect. A.6.1.2]. This
SNR is comparable to that of a beat signal obtained using a spectrum broadened in

67

Observation Time (s)
3

10

2

1

10

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

unlocked
10

10

10

0.30

5

0.25
0.20

0

0.15
0.10

locked-5

0.05

accumulated phase (rad)phase noise (rad2/ Hz)

10

0.00
10

-3

10

-2

10

-1

10

0

10

1

10

2

10

3

10

4

10

5

Frequency (Hz)Figure 5.7: Power spectral density versus frequency and the accumulated phase noise
versus observation time for the stabilized and unstabilized offset frequency derived fromthe octave spanning spectrum in Fig. 5.3 b).

MS fiber and using a 100 MHz repetition rate laser.7

In principle, given that a PMT

is a single photon detector, the detector noise should be very low. The SNR of themeasurement, we speculate, is limited by light noise from photons not contributing to
the beat signal. These photons may either result from insufficient filtering to ensure
spectral overlap, or from bad mode matching between the doubled 1160 nm light andfundamental light at 580 nm. By coupling the light, after the doubling crystal, intoa single mode fiber, the extra light noise due to mode mismatch should be taken out,
thereby increasing the SNR of the beat signal.
Stabilization of f0 is obtained with an accumulated phase error of 0.18 rad (1.65mHz - 102kHz), Fig. 5.7 b), following the error analysis discussed in Sect. 4.2. It is
important to point out that this is an in-loop measurement of the phase noise, andtherefore may not be used alone to characterize the noise on the output of the laser.
However, using the octave spectrum produced from the laser directly we altogether
eliminate phase noise generated by MS fiber. Also, because the prism based ν–to–2ν
7
Using higher repetition rate lasers, there is more energy per optical mode. Such systems typicallyallow for measured beat signals of 45 - 60 dB in a 100 kHz resolution bandwidth.

68interferometer used in the measurement of f0 , minimizes non-common mode mirrors,
the dominant source of out-of-loop phase noise on the laser should result from thephotomultiplier tube and the electronic amplifiers (this is because the noise spectrumof f0 is measured independent of the stabilization electronics).

5.2.2

Broadband Ti:sapphire oscillators as transfer oscillators for alloptical clocks

The ability to circumvent the use of MS fiber by generating the necessary continuum in the laser oscillator itself, makes broadband Ti:S lasers good candidates astransfer oscillators for all optical clocks. An experiment of interest would be to selfreference the offset frequency of the octave spanning spectrum. Stabilization of the laser
repetition rate would then be obtained by comparing of one tooth in the comb to an
optical standard. This comparison, used for measurement of the comb spacing, allows
for a much increased sensitivity to error in frep as compared to using a microwave reference. The laser stabilized in this manner would yield frep as the clock signal [12, 80],
which would allow for comparison of the Hydrogen maser [30].

69

Iodine optical clock
RF Output: frep= νopt /n

set νn = νopt
I2

ν

0

νn

ν2n

x2

set fo= 0
Figure 5.8: Schematic of an all optical clock based on molecular iodine. Stabilizationof the offset frequency is obtained via self-referencing. Stabilization of the n th comb
element, νn to the molecular iodine standard at νopt ∼ 1064 nm would then be used tofix frep = νopt /nChapter 6

Applications of the stabilized Ti:sapphire laser

A number of possible applications of the stabilized Ti:S laser have been discussed
in the introduction of this thesis. Of course, the two primary fields to which suchlasers may be applied are optical frequency metrology and ultrafast phenomena. In
this chapter I will discuss two experiment that make a small contribution to the hugebreadth of possible scientific applications. The first experiment discusses the applicationof stabilized Ti:S lasers to coherent control-type experiments in semiconductor physics.
The second, in contrast, explores the area of optical metrology, whereby synchronizationof a Ti:S laser and Erbium-doped fiber laser allow for coherent extension of the stabilized
comb to telecommunication wavelengths.

6.1

Quantum interference control in semiconductors: the matterinterferometer
Having demonstrated that stabilization of the laser results in a coherent pulse

train, one can begin to investigate phase sensitive physical processes. For example,
one may begin to consider coherent control-type processes in atomic [2], molecular[69] and semiconductor systems [31]. For example control of highly intense, ultrashortlight pulses reveals the phase sensitivity of physical processes in the strong-field regime[58, 6]. A recent illustration of this was presented by Baltuska et al. who demonstrated
influence over the spectrum of high-harmonic generated light by controlling the phase71a)b) E(t)

E

ν
ν

2ν

k

ν

pulse spectrum

2νtsin(φCE)1

T = 1 / f0t−1

Figure 6.1: a) Schematic depicting the quantum interference of one- and two- photon
pathways between the valence and conduction bands in LT-GaAs. Coherent controlof the injected photocurrent is obtained using an octave spanning spectrum with acontrolled carrier-envelope phase slip. b) The carrier-envelope phase change between
pulse to pulse evolves at an defined rate, T = 1/f0 .

of the driving field [6]. In the multi-photon regime, the photocurrent generated via
interaction of ultrashort pulses and a gold cathode was manipulated by controlling theelectric field of the pulse train [3].
Phase-stable modelocked lasers also have strong consequences in the low-fieldregime given the potential for deterministic control, for example, over the reaction
channels in chemical systems and population transfer between quantum states [9]. A
convenient medium with which to explore the latter is the semiconductor, whereby thecontinuum valence and conduction bands circumvent issues of parity that are often
common to atomic systems. In this section I will discuss how control over the carrierenvelope phase of the laser pulse train from a Ti:S laser may be used to determine thedirection of injected photocurrents in low temperature grown GaAs (LT-GaAs). I will
also discuss the potential for this technique as a means for solid state detection of thecarrier-envelope phase of a modelocked laser.
A simple experiment for exploring quantum interference in a semiconductor relies
on interfering one- and two- photon pathways between the continuum levels of the
conduction and valence bands [77]. In the presence of a two-colour light field with

72
frequencies ν and 2ν, there is quantum intereference between one- and two-photontransitions. The carrier population excited to conduction band states of a given crystal
momentum h̄k can be modulated by the phase parameter∆φ = 2φν − φ2ν ,(6.1)

where φν (2ν) is the phase of the ν(2ν) field. Such a modulation occurs at any point in
momentum space that energy conservation allows, although the modulation at different
points are not necessarily in phase. In fact, to a good approximation in GaAs, transitions
at k and -k are completely out of phase. To understand this let us examine the semiclassical interaction of light with vector potential A, interacting with an electron. The
Hamiltonian for matter-field interaction takes the form
Hint =

1(p + eA)(p + eA),
2m

(6.2)

where e is the electron charge and p = h̄i ∇ is the momentum operator. Because A andp do not commute we obtain
Hint =eeh̄
(p · A + A · p) =
A·∇
2m
im

(6.3)

The vector field has the form A(ν) = Aν cos(ω t + k y) for a plane wave travellingin the ẑ direction with corresponding electric field E = Aν ẑ sin(2 πν t + k y). The
transition rate from the valence band (ground state) to a state with a momentum
p = h̄k in the conduction band isW =dN (E)
2π|hψc |A · ∇|ψc i|2
,
h̄
dE

(E)
where dNdE
is the density of states. Recall that we are dealing with a two color fieldso A = A(ν) + A(2ν). The resonant non-interfering contribution from the one-photontransition is,
W2ν ∝

2π
|A(2ν)|2 pvc 2 .
h̄73
The two-photon transition is via an intermediate level, |ψI i,"#

2π |A(ν)|2 pvi 2
Wν ∝× |A(ν)|2 pic 2 .
h̄Γ(∆)
Here, Γ(∆) is a non-resonant denominator that is dependent on the detuning between
the energy of the intermediate level1 and the one-photon energy. The actual term ofinterest is the interfering term between the one- and two-photon transition pathways,[A(ν)pvi ][A(ν)pic ]2π[A(2ν)∗ pvc ]W2ν,ν ∝
h̄
Γ(∆)
½

¾

+ c.c.

