home Zoology History of photonic crystals. Mathematical model of a photonic crystal. From simple crystals to photonic

History of photonic crystals. Mathematical model of a photonic crystal. From simple crystals to photonic

Introduction Since ancient times, a person who has found a photonic crystal has been fascinated by a special iridescent play of light in it. It was found that iridescent overflows of scales and feathers of various animals and insects are due to the existence of superstructures on them, which received the name photonic crystals for their reflective properties. Photonic crystals are found in nature in/on: minerals (calcite, labradorite, opal); on the wings of butterflies; beetle shells; the eyes of some insects; algae; scales of fish; peacock feathers. 3

Photonic crystals This is a material whose structure is characterized by a periodic change in the refractive index in spatial directions Photonic crystal based on aluminum oxide. M. DEUBEL, G.V. FREYMANN, MARTIN WEGENER, SURESH PEREIRA, KURT BUSCH AND COSTAS M. SOUKOULIS “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications”// Nature materials Vol. 3, P

A bit of history… 1887 Rayleigh was the first to investigate the propagation of electromagnetic waves in periodic structures, which is analogous to the one-dimensional photonic crystal Photonic Crystals - the term was introduced in the late 1980s. to denote the optical analogue of semiconductors. These are artificial crystals made of a translucent dielectric in which air "holes" are created in an orderly manner. 5

Photonic crystals - the future of world energy High-temperature photonic crystals can act not only as a source of energy, but also as extremely high-quality detectors (energy, chemical) and sensors. Photonic crystals created by Massachusetts scientists are based on tungsten and tantalum. This connection is able to work satisfactorily at very high temperatures. Up to ˚С. In order for the photonic crystal to start converting one type of energy into another, convenient for use, any source (thermal, radio emission, hard radiation, sunlight, etc.) will do. 6

Dispersion law of electromagnetic waves in a photonic crystal (diagram of extended zones). The right side shows for a given direction in the crystal the relationship between the frequency? and the values of ReQ (solid curves) and ImQ (dashed curve in the stop zone omega -

Photonic Gap Theory It wasn't until 1987 when Eli Yablonovitch of Bell Communications Research (now a professor at UCLA) introduced the notion of an electromagnetic band gap. To expand horizons: Lecture by Eli Yablonovitch yablonovitch-uc-berkeley/view Lecture by John Pendry john-pendry-imperial-college/view 9

In nature, photonic crystals are also found: on the wings of African swallowtail butterflies, the mother-of-pearl coating of shells of mollusks, such as galiotis, barnacles of the sea mouse and bristles of the polychaete worm. Photo of an opal bracelet. Opal is a natural photonic crystal. It is called the "stone of deceptive hopes" 10

No heating and photochemical destruction of the pigment coating" title="(!LANG: Advantages of FA-based filters over the absorption mechanism (absorbing mechanism) for living organisms: Interference coloring does not require the absorption and dissipation of light energy, => no heating and photochemical destruction of the pigment coating" class="link_thumb"> 12 !} Advantages of FA-based filters over the absorption mechanism (absorbing mechanism) for living organisms: Interference coloring does not require absorption and dissipation of light energy, => no heating and photochemical destruction of the pigment coating. Butterflies living in hot climates have an iridescent wing pattern, and the structure of the photonic crystal on the surface has been found to reduce the absorption of light and, therefore, the heating of the wings. The sea mouse has been using photonic crystals for a long time. 12 no heating and photochemical destruction of the pigment coating "> no heating and photochemical destruction of the pigment coating. Butterflies living in a hot climate have an iridescent wing pattern, and the structure of the photonic crystal on the surface, as it turned out, reduces the absorption of light and, consequently, the heating of the wings. The sea mouse is already has been using photonic crystals in practice for a long time. , => no heating and photochemical destruction of the pigment"> title="Advantages of FA-based filters over the absorption mechanism (absorbing mechanism) for living organisms: Interference coloring does not require absorption and dissipation of light energy, => no heating and photochemical destruction of the pigment coating"> !}

Morpho didius iridescent butterfly and micrograph of its wing as an example of diffractive biological microstructure. Iridescent natural opal (semi-precious stone) and image of its microstructure, consisting of close-packed spheres of silicon dioxide. 13

