As a mathematical study, geometrical optics emerges as a short-
wavelength limit for solutions to
hyperbolic partial differential equations (Sommerfeld–Runge method) or as a property of propagation of field discontinuities according to Maxwell's equations (Luneburg method). In this short-wavelength limit, it is possible to approximate the solution locally by u(t,x) \approx a(t,x)e^{i(k\cdot x - \omega t)} where k, \omega satisfy a
dispersion relation, and the amplitude a(t,x) varies slowly. More precisely, the
leading order solution takes the form a_0(t,x) e^{i\varphi(t,x)/\varepsilon}. The phase \varphi(t,x)/\varepsilon can be linearized to recover large wavenumber k:= \nabla_x \varphi, and frequency \omega := -\partial_t \varphi. The amplitude a_0 satisfies a
transport equation. The small parameter \varepsilon\, enters the scene due to highly oscillatory initial conditions. Thus, when initial conditions oscillate much faster than the coefficients of the differential equation, solutions will be highly oscillatory, and transported along rays. Assuming coefficients in the differential equation are smooth, the rays will be too. In other words,
refraction does not take place. The motivation for this technique comes from studying the typical scenario of light propagation where short wavelength light travels along rays that minimize (more or less) its travel time. Its full application requires tools from
microlocal analysis.
Sommerfeld–Runge method 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. Their derivation was based on an oral remark by
Peter Debye. Consider a monochromatic scalar field \psi(\mathbf{r},t)=\phi(\mathbf{r})e^{i\omega t}, where \psi could be any of the components of
electric or
magnetic field and hence the function \phi satisfy the
wave equation \nabla^2\phi + k_o^2 n(\mathbf{r})^2 \phi =0 where k_o = \omega/c = 2\pi/\lambda_o with c being the
speed of light in vacuum. Here, n(\mathbf{r}) is the
refractive index of the medium. Without loss of generality, let us introduce \phi = A(k_o,\mathbf{r}) e^{i k_o S(\mathbf{r})} to convert the equation to -k_o^2 A[(\nabla S)^2 - n^2] + 2 i k_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 \lambda_o\sim k_o^{-1}\rightarrow 0, the following asymptotic series is assumed, A(k_o,\mathbf{r}) = \sum_{m=0}^\infty \frac{A_m(\mathbf{r})}{(ik_o)^m} For large but finite value of k_o, the series diverges, and one has to be careful in keeping only appropriate first few terms. For each value of 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. Substituting the series into the equation and collecting terms of different orders, one finds \begin{align} 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{align} in general, 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 S(\mathbf{r}) is a
Hamilton–Jacobi equation, written for example in Cartesian coordinates becomes \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 A_m(\mathbf{r}).
Luneburg method 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. It does not restrict the electromagnetic field to have a special form required by the Sommerfeld-Runge method which assumes the amplitude A(k_o,\mathbf{r}) and phase S(\mathbf{r}) satisfy the equation \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 \mathbf{E}(x, y, z, t) and \mathbf{H}(x, y, z, t) (in a linear isotropic medium described by dielectric constants \varepsilon(x, y, z) and \mu(x, y, z)) have finite discontinuities along a (moving) surface in \mathbf{R}^3 described by the equation Then Maxwell's equations in the integral form imply that \psi satisfies the
eikonal equation: \psi_x^2 + \psi_y^2 + \psi_z^2 = \varepsilon\mu = n^2, where n(x,y,z) 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: \begin{align} \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{align} 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. 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 \varphi(x, y, z, t) = 0 be a hypersurface (a 3-dimensional manifold) in spacetime \mathbf{R}^4 on which one or more of: \mathbf{E}(x, y, z, t), \mathbf{H}(x, y, z, t), \varepsilon(x, y, z), \mu(x, y, z), have a finite discontinuity. Then at each point of the hypersurface the following formulas hold: \begin{align} \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{align} where the \nabla operator acts in the xyz-space (for every fixed 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 \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): \begin{align} \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{align} Using Stokes' theorem in \mathbf{R}^4 one can conclude from the first of the above equations that for any domain D in \mathbf{R}^4 with a piecewise smooth (3-dimensional) boundary \Gamma the following is true: \oint_\Gamma (\mathbf{M} \cdot \varepsilon\mathbf{E}) \, dS = 0 where \mathbf{M} = (x_N, y_N, z_N) is the projection of the outward unit normal (x_N, y_N, z_N, t_N) of \Gamma onto the 3D slice t = \rm{const}, and dS is the volume 3-form on \Gamma. Similarly, one establishes the following from the remaining Maxwell's equations: \begin{align} \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{align} Now by considering arbitrary small sub-surfaces \Gamma_0 of \Gamma and setting up small neighbourhoods surrounding \Gamma_0 in \mathbf{R}^4, and subtracting the above integrals accordingly, one obtains: \begin{align} \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{align} where \nabla^{4D} denotes the gradient in the 4D xyzt-space. And since \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 \varepsilon and \mu are continuous, then the discontinuities of \mathbf{E} and \mathbf{H} satisfy: [\varepsilon\mathbf{E}] = \varepsilon[\mathbf{E}] and [\mu\mathbf{H}] = \mu[\mathbf{H}]. In this case the last two equations of the lemma can be written as: \begin{align} \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{align} Taking the cross product of the second equation with \nabla\varphi and substituting the first yields: \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 \mu and the second equation of the lemma imply: \nabla\varphi \cdot [\mathbf{H}] = 0, hence, for points lying on the surface \varphi = 0
only: \|\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 \varphi is of the following form: \varphi(x, y, z, t) = \psi(x, y, z) - ct, i.e. a 2D surface moving through space, modelled as level surfaces of \psi. (Mathematically \psi exists if \varphi_t \ne 0 by the
implicit function theorem.) The above equation written in terms of \psi becomes: \|\nabla\psi\|^2 = {\varepsilon\mu\over c^2} \, (-c)^2 = \varepsilon\mu = n^2 i.e., \psi_x^2 + \psi_y^2 + \psi_z^2 = n^2 which is the eikonal equation and it holds for all x, y, z, since the variable t is absent. Other laws of optics like
Snell's law and
Fresnel formulae can be similarly obtained by considering discontinuities in \varepsilon and \mu.
General equation using four-vector notation In
four-vector notation used in
special relativity, the wave equation can be written as \frac{\partial^2 \psi}{\partial x_i\partial x^i} = 0 and the substitution \psi= A e^{iS / \varepsilon} leads to -\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 \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 k_i = - \frac{\partial S}{\partial x^i}. ==See also==