What is a photonic crystal. Methods for making photonic crystals
Ilya Polishchuk, Doctor of Physical and Mathematical Sciences, Professor at the Moscow Institute of Physics and Technology, Leading Researcher, National Research Center "Kurchatov Institute"
The use of microelectronics in information processing and communication systems has fundamentally changed the world. There is no doubt that the consequences of the boom in research work in the field of physics of photonic crystals and devices based on them will be comparable in importance to the creation of integrated microelectronics more than half a century ago. Materials of a new type will make it possible to create optical microcircuits in the "image and likeness" of semiconductor electronics elements, and fundamentally new methods of transmitting, storing and processing information that are being developed today on photonic crystals, in turn, will find application in semiconductor electronics of the future. Not surprisingly, this area of research is one of the hottest in the world's largest scientific centers, high-tech companies and enterprises of the military-industrial complex. Russia, of course, is no exception. Moreover, photonic crystals are the subject of effective international cooperation. As an example, let us refer to more than ten years of cooperation between the Russian Kintech Lab LLC and the well-known American company General Electric.
History of photonic crystals
Historically, the theory of photon scattering on three-dimensional gratings began to develop intensively from the wavelength region ? ~ 0.01-1 nm, which lies in the X-ray range, where the nodes of the photonic crystal are the atoms themselves. In 1986, Eli Yablonovich from the University of California at Los Angeles proposed the idea of creating a three-dimensional dielectric structure, similar to ordinary crystals, in which electromagnetic waves of a certain spectral band could not propagate. Such structures are called photonic bandgap structures or photonic crystals. After 5 years, such a photonic crystal was made by drilling millimetric holes in a material with a high refractive index. Such an artificial crystal, later called yablonovite, did not transmit millimeter-wave radiation and actually realized a photonic structure with a band gap (by the way, phased antenna arrays can also be attributed to the same class of physical objects).
Photonic structures, in which the propagation of electromagnetic (in particular, optical) waves in a certain frequency band in one, two or three directions, is prohibited, can be used to create optical integrated devices for controlling these waves. At present, the ideology of photonic structures underlies the creation of non-threshold semiconductor lasers, lasers based on rare-earth ions, high-Q resonators, optical waveguides, spectral filters, and polarizers. The study of photonic crystals is now being carried out in more than two dozen countries, including Russia, and the number of publications in this area, as well as the number of symposiums and scientific conferences and schools, is growing exponentially.
To understand the processes occurring in a photonic crystal, it can be compared with a semiconductor crystal, and the propagation of photons with the movement of charge carriers - electrons and holes. For example, in ideal silicon, atoms are located in a diamond-like crystal structure, and, according to the band theory of a solid state, charged carriers, propagating through the crystal, interact with the periodic potential of the field of atomic nuclei. This is the reason for the formation of allowed and forbidden bands - quantum mechanics forbids the existence of electrons with energies corresponding to an energy range called the band gap. Similar to conventional crystals, photonic crystals contain a highly symmetrical unit cell structure. Moreover, if the structure of an ordinary crystal is determined by the positions of atoms in the crystal lattice, then the structure of a photonic crystal is determined by the periodic spatial modulation of the dielectric constant of the medium (the modulation scale is comparable to the wavelength of the interacting radiation).
Photonic conductors, insulators, semiconductors and superconductors
Continuing the analogy, photonic crystals can be divided into conductors, insulators, semiconductors, and superconductors.
Photonic conductors have wide allowed bands. These are transparent bodies in which light travels a long distance without being practically absorbed. Another class of photonic crystals, photonic insulators, has wide band gaps. This condition is satisfied, for example, by wide-range multilayer dielectric mirrors. Unlike ordinary opaque media, in which light quickly decays into heat, photonic insulators do not absorb light. As for photonic semiconductors, they have narrower band gaps compared to insulators.
Waveguides based on photonic crystals are used to make photonic textiles (pictured). Such textiles have just appeared, and even the scope of its application has not yet been fully realized. From it you can make, for example, interactive clothes, or you can make a soft display
Photo: emt-photoniccrystal.blogspot.com
Despite the fact that the idea of photonic bands and photonic crystals has been established in optics only in the last few years, the properties of structures with a layered change in the refractive index have long been known to physicists. One of the first practically important applications of such structures was the production of coatings with unique optical characteristics used to create highly efficient spectral filters and reduce unwanted reflections from optical elements (such optics are called coated) and dielectric mirrors with a reflection coefficient close to 100%. As another well-known example of 1D photonic structures, one can mention semiconductor lasers with distributed feedback, as well as optical waveguides with periodic longitudinal modulation of physical parameters (profile or refractive index).
As for ordinary crystals, nature gives us them very generously. Photonic crystals in nature are a rarity. Therefore, if we want to exploit the unique properties of photonic crystals, we are forced to develop various methods for growing them.
How to grow a photonic crystal
The creation of a three-dimensional photonic crystal in the visible wavelength range has been one of the top priorities in materials science over the past ten years, for which most researchers have focused on two fundamentally different approaches. One of them uses the seed template method (template) - the template method. This method creates the prerequisites for the self-organization of synthesized nanosystems. The second method is nanolithography.
