Optimization of chirped mirrors.

Abstract

We demonstrate that a highly efficient global optimization of chirped mirrors can be performed with the memetic algorithm. The inherently high sensitivity of chirped-mirror characteristics to manufacturing errors can be reduced significantly by means of the stochastic quasi-gradient algorithm. The applicability of these algorithms is not limited to chirped mirrors.

Full text

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Optimization of chirped mirrors
Vladislav Yakovlev and Gabriel Tempea
We demonstrate that a highly efficient global optimization of chirped mirrors can be performed with the
memetic algorithm.The inherently high sensitivity of chirped-mirror characteristics to manufacturing
errors can be reduced significantly by means of the stochastic quasi-gradient algorithm.
bility of these algorithms is not limited to chirped mirrors.
OCIS codes:
310.6860, 230.4170, 320.0320.
The applica-
© 2002 Optical Society of America
1.
Dispersive dielectric multilayer mirrors1?henceforth
called chirped mirrors ?CMs?? have contributed sig-
nificantly to enhancement of the performance, com-
pactness, and reliability of femtosecond laser sources.
The generation of sub-10-fs pulses directly from os-
cillators requires accurate, broadband, higher-order
dispersion control,2which can now be achieved rou-
tinely with CM dispersion-controlled oscillators.3
Progress in CM design and manufacturing in con-
junction with advanced oscillator architectures has
permitted the direct generation of sub-6-fs pulses.4–6
Novel spectral broadening techniques7–9allow spec-
tra to be produced that can extend over more than 1
opticaloctave;subsequentcompressionwithCMshas
led to pulses shorter than 5 fs.7,10,11,12
The first CM designs were obtained by computer
optimization and had a bandwidth of 200 nm at the
central wavelength of 800 nm.1
methods have been proposed for the calculation of
starting structures that, after limited computer opti-
mization,converge to
performance.13–16
Although the proposed design
methods are substantially different from one an-
other, the final designs perform comparably, as all
the designs suffer from a fundamental limitation:
To obtain a smooth dispersion curve on reflection, one
must match accurately the front section of the mirror
Introduction
Several predesign
designswithenhanced
to the medium of incidence.
plementations of CMs overcame this limitation by
preventing the beams reflected at the front interface
from interfering with the useful beam reflected
within the multilayer.17,18
mirrors that provide high reflectance and accurate
dispersion control over a full optical octave were dem-
onstrated.19,20
Mirrorsofthiskindmanufacturedby
optically attaching a thin glass wedge to the top of a
multilayer structure have been called tilted-front-
interface chirped mirrors.20
Although the recently found improved implemen-
tations and the analytical predesign methods were
essential for enhancing the performance of CMs, they
did not obviate the need for efficient computer opti-
mization techniques.Two major trends in CM de-
velopment can currently be identified:
mirror designs with even larger bandwidths, which
would permit the generation of nearly single-cycle
pulses for scientific applications, and the search for
robust CM designs that can be reliably manufactured
with a cheap coating technology and thus would be
suitable for industrial production.
geredthedevelopmentoftheoptimizationtechniques
described in this paper.
In Section 2 we describe an efficient global optimi-
zation algorithm that has been successfully tested for
CM design. One of the main problems with optimi-
zation of multilayer dielectric coatings is that the
merit function usually has many local extrema,21so
there is always a risk that a local optimization rou-
tine will get stuck far away from the best extremum.
Global optimization was recognized long ago as a
solution to this problem.
successful attempts to apply global optimization al-
gorithms to optical multilayer structures,21–23but
there is one common problem:
timization algorithms demand a large number of
evaluations of the merit function, and the optimiza-
Recently proposed im-
Drawing on this concept,
the quest for
These needs trig-
There have been many
All known global op-
V. Yakovlev ?vladislav.iakovlev@tuwien.ac.at? is with the Insti-
tute of Automation and Electrometry, Novosibirsk, Russia, and the
Instituteof Photonics,Vienna
Gusshausstrasse 27, A-1040 Vienna, Austria.
the Institute of Photonics, Vienna Institute of Technology,
Gusshausstrasse 27, A-1040 Vienna, Austria.
Received 18 February 2002; revised manuscript received 18 July
2002.
0003-6935?02?306514-07$15.00?0
© 2002 Optical Society of America
University of Technology,
G. Tempea is with
6514APPLIED OPTICS ? Vol. 41, No. 30 ? 20 October 2002

