Photonic crystal. Electrochemistry of Photonic Crystals From Simple Crystals to Photonic Crystals
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 is 3.4 in the near infrared region, an unprecedentedly high value unattainable by other synthesis methods. 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, such as 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 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. 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. According to its chemical composition, the natural mineral opal is a silicon dioxide hydrogel 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: natural sedimentation (precipitation of a dispersed phase in a liquid or gas under the action 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 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 is carried out by various methods: introduction from a solution in the 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
It has been shown that, depending on the polarity of the inclusion of photodiodes in the resonator, a frequency shift of the response occurs up or down in frequency with increasing illumination. It is proposed to use a system of coupled ring resonators to increase the sensitivity of the studied resonators to the illumination value. It is shown that for a fixed distance between coupled resonators, the frequency splitting of the system response into even (bright) and odd (dark) modes occurs with the help of light. We are confident that the proposed method for creating tunable ring resonators will make it possible to create a new class of light-controlled metamaterials.
This work was supported by the Ministry of Education of the Russian Federation (agreements no. 14.V37.21.1176 and no. 14.V37.21.1283), the Dynasty Foundation, the RFBR Foundation (project no. 13-02-00411), and the Scholarship of the President of the Russian Federation for young scientists and graduate students in 2012.
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Kapitonova Polina Vyacheslavovna - Saint Petersburg National Research University
Information Technology, Mechanics and Optics, Candidate of Technical Sciences, Researcher, [email protected], [email protected]
Belov Pavel Aleksandrovich - Saint Petersburg National Research University
Information Technology, Mechanics and Optics, Doctor of Phys.-Math. Sciences, Chief Researcher, [email protected]
ANALYSIS OF THE BAND STRUCTURE OF A PHOTONIC CRYSTAL WITH MULTIPLE OPTICAL LENGTHS OF LAYERS FOR THE TERAHERTZ RANGE
OH. Denisultanov, M.K. Khodzitsky
From the dispersion equation for an infinite photonic crystal, formulas are derived for the exact calculation of the band gap boundaries, the band gap width, and the exact position of the band gap centers of photonic crystals with multiple optical layer lengths in a two-layer cell for the terahertz frequency range from 0.1 to 1 THz. The formulas have been verified in the numerical simulation of photonic crystals by the transfer matrix method and by the time domain finite difference method for the first, second, and third optical length multiplicity in a two-layer cell of a photonic crystal. Formulas for the second multiplicity are confirmed experimentally. Keywords: photonic crystal, band gap, cutoff frequencies, multiple optical lengths, transmission matrix, metamaterial.
Introduction
In recent years, the study of artificial media with unusual properties ("metamaterials") has attracted the interest of a fairly large circle of scientists and engineers, which is due to the promising use of these media in the industrial and military industries in the development of new types of filters, phase shifters, superlenses, masking coatings, etc. .d. . One of the types of meta-materials is a photonic crystal, which is a layered structure with periodic
ski changing refractive index. Photonic crystals (PCs) are actively used in laser technologies, means of communication, filtering, due to such unique properties as the presence of a band structure in the spectrum, superresolution, superprism effect, etc. . Of particular interest is the study of photonic crystals in the terahertz (THz) range for spectroscopic, tomographic studies of new types of materials and biological objects. Researchers have already developed two-dimensional and three-dimensional PCs for the THz frequency range and studied their characteristics, but, unfortunately, at the moment there are no exact formulas for calculating the characteristics of the band structure of a photonic crystal, such as the band gap, band gap center, band gap boundaries. The purpose of this work is to obtain formulas for calculating the characteristics of a one-dimensional photonic crystal for the first, second, and third optical length multiplicity in a two-layer PC cell and to verify these formulas using numerical simulation using the transfer matrix method and the finite difference method in the time domain, as well as an experiment in the THz range frequencies.