(6.4)

Neglecting photon momentum, pcc = h̄k. As a result eqn. 6.4 yields a transition rate,
W2ν,ν ∝

2π pvc 2 k[|A2ν ||A2ν |sin(∆φ)].
h̄ Γ(∆)(6.5)

As a consequence, two-color light interacting with GaAs yields a population transfer
that is odd in k. That means that there are two asymmetric electron populations
created with opposite momentum, consequently yielding a current flow. To summarize,
the interaction of coherently related two-color light results in a phase-dependent injected
photo-current, J,
J̇(∆φ) ∝ (Aν )2 A2ν sin(∆φ) − J/τ,

(6.6)

where τ is a phenomenological decay time (For a rigorous derivation and explanationof photocurrent injection in semiconductors see ref. [5].)Quantum interference of photocurrents in a semiconductor, was first demonstrate
by Haché et al., who realized directional control over the injected photocurrent in bulk
GaAs using the fundamental light from an optical parametric amplifier and its second
harmonic [31]. The phase parameter between the one- and two-photon transitions,eqn. 6.1, was adjusted using a two-color interferometer and changing the relative phasebetween the fundamental and second harmonic light using a delay stage. In contrast,
1

From more rigorous theory is turns out that the most probable dipole transition for the two-photon
pathways is via an intermediate state that corresponds with the final state in the conduction band, thatis |ψi i = |ψc i. The large detuning makes the cross term’s transition rate weak as compared with thosedue diagonal elements.

74
for the experiments presented here, quantum interference control (QIC) in LT-GaAs is
obtained by adjusting the overall phase of light pulses emitted by a Ti:S laser. This
scheme allows for single-pulse control of the injected photocurrent in a semiconductor.
This is possible, once again, since Kerr-lens modelocking of the longitudinal cavity
modes transfers the phase information of the laser pulse train to the electric field of eachcomb element comprising the optical spectrum, E(νn ) = Eνn exp[−i2π(nfrep + f0 )t] =Eνn exp {−i[2πnfrep t + φCE (t)]}.
Control of the injected photocurrent generated in LT-GaAs requires coherently
related light with frequencies ν and 2ν. Because the octave spanning Ti:S spectrum in
Sect. 5.2 has insufficient power in the wings of the spectrum, the laser light is externallybroadened in microstructure (MS) fiber [62]. Injected photocurrent in LT-GaAs is
obtained by focusing the broadened light between two gold electrodes separated by
10 µm [see ref. [66]]. The generated photocurrent results in a space-charge effect that ismeasured as a voltage at the electrodes. Because of this effect, the faster recombination
time of LT-GaAs is preferred over that of GaAs.
To facilitate signal detection and avoid noise at DC, we modulate the carrierenvelope phase of the laser pulse train by stabilizing the laser offset frequency to 2.2
kHz. Notwithstanding that φCE is common to both light at ν and 2ν, it results in aphase difference in the transition pathway, ∆φ = φCE (t) = 2πf0 t + φ0 [eqn. 6.1]. As aconsequence we observe a QIC signal that varies sinusoidally at modulation frequency
f0 , with a phase offset φ0 [see. Fig. 6.1 b].
The creation of extra above gap carriers, not contributing to the interference signal, can in some instances reduce the detected photocurrent [32]. The loss in sensitivity
that results, however, is dependent on several parameters that are difficult to determine
exactly (e.g., pulse duration, pulse chirp and peak intensity). The production of incoherent carriers is of particular concern when using an extremely broadband source suchas that generated from broadening in MS fiber. To ascertain the net affect these carriers

75

Ti:S laser

EM

OC

PUMP

θ

MS



ν-to-2ν

locking
electronics

interferometerglass plateMS

lock-inamp.

samplegap 10 µm

fo
rf analyzer



30 µm


Figure 6.2: Experimental setup used for quantum interference control of injected photocurrents in LT-GaAs. The laser offset frequency is measured using a ν–to–2ν interferometer and stabilized via negative feedback to the laser. Microstructure fiber is used
for external broadening of the Ti:S laser spectrum for stabilization and QIC characterization. Because of fiber dispersion a prism sequence for time delay compensation, anda split mirror after the second prism is used for spectral filtering. Light at ν and 2ν arep-polarized such that the electric field oscillates transversely along the axis between thetwo gold electrodes on the LT-GaAs sample (inset).

have on the photocurrent signal we use incident light that is both filtered and unfiltered.
Figure 6.3 a) shows the spectrum of the photocurrent measured for the two different
incident optical spectra [see Fig. 6.3 b]. As observed in Fig. 6.3, the use of above gap
light (> 874 nm) does not influence the observed SNR, which is 40 dBm at a 10Hz
resolution bandwidth. The detected signal, measured using a lock-in amplifier, yields
a maximum signal strength that is slightly greater than 1 mV for a pulse repetition
rate of 93 MHz, with an average power (spot size) at ν and 2ν of 11.9 mW (10.2 ± 0.7

76
b)

a)
10 dB

10 dB

2000

2200

2400

2600

frequency (Hz)

400

600

800

1000

1200

wavelength (nm)Figure 6.3: a) Spectrum of the measured QIC signal as detected with a resolution
bandwidth of 10 Hz. The linewidth (resolution limited) of the measured QIC signal
indicates the stability of the laser offset frequency/carrier-envelope phase. The dottedand solid lines indicate the QIC signal strengths corresponding to the optical spectrain b). The solid black line shows the sample electronic baseline. b) The dip at ∼ 500
nm corresponds to unavoidable optical filtering due to the small gap between the split
mirrors in Fig. 2.

µm) and 1.53 mW (11.5 ± 0.7 µm), respectively. Currently, the SNR is dominated byelectronic pick-up from the sample, although distinct sidebands appear on the carrier
due to light noise, which we believe to result from inadequate phase noise suppression
in the stabilization feedback loop.
Next we use lock-in detection to demonstrate the sensitivity of quantum interference in the semiconductor system to small static shifts in the carrier-envelope phase,
φ0 . Lock-in detection of a stable amplitude signal, due to the narrow detection bandwidth, is strongly affected by the coherence of the laser offset frequency. As a result, forlock-in detection of injected photocurrents, coherence in the carrier-envelope phase is aprerequisite. To change φ0 we insert a 176.25 µm zinc borosilicate plate after the MS
fiber, before the interferometer used to lock the laser offset frequency. The small phasedifference due to dispersion of the glass plate facilitates fine tuning of φ0 . Rotation of
the glass plate results in shifts in φCE that are measured by the ν–to–2ν interferometer[see Fig. 6.2]. The stabilization loop compensates for the ensuing error by adjusting φ 0of the laser output that is then measured as an offset in the interference signal phase

77
a)

b)

1.5

phase (π-rad)35°

phase (π-rad)1.5

θ = 40°

1.0
30°
25°
20°

0.5
5°

10°

15°

1.0

0.5

0.0

0.0
0

20

40

60

time (s)

80

100

0

10

20

30

40

rotation angle (deg)Figure 6.4: a) Phase of the QIC signal as measured relative to the reference used forlaser stabilization. The time record of the QIC phase, measured with a time constantof 100 ms shows the phase jumps associated with rotations of the glass plate from 0
degrees to θ and back again, for eight different rotation angles. b) Comparison between
the phase change measured in a) (squares) and the calculated carrier-envelope phasechange (dashed line).

[Fig. 6.4 a]. The measured phase shifts in a) versus plate rotation are compared withthose calculated knowing the dispersion and thickness of the glass plate [see Fig. 6.4 b].
The measured and calculated changes correspond well up until a plate rotation angle of
30o, where the observed discrepancy may be the result of beam misalignment into theν–to–2ν interferometer.
Quantum interference in the semiconductor system is advantageous over morecommon phase-sensitive schemes in that detection suffers only minimal phase offsets
due to dispersion [46]. A significant improvement of the current SNR (> 30 dB SNR in
a 100 kHz resolution bandwidth) should be possible by better engineering the electrodegeometry. With increased SNR of the QIC signal, injected photocurrents in semiconductors could be utilized for stabilization of the laser offset frequency. It is our hopethat, in the near future, quantum interference of injected photocurrents will be used
as a phase sensitive photodetector. Given the potential for chirp free octave spanningspectra, solid state detection of the CE could replace the optical ν–to–2ν interferometer,
significantly reducing the complexity of the Ti:S stabilization loop [see Sect. 7].

