Photonic crystal
Overview
 
Photonic crystals are periodic optical nanostructure
Nanostructure
A 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 semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...

 crystal
Crystal
A 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 known as crystallography...

 affects the motion of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s. Photonic crystals occur in nature and in various forms have been studied scientifically for the last 100 years.
Photonic crystals are composed of periodic dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

 or metallo-dielectric nanostructure
Nanostructure
A 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 periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

 potential in a semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...

 crystal affects the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

 motion by defining allowed and forbidden electronic energy bands.
Encyclopedia
Photonic crystals are periodic optical nanostructure
Nanostructure
A 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 semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...

 crystal
Crystal
A 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 known as crystallography...

 affects the motion of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

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 dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

 or metallo-dielectric nanostructure
Nanostructure
A 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 periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

 potential in a semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...

 crystal affects the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

 motion by defining allowed and forbidden electronic energy bands. Essentially, photonic crystals contain regularly repeating internal regions of high and low dielectric constant
Dielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in 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 wavelength
Wavelength
In 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 gap
Band gap
In solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference between the top of the valence band and the...

s. This gives rise to distinct optical phenomena such as inhibition of spontaneous emission
Spontaneous emission
Spontaneous 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 a state with a lower energy, e.g., the ground state and emits a photon...

, high-reflecting omni-directional mirrors and low-loss-waveguiding
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...

, amongst others.

Since the basic physical phenomenon is based on diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

, 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. ~350 nm (blue) to 700 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 Yablonovitch
Eli Yablonovitch
Eli Yablonovitch along with Sajeev John, was one of the two applied physicists who invented the field of photonic crystals in 1987. He and his team were the first to create a 3-dimension structure that exhibited a full photonic bandgap, it is called Yablonovite...

 and Sajeev John
Sajeev John
Sajeev John is a Professor of Physics at the University of Toronto and 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, VCSEL
VCSEL
The vertical-cavity surface-emitting laser, or VCSEL , 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...

). A detailed theoretical study of one-dimensional optical structures was performed by Vladimir P. Bykov, 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. The concept of three-dimensional photonic crystals was then discussed by Ohtaka in 1979, who also developed a formalism for the calculation of the photonic band structure. 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 states
Density of states
In solid-state and condensed matter physics, the density of states of a system describes the number of states per interval of energy at each energy level that are available to be occupied by electrons. Unlike isolated systems, like atoms or molecules in gas phase, the density distributions are not...

, to control the spontaneous emission
Spontaneous emission
Spontaneous 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 a state with a lower energy, e.g., 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 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...

, 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. The structure that Yablonvitch was able to produce involved drilling an array of holes in a transparent material, where the holes of each layer form an inverse diamond structure - today it is known as Yablonovite
Yablonovite
Yablonovite is a photonic crystal structure that has an inverse cylindrical holes arranged in a diamond lattice. It was the first 3D photonic crystal to be fabricated with a complete photonic bandgap...

.

In 1996, Thomas Krauss
Thomas F Krauss
Thomas 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 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...

 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, 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 fibres
Photonic-crystal fiber
Photonic-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 Russell
Philip Russell
Philip 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 than 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. The first demonstration of such "inverse opal" structures came in 2000 by researchers at the University of Toronto, Canada. The ever expanding field of biomimetics
Biomimicry
Biomimicry or biomimetics is the examination of nature, its models, systems, processes, and elements to emulate or take inspiration from in order to solve human problems. The term biomimicry and biomimetics come from the Greek words bios, meaning life, and mimesis, meaning to imitate...

 - the study of structures from nature in order to better understand and use them in design - is also helping researchers in photonic crystals. For example, in 2006 a naturally-occurring photonic crystal was discovered in scales of a Brazilian beetle.

Construction strategies

The fabrication method depends on the number of dimensions that the photonic bandgap must exist in.

One-dimensional photonic crystals

In a one-dimensional photonic crystal, layers of different dielectric constant may be deposited or adhered together to form a band gap in a single direction. A Bragg grating is an example of this type of photonic crystal. One-dimensional photonic crystals can be either isotropic or anisotropic, with the latter having potential use as an optical switch
Optical switch
In telecommunication, an optical switch is a switch that enables signals in optical fibers or integrated optical circuits to be selectively switched from one circuit to another....

.

Two-dimensional photonic crystals

In two dimensions, holes may be drilled in a substrate that is transparent to the wavelength of radiation that the bandgap is designed to block. Triangular and square latices of holes have been successfully employed.

The Holey fiber or photonic crystal fiber can be made by taking cylindrical rods of glass in hexagonal lattice, and then heating and stretching them, the triangle-like airgaps between the glass rods become the holes that confine the modes.

