Studies of one-dimensional photonic crystals band structure. Photonic crystals

A huge number of works, and recently monographs, are devoted to the unusual properties of photonic crystals. Let us recall that photonic crystals are those artificial media in which, due to periodic changes in 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 between 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 states of photons is 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 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 their band gap. If there are “defects” in the band gap of a photonic crystal, then a photon can be “captured” by the “defect”, similar to how an electron or hole is captured by a 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 elementary 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 light range 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 gates, switches, multiplexers, etc. One of the extremely important structures from a practical point of view is photonic crystal optical fibers. They were first made from a set of glass capillaries collected in a dense pack, which was then subjected to conventional hood. The result was an optical fiber containing regularly spaced holes with a characteristic size of about 1 micron. Subsequently, optical photonic crystal light guides of various configurations and with different properties were obtained (Fig. 9).

A new drilling method for creating photonic crystal light guides has been developed at the Institute of Radio Engineering and Electronics and the Scientific Center for Fiber Optics of the Russian Academy of Sciences. First, mechanical holes with any matrix were drilled in a thick quartz workpiece, and then the workpiece was drawn. The result was a high-quality photonic crystal fiber. In such light guides it is easy to create defects of various shapes and sizes, so that several light modes can be excited simultaneously in them, the frequencies of which lie in the band gap of the photonic crystal. Defects, in particular, can take the form of a hollow channel, so that light will propagate not in quartz, but through the air, which can significantly reduce losses in long sections of photonic crystal light guides. The propagation of visible and infrared radiation in photonic crystal light guides is accompanied by various physical phenomena: Raman scattering, harmonic mixing, harmonic generation, which ultimately leads to the generation of supercontinuum.

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 widely different refractive indices. If an electromagnetic wave propagates along the normal, a band gap for certain frequencies appears in such a structure. If one of the layers of the structure is replaced with a substance with a different refractive index from the others 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 non-magnetic structure leads to a multiple increase in the Faraday rotation of the wave when propagating 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 coupling 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 constants. 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. This 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 Pointing 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 fulfilled 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 a 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 band gap near the center of the Brillouin zone. This is due to the effect of negative group velocity and, as a consequence, a negative refractive index for the wave. In fact, in this frequency range the waves become reversed.

Classification of methods for manufacturing photonic crystals. Photonic crystals are very rare in nature. They are distinguished by a special rainbow play of light - an optical phenomenon called iridescence (translated from Greek - rainbow). Such minerals include calcite, labradorite and opal SiO 2 ×n∙H 2 O with various inclusions. The most famous among them is opal - a semi-precious mineral, which is a colloidal crystal consisting of monodisperse spherical globules of silicon oxide. The play of light in the latter gives rise to the term opalescence, which denotes a special type of radiation scattering characteristic only of this crystal.

The main methods for manufacturing photonic crystals include methods that can be divided into three groups:

1. Methods using spontaneous formation of photonic crystals. This group of methods uses colloidal particles, such as monodisperse silicone or polystyrene particles, as well as other materials. Such particles, being in liquid vapor during evaporation, settle in a certain volume. As the particles deposit on each other, they form a three-dimensional photonic crystal, and are ordered predominantly into face-centered or hexagonal crystal lattices. The cellular method is also possible, which involves filtering a liquid containing particles through small spores. Although the honeycomb method allows the formation of a crystal at a relatively high speed, determined by the speed of liquid flow through the pores, however, defects form in such crystals when drying. There are other methods that use the spontaneous formation of photonic crystals, but each method has its own advantages and disadvantages. Most often, these methods are used to deposit spherical colloidal silicone particles, however, the resulting refractive index contrast is relatively small.

2. Methods using etching of objects. This group of methods uses a photoresist mask formed on the surface of the semiconductor, which sets the geometry of the etching area. Using such a mask, a simple photonic crystal is formed by etching the surface of a semiconductor not covered with photoresist. The disadvantage of this method is the need to use photolithography with high resolution at the level of tens and hundreds of nanometers. Beams of focused ions, such as Ga, are also used to produce photonic crystals by etching. Such ion beams make it possible to remove part of the material without the use of photolithography and additional etching. To increase the etching speed and improve its quality, as well as to deposit materials inside the etched areas, additional treatment with the necessary gases is used.

