Page 1

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

View online: http://dx.doi.org/10.1063/1.3058478

View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v130/i3

Published by the AIP Publishing LLC.

Additional information on J. Chem. Phys.Additional information on J. Chem. Phys.

Journal Homepage: http://jcp.aip.org/

Journal Information: http://jcp.aip.org/about/about_the_journal

Top downloads: http://jcp.aip.org/features/most_downloaded

Information for Authors: http://jcp.aip.org/authors

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 2

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

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 3

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?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

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?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 5

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?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 6

elated nature of most SLMs,15we limit the accessible time

window to approximately a quarter of the full shaper window

Wt. For our setup this amounts to roughly ?2 ps, that is, in

M1,2,3?x? we use ?t=2 ps as the worst case scenario. For the

linear phase modulation M1?x? the field is given by

E˜f1?x,?,?? ?

w0

?1 + ?2/zR

??kc

f2

2E˜i???

2exp?−w0

2??

2?1 + i?/zR?

4?1 + ?2/zR

+ i??x,?,?,1??,

2?

x + ?bf

f2

2

?21?

with the Rayleigh length zR?2f2

?,?,m????m?f/kc+?/?2kc??m?bf/f2?2+m?bfx/f2+????,

and the Gouy phase ????=arctan??/zR?. Figure 3 shows the

spectral intensity I?x,?? and the temporal intensity I?x,t? for

five equidistant positions within the Rayleigh length. To im-

prove the visibility of the pulses in the I?x,t? plot, we have

reduced the bandwidth to 5 nm. The x axis is normalized to

the beam waist w0? at the focus ?=0 and the z axis is normal-

ized to the Rayleigh length zR. When applying a linear phase

the spectrum as well as the temporal intensity move from

positive to negative x values by a little more than a beam

waist w0?. In general, this has no impact on an experiment

since all that happens is a minor change in the direction of

2/?kcw0

2?, the phase ??x,

propagation through the sample.

The situation changes when the double pulse is gener-

ated. For the transfer function M2?x? the field is

E˜f2?x,?,?? ?

w0

?1 + ?2/zR

+ i?????+ exp?−w0

+ i??x,?,?,1???.

Figure 4?a? shows the spectral and the temporal intensity

for five equidistant positions within the Rayleigh length.

While the time zero replica remains centered around x=0,

the delayed replica is substantially shifted along the x axis.

Consequently, the two replica hardly overlap in space at both

ends of the Rayleigh range and spectral interferences are

only present in the overlap region. Figures 4?b? and 4?c?

show the spectral and temporal intensities on axis ?x=0? as a

function of the longitudinal position. Within a tenth of the

Rayleigh length almost no effect is detectable.

When applying a sinusoidal phase modulation the field is

2E˜i???

2?exp?−w0

2?1 + i?/zR?

4?1 + ?2/zR

2??kc

2??kc

f2

x?

2

2?1 + i?/zR?

4?1 + ?2/zR

f2

x + ?bf

f2

2??

2

?22?

5

0

-5

t [ps]

-5

0

5

x/w ’

o

0

5

0

5

0

5

0

5

??zR

-1

-0.5

0

0.5

1

0.01

0

-0.01

?[rad/fs]

(a)

? [rad/fs]

-0.01

0

0.01

??zR

t [ps]

-0.1

0

0.1

-5

0

5

(b)

(c)

Intensity [a.u.]

1

0.5

0

Intensity [a.u.]

1

0.5

0

FIG. 4. ?a? Top row I?x,?? and bottom row I?x,t? at five positions within the Rayleigh length of the focusing element. ?b? Spectral intensity and ?c? temporal

intensity as a function of ? for x=0.

0.01

0

-0.01

?[rad/fs]

5

0

-5

t [ps]

-5

0

5

x/w ’

o

0

5

0

5

0

5

0

5

??zR

-1

-0.5

0

0.5

1

(a)

? [rad/fs]

-0.01

0

0.01

??zR

t [ps]

-0.1

0

0.1

-5

0

5

(b)

(c)

Intensity [a.u.]

1

0.5

0

Intensity [a.u.]

1

0.5

0

FIG. 5. ?a? Top row I?x,?? and bottom row I?x,t? at five positions within the Rayleigh length of the focusing element. ?b? Spectral intensity and ?c? temporal

intensity as a function of ? for x=0.

034302-5Space-time couplingJ. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 7

E˜f3?x,?,?? ?

w0

?1 + ?2/zR

?exp?−w0

+ i??x,?,?,m??.

2E˜i???

2?

m

Jm?A?eim?

2?1 + i?/zR?

4?1 + ?2/zR

2??kc

f2

x + m?bf

f2

2??

2

?23?

Figure 5?a? shows the spectral and the temporal intensity

for an amplitude of A=?. Depending on their order the tem-

poral replica are shifted by an incremental value parallel to

the x axis. Inspecting the spectral intensity within one-tenth

part of the Rayleigh length shows that even in this con-

stricted part of the Rayleigh length an amplitude modulation

appears where there should be none. The effect observed

here is similar to the Talbot effect. When moving the detector

away from perfect imaging geometry, i.e., ??0, a phase

modulation converts itself to an amplitude modulation.

III. EXPERIMENTAL VERIFICATION OF SPACE-TIME

COUPLING

The experiments were performed with a KML oscillator

delivering pulses at a repetition rate of 80 MHz and an en-

ergy of approximately 1 nJ. The oscillator pulses were am-

plitude and/or phase modulated in a standard pulse shaping

apparatus consisting of a double display SLM ?Jenoptik

SLM640-d? in the symmetry plane of a 4f zero-dispersion

compressor. All relevant experimental parameters are sum-

marized in Table I.

