Editing Total internal reflection (section)

== History ==

=== Discovery ===

The surprisingly comprehensive and largely correct explanations of the [[rainbow]] by [[Theodoric of Freiberg]] (written between 1304 and 1310)<ref>Boyer, 1959, p.{{nbsp}}110.</ref> and [[Kamāl al-Dīn al-Fārisī]] (completed by 1309),<ref>Kamāl al-Dīn al-Fārisī, [[:File:Autograph by Kamāl al-Dīn al-Fārisī 3.jpg|''Tanqih al-Manazir'']] (autograph manuscript, 708&nbsp;[[Hijri year|AH]] / 1309&nbsp;[[Common Era|CE]]), Adilnor Collection.</ref> although sometimes mentioned in connection with total internal reflection (TIR), are of dubious relevance because the internal reflection of sunlight in a spherical raindrop is ''not'' total.<ref group=Note>For an external ray incident on a spherical raindrop, the refracted ray is in the plane of the incident ray and the center of the drop, and the angle of refraction is less than the critical angle for water-air incidence; but this angle of refraction, by the spherical symmetry, is also the angle of incidence for the internal reflection, which is therefore less than total. Moreover, if that reflection were total, all subsequent internal reflections would have the same angle of incidence (due to the symmetry) and would also be total, so that the light would never escape to produce a visible bow.</ref> But, according to [[Carl Benjamin Boyer]], Theodoric's treatise on the rainbow also classified optical phenomena under five causes, the last of which was "a total reflection at the boundary of two transparent media".<ref>Boyer, 1959, pp.{{nbsp}}113, 114, 335.&nbsp; Boyer cites J.{{nbsp}}Würschmidt's edition of Theodoric's ''De iride et radialibus impressionibus'', in ''Beiträge zur Geschichte der Philosophie des Mittelalters'', vol.{{nbsp}}12, nos.{{nbsp}}5–6 (1914), at p.{{nbsp}}47.</ref> Theodoric's work was forgotten until it was rediscovered by [[Giovanni Battista Venturi]] in 1814.<ref>Boyer, 1959, pp.{{nbsp}}307, 335.</ref>

[[File:Johannes Kepler by Hans von Aachen.jpg|thumb|Johannes Kepler (1571–1630)]]

Theodoric having fallen into obscurity, the discovery of TIR was generally attributed to [[Johannes Kepler]], who published his findings in his ''[[Dioptrice]]'' in 1611. Although Kepler failed to find the true law of refraction, he showed by experiment that for air-to-glass incidence, the incident and refracted rays rotated in the same sense about the point of incidence, and that as the angle of incidence varied through ±90°, the angle of refraction (as we now call it) varied through ±42°. He was also aware that the incident and refracted rays were interchangeable. But these observations did not cover the case of a ray incident from glass to air at an angle beyond 42°, and Kepler promptly concluded that such a ray could only be ''reflected''.{{r|mach-2003}}

[[René Descartes]] rediscovered the law of refraction and published it in his ''[[Dioptrique]]'' of 1637. In the same work he mentioned the senses of rotation of the incident and refracted rays and the condition of TIR. But he neglected to discuss the limiting case, and consequently failed to give an expression for the critical angle, although he could easily have done so.{{r|sabra-1981}}

=== Huygens and Newton: Rival explanations ===

[[Christiaan Huygens]], in his ''[[Treatise on Light]]'' (1690), paid much attention to the threshold at which the incident ray is "unable to penetrate into the other transparent substance".<ref>Huygens, 1690, tr.&nbsp;Thompson, p.{{nbsp}}39.</ref> Although he gave neither a name nor an algebraic expression for the critical angle, he gave numerical examples for glass-to-air and water-to-air incidence, noted the large change in the angle of refraction for a small change in the angle of incidence near the critical angle, and cited this as the cause of the rapid increase in brightness of the reflected ray as the refracted ray approaches the tangent to the interface.<ref>Huygens, 1690, tr.&nbsp;Thompson, pp.{{nbsp}}40–41. Notice that Huygens' definition of the "angle of incidence" is the [[complementary angle|complement]] of the modern definition.</ref> Huygens' insight is confirmed by modern theory: in Eqs.{{nbsp}}({{EquationNote|13}}) and ({{EquationNote|15}}) above, there is nothing to say that the reflection coefficients increase exceptionally steeply as ''θ''<sub>t</sub> approaches 90°, except that, according to Snell's law, ''θ''<sub>t</sub>&nbsp;itself is an increasingly steep function of ''θ''<sub>i</sub>.

