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Spectral reflectance of AlAs/GaAs-Bragg mirrors

1. Reflectance of a quarter-wave stack  

Quarter-wave stack
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 tH=λ/(4nH) and tL=λ/(4nL) respectively.
nH and nL 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

  Formula reflectance reflectance eq.(1)  
  Formula admittance optical admittance of a quarter wave stack with (2p + 1) layers  
  In this equation the symbols and constants have the following meaning:  
  nH high index of refraction  
  nL low index of refraction up
  nS index of the substrate  
  (2p + 1) number of layers in the stack  
2. Approximations  

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

  Formula reflectance reflectance eq.(2)  
  Formula transmittance transmittance eq.(3)  

Reflectance of a Bragg mirror

Reflectance of a Bragg mirror according to eq. (2) with nH = nS = 3.5 and nL = 3.0.


Reflectance of a Bragg mirror

Reflectance of a Bragg mirror according to eq. (2) with nH = nS = 3.5 and nL = 3.0.

3. Spectral width of the high reflectance zone  

The spectral width Δλ of the high reflectance zone increases with increasing difference of the refractive indices nH - nL.
Δλ can be estimated with the design wavelength λ0 of the quarter-wave stack by

  Formula spectral width spectral width of high reflectance eq.(4) down

Relative spectral width

The relative spectral width w = Δλ/λ of the high reflectance zone as a function of the ratio of the refractive indices nH/nL according to eq. (4) is shown in the figure left.

4. Numerical examples for AlAs/GaAs-Bragg mirror  

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:

Number p of GaAs/AlAs film pairs Reflectance R at 1064 nm
5 0.80766
10 0.96104
15 0.99262
20 0.99862
25 0.99974
27 0.99987

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.

  Spectral reflectance of Bragg mirrors  
  Spectral reflectance of Bragg mirrors up