302 OPTICS FOR ENGINEERS
Thus, just as we obtain the coefcients in a Fourier series by multiplying the desired func-
tion by the appropriate sine or cosine term, here we multiply the desired beam prole by the
appropriate Hermite–Gaussian function. We need to repeat Equation 9.62 for each m
and n
.
Just as Fourier series is useful because coefcients of terms with high order approach zero, the
Hermite–Gaussian expansion is useful if
C
m
,n
→ 0form
→∞, n
→∞.
However, there is one major difference between the Fourier series and this expansion. In
expanding a periodic function, we know the period, so we know the fundamental frequency of
the sine and cosine waves. In the expansion of a beam prole, we do not have similar guidance
to help us choose w;