Photonic crystals are periodic optical
nanostructureA nanostructure is an object of intermediate size between molecular and microscopic structures.In describing nanostructures it is necessary to differentiate between the number of dimensions on the nanoscale. Nanotextured surfaces have one dimension on the nanoscale, i.e., only the thickness of the...
s that are designed to affect the motion of photons in a similar way that periodicity of a
semiconductorA semiconductor is a material that has an electrical resistivity between that of a conductor and an insulator, that is, generally in the range 10
3 Siemens/cm to 10
−8 S/cm. Devices made from semiconductor materials are the foundation of modern electronics, including radio,...
crystalA crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is crystallography...
affects the motion of
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
s. Photonic crystals occur in nature and in various forms have been studied scientifically for the last 100 years.
Introduction
Photonic crystals are composed of periodic
dielectricA dielectric is a nonconducting substance, i.e. an insulator. The term was coined by William Whewell in response to a request from Michael Faraday...
or metallo-dielectric
nanostructureA nanostructure is an object of intermediate size between molecular and microscopic structures.In describing nanostructures it is necessary to differentiate between the number of dimensions on the nanoscale. Nanotextured surfaces have one dimension on the nanoscale, i.e., only the thickness of the...
s that affect the propagation of electromagnetic waves (EM) in the same way as the
periodicPeriodicity is the quality of occurring at regular intervals or periods and can occur in different contexts:In timing devices:* A clock marks time at periodic intervals.* A metronome ticks at periodic intervals of time....
potential in a
semiconductorA semiconductor is a material that has an electrical resistivity between that of a conductor and an insulator, that is, generally in the range 10
3 Siemens/cm to 10
−8 S/cm. Devices made from semiconductor materials are the foundation of modern electronics, including radio,...
crystal affects the
electronAn electron is a subatomic particle that carries a negative electric charge. It has no known substructure and is believed to be a point particle. An electron has a mass that is approximately 1836 times less than that of the proton. The intrinsic angular momentum of the electron is a half integer...
motion by defining allowed and forbidden electronic energy bands. Essentially, photonic crystals contain regularly repeating internal regions of high and low
dielectric constantThe relative static permittivity of a material under given conditions is a measure of the extent to which it concentrates electrostatic lines of flux. It is the ratio of the amount of stored electrical energy when a potential is applied, relative to the permittivity of a vacuum...
. Photons (behaving as waves) propagate through this structure - or not - depending on their wavelength. Wavelengths of light that are allowed to travel are known as modes, and groups of allowed modes form bands. Disallowed bands of
wavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
s are called photonic
band gapIn solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states exist...
s. This gives rise to distinct optical phenomena such as inhibition of
spontaneous emissionSpontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to the ground state and emits a photon...
, high-reflecting omni-directional mirrors and low-loss-
waveguidingA waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguide for each type of wave...
, amongst others.
Since the basic physical phenomenon is based on
diffractionDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
, the periodicity of the photonic crystal structure has to be of the same length-scale as half the wavelength of the EM waves i.e. ~200 nm (blue) to 350 nm (red) for photonic crystals operating in the visible part of the spectrum - the repeating regions of high and low dielectric constants have to be of this dimension. This makes the fabrication of optical photonic crystals cumbersome and complex.
History of photonic crystals
Although photonic crystals have been studied in one form or another since 1887, the term “photonic crystal” was first used over 100 years later, after
Eli YablonovitchEli Yablonovitch along with Sajeev John, was one of the two applied physicists who invented the field of photonic crystals in 1987. In addition to pioneering photonic crystals, he was the first to recognize that a strained quantum well laser has a significantly reduced threshold current compared to...
and
Sajeev JohnSajeev John is a Professor of Physics at the University of Toronto and Government of Canada Research Chair holder.He received his bachelors degree in physics in 1979 from the Massachusetts Institute of Technology and his Ph.D. in physics at Harvard University in 1984. His Ph.D...
published two milestone papers on photonic crystals in 1987.
Before 1987, one-dimensional photonic crystals in the form of periodic multi-layers dielectric stacks (such as the Bragg mirror) were studied extensively. Lord Rayleigh started their study in 1887, by showing that such systems have a one-dimensional photonic band-gap, a spectral range of large reflectivity, known as a stop-band. Today, such structures are used in a diverse range of applications; from reflective coatings to enhancing the efficiency of LEDs to highly reflective mirrors in certain laser cavities (see, for example,
VCSELThe vertical-cavity surface-emitting laser is a type of semiconductor laser diode with laser beam emission perpendicular from the top surface, contrary to conventional edge-emitting semiconductor lasers which emit from surfaces formed by cleaving the individual chip out of a wafer.There are...
). A detailed theoretical study of one-dimensional optical structures was performed by
BykovBykov , or Bykova , is a common Russian last name and may refer to:-People:*Anatoly Bykov , a Russian businessman*Boris Bykov , a Soviet intelligence officer*Dmitrii L'vovich Bykov , a Russian writer...
