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Geometrical optics

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Model of optics describing light as geometric rays

Geometrical optics, orray optics, is a model ofoptics that describeslight propagation in terms ofrays. The ray in geometrical optics is anabstraction useful for approximating the paths along which light propagates under certain circumstances.

The simplifying assumptions of geometrical optics include that light rays:

propagate in straight-line paths as they travel in ahomogeneous medium
bend, and in particular circumstances may split in two, at theinterface between two dissimilarmedia
follow curved paths in a medium in which therefractive index changes
may be absorbed or reflected.

Geometrical optics does not account for certain optical effects such asdiffraction andinterference, which are considered inphysical optics. This simplification is useful in practice; it is an excellent approximation when the wavelength is small compared to the size of structures with which the light interacts. The techniques are particularly useful in describing geometrical aspects ofimaging, includingoptical aberrations.

Explanation

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Further information:Ray (optics) andRay tracing (physics)

As light travels through space, itoscillates inamplitude. In this image, each maximum amplitudecrest is marked with aplane to illustrate thewavefront. Theray is the arrowperpendicular to theseparallel surfaces.

A light ray is aline orcurve that isperpendicular to the light'swavefronts (and is thereforecollinear with thewave vector).A slightly more rigorous definition of a light ray follows fromFermat's principle, which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.^[1]

Geometrical optics is often simplified by making theparaxial approximation, or "small angle approximation". The mathematical behavior then becomeslinear, allowing optical components and systems to be described by simple matrices. This leads to the techniques ofGaussian optics andparaxialray tracing, which are used to find basic properties of optical systems, such as approximateimage and object positions andmagnifications.^[2]

Reflection

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Main article:Reflection (physics)

Glossy surfaces such asmirrors reflect light in a simple, predictable way. This allows for production of reflected images that can be associated with an actual (real) or extrapolated (virtual) location in space.

With such surfaces, the direction of the reflected ray is determined by the angle the incident ray makes with thesurface normal, a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal.^[3] This is known as theLaw of Reflection.

Forflat mirrors, the law of reflection implies that images of objects are upright and the same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size. (Themagnification of a flat mirror is equal to one.) The law also implies thatmirror images areparity inverted, which is perceived as a left-right inversion.

Mirrors with curved surfaces can be modeled byray tracing and using the law of reflection at each point on the surface. Formirrors with parabolic surfaces, parallel rays incident on the mirror produce reflected rays that converge at a commonfocus. Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing the focus to be smeared out in space. In particular, spherical mirrors exhibitspherical aberration. Curved mirrors can form images with magnification greater than or less than one, and the image can be upright or inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen.^[3]

Refraction

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This sectionshould include a summary of, or be summarized in, another article. SeeWikipedia:Summary style for information on how to incorporate it into this article's main text, or the main text of another article.(June 2009)

Main article:Refraction

Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction $n_{1}$ and another medium with index of refraction $n_{2}$ . In such situations,Snell's Law describes the resulting deflection of the light ray: $n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}$ where $\theta _{1}$ and $\theta _{2}$ are the angles between the normal (to the interface) and the incident and refracted waves, respectively. This phenomenon is also associated with a changing speed of light as seen from the definition of index of refraction provided above which implies: $v_{1}\sin \theta _{2}\ =v_{2}\sin \theta _{1}$ where $v_{1}$ and $v_{2}$ are the wave velocities through the respective media.^[3]

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 calledtotal internal reflection and allows forfiber 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 producepolarized light rays using a combination of reflection and refraction: When a refracted ray and the reflected ray form aright angle, the reflected ray has the property of "plane polarization". The angle of incidence required for such a scenario is known asBrewster's angle.^[3]

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 aprism 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 producedispersion spectra that appear as rainbows. The discovery of this phenomenon when passing light through a prism is famously attributed toIsaac Newton.^[3]

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 formirages 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 includingphotocopiers andscanners. The phenomenon is studied in the field ofgradient-index optics.^[4]

A ray tracing diagram for a simple converging lens

A device which produces converging or diverging light rays due to refraction is known as alens. Thin lenses produce focal points on either side that can be modeled using thelensmaker's equation.^[5] In general, two types of lenses exist:convex lenses, which cause parallel light rays to converge, andconcave lenses, which cause parallel light rays to diverge. The detailed prediction of how images are produced by these lenses can be made using ray-tracing similar to curved mirrors. Similarly to curved mirrors, thin lenses follow a simple equation that determines the location of the images given a particular focal length ( $f {\displaystyle f}$ ) and object distance( $S_{1}$ ): ${\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}}$ where $S_{2}$ is the distance associated with the image and is considered by convention to be negative if on the same side of the lens as the object and positive if on the opposite side of the lens.^[5] The focal length f is considered negative for concave lenses.

