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.


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?''


1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.