Among the first group of methods, the most widespread are those that use monodisperse colloidal spheres as templates for creating solids with a periodic system of pores. These methods make it possible to obtain photonic crystals based on metals, non-metals, oxides, semiconductors, polymers, etc. At the first stage, colloidal spheres of similar size are evenly "packed" in the form of three-dimensional (sometimes two-dimensional) frameworks, which subsequently act as templates as an analogue of natural opal. At the second stage, voids in the template structure are impregnated with liquid, which subsequently turns into a solid frame under various physical and chemical influences. Other methods for filling template voids with a substance are either electrochemical methods or the CVD (Chemical Vapor Deposition) method.
At the last stage, the template (colloidal spheres) is removed using, depending on its nature, the processes of dissolution or thermal decomposition. The resulting structures are often referred to as reverse replicas of the original colloidal crystals or "reverse opals".
For practical use, defect-free regions in a photonic crystal should not exceed 1000 µm2. Therefore, the problem of ordering quartz and polymer spherical particles is one of the most important in the creation of photonic crystals.
In the second group of methods, single-photon photolithography and two-photon photolithography allow the creation of three-dimensional photonic crystals with a resolution of 200 nm and use the property of some materials, such as polymers, which are sensitive to single- and two-photon irradiation and can change their properties under the influence of this radiation. Electron beam lithography is an expensive but high-precision technique for fabricating two-dimensional photonic crystals. In this method, a photoresist that changes its properties under the action of an electron beam is irradiated with the beam at specific locations to form a spatial mask. After irradiation, part of the photoresist is washed off, and the rest is used as a mask for etching in the subsequent technological cycle. The maximum resolution of this method is 10nm. Ion beam lithography is similar in principle, only an ion beam is used instead of an electron beam. The advantages of ion beam lithography over electron beam lithography are that the photoresist is more sensitive to ion beams than electron beams and there is no "proximity effect" that limits the smallest possible area size in electron beam lithography.
Let us also mention some other methods of growing photonic crystals. These include methods for the spontaneous formation of photonic crystals, etching methods, and holographic methods.
Photon future
Predictions are as dangerous as they are tempting. However, predictions about the future of photonic crystal devices are very optimistic. The field of application of photonic crystals is practically inexhaustible. Currently, devices or materials using the unique features of photonic crystals have already appeared on the world market (or will appear in the near future). These are lasers with photonic crystals (low-threshold and non-threshold lasers); waveguides based on photonic crystals (they are more compact and have lower losses compared to conventional fibers); materials with a negative refractive index, which make it possible to focus light to a point smaller than a wavelength; the dream of physicists - superprisms; optical storage and logical devices; displays based on photonic crystals. Photonic crystals will also perform color manipulation. A bendable large-format display on photonic crystals with a high spectral range, from infrared radiation to ultraviolet radiation, has already been developed, in which each pixel is a photonic crystal - an array of silicon microspheres located in space in a strictly defined way. Photonic superconductors are created. Such superconductors can be used to create optical temperature sensors, which, in turn, will operate at high frequencies and are compatible with photonic insulators and semiconductors.
Man is only planning the technological use of photonic crystals, and the sea mouse (Aphrodite aculeata) has been putting them into practice for a long time. The fur of this worm has such a pronounced phenomenon of iridescence that it is able to selectively reflect light with an efficiency close to 100% in the entire visible region of the spectrum - from red to green and blue. Such a specialized "on-board" optical computer helps this worm survive at a depth of up to 500 m. It can be said with certainty that human intelligence will go much further in using the unique properties of photonic crystals.
Photonic crystals (PCs) are structures characterized by a periodic change in the permittivity in space. The optical properties of PCs are very different from the optical properties of continuous media. The propagation of radiation inside a photonic crystal, due to the periodicity of the medium, becomes similar to the movement of an electron inside an ordinary crystal under the action of a periodic potential. As a result, electromagnetic waves in photonic crystals have a band spectrum and a coordinate dependence similar to the Bloch waves of electrons in ordinary crystals. Under certain conditions, gaps form in the band structure of a PC, similarly to forbidden electronic bands in natural crystals. Depending on the specific properties (the material of the elements, their size, and the grating period), the PC spectrum can form both completely frequency-forbidden zones, for which radiation propagation is impossible regardless of its polarization and direction, and partially forbidden (stop-zones), in which can spread only in selected directions.
Photonic crystals are of interest both from a fundamental point of view and for numerous applications. On the basis of photonic crystals, optical filters, waveguides (in particular, in fiber-optic communication lines), devices that allow controlling thermal radiation are created and developed, laser designs with a lower pump threshold have been proposed based on photonic crystals.
In addition to changing the reflection, transmission and absorption spectra, metal-dielectric photonic crystals have a specific density of photonic states. The changed density of states can significantly affect the lifetime of the excited state of an atom or molecule placed inside a photonic crystal and, consequently, change the nature of the luminescence. For example, if the transition frequency in an indicator molecule located in a photonic crystal falls into the band gap, then luminescence at this frequency will be suppressed.
FCs are divided into three types: one-dimensional, two-dimensional and three-dimensional.
One-, two- and three-dimensional photonic crystals. Different colors correspond to materials with different dielectric constants.
One-dimensional are PCs with alternating layers made of different materials.
Electron image of a one-dimensional PC used in a laser as a Bragg multilayer mirror.
