# Ray transfer matrix for non small angle

As I could understand, we can trace rays through optical elements using the ray transfer matrices . However, the references I could find online seems like it considers the angles to be small (Sin(t) = t). For example:

What should I change to make the transfer working for general case where the angle is not small ? should replace all the angles with its tangent?

Edit:

I made a small calculation for the propagation matrix:

$$\begin{pmatrix}1& d\\ 0& 1\end{pmatrix}$$ It gives: $$y_2 = y_1 + d\theta_1$$ However, the correct equation based on geometric principles should be: $$y_2 = y_1 + d\tan \theta_1$$ So, I supposed that in general case we should use $$\tan \theta$$ instead of $$\theta$$. Any confirmation?

In the linear case, $$1\cdot \theta_1 = 1.5\cdot \theta_2 \Rightarrow \theta_2 = \frac{1}{1.5} \theta_1$$, meaning the matrix is $$\begin{pmatrix} 1& 0 \\ 0& \frac{1}{1.5} \end{pmatrix}$$. But in more general, $$\theta_2 = \arcsin(\frac{1}{1.5} \sin \theta_1 )$$, so your solution of writing $$\tan \theta_2 = \frac{1}{1.5} \tan \theta_1$$ isn't correct.