Editing Total internal reflection (section)

=== 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}}