Editing Total internal reflection (section)

== Evanescent wave ==
{{further|Evanescent wave#Total internal reflection of light}}

Mathematically, waves are described in terms of time-varying [[field (physics)|fields]], a "field" being a function of location in space. A propagating wave requires an "effort" field and a "flow" field, the latter being a [[vector field|vector]] (if we are working in two or three dimensions). The product of effort and flow is related to [[power (physics)|power]] (see ''[[System equivalence]]''). For example, for sound waves in a [[viscosity|non-viscous]] fluid, we might take the effort field as the pressure (a&nbsp;scalar), and the flow field as the fluid velocity (a&nbsp;vector). The product of these two is [[intensity (physics)|intensity]] (power per unit area).{{r|pjs}}<ref group=Note>Power "per unit area" is appropriate for fields in three dimensions. In two dimensions, we might want the product of effort and flow to be power per unit ''length''. In one dimension, or in a [[lumped element model|lumped-element model]], we might want it to be simply power.</ref> For electromagnetic waves, we shall take the effort field as the [[electric&nbsp;field]]{{hsp}} {{math|'''E'''{{hsp}},}} and the flow field as the [[magnetic field#The H-field|magnetizing field]]{{hsp}} {{math|'''H'''}}. Both of these are vectors, and their [[cross product|vector&nbsp;product]] is again the intensity (see ''[[Poynting&nbsp;vector]]'').<ref>Stratton, 1941, pp.{{nnbsp}}131–7.</ref>

When a wave in (say) medium&nbsp;1 is reflected off the interface between medium&nbsp;1 and medium&nbsp;2, the flow field in medium&nbsp;1 is the vector sum of the flow fields due to the incident and reflected waves.<ref group=Note>We assume that the equations describing the fields are [[linearity#Physics|linear]].</ref>{{tsp}} If the reflection is oblique, the incident and reflected fields are not in opposite directions and therefore cannot cancel out at the interface; even if the reflection is total, either the normal component or the tangential component of the combined field (as a function of location and time) must be non-zero adjacent to the interface. Furthermore, the physical laws governing the fields will generally imply that one of the two components is ''[[continuous function|continuous]]'' across the interface (that is, it does not suddenly change as we cross the interface); for example, for electromagnetic waves, one of the [[interface conditions for electromagnetic fields|interface conditions]] is that the tangential component of {{math|'''H'''}} is continuous if there is no surface current.<ref>Stratton, 1941, p.{{hsp}}37.</ref> Hence, even if the reflection is total, there must be some penetration of the flow field into medium&nbsp;2; and this, in combination with the laws relating the effort and flow fields, implies that there will also be some penetration of the effort field. The same continuity condition implies that the variation ("waviness") of the field in medium&nbsp;2 will be synchronized with that of the incident and reflected waves in medium&nbsp;1.

[[File:Evanescent wave cropped.jpg|thumb|'''Fig.{{nnbsp}}9''':{{big|&nbsp;}}Depiction of an incident sinusoidal plane wave (bottom) and the associated evanescent wave (top), under conditions of total internal reflection. The reflected wave is not shown.]]

But, if the reflection is total, the spatial penetration of the fields into medium&nbsp;2 must be limited somehow, or else the total extent and hence the total energy of those fields would continue to increase, draining power from medium&nbsp;1. Total reflection of a continuing wavetrain permits some energy to be stored in medium&nbsp;2, but does not permit a ''continuing'' transfer of power from medium&nbsp;1 to medium&nbsp;2.

Thus, using mostly qualitative reasoning, we can conclude that total internal reflection must be accompanied by a wavelike field in the "external" medium, traveling along the interface in synchronism with the incident and reflected waves, but with some sort of limited spatial penetration into the "external" medium; such a field may be called an ''[[evanescent&nbsp;wave]]''.

Fig.{{nnbsp}}9 shows the basic idea. The incident wave is assumed to be [[plane wave|plane]] and [[sine wave|sinusoidal]]. The reflected wave, for simplicity, is not shown. The evanescent wave travels to the right in lock-step with the incident and reflected waves, but its amplitude falls off with increasing distance from the interface.

(Two features of the evanescent wave in Fig.{{nnbsp}}9 are to be explained later: first, that the evanescent wave crests are perpendicular to the interface; and second, that the evanescent wave is slightly ahead of the incident wave.)

