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Space-time coupling in femtosecond pulse shaping and its effects onSpace-time coupling in femtosecond pulse shaping and its effects on

coherent controlcoherent control

F. Frei, A. Galler, and T. Feurer

Citation: J. Chem. Phys. 130130, 034302 (2009); doi: 10.1063/1.3058478

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Space-time coupling in femtosecond pulse shaping and its effects

on coherent control

F. Frei, A. Galler,a?and T. Feurer

Institute of Applied Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland

?Received 14 July 2008; accepted 1 December 2008; published online 16 January 2009?

We present a Fourier optical analysis of a typical femtosecond pulse shaping apparatus and derive

analytic expressions for the space-time dependence of the emerging waveform after the pulse shaper

and in the focal volume of an additional focusing element. For both geometries the results are

verified experimentally. Hereafter, we analyze the influence of space-time coupling on nonlinear

processes, specifically second harmonic generation, resonant interaction with an atomic three-level

system, and resonant excitation of a diatomic molecule. © 2009 American Institute of Physics.

?DOI: 10.1063/1.3058478?

I. INTRODUCTION

Femtosecond pulse shaping has found many applications

in fundamental as well as in applied sciences1and had a

particular large influence on coherent control of quantum

systems.2Here, the electric field of a spectrally broadband

laser pulse is modulated in phase or amplitude or both so that

the tailored pulse steers the quantum system from an initial

state to a designated final state. To achieve the best possible

control efficiency requires maximal flexibility in designing

pulsed electric fields and present-day femtosecond pulse

shaping techniques draw near that goal.

In order to realize such tailored waveforms most experi-

ments have been using a spatial light modulator ?SLM? in the

symmetry plane of a 4f zero-dispersion compressor. It is

long known that such an optical arrangement does not only

modulate the temporal shape of a short laser pulse but also

affects its transverse spatial distribution. A detailed theoreti-

cal analysis of the so-called space-time coupling has been

published on the basis of Fourier optics3,4and by using

space-time Wigner distribution functions.5Very recently

Sussman et al.6published a detailed analysis of space-time

coupling, calculating the field distribution after a pulse

shaper in various geometries. Moreover, they discuss the in-

fluence of space-time coupling on nonlinear light matter in-

teraction in very general terms. To the present day there ex-

ists only a marginal number of experimental investigations.

Tanabe et al.7analyzed the space-time coupling after a per-

fectly aligned and a slightly misaligned pulse shaper by

spatial-spectral interferometry. They were able to reconstruct

the spatiotemporal amplitude and phase distribution from the

measured two-dimensional fringe patterns. In a later publica-

tion they have examined the spatial intensity distribution of a

focused shaped pulse through ablation experiments.8Other

short pulse diagnostic methods have been published, which

are able to measure the space-time distribution of laser

pulses, although they have not been specifically used to ana-

lyze space-time coupling in pulse shaping. They include

variations of the SPIDER technique,9,10a three-dimensional

version of second harmonic based frequency-resolved optical

gating,11

orsingle-shot

gating.12,13

Despite all the knowledge on space-time coupling in

femtosecond pulse shaping most of the theoretical work in

coherent control assumes a perfectly modulated pulse, whose

electric field depends on time ?or frequency? only. Some ex-

perimental demonstrations seem to agree well with those the-

oretical predictions, suggesting that the influence of space-

time coupling is not overwhelmingly important in certain

cases. One would expect that its impact is most severe in

nonlinear light-matter interactions as it is also suggested by

the analysis in Ref. 6. Here, we try to answer the question

why the effects of space-time coupling are hardly seen in

certain types of quantum control experiments while having a

considerable influence in others. The paper is organized as

follows. We begin with a comprehensive summary of the

Fourier optical description of pulse shaping and present ana-

lytic results for three different modulations which will be

used throughout the remainder of the paper. We then extend

the Fourier optical treatment and incorporate an additional

focusing element. This is motivated by the fact that almost

all experiments in coherent control are operated in such a

geometry. Next, we present experimental results; they verify

the Fourier optical calculations in both geometries. After

that, we proceed with investigating the influence of space-

time coupling on three different nonlinear effects. For a rela-

tively simple nonlinear interaction, namely, second harmonic

generation, we compare experimental results with simula-

tions. Based on simulations only, we then evaluate the effects

of space-time coupling on the interaction of laser pulses with

a resonant atomic three-level system and a resonant diatomic

molecule.