78

6.2

Synchronization between a fiber laser and a stabilized Ti:sapphirelaser: An absolute comb at 1.5 µm
With the technology available for absolute stabilization of modelocked lasers, anatural extension is to increase the bandwidth of an existing frequency comb by coherently stitching together spectra from separate oscillators, thus leading to a coherently
synthesized pulse. The ultra broadband comb generated by synchronizing multiple
modelocked laser sources can allow for extension of a frequency comb to access spectroscopic transitions in the far IR and the UV, and additionally for generation of single
cycle pulses.
Even though the coherent synthesis of multiple CW lasers is the basis of thefrequency chain, the first coherent synthesis of two modelocked lasers was demonstrated
only in 2001 by Shelton et al. [70]. The lasers in question were two Ti:S lasers operatinga different center wavelengths. Electronic referencing of the laser repetition rates andoptical referencing of the offset frequencies resulted in the coherent addition of the twolaser spectra. As a test of the coherence, an autocorrelation of synchronized pulse
trains revealed a shorter duration pulse than could have been achieved from each laser
separately [70].
Synchronization between two KLM Ti:S lasers has the advantage that the frequency combs of the lasers are known to be“well behaved”, due to the strong modelocking mechanism provided by the formation of Kerr lens [Sect. 5.1.3.1]. The viabilityof using of other modelocked laser sources as optical comb references, relies strongly on
the characteristics of the frequency combs produced by the lasers themselves. Questions
arise as to the intrinsic stability available using lasers that rely on different modelocking
mechanisms from the KLM Ti:S laser.
In this Section I will describe a well-behaved comb at 1.55 µm from a home-built
erbium-doped fiber laser and systematic control of the fiber laser comb parameters.

79This is achieved by synchronizing and frequency locking the fiber laser comb to that ofa phase-stabilized Ti:S laser. This capability for a fiber laser presents the possibility ofa wide bandwidth optical comb, located at telecommunications wavelengths.
Although Ti:S lasers are tremendously successful at producing optical frequency
combs, they suffer from several drawbacks that limit their use outside of the laboratory.
These includes critical alignment, size, pump laser requirements and expense. Thus itis desirable to examine other mode-locked lasers as possible sources of stable frequency
combs. Mode-locked fiber lasers are particularly attractive, being direct diode-pumped,
relatively compact and potentially much less expensive [55]. However, since fiber lasers
use a different mode-locking mechanism than do Ti:S lasers, it is not a priori obviousthat they will produce a clean frequency comb. Furthermore, fiber lasers present a
different set of challenges for controlling the frequency comb, particularly the offset
frequency.
The stretched-pulse mode-locked fiber laser here operates at a center wavelengthof 1.54 µm and a bandwidth of 40 nm [55]. Mode-locking is achieved using nonlinear
polarization rotation. The gain medium of the laser is provided by erbium doped fiber,
which is pumped by a standard 980 nm diode. At the fiber laser’s output, chirped
pulses of 550 fs are observed with an average power of 45 mW at a 33.33 MHz repetition
rate. The fiber laser is synchronized to the Ti:S laser described in Chap. 2. For theseexperiments, the Ti:S laser is operated at a center wavelength of 800 nm with an average
power of 400 mW at a 100 MHz repetition rate. For ease of stabilization, the fiber laser
is enclosed in a box to isolate it from air currents and acoustic noise.
Figure 6.5 depicts the all-electronic experimental setup used for synchronizing
the two modelocked lasers and the optical scheme used to determine the quality of thesynchronization. Determination of the fiber laser offset frequency is obtained via optical
heterodyne against the Ti:S laser [33]. A stable beat signal, however, first requires tightsynchronization between the laser repetition rates to maintain temporal overlap between

80
to laser diode current

Fiber laser

F

fref

V

FFT

fs syncφφ

OscilloscopeOpticalfilterSlow
PZT LFBBO

AOMFFT – signal analyzer
8 GHz loop

100 MHz loop

Fast
PZT LF

fAOM

LF – loop filter
BBO – β Barium Borate

AOM – acousto-optic
modulatorφ

Ti:S laser
Rep rate loop

phase shifter

Offset loop/ timing
jitter measurement

Figure 6.5: Schematic of the setup for synchronizing a fiber laser to a Ti:S. laser.
Measurement of f0 is obtained by angle tuning the BBO crystal such that it is phasematched for second harmonic generation at 1.55 µm. The optical filter used in the f 0
measurement is centered at 770 nm. Similarly, measurement of the timing jitter used
the BBO crystal for sum frequency generation and an optical filter centered at ∼520
nm to detect the sum frequency mixing between the fiber and Ti:S laser pulses.

the laser pulses.
Synchronization of the two lasers uses an electronic scheme similar to that usedpreviously to synchronize two Ti:S lasers [70]. The repetition rates of the two lasers aremeasured independently on fast photodetectors and are then compared using a doubly
balanced mixer that provides a phase sensitive error signal to the feedback loop. The
repetition rate servo loop to the fiber laser consists of two phase-locked loops. The first
loop compares the third harmonic of the fiber laser repetition rate to the fundamental of
the Ti:S laser at 100 MHz. Once phase lock is achieved with the 100 MHz loop, the phaselock is shifted to a second, higher phase sensitivity loop that compares higher harmonics
of the repetition rates of the two lasers at 8 GHz. In this experiment, negative feedback
to the repetition rate of the fiber laser is achieved by changing the cavity fiber length
using two fiber-stretching stages. A slow stage (∼3 kHz) is used to correct for slow
drifts in cavity length and a faster stage (∼ 6 kHz) is used to correct for the remaining

81
phase error. The slow stage consists of a low voltage, stacked piezo-electric transducer(PZT) with a range of 10 µm. A pre-tensioned section of fiber is glued to ends of this
PZT. The fast stage is also constructed by fixing a fiber under tension to two shearing
PZTs (with 1 µm travel), oriented in opposite directions.
Once the fiber and the Ti:S lasers are synchronized, a timing jitter measurement isperformed to determine the quality. By using sum frequency generation (SFG) between
the fiber and Ti:S laser pulses in a beta-Barium Borate (BBO) crystal an intensity
cross-correlation may be obtained. Amplitude noise in the SFG signal results fromtiming/synchronization errors between the two lasers. To obtain the best sensitivity
and linearity for the timing jitter measurement, the fiber and Ti:S lasers are offset intime by half a pulse width such that the SFG signal is generated at half of its peak value.
From this cross-correlation a time record is recorded with an rms timing jitter of 12.4
fs (0.5 Hz - 2.5 kHz BW) is obtained [63]. The main limitations to the repetition ratesynchronization are due to a relatively small servo bandwidth resulting from mechanicalresonance in the fast fiber stretching stage at ∼ 6 kHz.
Once the laser repetition rates are locked, the fiber laser f0 is obtained via optical heterodyne between the second harmonic of the fiber laser (∼ 770nm) with thefundamental of the Ti:S laser. As seen in Fig. 6.5, the two laser beams are spatiallyoverlapped at a dichroic beamsplitter before being passed through a BBO crystal. This
employs the same setup as was used to measure the timing jitter, with the exceptionthat the BBO crystal is tuned for phase matched frequency doubling at 1.55 µm. A 4
nm optical bandpass at ∼ 780 nm filters the optical signal before a fast photodetector
measures the RF heterodyne signal with components at frequencies given by±|(mfrep1 − 2nfrep2 ) + (f01 + f02 + fAOM )|.

(6.7)

Here the indices 1 and 2 represent the Ti:sapph and fiber lasers respectively. For thisexperiment f01 = 0. From equation 6.7 it can be seen that timing jitter (first term)

82
in the repetition rates of the two lasers appears directly as sidebands on the offset
frequency, indicating the importance the repetition rate synchronization. To stabilize
the fiber laser offset frequency, the RF heterodyne signal at mfrep1 = 2nfrep2 is detected
and filtered. The signal is then referenced to the AOM drive frequency, fAOM , using
a digital phase detector. Negative feedback for the fiber laser f0 lock is provided via
amplitude modulation of the diode pump power. A frequency-to-voltage converter, inconjunction with an fast Fourier transform analyzer, is used to measure the frequency
noise PSD of the heterodyne beat.

phase noise (Hz2/Hz)

12

800
10

10

600
10

locked

8

400
10

6

200
10

4

unlocked
0
10

0

10

1

10

2

10

3

10

4

10

Accumulated Frequency Error (kHz)105frequency (Hz)Figure 6.6: Fiber laser offset frequency noise PSD, Sfo (f ) (solid lines, left axis) andintegrated frequency noise (dotted lines, right axis) versus frequency offset from carrier. The grey (black) solid line is the PSD of the unlocked (locked) offset frequencies,
respectively.