Page 2

tion process is thus often extremely time consuming.
Here we show how the so-called memetic algorithm24
can be utilized for relatively fast global optimization
of chirped mirrors.
Furthermore, we present an algorithm that can
improve the robustness of CM designs.
delay dispersion ?GDD? of chirped mirrors is highly
sensitive to small discrepancies between the layer
thicknesses of a calculated design and those of the
manufactured mirror.This effect becomes increas-
ingly more pronounced as the required spectral range
gets broader. We show how the sensitivity of CMs to
manufacturing errors can be substantially reduced
with the aid of the so-called stochastic quasi-gradient
optimization.25,26
The group-
2.
The term “memetic algorithm” is used to describe the
class of evolutionary algorithms in which local search
plays a significant role.24
memetic algorithm ?which can be considered as an
improved genetic algorithm? can be utilized for rela-
tively fast global optimization of chirped mirrors.
Usually a type of genetic algorithm ?a particular type
of evolutionary algorithm? is used in conjunction with
local optimization.Employing terminology that is
specific to evolutionary computation, we call a set of
trial designs a population, and by the term “member
of the population” we shall mean a particular design
from this set.As in any genetic algorithm the
memetic optimization process goes from population
to population by means of crossover and mutation
operators. The difference between the genetic and
the memetic algorithms is that, in the latter, each
newly constructed member is locally optimized before
it is included in the new population; i.e., by construct-
ing a new set of trial designs from the old set we
locally optimize each design that is going to be added
to the new set.This local optimization does not nec-
essarily have to lead to the ultimate extremum:
most important thing is that it be fast but able to
considerably improve designs, bringing them closer
to local extrema.The effectiveness of a memetic al-
gorithmdependstoalargeextentontheeffectiveness
of the local optimization, which we would call partial
refinement.We show now how a fast partial refine-
ment of CMs can be implemented.
The properties of multilayer dielectric structures
are calculated by the transfer matrix method27?see
Appendix A?.To calculate the reflectivity of a mir-
ror consisting of n layers at a particular wavelength,
onehastocalculatetheproductofn transfermatrices
Mk.This is the most time-consuming part of any
CM optimization routine.
made faster when the merit function has to be eval-
uated for several designs that differ from some par-
ticular previously characterized design by only one
layer.Let us define n matrices Lk,
Lk??
?i?1
Memetic Algorithm
Here we show how the
The
The calculations can be
1
k ? 1
1 ? k ? n,
k?1Mi
(1)
and n matrices Rk,
Rk??
1
?i?k?1
k ? n
1 ? k ? n.
n
Mi
(2)
When all the matrices are calculated, one can quickly
find the reflectivity in the case when only the kth
layer is changed, because only one single matrix Mk
has to be calculated and only two matrix multiplica-
tions instead of n ? 1 have to be performed:
?
i?1
n
Mi? M1M2. . . Mk?1MkMk?1. . . Mn? LkMkRk.
(3)
This opens a way for quick evaluation of the gradient
of the merit function, quick optimization of one par-
ticular layer, and so on.
implement partial refinement by using this method.
For example, one can calculate the gradient and op-
timize the mirror along this direction ?hill climbing?
or optimize each layer one by one.
Many modifications of the memetic algorithm can
be developed. Their comparison goes beyond the
scope of this paper; we describe here a version of the
algorithm that has proved to be useful for CM opti-
mization. The merit function was defined such that
lower values correspond to better designs; thus we
aim to minimize the merit function.
ulation ?the set of initial designs? is constructed ran-
domly ?except, maybe, a few designs that can be
obtained from the analytical theory16or from previ-
ous optimizations?.All the designs have the same
number of layers made from the same materials; only
the layer thicknesses are different.
tion two members of the population are randomly
chosen according to their values of the merit function:
The lower the value is, the more chances there are
that the member will be selected ?rank selection was
used?.Then a new member is formed by the cross-
over:A part of its layers is taken from the first
design, the rest is taken from the second design, and
together they form the new member; the sizes of the
parts are determined randomly each time.
congestion of designs in a small number of local min-
ima we make a simple heuristic check on whether the
newly formed member is in the same local minimum
as one of the existing members:
designs we check whether the maximal difference
between layer thicknesses of the selected and the
newly constructed designs is smaller than 10 nm.
it is, we try to move the designs toward each other
step by step ?considering each design as a point in the
multidimensional parametric space?, allowing only
those steps that decrease the value of the merit func-
tion.If we manage to decrease the distance between
them to zero, one of the designs is excluded from the
population.Finally we perform a partial refinement
of the new member and form a new population by
adding the member to the old population.
step we kill the worst member of the new population
if the population size exceeds the prescribed limit.
There are a few ways to
The initial pop-
On each itera-
To avoid
For each of the
If
In a last
20 October 2002 ? Vol. 41, No. 30 ? APPLIED OPTICS6515

