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

secWKB

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:

eqhelmwkb

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

If $k(x)$ is a constant $k_{0}$ then solution of eqhelmwkb is:

$E=ce^{-jk_{0}x}$

General solution of equation eqhelmwkb as:

$E=a(x)e^{-jk_{0}L(x)}$

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

$\Delta E+k_{0}^{2}n^{2}(x)E=0$

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}

$\partial _{i}S_{0}\partial _{i}S_{0}=n^{2}$

that can also be written:

${\mbox{ grad }}^{2}S_{0}=n^{2}.$

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])) of$E$. Precision is not enough high in the exponential: If $S_{1}$ is neglected, phase of the wave is neglected. For terms in $k_{0}$:

$\partial _{i}\partial _{i}S_{0}+2\partial _{i}S_{0}\partial _{i}S_{1}=0$

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

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}.

$L=n_{1}{\vec {A_{1}I}}.{\vec {u}}_{1}+n_{2}{\vec {IA_{2}}}.{\vec {u}}_{2}$

where ${\vec {u}}_{1}={\frac {\vec {A_{1}I}}{|A_{1}I|}}$ and ${\vec {u}}_{2}={\frac {\vec {A_{2}I}}{|A_{2}I|}}$ 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:

$0=(n_{2}u_{2}-n_{1}u_{1}).d{\vec {I}}.$

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

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).

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 $r_{s}=|M_{s}M'|$ 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 $\Omega$, $\Delta U=-k^{2}U$:

$\int _{\Omega }U(M')(\Delta +k^{2})G_{M}(M')dM'=\int _{\Omega }(U(M')\Delta G_{M}(M')-G_{M}(M')\Delta U(M'))dM'.$

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

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

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 $r'_{01}=M_{s}M'$, $M'$ 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 length $\lambda$ 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 $M$.