# Evanescent wave

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Schematic representation of evanescent waves propagating along a metal-dielectric interface. The charge density oscillations, when associated with electromagnetic fields, are called surface plasmon-polariton waves. The exponential dependence of the electromagnetic field intensity on the distance away from the interface is shown on the right. These waves can be excited very efficiently with light in the visible range of the electromagnetic spectrum.

An evanescent wave is a near-field wave with an intensity that exhibits exponential decay without absorption as a function of the 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 wave motion properties, 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".[1][2]

## Evanescent wave applications

In optics and acoustics, evanescent waves are formed when waves traveling in a medium undergo total internal reflection at its boundary because they strike it at an angle greater than the so-called critical angle.[1][2] 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 was 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 to trap them for experimentation, or to cool them to very low temperatures, and to illuminate very small objects such as biological cells or single protein and DNA molecules 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 near-field region within one third of a 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 reabsorbed, while the remainder is radiated as EM waves.

Recently, a graphene-based Bragg grating (one-dimensional photonic crystal) has been fabricated and demonstrated its competence for excitation of surface electromagnetic waves in the periodic structure using a prism coupling technique.[3]

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

In microscopy, systems that capture the information contained in evanescent waves can be used to create super-resolution images. Matter radiates both propagating and evanescent electromagnetic waves. Conventional optical systems capture only the information in the propagating waves and hence are subject to the diffraction limit. Systems that capture the information contained in evanescent waves, such as the superlens and near field scanning optical microscopy, can overcome the diffraction limit; however these systems are then limited by the system's ability to accurately capture the evanescent waves.[4] The limitation on their resolution is given by

$k \propto \frac{1}{d} \ln{\frac{1}{\delta}}$,

where $k$ is the maximum wave vector that can be resolved, $d$ is the distance between the object and the sensor, and $\delta$ is a measure of the quality of the sensor.

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 traveling 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 traveling 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 traveling-wave solutions are allowed in the region of space between the two media. An example of this is so-called wave-mechanical tunnelling, and is known generally as evanescent wave coupling.

## Total internal reflection of light

Top to bottom: representation of a refracted incident wave and an evanescent wave at an interface.

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 E||, H||, Dy, and By. For the polarization considered in this example, the conditions on E|| and By 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 Hx components, however, superimpose constructively, so there can be no solution without a non-vanishing 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 non-vanishing 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 the wave vector of the transmitted wave has the form

$\mathbf{k} \ = \ k_y \hat{\mathbf{y}} + k_x \hat{\mathbf{x}} \ = \ i \alpha \hat{\mathbf{y}} + \beta \hat{\mathbf{x}},$

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

$\mathbf{E}(\mathbf{r},t) = \mathrm{Re} \left \{ E(\mathbf{r}) e^{ i \omega t } \right \} \mathbf{\hat{z}}$

where $\scriptstyle\mathbf{\hat{z}}$ 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:

$E(\mathbf{r}) = E_o e^{-i ( i \alpha y + \beta x ) } = E_o e^{\alpha y - i \beta x }$

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

## Evanescent-wave coupling

In optics, evanescent-wave coupling is a process by which electromagnetic waves are transmitted from one medium to another by means of the evanescent, exponentially decaying electromagnetic field.[5]

plot of 1/e-penetration depth of the evanescent wave against angle of incidence in units of wavelength for different refraction indices

Coupling is usually accomplished by placing two or more electromagnetic elements such as optical waveguides close together so that the evanescent field generated by one element does not decay much before it reaches the other element. With waveguides, if the receiving waveguide can support modes of the appropriate frequency, the evanescent field gives rise to propagating-wave modes, thereby connecting (or coupling) the wave from one waveguide to the next.

Evanescent-wave coupling is fundamentally identical to near field interaction in electromagnetic field theory. Depending on the impedance of the radiating source element, the evanescent wave is either predominantly electric (capacitive) or magnetic (inductive), unlike in the far field where these components of the wave eventually reach the ratio of the impedance of free space and the wave propagates radiatively. The evanescent wave coupling takes place in the non-radiative field near each medium and as such is always associated with matter; i.e., with the induced currents and charges within a partially reflecting surface. This coupling is directly analogous to the coupling between the primary and secondary coils of a transformer, or between the two plates of a capacitor. Mathematically, the process is the same as that of quantum tunneling, except with electromagnetic waves instead of quantum-mechanical wavefunctions.

## References

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