ⓘ Fermats principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original strong form, Fermats principle sta ..


ⓘ Fermats principle

Fermats principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermats principle states that the path taken by a ray between two given points is the path that can be traversed in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is "stationary" with respect to variations of the path - so that a deviation in the path causes, at most, a second-order change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in very close times. It can be shown that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam.

First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the ordinary law of refraction of light Fig. 1, Fermats principle was initially controversial because it seemed to ascribe knowledge and intent to nature. Not until the 19th century was it understood that natures ability to test alternative paths is merely a fundamental property of waves. If points A and B are given, a wavefront expanding from A sweeps all possible ray paths radiating from A, whether they pass through B or not. If the wavefront reaches point B, it sweeps not only the ray paths from A to B, but also an infinitude of nearby paths with the same endpoints. Fermats principle describes any ray that happens to reach point B ; there is no implication that the ray "knew" the shortest path or "intended" to take that path.

For the purpose of comparing traversal times, the time from one point to the next nominated point is taken as if the first point were a point-source. Without this condition, the traversal time would be ambiguous; for example, if the propagation time from P to P′ were reckoned from an arbitrary wavefront W containing P   Fig. 2, that time could be made arbitrarily small by suitably angling the wavefront.

Treating a point on the path as a source is the minimum requirement of Huygens principle, and is part of the explanation of Fermats principle. But it can also be shown that the geometric construction by which Huygens tried to apply his own principle as distinct from the principle itself is simply an invocation of Fermats principle. Hence all the conclusions that Huygens drew from that construction - including, without limitation, the laws of rectilinear propagation of light, ordinary reflection, ordinary refraction, and the extraordinary refraction of "Iceland crystal" calcite - are also consequences of Fermats principle.


1.1. Derivation Sufficient conditions

Let us suppose that:

1 A disturbance propagates sequentially through a medium a vacuum or some material, not necessarily homogeneous or isotropic, without action at a distance; 2 During propagation, the influence of the disturbance at any intermediate point P upon surrounding points has a non-zero angular spread as if P were a source, so that a disturbance originating at any point A arrives at any other point B via an infinitude of paths, by which B receives an infinitude of delayed versions of the disturbance at A ; and 3 These delayed versions of the disturbance will reinforce each other at B if they are synchronized within some tolerance.

Then the various propagation paths from A to B will help each other if their traversal times agree within the said tolerance. For a small tolerance in the limiting case, the permissible range of variations of the path is maximized if the path is such that its traversal time is stationary with respect to the variations, so that a variation of the path causes at most a second-order change in the traversal time.

The most obvious example of a stationarity in traversal time is a local or global minimum - that is, a path of least time, as in the "strong" form of Fermats principle. But that condition is not essential to the argument.

Having established that a path of stationary traversal time is reinforced by a maximally wide corridor of neighboring paths, we still need to explain how this reinforcement corresponds to intuitive notions of a ray. But, for brevity in the explanations, let us first define a ray path as a path of stationary traversal time.


1.2. Derivation A ray as a signal path line of sight

If the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly alter the disturbance reaching B from A - unlike a similar-sized obstruction outside any such corridor, blocking paths that do not reinforce each other. The former obstruction will significantly disrupt the signal reaching B from A, while the latter will not; thus the ray path marks a signal path. If the signal is visible light, the former obstruction will significantly affect the appearance of an object at A as seen by an observer at B, while the latter will not; so the ray path marks a line of sight.

In optical experiments, a line of sight is routinely assumed to be a ray path.


1.3. Derivation A ray as an energy path beam

If the corridor of paths reinforcing a ray path from A to B is substantially obstructed, this will significantly affect the energy reaching B from A - unlike a similar-sized obstruction outside any such corridor. Thus the ray path marks an energy path - as does a beam.

Suppose that a wavefront expanding from point A passes point P, which lies on a ray path from point A to point B. By definition, all points on the wavefront have the same propagation time from A. Now let the wavefront be blocked except for a window, centered on P, and small enough to lie within the corridor of paths that reinforce the ray path from A to B. Then all points on the unobstructed portion of the wavefront will have, nearly enough, equal propagation times to B, but not to points in other directions, so that B will be in the direction of peak intensity of the beam admitted through the window. So the ray path marks the beam. And in optical experiments, a beam is routinely considered as a collection of rays or if it is narrow as an approximation to a ray Fig. 3.


