Ray transfer matrix analysis

Some examples

Free space example

As one example, if there is free space between the two planes, the ray transfer matrix is given by: $${\displaystyle \mathbf{S}=\left[\begin{array}{cc}1& d\\ 0& 1\end{array}\right],}$$ where d is the separation distance (measured along the optical axis) between the two reference planes. The ray transfer equation thus becomes: $${\displaystyle \left[\begin{array}{c}{x}_2\\ {\theta }_2\end{array}\right]=\mathbf{S}\left[\begin{array}{c}{x}_1\\ {\theta }_1\end{array}\right],}$$ and this relates the parameters of the two rays as: $${\displaystyle \begin{array}{cc}{x}_2& =x_1+d{\theta }_1\\ {\theta }_2& =\phantom{x_1+d}{\theta }_1\end{array}}$$

Thin lens example

Another simple example is that of a thin lens. Its RTM is given by: $${\displaystyle \mathbf{L}=\left[\begin{array}{cc}1& 0\\ -\frac{1}{f}& 1\end{array}\right],}$$ where f is the focal length of the lens. To describe combinations of optical components, ray transfer matrices may be multiplied together to obtain an overall RTM for the compound optical system. For the example of free space of length d followed by a lens of focal length f: $${\displaystyle \mathbf{L}\mathbf{S}=\left[\begin{array}{cc}1& 0\\ -\frac{1}{f}& 1\end{array}\right]\left[\begin{array}{cc}1& d\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& d\\ -\frac{1}{f}& 1-\frac{d}{f}\end{array}\right].}$$

Note that, since the multiplication of matrices is non-commutative, this is not the same RTM as that for a lens followed by free space: $${\displaystyle \mathbf{S}\mathbf{L}=\left[\begin{array}{cc}1& d\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ -\frac{1}{f}& 1\end{array}\right]=\left[\begin{array}{cc}1-\frac{d}{f}& d\\ -\frac{1}{f}& 1\end{array}\right].}$$

Thus the matrices must be ordered appropriately, with the last matrix premultiplying the second last, and so on until the first matrix is premultiplied by the second. Other matrices can be constructed to represent interfaces with media of different refractive indices, reflection from mirrors, etc.

Eigenvalues

A ray transfer matrix can be regarded as a linear canonical transformation. According to the eigenvalues of the optical system, the system can be classified into several classes. Assume the ABCD matrix representing a system relates the output ray to the input according to $${\displaystyle \left[\begin{array}{c}{x}_2\\ {\theta }_2\end{array}\right]=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{c}{x}_1\\ {\theta }_1\end{array}\right]=\mathbf{T}\mathbf{v}.}$$

We compute the eigenvalues of the matrix ${\displaystyle \mathbf{T}}$ that satisfy eigenequation $${\displaystyle [\mathit{T}-\lambda I]\mathbf{v}=\left[\begin{array}{cc}A-\lambda & B\\ C& D-\lambda \end{array}\right]\mathbf{v}=0,}$$ by calculating the determinant $${\displaystyle \left|\begin{array}{cc}A-\lambda & B\\ C& D-\lambda \end{array}\right|={\lambda }^2-(A+D)\lambda +1=0.}$$

Let ${\displaystyle m=\frac{(A+D)}{2}}$, and we have eigenvalues ${\displaystyle {\lambda }_1,{\lambda }_2=m\pm \sqrt{m^2-1}}$.

According to the values of ${\displaystyle {\lambda }_1}$ and ${\displaystyle {\lambda }_2}$, there are several possible cases. For example:

1. A pair of real eigenvalues: ${\displaystyle r}$ and ${\displaystyle r^{-1}}$, where ${\displaystyle r\ne 1}$. This case represents a magnifier ${\displaystyle \left[\begin{array}{cc}r& 0\\ 0& r^{-1}\end{array}\right]}$
2. ${\displaystyle {\lambda }_1={\lambda }_2=1}$ or ${\displaystyle {\lambda }_1={\lambda }_2=-1}$. This case represents unity matrix (or with an additional coordinate reverter) ${\displaystyle \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]}$.
3. ${\displaystyle {\lambda }_1,{\lambda }_2=\pm 1}$. 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 ${\displaystyle e^{i\theta }}$ and ${\displaystyle e^{-i\theta }}$. This case is similar to a separable Fractional Fourier Transform.