Classification of photonic crystals 1. One-dimensional. In which the refractive index changes periodically in one spatial direction as shown in the figure. In this figure, the symbol Λ denotes the period of change of the refractive index, and the refractive indices of the 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. fourteen

2. Two-dimensional. In which the refractive index changes periodically in two spatial directions as shown in the figure. In this figure, a photonic crystal is created by rectangular regions with a refractive index of n1, which are in a medium with a refractive index of n2. In this case, the regions with the refractive index n1 are ordered in a two-dimensional cubic lattice. Such photonic crystals can exhibit their properties in two spatial directions, and the shape of regions with a refractive index n1 is not limited to rectangles, as in the figure, but can be any (circles, ellipses, arbitrary, etc.). The crystal lattice in which these regions are ordered can also be different, and not just cubic, as in the figure. fifteen

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. 16

Applications of Photonic Crystals The first application is spectral channel separation. In many cases, not one, but several light signals travel along an optical fiber. They sometimes need to be sorted - to send each one along a separate path. For example - an optical telephone cable, through which there are several conversations at the same time at different wavelengths. A photonic crystal is an ideal tool for "carving" the desired wavelength from the stream and directing it to where it is required. The second is a cross for light fluxes. Such a device, which protects light channels from mutual influence when they physically cross, is absolutely necessary when creating a light computer and light computer chips. 17

Photonic crystal in telecommunications Not so many years have passed since the beginning of the first developments, as it became clear to investors that photonic crystals are optical materials of a fundamentally new type and that they have a bright future. The output of the development of photonic crystals of the optical range to the level of commercial application, most likely, will occur in the field of telecommunications. eighteen

Advantages and disadvantages of lithographic and holographic methods for obtaining FC Pluses: high quality of the formed structure. Fast production speed Ease of mass production Disadvantages Expensive equipment required Possible deterioration of edge sharpness Difficulty in fabricating setups 22

A close-up on the bottom shows the remaining roughness of the order of 10 nm. The same roughness is visible on our SU-8 templates made by holographic lithography. This clearly shows that this roughness is not related to the fabrication process, but rather to the final resolution of the photoresist. 24

To move the fundamental PBGs wavelengths in the telecommunication mode from 1.5 µm and 1.3 µm, it is necessary to have a distance of the order of 1 µm or less in the plane of the rods. The fabricated samples have a problem: the rods begin to come into contact with each other, which leads to an undesirable large filling of the fraction. Solution: Reducing the diameter of the rod, hence filling the fraction, by etching in oxygen plasma 26

Optical properties of a PC Due to the periodicity of the medium, the propagation of radiation inside a photonic crystal becomes similar to the movement of an electron inside an ordinary crystal under the action of a periodic potential. Under certain conditions, gaps form in the band structure of a PC, similarly to forbidden electronic bands in natural crystals. 27

A two-dimensional periodic photonic crystal is obtained by forming a periodic structure of vertical dielectric rods planted in a square-nest manner on a silicon dioxide substrate. By placing "defects" in a photonic crystal, it is possible to create waveguides that, bent at any angle, give 100% transmission Two-dimensional photonic structures with a bandgap 28

A new method for obtaining a structure with polarization-sensitive photonic band gaps Development of an approach to combining the structure of a photonic band gap with other optical and optoelectronic devices Observation of the short- and long-wave band boundaries. Experience goal is: 29

The main factors that determine the properties of a photonic band gap (PBG) structure are the refractive contrast, the proportion of high and low material indices in the lattice, and the arrangement of the lattice elements. The configuration of the waveguide used is comparable to that of a semiconductor laser. The array is very small (100 nm in diameter) holes were etched on the core of the waveguide, forming a hexagonal grating 30

Fig.2a Sketch of the lattice and Brillouin zone illustrating the directions of symmetry in a horizontal close-packed lattice. b, c Measurement of transmission characteristics on a 19-nm photonic grating. 31 Brillouin zones with symmetrical directions

Fig.4 Snapshots electric field profiles of traveling waves corresponding to band 1 (a) and band 2 (b), near the K point for TM polarization. In a, the field has the same reflection symmetry with respect to y-z plane, which is the same as the plane wave, so it should easily interact with the incoming plane wave. In contrast, in b the field is asymmetric, which does not allow this interaction to occur. 33

Conclusions: PBG structures can be used as mirrors and elements for direct control of emission in semiconductor lasers. Demonstration of PBG concepts in waveguide geometry will allow the realization of very compact optical elements. that it will be possible to use non-linear effects 34

I cannot claim to be impartial in judging colors. I rejoice in sparkling hues and sincerely regret the skimpy browns. (Sir Winston Churchill).