Analytical and numerical modeling
Let us consider an infinite photonic crystal with refractive indices of the layers in a two-layer cell n1 and n2 and layer thicknesses d1 and d2, respectively. This structure is excited by a linearly polarized transverse electrical wave (TE wave). The wave vector k is directed perpendicular to the PC layers (Fig. 1). The dispersion equation for such a PC, obtained using the Floquet theorem and the continuity condition for the tangential field components at the layer boundary, has the following form:
C08 [kv (dx + d2)] = co8 [kg d ^] x co $ [k2 d2] -0.5)
s bt [kg e1] x bt [kg e2
where q is the Bloch wave number; k^ =
whether refraction; d1, d2 - layer thicknesses.
2 l x / x p1
; / - frequency; pg, p2 - indicator
Rice. 1. Layered-periodic structure under consideration
L. and L 1! I x. ] l! / l Peel! l "
and " and | Г ¡4 1 ! 1) 1 1 N V and | 1 У " 11
0,1 0,2 0,3 0,4 0,5 0,6
Frequency / THz
Rice. 2. Frequency dispersion of the complex Bloch wavenumber
The dispersion of the complex Bloch wave number obtained using Eq. (1) is shown in Fig. . 2. As can be seen from fig. 2, at the boundaries of the band gaps, the argument of the cosine q(d1 + d2) will take on the values either 0 or n. Therefore, based on this condition, we can calculate
to determine the values of the cutoff frequencies, band gaps, and band gap centers of the photonic crystal. However, for a photonic crystal with non-multiple optical lengths of layers inside a two-layer cell, these formulas can only be obtained in an implicit form. To obtain explicit formulas, multiple optical lengths must be used: nxx = n2e2; pyoh = 2хп2ё2; pyoh = 3xn2ё2... . The work considered formulas for the 1st, 2nd and 3rd multiplicity.
For a photonic crystal of the first multiplicity (nxx = n2d2), the formulas for boundary frequencies, widths
bandgap and the center of the bandgap have the following form:
(/n 1 L (/n "and 1 L
0.256-1.5. „arcso81---I + 2lt
a/ = /1 -/2; /33 = /+/2-; /pz =
/ 2a; /2 = i(t +1)
0.256-1.5. „, 1H -arsco81 ----- | + 2n(t +1)
where /1 and /2 - low-frequency and high-frequency boundaries of the forbidden zone, respectively; A/ - band gap; /33 is the center of the forbidden zone; c is the speed of light; / - center of allowed
o n n2 zone 6 = - + -;
For PC with layer parameters nx = 2.9; n2 = 1.445; ex = 540 µm; e2 = 1084 μm for the second band gap in the range 0.1-1 THz, the following parameters of the band structure take place: /1 = 0.1332 THz; /2 = 0.1541 THz; A/ = 0.0209 THz; /zz = 0.1437 THz.
For a PC, the optical lengths of the layers of which are related by the equality nxx = 2n2d2, the following formulas for the parameters of the band structure are obtained:
4 + v + U v2-4 6 + 3v-4v2 -4
4 + v-V v2 - 4 6 + 3v + ^v2 - 4
2 + in -V in2 - 4
2yt x s arcbo
B-#^4 2 + c + 4 c2 - 4
V-#^4 2 + v + l/v2 - 4
4 + v-Vv2 -4 6 + 3v + 4v2 - 4
4 + v + Uv2 - 4 6 + 3v-4v2 -4
where (/1 and /11), (/2 and /21), (/3 and /31), (/4 and /41) - low-frequency and high-frequency boundaries are prohibited
ny zones with numbers (4t + 1), (4t + 2), (4t + 3), (4t + 4), respectively; c is the speed of light; P= - + -;
m = 0.1.2,.... The band gap is calculated as A/ = /-/x; bandgap center
, / + /x. d /sz = ^ ; /pz - the center of the allowed zone.
For FC with parameters nx = 2.9; n2 = 1.445; ex = 540 µm; e2 = 541.87 μm for the second band gap in the range 0.1-1 THz, we have
/2 = 0.116 THz; /2x = 0.14 THz; A/ = 0.024 THz; /zz = 0.128 THz.
