Adaptive control of lasers and their interactions with matter using femtosecond pulse shaping

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Adaptive control of lasers and their interactions with matter using femtosecond pulse shaping
Efimov, Anatoly, 1971- ( Dissertant )
Reitze, David H. ( Thesis advisor )
Place of Publication:
Gainesville, Fla.
University of Florida
Publication Date:
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Subjects / Keywords:
Amplifiers ( jstor )
Electric pulses ( jstor )
Lasers ( jstor )
Optics ( jstor )
Phonons ( jstor )
Pixels ( jstor )
Pulses ( jstor )
Pumps ( jstor )
Shapers ( jstor )
Signals ( jstor )
Adaptive control systems ( lcsh )
Dissertations, Academic -- Physics -- UF
Physics thesis, Ph. D
government publication (state, provincial, terriorial, dependent) ( marcgt )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )


Coherent control of chemical reactions, atomic and molecular systems, lattice dynamics, and electronic motion rely on femtosecond laser sources capable of producing programmable arbitrarily shaped waveforms. To enter the time scale of natural dynamic processes in many systems, femtosecond pulse shaping techniques must be extended to the ultrashort pulse domain (<50 fs). Concurrently, reliable high-fidelity amplification of shaped waveforms is required in many applications. We demonstrate ultrabroad bandwidth pulse shaping of 13 fs pulses with Fourier-domain phase-only filtering using a liquid crystal array. We further demonstrate the amplification of shaped pulses in a multipass chirped pulse amplifier (CPA) system to produce millijoule-level optical waveforms with 30 fs resolution. Recently, a new approach to coherent control of physical systems was introduced, which, instead of relying on formidable theoretical calculations of complex system dynamics, makes use of an appropriate experimental feedback from the system itself to control its evolution. We apply this adaptive feedback approach for enhancement of ionization rates in a femtosecond plasma with the goal of minimization of phase distortions in the amplifier system. With the help of a learning algorithm and survival principles of nature, we teach our laser to control its own phase by using spectral blueshifting in a rapidly created plasma as a feedback to the algorithm. Control of lattice vibrations has long been sought as a means of studying phonon-related processes in solids. In addition, generation and control of large-amplitude optical phonon modes may open a path to femtosecond time-resolved studies of structural phase transitions and production of ultrashort shaped X-ray pulses. We perform pump-probe phase-resolved measurements and control of optical A1g mode in sapphire through shaped-pulse impulsive stimulated Raman scattering (ISRS). We chose this material as a candidate for possible nonlinear oscillations regime for its wide band gap and superior optical properties allowing for high-energy excitation. To enter a nonlinear regime, however, complex asymmetric multiple-pulse excitation is required. Therefore, we make a detailed proposal of the experimental adaptive feedback implementation for optimization of phonon amplitude based on the coherent probe scattering and a novel phase mask calculation algorithm for the real-time asymmetric pulse train generation. ( ,, )
femtosecond pulse shaping, amplified pulse shaping, coherent Control, adaptive feedback, adaptive control, coherent phonon, blueshift
Thesis (Ph. D.)--University of Florida, 2000.
Includes bibliographical references (p. 165-180).
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by Anatoly Efimov.

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ABSTRACT . . . . . . . . .. .. ... . ... . iv


1 INTRODUCTION .............. . . . ... 1

2 ULTRASHORT PULSE SHAPING .. . ... .. ... ... .. .... 7
2.1 Introduction .................................... 7
2.2 Ultrashort Pulse Generation and Characterization ....... .... ... 7
2.3 Titanium Sapphire Laser ................... ........ 13
2.4 Femtosecond Pulse Shaping ................... ...... 19
2.5 Ultrabroad-Bandwidth Pulse Shaping: Experimental Results . . ... 28
2.6 Phase Compensation with Deformable Mirror . . . . . ..... 38
2.7 Space-Time Coupling in Fourier-Domain Phase Compensation . . ... 43

3.1 Introduction ........ .. ... ... .. ...... ... ... .... . 52
3.2 Millijoule Amplifier System .................. ........ .. 52
3.3 Amplified Pulse Shaping .................. ........ .. .. 62

4.1 Adaptive Feedback Control: Overview ............... . . .. 80
4.2 Blueshifting as a Diagnostic of Pulse Phase ..... . . . . . 86
4.3 Experimental Details .................. ............ .. 91
4.4 Optimization Results .................. ............ .. 95
4.5 Discussion .................. .................. .. 104

5.1 Overview ........... .. ........... ....... 108
5.2 Coherent Phonon Excitation and Detection ..... . . . . . 111
5.3 Pump-Probe Experimental Considerations .................. .. 120
5.4 Arbitrary Pulse Sequence Generation ....... . . . . . .. 125
5.5 Coherent Phonon Generation and Control: Experimental Results . . . 134
5.6 Adaptive Feedback Control of Coherent Phonon Amplitude . . . ... 141

6 CONCLUSION . ......... .............. . 145


A SLM CALIBRATION . . . ............ .. 150



REFERENCES . . .. .................. .... . 165

BIOGRAPHICAL SKETCH ........... .. ....... . 181

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Anatoly Efimov

December 2000

Chairman: David Reitze
M i, .r Department: Physics

Coherent control of chemical reactions, atomic and molecular systems, lattice dy-

namics, and electronic motion rely on femtosecond laser sources capable of producing pro-

grammable arbitrarily shaped waveforms. To enter the time scale of natural dynamic pro-

cesses in many systems, femtosecond pulse shaping techniques must be extended to the ul-

trashort pulse domain (< 50 fs). Concurrently, reliable high-fidelity amplification of shaped

waveforms is required in many applications. We demonstrate ultrabroad bandwidth pulse

shaping of 13 fs pulses with Fourier-domain phase-only filtering using a liquid crystal array.

We further demonstrate the amplification of shaped pulses in a multipass chirped pulse

amplifier (CPA) --, -I. 1 to produce millijoule-level optical waveforms with 30 fs resolution.

Recently, a new approach to coherent control of pl,--, i, 1 systems was introduced,

which, instead of relying on formidable theoretical calculations of complex system dynamics,

makes use of an appropriate experimental feedback from the system itself to control its

evolution. We apply this adaptive feedback approach for enhancement of ionization rates

in a femtosecond plasma with the goal of minimization of phase distortions in the amplifier

-, i n With the help of a learning algorithm and survival principles of nature, we teach

our laser to control its own phase by using spectral blueshifting in a rapidly created plasma

as a feedback to the algorithm.

Control of lattice vibrations has long been sought as a means of studying phonon-

related processes in solids. In addition, generation and control of large-amplitude optical

phonon modes may open a path to femtosecond time-resolved studies of structural phase

transitions and production of ultrashort shaped X-ray pulses. We perform pump-probe

phase-resolved measurements and control of optical Alg mode in sapphire through shaped-

pulse impulsive stimulated Raman scattering (ISRS). We chose this material as a candidate

for possible nonlinear oscillations regime for its wide band gap and superior optical prop-

erties allowing for high-energy excitation. To enter a nonlinear regime, however, complex

asymmetric multiple-pulse excitation is required. Therefore, we make a detailed proposal of

the experimental adaptive feedback implementation for optimization of phonon amplitude

based on the coherent probe scattering and a novel phase mask calculation algorithm for

the real-time asymmetric pulse train generation.


The invention of the laser has created a revolution in p1!-, -ii 1 sciences comparable

to that of the invention of the transistor and its role in electronics. The first demonstra-

tion of successful action of stimulated emission from an optical resonator occurred in a

crystal of ruby, Cr3+ ions doped into sapphire, Al203 [1]. Rapid advances in generating

shorter and more intense pulses followed immediately through Q-switching to modelocking

of solid-state lasers [2, 3]. The latter technique remains the central method for generating

femtosecond pulses directly from the laser cavity. The first wide-bandwidth solid-state laser

at room temperature was realized in a new host material, alexandrite [4]. But until the last

decade of the 20th century, dye and color center lasers ruled the subpicosecond-time-scale

domain. Realization of the tremendous potential of titanium-doped sapphire both in large

gain bandwidth and energy storage capacity [5] quickly changed the situation in favor of

this convenient crystal material. Further reduction of the pulse width was made possible

by the introduction of dispersion compensating optics (such as a pair of prisms) inside the

optical cavity culminating in producing self-starting 6.5 fs pulses from a chirped-mirror and

prism pair phase controlled Ti:Sapphire laser [6].

In the last years, femtosecond lasers have become more user friendly, compact, and

commercially available from a number of manufacturers. This caused rapid growth and

expanded the spectrum of scientific and technological applications of the ultrafast lasers.

The primary use of ultrashort pulses is made in the field of time-resolved spectroscopy, since

many fundamental processes in atomic, molecular, and chemical systems occur on subpi-

cosecond time scales. Progress in this field was marked by awarding the 1999 Nobel Prize

to Ahmed Zewail for his studies of the transition states of chemical reactions using fem-

tosecond spectroscopy. Apart from spectroscopy, applications of femtosecond light sources

range from multiphoton imaging [7] to micromachining [8], from communications [9] to iso-

tope separation [10], and from high-speed microelectronic circuit testing [11] to prehistoric

fossil cleaning [12]. The art of ultrashort pulse generation made it possible to approach the

attosecond barrier [13]. When this barrier is broken, a new era of subatomic time-resolved

spectroscopy will emerge.

The pulse energy directly from a femtosecond laser is small, on the order of a few

nanojoules although the peak powers in the megawatt region can be obtained. To increase

output pulse energies, optical amplification methods are used routinely. However, direct

amplification of femtosecond pulses proved to be rather difficult because of undesirable

nonlinear effects in the gain medium of the amplifier. The dynamic increase of the peak

intensity of the pulse would inadvertently lead to deterioration and even damage to the

amplifier optics. The CPA technique [14] opened the way to multiterawatt amplifiers devel-

oped today in a number of leading laboratories around the world [15, 16, 17, 18, 19]. These

systems are almost as powerful as the Nova laser at the Lawrence Livermore National Lab-

oratory [20], yet occupy no more than a few standard optical tables and provide repetition

rates of at least ten hertz, or ten shots per second. The high peak intensity comes from

the very short temporal width of the pulse rather than the energy of the pulse. The use of

these modern amplifier systems is beneficial in many aspects, unless high pulse energy is a


A wide array of new 1!-, -i -, is waiting to be uncovered with the help of the tools

that are being created using the CPA technique. High-intensity laser-matter interaction

1r-, -i. is therefore being made in the reach of small laboratories around the world since

the capital cost of a tabletop terawatt --,- i ll, can be a few hundred thousand dollars.

Progress is now being made in large-amplitude coherent phonon generation [21, 22], in

laser-plasma interaction and plasma waveguides [23, 24, 25, 26], in ultrashort pulse X-ray

generation [27] and X-ray lasers [28], and in high-order harmonic generation [29, 30, 31].

Ideas were already put forward and undergo experimental verification for attosecond pulse

generation [32, 33, 34] and for table-top particle accelerators [35, 36, 37].

To make efficient use of femtosecond sources, pulse generation and amplification must

be supplemented with their precise characterization. This is, in fact, a nontrivial matter

because typical inverse pulse duration is far beyond the bandwidth of any electronic detec-

tion --,- in Originally, pulse characterization methods included spectrum and intensity

autocorrelation measurements [38, 39]. However, these methods provided only approximate

information on pulse duration and fidelity1. Nevertheless, autocorrelation is still the method

of choice for high-dynamic range measurements [40, 41, 42]. In recent decade, a plethora

of both amplitude- and phase-sensitive methods were developed [43, 44], which became as

common tools in the laboratories as the traditional autocorrelator.

Recent technological progress in developing reliable femtosecond pulse sources allowed

to shift the focus of experimental work from the mere observation of ultrafast dynamics in

complex p,!-,,-i, 1 systems to the control of these dynamics. Control of the outcome of

chemical reactions and selective excitation and dissociation of target chemical bonds in

complex molecules have 1.- '-,- been the Holy Grail of p1!-, -i, 1 chemistry, and achieving

this goal still remains elusive. Nevertheless, recent experiments [45] using femtosecond lasers

bundled with clever computer algorithms brought this goal closer than ever to our reach.

Just as tiny tips of the scanning atomic microscopes in .--1 i-, allow control of positions of

single atoms in space, ultrashort optical pulses prove to be indispensable tools for controlling

temporal dynamics of the microscopic world with resolution comparable to the time scale

of most fundamental processes.

In experiments on coherent control of atomic molecular and chemical systems, fem-

tosecond pulses of complex amplitude and phase structure are required [46]. The pulse

shaping techniques developed mostly over the last decade provides such capability. The

most well-developed method today is based on filtering the spectral amplitude and/or phase

of the femtosecond pulse [47, 48], termed Fourier-domain pulse shaping. Here, the desired

temporal profile of the electric field is connected to the spectral filter and pulse spectrum

by the Fourier transform, Chapter 2. Programmable pulse shaping is often needed, which

necessitates the use of computer controlled reconfigurable spectral masks. Such capability

is demonstrated in this work with near 10 fs resolution [49]. In the near future, we might

even expect to see a turn-key computer-controlled femtosecond optical waveform synthesizer

similar in functionality to RF synthesizers, but operating in the optical domain.
'See Section 2.2 in Chapter 2 for details on pulse measurement techniques.


Many control experiments which require pulse shaping can not be performed with

low pulse energies available directly from the femtosecond laser, Chapter 5. Therefore,

amplification of shaped pulses becomes important, Chapter 3. We show that the layout of a

typical CPA systems can easily host a pulse shaping apparatus without added complexity.

High-fidelity amplified pulse shaping with 30 fs resolution was demonstrated using this

technique [50].

Programmable femtosecond pulse shaping made possible the experimental develop-

ment of a revolutionary adaptive feedback approach, initially -I:_:_- -I .1 in 1992 [51], to

many common coherent control problems, Chapter 4. In particular, instead of trying

to synthesize the pulse shape based on a theoretical .1- 11-, ,-i of the -;, -I- i i for optimal

control, the --,-, i_, itself does the work by providing the appropriate feedback to the

- 11 lit" algorithm running on a computer, iteratively converging to the best solution in

the presence of real laboratory experimental conditions. A number of pilot experiments

using this approach have already been performed demonstrating the great potential of the

idea [52, 53, 54, 55, 56, 57, 45, 58, 59, 60, 61]. The most suitable set of problems to be

addressed with the adaptive feedback approach includes those which are not analytically

tractable or in which the solution can be very sensitive to the experimental uncertainties and

noise. One such problem is the enhancement of ionization rates in a femtosecond plasma.

The relation between the ionization rate, the transmitted pulse spectral shape and the exci-

tation pulse phase was previously : :_-. -1. .1 and numerically modeled [26]. In Chapter 4, we

address this problem experimentally using adaptive feedback approach. Correlation of the

spectral blueshifting to the pulse phase is established and provides the real-time feedback

for the learning loop.

A number of coherent control experiments require specific intensity rather than com-

plete electric field profiles, Chapter 5, leaving the temporal phase unspecified. Spectral

phase-only filtering, although incapable of delivering the exact intensity target, is proven

to yield adequate approximation in time domain, Chapter 2. Computation of the required

mask, however, relies on the time-consuming iterative procedure based on the global search

algorithm. In many cases, this slow-time operation precludes appropriate phase parame-

terization in adaptive feedback experiments, which would lead to the dramatic parameter

space reduction otherwise. In Section 5.4 Chapter 5, we address this problem by introducing

a real-time algorithm for computing the phase mask which yields the best approximation

of the temporal intensity target at the output of the pulse shaping apparatus. The algo-

rithm is still iterative, although the use of natural constrains of the problem accelerates

the convergence dramatically. We use phase masks generated with this algorithm in the

experiments on control of coherent phonons in solid dielectrics, Chapter 5. In addition, we

expect this approach to be widely used in adaptive feedback control experiments, which are

intrinsically real-time in nature.

Armed with the real-time programmable amplified pulse shaping tools we can now

approach a number of coherent control applications. In particular, we concern ourselves with

the problem of generating and controlling the coherent lattice oscillations in solid dielectrics

using shaped pulses. Multiple-pulse excitation can allow high-energy pumping and coherent

phonon amplitude increase without damage or other unwanted electronic nonlinear effects.

In addition, specially tailored intensity sequences are expected to be useful in controlling

the duration of the oscillations. Both of these effects are demonstrated in Chapter 5.

The present work, therefore, summarizes our contribution to both the field of Fourier-

domain pulse shaping and to adaptive feedback control as applied to optimization of the

pulse phase at the output of a multigigawatt amplifier system. First, we extend the ca-

pability of the pulse shaping technique toward 13 fs pulse durations (Chapter 2) using

an ultrabroadband pulse shaper with a computer-programmable liquid crystal spatial light

modulator (LC SLM) array. Second, we demonstrate complex phase-shaped waveforms

with millijoule total energies produced by amplification of shaped pulses in a custom-built

multipass CPA --, -1. i, (Chapter 3). We use the single-shot Frequency Resolved Optical

Gating (FROG) technique for complete amplitude and phase characterization of the re-

sultant waveforms and highlight difficulties associated with using this diagnostic method

with complex phase-shaped amplified pulses. Third, pioneering adaptive feedback exper-

iments based on blueshifting of laser spectrum during rapid femtosecond plasma creation

are reported in Chapter 4. There, we teach our laser-amplifier --, -I. ii, how to learn its best

phase through feedback and help from a personal computer. Finally, in Chapter 5, our

shaped-pulse amplifier --, -. 11' is put to use in the experiments on resonant optical coherent

phonon generation and control in solid dielectrics using multiple pulse excitation. In the

last chapter, we also make a detailed proposal for future experiments on adaptive control

of vibrational modes in crystals generated through multiple-pulse ISRS.


2.1 Introduction

An understanding of the methods for generating, (d i ,. I. i. i:_. and temporally tai-

loring femtosecond pulses is an essential prerequisite for understanding how these pulses

can interact with p1!,, -i, 1 systems. In this chapter and in Chapter 3, we lay out in detail

the experimental tools used in our research. We delve into the intricacies of ultrashort

pulse generation measurement and shaping. In Section 2.2, we give an overview of the im-

portant theoretical and experimental considerations that impact femtosecond pulse optics.

