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Eigenvalues of Ray Transfer Matrix

A ray transfer matrix can been ragarded as Linear canonical transformation. According to the eigenvalues of the optical system, the system can be classified into several classes. Assume the the ABCD matrix representing a system relates the output ray to the input according to

${\begin{bmatrix}x_{2}\\\theta _{2}\end{bmatrix}}={\begin{bmatrix}A&B\\C&D\end{bmatrix}}{\begin{bmatrix}x_{1}\\\theta _{1}\end{bmatrix}}=\mathbf {T} \mathbf {v}$.

We compute the eigenvlaues of the matrix $\mathbf {T}$ that satisfy eigenequation

$[{\boldsymbol {T}}-\lambda I]\mathbf {v} =\left[{\begin{array}{cc}A-\lambda &B\\C&D-\lambda \end{array}}\right]\mathbf {v} =0$,

by calculating the determinant

$\left|{\begin{array}{cc}A-\lambda &B\\C&D-\lambda \end{array}}\right|=\lambda ^{2}-(A+D)\lambda +1=0$.

Let $m={\frac {(A+D)}{2}}$, and we have eigenvalues $\lambda _{1},\lambda _{2}=m\pm {\sqrt {m^{2}-1}}$.

According to the values of $\lambda _{1}$ and $\lambda _{2}$, there are several possible cases. For example:

1. A pair of real eigenvalues: $r$ and $r^{-1}$, where $r\neq 1$. This case represents a magnifier ${\begin{bmatrix}r&0\\0&r^{-1}\end{bmatrix}}$
2. $\lambda _{1}=\lambda _{2}=1$ or $\lambda _{1}=\lambda _{2}=-1$. This case represents unity matrix (or with an additional coordinate reverter) ${\begin{bmatrix}1&0\\0&1\end{bmatrix}}$.
3. $\lambda _{1},\lambda _{2}=\pm 1$. This case occurs if but not only if the system is either a unity operator, a section of free space, or a lens
4. A pair of two unimodular, complex conjugated eigenvalues $e^{i\theta }$ and $e^{-i\theta }$. This case is similar to a separable Fractional Fourier Transformer.

Relation between geometrical ray optics and wave optics

The theory of Linear canonical transformation implies the relation between ray transfermatrix (geometrical optics) and wave optics.

Element	Matrix in geometrical optics	Operator in wave optics	Remarks
Scaling	${\begin{pmatrix}b^{-1}&0\\0&b\end{pmatrix}}$	${\mathcal {V}}[b]u(x)=u(bx)$
Quadratic phase factor	${\begin{pmatrix}1&0\\c&1\end{pmatrix}}$	$Q[c]=\exp j{\frac {k_{0}}{2}}cx^{2}$	$k_{0}$: wave number
Fresnel free-space-propagation operator	${\begin{pmatrix}1&d\\0&1\end{pmatrix}}$	${\mathcal {R}}[d]\left\{U\left(x_{1}\right)\right\}={\frac {1}{\sqrt {j\lambda d}}}\int _{-\infty }^{\infty }U\left(x_{1}\right)e^{j{\frac {k}{2d}}\left(x_{2}-x_{1}\right)^{2}}dx_{1}$	$x_{1}$: coordinate of the source $x_{2}$: coordinate of the goal
Normalized Fourier-transform operator	${\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$	${\mathcal {F}}=\left(j\lambda _{0}\right)^{-1/2}\int _{-\infty }^{\infty }dx\left[\exp \left(jk_{0}px\right)\right]\ldots$

Common Decomposition of Ray Transfer Matrix

There exist infinite ways to decompose a ray transfer matrix $\mathbf {T} ={\begin{bmatrix}A&B\\C&D\end{bmatrix}}$ into a concatenation of multiple transfer matrix. For example:

1. ${\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\left[{\begin{array}{ll}1&0\\D/B&1\end{array}}\right]\left[{\begin{array}{rr}B&0\\0&1/B\end{array}}\right]\left[{\begin{array}{ll}0&1\\-1&0\end{array}}\right]\left[{\begin{array}{ll}1&0\\A/B&1\end{array}}\right]$.
2. ${\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\left[{\begin{array}{ll}1&0\\C/A&1\end{array}}\right]\left[{\begin{array}{rr}A&0\\0&A^{-1}\end{array}}\right]\left[{\begin{array}{ll}1&B/A\\0&1\end{array}}\right]$
3. ${\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\left[{\begin{array}{ll}1&A/C\\0&1\end{array}}\right]\left[{\begin{array}{lr}-C^{-1}&0\\0&-C\end{array}}\right]\left[{\begin{array}{ll}0&1\\-1&0\end{array}}\right]\left[{\begin{array}{ll}1&D/C\\0&1\end{array}}\right]$
4. ${\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\left[{\begin{array}{ll}1&B/D\\0&1\end{array}}\right]\left[{\begin{array}{ll}D^{-1}&0\\0&D\end{array}}\right]\left[{\begin{array}{ll}1&0\\C/D&1\end{array}}\right]$