Editing Total internal reflection (section)

== Critical angle ==

The critical angle is the smallest angle of incidence that yields total reflection, or equivalently the largest angle for which a refracted ray exists.<ref>Jenkins & White, 1976, p.{{nbsp}}26.</ref> For light waves incident from an "internal" medium with a single [[refractive index]] {{math|''n''<sub>1</sub>}}, to an "external" medium with a single refractive index {{math|''n''<sub>2</sub>}}, the critical angle is given by <math>\theta_\text{c} = \arcsin(n_2/n_1)</math> and is defined if {{math|''n''<sub>2</sub> &le; ''n''<sub>1</sub>}}. For some other types of waves, it is more convenient to think in terms of propagation velocities rather than refractive indices. The explanation of the critical angle in terms of velocities is more general and will therefore be discussed first.

[[File:Wavefront refraction slow to fast.svg|thumb|'''Fig.{{nbsp}}4''': Refraction of a ''wavefront'' (red) from medium 1, with lower normal velocity ''v''<sub>1</sub>, to medium 2, with higher normal velocity ''v''<sub>2</sub>. The incident and refracted segments of the wavefront meet in a common line ''L'' (seen "end-on"), which travels along the interface at velocity ''u''.]]

When a [[wavefront]] is refracted from one medium to another, the incident (incoming) and refracted (outgoing) portions of the wavefront meet at a common line on the refracting surface (interface). Let this line, denoted by ''L'', move at velocity {{mvar|u}} across the surface,{{r|young-1814}}<ref>Born & Wolf, 1970, pp.{{nbsp}}12–13.</ref> where {{mvar|u}} is measured normal to&nbsp;''L'' (Fig.{{nbsp}}4). Let the incident and refracted wavefronts propagate with normal velocities <math>v_1</math> and <math>v_2</math> respectively, and let them make the [[dihedral angle]]s ''θ''<sub>1</sub> and ''θ''<sub>2</sub> respectively with the interface. From the geometry, <math>v_1</math> is the component of {{mvar|u}} in the direction normal to the incident wave, so that <math>v_1 = u \sin\theta_1.</math> Similarly, <math>v_2 = u \sin\theta_2.</math> Solving each equation for {{math|1/''u''}} and equating the results, we obtain the general law of refraction for waves:
{{NumBlk|:|<math>\frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2}.</math>|{{EquationRef|1}}}}
But the dihedral angle between two planes is also the angle between their normals. So ''θ''<sub>1</sub> is the angle between the normal to the incident wavefront and the normal to the interface, while ''θ''<sub>2</sub> is the angle between the normal to the refracted wavefront and the normal to the interface; and Eq.{{nbsp}}({{EquationNote|1}}) tells us that the sines of these angles are in the same ratio as the respective velocities.<ref>Huygens, 1690, tr.&nbsp;Thompson, p.{{nbsp}}38.</ref>

This result has the form of "[[Snell's law]]", except that we have not yet said that the ratio of velocities is constant, nor identified ''θ''<sub>1</sub> and ''θ''<sub>2</sub> with the angles of incidence and refraction (called ''θ''<sub>i</sub> and ''θ''<sub>t</sub> above). However, if we now suppose that the properties of the media are ''[[isotropy|isotropic]]'' (independent of direction), two further conclusions follow: first, the two velocities, and hence their ratio, are independent of their directions; and second, the wave-normal directions coincide with the ''ray'' directions, so that ''θ''<sub>1</sub> and ''θ''<sub>2</sub> coincide with the angles of incidence and refraction as defined above.<ref group=Note>[[birefringence|Birefringent]] media, such as [[calcite]], are non-isotropic (anisotropic). When we say that the extraordinary refraction of a calcite crystal "violates Snell's law", we mean that Snell's law does not apply to the extraordinary ''ray'', because the direction of this ray inside the crystal generally differs from that of the associated wave-normal (Huygens, 1690, tr.&nbsp;Thompson, p.{{nbsp}}65, Art.{{nbsp}}24), and because the wave-normal speed is itself dependent on direction. (Note that the cited passage contains a translation error: in the phrase "conjugate with respect to diameters which are not in the straight line AB", the word "not" is unsupported by [https://www.dbnl.org/tekst/huyg003oeuv19_01/huyg003oeuv19_01_0102.php Huygens' original French] and is geometrically incorrect.)</ref>

[[File:ReflexionTotal en.svg|thumb|'''Fig.{{nbsp}}5''': Behavior of a ray incident from a medium of higher refractive index ''n''<sub>1</sub> to a medium of lower refractive index ''n''<sub>2</sub>, at increasing angles of incidence<ref group=Note>According to Eqs.{{nbsp}}({{EquationNote|13}}) and ({{EquationNote|15}}), reflection is total for incidence ''at'' the critical angle. On that basis, Fig.{{nbsp}}5 ought to show a fully reflected ray, and no tangential ray, for incidence at ''θ''<sub>c</sub>. But, due to [[diffraction]], an incident beam of finite width cannot have a single angle of incidence; there must be some divergence of the beam. Moreover, the graph of the reflection coefficient vs. the angle of incidence becomes vertical at ''θ''<sub>c</sub> (Jenkins & White, 1976, p.{{nbsp}}527), so that a small divergence of the beam causes a large loss of reflection. Similarly, near the critical angle, a small divergence in the angle of incidence causes a large divergence in the angle of refraction (Huygens, 1690, tr.&nbsp;Thompson, p.{{nbsp}}41); the tangential refracted ray should therefore be taken only as a limiting case.</ref>]]

[[File:Total Internal Refraction diver.svg|thumb|'''Fig.{{nbsp}}6''': The angle of refraction for grazing incidence from air to water is the critical angle for incidence from water to air.]]

