Evanescent wave

Evanescent waves (Latin evanescere disappear ',' evaporate ') are in optics and acoustics, such as pipes or other conduits, known.

General Description

Does a wave on a medium in which they can not spread, so its amplitude does not fall directly behind the interface to zero. The amplitude sounds there instead exponentially and therefore at the interface does not change, but rather a steady course. The wave vector is complex-valued in the classically forbidden region. Such a wave is called evanescently. This effect can only be explained by wave-mechanics.

In quantum mechanics, this leads to the fact that particles can reside in a classically forbidden region, since the probabilities decrease exponentially in him ( as a probability interpretation of wave mechanics ), but still present. This allows, for example, the tunnel effect.

Evanescent wave occurs, for example in or behind surfaces on which waves are reflected. Since no energy is transported away, this also applies with full reflection and total reflection at an interface between two media.

Derivation of the wave pattern with

At the interface, behind which the evanescent field occurs, the continuity conditions apply to the tangential components of the E field:

The index e denotes the incident, the index r is the reflected and the index t the transmitted vector. Similarly, the refractive indices of the mediums are indicated on both sides of the interface with the indices of the associated wave vectors as follows. The interface is located and described by the plane. It is here that is a 2D problem treated.

If we calculate the scalar product in the above continuity conditions and sets for the component of the vector a, it follows that the components tangential to the interface ( in direction) are the same for all three wave vectors.

The component of the vector can also be described with the angle of incidence is measured from the perpendicular to the boundary surface. The magnitude of the vector is described by the dispersion relation.

The same is true for the vector of the transmitted wave:

If, to this equation for and sets for the above derived expression for, one obtains

The first factor in this product is positive. However, the second factor is negative, because the angle of incidence is greater than the critical angle of total reflection. This is imaginary.

Now it is to the transmitted beam of a plane wave with the amplitude at the interface:

With the term in the exponent describing the exponential decay of the amplitude of the further progresses the evanescent wave in direction. From can also be explicitly calculate how much the amplitude of the evanescent wave has already dropped at a certain distance behind the interface. For guidance here lends itself to the penetration depth, after which the amplitude of the wave decreases to 1 / e.

It should be noted that this is a drop in the amplitude, not to the intensity, the amount of the square of the amplitude. The 1/e-Eindringtiefe of intensity results from the absolute square of the wave function:

Quantum mechanical derivation

Evanescence can be treated quantum mechanically. Here we look at the interface where the total internal reflection occurs as a one-dimensional potential level at which a particle is reflected.

To obtain total reflection at all, the energy of the particle must be ground is less than the potential (). A suitable approach for the wave function of the particle is thus:

The from incoming wave has already been normalized to 1. and calculated with this approach from the Schrödinger equation. Because the potential is greater than the energy is imaginary and the new size it can be introduced.

Thus the exponent of the exponential function is real for negative and the wave function describes an exponential decay.

At the level of potential, both the wave functions themselves and their derivatives must be continuous. Substituting we get:

By equating the reflection and transmission coefficients and the probability wave can be determined.

The probability amplitude to meet the particle in is because non-zero. Nor, however, is different, the probability amplitude of the reflection exactly 1 the sum squares that are calculated as a multiplication by the complex conjugate.

It takes place 100 % instead of a reflection, yet the particles can penetrate to the probability in the barrier. From the energy conservation law is to be understood that the evanescent wave does not carry energy. Analogous to 1/e-Eindringtiefe in the optics can be calculated from the absolute square of the wave function in the barrier in quantum mechanics an x -dependent Eindringwahrscheinlichkeit.

The 1/e-Eindringtiefe is it. Derivation in the optical properties of the two media included in the relatively complex expression for. In this quantum mechanical derivation, the problem has been simplified in that the potential that is selected in the region of the incoming wave to 0, which corresponds to a refractive index of. In addition, normal incidence was assumed so that the angular dependence, which is itself not considered in the derivation of the wave pattern in the sine term in reflected.

Prevented detection by total reflection

If two glass prisms very close together (see figure), one can measure light where none should be, namely behind the second prism ( of transmitted light beam ). Due to the evanescent field behind the first prism, but the light can still be transmitted, if the second prism is immersed in the evanescent field. The intensity decreases exponentially with the distance between the prisms. This effect is called prevented or frustrated total reflection (english frustrated internal total reflection, FTIR ), because actually all light would be reflected upward. This is similar to the last high potential well in quantum mechanics, where the wave function in the forbidden region decays exponentially. Therefore, this effect is also known as an optical tunnel effect. In particular the beam splitters described effect is utilized, the ratio of intensities can be set very accurately between transmitted and reflected beam through the distance between the prisms.

The effect of frustrated total reflection is exploited in the ATR spectroscopy to make impurities and errors of surfaces and thin layers visible (see also: Evanescent Wave Scattering ). And the optical near field microscopy and the total internal reflection fluorescence microscopy (TIRF ) used evanescent waves.

In light waveguides are evanescent waves in low-index cladding (English cladding ) of the fiber. The casing prevents radiation emerging from the fiber core by preventing dirt or water near the evanescent field around the core and thus may disturb the total internal reflection.

Consisting of perforated metal door of microwave ovens must be protected by an additional disc, since the microwaves (wavelength in the centimeter range ) although can not get through the door in the furnace interior, but directly behind the holes generate evanescent fields, for example, a case approach Fingers to the decoupling of microwaves would result.