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.

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 $M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}$ 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.^[3]

Lenses suffer fromaberrations 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).^[3]

Underlying mathematics

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As a mathematical study, geometrical optics emerges as a short-wavelength limit for solutions tohyperbolic 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 adispersion relation, and the amplitude $a(t,x)$ varies slowly. More precisely, theleading 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 atransport 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 frommicrolocal analysis.

Sommerfeld–Runge method

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The method of obtaining equations of geometrical optics by taking the limit of zero wavelength was first described byArnold Sommerfeld and J. Runge in 1911.^[6] Their derivation was based on an oral remark byPeter Debye.^[7]^[8] Consider a monochromatic scalar field $\psi (\mathbf {r} ,t)=\phi (\mathbf {r} )e^{i\omega t}$ , where $\psi$ could be any of the components ofelectric ormagnetic 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 $c {\displaystyle c}$ being thespeed of light in vacuum. Here, $n(\mathbf {r} )$ is therefractive index of the medium. Without loss of generality, let us introduce $\phi =A(k_{o},\mathbf {r} )e^{ik_{o}S(\mathbf {r} )}$ to convert the equation to $-k_{o}^{2}A[(\nabla S)^{2}-n^{2}]+2ik_{o}(\nabla S\cdot \nabla A)+ik_{o}A\nabla ^{2}S+\nabla ^{2}A=0.$

Since the underlying principle of geometrical optics lies in the limit $\lambda _{o}\sim k_{o}^{-1}\rightarrow 0$ , 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.^[9] Substituting the series into the equation and collecting terms of different orders, one finds ${\begin{aligned}O(k_{o}^{2}):&\quad (\nabla S)^{2}=n^{2},\\[1ex]O(k_{o}):&\quad 2\nabla S\cdot \nabla A_{0}+A_{0}\nabla ^{2}S=0,\\[1ex]O(1):&\quad 2\nabla S\cdot \nabla A_{1}+A_{1}\nabla ^{2}S=-\nabla ^{2}A_{0},\end{aligned}}$ in general, $O(k_{o}^{1-m}):\quad 2\nabla S\cdot \nabla A_{m}+A_{m}\nabla ^{2}S=-\nabla ^{2}A_{m-1}.$

The first equation is known as theeikonal equation, which determines theeikonal $S(\mathbf {r} )$ is aHamilton–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

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The method of obtaining equations of geometrical optics by analysing surfaces of discontinuities of solutions to Maxwell's equations was first described byRudolf 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 $A(k_{o},\mathbf {r} )$ and phase $S(\mathbf {r} )$ 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 $\mathbf {E} (x,y,z,t)$ 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 $\psi (x,y,z)-ct=0$ . Then Maxwell's equations in the integral form imply that $\psi$ satisfies theeikonal equation: $\psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=\varepsilon \mu =n^{2},$ where $n {\displaystyle n}$ 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: ${\begin{aligned}\mathbf {E} ^{*}(x,y,z)&=\mathbf {E} (x,y,z,\psi (x,y,z)/c)\\[1ex]\mathbf {H} ^{*}(x,y,z)&=\mathbf {H} (x,y,z,\psi (x,y,z)/c)\end{aligned}}$

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 obeyFermat'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 $\varphi (x,y,z,t)=0$ 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: ${\begin{aligned}\nabla \varphi \cdot [\varepsilon \mathbf {E} ]&=0\\[1ex]\nabla \varphi \cdot [\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {E} ]+{\frac {1}{c}}\,\varphi _{t}\,[\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\frac {1}{c}}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]&=0\end{aligned}}$ where the $\nabla$ operator acts in the $x y z {\displaystyle xyz}$ -space (for every fixed $t {\displaystyle t}$ ) 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 $\nabla \varphi$ pointing in the direction of the quantities being subtractedfrom).