Two-dimensional FKs can have more diverse geometries. These include, for example, arrays of cylinders of infinite length (their transverse size is much smaller than the longitudinal one) or periodic systems of cylindrical holes.
Electronic images, two-dimensional forward and reverse FK with a triangular lattice.
The structures of three-dimensional PCs are very diverse. The most common in this category are artificial opals - ordered systems of spherical diffusers. There are two main types of opals: straight and reverse (inverse) opals. The transition from direct opal to reverse opal is carried out by replacing all spherical elements with cavities (usually air), while the space between these cavities is filled with some material.
Below is the surface of a PC, which is a straight opal with a cubic lattice based on self-organized spherical polystyrene microparticles.
The inner surface of a PC with a cubic lattice based on self-organized spherical polystyrene microparticles.
The next structure is an inverse opal synthesized as a result of a multi-stage chemical process: self-assembly of polymer spherical particles, impregnation of voids in the resulting material with a substance, and removal of the polymer matrix by chemical etching.
The surface of a quartz inverse opal. The photograph was obtained using scanning electron microscopy.
Another type of three-dimensional FCs are structures of the "woodpile" type (logpiles), formed by rectangular parallelepipeds crossed, as a rule, at right angles.
Electronic photo of PC from metal parallelepipeds.
Production Methods
The use of FCs in practice is significantly limited by the lack of universal and simple methods for their manufacture. In our time, several approaches to the creation of a FC have been implemented. Two main approaches are described below.
The first of these is the so-called self-organization or self-assembly method. When self-assembling a photonic crystal, colloidal particles are used (the most common are monodisperse silicon or polystyrene particles), which are in the liquid and, as the liquid evaporates, are deposited in the volume. As they "deposit" on each other, they form a three-dimensional PC and are ordered, depending on the conditions, into a cubic face-centered or hexagonal crystal lattice. This method is quite slow, the formation of FC may take several weeks. Also, its disadvantages include a poorly controlled percentage of the appearance of defects in the deposition process.
One of the varieties of the self-assembly method is the so-called honeycomb method. This method involves filtering the liquid in which the particles are located through small pores, and allows the formation of FC at a rate determined by the rate of flow of the liquid through these pores. Compared with the conventional deposition method, this method is much faster, however, the percentage of defects in its use is also higher.
The advantages of the described methods include the fact that they allow the formation of PC samples of large sizes (with an area of up to several square centimeters).
The second most popular method for the manufacture of FC is the etching method. Various etching methods are generally used to fabricate 2D PCs. These methods are based on the use of a photoresist mask (which defines, for example, an array of hemispheres) formed on the surface of a dielectric or metal and defining the geometry of the etched region. This mask can be obtained using the standard photolithography method, followed directly by chemical etching of the sample surface with photoresist. In this case, respectively, in the areas where the photoresist is located, the surface of the photoresist is etched, and in the areas without a photoresist, the dielectric or metal is etched. The process continues until the desired etch depth is reached, after which the photoresist is washed off.
The disadvantage of this method is the use of the photolithography process, the best spatial resolution of which is determined by the Rayleigh criterion. Therefore, this method is suitable for creating a PC with a band gap, which, as a rule, lies in the near infrared region of the spectrum. Most often, a combination of photolithography with electron beam lithography is used to achieve the desired resolution. This method is an expensive but highly accurate method for fabricating quasi-two-dimensional PCs. In this method, a photoresist that changes its properties under the action of an electron beam is irradiated at specific locations to form a spatial mask. After irradiation, part of the photoresist is washed off, and the remaining part is used as an etching mask in the subsequent technological cycle. The maximum resolution of this method is about 10 nm.
Parallels between electrodynamics and quantum mechanics
Any solution of Maxwell's equations , in the case of linear media and in the absence of free charges and current sources, can be represented as a superposition of functions harmonic in time with complex amplitudes depending on frequency: , where is either , or .
Since the fields are real, then , and can be written as a superposition of functions harmonic in time with a positive frequency: ,
Consideration of harmonic functions allows us to pass to the frequency form of Maxwell's equations, which does not contain time derivatives: ,
where the time dependence of the fields involved in these equations is represented as , . We assume that the media are isotropic and that the magnetic permeability is .
Explicitly expressing the field, taking the curl from both sides of the equations, and substituting the second equation into the first, we get:
where is the speed of light in vacuum.
In other words, we got an eigenvalue problem:
for the operator
where the dependence is determined by the structure under consideration.
The eigenfunctions (modes) of the resulting operator must satisfy the condition
Located as
In this case, the condition is met automatically, since the divergence of the rotor is always zero.
The operator is linear, which means that any linear combination of solutions to the eigenvalue problem with the same frequency will also be a solution. It can be shown that in the case this operator is Hermitian, i.e., for any vector functions
where the dot product is defined as
Since the operator is Hermitian it follows that its eigenvalues are real. It can also be shown that at 0" align="absmiddle">, the eigenvalues are non-negative, and hence the frequencies are real.
The scalar product of the eigenfunctions corresponding to different frequencies is always zero. In the case of equal frequencies, this is not necessarily the case, but it is always possible to work only with mutually orthogonal linear combinations of such eigenfunctions. Moreover, it is always possible to form a basis from mutually orthogonal eigenfunctions of the Hermitian operator .
