# Focused Laser Beams on Dielectric Mirrors

If you have a tightly focused laser beam, this can be considered as a superposition of a continuous range of plane waves with significantly different propagation directions; that is related to the beam divergence.

Now imagine that such a beam is reflected on a dielectric mirror, where it is known that the reflectance does not only depend on the wavelength, but also on the angle of incidence. You should not naïvely take the reflectance for the main direction; that would effectively mean that you assume the reflection of a simple plane wave, ignoring the beam divergence. So the question arises: could the angular dependence of the mirror possibly affect the reflection of such a beam in a significant way?

A first check on how relevant that issue is could be to calculate whether the reflectance varies substantially in the relevant angular range; that you can do with any simple thin-film coating software. Somewhat more is needed to actually calculate the resulting reflectance and possibly more details, such as the resulting reflected beam profile – how to get this done?

## Using Spatial Fourier Transforms

Mathematically, the issue is not too complicated as long as we are in the linear regime, i.e., as long as the involved optical intensities are not excessively high. Then we can simply

- consider the focused beam as a superposition of plane waves,
- calculate the reflection of each plane wave component according to its angle of incidence, and finally
- construct the resulting reflected beam from that superposition of reflected plane waves.

In other words, we apply a spatial Fourier transform (which essentially means to calculate the amplitudes of its plane wave components), multiply the Fourier transforms with the amplitude coefficients for the reflection, and transform back into real space if needed.

## How to Do It in Practice?

Imagine how you would do that with a simple thin-film coating software, which cannot do such calculations. You can start with some other numerical software, e.g. something like Matlab, in order to apply the spatial Fourier transform to your input beam as the first step in the explained algorithm. The trouble comes when you try to apply the reflection of the plane wave components; the reflectance calculation needs to be done not just once, but for each component – numerically possibly for 100 or more of them – with the thin-film coating software. So you would like to interface that with the other software, for example such that you remote-control that from Matlab. This may work in some cases, but normally you don't have such remote control features. Another approach would then be to calculate a table of reflection amplitude versus incidence angle, store that in a file and import that into Matlab. In any case, it's getting tedious. You then start thinking about doing the whole thing in Matlab, which is possible, but also tedious.

A much better approach is to do everything, including the Fourier transform stuff, in the thin-film software. You just need to have a sufficiently flexible product. Well, I can tell you one: my **RP Coating** software. I just made a demo case for that application. Because of the built-in script language, one requires only a few lines of script code to perform the task. It is great to have such flexibility, particularly if you want to do further things, for example for the detailed analysis of the situation.

## Flexibility is Essential for Real Work

This is just another example for how important it is to have software which cannot only do the basic calculations in a certain technical and scientific area, but is also flexible enough to be applied in a somewhat more sophisticated context. That's what happens all the time e.g. in the daily life of a technical consultant. Unfortunately, there are many nice-looking software packages which are quite deficient in that respect; you may get an easy start but then eventually hit the limits when it comes to doing some real piece of work.

Of course, it takes some technical know-how to do such things; software alone may not be sufficient. However, that software comes with (a) plenty of demo scripts from which you can learn such tricks and (b) with my personal technical support: if necessary, I just help you out with little more script code, tailored to your needs.

## Learning by Playing

By the way, it is not only fun but also instructive to play around with such a model, once you have it. You often expect certain outcomes, discover discrepancies with simulated results, understand those, etc. This is how real technical competence grows – not just by reading textbooks!

In our example case, you can discover the interesting physical detail that the common law “output angle equals incidence angle” can be violated (see the Figure): the mean angular direction of the output angle may *not* satisfy that law, although each involved plane wave component does it. This is simply because the mirror reflects certain angular components more than others.

This article is a posting of the RP Photonics Software News, authored by Dr. Rüdiger Paschotta. You may link to this page, because its location is permanent.

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