Figure 6.6 shows the in-loop measurement of the offset frequency for both theunlocked and locked cases. Integration of the locked frequency PSD in Fig. 6.6 yields
a fiber laser f0 linewidth of 355 kHz from 102 kHz to 0.5 Hz. Amplitude modulationof the pump diode suffers from a bandwidth-limiting steep roll-off beyond a few kHz,hence only a frequency lock of the offset frequency was achieved. Several physicalmechanisms may underlie the limited bandwidth. We hypothesize that it is phase lag,83
resulting from the lifetime of the Erbium doped fiber (∼ 1 ms) that is the limiting
factor in using pump amplitude modulation as an actuator. Improvement of the offset
frequency lock will require investigation into the transfer function of the current servo
loop and subsequent design of an electronic compensation system for the servo. Anotherpossibility of circumventing this limiting factor may lie in the modulation of the fiber
laser output by an AOM.
Servo limitations aside, the stretched-pulse mode-locked fiber laser produces aclean frequency comb that can be stabilized. A disadvantage of stabilizing the fiber
laser by means of comparison against another laser is that instabilities in synchronization
between the laser repetition rates results in a significant amount of noise on extra noiseon the offset frequency. The repetition rate lock may be tightened by using the opticaljitter measurement as the error signal instead of the high harmonic of the Ti:S f rep
signal. An even better possibility is to self-reference the fiber laser as a means to create
an absolute comb at 1.5 µm [72], along much the same lines as that for the Ti:S laser.
Broadening of the unamplified fiber laser output is achieved using MS fiber, designed
to work at telecommunication wavelengths [20]. Self-referencing for the fiber laser has
the definite advantage, over that for the Ti:S laser, in that the MS fiber used at 1.5 µm
may be spliced directly to the fiber laser output, thus eliminating alignment difficulties.

There is still much research to be carried out to resolve which modelocked source
will persist as the ultimate transfer oscillator for a primary optical standard. The choiceof laser will no doubt be somewhat dependent on ease of use, long term stability andpossibly on the choice of optical standard. In terms of simplicity, the broadband Ti:S
laser that yields a direct measurement of the laser offset frequency without the use of
MS fiber and the self-referenced fiber laser are both attractive candidates. The questionthat still needs to be addressed is the issue of stability. Both the Ti:S and fiber lasers

84
offer exceptional intrinsic stability of the comb parameters. Pump current control of thefiber laser offset frequency, necessary for this stabilization in the self-referencing scheme,
nevertheless, still suffers from the steep servo roll off at low frequencies. However, ahigher servo bandwidth for the fiber laser offset frequency may be obtain by inserting a
set of glass wedges into the laser, and using dispersion as a means for controlling f 0 .

Chapter 7

Conclusions and outlook

Stabilization of modelocked lasers via self-referencing heralds a new era for opticalfrequency metrology. The significant advances in phase stabilization of Ti:S lasers since2000 allow for absolute optical frequency measurements to be performed using a single oscillator directly referenced to the primary Cs microwave standard. Additionally,stabilization of the optical spectrum of a Ti:S laser, which naturally leads to control of
the optical carrier, now permits exploration into, and phase-determination of physicalprocesses that are sensitive to the field of an ultrashort light pulse. Phase sensitive
experiments of this type, however, are predicated on maintaining strong coherence inthe pulse phase. High phase stability lasers, however, require good engineering andthorough characterization of the stabilization system, as demonstrated here. Notwithstanding the technical challenges, the stabilized modelocked laser serves a relatively
simple and versatile tool for investigation of atomic, molecular and optical physics.
To summarize I would like to give a brief perspective from whence we have cometo hopefully where we are going. Figure 7.1 depicts an example of a harmonic frequency
chain used by the PTB (Physikalisch-Technischen Bundesanstalt) in Braunschweig, Germany used to perform an absolute optical measurement of Ca [67]. The details of the
figure are not important. What fig. 7.1 is meant to indicate is the level of complexity(12 oscillators and 14 PLL’s) involved in harmonic frequency synthesis, especially incomparison to Fig. 7.2 a), which shows the stabilized modelocked laser system that has

86replaced it (1 oscillator and 2 PLL’s).1 Although a stabilized Ti:S laser represents a
massive simplification over harmonic frequency chains, there is always room for further
reduction in complexity. In particular, toward the application to all-optical clocks, alimiting factor in terms of the reproducibility and long term stability of the Ti:S laser
is the use of miscrostructure fiber for measurement of the laser offset frequency. Asidefrom the issues of phase noise that we have been discussed here, complexities in coupling to MS fiber also can result in long-term loss in signal strength of f0 , which is not
acceptable in a possible clockwork.
Thankfully, the use of broadband Ti:S lasers can circumvent issues of MS fiber
altogether by generating sufficient bandwidth for direct measurement of f 0 using theunbroadened laser spectrum alone. Still, given the added simplification, there still exist
the complexity of nonlinear interferometers used for detection of f0 . Phase drifts dueto the optics and electronics are particularly problematic for maintaining long term CE
phase coherence. A further reduction in complexity to the phase stabilization scheme
may be obtained by using quantum interference in semiconductors as solid- state detection of the carrier-envelope phase [Sect. 6.1] instead of a ν–to–2ν interferometer. A
shown in Fig. 7.2 b), this technique, used in tandem with a chirp compensated octave
spanning laser spectrum, would allow for interferometer free optical phase detection ona chip. Solid state phase detection, via injected photocurrents, takes an optical input
and produces a voltage, V (t) ∝ sin(2π f0 t + φ0 ).2 For f0 6= 0, the output signal issensitive to the time evolution in the carrier-envelope phase, or in other words, measures the laser offset frequency. However, for f0 = 0, the voltage signal directly yields
the carrier-envelope phase [see Sect.4.1]. The quantum interference generated injected
photocurrent alone provides an error-signal of the CE phase without the need for com1

Typically optical frequency measurement uses higher repetition rate lasers that use only chirped
mirrors for intracavity dispersion compensation because of the higher frequency mode spacing andgreater power per optical mode.
2
This is exactly the signal produced from a phase comparison using a double-balanced mixer, except
operating at zero frequency.

87

Figure 7.1: PTB harmonic frequency chain to the Ca intercombination line.

parison to an external reference. By including amplification and filtering directly on thesame circuit as the semiconductor phase detector, one could condensed the entire ν–to–2ν interferometer and stabilization electronics onto a single chip. The final result would
be a phase sensitive photodetector! It is our hope that solid state detection of the CEphase will allow for this realization of this amazingly simple stabilization scheme. Sucha scheme would eliminate much of complexities due to drifts in optics and electronics
and should allow for a more reproducible user friendly system.

88a)
Ti:S Laser

PUMP

fref
MS

LF

1 oscillator, 2 PLL’s

530nm

1060nmfo

×2

ν-to-2ν Locking electronics
interferometerb)

Octave Laser

PUMPphase error out

light inε

LF

Optical phase detector

Figure 7.2: (a) Current experimental apparatus for stabilization of a Ti:S laser and (b)hopeful future experimental apparatus.

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Appendix A

Phase stabilization of Ti:sapphire lasers: the details

In this Appendix I hope to fill in many of the design considerations and details
for realization of a highly phase stable laser. Just as a reminder of the basic concept,
we wish to directly measure and the two characteristic frequencies of the optical comb,
the laser repetition rate, frep and the laser offset frequency, f0 . Recall that frep sets the
comb spacing and is inversely proportional to the laser cavity length, and that f 0 decides
the absolute comb position and is directly proportional to dispersion inside the laser
cavity. Once these two rf frequencies are measured, they are electronically compared toreference frequencies, necessary for stabilization. Negative feedback to the laser is thenused for error correction.

