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# Variable Reflectivity Mirrors

Acronym: VRM

Definition: mirrors with a spatial variation of the reflectivity (reflectance)

Alternative terms: variable reflection mirrors, graded reflectivity mirrors

More general term: mirrors

More specific term: Gaussian mirrors

Opposite term: uniform mirrors

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Variable reflectivity mirrors (also called variable reflection mirrors or graded reflectivity mirrors) are mirrors which exhibit a spatial variation of their reflectance (or reflectivity). Typically, the mirror design is radially symmetric, i.e., the reflectance depends only on the distance <$r$> from the center of the mirror. For example, there are Gaussian mirrors, where the reflectance is governed by a Gaussian function:

$$R(r) = {R_0}\exp \left( { - {{\left( {\frac{r}{w}} \right)}^2}} \right)$$

where <$w$> is a parameter determining the width of the Gaussian function. Other profiles are of course possible, for example supergaussian (where the exponent in the formula above is larger than 2), parabolic or Bessel function profiles. There are also mirrors where the reflectance depends on a linear coordinate.

## Operation Principles and Fabrication of Variable Reflectivity Mirrors

One possibility is to deposit a single relatively highly reflecting layer with variable thickness on a mirror substrate. For example, it could be a metallic coating or a high-index dielectric material. One can also make dielectric multilayer mirrors where the layer thickness is made spatially dependent.

The spatial dependence can be introduced e.g. by using some mask which causes a spatially dependent flux of material to be deposited. For making mirrors with a purely radial dependence of the reflectivity, one would usually have the substrate rotating during deposition.

Other fabrication techniques are also possible, for example the spatially dependent modification of reflection properties after production of originally uniform mirrors. If such surface modifications are applied with laser radiation, a high flexibility for making various types of reflectivity shapes is achieved.

Note that a simple variation of layer thickness values of a Bragg mirror design with a Gaussian variation of Bragg wavelength, for example, will not translate into a Gaussian reflectance profile, as shown in Figure 1. This is because the interference conditions in such a mirror lead to a complicated dependence of the reflectance on the Bragg wavelength. Figure 1: Reflectance profile at 1064 nm wavelength of a Bragg mirror, where the Bragg wavelength is varied according to a Gaussian function. It has been assumed that we have five SiO2/TiO2 Bragg layer pairs, with the Bragg wavelength being 1000 nm at the center. The diagram has been produced with the RP Coating software.

The variable reflectance may be associated with some deviation from a flat surface (or a predefined curved surface). However, such variations may be insignificant for many applications.

Note that besides the variable reflectance, there can be a variation of optical phase changes – not only due to thickness variations of the whole mirror, but also due to interference effects in dielectric coatings.

Only a small fraction of mirror manufacturers is able to produce variable reflectivity mirrors, and only a small fraction of used mirrors are of that type.

## Applications of Variable Reflectivity Mirrors

A typical application for variable reflectivity mirrors is as output couplers in lasers with unstable resonators. Here, the variable reflectivity serves to limit the laser beam diameter in a resonator where the beam would otherwise tend to expand more and more. Often, one uses Gaussian reflectivity mirrors. That approach allows one to construct lasers with relatively large mode areas and still close to diffraction-limited output beams, i.e., high beam quality. This is used for some high-power lasers, including e.g. various solid-state lasers and CO2 lasers.

Another application is in variable attenuators. For example, one may use a mirror where the reflectance depends only on some <$x$> coordinate, such that the attenuation of an incident beam can be varied by translating the mirror in the <$x$> direction.

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### Bibliography

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