Full text of DOI:10.1007/s003400000346

Appl. Phys. B 71, 277–284 (2000) / Digital Object Identifier (DOI) 10.1007/s003400000346
Applied Physics B
Lasers
and Optics
Optimal control of one- and two-photon transitions with
shaped femtosecond pulses and feedback
T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch, M. Motzkus∗
Max-Planck-Institut für Quantenoptik, 85740 Garching, Germany
Received: 3 November 1999/Published online: 5 July 2000 – Springer-Verlag 2000
Abstract. This paper reports two pump–probe experiments
in sodium where dynamically tailored ultrashort pulses from
a Ti:Sapphire-pumped optical parametric amplifier were em-
ployed. The first study focuses on the one-photon Na(3s →
3p) transition to derive sensitive criteria which judge the per-
formance of a frequency-domain pulse shaper using a spatial
light modulator. On the basis of the interpretation, follow-up
experiments are suggested to test their cogency. The sec-
ond experiment uses coherent quantum control by placing an
appropriate phase distribution on the incident beam to en-
hance or cancel the transition probability in the nonresonant
two-photon process Na(3s →→ 5s). Ignorant of the “ideal”
phase function, an evolutionary algorithm which uses a feed-
back derived from the experiment performs the optimization
and produces the desired bright or dark pulses within a few
minutes. Attention is given to the role of resonant 3s → 3p
transitions excited by the spectral wings of the pump pulse.
Different parametrizations of the phase distribution have been
examined. Two of these produced solutions which had not
previously been predicted by theory still meet the objective of
the experiment. The study represents the first successful ap-
plication of a feedback-organized self-learning algorithm to
the design of dark pulses.
occurrence of either constructive or destructive interference
depends on the spectral phase and amplitude distribution of
the pump beam. Optimal control of quantum interactions thus
relies on the effective manipulation of the coherence proper-
ties of the optical field.
The most effective techniques to shape ultrashort optical
pulses into desired waveforms rest upon filtering of the spec-
trally dispersed original pulse in the Fourier plane. The first
demonstration of frequency-domainpulse shaping in the fem-
tosecond (fs) regime by using a spatially patterned mask to
place an appropriate phase on the frequency spectrum has
been reported by Weiner et al. [8]. Weiner’s group, a few
years later, was also the first to demonstrate the production
of almost arbitrarily shaped fs pulses via programmable dy-
namic phase-only filtering [9–11]. Expanded flexibility is af-
forded by a shaper design which allows independent dynamic
control of phase and amplitude, as introduced by Wefers and
Nelson [12, 13]. A different approach to tailor fs pulses in
phase and amplitude is based on acousto-optical modulation
(AOM) and has been explored extensively by Warren and
co-workers [14]. Other techniques to shape ultrashort pulses
have been employed as well [15, 16].
The synthesis of waveforms which drive a quantum-
mechanical system towards a selected terminal state requires
complete knowledge of the Hamiltonian. With the exception
of di- and sometimes tri-atomic molecules this operator is
generally unknown, which complicates or even precludes the
direct calculation of the spectral filter creating the desired
output pulse. These restraints can be bypassed by using an
adaptive approach [7, 11, 17, 18] in the sense that an inter-
active dialogue is established between the experiment and
a self-learning algorithm which controls the phase mask. Sev-
eral successful implementations of this principle have been
reported since then [1–3,19–21].
PACS: 32.80.Qk; 42.50.Gy; 42.65.Re
The capability to tailor ultrashort femtosecond pulses and
thus synthesize almost arbitrary waveforms, which has been
a subject of intensive and successful work in the recent past,
has paved the way towards experiments which aim at a con-
trol of quantum-mechanical phenomena [1–3]. The concep-
tional scheme which has been proposed to this end, and which
has been termed coherent – or optimal control, tends to ma-
neuver an atomic or molecular system towards a desired tar-
get state through a suitable interaction with light [4–7]. This
objective can be achieved through quantum interference be-
tween distinct competing routes leading to this final state. The
The task of the algorithm which processes input from the
experiment is to optimize a merit function which indicates the
adequacy of the manipulated pulses to achieve a preset goal.