Obviously the angle of refraction cannot exceed 90°. In the limiting case, we put {{math|''θ''<sub>2</sub> {{=}} 90°}} and {{math|''θ''<sub>1</sub> {{=}} ''θ''<sub>c</sub>}} in Eq.{{nbsp}}({{EquationNote|1}}), and solve for the critical angle:
{{NumBlk|:|<math>\theta_\text{c} = \arcsin(v_1/v_2).</math>|{{EquationRef|2}}}}
In deriving this result, we retain the assumption of isotropic media in order to identify ''θ''<sub>1</sub> and ''θ''<sub>2</sub> with the angles of incidence and refraction.<ref group=Note>For non-isotropic media, Eq.{{nbsp}}({{EquationNote|1}}) still describes the law of refraction in terms of ''wave-normal'' directions and speeds, but the range of applicability of that law is determined by the constraints on the ''ray'' directions (Buchwald, 1989, p.{{nbsp}}29).</ref>

For [[electromagnetic waves]], and especially for light, it is customary to express the above results in terms of ''refractive indices''. The refractive index of a medium with normal velocity <math>v_1</math> is defined as <math>n_1 = c/v_1,</math> where ''c'' is the speed of light in vacuum.<ref>Born & Wolf, 1970, p.{{nbsp}}13; Jenkins & White, 1976, pp.{{nbsp}}9–10. This definition uses vacuum as the "reference medium". In principle, any isotropic medium can be chosen as the reference. For some purposes it is convenient to choose air, in which the speed of light is about 0.03% lower than in vacuum (Rutten and van Venrooij, 2002, pp.{{nbsp}}10, 352). The present article, however, chooses vacuum.</ref> Hence <math>v_1 = c/n_1.</math> Similarly, <math>v_2 = c/n_2.</math> Making these substitutions in Eqs.{{nbsp}}({{EquationNote|1}}) and ({{EquationNote|2}}), we obtain
{{NumBlk|:|<math>n_1 \sin\theta_1 = n_2 \sin\theta_2</math>|{{EquationRef|3}}}}
and
{{NumBlk|:|<math>\theta_\text{c} = \arcsin(n_2/n_1).</math>|{{EquationRef|4}}}}
Eq.{{nbsp}}({{EquationNote|3}}) is the law of refraction for general media, in terms of refractive indices, provided that ''θ''<sub>1</sub> and ''θ''<sub>2</sub> are taken as the dihedral angles; but if the media are ''isotropic'', then {{math|''n''<sub>1</sub>}} and {{math|''n''<sub>2</sub>}} become independent of direction, while ''θ''<sub>1</sub> and ''θ''<sub>2</sub> may be taken as the angles of incidence and refraction for the rays, and Eq.{{nbsp}}({{EquationNote|4}}) follows. So, for isotropic media, Eqs.{{nbsp}}({{EquationNote|3}}) and ({{EquationNote|4}}) together describe the behavior in Fig.{{nbsp}}5.

According to Eq.{{nbsp}}({{EquationNote|4}}), for incidence from water ({{math|''n''<sub>1</sub> ≈ 1.333}}) to air ({{math|''n''<sub>2</sub> ≈ 1}}), we have {{math|''θ''<sub>c</sub> ≈ 48.6°}}, whereas for incidence from common or acrylic glass ({{math|''n''<sub>1</sub> ≈ 1.50}}) to air ({{math|''n''<sub>2</sub> ≈ 1}}), we have {{math|''θ''<sub>c</sub> ≈ 41.8°}}.

The arcsin function yielding ''θ''<sub>c</sub> is defined only if {{math|''n''<sub>2</sub> &le; ''n''<sub>1</sub>}} <math>(v_2 \geq v_1).</math> Hence, for isotropic media, total internal reflection cannot occur if the second medium has a higher refractive index (lower normal velocity) than the first. For example, there cannot be TIR for incidence from air to water; rather, the critical angle for incidence from water to air is the angle of refraction at grazing incidence from air to water (Fig.{{nbsp}}6).<ref>Jenkins & White, 1976, p.{{nbsp}}25.</ref>

The medium with the higher refractive index is commonly described as optically ''denser'', and the one with the lower refractive index as optically ''rarer''.<ref>Jenkins & White, 1976, pp.{{nbsp}}10, 25.</ref> Hence it is said that total internal reflection is possible for "dense-to-rare" incidence, but not for "rare-to-dense" incidence.