The DBR's
reflectivity, R, for
intensity is approximately given by :R = \left[\frac{n_o (n_2)^{2N} - n_s(n_1)^{2N}}{n_o (n_2)^{2N} + n_s (n_1)^{2N}}\right]^2, where n_o,\ n_1,\ n_2 and n_s\, are the respective refractive indices of the originating medium, the two alternating materials, and the terminating medium (i.e. backing or substrate); and N is the number of repeated pairs of low/high refractive index material. This formula assumes the repeated pairs all have a quarter-wave thickness (that is n d = \lambda / 4, where n is the refractive index of the layer, d is the thickness of the layer, and \lambda is the wavelength of the light). The frequency
bandwidth \Delta f_0 of the photonic stop-band can be calculated by :\frac{\Delta f_0}{f_0} = \frac{4}{\pi}\arcsin\left(\frac{n_2 - n_1}{n_2 + n_1}\right), where f_o is the central frequency of the band. This configuration gives the largest possible ratio \frac{\Delta f_0}{f_0} that can be achieved with these two values of the refractive index. Increasing the number of pairs in a DBR increases the mirror reflectivity and increasing the refractive index contrast between the materials in the Bragg pairs increases both the reflectivity and the bandwidth. A common choice of materials for the stack is
titanium dioxide (
n ≈ 2.5) and
silica (
n ≈ 1.5). Substituting into the formula above gives a bandwidth of about 200 nm for 630 nm light. Distributed Bragg reflectors are critical components in
vertical cavity surface emitting lasers and other types of narrow-linewidth
laser diodes such as
distributed feedback (DFB) lasers and
distributed bragg reflector (DBR) lasers. They are also used to form the
cavity resonator (or
optical cavity) in
fiber lasers and
free electron lasers.
TE and TM mode reflectivity showed as a white dashed line, right half represents TE reflection. This section discusses the interaction of
transverse electric (TE) and
transverse magnetic (TM) polarized light with the DBR structure, over several wavelengths and incidence angles. This reflectivity of the DBR structure (described below) was calculated using the
transfer-matrix method (TMM), where the TE mode alone is highly reflected by this stack, while the TM modes are passed through. This also shows the DBR acting as a
polarizer. For TE and TM incidence we have the reflection spectra of a DBR stack, corresponding to a 6 layer stack of dielectric contrast of 11.5, between an air and dielectric layers. The thicknesses of the air and dielectric layers are 0.8 and 0.2 of the period, respectively. The wavelength in the figures below, corresponds to multiples of the cell period. This DBR is also a simple example of a 1D
photonic crystal. It has a complete TE band gap, but only a pseudo TM band gap. ==Bio-inspired Bragg reflectors==