A.1

Frequency difference measurement of the optical comb
The most readily available method for measurement of light is using a photodiode(PD). However, because PD’s are power law detectors they may only measure frequency
differences obtained via comparison between two optical carriers. In the case of boththe laser repetition rate and the laser offset frequency, optical comparison that yields a
difference frequency proportional to frep and f0 is provided within the comb itself via
heterodyne between adjacent comb components. This will be explain in detail shortly.
First let us examine a more simple example of optical comparison for difference
frequency measurement. Such a measurement may be obtained from the interference of96b) PD signal:a) Optical interference:2Re{exp [ i (ωc + ∆ω ) t ] + exp[ i (ωc − ∆ω ) t ]} | exp(i [ω c + ∆ω ) t ] + exp[i (ω c − ∆ω )t ]| =2[1 + cos(2∆ω t )]

E1+E2

I(t)
t
t
rf spectrum

Optical spectrum

νc
νc-∆ν

νc+∆ν

ν

0

2 ∆ν

f

Figure A.1: (a) Optical interference between two electric fields separated in frequency
by ∆ω. This results in sinusoidal modulation of the carrier-frequency, ωc , at 2∆ω. (b)Radio frequency spectrum that results from the current signal produced from measurement of the optical signal using a photodiode.

two coherent light sources. Interference between two collinearly propagating light waves
of different frequency, ωc +∆ω and ωc −∆ω, will result in a modulation of the light at thedifference frequency, 2∆ω, or in other words, in a time-dependent interference pattern[see Fig. A.1 a)]. If the two interfering wavelengths have a difference frequency within
the bandwidth of the PD, the PD measures the change in strength of the light due tothe travelling constructive and destructive interference pattern. Interference between
two wavelengths of light is the simplest example of optical beating. The amplitude
spectrum for this example is simply an rf spike at the difference frequency, 2∆ω = 2π∆ν[see Fig. A.1 b)].
A more complicated example of optical beating is the coherent addition of millionsof frequencies, which leads to the synthesis of complicated waveforms. This is thephysical mechanism behind pulse formation in the Ti:S laser [see Fig. A.2 a]. This is
the ultimate in optical interference and permits easy measurement of the characteristicfrequencies for the KLM Ti:S laser.

97a) Optical interference:n =5b) PD signal:[

E (t ) = ∑ exp i 2π (n f rep + f c )tn = −5E

n =10

I (t ) = E (t ) = 10 + ∑ cos (2π n f rep t )

]

2

n =1

trt

trt

I(t)
t

t
rf spectrum

Optical spectrum

fcν

frep =1/trt

0

frep =1/trt

fFigure A.2: a) Optical interference between multiple coherent optical fields separated
in frequency by ∆ω, at a carrier frequency, ωc . This results in severe AM modulationof the carrier-frequency, ωc , whereby the net result is pulse formation b) The current
signal produced by shining the light in a) on a photodetector (PD). The PD can only
measure the intensity of the light which is phase (carrier) insensitive.

A.2

Measurement of the laser repetition rate
The intrinsic amplitude modulation of a pulse train of a modelocked laser canyield direct measurement of the the laser repetition rate, frep . The laser pulse train
incident on a photodetector yields a series of amplitude spikes in time at trt = 1/frep .
Therefore the signal out of the PD is the AM modulation of the pulse train, without
the optical carrier, resulting in a current blip each time a pulse hits the detector at
an interval, t = trt [see Fig. A.2 a]. The Fourier transform of a voltage signal of delta
function spikes in time is a rf frequency spectrum starting at zero Hz with frequency
peaks at harmonics of frep = 1/trt [see Fig. A.2 b]. (If the PD were sensitive to theoptical carrier, the Fourier transform would yield the optical frequency spectrum of thelaser, centered at the optical carrier frequency.)
Of course, given that we are using modelocked lasers, everything can be thoughtof both in the time and frequency domains (deciphering the frequency and time domain

98
aspects of the light field produced by the modelocked laser is the main fun offered inthis field.) Generation of the rf spectrum can therefore also be explained by considering
every harmonic in the rf spectrum as being generated by beating between successive
components of the optical frequency comb, Fig. 2.3.

A.3

Measurement of the laser offset frequency, f0
Measurement of f0 relies on exactly the same principles of optical beating as doesmeasurement of frep . However, to generate a beat signal sensitive to f0 requires the useof a nonlinear interference. As described in Sect. 2.2, self-referenced measurement of theoffset frequency is obtained using nonlinear comparison. Measurement of f0 may also
be determined using the classic optical frequency metrology technique of comparison
against a secondary optical standard.
The offset frequency cannot be directly extracted from the pulse train in theway that frep may be. There are two ways to think of this. In the time domain,knowledge of f0 is contained within the carrier-frequency. Because the PD is insensitive
to the carrier it cannot provide direct measurement of f0 . In the frequency domain,
generation of an rf spectrum is obtained by the beating harmonics of the optical comb,|(m frep + f0 ) − (n frep + f0 )|, which results in perfect cancellation of f0 .
The nonlinear heterodyne that is utilized for measurement of f0 is obtained using a
ν–to–2ν interferometer that allows for optical comparison between the second harmonicof the low frequency comb components of an octave spectrum and the fundamental of
the high frequency comb components of the spectrum. The comparison at the output of
the interferometer may be thought of as interfering two pulse trains with a relative timedependent phase shift, φ(t) = 2π f0 t. The two pulse trains are temporally overlapped sothat interference is possible (interference occurs in free space, not only at the PD). If thetwo pulse trains were not shifted in frequency the PD would simply see the amplitude
modulation of a single pulse train. The interference due to a shift in frequency between

99
the two pulse trains results in an additional, slightly slower AM modulation of the pulse
train itself. The voltage signal out of the PD is then a series of delta function voltage
spikes with an additional sinusoidal modulation envelope at f0 . The Fourier transformof this voltage signal is then the rf frep spectrum with AM sidebands at f0 as in Fig. 2.4.
The bandwidth necessary for self-referencing of f0 is an optical octave. Typically,
the octave is obtained via external broadening of the Ti:S laser spectrum in MS fiber,
although, as shown in Chap. 5, it is preferable to generate the necessary bandwidth
within the oscillator directly.

A.4

Fun with microstructure fiber

Figure A.3: High-stability mounting of MS fiber.

Microstructure fiber is, by far, the most difficult element to work with in thestabilization loop. The worst aspect of the fiber is that changes in the input coupling
result in alterations in amplitude of the spectral output. Coupling problems of this sort
can be minimized by using very high quality coupling stages. We use Thorlabs flexture
stages (MDT616) each with differential micrometers for the x-, y- and z- translation
axes [see Fig. A.3]. Also when clamping MS fiber it is best to minimize the amountexcess fiber at the edge of the clamp. This will diminish pointing errors resulting fromvibrations of the dangling fiber end. Additionally, I have found that it is useful to add

100a thin piece of rubber sheet to secure the fiber in the clamp. All these precautions help
to minimize AM-to-PM conversion in MS fiber [see Sect. 4.3.1] and cut down the effectof long term coupling problems.
To allow for fine tuning of the generated continuum we use a broadband polarizer
and half-wave plate before the fiber. This allows us to control in the input polarizationof the light pulses and the coupled power. The input polarization to the input of thefiber is crucial because the fiber is slightly birefringent. For MS fiber from Lucent
Technologies, there exists a preferred axis which allows for maximum broadening aswell as preservation of the input polarization [see Sect. 3.2].

A.5

Self-referencing: the interferometer
There are many different configurations that may be used for the ν–to–2ν inter-

ferometer. The main design specifications, however, are always the same. The pointof the interferometer is to allow for generation of a self-referenced nonlinear opticalinterference signal of the optical frequency comb with itself. As explained above, interference is obtained by cross-referencing two pulse trains that differ in frequency by f 0 .
This requires temporal, spectral, polarization and spatial overlap of the beams of theinterferometer.

A.5.1

The Mach-Zender interferometer

One design for the ν–to–2ν interferometer may be obtained using a nonlinear
Mach-Zender interferometer. Light entering the interferometer is split into two arms
using a harmonic beamsplitter at 1064/532 nm that reflects 1064 nm light and transmits 532 nm light, we use Newport 10QM20HB.12 high energy Nd:Yag laser harmonic
beamplitter (R1064 = 99.5%, T532 = 80%). The choice of this beasmplitter satisfies the
spectral (ν and 2ν) interference aspect and is also a readily available dichroic beamsplitter.

101

AOM

λ/2


PBS

prisms

LBO

λ/2


glass

MS

Mach Zender Interferometer

Figure A.4: The Mach-Zender ν–to–2ν interferometer. The optical filter, polarizer andphotodetector are not shown in the figure. A big chunk of glass is used for fixed timedelay in the 1060 nm arm.

