Evanescent wave

An evanescent wave is a nearfield standing wave with an intensity that exhibits exponential decay with distance from the boundary at which the wave was formed. Evanescent waves are a general property of wave-equations, and can in principle occur in any context to which a wave-equation applies. They are formed at the boundary between two "media" with different properties in respect of wave motion, and are most intense within one-third of a wavelength from the surface of formation. In particular, evanescent waves can occur in the contexts of: optics and other forms of electromagnetic radiation, acoustics, quantum mechanics and "waves on strings".

In optics^[1] and acoustics, evanescent waves are formed when waves travelling in a medium undergo total internal reflection at its boundary because they hit it at an angle greater than the so-called critical angle. The physical explanation for the existence of the evanescent wave is that the electric and magnetic fields (or pressure gradients, in the case of acoustical waves) cannot be discontinuous at a boundary, as would be the case if there were no evanescent wave-field. In quantum mechanics, the physical explanation is exactly analogous - the Schrödinger wave-function representing particle motion normal to the boundary cannot be discontinuous at the boundary.

Electromagnetic evanescent waves have 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 and evanescent waves figure in the infrared spectroscopy technique known as attenuated total reflectance.

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, and a portion of the field energy is re-absorbed, while the remainder is radiated as EM waves.

In quantum mechanics the evanescent-wave solutions of the Schrödinger equation give rise to the phenomenon of wave-mechanical tunnelling.

More generally, practical applications of evanescent waves can be classified in the following way. (1) Those in which the energy associated with the wave is used to excite some other phenomenon within the region of space where the original travelling wave becomes evanescent (for example, as in the total internal reflection fluorescence microscope). (2) Those in which the evanescent wave "couples" two media in which travelling waves are allowed, and hence permits the transfer of energy or a particle between the media (depending on the wave-equation in use), even though no travelling-wave solutions are allowed in the region of space between the two media. An example of this is so-called "wave-mechanical tunnelling". This second type of application is known generally as "evanescent wave coupling".

Total internal reflection of light

Total internal reflection

For example, consider total internal reflection in two dimensions, with the interface between the media lying on the x axis, the normal along y, and the polarization along z. One might naively expect that for angles leading to total internal reflection, the solution would consist of an incident wave and a reflected wave, with no transmitted wave at all, but there is no such solution that obeys Maxwell's equations. Maxwell's equations in a dielectric medium impose a boundary condition of continuity for the components of the fields , , D_y, and B_y. For the polarization considered in this example, the conditions on and B_y are satisfied if the reflected wave has the same amplitude as the incident one, because these components of the incident and reflected waves superimpose destructively. Their H_x components, however, superimpose constructively, so there can be no solution without a nonvanishing transmitted wave. The transmitted wave cannot, however, be a sinusoidal wave, since it would then transport energy away from the boundary, but since the incident and reflected waves have equal energy, this would violate conservation of energy. We therefore conclude that the transmitted wave must be a nonvanishing solution to Maxwell's equations that is not a traveling wave, and the only such solutions in a dielectric are those that decay exponentially: evanescent waves.

Mathematically, evanescent waves can be characterized by a wave vector where one or more of the vector's components has an imaginary value. Because the vector has imaginary components, it may have a magnitude that is less than its real components. If the angle of incidence exceeds the critical angle, then wave vector of the transmitted wave has the form

which represents an evanescent wave because the y component is imaginary. (Here α and β are real and i represents the imaginary unit.)

For example, if the polarization is perpendicular to the plane of incidence, then the electric field of any of the waves (incident, reflected, or transmitted) can be expressed as

where is the unit vector in the z direction.

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

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

References

1. ^ Tineke Thio (2006), A Bright Future for Subwavelength Light Sources, 94, American Scientist, pp. pp. 40–47 

See also

• Snell's law
• Total internal reflection
• Total internal reflection fluorescence microscope
• Evanescent wave coupling
• Electromagnetic wave
• Quantum tunneling

External links

• Evanescent waves
• An animation from a product manufacturer that makes use of evanescent fields