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Exercise 4 ABCD matrix analysis (12 + 4 = 16) Following a confocal resonator is shown. This kind of resonator is often used in laser optics. It consists of two identical spherical mirrors $M_1$ and $M_2$ whose radii match the resonator's length. (a) Show, without first specifying the value of $d$, that after traversing the cavity two times the system matrix is $\begin{pmatrix} 1 - \frac{2d}{r} & 2d(1 - \frac{d}{r})\\ \frac{4}{r}(d - r^{-1}) & (2\frac{d}{r} - 1)^2 - \frac{2d}{r} \end{pmatrix}$. (b) Then for the specific case of $d = r$ show that after four reflections the system is back where it started and the light will retrace its original path.

          Exercise 4 ABCD matrix analysis (12 + 4 = 16)
Following a confocal resonator is shown. This kind of resonator is often used in laser
optics. It consists of two identical spherical mirrors $M_1$ and $M_2$ whose radii match the
resonator's length.
(a) Show, without first specifying the value of $d$, that after traversing the cavity two
times the system matrix is
$\begin{pmatrix} 1 - \frac{2d}{r} & 2d(1 - \frac{d}{r})\\ \frac{4}{r}(d - r^{-1}) & (2\frac{d}{r} - 1)^2 - \frac{2d}{r} \end{pmatrix}$.
(b) Then for the specific case of $d = r$ show that after four reflections the system is
back where it started and the light will retrace its original path.

Exercise 4 ABCD matrix analysis (12 + 4 = 16)
Following a confocal resonator is shown. This kind of resonator is often used in laser
optics. It consists of two identical spherical mirrors M1 and M2 whose radii match the
resonator's length.
(a) Show, without first specifying the value of d, that after traversing the cavity two
times the system matrix is
< p m a t r i x >.
(b) Then for the specific case of d = r show that after four reflections the system is
back where it started and the light will retrace its original path.

Added by Jared C.

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