# Microring resonator matrix

• I
• Rampart123
Rampart123
TL;DR Summary
Explaining the matrix elements.
Hello everyone,

A simple ring resonator with a bus waveguide is described by:
$$\begin{pmatrix} E_{t1}\\ E_{t2} \end{pmatrix} = \begin{pmatrix} t & k\\ -k^* & t^* \end{pmatrix} \begin{pmatrix} E_{i1}\\ E_{i2} \end{pmatrix}$$

I do not understand though why we have -k* and t*? Shouldn't they be also k and t?

I think the conjugation has to do with the phase of the circulating mode?

Rampart123 said:
TL;DR Summary: Explaining the matrix elements.

Hello everyone,

A simple ring resonator with a bus waveguide is described by:
$$\begin{pmatrix} E_{t1}\\ E_{t2} \end{pmatrix} = \begin{pmatrix} t & k\\ -k^* & t^* \end{pmatrix} \begin{pmatrix} E_{i1}\\ E_{i2} \end{pmatrix}$$

I do not understand though why we have -k* and t*? Shouldn't they be also k and t?

I think the conjugation has to do with the phase of the circulating mode?

The derivation appears in several papers, unfortunately some of these references are behind a paywall:

https://opg.optica.org/oe/fulltext.cfm?uri=oe-12-1-90&id=78458
https://digital-library.theiet.org/content/journals/10.1049/el_20000340
https://www.researchgate.net/public...GFnZSI6Il9kaXJlY3QiLCJwYWdlIjoiX2RpcmVjdCJ9fQ

Andy Resnick said:
Thank you for the reply. However, it seems to me that it is not explained in neither of these 3 papers that you mentioned.
In the first paper: It just uses the matrix but does not explain why we have the conjugate
In the second paper: The same as in the first.
In the third paper: The matrix does not have any conjugation, but rather the matrix consists of only t and k, which was also the question of mine.

Why it is
$$\begin{pmatrix} E_{t1}\\ E_{t2} \end{pmatrix} = \begin{pmatrix} t & k\\ -k^* & t^* \end{pmatrix} \begin{pmatrix} E_{i1}\\ E_{i2} \end{pmatrix}$$ and not

$$\begin{pmatrix} E_{t1}\\ E_{t2} \end{pmatrix} = \begin{pmatrix} t & k\\ k & t \end{pmatrix} \begin{pmatrix} E_{i1}\\ E_{i2} \end{pmatrix}$$

Most articles do not explain, they just use the matrix that they found in a book and then do some calculations.

Rampart123 said:
Thank you for the reply. However, it seems to me that it is not explained in neither of these 3 papers that you mentioned.
In the first paper: It just uses the matrix but does not explain why we have the conjugate
In the second paper: The same as in the first.
In the third paper: The matrix does not have any conjugation, but rather the matrix consists of only t and k, which was also the question of mine.

Why it is
$$\begin{pmatrix} E_{t1}\\ E_{t2} \end{pmatrix} = \begin{pmatrix} t & k\\ -k^* & t^* \end{pmatrix} \begin{pmatrix} E_{i1}\\ E_{i2} \end{pmatrix}$$ and not

$$\begin{pmatrix} E_{t1}\\ E_{t2} \end{pmatrix} = \begin{pmatrix} t & k\\ k & t \end{pmatrix} \begin{pmatrix} E_{i1}\\ E_{i2} \end{pmatrix}$$

Most articles do not explain, they just use the matrix that they found in a book and then do some calculations.

https://hal.science/hal-00474731/document

• Optics
Replies
5
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
914
• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
4
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Linear and Abstract Algebra
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
875
• General Math
Replies
9
Views
2K
• Linear and Abstract Algebra
Replies
1
Views
801