The [$ABCD$ matrix](http://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis) of a glass graded-index slab with refractive index $n(y)=n_0(1-\frac{1}{2}\alpha^{2}y^{2})$ and length $d$ is $A=\cos(\alpha d)$, $B=\frac{1}{\alpha}\sin(\alpha d)$, $C=-\alpha \sin(\alpha d)$, $D=\cos(\alpha d)$ for paraxial rays along the z axis. Usually, $\alpha$ is chosen to be sufficiently small so that $\alpha^{2}y^{2} << 1$. A [Gaussian beam](http://en.wikipedia.org/wiki/Gaussian_beam) of wavelength $\lambda_0$, waist radius $W_0$ in free space, and axis in the z direction enters the slab at its waist. How can I use the [$ABCD$ law](http://www.optique-ingenieur.org/en/courses/OPI_ang_M01_C03/co/Contenu_04.html) to get an expression for the beam width in the $y$ direction as a function of $d$?