frequency-resolvedoptical

II. FOURIER OPTICAL ANALYSIS

In a typical coherent control experiment, close to

bandwidth-limited pulses from an oscillator, an amplifier, or

a parametric amplifier are sent through a pulse shaper before

interacting with the quantum system of interest. Only few

a?Electronic mail: andreas.galler@iap.unibe.ch.

THE JOURNAL OF CHEMICAL PHYSICS 130, 034302 ?2009?

0021-9606/2009/130?3?/034302/14/$25.00© 2009 American Institute of Physics

130, 034302-1

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experiments use the pulses exiting the pulse shaper directly,

mostly they are focused to a small volume for various rea-

sons, for example, to increase the intensity. In the following

we summarize the Fourier optical description of a typical

pulse shaper, i.e., a pixelated SLM in the symmetry plane of

a 4f zero-dispersion compressor, without and with an addi-

tional focusing element. A schematic diagram together with

the coordinate system and the distances used throughout the

remainder of this paper is shown in Fig. 1.

It is sufficient to analyze the setup in a single transverse

dimension x only because the y component of the light field

remains unaffected by the pulse shaper. The incident laser

pulse is described through its slowly varying complex enve-

lope Ei?x,t? in time or E˜i?x,?? in frequency domain, with the

relative frequency ?=?−?cand the center frequency ?c.

Throughout the remainder of the analysis we use scalar fields

and the paraxial approximation assuming that the numerical

apertures of all lenses are sufficiently small. The transfer

function of the pulse shaper is derived from the transfer func-

tions of free space propagation, of an ideal lens, and of an

ideal grating with linear dispersion. Free space propagation

over a distance z is described through

E˜o?kx,?? = E˜i?kx,??exp?− ikz + iz

2kkx

2?,

?1?

where kxis the wave vector associated with x. A perfect lens

modifies the laser pulse according to

E˜o?x,?? = E˜i?x,??exp?ik

2fx2?,

?2?

where f is the wavelength independent focal length. Finally,

an ideal grating with linear dispersion affects a pulse through

E˜o?x,?? =?bE˜i?bx,??exp?i??x?,

?3?

with b=cos ?/cos ?c, ?=2?m/??cG cos ?c?, the diffraction

order m, the angle of incidence ?, the diffraction angle ?cat

center frequency ?c, and the grating constant G.14For broad-

band pulses the nonlinear dispersion of a real grating has to

be considered. However, its consequences are mainly seen in

the pulse replica15and may safely be neglected here.

The transfer function of the SLM depends on the actual

device used. Ideally, it allows for an arbitrary amplitude and

phase modulation M?x?. In practice, however, many devices

have technical limitations. Especially pixelated SLMs, where

a total of N pixels are separated by ?N−1? gap regions, have

quite complex transfer function,

N/2−1?rect?

+ rect?x − xn+ ?xp/2

?xg

M?x? = ?

n=−N/2

x − xn

?xp− ?xg?Mne−i?n

?Mge−i?g?,

?4?

where xnis the position of the nth pixel and rect?x? is the

rectangle function; ?xp??xg?, Mn?Mg?, and ?n??g? are the

width, the amplitude, and phase modulation applied by the

nth pixel ?gap?. The transfer function complicates even fur-

ther if the SLM introduces phase wraps or if the pixel-to-gap

transitions are not perfectly sharp.15A pixelated SLM has a

number of important consequences in pulse shaping. A prop-

erty of Fourier series, as Eq. ?4?, is that they repeat them-

selves with a period given by the reciprocal of the frequency

increment. Therefore, any waveform from the pulse shaper is

repeated infinitely in time with a period determined by the

frequency increment per pixel. These undesired repetitions of

a shaped waveform are labeled sampling replicas since they

are a direct consequence of the discrete sampling of the

SLM. The time increment between the minus and plus first

order replica pulses is usually called the shaping window Wt.