If intelligently parametrized, the procedure will eventually
zoom into an optimum which, however, need not necessarily
be the global extremum. An interpretation as to why control
is achieved is not straightforward.
∗Corresponding author. (E-mail: mcm@mpq.mpg.de)

278
This paper reports two pump–probe experiments per-
In the time domain the modulation of the frequency spec-
trum results in a pulse eout(t) which may be expressed as
a summed sequence of evenly spaced input pulses ein(t), each
characterized by an amplitude an and a phase Φn [23]:
ꢀ
formed on sodium. Data from the one-photon Na(3s → 3p)
study directly reflect the performance and fidelity of the pulse
shaper. Based on the results we may judge on principal lim-
itations of the shaping process and quantify its sensitivity to
a variation of the influencable parameters. The data also af-
ford a critical test of the validity of analytical descriptions
of coherently controlled one-photon excitation [22]. The non-
resonant two-photon process Na(3s →→ 5s) represents an
example of effectively controlled energy deposition in a two-
level system by tailoring the spectral phase of the pump
pulse, using an additionally implemented feedback scheme.
Three different parametrizations of the phase have been tested
and solutions of the “best” phase functions have been found
which had not previously been predicted by theory. In all
cases the global minimum of the cost function is asymptot-
ically approached within a short time. This is, at least to
the best of our knowledge, the first demonstration of dark
pulses that have been designed by a genetic algorithm which
only processes information received from the experiment,
i.e. free from any initial constraints dictated by theoretical
deliberations.
e
out(t) =
an exp(iΦn)ein(t +nτ) .
(1)
n
To perform efficient shaping the bandwidth of the incident
pulse must fit on the active area of the mask. Observing this
requirement, the parameter τ, which subsumes the dispersion
property of the reflective grating and the dimensional param-
eters of the mask (interpixel distance, total number of pixels)
is connected to the incident pulse width, tpulse, by τ ≈ 0.5×
tpulse [23]. The summation must be carried over the total num-
ber of pixels (128) and hence covers a maximum shaping
window of 128×τ. The amplitude and phase coefficients, an
and Φn, are related to the transmission and retardance values
placed on the modulator via a discrete complex Fourier trans-
formation. Since this is a fast procedure, any desired pulse as
expressed by (1) can be synthesized within less than one sec-
ond [29]. The concurrent manipulation of phase and ampli-
tude incurs a loss of pulse energy which may be unacceptable.
This drawback can be remedied to some extent by phase-only
filtering [30–32] which leaves the power spectrum and the
energy of the shaped pulse intact. The approach is, however,
disadvantageous since it limits the potential of modulation.
The optimization algorithm which is in control of the
pulse-shaping device is supposed to interpret a signal from
the experiment and develop a strategy which eventually pin-
points the global minimum of a multidimensional cost func-
tion. A coarse classification of the mathematical tools suited
for this problem discriminates between deterministic, and in-
deterministic approaches which exploit random principles to
define new search directions. Evolutionary strategies [33],
simulated annealing [34], and genetic algorithms [35] be-
long to this latter category. The present experiment uses an
evolutionary strategy which processes and tunes 48 vectors
of parameters for the LC voltages, starting from a random
seed [36]. Adaptive step-length control [33] has been imple-
mented to speed up the achievement of convergence.
1 Experimental setup
The femtosecond pulse source for both experiments was
a commercial Ti:sapphire laser system (CPA-1000, Clark
MXR Inc.) which supplied 1 mJ/100 fs/800 nm pulses at
a repetition rate of 1 kHz. Frequency conversion to the wave-
length interval between 580 nm and 700 nm in an optical
parametric amplifier (IR-OPA, Clark) yielded pulse energies
around 5 µJ. The programmable pulse-shaping apparatus is
a symmetric 4- f arrangement [8, 23] composed of one pair
each of reflective gratings (1800 lines/mm) and cylindrical
lenses ( f = 150 mm). Its active element – a liquid-crystal
(LC) mask – is installed in the common focal plane of both
lenses. Meticulous alignment must ensure zero net temporal
dispersion. This is achieved once the shapes of input and out-
put pulses match as long as the LC mask is turned off. The
technique of frequency-resolved optical gating (FROG) [24]
served to characterize the generated pulses.
