Influences of the Current Transverse Profile on the Transverse Mode Competition of VCSELs Diodes

Abstract

A spatially independent model of vertical-cavity surface-emitting lasers (VCSELs) is derived by integrating the spatially dependent rate equations over the cross section of the cavity of a VCSEL. The angular and radial non-uniformities of the injection current are taken into account. The well-known LP modes of a weakly-guiding cylindrical waveguide are employed to describe the transverse modal structure in the VCSEL cavity. This model is solved in a self-consistent way by using the 4th order Runge-Kutta method. The dependence of transverse mode competition on the current intensity, the angular and radial non-uniformities of the injection current, and the geometrical parameters of the electrical contact are thoroughly investigated and analyzed. The results are useful to the optimum design of the optical transverse modal structure of VCSELs.

Appendix

$$ d^{{s,c}}_{{ij}} = D_{n} {\left( {\frac{{\sigma _{i} }} {{r_{{cl}} }}} \right)}^{2} ,gnt^{{s,c}}_{{mn}} = 1,b_{0} = 1 $$

(A1)

$$ \begin{aligned} \xi ^{{s,c}}_{{ij}} = \frac{2} {{\pi r^{2}_{{cl}} J^{2}_{0} {\left( {\sigma _{i} } \right)}{\left( {1 + \delta _{{j0}} } \right)}}}{\int\limits_0^{2\pi } {{\int\limits_0^{r_{{cl}} } {J_{0} {\left( {\frac{{\sigma _{i} r}} {{r_{{cl}} }}} \right)} \cdot {\left( {\begin{array}{*{20}c} {{\sin {\left( {j\varphi } \right)}}} \\ {{\cos {\left( {j\varphi } \right)}}} \\ \end{array} } \right)}} }} } \\ \cdot j_{r} {\left( r \right)} \cdot j_{\varphi } {\left( \varphi \right)}rdrd\varphi \\ \end{aligned} $$

(A2)

$$ a^{{s,c}}_{{ii_{1} }} = \frac{{2D_{n} }} {{r^{2}_{{cl}} J^{2}_{0} {\left( {\sigma _{i} } \right)}}}{\int\limits_\delta ^{r_{{cl}} } {\frac{1} {r}J_{0} {\left( {\frac{{\sigma _{i} r}} {{r_{{cl}} }}} \right)}J_{0} {\left( {\frac{{\sigma _{{i_{1} }} r}} {{r_{{cl}} }}} \right)}} }dr $$

(A3)

$$ \begin{aligned} oss^{{s,c}}_{{ijmni_{1} j_{1} }} = \frac{2} {{\pi r^{2}_{{cl}} J^{2}_{0} {\left( {\sigma _{i} } \right)}}}{\int\limits_0^{2\pi } {{\int\limits_0^{r_{{cl}} } {J_{0} {\left( {\frac{{\sigma _{i} r}} {{r_{{cl}} }}} \right)}J_{0} {\left( {\frac{{\sigma _{{i_{1} }} r}} {{r_{{cl}} }}} \right)}} }} } \\ \cdot \sin {\left( {j\varphi } \right)}\sin {\left( {j_{1} \varphi } \right)} \cdot \psi ^{{s,c}}_{{mn}} {\left( {r,\varphi } \right)}rdrd\varphi \\ \end{aligned} $$

(A4)

$$ \begin{aligned} oss^{{s,c}}_{{ijmni_{1} j_{1} }} = \frac{2}{{\pi r^{2}_{{cl}} J^{2}_{0} {\left( {\sigma _{i} } \right)}(1 + \delta j0)}}{\int\limits_0^{2\pi } {{\int\limits_0^{r_{{cl}} } {J_{0} {\left( {\frac{{\sigma _{i} r}}{{r_{{cl}} }}} \right)}J_{0} {\left( {\frac{{\sigma _{{i_{1} }} r}}{{r_{{cl}} }}} \right)}} }} } \\ \cdot \cos {\left( {j\varphi } \right)}\cos {\left( {j_{1} \varphi } \right)} \cdot \psi ^{{s,c}}_{{mn}} {\left( {r,\varphi } \right)}rdrd\varphi \\ \end{aligned} $$

(A5)

$$ gnc^{{s,c}}_{{ijmn}} = \frac{1} {{\pi r^{2}_{{cl}} }}{\int\limits_0^{2\pi } {{\int\limits_0^{r_{{cl}} } {J_{0} {\left( {\frac{{\sigma _{i} r}} {{r_{{cl}} }}} \right)}\cos {\left( {j\varphi } \right)}\psi ^{{s,c}}_{{mn}} {\left( {r,\varphi } \right)}rdrd\varphi } }} } $$

(A6)