A. Slowly varying electric field after the pulse

shaper

When no additional focusing element is used, the slowly

varying electric field Esjust after the pulse shaping appara-

tus, i.e., at ?1, is found to be

bfe−4ikcf?

−?

?M˜?−kc

dx?Ei?− x?,t +?

?M˜?−kc

E˜s?x,?? =ikc

?

dx?E˜i?− x?,??

bf?x − x???ei???x−x??/b,

?5?

Es?x,t? =ikc

bfe−4ikcf?

−?

?

b?x − x???

bf?x − x???,

?6?

where M˜?kx? is the spatial Fourier transform of the modula-

tor’s transfer function M?x? and k?k??c??kc. The spatial

position x at the symmetry plane of the 4f compressor de-

pends linearly on the frequency through

x = −f?

kc

? ??x

???,

?7?

assuming that the carrier frequency, i.e., ?=0, passes

through the center of the SLM at x=0. The frequency-to-

space mapping ??/?x depends on the focal length of the

two lenses and the gratings used. To illustrate the properties

ffff

2w0

?f2

f2

?

?

x

z

?1

?2

FIG. 1. ?Color online? Typical pulse shaper in 4f geometry using a pixelated

SLM at the symmetry plane of the 4f setup. In most experiments the

shaper’s output waveform is focused by an additional focusing element to

the sample of interest.

034302-2Frei, Galler, and FeurerJ. Chem. Phys. 130, 034302 ?2009?

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Page 4

of the solution, we investigate three different transfer func-

tions which are frequently used in experiments. First, a linear

phase modulation, i.e., M1?x?=exp?i?x?, which leads to

Es1?x,t? ? Ei?− x +fb

kc

?,t +f?

kc

??.

?8?

The resulting pulse has the same slowly varying enve-

lope as the incident pulse, however, it is shifted both in time

and in space. This combined effect is known as space-time

coupling. The second example is an extension of the first,

namely, M2?x?=?1+exp?i?x??/2 which leads to

Es2?x,t? ? Ei?− x,t? + Ei?− x +fb

kc

?,t +f?

kc

??.

?9?

The resulting waveform consists of two replicas of the

incident pulse, one located at the time origin t=0 and the

second delayed in time. Space-time coupling is apparent; it

has no effect on the t=0 replica but spatially displaces the

time-delayed copy of the original pulse. The spatial displace-

ment increases with time delay and eventually the two rep-

licas no longer overlap in space. Third, we apply a sinusoidal

phase modulation M3?x?=exp?iA sin??x+???, which results

in

Jm?A?eim?Ei?− x + mfb

Es3?x,t? ? ?

m=−?

?

kc

?,t + mf?

kc

??.

?10?

The emerging waveform resembles a train of pulses

where each pulse m is a time-delayed replica of the original

pulse with an amplitude determined by Jm?A?. Again, each

pulse is not only delayed in time but also displaced along the

transverse direction. Note, while the linear and the sinusoidal

phase modulations are pure phase modulations, the double-

pulse transfer function requires a periodic amplitude modu-

lation in addition to a phase modulation. In all three cases we

find that the time delay or the pulse separation ?t is related

to ? through

?t = −f?

kc

?,

?11?

and, similarly, the spatial offset is given by

?x =fb

kc

? = −b

??t.

?12?

Thus, space-time coupling may be quantified by the ratio

?x/?t, which is determined mostly by the grating param-

eters. To illustrate the effect we examine two practical

examples.