3 Results
Sodium was evaporated in a heat pipe oven [25] pres-
surized with 10 mbar of argon as a buffer gas. The tem-
perature was set sufficiently low (520 K) to eliminate pulse-
propagation effects [26, 27]. Details of the excitation and
detection schemes will be supplied in the context of the re-
spective experiments.
3.1 The one-photon Na(3s → 3p) transition
In pursuit of the goal to control the one-photon transi-
tion in sodium (see Fig. 1a) we employed phase- and am-
plitude shaping of the incident spectrum centered around
589 nm [37] to generate a phase-related double-pulse se-
quence [38]. The resulting structures were characterized
by FROG [24] measurements (see Fig. 2). Moderate focus-
ing ( f = 300 mm) into the heat pipe resulted in a power
density of ≈ 1011 W/cm2 which was sufficient to saturate
the 3s → 3p transfer. The population induced in the 3p
state was probed with a narrow-band (∆ω = 0.2 cm−1) Nd-
YAG-pumped dye laser (20 µJ, 3 ns, 50 Hz) which was fired
synchronous with the Ti:sapphire system and tuned to the
3p1/2 → 5s and 3p3/2 → 5s transitions, respectively. Pump
and probe beams were aligned collinearly and diligent care
was taken to ensure that the probed volume was completely
overlapped by the pump.
2 Dynamic pulse shaping by spatial light modulation
The core of the pulse-shaper unit is an externally address-
able pair of one-dimensional liquid-crystal arrays sandwiched
between two parallel (horizontal) polarizers (SLM-256, Cam-
bridge Research & Instrumentation). Voltage to each indi-
vidual strip of this structure is supplied through a matching
pattern of indium–tin oxide (ITO) electrodes bonded to the
surfaces of the nematic-crystal arrays. This spatial light mod-
ulator (SLM) permits independent control of phase and am-
plitude for each of its n = 1 . . . 128 pixels [13, 28].

279
not seen as only 3p1/2 is probed. The phase of the initially ex-
cited population evolves freely in time as exp(iω12 t) and later
interferes with the different phase of the population induced
by the follow-up pulse. The description of a one-photon ab-
sorption in a first-order approximation yields a population of
a
)
G
h
p
e
i
a t
e
O
P A
0 0 f s
p
1
2
the probed excited state which is given by |c2(t)| , where
e
x
c
i
t
a
t
i
o
n
s
c
h
e
m
e
t
p
u
l
s
e
s
h
a
p
e
r
ꢁ
3
p
3
/
2
P
M
T
ꢂ ꢃ
ꢂ
ꢃ
2π
c2 (t) =
H1s2 tꢀ exp iω12tꢀ dtꢀ .
(2)
3
p
c
o
n
t
r
o
l
l
e
r
1
/
2
ih −∞
t
ꢂ ꢃ
H1s2 tꢀ is the interaction Hamiltonian which, assuming the
D
y
e
l
a
s
e
r
ꢂ ꢃ
dipole approximation, is given by H1s2 tꢀ = µe(tꢀ), where
e(tꢀ) symbolizes the electric field of the laser pulse. In the
slowly varying envelope limit a pulse is described as a time-
dependent envelope function e(tꢀ) including a carrier wave
with the central frequency of the laser field, ω0. This approx-
imation is valid for pulse durations down to a few femtosec-
onds. The phase of the laser pulse, Φ(t), is defined as the
position of the carrier wave relative to the envelope.
2
n
s
3
s
P
M
T
l
b
)
O
P A
0 f s
1
0
h
p
e
i
a
t
p
e
A sequence of pulses with envelopes en(t), phases Φn(t),
and common central frequency ω0 is described generally as:
#
5
s
P
M
T
4
p
p
u
l
s
e
s
h
a
p
e
r
ꢀ
o
a
p
l
t
i
m
i
t
z a t i o n
m
E(t) = exp(iω0t)
en(t −tn) exp(iΦn(t)) .
