![Electron density wave - plasmon excitations](https://upload.wikimedia.org/wikipedia/commons/1/16/Electron_density_wave_-_plasmon_excitations.png)
In
electromagnetics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
but whose energy is spatially concentrated in the vicinity of the source (oscillating charges and currents). Even when there is a propagating electromagnetic wave produced (e.g., by a transmitting
antenna
Antenna ( antennas or antennae) may refer to:
Science and engineering
* Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves
* Antennae Galaxies, the name of two collid ...
), one can still identify as an evanescent field the component of the electric or magnetic field that cannot be attributed to the propagating wave observed at a distance of many wavelengths (such as the
far field
The near field and far field are regions of the electromagnetic (EM) field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative ''near-field'' behaviors dominate close to the ant ...
of a transmitting antenna).
A hallmark of an evanescent field is that there is no net energy flow in that region. Since the net flow of electromagnetic energy is given by the average
Poynting vector
In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
, this means that the Poynting vector in these regions, as averaged over a complete oscillation cycle, is zero.
Use of the term
In many cases one cannot simply say that a field is or is not "evanescent": Having Poynting vector average to zero in some direction (or all directions).
In most cases where they exist, evanescent fields are simply thought of and referred to the same as all other electric or magnetic fields involved, without any special recognition of those fields' evanescence. The term's use is mostly limited to distinguishing a part of a field or solution in those cases where one might only expect the fields of a propagating wave.
For instance, in the illustration at the top of the article, energy is indeed carried in the horizontal direction. However, in the vertical direction, the field strength drops off exponentially with increasing distance above the surface. This leaves most of the field concentrated in a thin boundary layer very close to the interface; for that reason, it is referred to as a ''
surface wave
In physics, a surface wave is a mechanical wave that propagates along the Interface (chemistry), interface between differing media. A common example is gravity waves along the surface of liquids, such as ocean waves. Gravity waves can also occu ...
''. However, despite energy flowing horizontally, along the vertical there is ''no'' net propagation of energy ''away'' from (or toward) the surface, so that one could properly describe the field as being "evanescent in the vertical direction". This is one example of the context dependence of the term.
Everyday electronic devices and electrical appliances are surrounded by large fields which are evanescent; their operation involves alternating voltages (producing an electric field between them) and alternating currents (producing a magnetic field around them) which are expected to only carry power along internal wires, but not to the outsides of the devices. Even though the term "evanescent" is not mentioned in this ordinary context, the appliances' designers still may be concerned with ''maintaining'' evanescence, in order to prevent or limit production of a propagating electromagnetic wave, would lead to
radiation loss, since a propagating wave "steals" its power from the circuitry or donates unwanted
interference
Interference is the act of interfering, invading, or poaching. Interference may also refer to:
Communications
* Interference (communication), anything which alters, modifies, or disrupts a message
* Adjacent-channel interference, caused by extr ...
.
The term "evanescent field" ''does'' arise in various contexts where a propagating electromagnetic wave is involved (even if confined). The term then differentiates electromagnetic field components that accompany the propagating wave, but which do not themselves propagate. In other, similar cases, where a propagating electromagnetic wave would normally be expected (such as light
refracted
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
at the interface between glass and air), the term is invoked to describe that part of the field where the wave is suppressed (such as light traveling through glass, impinging on a glass-to-air interface but beyond the
critical angle
Critical angle may refer to:
*Critical angle (optics), the angle of incidence above which total internal reflection occurs
*Critical angle of attack
In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference lin ...
).
Although all electromagnetic fields are classically governed according to
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
, different technologies or problems have certain types of expected solutions, and when the primary solutions involve wave propagation the term ''evanescent'' is frequently applied to field components or solutions which do not share that property.
For instance, the
propagation constant
The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a cir ...
of a hollow metal
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
is a strong function of frequency (a so-called
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
). Below a certain frequency (the
cut-off frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather than ...
) the propagation constant becomes an imaginary number. A solution to the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
having an imaginary wavenumber does ''not'' propagate as a wave but falls off exponentially, so the field excited at that lower frequency is considered evanescent. It can also be simply said that propagation is "disallowed" for that frequency.
The formal solution to the wave equation can describe modes having an identical form, but the change of the propagation constant from real to imaginary as the frequency drops below the cut-off frequency totally changes the physical nature of the result. The solution may be described as a "cut-off mode" or an "evanescent mode";
while a different author will just state that no such mode exists. Since the evanescent field corresponding to the mode was computed as a solution to the wave equation, it is often discussed as being an "evanescent wave" even though its properties (such as carrying no energy) are inconsistent with the definition of
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
.
Although this article concentrates on electromagnetics, the term ''evanescent'' is used similarly in fields such as
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, where the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
arises from the physics involved. In these cases, solutions to the wave equation resulting in imaginary propagation constants are likewise called "evanescent", and have the essential property that no net energy is transferred, even though there is a non-zero field.