1.4. Derivation Analogies

According to the "strong" form of Fermats principle, the problem of finding the path of a light ray from point A in a medium of faster propagation, to point B in a medium of slower propagation Fig. 1, is analogous to the problem faced by a lifeguard in deciding where to enter the water in order to reach a drowning swimmer as soon as possible, given that the lifeguard can run faster than she can swim. But that analogy falls short of explaining the behavior of the light, because the lifeguard can think about the problem even if only for an instant whereas the light presumably cannot. The discovery that ants are capable of similar calculations does not bridge the gap between the animate and the inanimate.

In contrast, the above assumptions 1 to 3 hold for any wavelike disturbance and explain Fermats principle in purely mechanistic terms, without any imputation of knowledge or purpose.

The principle applies to waves in general, including e.g. sound waves in fluids and elastic waves in solids. In a modified form, it even works for matter waves: in quantum mechanics, the classical path of a particle is obtainable by applying Fermats principle to the associated wave - except that, because the frequency may vary with the path, the stationarity is in the phase shift or number of cycles and not necessarily in the time.

Fermats principle is most familiar, however, in the case of visible light: it is the link between geometrical optics, which describes certain optical phenomena in terms of rays, and the wave theory of light, which explains the same phenomena on the hypothesis that light consists of waves.


2. Equivalence to Huygens construction

In this article we distinguish between Huygens principle, which states that every point crossed by a traveling wave becomes the source of a secondary wave, and Huygens construction, which is described below.

Let the surface W be a wavefront at time t, and let the surface W′ be the same wavefront at the later time  t + Δt   Fig. 4. Let P be a general point on W. Then, according to Huygens construction,

a W′ is the envelope common tangent surface, on the forward side of W, of all the secondary wavefronts each of which would expand in time Δt from a point on W, and b if the secondary wavefront expanding from point P in time Δt touches the surface W′ at point P′, then P and P′ lie on a ray.

The construction may be repeated in order to find successive positions of the primary wavefront, and successive points on the ray.

The ray direction given by this construction is the radial direction of the secondary wavefront, and may differ from the normal of the secondary wavefront cf. Fig. 2, and therefore from the normal of the primary wavefront at the point of tangency. Hence the ray velocity, in magnitude and direction, is the radial velocity of an infinitesimal secondary wavefront, and is generally a function of location and direction.

Now let Q be a point on W close to P, and let Q′ be a point on W′ close to P′. Then, by the construction,

i the time taken for a secondary wavefront from P to reach Q′ has at most a second-order dependence on the displacement P′Q′, and ii the time taken for a secondary wavefront to reach P′ from Q has at most a second-order dependence on the displacement PQ.

By i, the ray path is a path of stationary traversal time from P to W′ ; and by ii, it is a path of stationary traversal time from a point on W to P′.

So Huygens construction implicitly defines a ray path as a path of stationary traversal time between successive positions of a wavefront, the time being reckoned from a point-source on the earlier wavefront. This conclusion remains valid if the secondary wavefronts are reflected or refracted by surfaces of discontinuity in the properties of the medium, provided that the comparison is restricted to the affect paths and the affected portions of the wavefronts.

Fermats principle, however, is conventionally expressed in point-to-point terms, not wavefront-to-wavefront terms. Accordingly, let us modify the example by supposing that the wavefront which becomes surface W at time t, and which becomes surface W′ at the later time t + Δt, is emitted from point A at time 0. Let P be a point on W as before, and B a point on W′. And let A,W, W′, and B be given, so that the problem is to find P.

If P satisfies Huygens construction, so that the secondary wavefront from P is tangential to W′ at B, then PB is a path of stationary traversal time from W to B. Adding the fixed time from A to W, we find that APB is the path of stationary traversal time from A to B possibly with a restricted domain of comparison, as noted above, in accordance with Fermats principle. The argument works just as well in the converse direction. Thus Huygens construction and Fermats principle are geometrically equivalent.

Through this equivalence, Fermats principle sustains Huygens construction and thence all the conclusions that Huygens was able to draw from that construction. In short, "The laws of geometrical optics may be derived from Fermats principle". With the exception of the Fermat-Huygens principle itself, these laws are special cases in the sense that they depend on further assumptions about the media. Two of them are mentioned under the next heading.


3.1. Special cases Isotropic media: Rays normal to wavefronts

In an isotropic medium, because the propagation speed is independent of direction, the secondary wavefronts that expand from points on a primary wavefront in a given infinitesimal time are spherical, so that their radii are normal to their common tangent surface at the points of tangency. But their radii mark the ray directions, and their common tangent surface is a general wavefront. Thus the rays are normal orthogonal to the wavefronts.

