Thin Film Design Methodology
Background
To find the optimum
design for a multilayer stack involves varying the thickness of each layer and
calculating the response. For a critical layer, a step change of as little as 1
nm can have a significant effect. Obviously, as the number of layers increases,
the number of possible combinations reaches astronomical proportions very
rapidly.
Thus, designing the best
thin film stack has always been a challenging task. Modern computer techniques
have helped, but considerable experience is still needed.
An experienced
designer will be able to make a good starting guess to get a good design. Some
programs then use mathematical methods to minimise the merit function, (where
the value of the merit function relates to the deviation between the design
found and the target).
Techniques Used
To find the best
solution close to the initial starting guess, one must solve a series of
differential equations. Various techniques used are the "Gradient", "Newton's
Method", "Least Square" and "Damped Least Squares".
The merit function
is a multivalley function, and these methods will only find a local minimum.
Whilst an experienced designer can make good guesses based on previous
experience for fewer than 5 layers, this process becomes much more difficult and
unreliable as the number of layers increases. These mathematical techniques are
valuable only for refining a existing design to a local optimum.
An alternative
approach used is the Simplex. This method depends on the calculation of the
merit function at several points. By searching neighbouring regions in a
systematic manner, it can jump out of a local minimum of the merit function in
some cases, but can never guarantee to find the global minimum.
Another approach
such as needle optimisation, adds an additional layer and alters its thickness
after each previous design has been optimised. This approach is very unlikely to
find the very best design, since the initial layer thickness guessed is critical
to the process. Additionally it assumes that the design which was found before
adding the next layer, was a good starting guess. This and other alternatives
are an improvement, but the final result is still dependent on the initial
starting conditions. These methods cannot guarantee finding the global minimum.
Scanning
Some people
approach the problem of finding the global minimum (and not just a local
minimum), by scanning the whole of the parameter space in a random or stepwise
manner. This approach will theoretically give the best answer, but it can take a
long time. For example, to scan a 5 layer design by stepping the optical layer
thicknesses in 2 nm steps over the range 10 to 350 nm, one would need to perform
170^{5} calculations
of the merit function. For a modern P.C., with each calculation taking 0.1
sec, the whole scan would take 450 years! If the step size is too large, then
many designs will be missed.
To reduce the
calculation time, some methods use random distributions of samples repeatedly,
to find the best result. However this method is also time consuming and does not
give repeatable answers.
OnlyFilm
A proprietary method (DGL Optimisation) has been developed over a 22
year period and some of the principals used in it have been published.
Whilst an early version has been sold to research institutions for the past 18
years, this is the first commercial version using Windows. OnlyFilm can
inherently guarantee global optimisation in affordable computing time. This
gives rise to a method which requires no starting guess, never produces nonsense
results, and will always find the best design within user selected deviations
from the target. Any combinations of refractive index can be chosen.
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Media Coverage of (OpTeFilm)
OnlyFilm Optical Thin Film Design Software
 Global optimization advances multivariable
thinfilm design, Laser Focus World,
May 1996
Article on Thin Film Design with Global Optimisation
(Published May 1996 in Laser Focus magazine)
Global optimization advances multivariable thinfilm design
by Dongguang Li and Burt Nathan
DONGGUANG LI is a senior researcher at the Faculty of Science, Technology
and Engineering, Edith Cowan University, Perth, Western Australia.
BURT NATHAN is managing director of Raylight Pty., Ltd., Aldinga, Adelaide,
South Australia, tel: +61 (85)574101/fax: 4071.
Optical thin films are used in a wide variety of optical components, with a
myriad of companies worldwide specializing in film design and manufacture.
Available multivariable software programs for designing the filmlayer
structure may not give optimum designs. To our knowledge, all current software
programs obtain their final designs either from optimizing a starting guess or
by techniques, which may or may not involve a pseudorandom process, that give
different answers every time, depending upon the initial conditions.
Next generation design programs offer true global optimization for thinfilm
design. A global optimization algorithm will always find the very best solution
possible within the boundary conditions stipulated. The possibility of creating
a true global optimization algorithm for a large number of interdependent
variables has been debated for decades. The OpTeFilm software and global
optimization technique developed by one of us, Dongguang Li, gives optical
designers a tool to solve multilayer thinfilm problems.
