Optical multilayer devices consist of a sequence of thin layers, usually only two transparent materials. They can perform a wide range of optical functions, including antireflection properties, wavelength filtering, and polarization-based light separation. Such functions are achieved by choosing an appropriate design; i.e. an appropriate sequence of layer thickness values.

A challenge is often to find a working design, or even to assess whether a design that meets the requirements of a specific application can exist. Very different design methods may be suitable, depending on the type of optical functions sought. In all cases, we assume that sufficiently powerful multi-coated optics design software is used. Along with proper software, a good idea of a proper design strategy is helpful. As a general rule, you shouldn’t expect powerful software to automatically deliver good designs for every conceivable purpose.

### What are the design goals?

Before starting any automated design optimization process, it is necessary to precisely define the required optical properties. More generally, there is a need to provide software with a means of determining how close different designs are to ideal performance. Only then will he have a chance to find models that will work well.

The commonly chosen method is to define a so-called Figure of Merit (FOM) function. This is a function that provides a single value, meant to be a reasonable measure of how close it is to an ideal design. A common method is to define the function so that it provides a zero result in the ideal case and positive values for less satisfactory designs. There will often be multiple criteria, all of which can add positive penalty points to the result. For example, a FOM function might have the following form, if only reflectance values at different wavelengths are of interest:

Simple software often uses a built-in FOM function of a fixed structure, where only parameters can be adjusted by the user. The user can enter several wavelengths and the corresponding desired reflectance values, for example, and the software can calculate the sum of the squares of the deviations from these reflectance values.

For greater flexibility, the software can offer several versions of the FOM functions. However, it is much better if it allows the user to freely define any type of FOM function by inputting a mathematical expression. This can refer to all possibly relevant optical properties of a structure: not only reflectance or transmission, but also wavelength-dependent group delay or group delay dispersion, and all for n any angle of incidence, polarization and wavelength. It allows the implementation of a large number of ideas. For example, differences in reflectance at different wavelengths could be used if the relative wavelength independence in a certain range is more important than the absolute values.

### Anti-reflective coatings

A typical task is to design an anti-reflective coating that works in a certain range of wavelengths, or possibly in several ranges. While the function of simple monolayer coatings is easy to understand, complex interference effects are relevant for multilayer coatings. Still, the task is relatively easy with suitable software. Since only a small number of layers are needed, a brute force approach could be taken, trying to find a numerical solution by trial and error, without any physical understanding of exactly how the design should work.

Often it will not be enough to start with a random multilayer sequence and improve performance by local numerical optimization, trying to reduce the FOM value as much as possible. Indeed, we operate in a multidimensional space (with one dimension for each layer), in which the FOM function has a large number of local optima. Local optimization easily crashes on one of them, which is far from enough. Typically, this problem is solved with a Monte Carlo approach: we try a large number of random layer thickness value combinations, take a subset of those with reasonably low FOM, and locally optimize them further. A regular desktop PC with suitable design software is normally sufficient to find a solution in a reasonable time, for example in less than a minute.

Figure 1 shows the reflectance versus wavelength of a sample pattern found in seconds with RP Coating software on an old PC. The goal was to achieve low reflectance at both 532 nm and 1064 nm. The FOM used contained reflectance penalties in a range around each wavelength.

Before using such a design, it is necessary to also evaluate its sensitivity to errors in layer thickness. For example, we can assume either systematic errors for all layers or a set of design versions with random errors for each layer separately. This can be integrated into a second version of FOM, just to check if the solution found is too sensitive.

### Dichroic mirrors

For some lasers, dichroic mirrors are needed when one has a high reflectance (>99.9%) at the laser wavelength and at the same time a high transmission at a shorter wavelength of radiation from pump. Such a device requires substantially more layers to achieve the high reflectance. As a result, the number of dimensions is usually too high for a brute force calculation approach.

A simple design idea would be to use a Bragg mirror (containing only λ/4 layers) so that the laser wavelength is in the Bragg reflection band while the pump wavelength lies outside, ideally between the side lobes in the correctly positioned reflectance spectrum. For significantly better performance, simple modifications are known, in which λ/8 layers are used at the beginning and at the end. Performance can be further improved by numerically optimizing all thickness values for even lower FOM.

To implement such a strategy, the software must allow the user to easily define the initial design, not by manually entering externally calculated thickness values, but so that the thickness values are automatically calculated by depending on a few parameters entered. Then it is convenient to experiment with different numbers of layer pairs, different coating materials, etc.

Figure 2 shows an example design for a laser wavelength of 1064 nm and a pump wavelength of 808 nm, as is common for Nd:YAG lasers. The dashed curve shows the initial design, while the thicker solid curve shows the digitally optimized design. Since only one local optimization is needed, the calculation is done in seconds.

### Double chip mirrors

Rather sophisticated design tasks arise when mirrors with controlled chromatic dispersion properties are required. This is particularly difficult when precise dispersion properties are required over a substantial spectral range, as is the case for wideband mode-locked lasers. For such applications, so-called double-chirped mirror designs were proposed by Franz Kärtner in 1997.^{1} Unlike an ordinary Bragg mirror, the Bragg wavelength is systematically increased in one direction; in a simplified image, the resulting chromatic dispersion comes from a frequency-dependent penetration depth into the structure. In addition, one must vary the relative thickness of the two materials and also place a broadband anti-reflective structure on top (unless one opts for a design used from Brewster’s angle).

With sufficiently flexible design software, the design process can start with an initial design, consisting of a parametrized mirror structure (controlled with a few design parameters) and a previously optimized antireflection structure. In a first step, one can effectively optimize the double-chirp structure thanks to the mentioned moderate number of design parameters. This can be followed by another local optimization, which can adjust all layer thickness values independently.

### conclusion

Quite different design strategies must be used, depending on the type of optical device to be designed. Especially for anti-reflective structures a brute force numerical approach works well, whereas in cases requiring many more layers there should be a reasonable initial design. Successful design work requires sufficiently flexible software, which allows the user to freely define merit functions for design goals and initial designs. Appropriate ideas are also needed for design strategies, which can come from past experience, examples shipped with the software, or knowledgeable technical support from the software vendor. On the other hand, the computing power requirements are then quite modest; a regular desktop PC is usually sufficient.

**REFERENCE**

1. FX Kartner et al., *Opt. Lett.*, 22, 11, 831 (1997); doi: 10.1364/ol.22.000831.