Question $1(50$ points$).$ This question is about ray$-$transfer matrices.

a$.(15$ points$)$ Determine the ray$-$transfer matrix $\backslash (\backslash$mathbf$\{$M$\}\backslash )$ for free space propagation.

b$.(15$ points$)$ Determine the ray$-$transfer matrix $\backslash (\backslash$mathbf$\{$M$\}\backslash )$ for a concave and convex lens of focal length $\backslash ($ f $\backslash )$ using the imaging equation

$\backslash [$

$\backslash$frac$\{1\}\{$z$\_\{1\}\}+\backslash$frac$\{1\}\{$z$\_\{2\}\}=\backslash$frac$\{1\}\{$f$\}$

$\backslash ]$

where $\backslash ($ f $\backslash )$ is the focal length satisfying $\backslash ($ f$>0\backslash )$ for convex lenses $($thin lens$)$ and $\backslash ($ f$<mn>0</mn><mtext></mtext><mi>\backslash </mi><mi>)</mi>$ for concave lenses $($thick lens$).$ You can assume that the heights of the rays entering and leaving the lens are the same and the paraxial approximation holds.

c$.(20$ points$)$ Using the results found in part $\backslash ($ a $\backslash )$ and $\backslash ($ b $\backslash ),$ determine the ray$-$transfer matrix $\backslash (\backslash$mathbf$\{$M$\}\backslash )$ for an optical system made of a thin convex lens of focal length $\backslash ($ f $\backslash )$ and a thin concave lens of focal length $\backslash (-$f $\backslash )$ separated by a distance of $\backslash ($ f $\backslash ).$ Discuss the imaging properties of this composite lens.