Thinfilm design overview
A multilayer stack consists of several layers of dielectric materials
deposited onto a substrate, each layer having a specified thickness. When an
electromagnetic wave strikes the layers in the stack, the phase and amplitude of
the wavefront is changed due to each layer's thickness. The resultant wave can
be calculated for both transmitted and reflected waves. In this way, a filter
with any arbitrary spectral response can be created, provided one uses enough
layers.
A merit function is used to evaluate how close each design gets to the target
requirement. The design process involves finding the design which corresponds to
the minimum of the merit function. Plotting the merit function against the layer
thickness of each layer, one axis per layer would be needed, plus the orthogonal
axis for the merit function.
The merit function plot would appear as a multipeak, multivariable plot.
Because there are an enormous number of interrelated possible layer
combinations, the best film design cannot be found by any simple process. It is
not obvious how to adjust the layers analytically to find the best solution. The
methods currently used in thinfilm design software, such as the Damped Least
Squares, Simplex, and needle optimizations, all depend on a starting condition
that is sometimes not obvious. Changing the initial conditions will give a
different result, and the user has no way of knowing how much improvement could
be effected.
With the global optimization algorithm embodied in OpTeFilm software, the
user knows that the design found is optimized there is no need to try other
starting conditions because there are no starting guesses. This efficiency has
powerful economic consequences. For example, previous designs needing excessive
numbers of layers can now be fabricated with fewer layers, lowering cost, to get
the same performance and better yields. A manufacturer can improve yields on
marginal designs by using a design with a greater margin of error, as well as
offering previously unavailable products.
Global optimization
The OpTeFilm global algorithm operates, metaphorically speaking, to discover
rather than synthesize the optimum solution. An analogy illustrates the
principles involved. Assume plotting the merit function against one of the film
layer's thicknesses with a goal of finding the layer thickness corresponding to
the minimum meritfunction value See fig.1.
With local optimization, (a fast method for a large number of layers), the
program finds the nearest minimum and stops. For some socalled
globaloptimization programs, the algorithm not only finds a local minimum but
can also find some neighboring minima. The process, however, is hit and miss,
because starting at a different place can result in different designs.
The global algorithm in OpTeFilm repeatedly narrows the region where the
global minimum is known to lie by using a special Latin Square algorithm that
operates simultaneously in all orthogonal dimensions, one for each layer to
find the optimum design. As the program runs, the user can observe the range of
thicknesses for each layer being reduced.
For example, consider a design of an input coupler for a Ti:sapphire laser.
This component requires low reflectance at 500nm to 550nm wavelength, and high
reflectance from 700 nm to 1000 nm. The basic design was optimized by another
commercial program to produce a 32layer design. With OpTeFilm, which only
required the boundary conditions (material refractive indices, substrate, number
of layers, and so forth), the resulting thinfilm design was obtained in two
hours on a standard IBM DX2 computer operating under Windows. While the
characteristics were similar, the OpTeFilm design accomplished these with nine
fewer layers (fig 2).
While the thinfilm design community is the first to benefit from this
method, the mathematical procedures used in OpTeFilm are also applicable to a
variety of other unsolved problems related to linked multivariable problems.
The earliest noncommercial application of this technique, was in the
optimisation of optical aberration in the design of a multilens system, future
applications can include image processing, and hydraulic flow in a complex
environment.
Footnote:
The shorthand descriptions of the designs shown below, can also be understood by
nonspecialists. The numbers refer to multiples (or fractions) of the
Quarterwave Optical Thickness (Refractive Index times physical thickness
divided by Quarter Ref. Wavelength), 'H' is the high value and 'L' the low value
of the film refractive index, 'G' is glass substrate, and 'A' is air. The
superscript refers to the number of repeats of the two layer thicknesses within
the brackets. Anyone with access to a design program can verify these results.
The conventional design used is:
(The numbers refer to the number of quarter wavelengths at 835 nm, G is glass
and A is air)
Substrate n=1.76, reference = 835 nm, High index = 2.35 and Low index =1.46.
with design G / (1.140H 1.130L) ^7 1.046H 1.008L (0.908H 0.894L) ^ 7 0.907H
0.430L / A.
The design produced by OpTeFilm for the same criteria is: G / 2.377L 1.026H
0.969L 2.137H 1.048L 0.845H 1.067L 2.159H 0.963L 0.995H 0.925L 0.993H 0.998L
0.983H 0.975L 0.975H 1.042L 0.992H 1.116L 1.050H 1.178L 1.169H 1.888L / A.