3. Holographic methods. Such methods are based on the application of holographic principles. Using holography, periodic changes in the refractive index in spatial directions are formed. To do this, the interference of two or more coherent waves is used, which creates a periodic distribution of the intensity of electromagnetic radiation. One-dimensional photonic crystals are created by the interference of two waves. Two-dimensional and three-dimensional photonic crystals are created by the interference of three or more waves.

The choice of a specific method for manufacturing photonic crystals is largely determined by the size of the structure that needs to be manufactured - one-dimensional, two-dimensional or three-dimensional.

One-dimensional periodic structures. The simplest and most common way to obtain one-dimensional periodic structures is vacuum layer-by-layer deposition of polycrystalline films from dielectric or semiconductor materials. This method has become widespread due to the use of periodic structures in the production of laser mirrors and interference filters. In such structures, when using materials with refractive indices that differ by approximately 2 times (for example, ZnSe and Na 3 AlF 6), it is possible to create spectral reflection bands (photonic band gaps) up to 300 nm wide, covering almost the entire visible region of the spectrum.

Advances in the synthesis of semiconductor heterostructures in recent decades have made it possible to create completely single-crystalline structures with a periodic change in the refractive index along the growth direction using molecular beam epitaxy or organometallic vapor deposition techniques. Currently, such structures are part of semiconductor lasers with vertical cavities. The maximum currently achievable ratio of refractive indices of materials apparently corresponds to the GaAs/Al 2 O 3 pair and is about 2. It should be noted the high perfection of the crystal structure of such mirrors and the accuracy of the formation of layer thicknesses at the level of one lattice period (about 0.5 nm).

Recently, the possibility of creating periodic one-dimensional semiconductor structures using a photolithographic mask and selective etching has been demonstrated. When etching silicon, it is possible to create structures with a period of the order of 1 micron or more, while the ratio of the refractive indices of silicon and air in the near-infrared region is 3.4 - an unprecedentedly large value unattainable by other synthesis methods. An example of a similar structure obtained at the Physico-Technical Institute named after. A.F. Ioffe RAS (St. Petersburg), shown in Fig. 3.96.

Rice. 3.96. Periodic structure of silicon - air, obtained by anisotropic etching using a photolithographic mask (structure period 8 μm)

Two-dimensional periodic structures. Two-dimensional periodic structures can be fabricated using selective etching of semiconductors, metals, and dielectrics. Selective etching technology has been developed for silicon and aluminum due to the widespread use of these materials in microelectronics. Porous silicon, for example, is considered a promising optical material that will allow the creation of highly integrated optoelectronic systems. The combination of advanced silicon technologies with quantum-size effects and the principles of the formation of photonic band gaps has led to the development of a new direction - silicon photonics.

The use of submicron lithography to form masks makes it possible to create silicon structures with a period of 300 nm or less. Due to their strong absorption of visible light, silicon photonic crystals can only be used in the near and mid-infrared regions of the spectrum. The combination of etching and oxidation, in principle, makes it possible to move to periodic silicon oxide-air structures, but at the same time the low refractive index ratio (1.45) does not allow the formation of a full band gap in two dimensions.

Two-dimensional periodic structures from semiconductor compounds A 3 B 5, also obtained by selective etching using lithographic masks or templates, seem promising. A 3 B 5 compounds are the main materials of modern optoelectronics. InP and GaAs compounds have larger band gaps than silicon and refractive index values that are as high as silicon, equal to 3.55 and 3.6, respectively.

Periodic structures based on aluminum oxide seem very interesting (Fig. 3.97a). They are obtained by electrochemical etching of aluminum metal, on the surface of which a mask is formed using lithography. Using electron lithographic templates, perfect two-dimensional periodic structures resembling a honeycomb with a pore diameter of less than 100 nm were obtained. It should be noted that selective etching of aluminum under a certain combination of etching conditions makes it possible to obtain regular structures even without the use of any masks or templates (Fig. 3.97b). The pore diameter can be only a few nanometers, which is unattainable with modern lithographic methods. The periodicity of pores is associated with the self-regulation of the process of aluminum oxidation during an electrochemical reaction. The initial conductive material (aluminum) is oxidized to Al 2 O 3 during the reaction. The aluminum oxide film, which is a dielectric, reduces the current and slows down the reaction. The combination of these processes allows a self-sustaining reaction regime to be achieved, in which continuous etching is made possible by the passage of current through the pores, and the reaction product forms a regular honeycomb structure. Some irregularity of the pores (Fig. 3.97b) is due to the granular structure of the original polycrystalline aluminum film.