Matrices for simple optical components

Relation between geometrical ray optics and wave optics

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

Common decomposition

There exist infinite ways to decompose a ray transfer matrix ${\displaystyle \mathbf{T}=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]}$ into a concatenation of multiple transfer matrices. For example in the special case when ${\displaystyle n_1=n_2}$:

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

Resonator stability

RTM analysis is particularly useful when modeling the behavior of light in optical resonators, such as those used in lasers. At its simplest, an optical resonator consists of two identical facing mirrors of 100% reflectivity and radius of curvature R, separated by some distance d. For the purposes of ray tracing, this is equivalent to a series of identical thin lenses of focal length f = R/2, each separated from the next by length d. This construction is known as a lens equivalent duct or lens equivalent waveguide. The RTM of each section of the waveguide is, as above, $${\displaystyle \mathbf{M}=\mathbf{L}\mathbf{S}=\left(\begin{array}{cc}1& d\\ \frac{-1}{f}& 1-\frac{d}{f}\end{array}\right).}$$

RTM analysis can now be used to determine the stability of the waveguide (and equivalently, the resonator). That is, it can be determined under what conditions light traveling down the waveguide will be periodically refocused and stay within the waveguide. To do so, we can find all the "eigenrays" of the system: the input ray vector at each of the mentioned sections of the waveguide times a real or complex factor λ is equal to the output one. This gives: $${\displaystyle \mathbf{M}\left[\begin{array}{c}{x}_1\\ {\theta }_1\end{array}\right]=\left[\begin{array}{c}{x}_2\\ {\theta }_2\end{array}\right]=\lambda \left[\begin{array}{c}{x}_1\\ {\theta }_1\end{array}\right].}$$ which is an eigenvalue equation: $${\displaystyle \left[\mathbf{M}-\lambda \mathbf{I}\right]\left[\begin{array}{c}{x}_1\\ {\theta }_1\end{array}\right]=0,}$$ where $\mathbf{I}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ is the 2 × 2 identity matrix.

We proceed to calculate the eigenvalues of the transfer matrix: $${\displaystyle det\left[\mathbf{M}-\lambda \mathbf{I}\right]=0,}$$ leading to the characteristic equation $${\displaystyle {\lambda }^2-\mathrm{tr}(\mathbf{M})\lambda +det(\mathbf{M})=0,}$$ where $${\displaystyle \mathrm{tr}(\mathbf{M})=A+D=2-\frac{d}{f}}$$ is the trace of the RTM, and $${\displaystyle det(\mathbf{M})=AD-BC=1}$$ is the determinant of the RTM. After one common substitution we have: $${\displaystyle {\lambda }^2-2g\lambda +1=0,}$$ where $${\displaystyle g\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{\mathrm{tr}(\mathbf{M})}{2}=1-\frac{d}{2f}}$$ is the stability parameter. The eigenvalues are the solutions of the characteristic equation. From the quadratic formula we find $${\displaystyle {\lambda }_{\pm }=g\pm \sqrt{g^2-1}.}$$

Now, consider a ray after N passes through the system: $${\displaystyle \left[\begin{array}{c}{x}_N\\ {\theta }_N\end{array}\right]={\lambda }^N\left[\begin{array}{c}{x}_1\\ {\theta }_1\end{array}\right].}$$

If the waveguide is stable, no ray should stray arbitrarily far from the main axis, that is, λ^N must not grow without limit. Suppose ${\displaystyle g^2>1}$. Then both eigenvalues are real. Since ${\displaystyle {\lambda }_{+}{\lambda }_{-}=1}$, one of them has to be bigger than 1 (in absolute value), which implies that the ray which corresponds to this eigenvector would not converge. Therefore, in a stable waveguide, ${\displaystyle g^2\le 1}$, and the eigenvalues can be represented by complex numbers: $${\displaystyle {\lambda }_{\pm }=g\pm i\sqrt{1-g^2}=\cos (\varphi )\pm i\sin (\varphi )=e^{\pm i\varphi },}$$ with the substitution g = cos(ϕ).

