Evanescent wave

# Evanescent wave

Discussion

Encyclopedia
An evanescent wave is a nearfield
Near and far field
The near field and far field and the transition zone are regions of the electromagnetic radiation field that emanates from a transmitting antenna, or as a result of radiation scattering off an object...

standing wave
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...

with an intensity
Intensity (physics)
In physics, intensity is a measure of the energy flux, averaged over the period of the wave. The word "intensity" here is not synonymous with "strength", "amplitude", or "level", as it sometimes is in colloquial speech...

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 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".

## Evanescent wave applications

In optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

and acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

, evanescent waves are formed when waves traveling in a medium undergo total internal reflection
Total internal reflection
Total internal reflection is an optical phenomenon that happens when a ray of light strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is...

at its boundary because they strike it at an angle greater than the so-called critical angle
Critical angle
Critical angle can refer to:*Critical angle the angle of incidence above which total internal reflection occurs*Critical angle of attack, in aerodynamics; the angle of attack which produces the maximum lift coefficient...

. 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
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, 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
Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. If absorbed, the pressure is the power flux density divided by the speed of light...

on small particles in order to trap them for experimentation, or to cool
Refrigeration
Refrigeration is a process in which work is done to move heat from one location to another. This work is traditionally done by mechanical work, but can also be done by magnetism, laser or other means...

them to very low temperatures, and to illuminate very small objects such as biological cells for microscopy
Microscopy
Microscopy is the technical field of using microscopes to view samples and objects that cannot be seen with the unaided eye...

(as in the total internal reflection fluorescence microscope
Total internal reflection fluorescence microscope
A total internal reflection fluorescence microscope is a type of microscope with which a thin region of a specimen, usually less than 200 nm, can be observed.-Background:...

). The evanescent wave from an optical fiber can be used in a gas sensor, and evanescent waves figure in the infrared spectroscopy
Infrared spectroscopy
Infrared spectroscopy is the spectroscopy that deals with the infrared region of the electromagnetic spectrum, that is light with a longer wavelength and lower frequency than visible light. It covers a range of techniques, mostly based on absorption spectroscopy. As with all spectroscopic...

technique known as attenuated total reflectance
Attenuated total reflectance
Attenuated total reflectance is a sampling technique used in conjunction with infrared spectroscopy which enables samples to be examined directly in the solid, liquid or gas state without further preparation....

.

In electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

, 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.

In quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, the evanescent-wave solutions of the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

give rise to the phenomenon of wave-mechanical tunneling
Quantum tunnelling
Quantum tunnelling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the sun, and has important...

.

In microscopy
Microscopy
Microscopy is the technical field of using microscopes to view samples and objects that cannot be seen with the unaided eye...

, 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
Superlens
A superlens, super lens or perfect lens is a lens which uses metamaterials to go beyond the diffraction limit. The diffraction limit is an inherent limitation in conventional optical devices or lenses. In 2000, a type of lens was proposed, consisting of a metamaterial that compensates for wave...

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. The limitation on their resolution is given by
,

where is the maximum wave vector
Wave vector
In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...

that can be resolved, is the distance between the object and the sensor, and is a measure of the quality
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....

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
Total internal reflection fluorescence microscope
A total internal reflection fluorescence microscope is a type of microscope with which a thin region of a specimen, usually less than 200 nm, can be observed.-Background:...

)
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
Quantum tunnelling
Quantum tunnelling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the sun, and has important...

, and is known generally as evanescent wave coupling.

## Total internal reflection of light

For example, consider total internal reflection
Total internal reflection
Total internal reflection is an optical phenomenon that happens when a ray of light strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is...

in two dimensions, with the interface between the media lying on the x axis, the normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

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
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.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
Wave vector
In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...

where one or more of the vector's components has an imaginary
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...

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
which represents an evanescent wave because the y component is imaginary. (Here α and β are real and i represents the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

.)

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.

## Evanescent-wave coupling

In optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, 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
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...

.

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 mode
Transverse mode
A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...

s of the appropriate frequency, the evanescent field gives rise to propagating-wave modes, thereby connecting (or coupling
Coupling (electronics)
In electronics and telecommunication, coupling is the desirable or undesirable transfer of energy from one medium, such as a metallic wire or an optical fiber, to another medium, including fortuitous transfer....

) the wave from one waveguide to the next.

Evanescent-wave coupling is fundamentally identical to near field
Near field
Near field may refer to:*Near-field , an algebraic structure*Near and far field, parts of an electromagnetic field*Near field communication, a set of short-range wireless technologies, typically requiring a distance of 4 cm or less...

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.

### Applications

• Evanescent wave coupling is commonly used in photonic and nanophotonic devices as waveguide sensors.
• Evanescent wave coupling is used to excite, for example, dielectric microsphere resonators.
• A typical application is resonant energy transfer
Resonant energy transfer
Resonant energy transfer may refer to:*Förster resonance energy transfer*Resonant inductive coupling...

, useful, for instance, for charging electronic gadgets without wires. A particular implementation of this is WiTricity
WiTricity
WiTricity, a portmanteau for "wireless electricity", is a trademark of WiTricity corporation referring to their devices and processes which use a form of wireless energy transfer including resonant energy transfer etc., the ability to provide electrical energy to remote objects without wires using...

; the same idea is also used in some Tesla coil
Tesla coil
A Tesla coil is a type of resonant transformer circuit invented by Nikola Tesla around 1891. It is used to produce high voltage, low current, high frequency alternating current electricity. Tesla coils produce higher current than the other source of high voltage discharges, electrostatic machines...

s.
• Evanescent coupling, as near field interaction, is one of the concerns in electromagnetic compatibility
Electromagnetic compatibility
Electromagnetic compatibility is the branch of electrical sciences which studies the unintentional generation, propagation and reception of electromagnetic energy with reference to the unwanted effects that such energy may induce...

.
• Evanescent wave coupling plays a major role in the theoretical explanation of extraordinary optical transmission
Extraordinary Optical Transmission
Extraordinary optical transmission is the phenomenon of greatly enhanced transmission of light through a subwavelength aperture in an otherwise opaque metallic film which has been patterned with a regularly repeating periodic structure...

.

• Coupling (electronics)
Coupling (electronics)
In electronics and telecommunication, coupling is the desirable or undesirable transfer of energy from one medium, such as a metallic wire or an optical fiber, to another medium, including fortuitous transfer....

• Electromagnetic wave
• Quantum tunneling
• Resonant energy transfer
Resonant energy transfer
Resonant energy transfer may refer to:*Förster resonance energy transfer*Resonant inductive coupling...

• Snell's law
Snell's law
In optics and physics, Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass...

• Total internal reflection
Total internal reflection
Total internal reflection is an optical phenomenon that happens when a ray of light strikes a medium boundary at an angle larger than a particular critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary and the incident angle is...

• Total internal reflection fluorescence microscope
Total internal reflection fluorescence microscope
A total internal reflection fluorescence microscope is a type of microscope with which a thin region of a specimen, usually less than 200 nm, can be observed.-Background:...

• Waveguide
Waveguide
A waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguides for each type of wave...