Rice. 3.97. Two-dimensional photonic crystal from Al 2 O 3: a) made using a lithographic mask; b) manufactured using self-regulation of the oxidation process

A study of the optical properties of nanoporous aluminum oxide showed an unusually high transparency of this material along the direction of the pores. The absence of Fresnel reflection, which inevitably exists at the interface between two continuous media, leads to transmittance values reaching 98%. In directions perpendicular to the pores, high reflection is observed with a reflection coefficient depending on the angle of incidence.

The relatively low values of the dielectric constant of aluminum oxide, in contrast to silicon, gallium arsenide and indium phosphide, do not allow the formation of a full band gap in two dimensions. However, despite this, the optical properties of porous aluminum oxide turn out to be quite interesting. For example, it has pronounced anisotropic light scattering, as well as birefringence, which makes it possible to use it to rotate the plane of polarization. Using various chemical methods, it is possible to fill pores with various oxides, as well as optically active materials, for example, nonlinear optical media, organic and inorganic phosphors, and electroluminescent compounds.

Three-dimensional periodic structures. Three-dimensional periodic structures are objects that have the greatest technological difficulties for experimental implementation. Historically, the first method of creating a three-dimensional photonic crystal is considered to be a method based on mechanical drilling of cylindrical holes in the bulk of the material, proposed by E. Yablonovich. Manufacturing such a three-dimensional periodic structure is a rather labor-intensive task, so many researchers have attempted to create a photonic crystal using other methods. Thus, in the Lin–Fleming method, a layer of silicon dioxide is applied to a silicon substrate, in which parallel strips are then formed and filled with polycrystalline silicon. Next, the process of applying silicon dioxide is repeated, but the stripes are formed in a perpendicular direction. After creating the required number of layers, the silicon oxide is removed by etching. As a result, a “woodpile” of polysilicon rods is formed (Fig. 3.98). It should be noted that the use of modern methods of submicron electron lithography and anisotropic ion etching makes it possible to obtain photonic crystals with a thickness of less than 10 structural cells.

Rice. 3.98. Three-dimensional photonic structure made of polysilicon rods

Methods for creating photonic crystals for the visible range, based on the use of self-organizing structures, have become widespread. The very idea of “assembling” photonic crystals from globules (balls) was borrowed from nature. It is known, for example, that natural opals have the properties of photonic crystals. The natural mineral opal in chemical composition is a silicon dioxide hydrogel SiO 2 × H 2 O with variable water content: SiO 2 – 65 – 90 wt. %; H 2 O – 4.5–20%; Al 2 O 3 – up to 9%; Fe 2 O 3 – up to 3%; TiO 2 – up to 5%. Using electron microscopy methods, it was established that natural opals are formed by densely packed spherical α-SiO 2 particles of uniform size with a diameter of 150 – 450 nm. Each particle consists of smaller globular formations with a diameter of 5–50 nm. The voids in the globule packing are filled with amorphous silicon oxide. The intensity of diffracted light is influenced by two factors: the first is the “ideality” of the densest packing of globules, the second is the difference in the refractive indices of amorphous and crystalline SiO 2 oxide. Noble black opals have the best play of light (for them the difference in refractive index values is ~0.02).

It is possible to create globular photonic crystals from colloidal particles in various ways: natural sedimentation (sedimentation of a dispersed phase in a liquid or gas under the influence of a gravitational field or centrifugal forces), centrifugation, filtration using membranes, electrophoresis, etc. Spherical particles act as colloidal particles polystyrene, polymethyl methacrylate, particles of silicon dioxide α-SiO 2.

The natural deposition method is a very slow process, requiring several weeks or even months. Centrifugation greatly accelerates the process of formation of colloidal crystals, but the materials obtained in this way are less ordered, since at a high sedimentation rate, separation of particles by size does not have time to occur. To speed up the sedimentation process, electrophoresis is used: they create a vertical electric field that “changes” the gravity of the particles depending on their size. Methods based on the use of capillary forces are also used. The basic idea is that under the action of capillary forces, crystallization occurs at the meniscus interface between the vertical substrate and the suspension, and as the solvent evaporates, a fine ordered structure is formed. Additionally, a vertical temperature gradient is used, which makes it possible to better optimize the speed of the process and the quality of the created crystal due to convection flows. In general, the choice of technique is determined by the requirements for the quality of the resulting crystals and the time required for their production.

