# Evanescence

Evanescence ( Latin: evanescere ' to disappear', 'to evaporate') describes the phenomenon that waves penetrate into a material in which they cannot propagate and decay exponentially below its surface. Evanescent waves occur, for example, in optics at totally reflective interfaces and in acoustics in pipes or other lines.

## general description

Evanescent fields behind the interface with total reflection. The directions of propagation of the waves are shown in yellow.

If a wave hits a medium in which it cannot propagate, its amplitude does not drop directly to zero after the boundary surface, but decays exponentially. This evanescent wave is called evanescent ; it can be described by a complex-valued wave vector .

In quantum mechanics, this means that particles can stay in a classically forbidden area, since the probabilities of being in it (as a probability interpretation of wave mechanics) drop exponentially, but are still present. This enables the tunnel effect , for example .

Evanescent waves occur e.g. B. in or behind surfaces on which waves are reflected. Since no energy is transported away, this also applies to complete reflection and total reflection at an interface between two media.

## Derivation in the wave image

At the interface behind which the evanescent field occurs, the continuity conditions for the tangential components of the E-field apply , and from these it follows:

${\ displaystyle \ left ({\ vec {k}} _ {e} - {\ vec {k}} _ {r} \ right) \ cdot {\ vec {r}} = 0}$
${\ displaystyle \ left ({\ vec {k}} _ {e} - {\ vec {k}} _ {t} \ right) \ cdot {\ vec {r}} = 0}$

The index e designates the incident, the index r the reflected and the index t the transmitted vector. The refractive indices of the media on both sides of the interface are also designated below with the indices of the associated wave vectors. The interface is located in the plane and described by . So a 2D problem is being dealt with here, that is, the wave vector of the incident wave lies in the plane. ${\ displaystyle k}$${\ displaystyle x / z}$${\ displaystyle y = 0}$${\ displaystyle x / y}$

If one calculates the scalar product in the above continuity conditions and substitutes for the -component of the -vector , the result is that the components tangential to the interface (in the -direction) are the same for all three wave vectors. ${\ displaystyle y}$${\ displaystyle r}$${\ displaystyle 0}$${\ displaystyle x}$

${\ displaystyle k_ {e, x} = k_ {r, x} = k_ {t, x}}$

The component of the vector can also be described with the angle of incidence , which is measured from the perpendicular to the interface. The magnitude of the vector is described by the dispersion relation . ${\ displaystyle x}$${\ displaystyle k}$ ${\ displaystyle \ varphi}$

${\ displaystyle k_ {e, x} = \ left | {\ vec {k}} _ {e} \ right | \ sin \ varphi = {\ frac {n_ {e} \ omega} {c}} \ sin \ varphi}$

The same applies to the vector of the transmitted wave: ${\ displaystyle k}$

${\ displaystyle \ left | {\ vec {k}} _ {t} \ right | = {\ frac {n_ {t} \ omega} {c}} = {\ sqrt {k_ {t, x} ^ {2 } + k_ {t, y} ^ {2}}}}$

If one converts this equation and substitutes for for the expression derived above , one obtains ${\ displaystyle k_ {t, y} ^ {2}}$${\ displaystyle k_ {t, x}}$${\ displaystyle k_ {e, x}}$

${\ displaystyle k_ {t, y} ^ {2} = \ left ({\ frac {n_ {t} \ omega} {c}} \ right) ^ {2} - \ overbrace {\ left ({\ frac { n_ {e} \ omega} {c}} \ right) ^ {2} \ sin ^ {2} \ varphi} ^ {k_ {t, x} ^ {2} = k_ {e, x} ^ {2} } = \ underbrace {\ left ({\ frac {n_ {t} \ omega} {c}} \ right) ^ {2}} _ {k_ {t} ^ {2}} \ cdot \ underbrace {\ left ( 1 - {\ frac {n_ {e} ^ {2}} {n_ {t} ^ {2}}} \ sin ^ {2} \ varphi \ right)} _ {<0}}$
The 1 / e penetration depth of the field as a function of the angle of incidence, given in units of the wavelength.