(3)
c
o
n
t
r
o
l
l
e
r
g
o
r
i
h
f
e
e
d
b
a
c
k
4
p
n
m
o
n
i
t
o
r
3
p
In frequency-domain pulse shaping the tailored pulse se-
quence originates from a Fourier-limited incident pulse hav-
ing no time-dependent phase. Hence each subpulse in (3)
has its own time-independent phase and amplitude-scaled en-
velope of the incident pulse (Wefers and Nelson [23]). The
shaping of two pulses of equal amplitude separated in time by
τ is hence described by:
3
s
Fig. 1a,b. Experimental setup. a collinear pump–probe arrangement to con-
trol the one- photon excitation of Na via a double-pulse sequence. The inset
illustrates the pertinent spectroscopic details. τ marks the time delay be-
tween both pulses. b Experimental layout and spectroscopic details of the
pump- and detection schemes of the two-photon experiment. Fluorescence
from 4p serves as feedback to the control algorithm
E(t) = exp(iω0t)e(t) exp(iΦ1)+exp(iω0t)e(t −τ) exp(iΦ2) .
(4)
In ideal pulse shaping the phase difference δΦ = Φ1 −Φ2 of
this double-pulse pair can be chosen arbitrarily and indepen-
dent of its separation in time, τ. Inserting this electric field
into (2) yields a population induced in the atomic system
given by [22]:
1
0
2
|c2| ∝ cos(δΦ +δω τ)
(5)
where δω = ω12 − ω0 stands for the detuning of the laser
frequency from 3s → 3p1/2. In this model a change of the
pulse-pair spacing while δΦ = const. induces a slow oscilla-
tion characterized by the detuning. Tuning the relative phase
δΦ of the pulse doublet which is fixed in τ, on the other
hand, again gives rise to a periodic (1 s−1) oscillation shifted
by δωτ.
-1
-2
-3
-200
0
200
400
time / fs
The experimental data presented in Fig. 3 show the popu-
lation of the 3p1/2 state in dependence on an exclusive varia-
tion of either δΦ (Fig. 3a) or τ (Fig. 3b), respectively. While
Fig. 3a agrees perfectly with Wefer’s pulse-shaping model,
a change of the pulse spacing seems to cause an ambiguous
picture. An oscillatory behavior of the population which ex-
ceeds the capability of time resolution of our pulse-shaping
setup is superimposed by a slow modulation approximately
proportional to the detuning. This is in distinct contrast to the
expected slow oscillation.
Fig. 2. Typical FROG calculation, in the time domain, of pulse envelope
(top) and phase of a generated double pulse
In the following we will give a theoretical description of
the response of this two-level system to the sequence of two
phase-related pump pulses. The treatment will be restricted to
the 3s(|1 ) and 3p1/2(|2 ) states and the temporal evolution of
the excited level as induced by the pulse pair. Coherences be-
tween the fine-split 3p levels due to broadband excitation are

280
A variation of the pulse-pair separation, τ, will induce rapid
oscillations of the probed signal with a frequency of ω12
=
2π/1.97 fs [37]. Such atomic oscillations were investigated
earlier in Cs by Blanchet et al. [40].
The experimental transient reveals a beating pattern which
is seemingly expressible as the sum of cosines with frequen-
cies δω, ω12, ω0. Such a transient would indeed appear if,
next to the two pulses of the same phase (ideal shaping, (5)),
a third pulse would act upon the system which shares a com-
mon carrier wave with the pulse varied in time τ. The phases
of these two pulses are coupled by ω0τ (6). In a detailed dis-
cussion Wefers and Nelson [23] have shown that the passively
transmitting gaps of the liquid-crystal array give rise to such
an additional pulse. Notwithstanding its low intensity it must
be considered in the regime of saturation where this experi-
ment was performed.
Another tentative explanation of the τ-transient rests on
the assumption of a general τ-dependence of the phase of
the second pulse. This would ascribe the displacement of the
phase of the second pulse to inhomogeneities in our shaper.