The space-time coupling effects were investigated di-

rectly after the pulse shaper at surface ?1in Fig. 1 and at the

focus of the last lens at surface ?2. First, the spatial intensity

distribution I?x???d??E˜j?x,???2was recorded with a charge

coupled device ?CCD? camera. The CCD camera ?DMK 21

AFO4 from Imaging Source? had quadratic pixels with di-

mensions of 5.6?5.6 ?m2. Second, the spectrally resolved

intensity distribution I?x,????E˜j?x,?+?c??2was measured

by moving the entrance slit of a spectrometer parallel to the

x coordinate. For several x positions spectra were recorded

and merged to a two-dimensional intensity plot. The spec-

trometer was an AvaSpec 2048 from Avantes with an en-

trance slit width of 10 ?m and a spectral resolution of 0.8

mn full width at half maximum.

A. Space-time coupling after the pulse shaper

First, the predictions of Eq. ?5? were verified by measur-

ing the spatial intensity profile I?x? with a CCD camera. The

spatial offset ?x was determined by fitting a Gaussian func-

tion to I?x?. Figure 6 shows the results for the three different

transfer functions, Eqs. ?8?–?10?.

In the case of the double pulse the spatial offset can only

be accurately determined once the peaks are clearly sepa-

rated in space, i.e., for approximately more than 0.4 mm. For

the sinusoidal phase modulation it is difficult to clearly iden-

tify the different maxima and a comparison with a simulated

beam profile assuming a Gaussian-shaped incident beam is

shown instead. All results agree very well with the theoreti-

cal space-time coupling constant of ?x/?t=0.600 mm/ps,

as predicted by Eq. ?12?. The spatial offset can be quite sub-

stantial and has to be considered when selecting optical ele-

ments, especially their size, in the setup following the pulse

shaper.

Figure 7 shows I?x,?? for the same three transfer func-

tions with the time delay ?t fixed to a particular value; in the

top row the experimental and in the bottom row the simu-

lated results assuming a Gaussian beam profile are shown.

For reference, the unshaped profiles are shown in Figs. 7?a?

and 7?e?.Alinear phase corresponding to a time delay of 1 ps

leads to a spatial offset of 0.6 mm, as seen in Figs. 7?b? and

7?f?. The spectrum, however, remains unaffected and is inde-

pendent of the transverse coordinate x. Figures 7?c? and 7?g?

confirm that within the double-pulse waveform one replica

remains unaffected and the second is shifted in time and

space; at a delay of 2.2 ps there is very little spatial overlap

between the two. When a sinusoidal phase with an interpulse

delay of 400 fs is applied to the pulse the different replica

still overlap to a large extent as seen in Figs. 7?d? and 7?h?,

and spectral modulations are visible in the overlap region. It

is worth emphasizing that spectral modulations can appear

even if the transfer function performs a phase modulation

only. For example, a sinusoidal phase modulation produces a

spectral amplitude modulation in spite of the fact that it is a

pure phase modulation.

TABLE I. Experimental parameters.

Number of pixels

Pixel width

Center wavelength

Spectral bandwidth

Beam waist

Focal length

Focal length

Grating constant

Diffraction order

Angle of incidence

Diffraction angle

N

640

?xp

?c

??

w0

f

f2

G

m

?

?c

b

?

100 ?m

820 nm

55 nm

1 mm

500 mm

400 mm

1.667 ?m

?1

11.3°

?43.4°

1.350

?2.251 ps/mm

6.807 THz/mm

0.600 mm/ps

Frequency-to-space mapping

Space-time coupling

??/?x

?x/?t

-202

-1.0

-0.5

0.0

0.5

1.0

?t [ps]

012

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(a)(b)(c)

-2-1012

0.0

0.2

0.4

0.6

0.8

1.0

Intensity [a.u.]

x [mm]

?t [ps]

?x [mm]

?x [mm]

FIG. 6. ?Color online? Spatial offset ?x as a function of the delay time ?t

for ?a? a linear phase and ?b? a double-pulse modulation. The solid lines

result from Eq. ?12?. ?c? Measured and simulated beam profile for a sinu-

soidal phase.

034302-6Frei, Galler, and FeurerJ. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 8

B. Space-time coupling in focusing geometry

The absence of space-time coupling at the focal plane of

an additional lens, as predicted by Eq. ?13?, was verified by

measuring the spatial intensity I?x? and the spatial-spectral

intensity distribution I?x,??. In practice, this was done by

magnifying the 134 ?m large focal spot to the CCD camera

or the entrance slit of the spectrometer. From the camera

images we extracted the center of mass of the intensity I?x?

and the results are shown in Fig. 8.

No space-time coupling is observed, i.e., the center of

mass is always at the same position irrespective of the trans-

fer function or the delay time.

Then, the CCD was replaced by the spectrometer and the

entrance slit was scanned across the image of the focal plane

yielding I?x,?? for a specific time delay. Again for reference

I?x,?? of an unshaped pulse is shown in Figs. 9?a? and 9?e?.

The simulations assume a Gaussian shape in space and time.

A linear phase modulation, as seen in Figs. 9?b? and 9?f?, has

no influence on the spatial-spectral intensity distribution.

Conversely, the double-pulse transfer function produces

spectral modulations with a contrast of one, as shown in

Figs. 9?c? and 9?g?, and their periodicity and shape corre-

spond to the amplitude modulations applied by the SLM.

Equation ?18? predicts that the intensity maxima should fall

on straight lines with a slope of approximately 11.5 THz/

mm, which is too small to be detected here. Lastly, Fig. 9?h?

shows that a sinusoidal phase modulations should leave

the spectral intensity unaffected. However, the measurements

in Fig. 9?d? show a residual intensity modulation ?10%?

which is probably due to a slight misalignment or spherical

aberration.

