Optical coating design approaches based on the needle
A. V. Tikhonravov, M. K. Trubetskov, and G. W. DeBell
Design approaches for optical thin films that recognize the key role of a design’s total optical thickness are
optimization procedures. Using the described design approaches, an optical coating engineer can obtain a
set of theoretical designs with different combinations of principal design metrics (merit function value,
number of layers, and total design optical thickness); this extends opportunities for choosing the most
nearly the same combinations of principal design metrics.
310.1620, 310.6860, 240.0310.
© 2007 Optical Society of America
Baumeister1introduced the idea of computational op-
timization in the design of optical coatings in 1958.
Since that time many other computational design
techniques have been proposed. Descriptions of many
of them can be found in various texts on optical thin-
film coatings.2–7Dobrowolski8has proposed a basic
classification scheme for computational design tech-
niques into local (refinement) and nonlocal (synthe-
sis) methods; this classification is useful in describing
modern design approaches.
Virtually all nonlocal design techniques use local
optimization in at least one step of the design pro-
cedure; thus computational performance of local
optimization procedures is of high importance.
Dobrowolski and Kemp9have studied and com-
pared the computational efficiency of different opti-
mization procedures. Unfortunately there is no single
universal local optimization procedure that is suit-
able in all situations. For this reason commercial
thin-film software packages typically provide the
user an opportunity to use several local optimization
techniques.10Experience has shown that local opti-
mization procedures that utilize merit function (MF)
derivatives achieve the highest accuracy and speed
when using analytical expressions for these deriva-
In this paper we consider nonlocal design ap-
proaches based on the needle optimization tech-
nique,12,13the needle optimization technique itself
being a nonlocal design procedure. The basic ideas
of the needle optimization technique and its general
description can be found in Refs. 4 and 13. Further
modifications of this technique are described in
In 1986 Dobrowolski15noted that “a certain mini-
mum overall optical thickness is required for a solu-
tion to a given problem no matter what design
method is employed.” The practical importance of
this statement is clearly demonstrated in publica-
tions devoted to Fourier-transform techniques16–20; it
was also confirmed by using design examples con-
structed using the needle optimization technique.14
Theoretical considerations related to the key role of
design total optical thickness are found in Refs. 14
and 21. The design approaches considered in this
paper are a further confirmation of the importance
that design total optical thickness plays in the design
We first consider a design approach aimed at ob-
taining a set of solutions with different combinations
of principal design metrics. These principal design
metrics are MF value, number of design layers, and
design total optical thickness. We will also consider
options for controlling the total number of design
layers. In Section 3, the generation of multiple solu-
tions with nearly the same combinations of principal
design metrics is discussed. To obtain such solutions,
the needle optimization technique is combined with
multiple optimizations of random starting designs.
A. V. Tikhonravov and M. K. Trubetskov (firstname.lastname@example.org) are
with the Research Computing Center, Moscow State University,
Leninskie Gory, 11992 Moscow, Russia. G. W. DeBell (gdebell1@
earthlink.net) is with MLD Technologies, 2672 Bayshore Parkway,
Suite 701, Mountain View, California 94043, USA.
Received 14 July 2006; accepted 8 September 2006; posted 9
October 2006 (Doc. ID 73046); published 25 January 2007.
© 2007 Optical Society of America
704APPLIED OPTICS ? Vol. 46, No. 5 ? 10 February 2007
Other design approaches aimed at obtaining multiple
solutions are also outlined.
Dobrowolski22first introduced the idea of gradual
evolution in 1965. He proposed to proceed with opti-
mization after the refinement procedure was finished
design and then to either reoptimize the new struc-
ture with respect to just the added layers or with
respect to all the layers of the new multilayer struc-
ture. A variation on the gradual evolution approach
can be implemented based on the needle optimization
technique by forcibly increasing a design’s total opti-
cal thickness at the end of each needle optimization
procedure. Figure 1 shows a schematic representa-
tion of this approach.
In principle, synthesis by gradual evolution can be
performed with no starting design at all. In this case
a one-layer design with zero layer thickness is spec-
ified at the first step of the design procedure. The zero
thickness layer is then forcibly increased by the re-
spective algorithm of the gradual evolution procedure
(see Fig. 1). The algorithm for increasing the design
total optical thickness is described in Ref. 14. Typi-
cally, the increase of design total optical thickness re-
sults in an increase of the MF value, but a subsequent
needle optimization procedure entirely compensates
for this increase and leads to a further decrease of the
merit function. As a result, the gradual evolution ap-
proach gives a series of optical coating designs with
Gradual Evolution Approach Based on the Needle
increasing design total optical thicknesses and corre-
spondingly decreasing MF values.