=== {{anchor|Frustrated total internal reflection}}{{anchor|Frustrated TIR}}Frustrated total internal reflection (FTIR) ===

If the internal reflection is to be total, there must be no diversion of the evanescent wave. Suppose, for example, that electromagnetic waves incident from glass (with a higher refractive index) to air (with a lower refractive index) at a certain angle of incidence are subject to TIR. And suppose that we have a third medium (often identical to the first) whose refractive index is sufficiently high that, if the third medium were to replace the second, we would get a standard transmitted wavetrain for the same angle of incidence. Then, if the third medium is brought within a distance of a few wavelengths from the surface of the first medium, where the evanescent wave has significant amplitude in the second medium, then the evanescent wave is effectively refracted into the third medium, giving non-zero transmission into the third medium, and therefore less than total reflection back into the first medium.{{r|harvard-ftir}} As the amplitude of the evanescent wave decays across the air gap, the transmitted waves are [[attenuation|attenuated]], so that there is less transmission, and therefore more reflection, than there would be with no gap; but as long as there is ''some'' transmission, the reflection is less than total. This phenomenon is called ''frustrated total internal reflection'' (where "frustrated" negates "total"), abbreviated "frustrated TIR" or "FTIR".

[[File:Drinking glass fingerprint FTIR.jpg|left|thumb|alt=A hand holding a glass of water with fingerprints visible from the inside.|'''Fig.{{nnbsp}}10''':{{big|&nbsp;}}Disembodied fingerprints visible from the inside of a glass of water, due to frustrated total internal reflection. The observed fingerprints are surrounded by white areas where total internal reflection occurs.]]

Frustrated TIR can be observed by looking into the top of a glass of water held in one's hand (Fig.{{nnbsp}}10). If&nbsp;the glass is held loosely, contact may not be sufficiently close and widespread to produce a noticeable effect. But if it is held more tightly, the ridges of one's [[fingerprint]]s interact strongly with the evanescent waves, allowing the ridges to be seen through the otherwise totally reflecting glass-air surface.{{r|ehrlich-1997}}

The same effect can be demonstrated with microwaves, using [[paraffin&nbsp;wax]] as the "internal" medium (where the incident and reflected waves exist). In this case the permitted gap width might be (e.g.) 1{{nnbsp}}cm or several cm, which is easily observable and adjustable.{{r|feynman-1963|bowley-2009}}

The term ''frustrated TIR'' also applies to the case in which the evanescent wave is [[scattering|scattered]] by an object sufficiently close to the reflecting interface. This effect, together with the strong dependence of the amount of scattered light on the distance from the interface, is exploited in ''[[total internal reflection microscopy]]''.{{r|ambrose-1956}}

The mechanism of FTIR is called ''[[evanescent-wave coupling]]'', and is a good analog to visualize [[quantum tunnelling|quantum tunneling]].<ref>{{Cite journal |last1=Van Rosum |first1=Aernout |last2=Van Den Berg |first2=Ed |date=May 2021 |title=Using frustrated internal reflection as an analog to quantum tunneling |journal=Journal of Physics: Conference Series |volume=1929 |issue=1 |page=012050 |doi=10.1088/1742-6596/1929/1/012050 |bibcode=2021JPhCS1929a2050V |s2cid=235591328 |doi-access=free }}</ref> Due to the wave nature of matter, an electron has a non-zero probability of "tunneling" through a barrier, even if [[classical mechanics]] would say that its energy is insufficient.{{r|harvard-ftir|ehrlich-1997}} Similarly, due to the wave nature of light, a [[photon]] has a non-zero probability of crossing a gap, even if [[geometrical optics|ray&nbsp;optics]] would say that its approach is too oblique.