Finally, we have measured the spectrum at the center of

the beam profile, i.e., at x=0, as a function of time delay ?t

and the results are shown in Fig. 10. All experiments agree

well with the simulations and show that in the case of pure

phase modulations, i.e., a linear and a sinusoidal phase, the

spectrum remains unchanged irrespective of the time delay.

The spectral modulations observed for a double-pulse trans-

fer function agree with those applied by the SLM and their

periodicity decreases with increasing time delay.

IV. EFFECTS OF SPACE-TIME COUPLING

ON COHERENT CONTROL

Most coherent control experiments involve nonlinear ex-

citation and possibly also probing processes and are expected

to be more susceptible to space-time coupling effects. In the

following we want to address two consequences of space-

time coupling in femtosecond pulse shaping which may have

a significant effect on coherent control experiments and on

comparing simulations to experimental results.

When an unshaped pulse is focused by an ideal focusing

element the maximum intensity varies as a function of spatial

position ?x,z? but the temporal profile is usually the same

irrespective of the position. Conversely, when a pulse exiting

a 4f pulse shaper is focused not only the maximum intensity

but also the pulse shape depends on the spatial position. In

an extreme case this could completely alter the outcome of a

coherent control experiment because the result of coherent

control depends on the spatial position within the focal vol-

ume. Since the signal detected, be it coherent or incoherent,

is usually a spatial average over all contributions from within

the focal volume, space-time coupling is very likely to have

an impact on the measurement.

In a first step we demonstrate experimentally that foot-

prints of space-time coupling can be found in a well under-

stood nonlinear process, namely, second harmonic genera-

tion. The second topic is related to modeling of coherent

?

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

(a)(b)

(c)

(d)

(e)(f)

(g)

(h)

2.4

2.3

2.2

2.1

? [rad/fs]

-1

0

1

x [mm]

2.4

2.3

2.2

2.1

-1

0

1

x [mm]

-1

0

1

x [mm]

-1

0

1

x [mm]

-1

0

1

x [mm]

-1

0

1

x [mm]

-1

0

1

x [mm]

-1

0

1

x [mm]

FIG. 7. Top row: experimentally measured and bottom row: simulated space-frequency distributions as a function of transverse coordinate x and frequency

?. ??a? and ?e?? Unshaped pulse, ??b? and ?f?? linear phase with ?t=1000 fs, ??c? and ?g?? double pulse with ?t=2200 fs, and ??d? and ?h?? sinusoidal phase

with ?t=400 fs.

-4-2024

-60

-40

-20

0

20

40

60

?t [ps]

(a)

-60

-40

-20

0

20

40

60

(b)(c)

?t [ps]

0.00.51.0

?t [ps]

-60

-40

-20

0

20

40

60

?

?

x [ m]

?

?

x [ m]

?

?

x [ m]

FIG. 8. ?Color online? Spatial offset ?x in the focus of a 400 mm lens as a

function of the delay time ?t for a ?a? linear phase, ?b? double-pulse modu-

lation, and ?c? sinusoidal phase. The solid lines are derived from Eq. ?13?.

034302-7 Space-time couplingJ. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 9

control experiments. Frequently it is impossible to simulate

the response of a quantum system for a spatially varying

pulse shape because it would simply be too time consuming.

What pulse shape should then be used for the simulations? A

reasonable choice might be the spatially averaged pulse

shape. A similar problem arises when simulations predict a

specific pulse shape which is best suited to achieve a certain

coherent control goal. In principle, space-time coupling al-

lows to realize that specific pulse shape only at a single spa-

tial position. Here, the question arises whether the spatial

variations of the pulse shape in the experiment will obscure

the coherent control result. First, we address this problem in

second harmonic generation. Specifically we compare the

spatially averaged second harmonic signal with that calcu-

lated from a spatially average fundamental pulse shape. Then

we turn to more complex systems and calculate the spatially

resolved material response for an atomic model system and a

molecular model system. These case studies are based on

coherent control results found in the literature.

A. Experimental verification of space-time coupling

in second harmonic generation

With a first set of experiments we aim to experimentally

demonstrate the appearance of space-time coupling effects in

second harmonic generation. A 50 ?m thick beta barium

borate ?BBO? crystal was placed at the focus of the last lens.

Since the focusing lens has a focal length of 400 mm we can

safely neglect the longitudinal effects. The second harmonic

radiation produced was imaged to the entrance slit of a 50

cm imaging spectrometer with a 3000 lines/mm grating.

Scanning the entrance slit across the image plane allowed

recording the second harmonic spectrum as a function of x.

Then, the spatiospectral intensity distribution ISHG?x,2?c

+?? was measured for the three transfer functions intro-

duced earlier and the results are compared to simulations

assuming a Gaussian in space and frequency, i.e., E˜i?kx,??

=F˜?kx?exp?−?2/??

For a linear phase modulation we find for the second

harmonic spectrum

2? with F˜?kx?=exp?−w0

2kx

2/4?.

ISHG1?x,?? ? F˜4?−kc

f2

x?exp?−?2

??

2?,

?24?

which is the same as for an unmodulated pulse. For the

double-pulse transfer function M2?x? the second harmonic

spectrum is

ISHG2?x,?? ? F˜4?−kc

f2

+ 2 exp?−??

+ 2 exp?−??

x?exp?−?2

8?cos??t

4??.

??

2?cos2???t

2? + ?bf

x?

f2

x?

2?t2

2? + ?bf

f2

2?t2

?25?

It is modulated with a periodicity that is determined by

the time delay, and the intensity maxima in the ?x,?? plane

follow straight lines with a slope of −2kcb/?f2??, here 23

THz/mm. To better visualize the spatial variation only a

small part of the measured second harmonic spectrum is

shown in Fig. 11?a?.

The measurement agrees well with the simulation in Fig.

