Geometrical optics

 ( Optics ) 

Various consequences of Snell's Law include the fact that for light rays traveling from a material with a high index of refraction to a material with a low index of refraction, it is possible for the interaction with the interface to result in zero transmission. This phenomenon is called total internal reflection and allows for fiber optics technology. As light signals travel down a fiber optic cable, they undergo total internal reflection allowing for essentially no light lost over the length of the cable. It is also possible to produce polarized light rays using a combination of reflection and refraction: When a refracted ray and the reflected ray form a right angle, the reflected ray has the property of "plane polarization". The angle of incidence required for such a scenario is known as Brewster's angle.

Snell's Law can be used to predict the deflection of light rays as they pass through "linear media" as long as the indexes of refraction and the geometry of the media are known. For example, the propagation of light through a prism results in the light ray being deflected depending on the shape and orientation of the prism. Additionally, since different frequencies of light have slightly different indexes of refraction in most materials, refraction can be used to produce dispersion spectrum that appear as rainbows. The discovery of this phenomenon when passing light through a prism is famously attributed to Isaac Newton.

Some media have an index of refraction which varies gradually with position and, thus, light rays curve through the medium rather than travel in straight lines. This effect is what is responsible for seen on hot days where the changing index of refraction of the air causes the light rays to bend creating the appearance of specular reflections in the distance (as if on the surface of a pool of water). Material that has a varying index of refraction is called a gradient-index (GRIN) material and has many useful properties used in modern optical scanning technologies including photocopiers and image scanner. The phenomenon is studied in the field of gradient-index optics.E. W. Marchand, Gradient Index Optics, New York, NY, Academic Press, 1978.

A device which produces converging or diverging light rays due to refraction is known as a lens. Thin lenses produce focal points on either side that can be modeled using the lensmaker's equation.

Incoming parallel rays are focused by a convex lens into an inverted real image one focal length from the lens, on the far side of the lens. Rays from an object at finite distance are focused further from the lens than the focal distance; the closer the object is to the lens, the further the image is from the lens. With concave lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at an upright virtual image one focal length from the lens, on the same side of the lens that the parallel rays are approaching on. Rays from an object at finite distance are associated with a virtual image that is closer to the lens than the focal length, and on the same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens.

Likewise, the magnification of a lens is given by $$M\; =\; -\; \backslash frac\{S\_2\}\{S\_1\}\; =\; \backslash frac\{f\}\{f\; -\; S\_1\}$$ where the negative sign is given, by convention, to indicate an upright object for positive values and an inverted object for negative values. Similar to mirrors, upright images produced by single lenses are virtual while inverted images are real.

Lenses suffer from aberrations that distort images and focal points. These are due to both to geometrical imperfections and due to the changing index of refraction for different wavelengths of light (chromatic aberration).

Since the underlying principle of geometrical optics lies in the limit $\backslash lambda\_o\backslash sim\; k\_o^\{-1\}\backslash rightarrow\; 0$, the following asymptotic series is assumed, $$A(k\_o,\backslash mathbf\{r\})\; =\; \backslash sum\_\{m=0\}^\backslash infty\; \backslash frac\{A\_m(\backslash mathbf\{r\})\}\{(ik\_o)^m\}$$

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.Borowitz, S. (1967). Fundamentals of quantum mechanics, particles, waves, and wave mechanics. Substituting the series into the equation and collecting terms of different orders, one finds in general, $$O(k\_o^\{1-m\}):\backslash quad\; 2\backslash nabla\; S\backslash cdot\; \backslash nabla\; A\_m\; +\; A\_m\backslash nabla^2\; S\; =-\backslash nabla^2\; A\_\{m-1\}.$$

The first equation is known as the eikonal equation, which determines the eikonal $S(\backslash mathbf\{r\})$ is a Hamilton–Jacobi equation, written for example in Cartesian coordinates becomes $$\backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; x\}\backslash right)^2\; +\; \backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; y\}\backslash right)^2\; +\; \backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; z\}\backslash right)^2\; =\; n^2.$$

The remaining equations determine the functions $A\_m(\backslash mathbf\{r\})$.