[[File:Christiaan-huygens4.jpg|left|thumb|Christiaan Huygens (1629–1695)]]

Huygens offered an explanation of TIR within the same framework as his explanations of the laws of rectilinear propagation, reflection, ordinary refraction, and even the extraordinary refraction of "[[Iceland spar|Iceland crystal]]" (calcite). That framework rested on two premises: first, every point crossed by a propagating wavefront becomes a source of secondary wavefronts ("Huygens' principle"); and second, given an initial wavefront, any subsequent position of the wavefront is the [[envelope (mathematics)|envelope]] (common tangent surface) of all the secondary wavefronts emitted from the initial position. All cases of reflection or refraction by a surface are then explained simply by considering the secondary waves emitted from that surface. In the case of refraction from a medium of slower propagation to a medium of faster propagation, there is a certain obliquity of incidence beyond which it is impossible for the secondary wavefronts to form a common tangent in the second medium;<ref>Huygens, 1690, tr.&nbsp;Thompson, pp.{{nbsp}}39–40.</ref> this is what we now call the critical angle. As the incident wavefront approaches this critical obliquity, the refracted wavefront becomes concentrated against the refracting surface, augmenting the secondary waves that produce the reflection back into the first medium.<ref>Huygens, 1690, tr.&nbsp;Thompson, pp.{{nbsp}}40–41.</ref>

Huygens' system even accommodated ''partial'' reflection at the interface between different media, albeit vaguely, by analogy with the laws of collisions between particles of different sizes.<ref>Huygens, 1690, tr.&nbsp;Thompson, pp.{{nbsp}}16, 42.</ref> However, as long as the wave theory continued to assume [[longitudinal wave]]s, it had no chance of accommodating polarization, hence no chance of explaining the polarization-dependence of extraordinary refraction,<ref>Huygens, 1690, tr.&nbsp;Thompson, pp.{{nbsp}}92–94.</ref> or of the partial reflection coefficient, or of the phase shift in TIR.

[[File:Portrait of Sir Isaac Newton, 1689.jpg|thumb|Isaac Newton (1642/3–1726/7)]]

[[Isaac Newton]] rejected the wave explanation of rectilinear propagation, believing that if light consisted of waves, it would "bend and spread every way" into the shadows.<ref>Newton, 1730, p.{{nbsp}}362.</ref> His corpuscular theory of light explained rectilinear propagation more simply, and it accounted for the ordinary laws of refraction and reflection, including TIR, on the hypothesis that the corpuscles of light were subject to a force acting perpendicular to the interface.<ref>Darrigol, 2012, pp.{{nbsp}}93–94, 103.</ref> In this model, for dense-to-rare incidence, the force was an attraction back towards the denser medium, and the critical angle was the angle of incidence at which the normal velocity of the approaching corpuscle was just enough to reach the far side of the force field; at more oblique incidence, the corpuscle would be turned back.<ref>Newton, 1730, pp.{{nbsp}}370–371.</ref> Newton gave what amounts to a formula for the critical angle, albeit in words: "as the Sines are which measure the Refraction, so is the Sine of Incidence at which the total Reflexion begins, to the Radius of the Circle".<ref>Newton, 1730, p.{{nbsp}}246. Notice that a "sine" meant the length of a side for a specified "radius" (hypotenuse), whereas nowadays we take the radius as unity or express the sine as a ratio.</ref>

Newton went beyond Huygens in two ways. First, not surprisingly, Newton pointed out the relationship between TIR and ''[[dispersion (optics)|dispersion]]'': when a beam of white light approaches a glass-to-air interface at increasing obliquity, the most strongly-refracted rays (violet) are the first to be "taken out" by "total Reflexion", followed by the less-refracted rays.<ref>Newton, 1730, pp.{{nbsp}}56–62, 264.</ref> Second, he observed that total reflection could be ''frustrated'' (as we now say) by laying together two prisms, one plane and the other slightly convex; and he explained this simply by noting that the corpuscles would be attracted not only to the first prism, but also to the second.<ref>Newton, 1730, pp.{{nbsp}}371–372.</ref>