First, we evaluate the space-time coupling constant

?x/?t as a function of focal length f for a number of differ-

ent grating constants and assume that the frequency-to-space

mapping ??/?x is fixed to a specific value. That is to say, a

laser spectrum with a given spectral bandwidth is always

dispersed across the same distance, for example, two-thirds

of the SLM’s active area. This boundary condition requires

that the angle of incidence has to be adjusted as the focal

length is varied. Figure 2?a? shows the space-time coupling

constant ?x/?t as a function of focal length f for a number

of different gratings with ??/?x fixed to 6.807 THz/mm.

The space-time coupling constant varies only slightly with

the focal length but strongly depends on the grating used. A

general trend to observe is that the higher the grating disper-

sion the smaller the space-time coupling constant, i.e., in the

limit of infinite dispersion space-time coupling disappears. In

practice the viable grating dispersion is limited by geometri-

cal constraints; either the focal length becomes too short or

the angle of incidence or the diffraction angle approach graz-

ing incidence, as shown in Fig. 2?b?.

Second, we assume that the gratings are used in Littrow

geometry, i.e., ?=?c. Then, the space-time coupling constant

is independent of the focal length f, however, the frequency-

to-space mapping ??/?x changes with the focal length. For

three out of four gratings the space-time coupling constant in

Littrow geometry is indicated by arrows in Fig. 2?a?.

B. Slowly varying electric field in focusing geometry

As stated above, in most experiments the pulse shaper’s

output is focused to a small interaction volume ?2by an

additional focusing element with a focal length of f2. In what

follows, we want to address two questions. First, does the

position of the focusing element influence the electric field

distribution in the focal plane and, second, does the electric

field distribution vary within the Rayleigh length?

We analyze the electric field distribution in the focal

plane for an arbitrary distance ?f2between the shaping setup

and the focusing element. For ?=1 the arrangement is a per-

fect 2f imaging geometry, and for ?→? the focusing ele-

ment moves further and further away from the pulse shaper’s

output. The electric field in the focal plane is found to be

(b)

?

1.0

0.5

0.0

50

0

-50

0100200

300

400

f [mm]

500600700800

?c

? ?

x/ t [mm/ps]

?? ? [deg]

c

300 1/mm

600 1/mm

800 1/mm

1200 1/mm

(a)

FIG. 2. ?Color online? ?a? Space-time coupling ?x/?t as a function of focal

length f given a constant frequency-to-space mapping ??/?x of 6.807

THz/mm; the gratings have 300 lines/mm ?dash-dotted curve?, 600 lines/mm

?dotted curve?, 800 lines/mm ?dashed curve?, and 1200 lines/mm ?solid

curve?, respectively. The arrows indicate the space-time coupling constant

for the Littrow geometry, i.e., ?=?c. ?b? Angle of incidence ? and diffrac-

tion angle at the center frequency ?cas a function of focal length.

034302-3Space-time couplingJ. Chem. Phys. 130, 034302 ?2009?

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E˜f?x,?? ? E˜i?−kc

f2

x,??M?f?

kc

? +fb

f2

x?

?exp?i? − 1

2

kc

f2

x2?.

?13?

The spatial distribution is now determined by the spatial

Fourier transform of the incident pulse E˜i?kx,?? and space-

time coupling seems to be absent. Nonetheless, the pulse

shape is a function of coordinate x because the argument of

the transfer function depends on x. It is important to note that

this x dependence introduces a frequency offset

???x? = −bkc

?f2

x

?14?

to the transfer function. For example, assume that the SLM is

programmed to introduce a ? phase jump at the center fre-

quency ?=0. The frequency offset will cause the ? jump to

appear at frequency ?=???x? depending on the position x at

the focal plane. A useful measure is the frequency offset that

corresponds to the beam size at the focal plane relative to the

spectral bandwidth. Assuming that the spectral bandwidth ??

is spread across one-half of the SLM’s active area D

=N?xpand that the spot size of a single frequency 2w0

the SLM is equal to the width of a single pixel ?xp, we find

SLMat

??