If, on the contrary, we express the field in terms of , we get a generalized eigenvalue problem:
in which operators are already present on both sides of the equation (in this case, after division by the operator on the left side of the equation, it becomes non-Hermitian). In some cases, this formulation is more convenient.
Note that when the equation is replaced by eigenvalues, the frequency will correspond to the new solution. This fact is called scalability and is of great practical importance. The production of photonic crystals with characteristic dimensions on the order of a micron is technically difficult. However, for testing purposes, it is possible to make a model of a photonic crystal with a period and an element size of the order of a centimeter that would operate in centimeter mode (in this case, materials should be used that would have approximately the same permittivity in the centimeter frequency range as the simulated materials).
Let us draw an analogy of the theory described above with quantum mechanics. In quantum mechanics, a scalar wave function is considered that takes complex values. In electrodynamics, it is vector, and the complex dependence is introduced only for convenience. A consequence of this fact, in particular, is that the band structures for photons in a photonic crystal will be different for waves with different polarizations, in contrast to the band structures for electrons.
Both in quantum mechanics and in electrodynamics, the problem is solved for the eigenvalues of the Hermitian operator. In quantum mechanics, Hermitian operators correspond to observables.
And finally, in quantum mechanics, if the operator is represented as a sum , the solution of the eigenvalue equation can be written as , that is, the problem is divided into three one-dimensional ones. In electrodynamics, this is impossible, since the operator "links" all three coordinates, even if they are separated in. For this reason, only a very limited number of problems in electrodynamics have analytical solutions. In particular, exact analytical solutions for the band spectrum of a PC are found mainly for one-dimensional PCs. That is why numerical simulation plays an important role in calculating the properties of photonic crystals.
Band structure
The photonic crystal is characterized by the periodicity of the function:
An arbitrary translation vector represented as
where are primitive translation vectors and are integers.
By Bloch's theorem, the eigenfunctions of an operator can be chosen in such a way that they have the form of a plane wave multiplied by a function that has the same periodicity as the FK:
where is a periodic function. In this case, the values can be selected in such a way that they belong to the first Brillouin zone.
Substituting this expression into the formulated eigenvalue problem, we obtain an eigenvalue equation
Eigenfunctions must be periodic and satisfy the condition .
It can be shown that each value of the vector corresponds to an infinite set of modes with a discrete set of frequencies , which we will number in ascending order with the index . Since the operator depends continuously on , the frequency at a fixed index on also depends continuously. The set of continuous functions constitutes the band structure of the FK. The study of the band structure of a photonic crystal makes it possible to obtain information about its optical properties. The presence of any additional symmetry in the FK allows us to confine ourselves to a certain subdomain of the Brillouin zone, which is called irreducible. The solutions for , which belongs to this irreducible zone, reproduce the solutions for the entire Brillouin zone.
Left: A 2D photonic crystal made up of cylinders packed into a square lattice. Right: The first Brillouin zone corresponding to a square lattice. The blue triangle corresponds to the irreducible Brillouin zone. G, M and X- points of high symmetry for a square lattice.
Frequency intervals that do not correspond to any modes for any real value of the wave vector are called band gaps. The width of such zones increases with an increase in the contrast of the permittivity in a PC (the ratio of the permittivities of the constituent elements of a photonic crystal). If radiation with a frequency lying inside the forbidden band is generated inside such a photonic crystal, it cannot propagate in it (it corresponds to the complex value of the wave vector). The amplitude of such a wave will decay exponentially inside the crystal (evanescent wave). One of the properties of a photonic crystal is based on this: the possibility of controlling spontaneous emission (in particular, its suppression). If such radiation is incident on the PC from outside, then it is completely reflected from the photonic crystal. This effect is the basis for the use of PC for reflective filters, as well as for resonators and waveguides with highly reflective walls.
As a rule, low-frequency modes are concentrated mainly in layers with a large dielectric constant, while high-frequency modes are mostly concentrated in layers with a lower dielectric constant. Therefore, the first zone is often called the dielectric zone, and the one following it is called the air zone.
Band structure of a one-dimensional PC corresponding to wave propagation perpendicular to the layers. In all three cases, each layer has a thickness of 0.5 a, where a- FC period. Left: Each layer has the same permittivity ε
= 13. Center: The permittivity of alternating layers has the values ε
= 12 and ε
= 13. Right: ε
= 1 and ε
= 13.
In the case of a PC with dimensions less than three, there are no complete band gaps for all directions, which is a consequence of the presence of one or two directions along which the PC is homogeneous. Intuitively, this can be explained by the fact that the wave does not experience multiple reflections along these directions, which is required for the formation of band gaps.
Despite this, it is possible to create one-dimensional PCs that would reflect waves incident on the PC at any angle.
The band structure of a one-dimensional PC with a period a, in which the thicknesses of alternating layers are 0.2 a and 0.8 a, and their permittivity - ε
= 13 and ε
= 1, respectively. The left part of the figure corresponds to the direction of wave propagation perpendicular to the layers (0, 0, k z), and the right one - in the direction along the layers (0, k y , 0). The band gap exists only for the direction perpendicular to the layers. Note that when k y > 0, the degeneracy is removed for two different polarizations.
