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

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secWKB

Ikonal equation, transport equation[edit]

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:

eqhelmwkb

If is a constant 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 the optical index.\index{optical index}

with . Let us develop using the following expansion (see ([#References|references]))

where is the small variable of the expansion (it corresponds to small wave lengths). Equalling terms in 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. Precision is not enough high in the exponential: If is neglected, phase of the wave is neglected. For terms in :

This equation is called transport equation.\index{transport equation} We have done the physical "optics approximation". We have now an asymptotic expansion of .

secFermat

Geometrical optics, Fermat principle[edit]

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

Principle: Fermat principle: trajectory followed by an optical ray minimizes the path integral:

where is the optical index\index{optical index} of the considered media. Functional is called optical path.\index{optical path

}

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

Theorem: Light ray trajectory equation is:

Proof: Let us parametrize optical path by some variable:

Setting:

yields:

Optical path can thus be written:

Let us calculate variations of :

Integrating by parts the second term:

Now we have:\footnote{ Indeed

and

}

and

so:

This is the light ray equation.

Remark:

Snell-Descartes laws\index{Snell--Descartes law} can be deduced from Fermat principle. Consider the space shared into two parts by a surface ; part above has index and part under has index . Let be a point of . Consider a point of medium and a point of medium . Let us introduce optical path\footnote{Inside each medium and , Fermat principle application shows that light propagates as a line}.

where and are unit vectors (see figure figfermat).

figfermat

Snell-Descartes laws can be deduced from fermat principle.

From Fermat principle, . As is unitary , and it yields:

This last equality is verified by each belonging to the surface:

where is tangent vector of surface. This is Snell-Descartes equation.

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

Theorem: Ikonal equation

is equivalent to light ray equation:

Proof:

Let us differentiate ikonal equation with respect to (see ([#References

Fermat principle is so a consequence of Maxwell equations.

secdiffra

Physical optics, Diffraction[edit]

Problem position[edit]

Consider a screen with a hole\index{diffraction} inside it. Complementar of in is noted (see figure figecran).

figecran

Names of the various surfaces for the considered diffraction problem.

The Electromagnetic signal that falls on is assumed not to be perturbed by the screen : value of each component of the electromagnetic field is the value of without any screen. The value of on the right hand side of is assumed to be zero. Let us state the diffraction problem ([#References|references]) (Rayleigh Sommerfeld diffraction problem):

Problem:

Given a function , find a function such that:

Elementary solution of Helmholtz operator in is

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

Problem:

Find such that:

This solution is:

eqgreendif

with where is the symmetrical of with respect to the screen. Thus:

Now using the fact that in , :

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

where is directed outwards surface . Integral over is reduced to an integral over if the {\it Sommerfeld radiation condition} \index{Sommerfeld radiation condition} is verified:

Sommerfeld radiation condition[edit]

Consider the particular case where surface is the portion of sphere centred en P with radius . Let us look for a condition for the integral defined by:

tends to zero when tends to infinity. We have:

thus

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

is satisfied, then is zero.

Remark:

If is a superposition of spherical waves, this condition is verified\footnote{ Indeed if is:

then

tends to zero when tends to infinity. }.

secHuyghens

Huyghens principle[edit]

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

Now:

where and , belonging to and being the symmetrical point of the point where field is evaluated with respect to the screen. Thus:

and

One can evaluate:

For large, it yields\footnote{Introducing the wave lenght defined by:

}:

This is the Huyghens principle  :

Principle:

  • Light propagates from close to close. Each surface element reached by it behaves like a secondary source that emits spherical wavelet with amplitude proportional to the element surface.
  • Complex amplitude of light vibration in one point is the sum of complex amplitudes produced by all secondary sources. It is said that vibrations interfere to create the vibration at considered point.

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

by

where , , . Then amplitude Fourier transform\index{Fourier transform} of light on is observed at .