Because much of the teaching of optics concentrates on isotropic media, treating anisotropic media as an optional topic, the assumption that the rays are normal to the wavefronts can become so pervasive that even Fermats principle is explained under that assumption, although in fact Fermats principle is more general.


3.2. Special cases Homogeneous media: Rectilinear propagation

In a homogeneous medium also called a uniform medium, all the secondary wavefronts that expand from a given primary wavefront W in a given time Δt are congruent and similarly oriented, so that their envelope W′ may be considered as the envelope of a single secondary wavefront which preserves its orientation while its center source moves over W. If P is its center while P′ is its point of tangency with W′, then P′ moves parallel to P, so that the plane tangential to W′ at P′ is parallel to the plane tangential to W at P. Let another congruent and similarly orientated secondary wavefront be centered on P′, moving with P, and let it meet its envelope W″ at point P″. Then, by the same reasoning, the plane tangential to W″ at P″ is parallel to the other two planes. Hence, due to the congruence and similar orientations, the ray directions PP′ and P′P″ are the same but not necessarily normal to the wavefronts, since the secondary wavefronts are not necessarily spherical. This construction can be repeated any number of times, giving a straight ray of any length. Thus a homogeneous medium admits rectilinear rays.


4.1. Modern version Formulation in terms of refractive index

Let a path Γ extend from point A to point B. Let s be the arc length measured along the path from A, and let t be the time taken to traverse that arc length at the ray speed v r {\displaystyle v_{\mathrm {r} }}. Then the traversal time of the entire path Γ is

where A and B simply denote the endpoints and are not to be construed as values of t or s. The condition for Γ to be a ray path is that the first-order change in T due to a change in Γ is zero; that is,

δ T = δ ∫ A B d s v r = 0 {\displaystyle \delta T=\,\delta \int _{A}^{B}{\frac {ds}{\,v_{\mathrm {r} }\,}}\,=\,0\,}.

Now let us define the optical length of a given path optical path length, OPL as the distance traversed by a ray in a homogeneous isotropic reference medium e.g., a vacuum in the same time that it takes to traverse the given path at the local ray velocity. Then, if c denotes the propagation speed in the reference medium e.g., the speed of light in a vacuum, the optical length of a path traversed in time dt   is ‍ dS = c dt, ‍ and the optical length of a path traversed in time T   is ‍ S = cT. So, multiplying equation 1 through by c , we obtain

S = ∫ A B d S = ∫ A B c v r d s = ∫ A B n r d s, {\displaystyle S\,=\int _{A}^{B}\!dS\,=\int _{A}^{B}{\frac {\,c\,}{v_{\mathrm {r} }}}\,ds\,=\int _{A}^{B}\!n_{\mathrm {r} }\,ds\,}

where n r = c / v r {\displaystyle \,n_{\mathrm {r} }=c/v_{\mathrm {r} }\,} is the ray index - that is, the refractive index calculated on the ray velocity instead of the usual phase velocity wave-normal velocity. For an infinitesimal path, we have d S = n r d s, {\displaystyle dS=n_{\mathrm {r} \,}ds\,\,} indicating that the optical length is the physical length multiplied by the ray index: the OPL is a notional geometric quantity, from which time has been factored out. In terms of OPL, the condition for Γ to be a ray path Fermats principle becomes

This has the form of Maupertuiss principle in classical mechanics for a single particle, with the ray index in optics taking the role of momentum or velocity in mechanics.

In an isotropic medium, for which the ray velocity is also the phase velocity, we may substitute the usual refractive index n for n r. 


4.2. Modern version Relation to Hamiltons principle

If x,y,z are Cartesian coordinates and an overdot denotes differentiation with respect to s , Fermats principle 2 may be written

δ S = δ ∫ A B n r d x 2 + d y 2 + d z 2 = δ ∫ A B n r x 2 + y 2 + z 2 d s = 0. {\displaystyle {\begin{aligned}\delta S&=\,\delta \int _{A}^{B}\!n_{\mathrm {r} }\,{\sqrt {dx^{2}+dy^{2}+dz^{2}}}\\&=\,\delta \int _{A}^{B}\!n_{\mathrm {r} }\,{\sqrt z,z{\big)}\,dz\,=\,0\,}.

