Reflection Spectrum of Tilted Dielectric Mirrors

Posted on 2006-11-02. Permanent link: http://www.rp-photonics.com/spotlight_2006_11_02.html

Ref.: encyclopedia articles on laser mirrors and dielectric mirrors

Dielectric multilayer mirrors can be highly reflecting only in a limited spectral bandwidth. When such a mirror, designed for use with normal incidence of the light, is tilted against an incident beam, the reflection features are shifted toward shorter wavelengths.

At a first glance, one might actually expect the opposite effect, following the (flawed) argument that the optical path length between different layers of the mirror coating is increased, leading to a larger phase delay, which could be compensated by using a longer wavelength. So the reflection features should be shifted toward longer wavelengths – just the opposite of what is observed! What is wrong here?

A closer look shows where the above argument is wrong: following a ray from one layer to the next one under some angle, and back to the first layer along a "reflected" beam, we actually don't get back to the place where we started, but rather to some offset position. But it doesn't make sense to compare the phases of two beams as measured at two different positions!

For a correct argument, we have to consider e.g. two beams as reflected from two subsequent layer interfaces, and compare their phases at the same point. Doing this, we find that the phase delay actually becomes smaller for angled incidence. Another way to understand the issue is to consider the phase shift along a direction perpendicular to the coating layers. This can be calculated using the component of the k vector perpendicular to the surface. And the magnitude of that is obviously reduced for tilted incidence.