# 2 \(\times\) 2 Matrices: Manifolds, Realizations, Applications
## 2 \(\times\) 2 Matrices Covering 3 \(\times\) 3 Matrices
### Matrices in Paraxial Geometric Optics
On the other and, when \(z_{1}=z_{2}=:\tan\frac{\pi}{4}\alpha\) and \(G=\sin\frac{\pi}{2}\alpha\), the matrix (13) becomes \(\mathbf{F}^{a}= \begin{pmatrix}\cos\frac{1}{2}\pi\alpha&-\sin\frac{1}{2}\pi \alpha\\ \sin\frac{1}{2}\pi\alpha&\cos\frac{1}{2}\pi\alpha\end{pmatrix} \), namely a _rotation_ of phase space \((q,p)\) by \(\alpha\), which is a fractional Fourier transform of power \(a:=-2\alpha/\pi\); for \(a=1\) this is the standard Fourier transform [14,15],  
\[\mathbf{F}:= \begin{pmatrix}0&1\\ -1&0\end{pmatrix} ,\quad\text{that acts as }\mathbf{F}:(q,p)\mapsto(p,-q), \tag{14}\]  
and the inverse is \(\mathbf{F}^{-1}=-\mathbf{F}= \begin{pmatrix}0&-1\\ 1&0\end{pmatrix} \). Fractional Fourier transformers (for \(a\) integer or non-integer) can be concatenated with products within the set; so \(\mathbf{F}^{2}=-\mathbf{1}\) and \(\mathbf{F}^{4}=\mathbf{1}\) brings us back to an 'identity' geometric optical system (there will be more on this in the next section). With a Fourier transformer piece, its inverse, and one lens, we can build:  
\[\mathbf{F}\,\mathbf{L}_{G}\mathbf{F}^{-1}= \begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}1&0\\ -G&1\end{pmatrix}\begin{pmatrix}0&-1\\ 1&0\end{pmatrix} = \begin{pmatrix}1&G\\ 0&1\end{pmatrix} =\mathbf{D}_{G}. \tag{15}\]  
When the lens is divergent (concave, \(G<0\)), this is negative free flight. An even better setup for negative free flight is:  
\[\mathbf{L}_{3/z}\mathbf{D}_{z}\mathbf{L}_{3/z}\mathbf{D}_{z}\mathbf{L}_{3/z}= \mathbf{D}_{-z}, \tag{16}\]  
because its total length is only \(2z\); it is also back-forth-symmetric so it minimizes aberrations. In (14) we saw that four concatenated geometrical Fourier transformers bring us back to the identity transformation, while the setup (15) cannot reach back to \(z=0\). A third setup to produce the identity can be 'built' with repeated blocks of one lens and a displacement,  
\[\mathbf{B}_{G,z} := \mathbf{L}_{G}\mathbf{D}_{z}= \begin{pmatrix}1&0\\ -G&1\end{pmatrix}\begin{pmatrix}1&z\\ 0&1\end{pmatrix} = \begin{pmatrix}1&z\\ -G&1{-}Gz\end{pmatrix} , \tag{17}\] \[\mathbf{B}_{G,z}^{2} = \begin{pmatrix}1{-}Gz&z(2{-}Gz)\\ -G(2{-}Gz)&(1{-}Gz)^{2}{-}Gz\end{pmatrix} \;\;\;\text{is } \begin{pmatrix}-1&0\\ 0&-1\end{pmatrix} \text{ for }Gz=2,\] (18) \[\mathbf{B}_{G,z}^{3} = \begin{pmatrix}(1{-}Gz)^{2}{-}Gz&z(1{-}Gz)(3{-}Gz)\\ -G(1{-}Gz)(3{-}Gz)&(1{-}Gz)^{3}{-}Gz(3{-}2 Gz)\end{pmatrix} \;\text{is }\begin{cases} \begin{pmatrix}1&0\\ 0&1\end{pmatrix} Gz{-}3,\\ \begin{pmatrix}-1&0\\ 0&-1\end{pmatrix} Gz{-}1.\end{pmatrix} \tag{19}\]  
Hence, the geometric optical transformation of inversion \( \begin{pmatrix}-1&0\\ 0&-1\end{pmatrix} \) can be obtained with \(\mathbf{F}^{2}\), \(\mathbf{B}_{2/z,z}^{2}\), or \(\mathbf{B}_{1/z,z}^{3}\) setups, while the identity \( \begin{pmatrix}1&0\\ 0&1\end{pmatrix} \) of linear transformations is built with \(\mathbf{F}^{4}\), or \(\mathbf{B}_{3/z,z}^{3}\) setups. This will be contrasted below with the inversion and unit integral canonical transformations in the linear wave model. It is finally a natural question whether any \(2\times 2\) matrix \( \begin{pmatrix}a&b\\ c&d\end{pmatrix} \) of unit determinant can be'represented' by a (paraxial) optical system. Part of the difficulty in proving this, is that the decomposition of generic matrices into left- and right-triangular ones is seldom used,compared with other decompositions such as the Euler angle, polar rotation, Iwasawa, or Bargmann, which are mathematically richer. Yet, the answer is in the affirmative: It requires up to three lenses separated by two free flights to realize all \(2\times 2\) symplectic matrices, except that we only miss a subset of zero measure as seen in (16), which can be bridged by just allowing one last displacement at either end; see [11] (Sect. 10.5) and [16]. The case \(N=2\) of three-dimensional setups with generally cylindrical lenses is addressed in Ref. [11] (Chs. 10 and 12), although the matter of minimal arrangements has been eschewed--as far as we know.