The technological process of growing synthetic opals using natural sedimentation can be divided into several stages. Initially, a monodisperse (~ 5% deviation in diameter) suspension of spherical globules of silicon oxide is prepared. The average particle diameter can vary over a wide range: from 200 to 1000 nm. The most well-known method for producing monodisperse colloidal microparticles of silicon dioxide is based on the hydrolysis of tetraethoxysilane Si(C 2 H 4 OH) 4 in an aqueous alcoholic medium in the presence of ammonium hydroxide as a catalyst. This method can produce particles with a smooth surface of an almost ideal spherical shape with a high degree of monodispersity (less than 3% deviation in diameter), as well as create particles with sizes less than 200 nm with a narrow size distribution. The internal structure of such particles is fractal: the particles consist of densely packed spheres of a smaller size (diameter of several tens of nanometers), and each such sphere is formed by polyhydroxy silicon complexes consisting of 10–100 atoms.

The next stage is the deposition of particles (Fig. 3.99). It can last for several months. Upon completion of the deposition stage, a close-packed periodic structure is formed. Next, the deposit is dried and annealed at a temperature of about 600 ºС. During the annealing process, softening and deformation of the spheres occurs at the points of contact. As a result, the porosity of synthetic opals is less than that for an ideal dense spherical packing. Perpendicular to the direction of the growth axis of the photonic crystal, the globules form highly ordered hexagonal close-packed layers.

Rice. 3.99. Stages of growing synthetic opals: a) deposition of particles;

b) drying the sediment; c) sample annealing

In Fig. Figure 3.100a shows a micrograph of synthetic opal obtained by scanning electron microscopy. The dimensions of the spheres are 855 nm. The presence of open porosity in synthetic opals allows the voids to be filled with various materials. Opal matrices are three-dimensional sublattices of interconnected nano-sized pores. The pore sizes are on the order of hundreds of nanometers; the sizes of the channels connecting the pores reach tens of nanometers. In this way, nanocomposites based on photonic crystals are obtained. The main requirement put forward when creating high-quality nanocomposites is the complete filling of the nanoporous space. Filling is carried out using various methods: injection from a solution into the melt; impregnation with concentrated solutions followed by evaporation of the solvent; electrochemical methods, chemical vapor deposition, etc.

Rice. 3.100. Microphotographs of photonic crystals: a) from synthetic opal;

b) from polystyrene microspheres

When selective etching of silicon oxide from such composites, spatially ordered nanostructures with high porosity (more than 74% of the volume), called reverse or inverted opals, are formed. This method of producing photonic crystals is called the template method. Not only silicon oxide particles, but also, for example, polymer particles can act as ordered monodisperse colloidal particles forming a photonic crystal. An example of a photonic crystal based on polystyrene microspheres is shown in Fig. 3.100b

Photonic crystals can be divided into three main classes according to the nature of the change in refractive index:

1. One-dimensional, in which the refractive index periodically changes in one spatial direction as shown in Figure 2. In this figure, the symbol L indicates the period of change in the refractive index, and and are the refractive indices of two materials (but in the general case, any number of materials may be present). Such photonic crystals consist of layers of different materials parallel to each other with different refractive indices and can exhibit their properties in one spatial direction, perpendicular to the layers.

Figure 1 - Schematic representation of a one-dimensional photonic crystal

2. Two-dimensional, in which the refractive index varies periodically in two spatial directions as shown in Figure 2. In this figure, the photonic crystal is created by rectangular regions with a refractive index, which are located in a medium with a refractive index. In this case, regions with a refractive index are ordered in a two-dimensional cubic lattice. Such photonic crystals can exhibit their properties in two spatial directions, and the shape of the regions with the refractive index is not limited to rectangles, as in the figure, but can be any (circles, ellipses, arbitrary, etc.). The crystal lattice in which these areas are ordered can also be different, and not just cubic, as in the above figure.

Figure - 2 Schematic representation of a two-dimensional photonic crystal

3. Three-dimensional, in which the refractive index periodically changes in three spatial directions. Such photonic crystals can exhibit their properties in three spatial directions, and they can be represented as an array of volumetric regions (spheres, cubes, etc.) ordered in a three-dimensional crystal lattice.

There are also resonant and non-resonant photonic crystals. Resonant photonic crystals differ from non-resonant ones in that they use materials whose dielectric constant (or refractive index) as a function of frequency has a pole at some resonant frequency.

