Paraxial approximation for spherical Green functions

Note: This post is just some notes for a discussion.

We can start by writing it simply as

\displaystyle f(\mathbf{r}) = \frac{e^{ikr}}{r}.

Now we use \mathbf{r} = [x\,y\,z]^T and assume |z| \gg |x|, |y|. After expanding the expression using the coordinates,

\displaystyle f(x, y, z) = \frac{\exp(ik\sqrt{z^2 + x^2 + y^2})}{\sqrt{z^2 + x^2 + y^2}},

we have to find now a way to get rid of the square roots. Taking z^2 as a common factor,

\displaystyle \sqrt{z^2 + x^2 + y^2} = \sqrt{z^2\left(1 + \frac{x^2 + y^2}{z^2}\right)}
\displaystyle = |z|\sqrt{1 + \frac{x^2 + y^2}{z^2}}.

As we know the argument of the square root is very close to 1, we can use a first order approximation, \sqrt{1 + x} \approx 1 + \frac{x}{2}:

\approx |z|\left(1 + \frac{x^2 + y^2}{2z^2}\right)

\approx |z|\left(1 + \frac{x^2 + y^2}{2z^2}\right)

\approx |z| + \frac{x^2 + y^2}{2|z|}.

Replacing in the original expression,

\displaystyle f(x, y, z) \approx \frac{1}{|z|}\exp\left(ik|z| + ik\frac{x^2+y^2}{2|z|}\right)

we get expressions matching the ones in the slides apart from some amplitude and phase conventions.


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