The band structure of a PC with an opal geometry is presented below. It can be seen that this PC has a total band gap at a wavelength of about 1.5 µm and one stop band, with a reflection maximum at a wavelength of 2.5 µm. By varying the etching time of the silicon matrix at one of the stages of inverse opal fabrication and thus by varying the diameter of the spheres, it is possible to localize the band gap in a certain wavelength range. The authors note that a structure with similar characteristics can be used in telecommunication technologies. Radiation at the band gap frequency can be localized inside the volume of the PC, and when the necessary channel is provided, it can propagate virtually without loss. Such a channel can be formed, for example, by removing photonic crystal elements along a certain line. When the channel is bent, the electromagnetic wave will also change direction, repeating the shape of the channel. Thus, such a PC is supposed to be used as a transmission unit between an emitting device and an optical microchip that processes the signal.
Comparison of the reflectance spectrum in the GL direction, measured experimentally, and the band structure calculated by the plane wave expansion method for an inverse silicon (Si) opal with a face-centered cubic lattice (the inset shows the first Brillouin zone). The volume fraction of silicon is 22%. Grating period 1.23 µm
In the case of one-dimensional PCs, even the smallest permittivity contrast is sufficient to form a band gap. It would seem that for three-dimensional dielectric PCs, a similar conclusion can be drawn: to assume the presence of a complete bandgap at any small contrast of dielectric permittivity in the case if, at the boundary of the Brillouin zone, the vector has the same moduli in all directions (which corresponds to the spherical Brillouin zone). However, three-dimensional crystals with a spherical Brillouin zone do not exist in nature. As a rule, it has a rather complex polygonal shape. Thus, it turns out that band gaps in different directions exist at different frequencies. Only if the dielectric contrast is large enough can the stop bands in different directions overlap and form a complete band gap in all directions. Closest to spherical (and thus most independent of the direction of the Bloch vector ) is the first Brillouin zone of the face-centered cubic (fcc) and diamond lattices, making 3D PCs with this structure most suitable for forming a total band gap in the spectrum. At the same time, for the appearance of total band gaps in the spectra of such PCs, a large contrast in the dielectric constant is required. If we denote the relative slit width as , then to achieve the values of 5\%" align="absmiddle">, a contrast is required for the diamond and fcc gratings, respectively. , bearing in mind that all PCs obtained in experiments are not ideal, and defects in the structure can significantly reduce the band gap.
The first Brillouin zone of a cubic face-centered lattice and points of high symmetry.
In conclusion, we note once again the similarity of the optical properties of PCs with the properties of electrons in quantum mechanics when considering the band structure of a solid. However, there is a significant difference between photons and electrons: electrons have a strong interaction with each other. Therefore, “electronic” problems, as a rule, require taking into account many-electron effects, which greatly increase the dimension of the problem, which often forces the use of insufficiently accurate approximations, while in a PC consisting of elements with a negligible nonlinear optical response, this difficulty is absent.
A promising area of modern optics is the control of radiation with the help of photonic crystals. In particular, log-piles PCs were studied at the Sandia Laboratory in order to achieve high selectivity of the emission of metal photonic crystals in the near infrared range, simultaneously with strong suppression of radiation in the mid-IR range (<20мкм). В этих работах было показано, что для таких ФК излучение в среднем ИК диапазоне сильно подавлено из-за наличия в спектре ФК полной фотонной щели. Однако качество полной фотонной щели падает с ростом температуры из-за увеличения поглощения в вольфраме, что приводит к низкой селективности излучения при высоких температурах.
According to Kirchhoff's law for radiation in thermal equilibrium, the emissivity of a gray body (or surface) is proportional to its absorptivity. Therefore, in order to obtain information on the emissivity of metallic PCs, one can study their absorption spectra. To achieve high selectivity of the emitting structure in the visible range (nm) containing PC, it is necessary to choose such conditions under which the absorption in the visible range is large, and in the IR is suppressed.
In our works, http, we analyzed in detail the change in the absorption spectrum of a photonic crystal with elements of tungsten and with the geometry of opal with a change in all its geometric parameters: lattice period, size of tungsten elements, and the number of layers in a PC sample. An analysis was also made of the influence on the absorption spectrum of defects in a PC that arise during its manufacture.
The idea of photonics of nanosized structures and photonic crystals was born while analyzing the possibility of creating an optical band structure. It was assumed that in the optical band structure, as well as in the semiconductor band structure, allowed and forbidden states for photons with different energies should exist. Theoretically, a model of the medium was proposed, in which periodic changes in the permittivity or refractive index of the medium were used as the periodic potential of the lattice. Thus, the concept of "photonic band gap" in a "photonic crystal" was introduced.
Photonic Crystal is a superlattice in which a field is artificially created, and its period is orders of magnitude greater than the period of the main lattice. A photonic crystal is a semitransparent dielectric with a certain periodic structure and unique optical properties.
The periodic structure is formed from the smallest holes, which periodically change the dielectric constant r. The diameter of these holes is such that light waves of a strictly defined length pass through them. All other waves are absorbed or reflected.
Photonic bands are formed in which the phase velocity of light propagation depends on e. In a crystal, light propagates coherently and forbidden frequencies appear, depending on the direction of propagation. Bragg diffraction for photonic crystals takes place in the optical wavelength range.
