Photonic crystal

A photonic crystal is a solid-state structure with a periodically varying dielectric constant or inhomogeneity, the period of which is comparable to the wavelength of light.

Definitions

This is a material whose structure is characterized by a periodic change in the refractive index in spatial directions ^[1] .
In another paper ^[2] , an extended definition of photonic crystals is encountered - “photonic crystals are called media in which the dielectric constant varies periodically in space with a period that allows Bragg light diffraction ”.
In the third paper ^[3] , the definition of photonic crystals in a different form is found - “for more than 10 years there has been a hearing about the“ structure with the photonic forbidden zone ”, which received the short name photonic crystals” .
Photonic crystals are spatially periodic solid - state structures whose dielectric constant is modulated with a period comparable to the wavelength of light ^[4]

General Information

Fig. 1. Photo bracelet with opal. Opal is a natural photonic crystal.

The photonic crystals, due to the periodic change in the refractive index , make it possible to obtain allowed and forbidden zones for photon energies, similarly to semiconductor materials , in which allowed and forbidden zones are observed for charge carrier energies ^[5] . Practically, this means that if a photon with an energy ( wavelength , frequency ) that corresponds to the forbidden zone of this photonic crystal falls on a photonic crystal, then it cannot propagate in the photonic crystal and is reflected back. And vice versa, this means that if a photon with energy (wavelength, frequency) that corresponds to the allowed zone of a given photonic crystal falls on a photonic crystal, then it can propagate in a photonic crystal. In other words, the photonic crystal performs the function of an optical filter , and it is precisely its properties that determine the bright and colorful colors of opal in the bracelet, which is shown in Fig. 1. In nature, photonic crystals are also found: on the wings of African sailfish butterflies ( Papilio nireus ) ^[6] ^[7] , the mother-of-pearl cover of the mollusk shells, such as abalone , marine mouse antennae , and bristle polychaetes.

Classification of photonic crystals

According to the nature of the change in the refractive index, photonic crystals can be divided into three main classes ^[5] :

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

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

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

2. two-dimensional, in which the refractive index varies periodically in two spatial directions as shown in Fig. 3. In this figure, the photonic crystal is created by rectangular areas with a refractive index $n_{1}$ ${\ displaystyle n_ {1}}$ that are in a medium with a refractive index $n_{2}$ ${\ displaystyle n_ {2}}$ . In this case, the region with the refractive index $n_{1}$ ${\ displaystyle n_ {1}}$ ordered in a two-dimensional cubic lattice . Such photonic crystals can manifest their properties in two spatial directions, and the shape of regions with a refractive index $n_{1}$ ${\ displaystyle n_ {1}}$ It 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 may also be different, and not just cubic, as shown in the figure.

3. three-dimensional, in which the refractive index varies periodically in three spatial directions. Such photonic crystals can manifest their properties in three spatial directions, and can be represented as an array of volume 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 small losses, dielectrics — almost perfect mirrors, semiconductors — substances capable of, for example, selectively reflecting photons of a certain wavelength and superconductors , in which, due to collective phenomena, photons are able to propagate almost unlimited distances.

Also distinguish between resonant and non-resonant photonic crystals ^[2] . Resonant photonic crystals differ from nonresonant ones in that they use materials whose dielectric constant (or refractive index) as a function of frequency has a pole at a certain resonant frequency.

Any non-uniformity in a photonic crystal (for example, the absence of one or several 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 based on photonic crystals.

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

Photonic crystals allow manipulations with electromagnetic waves of the optical range, and the characteristic dimensions of photonic crystals are often close to the wavelength value. 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 solution methods that are used to study the properties of photonic crystals most often due to their availability and easy adjustment to the tasks to be solved.

It is also relevant to mention that two main approaches are used to consider the properties of photonic crystals - methods for the time domain (which allow to obtain a solution of the problem depending on the time variable), and methods for the frequency domain (which provide a solution to the problem as a function of frequency) ^{[8 ]} .

The methods for the time domain are convenient in relation to dynamic problems that provide for the time dependence of the electromagnetic field on time. They can also be used to calculate the band structures of photonic crystals, but it is practically difficult to determine the position of zones in the output of such methods. In addition, when calculating the band diagrams of photonic crystals, the Fourier transform is used , whose frequency resolution depends on the total calculation time of the method. That is, to obtain a higher resolution in the zone diagram, you need to spend more time on performing the calculations. There is another problem - the time step of such methods should be proportional to the size of the spatial grid of the method. The requirement to increase the frequency resolution of the zone 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 (using the finite element method to solve Maxwell equations) ^[9] , RSOFT Fullwave (using the finite difference method ) ^[10] , self-developed software codes for finite element methods and differences, etc.

The methods for the frequency domain are convenient first of all because the solution of Maxwell's equations occurs immediately for a stationary system, and the frequencies of the optical modes of the system are determined directly from the solution; this allows faster calculation of photonic crystal band diagrams 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 method error numerically decreases exponentially with the number of iterations performed. The disadvantages of the method are the necessity of calculating 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 software package MPB ^[11] and the commercial package ^[12] . Both mentioned software packages cannot calculate the plots of photonic crystals, in which one or several 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 ^[13]

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 the above-mentioned approaches based on convenience and their availability, as well as the methods of the radiation transfer matrix ^[14] , the program for calculating transmittance and reflection of photonic crystals using this method ^[15] , the pdetool software package that is included in the package Matlab ^[16] and the Comsol Multiphysics package already mentioned above.

The theory of photonic bandages

Fig. 4. Distribution of the refractive index of the material in the two-dimensional photonic crystal under consideration.

As noted above, photonic crystals allow one to obtain allowed and forbidden zones for photon energies, similar to semiconductor materials , in which there are allowed and forbidden zones for charge carrier energies. In the literature ^{[17], the} appearance of forbidden zones 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 frequency of the forbidden zone are shifted to different regions of the photonic crystal. Thus, the field intensity of low-frequency waves is concentrated in areas with a high refractive index, and the field intensity of high-frequency waves - in areas with a lower refractive index. In ^[2] , another description of the nature of the forbidden zones in photonic crystals is found: “photonic crystals are called media in which the dielectric constant varies periodically in space with a period that allows Bragg diffraction of light”.

If radiation with a frequency of the forbidden band was generated inside such a photonic crystal, then it cannot propagate in it, but if such radiation is sent from the outside, then it is simply reflected from the photonic crystal. One-dimensional photonic crystals make it possible to obtain forbidden zones 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 zones for radiation propagating both in one, two directions, and in all directions of a given photonic crystal, which lie in the plane of Fig. 3. Three-dimensional photonic crystals can have forbidden zones 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 forms of regions with different refractive indices and a certain crystal symmetry ^[18] .

The number of forbidden zones, their position and width in the spectrum depends on both the geometrical parameters of the photonic crystal (the size of the regions with different refractive indices, their shape, the crystal lattice in which they are ordered) and the refractive indices. Therefore, the forbidden zones can be tunable, for example, due to the use of nonlinear materials with a pronounced Kerr effect ^[19] ^[20] , due to changes in the sizes of areas with different refractive indices ^[21] or due to changes in refractive indices under the influence of external fields ^[22] .

Fig. 5. Zone diagram for photon energies (TE polarization).

Fig. 6. Zone 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 (the main material, the refractive index n = 3.53, black areas in the figure) and air (which are filled with cylindrical holes are marked in white, n = 1 ). Holes have a diameter $d$ ${\ displaystyle d}$ and arranged in a hexagonal crystal lattice with a period (distance between the centers of adjacent cylinders) $\Lambda$ ${\ displaystyle \ Lambda}$ . In the photonic crystal under consideration, the ratio of the radius of the holes $r=d/2$ ${\ displaystyle r = d / 2}$ to the period $\Lambda$ ${\ displaystyle \ Lambda}$ equally $r/\Lambda =0.4$ ${\ displaystyle r / \ Lambda = 0.4}$ . Consider the band diagrams for TE (the electric field vector is parallel to the axes of the cylinders) and TM (the magnetic field vector is parallel to the axes of the cylinders) shown in Fig. 5 and 6, which were calculated for this photonic crystal using the free program MPB ^[23] . The X-axis shows wave vectors in the photonic crystal, the Y-axis represents the normalized frequency, $f_{n}=\Lambda /\lambda$ ${\ displaystyle f_ {n} = \ Lambda / \ lambda}$ ( $\lambda$ ${\ displaystyle \ lambda}$ - wavelength in vacuum) corresponding to the energy states. The blue and red solid curves in these figures represent the energy states in this photonic crystal for TE and TM polarized waves, respectively. Blue and pink areas show the forbidden zones for photons in this photonic crystal. Black broken lines are the so-called light lines (or light cone) of a given photonic crystal ^[24] ^[25] . 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 constructed using such photonic crystals with low losses are located. In other words, the light line defines the zone of the energy states of a given photonic crystal of interest to us. The first thing you should pay attention to is that this photonic crystal has two forbidden zones for TE-polarized waves and three wide forbidden zones for TM-polarized waves. The second is the band gaps for TE and TM-polarized waves lying in the region of small values of the normalized frequency $f_{n}=\Lambda /\lambda =0.3$ ${\ displaystyle f_ {n} = \ Lambda / \ lambda = 0.3}$ overlap, which means that this photonic crystal has a total forbidden zone in the region of overlapping of the forbidden zones of TE and TM waves not only in all directions, but also for waves of any polarization (TE or TM).

Fig. 7. The reflection spectrum of the photonic crystal under consideration (TE polarization).

Fig. 8. The reflection spectrum of the photonic crystal under consideration (TM polarization).

From the given dependencies, we can determine the geometrical parameters of a photonic crystal, the first forbidden zone with the value of the normalized frequency $\Lambda /\lambda =0.3$ ${\ displaystyle \ Lambda / \ lambda = 0.3}$ accounted for by wavelength $\lambda =980$ ${\ displaystyle \ lambda = 980}$ nm. The period of the photonic crystal is $\Lambda =0.3\lambda =294$ ${\ displaystyle \ Lambda = 0.3 \ lambda = 294}$ nm, hole radius is equal $r=0.4\Lambda =117.6$ ${\ displaystyle r = 0.4 \ Lambda = 117.6}$ nm. Fig. 7 and 8 show the reflectance spectra of a photonic crystal with the parameters defined above for the TE and TM waves, respectively. The spectra were calculated using the Translight program ^[15] , it was assumed that this photonic crystal consists of 8 pairs of layers of holes and the radiation propagates in the Γ-Κ direction. From the given dependencies, we can see the most well-known property of photonic crystals - electromagnetic waves with natural frequencies corresponding to the forbidden zones of the photonic crystal (Fig.5 and 6), are characterized by a reflection coefficient close to unity and are subjected to almost complete reflection from this photonic crystal. Электромагнитные волны с частотами вне запрещённых зон данного фотонного кристалла характеризуются меньшими коэффициентами отражения от фотонного кристалла и полностью или частично проходят через него.

Изготовление фотонных кристаллов

В настоящее время существует множество методов изготовления фотонных кристаллов, и новые методы продолжают появляться. Некоторые методы больше подходят для формирования одномерных фотонных кристаллов, другие удобны в отношении двумерных, третьи применимы чаще к трёхмерным фотонным кристаллам, четвёртые используются при изготовлении фотонных кристаллов на других оптических устройствах и т. д. Рассмотрим наиболее известные из этих методов.

Методы, использующие самопроизвольное формирование фотонных кристаллов

При самопроизвольном формировании фотонных кристаллов используются коллоидные частицы (чаще всего используются монодисперсные кварцевые или полистирольные частицы, но и другие материалы постепенно становятся доступными для использования по мере разработки технологических методов их получения ^[26] ^[27] ^[28] ^[29] ), которые находятся в жидкости и по мере испарения жидкости осаждаются в некотором объёме ^[30] . По мере их осаждения друг на друга, они формируют трёхмерный фотонный кристалл, и упорядочиваются преимущественно в гранецентрированную ^[31] или гексагональную ^[32] кристаллические решетки. Этот метод достаточно медленный, формирование фотонного кристалла может занять недели.

Другой метод самопроизвольного формирования фотонных кристаллов, называемый сотовым методом, предусматривает фильтрование жидкости, в которой находятся частицы, через маленькие поры. Этот метод представлен в работах ^[33] ^[34] , позволяет сформировать фотонный кристалл со скоростью, определённой скоростью течения жидкости через поры, но при высыхании такого кристалла образуются дефекты в кристалле ^[35] .

В работе ^[36] был предложен метод вертикального осаждения, который позволяет создавать высокоупорядоченные фотонные кристаллы большего размера, чем позволяют получить вышеописанные методы ^[37] .

Выше уже отмечалось, что в большинстве случаев требуется большой контраст коэффициента преломления в фотонном кристалле для получения запрещённых фотонных зон во всех направлениях. Упомянутые выше методы самопроизвольного формирования фотонного кристалла чаще всего применялись для осаждения сферических коллоидных частиц диоксида кремния, коэффициент преломления которого относительно мал, а значит мал и контраст коэффициента преломления. Для увеличения этого контраста, используется дополнительные технологические шаги (инвертирование), на которых сначала пространство между частицами заполняется материалом с большим коэффициентом преломления, а затем частицы вытравливаются ^[38] . Пошаговый метод формирования инверсного опала описан в методическом указании по выполнению лабораторной работы ^[39] .

Методы травления

Методы травления наиболее удобны для изготовления двухмерных фотонных кристаллов и являются широко используемыми технологическими методами при производстве полупроводниковых приборов. Эти методы основаны на применении маски из фоторезиста (которая задаёт, например, массив окружностей), осаждённой на поверхности полупроводника, которая задаёт геометрию области травления. Эта маска может быть получена в рамках стандартного фотолитографического процесса , за которым следует травление сухим или влажным методом поверхности образца с фоторезистом. При этом, в тех областях, в которых находится фоторезист, происходит травление поверхности фоторезиста, а в областях без фоторезиста — травление полупроводника. Так продолжается до тех пор, пока нужная глубина травления не будет достигнута и после этого фоторезист смывается. Таким образом формируется простейший фотонный кристалл. Недостатком данного метода является использование фотолитографии , наиболее распространённое разрешение которой составляет порядка одного микрона ^[40] . Как было показано выше в этой статье, фотонные кристаллы имеют характерные размеры порядка сотен нанометров, поэтому использование фотолитографии при производстве фотонных кристаллов с запрещёнными зонами ограниченно разрешением фотолитографического процесса. Тем не менее фотолитография используется, например в работе ^[41] . Чаще всего, для достижения нужного разрешения используется комбинация стандартного фотолитографического процесса с литографией при помощи электронного пучка ^[42] . Пучки сфокусированных ионов (чаще всего ионов Ga) также применяются при изготовлении фотонных кристаллов методом травления, они позволяют удалять часть материала без использования фотолитографии и дополнительного травления ^[43] . Современные системы, использующие сфокусированные ионные пучки, используют так называемую «карту травления», записанную в специального формата файлы, которая описывает, где пучок ионов будет работать, сколько импульсов ионный пучок должен послать в определённую точку и т. д. ^[44] Таким образом, создание фотонного кристалла при помощи таких систем максимально упрощено — достаточно создать такую «карту травления» (при помощи специального программного обеспечения), в которой будет определена периодическая область травления, загрузить её в компьютер, управляющий установкой сфокусированного ионного пучка и запустить процесс травления. Для большей скорости травления, повышения качества травления или же для осаждения материалов внутри вытравленных областей используются дополнительные газы. Материалы, осаждённые в вытравленные области, позволяют формировать фотонные кристаллы, с периодическим чередованием не только исходного материала и воздуха, но и исходного материала, воздуха и дополнительных материалов. Пример осаждения материалов при помощи данных систем можно найти в источниках ^[45] ^[46] ^[47] .

Голографические методы

Голографические методы создания фотонных кристаллов базируются на применении принципов голографии , для формирования периодического изменения коэффициента преломления в пространственных направлениях. Для этого используется интерференция двух или более когерентных волн, которая создаёт периодическое распределение интенсивности электрического поля ^[48] . Интерференция двух волн позволяет создавать одномерные фотонные кристаллы, трёх и более лучей — двухмерные и трёхмерные фотонные кристаллы ^[49] ^[50] .

Другие методы создания фотонных кристаллов

Однофотонная фотолитография и двухфотонная фотолитография позволяют создавать трёхмерные фотонные кристаллы с разрешением 200нм ^[37] и использует свойство некоторых материалов, таких как полимеры , которые чувствительны к одно- и двухфотонному облучению и могут изменять свои свойства под воздействием этого излучения ^[51] ^[52] . Литография при помощи пучка электронов ^[53] ^[54] является дорогим, но высокоточным методом для изготовления двумерных фотонных кристаллов ^[55] В этом методе, фоторезист, который меняет свои свойства под действием пучка электронов облучается пучком в определённых местах для формирования пространственной маски. После облучения, часть фоторезиста смывается, а оставшаяся часть используется как маска для травления в последующем технологическом цикле. Максимальное разрешение этого метода — 10нм ^[56] . Литография при помощи пучка ионов похожа по своему принципу, только вместо пучка электронов используется пучок ионов. Преимущества литографии при помощи пучка ионов над литографией при помощи пучка электроновThe reason is that the photoresist is more sensitive to ion beams than electrons and there is no “proximity effect”, which limits the minimum possible size of the area during lithography with an electron beam ^[57] ^[58] ^[59] .

Application

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

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

Lasers with photonic crystals will allow to obtain low-signal lasing, the so-called low-threshold and threshold-free lasers ;
Waveguides based on photonic crystals can be very compact and have low losses;
With the help of photonic crystals, it will be possible to create a medium with a negative refractive index , which will make it possible to focus the light into a point smaller than the wavelength (“ superlens ”);
Photonic crystals have significant dispersive properties (their properties depend on the wavelength of radiation passing through them), this will give the opportunity to create superprisms ;
A new class of displays in which pixel color manipulation is carried out using photonic crystals will partially or completely replace existing displays;
Due to the ordered nature of the phenomenon of photon retention in a photonic crystal, based on these media, it is possible to build optical memory devices and logic devices ^[60] ^[61] ;
Photon superconductors ^[62] ^[63] exhibit their superconducting properties at certain temperatures and can be used as fully optical temperature sensors capable of operating at high frequencies. and be combined with photon insulators and semiconductors.

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↑ Matlab - the language of technical calculations.
40 p. 40, JD Joannopoulos, RD Meade, and JN Winn, Photonic Crystals: Moldings, Princeton Univ. Press, 1995.
P. 241, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.
P. 246, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.
↑ D. Vujic and S. John . "Pulse reshaping in photonic crystal waveguides and microcavities with Kerr nonlinearity: Critical issues for all-optical switching," Physical Review A, Vol. 72, 2005, p. 013807. Archival copy of April 16, 2007 on the Wayback Machine
↑ http://www3.interscience.wiley.com/cgi-bin/fulltext/114286507/PDFSTART (not available link) J. Ge, Y. Hu, and Y. Yin, "Highly Tunable Superparamagnetic Colloidal Photonic Crystals," Angewandte Chemie International Edition, Vol. 46, No. 39, pp. 7428-7431.
↑ A. Figotin, YA Godin, and I. Vitebsky, "Two-dimensional tunable photonic crystals," Physical Review B, Vol. 57, 1998, p. 2841.
↑ MIT Photonic-Bands package, developed by Steven G. Johnson at MIT along with the Joannopoulos Ab Initio Physics group.
↑ http://www.elettra.trieste.it/experiments/beamlines/lilit/htdocs/people/luca/tesihtml/node14.html Fabrication of Photonic Band Gap Materials.
↑ P. Lalanne, "Electromagnetic Analysis of Photonic Crystal Waveguides", IEEE J. of Quentum Electronics, Vol. 38, No. 7, 2002, pp. 800-804. "
↑ A. Pucci, M. Bernabo, P. Elvati, LI Meza, F. Galembeck, CA de P. Leite, N. Tirelli, and G. Ruggeriab, "Induced by alcohol using polymers," J. Mater. Chem., Vol. 16, 2006, pp. 1058-1066.
A. Reinholdt, R. Detemple, AL Stepanov, TE Weirich, and U. Kreibig, "Novel nanoparticle matter: ZrN-nanoparticles," Applied Physics B: Lasers and Optics, Vol. 77, 2003, pp. 681–686. (inaccessible link)
↑ L. Maedler, WJ Stark, and SE Pratsinisa, "Simultaneous deposition of Au nanoparticles during the synthesis of TiO2 and SiO2," J. Mater. Res., Vol. 18, No. 1, 2003, pp. 115-120.
↑ KK Akurati, R. Dittmann, A. Vital, U. Klotz, P. Hug, T. Graule, and M. Winterer, “Synthesis-based composite oxide mixed-oxide nanoparticles from atmospheric pressure flame synthesis,” Journal of Nanoparticle Research Vol. 8, 2006, pp. 379-393.
25 p. 252, PN Prasad, Nanophotonics, John Wiley and Sons, 2004
↑ A.-P. Hynninen, JHJ Thijssen, ECM Vermolen, M. Dijkstra, and A. van Blaaderen, "Self-assembly route of the crystals," Nature Materials 6, 2007, pp. 202–205.
↑ X. Ma, W. Shi, Z. Yan, and B. Shen, "Core-shell colloidal photonic crystals," Fabrication of silica / zinc oxide, Applied Physics B: Lasers and Optics, Vol. 88, 2007, pp. 245–248.
↑ SH Park and Y. Xia, "Assembly of Mesoscale Particles over Large Particle Patterns" and Tuning Optical Filters, "Langmuir, Vol. 23, 1999, pp. 266-273.
↑ SH Park, B. Gates, Y. Xia, "A Three-Dimensional Photonic Crystal Operating in the Visible Region," Advanced Materials, 1999, Vol. 11, pp. 466-469. (inaccessible link)
P. 252, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.
↑ YA Vlasov, X.-Z. Bo, JC Sturm, and DJ Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature, Vol. 414, No. 6861, p. 289.
↑ ¹ ² p. 254, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.
M. Cai, R. Zong, B. Li, and J. Zhou, "Synthesis of inverse polymer films," Journal of Materials Science Letters, Vol. 22, No. 18, 2003, pp. 1295-1297.
↑ R. Schroden, N. Balakrishan, “Inverse opal photonic crystals. A laboratory guide, University of Minnesota. (Unsolved) (inaccessible link) . The date of circulation is October 22, 2007. Archived August 18, 2007.
↑ Virtual cleanroom, Georgia Institute of Technology. (Unsolved) (inaccessible link) . The appeal date is October 23, 2007. Archived December 23, 2016.
↑ P. Yao, GJ Schneider, DW Prather, ED Wetzel, and DJ O'Brien, "Fabrication of three-dimensional photonic crystals with multilayer photolithography," Optics Express, Vol. 13, No. 7, 2005, pp. 2370-2376.
↑ A. Jugessur, P. Pottier, and R. De La Rue, "photonic crystal microcavity filters", Optics Express, Vol. 12, No. 7, 2005, pp. 1304-1312.
↑ S. Khizroev, A. Lavrenov, N. Amos, R. Chomko, and D. Litvinov, “Focused Ion Beta as the Nanofabrication Tool for Rapid Prototyping of Nanomagnetic Devices,” Microsc Microanal 12 (Supp 2), 2006, pp. 128-129.
↑ Nanofabrication and rapid prototyping with DialBeam instruments. FEI Company.
↑ Y. Fu, N. Kok, A. Bryan, and ON Shing, "Integrated Micro-Cylindrical Fiber Coupling," IEEE Photonics Technology Letters, Vol. 12, No. 9, 2000, pp. 1213–1215. (Unsolved) (inaccessible link) . The date of circulation is October 23, 2007. Archived May 24, 2006.
↑ S. Matsui and Y. Ochiai, "Focused ion beam applications to solid state devices," Nanotechnology, Vol. 7, 1996, pp. 247-258.
↑ MW Phaneuf, "Applications (Fun and Practical) of FIB Nano-Deposition and Nano-Machining," Microsc. Microanal. 8 (Suppl. 2), 2002, pp. 568CD-569CD.
25 p. 257, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.
↑ GQ Liang, WD Mao, YY Pu, H. Zou, HZ Wang, and ZH Zeng, “Fabrication of a two-dimensional coupled photonic crystal resonator arrays by holographic lithography” (not available link) , Appl. Phys. Lett. Vol. 89, 2006, p. 041902.
D M. Duneau, F. Delyon, and M. Audier, “Holographic method of chemical vapor deposition” (unavailable link) , Journal of Applied Physics , Vol. 96, No. 5, 2004, pp. 2428–2436.
↑ BH Cumpston, SP Ananthavel, S. Barlow, DL Dyer, JE Ehrlich, LL Erskine, AA Heikal, SM Kuebler, I.-YSLee, D. McCord-Maughon, J. Qin, H. Roeckel, M. Rumi, X .-L. Wu, SR Marder, and JW Perry, "Two-photon polymerization initiators for optical fiber data," Nature, Vol. 398, No. 6722, 1999, pp. 51-54.
↑ S. Jeon, V. Malyarchuk, and JA Rogers, "Fabricating Three Dimensional Nanostructures Using an Optics Express, Vol. 14, No. 6, 2006, pp. 2300-2308.
↑ http://www.azonano.com/details.asp?ArticleID=1208 An article about lithography using an electron beam on the Azonano website.
↑ AS Gozdz, PSD Lin, A. Scherer, and SF Lee, "Fast Direct E-beam Lithographic Fabrication for 1.3μm DFB Lasers," IEEE Electronics Letters, Vol. 24, No. 2. 1988, pp. 123-125.
256 p. 256, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.
IN INEX association page, describing the parameters of the used lithography setup using an electron beam. (Unsolved) (inaccessible link) . The appeal date is October 23, 2007. Archived July 28, 2007
27 p. 277, J. Orloff, M. Utlaut, and Lynwood Swanson, High resolution focused ion beams. FIB and its applications, Kluwer Academic, 2003.
↑ K. Arshak, M. Mihov, A. Arshak, D. McDonagh, and D. Sutton, "Focused Ion Beam Lithography-Overview and New aproaches," Proc. 24th International Conference on Microelectronics (MIEL 2004), Vol. 2, 2004, pp. 459-462.
↑ K. Arshak, M. Mihov, A. Arshak, D. McDonagh, D. Sutton, and SB Newcomb, "Negative resist and ionism," J. Vac. Sci. Technol. B, Vol. 22, No. 1, 2004, pp. 189–195. (inaccessible link)
↑ K. Asakawa, Y. Sugimoto, Y. Watanabe, N. Ozaki, A. Mizutani, Y. Takata, Y. Kitagawa, H. Ishikawa, N. Ikeda, K. Awazu, X. Wang, A. Watanabe, S Nakamura, S. Ohkouchi, K. Inoue, M. Kristensen, O. Sigmund, PI Bore, and R. Baets, "Newtonic Physics" . 8, 2006, p. 208.
↑ P. Lodahl, AF van Driel, IS Nikolaev1, A. Irman, K. Overgaag, D. Vanmaekelbergh, and WL Vos, "Controlling the dynamics of nature," Nature, Vol. 430, No. 7000, 2004, p.
↑ CHR Ooi, TC Au Yeung, CH Kam, and TK Lim, "Photonic band gap in a superconductor-dielectric superlattice," Phys. Rev. B, Vol. 61, 2000, pp. 5920 - 5923. (inaccessible link)
↑ C.-J. Wu, M.-S. Chen, and T.-J. Yang, "Photonic band structure for a superconductor-dielectric superlattice," Physica C: Superconductivity, Vol. 432, 2005, pp. 133-113.

Literature

JD Joannopoulos, RD Meade, and JN Winn, Photonic Crystals: Moldings, Princeton Univ. Press, 1995.
PN Prasad, Nanophotonics, John Wiley and Sons, 2004.
J.-M. Lourtioz, H. Benistry, V. Berger, J.-M. Gerard, D. Maystre, A. Tchelnokov, Photonic Crystals. Towards Nanoscale Photonic Devices, Springer, 2005.
V.I. Belotelov, A.K. Zvezdin, Photonic crystals and other metamaterials. Quantum library. Issue 94. 2006

Links

[1] VI p. VI in the book Photonic Crystals, H. Benisty, V. Berger, J.-M. Gerard, D. Maystre, A. Tchelnokov, Springer 2005.

[ioffe.ru-2] ¹ ² ³ E.L. Ivchenko, A.N. Poddubny, "Resonant three-dimensional photonic crystals," Solid State Physics, 2006, Vol. 48, Iss. 3, pp. 540-547.

[3] V. A. Kosobukin, “Photonic Crystals,“ Window to Microworld ”, No. 4, 2002. (Neopr.) (Inaccessible link) . The date of circulation is October 22, 2007. Archived November 2, 2007.

[4] V.G. Fedotov, A.V. Selkin / MULTIWAVE BRAGG DIFFRACTION AND INTERFERENCE EFFECTS IN 3D PHOTO CRYSTALLINE FILMS . - Journal NRU ITMO. - Nanosystems: physics, chemistry, mathematics. - 2 (11). -UKK 538.958

[ab-initio.mit.edu-5] ² ¹ ² Photonic Crystals: Periodic Surprises in Electromagnetism

[6] CNews, Photonic crystals were the first to invent butterflies. (Unsolved) (inaccessible link) . The appeal date is June 26, 2019. Archived March 31, 2014.

[7] K S. Kinoshita, S. Yoshioka and K. Kawagoe "iridescent scale," Proc. R. Soc. Lond. B, Vol. 269, 2002, pp. 1417-1421.

[8] ttp://ab-initio.mit.edu/wiki/index.php/MPB_Introduction Steven Johnson, MPB manual.

[9] A software package for solving physical problems.

[10] ttp://optics.synopsys.com/rsoft/rsoft-passive-device-fullwave.html Software package for solving electrodynamic problems RSOFT Fullwave.

[11] Software package for calculating the band diagrams of photonic crystals MIT Photonic Bands.

[12] Software package for calculating band diagrams of photonic crystals RSOFT BandSolve.

[13] A. Reisinger, "Characteristics of lossy waveguides," Appl. Opt. Vol. 12, 1073, p. 1015.

[14] MH Eghlidi, K. Mehrany, and B. Rashidian, "Improved differential-transfer-matrix method for inhomogeneous one-dimensional photonic crystals," J. Opt. Soc. Am. B, Vol. 23, No. 7, 2006, pp. 1451-1459.

[Translight_P.M-15] ² ¹ ² Translight Program, Developers: Andrew L. Reynolds, The Glasgow University of Glasgow and the Initial Program Originator from Imperial College, London, Professor JB Pendry, Professor PM Bell, Dr. AJ Ward and Dr. L. Martin Moreno.

[16] Matlab - the language of technical calculations.

[17] 40 p. 40, JD Joannopoulos, RD Meade, and JN Winn, Photonic Crystals: Moldings, Princeton Univ. Press, 1995.

[18] P. 241, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.

[19] P. 246, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.

[20] D. Vujic and S. John . "Pulse reshaping in photonic crystal waveguides and microcavities with Kerr nonlinearity: Critical issues for all-optical switching," Physical Review A, Vol. 72, 2005, p. 013807. Archival copy of April 16, 2007 on the Wayback Machine

[21] ttp://www3.interscience.wiley.com/cgi-bin/fulltext/114286507/PDFSTART (not available link) J. Ge, Y. Hu, and Y. Yin, "Highly Tunable Superparamagnetic Colloidal Photonic Crystals," Angewandte Chemie International Edition, Vol. 46, No. 39, pp. 7428-7431.

[22] A. Figotin, YA Godin, and I. Vitebsky, "Two-dimensional tunable photonic crystals," Physical Review B, Vol. 57, 1998, p. 2841.

[23] MIT Photonic-Bands package, developed by Steven G. Johnson at MIT along with the Joannopoulos Ab Initio Physics group.

[24] ttp://www.elettra.trieste.it/experiments/beamlines/lilit/htdocs/people/luca/tesihtml/node14.html Fabrication of Photonic Band Gap Materials.

[25] P. Lalanne, "Electromagnetic Analysis of Photonic Crystal Waveguides", IEEE J. of Quentum Electronics, Vol. 38, No. 7, 2002, pp. 800-804. "

[26] A. Pucci, M. Bernabo, P. Elvati, LI Meza, F. Galembeck, CA de P. Leite, N. Tirelli, and G. Ruggeriab, "Induced by alcohol using polymers," J. Mater. Chem., Vol. 16, 2006, pp. 1058-1066.

[27] A. Reinholdt, R. Detemple, AL Stepanov, TE Weirich, and U. Kreibig, "Novel nanoparticle matter: ZrN-nanoparticles," Applied Physics B: Lasers and Optics, Vol. 77, 2003, pp. 681–686. (inaccessible link)

[28] L. Maedler, WJ Stark, and SE Pratsinisa, "Simultaneous deposition of Au nanoparticles during the synthesis of TiO2 and SiO2," J. Mater. Res., Vol. 18, No. 1, 2003, pp. 115-120.

[29] KK Akurati, R. Dittmann, A. Vital, U. Klotz, P. Hug, T. Graule, and M. Winterer, “Synthesis-based composite oxide mixed-oxide nanoparticles from atmospheric pressure flame synthesis,” Journal of Nanoparticle Research Vol. 8, 2006, pp. 379-393.

[30] 25 p. 252, PN Prasad, Nanophotonics, John Wiley and Sons, 2004

[31] A.-P. Hynninen, JHJ Thijssen, ECM Vermolen, M. Dijkstra, and A. van Blaaderen, "Self-assembly route of the crystals," Nature Materials 6, 2007, pp. 202–205.

[32] X. Ma, W. Shi, Z. Yan, and B. Shen, "Core-shell colloidal photonic crystals," Fabrication of silica / zinc oxide, Applied Physics B: Lasers and Optics, Vol. 88, 2007, pp. 245–248.

[33] SH Park and Y. Xia, "Assembly of Mesoscale Particles over Large Particle Patterns" and Tuning Optical Filters, "Langmuir, Vol. 23, 1999, pp. 266-273.

[34] SH Park, B. Gates, Y. Xia, "A Three-Dimensional Photonic Crystal Operating in the Visible Region," Advanced Materials, 1999, Vol. 11, pp. 466-469. (inaccessible link)

[35] P. 252, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.

[36] YA Vlasov, X.-Z. Bo, JC Sturm, and DJ Norris, "On-chip natural assembly of silicon photonic bandgap crystals," Nature, Vol. 414, No. 6861, p. 289.

[P.N._Prasad,_Nanophotonics_2004-37] ¹ ² p. 254, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.

[38] M. Cai, R. Zong, B. Li, and J. Zhou, "Synthesis of inverse polymer films," Journal of Materials Science Letters, Vol. 22, No. 18, 2003, pp. 1295-1297.

[39] R. Schroden, N. Balakrishan, “Inverse opal photonic crystals. A laboratory guide, University of Minnesota. (Unsolved) (inaccessible link) . The date of circulation is October 22, 2007. Archived August 18, 2007.

[40] Virtual cleanroom, Georgia Institute of Technology. (Unsolved) (inaccessible link) . The appeal date is October 23, 2007. Archived December 23, 2016.

[41] P. Yao, GJ Schneider, DW Prather, ED Wetzel, and DJ O'Brien, "Fabrication of three-dimensional photonic crystals with multilayer photolithography," Optics Express, Vol. 13, No. 7, 2005, pp. 2370-2376.

[42] A. Jugessur, P. Pottier, and R. De La Rue, "photonic crystal microcavity filters", Optics Express, Vol. 12, No. 7, 2005, pp. 1304-1312.

[43] S. Khizroev, A. Lavrenov, N. Amos, R. Chomko, and D. Litvinov, “Focused Ion Beta as the Nanofabrication Tool for Rapid Prototyping of Nanomagnetic Devices,” Microsc Microanal 12 (Supp 2), 2006, pp. 128-129.

[44] Nanofabrication and rapid prototyping with DialBeam instruments. FEI Company.

[45] Y. Fu, N. Kok, A. Bryan, and ON Shing, "Integrated Micro-Cylindrical Fiber Coupling," IEEE Photonics Technology Letters, Vol. 12, No. 9, 2000, pp. 1213–1215. (Unsolved) (inaccessible link) . The date of circulation is October 23, 2007. Archived May 24, 2006.

[46] S. Matsui and Y. Ochiai, "Focused ion beam applications to solid state devices," Nanotechnology, Vol. 7, 1996, pp. 247-258.

[47] MW Phaneuf, "Applications (Fun and Practical) of FIB Nano-Deposition and Nano-Machining," Microsc. Microanal. 8 (Suppl. 2), 2002, pp. 568CD-569CD.

[48] 25 p. 257, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.

[49] GQ Liang, WD Mao, YY Pu, H. Zou, HZ Wang, and ZH Zeng, “Fabrication of a two-dimensional coupled photonic crystal resonator arrays by holographic lithography” (not available link) , Appl. Phys. Lett. Vol. 89, 2006, p. 041902.

[50] D M. Duneau, F. Delyon, and M. Audier, “Holographic method of chemical vapor deposition” (unavailable link) , Journal of Applied Physics , Vol. 96, No. 5, 2004, pp. 2428–2436.

[51] BH Cumpston, SP Ananthavel, S. Barlow, DL Dyer, JE Ehrlich, LL Erskine, AA Heikal, SM Kuebler, I.-YSLee, D. McCord-Maughon, J. Qin, H. Roeckel, M. Rumi, X .-L. Wu, SR Marder, and JW Perry, "Two-photon polymerization initiators for optical fiber data," Nature, Vol. 398, No. 6722, 1999, pp. 51-54.

[52] S. Jeon, V. Malyarchuk, and JA Rogers, "Fabricating Three Dimensional Nanostructures Using an Optics Express, Vol. 14, No. 6, 2006, pp. 2300-2308.

[53] ttp://www.azonano.com/details.asp?ArticleID=1208 An article about lithography using an electron beam on the Azonano website.

[54] AS Gozdz, PSD Lin, A. Scherer, and SF Lee, "Fast Direct E-beam Lithographic Fabrication for 1.3μm DFB Lasers," IEEE Electronics Letters, Vol. 24, No. 2. 1988, pp. 123-125.

[55] 256 p. 256, PN Prasad, Nanophotonics, John Wiley and Sons, 2004.

[56] IN INEX association page, describing the parameters of the used lithography setup using an electron beam. (Unsolved) (inaccessible link) . The appeal date is October 23, 2007. Archived July 28, 2007

[57] 27 p. 277, J. Orloff, M. Utlaut, and Lynwood Swanson, High resolution focused ion beams. FIB and its applications, Kluwer Academic, 2003.

[58] K. Arshak, M. Mihov, A. Arshak, D. McDonagh, and D. Sutton, "Focused Ion Beam Lithography-Overview and New aproaches," Proc. 24th International Conference on Microelectronics (MIEL 2004), Vol. 2, 2004, pp. 459-462.

[59] K. Arshak, M. Mihov, A. Arshak, D. McDonagh, D. Sutton, and SB Newcomb, "Negative resist and ionism," J. Vac. Sci. Technol. B, Vol. 22, No. 1, 2004, pp. 189–195. (inaccessible link)

[60] K. Asakawa, Y. Sugimoto, Y. Watanabe, N. Ozaki, A. Mizutani, Y. Takata, Y. Kitagawa, H. Ishikawa, N. Ikeda, K. Awazu, X. Wang, A. Watanabe, S Nakamura, S. Ohkouchi, K. Inoue, M. Kristensen, O. Sigmund, PI Bore, and R. Baets, "Newtonic Physics" . 8, 2006, p. 208.

[61] P. Lodahl, AF van Driel, IS Nikolaev1, A. Irman, K. Overgaag, D. Vanmaekelbergh, and WL Vos, "Controlling the dynamics of nature," Nature, Vol. 430, No. 7000, 2004, p.

[62] CHR Ooi, TC Au Yeung, CH Kam, and TK Lim, "Photonic band gap in a superconductor-dielectric superlattice," Phys. Rev. B, Vol. 61, 2000, pp. 5920 - 5923. (inaccessible link)

[63] C.-J. Wu, M.-S. Chen, and T.-J. Yang, "Photonic band structure for a superconductor-dielectric superlattice," Physica C: Superconductivity, Vol. 432, 2005, pp. 133-113.