This has the form of Hamiltons principle in classical mechanics, except that the time dimension is missing: the third spatial coordinate in optics takes the role of time in mechanics. The optical Lagrangian is the function which, when integrated w.r.t. the parameter of the path, yields the OPL; it is the foundation of Lagrangian and Hamiltonian optics.


5.1. History Fermat vs. the Cartesians

If a ray follows a straight line, it obviously takes the path of least length. Hero of Alexandria, in his Catoptrics 1st century CE, showed that the ordinary law of reflection off a plane surface follows from the premise that the total length of the ray path is a minimum. In 1657, Pierre de Fermat received from Marin Cureau de la Chambre a copy of newly published treatise, in which La Chambre noted Heros principle and complained that it did not work for refraction.

Fermat replied that refraction might be brought into the same framework by supposing that light took the path of least resistance, and that different media offered different resistances. His eventual solution, described in a letter to La Chambre dated 1 January 1662, construed "resistance" as inversely proportional to speed, so that light took the path of least time. That premise yielded the ordinary law of refraction, provided that light traveled more slowly in the optically denser medium.

Fermats solution was a landmark in that it unified the then-known laws of geometrical optics under a variational principle or action principle, setting the precedent for the principle of least action in classical mechanics and the corresponding principles in other fields see History of variational principles in physics. It was the more notable because it used the method of adequality, which may be understood in retrospect as finding the point where the slope of an infinitesimally short chord is zero, without the intermediate step of finding a general expression for the slope the derivative.

It was also immediately controversial. The ordinary law of refraction was at that time attributed to Rene Descartes d. 1650, who had tried to explain it by supposing that light was a force that propagated instantaneously, or that light was analogous to a tennis ball that traveled faster in the denser medium, either premise being inconsistent with Fermats. Descartes most prominent defender, Claude Clerselier, criticized Fermat for apparently ascribing knowledge and intent to nature, and for failing to explain why nature should prefer to economize on time rather than distance. Clerselier wrote in part:

1. The principle that you take as the basis of your demonstration, namely that nature always acts in the shortest and simplest ways, is merely a moral principle and not a physical one; it is not, and cannot be, the cause of any effect in nature. For otherwise we would attribute knowledge to nature; but here, by "nature", we understand only this order and this law established in the world as it is, which acts without foresight, without choice, and by a necessary determination.

2. This same principle would make nature irresolute. For I ask you. when a ray of light must pass from a point in a rare medium to a point in a dense one, is there not reason for nature to hesitate if, by your principle, it must choose the straight line as soon as the bent one, since if the latter proves shorter in time, the former is shorter and simpler in length? Who will decide and who will pronounce? 

Fermat, being unaware of the mechanistic foundations of his own principle, was not well placed to defend it, except as a purely geometric and kinematic proposition. The wave theory of light, first proposed by Robert Hooke in the year of Fermats death, and rapidly improved by Ignace-Gaston Pardies and especially Christiaan Huygens, contained the necessary foundations; but the recognition of this fact was surprisingly slow.


5.2. History Huygenss oversight

Huygens repeatedly referred to the envelope of his secondary wavefronts as the termination of the movement, meaning that the later wavefront was the outer boundary that the disturbance could reach in a given time, which was therefore the minimum time in which each point on the later wavefront could be reached. But he did not argue that the direction of minimum time was that from the secondary source to the point of tangency; instead, he deduced the ray direction from the extent of the common tangent surface corresponding to a given extent of the initial wavefront. His only endorsement of Fermats principle was limited in scope: having derived the law of ordinary refraction, for which the rays are normal to the wavefronts, Huygens gave a geometric proof that a ray refracted according to this law takes the path of least time. He would hardly have thought this necessary if he had known that the principle of least time followed directly from the same common-tangent construction by which he had deduced not only the law of ordinary refraction, but also the laws of rectilinear propagation and ordinary reflection which were also known to follow from Fermats principle, and a previously unknown law of extraordinary refraction - the last by means of secondary wavefronts that were spheroidal rather than spherical, with the result that the rays were generally oblique to the wavefronts. It was as if Huygens had not noticed that his construction implied Fermats principle, and even as if he thought he had found an exception to that principle. Manuscript evidence cited by Alan E. Shapiro tends to confirm that Huygens believed the principle of least time to be invalid "in double refraction, where the rays are not normal to the wave fronts".

Shapiro further reports that the only three authorities who accepted "Huygens principle" in the 17th and 18th centuries, namely Philippe de La Hire, Denis Papin, and Gottfried Wilhelm Leibniz, did so because it accounted for the extraordinary refraction of "Iceland crystal" calcite in the same manner as the previously known laws of geometrical optics. But, for the time being, the corresponding extension of Fermats principle went unnoticed.