Like electrical media, depending on the width of the forbidden and allowed zones, photonic crystals can be divided into conductors - capable of conducting light over long distances with low losses, dielectrics - almost ideal mirrors, semiconductors - substances capable, for example, of selectively reflecting photons of a certain wavelength and superconductors, in which, thanks to collective phenomena, photons are able to propagate over almost unlimited distances. There are also resonant and non-resonant photonic crystals. Resonant photonic crystals differ from non-resonant ones in that they use materials whose dielectric constant (or refractive index) as a function of frequency has a pole at some resonant frequency.

Any inhomogeneity in a photonic crystal is called a photonic crystal defect. The electromagnetic field is often concentrated in such areas, which is used in microcavities and waveguides built on the basis of photonic crystals. There are a number of analogies when describing the propagation of electromagnetic waves in photonic crystals and the electronic properties of crystals. Let's list some of them.

1. The state of the electron inside the crystal (the law of motion) is given by solving the Schrldinger equation; the propagation of light in a photonic crystal obeys the wave equation, which is a consequence of Maxwell’s equations:

2. The state of the electron is described by the scalar wave function w(r,t), the state of the electromagnetic wave is described by vector fields - the strength of the magnetic or electrical components, H (r,t) or E(r,t).
3. The electron wave function w(r,t) can be expanded into a series of eigenstates wE(r), each of which has its own energy E. The electromagnetic field strength H(r,t) can be represented by a superposition of monochromatic components (modes) electromagnetic field Hsh(r), each of which corresponds to its own value - the mode frequency u:

4. Atomic potential U(r) and dielectric constant e(r), appearing in the Schrldinger and Maxwell equations, are periodic functions with periods equal to any vectors R of the crystal lattice and photonic crystal, respectively:

U(r) = U(r + R), (3)

5. For the electron wave function and the electromagnetic field strength, Bloch’s theorem is satisfied with periodic functions u k and u k.

6. Possible values of wave vectors k fill the Brillouin zone of the crystal lattice or unit cell of a photonic crystal, defined in the space of inverse vectors.
7. The electron energy E, which is the eigenvalue of the Schrldinger equation, and the eigenvalue of the wave equation (consequences of Maxwell’s equations) - the mode frequency u - are related to the values of the wave vectors k of the Bloch functions (4) by the dispersion law E(k) and u(k).
8. An impurity atom that violates the translational symmetry of the atomic potential is a crystal defect and can create an impurity electronic state localized in the vicinity of the defect. Changes in the dielectric constant in a certain region of the photonic crystal break the translational symmetry e(r) and lead to the appearance of a permitted mode inside the photonic band gap, localized in its spatial vicinity.

Rice. 2. Schematic representation of a one-dimensional photonic crystal.

1. one-dimensional, in which the refractive index periodically changes in one spatial direction as shown in Fig. 2. In this figure, the symbol Λ indicates the period of change of the refractive index, and - the refractive indices of two materials (but in general, any number of materials can be present). Such photonic crystals consist of layers of different materials parallel to each other with different refractive indices and can exhibit their properties in one spatial direction, perpendicular to the layers.

Rice. 3. Schematic representation of a two-dimensional photonic crystal.

2. two-dimensional, in which the refractive index periodically changes in two spatial directions as shown in Fig. 3. In this figure, a photonic crystal is created by rectangular regions of refractive index , which are in a medium of refractive index. In this case, regions with a refractive index are ordered in a two-dimensional cubic lattice. Such photonic crystals can exhibit their properties in two spatial directions, and the shape of the regions with the refractive index is not limited to rectangles, as in the figure, but can be any (circles, ellipses, arbitrary, etc.). The crystal lattice in which these areas are ordered can also be different, and not just cubic, as in the above figure.

3. three-dimensional, in which the refractive index periodically changes in three spatial directions. Such photonic crystals can exhibit their properties in three spatial directions, and they can be represented as an array of volumetric regions (spheres, cubes, etc.) ordered in a three-dimensional crystal lattice.

A distinction is also made between resonant and non-resonant photonic crystals. Resonant photonic crystals differ from non-resonant ones in that they use materials whose dielectric constant (or refractive index) as a function of frequency has a pole at some resonant frequency.

Any inhomogeneity in a photonic crystal (for example, the absence of one or more squares in Fig. 3, their larger or smaller size relative to the squares of the original photonic crystal, etc.) is called a defect in the photonic crystal. The electromagnetic field is often concentrated in such areas, which is used in microcavities and waveguides built on the basis of photonic crystals.