The description of our home-built laser oscillator and its performance parameters are given

in Section 2.3. We devote Section 2.4 to ultrabroad bandwidth femtosecond pulse shaping

and present our experimental results in Section 2.5. One of the specific applications of

pulse shaping technique is compensation and correction of spectral phase distortions, which

occur primarily due to frequency dispersion in complex optical systems. In Section 2.6, we

develop theoretically the specialized pulse shaping techniques for spectral phase compensa-

tion based on spectral adaptive optics. Finally, Section 2.7 presents theoretical treatment

of spatio-temporal coupling effects .'...i 1' l in:_< Fourier-domain phase compensation.

2.2 Ultrashort Pulse Generation and Characterization

The generation of ultrashort pulses on the time scale of a few femtoseconds requires

special consideration with regard to the pulse spectral bandwidth and spectral phase dis-

tortions caused by material and geometric dispersion. Requirements on the large spectral

width stem from the time-bandwidth product relationship AVAr = const, fundamental to

the Fourier transform between the time and frequency domains. The value of the const is

dependent on the actual shape of the pulse and is calculated elsewhere [62]. Figure 2.1

illustrates typical bandwidth requirements for the generation of extremely short pulses. As


- 45

20 30 40 50 60 70 80 90 100 110 120
Bandwidth (nm)

Figure 2.1: Relation between the pulse width and the bandwidth for the transform-limited
pulse of gaussian, exp ( 21n 2 ) and hyperbolic secant, sech2 (1.7627t) shapes, where
r is intensity full width at half-maximum (FWHM).

shown in the figure, bandwidths of over 100 nm (Aw/wo = 0.125 at Ao = 800 nm) for

Gaussian and over 70 nm (Ac/co = 0.09) for hyperbolic secant pulse shapes are required

to generate sub-10 fs pulses and grow rapidly as shorter pulses are desired. The figure

points to the inherent difficulty in generating femtosecond pulses, namely the need for large

gain bandwidth from the laser medium and broadband throughput of all its passive com-

ponents. It comes as no surprise, therefore, that very few specific materials are used in

femtosecond-pulse lasers. A specially designed set of optical components is 1.-- i,- required

as well.

It is well known that even if the width of the spectrum is large enough to support the

pulse of duration determined by the uncertainty relation for the given pulse shape (so called

i I I-in!, I_ -limited pui'I- ), in practice such pulses are seldom produced. The uncer' iil-,r

relation holds true only when the individual frequency constituents are synchronized with

each other, or in other words, when the spectral phase is no more than a linear function of

- - Hyperbolic Sech

---------- I -------- --------- ----------

i i Zi~ t !

I - ,- - - - -, , --- --


E (w) A (w) ei() (2.1)

S(w) = 0 + i1 (W Uo) transform limited

where E(w) is the electric field in spectral domain with frequency-dependent amplitude

A(w) and phase O(w). Frequency synchronization is achieved in a laser by phase locking

the longitudinal modes of the open resonator under the gain curve of the active medium. A

number of modelocking techniques were exploited for femtosecond pulse generation includ-

ing active, passive and self-modelocking [63, 64, 65, 66, 67, 68]. The latter technique is used

widely in Ti:sapphire lasers and is based on the nonlinear lensing effect of the high peak in-

tensity femtosecond pulse inside the crystal (hence the term Kerr-Lens modelocking). This

technique is used in our laser described in Section 2.3.

After a pulse leaves the laser cavity, its spectral phase can be deformed into a compli-

cated function of frequency due to material dispersion or artificially introduced geometrical

dispersion (grating or prism pairs, chirped mirrors, etc.). In optical fibers and integrated

optical devices the situation is complicated by waveguide and modal dispersion as well [69].

In general, spectral phase can be expressed in a polynomial form:

1 wo) (2.2)

where 0o is a central frequency chosen arbitrarily within the spectrum of the pulse. This

expansion is a good approximation for well-behaved differentiable functions which 0 (w)

is, at least theoretically. This form of writing the phase function has become customary

because the compensation methods differ somewhat for each term of the polynomial.

When considered in terms of geometrical optics, the group delay of an optical pulse

(the average time it takes for the pulse to travel a unit distance) is given by the second term1

in the expansion (2.2), 0' (wo). The higher order terms express the linear and nonlinear time

delay (or advance) that each frequency component suffers relative to the central frequency
1The first (constant) term p (wo) is related to the phase of the carrier wave under the amplitude envelope
of the pulse and is an important factor in interaction of sub-10 fs pulses with matter.


Wo and, hence, determine the amount of pulse broadening and distortion that occurs. Group

delay dispersion2 (GDD), the '" (wo) term in the expansion (2.2), is the most influential

cause for the temporal pulse broadening. Due to GDD, each frequency component that

comprises the spectrum of the pulse experiences the delay linearly proportional to the

offset from the central frequency wo, hence, the term In,. i chirp" (Figure 2.2). Since

a) b)

t t

Figure 2.2: Illustration of a Gaussian pulse with positive (a) and negative (b) linear chirp.

all optical glass materials exhibit GDD of the same sign near 800 nm (time delay is a

decreasing function of wavelength), no combination of different glasses can eliminate this

term3. Fortunately, as was first shown by Treacy [70], a pair of parallel diffraction gratings

can be used to compensate for the excess of GDD producing the time delay which is an

increasing function of wavelength. In recent years, a great number of higher-order phase

compensation methods have been developed [71, 72, 73, 74, 75, 76, 77, 78], which eventually

allowed the generation of ~ 4 fs pulses directly from the laser cavity.

Because of the extremely short temporal duration of the femtosecond pulse there is

generally no direct way of measuring its width, or its complete temporal intensity profile.

The fastest photo-multiplier tubes (PMT) on the market have characteristic rise times of no

better than 0.1 nanosecond. Photodiode detectors sacrifice sensitivity for speed to obtain an

order-of-magnitude better temporal resolution; however it is still limited to approximately

10 picoseconds. Finally, streak cameras achieve subpicosecond resolution, which is still

insufficient for ultrashort pulsewidth measurements below 500 fs.
2Also sometimes referred to as "quadratic phase dispersion" or simply "quadratic term."
3In fiber optics both regions of positive and negative GDD are available with zero dispersion point around
1.3 pm. Dispersion shifted fibers allow this point to be moved towards shorter or longer wavelengths.

The autocorrelation technique [38] offers virtually unlimited resolution, robustness,

and ease of use and is based on the idea of measuring the pulse against itself through obtain-

ing the mathematical autocorrelation function of the original waveform. The second-order

intensity autocorrelation offers simple real-time operation with pulse width estimate (exact

rms width regardless of pulse shape [79]), but provides little information on pulse shape and

phase4. Briefly, when two replicas of the input pulse at the fundamental frequency w are

crossed noncollinearly in space and time in a nonlinear medium under phase-matched geom-

etry for second harmonic generation (SHG), the intensity I2w of the resultant upconverted

pulse is

I2,(t) I(t)I(t T) (2.3)

where 7 is the delay between replica pulses. The second harmonic signal is, therefore,

related to the background-free second order autocorrelation function G(r):

G(r) fJ I(t)I(t 7) dt
G(7)(= (2.4)
f12 (t) dt

Because pulse intensities are involved in G(7), no direct phase information can be extracted

from the autocorrelation measurements. Presence of phase, however, still affects the tem-

poral intensity profile (e.g., chirping stretches the pulse I(t)) and hence the autocorrelation.

The duration of the pulse can be obtained from the autocorrelation measurements only if

a particular shape of the pulse is assumed. For example, FWHM of the autocorrelation

trace and pulse intensity are related as 7Tuise/Tautocorr = '.'1- 2 for hyperbolic sech and

Tpulse/Tautocorr = 0.7071 for Gaussian intensity pulse shapes [62].
If the two replica pulses are crossed in a nonlinear medium in collinear fashion,

the detected SHG signal I 2w, (E (t) + E (t ))2 2 will contain constant background,

I2w,bkg 12 (t) + 12(t ), independent of the delay 7 due to the SHG by individual pulses.
I ii .i. i~. autocorrelation (second order autocorrelation) should be clearly distinguished from field auto-
correlation (linear autocorrelation). The latter yields only information equivalent to the intensity spectrum
but is often used in applications involving short-coherence length continuous sources and Fourier transform
infrared (FTIR) spectroscopy.

This signal gives the interferometric second order autocorrelation:

) f (E(t) + E (t )) d(2.5)
2 f E2 (t) 12

S (A (t) eiwt+'i(t) + A (t- T) ei(t-T)+(t-T) 2 dt

and, hence, contains the phase information of the measured pulse. It was shown that in-

terferometric autocorrelation in (..i i imI i. .1i with the intensity spectrum measurement can

yield insight into the phase distortions of the pulse [80]. Recently, to overcome phase-

matching bandwidth limitations of the nonlinear SHG process, two-photon current gener-

ation in a regular photodiode was used as effective wide-bandwidth nonlinearity for auto-

correlation measurements [81].

Inherent to the second-order autocorrelation, because of its symmetry, is the ambi-

guity in the direction of time. The third-order autocorrelation eliminates this ambiguity

and just as the second-order autocorrelation can yield a large dynamic range of over six

orders of magnitude, which is beneficial, for example, in measuring the contrast of the out-

put pulse from a terawatt amplifier [82]. Reports were published recently on the full field

reconstruction from the spectral intensity and the background-free autocorrelation [83] or

cross-correlation [84] measurements.

A number of spectrally resolved pulse measurement techniques for complete intensity

and phase characterization have blossomed in recent years. Several variations of the FROG

technique5 are based on spectrally resolving the correlation functions and applying a sophis-

ticated iterative inversion algorithm to the experimental 2D trace for intensity and phase

extraction [43, 85, 86, 87, 88]. The experimentally recorded signal in the Polarization Gate

(PG) FROG, for example, is related to the spectrally resolved third-order autocorrelation:

r 2
I(w, ) Et) ( E (t )|2 exp (-it) dt (2.6)

where the gate function EE (t )|12 is derived from the field E(t) to be measured. Theoret-

ically, the FROG algorithm 11'- -, yields a unique solution for the intensity and phase (in
5Refer to appendix B for more details on single-shot SHG FROG

certain cases ambiguities exist, which are known and can be dealt with). However, under

experimental conditions, FROG can sometimes be unreliable for recovering complex phase

profiles such as those obtained with Fourier-domain pulse shaping (Section 3). Another

recently developed phase-sensitive technique is Spectral Phase Interferometry for Direct

Electric-Field Reconstruction (SPIDER), introduced by laconis and Walmsley [44, 89]. In

contrast to FROG, in using SPIDER one records only a one-dimensional trace and the recov-

ery algorithm is not iterative, which is very attractive for real-time pulse characterization.

Recently, however, real-time operation of the FROG technique was reported [90, 91].

2.3 Titanium Sapphire Laser

Since about 1990, the Titanium Sapphire laser [5] has been a workhorse in femtosec-

ond p.1i, -i. laboratories around the world. Transition metals in different host lattices have

been used since the beginning of the laser era, and it comes as somewhat of a surprise

that it took so many years to discover the best dopant-host combination for ultrashort

pulse generation. Thus far, no other material has been found to surpass the performance of

Titanium-doped Sapphire in both the gain bandwidth (3200 cm-1) and the peak emission

crossection (3 x 10-19 cm2).

The Ti:Sapphire is a four-level --, -. wi_. which gives it comparatively high efficiency.

Its wide fluorescence bandwidth, over 200 nm [92], can support pulse durations of a few

femtoseconds at the 800 nm central wavelength. In addition, its substantial energy storage

capacity is due to the long lifetime of the upper lasing level, estimated to be 3.2 pts. As

a result, Ti:sapphire lasers produce relatively high pulse energies as compared with other

ultrashort-pulse laser gain media. An appreciable Kerr nonlinearity makes it relatively

easy to self-modelock the laser. Much theoretical work has been devoted to the 1p-, -i. of

the Ti:Sapphire laser [66, 93, 94, 95, 96], but a complete understanding has not yet been

reached because of the frustratingly complicated interplay among the linear and nonlinear

phase effects, nonlinear temporal and spatial coupling, and saturated gain.

The design of our laser is shown schematically in Figure 2.3. It is an astigmatically

compensated double-folded cavity of X-configuration defined by four mirrors M1, M2, M3

and the output coupler M4. The Ti:sapphire crystal is located at the common focal plane of

Retro reflector

/ Output:
S0 = 805 nm
A > 60 nm
Phase Compensator 1 ~ -nJ

1 M4


M2 M3 Pump

Figure 2.3: The layout of the Ti:Sapphire Laser and the external phase compensator. M1,
M2, M3 and M4 are cavity mirrors, M4 being the output coupler. The slit near M1 is
used for wavelength tuning and spectrum control, while the iris near output coupler M4
contributes significantly to the cleaning of the output mode eliminating completely the
transverse structure at the laser output beam. This iris is not used for modelocking.

two 10-cm spherical mirrors M2 and M3 and is CW-pumped by focusing 4.7 W of multiline

output from the Argon-Ion laser (Coherent Innova 310). For intracavity phase compensa-

tion, we use two brewster-cut fused silica [65] prisms with apex separation experimentally

optimized for the shortest output pulse. The output from the laser is taken through the

.' wedged output coupler. The linear dispersion introduced by the output coupler as well

as subsequent mode cleaner-collimator is partially compensated with the external prism

compensator also shown in Figure 2.3.

Modelocking is initiated by instantaneously moving the second intracavity prism out

of the beam. When the laser enters the modelocked regime, the average power increases

by about 50% and reaches 250-400 mW depending on alignment. At a repetition rate of

91 MHz, the output pulse energy is typically about 3 nJ. The average output power can

be increased easily to beyond 500 mW at the expense of the pulsewidth. The long-time

stability of the laser is assisted largely by cooling the Ti:Sapphire crystal with 15 0C chilled

water passing through the water jackets in the crystal's copper housing. Stable modelocked


operation of the laser was demonstrated experimentally for over 50 hours while testing the

long-time stability of the pump-probe zero delay position (see Section 5.3).

Depending on a particular application, the laser was operated in two different regimes

with different output pulse characteristics. For the ultrafast pulse shaping experiments,

reported in this chapter, Section 2.5, we minimized the FWHM of the second-order auto-

correlation function, Figure 2.4, so as to optimize the duration of the pulse. Resultant pulse

width was estimated to be 13.8 fs assuming sech2 pulse shape. On the other hand, for

600 700 800 900 1000
SWavelength (nm)


-80 -40 0 40 80
Time (fs)

Figure 2.4: Second order intensity autocorrelation of a 13.8 fs pulse from Ti:Sapphire laser,
assuming sech2 pulse shape. Spectrum of the pulse is shown in the inset.

the amplified pulse shaping and pump-probe experiments, C'!i II. 3 and 5, the pulse was

optimized for the large spectral bandwidth of about 60 nm to partially compensate for the

spectral gain narrowing in the amplifier. In this case, because of the large asymmetry of

the intensity spectrum and some phase distortions, the pulse width was measured 17-19 fs

compared to 17 fs transform-limited value based on the spectrum alone. The amplifier seed

pulse experimental and recovered FROG traces are shown in Figure 2.5, recovered spec-

trum and phase-in Figure 2.6, and the temporal intensity profile-in Figure 2.7. The

FROG trace, Figure 2.5, exhibits classical higher-order phase-limited shape [97]. Indeed,



t 420


-200 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 200
Delay (fs) Delay (fs)

Figure 2.5: Experimental (a) and recovered (b) FROG traces of 19 fs pulse directly from
laser. Intensity color coding is done on a logarithmic scale over three orders of magnitude
blue to red.





750 800 850 900
Wavelength (nm)

Figure 2.6: Spectral intensity (solid curve) and phase (circles) recovered from FROG trace
shown in Figure 2.5a. Dashed curve is an independently measured fundamental spectrum.
FROG error was 0.004 on 256 x 256 grid.

'" 0.3

-200 -100 0 100 200
Time (fs)

Figure 2.7: Temporal intensity profile of the pulse recovered from FROG trace shown in
Figure 2.5a. Small oscillations are characteristic for cubic phase distortion.

the recovered spectral phase, Figure 2.6, resembles a third-order polynomial curve near the

center of the spectrum, and the temporal intensity, Figure 2.7, exhibits characteristic cubic

phase-induced oscillations on one side.

We should note that spectra in excess of 90 nm can be obtained from our laser with

the higher-reflectance output coupler, as shown in Figure 2.8. Operation of the laser in

this regime, however, was not found to be sufficiently stable.

It is interesting to note that our laser could, in fact, operate bistably [96], i.e. when

the pulse train at the laser output exhibits period doubling in intensity, as predicted by

Kalashnikov, and in spectral characteristics as well. In this regime, the individual pulses

separated by twice the repetition period are identical to each other, but each pulse in

the train is distinctly different from its nearest neighbor. This situation is fundamentally

different from what is called I. .Il.. p 1l-.i:_.." where two (or more) pulses are closely spaced

(separated by a few femtoseconds) and are easily observed on the autocorrelation or FROG

trace. In the former case, the FROG trace (which is the average of many traces generated by

individual pulses in the megahertz pulse train) may look valid, but the recovery algorithm

will fail to converge. For example, Figure 2.9 contrasts FROG traces from the bistable (a)

and stable (b) laser, however, the FROG error6 in the first case was 0.009 on 256 x




c 0.4


600 650 700 750 800 850 900 950 1000
Wavelength (nm)

Figure 2.8: Intensity spectrum obtained from the laser with the 90 % output coupler.
Spectral FWHM exceeds 90 nm.



-200 -150 -100 -50 0 50 100 150
Delay (fs)

-150 -100 -50 0 50 100 150 200
Delay (fs)

Figure 2.9: FROG trace directly from laser operating in bistable regime, (a), compared
to regular single-pulse regime, (b). Although signal-to-noise ratios (S/N) are similar in
both cases, trace (a) does not yield convergence of the FROG recovery algorithm since it
effectively consists of two different overlapping traces for two different pulses. FROG error
of trace (b) is excellent.


