# Dictionary Definition

evanescent adj : tending to vanish like vapor; "evanescent beauty"

# Extensive Definition

An evanescent wave is a nearfield standing wave exhibiting exponential decay with distance. Evanescent waves are always associated with matter, and are most intense within one-third wavelength from any acoustic, optical, or electromagnetic transducer. Optical evanescent waves are commonly found during total internal reflection.
The effect has been used to exert optical radiation pressure on small particles in order to trap them for experimentation, or to cool them to very low temperatures, and to illuminate very small objects such as biological cells for microscopy (as in the total internal reflection fluorescence microscope). The evanescent wave from an optical fiber can be used in a gas sensor.
In optics, evanescent waves are formed when sinusoidal waves are (internally) reflected off an interface at an angle greater than the critical angle so that total internal reflection occurs. The physical explanation for their existence is that the electric and magnetic fields cannot be discontinuous at a boundary, as would be the case if there were no evanescent field.
In electrical engineering, evanescent waves are found in the nearfield region within one-third wavelength of any radio antenna. During normal operation, an antenna emits electromagnetic fields into the surrounding nearfield region, then a portion of the field energy is re-absorbed, while the remainder is radiated as EM waves.
"Evanescent" means "tends to vanish", which is appropriate because the intensity of evanescent waves decays exponentially with the distance from the interface at which they are formed.

## Total internal reflection

Mathematically, evanescent waves are characterized by a wave vector where one or more of the vector's components has an imaginary value.
For example, the wave vector defined by
\mathbf \ = \ k_y \hat + k_z \hat
\ = \ i \alpha \hat + \beta \hat
represents an evanescent wave because the vector's y component is an imaginary number. In this equation, i represents the imaginary unit:
i^2 = -1. \,
This type of evanescent wave is created when an electromagnetic wave, incident upon the interface between two dielectric media of different refractive indices, experiences total internal reflection. If the angle of incidence exceeds the critical angle, then the z component kz of the wave vector becomes larger than the overall magnitude k of the wave vector:
k_z \ > \ k
where we are assuming, without loss of generality, that the interface is a planar surface with normal parallel to the y-axis.
From the definition of a vector's magnitude,
k^2 \ = \ | \mathbf |^2 = k_y^2 + k_z^2.
Solving for ky, we find
k_y \ = \ \pm \sqrt \ = \ \pm i \sqrt \ = \ \pm i \alpha.

## Electric field

In sinusoidal steady-state, the electric field in the transverse direction is the real part of a complex exponential:
\mathbf(\mathbf,t) = \mathrm \left \
where
E(\mathbf) = E_o e^
and
\mathbf
is the unit vector in the x direction.
Substituting the evanescent form of the wave vector k (as given above), we find:
E(\mathbf) = E_o e^ = E_o e^
where \alpha is the attenuation constant and \beta is the propagation constant.

## References

evanescent in German: Evaneszenz
evanescent in French: Onde évanescente
evanescent in Japanese: エバネッセント場