??

Equation ?15? provides a reasonable estimate and shows

that the frequency offset relative to the spectral bandwidth is

typically on the order of a percent and that it decreases in-

versely proportional to the number of pixels N. For our ex-

perimental setup the exact value turns out to be 1.5%.

Equation ?13? also exhibits a quadratic spatial phase

which disappears for ?=1, i.e., for a 2f imaging geometry.

Generally, it has to be checked from case to case whether this

phase will distort the waveform enough to influence the mea-

surement. The easiest way to avoid the quadratic spatial

phase all together is to use the focusing element in 2f geom-

etry or, should that not be possible, to relay image the

shaper’s output to the back focal plane of the focusing

element.

With Eq. ?13? one can now derive analytic expressions

for the three different modulator transfer functions discussed

above in 2f focusing geometry, i.e., ?=1. For the linear

phase modulation we find

Ef1?x,t? ? E˜i?−kc

f2

kc

? 4bw0

SLM

D

=2b

N.

?15?

x,t +f?

??exp?i?fb

f2

x?.

?16?

Space-time coupling as discussed in Sec. II A has disap-

peared, however, the pulse acquires a linear spatial phase

which is proportional to ?. Naturally, this linear phase is a

consequence of the transverse beam displacement before the

focusing element; from Fourier theorems it is known that a

displacement along the x axis leads to a linear phase in the

Fourier domain, i.e., in kxspace. When a double pulse with a

variable time delay is generated the resulting waveform is

Ef2?x,t? ? E˜i?−kc

f2

x,t?+ E˜i?−kc

f2

x,t +f?

kc

??exp?i?fb

f2

x?.

?17?

Both replicas overlap perfectly in space irrespective of

the time delay, but the time-delayed replica acquires a linear

spatial phase. That is, we expect to observe a spatially vary-

ing interference pattern which becomes obvious when calcu-

latingthespectral intensity

in Eq. ?17?,

If2?x,?? ??E˜i?−kc

f2

fortheelectricfield

x,???

2

cos2??

2?f?

kc

? +fb

f2

x??.

?18?

The spectrum is that of the original pulse but modulated

as a function of frequency ? and spatial coordinate x. The

maxima fall on straight lines each having a slope of

−kcb/?f2??. Lastly, a sinusoidal phase modulation yields

Jm?A?E˜i?−kc

?exp?im??fb

f2

Ef3?x,t? ? ?

m=−?

?

f2

x,t + mf?

x + ???.

kc

??

?19?

Again, there is perfect spatial overlap between all pulses

within the pulse train and each pulse m acquires a linear

spatial phase which is proportional to m.

For the following discussion on longitudinal effects we

assume either that the focusing element is positioned f2away

from the pulse shaper’s output or that the output is relay

imaged to the back focal plane. For such a geometry we find

for the electric field in the x-z plane

E˜f?x,?,?? ??

−?

kc

?

dkxE˜s?f2

kx,??exp?i

?

2kc

kx

2?e−ikxx, ?20?

where ? is measured relative to the focal plane ?see Fig. 1?

and E˜s?x,?? is given by Eq. ?5?. Next, we present analytic

results for the three previously introduced transfer functions

assuming a Gaussian spatial distribution and show the corre-

sponding spectral and temporal distributions at a number of

different positions within the Rayleigh length. In order to

minimize the influence of waveform replicas due to the pix-

5

0

-5

t [ps]

-5

0

5

x/w ’

o

0

5

0

5

0

5

0

5

0.01

0

-0.01

?[rad/fs]

??zR

-1

-0.5

0

0.5

1

FIG. 3. Top row I?x,?? and bottom row I?x,t? at five positions within the

Rayleigh length of the focusing element.

034302-4Frei, Galler, and FeurerJ. Chem. Phys. 130, 034302 ?2009?

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