For ${\displaystyle g^2<1}$ let ${\displaystyle r_{+}}$ and ${\displaystyle r_{-}}$ be the eigenvectors with respect to the eigenvalues ${\displaystyle {\lambda }_{+}}$ and ${\displaystyle {\lambda }_{-}}$ respectively, which span all the vector space because they are orthogonal, the latter due to ${\displaystyle {\lambda }_{+}\ne {\lambda }_{-}}$. The input vector can therefore be written as $${\displaystyle c_{+}{r}_{+}+c_{-}{r}_{-},}$$ for some constants ${\displaystyle c_{+}}$ and ${\displaystyle c_{-}}$.

After N waveguide sectors, the output reads $${\displaystyle {\mathbf{M}}^N(c_{+}{r}_{+}+c_{-}{r}_{-})={\lambda }_{+}^N{c}_{+}{r}_{+}+{\lambda }_{-}^N{c}_{-}{r}_{-}=e^{iN\varphi }{c}_{+}{r}_{+}+e^{-iN\varphi }{c}_{-}{r}_{-},}$$ which represents a periodic function.

Gaussian beams

The same matrices can also be used to calculate the evolution of Gaussian beams propagating through optical components described by the same transmission matrices. If we have a Gaussian beam of wavelength ${\displaystyle {\lambda }_0}$, radius of curvature R (positive for diverging, negative for converging), beam spot size w and refractive index n, it is possible to define a complex beam parameter q by: $${\displaystyle \frac{1}{q}=\frac{1}{R}-\frac{i{\lambda }_0}{\pi n{w}^2}.}$$

(R, w, and q are functions of position.) If the beam axis is in the z direction, with waist at z₀ and Rayleigh range z_R, this can be equivalently written as $${\displaystyle q=(z-z_0)+i{z}_R.}$$

This beam can be propagated through an optical system with a given ray transfer matrix by using the equation: $${\displaystyle \left[\begin{array}{c}{q}_2\\ 1\end{array}\right]=k\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\left[\begin{array}{c}{q}_1\\ 1\end{array}\right],}$$ where k is a normalization constant chosen to keep the second component of the ray vector equal to 1. Using matrix multiplication, this equation expands as $${\displaystyle \begin{array}{cc}{q}_2& =k(A{q}_1+B)\\ 1& =k(C{q}_1+D).\end{array}}$$

Dividing the first equation by the second eliminates the normalization constant: $${\displaystyle q_2=\frac{A{q}_1+B}{C{q}_1+D},}$$

It is often convenient to express this last equation in reciprocal form: $${\displaystyle \frac{1}{q_2}=\frac{C+D/q_1}{A+B/q_1}.}$$

Example: Free space

Consider a beam traveling a distance d through free space, the ray transfer matrix is $${\displaystyle \left[\begin{array}{cc}A& B\\ C& D\end{array}\right]=\left[\begin{array}{cc}1& d\\ 0& 1\end{array}\right].}$$ and so $${\displaystyle q_2=\frac{A{q}_1+B}{C{q}_1+D}=\frac{q_1+d}{1}=q_1+d}$$ consistent with the expression above for ordinary Gaussian beam propagation, i.e. ${\displaystyle q=(z-z_0)+i{z}_R}$. As the beam propagates, both the radius and waist change.

Example: Thin lens

Consider a beam traveling through a thin lens with focal length f. The ray transfer matrix is $${\displaystyle \left[\begin{array}{cc}A& B\\ C& D\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -1/f& 1\end{array}\right].}$$ and so $${\displaystyle q_2=\frac{A{q}_1+B}{C{q}_1+D}=\frac{q_1}{-\frac{q_1}{f}+1}}$$ $${\displaystyle \frac{1}{q_2}=\frac{-\frac{q_1}{f}+1}{q_1}=\frac{1}{q_1}-\frac{1}{f}.}$$ Only the real part of 1/q is affected: the wavefront curvature 1/R is reduced by the power of the lens 1/f, while the lateral beam size w remains unchanged upon exiting the thin lens.

Higher rank matrices

Methods using transfer matrices of higher dimensionality, that is 3 × 3, 4 × 4, and 6 × 6, are also used in optical analysis. In particular, 4 × 4 propagation matrices are used in the design and analysis of prism sequences for pulse compression in femtosecond lasers.