5.3. History Laplace, Young, Fresnel, and Lorentz

On 30 January 1809, Pierre-Simon Laplace, reporting on the work of his protege Etienne-Louis Malus, claimed that the extraordinary refraction of calcite could be explained under the corpuscular theory of light with the aid of Maupertuiss principle of least action: that the integral of speed with respect to distance was a minimum. The corpuscular speed that satisfied this principle was proportional to the reciprocal of the ray speed given by the radius of Huygens spheroid. Laplace continued:

According to Huygens, the velocity of the extraordinary ray, in the crystal, is simply expressed by the radius of the spheroid; consequently his hypothesis does not agree with the principle of the least action: but it is remarkable that it agrees with the principle of Fermat, which is, that light passes, from a given point without the crystal, to a given point within it, in the least possible time; for it is easy to see that this principle coincides with that of the least action, if we invert the expression of the velocity.

Laplaces report was the subject of a wide-ranging rebuttal by Thomas Young, who wrote in part:

The principle of Fermat, although it was assumed by that mathematician on hypothetical, or even imaginary grounds, is in fact a fundamental law with respect to undulatory motion, and is explicitly the basis of every determination in the Huygenian theory. Mr. Laplace seems to be unacquainted with this most essential principle of one of the two theories which he compares; for he says, that "it is remarkable," that the Huygenian law of extraordinary refraction agrees with the principle of Fermat; which he would scarcely have observed, if he had been aware that the law was an immediate consequence of the principle.

In fact Laplace was aware that Fermats principle follows from Huygens construction in the case of refraction from an isotropic medium to an anisotropic one; a geometric proof was contained in the long version of Laplaces report, printed in 1810.

Youngs claim was more general than Laplaces, and likewise upheld Fermats principle even in the case of extraordinary refraction, in which the rays are generally not perpendicular to the wavefronts. Unfortunately, however, the omitted middle sentence of the quoted paragraph by Young began "The motion of every undulation must necessarily be in a direction perpendicular to its surface." emphasis added, and was therefore bound to sow confusion rather than clarity.

No such confusion subsists in Augustin-Jean Fresnels "Second Memoir" on double refraction Fresnel, 1827, which addresses Fermats principle in several places without naming Fermat, proceeding from the special case in which rays are normal to wavefronts, to the general case in which rays are paths of least time or stationary time. In the following summary, page numbers refer to Alfred W. Hobsons translation.

  • For refraction of a plane wave at parallel incidence on one face of an anisotropic crystalline wedge pp. 291–2, in order to find the "first ray arrived" at an observation point beyond the other face of the wedge, it suffices to treat the rays outside the crystal as normal to the wavefronts, and within the crystal to consider only the parallel wavefronts whatever the ray direction. So in this case, Fresnel does not attempt to trace the complete ray path.
  • Next, Fresnel considers a ray refracted from a point-source M inside a crystal, through a point A on the surface, to an observation point B outside pp. 294–6. The surface passing through B and given by the "locus of the disturbances which arrive first" is, according to Huygens construction, normal to "the ray AB of swiftest arrival". But this construction requires knowledge of the "surface of the wave" that is, the secondary wavefront within the crystal.
  • Then he considers a plane wavefront propagating in a medium with non-spherical secondary wavefronts, oriented so that the ray path given by Huygens construction - from the source of the secondary wavefront to its point of tangency with the subsequent primary wavefront - is not normal to the primary wavefronts p. 296. He shows that this path is nevertheless "the path of quickest arrival of the disturbance" from the earlier primary wavefront to the point of tangency.
  • In a later heading p. 305 he declares that "The construction of Huygens, which determines the path of swiftest arrival," is applicable to secondary wavefronts of any shape. He then notes that when we apply Huygens construction to refraction into a crystal with a two-sheeted secondary wavefront, and draw the lines from the two points of tangency to the center of the secondary wavefront, "we shall have the directions of the two paths of swiftest arrival, and consequently of the ordinary and of the extraordinary ray."
  • Under the heading "Definition of the word Ray p. 309, he concludes that this term must be applied to the line which joins the center of the secondary wave to a point on its surface, whatever the inclination of this line to the surface.
  • As a "new consideration" pp. 310–11, he notes that if a plane wavefront is passed through a small hole centered on point E, then the direction ED of maximum intensity of the resulting beam will be that in which the secondary wave starting from E will "arrive there the first", and the secondary wavefronts from opposite sides of the hole equidistant from E will "arrive at D in the same time" as each other. This direction is not assumed to be normal to any wavefront.