Measurements which show the feedback of the controlled
one-photon excitation to a variation of τ represent an ex-
tremely sensitive criterion of the quality of a pulse shaper
incorporating a discrete mask and could serve to quantify
the deviation from ideality, since an “ideal” shaper satis-
fies the condition formulated in (5). We propose to record
the τ-transient with enhanced temporal resolution by using
shorter pulses, by increasing the number of pixels, and by
performing an analogous experiment addressing an atomic
transition in the IR (smaller ω12). This should provide us with
a deeper understanding of the physical reasons which are be-
hind these surprising results.
Fig. 3a,b. Population of Na(3p1/2) vs. characteristics of double pulse. a φ-
transient. The relative phase is varied and plotted for three different pulse
separations τ (1.2, 1.6, and 2.0 ps). b τ-transient. The pulses are set to equal
phase while the time separation τ is changed. The resolution is 2×40 fs for
the large graph and 1×40 fs for the inset
3.2 The two-photon Na(3s →→ 5s) transition
The objective of the study that will be presented in this sec-
tion is the coherent control of a nonresonant two-photon
process through spectral phase manipulation via feedback op-
timization steered by an evolutionary algorithm. A schematic
diagram of the experimental layout as well as the relevant
spectroscopic details of the employed pump and detection
scheme are displayed in Fig. 1b. The exciting laser was
tuned to λ = 598 nm which is close to the 3s →→ 5s reson-
ance, and focused to provide a maximum power density of
≈ 1×1011 W/cm2 inside the heat pipe. The population of the
5s target level optically decays to 3p or undergoes collisional
relaxation to the 4p state. Both levels are monitored sepa-
rately via their fluorescence to the 3s ground state at 589 nm
and 330 nm, respectively.
Due to the spectral width of the ultrashort 598 nm pulses
a competitive (1 + 1)-photon excitation of 5s via 3p (at
589 nm) can not be excluded rightaway. We will thus have to
offer evidence that the 5s level is indeed populated as the re-
sult of a nonresonant two-photon absorption. In a theoretical
treatment of the quantum control of multiphoton transitions
by shaped ultrashort pulses which excludes strong-field ef-
fects, Meshulach et al. [41] have calculated the effect of a π
phase step on the probability of N-photon absorption in a two-
level system. The plots of this quantity vs. the normalized
step position peak at the frequency of the N-photon absorp-
tion. They are symmetric with respect to this maximum and
If the phase of each pulse could be fixed regardless of
the time difference τ, as is presumed when pulse shaping is
performed, only a slow oscillation should show. Referring to
the experimental data we conclude that our shaper can not
fix the phase of both pulses to the same value for any pulse
spacing τ. As it seems, the phase of the second pulse is still
dependent on its delay with respect to the first. Consequently
the second pulse will induce a population with an initial phase
given by this τ-dependent optical phase and the transient of τ
will reproduce the superposition of the mutually independent
evolutions of optical phase and free atomic phase which both
depend on τ. A clear-cut discrimination of these two phase
dynamics as in ideal pulse shaping, where no phase evolution
of the laser field is allowed, can therefore not be expected.
Rather than resorting to pulse shaping, phase-correlated
pulses may be created by interferometric superposition of two
coherent beams [39, 40]. The difference in phase between the
correlated pulses can not be chosen arbitrarily since they are
coupled through their common carrier wave in the sense that
δΦ = ω0τ. Inserting this condition into (5) leads to an alto-
gether different dynamics in the τ-transient:
2
|c2| ∝ cos(ω0τ +δω τ) = cos(ω12τ) .
(6)

281
vanish for N values of the phase step position. The number
of minima is thus indicative of the order of the absorption
process.
Na(3P)
Na(4P)
Figure 4 shows our experimental result for the 3s →→ 5s
transition as a function of the step position induced by the
SLM. The position of the maximum and the occurrence of
two symmetrically arranged minima suggest a two-photon
process induced by a wavelength of ≈ 602 nm. This number
is directly read from a spectrum of the laser pulse which was
taken while pixel #43 (maximum) was blocked (see Fig. 4).
