Editing Total internal reflection (section)

=== 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.