A.5.1.1

Doubling crystal

Once the light is split at the dichroic beamsplitter, the 1064 nm arm is doubled
in a nonlinear crystal. One has to consider, however, that one only has 10’s of mW ofpower for doubling, and that the light is also chirped because of propagation through the
MS fiber. As a result a longer crystal can be used to increase the nonlinear interaction
length. Of course there are always trade-offs to be made. The longer the crystal, thenarrower the phase matching bandwidth, and the smaller the acceptance angle. The
latter may result in aperturing of the beam to a narrow line of second harmonic light,
thereby decreasing the mode matching between the 2ν and ν light.
In the experiments here, we have used two nonlinear crystals, 4 mm BBO and 1
cm long LBO. Both crystals are cut for type I phase matching at 1064 nm and the input
face should be anti-reflection coated at this wavelength. BBO has a higher nonlinear102a) Frequency shifter

PZT

f0 + 2 fAOM
f0 + fAOM
f0
f0 - fAOMb) Noise eaterf0 - 2 fAOM

fAOM

∆P
P0
P 0 - ∆P
PZT

∆A cos(ωΑΟΜ t)

Figure A.5: a) Configuration for using an AOM as a phase sifter. The AOM is driven
by a constant amplitude sinusoidal rf signal of frequency fAOM that creates a pressure
grating in the AOM material, which causes diffraction of the input light by harmonics
of fAOM . b) Configuration for using an AOM as a noise eater. Varying the amplitudeof the driving signal proportionally changes the light power in the first diffracted order,
thereby modulating the power on the undiffracted beam.

coefficient than LBO and thus can be shorter. However, using a longer crystal of LBO
may allow for focussing of the beam within the acceptance angle, which better preserves
the mode matching to the fundamental beam.
Because of the polarization rotation caused by doubling (there is a rotation of
the second harmonic light by 90 deg with respect to the fundamental for type I phasematching), and also because of the random array of polarizations produced by the MS
fiber (MS fiber is slightly birefringent, and is also susceptible to nonlinear polarization
rotation [see Sect. 3.2], half-wave plates at 532 and 1064 nm are used to adjust thepolarization in both arms of the interferometer.

A.5.1.2

The AOM

In the green arm of the interferometer an AOM is used to shift the frequency
comb, of that arm only, by fAOM . This is used so that the offset frequency may belocked to zero Hz. From the previous section, if the offset frequency is zero, there is nooffset sideband on the harmonic spectrum of frep . To circumvent this, the AOM shifts
the entire frequency spectrum of the green arm by fAOM , such that when f0 = 0, there

103
is still a sideband beat note at fAOM [see Fig. A.5 a].
The AOM used here is from Isomet, model #1205C-2. The AOM uses high index(n ∼ 3) material to which a fast PZT is attached. By applying an amplified rf frequency(28 dBm , this requires a power amplifier) to the PZT, the pressure wave caused by the
PZT results in the formation of an index grating transverse to the propagation directionof the light. Light diffracted off the grating is doppler shifted by k fAOM , where k is thediffraction order. Therefore, to align the AOM requires that it is tuned such that the
majority of the light is diffracted into the first order. If light is diffracted into k = +1,
then the comb is shifted by +fAOM . If light is diffracted into order k = −1, then the
comb is shifted by −fAOM .
A.5.1.3

Temporal, polarization, spectral and spatial overlap

Light from the two arms of the interferometer is re-combined at a polarizing
beamsplitter (PBS). Such a beamsplitter transmits vertically polarized light and reflects
horizontally polarized light. This is practical, since in general, light in the green arm
is horizontally polarized, and that in the IR arm, because of the doubling crystal, isvertically polarized.
Good spatial mode-matching between the ν and 2ν beams is very importantin obtaining a strong beat signal. This is because the interference results between the
wavefronts of the doubled ν and fundamental 2ν beams. To optimize the interference
requires that the mode size, wavefront curvature and propagation be same for bothbeams. Overlap between the beams can be ensured by verifying that the two beams
remain overlapped over a large distance. This is done by checking the overlap just after
the re-combining beamsplitter and then about 1 m away.
Before being recombined, however, the path lengths of the interferometer arms
are balanced such that temporal overlap is obtained between the two pulse trains.
This may be achieved by using a corner cube or by using glass and a wedge pair as

104shown in Fig. A.4.
To satisfy the conditions for interference, a polarizer is used to ensure the polarizations of the 2ν and doubled ν light projected along the same axis. The polarizer isalso a convenient means for selecting the relative power from the two arms to produce
the strongest beat signal.
After the polarizer, we use a spectral interference filter to ensure both spectral overlap and purity. The interference filter should be chosen such that its bandwidth
matches the phase matching bandwidth of the nonlinear crystal and has a center wavelength where there is best nonlinear spectral overlap. We use an interference filter
centered at 530 nm with a 10 nm bandwidth.
Aside from ensuring spectral overlap, the interference filter is used to filter out any
light not contributing to the interference signal, which would act to raise the detection
noise floor. Filtering of the signal also minimizes saturation of the PD which can cause
horrible oscillations in the PD amplifier circuit, or result in complete insensitivity to themeasurement of f0 . The photodetector that is used uses a visible Si PIN photodiodethat has a bandwidth of 1 GHz (S5973 - Hamamatsu). This diode is used in a low noise
2 GHz photodetector amplifier circuit (nearly all surface mount components) designedby Terry Brown.

A.5.2

The prism interferometerA second interferometer design may be used that is much more simple that minimizes non-common mode mirrors is the prims interferometer. Its clever design wasconceived by Henry Kapteyn. The interferometer uses a compressor prism pair, notfor pulse compression but for spectral dispersion of the octave spectrum. The prisms
are aligned such they diffract the input light to minimum deviation and chosen suchthat they give a high second order dispersion (SF10, SF18) for high spatial resolution
for a short prism separation. At the exit of the second prism the light is horizontally

105

MS

PD
split mirror
polarizer

filter×2

Prism interferometer

Figure A.6: The prism ν–to–2ν interferometer.

dispersed such that a split mirror after the second prism may be used for temporal
overlap between the ν and 2ν portions of the octave spectrum. The position of the split
mirror is chosen such that green light is reflected by one half of the mirror, and IR lightis reflected by the other half. This serves two purposes. If one mirror is put on an x-ztranslation stage, the z-stage can be used to adjust the path length of the green withrespect to the IR arm. The x-stage may be used to increase the mirror separation, for
optical filtering. We use a SF10 prisms with a separation of ∼ 5 cm. At this separation
the distance difference between the split mirrors is ∼ 1 mm, where the mirror reflecting
the green light is pushed in further than that reflecting the IR.
Light is reflected from the two mirrors and sent back through the prism pair, but
at a slight downward angle from the original beam. The return trip through the prism
pair acts to re-combine the red and green portions of the light, which are then picked
off and focused into a nonlinear crystal. The crystal is tuned for phase matching the IR
light only. As in the Mach-Zender interferometer, a polarizer and optical filter are used

106
for filtering and a fast PD is used for detection of the offset beat signal.
Once the offset frequency beat signal and repetition rate are measured, the next
job is to phase lock them to a reference frequency. This is done using a series ofelectronics that provide comparison to a reference and then negative feedback to thelaser for error correction [see Fig. A.7].

A.6

The laser stabilization loop
In this section I will primarily discuss the feedback electronics and briefly discuss

the feedback actuators for frep and f0 .
synthesizer

Ti:sapphire laser

PUMP

HV amp(orthogonalizer)±150 VEM

couplermixer
amplifier

HV driver(1 kV)JILA LF

fref

frep
stabilization

BP (10 GHz)
(10 MHz BW)

frep-10 dBm+ 28 dBm

power amp
AOM

fs
synchronizer 0 dBm
fAOM= 7 frep
8

tracking
filterfAOM+fo- 20 dBm
BP (50-100MHz)(10 MHz BW)+ 60 dB
0 dBm

rf analyzer

frep

f0

2 frep3 frepf0
stabilizationn frep+ 7dBm

F-toV

SRA-1

FFT

f0
characterization(244 µHz – 102kHz)

Figure A.7: The electronic diagram for phase lock of the offset frequency and therepetition rate. Details about the electronics may be found in the sections below. Notethat in this schematic, the fs-synchronizer provides the reference frequency and provides
the phase comparison (mixer in Fig. 2.7.)

