What's the mathematical background to the representation for Gaussian beams?

Question

Background

A general optical system (not necessarily having an axis of rotational symmetry) can be represented, for small deviations from a base ray, by the matrix transfer equation, $$ \left[\begin{array}{c} n'L'\\ n'M'\\ x'\\ y' \end{array}\right] = \left[\begin{array}{cc} B & -A\\ -D & C \end{array}\right] \left[\begin{array}{c} nL\\ nM\\ x\\ y \end{array}\right] = D(g) \left[\begin{array}{c} nL\\ nM\\ x\\ y \end{array}\right] $$ where $L,M,N$ are the direction cosines of a ray, $n$ is refractive index, $x,y$ are coords on a reference plane, $A,B,C,D$ are $2\times 2$ matrices and a prime (') denotes the image space and unprimed quantities are in the object space. The sign convention for the matrix sub-blocks is from ``The Ray and Wave Theory of Lenses'' by A. Walther.

The $4\times 4$ transfer matrices $D(g)$ are the defining represention of the symplectic group $Sp(4,R)$ carried on a 4-d vector space $V_{4}$ . They obey, $$ D(g)^{T} \left[\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right] D(g) = \left[\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right] $$ where the $0$ is $2\times 2$ and $1$ is the unit $2\times 2$ matrix. Thus one can say that geometric optics is the study of a finite dimensional representation of $Sp(4,R)$ and each optical system is a $g\in Sp(4,R)$. Furthermore, two optical systems $g$ followed by $h$ are equivalent to the single optical system $hg$.

It turns out (see Guillemin and Sternberg ``Symplectic Techniques in Physics''), provided the optical system is lossless, that diffractive optics in the Fresnel approximation is the study of an infinite-dimensional projective representation of $Sp(4,R)$ carried on a Hilbert space. In this case, the complex amplitude $u(x,y)$ on a reference plane is a vector $|u\rangle$ and the transfer equation between object and images reference planes is now, $$ |u'\rangle=K(g)|u\rangle $$ where $K(g)$ is a kernel which is actually a projective representation of $Sp(4,R)$ carried on the Hilbert space. Note that, in a book on diffraction, like Walther's, the transfer equation would be appear as, $$ u'(x',y')=\int K(x',y'|x,y)u(x,y)dxdy $$ and the kernel $K(x'y'|x,y)$ depends on the matrices $A,B,C,D$ via a function called the eikonal. The details are not important for this question.

Question

If one studies the diffraction of a Gaussian beam, $$ u(x,y)=\exp{\left\{\frac{i}{2} \left[\begin{array}{cc} x & y \end{array}\right]\left(\frac{q}{n}\right)^{-1} \left[ \begin{array}{c} x \\ y \end{array} \right]\right\}} $$ by an optical system in the Fresnel approximation, then the output beam is also Gaussian and the relation between the $2\times 2$ input and output complex beam parameter matrices is: $$ \frac{q'}{n'}=\left(C\frac{q}{n}-D\right)\left(B-A\frac{q}{n}\right)^{-1} $$ This is another representation (realization) of $Sp(4,R)$ carried on the space of symmetric $2\times 2$ matrices like $q/n$. It's some sort of conformal transformation because, for cylindrical lenses, $q/n$ is a complex number and $A,B,C,D$ are real numbers, so the above equation is a conformal transformation.

The existence of this representation was a complete surprise for me; it's clearly related to the infinite-dimensional projective representation of $Sp(4,R)$ because the derivation proceeds via Fresnel theory, but it's carried on the 3-d space of symmetric $2\times 2$ matrices whilst the projective representation has no irreducible subspaces other that the even and odd parity irreps.

My question is, ``What is the mathematical background to the representation for Gaussian beams?''

Answer 1

After some searching, it turns out that mathematicians http://ptmat.fc.ul.pt/~pedro/thesis.pdf know about this stuff. Set up, $$ Z=X+iY $$ where $X$ and $Y$ are $m\times m$ real symmetric matrices and $Y$ is positive definite. The space of matrices like $Z$ is a generalization of the upper half of the complex plane which is called the Siegel Upper Half Plane ($SUHP$). The conformal realization of $SL(2,R)=Sp(2,R)$ on the upper half complex plane generalizes as the realization of the symplectic group $Sp(2m,R)$ carried on the $SUHP$. Let, $$ D(g)= \left[ \begin{array}{cc} A & B\\ C & D \end{array} \right] $$
be a $2m\times 2m$ matrix in the defining rep of the symplectic group $Sp(2m,R)$. The conformal realization of $Sp(2m,R)$ carried on the $SUHP$ is the map of the $SUHP$ to itself, $$ \rho_{g}(Z)=\left(AZ+B\right)\left(CZ+D\right)^{-1} $$ and it's a realization of the group because, $$ \rho_{hg}(Z)=\rho_{h}(\rho_{g}(Z)) $$ for $g,h\in Sp(2m,R)$.

It's interesting that the Gaussian beam parameter matrix $q/n$ lives on the $SUHP$ (actually, the $SLHP$ in the usual way it's defined). However, I still don't understand the relation between the conformal realization of $Sp(2m,R)$ carried on the $SUHP$ and the infinite-dimensional projective rep of $Sp(2m,R)$ carried on the Hilbert space that is Fresnel optics.