As an ideal light source for 3D cameras, 940 nm vertical cavity surface emitting lasers (VCSELs) have broad application prospects and can be used in virtual reality and car-assisted driving. The optimal design of the distributed Bragg reflector (DBR) is crucial for improving the performance characteristics of 940 nm VCSELs. In traditional research, there are few studies on DBRs in the 940 nm band. To provide guidance for the design and optimization of DBR structures in 940 nm VCSELs, we systematically study the reflection properties of various DBR structures. In the present study, we apply the transmission matrix method (TMM) to multilayer dielectric films to calculate and analyze the influence of different stacking methods and periods on the DBR reflection spectrum. The model is modified and the influence of the incident angle on the DBR reflection spectrum is calculated and analyzed. A linear fitting model between the refractive index of Al_xGaAs and Al atomic fraction x at a wavelength of 940 nm is established. The effect of the gradient layer on the reflectance spectrum characteristics of the DBR is calculated and analyzed using the multilayer division equivalent method. Through our research analysis, the relationship between the DBR structure and its reflective properties can be understood more clearly.

In this study, we choose Al_0.89GaAs as the DBR low-refractive-index layer material (L) and Al_0.09GaAs as the DBR high-refractive-index layer material (H) with refractive indices of 3.497 and 3.040, respectively. The reflection characteristics of different DBR structures are analyzed using the TMM. First, we use the transmission matrix of the multilayer dielectric film to study the effects of different stacking methods and periods on the reflection characteristics of the DBR structure. Subsequently, considering the influence of the incident angle on the reflection spectrum, we modify the transmission matrix and study the influence of different incident angles on the reflection characteristics of the DBR. Finally, to simplify the calculation of reflection spectrum characteristics of the gradient layer structure, we linearly fit the relationship between the refractive index of Al_xGaAs material and the aluminum atomic fraction x, and we adopt the multilayer division equivalent method by dividing the Alx1-x2GaAs gradient layer into sufficiently small ultra-thin and equal thickness layers; when the divisions are sufficiently large, this stepped layered structure can truly replace the gradient layer structure.

The highest reflectivity (99.86%) of the DBR structure arranged using the LH stacking method is significantly greater than that (98.32%) of the structure arranged using the HL stacking method, but the reflection spectral bandwidths of the two structures are basically the same (Fig. 5). When the number of DBR periods is 15, the reflectivity can reach 98.3%; when the number of periods is >17, the reflectivity of the DBR is >99%; when the number of periods is >20, the reflectivity is >99.5%; and when the number of periods is >40, the reflectivity is >99.99% (Fig. 6). As the incident angle increases, the optical path difference of the dielectric layer decreases, and the DBR reflection spectrum shifts to the short-wavelength direction as a whole. When the incident angle is 0 (normal incidence), the central wavelength of the DBR reflection spectrum is ~940 nm, and when the incident angle is π/3, the central wavelength of the DBR reflection spectrum shifts to 910 nm; that is, the central wavelength is greatly affected by the incident angle (Fig. 7). When the reflectivity is >99.4%, the stop bandwidth of the mutant DBR (D=0 nm) is 89 nm, the stop bandwidth of the DBR with D=10 nm is 88 nm, the stop bandwidth of the DBR with D=20 nm is 85 nm, the stop bandwidth of the DBR with D=30 nm is 81 nm, and the stop bandwidth of the DBR with D=40 nm is 75 nm (Fig. 9). The maximum reflectivity of the gradient DBR gradually decreases with an increase in the thickness of the gradient layer. The highest reflectivity of the mutant DBR exceeds 99.85%, and the highest reflectivity of the DBR with D=40 nm is still >99.6% (Fig. 10).

In this study, using the transfer matrix model, the effects of the DBR stacking method, number of DBR periods, incident angle, and thickness of the gradient layer on the reflectance characteristics of an Al_0.89GaAs/Al_0.09GaAs DBR are investigated. At a wavelength of 940 nm, the refractive index of Al_xGaAs has a linear relationship with the aluminum atomic fraction x, which can be expressed as nAlxGaAs=-0.572x+3.550, which is consistent with the calculation result of the Sellmerier formula. When the incident medium is GaAs and the output medium is air, the DBR with the LH stacking method has greater reflectivity. To study the relationship between the incident angle and the DBR reflection spectrum, the TMM is modified. It is found that, with an increase in the incident angle, the reflection spectrum of the DBR structure moves in the short-wavelength direction, and the reflectivity of the DBR increases. Using the multilayer division equivalent method, the refractive index gradient structure is replaced by a refractive index stepped structure, and the reflection spectrum characteristics of the gradient DBR are analyzed. It is found that, with an increase in the thickness of the gradient layer, the reflection bandwidth of the DBR narrows and the reflectivity at the center wavelength is essentially unaffected. In follow-up research and device preparation, our calculation results can provide a useful guide for the design and optimization of the DBR structure in 940 nm VCSELs.