Methods for theoretical study of photonic crystals, numerical methods and software

Photonic crystals allow manipulation of electromagnetic waves in the optical range, and the characteristic dimensions of photonic crystals are often close to the wavelength. Therefore, the methods of ray theory are not applicable to them, but wave theory and the solution of Maxwell's equations are used. Maxwell's equations can be solved analytically and numerically, but it is numerical solution methods that are most often used to study the properties of photonic crystals due to their availability and easy adjustment to the problems being solved.

It is also appropriate to mention that two main approaches are used to consider the properties of photonic crystals - methods for the time domain (which provide a solution to the problem depending on the time variable), and methods for the frequency domain (which provide the solution to the problem as a function of frequency).

Time domain methods are convenient for dynamic problems that involve the time dependence of the electromagnetic field. They can also be used to calculate the band structures of photonic crystals, but it is practically difficult to identify the band positions in the output of such methods. In addition, when calculating band diagrams of photonic crystals, the Fourier transform is used, the frequency resolution of which depends on the total calculation time of the method. That is, to obtain greater resolution in the band diagram, you need to spend more time performing calculations. There is also another problem - the time step of such methods must be proportional to the size of the spatial grid of the method. The requirement to increase the frequency resolution of band diagrams requires a reduction in the time step, and therefore the size of the spatial grid, an increase in the number of iterations, the required computer memory and calculation time. Such methods are implemented in well-known commercial modeling packages Comsol Multiphysics (uses the finite element method to solve Maxwell’s equations), RSOFT Fullwave (uses the finite difference method), independently developed program codes for finite element and difference methods, etc.

Methods for the frequency domain are convenient primarily because the solution of Maxwell’s equations occurs immediately for a stationary system and the frequencies of the optical modes of the system are determined directly from the solution; this makes it possible to calculate band diagrams of photonic crystals faster than using methods for the time domain. Their advantages include the number of iterations, which is practically independent of the resolution of the spatial grid of the method and the fact that the error of the method numerically decreases exponentially with the number of iterations performed. The disadvantages of the method are the need to calculate the natural frequencies of the optical modes of the system in the low-frequency region in order to calculate frequencies in the higher-frequency region, and, naturally, the impossibility of describing the dynamics of the development of optical oscillations in the system. These methods are implemented in the free MPB software package and the commercial package. Both software packages mentioned cannot calculate the band diagrams of photonic crystals in which one or more materials have complex refractive index values. To study such photonic crystals, a combination of two RSOFT packages - BandSolve and FullWAVE - is used, or the perturbation method is used

Of course, theoretical studies of photonic crystals are not limited only to the calculation of band diagrams, but also require knowledge about stationary processes during the propagation of electromagnetic waves through photonic crystals. An example is the problem of studying the transmission spectrum of photonic crystals. For such problems, you can use both of the approaches mentioned above based on convenience and their availability, as well as radiative transfer matrix methods, a program for calculating the transmission and reflection spectra of photonic crystals using this method, the pdetool software package which is part of the Matlab package and the package already mentioned above Comsol Multiphysics.

Photonic band gap theory

As noted above, photonic crystals make it possible to obtain allowed and forbidden bands for photon energies, similar to semiconductor materials, in which there are allowed and forbidden bands for charge carrier energies. In the literature, the appearance of band gaps is explained by the fact that under certain conditions, the intensities of the electric field of standing waves of a photonic crystal with frequencies close to the frequency of the band gap are shifted to different regions of the photonic crystal. Thus, the field intensity of low-frequency waves is concentrated in areas with a high refractive index, and the field intensity of high-frequency waves is concentrated in areas with a lower refractive index. The work contains another description of the nature of band gaps in photonic crystals: “photonic crystals are usually called media in which the dielectric constant changes periodically in space with a period allowing Bragg diffraction of light.”

If radiation with a band gap frequency was generated inside such a photonic crystal, then it cannot propagate in it, but if such radiation is sent from outside, then it is simply reflected from the photonic crystal. One-dimensional photonic crystals make it possible to obtain band gaps and filtering properties for radiation propagating in one direction, perpendicular to the layers of materials shown in Fig. 2. Two-dimensional photonic crystals can have band gaps for radiation propagating in one, two directions, or in all directions of a given photonic crystal, which lie in the plane of Fig. 3. Three-dimensional photonic crystals can have band gaps in one, several, or all directions. Forbidden zones exist for all directions in a photonic crystal with a large difference in the refractive indices of the materials that make up the photonic crystal, certain shapes of regions with different refractive indices and a certain crystal symmetry.