To illustrate the gradual evolution approach we use
a model design problem with the following target at-
tributes. The target transmittance is specified in the
spectral region from 440 to 700 nm. It has four 100%
transmittance zones, each of them with a 10 nm
width (490–500 nm, 540–550 nm, 590–600 nm, and
640–650 nm). In other zones of the specified spectral
region the target transmittance is 0%. The high re-
flection zones are separated from the high transmis-
sion zones by 2 nm gaps. For this model design
example we use a substrate with a refractive index of
1.52; an incident medium of index 1.0; and high- and
low-index nondispersive layer materials with refrac-
tive indices of 2.2 and 1.46, respectively. Normal light
incidence is assumed throughout this paper.
The MF for the design problem we are considering
was specified in the form
where Tˆ??? is the target transmittance specified in
the model design problem, T??? is the actual trans-
mittance of the designed coating shown as a percent-
age, ??j? is the wavelength grid where the target
transmittance is specified, L is the total number of
wavelength grid points, and ?Tjrepresents the MF
tolerances that were all set equal to 1%. It is worth
mentioning that with 1% tolerances, the MF repre-
Fig. 1.Schematic of the gradual evolution approach enhanced with the needle optimization technique.
10 February 2007 ? Vol. 46, No. 5 ? APPLIED OPTICS705
sents a rms deviation of the actual transmittance
from the target transmittance in percentage.
In all computational examples of this section, we
use 485 wavelength grid points, the wavelength grid
is nonlinear, being denser at shorter wavelength val-
ues and sparser at longer wavelength values.
Hypothetically, gradual evolution based on the nee-
dle optimization technique is a never-ending design
procedure; thus some termination criteria should be
specified (see Fig. 1). It is natural to terminate the
gradual evolution procedure if the required level of the
MF is achieved, if the total number of design layers
exceeds a predetermined value or if the total optical
a specified value. Figure 2 illustrates the progression
of the gradual evolution procedure when only one ter-
mination criterion, namely, an upper limit of 30 ?m
for the design total optical thickness, was specified,
and there was no initial starting design.
Each circle in Fig. 2 corresponds to one of the de-
signs obtained in the course of the gradual evolution
procedure. Note that MF values achieved with the
designs obtained are presented on a logarithmic
scale. One can see that these values are decreased
when design total optical thicknesses are increased.
The number of design layers are not shown in Fig. 2,
but as a whole they are increasing with the increase
of the design optical thicknesses. A general impres-
sion about the numbers of layers in the designs ob-
tained with this procedure and about their spectral
properties at different design total optical thick-
nesses can be drawn from Fig. 3, where results for
four designs are depicted in more detail.
Figure 3 clearly demonstrates qualitative varia-
tions in the spectral properties of designs with
larger design total optical thicknesses. Figure 2 il-
thicknesses: Each circle corresponds to one of the designs obtained
in the course of the gradual evolution procedure; the four dark
triangles correspond to the designs with spectral transmittances
shown in Fig. 3.
Achieved merit function values versus design total optical
MF, merit function value; N, number of design layers; and TOT, design total optical thickness in nanometers.
Spectral transmittances of several designs obtained in the course of the gradual evolution process. Principal design metrics are
706APPLIED OPTICS ? Vol. 46, No. 5 ? 10 February 2007
lustrates typical variations in MF values; note that
there are some critical total optical thicknesses at
which an especially rapid decrease of MF values is
observed. This is further confirmation of Dobrowolski’s
observation of the key role a design’s total optical
thickness plays in achieving the desired spectral
The gradual evolution approach based on needle
optimization provides an optical coating engineer
with expanded opportunities for choosing the most
practical design. The generation of a set of theoret-
ical designs with various combinations of principal
design metrics facilitates the choice of a design with
an optical thickness required to achieve an appro-
priate approximation of desired spectral character-
The number of design layers generally increases
with the growing design total optical thickness. How-
ever, there are approaches that can be used to limit
the number of layers in a design. One approach is to
stop the gradual evolution procedure when the MF
values are lower than the value that is actually re-
ber of design layers.
