How to Design Multilayer Optical Devices
When designing some multilayer optical device such as a laser mirror, an anti-reflection coating or an optical filter coating, the first question is usually what general design strategy to choose. In many cases, one requires some design strategy, and also sometimes a rather flexible multilayer optics design software.
The very first step is to understand and clearly formulate what features the wanted design should have. After all, the best design software can optimize designs only if it is clearly defined how good a given design is based on its calculated optical properties.
The common approach is to define some figure of merit (FOM) – a value which can be computed for arbitrary designs and is a clear measure for how good it is. For example, I usually do it such that an ideal design – perfectly fulfilling all requirements – would obtain an FOM of zero, while any deviations from perfect properties incur some positive penalties. There are often multiple requirements concerning optical properties at different wavelengths, for example, and a good way is to add up the squares of all design imperfections – e.g. deviations of the reflectance or transmittance from the ideal value. It is often not possible to find a design with a zero FOM of that type, but the end goal is then to find the design with the minimum possible FOM, or at least one with an sufficiently small FOM value.
As an example, I show the definition of an FOM function for an anti-reflection coating for normal incidence with a certain bandwidth, as made in our software RP Coating:
FOM() := sqrt(sum(d := -0.5 to +0.5 step 0.1, begin var dl; dl := d * dl_AR; R(l_AR1 + dl)^2 + R(l_AR2 + dl)^2; end));
Here, I took into account reflectivities in some ranges around two different wavelengths for which zero reflectance is desired.
It would not be difficult to extend that, for example for more wavelengths or for non-normal incidence with some range of angles. In other cases, one may want to take into account additional optical properties, e.g. concerning chromatic dispersion.
The Initial Configuration
One typically has a limited choice of coating materials which one can use for a design, and possibly also some constraints concerning the number of layers or the total coating thickness. One may sometimes want to try with different coating materials in order to decide which ones are most appropriate for the purpose.
The next question is whether one should start with some initial design according to some design idea, or rather just do the whole optimization numerically. That depends on the situation; several typical example cases are discussed in the following.
Anti-reflection coatings of the simplest design consist of a single layer, the refractive index of which should be close to the square root of that of the substrate material. That is almost too trivial to use a special design software.
If the mentioned condition for the refractive index cannot be met, or some other features are required, one usually uses a design with a couple of layers, not having an intuitively understandable structure. This is a case where a purely numerical approach is appropriate. However, it is usually not sufficient to start with a random sequence of layer thickness values and apply a local optimization, i.e., a numerical “tweaking” of parameters to get the FOM as small as possible because the multi-dimensional parameter space contains a large number of local extrema, many of which are not satisfactory. A local optimization may easily get stuck in a non-satisfactory local optimum.
Therefore, one usually applies something like a Monte-Carlo algorithm: apply local optimizations to many different randomly chosen starting designs and take the best of those results. Because of the limited number of dimensions of the parameter space, one can usually find a good design within a reasonable time, just using an ordinary personal computer.
The figure below shows an example, where zero reflectance at 1064 nm and 532 nm, each time in a bandwidth of 10 nm, was requested.
Simple Laser Mirrors
Simple laser mirrors, supposed to provide a high reflectance in a certain wavelength range, are usually made as Bragg mirrors. So one naturally starts with such a simple Bragg mirror design, being a sequence of <$\lambda /4$> layers of two different materials. Design software may then simply be used to check whether that works sufficiently well e.g. in some range of wavelengths and angles of incidence. In some cases, one may apply a local optimization for further minimizing the obtained FOM.
Somewhat more difficult is the design of dichroic mirrors, exhibiting specified properties at two different wavelengths. For example, one may design a folding mirror for a diode-pumped bulk laser with an ytterbium-doped laser gain medium, through which pump radiation at a somewhat shorter wavelength can be injected.
Here, it is usually not sufficient to use a simple Bragg mirror design, since that exhibit substantial reflectance peaks outside the central reflecting region. Instead, one uses a short-pass filter design which should have quite low reflectance in a wavelength region below the central reflecting region. That can be achieved with a slight modification of the Bragg mirror design: for the last layer, consisting of the low-index material, one chooses a thickness of <$\lambda /8$> instead of <$\lambda /4$>. That already gives good results, but one again improve it further with a local optimization. As only a single optimization of that type is required, that also works fairly quickly – e.g. in a few seconds. A Monte-Carlo approach, however, would usually be far too slow due to the high-dimensional parameter space.
In our software RP Coating, such a short-pass filter design could be defined with the following code (after defining variables for the substrate, low- and high-index materials, also for the number of layer pairs and the Bragg wavelength):
beam from superstrate substrate: (material_s$) * (material_l$), l/8 at l_Bragg for j := 1 to N_Bragg do begin * (material_h$), l/4 at l_Bragg * (material_l$), l/4 at l_Bragg end * (material_h$), l/4 at l_Bragg * (material_l$), l/8 at l_Bragg superstrate: air
The figure below shows an example case. With the local numerical optimization, quite low reflectance around 808 nm has been achieved, where the basic start design (fine dashed curve) would still have exhibited some substantial wiggles.
The solid curve applies to the numerically optimized version, and the fine dashed curve to the original “raw” design of a modified Bragg mirror.
A substantially more difficult case is the design of double-chirped mirrors. Here, the design goal is (a) a specified wavelength-dependent group delay dispersion in some range and (b) a high reflectance in the same range. The basic idea is to achieve that by making a kind of Bragg mirror, where however the Bragg wavelength is systematically varied within the design, such that different wavelength components are essentially reflected at different positions within the coding structure. In addition, one also needs to modulate the duty cycle (thickness ratio) of the layer thickness values in each layer pair, and apply a broadband anti-reflection coating on top. See my encyclopedia article on chirped mirrors (and the literature cited there, e.g. the initial paper by Franz Kärtner) for more details of such sophisticated mirror designs.
For efficient optimization, it is advisable to describe the whole structure with a small number of parameters, and particularly describing the (possibly nonlinear) chirp. One can then apply a local optimization to those parameters, instead to the full set of layer thickness values. That way, one substantially reduces the number of dimensions, gaining a lot in terms of efficiency. In the end, one may apply a single local optimization to get some further improvement.
Of course, the used design software must be sufficiently flexible for applying such tricks. (That is the case for our product RP Coating.) That flexibility brings such a huge gain in efficiency that the efficiency of the algorithm for calculating multilayer optical properties, or in fact the power of the used CPU, is of comparatively minor importance. Quite sophisticated designs can be found within a rather short time even on an ordinary PC.
The two figures below are from an example case where anomalous dispersion is required, with an increasing strength for shorter wavelengths.
We have seen that flexibility of multilayer design software can be highly desirable in many cases. A first issue is that it allows one to define basically any design goals through a freely defined figure-of-merit function. Note that some commercial software uses a hard-wired function of that type, where only certain parameters can be adjusted. In contrast, in RP Coating you can use any structure or function, therefore not relying on a suitable function being predefined by the software developer.
The last example (double-chirped mirror) shows that it can also be very beneficial to not necessarily apply numerical optimization to the layer thickness values directly, but rather to a reduced set of parameters determining those. That can result in an enormous gain of speed in some situations.
Another aspect is that flexible software allows you to calculate all sorts of properties and to display those in a suitable way with diagrams which you can define yourself – not just selecting between a limited number of predefined diagram types.
Even with rather powerful software, multilayer coating design remains a non-trivial issue, and it can be important particularly for newcomers to get competent and helpful advice within the technical support. That can begin with proposing an appropriate design strategy for a given case, for example.
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