Analytical solutions of coupled-mode equations for microring resonators

Abstract

We present a study on analytical solutions of coupled-mode equations for microring resonators with an emphasis on occurrence of all-optical EIT phenomenon, obtained by using a cofactor. As concrete examples, analytical solutions for a 3×3 linearly distributed coupler and a circularly distributed coupler are obtained. The former corresponds to a non-degenerate eigenvalue problem and the latter corresponds to a degenerate eigenvalue problem. For comparison and without loss of generality, analytical solution for a 4×4 linearly distributed coupler is also obtained. This paper may be of interest to optical physics and integrated photonics communities.

Appendix A:

Comparing

$$\left( {{\begin{array}{ccc} x & \xi & \sigma \\ \xi & x & \eta \\ \sigma & \eta & x \end{array} }} \right)\left( {\begin{array}{l} \varphi_{1} \\ \varphi_{2} \\ \varphi_{3} \end{array}} \right)=0 $$

with

$$\left( {{\begin{array}{ccc} {a_{11} } & {a_{12} } & {a_{13} } \\ {a_{21} } & {a_{22} } & {a_{23} } \\ {a_{31} } & {a_{32} } & {a_{33} } \end{array} }} \right)\left( {\begin{array}{l} A_{11} \\ A_{21} \\ A_{31} \end{array}} \right)=0, $$

we have

$$\left( {{\begin{array}{ccc} x & \xi & \sigma \\ \xi & x & \eta \\ \sigma & \eta & x \end{array} }} \right)=\left( {{\begin{array}{ccc} {a_{11} } & {a_{12} } & {a_{13} } \\ {a_{21} } & {a_{22} } & {a_{23} } \\ {a_{31} } & {a_{32} } & {a_{33} } \end{array} }} \right), \quad \varphi_{1} =A_{11} , \quad \varphi_{2} =A_{21} , \quad \varphi_{3} =A_{31} . $$

(A.1)

and

$$\left| {{\begin{array}{ccc} {a_{11} } & {a_{12} } & {a_{13} } \\ {a_{21} } & {a_{22} } & {a_{23} } \\ {a_{31} } & {a_{32} } & {a_{33} } \end{array} }} \right|=a_{11} A_{11} +a_{12} A_{21} +a_{13} A_{31} , $$

where

$$ A_{11} =\left| {{\begin{array}{cc} {a_{22} } & {a_{23} } \\ {a_{32} } & {a_{33} } \end{array} }} \right|, \quad A_{21} =-\left| {{\begin{array}{cc} {a_{21} } & {a_{23} } \\ {a_{31} } & {a_{33} } \end{array} }} \right|, \quad A_{31} =\left| {{\begin{array}{cc} {a_{21} } & {a_{22} } \\ {a_{31} } & {a_{32} } \end{array} }} \right|. $$

(A.2)