# Does the ray transfer matrix for a gap change, based on the index of refraction of the medium?

The ray transfer matrix for a gap is typically:

$$\begin{bmatrix} 1 & d \\ 0 & 1\\ \end{bmatrix}$$

If I know that my glass is a thickness $L$ does the ray of light that goes into it travel a distance $n' \, L$ where $n'$ is the refractive index of the glass? So would the $d$ inside the matrix change to $n' \, L$?

Close.

The transfer matrix upon entering the glass is:

$$\begin{bmatrix} 1 & 0 \\ 0 & \frac1{n'}\\ \end{bmatrix}$$

The matrix inside the glass is:

$$\begin{bmatrix} 1 & d \\ 0 & 1\\ \end{bmatrix}$$

And upon exiting the glass it's:

$$\begin{bmatrix} 1 & 0 \\ 0 & \frac{n'}1\\ \end{bmatrix}$$

So multiplying together you get:

$$\begin{bmatrix} 1 & 0 \\ 0 & \frac{n'}1\\ \end{bmatrix} \begin{bmatrix} 1 & d \\ 0 & 1\\ \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & \frac1{n'}\\ \end{bmatrix}= \begin{bmatrix} 1 & \frac{d}{n'} \\ 0 & 1\\ \end{bmatrix}$$

This makes sense intuitively as the higher the index of refraction of your glass the less displacement you'll get.

If your starting and/or ending medium have index of refractions other than 1 then you'd need to modify the 1 in the corresponding fractions to get the corresponding transfer matrix.

• This answer does a better job of putting it all together than my answer, since I was only considering the center matrix, where you still use $d$. Combining the propagation matrices with the center one produces a result that is more readily applicable. Commented Oct 19, 2015 at 22:54

No, you would use the actual thickness of the glass $L$. This is because your ray matrix should change the angle of the incoming ray at the boundary using Snell's law, and to calculate the propagation accurately beyond that, you will want to use the distance covered by the ray in the lab frame.