Introduction to Mathematical Physics/Electromagnetism/Optics, particular case of electromagnetism

Ikonal equation, transport equation

WKB (Wentzel-Kramers-Brillouin) method\index{WKB method} is used to show how electromagnetism (Helmholtz equation) implies geometric and physical optical. Let us consider Helmholtz equation: Template:IMP/label Template:IMP/eq If $k(x)$ is a constant $k_0$ then solution of eqhelmwkb is: Template:IMP/eq General solution of equation eqhelmwkb as: Template:IMP/eq This is variation of constants method. Let us write Helmholtz equation\index{Helmholtz equation} using $n(x)$ the optical index.\index{optical index} Template:IMP/eq with $n=v_0/v$. Let us develop $E$ using the following expansion (see ([#References|references])) Template:IMP/eq where $\frac{1}{j{k}_0}$ is the small variable of the expansion (it corresponds to small wave lengths). Equalling terms in $k_0^2$ yields to {\it ikonal equation

}\index{ikonal equation}

Template:IMP/eq that can also be written: Template:IMP/eq It is said that we have used the "geometrical approximation"\footnote{ Fermat principle can be shown from ikonal equation. Fermat principle is in fact just the variational form of ikonal equation. } . If expansion is limited at this first order, it is not an asymptotic development (see ([#References|references])) of$E$. Precision is not enough high in the exponential: If $S_1$ is neglected, phase of the wave is neglected. For terms in $k_0$: Template:IMP/eq This equation is called transport equation.\index{transport equation} We have done the physical "optics approximation". We have now an asymptotic expansion of $E$.

Geometrical optics, Fermat principle

Geometrical optics laws can be expressed in a variational form \index{Fermat principle} {\it via} Fermat principle (see ([#References|references])): Template:IMP/prin}

Fermat principle allows to derive the light ray equation \index{light ray equation} as a consequence of Maxwell equations: Template:IMP/thm Template:IMP/pf Template:IMP/rem Another equation of geometrical optics is ikonal equation.\index{ikonal equation} Template:IMP/thm Template:IMP/pf Fermat principle is so a consequence of Maxwell equations.

Physical optics, Diffraction

Problem position

Consider a screen $S_1$ with a hole\index{diffraction} $\mathrm{\Sigma }$ inside it. Complementar of $\mathrm{\Sigma }$ in $S_1$ is noted ${\mathrm{\Sigma }}^c$ (see figure figecran). Template:IMP/label

Names of the various surfaces for the considered diffraction problem.

The Electromagnetic signal that falls on $\mathrm{\Sigma }$ is assumed not to be perturbed by the screen $S_c$: value of each component $U$ of the electromagnetic field is the value $U_{free}$ of $U$ without any screen. The value of $U$ on the right hand side of $S_c$ is assumed to be zero. Let us state the diffraction problem ([#References|references]) (Rayleigh Sommerfeld diffraction problem): Template:IMP/prob Elementary solution of Helmholtz operator $\mathrm{\Delta }+k^2$ in $R^3$ is Template:IMP/eq where $r=|M{M}^{\prime }|$. Green solution for our screen problem is obtained using images method\index{images method} (see section secimage). It is solution of following problem: Template:IMP/prob This solution is:

with $r_s=|M_s{M}^{\prime }|$ where $M_s$ is the symmetrical of $M$ with respect to the screen. Thus: Template:IMP/eq Now using the fact that in $\mathrm{\Omega }$, $\mathrm{\Delta }U=-k^{2}U$: Template:IMP/eq Applying Green's theorem, volume integral can be transformed to a surface integral: Template:IMP/eq where $n$ is directed outwards surface $𝒮$. Integral over $S=S_1+S_2$ is reduced to an integral over $S_1$ if the {\it Sommerfeld radiation condition} \index{Sommerfeld radiation condition} is verified:

Sommerfeld radiation condition

Consider the particular case where surface $S_2$ is the portion of sphere centred en P with radius $R$. Let us look for a condition for the integral $I$ defined by: Template:IMP/eq tends to zero when $R$ tends to infinity. We have: Template:IMP/eq thus Template:IMP/eq where $\omega$ is the solid angle. If, in all directions, condition: Template:IMP/eq is satisfied, then $I$ is zero. Template:IMP/rem

Huyghens principle

From equation eqgreendif, $G$ is zero on $S_1$. \index{Huyghens principle} We thus have: Template:IMP/eq Now:

$\begin{array}{ccc}\frac{\partial G}{\partial n}& =& \cos (n,r_{01})(jk-\frac{1}{r_{01}})\frac{e^{jk{r}_{01}}}{r_{01}}\\ & & -\cos (n,r{\text{'}}_{01})(jk-\frac{1}{r{\text{'}}_{01}})\frac{e^{jkr{\text{'}}_{01}}}{r{\text{'}}_{01}}\end{array}$

where $r_{01}=M{M}^{\prime }$ and $r{\text{'}}_{01}=M_s{M}^{\prime }$, $M^{\prime }$ belonging to $\mathrm{\Sigma }$ and $M_s$ being the symmetrical point of the point $M$ where field $U$ is evaluated with respect to the screen. Thus: Template:IMP/eq and Template:IMP/eq One can evaluate: Template:IMP/eq For $r_{01}$ large, it yields\footnote{Introducing the wave length $\lambda$ defined by: Template:IMP/eq }: Template:IMP/eq This is the Huyghens principle  : Template:IMP/prin

Let $O$ a point on $S_1$. Fraunhoffer approximation \index{Fraunhoffer approximation} consists in approximating: Template:IMP/eq by Template:IMP/eq where $R=OM$, $R_m=O{M}^{\prime }$, $R_M=OM$. Then amplitude Fourier transform\index{Fourier transform} of light on $S_1$ is observed at $M$.