The number of band gaps, their position and width in the spectrum depends both on the geometric parameters of the photonic crystal (the size of regions with different refractive indices, their shape, the crystal lattice in which they are ordered) and on the refractive indices. Therefore, forbidden zones can be tunable, for example, due to the use of nonlinear materials with a pronounced Kerr effect, due to changes in the sizes of areas with different refractive indexes, or due to changes in refractive indices under the influence of external fields.

Rice. 5. Band diagram for photon energies (TE polarization).

Rice. 6. Band diagram for photon energies (TM polarization).

Let us consider the band diagrams of the photonic crystal shown in Fig. 4. This two-dimensional photonic crystal consists of two materials alternating in the plane - gallium arsenide GaAs (base material, refractive index n=3.53, black areas in the figure) and air (with which the cylindrical holes are filled, indicated in white, n=1 ). The holes have a diameter and are ordered in a hexagonal crystal lattice with a period (the distance between the centers of adjacent cylinders). In the photonic crystal under consideration, the ratio of the hole radius to the period is equal to . Let's consider the band diagrams for TE (the electric field vector is directed parallel to the axes of the cylinders) and TM (the magnetic field vector is directed parallel to the axes of the cylinders) shown in Fig. 5 and 6, which were calculated for this photonic crystal using the free MPB program. The X axis shows the wave vectors in the photonic crystal, and the Y axis shows the normalized frequency ( - wavelength in vacuum) corresponding to the energy states. The blue and red solid curves in these figures represent the energy states in a given photonic crystal for TE and TM polarized waves, respectively. The blue and pink areas show the photon band gaps in a given photonic crystal. The black dashed lines are the so-called light lines (or light cone) of a given photonic crystal. One of the main applications of these photonic crystals is optical waveguides, and the light line defines the region within which the waveguide modes of the low-loss waveguides built using such photonic crystals are located. In other words, the light line defines the zone of energy states of interest to us for a given photonic crystal. The first thing worth paying attention to is that this photonic crystal has two band gaps for TE-polarized waves and three wide band gaps for TM-polarized waves. Secondly, the forbidden zones for TE and TM-polarized waves, lying in the region of small values of the normalized frequency, overlap, which means that a given photonic crystal has a complete forbidden zone in the region of overlap of the forbidden zones of TE and TM waves, not only in all directions, but also for waves of any polarization (TE or TM).

Rice. 7. Reflection spectrum of the photonic crystal under consideration (TE polarization).

Rice. 8. Reflection spectrum of the photonic crystal under consideration (TM polarization).

From the given dependencies we can determine the geometric parameters of a photonic crystal, the first band gap of which, with the value of the normalized frequency, falls on the wavelength nm. The period of the photonic crystal is nm, the radius of the holes is nm. Rice. 7 and 8 show the reflectance spectra of a photonic crystal with the parameters defined above for TE and TM waves, respectively. The spectra were calculated using the Translight program, it was assumed that this photonic crystal consists of 8 pairs of layers of holes and the radiation propagates in the Γ-K direction. From the above dependencies we can see the most well-known property of photonic crystals - electromagnetic waves with natural frequencies corresponding to the band gaps of the photonic crystal (Fig. 5 and 6) are characterized by a reflection coefficient close to unity and are subject to almost complete reflection from a given photonic crystal. Electromagnetic waves with frequencies outside the band gaps of a given photonic crystal are characterized by lower reflection coefficients from the photonic crystal and pass through it completely or partially.

Fabrication of photonic crystals

There are currently many methods for making photonic crystals, and new methods continue to emerge. Some methods are more suitable for the formation of one-dimensional photonic crystals, others are convenient for two-dimensional ones, others are more often applicable to three-dimensional photonic crystals, others are used in the production of photonic crystals on other optical devices, etc. Let's consider the most famous of these methods.

Methods using spontaneous formation of photonic crystals

In the spontaneous formation of photonic crystals, colloidal particles are used (most often monodisperse silicone or polystyrene particles are used, but other materials are gradually becoming available for use as technological methods for their production are developed), which are located in a liquid and, as the liquid evaporates, settle in a certain volume. As they deposit on each other, they form a three-dimensional photonic crystal, and are ordered predominantly into face-centered or hexagonal crystal lattices. This method is quite slow and can take weeks to form a photonic crystal.