For a photonic crystal whose optical lengths are related by the equality nxx = 3n2d2, the following formulas for the parameters of the band structure are obtained:
1 -0.5ß + ^/2.25ß2 -ß-7 3 + 2.5ß-^/ 2.25ß2-ß-7
1 -0.5ß-^2.25ß2 -ß-7 3 + 2.5ß + V 2.25ß2-ß-7
1 -0.5ß-J2.25ß2 -ß-7 3 + 2.5ß + yl2.25ß2 - ß - 7
1 - 0.5ß + 72.25ß2 - ß - 7 3 + 2.5ß-sj2.25ß2 -ß-7
where (/1 and /11), (/2 and /2), (/3 and /) are the low-frequency and high-frequency band gaps with
numbers (3m+1), (3m+2), (3m+3), respectively; c is the speed of light; p = - + -; t = 0,1,2,.... Width
band gap is calculated as D/ = / - /1; bandgap center /zz =
permitted zone.
For a PC with parameters n1 = 2.9; n2 = 1.445; = 540 µm; d2 = 361.24 μm for the second band gap in the range 0.1-1 THz, we have
/2 = 0.1283 THz; = 0.1591 THz; D/ = 0.0308 THz; /zz = 0.1437 THz.
To simulate a PC of finite length, it is necessary to use the method of transfer matrices, which allows you to calculate the value of the electromagnetic field of a wave passing through a photonic crystal at an arbitrary point of the 2nd layer. The transfer matrix for one layer is as follows:
cos(k0 x n x p x sin(k0
: z x cos 0) x n x z x cos 0)
(-i / p) x sin(k0 x n x z x cos 0)
where k0 = -; p = - cos 0 ; n = ; z - coordinate on the Oz axis; 0 - angle of incidence of the wave on the first layer.
Using the method of transfer matrices, in the mathematical package MATLAB, the band structure of a photonic crystal was constructed for the optical lengths of the layers in a two-layer cell of the 1st, 2nd and 3rd multiplicity), in the THz frequency range (for 0=0) with 10 unit cells with the layer parameters indicated above (Fig. 3).
As can be seen from fig. 3, in the transmission spectrum of PCs of the 1st, 2nd, and 3rd multiplicity, there are band gaps that are multiples of two, three, and four, respectively, compared with the band structure of PCs with non-multiple optical lengths of the layers inside the unit cell. For all three cases of multiplicity, the relative error in calculating the parameters of the band structure of the final PC does not exceed 1% compared to the formulas for the infinite PC (the band gap was calculated at the level of 0.5 of the transmittance for the final PC).
Also, the structure of a one-dimensional PC was calculated by the finite difference method in the time domain using the CST Microwave Studio three-dimensional modeling software package (Fig. 4). One can see the same behavior of the band structure of the final PC as for the transmission spectra obtained by the transfer matrix method. The relative error in calculating the parameters of the band structure of a finite PC in this simulation package does not exceed 3% compared to the formulas for an infinite PC.
Tszh.M.
pShshShSh) sschm
pxx=3n2ё2 Frequency / THz
Rice. Fig. 3. Band structure of a photonic crystal for three multiplicities, optical lengths of layers in a two-layer cell in the THz frequency range (numbers indicate the band gap number, arrows indicate drop-down
prohibited areas)
I-e-e t o
pyoh \u003d 2p2ё2 -YES / ut1
pxx=3n2ё2 Frequency, THz
Rice. Fig. 4. Three-dimensional model of the PC in the MA (a) and the transmittance of the PC for three multiplicity (b)
experimental part
The case of the 2nd multiplicity was verified experimentally by the method of continuous THz spectroscopy in the range of 0.1-1 THz. The method of mixing frequencies of infrared radiation on a photoconductive (FC) antenna was used to generate THz radiation. The second FP antenna was used as a receiver. An assembled PC was installed between the transmitting and receiving PC antennas (Fig. 5).
The investigated photonic crystal has the following parameters: the number of bilayer cells -3; the refractive indices of the layers - nx = 2.9 and n2 = 1.445; layer thicknesses - ех = 540 μm and е2 = 520 μm (е2 is 21 μm less than for the case of the ideal 2nd multiplicity). On fig. 5 shows a comparison of the experimental and theoretical spectra for 4 and 5 band gaps. As can be seen from the experimental graph, as well as for simulation, a band gap that is a multiple of three is observed in comparison with the band structure of a PC with non-multiple optical lengths of the layers inside the unit cell. A slight discrepancy between the positions of the centers of forbidden zones in the experimental and theoretical
tic spectrum is due to the difference in the thickness of the Teflon layers in the experiment from the ideal 2nd multiplicity.