Another reason why internal reflection may be less than total, even beyond the critical angle, is that the external medium may be "lossy" (less than perfectly transparent), in which case the external medium will absorb energy from the evanescent wave, so that the maintenance of the evanescent wave will draw power from the incident wave. The consequent less-than-total reflection is called ''attenuated total reflectance'' (ATR). This effect, and especially the frequency-dependence of the absorption, can be used to study the composition of an unknown external medium.{{r|thermo-fisher}}

{{clear}}

=== Derivation of evanescent wave ===

In a uniform plane sinusoidal electromagnetic wave, the electric field {{math|'''E'''}} has the form
{{NumBlk|:|<math>\mathbf{E_k} e^{i(\mathbf{k\cdot r} - \omega t)},</math>|{{EquationRef|5}}}}
where {{math|'''E<sub>k</sub>'''}} is the (constant) [[complex number|complex]] amplitude vector, {{math|''i''}} is the [[imaginary unit]], {{math|'''k'''}} is the [[wave vector]] (whose magnitude {{math|''k''}} is the angular [[wavenumber]]), {{math|'''r'''}} is the [[position (vector)|position vector]], ''ω'' is the [[angular frequency]], {{math|''t''}} is time, and it is understood that the ''real part'' of the expression is the physical field.<ref group="Note">The above form ({{EquationNote|5}}) is typically used by physicists. [[electrical engineering|Electrical engineers]] typically prefer the form <math>\mathbf{E_k} e^{j(\omega t - \mathbf{k \cdot r})};</math> that is, they not only use {{math|''j''}} instead of {{math|''i''}} for the imaginary unit, but also change the sign of the exponent, with the result that the whole expression is replaced by its [[complex conjugate]], leaving the real part unchanged. The electrical engineers' form and the formulae derived therefrom may be converted to the physicists' convention by substituting {{math|''−i''}} for {{math|''j''}} (Stratton, 1941, pp.{{nbsp}}vii–viii).</ref> The magnetizing field {{math|'''H'''}} has the same form with the same {{math|'''k'''}} and ''ω''. The value of the expression is unchanged if the position {{math|'''r'''}} varies in a direction normal to {{math|'''k'''}}; hence {{math|'''k'''}} ''is normal to the wavefronts''.

If ''ℓ'' is the component of {{math|'''r'''}} in the direction of {{math|'''k''',}} the field ({{EquationNote|5}}) can be written <math>\mathbf{E_k}e^{i(k\ell-\omega t)}.</math> If the [[argument (complex analysis)|argument]] of <math>e^{i(\cdots)}</math> is to be constant, ''ℓ'' must increase at the velocity <math>\omega/k,</math> known as the ''[[phase velocity]]''.<ref>Jenkins & White, 1976, p.{{nbsp}}228.</ref> This in turn is equal to <math>c/n,</math> where {{math|''c''}} is the phase velocity in the reference medium (taken as vacuum), and {{math|''n''}} is the local refractive index w.r.t. the reference medium. Solving for {{math|''k''}} gives <math>k = n\omega/c,</math> i.e.
{{NumBlk|:|<math>k = nk_0,</math>|{{EquationRef|6}}}}
where <math>k_0 = \omega/c</math> is the wavenumber in vacuum.<ref>Born & Wolf, 1970, pp.{{nbsp}}16–17, eqs.{{nbsp}}(20), (21).</ref><ref group=Note>We assume that there are no [[Doppler effect|Doppler shifts]], so that ''ω'' does not change at interfaces between media.</ref>

From ({{EquationNote|5}}), the electric field in the "external" medium has the form
{{NumBlk|:|<math>\mathbf{E}_\text{t} = \mathbf{E}_{\mathbf{k}\text{t}} e^{i(\mathbf{k_\text{t} \cdot r} - \omega t)},</math>|{{EquationRef|7}}}}
where {{math|'''k'''<sub>t</sub>}} is the wave vector for the transmitted wave (we assume isotropic media, but the transmitted wave is not ''yet'' assumed to be evanescent).

[[File:Wave vectors dense to rare n1 to n2.svg|thumb|'''Fig.{{nbsp}}11''': Incident, reflected, and transmitted wave vectors ({{math|'''k'''<sub>i</sub>, '''k'''<sub>r</sub>,}} and {{math|'''k'''<sub>t</sub>}}) for incidence from a medium with higher refractive index {{math|''n''<sub>1</sub>}} to a medium with lower refractive index {{math|''n''<sub>2</sub>}}. The red arrows are perpendicular to the wave vectors and therefore parallel to the respective wavefronts.]]