Origin of photonic crystals

Looking at the wings of a butterfly or the mother-of-pearl coating of shells (Figure 1), one wonders how Nature - even for many hundreds of thousands or millions of years - could create such amazing biostructures. However, not only in the bioworld there are similar structures with iridescent color, which are an example of the almost limitless creative possibilities of Nature. For example, the semi-precious stone opal has fascinated people since ancient times with its brilliance (Figure 2).

Today, every ninth grader knows that not only the processes of absorption and reflection of light lead to what we call the coloring of the world, but also the processes of diffraction and interference. Diffraction gratings that we can find in nature are structures with a periodically changing dielectric constant, while their period is commensurate with a long wavelength of light (Figure 3). These can be 1D lattices, as in the mother-of-pearl coating of mollusk shells such as galiotis, 2D lattices, like barnacles of sea mice, polychaete worms, and 3D lattices, which give iridescent blue coloring to butterflies from Peru, as well as opal.

In this case, Nature, as undoubtedly the most experienced materials chemist, is pushing us to the next exit: three-dimensional optical diffraction gratings can be synthesized by creating dielectric gratings that are geometrically complementary to each other, i.e. one is inverse to the other. And since Jean-Marie Lehn said the famous phrase: “If something exists, then it can be synthesized,” we simply must implement this conclusion on practice.

Photonic Semiconductors and the Photonic Gap

So, in a simple formulation, a photonic crystal is a material whose structure is characterized by a periodic change in the refractive index in spatial directions, which leads to the formation of a photonic band gap. Usually, in order to understand the meaning of the terms "photonic crystal" and "photonic bandgap", such a material is considered as an optical analogy to semiconductors. The solution of Maxwell's equations for the propagation of light in a dielectric grating shows that, due to Bragg diffraction, the distribution of photons over frequencies ω(k) depending on the wave vector k (2π/λ) will have discontinuity regions. This statement is graphically presented in Figure 4, which shows an analogy between the propagation of an electron in a 1D crystal lattice and a photon in a 1D photonic lattice. The continuous density of states of both a free electron and a photon in vacuum undergo a break inside, respectively, the crystal and photon lattices in the so-called "stop zones" at the value of the wave vector k (i.e. momentum), which corresponds to a standing wave. This is the condition for the Bragg diffraction of an electron and a photon.

The photonic band gap is a frequency range ω(k) in the reciprocal space of wave vectors k, where the propagation of light of a certain frequency (or wavelength) is prohibited in the photonic crystal in all directions, while the light incident on the photonic crystal is completely reflected from it. If the light "arises" inside the photonic crystal, then it will be "frozen" into it. The zone itself may be incomplete, the so-called stop zone. Figure 5 shows 1D, 2D and 3D photonic crystals in real space and the photon density of states in reciprocal space.

The photonic band gap of a three-dimensional photonic crystal is some analogy of the electronic band gap in a silicon crystal. Therefore, the photonic bandgap "controls" the light flux in a silicon photonic crystal in the same way as the transport of charge carriers occurs in a silicon crystal. In these two cases, the band gap is caused by standing waves of photons or electrons, respectively.

Make a photonic crystal yourself

Oddly enough, but the Maxwellian equations for photonic crystals are not sensitive to scaling, in contrast to the Schrödinger equation in the case of electronic crystals. This is due to the fact that the wavelength of an electron in a "normal" crystal is more or less fixed at a level of a few angstroms, while the dimensional scale of the wavelength of light in photonic crystals can vary from ultraviolet to microwave radiation, solely due to a change in the dimension of the photon components. gratings. This leads to truly inexhaustible possibilities for fine-tuning the properties of a photonic crystal.

