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


\Delta E+k^2(x)E=0

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


E=ce^{-jk_0x}

General solution of equation eqhelmwkb as:


E=a(x)e^{-jk_0L(x)}

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


\Delta E+k_0^2n^2(x)E=0

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


E(x)=e^{jk_0[S_0+\frac{S_1}{jk_0}+\dots]}

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}


\partial_iS_0\partial_iS_0=n^2

that can also be written:


\mbox{ grad }^2S_0=n^2.

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])) ofE. Precision is not enough high in the exponential: If S_1 is neglected, phase of the wave is neglected. For terms in k_0:


\partial_i\partial_iS_0+2\partial_iS_0\partial_iS_1=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.

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:


L=\int n(\vec r) ds

where n(r) is the optical index\index{optical index} of the considered media. Functional L 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:


\frac{d}{ds}(n\frac{dr}{ds})=\mbox{ grad } n.

Proof: Let us parametrize optical path by some t variable:


L=\int_{t_1}^{t_2} n(\vec r) \frac{ds}{dt}dt

Setting:


M(\dot{\vec r})=\frac{ds}{dt}

yields:


L(\vec r)=\int_{t_1}^{t_2} n(\vec r) M(\dot{\vec r})dt.

Optical path L can thus be written:


L=\int_{t_1}^{t_2} F(\vec r,\dot{\vec r})dt

Let us calculate variations of L:


0=L(\vec r + \vec u)-L(\vec r )=\int_{t_1}^{t_2} (\frac{\partial
n}{\partial \vec r} M(\dot{\vec r}) \vec u+n(\vec r) \frac{\partial
M}{\partial \dot{\vec r}}\dot{\vec u}) dt

Integrating by parts the second term:


\int_{t_1}^{t_2}n(\vec r) \frac{\partial M}{\partial \dot{\vec
r}}\dot{\vec u} dt=[]+\frac{d}{dt}( \frac{\partial M}{\partial \dot{\vec
r}})\vec u

Now we have:\footnote{ Indeed


M(\dot x,\dot y, \dot z)=\sqrt{\dot x^2+\dot y^2+\dot z^2}

and


\frac{\partial M}{\partial \dot x}=\frac{\dot x}{\sqrt{\dot x^2+\dot y^2+\dot z^2}}=\frac{dx}{ds}

}


\frac{\partial M}{\partial \dot{\vec r}}=\frac{d\vec r}{ds}

and


\frac{d}{dt}={M(\dot{\vec r})}\frac{1}{ds}

so:


\frac{d}{ds}(n\frac{dr}{ds})=\mbox{ grad } n

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 S; part above S has index n_1 and part under S has index n_2. Let I be a point of S. Consider A_1 a point of medium 1 and A_2 a point of medium 2. Let us introduce optical path\footnote{Inside each medium 1 and 2, Fermat principle application shows that light propagates as a line}.


L=n_1\vec{A_1I}.\vec{u}_1+n_2\vec{IA_2}.\vec{u}_2

where \vec{u}_1=\frac {\vec{A_1I}}{|A_1I|} and \vec{u}_2=\frac
{\vec{A_2I}}{|A_2I|} are unit vectors (see figure figfermat).

figfermat

Snell-Descartes laws can be deduced from fermat principle.

From Fermat principle, dL=0. As u_1 is unitary \vec{u}_1.d\vec{u}_1=0, and it yields:


0=(n_2u_2-n_1u_1).d\vec{I}.

This last equality is verified by each d\vec{I} belonging to the surface:


(n_2\vec{u}_2-n_1\vec{u}_1).\vec t=0

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

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

Theorem: Ikonal equation


n\frac{dr}{ds}=\mbox{ grad } L

is equivalent to light ray equation:


\frac{d}{ds}(n\frac{dr}{ds})=\mbox{ grad } n

Proof:

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

Fermat principle is so a consequence of Maxwell equations.

secdiffra

Physical optics, Diffraction[edit]

Problem position[edit]

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).

figecran

Names of the various surfaces for the considered diffraction problem.

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):

Problem:

Given a function U_{free}, find a function U such that:


(\Delta +k^2)U=0\mbox{ in  }\Omega


U=U_{free} \mbox{ on  }\Sigma


U=0\mbox{ on  }\Sigma^c

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


G_M(M')=\frac{e^{jkr}}{4\pi r}

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:

Problem:

Find u such that:


(\Delta +k^2)U=\delta_M\mbox{ in  }\Omega


U=0\mbox{ on  }S_1=\Sigma^c\cup\Sigma^c

This solution is:

eqgreendif

G_M(M')=\frac{1}{4\pi}(\frac{e^{jkr}}{r}+\frac{e^{jkr^s}}{r^s})

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


U(M)=\int_\Omega u(M')\delta_M(M') dM'=\int_\Omega
u(M')(\Delta+k^2)G_M(M')dM'

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


\int_{\Omega}U(M')(\Delta+k^2)G_M(M') dM'=\int_{\Omega}(U(M')\Delta
G_M(M')-G_M(M')\Delta U(M')) dM'.

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


\int_{\Omega}(U\Delta G_M-G_M\Delta U) dM'=\int_{\mathcal S}(U\frac{\partial
G_M}{\partial n}-G_M\frac{\partial U}{\partial n}) ds'

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[edit]

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:


I=\int_{S_2}(U\frac{\partial G}{\partial n}-G\frac{\partial
U}{\partial n}) ds'

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


\frac{\partial G}{\partial n}=(jk-\frac{1}{R})\frac{e^{jkR}}{R}\sim jkG,

thus


I=\int_{\omega}\frac{e^{jkR}}{R}(\frac{\partial U}{\partial n}-jkU)R^2
d\omega

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


\lim_{R\rightarrow\infty}R(\frac{\partial U}{\partial n}-jkU)=0

is satisfied, then I is zero.

Remark:

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


U=\frac{e^{jkR}}{R}

then


R(\frac{\partial U}{\partial n}-jkU)=-\frac{e^{jkR}}{R}

tends to zero when R tends to infinity. }.

secHuyghens

Huyghens principle[edit]

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


U(M)=\frac{1}{4\pi}\int_{S_1}U(M')\frac{\partial G_M(M')}{\partial n}ds'

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:


r_{01}=r'_{01}

and


\cos(n,r'_{01})=-\cos(n,r_{01})

One can evaluate:


\frac{\partial G_M}{\partial n}=
2\cos(n,r_{01})(jk-\frac{1}{r_{01}})\frac{e^{jkr_{01}}}{r_{01}}

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


k=\frac{2\pi}{\lambda}

}:


U(M)=\int_{S_1}U(M')\frac{j}{\lambda}\cos(n,r_{01})
\frac{e^{jkr_{01}}}{r_{01}}ds'

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


\frac{e^{jkr_{01}}}{r_{01}}

by


\frac{e^{jkR}}{R}e^{jk\vec R_M.\vec R_m /R}.

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.