256 grid compared to 0.0019 on 128 x 128 which is dramatically different for the traces

of equivalent signal-to-noise ratio. More complicated quasi-periodic and chaotic behavior

was also observed, but was not characterized in great detail. Finally, regular double- and

multiple-pulsing was observed sometimes, and examples are shown in Figures 2.10 and 2.11.

5.0 a)





-400 -300 -200 -100 0 100 200 300 -300 -200 -100 0 100 200 300 400
Delay (fs) Delay (fs)

Figure 2.10: Experimental (a) and recovered (b) FROG traces of the double pulse directly
from laser.

2.4 Femtosecond Pulse Shaping

Shaping of optical femtosecond pulses [48] promises great advantages to the fields of

p1!-, -i, 1 chemistry, communications, ultrafast spectroscopy, and high-energy field ]!-, -1.

It has led already to many applications such as observation of the fundamental dark soliton

in optical fiber [98], mode-selective excitation of coherent phonons [99, 100], engineering of

Rydberg wave packets [101], and measurement of their amplitude and phase [102], as well

as information storage and retrieval through quantum phase of the Rydberg atom [103].

A particularly powerful application of the femtosecond pulse shaping lies in the field

of coherent control of atomic, molecular, and chemical systems [104, 105, 106, 107, 108],

in which the goal is to drive the --, -.I i1 into a user-specified final quantum state by use of

tailored optical fields. Similar issues can be raised in the field of optoelectronics [109] and
6Frog error is typically used to illustrate the degree of convergence of the recovery algorithm and to
indicate whether the recovered data can, in fact, be trusted. For more formal definition, see appendix B.

Wavelength (nm)

inII I I I[ \/ I 71

-300 -200 -100 0 100 200 300
Time (fs)

Figure 2.11: Recovered intensity and phase of the double pulse in spectral (top) and tem-
poral (bottom) domains.

laser-plasma interaction: It has been proposed that a complex asymmetric chirped-pulse

train can resonantly excite large-amplitude plasma wake-fields [110, 111] for the new class

of the laser-based charged particle accelerators. Finally, in recent years, a new adaptive

feedback approach to coherent control experiments has developed (see Chapter 4), where

a 1!-, 4-i 1 system under study performs self-optimization with use of a learning computer

algorithm. Here again the pulse shaping technique, particularly in its programmable incar-

nation, p1 ,-, a central role.

Currently, there exist two major approaches to femtosecond pulse shaping: Fourier-

domain and direct in-time. In the first technique, shaping is performed by experimentally

Fourier-transforming the pulse from the time to the frequency domain and operating on

the spectral intensity and/or spectral phase of the pulse. Because of the extremely short

duration of the femtosecond pulse, this has been the most convenient way for shaping and

the technique is well-developed by now.


The experimental apparatus typically consists of two grating-lens (or mirrors) pairs

and an amplitude and/or phase mask in the Fourier plane, Figure 2.12. The individual

Input Pulse

Output Pulse


I1< f

f >1

Figure 2.12: Fourier-domain pulse shaper. Spherical mirrors are used to minimize disper-
sion, however, incident angles should be kept small to reduce astigmatism.

frequency components within the input pulse are angularly dispersed by the input side

diffraction grating. The following focusing element (mirror or lens) converts the angularly

diverging beam into a collimated one, however, each individual frequency component is

focused to a diffraction limited spot on the back focal plane, where the frequency components

are spatially separated along one dimension. Essentially, the first grating-mirror pair acts

as a simple spectrometer and, similarly, a second pair serves as an ir.. I spectrometer

to recombine the frequency components into a single beam at the output.

The electric field in the back focal plane of the input lens or mirror' in space-frequency

domain can be expressed as


Ebeforemask(X, w) ) E(w)e-(x-. i -/w
7This is the masking or Fourier plane


where E(w) is the input pulse -" i1 9',_ and the exponent factor describes the mapping

of the frequency w onto transverse spatial coordinate x. The horizontal beam radius wo in

the masking plane is related to the input beam radius as

cosOi f\
wo = W
COS Od 7TWin


where the first factor describes the horizontal beam size change due to the difference in the

input, Oi, and diffraction, Od angles, f is the focal length, and win 1 mm typically.

The g[w] function which describes the space-frequency mapping can be derived from

the first-order grating equation,

sin Oi + sin(Q + Od,o)


where d is the grating grove period and we refer to Figure 2.13 for definitions of the angular

parameters. The position of the focal spot center in the masking plane for the frequency

> m +X

Figure 2.13: Beam geometry at the pulse shaper input, where Oi is the incidence angle, Od
and 0d,o are the diffraction angles for the w and wo frequency components, and Q = Od Odo

component w, from geometry, is given by

x =g[w] f tan [w] (2.10)
8For ultrabroad-bandwidth pulses E(cL) should take into account the profiles of the grating and lens (or
mirror) efficiency curves.

We must note that the mapping of the frequency onto space x, described by the relationship

(2.10), is not linear, however the mapping of the wavelength onto x is mostly linear. This

can be seen by calculating the first coefficients in the expansion

A(x) = Ao + Ax) Ax + A(2) Ax2 + XAAx3 + (2.11)
2! 3!

where A) = (d"1/dx")A Ao, Ao = 2rc/wo is the center wavelength, and Ax is the offset

from the mask center, such that A(Ax = 0) = Ao. For the first three terms we obtain

(dA) dcosOd,o (2.12)
dx) \o f
(d2 A dsin Od, (2.13)
d2A0 f2
(d3A 3d cos Od,o
dx3 f3

It can be seen, that the relative contributions of the second and third terms in (2.11) scale


A(2)x (2.15)

A(3)A2 Ax2
S(3 (1 ) (2.16)

for reasonable diffraction angles 0d,o 0, and are small if Ax/f < 1. In our case, Axz/ f

0.1 and the deviation from linearity is negligible over a bandwidth of 120 nm, which is the

baseband spectrum of our pulses. If the assumption of linear A <-4 x mapping can be safely

made even for ultrabroad-bandwidth pulses, an often made assumption of linear w - x

mapping is clearly erroneous. In Figure 2.14, we plot the space-frequency relationship

for our pulse shaper. For our experimental conditions, Aw/wo = 0.16 and the derivative

dw/dx varies over Il' over the baseband spectrum of our pulses, even though variation of

dA/dx is only 1.' For pixelated masks, this dramatic nonlinearity also leads to unequal

bandwidth seen by each pixel.





400 -------- --------
200 -------------- ---------
------- ------ -- 110
a 0- -




-20 -10 0 10 20
Mask Plane position (mm)

Figure 2.14: Spatial frequency dispersion in the masking plane of the pulse shaper. Spectral
widths for 21, 10, and 5 fs pulses are shown for reference.

Without the mask, the device acts as a !,-dl'-il i--i,, delay line, meaning that

ideally the output pulse is identical to the input pulse. This, of course, is only an approxi-

mation since deterioration of the output pulse '1.-- occurs, however small, due to limited

spectral bandwidth of the gratings, dispersion in the lenses (if used instead of mirrors) and

mirror coatings, astigmatism, and slight misalignment.

Immediately after the mask, the electric field E(x, w) is obtained by multiplying (2.7)

by the spatial mask M(x):

Eaftermask(X, ) ~ E(w)e-(x-g[]) w M(x) (2.17)

In frequency domain, the action of a pulse shaper can be described as linear filtering


E,,t(u) = Ei,(wu)M(uw)

of the input pulse spectrum Eij(w) with the spectral filter M(w), which corresponds to the

spatial mask M(x):

M(w) 2) I M(x)e-2(x-g[w])2/w dx (2.19)

Equation (2.19) shows that the effective filter in the frequency domain is the spatial mask

function M(x) convolved with the intensity profile of the beam in the Fourier plane. The

main effect of this convolution is to limit the spectral resolution of the pulse shaper to

6w v n2wo/ (dx/dwo) FWHM.

Based on the type of mask used in the Fourier plane of the pulse shaper, we can

distinguish phase-only, amplitude-only, or combined phase-and-amplitude shaping meth-

ods. In the first case, the mask modulates only the spectral phase of the pulse. This

has the advantage of high throughput, and it is this technique of programmable phase-

only pulse shaping that is used in present work. Clearly, completely arbitrary waveforms

can not be generated with the phase-only method (or with the amplitude-only method).

Nevertheless, for a number of experiments, it is the temporal intensity profile, which is

of central importance [99, 111], and we show below and in Chapter 5 that, in fact, quite

complex outputs can be produced with the phase-only method. In the last case, a combined

phase-amplitude technique can bring true arbitrary waveforms limited by resolution of the

device, but involves more complex experimental setups. Amplitude filtering is not desirable

if amplification of shaped pulses is considered (Chapter 3).

Many types of masks are used in the Fourier plane of the shaper including fixed (non-

programmable) microlithographically patterned amplitude and reactive ion etched phase

masks [112], programmable LC SLMs [113, 114, 115, 49], optically addressed SLMs [116],

commercial 2D LC displays, and acousto-optical modulators (AOM) [117]. The mask can

also be made reflective, in which case the role of the i n'. -- spectrometer is p1 ',. .1

by the input grating-lens pair, through which the beam retraces itself with vertical offset

on the way out. Superior performance of arbitrary deformable mirrors was predicted for

the purpose of phase compensation by Efimov and Reitze [75] and was recently confirmed

experimentally [118]. Also, programmable Micro-Optical Mechanical Systems (MOEMS)

promise advantages for amplitude and phase pulse shaping.

For a discrete N-pixel filter, such as the LC SLM used in this work, the ]1,!, -, ,1 mask

can be written as [119]

M(x) C (x) K 6 (x nwp) rect (x/Wp) (2.20)

where C(x) is the continuous spatial mask that corresponds to the desired frequency filter,

6(x) is the Dirac delta function, w, is the p1!-, -i1, 1 width of each pixel, the function rect(a) =

1 for Ix| < 1/2 and 0 otherwise, and "*" denotes convolution. From Equation (2.19) the

corresponding spectral filter M(w) can be obtained, and for small focal spots in the masking

plane, wo < wp, is given approximately by

M(w) C (g [w]) 6 (g [w] nw) rect (g [] /w~) (2.21)

The shaped pulse in the time domain is obtained by Fourier-transforming Equation 2.18 to


Eo.t(t) = Ein(t) M(t) (2.22)

where M(t) is the Fourier transform of the spectral filter (2.19) which gives the effective

impulse response of the pulse shaper.

An analytic expression for the output electric field for a discrete mask can be obtained

if liner space-to-frequency mapping is assumed:

x = g[[w] = a( wo) (2.23)

where a = (dx/dw),=_o is given by

a = (2.24)
Ujod cos 0d,o

In this case, assuming that the bandwidth of the pulse shaper exceeds the bandwidth of the

input pulse, the electric field Eout(t) of the shaped pulse is expressed as [119]

Eo,,t(t) exp Ei,(t) C* t -n j sine (t t6f) (2.25)

where the desired impulse response function C(t) is given by the Fourier transform of C(w),

6f = 6w/27 is the frequency bandwidth per pixel, and since( ) = sin(/)/{. The result of the

pixelation of the mask is now evident from the Equation (2.25). The output electric field

profile Eout(t) consists of the main pulse, which is the convolution of the input pulse with

the desired response function C(t), supplemented by a number of replica pulses which result

from the convolution of Ei,(t) with time-shifted functions C(t n/6f). The entire result is

weighted by the sinc(rt 6f) function which is due to pixelation and a Gaussian envelope due

to the finite focal spot size in the masking plane. Both these weighting factors contribute to

the suppression of the replica pulses. If the desired response function is well localized near

zero time, in the region Itl < 1/6f, then the actual pulse shape will closely approximate

the desired shaped pulse, except for a pair or two of low-amplitude replica pulses centered

at times t = In/6f. On the other hand, in case C(t) extends out to times t 1/6f, then

n = 1 replica waveforms will alias onto the main pulse near t = 0 and blend in with the

main pulse.

In the more realistic case of nonlinear x +-4 w mapping g [w], the electric field amplitude

needs to be evaluated numerically. Intensity of the output waveform can also be computed


out(t) = d)' ei' du E*1t(w)Eo,,(a + ') (2.26)

For the sake of completeness, we must mention the second major approach to

pulse shaping: the direct in-time approach, represented mainly by the recently developed

and very promising technique based on the acousto-optic programmable dispersive filter

(AOPDF) [120, 121]. Here, the pulse is not spatially dispersed and, in fact, no special ap-

paratus is needed, instead, the acoustic wave is launched into a thick AO crystal collinearly

with the light beam. The acoustic wave is modulated in such a way as to create a local phase-


matching condition for light scattering into an orthogonal polarization inside the crystal at

different depths for the different wavelengths, Figure 2.15. Then, due to the birefringence

of the crystal, group delays of these frequency components will be different as controlled by

the user through modulation of the RF voltage applied to the transducer, which drives the

acoustic wave. The device allows independent control of the amplitude (through acoustic


short pulse W W

mode 1


Figure 2.15: AOPDF principle. The acoustic wave and the incident and diffracted optical
waves are collinear and propagate along the z axis [120].

wave amplitude) and phase (through frequency modulation) of the femtosecond pulse with

reportedly high efficiency and resolution.

2.5 Ultrabroad-Bandwidth Pulse Shaping: Experimental Results

All the pulse shaping experiments reported in this section were performed using a

commercially available one-dimensional LC SLM (\I. .. I.-.I I1: Optics SLM2256), which is

described in more detail below. Our pulse shaper configuration is shown in Figure 2.12. It

consists of a pair of 600 lines/mm gratings placed at the focal planes of a unit-magnification

confocal pair of concave 12.5 cm focal length gold spherical mirrors. In the Fourier plane,

midway through the apparatus, the optical wavelengths are spatially separated with a near-

linear dispersion (2.12) with dx/dAo a 0.085 mm/nm, where Od,o = 28.10 is the diffraction

angle of the central wavelength component.

The SLM positioned in the masking plane consists of 128 individually addressable

elements 100 pm wide with a 2 pm gap between pixels. The liquid crystal pixels are

mounted in a 4.6 mm thick fused silica housing. Driver circuitry provides independent

voltage control of each pixel with 12 bit resolution. Phase modulation by each pixel results

from the voltage-dependent change in the refraction index of the liquid crystal

2wn (V,) L
A0 = (2.27)

where L is the depth of the LC layer along the light propagation direction and is typically

a few microns thick. The refraction index change n(Vp) results from the rotation of the

rod-like LC molecules to partially align themselves along the direction of the applied DC

field9 (propagation direction).

An important consideration in our experiments was the variation of the amount

of retardation (phase) as a function of wavelength. Phase shifts in excess of 2w at all

wavelengths are required in order to utilize the full capability of the pulse shaper. We,

therefore, performed a thorough calibration of the modulator at 730, 800, and 860 nm

wavelengths, which effectively span the mode-locked spectrum of the pulse (Appendix A).

The maximum phase shift occurs at shorter wavelengths as expected from Equation (2.27)

and was measured to be almost 3w at 730 nm. We also tested pixel uniformity of the device

at a single wavelength by operating the modulator at a fixed voltage and scanning across

the pixels. Sli,!ii variations of transmitted intensity (<5%) were observed.

Temporally shaped pulse profiles were measured using noncollinear cross-correlation

techniques. A small portion of the beam was split off before the pulse shaper and delayed to

serve as a reference. The signal and reference beams were subsequently mixed in a 0.1-mm

KDP crystal. Because of the high repetition rate and average power of the Ti:sapphire

laser, we were able to obtain all of our data in single scans. Figure 2.16 displays a cross

correlation of an unshaped pulse propagating through the shaper (blue curve) and the

intensity autocorrelation of the pulse directly from laser (red curve). The autocorrelation

and cross-correlation widths are 19.8 and 20.3 fs, respectively.

One of the simplest pulses which can be synthesized using pure phase filtering is the

odd pulse. The odd pulse is generated when a r-phase shift of the carrier frequency is
9The LC pixels are usually driven by a few kilohertz square wave rather than DC to prevent electromi-
garion effects in the liquid crystal.

I3 I I
-100 -80 -60 -40 -20 0 20 40 60 80 100
Time (fs)

Figure 2.16: Cross correlation (blue curve) of an unshaped pulse after it propagates through
the shaper. The sech2 deconvolved pulse width is 13 fs. For comparison, an autocorrelation
directly from the oscillator is displayed (red curve).

imposed upon one half of the spectrum about its symmetry point. The term "odd p'II-

reflects the antisymmetric functional dependence of the electric field envelope on time and

is a special type of a zero-area (07) pulse, which is of fundamental significance in the field

of coherent optics [122]. The resonant interaction of an odd pulse with a two-level --, i,

results in initial excitation from the ground state to the excited state during the leading

part of the pulse followed by deexcitation back to the ground state during the latter part

of the pulse as a result of the abrupt 7r-phase shift. Odd pulses could also prove useful for

enhancing terahertz emissions from asymmetric coupled quantum wells. A cross correlation

of an odd pulse generated from a 13-fs pulse is shown on Figure 2.17. Unlike a pure odd

pulse, which possesses symmetric intensity profile, our pulse displays slight asymmetry in

the widths of the peaks. These deviations come primarily from two sources. First, the

frequency spectrum of the Ti:sapphire laser is not symmetric and, therefore, does not truly

conform to the criteria for an odd pulse. In addition, uncompensated cubic and quartic

phase dispersion from the SLM is present in the shaped output pulse. Nevertheless, the

overall fidelity of the odd pulse is quite good.

-300 -200 -100 0 100 200 300
Time (fs)

Figure 2.17: Cross correlation of an odd pulse.