The first procedure consists of removing thin layers
with subsequent reoptimizations of the modified de-
signs. For example, all layers with thicknesses less
than 10 nm can be removed one by one, thinnest
layer first. It is essential to remove layers in such a
way as to keep design total optical thicknesses un-
changed. This can be done by changing the refractive
index of a thin layer to that of the preceding or fol-
lowing layer, i.e., H for L or L for H in standard
notation for two material designs. Practical experi-
ence shows that layers thinner than 10 nm can usu-
ally be removed without a large degradation of the
The second procedure is called design cleaning. In
this procedure we first specify an allowed increase of
the MF value; then we try to remove design layers
one by one while keeping the design total optical
thickness unchanged. This is done in the same man-
ner described in the first procedure. The layer caus-
ing the smallest degradation of the MF value is
removed, and the resulting design is reoptimized.
This procedure is repeated until the predetermined
MF increase is exceeded.
Figure 4 illustrates an application of the design
cleaning procedure to the 146-layer design obtained
in the course of the gradual evolution procedure
shown in Fig. 3(d). The initial design has an optical
thickness of 23,665 nm and a MF value of 0.114. An
increase of three times the MF value was allowed
for the design cleaning procedure. As a result, the
number of design layers was reduced from 146 to 93.
The total optical thickness of the cleaned design is
23,104 nm, and the new MF value is 0.306.
Other design approaches aimed at obtaining de-
signs with various combinations of principal design
metrics are discussed in the next section.
Solutions to Design Problems
The recognition of the key role of design total optical
thickness leads to a new philosophy of choosing a
design strategy. Once the designer has a technique
such as needle optimization at his disposal, his at-
tention can switch from mere optimization of the MF
for choosing a design strategy that can provide the
most manufacturable design. Needle-enhanced grad-
ual evolution facilitates a convenient estimate of the
minimum total optical thickness required to achieve
desired spectral characteristics. Knowing this value,
we can perform a search for suitable starting designs.
A useful approach for such a search combines the
needle optimization technique with a random design
Let us denote the desired design total optical thick-
can be generated utilizing the following strategy.
Let N0be the total number of layers for each design
If we specify the layer optical thicknesses in each
design as random values with a mathematical expec-
tation equal to TOT?N0, then the total optical thick-
nesses of each starting design should be close to the
Each random starting design is refined by using
one of the more effective local optimization tech-
niques.23The refined designs are then ordered in
accordance with their respective MF values. Typi-
cally, even the best designs obtained by the refine-
ment procedure can be further improved by using the
needle optimization procedure. In this way, we obtain
a set of designs whose total optical thicknesses do not
differ too much from the initially specified TOT value.
This property of the resulting designs is connected
with the fact that design total optical thickness is a
Design Approaches Aimed at Obtaining Multiple
design cleaning process from the 146-layer design with spectral
characteristic depicted in Fig. 3(d).
Transmittance of the 93-layer design obtained with the
10 February 2007 ? Vol. 46, No. 5 ? APPLIED OPTICS707
rather stable parameter that does not differ too much
from its initial value either in the course of the re-
finement procedure or during the subsequent needle
We use the design example of the previous section
to illustrate the approach described above. Columns
2–4 of Table 1 show the principal metrics (MF, N,
and TOT) of the ten best designs according to their
MF values. This set of designs was selected from the
designs obtained by the refinement of 50 random
starting designs, all having 200 layers with layer
optical thicknesses being randomly chosen such that
each has an equal probability of having an optical
thickness between 0 and 240 nm, and accordingly,
having a mathematical expectation value equal to
120 nm. Thus the initial total optical thicknesses of
the starting designs were close to 24,000 nm. This
value was chosen because it was just above the thick-
ness value where a rapid decrease in the MF is ob-
served (see Fig. 2). Table 1 demonstrates (see column
4) that all designs obtained by the refinement pro-
cedure have total optical thicknesses close to the
initially chosen value of 24,000 nm.
Table 1 also shows that needle optimization im-
proved the MFs of all the refined random starting
designs. Note that although the needle optimization
process increased the number of design layers, the
change in total optical thickness of the designs is
insignificant. The manufacturability of these designs
can be improved by using the thin layer removal and
design cleaning procedures described above.
Practical experience gained from application of the
design approach described above has shown that in
many cases multiple solutions with nearly the same
combinations of principal design metrics can be ob-
tained. We demonstrate this by using two different
We first consider a partitioned reflector–
transmitter with two high-transmission zones and
two high-reflection zones. The target transmittance
is equal to 100% in the spectral zones from 588 to
622 nm and from 1520 to 1570 nm, and it is equal to
0% in the spectral zones from 738 to 762 nm and from
1040 to 1090 nm. The step of the wavelength grid
used for the definition of the MF [see Eq. (1)] is 1 nm
in the first and third zones and 2 nm in the second
and fourth zones. Thus the total number of wave-
length grid points in Eq. (1) is 112. As in the previous
examples, all tolerances in Eq. (1) were set equal to
1%, which means that the MF represents a rms de-
viation of actual transmittance from the target trans-
mittance expressed in percentage. The substrate
refractive index is 1.52, the incident medium is air,
and the layer material refractive indices are 2.2 and
1.46 as before.
