Geometrical optics

The method of obtaining equations of geometrical optics by taking the limit of zero wavelength was first described by Arnold Sommerfeld and J. Runge in 1911.[6] Their derivation was based on an oral remark by Peter Debye.[7][8] Consider a monochromatic scalar field ${\displaystyle \psi (\mathbf {r} ,t)=\phi (\mathbf {r} )e^{i\omega t}}$, where ${\displaystyle \psi }$ could be any of the components of electric or magnetic field and hence the function ${\displaystyle \phi }$ satisfy the wave equation

$${\displaystyle \nabla ^{2}\phi +k_{o}^{2}n(\mathbf {r} )^{2}\phi =0}$$

where ${\displaystyle k_{o}=\omega /c=2\pi /\lambda _{o}}$ with ${\displaystyle c}$ being the speed of light in vacuum. Here, ${\displaystyle n(\mathbf {r} )}$ is the refractive index of the medium. Without loss of generality, let us introduce ${\displaystyle \phi =A(k_{o},\mathbf {r} )e^{ik_{o}S(\mathbf {r} )}}$ to convert the equation to

$${\displaystyle -k_{o}^{2}A[(\nabla S)^{2}-n^{2}]+2ik_{o}(\nabla S\cdot \nabla A)+ik_{o}A\nabla ^{2}S+\nabla ^{2}A=0.}$$

Since the underlying principle of geometrical optics lies in the limit ${\displaystyle \lambda _{o}\sim k_{o}^{-1}\rightarrow 0}$, the following asymptotic series is assumed,

$${\displaystyle A(k_{o},\mathbf {r} )=\sum _{m=0}^{\infty }{\frac {A_{m}(\mathbf {r} )}{(ik_{o})^{m}}}}$$

For large but finite value of ${\displaystyle k_{o}}$, the series diverges, and one has to be careful in keeping only appropriate first few terms. For each value of ${\displaystyle k_{o}}$, one can find an optimum number of terms to be kept and adding more terms than the optimum number might result in a poorer approximation.[9] Substituting the series into the equation and collecting terms of different orders, one finds

$${\displaystyle {\begin{aligned}O(k_{o}^{2}):&\quad (\nabla S)^{2}=n^{2},\\[1ex]O(k_{o}):&\quad 2\nabla S\cdot \nabla A_{0}+A_{0}\nabla ^{2}S=0,\\[1ex]O(1):&\quad 2\nabla S\cdot \nabla A_{1}+A_{1}\nabla ^{2}S=-\nabla ^{2}A_{0},\end{aligned}}}$$

in general,

$${\displaystyle O(k_{o}^{1-m}):\quad 2\nabla S\cdot \nabla A_{m}+A_{m}\nabla ^{2}S=-\nabla ^{2}A_{m-1}.}$$

The first equation is known as the eikonal equation, which determines the eikonal ${\displaystyle S(\mathbf {r} )}$ is a Hamilton–Jacobi equation, written for example in Cartesian coordinates becomes

$${\displaystyle \left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}+\left({\frac {\partial S}{\partial z}}\right)^{2}=n^{2}.}$$

The remaining equations determine the functions ${\displaystyle A_{m}(\mathbf {r} )}$.

The method of obtaining equations of geometrical optics by analysing surfaces of discontinuities of solutions to Maxwell's equations was first described by Rudolf Karl Luneburg in 1944.[10] It does not restrict the electromagnetic field to have a special form required by the Sommerfeld-Runge method which assumes the amplitude ${\displaystyle A(k_{o},\mathbf {r} )}$ and phase ${\displaystyle S(\mathbf {r} )}$ satisfy the equation ${\textstyle \lim _{k_{0}\to \infty }{\frac {1}{k_{0}}}\left({\frac {1}{A}}\,\nabla S\cdot \nabla A+{\frac {1}{2}}\nabla ^{2}S\right)=0}$. This condition is satisfied by e.g. plane waves but is not additive.

The main conclusion of Luneburg's approach is the following:

Theorem. Suppose the fields ${\displaystyle \mathbf {E} (x,y,z,t)}$ and ${\displaystyle \mathbf {H} (x,y,z,t)}$ (in a linear isotropic medium described by dielectric constants ${\displaystyle \varepsilon (x,y,z)}$ and ${\displaystyle \mu (x,y,z)}$) have finite discontinuities along a (moving) surface in ${\displaystyle \mathbf {R} ^{3}}$ described by the equation ${\displaystyle \psi (x,y,z)-ct=0}$. Then Maxwell's equations in the integral form imply that ${\displaystyle \psi }$ satisfies the eikonal equation:

$${\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=\varepsilon \mu =n^{2},}$$

where ${\displaystyle n}$ is the index of refraction of the medium (Gaussian units).

An example of such surface of discontinuity is the initial wave front emanating from a source that starts radiating at a certain instant of time.