The first factor in this product is positive. The second factor, however, becomes negative because the angle of incidence is greater than the critical angle of total reflection . This becomes imaginary. ${\ displaystyle \ varphi}$${\ displaystyle k_ {t, y}}$

${\ displaystyle k_ {t, y} = \ pm \ imath k_ {t} \ underbrace {\ sqrt {{\ frac {n_ {e} ^ {2}} {n_ {t} ^ {2}}} \ sin ^ {2} \ varphi -1}} _ {: = \ beta} = \ pm \ imath \ beta k_ {t}}$

Now we apply a plane wave with amplitude at the interface for the transmitted beam : ${\ displaystyle A}$

${\ displaystyle {\ vec {E}} _ {t} = {\ vec {A}} e ^ {\ mathrm {i} \ left ({\ vec {k}} \ cdot {\ vec {r}} - \ omega t \ right)} = {\ vec {A}} e ^ {- \ beta k_ {t} y} e ^ {\ mathrm {i} k_ {t, x} x- \ mathrm {i} \ omega t}}$

The term with in the exponent describes the exponential decrease in amplitude the further the evanescent wave advances in the direction. It is also possible to explicitly calculate how much the amplitude of the evanescent wave has already dropped at a certain distance behind the interface. The penetration depth after which the amplitude of the wave has dropped to 1 / e is useful for orientation . ${\ displaystyle \ beta}$${\ displaystyle y}$${\ displaystyle \ beta}$

${\ displaystyle y _ {{\ frac {1} {e}}, {\ text {E}}} = {\ frac {1} {k_ {t} \ beta}}}$

Note that this is a decrease in amplitude, not intensity, i.e. the square of the magnitude of the amplitude. The 1 / e penetration depth of the intensity results from the square of the magnitude of the wave function:

${\ displaystyle y _ {{\ frac {1} {e}}, {\ text {I}}} = {\ frac {1} {2k_ {t} \ beta}}}$

## Quantum mechanical derivation

Evanescence can be treated quantum mechanically. The interface at which total reflection occurs is viewed as a one-dimensional potential level at which a particle is reflected.

${\ displaystyle V (x) = {\ begin {cases} 0 & {\ mbox {falls}} x <0 \\ V_ {0} & {\ mbox {falls}} x \ geq 0 \ end {cases}}}$

To get total reflection at all, the energy of the particle with mass must be smaller than that of the potential ( ). A suitable approach for the wave function of the particle is thus: ${\ displaystyle E}$${\ displaystyle m}$${\ displaystyle 0

${\ displaystyle \ Psi (x) = {\ begin {cases} e ^ {\ mathrm {i} k_ {e} x} + r \ cdot e ^ {- \ mathrm {i} k_ {e} x} & { \ mbox {falls}} x <0 \\ t \ cdot e ^ {\ mathrm {i} k_ {t} x} & {\ mbox {falls}} x \ geq 0 \ end {cases}}}$

The incoming wave has already been normalized to 1 here. and are calculated using this approach from the Schrödinger equation . Because the potential is greater than the energy it becomes imaginary and the new quantity can be introduced. ${\ displaystyle - \ infty}$${\ displaystyle k_ {e}}$${\ displaystyle k_ {t}}$${\ displaystyle k_ {t}}$${\ displaystyle \ kappa _ {t}}$

${\ displaystyle k = {\ begin {cases} k_ {e} = {\ frac {1} {\ hbar}} {\ sqrt {2mE}} & {\ mbox {if}} x <0 \\ k_ {t } = {\ frac {1} {\ hbar}} {\ sqrt {2m (EV)}} = {\ frac {\ mathrm {i}} {\ hbar}} {\ sqrt {2m | EV |}} = : \ mathrm {i} \ kappa _ {t} {\ mbox {mit}} \ kappa _ {t} \ in \ mathbb {R} & {\ mbox {falls}} x \ geq 0 \ end {cases}} }$

This makes the exponent of the exponential function negative real and the wave function describes an exponential decrease. ${\ displaystyle x \ geq 0}$

At the potential step , both the wave functions themselves and their derivatives must be continuous. Substituting in one obtains: ${\ displaystyle \ Psi}$${\ displaystyle \ Psi '}$${\ displaystyle x = 0}$