Such crystals are called photonic bandgap materials (PBGs). From the point of view of quantum electronics, Einstein's law for stimulated emission does not hold in such active media. In accordance with this law, the rates of induced emission and absorption are equal and the sum of the excited N 2 and unexcited
atoms JV is A, + N., = N. Then or 50%.
In photonic crystals, a 100% level population inversion is possible. This makes it possible to reduce the pump power and reduce the unnecessary heating of the crystal.
If the crystal is affected by sound waves, then the length of the light wave and the direction of movement of the light wave, characteristic of the crystal, can change. A distinctive property of photonic crystals is the proportionality of the reflection coefficient R light in the long-wavelength part of the spectrum to its frequency squared co 2, and not as for Rayleigh scattering R~ from 4 . The short-wave component of the optical spectrum is described by the laws of geometric optics.
In the industrial creation of photonic crystals, it is necessary to find a technology for creating three-dimensional superlattices. This is a very difficult task, since standard replication techniques using lithography methods are unacceptable for creating 3D nanostructures.
The attention of researchers was attracted by noble opal (Fig. 2.23). Is it a mineral Si() 2 ? P 1.0 hydroxide subclass. In natural opals, the voids of the globules are filled with silica and molecular water. From the point of view of nanoelectronics, opals are close-packed (mainly according to the cubic law) nanospheres (globules) of silica. As a rule, the diameter of nanospheres is in the range of 200–600 nm. The packing of silica globules forms a three-dimensional lattice. Such superlattices contain structural voids 140–400 nm in size, which can be filled with semiconductor, optically active, and magnetic materials. In an opal-like structure, it is possible to create a three-dimensional lattice with a nanoscale structure. The optical opal matrix structure can serve as a 3E photonic crystal.
The technology of oxidized macroporous silicon has been developed. Based on this technological process, three-dimensional structures in the form of silicon dioxide pins were created (Fig. 2.24).
Photonic band gaps were found in these structures. The band gap parameters can be changed at the stage of lithographic processes or by filling the pin structure with other materials.
Various designs of lasers have been developed on the basis of photonic crystals. Another class of optical elements based on photonic crystals is photonic crystal fibers(FKV). They have
Rice. 2.23. Structure of synthetic opal (a) and natural opals (b)"
" Source: Gudilin E. A.[and etc.]. Wealth of the Nanoworld. Photo essay from the depths of matter; ed. Yu. D. Tretyakova. M.: BINOM. Knowledge Lab, 2010.
Rice. 2.24.
band gap in a given wavelength range. Unlike conventional optical fibers, photonic bandgap fibers have the ability to shift the zero dispersion wavelength to the visible region of the spectrum. In this case, the conditions for soliton regimes of visible light propagation are provided.
By changing the size of the air tubes and, accordingly, the size of the core, it is possible to increase the concentration of the power of light radiation, the nonlinear properties of the fibers. By varying the fiber and cladding geometry, an optimal combination of strong non-linearity and low dispersion can be obtained in the desired wavelength range.
On fig. 2.25 are presented to the FCF. They are divided into two types. The first type is referred to FKV with a continuous light-guiding core. Structurally, such a fiber is made in the form of a core of quartz glass in a shell of a photonic crystal. The wave properties of such fibers are provided both by the effect of total internal reflection and by the band properties of the photonic crystal. Therefore, low-order modes propagate in such fibers in a wide spectral range. High-order modes are shifted into the shell and decay there. In this case, the waveguiding properties of the crystal for zero-order modes are determined by the effect of total internal reflection. The band structure of a photonic crystal manifests itself only indirectly.
The second type of FKV has a hollow light-guiding core. Light can propagate both through the core of the fiber and through the cladding. At the core of
Rice. 2.25.
a - section with a continuous light-guiding core;
6 - section with a hollow light-guiding residential strand, the refractive index is less than the average refractive index of the shell. This makes it possible to significantly increase the power of the transported radiation. At present, fibers have been created that have a loss of 0.58 dB / km at a wavelength X= 1.55 µm, which is close to the loss in standard single-mode fiber (0.2 dB/km).
Among other advantages of photonic crystal fibers, we note the following:
- single-mode mode for all calculated wavelengths;
- wide range of main fashion spot change;
- constant and high value of the dispersion coefficient for wavelengths of 1.3-1.5 μm and zero dispersion for wavelengths in the visible spectrum;
- controlled polarization values, group velocity dispersions, transmission spectrum.
Fibers with a photonic crystal cladding are widely used to solve problems in optics, laser physics, and especially in telecommunications systems. Recently, interest has been attracted by various resonances arising in photonic crystals. Polariton effects in photonic crystals take place during the interaction of electron and photon resonances. When creating metal-dielectric nanostructures with a period much smaller than the optical wavelength, it is possible to realize a situation in which the conditions r
A very significant product of the development of photonics are telecommunication fiber-optic systems. Their functioning is based on the processes of electro-optical conversion of an information signal, transmission of a modulated optical signal to a fiber optic light guide, and inverse opto-electronic conversion.
In the last decade, the development of microelectronics has slowed down, since the limits on the speed of standard semiconductor devices have already been practically reached. An increasing number of studies are devoted to the development of areas alternative to semiconductor electronics - these are spintronics, microelectronics with superconducting elements, photonics, and some others.