The surfaces of field discontinuity thus become geometrical optics wave fronts with the corresponding geometrical optics fields defined as:

$${\displaystyle {\begin{aligned}\mathbf {E} ^{*}(x,y,z)&=\mathbf {E} (x,y,z,\psi (x,y,z)/c)\\[1ex]\mathbf {H} ^{*}(x,y,z)&=\mathbf {H} (x,y,z,\psi (x,y,z)/c)\end{aligned}}}$$

Those fields obey transport equations consistent with the transport equations of the Sommerfeld-Runge approach. Light rays in Luneburg's theory are defined as trajectories orthogonal to the discontinuity surfaces and can be shown to obey Fermat's principle of least time thus establishing the identity of those rays with light rays of standard optics.

The above developments can be generalised to anisotropic media.[11]

The proof of Luneburg's theorem is based on investigating how Maxwell's equations govern the propagation of discontinuities of solutions. The basic technical lemma is as follows:

A technical lemma. Let ${\displaystyle \varphi (x,y,z,t)=0}$ be a hypersurface (a 3-dimensional manifold) in spacetime ${\displaystyle \mathbf {R} ^{4}}$ on which one or more of: ${\displaystyle \mathbf {E} (x,y,z,t)}$, ${\displaystyle \mathbf {H} (x,y,z,t)}$, ${\displaystyle \varepsilon (x,y,z)}$, ${\displaystyle \mu (x,y,z)}$, have a finite discontinuity. Then at each point of the hypersurface the following formulas hold:

$${\displaystyle {\begin{aligned}\nabla \varphi \cdot [\varepsilon \mathbf {E} ]&=0\\[1ex]\nabla \varphi \cdot [\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {E} ]+{\frac {1}{c}}\,\varphi _{t}\,[\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\frac {1}{c}}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]&=0\end{aligned}}}$$

where the ${\displaystyle \nabla }$ operator acts in the ${\displaystyle xyz}$-space (for every fixed ${\displaystyle t}$) and the square brackets denote the difference in values on both sides of the discontinuity surface (set up according to an arbitrary but fixed convention, e.g. the gradient ${\displaystyle \nabla \varphi }$ pointing in the direction of the quantities being subtracted from).

Sketch of proof. Start with Maxwell's equations away from the sources (Gaussian units):

$${\displaystyle {\begin{aligned}\nabla \cdot \varepsilon \mathbf {E} =0\\[1ex]\nabla \cdot \mu \mathbf {H} =0\\[1ex]\nabla \times \mathbf {E} +{\tfrac {\mu }{c}}\,\mathbf {H} _{t}=0\\[1ex]\nabla \times \mathbf {H} -{\tfrac {\varepsilon }{c}}\,\mathbf {E} _{t}=0\end{aligned}}}$$

Using Stokes' theorem in ${\displaystyle \mathbf {R} ^{4}}$ one can conclude from the first of the above equations that for any domain ${\displaystyle D}$ in ${\displaystyle \mathbf {R} ^{4}}$ with a piecewise smooth (3-dimensional) boundary ${\displaystyle \Gamma }$ the following is true:

$${\displaystyle \oint _{\Gamma }(\mathbf {M} \cdot \varepsilon \mathbf {E} )\,dS=0}$$

where ${\displaystyle \mathbf {M} =(x_{N},y_{N},z_{N})}$ is the projection of the outward unit normal ${\displaystyle (x_{N},y_{N},z_{N},t_{N})}$ of ${\displaystyle \Gamma }$ onto the 3D slice ${\displaystyle t={\rm {const}}}$, and ${\displaystyle dS}$ is the volume 3-form on ${\displaystyle \Gamma }$. Similarly, one establishes the following from the remaining Maxwell's equations:

$${\displaystyle {\begin{aligned}\oint _{\Gamma }\left(\mathbf {M} \cdot \mu \mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {E} +{\frac {\mu }{c}}\,t_{N}\,\mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {H} -{\frac {\varepsilon }{c}}\,t_{N}\,\mathbf {E} \right)dS&=0\end{aligned}}}$$

Now by considering arbitrary small sub-surfaces ${\displaystyle \Gamma _{0}}$ of ${\displaystyle \Gamma }$ and setting up small neighbourhoods surrounding ${\displaystyle \Gamma _{0}}$ in ${\displaystyle \mathbf {R} ^{4}}$, and subtracting the above integrals accordingly, one obtains:

$${\displaystyle {\begin{aligned}\int _{\Gamma _{0}}(\nabla \varphi \cdot [\varepsilon \mathbf {E} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}(\nabla \varphi \cdot [\mu \mathbf {H} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {E} ]+{1 \over c}\,\varphi _{t}\,[\mu \mathbf {H} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {H} ]-{1 \over c}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\end{aligned}}}$$

where ${\displaystyle \nabla ^{4D}}$ denotes the gradient in the 4D ${\displaystyle xyzt}$-space. And since ${\displaystyle \Gamma _{0}}$ is arbitrary, the integrands must be equal to 0 which proves the lemma.

