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:

If $$k(x)$$ is a constant $$k_0$$ then solution of eqhelmwkb is:

General solution of equation eqhelmwkb as:

This is variation of constants method. Let us write Helmholtz equation\index{Helmholtz equation} using $$n(x)$$ the optical index.\index{optical index}

with $$n=v_0/v$$. Let us develop $$E$$ using the following expansion (see ([#References|references]))

where $$\frac{1}{jk_0}$$ 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}

that can also be written:

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 $$k_0$$:

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])):

Fermat principle allows to derive the light ray equation \index{light ray equation} as a consequence of Maxwell equations:

Another equation of geometrical optics is ikonal equation.\index{ikonal equation}

Fermat principle is so a consequence of Maxwell equations.

Problem position
Consider a screen $$S_1$$ with a hole\index{diffraction} $$\Sigma$$ inside it. Complementar of $$\Sigma$$ in $$S_1$$ is noted $$\Sigma^c$$ (see figure figecran).

The Electromagnetic signal that falls on $$\Sigma$$ 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):

Elementary solution of Helmholtz operator $$\Delta +k^2$$ in $$R^3$$ is

where $$r=|MM'|$$. Green solution for our screen problem is obtained using images method\index{images method} (see section secimage). It is solution of following problem:

This solution is:

with $$r_s=|M_sM'|$$ where $$M_s$$ is the symmetrical of $$M$$ with respect to the screen. Thus:

Now using the fact that in $$\Omega$$, $$\Delta U=-k^2U$$:

Applying Green's theorem, volume integral can be transformed to a surface integral:

where $$n$$ is directed outwards surface $${\mathcal S}$$. 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:

tends to zero when $$R$$ tends to infinity. We have:

thus

where $$\omega$$ is the solid angle. If, in all directions, condition:

is satisfied, then $$I$$ is zero.

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

Now: $$\begin{matrix} \frac{\partial G}{\partial n}&=&\cos (n,r_{01})(jk-\frac{1}{r_{01}})\frac{e^{jkr_{01}}}{r_{01}}\\ &&-\cos(n,r'_{01})(jk-\frac{1}{r'_{01}})\frac{e^{jkr'_{01}}}{r'_{01}} \end{matrix}$$ where $$r_{01}=MM'$$ and $$r'_{01}=M_sM'$$, $$M'$$ belonging to $$\Sigma$$ and $$M_s$$ being the symmetrical point of the point $$M$$ where field $$U$$ is evaluated with respect to the screen. Thus:

and

One can evaluate:

For $$r_{01}$$ large, it yields\footnote{Introducing the wave length $$\lambda$$ defined by:

}:

This is the Huyghens principle :

Let $$O$$ a point on $$S_1$$. Fraunhoffer approximation \index{Fraunhoffer approximation} consists in approximating:

by

where $$R=OM$$, $$R_m=OM'$$, $$R_M=OM$$. Then amplitude Fourier transform\index{Fourier transform} of light on $$S_1$$ is observed at $$M$$.