, who was the first to investigate the effect of a photonic band-gap on the spontaneous emission from atoms and molecules embedded within the photonic structure. Bykov also speculated as to what could happen if two- or three-dimensional periodic optical structures were used. However, these ideas did not take off until after the publication of two milestone papers in 1987 by Yablonovitch and John. Both these papers concerned high dimensional periodic optical structures – photonic crystals. Yablonovitch’s main motivation was to engineer the photonic
density of statesIn statistical and condensed matter physics, the density of states of a system describes the number of states at each energy level that are available to be occupied. A high DOS at a specific energy level means that there are many states available for occupation...
, in order to control the
spontaneous emissionSpontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to the ground state and emits a photon...
of materials embedded within the photonic crystal; John’s idea was to use photonic crystals to affect the localisation and control of light.
After 1987, the number of research papers concerning photonic crystals began to grow exponentially. However, due to the difficulty of actually fabricating these structures at optical scales (see Fabrication Challenges), early studies were either theoretical or in the microwave regime, where photonic crystals can be built on the far more readily accessible centimetre scale. (This fact is due to a property of the electromagnetic fields known as scale invariance – in essence, the electromagnetic fields, as the solutions to
Maxwell's equationsMaxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined to show that light is an electromagnetic wave...
, has no natural length scale, and so solutions for centimetre scale structure at microwave frequencies are the same as for nanometre scale structures at optical frequencies.) By 1991, Yablonovitch had demonstrated the first three-dimensional photonic band-gap in the microwave regime.
In 1996,
Thomas KraussThomas F Krauss is a physics researcher at the University of St Andrews. He has several research interests, but is mostly known for his work in the field of photonic crystals, where he made the first demonstration of two-dimensional photonic band-gap effects at optical wavelengths...
made the first demonstration of a two-dimensional photonic crystal at optical wavelengths. This opened up the way for photonic crystals to be fabricated in semiconductor materials by borrowing the methods used in the semiconductor industry. Today, such techniques use photonic crystal slabs, which are two dimensional photonic crystals “etched” into slabs of semiconductor;
total internal reflectionTotal internal reflection is an optical phenomenon that occurs 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, no light can pass through...
confines light to the slab, and allows photonic crystal effects, such as engineering the photonic dispersion to be used in the slab. Research is underway around the world to use photonic crystal slabs in integrated computer chips, in order to improve the optical processing of communications both on-chip and between chips.
Although such techniques have yet to mature into commercial applications, two-dimensional photonic crystals have found commercial use in the form of
photonic crystal fibresPhotonic-crystal fiber is a new class of optical fiber based on the properties of photonic crystals. Because of its ability to confine light in hollow cores or with confinement characteristics not possible in conventional optical fiber, PCF is now finding applications in fiber-optic...
(otherwise known as holey fibres, because of the air holes that run through them). Photonic crystal fibres were first developed by
Philip RussellPhilip St. John Russell, FRS, is the Director of the third division of the Max Planck Research Group at the Institute of Optics, Information and Photonics at the University of Erlangen-Nuremberg. His area of research is "photonics and new materials"...
in 1998, and can be designed to possess enhanced properties over (normal) optical fibres.
The study of three-dimensional photonic crystals has proceeded more slowly then their two-dimensional counterparts. This is because of the increased difficulty in fabrication; there was no inheritance of readily applicable techniques from the semiconductor industry for fabricators of three-dimensional photonic crystals to draw on. Attempts have been made, however, to adapt some of the same techniques, and quite advanced examples have been demonstrated, for example in the construction of "woodpile" structures constructed on a planar layer-by-layer basis. Another strand of research has been to try and construct three-dimensional photonic structures from self-assembly – essentially allowing a mixture of dielectric nano-spheres to settle from solution into three-dimensionally periodic structures possessing photonic band-gaps (see
colloidal crystals, below).
Fabrication challenges
The major challenge for higher dimensional photonic crystals is in fabrication of these structures, with sufficient precision to prevent scattering losses blurring the crystal properties and with processes that can be robustly mass produced. One promising method of fabrication for two-dimensionally periodic photonic crystals is a
photonic-crystal fiberPhotonic-crystal fiber is a new class of optical fiber based on the properties of photonic crystals. Because of its ability to confine light in hollow cores or with confinement characteristics not possible in conventional optical fiber, PCF is now finding applications in fiber-optic...
, such as a "holey fiber". Using fiber draw techniques developed for
communications fiberAn optical fiber is a glass or plastic fiber that carries light along its length. Fiber optics is the overlap of applied science and engineering concerned with the design and application of optical fibers...
it meets these two requirements, and photonic crystal fibres are commercially available. Another promising method for developing two-dimensional photonic crystals is the so-called photonic crystal slab. These structures consist of a slab of material (such as
siliconSilicon is the most common metalloid. It is a chemical element, which has the symbol Si and atomic number 14. A tetravalent metalloid, silicon is less reactive than its chemical analog carbon...
