Editing Total internal reflection (section)

=== 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>