11?b? and shows that spectral variations across the focal

plane can be quite substantial. When applying a sinusoidal

phase modulation we find

?

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

(a)(b)

(c)

(d)

(e)(f)

(g)

(h)

2.4

2.3

2.2

2.1

? [rad/fs]

-1000

?

100

x [ m]

-1000

?

100

x [ m]

-1000

?

100

x [ m]

-1000

?

100

x [ m]

-1000

?

100

x [ m]

-1000

?

100

x [ m]

-1000

?

100

x [ m]

-1000

?

100

x [ m]

2.4

2.3

2.2

2.1

FIG. 9. Top row: experimentally measured; bottom row: simulated space-frequency distributions as a function of transverse coordinate x and frequency ?. ??a?

and ?e?? Unshaped pulse ??b? and ?f?? linear phase with ?t=1000 fs, ??c? and ?g?? double pulse with ?t=200 fs, and ??d? and ?h?? sinusoidal phase with

?t=200 fs.

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

(a)(b)

(c)

(d)(e)(f)

2.4

2.3

2.2

2.1

? [rad/fs]

2.4

2.3

2.2

2.1

? [rad/fs]

-5

0

5

?t [ps]

0.5

?t [ps]

0

1

0.5

0

1

?t [ps]

FIG. 10. Measured ?top row? and simulated ?bottom row? spectra at the

center of the focal plane, i.e., at x=0, as a function of time delay ?t and

frequency ?. ??a? and ?d?? Linear, ??b? and ?e?? double pulse, and ??c? and

?f?? sinusoidal phase modulation.

034302-8Frei, Galler, and FeurerJ. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 10

ISHG3?x,?? ? F˜4?−kc

f2

x + ???exp?−?2

x?J0

2?2A sin???f

2kc

2?,

?

+ ?bf

f2

??

?26?

where we have assumed that ???f?/kc?1 and, as a conse-

quence, only the zeroth order Bessel function contributes to

the resulting spectrum. Experimental results for A=0.7? and

an interpulse time delay of ?t=1000 fs are shown in Fig.

12?a?. The resulting second harmonic spectrum is amplitude

modulated, as it is known from literature,16but the modula-

tions are tilted in the ?x,?? plane.

The slope agrees well with the analytic prediction of 23

THz/mm, as seen in Fig. 12?b?. The somewhat larger width

in the measurement is due to the finite resolution of the spec-

trometer.

B. Spatially averaged fields

After having verified footprints of space-time coupling

in second harmonic generation, we come back to the prob-

lem of what pulse shape should best be used to simulate

coherent control experiments when due to time constraints

spatial variations cannot be considered. As said above the

most appropriate choice seems to be the spatially averaged

field

E˜o??? ??

V

dVE˜o?x,?,??,

?27?

where the integration is over the interaction volume V. In the

following we consider cases where longitudinal variations

can be neglected, that is, the shaped pulses interact with a

solid surface, a thin crystal, a liquid in a thin cuvette, etc.,

and averaging is with respect to x only. Further assuming that

the incident field may be written as a product of two func-

tions, where one describes the spectral content of the pulse

and the other its spatial variation, i.e., E˜i?x,??=F?x?E˜i???,

we obtain from Eq. ?13?

E˜o??? ? E˜i???M???,

with

M??? =?dx M?x?F˜??

?28?

b? −kc

bfx?

?29?

and F˜?kx? being the spatial Fourier transform of F?x?. Incor-

porating the frequency-to-space mapping, i.e., Eq ?7?, the

spatial convolution can be rewritten as a convolution with

respect to frequency,

M??? =?d?1M?f?

kc

?1?F˜??

b?? − ?1??.

?30?

Equation ?30? performs a convolution of the modulator

transfer function, M?x?, with the function F˜?kx? representing

the spot size of the spectral component ?. Equation ?28?

together with Eq. ?30? has successfully been used to simulate

the space averaged electric field in the focus of a lens, the

appearance of pixel or wrap replica in pulse shaping, etc. But

are they sufficient to simulate nonlinear interactions? We at-

tempt to give an answer to this question by analyzing three

different case studies. The procedure in all three cases is the

same. First, the spatially resolved nonlinear response is cal-

culated from a spatially varying driving field and the result-

ing signal is obtained from a spatial average over the nonlin-

ear response. Alternatively, the nonlinear signal is calculated

from the spatial average of the driving field and then both

results are compared. While the first route involves a large

number of computations, the second route requires only a

single simulation.

C. Second harmonic generation

We assume that the second harmonic spectrum ISHG???

of a shaped pulse has been measured by collecting all second

harmonic field contributions along the x coordinate at the

entrance slit of a spectrometer. Such a coherent superposition

is described through Eq. ?31?,

(a)

0.005

-0.005

0

-2000

?

200

x [ m]

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

(b)

0.005

-0.005

0

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

FIG. 11. Spatially resolved second harmonic spectrum of a double pulse

with a time delay of ?t=1000 fs. ?a? Measurement and ?b? analytical solu-

tion, Eq. ?25?.

(a)

0.005

-0.005

0

-2000

?

200

x [ m]

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

(b)

0.005

-0.005

0

Intensity [a.u.]

1

0.8

0.6

0.4

0.2

0

FIG. 12. Spatially resolved second harmonic spectrum for a sinusoidal

phase modulation. ?a? Measurement and ?b? analytical solution, Eq. ?26?.

034302-9Space-time couplingJ. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 11

ISHG??? =??dxE˜F?x,?? ? E˜F?x,???

2

,

?31?

ISHG??? = ?E˜F??? ? E˜F????2.

The convolution ? is with respect to the frequency ?.