1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3
0.3 0.35 0.4 0.45 0.5 Frequency, THz
Experiment
Modeling
Rice. Fig. 5. Photograph of the setup, photograph of the photonic crystal model (a) and a comparative graph of the experimental and theoretical transmittance of a photonic crystal with three elementary
cells (b)
Conclusion
Thus, exact formulas were obtained for calculating the band structure parameters (band gap, band gap boundaries, and band gap center) of one-dimensional photonic crystals with multiple optical layer lengths inside a two-layer unit cell for the case of a TE wave with a wave vector perpendicular to the planes of the photonic layers. crystal. It was demonstrated for a photonic crystal of the 1st, 2nd and 3rd multiplicity the disappearance of band gaps, a multiple of two, three, four, respectively, in comparison with the band structure of photonic crystals with non-multiple optical lengths of the layers inside the unit cell. The formulas for the 1st, 2nd and 3rd multiplicity were tested using the transfer matrix method and 3D finite difference numerical simulations in the time domain. The case of the 2nd multiplicity was verified experimentally in the THz frequency range from 0.1 to 1 THz. The obtained formulas can be used to develop broadband filters based on photonic crystals for industrial, military and medical applications without the need to model the band structure of a photonic crystal in various mathematical packages.
The work was partially supported by grant No. 14.132.21.1421 within the framework of the Federal Target Program "Scientific and Scientific-Pedagogical Personnel of Innovative Russia" for 2009-2013.
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Denisultanov Alaudi Khozhbaudievich
Khodzitsky Mikhail Konstantinovich
St. Petersburg National Research University of Information Technologies, Mechanics and Optics, student, [email protected]
St. Petersburg National Research University of Information Technologies, Mechanics and Optics, Candidate of Phys.-Math. sciences, assistant, [email protected]
) — a material whose structure is characterized by a periodic change in the refractive index in 1, 2 or 3 spatial directions.
Description
A distinctive feature of photonic crystals (PC) is the presence of a spatially periodic change in the refractive index. Depending on the number of spatial directions along which the refractive index changes periodically, photonic crystals are called one-dimensional, two-dimensional and three-dimensional, or abbreviated as 1D PC, 2D PC and 3D PC (D - from the English dimension), respectively. Conventionally, the structure of 2D PC and 3D PC is shown in fig.
The most striking feature of photonic crystals is the existence in a 3D PC with a sufficiently large contrast in the refractive indices of the components of certain spectral regions, called total photonic band gaps (PBGs): the existence of radiation with photon energy belonging to the PBG in such crystals is impossible. In particular, radiation whose spectrum belongs to the PBG does not penetrate into the PC from the outside, cannot exist in it, and is completely reflected from the boundary. The prohibition is violated only if there are structural defects or if the size of the PC is limited. In this case, purposefully created linear defects are with small bending losses (up to micron radii of curvature), point defects are miniature resonators. The practical implementation of the potential possibilities of 3D PC based on the wide possibilities of controlling the characteristics of light (photon) beams is just beginning. It is hampered by the lack of effective methods for creating high-quality 3D PCs, methods for the targeted formation of local inhomogeneities, linear and point defects in them, as well as methods for interfacing with other photonic and electronic devices.
Significantly greater progress has been made towards the practical application of 2D PCs, which are used, as a rule, in the form of planar (film) photonic crystals or in the form of (PCF) (see details in the relevant articles).
PCFs are a two-dimensional structure with a defect in the central part, elongated in the perpendicular direction. Being a fundamentally new type of optical fibers, PCFs provide opportunities for transporting light waves and controlling light signals that are inaccessible to other types.