Page 3

When the global optimization is completed ?after the
specified number of iterations is reached or the con-
vergence criterion is fulfilled?, a few best members
are chosen and further optimized with the aid of a
full-power local optimization ?conjugate gradients al-
gorithm,28for example?.
3.
Any manufacturing process introduces systematic
and random errors into the optical thicknesses of
layers.The systematic errors ?e.g., a uniform in-
crease in the thicknesses of all layers? are usually not
critical for chirped mirrors, because they only slightly
change the reflectance and dispersion characteristics
or shift them in the spectral domain, but the random
errors pose a serious problem, causing large oscilla-
tions of the mirror’s GDD.
algorithm described in Section 2 leads to a set of
designs of comparable quality.
designs, one may notice that they have different sen-
sitivities to such random perturbations of layer thick-
nesses.This raises the question:
optimize this sensitivity?
to do so; for example, sensitivity optimization is in-
cluded in commercial software29?where it is known to
work quite slowly?.Still, as far as we know, nobody
has yet reported how algorithms of optimization of
noisy functions can be applied to the design of chirped
mirrors.
Why should we make the merit function noisy and
how can we do it?If one were able to evaluate the
merit function for each of the trial designs by manu-
facturingthecorrespondingmirrorandmeasuringits
properties, the function would look as if some noise
were added to it. To find a minimum of such a func-
tion would mean to find a point where its average
value is minimal, and this solution would obviously
give us the most robust design.
this noise, calculating the merit function each time
not exactly for the given layer thicknesses but for
randomly perturbed thicknesses for which the ran-
dom perturbation simulates the perturbations intro-
duced by the manufacturing process.
The classic optimization algorithms are barely ap-
plicable when the function values are corrupted with
noise,becausetheyrelyondeterminationofthemerit
function.One of the approaches that are suitable
for this case is called stochastic quasi-gradient meth-
ods.25
The underlying idea is simple:
ation we estimate the gradient of the merit function
and make a step in the direction opposite the gradi-
ent. It resembles the steepest-descent method, but
the stochastic nature of the merit function requires
specialized techniques to be able to estimate the gra-
dient and places certain restrictions on the sequence
of step sizes. If all the necessary mathematical con-
ditions ?thoroughly described in Ref. 25? are met, the
algorithm is guaranteed to converge to a stationary
point with probability 1, though the convergence may
be slow. One of the advantages of this method is
that the sensitivity does not have to be explicitly
calculated to be optimized.
Stochastic Quasi-Gradient Algorithm
The global optimization
In analyzing these
Is it possible to
There have been attempts
We can simulate
At each iter-
The algorithm that we
implemented is analogous to the one used for feed-
back control of adaptive optics,26for which it is
termed parallel stochastic perturbative gradient de-
scent.To estimate the gradient of the merit func-
tionweused thesimultaneous
approximation.30
Let ?x1, x2, . . . , xn? be a vector of
layer thicknesses.The noisy merit function J˜?x1,
x2, . . . , xn? is connected to the exact merit function
J?x1, x2, . . . , xn? by application of random technolog-
ical perturbations to the layer thicknesses:
perturbation
J˜?x1, x2, . . . , xn? ? J?T1?x1?, T2?x2?, . . . , Tn?xn??.
(4)
To estimate the gradient of this function we introduce
m vectors of n mutually independent mean-zero ran-
dom variables ?pk1, pk2, . . . , pkn?, k ? 1, . . . , m.
the variables satisfy certain mathematical condi-
tions,30the most important of which is that E?pki
or some higher inverse moment of pkimust be
bounded, then the gradient of J˜?x1, x2, . . . , xn? may
be estimated from 2m measurements:
m?
where
If
?1?
?J˜?1
m?
k?1
J˜k
?? J˜k
2ckpk1
···
J˜k
?
?? J˜k
?
2ckpkn?
, (5)
J˜k
?? J?T1?x1? ckpk1?, T2?x2? ckpk2?, . . . ,
Tn?xn? ckpkn??,
J˜k
?? J?T1?x1? ckpk1?, T2?x2? ckpk2?, . . . ,
Tn?xn? ckpkn??,
and ckis a positive scalar that determines the step
size.It is common to take the random variables pki
to be symmetrically Bernoulli distributed.
number of iterations m provides a more accurate es-
timation of the gradient, improving the convergence
of the algorithm, but requires more evaluations of the
merit function, increasing the time necessary for one
iteration.A compromise should be found that will
produce the best performance.
ture of this algorithm is that the optimal value of m
is often much smaller than the number of variables n,
which makes the algorithm faster than the one based
on the classic estimation of the gradient.
A larger
The remarkable fea-
4.
Global optimization outperforms local optimization
algorithms, particularly if no good starting design is
available. Despite progress in analytical design
techniques for chirped mirrors, for several dispersive
mirror design problems no direct synthesis is cur-
rently possible.One of these problems is the design
of dichroic CMs ?input couplers? used in femtosecond
lasers, where they exhibit high reflectance and con-
trolled negative dispersion over the fluorescence
spectrum of the active laser medium and high trans-
Results
6516 APPLIED OPTICS ? Vol. 41, No. 30 ? 20 October 2002

