Microring resonator matrix

[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?

Thank you in advance!

 

[Andy Resnick]

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?

Thank you in advance!

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

 

[Rampart123]

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

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.

 

[Andy Resnick]

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.

Ok, so a little more digging is required. How about this:

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