One-dimensional PCs (1D PCs) are a multilayer structure of alternating layers with different refractive indices. In classical optics, long before the appearance of the term "photonic crystal", it was well known that in such periodic structures the nature of the propagation of light waves changes significantly due to the phenomena of interference and diffraction. For example, multilayer reflective coatings have long been widely used for the manufacture of mirrors and film interference filters, and volumetric Bragg gratings as spectral selectors and filters. After the term PC became widely used, such layered media, in which the refractive index periodically changes along one direction, began to be attributed to the class of one-dimensional photonic crystals. With perpendicular light incidence, the spectral dependence of the reflection coefficient from multilayer coatings is the so-called "Bragg table" - at certain wavelengths, the reflection coefficient rapidly approaches unity with an increase in the number of layers. Light waves falling in the spectral range shown in fig. b arrow, are almost completely reflected from the periodic structure. According to the terminology of the FK, this range of wavelengths and the corresponding range of photon energies (or the energy band) are forbidden for light waves propagating perpendicular to the layers.
The potential for practical applications of PCs is enormous due to the unique possibilities of controlling photons and has not yet been fully explored. There is no doubt that in the coming years new devices and structural elements will be proposed, possibly fundamentally different from those that are used or developed today.
Huge prospects for the use of PCs in photonics were realized after the publication of an article by E. Yablonovich, in which it was proposed to use PCs with full PBGs to control the spontaneous emission spectrum.
Among the photonic devices that can be expected in the near future are the following:
- ultra-small low-threshold FK lasers;
- superbright PCs with a controlled emission spectrum;
- subminiature FK waveguides with micron bending radius;
- photonic integrated circuits with a high degree of integration based on planar PCs;
- miniature FK spectral filters, including tunable ones;
- FK devices of random access optical memory;
- FK optical signal processing devices;
- means for delivering high-power laser radiation based on PCF with a hollow core.
The most tempting, but also the most difficult to implement application of three-dimensional PCs is the creation of super-large volumetrically integrated complexes of photonic and electronic devices for information processing.
Other potential uses for 3D photonic crystals include the manufacture of artificial opal-based jewelry.
Photonic crystals are also found in nature, giving additional shades of color to the world around us. Thus, the mother-of-pearl coating of shells of mollusks, such as haliotis, has a 1D FC structure, the antennae of a sea mouse and the bristles of a polychaete worm are 2D FC, and natural semiprecious opals and wings of African swallowtail butterflies (Papilio ulysses) are natural three-dimensional photonic crystals.
Illustrations
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a– structure of two-dimensional (top) and three-dimensional (bottom) PC; b is the band gap of a one-dimensional PC formed by quarter-wavelength GaAs/AlxOy layers (the band gap is shown by an arrow); in is the inverted nickel FC, obtained by the staff of the FNM Moscow State University. M.V. Lomonosova N.A. Sapolotova, K.S. Napolsky and A.A. Eliseev |
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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 compound is capable of operating 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
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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
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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
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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 Photographs of the electric field of the 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 reflective symmetry about the y-z plane 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
In the last decade, the development of microelectronics has slowed down, since the limits on the speed of standard semiconductor devices have already been practically reached. An increasing number of studies are devoted to the development of areas alternative to semiconductor electronics - these are spintronics, microelectronics with superconducting elements, photonics, and some others.
The new principle of transmission and processing of information using a light signal, rather than an electrical signal, can accelerate the onset of a new stage in the information age.
From simple crystals to photonic
The basis of electronic devices of the future can be photonic crystals - these are synthetic ordered materials in which the dielectric constant changes periodically inside the structure. In the crystal lattice of a traditional semiconductor, the regularity, the periodicity of the arrangement of atoms leads to the formation of the so-called band energy structure - with allowed and forbidden zones. An electron whose energy falls into the allowed band can move through the crystal, while an electron with energy in the band gap is "locked".
By analogy with an ordinary crystal, the idea of a photonic crystal arose. In it, the periodicity of the permittivity causes the appearance of photonic zones, in particular, the forbidden zone, within which the propagation of light with a certain wavelength is suppressed. That is, being transparent to a wide spectrum of electromagnetic radiation, photonic crystals do not transmit light with a selected wavelength (equal to twice the period of the structure along the length of the optical path).