Three-dimensional photonic crystals

There are several structure types that have been constructed:
  • Spheres in a diamond lattice

  • Yablonovite
    Yablonovite
    Yablonovite is a photonic crystal structure that has an inverse cylindrical holes arranged in a diamond lattice. It was the first 3D photonic crystal to be fabricated with a complete photonic bandgap...


  • The Woodpile Structure – "rods" are repeatedly etched using beam lithography, filled in and new material is then deposited thereon, and the process is repeated on the next layer with etched channels that are perpendicular to the layer below, and parallel to and out of phase with the channels two layers below. The process is repeated until the structure is of the desired height. The fill-in material is then dissolved using an agent that can dissolve the fill in material but not the deposition material. It is generally hard to introduce defects into this structure.

  • Inverse opals or Inverse Colloidal Crystals-Spheres (such as polystyrene
    Polystyrene
    Polystyrene ) also known as Thermocole, abbreviated following ISO Standard PS, is an aromatic polymer made from the monomer styrene, a liquid hydrocarbon that is manufactured from petroleum by the chemical industry...

    ) can be allowed to deposit into a cubic close packed lattice suspended in a solvent
    Solvent
    A solvent is a liquid, solid, or gas that dissolves another solid, liquid, or gaseous solute, resulting in a solution that is soluble in a certain volume of solvent at a specified temperature...

    . Then a hardener is introduced which makes a transparent solid out of the volume occupied by the solvent. The spheres are then dissolved using an acid such as Hydrochloric acid
    Hydrochloric acid
    Hydrochloric acid is a solution of hydrogen chloride in water, that is a highly corrosive, strong mineral acid with many industrial uses. It is found naturally in gastric acid....

    .

  • A stack of two-dimensional crystals - This is a more general class of photonic crystals than Yablonivite, but the original implementation of Yablonovite was created using this method.

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 fiber
Photonic-crystal fiber
Photonic-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 fiber
Optical fiber
An optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of...

 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 silicon
Silicon
Silicon is a chemical element with the symbol Si and atomic number 14. A tetravalent metalloid, it is less reactive than its chemical analog carbon, the nonmetal directly above it in the periodic table, but more reactive than germanium, the metalloid directly below it in the table...

) 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 photolithography
Photolithography
Photolithography 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 photomask to a light-sensitive chemical "photoresist", or simply "resist," on the substrate...

 and etching techniques similar to those used for integrated circuits. Some of these techniques are already commercially available. To circumvent nanotechnological methods
Nanotechnology
Nanotechnology is the study of manipulating matter on an atomic and molecular scale. Generally, nanotechnology deals with developing materials, devices, or other structures possessing at least one dimension sized from 1 to 100 nanometres...

 with their complex machinery, alternate approaches have been followed to grow photonic crystals as self-assembled structures from colloidal crystal
Colloidal crystal
A colloidal crystal is an ordered array of colloid 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...

s.

Mass-scale 3D photonic crystal films and fibres can now be produced using a shear-assembly technique which stacks 200-300 nm colloidal polymer spheres into perfect films of fcc lattice. Because the particles have a softer transparent rubber coating the films can be stretched and molded, tuning the photonic bandgaps and producing striking structural color
Color
Color or colour is the visual perceptual property corresponding in humans to the categories called red, green, blue and others. Color derives from the spectrum of light interacting in the eye with the spectral sensitivities of the light receptors...

 effects.

Computing photonic band structure

The photonic band gap (PBG) is essentially the gap between the air-line and the dielectric-line in the dispersion relation
Dispersion (optics)
In 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.
  1. Plane wave expansion method
    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 popular among the photonic crystal community as a method of solving for the band structure of specific...

  2. Finite element method
    Finite element method
    The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...

    .
  3. Finite difference time domain method
  4. Order-n spectral method
  5. KKR method
    Muffin-tin approximation
    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. Many modern electronic structure methods...

  6. Bloch wave – MoM method
    Bloch wave – MoM method
    Bloch wave – MoM is a technique for determining the band structure of triply-periodic electromagnetic media such as photonic crystals. This technique uses the method of moments in combination with a Bloch wave expansion of the electromagnetic field in the structure...



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 DBR
Distributed Bragg reflector
A 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 zone
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...

.

In order to speed up the calculation of the frequency band structure, the Reduced Bloch Mode Expansion (RBME) method can be used. The RBME method applies "on top" of any of the primary expansion methods mentioned above. For large unit cell models, the RBME method can reduce the time for computing the band structure by up to two orders of magnitude.

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 optics
Thin-film optics
Thin-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 paints
ChromaFlair
ChromaFlair 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 inks
Security printing
Security printing is the field of the printing industry that deals with the printing of items such as banknotes, passports, tamper-evident labels, product authentication, 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 fiber
Photonic-crystal fiber
Photonic-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 fiber
Optical fiber
An optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of...

 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.

External links

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