# 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.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 12, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9446398615837097, "perplexity": 258.94490206499563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188717.24/warc/CC-MAIN-20170322212948-00201-ip-10-233-31-227.ec2.internal.warc.gz"}