The implementation of a feedback-controlled optimiza-
tion routine requires to identify an observable which is
uniquely tied to the quantity to be controlled. Population of
5s gives rise to fluorescence from the 3p and 4p levels. 3p
may, however, also be pumped in a 589 nm one-photon step
from 3s. The text to follow describes two experiments which
address and settle this tentativeness.
without
589 nm
with
589 nm
with
589 nm
without
589 nm
Fig. 5. Response of 3p and 4p fluorescence to the presence or absence of
589 nm light (one-photon resonance)
The data of the first test are illustrated in Fig. 5 and show
the fluorescence from the 3p and 4p levels, following excita-
tion of 5s by 1 mW of unchanged or modified pump pulses.
The latter were obtained by clipping, in the Fourier plane, the
blue wings (< 591 nm) of the frequency spectrum. The en-
suing pulse spectrum is shown in the right panel of Fig. 4.
Fluorescence from 4p appears with equal intensity for either
excitation condition. The 3p analog, however, is drastically
diminished in the absence of the wavelength matching the
one-photon resonance.
m = 3/2
a) Na(3P)
b) Na(4P)
m = 1
m = 1
The previous measurement strongly indicates that 5s,
which is the precursor to 4p, is accessed nonresonantly, rather
than by a (1+1)-sequence. Supporting evidence comes from
an examination of the fluorescence intensities vs. laser power,
which is displayed in Fig. 6. Again, the 4p signal appears
unimpressed by the particularities of the pump laser’s fre-
quency profile and exhibits a quadratic slope, indicative of
a two-photon process. The 3p data are more complex. In the
presence of 589 nm the signal behaves linearly for low laser
intensity and scales ∝ I3/2 above approximately 0.2 mW,
pointing to saturation [42]. Blocking the resonant wavelength
produces the same low-intensity behavior, but a quadratic
slope beyond 0.2 mW.
The bottom line of the conclusions which may be drawn
from both checks is as follows: given the conditions of our
experiment (pump ≈ 1 mW), 4p is exclusively fed from 5s
which owes its population to a nonresonant two-photon ex-
citation. The 3p state draws to some extent from 5s, but is
predominantly pumped in a resonant single step when the
m = 2
m = 2
with 589 nm
without 589 nm
0,1
1
0,1
1
laser power / mW
Fig. 6. Power dependence of 3p and 4p fluorescence with or without
589 nm light
pulse is left unmodified. We may thus apply Meshulach’s
model [41] to describe the coherently controlled population
of Na(5s) and we have identified 4p fluorescence as a directly
linked criterion which is suited to serve as input to the steer-
ing algorithm which updates the modulator.
The nonresonant two-photon interaction of an ultrashort
pulse with a two-level system induces a transition with
pixel 43
spectral
block
Fig. 4. Left: phase step shifted across
the mask. Fluorescence from collisionally
populated 4p shows symmetry around
601,85 nm
pixel #43. Right: OPA spectrum behind
SLM observed with pixel #43 shut and
left wing clipped by blocking part of
spectrum
30
40
50
60
590
595
600
605
610
pixel number
wavelength / nm

282
a probability S2 [43]:
4p dark pulses
ꢄ
ꢄ
ꢄ
ꢄ
ꢁ
ꢅ
ꢆ
ꢅ
ꢆ
ω0
ω0
2
S2 =
A
+Ω A
ꢅ
−Ω
2
ꢄ
ꢄ
ꢄ
ꢄ
ꢇ
ꢌꢍ
2
ꢆ
ꢅ
ꢆ
ω0
ω0
×exp i Φ
+Ω +Φ
interference term
−Ω
dΩ
(7)
2
2
4p bright pulses
ꢈ
ꢉꢊ
ꢋ
where ω0 is the energy of the 3s → 5s transition which cor-
responds to 301 nm [37]. Two-photon transitions occur for
all pairs of photons which satisfy the condition ω1 + ω2 =
ω0. The detuning of frequencies ω1, ω2 from ω0/2 is de-
noted by Ω. Control of the excitation process is exercised
via the interference term and can either maximize or min-
imize the probability S2, as Meshulach et al. [41, 43] have
recently demonstrated for the nonresonant two-photon tran-
sition of caesium. Maximization is obviously achieved if the
interference term vanishes, which describes the minimum-
duration transform-limited pulse. This solution is not singu-
lar, however, since any shaped pulse with the same power
spectrum A(ω) but with an antisymmetric phase function,
Φ (ω0/2−Ω) = −Φ (ω0/2+Ω), will yield the same result,
irrespective of the particular appearance of the phase dis-
tribution. This result is counterintuitive since longer, i.e.
less-intense, pulses should be less effective in transferring
population. In their paper, Meshulach et al. [43] have also
formulated phase requirements to produce so-called dark
pulses which altogether cancel the two-photon pumping prob-
ability. No net transitions are induced as long as Φ (Ω) =
cos (βΩ).The total of solutions, discriminated by virtue of
the parameter β, is symmetric with respect to the center fre-
quency ω0/2.