Sketch of proof. Start with Maxwell's equations away from the sources (Gaussian units): ${\begin{aligned}\nabla \cdot \varepsilon \mathbf {E} =0\\[1ex]\nabla \cdot \mu \mathbf {H} =0\\[1ex]\nabla \times \mathbf {E} +{\tfrac {\mu }{c}}\,\mathbf {H} _{t}=0\\[1ex]\nabla \times \mathbf {H} -{\tfrac {\varepsilon }{c}}\,\mathbf {E} _{t}=0\end{aligned}}$

Using Stokes' theorem in $\mathbf {R} ^{4}$ one can conclude from the first of the above equations that for any domain $D {\displaystyle D}$ 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 $d S {\displaystyle dS}$ is the volume 3-form on $\Gamma$ . Similarly, one establishes the following from the remaining Maxwell's equations: ${\begin{aligned}\oint _{\Gamma }\left(\mathbf {M} \cdot \mu \mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {E} +{\frac {\mu }{c}}\,t_{N}\,\mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {H} -{\frac {\varepsilon }{c}}\,t_{N}\,\mathbf {E} \right)dS&=0\end{aligned}}$

Now by considering arbitrary small sub-surfaces $\Gamma _{0}$ of $\Gamma$ and setting up small neighbourhoods surrounding $\Gamma _{0}$ in $\mathbf {R} ^{4}$ , and subtracting the above integrals accordingly, one obtains: ${\begin{aligned}\int _{\Gamma _{0}}(\nabla \varphi \cdot [\varepsilon \mathbf {E} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}(\nabla \varphi \cdot [\mu \mathbf {H} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {E} ]+{1 \over c}\,\varphi _{t}\,[\mu \mathbf {H} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {H} ]-{1 \over c}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\end{aligned}}$ where $\nabla ^{4D}$ denotes the gradient in the 4D $x y z t {\displaystyle xyzt}$ -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:

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

Taking the cross product of the second equation with $\nabla \varphi$ 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 $\varphi _{t}\neq 0$ by theimplicit 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 $x {\displaystyle x}$ , $y {\displaystyle y}$ , $z {\displaystyle z}$ , since the variable $t {\displaystyle t}$ is absent. Other laws of optics likeSnell's law andFresnel formulae can be similarly obtained by considering discontinuities in $\varepsilon$ and $\mu$ .

General equation using four-vector notation

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Infour-vector notation used inspecial relativity, the wave equation can be written as ${\frac {\partial ^{2}\psi }{\partial x_{i}\partial x^{i}}}=0$

and the substitution $\psi =Ae^{iS/\varepsilon }$ leads to^[12] $-{\frac {A}{\varepsilon ^{2}}}{\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {2i}{\varepsilon }}{\frac {\partial A}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {iA}{\varepsilon }}{\frac {\partial ^{2}S}{\partial x_{i}\partial x^{i}}}+{\frac {\partial ^{2}A}{\partial x_{i}\partial x^{i}}}=0.$

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

References

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^Arthur Schuster,An Introduction to the Theory of Optics, London: Edward Arnold, 1904online.
^Greivenkamp, John E. (2004).Field Guide to Geometrical Optics. SPIE Field Guides. Vol. 1.SPIE. pp. 19–20.ISBN 0-8194-5294-7.
^^a ^b ^c ^d ^e ^f ^gHugh D. Young (1992).University Physics 8e. Addison-Wesley.ISBN 0-201-52981-5. Chapter 35.
^E. W. Marchand, Gradient Index Optics, New York, NY, Academic Press, 1978.
^^a ^bHecht, Eugene (1987).Optics (2nd ed.). Addison Wesley.ISBN 0-201-11609-X. Chapters 5 & 6.
^Sommerfeld, A., & Runge, J. (1911). Anwendung der Vektorrechnung auf die Grundlagen der geometrischen Optik. Annalen der Physik, 340(7), 277-298.
^Born, M., & Wolf, E. (2013).Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. Elsevier.
^Sommerfield, A.; J., Runge."The application of vector calculus to the foundations of geometrical optics"(PDF).Neo-classical physics. Translated by D. H. Delphenich. Retrieved3 November 2023.
^Borowitz, S. (1967). Fundamentals of quantum mechanics, particles, waves, and wave mechanics.
^Luneburg, R. K.,Mathematical Theory of Optics, Brown University Press 1944 [mimeographed notes], University of California Press 1964
^Kline, M., Kay, I. W.,Electromagnetic Theory and Geometrical Optics, Interscience Publishers 1965
^Landau, L. D., & Lifshitz, E. M. (1975). The classical theory of fields.

External links

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Feynman's lecture on Geometrical Optics

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