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

Theorem. Suppose the fields $\backslash mathbf\{E\}(x,\; y,\; z,\; t)$ and $\backslash mathbf\{H\}(x,\; y,\; z,\; t)$ (in a linear isotropic medium described by dielectric constants $\backslash varepsilon(x,\; y,\; z)$ and $\backslash mu(x,\; y,\; z)$) have finite discontinuities along a (moving) surface in $\backslash mathbf\{R\}^3$ described by the equation Then Maxwell's equations in the integral form imply that $\backslash psi$ satisfies the eikonal equation: $$\backslash psi\_x^2\; +\; \backslash psi\_y^2\; +\; \backslash psi\_z^2\; =\; \backslash varepsilon\backslash mu\; =\; n^2,$$ where $n$ 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:

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.Kline, M., Kay, I. W., Electromagnetic Theory and Geometrical Optics, Interscience Publishers 1965

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 $\backslash varphi(x,\; y,\; z,\; t)\; =\; 0$ be a hypersurface (a 3-dimensional manifold) in spacetime $\backslash mathbf\{R\}^4$ on which one or more of: $\backslash mathbf\{E\}(x,\; y,\; z,\; t)$, $\backslash mathbf\{H\}(x,\; y,\; z,\; t)$, $\backslash varepsilon(x,\; y,\; z)$, $\backslash mu(x,\; y,\; z)$, have a finite discontinuity. Then at each point of the hypersurface the following formulas hold: where the $\backslash nabla$ 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 $\backslash nabla\backslash varphi$ pointing in the direction of the quantities being subtracted from).

Sketch of proof. Start with Maxwell's equations away from the sources (Gaussian units): $$\backslash begin\{align\}\; \backslash nabla\; \backslash cdot\; \backslash varepsilon\backslash mathbf\{E\}\; =\; 0\; \backslash \backslash 1ex\; \backslash nabla\; \backslash cdot\; \backslash mu\; \backslash mathbf\{H\}\; =\; 0\; \backslash \backslash 1ex\; \backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; +\; \backslash tfrac\{\backslash mu\}\{c\}\; \backslash ,\; \backslash mathbf\{H\}\_t\; =\; 0\; \backslash \backslash 1ex\; \backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; -\; \backslash tfrac\{\backslash varepsilon\}\{c\}\; \backslash ,\; \backslash mathbf\{E\}\_t\; =\; 0\; \backslash end\{align\}$$

Using Stokes' theorem in $\backslash mathbf\{R\}^4$ one can conclude from the first of the above equations that for any domain $D$ in $\backslash mathbf\{R\}^4$ with a piecewise smooth (3-dimensional) boundary $\backslash Gamma$ the following is true: $$\backslash oint\_\backslash Gamma\; (\backslash mathbf\{M\}\; \backslash cdot\; \backslash varepsilon\backslash mathbf\{E\})\; \backslash ,\; dS\; =\; 0$$ where $\backslash mathbf\{M\}\; =\; (x\_N,\; y\_N,\; z\_N)$ is the projection of the outward unit normal $(x\_N,\; y\_N,\; z\_N,\; t\_N)$ of $\backslash Gamma$ onto the 3D slice $t\; =\; \backslash rm\{const\}$, and $dS$ is the volume 3-form on $\backslash Gamma$. Similarly, one establishes the following from the remaining Maxwell's equations:

Now by considering arbitrary small sub-surfaces $\backslash Gamma\_0$ of $\backslash Gamma$ and setting up small neighbourhoods surrounding $\backslash Gamma\_0$ in $\backslash mathbf\{R\}^4$, and subtracting the above integrals accordingly, one obtains: where $\backslash nabla^\{4D\}$ denotes the gradient in the 4D $xyzt$-space. And since $\backslash Gamma\_0$ 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 $\backslash varepsilon$ and $\backslash mu$ are continuous, then the discontinuities of $\backslash mathbf\{E\}$ and $\backslash mathbf\{H\}$ satisfy: $\backslash varepsilon\backslash mathbf\{E\}\; =\; \backslash varepsilon\backslash mathbf\{E\}$ and $\backslash mu\backslash mathbf\{H\}\; =\; \backslash mu\backslash mathbf\{H\}$. In this case the last two equations of the lemma can be written as:

Taking the cross product of the second equation with $\backslash nabla\backslash varphi$ and substituting the first yields: $$\backslash nabla\backslash varphi\; \backslash times\; (\backslash nabla\backslash varphi\; \backslash times\; \backslash mathbf\{H\})\; -\; \{\backslash varepsilon\backslash over\; c\}\; \backslash ,\; \backslash varphi\_t\; \backslash ,\; (\backslash nabla\backslash varphi\; \backslash times\; \backslash mathbf\{E\})\; =\; (\backslash nabla\backslash varphi\; \backslash cdot\; \backslash mathbf\{H\})\; \backslash ,\; \backslash nabla\backslash varphi\; -\; \backslash |\backslash nabla\backslash varphi\backslash |^2\; \backslash ,\; \backslash mathbf\{H\}\; +\; \{\backslash varepsilon\backslash mu\backslash over\; c^2\}\; \backslash varphi\_t^2\; \backslash ,\; \backslash mathbf\{H\}\; =\; 0$$

The continuity of $\backslash mu$ and the second equation of the lemma imply: $\backslash nabla\backslash varphi\; \backslash cdot\; \backslash mathbf\{H\}\; =\; 0$, hence, for points lying on the surface $\backslash varphi\; =\; 0$ only: $$\backslash |\backslash nabla\backslash varphi\backslash |^2\; =\; \{\backslash varepsilon\backslash mu\backslash over\; c^2\}\; \backslash varphi\_t^2$$

(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 $\backslash varphi$ is of the following form: $\backslash varphi(x,\; y,\; z,\; t)\; =\; \backslash psi(x,\; y,\; z)\; -\; ct$, i.e. a 2D surface moving through space, modelled as level surfaces of $\backslash psi$. (Mathematically $\backslash psi$ exists if $\backslash varphi\_t\; \backslash ne\; 0$ by the implicit function theorem.) The above equation written in terms of $\backslash psi$ becomes: $$\backslash |\backslash nabla\backslash psi\backslash |^2\; =\; \{\backslash varepsilon\backslash mu\backslash over\; c^2\}\; \backslash ,\; (-c)^2\; =\; \backslash varepsilon\backslash mu\; =\; n^2$$ i.e., $$\backslash psi\_x^2\; +\; \backslash psi\_y^2\; +\; \backslash psi\_z^2\; =\; n^2$$ 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 $\backslash varepsilon$ and $\backslash mu$.

and the substitution $\backslash psi=\; A\; e^\{iS\; /\; \backslash varepsilon\}$ leads toLandau, L. D., & Lifshitz, E. M. (1975). The classical theory of fields. $$-\backslash frac\{A\}\{\backslash varepsilon^2\}\backslash frac\{\backslash partial\; S\}\{\backslash partial\; x\_i\}\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; x^i\}\; +\; \backslash frac\{2i\}\{\backslash varepsilon\}\; \backslash frac\{\backslash partial\; A\}\{\backslash partial\; x\_i\}\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; x^i\}\; +\; \backslash frac\{iA\}\{\backslash varepsilon\}\; \backslash frac\{\backslash partial^2\; S\}\{\backslash partial\; x\_i\backslash partial\; x^i\}\; +\; \backslash frac\{\backslash partial^2\; A\}\{\backslash partial\; x\_i\backslash partial\; x^i\}\; =\; 0.$$

Therefore, the eikonal equation is given by $$\backslash frac\{\backslash partial\; S\}\{\backslash partial\; x\_i\}\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; x^i\}\; =\; 0.$$

Once eikonal is found by solving the above equation, the wave four-vector can be found from $$k\_i\; =\; -\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; x^i\}.$$

• Hamiltonian optics
• Geometrical acoustics

• Robert Alfred Herman (1900) A Treatise on Geometrical optics from Archive.org.
• "The Light of the Eyes and the Enlightened Landscape of Vision" is a manuscript, in Arabic, about geometrical optics, dating from the 16th century.
• Theory of Systems of Rays – W.R. Hamilton in Transactions of the Royal Irish Academy, Vol. XV, 1828.

• 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"

• Feynman's lecture on Geometrical Optics