In two other ways, however, Newton's system was less coherent. First, his explanation of ''partial'' reflection depended not only on the supposed forces of attraction between corpuscles and media, but also on the more nebulous hypothesis of "Fits of easy Reflexion" and "Fits of easy Transmission".<ref>Newton, 1730, p.{{nbsp}}281.</ref> Second, although his corpuscles could conceivably have "sides" or "poles", whose orientations could conceivably determine whether the corpuscles suffered ordinary or extraordinary refraction in "Island-Crystal",<ref>Newton, 1730, p.{{nbsp}}373.</ref> his geometric description of the extraordinary refraction<ref>Newton, 1730, p.{{nbsp}}356.</ref> was theoretically unsupported<ref>Buchwald, 1980, pp.{{nbsp}}327, 331–332.</ref> and empirically inaccurate.<ref>Buchwald, 1980, pp.{{nbsp}}335–336, 364; Buchwald, 1989, pp.{{nbsp}}9–10, 13.</ref>

=== Laplace, Malus, and attenuated total reflectance (ATR) ===

[[William Hyde Wollaston]], in the first of a pair of papers read to the [[Royal Society]] of London in 1802,{{r|wollaston-1802a}} reported his invention of a [[refractometer]] based on the critical angle of incidence from an internal medium of known "refractive power" (refractive index) to an external medium whose index was to be measured.<ref>Buchwald, 1989, pp.{{nbsp}}19–21.</ref> With this device, Wollaston measured the "refractive powers" of numerous materials, some of which were too opaque to permit direct measurement of an angle of refraction. Translations of his papers were published in France in 1803, and apparently came to the attention of [[Pierre-Simon Laplace]].<ref>Buchwald, 1989, p.{{nbsp}}28.</ref>

[[File:Pierre-Simon de Laplace by Johann Ernst Heinsius (1775).jpg|left|thumb|Pierre-Simon Laplace (1749–1827)]]

According to Laplace's elaboration of Newton's theory of refraction, a corpuscle incident on a plane interface between two homogeneous isotropic media was subject to a force field that was symmetrical about the interface. If both media were transparent, total reflection would occur if the corpuscle were turned back before it exited the field in the second medium. But if the second medium were opaque, reflection would not be total unless the corpuscle were turned back before it left the ''first'' medium; this required a larger critical angle than the one given by Snell's law, and consequently impugned the validity of Wollaston's method for opaque media.<ref>Darrigol, 2012, pp.{{nbsp}}187–188.</ref> Laplace combined the two cases into a single formula for the relative refractive index in terms of the critical angle (minimum angle of incidence for TIR). The formula contained a parameter which took one value for a transparent external medium and another value for an opaque external medium. Laplace's theory further predicted a relationship between refractive index and density for a given substance.<ref>Buchwald, 1989, p.{{nbsp}}30.</ref>

[[File:Malus by Boilly 1810.jpg|thumb|Étienne-Louis Malus (1775–1812)]]

In 1807, Laplace's theory was tested experimentally by his protégé, [[Étienne-Louis Malus]]. Taking Laplace's formula for the refractive index as given, and using it to measure the refractive index of beeswax in the liquid (transparent) state and the solid (opaque) state at various temperatures (hence various densities), Malus verified Laplace's relationship between refractive index and density.<ref>Buchwald, 1980, pp.{{nbsp}}29–31.</ref>{{r|frankel-1976}}

But Laplace's theory implied that if the angle of incidence exceeded his modified critical angle, the reflection would be total even if the external medium was absorbent. Clearly this was wrong: in Eqs.{{nbsp}}({{EquationNote|12}}) above, there is no threshold value of the angle ''θ''<sub>i</sub> beyond which ''κ'' becomes infinite; so the penetration depth of the evanescent wave (1/''κ'') is always non-zero, and the external medium, if it is at all lossy, will attenuate the reflection. As to why Malus apparently observed such an angle for opaque wax, we must infer that there was a certain angle beyond which the attenuation of the reflection was so small that [[attenuated total reflectance|ATR]] was visually indistinguishable from TIR.<ref>Buchwald, 1989, p.{{nbsp}}30 (quoting Malus).</ref>