It's now easy to show that as they propagate through a continuous medium, the discontinuity surfaces obey the eikonal equation. Specifically, if ${\displaystyle \varepsilon }$ and ${\displaystyle \mu }$ are continuous, then the discontinuities of ${\displaystyle \mathbf {E} }$ and ${\displaystyle \mathbf {H} }$ satisfy: ${\displaystyle [\varepsilon \mathbf {E} ]=\varepsilon [\mathbf {E} ]}$ and ${\displaystyle [\mu \mathbf {H} ]=\mu [\mathbf {H} ]}$. In this case the last two equations of the lemma can be written as:

$${\displaystyle {\begin{aligned}\nabla \varphi \times [\mathbf {E} ]+{\mu \over c}\,\varphi _{t}\,[\mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\varepsilon \over c}\,\varphi _{t}\,[\mathbf {E} ]&=0\end{aligned}}}$$

Taking the cross product of the second equation with ${\displaystyle \nabla \varphi }$ and substituting the first yields:

$${\displaystyle \nabla \varphi \times (\nabla \varphi \times [\mathbf {H} ])-{\varepsilon \over c}\,\varphi _{t}\,(\nabla \varphi \times [\mathbf {E} ])=(\nabla \varphi \cdot [\mathbf {H} ])\,\nabla \varphi -\|\nabla \varphi \|^{2}\,[\mathbf {H} ]+{\varepsilon \mu \over c^{2}}\varphi _{t}^{2}\,[\mathbf {H} ]=0}$$

The continuity of ${\displaystyle \mu }$ and the second equation of the lemma imply: ${\displaystyle \nabla \varphi \cdot [\mathbf {H} ]=0}$, hence, for points lying on the surface ${\displaystyle \varphi =0}$ only:

$${\displaystyle \|\nabla \varphi \|^{2}={\varepsilon \mu \over c^{2}}\varphi _{t}^{2}}$$

(Notice the presence of the discontinuity is essential in this step as we'd be dividing by zero otherwise.)

Because of the physical considerations one can assume without loss of generality that ${\displaystyle \varphi }$ is of the following form: ${\displaystyle \varphi (x,y,z,t)=\psi (x,y,z)-ct}$, i.e. a 2D surface moving through space, modelled as level surfaces of ${\displaystyle \psi }$. (Mathematically ${\displaystyle \psi }$ exists if ${\displaystyle \varphi _{t}\neq 0}$ by the implicit function theorem.) The above equation written in terms of ${\displaystyle \psi }$ becomes:

$${\displaystyle \|\nabla \psi \|^{2}={\varepsilon \mu \over c^{2}}\,(-c)^{2}=\varepsilon \mu =n^{2}}$$

i.e.,

$${\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=n^{2}}$$

which is the eikonal equation and it holds for all ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$, since the variable ${\displaystyle t}$ is absent. Other laws of optics like Snell's law and Fresnel formulae can be similarly obtained by considering discontinuities in ${\displaystyle \varepsilon }$ and ${\displaystyle \mu }$.

In four-vector notation used in special relativity, the wave equation can be written as

$${\displaystyle {\frac {\partial ^{2}\psi }{\partial x_{i}\partial x^{i}}}=0}$$

and the substitution ${\displaystyle \psi =Ae^{iS/\varepsilon }}$ leads to[12]

$${\displaystyle -{\frac {A}{\varepsilon ^{2}}}{\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {2i}{\varepsilon }}{\frac {\partial A}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {iA}{\varepsilon }}{\frac {\partial ^{2}S}{\partial x_{i}\partial x^{i}}}+{\frac {\partial ^{2}A}{\partial x_{i}\partial x^{i}}}=0.}$$

Therefore the eikonal equation is given by

$${\displaystyle {\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}=0.}$$

Once eikonal is found by solving the above equation, the wave four-vector can be found from

$${\displaystyle k_{i}=-{\frac {\partial S}{\partial x^{i}}}.}$$

• H. Bruns, "Das Eikonal"
• M. Malus, "Optique"
• J. Plucker, "Discussion of the general form for light waves"
• E. Kummer, "General theory of rectilinear ray systems"
• E. Kummer, presentation on optically-realizable rectilinear ray systems
• R. Meibauer, "Theory of rectilinear systems of light rays"
• M. Pasch, "On the focal surfaces of ray systems and the singularity surfaces of complexes"
• A. Levistal, "Research in geometrical optics"
• F. Klein, "On the Bruns eikonal"
• R. Dontot, "On integral invariants and some points of geometrical optics"
• T. de Donder, "On the integral invariants of optics"