Another important phase filter for generating a train of equally spaced pulses is based

on so called maximal length sequences [123] (I--. i. 1". ). This method has previously

been used to generate pulse trains with repetition rates of up to 12.5 THz [124]. Pulse

trains such as these have been used, for example, in experiments on resonant excitation of

optical phonon modes in molecular crystals by ISRS [99, 100]. These masks are periodic

binary phase masks with spectral period 6F. Each period is divided into P pixels with the

phase of each pixel being 0 or AO as determined by the M-sequence. The output pulse

train then consists of P pulses under Gaussian envelope with repetition rate 6F. For these

experiments we used the length 7 M-sequence {AO, AO, 0, AO, 0, 0, 0} with A 1.1r. The

results are shown in Figure 2.18, which displays cross correlations of the pulse trains with

repetition rates 8.8, 16 and 23.6 terahertz. We note that 23.6 THz is, to our knowledge, the

highest modulation frequency ever imposed on a lightwave by a linear filtering technique.

Both 8.8 THz and 16 THZ trains have well-resolved individual peaks. Pulses in a 23.6 THZ

train are not as well resolved, but one should take into account the wash-out effect of the

cross correlation in interpreting these trains: the signal intensity peaks are factor of r 1.5

narrower than the widths of the corresponding cross correlation peaks. The choice of the

repetition rates was dictated by fixed pixel sizes and fixed spectral dispersion in the Fourier

-1.0 -0.5 0.0 0.5 1.0

Time (ps)

Figure 2.18: Cross correlations of the phase-only shaped terahertz-rate pulse trains gen-
erated by length 7 M-sequences with repetition rates of (a) 8.8 THz, (b) 16 THz, and
(c) 23.6 THz.


plane of the shaper. Continuous adjustment of the spatial frequency dispersion in the

Fourier plane requires either adjustment of the diffraction angle of the shaper gratings or

incorporation of adjustable focal-length optics within the shaper0l. Note, however, that the

repetition periods of the pulse trains generated in these experiments are not simple integer

multiples of each other because of the higher-order spatial dispersion.

Many p.!i, -i4 1 applications demand complex pulse shapes that, for instance, possess

asymmetric temporal profiles. As an example, impulsive resonant Raman excitation of

large-amplitude (anharmonic) optical phonons requires tailoring an optical pulse train to

first harmonically excite vibrational modes with a sequence of equi-spaced pulses. As the

amplitude of the vibration becomes sufficiently large, inter-pulse spacing at the end of

the train should be adjusted to follow the phonon out of the harmonic well. Such pulse

trains may also be useful in driving large-amplitude plasma wakefield oscillations in low-

density laser-produced plasma [111]. In principle, any arbitrarily shaped waveform can be

synthesized with both amplitude and phase shaping. The appropriate complex frequency

filter is approximately given by complex M (w) = A (w) exp [if ()] = Eout (a) /Ei, (w).

However, as noted previously, pure phase filtering is advantageous when (i) both phase and

amplitude filtering cannot be experimentally implemented, or (ii) the reduction in energy

that necessarily accompanies amplitude filtering is undesirable. Binary phase-only filters

necessarily generate waveforms which have symmetric intensity profiles and cannot be used

for the generation of more complex pulses. We, therefore, attempted to synthesize more

complex (asymmetric) pulse trains using gray-level phase masks. For these experiments,

phase filters were designed using simulated annealing optimization codes [125]. We adopted

a modest strategy of selecting targets of the form:

Etarget (t) =u (t)* A exp [ia 2] 6 (t kT) (2.28)

i.e. targets consisting of trains of pulses in which the spacing between pulses and the pulse

durations were varied. Here, u (t) is the field profile of the unshaped pulse, 0 < Ak < 1 is
10We show later on that simple and complex pulse trains of any repetition rate not exceeding the inverse
duration of the pulse can, in fact, be produced with help of computer algorithms, such as Simulated Annealing
or Gerberg-Saxton.

real and defines the field amplitude for the pulse at time t kT, a() is the quadratic chirp

parameter for the kth pulse, and "*" is the convolution operator. Briefly, phase masks with

64 gray levels were randomly generated and multiplied with the experimentally measured

input spectrum to generate an output field spectrum Eguess (w) = Ei, (w) exp [i5i (w)]. The

resulting temporal waveform Eguess (t) was computed by an inverse Fourier transform and

compared with a specified target waveform Etarget (t) by computation of a simple cost

function J:

J [2 (w)] Eess (i) E2aret () (2.29)

which minimizes the differences in intensity between the generated and target fields. The

minimization of J proceeds by the computation of AJ J= Jrrent Jprevios, in which the

current guess is l'.---,- accepted if AJ < 0 and is accepted with probability exp (-AJ/T)

if AJ > 0. The i' -iq' i i ,re" T is initially set to a value well in excess of AJ. In this

way, the cost function is initially free to move about its entire parameter space and seek out

the global minimum. As the annealing proceeds, the temperature is reduced and the cost

function descends into the global minimum. On a modest computer (450 MHz AMD K6-

III), this procedure may require many hours to converge. A much more effective procedure

is realized with the Gerberg-Saxton algorithm (see Section 5.4 in Chapter 5).

Example results for three different trains are shown in Figure 2.19 The first, Fig-

ure 2.19a, is a train of three pulses in which the amplitudes and the interpulse spacings

are varied. The agreement between the target train (dashed blue curve), the numerically

synthesized pulse train, in which the optimal mask is used (green dotted-dashed curve),

and the experimental cross correlation (solid red curve) is excellent, with slight deviations

in the amplitudes (<10%) and positions (<5 fs) of the experimentally synthesized train.

Approximately 92' of the initial pulse energy resides in the target pulses in good agree-

ment with numerical results. Figure 2.19b depicts a train of three equal amplitude pulses in

which both interpulse spacings and pulse durations are varied. Again, we find reasonably

good agreement between the target and the cross correlation. The temporal positions agree

to within 10 fs, and, with the exception of the first pulse in the train, pulse durations are

within 10% of the target. In the worst case, the amplitudes vary by 2'' of the target

/ .
-- -- -- -

I I t II f I
I I / I \ I I. i I;

-400 -200 0 200 400

Time (fs)

Figure 2.19: Cross correlations of asymmetric, chirped trains of femtosecond pulses gener-
ated using 64 gray-level phase masks designed with simulated annealing. Each panel shows
the target (dashed curve), the numerical result (dotted-dashed curve) and the experimental
cross correlation (solid curve).

value. Finally, in Figure 2.19c, we display a chirped pulse train consisting of six replica

pulses. Once again, we observe minimal deviations (<10 fs) from the peak target pulse

positions. Excluding the initial pulse, the experimental cross correlation amplitudes are

within 15% of their target values. The initial pulse has a peak amplitude of approximately

1' of the target value, and slight variations in pulse durations are observed.

Compensation of the high-order phase dispersion is another potential application of

the programmable pulse shaping. The propagation of ultrabroad bandwidth pulses through

dispersive media results in severe pulse broadening as frequency dependent phase shifts

accumulate. For optical pulses which are not too broad (i.e., which satisfy Aw/:wo << 1)

the phase of an optical pulse propagating through dispersive media can be conveniently

expressed in a Taylor series 2.2, Section 2.2:

P (w) = +(wo) + () () wo) + (2) ( -o)2 + (2.30)

)(3) (w wo)3 + (4) ( w4 + (2.31)

where we defined (') = (d'0/dw ), 0 as the nth-order derivative of the phase evaluated

at o0. Compensation of large amounts of quadratic phase dispersion can be accomplished

through the use of grating or prism pairs [70, 126]; however, for pulse compression of sub-

10 fs pulses [72] or chirped pulse amplification of sub-100 fs pulses [127, 128], compensation

of higher-order phase dispersion (i(3), &(4), etc.) is essential for optimizing the fidelity and

duration of the pulse. Here we show that large cubic (V(3)) and quartic (V(4)) phase shifts

(over 120r) can be imprinted on pulses by the programmable pulse shaper.

Because the liquid-crystal SLM is restricted to phase shifts of 2r, larger phase shifts

are accomplished by folding the phase back into the range -r < 0 < starting at some

point on w-axis (x-axis in Fourier plane) and proceeding until the phase goes out of limits

again, at which point the next-order folding occurs and so on. Moreover, since the SLM

is pixelated, it cannot imprint a smooth continuous-phase profile. Instead, the spectrum

is sampled by the modulator at discrete points, 0n (x) = (xn) where z, is the position

of the nth pixel. As discussed by Weiner et al., these sampling criteria place an upper

limit of A0p wT between pixels, thus limiting the total amount of phase shift that can


be imparted to a pulse [113, 114]. In our experiments, this places an upper limitation

of approximately 1307 of total phase shift. ('I. I;y, any increase in pixel number of the

SLM would allow for larger phase shifts". In Figure 2.20a, we show the cross correlation

a) ,
- I

- - - -

-300 -200 -100 0 100 200
-300 -200 -100 0 100 200

-800 -600 -400 -200
Time (fs)

0 200

-800 -600 -400 -200 0 200 400 600 800
Time (fs)

Figure 2.20: Experimental cross correlations (solid curves) and numerical simulation results
(dashed curves) of a phase-modulated pulse with (a) 4 (3) 6 x 104 fs3, (b) 4)(3) 6 x 105 fs3
of cubic phase, and (c) )(4) 1.1 x 105 fs4, and (d) (4) = 1.1 x 106 fs4 of quartic phase.

of a pulse with a cubic chirp of T(3) = 6 x 104 fs3, which corresponds to a total phase

shift of ~ 12r over a 100-nm bandwidth. The experimental cross correlation (solid curve)

shows remarkable fidelity and agrees well with the theoretical intensity profile (dashed

curve). The temporally chirped, oscillatory prepulse is characteristic of cubic chirp, in

which equally temporally advanced higher and lower frequency components lead the pulse

and interfere with each other. The effects of sampling are clearly evident in Figure 2.20b,

which shows a cross correlation for a cubic chirp of T(3) = 6 x 105 fs3 (a phase shift of 121r).
"Modulators with 640 pixels(JENOPTIK, http://ww 1..... p 1i.-1d .I. /) were recently used for adaptive
pulse compression from the OPA system (M. Motzkus, personal communication).

-400 -300 -200 -100 0 100 200 300 400

The main lobe shows three distinct peaks and a pulse substructure is seen on the trailing

edge of the pulse. Similar results are seen for the quartic phase profiles which are shown in

Figure 2.20c and 2.20d. Figure 2.20c displays an experimental cross correlation (solid curve)

and a theoretical intensity profile (both plotted on a linear scale) for &(4) = 1.1 x 105 fs4

(a phase shift of 40r). Good agreement in the main pulse and the first sidelobe are seen,

with deviations occurring in the wings. In particular, the experimental trace shows some

asymmetry. SlI,!_l oscillations of the shaped pulse appear at negative time delays. These

deviations are more pronounced in Figure 2.20d, which corresponds to a quartic phase of

#(4) = 1.1 x 106 fs4 (a phase shift of 400r). These asymmetries are caused by the residual

cubic-phase dispersion present on the pulse, because imperfect alignment of the spectrum

with the filter will necessarily result in the introduction of lower-order terms in the phase

dispersion, i.e. A (w) = (1/4!) #(4) (o) [ Awo )]4 M A (w wo)4 + A (w wo)3,

where A and B are coefficients that depend on Aw. Moreover, any nonlinearity in the

spatial frequency dispersion will also introduce lower-order terms into the phase expansion.

2.6 Phase Compensation with Deformable Mirror

The recent success in generating amplified sub-20 fs pulses using regenerative pulse

shaping [129, 130] demonstrated a practical way of reducing gain narrowing effects in the

amplifier and obtaining wide spectral bandwidth at the output. To fully utilize this band-

width, careful phase control of the amplified pulses is required.

Pixelated devices are not very well suited for phase compensation in amplifiers where

one is trying to flatten out the smooth phase variation, caused by, for example, material

dispersion in the amplifier --, -. i i This is not only because pixelated devices ,1'.- -, produce

staircase-like approximation to the desired phase and replica pulses, but also because it can

be difficult to find an optimal placement of the modulator [131]. After the amplifier, the

modulator will be susceptible to damage and introduce large amount of loss, and before the

amplifier pixelated devices can produce spectral intensity modulations damaging for the

amplifier optics (see Chapter 3).

It is therefore desirable to find a non-pixelated mask which would allow arbitrary

and programmable phase compensation of smooth phase distortion of an ultra high band-

width pulse. Here we propose and numerically verify the new phase compensation method

based on adaptive optics, "Spectral Adaptive Optics." We derive our idea from the ele-

gant method for placing purely cubic phase modulation on femtosecond pulse proposed by

Heritage et al. [132].

As pointed out by Treacy [70] and Kolner [133], the 1!-r, -i 1 descriptions of paraxial

diffraction (spatial optics) and temporal dispersion (temporal optics) are essentially identi-

cal. Thus, there is a direct analogy between spatial and temporal optics. Methods designed

to compensate for spatial phase distortion and diffraction in optical systems should, there-

fore, apply, with suitable modification, to optical systems in which temporal dispersion is

a concern. Spatial adaptive optics [134] has been developed to compensate arbitrary and

time varying spatial frequency dispersion and minimize aberrations in ground-based as-

tronomical telescopes, which occur as light rays propagate through turbulent atmospheric

conditions. This is normally accomplished using two-dimensional deformable mirrors whose

shape can change to correct for phase aberrations during propagation and produce nearly

diffraction-limited spatial images. In this same manner, the spectral adaptive optics can be

used to minimize arbitrary phase dispersion in ultrashort pulse optical systems and produce

nearly dispersion free temporal images. An added advantage of using deformable mirrors

is the simplicity of use and alignment supplemented by 100% throughput if a broadband

dielectric coating is used. Moreover, in contrast to AOM modulators, a deformable mirror

is a static device and can be used with laser systems operating at any repetition rate.

Spectral adaptive optics operates by placing a one-dimensional continuously de-

formable mirror at the Fourier plane of a modified pulse shaper or stretcher as in Fig-

ure 2.21. The phase value for a particular frequency component comes directly from the

local retardation introduced by the mirror deformation. The mirror accommodates the

entire spectrum of the optical pulse and is deformed via a linear array of electronically pro-

grammable actuators, which provide push-pull action parallel to the propagation direction

of the spatially dispersed beam. If the number of actuators, which control the shape of the

mirror, is N + 1, the surface of the mirror can assume the profile described by Nth order

polynomial, which corrects phase distortions up to Nth order, Equation (2.2).


Spherical Deformab
Mirror Mirror

It f >1

Figure 2.21: Schematic design of the spectral adaptive optics pulse shaper. For separating
the output beam from the input beam, a vertical offset is introduced by slight tilting of the
mirror. This shaper can be modified into a stretcher by setting the distance between the
grating and the spherical mirror to be less than f and double passing through the device,
desirably with the image inversion before the second pass.

For assessing the feasibility of this method, we have developed a custom two-

dimensional ray-tracing code capable of tracking the phase of each frequency component

of the input spectrum through a two-dimensional optical -, -I ii, which consists of an ar-

bitrary number of diffraction gratings, spherical mirrors, parabolic mirrors, polynomial

mirrors, prisms glass slabs, and lenses12. The target of the modeling was 10 fs chirped pulse

amplifier, similar to one described in Chapter 3. Our numerical code was tested by com-

parison with analytical phase results for simple optical systems, as well as expected spatial

and angular beam distributions. Issues addressed in our simulations include the degree to

which the phase dispersion can be minimized through spectral adaptive optics, effects asso-

ciated with beam divergence due to mirror deformation, and the extent to which transverse

spatial effects compromise the pulse duration after propagation through the amplifier. The

modeling was performed assuming 600 lines/mm grating in pulse stretcher configuration
12The source code is available from the author upon request


with 170 incident angle and effective grating separation of 105.5 cm. The clear aperture of

all components was chosen to accommodate 150 nm of spectrum.

The modeling results are presented in Figure 2.22, where spectral phase accumulated



100 -
50 -
-0.1 -50
-150 -
-200 _ _ _, _
0.7 0.8 0.8 0.8 0.9 0.9
-0.2 -
0.70 0.75 0.80 0.85 0.90
Wavelength (jtm)

Figure 2.22: Wavelength-dependent phase distortion of a chirped pulse amplifier designed
with a deformable mirror stretcher. The phase is essentially flat over 120 nm of bandwidth.
Inset: phase dispersion for a flat mirror stretcher of the same design (note the different
vertical scale).

through the system is shown. For comparison, the total phase for the flat mirror stretcher

is displayed as an inset. For the case of optimally shaped deformable mirror, no more

than 0.2 rad of phase accumulates across the entire spectrum, and the phase variation

is essentially zero in the range 760-860 nm. The corresponding output pulse maintains

the shape of the input pulse over many orders of magnitude. A natural question arises

concerning the angular divergence of the beam upon exiting the stretcher. The bending

of the mirror will necessarily introduce a change in the reflected angle as a function of

frequency. However, the amount of deflection needed for phase compensation is only ~ 1pm

over the entire length of the mirror (7 cm), giving a rough estimate for the maximum angular

divergence of ~ 10 prad in the low intensity wings of the pulse spectrum. This estimate is

born out by our ray tracing results, which predict a 20 prad beam divergence at the output.


Another consideration is the variation of the pulse phase and intensity across the

transverse beam profile. Frequency components which are located off beam center will not

propagate along the same optical path as those in the center. Therefore, any beam with finite

transverse spatial dimension will have a group delay that depends both on frequency and

on transverse spatial position in the diffracting dimension of the beam. To examine these

effects, we have computed the spatial dependence of the phase, and intensity, Figure 2.23a,

for our adaptive pulse stretcher. It can be seen that the absolute accumulated phase is

a) Phase b) Output Spatio-Temporal Pulse Profile


.. ...... ....... .... . .. '" .

Figure 2.23: Ray tracing calculation results of spatio-temporal effects in spectral adaptive
optics pulse stretcher. The phase of the pulse as a function of wavelength and transverse
spatial position (a) and the resultant spatio-temporal intensity pulse profile (b).

less than 7r/5 across the entire spatial profile of the beam which has negligible effect upon

output pulse shape fidelity, Figure 2.23b. in which the full spatial dependence of the pulse

intensity, I (t, x), is shown for a Gaussian profile with 1.5 mm radius.

