rus
Issue #8/2018
A.A.Kraysky, A.V.Kraysky, M.A.Kazaryan, R.A.Zakharyan
Resonant increase in the intense field inside the photonic crystal in the transparency windows near the forbidden zone and some of its applications
Resonant increase in the intense field inside the photonic crystal in the transparency windows near the forbidden zone and some of its applications
The review describes the effect of a resonant increase in the intense light field inside a finite-size photonic crystal medium, when the wavelength of this radiation is in the transparency windows of the medium adjacent to the edge of the band gap of the medium. Some applications of this effect are described: the creation of lasers with a low lasing threshold; observation of nonlinear optical effects at low radiation power; creation of sensitive sensors. A review of the theoretical studies of this effect is given.
DOI: 10.22184/1993-7296.2018.12.8.802.822
DOI: 10.22184/1993-7296.2018.12.8.802.822
Теги: band gap increase in intensity lowering threshold perturbation theory photonic crystal sensors transparency window zone edge запрещенная зона край зоны окно прозрачности понижение порога сенсоры теория возмущений увеличение интенсивности фотонный кристалл
1. INTRODUCTION
The number of publications on photonic crystals (PC) according to the GOOGLE SCOLAR search engine experienced an increase to the beginning of the 2010s. The interst was growing over the past decade, and has firmly stabilized at an average level of 582 publications per year with a standard deviation of 22 (Fig. 1.). The interest in this direction is caused by the progress in materials science and the possibility of creating precisely specified structures of materials with the necessary properties. The possibility of limiting miniaturization of devices and the possibility of obtaining high performance properties when using such materials also contributes to the stimulation of interest.
This article presents an overview of research related to the field enhancement effect in transparency windows near the forbidden zone (FZ) of the PC and some important applications of this effect from our point of view.
The term "PC" is an English translation of the "photonic crystal" and was fairly confidently established in Russian, although it was criticized, apparently with sufficient reason. The English term was introduced in [1].
2.
PROPAGATION OF LIGHT IN PERIODIC STRUCTURES
PCs are the media with spatially periodic optical properties, when the period is comparable with the radiation wavelength. For a three-dimensional medium, the periodicity can be in all three dimensions similar to the arrangement of structural elements in natural crystals. In this case, they are called three-dimensional crystals (3-d crystals). Flat-layered structure is often called one-dimensional PC. In planar media with periodically structured optical properties of the waveguide, in two or one dimension it is also possible to speak of two-dimensional and one-dimensional crystals. And for fiber, actually one-dimensional structures, there can also exist photonic crystal structures, e. g., Bragg mirrors, when the optical properties of a fiber waveguide periodically change in space. A common feature of all such structures of properties is the presence of a "forbidden zone," i. e., the regions of wavelengths in the spectrum that cannot propagate in any range of directions.
The propagation of waves of different nature in periodic structures was considered in detail for a long time (see, e. g., [2, 3]) and continues to be the subject of research till the present. Moreover, according to the same GOOGLE SCOLAR system, this trend in science, since the mid‑90s of the last century from a level of less than 100 publications per year, is constantly increasing the number of publications, reaching last year the level of more than 1,400 papers per year. For the first time these media were created and studied in the optics as early as the 19th century: photography in natural colors, discovered by E. Becquerel in the second third of the century, and brought to a finished look by Lippmann [4, 5] at the end of the century, for which he received Nobel Prize in 1908. Later, it laid the foundation to the Denisyuk’s holography. The first phenomenon observed by Becquerel was explained by Zenker [4, 5], then his theory was developed by Rayleigh [6–7]. In acoustooptics, the diffraction of light by ultrasound was considered (e. g., [8–9]). In crystallography, X-ray diffraction on the crystal grating was considered (see, e. g., [10–11]). In a solid, the transfer phenomena are associated with the motion of electrons in a periodic crystal structure. The concept of a band structure, important for such media, with forbidden zones in the energy spectrum, within the limits of which a uniform wave cannot propagate, was developed. These concepts were also used in the studies of photonic crystals (see, e. g., [1]). The concept of FZ is one of the key in the PC science. For the first time the possibility of the formation of the FZ was established by V. P. Bykov [12]. Later, their properties were studied independently of this work [13, 14]. The concept of such a forbidden zone refers to an infinite crystal, in the case of a one-dimensional flat-layered medium – to a PC of infinite thickness. For these, for any value, the modulation amplitude of the variable part of the optical properties of a wave with frequencies lying inside the FZ corresponding to such optical characteristics, cannot exist.
3.
SOME PROPERTIES OF ONE-DIMENSIONAL PHOTONIC CRYSTALS
The actual PCs have finite dimensions, which lead to some differences in their properties. The easiest thing is that the presence of the FZ and the essence of the effect can be shown by the example of a multilayer periodic structure, when the optical characteristics change only along the depth of the layer. Before considering the properties of a PC of finite thickness, we will show the situation with wave propagation in a one-dimensional periodic medium.
It is well known that in the reflection spectrum of a periodic structure for small values of the variable part of the refractive index and a not too thick layer, there is a resonant wavelength for which the reflection coefficient is maximal. The reflection coefficient falls as the frequency of the incident light is detuned from the resonant frequency, and the spectrum of the reflection coefficient of such a structure emerges (Fig. 2).
The calculation was carried out for harmonic change of the refractive index n = n0(1 + v cos (2px / d)), where the average value of the refractive index is n0 = 1,33, the relative amplitude of the variable part of the refractive index is v = 0,002, the period is 167.6 nm, the resonant wavelength is λр = 600 nm, layer thickness 20 microns. For details on the little-known features of this spectrum, see [15].
When the reflection coefficient increases to 0.1 for the resonant wavelength, e. g., due to an increase in the relative amplitude of the variable part of the refractive index, the shape of the spectral maximum of the reflection coefficient remains almost unchanged. With a further increase in the resonance reflection coefficient, the shape of the spectral peak begins to change, it broadens and gradually takes on the form characteristic for a PC [15].
Fig. 3 shows the transmission spectrum of such a medium calculated by us with a normal incidence of a plane wave on the layer. The FZ centered at 600 nm is visible. Within this FZ, transmission is practically absent. At wavelengths of λn = λр / n (n = 1, 2, …), transmission dips may also appear, i. e. FZ of a higher order on the frequency scale (ωn = n · ωр, where ωр = 2 π c / λр). The interference structure, determined by the number of the PC periods, is visible near the FZ. With an increase in the amplitude of the variable part of the refractive index, the transmittance in the minima of this structure near the FZ decreases significantly. Thus, narrow bandwidths are formed. Two such zones, the 1st and the 2nd, are shown in Fig. 4a. We refer to such spectral structures as transparency windows. The first windows to the right and to the left limit the FZ (see fig.3a). In these windows, the field amplitude can significantly increase [16–26]. The fig. 3b shows the spatial pattern of the distribution of the amplitude of the field in depth for a wavelength of 565.76 nm, which corresponds to the maximum of the first short-wave transparency window. This is a typical picture of the interference of two opposite waves, which are the Bloch waves in the PC. On the input plane (left), the field amplitude is 1, i. e. is equal to the input field, since the reflection coefficient is equal to 0. On the output plane, the field amplitude is also equal to 1, since the transmission is complete.
The calculation shows that for radiation with a wavelength of 600 nm (the center of the zone), a reflected wave of the same intensity as that incident on the layer is observed on the input plane, since the reflection coefficient is almost 1. Therefore, the field amplitude increases by 2 times (intensity – by 4 times) in the vicinity of the nearest interference maximum. Further in depth, the maximum amplitude decreases exponentially with a damping depth by a factor of e near 1 µm. For radiation at the maximum of the short-wave transparency window (λ = 566.06 nm), the distribution of the field amplitude in depth is shown in the fig. 3b. The increase in amplitude is maximum at the center of the layer and is 3.45, which corresponds to an increase in the intensity of radiation by a factor of 12.
Fig. 4a the first and second transparency windows for the short-wave edge of the FZ for the PC layer with a thickness of 10.076 µm. The depth distribution of the amplitude of the field for the first and second transparency windows is shown (fig. 4b).
Another well-known significant property of light propagation in the PCs in transparency windows is that the field with a wavelength lying in the region of the long-wave transparency window concentrates mainly in optically more dense layers, and radiation with wave dynamics from a short-wave transparency window is predominantly in less dense layers (Fig. 5).
A thinner layer has a higher density and there are the field maxima for a long-wave edge (fig. 5a). For the short-wave edge (fig. 5b), the field maxima are located in a thick, less dense layer. When using this effect, this property must be borne in mind. In particular, this is important for nonlinear processes, when two fields interacting with one of the substances that make up the period are located in different spectral intervals.
4.
SOME APPLICATIONS OF THE PROPERTIES OF PCS NEAR THE FZ EDGE
The effect discussed can be applied widely. It forms the basis for the use of the PCs for delaying short light pulses [27], for creating low-threshold lasers [29–30], for lowering the radiation power level in nonlinear effects, in particular, for lowering the SRS threshold [31], for increasing the secondary radiation power – luminescence and Raman scattering [32–35], for increasing the conversion of light in solar cells [36], for use as sensors [37–44], parametric frequency converters [18], harmonic generators [19–21] and many others applications.
4.1. Slow light and delay lines
For a one-dimensional PC, the dispersion curve for frequencies between two FZs is shown in Fig.6. By analogy with the concept of quasiparticles in the theory of a solid body, taking into account the fact that the group velocity of light in an optically dense medium is given by the expression
Vg = ∂ω / ∂q,
for quite a long time, many researchers have been using the concept of "slow light" for radiation with a wavelength located near the edge of the FZ, because at the edge of the zone this derivative of the dispersion curve turns to 0. The increase in efficiency interactions of light with the medium in absorption, scattering, light conversion, etc. observed in the experiments conducted on the basis of these ideas, in this case are explained by an increase in the power density due to the light slowing down (see, e. g., [13]). It should, however, be borne in mind that near the FZ edge the regions of low group velocity have a very narrow spectral length. Therefore, when explaining the observed effect in this way, the question arises whether it is correct to use the concept of group velocity with extremely strong dispersion with respect to a finite width pulse.
In [27], the passage of short light pulses through periodic structures with a complex unit cell with several types of excitonic quantum wells on the period was theoretically studied. The structures with 70–80 periods, having a FZ, were considered. The authors report that the parameters of this structure were optimized for maximum deceleration with minimal losses and small distortions of the pulse shape. A calculated delay of 2 ps was obtained, which corresponds to a deceleration of the speed of light compared to a vacuum by 50 times. The intensity of the pulse during the passage of such a structure according to calculations decreases by 4 times. This is possible only for pulses with a rather narrow spectrum with a duration of more than 10 ps.
4.2.
LOW-THRESHOLD LASERS AND NONLINEAR OPTICS
The advantage of photonic crystals is an increase in the power density of radiation in specific areas of space when the radiation spectrum is near the edge of the forbidden zone in the transparency windows. This can lower the thresholds of non-linear phenomena. In particular, with the introduction of an amplifying medium into the region of increased power, low-threshold lasing can be obtained [29, 30].
In [29], the surface (Tamm) state in the band gap of a one-dimensional periodic structure with a defect was used to control the low-threshold generation wavelength. This is the name of a state with a transparency window, which occurs in the FZ of a periodic structure that has a defect in periodicity. At the same time, for radiation in this transparency window, there is a strong amplification of the field amplitude, as a rule, a much stronger one than that for an ideal crystal transparency window adjacent to the FZ. In the vicinity of the defect, the spatial localization of radiation is observed. In the case of a thick defect layer, this is essentially the Fabry-Perot interferometer. In [29], the defect was filled with a nematic liquid crystal to control the Tamm state.
The multilayer structure was composed of SiO2 and TiO2 layers deposited on an In-Sn-coated oxide glass substrate. The refractive indices of SiO2 and TiO2 were 1.45 and 2.35, respectively. The thicknesses of the SiO2 and TiO2 layers were 103 nm and 64 nm, respectively. As a result, the center of the FZ was at a wavelength of 600 nm. Five pairs of SiO2 and TiO2 layers were deposited on the substrate. Two such samples were superimposed on each other so that the coatings were facing each other, separated by a gap of 1 µm. This gap was filled with a liquid crystal (LC), which was oriented parallel to the surfaces of the samples. The refractive indices of ordinary and extraordinary rays are no = 1.5 and ne = 1.7, respectively. When a voltage was applied for radiation, the electric field of which is oriented along the y axis, all 4 transparency windows inside the FZ, when exceeding 1 V, began to shift to the shortwave region and shifted by 40 nm at a voltage of 8 V.
For a laser experiment in a similar pattern, the LC was mixed with a dye as an active medium to fill in the gap. The medium was excited by an 8 ns pulse of the 2nd harmonic of a neodymium laser (Nd: YAG). According to the authors, the threshold for the appearance of strong radiation was 5 µJ / pulse. The emission line offset was 25 nm with the application of voltage 2 V.
In [30] a scheme in which, according to the authors’ calculations, the generation threshold can be reduced according to the N‑5 law, where N is the number of periods of the photon structure, is proposed.
The effect of increasing the light emission power in the PC near the FZ edge was used in the study of SRS. When saturation of artificial opal with nitrobenzene was observed in [31], the generation threshold in nitrobenzene was lowered by no less than 20 times.
4.3.
OBSERVATION OF LUMINESCENCE AND RAMAN SCATTERING
Also, the effect of increasing the power density of light radiation near the edge of the FZ was used to increase the signal in studies of Raman scattering and luminescence, when there was either excitation or emitted radiation near the FZ edge. An increase in the luminescence intensity of DNA introduced into artificial opal has been reported in [32].
In [33], the possibility of modifying the ferroelectric properties of materials when they are introduced into the pores of artificial opals is discussed, and the characteristics of the physical properties of artificial opals when the conductive media are introduced in their pores: mercury, amorphous carbon, silver, gold, etc. are considered. The transmission and reflection spectra of broadband radiation are analyzed to determine the characteristics of the stop zones depending on the diameter of the globules, the type of ferroelectric or metal introduced into the opal pores, the conditions of temperature annealing and etc. It is reported on the radiation conditions of slow electromagnetic waves in artificial opals and their characteristic properties. The possibilities of increasing the efficiency of combinational and nonlinear-optical processes in photonic crystals filled with ferroelectrics and metals are analyzed.
In [34], the amplification of Raman scattering when excited by light with λ = 1.06 µm in the PC structures of the PS in the photonic band gap of 0.9–1.2 µm is described.
In [35], a PC structure was modeled, formed by 21 pairs of layers of porous silicon (PS) with thicknesses of d1 = 100 nm (n1 = 2.36) and d2 = 130 nm (n2 = 1.91) and a uniform PS layer with a thickness equal to the thickness of the model structure (n = 2.36). It is shown, in particular, that there is a resonant penetration of the exciting field into the structure. The spectrum of the efficiency of the emission of the Raman scattering is shifted by 520 cm‑1 to the maximum frequency of the optical phonon in crystalline silicon, i. e. the presence of the resonant exit of the scattered radiation from the structure is shown. An explanation of the experimentally observed effect of multiple amplification of the Stokes component of scattered light is given. The possibility of enhancing the anti-Stokes component of the Raman scattering when excited near the edges of the photonic crystal band gap is predicted.
In [36], based on the theoretical studies of the optical properties of photonic-crystalline catalysts based on TiO2 in the area of the FZ, the authors conclude that although the short-wave (blue) neighborhood of the FZ was previously observed in inverted opals based on TiO2, it was not adjusted to the characteristics of the material. According to the calculations, it was concluded that optimization of not only the red border of the reference zone, but also the blue one leads to enhanced absorption and, therefore, will increase the yield of the photocatalytic reaction. The authors believe that their theoretical results will help to improve the photocatalytic activity of inverted opals based on TiO2, as well as cause new developments in other related fields, such as photovoltaic.
4.4. APPLICATIONS IN SENSORS
Let us discuss the sensors working with the use of this effect. The advantage of such sensors are their small sizes, coupled with high accuracy measurements. Often, such sensors are created for embedding in a chip.
In [37], the sensors using the frequency range near the discharge zone were investigated. In this paper, it was proposed to use the Bragg grating in contact with the test gas as a sensor. The light entered from the side of the grating, and the back reflection was viewed. The position of the FZ in the formed one-dimensional PC depends on the refractive index of the medium in contact with the grating. The position of the midpoint of the FZ determines the value of the refractive index of the medium under study, from which the concentration of the component of interest was judged. The measurement accuracy was 2Ч10–5, with a sample size of 1.6 mm.
A similar sensor geometry was proposed in [38], where the intensity of the wave not reflected, but transmitted, was measured.
In [39], a sensor based on the Bragg grating was also proposed. Light also entered the end. The frequency of the light was chosen close to the FZ boundary of the grating, which acted as a waveguide. The intensity of the transmitted light was measured at selected frequencies near the FZ. With a slight change in the refractive index of the medium in contact with the grating, the position of the FZ slightly shifted in frequency. The slope of the dependence of the transmittance on the frequency of the incident light on the border of the FZ is very large, and with a slight change in the refractive index, the intensity of the transmitted radiation changed significantly. Thus, a sufficiently sensitive sensor was obtained (it senses a change in the refractive index of the medium 4 Ч 10–4) with a grating size of 76.19 µm (401 periods).
An improvement was made in [40] that increased the sensitivity of the instrument. The intensity of the light transmitted along the Bragg grating, which worked as a one-dimensional photonic crystal waveguide, was also studied. However, they did not follow the change in intensity at a selected wavelength near the FZ, which occurred due to a change in the position of the FZ with a change in the refractive index of the test gas, but the movement of the transparency window closest to the FZ. Since with an increase in the thickness of the PC, this window can be made very narrow [26], the measurement accuracy of the refractive index of the gas, and, therefore, its concentration can be made quite high. In the experiment, the shift of the window by 3.5 nm occurred when the refractive index changed by 0.01, and the sample thickness was 1200 nm.
The field enhancement effect in the transparency windows operates effectively when measuring the concentration of weakly absorbing substances. With the passage of light through a gas, the frequencies that are inside three gas absorption lines will be attenuated. However, this attenuation may be too small due to the relatively low densities of gases at atmospheric pressure. In sensors, where the light travels directly through the gas (without PC), a significant path length is needed along which the light interacts with the test substance to be able to register these changes. We note that the loss in the intensity of radiation due to absorption is proportional to this intensity and, accordingly, to the modulus of the field in the square. If we combine the gas absorption line and the transparency window, then due to the presence of the periodic structure of the photonic crystal, its inner field will increase [15, 16], and by a significant amount proportional to the number of PC periods. Losses will increase accordingly. Therefore, it is possible to make smaller sensors with greater sensitivity [41–43]. In [42], this effect made it possible to create a probe of micrometric dimensions, which is important, e. g., in miniature devices. Oxygen absorption in ambient air on the 760 nm line is about 2.6% with an optical path length of about 1 m. This line is used because its O2 absorption spectrum is minimally sensitive to temperature changes and is influenced only by pressure changes. The lowest oxygen concentration that can be measured is about 20–50 ppm, which corresponds to a change in transmittance of about 0.0005. The smallest experimentally observed absorption in the experiment is about 3 times higher than the limit level of shot noise and is about 10–6. When measured in free space, this corresponds to a concentration of about 8 ppm with a path length of 1 m. Note that in a certain sense, the use of PCs with such properties can be interpreted as a long-known method of using multipass cells.
In the sensor proposed in [44], the effect of the periodic structure of the photonic crystal on the photoluminescence of a substance that fills a one-dimensional photonic crystal is studied. It is a waveguide formed by two parallel rows of holes and a gap between them. The peak in the luminescence spectrum occurs at the frequency in the center of the transparency window closest to the FZ, when the field envelope has one maximum over the sample length. The position of the transparency window is highly dependent on the refractive index of the substance filling the sensor cavities. By determining the position of the transparency window, it is possible to determine the refractive index of the substance under study. The sensitivity of the method is 7Ч102 nm per unit of the refractive index.
5.
OVERVIEW OF THE FIELD AMPLIFICATION EFFECT
For the first time, the field amplification phenomenon inside a PC at a wavelength inside the transparency window was theoretically shown in [16] for a layered periodic medium with a rectangular modulation of the refractive index profile. For this particular case, expressions were obtained for the field in the medium. Later in [19], the effect of a one-dimensional Bragg grating on the parametric amplification of light was investigated. It was assumed that the refractive index of the grating varies by thickness according to a harmonic law. The calculation followed the coupled-mode equations. Following this method, the field is considered to be the sum of several waves (in the case of [19], signal, pumping and idle, propagating both forward and backward), each of which is a carrier-frequency wave modulated by a slow envelope. Thus, an approximate distribution of the field in the medium was found. Note that the coupled-mode equation gives a correct result only with small variations of the dielectric permeability (DP) of the medium. With a DP variation of the order of 10% of the average value, the position of the transparency windows and, accordingly, the frequency maxima of the field in the medium is noticeably distorted by this method, as shown by a comparison with the exact calculation [26]. Later in [17], the passage of light through a repeated N-fold flat layer of matter was investigated. For calculation, the matrix method was used. Following it, the reflection and transmission coefficients through a periodic medium, as well as the density of states were expressed through the reflection and transmission coefficients of a single layer. Moreover, these coefficients for one layer must be calculated for each frequency of light. In [20–22], the effect of field amplification at the edge of the FZ during the generation of the 2nd harmonic was considered. The calculation in these three papers was carried out by the coupled-mode equations (CME). In [20], a medium with a harmonic (by thickness) DP variation was considered. In the research, expressions for the field and transmission and reflection coefficients are obtained. It was shown that the maximum field value is proportional to the number of layers and the amplitude of the change in the dielectric permeability (DP) of the medium. Only adjacent transparency windows were considered in this paper. In [21], the medium was assumed to consist of flat homogeneous layers with sharp boundaries. As in [20], the CME equations were used. It was found in the research that the width of the first TW is directly proportional to the square of the variation of the DP and inversely proportional to the period number cube. In [22], regardless of [20], a medium with harmonic variation of the DP was considered, and expressions for the reflection and transmission coefficients and fields in the medium were also obtained. A linear increase in the maximum field value in the TW with an increase in the number of layers was also established. In all these researches, only those TWs closest to the FZ were considered. Only in [24] it was mentioned that the quality factor of the resonance (in other words, the TW width) decreases with increasing its number. The authors of [24] consider the Bragg grating as a waveguide and, in the resulting one-dimensional PC, they analyze the field amplification in the TW. In this paper, using a numerical calculation, an increase in the maximum of the field value in the transparency window was shown in direct proportion to the increase in the number of layers and the proportionality of the quality factor to the number of layers in a cube. In [25], the expressions for the field inside the crystal were obtained, it was shown that the maximum amplitude of the field is proportional to the number of layers, the width of the first transparency window is inversely proportional to the cube of the number of layers, the dependence of the position of the transparency window on its number and PC thickness is calculated. But some of these dependencies (the relationship of the width of the first resonance and the number of layers) is given without output or reference. In many papers, this effect is numerically calculated (e. g., [45, 46, 47]). They used the matrix method, the coupled-mode equations, the finite element method. In most of the papers, the field enhancement effect in the vicinity of the FZ was considered in connection with other (as a rule, nonlinear) effects.
In [26], a more universal approach was proposed to the problem of finding the field in the PC near the FZ. Under normal incidence of a light wave with a frequency ω1 on a crystal with thickness H, the field in it represents the sum of two Bloch waves (BW) (isotropic media were considered in which the solution is degenerate with polarization twice, and the conclusions will be related to each polarization separately) with quasi-impulses q1 and q2 (ω (q1) = ω (q2) = ω1) (see figure 6). The detunings Δq=|q0–q1| and Δq=|q0–q2–2π / d| (d is the period of the structure) of the WB with the frequency of the position of the middle of the transparency window, with a sufficiently thick crystal will be small (~ 1 / H) [26]. Therefore, the corresponding terms of the light propagation equation in [26] can be considered as small corrections. Then (by analogy with [49]) using the perturbation theory, the field in a close neighborhood of the edge of the FZ (w0; q0) can be expressed in terms of solutions at the point q0. Thus, the solution of the problem is obtained through an unknown set of eigenfunctions uq0, j (x), where j is the zone number (j = 1, 2, …), which corresponds to the border of the zone. This, in [26], allowed, in particular, to obtain dependences of window parameters (field increase, position of maxima, window width) on parameters such as layer thickness and window number without specifying the optical properties profile on the period. To obtain numerical results for a particular structure, it is necessary to find this set. Based on the obtained dependences, it was also possible to obtain the dependence of the position of the edges and the band gap of an infinitely long PC from the measured position of the FZ edges of the crystal of finite size.
The results of calculations for transparency windows from the shortwave side of the FZ are shown in Fig.7 [26]. A medium with an average value of the refractive index of n0 = 1.79, a relative modulation amplitude of the refractive index of α = 0.33 with a period of 167.6 nm, a resonant wavelength of 600 nm was considered. The lines on all graphs are linear approximations of the corresponding points. In [26], for the window with the number n, the following is obtained:
Emax = const · H / n. (1)
The magnification factors of the maximum field amplitude divided by the window number are shown for different windows depending on the thickness of the PC (Fig. 7a, c). In accordance with expression (1), they lay down well on the straight line common to all windows.
The dependence of the window frequency on its number and the PC thickness is [26]:
ω = ω0 + const · n2 / H2. (2)
The calculated values for different windows depending on the square of the reverse thickness of the window are shown in Fig.7b on the right above. It can be seen that, in accordance with expression (2), they are also well approximated by straight lines. Calculated values show similar behavior depending on the square of the window number. In this case, from the linear approximation of the dependence of the position of the resonance on the square of its number for one crystal thickness value, it is possible to determine the exact frequency of the edge edge for an infinite PC. From (2) it can be seen that for an infinitely thick crystal, the frequency of the edge of the zone is equal to the free term. The same procedure can be carried out for the other edge and thus the width of the forbidden zone is found and the position of its center can be estimated. In [26], it is shown that the width of the transparency window is
δω = const · n2 / H3. (3)
In Fig. 7d, the lower right image shows the widths for different windows as a function of the inverse thickness cube.
6. CONCLUSION
Among the properties of photonic crystals, an important role is played by their resonant properties and, above all, by a strong increase in the power of light emission within the PC, with wavelengths, inside the transparency windows in the spectrum located near the forbidden zones. The calculations show that this increase can be several orders of magnitude. The experiments confirm the fact of an increase in the power density of the radiation wavelengths located near the edge of the FZ. The explanation of this effect is given by the researchers both on the basis of the resonant properties of such a structure and on the basis of ideas about the slowing down of the group speed of light in this area.
This property is used for various purposes: to create miniature lasers with a low lasing threshold and a controlled wavelength phenomena at reduced radiation power and the possibility of creating on this basis generators of radiation harmonics, parametric frequency converters, radiation converters based on combined light scattering; in the observation of weak radiation in the analysis of substances (photoluminescence, Raman scattering of light); in solar batteries to increase the conversion rate of solar energy into electrical energy; it is widely used in sensors of various types, since the parameters of transparency windows are sensitive to small changes in the refractive index of substances that fill porous PCs or bordering waveguide PCs.
The theoretical foundations of the resonance properties of PCs have been studied for a long time, their basic properties are known, but they still continue to improve till now, which witnesses the ongoing process of learning this phenomenon.
The number of publications on photonic crystals (PC) according to the GOOGLE SCOLAR search engine experienced an increase to the beginning of the 2010s. The interst was growing over the past decade, and has firmly stabilized at an average level of 582 publications per year with a standard deviation of 22 (Fig. 1.). The interest in this direction is caused by the progress in materials science and the possibility of creating precisely specified structures of materials with the necessary properties. The possibility of limiting miniaturization of devices and the possibility of obtaining high performance properties when using such materials also contributes to the stimulation of interest.
This article presents an overview of research related to the field enhancement effect in transparency windows near the forbidden zone (FZ) of the PC and some important applications of this effect from our point of view.
The term "PC" is an English translation of the "photonic crystal" and was fairly confidently established in Russian, although it was criticized, apparently with sufficient reason. The English term was introduced in [1].
2.
PROPAGATION OF LIGHT IN PERIODIC STRUCTURES
PCs are the media with spatially periodic optical properties, when the period is comparable with the radiation wavelength. For a three-dimensional medium, the periodicity can be in all three dimensions similar to the arrangement of structural elements in natural crystals. In this case, they are called three-dimensional crystals (3-d crystals). Flat-layered structure is often called one-dimensional PC. In planar media with periodically structured optical properties of the waveguide, in two or one dimension it is also possible to speak of two-dimensional and one-dimensional crystals. And for fiber, actually one-dimensional structures, there can also exist photonic crystal structures, e. g., Bragg mirrors, when the optical properties of a fiber waveguide periodically change in space. A common feature of all such structures of properties is the presence of a "forbidden zone," i. e., the regions of wavelengths in the spectrum that cannot propagate in any range of directions.
The propagation of waves of different nature in periodic structures was considered in detail for a long time (see, e. g., [2, 3]) and continues to be the subject of research till the present. Moreover, according to the same GOOGLE SCOLAR system, this trend in science, since the mid‑90s of the last century from a level of less than 100 publications per year, is constantly increasing the number of publications, reaching last year the level of more than 1,400 papers per year. For the first time these media were created and studied in the optics as early as the 19th century: photography in natural colors, discovered by E. Becquerel in the second third of the century, and brought to a finished look by Lippmann [4, 5] at the end of the century, for which he received Nobel Prize in 1908. Later, it laid the foundation to the Denisyuk’s holography. The first phenomenon observed by Becquerel was explained by Zenker [4, 5], then his theory was developed by Rayleigh [6–7]. In acoustooptics, the diffraction of light by ultrasound was considered (e. g., [8–9]). In crystallography, X-ray diffraction on the crystal grating was considered (see, e. g., [10–11]). In a solid, the transfer phenomena are associated with the motion of electrons in a periodic crystal structure. The concept of a band structure, important for such media, with forbidden zones in the energy spectrum, within the limits of which a uniform wave cannot propagate, was developed. These concepts were also used in the studies of photonic crystals (see, e. g., [1]). The concept of FZ is one of the key in the PC science. For the first time the possibility of the formation of the FZ was established by V. P. Bykov [12]. Later, their properties were studied independently of this work [13, 14]. The concept of such a forbidden zone refers to an infinite crystal, in the case of a one-dimensional flat-layered medium – to a PC of infinite thickness. For these, for any value, the modulation amplitude of the variable part of the optical properties of a wave with frequencies lying inside the FZ corresponding to such optical characteristics, cannot exist.
3.
SOME PROPERTIES OF ONE-DIMENSIONAL PHOTONIC CRYSTALS
The actual PCs have finite dimensions, which lead to some differences in their properties. The easiest thing is that the presence of the FZ and the essence of the effect can be shown by the example of a multilayer periodic structure, when the optical characteristics change only along the depth of the layer. Before considering the properties of a PC of finite thickness, we will show the situation with wave propagation in a one-dimensional periodic medium.
It is well known that in the reflection spectrum of a periodic structure for small values of the variable part of the refractive index and a not too thick layer, there is a resonant wavelength for which the reflection coefficient is maximal. The reflection coefficient falls as the frequency of the incident light is detuned from the resonant frequency, and the spectrum of the reflection coefficient of such a structure emerges (Fig. 2).
The calculation was carried out for harmonic change of the refractive index n = n0(1 + v cos (2px / d)), where the average value of the refractive index is n0 = 1,33, the relative amplitude of the variable part of the refractive index is v = 0,002, the period is 167.6 nm, the resonant wavelength is λр = 600 nm, layer thickness 20 microns. For details on the little-known features of this spectrum, see [15].
When the reflection coefficient increases to 0.1 for the resonant wavelength, e. g., due to an increase in the relative amplitude of the variable part of the refractive index, the shape of the spectral maximum of the reflection coefficient remains almost unchanged. With a further increase in the resonance reflection coefficient, the shape of the spectral peak begins to change, it broadens and gradually takes on the form characteristic for a PC [15].
Fig. 3 shows the transmission spectrum of such a medium calculated by us with a normal incidence of a plane wave on the layer. The FZ centered at 600 nm is visible. Within this FZ, transmission is practically absent. At wavelengths of λn = λр / n (n = 1, 2, …), transmission dips may also appear, i. e. FZ of a higher order on the frequency scale (ωn = n · ωр, where ωр = 2 π c / λр). The interference structure, determined by the number of the PC periods, is visible near the FZ. With an increase in the amplitude of the variable part of the refractive index, the transmittance in the minima of this structure near the FZ decreases significantly. Thus, narrow bandwidths are formed. Two such zones, the 1st and the 2nd, are shown in Fig. 4a. We refer to such spectral structures as transparency windows. The first windows to the right and to the left limit the FZ (see fig.3a). In these windows, the field amplitude can significantly increase [16–26]. The fig. 3b shows the spatial pattern of the distribution of the amplitude of the field in depth for a wavelength of 565.76 nm, which corresponds to the maximum of the first short-wave transparency window. This is a typical picture of the interference of two opposite waves, which are the Bloch waves in the PC. On the input plane (left), the field amplitude is 1, i. e. is equal to the input field, since the reflection coefficient is equal to 0. On the output plane, the field amplitude is also equal to 1, since the transmission is complete.
The calculation shows that for radiation with a wavelength of 600 nm (the center of the zone), a reflected wave of the same intensity as that incident on the layer is observed on the input plane, since the reflection coefficient is almost 1. Therefore, the field amplitude increases by 2 times (intensity – by 4 times) in the vicinity of the nearest interference maximum. Further in depth, the maximum amplitude decreases exponentially with a damping depth by a factor of e near 1 µm. For radiation at the maximum of the short-wave transparency window (λ = 566.06 nm), the distribution of the field amplitude in depth is shown in the fig. 3b. The increase in amplitude is maximum at the center of the layer and is 3.45, which corresponds to an increase in the intensity of radiation by a factor of 12.
Fig. 4a the first and second transparency windows for the short-wave edge of the FZ for the PC layer with a thickness of 10.076 µm. The depth distribution of the amplitude of the field for the first and second transparency windows is shown (fig. 4b).
Another well-known significant property of light propagation in the PCs in transparency windows is that the field with a wavelength lying in the region of the long-wave transparency window concentrates mainly in optically more dense layers, and radiation with wave dynamics from a short-wave transparency window is predominantly in less dense layers (Fig. 5).
A thinner layer has a higher density and there are the field maxima for a long-wave edge (fig. 5a). For the short-wave edge (fig. 5b), the field maxima are located in a thick, less dense layer. When using this effect, this property must be borne in mind. In particular, this is important for nonlinear processes, when two fields interacting with one of the substances that make up the period are located in different spectral intervals.
4.
SOME APPLICATIONS OF THE PROPERTIES OF PCS NEAR THE FZ EDGE
The effect discussed can be applied widely. It forms the basis for the use of the PCs for delaying short light pulses [27], for creating low-threshold lasers [29–30], for lowering the radiation power level in nonlinear effects, in particular, for lowering the SRS threshold [31], for increasing the secondary radiation power – luminescence and Raman scattering [32–35], for increasing the conversion of light in solar cells [36], for use as sensors [37–44], parametric frequency converters [18], harmonic generators [19–21] and many others applications.
4.1. Slow light and delay lines
For a one-dimensional PC, the dispersion curve for frequencies between two FZs is shown in Fig.6. By analogy with the concept of quasiparticles in the theory of a solid body, taking into account the fact that the group velocity of light in an optically dense medium is given by the expression
Vg = ∂ω / ∂q,
for quite a long time, many researchers have been using the concept of "slow light" for radiation with a wavelength located near the edge of the FZ, because at the edge of the zone this derivative of the dispersion curve turns to 0. The increase in efficiency interactions of light with the medium in absorption, scattering, light conversion, etc. observed in the experiments conducted on the basis of these ideas, in this case are explained by an increase in the power density due to the light slowing down (see, e. g., [13]). It should, however, be borne in mind that near the FZ edge the regions of low group velocity have a very narrow spectral length. Therefore, when explaining the observed effect in this way, the question arises whether it is correct to use the concept of group velocity with extremely strong dispersion with respect to a finite width pulse.
In [27], the passage of short light pulses through periodic structures with a complex unit cell with several types of excitonic quantum wells on the period was theoretically studied. The structures with 70–80 periods, having a FZ, were considered. The authors report that the parameters of this structure were optimized for maximum deceleration with minimal losses and small distortions of the pulse shape. A calculated delay of 2 ps was obtained, which corresponds to a deceleration of the speed of light compared to a vacuum by 50 times. The intensity of the pulse during the passage of such a structure according to calculations decreases by 4 times. This is possible only for pulses with a rather narrow spectrum with a duration of more than 10 ps.
4.2.
LOW-THRESHOLD LASERS AND NONLINEAR OPTICS
The advantage of photonic crystals is an increase in the power density of radiation in specific areas of space when the radiation spectrum is near the edge of the forbidden zone in the transparency windows. This can lower the thresholds of non-linear phenomena. In particular, with the introduction of an amplifying medium into the region of increased power, low-threshold lasing can be obtained [29, 30].
In [29], the surface (Tamm) state in the band gap of a one-dimensional periodic structure with a defect was used to control the low-threshold generation wavelength. This is the name of a state with a transparency window, which occurs in the FZ of a periodic structure that has a defect in periodicity. At the same time, for radiation in this transparency window, there is a strong amplification of the field amplitude, as a rule, a much stronger one than that for an ideal crystal transparency window adjacent to the FZ. In the vicinity of the defect, the spatial localization of radiation is observed. In the case of a thick defect layer, this is essentially the Fabry-Perot interferometer. In [29], the defect was filled with a nematic liquid crystal to control the Tamm state.
The multilayer structure was composed of SiO2 and TiO2 layers deposited on an In-Sn-coated oxide glass substrate. The refractive indices of SiO2 and TiO2 were 1.45 and 2.35, respectively. The thicknesses of the SiO2 and TiO2 layers were 103 nm and 64 nm, respectively. As a result, the center of the FZ was at a wavelength of 600 nm. Five pairs of SiO2 and TiO2 layers were deposited on the substrate. Two such samples were superimposed on each other so that the coatings were facing each other, separated by a gap of 1 µm. This gap was filled with a liquid crystal (LC), which was oriented parallel to the surfaces of the samples. The refractive indices of ordinary and extraordinary rays are no = 1.5 and ne = 1.7, respectively. When a voltage was applied for radiation, the electric field of which is oriented along the y axis, all 4 transparency windows inside the FZ, when exceeding 1 V, began to shift to the shortwave region and shifted by 40 nm at a voltage of 8 V.
For a laser experiment in a similar pattern, the LC was mixed with a dye as an active medium to fill in the gap. The medium was excited by an 8 ns pulse of the 2nd harmonic of a neodymium laser (Nd: YAG). According to the authors, the threshold for the appearance of strong radiation was 5 µJ / pulse. The emission line offset was 25 nm with the application of voltage 2 V.
In [30] a scheme in which, according to the authors’ calculations, the generation threshold can be reduced according to the N‑5 law, where N is the number of periods of the photon structure, is proposed.
The effect of increasing the light emission power in the PC near the FZ edge was used in the study of SRS. When saturation of artificial opal with nitrobenzene was observed in [31], the generation threshold in nitrobenzene was lowered by no less than 20 times.
4.3.
OBSERVATION OF LUMINESCENCE AND RAMAN SCATTERING
Also, the effect of increasing the power density of light radiation near the edge of the FZ was used to increase the signal in studies of Raman scattering and luminescence, when there was either excitation or emitted radiation near the FZ edge. An increase in the luminescence intensity of DNA introduced into artificial opal has been reported in [32].
In [33], the possibility of modifying the ferroelectric properties of materials when they are introduced into the pores of artificial opals is discussed, and the characteristics of the physical properties of artificial opals when the conductive media are introduced in their pores: mercury, amorphous carbon, silver, gold, etc. are considered. The transmission and reflection spectra of broadband radiation are analyzed to determine the characteristics of the stop zones depending on the diameter of the globules, the type of ferroelectric or metal introduced into the opal pores, the conditions of temperature annealing and etc. It is reported on the radiation conditions of slow electromagnetic waves in artificial opals and their characteristic properties. The possibilities of increasing the efficiency of combinational and nonlinear-optical processes in photonic crystals filled with ferroelectrics and metals are analyzed.
In [34], the amplification of Raman scattering when excited by light with λ = 1.06 µm in the PC structures of the PS in the photonic band gap of 0.9–1.2 µm is described.
In [35], a PC structure was modeled, formed by 21 pairs of layers of porous silicon (PS) with thicknesses of d1 = 100 nm (n1 = 2.36) and d2 = 130 nm (n2 = 1.91) and a uniform PS layer with a thickness equal to the thickness of the model structure (n = 2.36). It is shown, in particular, that there is a resonant penetration of the exciting field into the structure. The spectrum of the efficiency of the emission of the Raman scattering is shifted by 520 cm‑1 to the maximum frequency of the optical phonon in crystalline silicon, i. e. the presence of the resonant exit of the scattered radiation from the structure is shown. An explanation of the experimentally observed effect of multiple amplification of the Stokes component of scattered light is given. The possibility of enhancing the anti-Stokes component of the Raman scattering when excited near the edges of the photonic crystal band gap is predicted.
In [36], based on the theoretical studies of the optical properties of photonic-crystalline catalysts based on TiO2 in the area of the FZ, the authors conclude that although the short-wave (blue) neighborhood of the FZ was previously observed in inverted opals based on TiO2, it was not adjusted to the characteristics of the material. According to the calculations, it was concluded that optimization of not only the red border of the reference zone, but also the blue one leads to enhanced absorption and, therefore, will increase the yield of the photocatalytic reaction. The authors believe that their theoretical results will help to improve the photocatalytic activity of inverted opals based on TiO2, as well as cause new developments in other related fields, such as photovoltaic.
4.4. APPLICATIONS IN SENSORS
Let us discuss the sensors working with the use of this effect. The advantage of such sensors are their small sizes, coupled with high accuracy measurements. Often, such sensors are created for embedding in a chip.
In [37], the sensors using the frequency range near the discharge zone were investigated. In this paper, it was proposed to use the Bragg grating in contact with the test gas as a sensor. The light entered from the side of the grating, and the back reflection was viewed. The position of the FZ in the formed one-dimensional PC depends on the refractive index of the medium in contact with the grating. The position of the midpoint of the FZ determines the value of the refractive index of the medium under study, from which the concentration of the component of interest was judged. The measurement accuracy was 2Ч10–5, with a sample size of 1.6 mm.
A similar sensor geometry was proposed in [38], where the intensity of the wave not reflected, but transmitted, was measured.
In [39], a sensor based on the Bragg grating was also proposed. Light also entered the end. The frequency of the light was chosen close to the FZ boundary of the grating, which acted as a waveguide. The intensity of the transmitted light was measured at selected frequencies near the FZ. With a slight change in the refractive index of the medium in contact with the grating, the position of the FZ slightly shifted in frequency. The slope of the dependence of the transmittance on the frequency of the incident light on the border of the FZ is very large, and with a slight change in the refractive index, the intensity of the transmitted radiation changed significantly. Thus, a sufficiently sensitive sensor was obtained (it senses a change in the refractive index of the medium 4 Ч 10–4) with a grating size of 76.19 µm (401 periods).
An improvement was made in [40] that increased the sensitivity of the instrument. The intensity of the light transmitted along the Bragg grating, which worked as a one-dimensional photonic crystal waveguide, was also studied. However, they did not follow the change in intensity at a selected wavelength near the FZ, which occurred due to a change in the position of the FZ with a change in the refractive index of the test gas, but the movement of the transparency window closest to the FZ. Since with an increase in the thickness of the PC, this window can be made very narrow [26], the measurement accuracy of the refractive index of the gas, and, therefore, its concentration can be made quite high. In the experiment, the shift of the window by 3.5 nm occurred when the refractive index changed by 0.01, and the sample thickness was 1200 nm.
The field enhancement effect in the transparency windows operates effectively when measuring the concentration of weakly absorbing substances. With the passage of light through a gas, the frequencies that are inside three gas absorption lines will be attenuated. However, this attenuation may be too small due to the relatively low densities of gases at atmospheric pressure. In sensors, where the light travels directly through the gas (without PC), a significant path length is needed along which the light interacts with the test substance to be able to register these changes. We note that the loss in the intensity of radiation due to absorption is proportional to this intensity and, accordingly, to the modulus of the field in the square. If we combine the gas absorption line and the transparency window, then due to the presence of the periodic structure of the photonic crystal, its inner field will increase [15, 16], and by a significant amount proportional to the number of PC periods. Losses will increase accordingly. Therefore, it is possible to make smaller sensors with greater sensitivity [41–43]. In [42], this effect made it possible to create a probe of micrometric dimensions, which is important, e. g., in miniature devices. Oxygen absorption in ambient air on the 760 nm line is about 2.6% with an optical path length of about 1 m. This line is used because its O2 absorption spectrum is minimally sensitive to temperature changes and is influenced only by pressure changes. The lowest oxygen concentration that can be measured is about 20–50 ppm, which corresponds to a change in transmittance of about 0.0005. The smallest experimentally observed absorption in the experiment is about 3 times higher than the limit level of shot noise and is about 10–6. When measured in free space, this corresponds to a concentration of about 8 ppm with a path length of 1 m. Note that in a certain sense, the use of PCs with such properties can be interpreted as a long-known method of using multipass cells.
In the sensor proposed in [44], the effect of the periodic structure of the photonic crystal on the photoluminescence of a substance that fills a one-dimensional photonic crystal is studied. It is a waveguide formed by two parallel rows of holes and a gap between them. The peak in the luminescence spectrum occurs at the frequency in the center of the transparency window closest to the FZ, when the field envelope has one maximum over the sample length. The position of the transparency window is highly dependent on the refractive index of the substance filling the sensor cavities. By determining the position of the transparency window, it is possible to determine the refractive index of the substance under study. The sensitivity of the method is 7Ч102 nm per unit of the refractive index.
5.
OVERVIEW OF THE FIELD AMPLIFICATION EFFECT
For the first time, the field amplification phenomenon inside a PC at a wavelength inside the transparency window was theoretically shown in [16] for a layered periodic medium with a rectangular modulation of the refractive index profile. For this particular case, expressions were obtained for the field in the medium. Later in [19], the effect of a one-dimensional Bragg grating on the parametric amplification of light was investigated. It was assumed that the refractive index of the grating varies by thickness according to a harmonic law. The calculation followed the coupled-mode equations. Following this method, the field is considered to be the sum of several waves (in the case of [19], signal, pumping and idle, propagating both forward and backward), each of which is a carrier-frequency wave modulated by a slow envelope. Thus, an approximate distribution of the field in the medium was found. Note that the coupled-mode equation gives a correct result only with small variations of the dielectric permeability (DP) of the medium. With a DP variation of the order of 10% of the average value, the position of the transparency windows and, accordingly, the frequency maxima of the field in the medium is noticeably distorted by this method, as shown by a comparison with the exact calculation [26]. Later in [17], the passage of light through a repeated N-fold flat layer of matter was investigated. For calculation, the matrix method was used. Following it, the reflection and transmission coefficients through a periodic medium, as well as the density of states were expressed through the reflection and transmission coefficients of a single layer. Moreover, these coefficients for one layer must be calculated for each frequency of light. In [20–22], the effect of field amplification at the edge of the FZ during the generation of the 2nd harmonic was considered. The calculation in these three papers was carried out by the coupled-mode equations (CME). In [20], a medium with a harmonic (by thickness) DP variation was considered. In the research, expressions for the field and transmission and reflection coefficients are obtained. It was shown that the maximum field value is proportional to the number of layers and the amplitude of the change in the dielectric permeability (DP) of the medium. Only adjacent transparency windows were considered in this paper. In [21], the medium was assumed to consist of flat homogeneous layers with sharp boundaries. As in [20], the CME equations were used. It was found in the research that the width of the first TW is directly proportional to the square of the variation of the DP and inversely proportional to the period number cube. In [22], regardless of [20], a medium with harmonic variation of the DP was considered, and expressions for the reflection and transmission coefficients and fields in the medium were also obtained. A linear increase in the maximum field value in the TW with an increase in the number of layers was also established. In all these researches, only those TWs closest to the FZ were considered. Only in [24] it was mentioned that the quality factor of the resonance (in other words, the TW width) decreases with increasing its number. The authors of [24] consider the Bragg grating as a waveguide and, in the resulting one-dimensional PC, they analyze the field amplification in the TW. In this paper, using a numerical calculation, an increase in the maximum of the field value in the transparency window was shown in direct proportion to the increase in the number of layers and the proportionality of the quality factor to the number of layers in a cube. In [25], the expressions for the field inside the crystal were obtained, it was shown that the maximum amplitude of the field is proportional to the number of layers, the width of the first transparency window is inversely proportional to the cube of the number of layers, the dependence of the position of the transparency window on its number and PC thickness is calculated. But some of these dependencies (the relationship of the width of the first resonance and the number of layers) is given without output or reference. In many papers, this effect is numerically calculated (e. g., [45, 46, 47]). They used the matrix method, the coupled-mode equations, the finite element method. In most of the papers, the field enhancement effect in the vicinity of the FZ was considered in connection with other (as a rule, nonlinear) effects.
In [26], a more universal approach was proposed to the problem of finding the field in the PC near the FZ. Under normal incidence of a light wave with a frequency ω1 on a crystal with thickness H, the field in it represents the sum of two Bloch waves (BW) (isotropic media were considered in which the solution is degenerate with polarization twice, and the conclusions will be related to each polarization separately) with quasi-impulses q1 and q2 (ω (q1) = ω (q2) = ω1) (see figure 6). The detunings Δq=|q0–q1| and Δq=|q0–q2–2π / d| (d is the period of the structure) of the WB with the frequency of the position of the middle of the transparency window, with a sufficiently thick crystal will be small (~ 1 / H) [26]. Therefore, the corresponding terms of the light propagation equation in [26] can be considered as small corrections. Then (by analogy with [49]) using the perturbation theory, the field in a close neighborhood of the edge of the FZ (w0; q0) can be expressed in terms of solutions at the point q0. Thus, the solution of the problem is obtained through an unknown set of eigenfunctions uq0, j (x), where j is the zone number (j = 1, 2, …), which corresponds to the border of the zone. This, in [26], allowed, in particular, to obtain dependences of window parameters (field increase, position of maxima, window width) on parameters such as layer thickness and window number without specifying the optical properties profile on the period. To obtain numerical results for a particular structure, it is necessary to find this set. Based on the obtained dependences, it was also possible to obtain the dependence of the position of the edges and the band gap of an infinitely long PC from the measured position of the FZ edges of the crystal of finite size.
The results of calculations for transparency windows from the shortwave side of the FZ are shown in Fig.7 [26]. A medium with an average value of the refractive index of n0 = 1.79, a relative modulation amplitude of the refractive index of α = 0.33 with a period of 167.6 nm, a resonant wavelength of 600 nm was considered. The lines on all graphs are linear approximations of the corresponding points. In [26], for the window with the number n, the following is obtained:
Emax = const · H / n. (1)
The magnification factors of the maximum field amplitude divided by the window number are shown for different windows depending on the thickness of the PC (Fig. 7a, c). In accordance with expression (1), they lay down well on the straight line common to all windows.
The dependence of the window frequency on its number and the PC thickness is [26]:
ω = ω0 + const · n2 / H2. (2)
The calculated values for different windows depending on the square of the reverse thickness of the window are shown in Fig.7b on the right above. It can be seen that, in accordance with expression (2), they are also well approximated by straight lines. Calculated values show similar behavior depending on the square of the window number. In this case, from the linear approximation of the dependence of the position of the resonance on the square of its number for one crystal thickness value, it is possible to determine the exact frequency of the edge edge for an infinite PC. From (2) it can be seen that for an infinitely thick crystal, the frequency of the edge of the zone is equal to the free term. The same procedure can be carried out for the other edge and thus the width of the forbidden zone is found and the position of its center can be estimated. In [26], it is shown that the width of the transparency window is
δω = const · n2 / H3. (3)
In Fig. 7d, the lower right image shows the widths for different windows as a function of the inverse thickness cube.
6. CONCLUSION
Among the properties of photonic crystals, an important role is played by their resonant properties and, above all, by a strong increase in the power of light emission within the PC, with wavelengths, inside the transparency windows in the spectrum located near the forbidden zones. The calculations show that this increase can be several orders of magnitude. The experiments confirm the fact of an increase in the power density of the radiation wavelengths located near the edge of the FZ. The explanation of this effect is given by the researchers both on the basis of the resonant properties of such a structure and on the basis of ideas about the slowing down of the group speed of light in this area.
This property is used for various purposes: to create miniature lasers with a low lasing threshold and a controlled wavelength phenomena at reduced radiation power and the possibility of creating on this basis generators of radiation harmonics, parametric frequency converters, radiation converters based on combined light scattering; in the observation of weak radiation in the analysis of substances (photoluminescence, Raman scattering of light); in solar batteries to increase the conversion rate of solar energy into electrical energy; it is widely used in sensors of various types, since the parameters of transparency windows are sensitive to small changes in the refractive index of substances that fill porous PCs or bordering waveguide PCs.
The theoretical foundations of the resonance properties of PCs have been studied for a long time, their basic properties are known, but they still continue to improve till now, which witnesses the ongoing process of learning this phenomenon.
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