Table 2 shows the principal design metrics of three
partitioned reflector–transmitter designs. All three
designs have 27 layers and nearly the same total
optical thicknesses. Their MF values are also nearly
the same. However, the designs are different, which
is clearly seen from their refractive index profiles in
Fig. 5. Their spectral properties outside the target
region are also quite different. Figure 6 shows the
transmittances of the first two designs from Table 2.
transmitter designs. Design metrics for these designs are pre-
sented in Table 2.
Refractive index profiles of three partitioned reflector–
Reflector–Transmitter Designs with Refractive Index Profiles
Shown in Fig. 5
Principal Design Metrics of the Three Partitioned
Table 1.Principal Design Metrics of the Ten Best Designsa
After Refinement of
aThese designs are obtained by the refinement of 50 starting
designs having 200 layers with random layer optical thicknesses
having mathematical expectations that are equal to 120 nm (col-
umns 2–4). The design metrics of these designs after the applica-
tion of the needle optimization procedure (columns 5–7).
708 APPLIED OPTICS ? Vol. 46, No. 5 ? 10 February 2007
The transmittance of the third design is not shown
because it clutters the figure.
Multiple solutions for a hot-mirror design with
nearly the same principal design metrics are shown
in Table 3 and Figs. 7 and 8. In this case, target
transmittance values were specified at 392 evenly
distributed wavelength points with a 2 nm spacing.
They are equal to 100% in the spectral region from
400 to 690 nm and 0% in the spectral region from 710
to 1200 nm. In other respects, the MF is the same as
before. For this design problem the layer material
refractive indices are equal to 2.35 and 1.45, the sub-
strate refractive index is 1.52, and the incident me-
dium is air.
Figure 7 shows the refractive index profiles of three
hot-mirror designs whose principal design metrics
are given in Table 3. The transmittance of the design
with its refractive index profile shown in Fig. 7(a) is
shown in Fig. 8. The transmittances of the two other
designs are nearly the same and thus are not shown
in Fig. 8.
of solutions to a design problem. A direct needle op-
timization of a series of starting designs with various
total optical thicknesses also provides a set of solu-
tions with various combinations of principal design
metrics. Single layers with different thicknesses can
be used as starting designs when followed by the use
of the needle optimization procedure. Once the re-
quired total optical thickness is established, one can
perform a search for multiple solutions by using var-
ious starting designs with the appropriate total op-
tical thickness. Quarter-wave mirrors and various
combinations of quarter-wave mirrors can be useful
for this purpose.
similar combinations of principal design metrics. Nev-
ertheless, in many cases such solutions can be ob-
tained. This obviously presents opportunities for an
optical coating engineer to choose the most manufac-
turable design. Preproduction estimation of manufac-
designs: solid curve, transmittance of the design with the refrac-
tive index profile a in Fig. 5; dotted curve, transmittance of the
design with the refractive index profile b in Fig. 5.
Transmittances of two partitioned reflector–transmitter
sign metrics for these designs are presented in Table 3.
Refractive index profiles of three hot-mirror designs. De-
index profile shown in Fig. 7(a).
Transmittance of the hot-mirror design with the refractive
Table 3.Principal Design Metrics of the Three Hot Mirror Designs with
Refractive Index Profiles Shown in Fig. 7
10 February 2007 ? Vol. 46, No. 5 ? APPLIED OPTICS709
turing errors24and computational manufacturing of Download full-text
optical coatings25–27are powerful tools for supporting
such a choice, but these topics are outside the scope of
The recognition of the key role of design total optical
thickness has given rise to new design approaches
based on the needle optimization technique. The
gradual evolution approach, in combination with the
needle optimization algorithm, facilitates the gener-
ation of a set of designs with various combinations of
principal design metrics: merit function value, num-
ber of design layers, and design total optical thick-
ness. It provides an optical coating engineer with
additional opportunities to choose the most practical
design from a set of theoretical designs with various
Multiple solutions with nearly the same combina-
tions of principal design metrics can be generated for
the most manufacturable design. Combining the nee-
dle optimization technique with multiple optimiza-
tions of random starting designs is an effective
approach for finding such solutions. Other design ap-
proaches aimed at obtaining multiple solutions have
also been described.
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