We should note that recent calculations by Wefers and Nelson [135] on the effects of

phase filtering in a zero-dispersion pulse shaper 1:_:_. -. that spatio-temporal distortions of

the pulse occur, which would not be detectable by a simple ray tracing .,!1 1, --i It was

shown that any type of Fourier-domain pulse shaping is accompanied by this spatio-temporal

coupling. This is manifested, for example, in a time-dependent lateral displacement of each

peak in a pulse train, similar to one shown in Figure 2.18, from the propagation axis.

In the next section, we extend this ,&1, 1-,i- on the Fourier-domain phase compensation

with arbitrary phase masks in a zero dispersion pulse shaper. There, we conclude that

in contrast to pulse shaping, compensation of reasonable amounts of phase leads to only

slight distortions in the output pulse spatial profile. This is because the output pulse is

well localized in time which is the purpose of phase compensation. Extended pulses, usually

sought in pulse shaping experiments, on the other hand, may posses a noticeable space-time


Finally, we note that our proposal was recently put to an experimental test, where

transform-limited pulses were obtained from the chirped-pulse amplifier with the use of a

zero-dispersion pulse shaper incorporating electrostatically controlled deformable mirror in

its Fourier plane [118, 55].

2.7 Space-Time Coupling in Fourier-Domain Phase Compensation

It has already been demonstrated theoretically that Fourier domain pulse shaping

with any type of mask leads to spatio-temporal coupling in the resultant waveform [135, 136].

In particular, emerging pulse acquires a local time-dependent shift of the transverse intensity

profile. A pulse train, for instance, will have its leading peaks shifted in the horizontal

plane to one side of the propagation axis while the trailing peaks will be shifted to the

other (see Figure 5 in [135]). One can think of this effect as a tilt of the axis along which

pulse shaping occurs relative to the propagation (t) axis. On (x, t) plane, where x is the

transversal coordinate, the angle of this tilt is given by

Ox cd cos Oi
tan = (2.32)

where c is the speed of light, d is the diffraction grating grove spacing, Oi is the grating

incident angle and A is the wavelength of light. This coupling effect is fundamental in that

it depends only on pulse shaper design parameters and is 1'.-- -,- present, unless Oi = r/2,

which is practically impossible. The origin of the spatio-temporal coupling can be traced to

the space-time entanglement produced by the grating-lens pair in the Fourier plane of the

shaper. Any shaping in the time-frequency domain, therefore, affects the spatial domain as


The question then arises, what similar effects do we expect to observe when the pulse

shaper is used for phase compensation, i.e., phase profile inverted with respect to the input

pulse phase is placed in the Fourier plane of the pulse shaper. The output is expected to

be near transform limit, but the amount of phase present in the masking plane can still be

large if the input pulse is severely distorted.

The simplest case to address is pure quadratic phase (linear chirp) compensation.

This situation, however, is of limited use practically, since linear chirp is usually dealt with

a simple pair of gratings, prisms, or a slab of glass. On the other hand, it is beneficial that

quadratic phase case is fully analytic. The equations that we have developed for this case

were used to verify the validity of the numerical model described below.

Since higher-order phase compensation can not be treated analytically in full, we

have performed numerical ., _1-, ,-i, of the problem. We considered a zero-dispersion pulse

shaper in two dimensions, similar to that shown in Figure 2.12, except no astigmatic effects

were taken into account and the mask was considered to be continuous. The input electric

field with initial phase distortion in (w, x) domain is specified as

(wa, x) exp 2 + i (0) (2.33)

where Ax and A, are spatial and spectral pulse widths, Q = a wo measures the offset

from the central frequency wo, and 1(Q) signifies the initial spectral phase distortion of the

pulse to be compensated with the pulse shaper. The following notation is used to describe

the domain in which electric field is written: E(x, t), E(k, t), E(k, w), and E(x, w), where

(x, k) and (t, w) are pairs of (. -iill:- ,Il variables. Although arbitrary phase profiles can be
examined with our .,11 i', -i-. in the following, we mainly focus on cubic and quartic terms

in the Taylor series expansion (2.2).

After the first grating, the electric field becomes

622 Q2
(w, x) exp L 2- + +iQx+ i () (2.34)

where 6 = cos i/cos Od is the ratio of the incident and diffracted angle cosines and =

A/cd cos Od with d being the grove spacing. The action of a lens between the focal planes is

described by the spatial Fourier transform:

E2 (k) i El -kf

E(X) = El o (2.35)

where indices 1 and 2 refer to the front and back focal planes of the lens in the direction of

beam propagation, f is the focal length, and ko = 27/A. When the field reaches the Fourier

plane of the shaper, it is converted to the spatial domain, after which a phase-only mask

M(x) is applied:

M(x) =M(Q), with a = -d (2.36)
27rcd cos Oed

After the mask, the electric field is propagated similarly through the second lens and the

grating and is cast into any of the four domains (x, t), (x, w), (k, t), or (k, w) at the output.

In our im 11-,--. we model a typical pulse shaper with 600 lines/mm diffraction gratings and

12 cm focal-length lenses. Incidence angle Oi is taken to be zero for simplicityl3

We start with a Gaussian pulse of 13 fs duration and 1 mm diameter (both are inten-

sity FWHM), shown in Figure 2.24. As expected, in all four (,-IinI:- ,I. domains the field

amplitude contours have elliptic shapes being the cross-sections of a Gaussian surface at dif-

ferent height levels. If the pulse carries certain amount of cubic phase, oscillating structure

appears on its temporal leading or trailing edge due to interference of equally advanced or

delayed frequency components from the opposite ends of the pulse spectrum. An example

is shown in Figure 2.25a for #(3) = 120, 000 fs3. One should note the symmetry of the plot

with respect to the propagation axis (t-axis). In comparison, if the same amount of phase is

incurred upon a transform limited pulse by a pulse shaper with an appropriate phase mask

in its Fourier plane, the field amplitude profile shown in Figure 2.25b results. The asym-

metry of the second plot is an illustration of spatio-temporal coupling, Equation (2.32). To

compensate the phase of Figure 2.25a, the mask in the pulse shaper should present a -)(3)

phase to the pulse, but due to the tilt of the shaping axis, we expect the compensation to

be spatially non-uniform, resulting in an asymmetric spatio-temporal profile of the pulse.
13The source code is available from the author upon request.



-1 -


5 kt
0 -
5 -
5 -
0 -
5 -

-100 -60 -20 20 60



).5 -0.25 0 0.25 0.5

Q2( rad/s )

Figure 2.24: Input 13 fs, 1 mm pulse shown in four conjugate domains. Normalized to
unity, electric field amplitude is linearly mapped between 0 and 1 with contour separation
of 0.1.




---- -

-1-- -- -- -- -- -- -- -- -- --- -- - --- ---

-1400 -1200 -1000 -800 -600 -400 -200 0 200
Time (fs)

Figure 2.25: Positive cubic phase-distorted pulse (a) compared to a Fourier-domain shaped
pulse (b). Both pulses carry equal amounts of cubic phase <)(3) 120,000 fs3, however, the
shaped pulse is asymmetric with respect to the propagation axis.


Indeed, such asymmetry is observed in the numerical results, Figure 2.26. In (x, t)













-100 -60 -20 20 60 -0.5 -0.25 0 0.25 0.5
t(fs) Q(rad/s)

Figure 2.26: Phase compensation of (3) = 120, 000 fs3 cubic phase. Field amplitude con-
tours express complicated structure in all four domains. Transverse spatial dependence of
the spectrum, (x, w), should be easily obsevable.

domain, the field amplitude profile displays a complex structure on one side of the spatial

mode. This structure is the result of the interference of the frequency components from the

oscillating tail of the input pulse with the components from the center of the pulse spectrum.

The field profile in (x, w) domain demonstrates that opposite sides of the spatial mode of the

pulse possess different spectral content. This spatio-spectral dependence should be easily

observable by simply scanning the beam across a spectrometer slit in the horizontal plane.

For comparison, compensation of the same amount of cubic phase, but with the opposite

sign, is shown in Figure 2.27. In this case, the shape of the contours is flipped about

the propagation axis compared to Figure 2.26. This is expected, since the shaping axis









-100 -60 -20 20
t (fs )

60 -0.5 -0.25 0 0.25
Q( rad/s )

Figure 2.27: Phase compensation of T(3) = -120, 000 fs3 cubic phase. The contour structure
in (x, t) and (x, w) domains is inverted about the t-axis compared to Figure 2.26.

X ()

ko -




tilt is fundamental to the pulse shaper and is not related to pulse parameters, whereas the

oscillating tail of the distorted pulse appears on the opposite side of the main t = 0 peak.

2 1 Y

xt xco




15 kt
10 -
0 -e
-10 -

-100 -60 -20 20 60 -0.5 -0.3 0 0.3 0.5
t(fs) Q(rad/s)

Figure 2.28: Phase compensation of #(4) = 1.2 x 106 fs4 quartic phase. Symmetric amplitude
profile in (x, t) domain results and center of symmetry is present in (x, w) domain.

Similar results are obtained for quartic phase compensation, Figure 2.28, where & (4)

1.2 x 106 fs4 of phase is compensated with the pulse shaper. Here, because initial distortion of

the pulse is symmetric, in contrast to cubic case, the resultant pulse profile in (x, t) domain

is symmetric about the propagation axis. However, field amplitude contours are not at all

elliptic as one would expect for perfect compensation. Easily observable structure again

results in (x, w) domain, which, unlike the cubic phase, possesses the center of symmetry.

To conclude, we must note that the effects discussed above are substantial only for

ultrashort and spatially small pulses. This is because the shaping axis tilt (2.32) is indepen-

dent of pulse parameters, and its effect on the pulse is easily masked by a large (spatially

and temporally) pulse. Indeed, to demonstrate the existence of the effects just described


we had to use quite large amounts of higher-order phase. For example, as we showed pre-

viously, cubic phases larger than ~ 100, 000 fs(3) could not be placed on the SLM without

undersampling. In contrast, typical amounts of higher-order phase present in most CPA

systems, for example, do not exceed 1 3 x 104 fs3 and 1 3 x 105 fs4. On the other hand,

when experimental pulse shaping enters sub-10 fs domain, and masks with large number of

pixels (or masks which are continuous) are used, the phase compensation effects described

here will likely 1p1 I, an important role and be easily observable in experiments.


3.1 Introduction

In the previous chapter, we presented the fundamental techniques for generation,

measurement and shaping of femtosecond pulses. For many experimental applications, in-

cluding those described in the subsequent chapters, we require pulse intensities greater

than can be produced by laser oscillators alone. For these applications, pulse amplifica-

tion is required. In this chapter, we discuss in detail techniques and considerations for

the amplification of femtosecond shaped pulses. In particular, we describe our home-built

millijoule amplifier --, -1. in for producing complex shaped and phase compensated pulses

with near-30 fs resolution, Section 3.2. Particular problems and issues associated with the

amplification of phase-shaped pulses are discussed. In Section 3.3, we present experimental

SHG FROG measurements of a number of amplified shaped pulses useful in coherent control

experiments (Chapter 5).

3.2 Millijoule Amplifier System

Output pulse energies from a femtosecond laser typically do not exceed a few nano-

joules. Substantial effort is required to achieve energies in the range of tens of nanojoules

directly from laser either by cavity dumping [137, 138, 139] or by reducing the repetition

rate of the laser by extending the cavity length [140]. Even then, pulse shaping will claim at

least 50% of the pulse energy due to grating loss and scattering from the mask. At the same

time, a great number of applications require shaped pulses with energies in the range of tens

of microjoules and above. Therefore, we have developed a custom multipass chirped pulse

amplifier --, -I. ii, with built-in pulse shaping and phase compensation capabilities, which we

describe in this section.

The past ten years have witnessed the dramatic improvement in amplification of

ultrashort pulses to millijoule and even to joule levels [141, 142], corresponding to multit-

erawatt peak powers. Several different amplifier designs have been used in the past for this

purpose. Multipass amplifiers without intentional pulse stretching were shown to produce

sub-20 fs pulses with kilohertz repetition rates [143]. The amplifier reported in [143] was

seeded with 10 fs pulses which expanded temporally in the input switching optics to the

duration of a few picoseconds due to natural material dispersion. This amount of pulse

stretching was sufficient to operate the amplifier below thresholds for nonlinear effects with

output energies of up to 100 /J. Although the complexity of the --, -1. 11, was greatly reduced

in comparison to the systems incorporating stretchers, the crucial performance parameters

were also diminished. In addition to the limited pulse energy, the amplified spontaneous

emission (ASE) at the output was comparatively high (over 10%), and very sensitive to

the alignment of the amplifier optics and the pump energy level. Recently, however, this

-, -1. il was improved to produce millijoule-level pulses with enhancements in other pulse

characteristics as well [144].

Notable in many aspects is another technique for short-pulse amplification based

on spatially dispersing the frequency components in the amplifying medium [145]. This

technique has a potential advantage of providing control over gain narrowing, but so far

earned little popularity because of the high losses and geometrical limitations.

Currently, amplification of the femtosecond pulses is normally done using the Chir-

ped Pulse Amplification (CPA) technique [14] where input pulse is first stretched in time,

amplified in energy in an amplifier cavity and finally compressed back to near transform-

limited duration, Figure 3.1. These steps are necessary to reduce the peak intensity of the

pulse during amplification and avoid deleterious nonlinear effects within amplifier optics

such as self-phase modulation and self-focusing leading to damage [146, 147].

Two of the most successful high-power CPA schemes are based on the multipass

configuration [148] and regenerative amplification [15, 149, 150]. Quite often in terawatt

systems, these schemes are used together with first stage being the regenerative amplifier

and multipass configuration used for the output power-boost stage. The regenerative ampli-

fier has a number of advantages which made it popular for amplification of > 50 fs pulses.

Laser Stretcher Amplifier Compressor

Figure 3.1: The principle of CPA technique. The seed pulse from the laser is temporally
stretched in a stretcher by separating in time its frequency components. Next, stretched
pulse energy is increased in the amplifier. Finally, the compressor reverses the stretcher by
putting the pulse frequency components back together.

In particular, the ASE can be suppressed on every pass by the intracavity polarization

switching optics. Since the beam follows the same path while in the cavity, no realignment

is necessary to adjust the number of passes, which simplifies the operation and mainte-

nance considerably and constitutes another advantage of the regenerative amplifier. On

the other hand, the amount of optical material traversed by the beam is much greater for

the regenerative amplifier which necessarily leads to large amounts of higher-order phase

distortions. This last point made the multipass configuration conceivably more suitable for
amplification of the ultrabroad-bandwidth femtosecond pulses. This usually comes at the

expense of increased level of ASE and rather complicated alignment procedure.

Using CPA technique, pulses as short as 19 fs have been produced with 100 TW peak

power levels [151]. The use of programmable frequency-dependent filters for pulse shaping

has also been demonstrated in chirped pulse amplifiers with 500 fs resolution [152]. Recently,

Dugan et al. showed that >100 fs pulses can be shaped and dispersion compensated by use

of an acousto-optic modulator in a separate pulse shaper placed before the stretcher [153].

Also, the use of a novel direct-in-beam acousto-optic programmable dispersive filter has

been demonstrated recently for phase compensation of 17 fs amplified pulses [121, 120].

In this section we describe our chirped pulse amplification --, -I. 1,i which incorporates

an LC SLM in the Fourier plane of the pulse stretcher to provide both phase compensation

and complex pulse shaping of the output millijoule pulses with 30 fs resolution [50]. For a

millijoule class CPA incorporating a programmable mask directly into the pulse stretcher

allows for many additional degrees of freedom in controlling the amplified output waveform.

In particular one can compensate for nearly arbitrary linear and nonlinear phase dispersion

by simply placing the appropriate phase on the frequency spectrum of the pulse. Addition-

ally one can apply complex phase masks to the SLM to achieve temporal shaping of the

pulse before amplification. Although it is possible to shape pulses after amplification, doing

so would incur substantial energy loss and a risk of damage to the mask since focusing on

the mask at least in one dimension is required within the pulse shaper1

The simplified schematic of the amplifier system is shown in Figure 3.2 with all pump

optics omitted. A seed pulse train at 90 MHz repetition rate from the master oscillator is

first downcollimated in a mode cleaner comprised of two plano-convex lenses L1 (15 cm focal

length) and L2 (10 cm focal length) with a 100 pm-diameter pinhole PHI placed in their

common focal point. The diameter of the pinhole was experimentally chosen to provide more

than '.' throughput and reasonable mode quality at the output in far field. The additional

dispersion caused by the bulk glass of the lenses L1 and L2 was partially precompensated

by the external prism compensator (see Figure 2.3). The additional uncompensated higher-

order phase due to the mode cleaner was proven to cause pulse broadening by no more than

1 fs at the benefit of greatly improved transverse mode quality and reduced divergence of

the beam.

Since the amplifier pump laser operates at 10 Hz repetition rate, input pulse picking

at this frequency is performed with the Pockels switch consisting of Glan-Thompson crossed

polarizers P1 and P2 and a Pockels cell PC1 (i. .1..:: E.-O., Inc.). A high voltage pulse

properly synchronized with the master oscillator at 10 Hz controls the switching window

when rapid 900 polarization rotation occurs allowing the single input pulse to pass through

the output polarizer. The transient window has a width of about 13 ns in time allowing

selection of only one pulse from the train with the selectivity of better than 30 dB.

Pulse stretching is accomplished with a Martinez-type grating-lens pulse stretcher,

Figure 3.3 [154] (gratings G1, G2; lenses L3, L4, and retroreflector RM1 in Figure 3.2). The
'Damage threshold of the liquid crystal or ITO conductive layer was estimated to be ~ 1 mJ/cm2 by the



1 mJ, 26 fs
at 10 Hz



Ti:Sa PH2

M9 M13V\ M8

_M7 M6







M3 P2 PC1 P1 M2

L1 PH1I L2\ PR1
Ti:Sapphire I

Figure 3.2: Setup schematics of the multipass amplifier system. Green arrows show the
direction of pump beams however pump optics is not shown.