The new principle of transmission and processing of information using a light signal, rather than an electrical signal, can accelerate the onset of a new stage in the information age.
From simple crystals to photonic
The basis of electronic devices of the future can be photonic crystals - these are synthetic ordered materials in which the dielectric constant changes periodically inside the structure. In the crystal lattice of a traditional semiconductor, the regularity, the periodicity of the arrangement of atoms leads to the formation of the so-called band energy structure - with allowed and forbidden zones. An electron whose energy falls into the allowed band can move through the crystal, while an electron with energy in the band gap is "locked".
By analogy with an ordinary crystal, the idea of a photonic crystal arose. In it, the periodicity of the permittivity causes the appearance of photonic zones, in particular, the forbidden zone, within which the propagation of light with a certain wavelength is suppressed. That is, being transparent to a wide spectrum of electromagnetic radiation, photonic crystals do not transmit light with a selected wavelength (equal to twice the period of the structure along the length of the optical path).
Photonic crystals can have different dimensions. One-dimensional (1D) crystals are a multilayer structure of alternating layers with different refractive indices. Two-dimensional photonic crystals (2D) can be represented as a periodic structure of rods with different permittivities. The first synthetic prototypes of photonic crystals were three-dimensional and were created in the early 1990s by the staff of the research center Bell Labs(USA). To obtain a periodic lattice in a dielectric material, American scientists drilled cylindrical holes in such a way as to obtain a three-dimensional network of voids. In order for the material to become a photonic crystal, its permittivity was modulated with a period of 1 centimeter in all three dimensions.
Natural analogues of photonic crystals are mother-of-pearl coatings of shells (1D), antennae of a sea mouse, polychaete worm (2D), wings of an African sailboat butterfly and semi-precious stones, such as opal (3D).
But even today, even with the help of the most modern and expensive methods of electron lithography and anisotropic ion etching, it is difficult to produce defect-free three-dimensional photonic crystals with a thickness of more than 10 structural cells.
Photonic crystals should find wide application in photonic integrated technologies, which in the future will replace electrical integrated circuits in computers. When information is transmitted using photons instead of electrons, power consumption will be sharply reduced, clock frequencies and information transfer rates will increase.
Titanium oxide photonic crystal
Titanium oxide TiO 2 has a set of unique characteristics such as high refractive index, chemical stability and low toxicity, which makes it the most promising material for creating one-dimensional photonic crystals. If we consider photonic crystals for solar cells, then titanium oxide wins here because of its semiconductor properties. An increase in the efficiency of solar cells using a semiconductor layer with a periodic photonic crystal structure, including titanium oxide photonic crystals, has been previously demonstrated.
But so far, the use of photonic crystals based on titanium dioxide is limited by the lack of a reproducible and inexpensive technology for their creation.
Nina Sapoletova, Sergei Kushnir and Kirill Napolsky, members of the Faculty of Chemistry and the Faculty of Materials Sciences of Moscow State University, have improved the synthesis of one-dimensional photonic crystals based on porous titanium oxide films.
“Anodizing (electrochemical oxidation) of valve metals, including aluminum and titanium, is an effective method for obtaining porous oxide films with nanometer-sized channels,” explained Kirill Napolsky, head of the electrochemical nanostructuring group, Candidate of Chemical Sciences.
Anodizing is usually carried out in a two-electrode electrochemical cell. Two metal plates, a cathode and an anode, are lowered into the electrolyte solution, and an electric voltage is applied. Hydrogen is released at the cathode, and electrochemical oxidation of the metal occurs at the anode. If the voltage applied to the cell is periodically changed, then a porous film with a porosity specified in thickness is formed on the anode.
The effective refractive index will be modulated if the pore diameter changes periodically within the structure. The titanium anodizing techniques developed earlier did not allow obtaining materials with a high degree of structure periodicity. Chemists from Moscow State University have developed a new method of metal anodizing with voltage modulation depending on the anodizing charge, which allows creating porous anodic metal oxides with high accuracy. The possibilities of the new technique were demonstrated by the chemists using one-dimensional photonic crystals from anodic titanium oxide as an example.
As a result of changing the anodizing voltage according to a sinusoidal law in the range of 40–60 Volts, scientists obtained nanotubes of anodic titanium oxide with a constant outer diameter and a periodically changing inner diameter (see figure).
“The anodizing methods used earlier did not allow obtaining materials with a high degree of structure periodicity. We have developed a new methodology, the key component of which is in situ(immediately during synthesis) measurement of the anodizing charge, which makes it possible to control with high accuracy the thickness of layers with different porosity in the formed oxide film, ”explained one of the authors of the work, candidate of chemical sciences Sergey Kushnir.
The developed technique will simplify the creation of new materials with a modulated structure based on anodic metal oxides. “If we consider the use of photonic crystals from anodic titanium oxide in solar cells as a practical application of the technique, then a systematic study of the influence of the structural parameters of such photonic crystals on the efficiency of light conversion in solar cells remains to be carried out,” Sergey Kushnir specified.