At present, there are many methods for manufacturing photonic crystals. Some of them are more suitable for the formation of one-dimensional photonic crystals, others are convenient for two-dimensional ones, others are more applicable to three-dimensional photonic crystals, the fourth are used in the manufacture of photonic crystals on other optical devices, etc. However, not everything is limited only by varying the dimension structural elements. Photonic crystals can also be created due to optical non-linearity, metal-non-metal transition, liquid crystal state, ferroelectric birefringence, swelling and shrinking of polymer gels, and so on, the main thing is to change the refractive index.

Where without defects?!

There are practically no materials in the world in which there would be no defects, and this is good. It is precisely the defects in solid-phase materials in b about to a greater extent than the crystal structure itself, influence the various properties of materials and, ultimately, their functional characteristics, as well as possible applications. A similar statement is also true in the case of photonic crystals. From a theoretical consideration, it follows that the introduction of defects (point, extended - dislocations - or bending) at the microlevel into an ideal photonic lattice allows you to create certain states inside the photonic bandgap, on which light can be localized, and the propagation of light can be limited or, on the contrary, enhanced along and around a very small waveguide (Figure 6). If we draw an analogy with semiconductors, then these states resemble impurity levels in semiconductors. Photonic crystals with such a “controlled imperfection” can be used to create all-optical devices and circuits of a new generation of optical telecommunication technologies.

Lighting informatics

Figure 7 shows one of the futuristic images of the all-light chip of the future, which has undoubtedly excited the imagination of chemists, physicists and materials scientists for a decade. The all-optical chip consists of integrated micro-sized photonic crystals with 1D, 2D and 3D periodicity, which can play the role of switches, filters, low-threshold lasers, etc., while light is transmitted between them through waveguides solely due to the defective structure. And although the topic of photonic crystals exists in " road maps» development of photonic technologies, research and practical application of these materials are still at the earliest stages of their development. This is the subject of future discoveries that may lead to the creation of all-light ultrafast computers, as well as quantum computers. However, in order for the dreams of science fiction writers and many scientists who have devoted their lives to studying such interesting and practically significant materials as photonic crystals to come true, a number of questions must be answered. For example, such as: what needs to be changed in the materials themselves in order to solve the problem associated with the reduction of such integrated chips from micro-sized photonic crystals for wide application in practice? Is it possible to use microdesign (“top-down”), or self-assembly (“bottom-up”), or some fusion of these two methods (for example, directed self-assembly) to commercialize the production of chips from microsized photonic crystals? Is the science of computers based on light chips made of microphotonic crystals a reality, or is it still a futurist fantasy?

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 range 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 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 effective method 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 as practical use methods to consider the use of photonic crystals from anodic titanium oxide in solar cells, then a systematic study of the influence of the structural parameters of such photonic crystals on the efficiency of light conversion in solar cells has yet to be carried out,” Sergey Kushnir specified.

Classification of methods for manufacturing photonic crystals. Photonic crystals in nature are a rarity. They are distinguished by a special iridescent play of light - an optical phenomenon called irization (translated from Greek - rainbow). These 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 silicon oxide globules. From the play of light in the latter comes the term opalescence, denoting a special type of radiation scattering characteristic only for this crystal.

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

1. Methods using the 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, are deposited in a certain volume. As the particles settle on top of each other, they form a three-dimensional photonic crystal, and are ordered predominantly in a face-centered or hexagonal crystal lattice. A honeycomb method is also possible, which is based on filtering the liquid in which the particles are located through small spores. Although the honeycomb method makes it possible to form a crystal at a relatively high rate, determined by the rate of liquid flow through the pores, however, defects are formed in such crystals upon 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 object etching. This group of methods uses a photoresist mask formed on the semiconductor surface, which defines the geometry of the etching region. Using such a mask, the simplest photonic crystal is formed by etching the surface of a semiconductor that is not covered with a photoresist. The disadvantage of this method is the need to use photolithography with high resolution at the level of tens and hundreds of nanometers. Also, beams of focused ions, such as Ga, are used to make 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 rate 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 the principles of holography. With the help of holography, periodic changes in the refractive index in spatial directions are formed. To do this, use the interference of two or more coherent waves, 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 circumstance of what dimension the structure 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 the vacuum layer-by-layer deposition of polycrystalline films from dielectric or semiconductor materials. This method has become widespread in connection with 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 about 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 make it possible to create completely single-crystal structures with a periodic change in the refractive index along the growth direction using molecular beam epitaxy or vapor deposition using organometallic compounds. At present, such structures are part of semiconductor lasers with vertical cavities. The maximum achievable presently ratio of the 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 formation of the layer thickness at the level of one grating 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 μm or more, while the ratio of the refractive indices of silicon and air in the near infrared region is 3.4 - unprecedented great importance, unattainable by other methods of synthesis. An example of a similar structure obtained at the Physico-Technical Institute. A. F. Ioffe RAS (St. Petersburg), is shown in fig. 3.96.