107
A.6.1

Stabilization of the laser offset frequency

Once f0 is measured using a ν–to–2ν interferometer, electronic filtering and comparison are necessary for feedback stabilization to the laser. The following section givesa break-down off the stabilization loop schematic in Fig. A.7. Before the f 0 beat signal
is filtered and amplified, a small portion of the signal is split using a directional coupler.
This allows for monitoring of the rf spectrum to ensure that the f0 signal strength ismaintained over time. The typical signal strength of the offset sidebands is ∼ -60 dBm
with a SNR of 35-40 dB at a 100 kHz resolution bandwidth (res BW). Comparisonof the f0 signal to a reference frequency requires a signal strength of at least 0 dBm,
achieved via amplification. However, before the signal is amplified, the rf spectrum isfiltered such that only the one sideband that contains information about the f 0 alone[see Fig. A.7] is used for comparison. For filtering we use a simple 100 MHz low passfilter in series with a sharp bandpass filter. The filter we use is produced by K & L
and yields a nearly perfect square bandpass, with a passband of ∼20 MHz and a 4 pole
rejection roll off. 1
Strong filtering of the beat note before amplification is necessary so that theamplifiers are not saturated by the other rf components of the spectrum. We use three
Mini-circuits ZFL-500HLN amplifiers in series, each providing 24 dB gain. Once filteredand amplified the signal is split using a 3 dB coupler, whereby one half of the signal
is used for locking [see below] and the other used for an in-loop measurement of thestability of f0 [see Sect. A.7].

A.6.1.1

The fs-synchronizer: offset frequency phase-locked-loop

The phase-locked loop performs a comparison between a reference (master) signal
and the signal to stabilized (slave). The action of the loop is to force the slave to take up
1n pole rejection indicates the slope of the roll off. 1 pole = 10 dB per decade, where one decade =1 order of magnitude in frequency change.

.. Nfrom tracking VCO(filtered slave signal)

.. M
.. 8

REP RATE
INPUT
frep

108
phase monitor

.. M
.. 2

frep
8

LF

7 frep8Servo out

OUT to AOM

SSB mixer
SSB – single sideband mixer
DPD – digital phase detector.. M Divider strength .. Nreference
multiplier

Figure A.8: Block diagram of the fs synchronizer. The reference “multiplier” allows forfrep
f0 to be locked to 16 different reference frequencies, fref = N ( 16). In reality divisionof the reference frequency by 16 and division of f0 by N , performs the same task.

the frequency and phase information of the master through negative feedback actuatedby a voltage-to-frequency converter (the laser) [see Fig. 2.7]. The stabilization circuit at
the heart of the offset frequency stabilization loop is the“fs-synchronize” [see Fig. A.8].
The synchronizer is a wonderful device designed by John Hall. It serves several purposes:
it 1) derives a reference frequency from an frep input, 2) produces a signal to drive theAOM to lock f0 to zero, and 3) provides the phase comparison and error signal shaping.
The advantage of using the synchronizer over a mixer for stabilization of f 0 is that
it utilizes a digital rather than analogue detector (double-balanced mixer) for comparison between the slave and master signal. The digital phase detector acts to produce anerror signal proportional to the frequency difference between the master and slave, thisallows for a frequency lock. Once the two signals differ only in phase, the digital phasedetector (DPD) acts as a phase lock [see Fig. A.9]. The combination of frequency/phaselock allows for a larger capture range than a double balanced mixer, which can only
produce an error signal if the master and slave are close in frequency. Therefore, if a
perturbation knocks the laser out of lock, the DPD can recapture the slave and then

109
a)
S1

∆φ

S1 – S2∆Q
T1

DPD
S2 – S1

S2

∆Q
T2
Rf
b)

R

Ri

R

Ri

S1
DPD
S2
C

C

+
_
Rf

V+ - VV+ ∝ 1/ T1(t) = f1 (t)V- ∝ 1/ T2(t) = f2 (t)Figure A.9: a) Schematic of error detection using a digital phase detector (DPD). The
DPD takes two input signals, S1 and S2 (50 % duty cycle produced from a flip flop that
divides the original input frequency by 2), and produces two output signals, S1−S2 andS2−S1. Here S1 and S2 differ only by a constant phase offset. b) Schematic of how thedigital phase detector is used for producing an error signal linearly proportional to thefrequency difference between the square wave signals, S1 and S2. Given the two signalsin a), the output of the differential amplifier would be a constant voltage proportional
to the phase difference between S1 and S2. this scheme, however, produces a zero volt
output if the phase difference is zero or 180 degrees. If the phase difference were timedependent, ∆φ = 2πf (t), the output signal would evolve as f (t).

re-phase lock it. The range of the DPD is determined by a divider stage. The larger
the division, the larger the range and the looser the lock (the weaker the slope with frequency/phase). A drawback of using the DPD, however, is that it uses a comparator,
which requires a SNR for the slave signal > 30 dB (100 kHz res. BW) such that thecomparator output is not dominated by zero crossings resulting from low SNR.

A.6.1.2

The tracking filter

If the f0 beat signal exhibits low SNR or a strong amplitude modulation, a tracking filter is used to clean up and amplify the signal. Typically, a tracking filter is usedin conjunction with the fs-synchronizer for low SNR f0 beat signals. (A tracking filter is

110

f0

phase monitor
LF

VCO(error signal
for frequency lock)amplified / cleanf0
Figure A.10: Block diagram of the tracking filter used to produce an amplitude stablerf signal that tracks the phase information of the low SNR beat signal, f 0 .

not typically necessary if a double balanced mixer is used instead of the fs-synchronizer.)
The tracking filter is a PLL that locks an rf voltage controlled oscillator (VCO)
to a noisy rf signal. The VCO acts as a very narrow, dynamic filter that tracks thefrequency and phase information of the input signal, but with a strong stable amplitude[see Fig. A.10]. Because the tracking filter is an electronic PLL its speed is limited onlyby that of the electronics. Typically, if the beat signal for f0 is weak (< 30 dB SNR
in 100 kHz res BW), a tracking filter is used to increase the offset frequency’s SNR for
used with a DPD.

A.6.2

Negative feedback

Negative feedback to the laser for error correction of f0 may be actuated either
by tilting the laser end mirror or via modulation of the laser pump power. Of course,
each different feedback mechanism requires a different driver to make the feedback error
signal compatible with the actuator. In the case of the PZT, a high voltage driver must
be used. The driver for the AOM is depicted in Fig. A.5 a.

A.6.2.1

Feedback using a PZT

Tilting of the laser end mirror is actuated using the PZT scheme in Fig. A.11.
Tilting of the laser end mirror is achieved using two fast (100 kHz) PZT’s driven in

111
opposition using the amplified error signal derived from the fs-synchronizer (maximum± 150 V). Fast PZT’s may be used in this configuration if the laser is passively stable.
Our laser results in long time scale drifts in f0 of around 1 MHz per minute. The rangeof the system is on the order of 50 MHz and the combined PZT + mirror system canyield a bandwidth up to 30 kHz if the mirror size is minimized.

A.6.2.2

Feedback using an AOM

Placing an AOM in the path of the pump beam allows for control of the offset
frequency by inducing amplitude–to–phase modulation in the laser crystal. The AOM(IntraAction # ASM-803B47) is set up in a noise eater configuration [see Fig. A.11 b)].
This is different from the configuration used in the Mach-Zender interferometer thatuses the AOM for frequency shifting the first order diffracted light. Here, in contrast,
the first order diffracted light is used for low-level amplitude modulation of the zeroorder transmitted beam.

Figure A.11: The disc PZTs + mirror used for stabilization for fine tilting of the laser
end mirror, EM. Two fast PZT’s are attached to a blank mirror substrate using vacuum
epoxy (Torr Seal) and the mirror is centered on the PZT’s using sturdy wax for easy
removal. Two small cuts are made in the mirror blank to allow room for contact
wires. The mounted PZT system is held in a heavy copper mirror mount to minimize
momentum transfer from the PZT to the mirror mount.

112

Figure A.12: The tube PZT + mirror used for stabilization of the laser cavity length.
The PZT is attached to a blank mirror substrate using vacuum epoxy (Torr Seal) andthe mirror is mounted on the PZT using sturdy wax for easy removal. The mounted
PZT is held in a heavy copper mirror mount to minimize momentum transfer from the
PZT to the mirror mount.

A.6.3

Stabilization of the laser repetition rate

Comparison of frep to a reference frequency is obtained using an SRA-1 doublebalanced mixer. The mixer requires a signal strength of 0 dBm for the rf port, thus
requiring filtering and amplification of the frep signal. The error signal generated via
comparison to the synthesized reference is filtered and amplified using a loop filter. We
use a JILA designed loop filter that allows for variable PI (proportional-integrator) corners to match the transfer functions of different feedback actuators. In the experiments
here, only a slow tube PZT is used for cavity length correction [see Fig. A.12]. The large
range required of the PZT results in a relatively low bandwidth (∼ 1 kHz). More tightsynchronization of frep requires using an additional fast PZT, and therefore splitting
the error signal between a fast and slow feedback loop. Details about synchronizationof frep may be found in [70].