In Cartesian coordinates {{math|(''x'', ''y'', ''z'')}}, let the region {{math|''y'' < 0}} have refractive index {{math|''n''<sub>1</sub>,}} and let the region {{math|''y'' > 0}} have refractive index {{math|''n''<sub>2</sub>}}. Then the {{math|''xz''}} plane is the interface, and the {{math|''y''}} axis is normal to the interface (Fig.{{nbsp}}11). Let {{math|'''i'''}} and {{math|'''j'''}} be the unit vectors in the {{math|''x''}} and {{math|''y''}} directions respectively. Let the [[plane of incidence]] (containing the incident wave-normal and the normal to the interface) be the {{math|''xy''}} plane (the plane of the page), with the angle of incidence ''θ''<sub>i</sub> measured from {{math|'''j'''}} towards {{math|'''i'''}}. Let the angle of refraction, measured in the same sense, be ''θ''<sub>t</sub> ("t" for ''transmitted'', reserving "r" for ''reflected'').

From ({{EquationNote|6}}), the transmitted wave vector {{math|'''k'''<sub>t</sub>}} has magnitude {{math|''n''<sub>2</sub>''k''<sub>0</sub>}}. Hence, from the geometry,
<math display=block>
 \mathbf{k}_\text{t} = n_2 k_0 (\mathbf{i} \sin\theta_\text{t} + \mathbf{j} \cos\theta_\text{t}) = k_0 (\mathbf{i}\,n_1 \sin\theta_\text{i} + \mathbf{j}\,n_2 \cos\theta_\text{t}),
</math>
where the last step uses Snell's law. Taking the [[dot product]] with the position vector, we get
<math display=block>
 \mathbf{k}_\text{t} \cdot \mathbf{r} = k_0 (n_1 x \sin\theta_\text{i} + n_2 y \cos\theta_\text{t}),
</math>
so that Eq.{{nbsp}}({{EquationNote|7}}) becomes
{{NumBlk|:|<math>\mathbf{E}_\text{t} = \mathbf{E}_{\mathbf{k}\text{t}} \exp[i(n_1 k_0 x \sin\theta_\text{i} + n_2 k_0 y \cos\theta_\text{t} - \omega t)].</math>|{{EquationRef|8}}}}

In the case of TIR, the angle ''θ''<sub>t</sub> does not exist in the usual sense. But we can still interpret ({{EquationNote|8}}) for the transmitted (evanescent) wave by allowing {{math|cos{{tsp}}''θ''<sub>t</sub>}} to be ''complex''. This becomes necessary when we write {{math|cos{{tsp}}''θ''<sub>t</sub>}} in terms of {{math|sin{{tsp}}''θ''<sub>t</sub>,}} and thence in terms of {{math|sin{{tsp}}''θ''<sub>i</sub>}} using Snell's law:
<math display=block>
 \cos\theta_\text{t} = \sqrt{1 - \sin^2\theta_\text{t}} = \sqrt{1 - (n_1/n_2)^2 \sin^2\theta_\text{i}}.
</math>
For ''θ''<sub>i</sub> greater than the critical angle, the value under the square-root symbol is negative, so that<ref>Born & Wolf, 1970, p.{{nbsp}}47, eq.{{nbsp}}(54), where their {{mvar|n}} is our <math>n_2/n_1</math> (''not'' our <math>n_1/n_2</math>).</ref>
{{NumBlk|:|<math>\cos\theta_\text{t} = \pm i\,\sqrt{(n_1/n_2)^2 \sin^2\theta_\text{i} - 1}.</math>|{{EquationRef|9}}}}
To determine which sign is applicable, we substitute ({{EquationNote|9}}) into ({{EquationNote|8}}), obtaining
{{NumBlk|:|<math>\mathbf{E}_\text{t} = \mathbf{E}_{\mathbf{k}\text{t}} e^{\mp\sqrt{n_1^2\sin^2\theta_\text{i} - n_2^2}\, k_0 y} e^{i[(n_1 k_0 \sin\theta_\text{i})x - \omega t]},</math>|{{EquationRef|10}}}}
where the undetermined sign is the opposite of that in ({{EquationNote|9}}). For an ''evanescent'' transmitted wave{{snd}} that is, one whose amplitude decays as {{math|''y''}} increases{{snd}} the undetermined sign in ({{EquationNote|10}}) must be ''minus'', so the undetermined sign in ({{EquationNote|9}}) must be ''plus''.<ref group=Note>If we correctly convert this to the electrical engineering convention, we get {{math|''−j''{{px2}}{{radic|⋯}}}} on the right-hand side of ({{EquationNote|9}}), which is ''not'' the principal square root. So it is not valid to assume, ''a&nbsp;priori'', that what mathematicians call the "[[square root#Principal square root of a complex number|principal square root]]" is the physically applicable one.</ref>