Figure 3.3: Schematic view of the pulse stretcher with SLM in the Fourier plane.

displacement of the grating G2 from the focal plane of the lens L4 leads to a large amount of

positive linear chirp for the input pulse (linearly proportional to the displacement). The sign

of the quadratic phase is determined by the direction of grating displacement, being positive

(same as due to dispersion in glass in near infrared region) for the grating positioned between

the lens and external focus. We used a scanning autocorrelation technique to confirm that

the input pulse was stretched to ~ 10 ps upon two passes through the stretcher.

After the first gi I ii-:. each spectrally dispersed frequency component is brought to

focus by the spherical lens L3 in the SLM working area. The focal spot radius can be

estimated as wf = Af/'rwi w 50 tim where wi and wf are 1/e2 beam radii at the input

and at the focus correspondingly, f = 12 cm is a lens focal length, and A = 810 nm is the

central wavelength. Optimal operation for best resolution is therefore achieved [135] since

the SLM pixel has a width of 100 ptm. The second lens focuses the beam in horizontal

plane and collimates it in a vertical plane. Since the grating G2 is positioned less then one

focal length away the diffracted beam is not completely spectrally recombined and requires

a second pass through the stretcher. Retroreflector RM1 steps the beam up and sends it

back through G2, L4, SLM, L3, and G1 after which it appears parallel to the input beam

and slightly displaced vertically down for pickup with the mirror M5.


We should note that because the efficiency of the gratings is quite low, the overall

throughput of the stretcher is reduced to about 15%. Also, although using mirrors instead

of lenses inside the stretcher would be beneficial from several aspects, the geometrical re-

strictions on placing optical components would require mirror incident angle to be too large

causing uncompensatable astigmatism. We therefore used achromatic lenses L3 and L4.

After the stretcher, the seed pulse enters the amplifier four-mirror cavity (spherical

mirrors M9, M10, Mil, and M12, 1 meter focal length) through steering mirrors M7 and

M8. The beam on each pass is vertically displaced allowing for convenient pick-off of the

amplified pulse with mirror M13. The number of passes through the amplifier is limited

to about 16 by the tightness of the beams in the system. We have found, however, that

10 passes is quite sufficient for millijoule-level amplification at saturation.

The crucial part in ASE suppression in our amplifier is I .1-, .1 by two arrays of

pinholes (PH2) located about 10 cm behind the Ti:sapphire crystal. The natural divergence

of the input beam causes the individual beams on every pass be focused slightly further

away from the cross point of all the beams inside the crystal which in turn is determined

by the parallelism of the beams in the collimated arms of the amplifier between mirrors M9

and M12, M10 and Mil. This allows for spatial filtering with pinholes PH2 which is best

done at the beam waist. We exploit the spatial incoherence property of the ASE for filtering

it out. Since the ASE is incoherent, it focuses to a spot somewhat larger on the pinhole

and suffers reduced transmission. In addition, larger spot size inside the crystal reduces

the peak intensity and therefore reduces the unwanted nonlinear effects including the risk

of catastrophic damage due to self focusing. The presence of the spatial filter inside the

amplifier cavity increases the demand for precise alignment, but rewards us with minimal

ASE content at the output signal. The design and the material for the pinhole arrays must

be carefully chosen and optimized for best performance and high damage threshold. In

fact, the maximum achievable energy of the output pulse is now limited by the damage to

the pinhole's circumference with associated plasma production and subsequent performance


The intracavity spatial filter by itself provides for suppression of spontaneous radia-

tion at the output down to approximately 1%. This is considered quite adequate for many

Table 3.1: Amplifier output pulse parameters.

Parameter Value
Pulse energy >1 mJ
Pulse width <30 fs
Pulse-to-pulse energy stability 10% rms
ASE energy content 0.1%

applications. But to reduce the ASE even further, the output Pockels cell switch (polarizers

P3, P4, and Pockels cell PC2) is used, which operates in the same mode as the input switch

by opening for 13 ns with a high-voltage pulse. The upper state of the lasing transition in

Ti:sapphire has a life time of ~ 3 pfs so that after the second Pockels cell the ASE content

is usually reduced to below 0.1% in energy.

Pumping of the amplifier is done with a 10 Hz Nd:YAG laser (Spectra Physics

GCR 170). The optical pulse from the Nd:YAG cavity is frequency doubled to 532 nm

in an external type-II second harmonic generator and then split into two equal energy arms

to pump both sides of the Ti:sapphire crystal. Each pump arm was designed to provide

imaging of the plane located in the middle of the laser cavity onto the Ti:sapphire crystal.

This pumping technique eliminates problems with pump laser pointing instability and drift.

If simple loose focusing is used instead of imaging, our pump laser required warm up periods

of over 3 hours to stabilize which was unacceptable from a work-a-day prospective.

Temporal recompression of the amplified pulse is done in a standard parallel grating

compressor [70] with throughput of over I .' Some of the output pulse parameters are

summarized in the Table 3.1.

Information on amplified pulse characteristics was obtained through spectral, single-

shot autocorrelation and single-shot FROG measurements (see appendix B). Mode quality

was examined with the custom built mode profilometer based on water cooled 16-bit CCD

camera (Photometrics CH250).

Amplified pulse single-shot spectra for a number of output energies are shown in

Figure 3.4. One can see that the pulse spectrum narrows considerably upon transmission

through the amplifier --, -I. ill even when not pumped (green curve). Primarily, this is the


effect of the diffraction grating efficiency curves [155]. When low-level pumping is activated

(pink curve) the gain narrowing sets in and narrows the spectrum even further. However,

as the pump pulse energy is increased the amplified pulse spectrum widens again, predom-

inantly on the longer-wavelength side dictated by the sign of stretched pulse chirp and a

local slope of the Ti:sapphire gain curve [156]. At near saturation (black curve, pulse energy

S2mJ) the spectrum is widest and supports near 25 fs transform-limited pulse duration.

The spectrum shape is near-Gaussian now2 II:_:_. -li:_ also Gaussian temporal intensity

pulse profile. Single-shot FROG trace of the amplified pulse is shown in Figure 3.5a along-

-0 04mJ
-030 mJ

720 740 760 780 800 820 840 860 880 900 920
Wavelength (nm)

Figure 3.4: Pulse spectra through the amplifier system running at different energies. Note
that the spectrum from the oscillator is the widest and canonically asymmetric that cor-
responds to sech2 pulseshape. The amplified spectra are pulled towards longer wavelength
and are more symmetric.

side with the recovered trace, Figure 3.5b. Recovered Intensity spectrum and phase are

plotted in Figure 3.6. The recovery error was 0.002 on 256 x 256 grid. Since the ampli-

fier was running not at saturation, the spectral FWHM was only 33 nm. Due to obvious

phase distortions, the recovered pulse duration was 32 fs, Figure 3.7 Although not optimal,

2Note: asymmetry on the linear wavelength axis is expected, it is only on a linear frequency axis that
Gaussian is symmetric.






-200 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 200
Delay (fs) Delay (fs)

Figure 3.5: Amplified pulse FROG trace (a) and recovered trace (b) for the amplifier running
at ~ 0.7 mJ. Intensity scale is logarithmic spanning three orders of magnitude from blue
to red.

this regime of low-energy operation was required for our experiments on coherent phonon

generation in solid dielectrics (Chapter 5), however for experiments on adaptive feedback

control of pulse phase, Chapter 4, the amplifier was run at full saturation, resulting in

wider spectrum, higher nonlinear phase distortions, and with larger amount of linear phase

distortion uncovered by wider spectrum.

In addition to bandwidth and temporal shape, the quality of the spatial profile of the

amplified output beam is also an important consideration. In order to quantify the mode

profile, we examined the spatial intensity distribution using the all-reflective setup shown in

Figure 3.8. These measurements are particularly important for our amplifier, because the

presence of the SLM in the pulse stretcher can compromise the spatial profile owning to the

spatio-temporal effects discussed in Section 2.7, Chapter 2. When the amplifier runs with

SLM off the mode quality is Gaussian at energies up to 1.1 mJ (Figure 3.9a) after which

it starts to deteriorate (Figure 3.9b) mainly due to the nonlinear lensing in the amplifier

crystal and interaction with ASE-suppressing pinholes. As a result of lensing, the beam

waist shifts from the plane of the spatial filter and if energy is increased further, damage to

the pinholes occurs with plasma production on the edges. We observe similar effects when

the SLM is turned on and zero phase is loaded, Figure 3.10, however mode degradation



/ 6


740 760 780 800 820 840 860 880
Wavelength (nm)

Figure 3.6: FROG recovered intensity (blue curve) and phase (red curve) of the amplified
pulse, Figure 3.5. FROG error was 0.002 on 256 x 256 grid.

occurs at lower energy consistent with the data on spectral distortions described in the

next section.

3.3 Amplified Pulse Shaping

Pulse shaping of amplified millijoule pulses was done by activating the SLM in the

pulse stretcher. In Figure 3.11, we compare two single-shot SHG FROG traces obtained

(a) with SLM having each pixel at zero volts (which is equivalent to SLM switched off)

and (b) with SLM biased with pixel dependent voltage corresponding to zero phase (which

is nearly equivalent to biasing each pixel with the same voltage near 1600 mV). It can be

seen that in the second case a wing structure results at large delays from the main intensity

peak which is the result of the pixelated nature of the mask. The wings can also be seen

on the recovered intensity profile, Figure 3.11c at a 10-2 10-3 level. This structure can

be seen on all FROG traces and auto- and cross-correlation data presented here and in the

literature and presents a definite problem for some applications. Any improvement in this

regard seems to be possible only through a trade off with the pulse shaping resolution and

can be achieved by increasing the focal spot size of individual frequency components in the



5 4

S- *2
0 ,

-200 -150 -100 -50 0 50 100 150 200

Figure 3.7: FROG recovered temporal intensity (solid curve) and phase (circles) of the
amplified pulse, Figure 3.5. FROG error: 0.002 on 256 x 256 grid. Recovered intensity
FWHM is 32 fs

Fourier plane. Alternatively, the SLM could be moved away from the Fourier plane where

beam spots are larger or input beam could be made diverging or converging to shift the

focal plane axially from the plane of the SLM.

The pixelation of the mask has another unfortunate effect. When SLM is ol ,I i;1 :-.

there is an abrupt index change at pixel-gap interface which is a source of diffraction. A

substantial amount of light power can get scattered away on these interfaces. Accompa-

nying this loss is spectral amplitude modulation. The modulation depth again depends on

the spot size at the masking plane and can be near 100%. This modulation of the spec-

trum leads to replica pulses and presents certain problems [48] in regular pulse shaping.

However in our case the shaped pulse is also temporally stretched at the same time before

amplification. Since linear stretching maps the spectrum onto time, the stretched pulse will

become modulated in time as shown in Figure 3.12a, red curve. Severe modulation of

the stretched pulse can lead to nonlinear distortions during amplification simply because

the width of each peak riding on the main stretched pulse is small and therefore has high

peak intensity. In our -, -.I ii. the threshold is reached when amplified pulse energy exceeds

~ 0.7 mJ. This threshold places an upper limit on the achievable output energies from

M3 M2

ND 3.0
2 wedge



S2 wedge

Figure 3.8: Optical setup for measuring beam intensity profile of the amplifier output.
Front surface reflections off two quartz wedges and one transmission through ND3 reflective
filter are used for attenuation. Separation between mirror M3 and CCD is about 30 cm to
eliminate diffraction rings produced by dust particles on the optical surfaces.

our ,- in The problem can be alleviated at the expense of pulseshaping resolution as

mentioned above.

Initial characterization of amplified shaped waveforms was performed with the scan-

ning and single-shot autocorrelation technique. For example, Figure 3.13 shows an odd

pulse (Or pulse) generated by application of an abrupt rt phase shift at the midpoint of the

pulse frequency spectrum (circles). For comparison, a numerical autocorrelation trace of

an odd pulse is also shown (solid curve) where residual spectral phase due to dispersion in

the amplifier was taken into account. Single-shot autocorrelation of a 10 THz repetition

rate pulse train is shown in Figure 3.14, which was generated with the binary length 3

M-sequence phase applied to the SLM. The fidelity of the trace is high, however noticeable

wing structure is present at large delays.

Further and more detailed measurements of the amplified shaped pulses were per-

formed using a single-shot Second Harmonic Generation Frequency Resolved Gating tech-

nique [131]. For all of the experiments we used a custom-built FROG setup with

100 ptm thick KDP crystal (see appendix B).

8 2

Figure 3.9: Spatial intensity beam profile for (a) 1.1 mJ and (b) 1.3 mJ output pulse energies
with SLM off. Axes are vertical and horizontal dimensions scaled in millimeters.

It is known that FROG works well on pulses with smooth phase variations. However,

still under debate is the capability of FROG technique to recover complex phase profiles

like the ones produced by pulse shaping. Thus far, there have been no detailed studies

done on the FROG characterization of SLM-shaped pulses on near 10-fs time scale (this

being even more true for the amplified shaped waveforms), although both techniques are

well developed. We address this problem in the rest of this chapter.

An example single-shot experimental and recovered FROG trace of the amplified

odd pulse are shown in Figure 3.15a and 3.15b. We obtained a number of similar traces

for different amplitudes of the phase jump at 801 nm, near the center of the spectrum.

Figure 3.16 summarizes the recovery results for one set of data. Each panel shows recovered

intensity spectrum (solid blue curve), the independently measured fundamental spectrum

(solid red curve), and the spectral phase (blue circles). The topmost panel shows the

unshaped pulse and the amplitude of the phase jump is increased in steps of r/5 from top

to bottom as indicated in the figure. Several features are apparent: The phase jumps are

clearly seen at the exact expected position. The amplitude of the jump is seen to increase

as expected, however the exact phase jump value differs slightly from the one applied to

the SLM. The big discrepancy is that each phase jump is accompanied by an increasing dip


8 2

a) r b)
8 8
6 6
4 4 4
6 6
8 2 8 2

Figure 3.10: Spatial intensity beam profile for (a) 0.5 mJ and (b) 1 mJ output pulse energies
with SLM on and zero phase loaded. Axes are vertical and horizontal dimensions scaled in

in the intensity spectrum which is not observed on the independent fundamental spectrum

measurement. We do not yet have complete understanding of this feature. The recovered

spectra look more trustworthy for the following reason: The phase jump was applied in such

a way that longer wavelength components propagate through pixels at zero volts whereas

short wavelength part of the spectrum is at specific voltage determined by the phase and

therefore experience amplitude scattering due to pixel-gap interface. Indeed, it can be seen

that the left part of the recovered spectrum has lower magnitude. Interestingly, when the

phase jump exceeds 7r the recovered phase is folded down by 2r, that is

Recovered = |27 AaOpplied| (3.1)

For example, the amplitude of the phase jump on the last panel of Figure 3.16 is closer to

6.28 3.8 = 2.48 rather than 3.8. Finally, we observe the inherent phase reversal ambiguity

of the SHG FROG: phase profiles on the fifth and sixth panels are flipped around horizontal


Similar results are obtained when the sides of the phase step are reversed, Figure 3.17.

Here, the longer-wavelength part of the spectrum propagates through SLM pixels which

300 300

200 a) b) 200

100 100

0 0

-100 -100

-200 -200

-300 -300
380 390 400 410 420 430 390 400 410 420 430
Wavelength (nm) Wavelength (nm)

C) SLM off
4* SLM zero phase

0 -

-200 -100 0 100 200

Time (fs)

Figure 3.11: SLM turning on effects: (a) SLM off, (b) zero phase loaded into SLM, logarith-
mic intensity scale spanning three orders of magnitude; (c) recovered intensity (log scale)
and phase for above two cases. Frog error: 0.002.

-15 -10 -5 0 5 10
Time (ps)

15 20 25

760 800 840 880 920

760 800 840 880 920
Wavelength (nm)

Figure 3.12: (a) Stretched pulse intensity with SLM off (blue curve) and SLM at zero phase
(red curve), inset: details of intensity modulation of the stretched pulse when SLM is on;
(b) amplified pulse spectrum below threshold of nonlinear distortions and (c) above the

-200 -100 0 100 200
Time (fs)

Figure 3.13: Autocorrelation of amplified odd pulse (circles) and theoretical trace (solid
curve) based on the pulse spectrum and residual spectral phase.

are biased and therefore experiences -I Irlin-:_. observed again on the recovered intensity

spectrum. Finally, we must mention that in time, the recovered intensity profiles behave as

expected, showing double-peaked structure with phase jump size-dependent relative peak


We next examine the potential spectral resolution of our measurement technique by

applying fixed-size phase jumps positioned at different neighboring wavelengths, Figure 3.18.

It can be seen that phase jump positions are nicely resolved although the step is not as sharp

as one would expect, however, recall that the spectral filter M(w) is a convolution of the

spatial mask M(x) and the focused beam spot at the masking plane, Equation (2.19),

Chapter 2. Therefore, the recovered phase jump steepness is in fact not expected to be


The jump edge at pixel 85 (black curve in Figure 3.18) is shifted slightly towards

longer wavelength from its expected position. This is because the neighboring pixel 86 was

not operational as was established postfactum. With this in mind, the separation between

-600 -400 -200 0 200 400 600

Time (fs)

Figure 3.14: Single-shot autocorrelation of 10 THz amplified pulse train generated with
binary M3 phase mask.


S 420

a 410


390 -

-200 0 200 400 -200
Delay (fs)

Figure 3.15: Experimental single-shot
recovered trace (b). Intensity scaling is

200 400

Delay (fs)

FROG trace of the amplified odd pulse (a) and
logarithmic with three orders of magnitude range.

- Recovered Intensity
- Measured Intensity

Recoverd phase

A i
.I y \ -\.l 4 /


760 780 800 820 840 760 780 800 820 840 860
Wavelength (nm) Wavelength (nm)

Figure 3.16: FROG reconstruction results for variable amplitude phase jumps (see text for
details). Typical error is 0.0045 on 256 x 256 grid.