Unusual properties of photonic crystals have been the subject of a huge number of works and, more recently, monographs. Recall that photonic crystals are such artificial media in which, due to a periodic change in the dielectric parameters (meaning the refractive index), the properties of propagating electromagnetic waves (light) become similar to the properties of electrons propagating in real crystals. Accordingly, the term "photonic crystal" emphasizes the similarity of photons and electrons. Quantization of the properties of photons leads to the fact that in the spectrum of an electromagnetic wave propagating in a photonic crystal, forbidden bands can appear, in which the density of photon states is equal to zero.
A three-dimensional photonic crystal with an absolute bandgap was first realized for electromagnetic waves in the microwave range. The existence of an absolute band gap means that electromagnetic waves in a certain frequency band cannot propagate in a given crystal in any direction, since the density of state of photons whose energy corresponds to this frequency band is equal to zero at any point in the crystal. Like real crystals, photonic crystals can be conductors, semiconductors, insulators, and superconductors in terms of the presence and properties of the band gap. If there are "defects" in the band gap of a photonic crystal, then a "capture" of a photon by a "defect" is possible, similar to how an electron or a hole is captured by the corresponding impurity located in the band gap of a semiconductor.
Such propagating waves with energy located inside the band gap are called defect modes.
photonic crystal metamaterial refraction
As already noted, unusual properties of a photonic crystal are observed when the dimensions of the unit cell of the crystal are of the order of the length of the wave propagating in it. It is clear that ideal photonic crystals in the visible range of light can only be produced using submicron technologies. The level of modern science and technology makes it possible to create such three-dimensional crystals.
The applications of photonic crystals are quite numerous - optical isolators, optical isolators, switches, multiplexers, etc. From a practical point of view, one of the extremely important structures is photonic-crystal optical fibers. They were first made from a set of glass capillaries assembled into a dense pack, which was then subjected to conventional drawing. The result was an optical fiber containing regularly spaced holes with a characteristic size of about 1 μm. Subsequently, optical photonic-crystal fibers of various configurations and with various properties were obtained (Fig. 9).
A new drilling method has been developed at the Institute of Radio Engineering and Electronics and at the Research Center for Fiber Optics of the Russian Academy of Sciences to create photonic-crystal light guides. First, mechanical holes with any matrix were drilled in a thick quartz workpiece, and then the workpiece was drawn. As a result, a high quality photonic crystal fiber was obtained. In such fibers, it is easy to create defects of various shapes and sizes, so that several modes of light can be simultaneously excited in them, the frequencies of which lie in the band gap of a photonic crystal. Defects, in particular, can have the form of a hollow channel, so that light will propagate not in quartz, but through air, which can significantly reduce losses in long sections of photonic crystal fibers. The propagation of visible and infrared radiation in photonic crystal fibers is accompanied by a variety of physical phenomena: Raman scattering, harmonic mixing, harmonic generation, which ultimately leads to supercontinuum generation.
No less interesting, from the point of view of studying physical effects and possible applications, are one- and two-dimensional photonic crystals. Strictly speaking, these structures are not photonic crystals, but they can be considered as such when electromagnetic waves propagate in certain directions. A typical one-dimensional photonic crystal is a multilayer periodic structure, consisting of layers of at least two substances with very different refractive indices. If an electromagnetic wave propagates along the normal, a forbidden band appears in such a structure for certain frequencies. If one of the layers of the structure is replaced by a substance with a different refractive index or the thickness of one layer is changed, then such a layer will be a defect capable of capturing a wave whose frequency is in the band gap.
The presence of a magnetic defect layer in a dielectric nonmagnetic structure leads to a multiple increase in the Faraday rotation of the wave during propagation in such a structure and to an increase in the optical transparency of the medium.
Generally speaking, the presence of magnetic layers in photonic crystals can significantly change their properties, primarily in the microwave range. The fact is that in the microwave range, the magnetic permeability of ferromagnets in a certain frequency band is negative, which facilitates their use in the creation of metamaterials. By conjugating such substances with metallic non-magnetic layers or structures consisting of individual conductors or periodic structures of conductors, it is possible to produce structures with negative values of magnetic and dielectric permittivity. An example is the structures created at the Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, designed to detect "negative" reflection and refraction of magnetostatic spin waves. Such a structure is a film of yttrium iron garnet with metal conductors on its surface. The properties of magnetostatic spin waves propagating in thin ferromagnetic films strongly depend on the external magnetic field. In the general case, one of the types of such waves is a backward wave, so the scalar product of the wave vector and the Poynting vector for this type of wave is negative.
The existence of backward waves in photonic crystals is also due to the periodicity of the properties of the crystal itself. In particular, for waves whose wave vectors lie in the first Brillouin zone, the propagation condition can be satisfied as for direct waves, and for the same waves in the second Brillouin zone, as for backward ones. Like metamaterials, photonic crystals can also exhibit unusual properties in propagating waves, such as "negative" refraction.
However, photonic crystals can be the metamaterial for which the phenomenon of "negative" refraction is possible not only in the microwave range, but also in the optical frequency range. Experiments confirm the existence of "negative" refraction in photonic crystals for waves with frequencies higher than the frequency of the first forbidden zone near the center of the Brillouin zone. This is due to the effect of the negative group velocity and, as a consequence, the negative refractive index for the wave. In fact, in this frequency range, the waves become backward.