Page 4

mittance over the relatively narrow wavelength
range at which the medium efficiently absorbs the
pump radiation. Although double-chirped mirrors
inherently tend to exhibit large transmittance at
wavelengths below the high-reflectance range, pa-
rameters such as the position, the width, and the
losses of the high-transmittance band cannot be ex-
plicitly taken into account by analytical predesign;
thus the use of a global optimization technique is
called for.
To demonstrate the powerfulness of the memetic
algorithm we present a CM designed to be used as an
input coupler of a Ti:sapphire oscillator ?Fig. 1? that
has a high reflectance and a nearly constant GDD in
the range 650–900 nm ?the target GDD was set to
?35 fs2? and transmission as high as possible in the
range 527–537 nm. The mirror consists of 60 layers
made from two materials:
material is SiO2, and the refractive index of the sec-
ond material is 2.25 at 800 nm.
designed, first, with the conjugate gradients algo-
rithm starting from a double-chirped design provided
by the analytical theory16and, second, by use of the
memetic algorithm starting from the same initial
stack formula. The population for the memetic al-
gorithm consisted of 100 members; 3000 iterations of
the algorithm were performed.
Fig. 1, the global optimization found a better solution
than the local refinement.
The memetic algorithm has also proved to be effi-
cient in the design of ultrabroadband, octave-
spanning chirped mirrors such as the one depicted in
Fig. 2. The mirror consists of 70 layers made from
the same materials as the input coupler presented
above.The incidence medium for this design ?fused
silica? has a refractive index close to that of the low-
refractive-index material; this is the so-called “back-
side-coated mirror”19 or “tilted front-interface”20
the low-refractive-index
The mirror was
As can be seen from
approach—a powerful technique for suppressing the
GDD oscillations and increasing the mirrors’ band-
width.To design such a broadband mirror we used
a merit function based on analysis of the reflection of
a trial pulse to properly weight the reflectance
against GDD during optimization.
As stated above, without the partial refinement the
memetic algorithm that we implemented would be
just a genetic algorithm.
memetic approach is demonstrated by a comparison
of optimizations with and without the partial refine-
ment. Such a comparison is shown in Fig. 3.
cannot compare the algorithms in terms of the num-
ber of merit function evaluations because the
memetic algorithm uses two different ways to calcu-
late the function, so we compare them in terms of the
processor time required by the optimization ?a PC
with a 600-MHz Athlon processor was used?.
though the partial refinement slows down the opti-
mization process at the initial stage, it allows the
memetic algorithm to outperform its genetic counter-
part on the long-term run.
A result of the sensitivity optimization is shown in
Fig. 4. A mirror consisting of 49 layers was opti-
mized in the wavelength range 650–900 nm for nor-
mal incidence of light from the air.
obtained was further optimized with the aid of a sim-
ple implementation of the stochastic quasi-gradient
algorithm, in which the step size was kept constant
?equal to 0.5 nm?, the random variables pkiwere al-
lowed to take only values ?1 and ?1 with equal
probabilities, ckwere chosen to be equal to 2 nm; we
performed 10,000 iterations to get the final design.
The layer thicknesses before and after the sensitivity
optimization are compared in Table 1.
assumptions were made regarding the error distribu-
tion of layer thicknesses introduced by the manufac-
turing process:First,
The effectiveness of the
We
Al-
The design
Two major
the perturbationswere
Fig. 1.
using the conjugate-gradient algorithm ?dashed curves?.
Large transmittance in the wavelength range 527–537 nm as well as high reflectance in the range 650–900 nm together with a target
dispersion of ?35 fs2were required. The error bars correspond to perturbation of layer thicknesses with a standard deviation of 1 nm.
Comparison of an input-coupler design found by the memetic algorithm ?solid curves? with that obtained by local optimization
In both cases the starting design was obtained by the double-chirping method.
20 October 2002 ? Vol. 41, No. 30 ? APPLIED OPTICS6517

Page 5

simulated by an additive noise that affected all the
layers in the same way, and, second, the distribution
was taken to be Gaussian with a standard deviation
of 1 nm.These approximations agree qualitatively
with the experimental measurements of available
mirrors, though we cannot rigorously prove their va-
lidity.It was observed that the sensitivity optimi-
zationsubstantiallyshortens
Although it tends to decrease the quality of the ide-
ally manufactured design ?with the layer thicknesses
exactly equal to the prescribed ones?, this effect is
overcompensated for by the improved design robust-
theerror bars.
ness, provided that the technological perturbations
have been well enough approximated.
5.
Using the memetic approach to global optimization,
we constructed an algorithm that showed good re-
sults when it was applied to chirped-mirror design.
The key component of this algorithm is the fast par-
tial refinement of trial designs.
quasi-gradientalgorithmcanbeutilizedtoreducethe
mirror’s sensitivity to the random perturbations of
layer thicknesses that are introduced by any manu-
facturing process. We do not see any obstacle to
Conclusions
The stochastic
Fig. 2.
optimizationproduceddesignsofcomparablequalityinthiscase,probablybecausetheinitialdesignwasclosetotheglobaloptimum.
mirror consists of 70 layers, the incidence medium is fused silica, the angle of incidence is 5°, and the light is p polarized.
reflectance of the fused-silica wedge to air nor the dispersion of the wedged plate was taken into account in the figure.
Example of a broadband TFI chirped mirror designed for the wavelength range 560–1130 nm.The global and the local
The
Neither the
Fig. 3.
algorithm formed by exclusion of the partial refinement.
axis is the value of the merit function of the best member in the
population.Each curve is the result of averaging over 11 runs of
the optimization code.
Comparison of the memetic algorithm with the genetic
The y
Fig. 4.
chirped mirror.
medium is air; normal incidence.
for a standard deviation of technological errors equal to 1 nm.
Result of sensitivity optimization of a modest-bandwidth
The mirror consists of 49 layers and the incidence
The error bars were obtained
6518 APPLIED OPTICS ? Vol. 41, No. 30 ? 20 October 2002