Evanescent wave

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An evanescent wave is a wave exhibiting exponential decay with distance. Evanescent waves are observed in total internal reflection.

The effect has been used to exert 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.

"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.

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[edit] 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} \ = \ k_y \hat{\mathbf{y}} + k_z \hat{\mathbf{z}} \ = \ j \alpha \hat{\mathbf{y}} + \beta \hat{\mathbf{z}}

represents an evanescent wave because the vector's y component is an imaginary number. In this equation, j represents the imaginary unit:

j^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{k} |^2 = k_y^2 + k_z^2.

Solving for ky, we find

k_y \ = \ \pm \sqrt{k^2 - k_z^2} \ = \ \pm j \sqrt{k_z^2 - k^2} \ = \ \pm j \alpha.

[edit] Electric field

In sinusoidal steady-state, the electric field in the transverse direction is the real part of a complex exponential:

\mathbf{E}(\mathbf{r},t) = \mathrm{Re} \left \{ \mathbf{\hat{x}} \cdot E(\mathbf{r}) \cdot e^{ j \omega t } \right \}

where

E(\mathbf{r}) = E_o e^{-j \mathbf{k} \cdot \mathbf{r}}

and

\mathbf{ \hat{x} }

is the unit vector in the x direction .

Substituting the evanescent form of the wave vector k (as given above), we find:

E(\mathbf{r}) = E_o e^{-j ( j \alpha y + \beta z ) } = E_o e^{\alpha y - j \beta z }

where α is the attenuation constant and β is the propagation constant.

[edit] See also

[edit] External link

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