Editing Total internal reflection (section)

=={{anchor|Phase shift upon total internal reflection}}Phase shifts ==

Between 1817 and 1823, [[Augustin-Jean Fresnel]] discovered that total internal reflection is accompanied by a non-trivial [[phase (waves)|phase]] shift (that is, a phase shift that is not restricted to 0° or 180°), as the [[Fresnel equations|Fresnel reflection coefficient]] acquires a non-zero [[complex number|imaginary part]].<ref>Whittaker, 1910, pp.{{nbsp}}132, 135–136.</ref> We shall now explain this effect for electromagnetic waves in the case of [[linearity|linear]], [[homogeneity (physics)|homogeneous]], isotropic, non-magnetic media. The phase shift turns out to be an ''advance'', which grows as the incidence angle increases beyond the critical angle, but which depends on the polarization of the incident wave.

In equations ({{EquationNote|5}}), ({{EquationNote|7}}), ({{EquationNote|8}}), ({{EquationNote|10}}), and ({{EquationNote|11}}), we advance the phase by the angle ''ϕ'' if we replace {{math|''ωt''}} by {{math|''ωt''&nbsp;+&nbsp;''ϕ''}} (that is, if we replace {{math|−''ωt''}} by {{math|−''ωt''&nbsp;−&nbsp;''ϕ''}}), with the result that the (complex) field is multiplied by {{math|''e''<sup>−''iϕ''</sup>}}. So a phase ''advance'' is equivalent to multiplication by a complex constant with a ''negative'' [[argument (complex analysis)|argument]]. This becomes more obvious when (e.g.) the field ({{EquationNote|5}}) is factored as <math>\mathbf{E_k} e^{i\mathbf{k\cdot r}} e^{-i\omega t},</math> where the last factor contains the time dependence.<ref group=Note>In the electrical engineering convention, the time-dependent factor is {{math|''e<sup>jωt</sup>''}}, so that a phase advance corresponds to multiplication by a complex constant with a ''positive'' argument. This article, however, uses the physics convention, with the time-dependent factor {{math|''e''<sup>−''iωt''</sup>}}.</ref>

To represent the polarization of the incident, reflected, or transmitted wave, the electric field adjacent to an interface can be resolved into two perpendicular components, known as the [[Polarization (waves)#s and p designations|''s'' and ''p'']] components, which are parallel to the ''surface'' and the ''plane'' of incidence respectively; in other words, the ''s'' and ''p'' components are respectively ''square'' and ''parallel'' to the plane of incidence.<ref group=Note>The ''s'' originally comes from the German {{lang|de|senkrecht}}, meaning "perpendicular" (to the plane of incidence). The alternative mnemonics in the text are perhaps more suitable for English speakers.</ref>

For each component of polarization, the incident, reflected, or transmitted electric field ({{math|'''E'''}} in Eq.{{nbsp}}({{EquationNote|5}})) has a certain direction and can be represented by its (complex) scalar component in that direction. The reflection or transmission coefficient can then be defined as a ''ratio'' of complex components at the same point, or at infinitesimally separated points on opposite sides of the interface. But, in order to fix the ''signs'' of the coefficients, we must choose positive senses for the "directions". For the ''s'' components, the obvious choice is to say that the positive directions of the incident, reflected, and transmitted fields are all the same (e.g., the {{mvar|z}} direction in Fig.{{nbsp}}11). For the ''p'' components, this article adopts the convention that the positive directions of the incident, reflected, and transmitted fields are inclined towards the same medium (that is, towards the same side of the interface, e.g. like the red arrows in Fig.{{nbsp}}11).<ref group=Note>In other words, for ''both'' polarizations, this article uses the convention that the positive directions of the incident, reflected, and transmitted fields are all the same for whichever field is normal to the plane of incidence; this is the {{math|'''E'''}} field for the ''s'' polarization, and the {{math|'''H'''}} field for the ''p'' polarization.</ref> But the reader should be warned that some books use a different convention for the ''p'' components, causing a different sign in the resulting formula for the reflection coefficient.<ref>One notable authority that uses the "different" convention (but without taking it very far) is ''The Feynman Lectures on Physics'', at volume&nbsp;I, eq.{{nbsp}}(33.8) (for {{mvar|B}}) and volume&nbsp;II, Figs.{{nbsp}}33-6 and 33-7.</ref>

