2025-05-23 @ 03:46
2025-05-23 @ 03:46
Can I insert $\int_a^b f(x) dx$ into blogs? Ok!
I've been meaning to work on my website for ages, but keep fretting over some website formating updates. Well, I can worry about that later, I am going to start typing up some notes of random subjects that have been piling up. I'll start with modeling atmospheric seeing using ray tracing, as I derived a lot of mysterious diffraction integrals that are widely applicable to electromagnetic theory.
Helmholtz-Kirchoff Diffraction Theory
Green's second identity Start with the divergence theorem $$ \int_V \div \vec{F} dV = \oint_S \vec{F} \dot d\vec{S} $$ where $d\vec{S} = \un dS$. Apply the divergence theorem to a vector-valued function $\vec{F}_1$ constructed from scalar functions $u(\r) : \setR^3 \rightarrow \setR$ and $v(\r) : \setR^3 \rightarrow \setR$ as $$ \vec{F}_1 = u \grad v \Rightarrow \int_V \bb{\grad u \dot \grad v + u \grad^2 v} dV = \oint_S u \pd{v}{n} dS $$ where $\de v / \de n = \grad v \dot \un$. Next, apply the divergence theorem to the vector-valued function $\vec{F}_2$ with the special form $$ \vec{F}_2 = v \grad u \Rightarrow \int_V \bb{\grad v \dot \grad u + v \grad^2 u} dV = \oint_S v \pd{u}{n} dS $$ subtracting these equations we get Green's second identity $$ \boxed{ \int_V \bb{u \grad^2 v - v \grad^2 u} dV = \oint_S \bb{u \pd{v}{n} - v \pd{u}{n}} dS } $$
Let $u$ and $v$ satisfy Helmholtz equations with the same wavenumber $k$, so $$ \grad^2 u + k^2 u = 0 $$ and $$ \grad^2 v + k^2 v = 0 $$ insert these into Green's second identity $$ \int_V \bb{u \bb{-k^2 v} - v \bb{-k^2 u}} dV = \oint_S \bb{u \pd{v}{n} - v \pd{u}{n}} dS $$ $$ \oint_S \bb{u \pd{v}{n} - v \pd{u}{n}} dS = 0 $$
Apply to a spherical wave $$ v = \q{e^{i k r}}{4 \pi r} $$ $$ \grad^2 v + k^2 v =0 $$ $$ \pd{v}{n} = \pd{}{r} \bb{\q{e^{i k r}}{4 \pi r}} = \bb{i k - \q{1}{r}} \q{e^{i k r}}{4 \pi r} $$ $$ \oint_S \bb{u \pd{v}{n} - v \pd{u}{n}} dS = 0 $$ $$ \oint_{S'} \bb{u \pd{}{n'} \bb{\q{e^{i k r'}}{4 \pi r'}} - \bb{\q{e^{i k r'}}{4 \pi r'}} \pd{u}{n'}} dS' + \oint_{S''} \bb{u \bb{i k - \q{1}{r''}} \q{e^{i k r''}}{4 \pi r''} - \bb{\q{e^{i k r''}}{4 \pi r''}} \pd{u}{n''}} dS'' = 0 $$ $$ \lim_{a \rightarrow 0} \oint_{S''} \bb{u \bb{i k - \q{1}{r''}} \q{e^{i k r''}}{4 \pi r''} - \bb{\q{e^{i k r''}}{4 \pi r''}} \pd{u}{n''}} dS'' = \lim_{a \rightarrow 0} -\oint_{S''} u \q{e^{i k r''}}{4 \pi r''^2} dS'' = -u(\vec{0}) \q{e^{i k 0}}{4 \pi a^2} 4 \pi a^2 = -u(\vec{0}) $$ $$ u(\vec{0}) = \oint_{S'} \bb{u \pd{}{n'} \bb{\q{e^{i k r'}}{4 \pi r'}} - \bb{\q{e^{i k r'}}{4 \pi r'}} \pd{u}{n'}} dS' $$ $$ \boxed{ u(\r) = \oint_{S'} \bb{u \pd{}{n'} \bb{\q{e^{i k R}}{4 \pi R}} - \bb{\q{e^{i k R}}{4 \pi R}} \pd{u}{n'}} dS' } $$ where $\R \equiv \r - \r'$ and $R = \norm{\R}$
