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

secWKB

Ikonal equation, transport equationEdit

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 principleEdit

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, DiffractionEdit

Problem positionEdit

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 conditionEdit

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 principleEdit

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  .