-- Recovered Intensity
-- Measured Intensity

Recoverd phase

+^ -- -- '.- --*- y- \ \ '. ."' "-"

SA = 37i/5 1 .9_ *.*'.. A = 61/5 = 3.8

760 780 800 820 840 760 780 800 820 840 860
Wavelength (nm) Wavelength (nm)

Figure 3.17: Same as Figure 3.16, but sides of the phase step are reversed: longer-wavelength
part of the spectrum propagates through voltage biased SLM pixels.


A\ = 4/3n Pixel 85
_^ Pixel 90
\ Pixel 95

760 780 800 820 840 860
Wavelength (nm)

Figure 3.18: FROG recovered spectral phase of SLM-shaped amplified pulse with phase
jumps of 4/3r at pixels 85, 90 and 95. Phase jump positions are nicely resolved.

neighboring jump edge pairs is measured to be 6.5 nm so the derived pixel pitch is 1.3 nm,

exactly as expected based on separate measurements of spatial wavelength dispersion.

It is interesting to note once again that although the loaded phase jump amplitude

was 4/3r, the recovered amplitude is only 2/37 demonstrating once again the phase 2r-

folding, Equation (3.1), by the FROG algorithm.

The ability to generate millijoule high-repetition rate pulse trains is of major impor-

tance (see Chapter 5). Example FROG traces of binary phase shaped amplified trains are

shown in Figure 3.19. Only M3 sequences were attempted because of the narrow available

delay range of our single-shot FROG apparatus (Appendix B). The phase mask applied to

the SLM consisted of sequential repetitions of pattern P = {A0/2, -AO/2, -Ap/2} which

consisted of total of 24 SLM pixels with step size A0 varying from 1/37 to 2r in steps of

1/3r. On these FROG spectrograms observe interesting spectral structure of the peaks on

the autocorrelation (taken by mentally integrating the trace in the horizontal dimension).

Recovered temporal profiles and spectral phases of M3 pulse trains are shown in Figure 3.20.

The fidelity of the trains on the left pane is quite good. We believe that 14 THz repetition

rate is the highest achieved so far with millijoule amplified shaped pulses. Note that the

multiple pulse structure disappears completely for A0 = 2r as expected. As in the case of

a single phase jump before, each phase jump is accompanied by the dip in the recovered

300 300

200 a b) 200

100 100

0 0

-100 -100

-200 -200



^ 100


4 -100










300 300

200 e200

100 100

0 0

-100 -100

-200 -200

-300 -300
380 390 400 410 420 430380 390 400 410 420 430
Wavelength (nm)

Figure 3.19: Experimental single-shot FROG traces of amplified 14 THz M3 pulse trains
with variable size of phase steps: (a) 1/37, (b) 2/37, (c) 7, (d) 4/37, (e) 5/37, (a) 27.
Intensity scaling is linear spanning three orders of magnitude from blue to white.

C) []

^^^^^H i^^^^^^0




-300 -150

0 150 300 780 800 820 840 860
0 150 300 780 800 820 840 860

Time (fs)

Wavelength (nm)

Figure 3.20: Recovered temporal intensity (left pane) and spectral intensity and phase (right
pane) of amplified 14 THz M3 pulse trains with variable Ap.


amplitude spectrum as can be seen on the right pane of Figure 3.20. The sizes of the phase

steps are close to expected, and 27-folding is also observed.

\ Recovered
Sa) \ ----Loaded C)

--A------ - --- 0

b) .) A

760 780 800 820 840 860 -300 -200 -100 0 100 200 300
Wavelength (nm) Time (fs)

Figure 3.21: Recovered spectral intensity and phase of Amplified shaped pulse with Gaus-
sian bump written on the phase of height r/2 (a) and 7r (b). Green dashed curves are
independently measured fundamental spectra in both cases. Figures (c) and (d) are corre-
sponding temporal intensity profiles.

Even more involved is the following example of applying Gaussian bumps to the

amplified pulse spectral phase. The FROG recovery results are shown in Figure 3.21. In

case of Figure 3.21(a), the modulation height was r/2 (dashed blue curve). The recovered

phase does show a structure where the Gaussian perturbation is expected with the right

height but slightly larger width. Again, the recovered intensity spectrum shows two dips

coincident with the edges of the bump. A similar situation exists for the bump of size 7r

which is recovered rather well in position, height and width. Figures 3.21(c) and (d) display

recovered intensity in time; as expected, structure in the second case is more pronounced.

Finally, we have performed a series of experiments on amplification of truly complex

waveforms generated with the simulated annealing algorithm. In most of the cases, the re-

Nonshaped 30 fs pulse Triple pulse
300 300
200 200
100 100

0 0
S-100 -100
-200 -200

-300 -300
380 390 400 410 420 430 380 390 400 410 420 430
Unequally spaced triple pulse Unequally spaced, variable width
300 300
200 200
100 100

0 0
S-100 -100
-200 -200

-300 -300
380 390 400 410 420 430 380 390 400 410 420 430
Wavelength (nm) Wavelength (nm)

Figure 3.22: Experimental FROG traces of complex shaped amplified waveforms for which
phase masks were created with the Simulated Annealing algorithm. Traces (c) and (d) are
clipped in delay by the spectrometer CCD matrix vertical size and can not be recovered
with confidence.

sultant waveforms3 extend further than 700 fs delay window available for FROG single-shot

measurements. Examples are shown in Figure 3.22 where FROG traces of triple pulse (b),

unequally spaced triple pulse (c), and unequally spaced variable width triple pulse (c) are

displayed. Since traces (b), (c), and (d) can not be recovered by FROG algorithm, cross-

correlations of these waveforms were measured, Figure 3.23.

Another important application of the pulse shaping capability of our amplifier --,- i,,

is in phase compensation of the amplified pulse. We describe this in great detail in the

following chapter.

From the results presented here the following conclusions can be drawn:

Generation of shaped waveforms at 1 mJ energy level is possible as demonstrated above

but achieving even higher pulse energies will require using a nonpixelated mask. This

3Autocorrelations, strictly speaking

-100 0 100 200

-600 -400 -200 0 200 400 600

-600 -400 -200 0 200 400 600
Time (fs)

Figure 3.23: Experimental cross-correlations of complex
the Simulated Annealing algorithm: (a) triple pulse, (b)
(c) unequally spaced variable width triple pulse.

shaped waveforms produced with
unequally spaced triple pulse and

-300 -200


is dictated by the severe intensity spectrum modulation during pulse shaping observed

in our experiments.

* Incorporation of the SLM into the pulse stretcher has many advantages and is demon-

strated to work very well, however the amount of stretching is limited by the size of

the SLM widow in spectral direction. SLMs with much wider windows are desired.

* Applying any phase to the SLM (even zero phase) leads to the noticeable wing struc-

ture of the pulse at two-three orders down from the main peak. This can be a problem

for a number of applications. At the same time, the threshold for nonlinear spectral

and modal distortions is lowered approximately 2 times in our system.

* Saturating the amplifier is beneficial for obtaining the widest spectrum possible, how-

ever this was hardly possible in our system when any phase was applied to the SLM

(see above).

* Pinhole filtering of spatial beam profile on every pass in the amplifier cavity pro-

vides superior ASE suppression. However, damage of the aperture circumference and

subsequent mode structure degradation can occur at high energies.

* FROG characterization of the shaped waveforms was proven adequate, however, a

number of features were not completely resolved, such as the dips in the recovered

intensity spectra that accompany any abrupt phase jump. It was verified that the am-

plified output beam carried some spatio-spectral dependence which could be important

in single-shot FROG techniques and could lead to observed discrepancies. Moreover,

spatiotemporal coupling inherent to any type of Fourier domain pulse shaping (Sec-

tion 2.7) may also p1 i-, an important role.


4.1 Adaptive Feedback Control: Overview

Femtosecond pulse sources together with now sophisticated pulse shaping techniques

opened the way to controlling the dynamics and achievable states of quantum atomic,

molecular or more complex systems. A typical coherent control experiment has the goal of

driving the quantum --, -1. in into a particular final state and usually consists of a purely

theoretical computation of the required excitation field based on the Hamiltonian of the

-,I -w. i1 [46]. Next, major efforts are spent in the laboratory in trying to synthesize the

required field (if at all possible) with any of the existing pulse shaping techniques. The

actual experiment is then performed with the results possibly -,,:.:.. -.i_:_. a way to improve

the computation, Figure 4.1a.

There is a number of problems in this approach which were recognized by the pioneers

of the experimental adaptive learning [51]:

Much theoretical work ... has gone into the design of laser sequences that
can drive a reaction into a desired, thermally inaccessible state, but successful
experimental implementation of these ideas has been an elusive goal. A ma-
jor stumbling block is the complicated nature of molecular Hamiltonians which
typically have many degrees of freedom tightly coupled together, all of which
may have to be simultaneously controlled. It has been shown theoretically, un-
der appropriate conditions, that molecules can be controlled, i.e., fields can be
designed to drive them into desired final states... However, translating these
results from theory to experiment has not been possible until now. An underly-
ing problem is that the methods for designing fields requires full knowledge of
the molecular Hamiltonian which is known only approximately for systems with
more than two or three atoms. In addition, laboratory uncertainties can arise
due to optical pulse generation errors of various types. Fields designed theoret-
ically on the basis of an appropriate Hamiltonian may not be sufficiently robust
to tolerate errors arising from the Hamiltonian as well as laboratory introduced

R. S. Judson and H. Rabitz, Physical Review Letters, 68(10), page 1500.

a) b)

O/ Smart computer

'I -

Typical laser

Sensitive V --'
Enlightened detector
theorist Experimentalist

Figure 4.1: Coherent control experiments of the past, (a). Adaptive learning experiments
of the future, (b).

The authors of the above quotation therefore .:_ :_ -1. ,1 to leapfrog these difficulties
by using the actual laboratory experiment for -. Ii\ 11,' the underlying- --1. i dynamics
equations under the real-life conditions, Figure 4.1b. In this approach, all that is required
is essentially a feedback signal which strongly correlates with the goal, therefore, indicat-
ing the level of achievement of that goal. The computer optimization algorithm monitors
this signal and provides stimuli for programmable pulse shaping apparatus in the effort to
enhance the excitation field and increase (or decrease) the feedback signal. The procedure
therefore is iterative and provided that the choice of the algorithm and its configuration are
optimally done, leads to the most optimal laser field profile under all existing experimental
uncertainties and noise.
This idea is simple yet powerful and nearly any experimental optimization problem
can be addressed this way in many different fields of science. Its robustness has already
been demonstrated in a number of experiments, and we can expect more developments in
the future.

Adaptive feedback can be used, for example, to make better lasers as was first demon-

strated by Yelin et al. [157]. In this work, pulses were used directly from the Ti:sapphire

laser adjusted for wide spectrum operation. In this regime, initial pulse duration was 80 fs

due to severe phase distortions. The SHG feedback signal, which is intensity and, therefore,

spectral phase-dependent, was fed into a computer running simulated annealing algorithm.

Each pixel of 128-element LC SLM was independently controlled and convergence from

80 fs down to 14 fs was achieved in only 1000 iterations. Remarkably, nearly transform

limited pulses were obtained without prior knowledge of the input pulse parameters by the

algorithm. Analogous results were obtained by Baumert et al. [57], however input pulses

in this work were intentionally predistorted by propagation through 150 mm of SF10 glass.

An evolutionary algorithm monitored the SHG signal from focusing the output pulses from

the LC SLM-based pulse shaper into a nonlinear crystal. No phase-voltage calibration was

needed since individual pixel voltages were used directly in the optimization. Adaptive

compression from 1.2 ps down to 88 fs was demonstrated.

Adaptive phase control of pulses from a C'!i i. I Pulse Amplifier was first performed

by Efimov et al. [78, 158, 52, 53] and is detailed in this chapter. Independently, similar

experiments were carried out by Brixner et al. [56]. In this work the separate pulse shaper

was used behind the kilohertz regenerative amplifier. This placement decision was justi-

fiably made fearing the possible amplifier component damage due to spectral amplitude

modulation caused by the SLM (see Section 3.3). However, even though cylindrical, instead

of spherical, focusing was used in the pulse shaper, output energy was limited to 200 pJ by

the damage threshold of the SLM. The authors demonstrated automatic pulse compression

from 195 to 103 fs. Remarkable results were obtained by Zeek et al. [55] using precisely

our previous idea of Spectral Adaptive Optics (see Section 2.6 and reference [75]) for phase

compensation in chirped pulse amplifiers. In this work, a deformable-mirror pulse shaper

positioned between stretcher and amplifier [118] and controlled by evolutionary algorithm

was used to adaptively optimize millijoule pulses down to 15 fs transform limit. Using

SHG signal as a feedback, exceptional intensity contrast of near four orders, unlike that

produced with SLM-based pulse shapers, was demonstrated with the use of SHG FROG

diagnostics. Adaptive pulse compression at the output of a tunable Optical Parametric

Amplifier (OPA), but at a fixed wavelength, was demonstrated by Zeidler et al. [60]. The

authors used LC SLM-based pulse shaper in a standard 4f setup, however, the phase was

parameterized by Taylor series, Equation (2.2), to include 2nd and 3rd orders. Mostly

quadratic phase compression from over 400 fs to 16 fs was shown. Apart from temporal

phase correction, feedback loops were demonstrated for spatial wavefront correction of the

amplified femtosecond pulses [159]. Here, a deformable mirror was used to introduce small

changes to spatial beam phase which was monitored with Shack-Hartman detector. As a

result, laser beam brightness was improved about 5 times.

A natural way of extending adaptive phase compensation is adaptive pulse shaping

and this step was first taken by Meshulach et al. [54]. In this work a more sophisticated

feedback was required since it was not just the peak intensity that was optimized, but the

whole temporal intensity profile needed to be as close to the target as possible. Authors

used, therefore, a rapid-scanning SHG cross-correlation technique to digitally acquire the

whole trace and use it in evaluating the cost function minimizing the intensity differences

between the experimental and the target cross-correlation traces at a number of fixed points.

A simulated annealing algorithm was used to control the LC SLM in a standard 4f pulse

shaper. Multipulse and quasi-square waveforms were adaptively sculptured with ~ 30 fs

resolution. Some simple adaptive pulse shaping was attempted in [160] where the intensity

profile recovered with spectral interferometry was used as a feedback to be compared with

the target in a simple cost function.

The first application of feedback approach to controlling complex p1!-, l 1 systems

with tailored femtosecond pulses was reported by Bardeen et al. [161] in 1997. In this work,

an AOM-based pulses shaper was used to control the phase of amplified pulses of near 50 fs

duration. The shape of the RF waveform applied to the AO transducer was controlled

through parameterization of its central time, width, and chirp, so no calibration of the de-

vice was used. Shaped pulses were focused into the solution of laser dye molecule IR125 in

methanol. Optimization of fluorescence following electronic excitation to the upper level was

sought adaptively with help of the Genetic Algorithm (GA). It was shown that depending

on the exact optimization parameter GA converges either to the spectrally narrowed optical

pulse with low energy and centered near the absorption maximum of IR125 (fluorescence

efficiency optimized), or to the strongly positively chirped, wide bandwidth pulse (fluo-

rescence effectiveness optimized) as was expected based on the dynamically Stokes-shifted

population created in the experiment.

Adaptive feedback in femtochemistry was recently employed by G. Gerber's group for

optimization of different product branching ratios in dissociation reaction of iron pentacar-

bonyl [45, 58]. The feedback signal for the evolutionary algorithm in this work was derived

from the time-of-flight mass spectrometer measuring the absolute amounts of reaction frag-

ments after laser pulse excitation. A number of dissociation and ionization channels of

Fe(CO)5 occur upon illumination with the femtosecond pulse; maximization and minimiza-

tion of Fe(CO)t/Fe+ ratio was demonstrated using learning loop method. For example,

in one instance, this ratio was as high as 4.8 when maximized and as low as 0.06 when

minimized where the maximization lead to near transform limited excitation pulse, while

minimization yielded double-peaked intensity profile with 500 fs pulse separation.

Feedback approach can be used not only for optimal control as the only goal of the

experiment but also to yield insight into underlying principles of light-matter interaction.

For example, in the work by Hornung, et al. [59] unparameterized phase search with evolu-

tionary algorithm lead to unexpected new phase structures while seeking minimization of

two-photon excitation in Na. At the same time Ii !ii and "dark" pulses were produced

with phase masks with expected symmetry when modest parameterization was inflicted.

The work was done with a 128-pixel LC SLM-based pulse shaper and a commercial CPA

and OPA systems at 1 kHz repetition rate. Fluorescence from 4p level following 3s 5s

two-photon excitation served as a feedback signal for the algorithm. In another work of

this group [61], optimization of either of the two vibrational modes of K2 in vapor was

demonstrated with use of ~80 fs pulses in a Degenerate Four Wave Mixing arrangement.

One or two interacting beams were derived at the output of both amplitude and phase LC

SLM-based pulse shaper. The authors raise an important issue about the value of different

parameterizations in frequency and time domain arguing that in general the solutions found

by the algorithm can be quite complex and hide the 1,!-, -i, d1 reason for obtained control.

But with the 1...--1~iil-, to chose between different parameterizations, each representing

a certain control mechanism, the important process can be extracted, other mechanisms


excluded, and the number of independent parameters can be substantially reduced. Using

this approach it was possible to establish the importance of phase (in addition to intensity)

in multiple-pulse optimization of X state through ISRS.

A remarkable example of using feedback optimization without any pulse shapers

was the report on enhancing the X-ray emission from a femtosecond laser-produced solid

plasma [162]. In this experiment, two pulses in the UV spectral region were used: the

plasma forming prepulse followed by the main pulse some few hundred picoseconds later.

There were three parameters under optimization: (i) delay between pulses, (ii) the energy

ratio between two pulses, and (iii) the axial position of the focusing lens. All three param-

eters were controlled with simple motors by either simplex downhill, simulated ,mi, ilh:_.

or evolutionary algorithm. Not surprisingly, the simplex downhill method was shown to

fail completely in the presence of 10% experimental noise unlike the other two, which were

seemingly unaffected by noise. As a result of the optimization, approximately 27-fold in-

crease in X-ray yield was obtained. The final position of all three motors was arrived at

within 5% uncer' hii -,l in a number of subsequent experimental runs.