Kirchoff Approximation
Apply Helmholtz-Kirchoff integral to infinite plane $$ u(\r) = \oint_{S'} \bb{u \pd{}{n'} \bb{\q{e^{i k R}}{4 \pi R}} - \bb{\q{e^{i k R}}{4 \pi R}} \pd{u}{n'}} dS' $$ $$ u(\r) = \int_{S'} \bb{u \pd{}{z'} \bb{\q{e^{i k R}}{4 \pi R}} - \bb{\q{e^{i k R}}{4 \pi R}} \pd{u}{z'}} dS' $$ $$ \pd{}{z'} \bb{\q{e^{i k R}}{4 \pi R}} = \q{1}{4 \pi} \bb{\q{1}{R^3} \bb{z - z'} e^{i k R} - \q{1}{R} e^{i k R} i k \q{1}{R} \bb{z - z'}} = \q{e^{i k R}}{4 \pi R} \bb{\q{1}{R} - i k} \q{z - z'}{R} $$ $$ \pd{}{z'} \bb{\q{e^{i k R}}{4 \pi R}} \approx -i k \q{e^{i k R}}{4 \pi R} \q{z - z'}{R} $$ for $z \gg \q{1}{k}$ and $R \gg \q{1}{k}$. $$ u(\r) \approx \int_{S'} \bb{-i k u \q{e^{i k R}}{4 \pi R} \q{z - z'}{R} - \bb{\q{e^{i k R}}{4 \pi R}} \pd{u}{z'}} dS' $$ $$ u(\r) \approx -\int_{S'} \q{e^{i k R}}{4 \pi R} \bb{i k u \q{z - z'}{R} + \pd{u}{z'}} dS' $$ use the approximation $\de u / \de z' \approx i k u$ (under the assumptions of $z$ and $R$, $u$ is almost a plane wave) $$ \boxed{ u(\r) \approx -i k \int_{S'} u \q{e^{i k R}}{4 \pi R} \bb{\q{z - z'}{R} + 1} dS' } $$ note, the following lemma was used \begin{equation} \begin{split} \pd{}{z'} R^n &= n R^{n - 1} \pd{}{z'} \bb{\bb{x - x'}^2 + \bb{y - y'}^2 + \bb{z - z'}^2}^\q{1}{2} \\ &= n R^{n - 1} \q{1}{2 R} \pd{}{z'} \bb{\bb{x - x'}^2 + \bb{y - y'}^2 + \bb{z - z'}^2} \\ &= -n R^{n - 1} \q{1}{2 R} 2 \bb{z - z'} \\ &= -n R^{n - 2} \bb{z - z'} \end{split} \end{equation}
Rayleigh-Sommerfeld Diffration Integral
Decompose Kirchoff approximation into two parts $$ u(\r) = \int_{S'} \bb{u \pd{}{z'} \bb{\q{e^{i k R}}{4 \pi R}} - \bb{\q{e^{i k R}}{4 \pi R}} \pd{u}{z'}} dS' $$ $$ \boxed{ u(\r) = \q{1}{2} \bb{u_{I} + u_{II}} } $$ where $$ u_{I}(\r) \equiv 2 \int_{S'} u \pd{}{z'} \bb{\q{e^{i k R}}{4 \pi R}} dS' $$ and $$ u_{II}(\r) \equiv -2 \int_{S'} \bb{\q{e^{i k R}}{4 \pi R}} \pd{u}{z'} dS' $$ expand the derivatives to get $$ u(\r) \approx -i k \int_{S'} u \q{e^{i k R}}{4 \pi R} \bb{\q{z - z'}{R} + 1} dS' $$ $$ \boxed{ u_{I}(\r) \approx -2 i k \int_{S'} u \q{e^{i k R}}{4 \pi R} \q{z - z'}{R} dS' } $$ $$ \boxed{ u_{II}(\r) \approx -2 i k \int_{S'} u \q{e^{i k R}}{4 \pi R} dS' } $$
Fresnel Diffration Integral