Rice. 3.96. Silicon-air periodic structure 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. The technology of selective etching has been developed for silicon and aluminum due to the wide use of these materials in microelectronics. Porous silicon, for example, is considered as a promising optical material that will make it possible to create integrated optoelectronic systems with a high degree of integration. The combination of advanced silicon technologies with quantum size effects and the principles of formation of photonic band gaps has led to the development of a new direction - silicon photonics.

The use of submicron lithography for the formation of masks makes it possible to create silicon structures with a period of 300 nm or less. Due to the strong absorption of visible radiation, 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 proceed to periodic silicon oxide–air structures, but at the same time, the low refractive index ratio (component 1.45) does not allow the formation of a full-fledged band gap in two dimensions.

Two-dimensional periodic structures of A 3 B 5 semiconductor compounds, which are 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 a larger band gap than silicon and the same high refractive index values as silicon, equal to 3.55 and 3.6, respectively.

Very interesting are periodic structures based on aluminum oxide (Fig. 3.97a). They are obtained by electrochemical etching of metallic aluminum, on the surface of which a mask is formed using lithography. Using electron lithographic templates, perfect two-dimensional periodic structures resembling honeycombs 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). In this case, the pore diameter can be only a few nanometers, which is unattainable for modern lithographic methods. The periodicity of the pores is associated with the self-regulation of the aluminum oxidation process during the electrochemical reaction. The initial conductive material (aluminum) during the reaction is oxidized to Al 2 O 3 . The aluminum oxide film, which is a dielectric, reduces the current and slows down the reaction. The combination of these processes makes it possible to achieve a self-sustaining reaction mode, 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 of Al 2 O 3: a) made using a lithographic mask; b) made with the help of self-regulation of the oxidation process

A study of the optical properties of nanoporous alumina showed an unusually high transparency of this material along the pore direction. 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, a high reflection is observed with a reflection coefficient depending on the angle of incidence.

The relatively low values of the permittivity of aluminum oxide, in contrast to silicon, gallium arsenide, and indium phosphide, do not allow the formation of a full-fledged band gap in two dimensions. However, despite this, the optical properties of porous alumina are quite interesting. For example, it has a pronounced anisotropic light scattering, as well as birefringence, which allows it to be used to rotate the polarization plane. Using various chemical methods, it is possible to fill the pores with various oxides, as well as optically active materials, for example, nonlinear optical media, organic and inorganic luminophores, 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 way to create a three-dimensional photonic crystal is considered to be the method based on the mechanical drilling of cylindrical holes in the volume of the material, proposed by E. Yablonovich. The fabrication of such a three-dimensional periodic structure is a rather laborious task; therefore, many researchers have attempted to create a photonic crystal by 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, filled with polycrystalline silicon. Further, the process of applying silicon dioxide is repeated, but the strips are formed in a perpendicular direction. After creating the required number of layers, 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 modern methods 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. 3D photonic structure from 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) is borrowed from nature. It is known, for example, that natural opals have the properties of photonic crystals. Natural mineral opal chemical composition is a hydrogel of silicon dioxide SiO 2 × H 2 O with a 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, it was found that natural opals are formed by close-packed spherical particles of α-SiO 2 , uniform in size, 150–450 nm in diameter. Each particle consists of smaller globular formations with a diameter of 5–50 nm. The globule packing voids are filled with amorphous silicon oxide. The intensity of diffracted light is influenced by two factors: the first is the "ideal" dense packing of globules, the second is the difference in the refractive indices of amorphous and crystalline oxide SiO 2 . 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: by natural sedimentation (precipitation of a dispersed phase in a liquid or gas under the action of gravitational field or centrifugal forces), centrifugation, filtration using membranes, electrophoresis, etc. Spherical particles of polystyrene, polymethyl methacrylate, particles of silicon dioxide α-SiO 2 act as colloidal particles.

