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Thin Film Design Methodology

Background

 To find the optimum design for a multi-layer 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 multi-valley 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 1705 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 thin-film design, Laser Focus World, May 1996

           

Article on Thin Film Design with Global Optimisation

(Published May 1996 in Laser Focus magazine)

Global optimization advances multi-variable thin-film 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)-57-4101/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 multi-variable software programs for designing the film-layer 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 pseudo-random process, that give different answers every time, depending upon the initial conditions.

Next generation design programs offer true global optimization for thin-film 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 inter-dependent 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 thin-film problems.

Thin-film 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 multi-peak, multi-variable plot. Because there are an enormous number of inter-related 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 thin-film 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 merit-function 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 so-called global-optimization 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 500-nm to 550-nm wavelength, and high reflectance from 700 nm to 1000 nm. The basic design was optimized by another commercial program to produce a 32-layer design. With OpTeFilm, which only required the boundary conditions (material refractive indices, substrate, number of layers, and so forth), the resulting thin-film design was obtained in two hours on a standard IBM DX-2 computer operating under Windows. While the characteristics were similar, the OpTeFilm design accomplished these with nine fewer layers (fig 2).

While the thin-film 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 multi-variable problems. The earliest non-commercial application of this technique, was in the optimisation of optical aberration in the design of a multi-lens 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 non-specialists. The numbers refer to multiples (or fractions) of the Quarter-wave 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.

 

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