=== Fresnel and the phase shift ===

[[Fresnel]] came to the study of total internal reflection through his research on polarization. In 1811, [[François Arago]] discovered that polarized light was apparently "depolarized" in an orientation-dependent and color-dependent manner when passed through a slice of doubly-refractive crystal: the emerging light showed colors when viewed through an analyzer (second polarizer). ''Chromatic polarization'', as this phenomenon came to be called, was more thoroughly investigated in 1812 by [[Jean-Baptiste Biot]]. In 1813, Biot established that one case studied by Arago, namely [[quartz]] cut perpendicular to its [[optic axis of a crystal|optic axis]], was actually a gradual rotation of the [[plane of polarization]] with distance.<ref>Darrigol, 2012, pp.{{nbsp}}193–196, 290.</ref>

[[File:Augustin Fresnel.jpg|thumb|left|Augustin-Jean Fresnel (1788–1827)]]

In 1816, Fresnel offered his first attempt at a ''wave-based'' theory of chromatic polarization. Without (yet) explicitly invoking [[transverse wave]]s, his theory treated the light as consisting of two perpendicularly polarized components.<ref>Darrigol, 2012, p.{{nbsp}}206.</ref> In 1817 he noticed that plane-polarized light seemed to be partly depolarized by total internal reflection, if initially polarized at an acute angle to the plane of incidence.{{r|brewster-priority}} By including total internal reflection in a chromatic-polarization experiment, he found that the apparently depolarized light was a mixture of components polarized parallel and perpendicular to the plane of incidence, and that the total reflection introduced a phase difference between them.<ref>Darrigol, 2012, p.{{nbsp}}207.</ref> Choosing an appropriate angle of incidence (not yet exactly specified) gave a phase difference of 1/8 of a cycle. Two such reflections from the "parallel faces" of "two coupled prisms" gave a phase difference of 1/4 of a cycle. In that case, if the light was initially polarized at 45° to the plane of incidence and reflection, it appeared to be ''completely'' depolarized after the two reflections. These findings were reported in a memoir submitted and read to the [[French Academy of Sciences]] in November 1817.{{r|fresnel-1817}}

In 1821, Fresnel derived formulae equivalent to his sine and tangent laws (Eqs.{{nbsp}}({{EquationNote|19}}) and ({{EquationNote|20}}), above) by modeling light waves as [[S-wave|transverse elastic waves]] with vibrations perpendicular to what had previously been called the [[plane of polarization]].<ref>Darrigol, 2012, p.{{nbsp}}212.</ref><ref group=Note>Hence, where Fresnel says that after total internal reflection at the appropriate incidence, the wave polarized parallel to the plane of incidence is "behind" by 1/8 of a cycle (quoted by Buchwald, 1989, p.{{nnbsp}}381), he refers to the wave whose plane of polarization is parallel to the plane of incidence, i.e. the wave whose vibration is ''perpendicular'' to that plane, i.e. what we now call the ''s'' component.</ref> Using old experimental data, he promptly confirmed that the equations correctly predicted the direction of polarization of the reflected beam when the incident beam was polarized at 45° to the plane of incidence, for light incident from air onto glass or water.<ref>Buchwald, 1989, pp.{{nbsp}}390–391; Fresnel, 1866, pp.{{nbsp}}646–648.</ref> The experimental confirmation was reported in a "postscript" to the work in which Fresnel expounded his mature theory of chromatic polarization, introducing transverse waves.{{r|fresnel-1821a}} Details of the derivation were given later, in a memoir read to the academy in January 1823.{{r|fresnel-1823a}} The derivation combined conservation of energy with continuity of the ''tangential'' vibration at the interface, but failed to allow for any condition on the ''normal'' component of vibration.<ref>Buchwald, 1989, pp.{{nbsp}}391–393; Darrigol, 2012, pp.{{nbsp}}212–313; Whittaker, 1910, pp.{{nbsp}}133–135.</ref>

Meanwhile, in a memoir submitted in December 1822,{{r|fresnel-1822z}} Fresnel coined the terms ''[[linear polarization]]'', ''[[circular polarization]]'', and ''[[elliptical polarization]]''.<ref>Buchwald, 1989, pp.{{nbsp}}230–231; Fresnel, 1866, p.{{nbsp}}744.</ref> For ''circular'' polarization, the two perpendicular components were a quarter-cycle (±90°) out of phase.