In the present experiment the designed pulses were cre-
ated by phase-only modulation [38]. The task to pinpoint
the conditions which either maximize or cancel S2 was left
to an evolutionary strategy which was integrated in a feed-
back loop. Unbiased by any a-priori modeling the algorithm
set out from a phase filter Φ (n) = a cos (bn +c) with n as
the variable which numbers the LC pixels, and a, b, and c
as free parameters to be optimized. This approach is still
tractable but sufficiently general to comprise Meshulach’s so-
lution [43]. The experiment was run repeatedly for either
objective and achieved convergence within five generations.
The phase filters which were retrieved as a result of the op-
timization procedure are symmetric (cosine) in the case of
extinction, and antisymmetric (sine) in the case of enhance-
ment of fluorescence. Symmetry persists with reference to
the center frequency ω0/2 which impinges on strip #43 (see
Fig. 7). This good agreement with theory which this three-
parameter optimization produces requires to impose upper
and lower restrictions on the parameter b. In the bright-
pulse case b must be sufficiently large to allow at least four
oscillations of the phase over the width of the mask. In
the absence of this lower limit the algorithm would merely
compensate the chirp of the incoming pulse to produce the
Fourier-limited shape, i.e. the pulse having the minimum
time duration, which obviously maximizes S2. To optimize
the dark pulses b has been limited to yield a maximum of
eight phase oscillations. Lifting this restriction would re-
sult in very long pulses which are dark due to insufficient
intensity.
0
20
40
60
80
100
120
pixel number
Fig. 7. Periodic phase functions obtained from three (top) and two consecu-
tive optimization runs. In accordance with theory, traces show symmetry for
dark and antisymmetry for bright pulses. Dotted line marks pixel #43
In a further experiment we lifted the restriction on the
dimensionality of the parameter space and tried a model of
the phase filter which permitted an unbiased choice of pa-
rameters. Aiming at the generation of dark pulses we in-
troduced a phase function defined by the minimum num-
ber of sampling points connected by a linear interpolation.
Each of these points may assume 64 discrete values within
a range from 0 to 2π. Six parameters proved sufficient to
achieve this goal. The dark pulse retrieved by the algorithm
is shown in Fig. 8a whereas Fig. 8b represents the phase set-
ting of the mask. The property of being “dark” is indeed
phase-related, which is convincingly shown by comparison
with the effect induced by a chirped pulse of equivalent en-
ergy and duration. The evolution of a dark pulse as mirrored
by the decrease of the 4p fluorescence feedback signal is
shown in the right column of Fig. 9. Compared to an un-
modulated pulse the 5s population is reduced to < 3%. The
left panel proves the insensitivity of the one-photon 3s →
3p transition to a phase-only modulation. Once the reso-
nant pumping of 3p is suppressed by blocking the relevant
wavelength the fluorescence from this level perfectly matches
that of 4p (Fig. 9, bottom row). 3p is now populated via
radiative decay of 5s and is hence equally suited as feed-
back input.