Thus Fresnel showed, even for anisotropic media, that the ray path given by Huygens construction is the path of least time between successive positions of a plane or diverging wavefront, that the ray velocities are the radii of the secondary "wave surface" after unit time, and that a stationary traversal time accounts for the direction of maximum intensity of a beam. However, establishing the general equivalence between Huygens construction and Fermats principle would have required further consideration of Fermats principle in point-to-point terms.

According to Adriaan J. de Witte 1959, the equivalence of Huygens construction and Fermats principle was discussed by Hendrik Lorentz in 1907. De Witte offers his own proof, which "although in essence the same, is believed to be more cogent and more general." But, he notes, "The matter seems to have escaped treatment in textbooks."


6. Bibliography

  • A.E. Shapiro, 1973, "Kinematic optics: A study of the wave theory of light in the seventeenth century", Archive for History of Exact Sciences, vol. 11, no. 2/3 June 1973, pp. 134–266, doi:10.1007/BF00343533.
  • I. Newton, 1730, Opticks: or, a Treatise of the Reflections, Refractions, Inflections, and Colours of Light, 4th Ed. ; republished with Foreword by A. Einstein and Introduction by E.T. Whittaker London: George Bell & Sons, 1931; reprinted with additional Preface by I.B. Cohen and Analytical Table of Contents by D.H.D. Roller, Mineola, NY: Dover, 1952, 1979 with revised preface, 2012. Cited page numbers match the Gutenberg HTML edition and the Dover editions.
  • P. Mihas, 2006, "Developing ideas of refraction, lenses and rainbow through the use of historical resources", Science & Education, vol. 17, no. 7 August 2008, pp. 751–777 online 6 September 2006, doi:10.1007/s11191-006-9044-8.
  • A.J. de Witte, 1959, "Equivalence of Huygens principle and Fermats principle in ray geometry", American Journal of Physics, vol. 27, no. 5 May 1959, pp. 293–301, doi:10.1119/1.1934839. Erratum: In Fig. 7b, each instance of "ray" should be "normal".
  • T. Young, 1809, Article X in the Quarterly Review, vol. 2, no. 4 November 1809, pp. 337–48.
  • M. Born and E. Wolf, 1970, Principles of Optics, 4th Ed., Oxford: Pergamon Press.
  • C. Huygens, 1690, Traite de la Lumiere Leiden: Van der Aa, translated by S.P. Thompson as Treatise on Light, University of Chicago Press, 1912; Project Gutenberg, 2005. Cited page numbers match the 1912 edition and the Gutenberg HTML edition.
  • O. Darrigol, 2012, A History of Optics: From Greek Antiquity to the Nineteenth Century, Oxford, ISBN 978-0-19-964437-7.
  • J. Chaves, 2016, Introduction to Nonimaging Optics, 2nd Ed., Boca Raton, FL: CRC Press, ISBN 978-1-4822-0674-6.
  • A.I. Sabra, 1981, Theories of Light: From Descartes to Newton London: Oldbourne Book Co., 1967, reprinted Cambridge University Press, 1981, ISBN 0-521-28436-8.
  • A. Ziggelaar, 1980, "The sine law of refraction derived from the principle of Fermat - prior to Fermat? The theses of Wilhelm Boelmans S.J. in 1634", Centaurus, vol. 24, no. 1 September 1980, pp. 246–62, doi:10.1111/j.1600-0498.1980.tb00377.x.
  • A. Fresnel, 1827, "Memoire sur la double refraction", Memoires de lAcademie Royale des Sciences de lInstitut de France, vol.  VII for 1824, printed 1827, pp. 45–176; reprinted as Second memoire…" in Oeuvres completes dAugustin Fresnel, vol. 2 Paris: Imprimerie Imperiale, 1868, pp. 479–596; translated by A.W. Hobson as "Memoir on double refraction", in R. Taylor ed., Scientific Memoirs, vol.  V London: Taylor & Francis, 1852, pp. 238–333. Cited page numbers are from the translation.
  • E. Frankel, 1974, "The search for a corpuscular theory of double refraction: Malus, Laplace and the price competition of 1808", Centaurus, vol. 18, no. 3 September 1974, pp. 223–245, doi:10.1111/j.1600-0498.1974.tb00298.x.
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