The new terminology was useful in the memoir of January 1823,{{r|fresnel-1823a}} containing the detailed derivations of the sine and tangent laws: in that same memoir, Fresnel found that for angles of incidence greater than the critical angle, the resulting reflection coefficients were complex with unit magnitude. Noting that the magnitude represented the amplitude ratio as usual, he guessed that the argument represented the phase shift, and verified the hypothesis by experiment.<ref>Lloyd, 1834, pp.{{nbsp}}369–370; Buchwald, 1989, pp.{{nbsp}}393–394, 453; Fresnel, 1866, pp.{{nbsp}}781–796.</ref> The verification involved
* calculating the angle of incidence that would introduce a total phase difference of 90° between the ''s'' and ''p'' components, for various numbers of total internal reflections at that angle (generally there were two solutions),
* subjecting light to that number of total internal reflections at that angle of incidence, with an initial linear polarization at 45° to the plane of incidence, and
* checking that the final polarization was circular.<ref>Fresnel, 1866, pp.{{nbsp}}760–761, 792–796; Whewell, 1857, p.{{nbsp}}359.</ref>
This procedure was necessary because, with the technology of the time, one could not measure the ''s'' and ''p'' phase-shifts directly, and one could not measure an arbitrary degree of ellipticality of polarization, such as might be caused by the difference between the phase shifts. But one could verify that the polarization was ''circular'', because the brightness of the light was then insensitive to the orientation of the analyzer.

For glass with a refractive index of 1.51, Fresnel calculated that a 45° phase difference between the two reflection coefficients (hence a 90° difference after two reflections) required an angle of incidence of 48°37' or 54°37'. He cut a rhomb to the latter angle and found that it performed as expected.<ref>Fresnel, 1866, pp.{{nbsp}}760–761, 792–793.</ref> Thus the specification of the Fresnel rhomb was completed. Similarly, Fresnel calculated and verified the angle of incidence that would give a 90° phase difference after ''three'' reflections at the same angle, and ''four'' reflections at the same angle. In each case there were two solutions, and in each case he reported that the larger angle of incidence gave an accurate circular polarization (for an initial linear polarization at 45° to the plane of reflection). For the case of three reflections he also tested the smaller angle, but found that it gave some coloration due to the proximity of the critical angle and its slight dependence on wavelength. (Compare Fig.{{nnbsp}}13 above, which shows that the phase difference {{mvar|δ}} is more sensitive to the refractive index for smaller angles of incidence.)

For added confidence, Fresnel predicted and verified that four total internal reflections at 68°27' would give an accurate circular polarization if two of the reflections had water as the external medium while the other two had air, but not if the reflecting surfaces were all wet or all dry.<ref>Fresnel, 1866, pp.{{nbsp}}761, 793–796; Whewell, 1857, p.{{nbsp}}359.</ref>

Fresnel's deduction of the phase shift in TIR is thought to have been the first occasion on which a physical meaning was attached to the argument of a complex number. Although this reasoning was applied without the benefit of knowing that light waves were electromagnetic, it passed the test of experiment, and survived remarkably intact after [[James Clerk Maxwell]] changed the presumed nature of the waves.<ref>Bochner, 1963, pp.{{nbsp}}198–200.</ref> Meanwhile, Fresnel's success inspired [[James MacCullagh]] and [[Augustin-Louis Cauchy]], beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a [[refractive index#Complex refractive index|complex refractive index]].<ref>Whittaker, 1910, pp.{{tsp}}177–9.</ref> The imaginary part of the complex index represents absorption.<ref>Bochner, 1963, p.{{nbsp}}200; Born & Wolf, 1970, p.{{nbsp}}613.</ref>

The term ''critical angle'', used for convenience in the above narrative, is anachronistic: it apparently dates from 1873.{{r|merriamW-ca}}

In the 20th century, [[quantum electrodynamics]] reinterpreted the amplitude of an electromagnetic wave in terms of the probability of finding a photon.{{r|feynman-1988}} In this framework, partial transmission and frustrated TIR concern the probability of a photon crossing a boundary, and attenuated total reflectance concerns the probability of a photon being absorbed on the other side.

Research into the more subtle aspects of the phase shift in TIR, including the Goos–Hänchen and Imbert–Fedorov effects and their quantum interpretations, has continued into the 21st century.{{r|bliokh-aiello-2013}}