For the ''s'' polarization, let the reflection and transmission coefficients be {{mvar|r<sub>s</sub>}} and {{mvar|t<sub>s</sub>}} respectively. For the ''p'' polarization, let the corresponding coefficients be {{mvar|r<sub>p</sub>}} and {{mvar|t<sub>p</sub>{{hsp}}}}. Then, for ''[[linearity|linear]], [[homogeneity (physics)|homogeneous]], isotropic, non-magnetic'' media, the coefficients are given by<ref>Born & Wolf, 1970, p.{{nbsp}}40, eqs.{{nbsp}}(20), (21), where the subscript ⊥ corresponds to ''s'', and ∥ to ''p''.</ref>
<!-- PLEASE DON'T CHANGE THESE EQUATIONS JUST BECAUSE YOUR FAVORITE TEXTBOOK USES A DIFFERENT SIGN CONVENTION, OR BECAUSE IT USES IMPEDANCES INSTEAD OF REFRACTIVE INDICES: -->
{{NumBlk|:|<math>r_s = \frac{n_1\cos\theta_\text{i} - n_2\cos\theta_\text{t}}{n_1\cos\theta_\text{i} + n_2\cos\theta_\text{t}},</math>|{{EquationRef|13}}}}
{{NumBlk|:|<math>t_s = \frac{2n_1\cos\theta_\text{i}}{n_1\cos\theta_\text{i} + n_2\cos\theta_\text{t}},</math>|{{EquationRef|14}}}}
{{NumBlk|:|<math>r_p = \frac{n_2\cos\theta_\text{i} - n_1\cos\theta_\text{t}}{n_2\cos\theta_\text{i} + n_1\cos\theta_\text{t}},</math>|{{EquationRef|15}}}}
{{NumBlk|:|<math>t_p = \frac{2n_1\cos\theta_\text{i}}{n_2\cos\theta_\text{i} + n_1\cos\theta_\text{t}}.</math>|{{EquationRef|16}}}}
<!-- PLEASE DON'T CHANGE THESE EQUATIONS JUST BECAUSE YOUR FAVORITE TEXTBOOK USES A DIFFERENT SIGN CONVENTION, OR BECAUSE IT USES IMPEDANCES INSTEAD OF REFRACTIVE INDICES! -->
(For a derivation of the above, see {{slink|Fresnel equations#Theory}}.)

Now we suppose that the transmitted wave is evanescent. With the correct sign (+), substituting ({{EquationNote|9}}) into ({{EquationNote|13}}) gives
<math display=block>
 r_s = \frac{n\cos\theta_\text{i} - i\sqrt{n^2\sin^2\theta_\text{i} - 1}}{n\cos\theta_\text{i} + i\sqrt{n^2\sin^2\theta_\text{i} - 1}},
</math><!-- PLEASE DON'T CHANGE THIS EQUATION JUST BECAUSE YOUR FAVORITE TEXTBOOK USES THE RECIPROCAL REFRACTIVE INDEX. -->
where
<math display=block>
 n = n_1/n_2;
</math>
that is, {{mvar|n}} is the index of the "internal" medium relative to the "external" one, or the index of the internal medium if the external one is vacuum.<ref group=Note>This nomenclature follows Jenkins & White, 1976, pp.{{nbsp}}526–529. Some authors, however, use the ''reciprocal'' refractive index and therefore obtain different forms for our Eqs.{{nbsp}}({{EquationNote|17}}) and ({{EquationNote|18}}). Examples include Born & Wolf {{bracket|1970, p.{{nbsp}}49, eqs.{{nbsp}}(60)}} and Stratton {{bracket|1941, p.{{nbsp}}499, eqs.{{nbsp}}(43)}}. Furthermore, Born & Wolf define {{math|''δ''<sub>⊥</sub>}} and {{math|''δ''<sub>∥</sub>}} as arguments rather than phase shifts, causing a change of sign.</ref> So the magnitude of {{mvar|r<sub>s</sub>}} is 1, and the ''[[argument (complex analysis)|argument]]'' of {{mvar|r<sub>s</sub>}} is
<math display=block>
 -2\arctan\frac{\sqrt{n^2\sin^2\theta_\text{i} - 1}}{n\cos\theta_\text{i}},
</math><!-- PLEASE DON'T CHANGE THIS EQUATION JUST BECAUSE YOUR FAVORITE TEXTBOOK USES THE RECIPROCAL REFRACTIVE INDEX. -->
which gives a phase ''advance'' of<ref name=jw529>Cf. Jenkins & White, 1976, p.{{nbsp}}529.</ref>
{{NumBlk|:|<math>\delta_s = 2\arctan\frac{\sqrt{n^2\sin^2\theta_\text{i} - 1}}{n\cos\theta_\text{i}}.</math>|{{EquationRef|17}}}}<!-- PLEASE DON'T CHANGE THIS EQUATION JUST BECAUSE YOUR FAVORITE TEXTBOOK USES THE RECIPROCAL REFRACTIVE INDEX OR DEFINES \delta_s AS THE ARGUMENT RATHER THAN THE PHASE SHIFT. -->

