# 2 \(\times\) 2 Matrices: Manifolds, Realizations, Applications
## 2 \(\times\) 2 Matrices Covering 3 \(\times\) 3 Matrices
### Matrices in Paraxial Geometric Optics  
Geometric optics uses symplectic matrices to keep track of linear transformations between ray positions, registered as the points \(\mathbf{q}\) where the ray crosses the standard plane screen \(z=0\), and the ray momentum, which is the projection \(\mathbf{p}\) on the screen plane of a vector \(\vec{n}\) along the ray, whose magnitude is the refraction index \(n:=|\vec{n}|\) (\(n=1\) in vacuum). In plane optics \(N=1\), these are coordinates \(q\in R\) and \(p=n\sin\theta\in n[-1,1]\), where \(\theta\) is the angle between the ray \(\vec{n}\) and the normal to the screen. The coordinates \((q,p)\) are said to be _canonically conjugate_; they enter the Hamilton Equation (9) with a Hamiltonian \(h=n\cos\theta\), where we note that \(p^{2}+h^{2}=n^{2}\) lies on the so-called _Descartes_ sphere (for \(N=1\) a circle). We thus interpret \(h\) as the component of the ray vector that is normal to the screen, in the same sense that \(z\) is the coordinate normal to it and becomes the evolution parameter of the ray \(q(z)\) through a setup of free spaces and lenses. Symplectic matrices are used to describe linear canonical transformations of phase space in a model of geometric optics, called _paraxial_, where the angle \(\theta\) is assumed to be conveniently _small_, so that \(\sin\theta\approx\theta\) and \(\cos\theta\approx 1+\frac{1}{2}\theta^{2}\), but extending \(\theta\in R\). For \(N=1\), paraxial optical systems have thus phase space coordinates in a plane \((q,p)\in R^{2}\) that can be subject to linear transformations. A basis for all linear transformations in this paraxial model are 'free' spaces for displacements by \(z\geq 0\), and 'flat' lenses of Gaussian power \(G\) (i.e., focal distance \(1/G\)), which are represented through the following matrices and their actions,  
displacement  
\[z\]  
, \[\mathbf{D}_{z}:= \begin{pmatrix}1&z\\ 0&1\end{pmatrix} \]  
, \[ \begin{pmatrix}q^{\prime}\\ p^{\prime}\end{pmatrix} =\mathbf{D}_{z} \begin{pmatrix}q\\ p\end{pmatrix} = \begin{pmatrix}q+zp\\ p\end{pmatrix} \]  
, (11) lens power  
\[G\]  
, \[\mathbf{L}_{G}:= \begin{pmatrix}1&0\\ -G&1\end{pmatrix} \]  
, \[ \begin{pmatrix}q^{\prime}\\ p^{\prime}\end{pmatrix} =\mathbf{L}_{G} \begin{pmatrix}q\\ p\end{pmatrix} = \begin{pmatrix}q\\ p-Gq\end{pmatrix} \]  
. (12)  
To corroborate these identifications: Displacement skews phase space, translating the point on the screen \(q\) to \(q^{\prime}\) by \(2n\theta=zp\), and leaving the ray direction \(\theta\) invariant. A lens of power \(G>0\) is convex and convergent by redirecting rays of inclination \(p\) to \(p^{\prime}\) towards the optical axis by angles \(-Gq/n\) to concentrate them at the focal distance \(F=1/G\), while if \(G<0\), the lens is concave and divergent. Representing the optical elements of the paraxial model by matrices allows us to design compound systems with little effort [12, 13]. As we picture light rays advancing from left to right, one lens \(G\) between a free space \(z_{1}\geq 0\) at its left, and another \(z_{2}\geq 0\) to its right, three-element setups are represented:  
\[\mathbf{D}_{z_{2}}\mathbf{L}_{G}\mathbf{D}_{z_{1}}= \begin{pmatrix}1&z_{2}\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ -G&1\end{pmatrix}\begin{pmatrix}1&z_{1}\\ 0&1\end{pmatrix} = \begin{pmatrix}1{-}z_{2}G&z_{2}{+}z_{1}{-}z_{2}Gz_{1}\\ -G&1{-}Gz_{1}\end{pmatrix} . \tag{13}\]  
The order is important because the matrices act on \( \begin{pmatrix}q\\ p\end{pmatrix} \), the phase space vector to their right: First the light ray is acted by the \(z_{1}\) flight, then by the lens, and last by the \(z_{2}\) flight. When the 1-2 element of the product matrix (13) is zero, the image \(q^{\prime}(q)=\mu q\) is independent of the incoming ray direction, so the system satisfies the focal condition: \(z_{2}+z_{1}=z_{2}Gz_{1}\) (i.e., \(1/F=1/z_{1}+1/z_{2}\)), and is an _imager_ with magnification \(\mu:=1-z_{2}G=(1-Gz_{1})^{-1}=-z_{2}/z_{1}\); being negative, the image will be inverted.