# Distributed Bragg Reflectors for GaN-Based Vertical-Cavity Surface-Emitting Lasers
## 2 Fabrication Techniques and Design Considerations of Distributed Bragg Reflectors (DBRs)
### Optical Design Consideration  
The optical principle of a DBR mirror is based on successive Fresnel reflection at normal incidence at interfaces among two alternating layers with refractive indices \(n_{1}\) and \(n_{2}\): \(r=\frac{n_{1}-n_{2}}{n_{1}+n_{2}}\). When the optical thickness of each layer is maintained at a quarter wavelength (\(\lambda\)/4) thick, the path difference between reflections from successive interfaces equals to half the wavelength (\(\lambda\)/2), or 180\({}^{\circ}\) out of phase. However, since reflections (\(r\)) at successive interfaces have alternating signs to compensate the 180\({}^{\circ}\) phase shift from the differences in path length, all the reflected components interfere constructively, and the cumulative reflection can be enhanced by design. The reflectance spectrum of a DBR can be calculated using the transmission matrix theory [43]; an example of the calculated spectrum of an AlN (\(n=2.15\))/GaN (\(n=2.52\)) DBR with different number of pairs is shown in Figure 1a. As the number of pairs increases, the shape of the reflectance spectrum evolves from a rounded peak to a perfect rectangular spectral band, with the peak of reflectance increasing according to the following equation [45]:  
Figure 1: (**a**) Calculated reflectance spectrums of an AlN/GaN distributed Bragg reflectors (DBRs) with a Bragg wavelength \(\lambda\) of 450 nm. The number of DBR pairs is increased from 10 to 500 pairs. See text for the explanation of \(\Delta\lambda_{\text{max}}\). (**b**) Plots of peak mirror reflectance, \(R\), versus the number of DBR pairs for AlAs/GaAs mirrors (2.95,3.52), InP/InGaAsP lattice matched mirrors (3.17,3.40), AlN/GaN mirrors (2.15,2.52), and AlInN/GaN lattice matched mirrors (2.33,2.52) [44]. Reflectance from a dielectric DBR is also calculated and plotted. \[R=\left[\frac{1-\left(\frac{n_{L}}{n_{H}}\right)^{2m}}{1+\left(\frac{n_{L}}{n_{H}} \right)^{2m}}\right]^{2}, \tag{1}\]  
where \(m\) is the number of layer pairs, and \(n_{L}\) and \(n_{H}\) are the optical indices of the materials with lower and higher indices of refraction, respectively. According to Equation (1), we plot in Figure 1b the peak reflectivity versus the number of DBR pairs m for several epitaxial systems consisting of GaAs, InP, and GaN, and a dielectric DBR from SiO\({}_{2}\)/HfO\({}_{2}\) for comparison. For a low-threshold VCSEL operation, a mirror reflectivity of \(>99.5\%\) is often required to reduce the round-trip mirror loss. Dielectric DBRs in general provides the most ideal reflectance curve with relatively few numbers of DBR pairs (m \(\sim\) 10). AlAs/GaAs and AlN/GaN have a sufficient index contrast to reach high reflectance in \(\sim\)20 pairs of DBR. In the other two cases epitaxial DBRs requires more than 40 pairs for high reflectance. In practice, the number of pairs calculated based on Equation (1) gives a lower bound in experimental design; additional pairs are often required to offset other non-idealities such as surface roughness or thickness variations. As seen in Figure 1a, the width of the high-reflectance band of DBR mirrors, \(\Delta\lambda_{max}\), is an important design parameter for VCSEL. The stopband width of DBR mirror is expressed as [45]:  
\[\Delta\lambda_{max}=\frac{4\lambda}{\pi}\sin^{-1}\left(\frac{\Delta n}{n_{H}+n _{L}}\right), \tag{2}\]  
The \(\Delta\lambda_{max}\) given in Equation (2) specifies the full width at half maxima of the reflectance of a DBR mirror when the number of pairs approaches infinity. \(\Delta\lambda_{\max}\) is proportional to the Bragg wavelength \(\lambda\) and is sensitively affected by the index contrast (\(\Delta n=n_{H}-n_{L}\)). Thus, a higher \(\Delta n\) is very desirable in DBR fabrication for both a high peak reflectance (from Equation (1)) and a wide stopband width (from Equation (2)).