Making the same substitution in ({{EquationNote|14}}), we find that {{mvar|t<sub>s</sub>}} has the same denominator as {{mvar|r<sub>s</sub>}} with a positive real numerator (instead of a complex conjugate numerator) and therefore has ''half'' the argument of {{math|''r<sub>s</sub>''}}, so that ''the phase advance of the evanescent wave is half that of the reflected wave''.

With the same choice of sign,<ref group=Note>It is merely fortuitous that the principal square root turns out to be the correct one in the present situation, and only because we use the time-dependent factor {{math|''e''<sup>−''iωt''</sup>}}. If we instead used the electrical engineers' time-dependent factor {{math|''e<sup>jωt</sup>''}}, choosing the principal square root would yield the same argument for the reflection coefficient, but this would be interpreted as the ''opposite'' phase shift, which would be wrong. But if we choose the square root so that the transmitted field is evanescent, we get the right phase shift with either time-dependent factor.</ref> substituting ({{EquationNote|9}}) into ({{EquationNote|15}}) gives
<math display=block>
 r_p = \frac{\cos\theta_\text{i} - in\sqrt{n^2\sin^2\theta_\text{i} - 1}}{\cos\theta_\text{i} + in\sqrt{n^2\sin^2\theta_\text{i} - 1}},
</math><!-- PLEASE DON'T CHANGE THIS EQUATION JUST BECAUSE YOUR FAVORITE TEXTBOOK USES THE RECIPROCAL REFRACTIVE INDEX. -->
whose magnitude is 1, and whose argument is
<math display=block>
 -2\arctan\frac{n\sqrt{n^2\sin^2\theta_\text{i} - 1}}{\cos\theta_\text{i}},
</math><!-- PLEASE DON'T CHANGE THIS EQUATION JUST BECAUSE YOUR FAVORITE TEXTBOOK USES THE RECIPROCAL REFRACTIVE INDEX. -->
which gives a phase ''advance'' of<ref name=jw529/>
{{NumBlk|:|<math>\delta_p = 2\arctan\frac{n\sqrt{n^2\sin^2\theta_\text{i} - 1}}{\cos\theta_\text{i}}.</math>|{{EquationRef|18}}}}<!-- PLEASE DON'T CHANGE THIS EQUATION JUST BECAUSE YOUR FAVORITE TEXTBOOK USES THE RECIPROCAL REFRACTIVE INDEX OR DEFINES \delta_p AS THE ARGUMENT RATHER THAN THE PHASE SHIFT. -->

Making the same substitution in ({{EquationNote|16}}), we again find that the phase advance of the evanescent wave is ''half'' that of the reflected wave.