Femtosecond pulse shaping has proven, however, its usefulness in optimizing the

high harmonic generation in gases in XUV and soft X-ray regions [30, 163]. In these

experiments, femtosecond pulses shaped in a deformable mirror-based pulse shaper were

amplified to millijoule level and focused into a hollow-core capillary filled with inert gas.

By changing the gas pressure inside the capillary optimal phase matching between the pump

and upconverted pulses can be tuned up leading to substantial efficiency increase in X-ray

production [164]. Moreover, 1 '.1-, i:_: temporal shaping to the excitation pulse can further

increase the output of the selected harmonic. Adaptive feedback enhancement of factor of

8-10 was indeed demonstrated, again ahead of the corresponding theoretical development.

In this chapter, we apply the adaptive feedback to the problem of enhancement of

ionization rates in the femtosecond plasma which is created rapidly when a short intense

pulse is focused in a gas at a near-atmospheric pressure. Spectral changes of the pulse

transmitted through plasma region are monitored and the amount of blueshifting is used as

a feedback signal to the learning algorithm. Numerical simulations -:_-:_- -I that the spectral

phase of the excitation pulse strongly correlates to the shape of the transmitted pulse

spectrum and the amount of blueshifting that occurs (see section 4.2 and [26]). Maximal

blueshifting is expected for the transform-limited pulse, which is indeed demonstrated in

our experiments.

4.2 Blueshifting as a Diagnostic of Pulse Phase

In all previous work on adaptive phase compensation done by other groups, SHG

was used as a feedback signal. It must be noted, however, that in all cases the stability

of the pulse source was rather good or multiple-shot averaging was employed to obtain

reasonable signal-to-noise ratios. High-energy systems, on the other hand, usually operate

at low repetition rates where averaging can be time consuming, shot-to-shot fluctuations

can also be as high as 50% if the amplifier operates far from saturation. Therefore, we seek

an alternative feedback signal which would be more sensitive to pulse phase and have a

large dynamic range desirablyy background-free) for the optimization algorithm to lock to.

Note also that the output pulses from our amplifier --, -. i i already have only modest phase

distortions and we are looking to further improve the fidelity of the pulse (enhance wing

structure, etc.) The changes in the peak intensity are therefore expected to be rather small,

possibly smaller than shot-to-shot fluctuations rendering SHG feedback inappropriate. As

a feedback signal we therefore used spectral blueshifting in the air breakdown in the focus

of the amplifier output beam as described below. This choice is justified in our case, since

minor phase errors manifest themselves primarily at the wings of the femtosecond pulse.

The nature of the spectral blue shifting can be understood simply from the temporal

dynamics of the pulse phase due to the rapidly created plasma. The instant carrier frequency

a of the pulse is given by the derivative of the time-dependent phase:

Of wo -L On
a) U o - (t, z) dz (4.1)
ot co C0o

where uo is the unperturbed carrier frequency, L is the interaction length, and n is the

dynamically changing refraction index of the plasma. On the femtosecond time scales, the

atmospheric-density plasma can be considered collisionless, so the refraction index is

n ()= 1 V (2) (4.2)

where the plasma frequency uwp = V/4N (t) e2/m is increasing in time during the pulse

due to the rapid growth of the electron concentration N, caused by the multiphoton ion-

ization. The index of refraction, therefore, decreases leading to the positive frequency shift

in Equation (4.1),

On 2 2e2 ONe 27e2 NNe
T_ (4.3)
t me2 (1 4ie2Ne/meW2) 2 Ot mew2 Ot

It is interesting to compare the sensitivity of the SHG and blueshift signals to the

spectral chirp of the femtosecond pulse. Consider, for example, a linearly-chirped femtosec-

ond pulse with envelope amplitude p (t) written as follows [165]

() exp 2 (4.4)
V1/2 2 V2 02

where po is the peak amplitude, and To is the duration of the unchirped pulse. The chirp

parameter V2 1 + 02/04 measures the change in pulse duration and peak amplitude due

to the quadratic spectral phase term 0 responsible for linear pulse -lr Il I1, i,:_. (Chapter 2)1.

Assuming the perfect phase-matching conditions, the intensity of the SHG pulse I2w is

proportional to the square of the fundamental intensity I p2:

-2w exp -22 (4.5)
V2 V2 0

In experiment, the measured (iintitt, is the integrated intensity, or energy, of the SHG

pulse, so that the SHG signal W2 is the result of integration

pi4 2 2
n4 rp 92 0 n4
W2w exp 2 dt (4.6)
h V = 2 t p
'The amount of chirp /T 1, or V 2 stretches the pulse by I,,' .

This result contains the dependence of the SHG signal on both the chirp parameter V as well

as the amplitude po. Fluctuations of the latter typically comprise the main source of noise

in the experiment, obscuring the signal dependence on phase p. One can see immediately

from Equation (4.6) that the SHG signal is much more sensitive to amplitude fluctuations

than to the change in chirp. Figure 4.2 illustrates the weak dependence of the SHG signal

Amplitude po
1 0.985 0.946 0.894 0.841 0.790 0.745

0.9 SHG
a 0.7
0.6 -



=-i- pp = const, vary chirp, p
3 = const, vary po

0.00 0.25 0.50 0.75 1.00 1.25 1.50

Chirp 0/,02

Figure 4.2: SHG signal as a function of quadratic phase 0 in units of r2 keeping po = const
(squares), and as a function of po, keeping chirp parameter V = const (circles). Respective
points on each curve correspond to the same peak amplitude as shown in the inset.

on the pulse phase as well as the dominance of the amplitude fluctuations. The ideal case of

no amplitude fluctuations (po = const) is shown as squares, where the SHG signal changes

by only :;,' when the chirp 0/T2 is varied from 0 to 1. For comparison, if the amplitude of

the femtosecond pulse is varied (keeping the chirp constant) in the same range as that due

to chirp, the red curve results (circles), illustrating the superior sensitivity of the SHG signal

to amplitude fluctuations. It follows, therefore, that the SHG feedback signal is suitable

only for weakly fluctuating pulse sources and for monitoring chirp values in a large range

of change.

A completely different situation occurs when we consider the spectral blueshifting as

a feedback of the pulse phase. Because of the complexity of the problem we have to consider

only a simple model of the ionization and plasma dynamics. The balance equations for the

electron and neutral atom concentrations N, and No assuming single-stage ionization can

be expressed as

S w (t) No (4.7)
-w (t) No (4.8)

such that the maximum number of electrons created is limited by No (t = -oo). The ion-

ization rate w (t) is a very nonlinear function of electric field amplitude given, for example,

by Keldysh formula

1 2 Ea
w ()2 exp [ ], P < E (4.9)
p (t) 1/ 3 p (t)

where p (t) is the time-dependent electric field amplitude envelope and Ea is the atomic

field which defines the ionization potential of the atom2. Integration of the Equations (4.7)

and (4.8) gives the dynamics of the electron concentration in time:

Ne (1) w (t) No(0) exp w (t') dt' dt" (4.10)

Using Equations (4.2), (4.3), and (4.10), the temporal dynamics of the instantaneous carrier

frequency, Equation (4.1), can now be numerically evaluated3. Figure 4.3 shows the electron

concentration (dashed curves) and frequency (dotted curves) dynamics for the three values

of the chirp parameter 4/r02 = 0, 0.5, and 1.5. In spite of only minor chirping of the input

pulse, the peak shifted frequency can be seen to change by nearly 4 times, demonstrating

the dramatic sensitivity of the blueshift signal to the pulse chirp. This is seen more clearly

when the spectrum of the frequency-shifted pulse is taken, Figure 4.4. The inset shows

the spectra of the frequency-shifted pulse transmitted through plasma for three values of
2In the following numerical modeling we assume modest electric field amplitude po/Ea 0.1.
3The author is grateful to Dr. Vladimir Malinovsky for assistance in numerical modeling of the femtosec-
ond plasma ionization dynamics and spectral blueshifting by linearly chirped pulses.

1.0 1 1
S "-------------------------

---- 2 = 0 4

0.8- o2 = 05


a 0.6 -
(t / IN i \

S0.4 '0

,2' ./ \ Z 1

2 4 6 8 10 12
Time t/k

Figure 4.3: Electron concentration (dashed curves), and instantaneous frequency (dotted
curves) as functions of time for three values of the chirp parameter 0/702. Solid curves show
the field amplitude profiles.

the chirp. Clearly, the maximum blueshifting occurs for the transform-limited pulse (red

curve). The vertical arrow marks the position of the "detector"-far on the blue side of the

spectrum. Just like in Figure 4.2 we plot the blueshift signal as a function of chirp 1 /T

assuming no amplitude fluctuations (squares) and as a function of peak amplitude keeping

the chirp constant (circles) where in both cases the amplitudes of the corresponding points

on the graph are the same. Two observations can be made. First, in contrast to the SHG

case, the sensitivity of the blueshifting to chirp is dramatic: the signal changes by 3 orders

of magnitude when pulse is chirped to ~ Il' of its duration. Second, it can be seen that

the relative position of the ( = const and po = const curves is interchanged compared

to the SHG case, Figure 4.2, confirming the improved immunity of the blueshift signal to

the amplitude fluctuations of the excitation pulse source. This improved immunity can be

easily visualized using Figure 4.3: The amount of blueshifting is proportional to the slope

of Ne(t) curve. Chirping of the pump pulse leads to both vertical shrinking and horizontal
of N (t) curve. Chirping of the pump pulse leads to both vertical shrinking and horizontal

Amplitude po
1 0.985 0.946 0.894 0.841 0.79 0.745

1 --- Po = const, vary chirp, p
S----4 = const, vary po
S0.01 2
1 ~og = 0.5
.3 2
1E-3 -- /To

E 1E-4


1 E-6
1E-7 I
0.00 0.25 0.50 0.75 1.00 1.25 1.50
Chirp /T02

Figure 4.4: Inset: spectra of the blueshifted pulse for three values of the chirp parameter.
The maximal blue shifting corresponds to the transform-limited pulse (red curve). The
arrow marks the position of the "detector". Main graph: blueshift signal as a function of
/702o (po const), squares, and as a function of po, (V const), circles. Respective points
on each curve correspond to the same peak amplitude as in Figure 4.2.

stretching of the Figure, whereas amplitude fluctuations lead only to vertical shrinking. It

is clear, therefore, that change of slope will be larger in the first case.

4.3 Experimental Details

Our adaptive feedback experiments were performed using the multipass amplifier

-b, -. in described in Section 3.2, however at the time of the experiments the uncompensated

pulse duration was slightly longer due to larger amount of cubic phase. We start, depending

on alignment and amplifier regime, with 37-42 fs linearly polarized pulses which are focused

by a 10 cm focal lens off-axis parabolic mirror in air, nitrogen, or argon at the atmospheric

pressure to a spot diameter of approximately 20 ptm and a maximum pulse intensity of

> 1016 W/cm2. The light from the interaction region was collected using a large lens

and sent to a diffraction grating for spectral filtering. To obtain an unambiguous signal,

wavelength components in the range 45010 nm were collected and measured using a photo-

multiplier tube (PMT). In the absence of any phase compensation, no light was detected

in this spectral region. To minimize noise in the data due to pulse-to-pulse fluctuations in

energy, spectrum, and/or temporal duration, a small amount of light was picked off before

focusing and sent to a reference PMT. Because the plasma breakdown is a highly nonlinear

process, we normalized the signal pulse by the square of the reference pulse energy. A

schematic of the experimental apparatus is shown in Figure 4.5.

Figure 4.5: Schematic layout of the phase optimization experiment. The learning loop
consists of Ti:sapphire laser, the pulse stretcher which incorporates a programmable LC
SLM, the amplifier, the compressor, the ionization in a gas cell, the spectral filter and
detector, and the computer-based genetic algorithm which iteratively reprograms the SLM.

During the experiment, the phase is optimized by programming the SLM to provide

a parameterized phase compensation of the form:

S(w) O (woj) + (0) (w wo) + 1(2) (w U)2 +


(3) )3 + (4) ( 4 + NL ( o) (4.11)

where the pulse phase is defined as exp(-i((w)). The goal of the experiment is to deter-

mine the set of coefficients )(i), )NL that produces the most optimal pulse in a sense of

Section 4.2 The first three coefficients, d(i)d(i), are the quadratic, cubic, and quartic

higher-order dispersion terms. Numerical .111 i', -i- of our amplifier leads us to believe that

cubic dispersion makes the dominant contribution to the total dispersion. Nevertheless,

we expect some quartic and nonlinear phase to be present, so we include them as well

as quadratic phase in the optimization. The last term in Equation (4.11), (NL( Wo),

reflects nonlinear phase shifts arising from the propagation of the amplified pulse through

transmissive elements in the amplifier chain [166]. To a good approximation, the large chirp

imposed by the stretcher linearly maps the frequency of the pulse onto time such that the

temporal intensity profile of the stretched pulse closely approximates the frequency inten-

sity profile, Istretched(t) -I(au)), where a is the quadratic chirp parameter. Thus, the self

phase modulation resulting from the amplification takes the form:

SNL(t) = I(a~ ) (4.12)

where n2 is the nonlinear index of refraction and L is propagation length. Here, we assume

a Gaussian dependence for the frequency spectrum,

] (4.13)
coNL( Uo) ONL,max exp r(a)2 (4.13)

validated by our experimental measurements of amplified pulse spectra (Figure 3.4) and

parameterize the constants in (4.12) by a single phase 'NL,max4

The optimization was performed using GENESIS, a genetic algorithm (GA)5. The

GA is a global optimization method that mimics the paradigms of biological evolution to

efficiently search a large dimensional parameter space. The optimization begins with a

random sample population of individuals (trials) comprised of the phase coefficients to be
4Further referred to as )NL for compactness
5The code is available at


optimized, in this case the set {d2 /dW2, d3 /dw3, d4 /dw4, 4NL} or a subset thereof.

Each coefficient is parameterized to 1024 possible values. In the GA, each parameter is

represented by a ,. in ," an N-bit binary string that uniquely specifies its value, Figure 4.6.

Each individual (chromosome) in our optimization is composed of 4 genes. A population


.d2. d I3 d4_ .. Crossover Mutation
dW2 do3 do)4

I -- I i
0 1 1 0 11 1 1 i 1 -

Figure 4.6: Basic operation of the genetic algorithm. The phase coefficients are parameter-
ized in N-bit strings and "evaluated" in the experiment. The algorithm generates a new
set of individuals using the crossover and mutation operations. The evolution of the phase
is continued in this way until an optimal solution is reached.
of 50 randomly chosen individuals is selected to begin the optimization. After evaluating
each individual in the initial population in the test function in our case, by measuring

the signal from the blueshifting experiment), individuals are chosen to propagate to the
1 0 0 0 0 0 0 0 0 0
S 0 0 1 1 1 1 1 1

next generation by a random selection process such that the expected number of times an
ZZZZZZZZZZZZZZ ------ ---------------------------------------------

Figure4.6: Bindividual appears in the next genetion is proportional to the coefficistrength, or fitness, of that
individual. A large fraction of individuals in the experiment. The algorithm generation is s a another
set of individual in the population. The crossover and mutation operation takes a subsetvolution of N-m bits from one
is continued in this way until an optimal solution is reached.

parent 50 rand m bits from the second parent to form one offspring. The remaining m bits fromevaluating
each individual in the initial population in the test function (in our case, by measuring

the signal from the blueshifting experiment), individuals are chosen to propagate to the

next generation by a random selection process such that the expected number of times an

individual appears in the next generation is proportional to the strength, or fitness, of that

individual. A large fraction of individuals in the new generation is !-- .I' with another

individual in the population. The crossover operation takes a subset of N-m bits from one

parent and m bits from the second parent to form one offspring. The remaining m bits from

parent 1 and N-m bits from parent 2 are combined to form a second offspring. Because

particularly strong genes will be replicated many times in a subsequent generation, they may

survive intact. A small fraction of the genes are mutated, i.e., a bit is flipped randomly as the

propagation occurs. Mutation insures genetic diversity and guards against a particularly

fit individual dominating the entire population. It also prevents the optimization from

becoming trapped in a local minimum.

4.4 Optimization Results

In the first experiment, the laser -,-1.1 1i was programmed to optimize the full

set {d24/dWa2, d3 /dw3, d4 l/da4, 4NL} of the parameters in the range (1,000 fs2,

50,000 fs3, 1,000,000 fs4, 5 rad). The entire parameter space corresponds to approx-

imately 1012 possible phase combinations. For each trial, four laser shots were averaged,

with the GA seeking to maximize the amount of light spectrally shifted into the 440-460

nm wavelength band. The optimization proceeded until either 1000 trials had taken place

(corresponding to 20 generations) or the GA determined that an optimal solution had been

achieved. Each run took up to 30 minutes.

Figure 4.7 displays how the blueshift signal evolves as the experiment progresses.

In this case, the achievement measures an increase in the blueshift signal. The squares

denote the best single result in a particular generation, i.e. the largest blueshift signal

corresponding to a specific set of phase coefficients. The triangles represent the average

value of the blueshift signal for all of the trials in that generation. The average achievement

monitors the rate of convergence of the optimization, increasing as the -ii i,:, individuals

in each trial are subsequently propagated to the successive generation. In generation 0 of

the experiment, one of the randomly selected individuals has increased the blueshift to a

non-zero level6. As the experiment progresses through 5 generations, the best blueshift

signal rapidly increases in magnitude to 8 times its initial value. After the 5th generation,

the blueshift increases at a slower rate with pronounced fluctuations from generation to

generation. The average value also shows the same trend. After 1000 trials, the experiment

is terminated by the GA, although the average achievement has not yet reached a plateau.

Figure 4.8 displays the evolution of the phase coefficients during the experiment.

The points in each graph represent the average value of a particular phase coefficient in
6No blueshift signal is observed in the absence of any phase compensation.