With the correct sign, the result ({{EquationNote|10}}) can be abbreviated
{{NumBlk|:|<math>\mathbf{E}_\text{t} \propto e^{-\kappa y} e^{i(k_x x - \omega t)},</math>|{{EquationRef|11}}}}
where
{{NumBlk|:|<math>\begin{align}
  \kappa &= k_0 \sqrt{n_1^2 \sin^2\theta_\text{i} - n_2^2}, \\
  k_x    &= n_1 k_0 \sin\theta_\text{i},
\end{align}</math>|{{EquationRef|12}}}}
and {{math|''k''<sub>0</sub>}} is the wavenumber in vacuum, i.e.&nbsp;<math>\omega/c.</math>

So the evanescent wave is a plane sinewave traveling in the {{mvar|x}} direction, with an amplitude that decays exponentially in the {{mvar|y}} direction (Fig.{{nbsp}}9). It is evident that the energy stored in this wave likewise travels in the {{mvar|x}} direction and does not cross the interface. Hence the [[Poynting vector]] generally has a component in the {{mvar|x}} direction, but its {{mvar|y}} component averages to zero (although its instantaneous {{mvar|y}} component is not ''identically'' zero).<ref>Stratton, 1941, p.{{nbsp}}499; Born & Wolf, 1970, p.{{nbsp}}48.</ref>{{r|coldatoms}}

[[File:FITR penetration depth.svg|thumb|'''Fig.{{nbsp}}12''': Penetration depth of the evanescent wave (in&nbsp;wavelengths) vs.&nbsp;angle of incidence, for various values of the relative refractive index (internal w.r.t. external)]]

Eq.{{nbsp}}({{EquationNote|11}}) indicates that the amplitude of the evanescent wave falls off by a factor {{mvar|e}} as the coordinate {{mvar|y}} (measured from the interface) increases by the distance <math>d = 1/\kappa,</math> commonly called the "penetration depth" of the evanescent wave.<ref>Hecht, 2017, p.{{nbsp}}136.</ref> Taking reciprocals of the first equation of ({{EquationNote|12}}), we find that the penetration depth is{{r|coldatoms}}
<math display=block>
 d = \frac{\lambda_0}{2\pi \sqrt{n_1^2 \sin^2\theta_\text{i} - n_2^2}},
</math>
where ''λ''<sub>0</sub> is the wavelength in vacuum, i.e. <math>2\pi/k_0.</math><ref>Born & Wolf, 1970, p.{{nbsp}}16.</ref> Dividing the numerator and denominator by {{math|''n''<sub>2</sub>}} yields
<math display=block>
 d = \frac{\lambda_2}{2\pi \sqrt{(n_1/n_2)^2 \sin^2\theta_\text{i} - 1}},
</math>
where <math>\lambda_2 = \lambda_0/n_2</math> is the wavelength in the second (external) medium. Hence we can plot {{mvar|d}} in units of ''λ''<sub>2</sub> as a function of the angle of incidence for various values of <math>n_1/n_2</math> (Fig.{{nbsp}}12). As&nbsp;''θ''<sub>i</sub> decreases towards the critical angle, the denominator approaches zero, so that {{mvar|d}} increases without limit{{snd}} as is to be expected, because as soon as ''θ''<sub>i</sub> is ''less'' than critical, uniform plane waves are permitted in the external medium. As ''θ''<sub>i</sub> approaches 90° (grazing incidence), {{mvar|d}} approaches a minimum
<math display=block>
 d_\text{min} = \frac{\lambda_2}{2\pi \sqrt{(n_1/n_2)^2 - 1}}.
</math>
For incidence from water to air, or common glass to air, {{math|''d''<sub>min</sub>}} is not much different from ''λ''<sub>2</sub>/(2''π''). But {{mvar|d}} is larger at smaller angles of incidence (Fig.{{nbsp}}12), and the amplitude may still be significant at distances of several times {{mvar|d}}; for example, because {{math|e<sup>−4.6</sup>}} is just greater than 0.01, the evanescent wave amplitude within a distance {{math|4.6{{px2}}''d''}} of the interface is at least 1% of its value at the interface. Hence, speaking loosely, we tend to say that the evanescent wave amplitude is significant within "a few wavelengths" of the interface.