Equations ({{EquationNote|17}}) and ({{EquationNote|18}}) apply when {{math|''θ''<sub>c</sub> ≤ ''θ''<sub>i</sub> < 90°}}, where ''θ''<sub>i</sub> is the angle of incidence, and ''θ''<sub>c</sub> is the critical angle {{math|arcsin{{tsp}}(1/''n'')}}. These equations show that
* each phase advance is zero at the critical angle (for which the numerator is zero);
* each phase advance approaches 180° as {{math|''θ''<sub>i</sub> → 90°}}; and
* {{math|''δ<sub>p</sub> > δ<sub>s</sub>''}} at intermediate values of ''θ''<sub>i</sub> (because the factor {{math|''n''}} is in the numerator of ({{EquationNote|18}}) and the denominator of ({{EquationNote|17}})).<ref>"The phase of the polarization in which the ''magnetic'' field is parallel to the interface is advanced with respect to that of the other polarization."{{snd}} Fitzpatrick, 2013, p.{{nbsp}}140; Fitzpatrick, 2013a; emphasis added.</ref>

For {{math|''θ''<sub>i</sub> ≤ ''θ''<sub>c</sub>}}, the reflection coefficients are given by equations ({{EquationNote|13}}) and ({{EquationNote|15}}) and are ''real'', so that the phase shift is either 0° (if the coefficient is positive) or 180° (if the coefficient is negative).

In ({{EquationNote|13}}), if we put <math>n_2 = n_1\sin\theta_\text{i}/\sin\theta_\text{t}</math> (Snell's law) and multiply the numerator and denominator by {{math|{{sfrac|1|''n''<sub>1</sub>}}{{tsp}}sin{{tsp}}''θ''<sub>t</sub>}}, we obtain<ref>Fresnel, 1866, pp.{{nbsp}}773, 789n.</ref><ref>Born & Wolf, 1970, p.{{nbsp}}40, eqs.{{nbsp}}(21a); Hecht, 2017, p.{{nbsp}}125, eq.{{nbsp}}(4.42); Jenkins & White, 1976, p.{{nbsp}}524, eqs.{{nbsp}}(25a).</ref>
{{NumBlk|:|<math>r_s = -\frac{\sin(\theta_\text{i} - \theta_\text{t})}{\sin(\theta_\text{i} + \theta_\text{t})},</math>|{{EquationRef|19}}}}
which is positive for all angles of incidence with a transmitted ray (since {{math|''θ''<sub>t</sub> > ''θ''<sub>i</sub>}}), giving a phase shift {{mvar|δ<sub>s</sub>}} of zero.

If we do likewise with ({{EquationNote|15}}), the result is easily shown to be equivalent to<ref>Fresnel, 1866, pp.{{nbsp}}757,{{nbsp}}789n.</ref><ref>Born & Wolf, 1970, p.{{nbsp}}40, eqs.{{nbsp}}(21a); Hecht, 2017, p.{{nbsp}}125, eq.{{nbsp}}(4.43); Jenkins & White, 1976, p.{{nbsp}}524, eqs.{{nbsp}}(25a).</ref>
{{NumBlk|:|<math>r_p = \frac{\tan(\theta_\text{i} - \theta_\text{t})}{\tan(\theta_\text{i} + \theta_\text{t})},</math>|{{EquationRef|20}}}}<!-- PLEASE DON'T CHANGE THE SIGN OF THIS RESULT JUST BECAUSE YOUR FAVORITE TEXTBOOK USES A DIFFERENT SIGN CONVENTION FROM THAT USED IN THIS ARTICLE AND ITS REFERENCES! -->
which is negative for small angles (that is, near normal incidence), but changes sign at ''[[Brewster's angle]]'', where ''θ''<sub>i</sub> and ''θ''<sub>t</sub> are complementary. Thus the phase shift {{mvar|δ<sub>p</sub>}} is 180° for small ''θ''<sub>i</sub> but switches to 0° at Brewster's angle. Combining the complementarity with Snell's law yields {{math|''θ''<sub>i</sub> {{=}} arctan{{tsp}}(1/''n'')}} as Brewster's angle for dense-to-rare incidence.<ref group=Note>The more familiar formula {{math|arctan{{tsp}}''n''}} is for rare-to-dense incidence. In both cases, {{mvar|n}} is the refractive index of the denser medium relative to the rarer medium.</ref>

(Equations ({{EquationNote|19}}) and ({{EquationNote|20}}) are known as ''Fresnel's sine law'' and ''Fresnel's tangent law''.<ref>Whittaker, 1910, p.{{nbsp}}134; Darrigol, 2012, p.{{nbsp}}213.</ref> Both reduce to 0/0 at normal incidence, but yield the correct results in the [[limit (mathematics)|limit]] as {{math|''θ''<sub>i</sub> → 0}}. That they have opposite signs as we approach normal incidence is an obvious disadvantage of the sign convention used in this article; the corresponding advantage is that they have the same signs at grazing incidence.)