Photonic crystals can have different dimensions. One-dimensional (1D) crystals are a multilayer structure of alternating layers with different refractive indices. Two-dimensional photonic crystals (2D) can be represented as a periodic structure of rods with different permittivities. The first synthetic prototypes of photonic crystals were three-dimensional and were created in the early 1990s by the staff of the research center Bell Labs(USA). To obtain a periodic lattice in a dielectric material, American scientists drilled cylindrical holes in such a way as to obtain a three-dimensional network of voids. In order for the material to become a photonic crystal, its permittivity was modulated with a period of 1 centimeter in all three dimensions.
Natural analogues of photonic crystals are mother-of-pearl coatings of shells (1D), antennae of a sea mouse, polychaete worm (2D), wings of an African sailboat butterfly and semi-precious stones, such as opal (3D).
But even today, even with the help of the most modern and expensive methods of electron lithography and anisotropic ion etching, it is difficult to produce defect-free three-dimensional photonic crystals with a thickness of more than 10 structural cells.
Photonic crystals should find wide application in photonic integrated technologies, which in the future will replace electrical integrated circuits in computers. When information is transmitted using photons instead of electrons, power consumption will be sharply reduced, clock frequencies and information transfer rates will increase.
Titanium oxide photonic crystal
Titanium oxide TiO 2 has a set of unique characteristics such as high refractive index, chemical stability and low toxicity, which makes it the most promising material for creating one-dimensional photonic crystals. If we consider photonic crystals for solar cells, then titanium oxide wins here because of its semiconductor properties. An increase in the efficiency of solar cells using a semiconductor layer with a periodic photonic crystal structure, including titanium oxide photonic crystals, has been previously demonstrated.
But so far, the use of photonic crystals based on titanium dioxide is limited by the lack of a reproducible and inexpensive technology for their creation.
Nina Sapoletova, Sergei Kushnir and Kirill Napolsky, members of the Faculty of Chemistry and the Faculty of Materials Sciences of Moscow State University, have improved the synthesis of one-dimensional photonic crystals based on porous titanium oxide films.
“Anodizing (electrochemical oxidation) of valve metals, including aluminum and titanium, is an effective method for obtaining porous oxide films with nanometer-sized channels,” explained Kirill Napolsky, head of the electrochemical nanostructuring group, Candidate of Chemical Sciences.
Anodizing is usually carried out in a two-electrode electrochemical cell. Two metal plates, a cathode and an anode, are lowered into the electrolyte solution, and an electric voltage is applied. Hydrogen is released at the cathode, and electrochemical oxidation of the metal occurs at the anode. If the voltage applied to the cell is periodically changed, then a porous film with a porosity specified in thickness is formed on the anode.
The effective refractive index will be modulated if the pore diameter changes periodically within the structure. The titanium anodizing techniques developed earlier did not allow obtaining materials with a high degree of structure periodicity. Chemists from Moscow State University have developed a new method of metal anodizing with voltage modulation depending on the anodizing charge, which allows creating porous anodic metal oxides with high accuracy. The possibilities of the new technique were demonstrated by the chemists using one-dimensional photonic crystals from anodic titanium oxide as an example.
As a result of changing the anodizing voltage according to a sinusoidal law in the range of 40–60 Volts, scientists obtained nanotubes of anodic titanium oxide with a constant outer diameter and a periodically changing inner diameter (see figure).
“The anodizing methods used earlier did not allow obtaining materials with a high degree of structure periodicity. We have developed a new methodology, the key component of which is in situ(immediately during synthesis) measurement of the anodizing charge, which makes it possible to control with high accuracy the thickness of layers with different porosity in the formed oxide film, ”explained one of the authors of the work, candidate of chemical sciences Sergey Kushnir.
The developed technique will simplify the creation of new materials with a modulated structure based on anodic metal oxides. “If we consider the use of photonic crystals from anodic titanium oxide in solar cells as a practical application of the technique, then a systematic study of the influence of the structural parameters of such photonic crystals on the efficiency of light conversion in solar cells remains to be carried out,” Sergey Kushnir specified.