The natural precipitation method is a very slow process, requiring several weeks or even months. To a large extent, centrifugation accelerates the process of formation of colloidal crystals, but the materials obtained in this way are less ordered, since at a high deposition rate, separation of particles by size does not have time to occur. To accelerate the sedimentation process, electrophoresis is used: a vertical electric field is created, which “changes” the gravity of the particles depending on their size. Methods based on the use of capillary forces are also used. The main idea is that, under the action of capillary forces, crystallization occurs at the meniscus boundary 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 currents. In general, the choice of technique is determined by the requirements for the quality of the resulting crystals and the time spent on their manufacture.

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

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

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

b) drying the precipitate; c) sample annealing

On fig. 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 makes it possible to fill voids with various materials. Opal matrices are three-dimensional sublattices of interconnected nanosized pores. The pore sizes are on the order of hundreds of nanometers, and 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 in the creation of high-quality nanocomposites is the completeness of the filling of the nanoporous space. Filling spend various methods: intrusion from solution in melt; impregnation with concentrated solutions followed by evaporation of the solvent; electrochemical methods, chemical vapor deposition, etc.

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

b) from polystyrene microspheres

The selective etching of silicon oxide from such composites results in the formation of spatially ordered nanostructures with high porosity (more than 74% of the volume), called reversed or inverted opals. This method of obtaining photonic crystals is called the template method. As ordered monodisperse colloidal particles forming a photonic crystal, not only silicon oxide particles, but also, for example, polymer ones can act. An example of a photonic crystal based on polystyrene microspheres is shown in fig. 3.100b

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 the 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 picture, a photonic crystal is created by rectangular regions with a refractive index that are in a medium with a refractive index. In this case, the 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 regions 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.

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 perfect 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 to almost unlimited distances.

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 permittivity (or refractive index) as a function of frequency has a pole at some resonant frequency.

Any inhomogeneity in the 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 photonic crystal defect. In such areas, the electromagnetic field is often concentrated, which is used in microresonators and waveguides built on the basis of photonic crystals.

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

Photonic crystals allow manipulations with 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 the wave theory and the solution of Maxwell's equations are used. Maxwell's equations can be solved analytically and numerically, but it is numerical methods of solution that are most often used to study the properties of photonic crystals due to their availability and easy adjustment to the tasks being solved.

It is also worth mentioning that two main approaches to considering the properties of photonic crystals are used - time domain methods (which allow you to get a solution to the problem depending on the time variable), and frequency domain methods (which provide a solution to the problem as a function of frequency) .

Time domain methods are convenient for dynamic problems that involve the time dependence of the electrical magnetic field from time. They can also be used to calculate the band structures of photonic crystals, however, it is practically difficult to determine the position of the bands in the output data of such methods. In addition, when calculating the 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 a higher resolution in the band diagram, you need to spend more time performing calculations. There is 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 decrease in the time step, and hence the size of the spatial grid, an increase in the number of iterations, the required computer RAM, and the calculation time. Such methods are implemented in the well-known commercial modeling packages Comsol Multiphysics (the finite element method is used to solve Maxwell's equations), RSOFT Fullwave (the finite difference method is used), software codes for finite element and difference methods independently developed by researchers, etc.

Methods for the frequency domain are convenient, first of all, because the solution of the Maxwell equations occurs immediately for a stationary system and the frequencies of the optical modes of the system are determined directly from the solution, this allows you to quickly calculate the band diagrams of photonic crystals than using methods for the time domain. Their advantages include the number of iterations, which practically does not depend on the resolution of the spatial grid of the method, and the fact that the error of the method decreases numerically 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 the 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 mentioned software packages cannot calculate 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 to the calculation of band diagrams, but also require knowledge of 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 tasks, you can use both of the above approaches based on convenience and their availability, as well as methods of the radiative transfer matrix, 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.

