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- Summary
- Why is the transformation matrix of a beam splitter unitary?

The transformation matrix for a beam splitter relates the four E-fields involved as follows:

$$

\left(\begin{array}{c}

E_{1}\\

E_{2}

\end{array}\right)=\left(\begin{array}{cc}

T & R\\

R & T

\end{array}\right)\left(\begin{array}{c}

E_{3}\\

E_{4}

\end{array}\right)

\tag{1}$$

Here, the amplitude transmission and reflection coefficients are given by ##T=|T|e^{i\theta}## and ##R=|R|e^{i\varphi}##. In various textbooks we are told that for a lossless beam splitter these two quantities are subject to:

$$|T|^2 + |R|^2 =1 \tag{2}$$

I used to believe that this is because ##T^2## and ##R^2## represent transmittance and reflectanace respectively, so in the presence of loss (e.g. absorptance ##A##), one would have ##T^2 + R^2 + A = 1##.

However, according to this, the reason is because the matrix is unitary, which means that we can write:

$$\left(\begin{array}{cc}

T & R\\

R & T

\end{array}\right)\left(\begin{array}{cc}

T^{*} & R^{*}\\

R^{*} & T^{*}

\end{array}\right)=\left(\begin{array}{cc}

1 & 0\\

0 & 1

\end{array}\right) \tag{3}$$

Indeed one of the two relations you get from the above is equation (2). So, how did they know that the transformation matrix must be unitary? Did they assume equation (2) to be true in advance? Or, is equation (2) a consequence of the fact that the matrix is unitary?

Any explanation would be greatly appreciated.

$$

\left(\begin{array}{c}

E_{1}\\

E_{2}

\end{array}\right)=\left(\begin{array}{cc}

T & R\\

R & T

\end{array}\right)\left(\begin{array}{c}

E_{3}\\

E_{4}

\end{array}\right)

\tag{1}$$

Here, the amplitude transmission and reflection coefficients are given by ##T=|T|e^{i\theta}## and ##R=|R|e^{i\varphi}##. In various textbooks we are told that for a lossless beam splitter these two quantities are subject to:

$$|T|^2 + |R|^2 =1 \tag{2}$$

I used to believe that this is because ##T^2## and ##R^2## represent transmittance and reflectanace respectively, so in the presence of loss (e.g. absorptance ##A##), one would have ##T^2 + R^2 + A = 1##.

However, according to this, the reason is because the matrix is unitary, which means that we can write:

$$\left(\begin{array}{cc}

T & R\\

R & T

\end{array}\right)\left(\begin{array}{cc}

T^{*} & R^{*}\\

R^{*} & T^{*}

\end{array}\right)=\left(\begin{array}{cc}

1 & 0\\

0 & 1

\end{array}\right) \tag{3}$$

Indeed one of the two relations you get from the above is equation (2). So, how did they know that the transformation matrix must be unitary? Did they assume equation (2) to be true in advance? Or, is equation (2) a consequence of the fact that the matrix is unitary?

Any explanation would be greatly appreciated.