• 1. Reflectance of a quarter-wave stack
• 2. Approximations
• 3. Spectral width of the high reflectance zone
• 4. Numerical examples for AlAs/GaAs-Bragg mirror

A thin film Bragg reflector consists of a multilayer-stack of alternate high- and low-index films, all one quarter wavelength thick (see figure left). The geometrical thicknesses of the high- und low-index films are t_H=l/(4n_H) and t_L=l/(4n_L) respectively.
n_H and n_L are the indices of refraction of the high- and low-index films, respectively and l is the center wavelength of the Bragg mirror.
On every interface in the stack a part of the incident beam is reflected. The reflected parts have a phase shift of 180° only if the incident light goes from low-index medium in a high-index medium. The relative phase difference of all reflected beams is zero or a multiple of 360° and therefore they interfere constructively.
The intensity of the incident light beam decreases during his travel trough the quarter-wave stack and at the same time the reflected light increases, if the absorptance A of the stack is negligible.

The reflectance R of a quarter-wave stack in air or free space with high-index layers outermost on both sides is given by

If the number (2p + 1) of films in the quarter-wave stack is large and the absorption can be neglected then

Reflectance of a Bragg mirror according to eq. (2) with n_H = n_S = 3.5 and n_L = 3.0.

Reflectance of a Bragg mirror according to eq. (2) with n_H = n_S = 3.5 and n_L = 3.0.

The spectral width Dl of the high reflectance zone increases with increasing difference of the refractive indices n_H - n_L.
Dl can be estimated with the design wavelength l₀ of the quarter-wave stack by

The relative spectral width w = D l/l of the high reflectance zone as a function of the ratio of the refractive indices n_H/n_L according to eq. (4) is shown in the figure left.

For the wavelength 1064 nm (photon energy = 1.165 eV) the refractive indices of the materials GaAs and AlAs are due to equation (1) given on the page n(AlGaAs) n(GaAs) = 3.49 and n(AlAs) = 2.95 respectively.
With this values the reflectance R of a Bragg-mirror consisting of p pairs of quarter-wave layers on a GaAs substrate can be calculated with the above mentioned eq.(1) as follows:

Spectral reflectance of Bragg-mirrors consisting of AlAs/GaAs quarter-wave stacks with the center wavelength of l₀ = 1064 nm and different numbers of AlAs/GaAs pairs.