) which can be patterned using techniques borrowed from the semiconductor industry. Such chips offer the potential to combine photonic processing with electronic processing on a single chip.
For three dimensional photonic crystals various techniques have been used including
photolithographyOptical lithography, is a process used in microfabrication to selectively remove parts of a thin film or the bulk of a substrate. It uses light to transfer a geometric pattern from a photo mask to a light-sensitive chemical photo resist, or simply "resist," on the substrate. A series of chemical...
and etching techniques similar to those used for integrated circuits. Some of these techniques are already commercially available like Nanoscribe's Direct Laser Writing system. To circumvent
nanotechnological methodsNanotechnology, shortened to "nanotech", is the study of the control of matter on an atomic and molecular scale. Generally nanotechnology deals with structures of the size 100 nanometers or smaller, and involves developing materials or devices within that size.Nanotechnology is very diverse,...
with their complex machinery, alternate approaches have been followed to grow photonic crystals as self-assembled structures from
colloidal crystalA colloidal crystal is an ordered array of particles, analogous to a standard crystal whose repeating subunits are atoms or molecules. A natural example of this phenomenon can be found in the gem opal, where spheres of silica assume a close-packed locally periodic structure under moderate compression...
s.
Computing photonic band structure
The photonic band gap (PBG) is essentially the gap between the air-line and the dielectric-line in the
dispersion relationIn optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...
of the PBG system. To design photonic crystal systems, it is essential to engineer the location and size of the bandgap; this is done by computational modeling using any of the following methods.
- Plane wave expansion method
Plane wave expansion method refers to a computational technique in electromagnetics to solve the Maxwell's equations by formulating an eigenvalue problem out of the equation. This method is a popular among the photonic crystal community as a method of solving for the band structure of specific...
.
- Finite Difference Time Domain method
- Order-N spectral method
- KKR method
The muffin-tin approximation is a shape approximation of the potential field in an atomistic environment. It is most commonly employed in quantum mechanical simulations of electronic band structure in solids. The approximation was proposed by John C. Slater....
Essentially these methods solve for the frequencies (normal models) of the photonic crystal for each value of the propagation direction given by the wave vector, or vice-versa. The various lines in the band structure, correspond to the different cases of
n, the band index. For an introduction to photonic band structure, see Joannopoulos.
The
plane wave expansion method, can be used to calculate the band structure using an eigen formulation of the Maxwell's equations, and thus solving for the eigen frequencies for each of the propagation directions, of the wave vectors. It directly solves for the dispersion diagram. Electric field strength values can also be calculated over the spatial domain of the problem using the eigen vectors of the same problem. For the picture shown to the right, corresponds to the band-structure of a 1D
DBRA distributed Bragg reflector is a reflector used in waveguides, such as optical fibers. It is a structure formed from multiple layers of alternating materials with varying refractive index, or by periodic variation of some characteristic of a dielectric waveguide, resulting in periodic variation...
with air-core interleaved with a dielectric material of relative permittivity 12.25, and a lattice period to air-core thickness ratio (d/a) of 0.8, is solved using 101 planewaves over the first irreducible
Brillouin zoneIn mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
.
Applications
Photonic crystals are attractive optical materials for controlling and manipulating the flow of light. One dimensional photonic crystals are already in widespread use in the form of
thin-film opticsThin-film optics is the branch of optics that deals with very thin structured layers of different materials. In order to exhibit thin-film optics, the thickness of the layers of material must be on the order of the wavelengths of visible light...
with applications ranging from low and high reflection coatings on lenses and mirrors to
colour changing paintsChromaFlair is the registered trademark for a pigment used in paint systems, primarily for automobiles. When the paint is applied, it changes color depending on the light source and viewing angle...
and
inksSecurity printing is the field of the printing industry that deals with the printing of items such as banknotes, passports, tamper-evident labels, stock certificates, postage stamps and identity cards...
. Higher dimensional photonic crystals are of great interest for both fundamental and applied research, and the two dimensional ones are beginning to find commercial applications. The first commercial products involving two-dimensionally periodic photonic crystals are already available in the form of
photonic-crystal fiberPhotonic-crystal fiber is a new class of optical fiber based on the properties of photonic crystals. Because of its ability to confine light in hollow cores or with confinement characteristics not possible in conventional optical fiber, PCF is now finding applications in fiber-optic...
s, which use a microscale structure to confine light with radically different characteristics compared to conventional
optical fiberAn optical fiber is a glass or plastic fiber that carries light along its length. Fiber optics is the overlap of applied science and engineering concerned with the design and application of optical fibers...
for applications in nonlinear devices and guiding exotic wavelengths. The three-dimensional counterparts are still far from commercialization but offer additional features possibly leading to new device concepts (e.g. optical computers), when some technological aspects such as manufacturability and principal difficulties such as disorder are under control.
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