Mathematical Optics

Exercise 5.6  With the help of the generating function for Hermite polynomials

•  Prove the Mehler identity (Prudnikov et al. 1986b, Equation 5.12.2.1):

$\sum _{n=0}^{\mathrm{\infty }}\mathrm{H}{\mathrm{G}}_{n,n}(\mathbf{r},0)\frac{{(t/2)}^n}{n!}=\frac{1}{\sqrt{1-t^2}}\exp \hspace{0.17em}\left(\frac{2xyt}{1-t^2}-\frac{r^2}{2}\cdot \frac{1+t^2}{1-t^2}\right).$

(5.40)

•  Prove that for each real z, the family of HG beams {HG_m,n (r, z), m, n = 0, 1, …} constitute an orthogonal basis of the space $L_2({\mathbb{R}}^2)$, where the orthogonal property is

$\begin{array}{ll}\underset{{\mathbb{R}}_2}{\iint }\mathrm{H}{\mathrm{G}}_{m,n}(\mathbf{r},z){\mathrm{H}\mathrm{G}}_{M,N}^{\ast }(\mathbf{r},z)d\mathrm{r}& \stackrel{(5.8)}{=}\hspace{0.17em}\underset{{\mathbb{R}}_2}{\iint }\mathrm{H}{\mathrm{G}}_{m,n}(\mathrm{r},0)\mathrm{H}{\mathrm{G}}_{M,N}(\mathrm{r},0)d\mathrm{r}\\ =\pi 2^{m+n}m!n!{\delta }_{m,M}{\delta }_{n,N}.\end{array}$

(5.41)

•  Calculate the Fresnel integral of HG_m,n (r, 0) and prove Equation 5.39.

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It is obvious that the second and third statements in this exercise can be reduced to a lD case. For example, to prove Equation 5.41 it is sufficient to consider the 1D counterparts of HG beams, ...