The theory of photonic band gaps

As noted above, photonic crystals make it possible to obtain allowed and forbidden bands for photon energies, similarly to semiconductor materials, in which there are allowed and forbidden bands for charge carrier energies. In the literary source, the appearance of forbidden bands is explained by the fact that under certain conditions, the intensity of the electric field of standing waves of a photonic crystal with frequencies close to the band gap frequency shifts to different areas of the photonic crystal. Thus, the intensity of the field of low-frequency waves is concentrated in areas with a large refractive index, and the intensity of the field of high-frequency waves is concentrated in areas with a lower refractive index. Another description of the nature of forbidden bands in photonic crystals is found in the work: “photonic crystals are usually called media in which the permittivity periodically changes in space with a period that allows Bragg light diffraction.”

If radiation with the forbidden band 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 allow obtaining 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 forbidden bands for radiation propagating both in one, two directions, and in all directions of a given photonic crystal, which lie in the plane. 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 forbidden bands, 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, the forbidden bands can be tunable, for example, due to the use of nonlinear materials with a pronounced Kerr effect, due to a change in the size of regions with different refractive indices, or due to a change in the 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).

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 cylindrical holes are filled, marked 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 neighboring cylinders) . In the photonic crystal under consideration, the ratio of the hole radius to the period is equal to . 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. Wave vectors in a photonic crystal are plotted along the X axis, and the normalized frequency is plotted along the Y axis, ( - wavelength in vacuum) corresponding to 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 band gaps for photons in a given photonic crystal. Black dashed lines are the so-called light lines (or light cone) of this photonic crystal. One of the main areas of application of these photonic crystals is optical waveguides, and the light line defines the region within which the waveguide modes of waveguides built using such photonic crystals with low losses are located. In other words, the light line determines the zone of energy states of interest to us in a given photonic crystal. The first thing you should pay 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 band gaps for TE and TM polarized waves, which lie in the region of small values of the normalized frequency , overlap, which means that this photonic crystal has a complete band gap in the region of overlapping of the band gaps 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 considered photonic crystal (TE polarization).

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

From the above dependences, we can determine the geometric parameters of a photonic crystal, the first forbidden zone of which, with the value of the normalized frequency , falls on the wavelength nm. The period of the photonic crystal is equal to nm, the radius of the holes is equal to 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 hole layers and the radiation propagates in the Γ-Κ direction. From the above dependences, we can see the most well-known property of photonic crystals - electromagnetic waves with natural frequencies corresponding to the forbidden bands of a photonic crystal (Fig. 5 and 6), are characterized by a reflection coefficient close to unity and are subjected to almost complete reflection from a given photonic crystal. Electromagnetic waves with frequencies outside the forbidden bands of a given photonic crystal are characterized by lower reflection coefficients from the photonic crystal and completely or partially pass through it.

Making photonic crystals

Currently, there are 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 manufacture of photonic crystals on other optical devices, etc. Let us consider the most famous of these methods.

Methods using the 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 in the liquid and, as the liquid evaporates, are deposited in a certain volume. As they are deposited on each other, they form a three-dimensional photonic crystal, and are ordered predominantly in a face-centered or hexagonal crystal lattice. This method is quite slow, the formation of a photonic crystal can take weeks.

Another method of spontaneous formation of photonic crystals, called the honeycomb method, involves filtering the liquid in which the particles are located through small pores. This method is presented in the works, allows you to form a photonic crystal at a speed certain speed liquid flows through the pores, but when such a crystal dries, defects are formed in the crystal.

It has already been noted above that in most cases a large refractive index contrast in a photonic crystal is required to obtain forbidden photonic bands in all directions. The methods of spontaneous formation of a photonic crystal mentioned above were most often used for the deposition of spherical colloidal particles of silicone, the refractive index of which is small, and hence 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 away. A step-by-step method for forming an inverse opal is described in guidelines for the implementation laboratory work.

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. For this, the interference of two or more coherent waves is used, which creates a periodic distribution of the intensity of the electric field. 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 allow the creation of three-dimensional photonic crystals with a resolution of 200nm and use the property of some materials, such as polymers, which are sensitive to single- and two-photon radiation and can change their properties under the influence of 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 with a beam at certain places 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 beam lithography. electrons.

Application

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

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

Research groups in 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
Lomonosov Moscow State University
Institute of Radio Engineering and Electronics RAS
Dnepropetrovsk National University named after Oles Gonchar
Sumy State University