Alternatively, Eq. ?32? represents the commonly used ap-

proximation outlined above, i.e., the second harmonic spec-

trum is calculated from the spatially averaged fundamental

field given by Eq. ?28?. Is there a detectable difference be-

tween the two?

As an example, we explicitly present analytic results for

the transfer function M2?x?, which yields a double pulse in

the time domain. We assume a Gaussian dependence in space

and in frequency, i.e.,

E˜i?x,?? = exp?−x2

w0

??

?32?

2−?2

2?,

?33?

with the beam waist w0and the spectral width ??. In both

cases, i.e., for Eqs. ?31? and ?32?, the solution is of the form

ISHG??? = exp??2

??

+ ?3cos??t???,

2??1 + ?1+ ?2cos??t?

2?

?34?

only the constants ?1,2,3differ. Figures 13?a?–13?c? show the

three constants as a function of time delay. As long as the

time delay is comparable to the pulse duration all constants

are to a good approximation identical; differences appear for

larger time delays but tend to disappear in the signal because

the absolute magnitude of the three constants becomes very

small. While ?1only affects the overall amplitude, ?2and ?3

modify the modulation depth; ?2, however, is too small to

have a measurable effect. For two different time delays,

namely, 400 and 1400 fs, which are indicated by the dashed

vertical lines, the calculated spectral intensities are shown in

Figs. 13?d? and 13?e?. The differences in Fig. 13?d? are neg-

ligible. For a time delay of 1400 fs, as shown in Fig. 13?e?,

the modulation depth is substantially different, but whether

this difference is detected at all depends on the resolution of

the spectrometer.

In summary, space-time coupling effects can be detected

in second harmonic generation, but in most cases, i.e., for

most transfer functions, we found that the spatially averaged

second harmonic spectrum can be predicted with good accu-

racy from the spatially averaged driving field. A possible

reason might be that the acceptance function of second har-

monic generation in a thin crystal is broader than the funda-

mental spectrum and the frequency offset due to space-time

coupling.

D. Resonant interaction with an atomic three-level

system

The influence of space-time coupling may change dra-

matically when the acceptance function of the system is

smaller than the frequency offset given by Eq. ?15?. To ana-

lyze such a situation we exemplarily simulate the interaction

of shaped pulses with an atomic three-level system. Of spe-

cial interest are two coherent control scenarios which have

been published in the literature, namely, adiabatic population

transfer through strongly chirped pulses17and selectivity in

coherentpopulationtransferinducedby optical

Eq. 23

Eq. 24

(a)

10

0

10

1

10

-1

10

-2

10

-3

10

0

10

2

?t [fs]

?1

(b)

?2

(c)

?3

10

0

10

1

10

-1

10

-2

10

-3

10

0

10

1

10

-1

10

-2

10

-3

10

0

10

2

?t [fs]

10

0

10

2

?t [fs]

(d)

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

(e)

00.2

? [rad/fs]

Intensity [a.u.]

0.1

-0.1

-0.200.20.1-0.1

1.0

0.8

0.6

0.4

0.2

0.0

Intensity [a.u.]

? [rad/fs]

Eq. 23

Eq. 24

Eq. 23

Eq. 24

FIG. 13. The three constants ?a? ?1, ?b? ?2, and ?c? ?3as a function of time

delay in a double logarithmic plot. Spectral intensity for two different time

delays, namely, ?d? 400 fs and ?e? 1400 fs.

0

1

2

794.67 nm

780.03 nm

(a)(b)

Population

1.0

0.5

0.0

0

1

2

-1000

?

100

x [ m]

FIG. 14. ?Color online? ?a? Rubidium V-type three-level system. ?b? Spa-

tially resolved population distribution after interaction with a Gaussian-

shaped bandwidth-limited pulse with a maximum fluence of 2 J/m2.

Ground state: black solid curve; lower excited state: red dotted curve; and

upper excited state: blue dashed curve.

(c)(d)

?P

(a)

Population

1.0

0.5

0.0

-1000

?

100

x [ m]

-1000

?

100

x [ m]

(b)

0.001

-0.001

0.000

0

1

2

FIG. 15. ?Color online? Spatially resolved final population of the Rb three-

level system after excitation with ?a? a downchirped pulse ?−10 000 fs2?

and ?b? an upchirped pulse ?+10 000 fs2?, respectively. ?c? and ?d? show the

difference between the simulations in ?a? and ?b? and the corresponding ones

without space-time coupling. Ground state: black solid curve; lower excited

state: red dotted curve; and upper excited state: blue dashed curve.

034302-10Frei, Galler, and FeurerJ. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 12

interference.18We assume two upper states which are con-

nected to a common ground state ?V-type system?. In con-

trast to Sec. IV C the excitation is a one-photon process,

however, nonlinearities arise from a substantial population

transfer. In both scenarios the task is to coherently control

the population dynamics, and we assume that the final ex-

cited state populations are detected through spontaneous

emission from these levels. That is, the detection channel is

incoherent and the relative amplitudes of the observed spon-

taneous emission peaks can be used to deduce the relative

population distribution in the upper states. As in Sec. IV C, it

is our intention to explore whether space-time coupling has

any measurable effect on the incoherent signal detected.

More precisely, is it sufficient to use Eqs. ?28? and ?30? to

calculate the fluorescence signal or is it mandatory to con-

sider space-time coupling effects?

As a model system we use the fine structure doublet of

the rubidium atom with the two resonant transitions at

780.03 nm and 794.76 nm, as shown in Fig. 14?a?. The cor-

responding transition dipole moments are 2.09?10−29Asm

and 1.48?10−29Asm. Since the simulations were carried

out in the regime of the extreme sharp-line limit, the exact

shape of the absorption line profile is irrelevant. To calculate

the single atom response to a shaped pulse, the density ma-

trix was integrated in the rotating wave approximation.19,20

The diagonal elements at times much larger than the pulse

duration yield the final relative population distribution.