Both, the three- as well as the six-parameter approach
converge after less than 10 generations, i.e. within less than
five minutes. A comparative inspection of the phase functions
returned by either method raised the question of the exis-
tence of additional solutions which are of altogether differ-
ent character. We thus expanded the previous parametrization
a)
b)
5
0
-1,0
-0,5
0,0
0,5
20
40
60
80 100 120
pixel number
time / ps
Fig. 8a,b. Free optimization using a six-parameter phase function with lin-
ear interpolation. a Cross-correlation of a typical dark pulse. b Phase values
as achieved in three different runs

283
Na(3P)
Na(4P)
a)
best
worst
mean
0
with 589 nm
with 589 nm
1
best
worst
mean
1
BEST
WORST
MEAN
0
BEST
WORST
0
5
10
0
10
20
30
40
50
generation
generation
b)
6
without 589 nm
without 589 nm
BEST
WORST
MEAN
BEST
WORST
4
2
0
0
10
20
0
10
20
20 40 60 80 100 120
pixel number
20 40 60 80 100 120
pixel number
generations
Fig. 9. Convergence data of the six-parameter search for the dark pulse. Fig-
ure shows the best and worst mask patterns for each generation. As long as
589 nm light is blocked, 3p and 4p fluorescence are equally suited as feed-
back signal. This is not the case if one-photon transitions are admitted since
the direct excitation of 3p is phase insensitive
c)
4
2
to 128 sample points, each falling between 0 and 2π as be-
fore. Figure 10 documents the convergence towards the dark
(left) and the bright pulse (right) which was attained after
≈ 10 generations. In accordance with theory an antisymmet-
ric phase function causes population enhancement (Fig. 10b).
No likewise apparent symmetry properties, however, charac-
terize the suppression of two-photon pumping. Re-runs of the
optimization procedure produced identical experimental re-
sults but differing phase functions. The solutions which the
algorithm produced bore no resemblance with the prediction
of theory. The shaped pulses show a complex phase- and
amplitude-time structure of comparable duration (≈ 2 ps). It
is thus not their peak power but rather their phase distribution
which produces qualities such as “bright” or “dark”.
0
-2
-4
-1500
0
1500
-1500
0
1500
time / fs
time / fs
Fig. 10a–c. Convergence data of the 128-parameter search for the dark (left)
and bright (right) pulse. a Normalized fluorescence intensity to document
convergence. Dashed lines mark “no signal” (0) and “unshaped reference
pulse” (1). b Right: phase structure of bright pulse showing antisymmetry.
Reference position has shifted to pixel 64 (=ꢎ 602 nm) due to re-alignment
of optical setup. c Pulse shape and phase structure in the time domain.
Bright and dark pulses show a complex structure, but note their similar
durations
new phase structures in addition to known analytic solutions.
These structures are not intuitively understandable and call
for further theoretical studies.
4 Conclusion
In this paper we studied the influence of phase-modulated fs
laser pulses on one- and two-photon transitions in an atomic
prototype system and tested the implementation of a feedback
loop using evolutionary algorithms.
The one-photon transition presents an excellent tool to test
the quality of phase-related pulses, as the excitation of the 3p
level in sodium depends critically on the relative phase of the
double pulse. It has been shown that the experimental out-
come could be explained by a combination of two limiting
cases. The relative contribution depends on the nature of the
phase coupling within the pulse sequence, which is influenced
by the experimental conditions of the setup.
The phase-modulated excitation of the two-photon tran-
sition shows that the feedback approach can be success-
fully used to find fs laser pulses for different control objec-
tives, even without an intelligent initial guess supplied by
theory. The best solutions for both extremes were obtained
within five generations. By allowing the feedback algorithm
to search in an extended parameter space the algorithm found
Since the approach of feedback optimization using evolu-
tionary algorithms works well in an atomic prototype system
the path ahead is now open to find optical pulses for differ-
ent objectives in even more complex systems without prior
knowledge of the spectroscopic details [1–3]. As an example,
it should now be possible to find laser pulses which give rise
to light-induced transparency, as well as lasers without inver-
sion, even for complex multi-level systems. The feedback ap-
proach also offers the possibility to suppress unwanted com-
petitive multiphoton absorptions while a specific transition is
excited with high intensity. This might be important when the
collision complex of a chemical reaction is investigated with
broadband fs laser pulses in real time via photoassociation
processes, where the excitation of the complex occurs near
a one-photon resonance and high intensities are necessary.
Acknowledgements. The authors thank T. Lang and H. Bauer for their tech-
nical assistance. Financial support by the DFG-Schwerpunkt “Femtosekun-
den-Spektroskopie” is gratefully acknowledged.

284
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