Evanescent wave applications
In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
and
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, evanescent waves are formed when waves traveling in a medium undergo
total internal reflection
Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflected b ...
at its boundary because they strike it at an angle greater than the so-called ''
critical angle
Critical angle may refer to:
*Critical angle (optics), the angle of incidence above which total internal reflection occurs
*Critical angle of attack
In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference lin ...
''.
The physical explanation for the existence of the evanescent wave is that the electric and magnetic fields (or
pressure gradients
In atmospheric science, the pressure gradient (typically of air but more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular location. The pr ...
, 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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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 mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
on small particles to trap them for experimentation, or to
cool
Cool commonly refers to:
* Cool, a moderately low temperature
* Cool (aesthetic), an aesthetic of attitude, behavior, and style
Cool or COOL may also refer to:
Economics
* Country of origin labelling
* mCOOL - US consumer legislation to enforc ...
them to very low temperatures, and to illuminate very small objects such as
biological cell
The cell is the basic structural and functional unit of life forms. Every cell consists of a cytoplasm enclosed within a membrane, and contains many biomolecules such as proteins, DNA and RNA, as well as many small molecules of nutrients and ...
s or
single protein and DNA molecules for
microscopy
Microscopy is the technical field of using microscopes to view objects and areas of objects that cannot be seen with the naked eye (objects that are not within the resolution range of the normal eye). There are three well-known branches of micr ...
(as in the
total internal reflection fluorescence microscope A total internal reflection fluorescence microscope (TIRFM) is a type of microscope with which a thin region of a specimen, usually less than 200 nanometers can be observed.
TIRFM is an imaging modality which uses the excitation of fluorescent cel ...
). The evanescent wave from an
optical fiber
An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
can be used in a gas sensor, and evanescent waves figure in the
infrared spectroscopy
Infrared spectroscopy (IR spectroscopy or vibrational spectroscopy) is the measurement of the interaction of infrared radiation with matter by absorption, emission, or reflection. It is used to study and identify chemical substances or function ...
technique known as
attenuated total reflectance
Attenuated total reflection (ATR) is a sampling technique used in conjunction with infrared spectroscopy which enables samples to be examined directly in the solid or liquid state without further preparation.
ATR uses a property of total inter ...
.
In
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, 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
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of Crystal structure, natural crystals gives rise to X-ray crystallograp ...
) has been fabricated and demonstrated its competence for excitation of surface electromagnetic waves in the periodic structure using a
prism coupling technique.
In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the evanescent-wave solutions of the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
give rise to the phenomenon of
wave-mechanical tunneling.
In
microscopy
Microscopy is the technical field of using microscopes to view objects and areas of objects that cannot be seen with the naked eye (objects that are not within the resolution range of the normal eye). There are three well-known branches of micr ...
, 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
The resolution of an optical imaging system a microscope, telescope, or camera can be limited by factors such as imperfections in the lenses or misalignment. However, there is a principal limit to the resolution of any optical system, due to th ...
. Systems that capture the information contained in evanescent waves, such as the
superlens A superlens, or super lens, is a lens (optics), lens which uses metamaterials to go beyond the diffraction limit. For example, in 1995, Guerra combined a transparent grating having 50nm lines and spaces (the "metamaterial") with a conventional micro ...
and
near field scanning optical microscopy
Near-field scanning optical microscopy (NSOM) or scanning near-field optical microscopy (SNOM) is a microscopy technique for nanostructure investigation that breaks the far field Diffraction-limited system, resolution limit by exploiting the prop ...
, 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 maximal
wave vector
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
that can be resolved,
is the distance between the object and the sensor, and
is a measure of the
quality
Quality may refer to:
Concepts
*Quality (business), the ''non-inferiority'' or ''superiority'' of something
*Quality (philosophy), an attribute or a property
*Quality (physics), in response theory
*Energy quality, used in various science discipli ...
of the sensor.
More generally, practical applications of evanescent waves can be classified as (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 A total internal reflection fluorescence microscope (TIRFM) is a type of microscope with which a thin region of a specimen, usually less than 200 nanometers can be observed.
TIRFM is an imaging modality which uses the excitation of fluorescent cel ...
) or (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
In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillati ...
''.
Total internal reflection of light
![Evanescent wave](https://upload.wikimedia.org/wikipedia/commons/3/3a/Evanescent_wave.jpg)
For example, consider
total internal reflection
Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflected b ...
in two dimensions, with the interface between the media lying on the x axis, the
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
along y, and the
polarization along z. One might 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, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
. Maxwell's equations in a dielectric medium impose a boundary condition of continuity for the components of the fields ''E
, , , H
, , , D
y'', and ''B
y''. For the polarization considered in this example, the conditions on ''E
, , '' 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 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
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
. 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
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
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.