113

A.7

Measurement of the offset frequency stability
In a typical PLL, the gain of the servo system drops with increasing frequency,

whereby the rate of this drop is determined by the design of the servo loop (electronic
response and the transfer function of the negative feedback actuator). As a result, thefeedback loop has a much stronger ability to suppress noise at low frequency than athigh frequency. Because of this characteristic, suppression of drift is by far the easiest
task. Suppression of high frequency noise means hard work must be put into the designof the entire feedback system, at which point a limit is encountered. This limit isoften determined by the point when accumulated phase shifts in the loop (electronic
and actuator) result in positive feedback and hence oscillations in the loop. This point
determines the unity gain of the servo or in other words the bandwidth of the feedback
system.
Residual noise on f0 due to imperfect stabilization is obtained in the frequency
domain by analyzing f0 ’s linewidth. Remember, phase and frequency noise differentiate
by only a time derivative, and thus may be considered one in the same. The noise spectrum of f0 ’s lineshape may be used to determine problem areas, and helps in identifying
the causes of noise.
One important issue in measuring the noise spectrum of a signal is in distinguishing between amplitude and phase noise. In the Fourier domain, by naively looking at
the linewidth of a signal on an rf analyzer, the sidebands on the carrier frequency arecomposed from contributions of both. Proper noise analysis, for determination of thelinewidth of signal requires the separation of the two.

114a) Sφ(ν)

J0(β)

b)

SΑ(ν)

Jn(β)
fo-2fm
fo

α/2

α/2

fo-fm

fo fo+fm

ν

fo+2fmJn(β)ν

Figure A.13: a) Frequency spectrum of a carrier frequency, f0 phase modulated at fm
with a modulation strength β. The conjugate sidebands on the carrier are shift in phaseby π. b) Frequency spectrum of a carrier frequency, f0 amplitude modulated at fm witha modulation strength α.

A.7.1
A.7.1.1

Phase noise vs. amplitude noise
Amplitude Modulation

Amplitude modulation of a carrier frequency, ω0 , with amplitude modulation
strength α and modulation frequency, ωm is described by1
V (t) = δA(t) cos(ω0 t) = cos(ω0 t) + α {cos [(ωm + ω0 ) t] + cos [(ωm − ω0 ) t]} ,
2
¸

·

(A.1)
where δA(t) = [1 + α cos(ωm t)].
Given eqn. A.1, AM modulation results in the production of conjugate AM sidebands, where the height of each sideband is given by α2 [see Fig. A.13 a]. Because ofthis, pure AM noise, δA, on a carrier frequency is very easy to analyze. Integration of
the power spectral density of the noise spectrum, SA (f ), infinitely far from the carrier,to the carrier itself, will yield the total rms amplitude noise for the spectrum.
∆Arms =

s Z
0

2

−∞

SA (f )df

(A.2)

¯
¯
¯ F {δA(t)} ¯2¯ , where ∆fRB is the resolution bandwidth and ν is the fre¯√Here SA (f ) = ¯∆fRB ¯quency relative to the carrier. Consequently, SA (ν) is measured in V 2 /Hz. The factorof two takes into account single sideband integration of the noise.

115
A.7.1.2

Phase modulation

Phase modulation of a carrier frequency, ω0 , with strength β and modulation
frequency, ωm is described as
V (t) = cos [ω0 t + δφ(t)] =∞X

Jn (β)cos [(ω0 + n ωm )t] ,

(A.3)

n=0

where δφ(t) = β cos(ωm t). As seen in eqn. A.3, phase noise is decomposed into harmonics of ωm , whose strength is given by the Bessel function coefficients, Jn (β) [seeFig. A.13].
Because of the nature of PM modulation, at strong modulation strengths (β >>1), the contribution of the rms phase noise from several modulation frequencies cannot
be determined directly from a phase noise spectrum. At weak modulation strength(β << 1), however, the first Bessel function sideband dominates the noise spectrum. In
this limiting case, the strength of the J1 (β) sideband is ∼ β2 , thereby allowing for thestraight forward noise analysis as used for AM modulation. In this limit eqn. 4.6 and4.5 hold.

A.7.2

Measurement of the phase noise spectrum

Different devices may be used that allow for separation of phase and amplitude
noise from a noisy signal. Both those presented in this section allow for a measurementof the single sideband spectrum of the noise on a carrier frequency. As in every device,
the amplitude noise rejection is subject to study. Also, such devices may be prone to
AM–to–PM conversion if the signal being measured exhibits strong AM noise.

A.7.2.1

The mixer

A double balanced mixer, SRA-1, may used for measuring the phase noise spectrum of a quiet signal. As a phase detector, the mixer allows for a phase comparisonof two signals, with close carrier frequencies. The DC output of the mixer is a voltage

116
proportional to sinusoidal phase difference, δφ, between the two signals. Because ofthis, the mixer output yields a linear relationship between phase and voltage only forsmall phase deviations (∆φ << 1). In this limit the generated phase noise spectrum
can be assumed to be dominated by the first order sidebands (see Sect. A.7.1).
The noise spectrum produced from a mixer is roughly calibrated by determining
the peak-to-peak voltage due to a phase excursion, ∆φ, equal to π radians. An importantconsideration in using a mixer as a phase detector is that the phase offset between thetwo signals may make it such that, the mixer output is offset outside its linear range.
This may result in clipping of the noise signal, thus giving a severe underestimation of
the rms phase noise. To avoid this, a phase shifter2 may be used and the mixer output
should be monitored on a scope since even slight changes in temperature of the BNC
cables can result in phase offsets. Also because the mixer essentially multiplies two sine
waves, the DC output of the mixer is sensitive to amplitude modulation on both signals.

A.7.2.2a)

The Frequency-to-Voltage converter

S(t)

b)

∆t

Ri

R

∆V

S(t)

_

Rf
V(t)

C

+

T(t)
T(t)= T0 + δT = 1/(f0 +δ f)

V(t) = V0 + δV
∝ f0 + δ f(t)Figure A.14: a) Chirped square wave signal S(t). b) Schematic of the basic operationof a frequency-to-voltage converter. b) Here the input signal exhibits a charge spikes
separated in time by T (t). This signal is low pass filtered (integrated) and then amplified
using an operational amplifier. The output of the operational amplifier is a voltage
R
V0 = R Rfi C TV0∆t ∝ f0 . Frequency deviations, δf , away from f0 appear as a change inRvoltage at the output of the amplifier, δV = R(C V ∆t) Rfi δf .

For phase noisy signals a frequency-to-voltage converter (F-to-V) may be used2

The simplest phase shifter is made by simply adding a length of cable to either the reference signal.
This is much like adjusting the length of an interferometer. For standard 50 Ω BNC, 1 ft of cable = 1.5
ns time shift.

117
for an accurate determination of a noise spectrum. The F-to-V is an active electronic
device that yields a voltage that is in proportion to the frequency of an electronic signal.
The basic operation of an F-to-V converter is shown in Fig. A.14. The input
signal, with carrier frequency, f0 , and frequency noise, δf , is first converted to a square1
wave signal using a comparator with a period T (t) = (f0 +δf) . The amplitude of thesquare wave, ∆V , is a fixed and regulated thus removing amplitude noise from thecarrier. On one half cycle, the square wave charges a capacitor, Q = I∆t = CV ,and on the other half cycle, discharges it through a resistor, R. On the positive halfcycle the charge on the capacitor results in a current, V = IR input to an operational
amplifier that is directly proportional to positive duration of the square pulse, V = V 0 +δV = CVT R = R (f0 + δf ) CV . Noise on the carrier frequency results in deviations in
output voltage, δV , from V0 . The frequency spectrum of such a signal results in carriersidebands that are directly proportional to the strength of the frequency modulation,
where δVV(ν)= δff0 . In essence, the F-to-V converter converts phase noise to amplitude0
noise thus removing the Bessel function sidebands intrinsic to phase modulation [seeSect. A.7.1]. The resulting noise spectrum produced by the F-to-V is straight forward
to analyze using eqn. 4.6.