The simulations use a Gaussian spatial beam profile. In

order to separate intensity effects from those caused by a

spatially varying pulse shape, we always calculate the final

population distribution as a function of x for two cases; first,

for a pulse whose shape is independent of x and, second, for

the pulse whose shape given by Eq. ?13? ?with ?=1?. An

example is shown in Fig. 14?b? where a Rb atom has been

illuminated with a bandwidth-limited pulse. Close to the

beam center the ground state is strongly depleted and the two

excited states have almost the same population.

A very powerful tool to selectively populate one of the

excited states and, thus, to coherently control the population

dynamics is the technique of adiabatic rapid passage. The

effect is best understood in the frame of dressed states, which

are eigenstates of the field-matter Hamiltonian.21The adia-

baticity theorem in quantum mechanics states that a system

remains in a dressed state if the slowly varying amplitude of

the electric field varies more slowly than the relevant internal

time scales of the system.22As a consequence, the system is

able to dynamically adjust itself to variations in the external

parameters, such as intensity or instantaneous frequency. If,

for example, the instantaneous frequency of a shaped pulse

sweeps across the resonance frequency of a two-level sys-

tem, then this system can be excited very efficiently.23While

in a two-level system the sign of the frequency sweep is

irrelevant, in a three-level system it determines which of the

two excited states is populated. The two excited states can be

more or less completely decoupled by a suitably large fre-

quency sweep, even if both transitions are contained within

the spectral bandwidth of the incident field. While an up-

chirped field favors population transfer to the lower level, a

downchirped field preferentially populates the upper level,

depending on which resonance is crossed first.

Figures 15?a? and 15?b? show the corresponding simula-

tions for the transfer function M???=exp?−i?2?2/2?. All

relevant parameters are summarized in Table II. If the pulse

has a downchirp of ?2=−10 000 fs2, as shown in Fig. 15?a?,

the upper lying excited state is populated more efficiently

than the lower lying excited state. For an upchirp of the same

magnitude the situation is reversed, as seen in Fig. 15?b?. In

Figs. 15?c? and 15?d? the difference in population between

the simulations in Figs. 15?a? and 15?b? and those without

space-time coupling is shown. For all spatial positions the

differences are vanishingly small. This is not surprising be-

cause the only effect of space-time coupling is that the mini-

TABLE II. Simulation parameters.

Center wavelength

Spectral bandwidth

Maximum fluence

Beam waist

Focal length

?c

??

F

w0

f2

790 nm

24 nm

39 J/m2

0.48 mm

200 mm

TABLE III. Spatially averaged population for a pulse with no ?Pm?E?x??? and with ?Pst?E?x??? space-time

coupling. Final populations calculated from spatially averaged fields with and with no space-time coupling.

Populations for the maximum field at x=0.

?Pm?E?x???

0.4705

0.0834

0.4461

0.4705

0.4082

0.1213

?Pst?E?x???

0.4705

0.0834

0.4461

0.4705

0.4082

0.1213

Pm??E?x???

0.2956

0.4121

0.2922

0.2956

0.2069

0.4975

Pst??E?x???

0.3318

0.4030

0.2653

0.3318

0.2236

0.4446

P?E?x=0??

0.0020

0.0310

0.9670

0.0020

0.9476

0.0504

Fig. 15?a??0?

?1?

?2?

?0?

?1?

?2?

Fig. 15?b?

Fig. 16?a??0?

?1?

?2?

?0?

?1?

?2?

0.4451

0.0000

0.5549

0.5125

0.4841

0.0034

0.4452

0.0003

0.5545

0.5124

0.4829

0.0047

0.2245

0.0000

0.7755

0.4087

0.5891

0.0022

0.2251

0.0000

0.7749

0.4094

0.5884

0.0022

0.0004

0.0000

0.9996

0.0097

0.9794

0.0109

Fig. 16?b?

034302-11Space-time couplingJ. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 13

mum of the quadratic spectral phase shifts along the fre-

quency axis as x varies and the excitation process is more or

less insensitive to such a shift.

The space-averaged populations are summarized in

Table III and compared to the populations calculated from

the space-averaged pulses. Space-time coupling has no ap-

parent influence and discrepancies are dominated by inten-

sity effects. In other words, using the spatially averaged

pulse to calculate the nonlinear material response is not at all

the same as averaging the spatially resolved response, but the

differences stem from intensity effects rather than space-time

coupling. In Ref. 17 the probe beam was much smaller than

the pump, in which case the population distribution at x=0 is

probed and no averaging is necessary.

In an alternative coherent control scenario the selectivity

inthepopulationtransfer

interference.18Here, two chirped pulses interfere with each

other such that the spectral intensity at one transition is

maximized, while the spectral intensity at the other transition

is zero due to destructive interference. By changing the time

delay between the two pulses the spectral interference pattern

can be shifted and the two transitions can be switched on and

off at will. Unlike the previous scheme, this approach can be

applied to more than two upper state levels, does not need to

operate in the strong field regime, and requires no change in

the sign of the chirp. However, this scheme involves spectral

interferences with the intention to transmit or block specific

frequency components. Shifting the interference pattern

across the spectrum changes the relative amplitudes of

isinduced byoptical

those frequencies we wanted to transmit or block, and, thus,

we expect that space-time coupling has a much stronger

influence.

The corresponding simulations use the transfer function

M???=1/2?1+exp?−i?t??c+???? to modulate an incident

chirped pulse with a quadratic phase modulation of exp

?−i?2?2/2?. The results are shown in Figs. 16?a? and 16?b?.