${\ displaystyle 1 + r = t}$
${\ displaystyle \ mathrm {i} k_ {e} - \ mathrm {i} rk_ {e} = - t \ kappa _ {t}}$

By equating, the reflection and transmission coefficients and the probability wave can be determined. ${\ displaystyle r}$${\ displaystyle t}$

${\ displaystyle r = {\ frac {\ mathrm {i} k_ {e} + \ kappa _ {t}} {\ mathrm {i} k_ {e} - \ kappa _ {t}}}}$
${\ displaystyle t = 1 + r = {\ frac {2 \ mathrm {i} k_ {e}} {\ mathrm {i} k_ {e} - \ kappa _ {t}}}}$

The probability amplitude of encountering the particle at is not zero because of . The probability amplitude of the reflection is also not exactly 1. However, the squares of absolute values ​​are different , which are calculated by multiplying with the complex conjugate . ${\ displaystyle x \ geq 0}$${\ displaystyle t \ neq 0}$${\ displaystyle \ mathbb {C}}$

${\ displaystyle R = | r | ^ {2} = 1}$
${\ displaystyle T = | t | ^ {2} = {\ frac {4k_ {e} ^ {2}} {\ kappa _ {t} ^ {2} + k_ {e} ^ {2}}}}$

So there is 100% reflection and yet there is a probability that the particle can penetrate the barrier. From the law of conservation of energy it becomes clear to me that the evanescent wave does not transport any energy. Analogous to the 1 / e penetration depth in optics, an x-dependent penetration probability can be calculated in quantum mechanics from the square of the magnitude of the wave function in the area of ​​the barrier. ${\ displaystyle T}$${\ displaystyle R = 1}$${\ displaystyle W}$

${\ displaystyle W (x \ geq 0) = T \ cdot e ^ {- 2 \ kappa _ {t} x} = {\ frac {4k_ {e} ^ {2}} {2mV_ {0} / \ hbar ^ {2}}} \ cdot e ^ {- 2 \ kappa _ {t} x}}$

The 1 / e penetration depth is thus . In the optical derivation, the properties of the two media flowed into the comparatively complicated expression for . In this quantum mechanical derivation, the problem was simplified in that the potential in the area of ​​the incoming wave was chosen as zero, which corresponds to a refractive index of . In addition, perpendicular incidence was assumed, so that the angle dependence, which is reflected in the derivation of the wave image in the sine term , is not taken into account. ${\ displaystyle x _ {\ frac {1} {e}} = {\ frac {1} {2 \ kappa _ {t}}}}$${\ displaystyle \ beta}$${\ displaystyle n_ {e} = 1}$${\ displaystyle \ beta}$

## Evidence and practical relevance

### Disturbed total reflection

Total reflection in the prism prevented due to decoupling of evanescent fields

If you bring two glass prisms very close together (see illustration), you can measure light where none should be, namely behind the second prism (transmitted light beam): due to the evanescent field behind the first prism, light can be transmitted if the second prism is in the evanescent field is immersed. The intensity decreases exponentially with the distance between the prisms. This effect is called prevented or disturbed total reflection ( English frustrated internal total reflection, FITR ), since actually all light should be reflected upwards. This resembles the finitely high potential well in quantum mechanics , where the wave function decays exponentially in the forbidden area. Therefore this effect is also known as the optical tunnel effect. In the case of special beam splitters , the described effect is used, whereby the ratio of the intensities between the transmitted and reflected beam can be set very precisely through the spacing of the prisms.

The effect of disturbed total reflection is used in ATR spectroscopy to make impurities and defects in surfaces and thin layers visible ( see also: Evanescent Wave Scattering ). Near-field optical microscopy and internal total reflection fluorescence microscopy (TIRF) also use evanescent waves.

### Secure the reflection

In light waveguides are evanescent waves in the low-index cladding ( English cladding ) of the fiber. The cladding prevents radiation from escaping from the fiber core by preventing dirt or water from approaching the evanescent field around the core and thus disrupting the total reflection.

The perforated sheet metal door of microwave ovens must be protected by an additional pane, as the microwaves ( wavelengths in the centimeter range) inside the oven cannot pass through the door, but generate evanescent fields immediately behind the holes. B. a finger would lead to the decoupling of microwaves.