Another method for spontaneously forming photonic crystals, called the honeycomb method, involves filtering a liquid containing particles through small pores. This method, presented in the works, makes it possible to form a photonic crystal at a speed determined by the speed of liquid flow through the pores, but when such a crystal dries, defects are formed in the crystal.

It was already noted above that in most cases a large refractive index contrast in a photonic crystal is required to obtain photonic band gaps in all directions. The above-mentioned methods of spontaneous formation of a photonic crystal were most often used to deposit spherical colloidal particles of silicone, the refractive index of which is small, and therefore the refractive index contrast is also small. To increase this contrast, additional technological steps are used in which the space between the particles is first filled with a material with a high refractive index, and then the particles are etched. The step-by-step method for forming inverse opal is described in the laboratory work instructions.

Etching methods

Holographic methods

Holographic methods for creating photonic crystals are based on the application of the principles of holography to form a periodic change in the refractive index in spatial directions. This uses the interference of two or more coherent waves, which creates a periodic distribution of electric field intensity. The interference of two waves allows you to create one-dimensional photonic crystals, three or more beams - two-dimensional and three-dimensional photonic crystals.

Other methods for creating photonic crystals

Single-photon photolithography and two-photon photolithography create three-dimensional photonic crystals with a resolution of 200 nm and take advantage of the properties of some materials, such as polymers, that are sensitive to one- and two-photon radiation and can change their properties when exposed to this radiation. Electron beam lithography is an expensive but highly accurate method for fabricating two-dimensional photonic crystals. In this method, a photoresist that changes its properties under the action of an electron beam is irradiated by the beam at specific locations to form a spatial mask. After irradiation, part of the photoresist is washed off, and the remaining part 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, but instead of an electron beam, an ion beam is used. The advantages of ion beam lithography over electron beam lithography are that the photoresist is more sensitive to ion beams than to electron beams and there is no “proximity effect” that limits the smallest possible area size in beam lithography electrons

Application

The distributed Bragg reflector is an already widely used and well-known example of a one-dimensional photonic crystal.

The future of modern electronics is associated with photonic crystals. At the moment, there is an intensive study of the properties of photonic crystals, the development of theoretical methods for their study, the development and research of various devices with photonic crystals, the practical implementation of theoretically predicted effects in photonic crystals, and it is assumed that:

Research groups around the world

Research on photonic crystals is carried out in many laboratories of institutes and companies involved in electronics. For example:

Moscow State Technical University named after N. E. Bauman
Moscow State University named after M.V. Lomonosov
Institute of Radio Engineering and Electronics RAS
Dnipropetrovsk National University named after Oles Gonchar
Sumy State University

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2014 G.

Photonic crystals

Photonic crystals (PCs) are structures characterized by a periodic change in dielectric constant 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 influence of a periodic potential. As a result, electromagnetic waves in photonic crystals have a band spectrum and coordinate dependence similar to Bloch waves of electrons in ordinary crystals. Under certain conditions, gaps form in the band structure of PCs, similar to forbidden electronic bands in natural crystals. Depending on the specific properties (material of the elements, their size and lattice period), both completely forbidden frequency zones, for which the propagation of radiation is impossible regardless of its polarization and direction, and partially forbidden (stop zones), in which distribution is possible only in selected directions.

Photonic crystals are interesting both from a fundamental point of view and for numerous applications. Based on photonic crystals, optical filters, waveguides (in particular, in fiber-optic communication lines), and devices that allow the control of thermal radiation are created and developed; laser designs with a reduced 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 character of 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.

FCs with alternating layers made of different materials are one-dimensional.

Electron image of a one-dimensional PC used in a laser as a Bragg multilayer mirror.

Two-dimensional PCs can have more diverse geometries. These, for example, include arrays of cylinders of infinite length (their transverse size is much smaller than the longitudinal one) or periodic systems of cylindrical holes.

Electronic images of two-dimensional forward and inverse photonic crystals 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: direct and 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 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 following structure is an inverse opal synthesized as a result of a multi-step chemical process: self-assembly of polymer spherical particles, impregnation of the voids of the resulting material with a substance, and removal of the polymer matrix by chemical etching.

Surface of quartz inverse opal. The photograph was obtained using scanning electron microscopy.

Another type of three-dimensional PCs are logpiles-type structures formed by rectangular parallelepipeds crossed, usually at right angles.

Electronic photograph of a FC made of metal parallelepipeds.