\begin{equation} \begin{split} u_{I}(\r) &\approx -2 i k \int_{S'} u \q{e^{i k R}}{4 \pi R} \q{z - z'}{R} dS' \\ &\approx -2 i k \int_{S'} u \q{e^{i k R}}{4 \pi R} dS' \quad\text{for}\quad z' = 0,\quad R \approx 0,\quad z \gg x, y \\ &\approx -2 i k \int_{S'} u \bb{\q{e^{i k \bb{z + \q{1}{2} \q{\bb{x - x'}^2 + \bb{y - y'}^2}{z}}}}{4 \pi \bb{z + \q{1}{2} \q{\bb{x - x'}^2 + \bb{y - y'}^2}{z}}}} dS' \end{split} \end{equation} $$ \boxed{ u(\r) \approx -\q{i k e^{i k z}}{2 \pi z} \int_{S'} u e^{\q{i k}{2 z} \bb{\bb{x - x'}^2 + \bb{y - y'}^2}} dS' } $$ where the following power expansion of $R$ and approximation was used \begin{equation} \begin{split} R &= \bb{\bb{x - x'}^2 + \bb{y - y'}^2 + \bb{z - z'}^2}^\q{1}{2} \\ &= \bb{\bb{x - x'}^2 + \bb{y - y'}^2 + \bb{z}^2}^\q{1}{2} \quad\text{for}\quad z' = 0 \\ &= z \bb{\q{\bb{x - x'}^2 + \bb{y - y'}^2}{z^2} + 1}^\q{1}{2} \\ &= z \bb{1 + \epsilon}^\q{1}{2} \quad\text{where}\quad \epsilon = \q{\bb{x - x'}^2 + \bb{y - y'}^2}{z^2} \\ &\approx z \bb{1 + \q{1}{2} \epsilon} \\ &= z + \q{1}{2} \q{\bb{x - x'}^2 + \bb{y - y}^2}{z} \end{split} \end{equation}
Sanity Check
As a sanity check, we can compare these results against those quoted on wikipedia $$ u_{I}(\r) \approx -2 i k \int_{S'} u \q{e^{i k R}}{4 \pi R} \q{z - z'}{R} dS' $$ with $E = u$ and $z' = 0$ becomes $$ E(x, y, z) = \q{1}{i \lambda} \int_{-\infty}^\infty \int_{-\infty}^\infty E(x', y', 0) \q{e^{i k r}}{r} \q{z}{r} dx' dy' $$ which checks out with the Rayleigh Sommerfeld Diffraction Integral page. Also, $$ u(\r) \approx -\q{i k e^{i k z}}{2 \pi z} \int_{S'} u e^{\q{i k}{2 z} \bb{\bb{x - x'}^2 + \bb{y - y'}^2}} dS' $$ with $E = u$, $k = 2 \pi / \lambda$, and $-i = 1 / i$ becomes $$ E(x, y, z) = \q{e^{i k z}}{i \lambda z} \int_{-\infty}^\infty \int_{-\infty}^\infty E(x', y', 0) e^{\q{i k}{2 z} \bb{\bb{x - x'}^2 + \bb{y - y'}^2}} dx' dy' $$ which also matches the wikipedia page on the Fresnel Diffraction Integral.
Of course, I used none of this for the refraction model at the end of the day! But this stuff is still good to have in one's back pocket. I'll transcribe the variational approach I used in another post. I'll also cheat and lightly edit this post at another time.
2023-06-14 @ 03:10
Roughly imported and formatted equations for several more Study topics. These pages need a lot of work, both in selecting and organizing topics, and adding a significant amount of explanatory text.
2023-06-13 @ 01:36
I've been adding things to the Study pages recently. A lot of the subjects are under construction, and therefore incomplete, but this will improve over time
This evening I started adding information about Computer Graphics.
I need to adjust the CSS, and decide how best to break up long subjects into smaller pieces.
2023-04-05 @ 15:02
Updated website.