The chirp was ?2=210 000 fs2and all other simulation pa-

rameters are from Table II. When the time delay is set to

?t=51.7 fs the resulting spectrum has no intensity around

795 nm ?see Fig. 16?e?? and, as a consequence, no population

is found in the lower excited state. Conversely, for a time

delay of ?t=53.3 fs destructive interference causes the spec-

trum to have a null around 780 nm ?see Fig. 16?f?? and no

population is transferred to the upper excited state. When the

two simulations are compared to those without space-time

coupling effects, then the results are different and the effect

of space-time coupling is clearly observable, as seen in Figs.

16?c? and 16?d?. In contrast to the previous control scheme,

the outcome of the control experiment here relies on an am-

plitude modulation, and as x is varied the amplitude modu-

lation shifts along the frequency axis, as seen in Figs. 16?e?

and 16?f?. As a consequence, the relative spectral amplitudes

at the two transition frequencies and, therefore, the final

population distribution change.

Again, the space-averaged populations are summarized

in Table III and compared to the populations calculated from

the space-averaged fields. Because the deviations in Figs.

16?c? and 16?d? are almost perfectly antisymmetric with re-

spect to x=0, there is only very little effect on the spatially

averaged signals. When using the spatially averaged fields to

calculate the final population distribution the desired effect is

observed but the absolute numbers deviate from the space-

averaged populations.

These two case studies confirm that space-time coupling

influences the spatially resolved signal if well localized

modulation features shift with respect to one or more narrow

acceptance functions of a quantum system. The rather large

discrepancies with respect to the results obtained from the

spatially averaged fields are mostly due to intensity varia-

tions rather than variations in the pulse shape.

E. Resonant interaction with a diatomic molecule

Other important classes of quantum systems in coherent

control studies are those which show internal dynamics on

the same time scale as the slowly varying envelope of the

(a)(b)

(c)(d)

(e)

(f)

?P

-1000

?

100

x [ m]

-1000

?

100

x [ m]

-1000

?

100

x [ m]

-1000

?

100

x [ m]

Population

1.0

0.5

0.0

0.02

0.00

-0.02

2.45

2.40

2.35

2.30

? [rad/fs]

0

1

2

FIG. 16. ?Color online? Spatially resolved final population of the Rb three-

level system. The top row ??a? and ?b?? shows simulations with space-time

coupling and the center row ??c? and ?d?? the difference with respect to

simulations without space-time coupling. The bottom row ??e? and ?f??

shows the spatially varying spectral intensity as a function of x and the two

horizontal dashed lines indicate the two resonance frequencies. For the left

column the time delay is 51.7 fs and for the right column the time delay is

53.3 fs.

Internuclear separation [nm]

Potential energy [eV]

?

1 +

g

?

1 +

u

825 nm

(a)

A

X

(b)

Population

0

1

2

0.40.60.8

-2000200

0.0

0.5

1.0

A

X

x [ m]

?

FIG. 17. ?Color online? ?a? Ground and excited state potential curves of K2.

?b? Spatially resolved population in the ground ?solid curve? and the excited

state ?dashed curve? after the interaction with a bandwidth-limited pulse.

034302-12Frei, Galler, and Feurer J. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 14

shaped light pulses. Exemplarily, we investigate the influ-

ence of space-time coupling in a typical coherent control

scenario, namely, for a double-pulse excitation of a simple

molecule. When a double-pulse excitation scheme is used the

timing between the two pulses and their relative phase have

a strong influence on the final population distribution. In

other words, the optical interference between the two pulses

strongly influences the wavepacket being transferred to the

excited state, and we may expect a relatively strong influence

of space-time coupling.

As a model system we choose the potassium dimer K2

and the simulations refer to experimental results obtained by

Hornung et al.24

We numerically integrate the one-

dimensional Schrödinger equation, and the two potential

curves of interest of K2are shown in Fig. 17?a?.According to

Ref. 24 the simulations were performed at a center wave-

length of 825 nm and a spectral bandwidth of 10 nm.

Throughout all simulations the maximum fluence was fixed

at 3.34 J/m2and the focal length was f2=400 mm. For ref-

erence, Fig. 17?b? shows the population in the ground and

the excited state after the interaction with a single

bandwidth-limited pulse; the population transfer is roughly

25% and varies as a function of x because of the Gaussian

beam profile.

In Ref. 24 it was found that a double pulse with an

interpulse time delay of ?=720 fs ??=560 fs? and a relative

phase of 0 ?2.3 rad? decreases ?enhances? the excited state

population compared to a single transform-limited pulse. The

simulations including space-time coupling, as shown in Figs.

18?a? and 18?b?, confirm this finding; the population in the

excited state is lower in Fig. 18?a? than in the transform-

limited case and it is higher in Fig. 18?b? than in the

transform-limited case. When comparing the results to those

without space-time coupling, we see that in the first case, i.e.,

in Fig. 18?c?, the differences are very small. However, when

the pulse sequence is maximizing the excited state popula-

tion the differences become very large; the maximal devia-

tion in Fig. 18?d? is roughly 10%.

Looking at the results summarized in Table IV a similar

behavior is seen as for the rubidium simulations above, see

Table III. Although discrepancies are observed in the spa-

tially resolved populations, the averaged populations show

almost no sign of it. The results obtained for the spatially

averaged fields differ in absolute numbers but show the same

trend and confirm the control goal.