For the plane of incidence as the
plane at
and the interface of the two mediums as the
plane at
, the wave vector of the transmitted wave has the form
:
with
and
, where
is the magnitude of the wave vector of the transmitted wave (so the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
),
is the angle of refraction, and
and
are the unit vectors along the
axis direction and the
axis direction respectively.
By using the
Snell's law
Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
where
,
, and
are the refractive index of the medium where the incident wave and the reflected wave exist, the refractive index of the medium where the transmitted wave exists, and the angle of incidence respectively,
:
.
with
.
If a part of the condition of the
total internal reflection
Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflected b ...
as
, is satisfied, then
:
.
If the
polarization is perpendicular to the plane of incidence (along the
direction), then the electric field of any of the waves (incident, reflected, or transmitted) can be expressed as
:
where
is the
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
in the
axis direction.
By assuming plane waves as
, and substituting the transmitted wave vector
into
, we find for the transmitted wave:
:
where
is the
attenuation constant
The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a ci ...
, and
is the
phase constant
The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a ci ...
.
is ignored since it does not physically make sense (the wave amplification along ''y'' the direction in this case).
Evanescent-wave coupling
![FITR penetration depth](https://upload.wikimedia.org/wikipedia/commons/6/64/FITR_penetration_depth.svg)
Especially in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, ''evanescent-wave coupling'' refers to the coupling between two waves due to physical overlap of what would otherwise be described as the evanescent fields corresponding to the propagating waves.
One classical example is
FTIR (Frustrated Total Internal Reflection) in which the evanescent field very close (see graph) to the surface of a dense medium at which a wave normally undergoes
total internal reflection
Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflected b ...
overlaps another dense medium in the vicinity. This disrupts the totality of the reflection, diverting some power into the second medium.
Coupling between two
optical waveguide
An optical waveguide is a physical structure that guides electromagnetic waves in the optical spectrum. Common types of optical waveguides include optical fiber waveguides, transparent dielectric waveguides made of plastic and glass, liquid light g ...
s may be effected by placing the fiber cores close together so that the evanescent field generated by one element excites a wave in the other fiber. This is used to produce
fiber-optic splitters and in
fiber tapping
Fiber tapping uses a network tap method that extracts signal from an optical fiber without breaking the connection. Tapping of optical fiber allows diverting some of the signal being transmitted in the core of the fiber into another fiber or a ...
. At radio (and even optical) frequencies, such a device is called a
directional coupler. The device is usually called a power divider in the case of microwave transmission and modulation.
Evanescent-wave coupling is synonymous with
near field interaction in electromagnetic field theory. Depending on the nature of the source element, the evanescent field involved is either predominantly electric (capacitive) or magnetic (inductive), unlike (propagating) waves in the far field where these components are connected (identical phase, in the ratio of the
impedance of free space The impedance of free space, , is a physical constant relating the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space. That is, , where is the electric field strength and is the magnetic fiel ...
). 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. In quantum mechanics the wave function interaction may be discussed in terms of particles and described as
quantum tunneling
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
.
Applications
Evanescent wave coupling is commonly used in photonic and nanophotonic devices as waveguide sensors or couplers (see e.g.,
prism coupler
A prism coupler is a prism designed to couple a substantial fraction of the power contained in a beam of light (e.g., a laser beam) into a thin film to be used as a waveguide without the need for precision polishing of the edge of the film, witho ...
).
Evanescent wave coupling is used to excite, for example, dielectric microsphere resonators.
Evanescent coupling, as near field interaction, is one of the concerns in
electromagnetic compatibility
Electromagnetic compatibility (EMC) is the ability of electrical equipment and systems to function acceptably in their electromagnetic environment, by limiting the unintentional generation, propagation and reception of electromagnetic energy whic ...
.
Coupling of optical fibers without loss for
fiber tapping
Fiber tapping uses a network tap method that extracts signal from an optical fiber without breaking the connection. Tapping of optical fiber allows diverting some of the signal being transmitted in the core of the fiber into another fiber or a ...
.
Evanescent wave coupling plays a major role in the theoretical explanation of
extraordinary optical transmission.
Evanescent wave coupling is used in powering devices wirelessly.
A
total internal reflection fluorescence microscope A total internal reflection fluorescence microscope (TIRFM) is a type of microscope with which a thin region of a specimen, usually less than 200 nanometers can be observed.
TIRFM is an imaging modality which uses the excitation of fluorescent cel ...
uses the evanescent wave produced by
total internal reflection
Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflected b ...
to excite fluorophores close to a surface. This is useful when surface properties of biological samples need to be studied.
See also
Notes
References
External links
Evanescent waves
Evanescent and propagating waves animation on Youtube.com
{{DEFAULTSORT:Evanescent Wave
Electromagnetism
Optics
Metamaterials
Materials science
Nanotechnology
Quantum mechanics