[[File:Phase advance at internal reflection.svg|thumb|'''Fig. 13''': Phase advance at "internal" reflections for refractive indices of 1.55, 1.5, and 1.45 ("internal" relative to "external"). Beyond the critical angle, the ''p''&nbsp;(red) and ''s''&nbsp;(blue) polarizations undergo unequal phase shifts on ''total'' internal reflection; the macroscopically observable difference between these shifts is plotted in black.]]

That completes the information needed to plot {{mvar|δ<sub>s</sub>}} and {{mvar|δ<sub>p</sub>}} for all angles of incidence. This is done in Fig.{{nbsp}}13,<ref name=jw529/> with {{mvar|δ<sub>p</sub>}} in red and {{mvar|δ<sub>s</sub>}} in blue, for three refractive indices. On the angle-of-incidence scale (horizontal axis), Brewster's angle is where {{mvar|δ<sub>p</sub>}} (red) falls from 180° to 0°, and the critical angle is where both {{mvar|δ<sub>p</sub>}} and {{mvar|δ<sub>s</sub>}} (red and blue) start to rise again. To the left of the critical angle is the region of ''partial'' reflection, where both reflection coefficients are real (phase 0° or 180°) with magnitudes less than&nbsp;1. To the right of the critical angle is the region of ''total'' reflection, where both reflection coefficients are complex with magnitudes equal to&nbsp;1. In that region, the black curves show the phase advance of the ''p''&nbsp;component relative to the ''s''&nbsp;component:<ref>Stratton, 1941, p.{{nbsp}}500, eq.{{nbsp}}(44). The corresponding expression in Born & Wolf (1970, p.{{nbsp}}50) is the other way around because the terms represent arguments rather than phase shifts.</ref>
<math display=block>
 \delta = \delta_p - \delta_s.
</math>
It can be seen that a refractive index of 1.45 is not enough to give a 45° phase difference, whereas a refractive index of 1.5 is enough (by a slim margin) to give a 45° phase difference at two angles of incidence: about 50.2° and 53.3°.

This 45° relative shift is employed in Fresnel's invention, now known as the [[Fresnel rhomb]], in which the angles of incidence are chosen such that the two internal reflections cause a total relative phase shift of 90° between the two polarizations of an incident wave. This device performs the same function as a birefringent [[quarter-wave plate]], but is more achromatic (that is, the phase shift of the rhomb is less sensitive to [[wavelength]]). Either device may be used, for instance, to transform [[linear polarization]] to [[circular polarization]] (which Fresnel also discovered) and conversely.

{{further|Fresnel rhomb}}

In Fig.{{nbsp}}13, {{mvar|δ}} is computed by a final subtraction; but there are other ways of expressing it. Fresnel himself, in 1823,<ref>Buchwald, 1989, pp.{{nbsp}}394, 453; Fresnel, 1866, pp.{{nbsp}}759, 786–787, 790.</ref> gave a formula for {{math|cos{{tsp}}''δ''}}. Born and Wolf (1970, p.{{nbsp}}50) derive an expression for {{math|tan{{hsp}}(''δ''/2)}} and find its maximum analytically.

For TIR of a beam with finite width, the variation in the phase shift with the angle of incidence gives rise to the ''[[Goos–Hänchen effect]]'', which is a lateral shift of the reflected beam within the plane of incidence.{{r|coldatoms}}{{r|berman-2012}} This effect applies to linear polarization in the ''s'' or ''p'' direction. The ''[[Imbert–Fedorov effect]]'' is an analogous effect for circular or [[elliptical polarization]] and produces a shift perpendicular to the plane of incidence.{{r|bliokh-aiello-2013}}