V. CONCLUSION

We presented a Fourier optical description of a 4f fem-

tosecond pulse shaping apparatus and analytic expressions to

calculate the space-time dependence of the emerging wave-

form just after the pulse shaper and in the focal volume of an

additional focusing element. For both geometries the results

were experimentally verified with three different, frequently

used transfer functions. Just after the pulse shaper space-time

coupling can lead to a substantial spatial beam displacement

which may be minimized by using a highly dispersive grat-

ing. The immediate consequence is that the aperture of the

all optical elements following the pulse shaper has to be

large enough in order to prevent vignetting. The situation

changes when the shaper’s output is focused by an additional

lens. It turns out that the position of the lens is important,

and ideally it should be located one focal length away from

the shaper’s last grating. If for some reason such a geometry

is not feasible then we recommend to relay image the last

grating to the back focal plane of the lens. Even for an ideal

2f focusing geometry the pulse shape is a function of posi-

tion within the focal volume and depends on the longitudinal

position ? and the transverse position x. Longitudinal varia-

tions may be neglected if the sample thickness is smaller

than about a tenth of the Rayleigh length.

Next, we have investigated the influence of space-time

coupling on nonlinear processes. With second harmonic gen-

eration in a thin BBO crystal, we have shown experimentally

that space-time coupling is actually observable. With numeri-

cal simulations we have investigated a small selection of

(a)(b)

(c)(d)

Population

1.0

0.5

0.0

?P

0.01

0.00

-0.01

-2000

?

200

x [ m]

-2000

?

200

x [ m]

1.0

0.5

0.0

0.1

0.0

-0.1

A

X

FIG. 18. ?Color online? Spatially resolved population in the ground ?solid

curve? and in the excited state ?dashed curve? of K2after interacting with a

double pulse. ?a? Time delay 720 fs and relative phase 0 and ?b? time delay

of 560 fs and relative phase of 2.3 rad. The differences with respect to

simulations neglecting space-time coupling are shown in ?c? and ?d?.

TABLE IV. Spatially averaged population for a pulse with no ?Pm?E?x??? and with ?Pst?E?x??? space-time

coupling. Final populations calculated from spatially averaged fields with and with no space-time coupling.

?Pm?E?x???

0.8949

0.1051

0.9066

0.0934

0.8554

0.1446

?Pst?E?x???

¯

¯

0.9057

0.0943

0.8580

0.1420

Pm??E?x???

0.9247

0.0753

0.9331

0.0669

0.9005

0.0995

Pst??E?x???

¯

¯

0.9436

0.0564

0.9108

0.0892

Fig. 17?b??X?

?A?

?X?

?A?

?X?

?A?

Fig. 18?a?

Fig. 18?b?

034302-13Space-time couplingJ. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Page 15

coherent control scenarios in order to evaluate the influence

of space-time coupling in such experiments. We would like

to emphasize that it is very difficult to predict the influence

of space-time coupling on the spatially integrated nonlinear

signal of interest and it is advisable to check its importance

from case to case. The only rule of thumb is that space-time

coupling becomes relevant when both the quantum system

and the modulator’s transfer function exhibit narrow spectral

features which are comparable in width to the space-time

coupling constant Eq. ?15?.

ACKNOWLEDGMENTS

This work was supported by the National Center of

Competence in Research: Quantum Photonics, Research In-

strument of the Swiss National Science Foundation ?SNSF?.

1A. M. Weiner, Rev. Sci. Instrum. 71, 1929 ?2000?.

2S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics ?Wiley,

New York, 2000?.

3M. B. Danailov and I. P. Christov, J. Mod. Opt. 36, 725 ?1989?.

4M. M. Wefers and K. A. Nelson, IEEE J. Quantum Electron. 32, 161

?1996?.

5J. Paye and A. Migus, J. Opt. Soc. Am. B 12, 1480 ?1995?.

6B. J. Sussman, R. Lausten, and A. Stolow, Phys. Rev. A 77, 043416

?2008?.

7T. Tanabe, H. Tanabe, Y. Teramura, and F. Kannari, J. Opt. Soc. Am. B

19, 2795 ?2002?.

8T. Tanabe, F. Kannari, F. Korte, J. Koch, and B. Chichkov, Appl. Opt.

44, 1092 ?2005?.

9C. Dorrer and I. A. Walmsley, Opt. Lett. 27, 1947 ?2002?.

10C. Dorrer, E. M. Kosik, and I. A. Walmsley, Appl. Phys. B: Lasers Opt.

74, s209 ?2002?.

11J. C. Vaughan, T. Feurer, and K. A. Nelson, Opt. Lett. 28, 2408 ?2003?.

12S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, Opt. Express 11, 68

?2003?.

13S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, Opt. Express 11, 491

?2003?.

14O. E. Martinez, J. Opt. Soc. Am. B 3, 929 ?1986?.

15J. C. Vaughan, T. Feurer, K. W. Stone, and K. A. Nelson, Opt. Express

14, 1314 ?2006?.

16M. Hacker, R. Netz, M. Roth, G. Stobrawa, T. Feurer, and R. Sauerbrey,

Appl. Phys. B: Lasers Opt. 73, 273 ?2001?.

17R. Netz, T. Feurer, G. Roberts, and R. Sauerbrey, Phys. Rev. A 65,

043406 ?2002?.

18R. Netz, A. Nazarkin, and R. Sauerbrey, Phys. Rev. Lett. 90, 063001

?2003?.

19L. Mandel and E. Wolf, Optical Coherence and Quantum Optics ?Cam-

bridge University Press, Cambridge, 1995?.

20L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms

?Wiley, New York, 1975?.

21G. L. Peterson and C. D. Cantrell, Phys. Rev. A 31, 807 ?1985?.

22A. Messiah, Quantenmechanik ?Walter de Gruyter, Berlin, 1990?.

23J. S. Melinger, S. R. Gandhi, A. Hariharan, J. X. Tull, and W. S. Warren,

Phys. Rev. Lett. 68, 2000 ?1992?.

24T. Hornung, R. Meier, and M. Motzkus, Chem. Phys. Lett. 326, 445

?2000?.

034302-14Frei, Galler, and FeurerJ. Chem. Phys. 130, 034302 ?2009?

Downloaded 15 Sep 2013 to 202.116.1.148. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions