April 30, 2010 / Vol. 8, Supplement / CHINESE OPTICS LETTERS 21
High-dimensional sensitivity analysis of complex optronic
systems by experimental design: applications to the case of
the design and the robustness of optical coatings
Olivier Vasseur
1∗
, Magalie Claeys-Bruno
2
, Michel Cathelinaud
3
, and Michelle Sergent
2
1
Office National d’Etudes et de Recherches A´erospatiales, Palaiseau Cedex 97761, France
2
Universit´e Paul C´ezanne Aix Marseille I II, LMRE, Marseille Cedex 20 13397, France
3
Missions des Ressources et Comp´etences Technologiques, Meudon 92135, France
∗
E-mail: Olivier.Vasseur@onera.fr
Received Novemb er 22, 2009
We present the advantages of exp erimental design in the sensitivity analysis of optical coatings with a high
numb er of layers by limited numbers of runs of the code. This methodology is effective in studying the
uncertainties propagation, and to qualify the interactions between the layers. The results are illustrated by
various types of filters and by the influence of two monitoring techniques on filter quality. The sensitivity
analysis by exp erimental design of optical coatings is useful to assess the potential robustness of filters and
give clues to study complex optronic systems.
OCIS co des: 310.0310, 220.0220, 120.0120.
doi: 10.3788/COL201008S1.0021.
The study of complex optronic systems entails sensitiv-
ity analysis with a large number of parameters. Very
often the response depends on synergies or interactions
between these parameters. Due to interference charac-
teristics of multilayer filters, optical coatings make pos-
sible the evaluation of methods that can explore high-
dimensional space parameters and the presence of inter-
actions between parts of these parameters. For coatings
production with a high number of layers, sensitivity anal-
ysis is an efficient way to determine the most critical
layers of an optical coating
[1]
. Refractive index errors or
thickness errors during the manufacturing of these layers
can induce dramatic consequences on the desired optical
properties
[2]
.
We present the advantages of using the method of ex-
perimental design
[3]
, which is used for metamodel con-
structions and high-dimensional code explorations with
limited numbers of runs of the code, particularly in the
case of coatings with a high number of layers. This
methodology is more effective in studying uncertainties
propagation (refractive index or thickness values) to de-
termine the influence of errors on the optical properties,
and to quantify the interactions between the errors of
each layer. The results are illustrated by various types
of filters, particularly bandpass filters and multiple half-
wave filters. Different designs such as factorial, fractional
factorial, and space-filling designs are used to present the
results.
Furthermore, we study the influence of two monitoring
techniques, and show the most critical coating layers and
the dep endency of these layers with future manufactur-
ing.
The results show that the study of thin-film filters
is very useful in examining the interactions of high-
dimensional systems due to the filter’s adjustable number
of layers, and the existence of interactions between these
layers.
Finally, we demonstrate that sensitivity analysis of op-
tical coatings by experimental design is useful in assess-
ing the potential robustness of filters, and gives clues to
study complex optronic systems.
The codes to study complex phenomena become more
and more realistic with a larger input data set. However,
due to the complexity of the mathematical system un-
derlying the computer simulation tools, there are often
no explicit input-output formulas. Although computer
power has significantly increased in the past years, the
evaluation of a particular setting of the design parame-
ters may still be very time-consuming. The simulator is
often replaced by a metamodel to approximate the re-
lationship between the code and the design parameters.
These metamodels are built using numerical designs of
experiments that can indicate interactions between the
parameters. The choice of an underlying empirical model
(depending on accuracy and interactions level) can be
written as
Y = Cste +
i
b
i
X
i
+
i<j
b
i,j
X
i
X
j
+
i<j<k
b
i,j,k
X
i
X
j
X
k
+ . . ., (1)
where Y is the response of the model, X
i
is the ith pa-
rameter, b
i
is the effect of the ith parameter, and b
i,j,
the interaction between the ith and jth parameters. This
model is valid for the levels −1 and +1 of the undimen-
sional variables (X
i
). In our case, parameters X
i
can be
thickness, refractive index, or optical thickness.
In our study, we use factorial and fractional factorial
designs at two levels for each parameter (low level: −1
or −; high level: +1 or +). The number of runs is 2
n
with n parameters for a full factorial design, and 2
n−p
for a fractional factorial design corresp onding to a subset
that is 1/2
p
of the full factorial design 2
n
where p is the
1671-7694/2010/S10021-04
c
° 2010 Chinese Optics Letters