One-Dimensional Photonic Crystals With the Superconducting Defects

During the last 27 years, starting from pioneering papers by E. Yablonovich [1] and S. John [2], photonic crystals (PCs) or photonic band gap (PBG) materials are the objects of intensive theoretical and experimental research, because of promising applications in modern photonics and related areas [3–6]. Usually, PCs are artificial one-, twoor three-dimensional (1D, 2D, and 3D) periodic structures with periods which are comparable with the wavelengths of electro‐ magnetic waves (EMWs) and constructed of materials with different refractive indices [3–6]. The PBGs are forbidden regions in the dispersion law and in transmittivity spectra, where EMWs with selected frequencies cannot propagate through the PC. A lot of attention has been directed to the study of conventional PCs on the base of dielectrics, semiconductors, and normal metals as well as on the base of functional materials like magnetics [7, 8], ferroelectrics [9, 10] and liquid crystals [6, 11, 12] which can be controlled by external magnetic or electric fields. Another interesting group of PCs are those with superconducting (SC) constituents, socalled superconducting PCs. Scientific activity in different aspects of SC PC is described in several original and review papers by S. Savel'ev et al. [13 – 15], and in the review paper by S. Anlage [16]. The majority of papers about SC PC is devoted to regular periodic photonic structures. But, if the periodicity of a PC is destroyed by the introduction of a so-called defect unit, for example a defect layer into a 1D PC, the transmittivity spectra are drastically changed. In that case, defect modes (DMs) emerge inside the PBG because of localization of EMWs around the defect unit.


Introduction
During the last 27 years, starting from pioneering papers by E. Yablonovich [1] and S. John [2], photonic crystals (PCs) or photonic band gap (PBG) materials are the objects of intensive theoretical and experimental research, because of promising applications in modern photonics and related areas [3][4][5][6]. Usually, PCs are artificial one-, two-or three-dimensional (1D, 2D, and 3D) periodic structures with periods which are comparable with the wavelengths of electromagnetic waves (EMWs) and constructed of materials with different refractive indices [3][4][5][6]. The PBGs are forbidden regions in the dispersion law and in transmittivity spectra, where EMWs with selected frequencies cannot propagate through the PC. A lot of attention has been directed to the study of conventional PCs on the base of dielectrics, semiconductors, and normal metals as well as on the base of functional materials like magnetics [7,8], ferroelectrics [9,10] and liquid crystals [6,11,12] which can be controlled by external magnetic or electric fields. Another interesting group of PCs are those with superconducting (SC) constituents, socalled superconducting PCs. Scientific activity in different aspects of SC PC is described in several original and review papers by S. Savel'ev et al. [13 -15], and in the review paper by S. Anlage [16]. The majority of papers about SC PC is devoted to regular periodic photonic structures. But, if the periodicity of a PC is destroyed by the introduction of a so-called defect unit, for example a defect layer into a 1D PC, the transmittivity spectra are drastically changed. In that case, defect modes (DMs) emerge inside the PBG because of localization of EMWs around the defect unit.
These DMs can be observed as very narrow peaks (in comparison with the PBG widths) with sufficiently large transmittivity. If such defects are made of functional materials which are sensitive to the action of various external factors like electric or magnetic fields or temperature, it is possible to control the transmittivity of PCs at the DM frequencies.
In our papers [17,18] we investigated the influence of a complex defect layer constructed from dielectric and SC sublayers on the transmittivity of PCs for different thicknesses of the SC constituent and temperatures for normal [17] and oblique [18] incidence of EMWs on the PC.
In [19] it was shown that a thin SC layer on the top of a dielectric PC leads to selective transmittivity for different polarizations of incident EMWs. The influence of a complex defect layer (composed of SC and dielectric sublayers) as a spacer between two different PCs was studied in [20]. The localized modes in a metamaterial-dielectric PC with a dielectric-SC pair defect were investigated in [21]. The behavior of a DM inside a PBG depending on position (SC-dielectric and dielectric-SC) in a 1D PC was analyzed in [22]. In a recent paper [23], the normal and oblique incidence of an EMW on 1D dielectric PC with a thin SC defect layer was theoretically investigated and the obtained results show qualitative agreement with our calculations [17,18]. An interesting situation takes place when two (or more) defects are introduced into a PC. In the case of two defects two DMs appear inside PBG and their mutual positions strongly depend on the distance between the defect units. The linear and nonlinear optical, as well as, magneto-optical properties of 1D PCs with two magnetic defects were theoretically studied in a few papers [24 -27]. In these papers some interesting peculiarities of the spectra of DMs inside the PBGs, like the possibilities to obtain relatively wide peaks formed by two DMs, were obtained and discussed. One expect that in PCs with two complex defects composed of SC and dielectric sublayers one can obtain new peculiarities as well.
In this chapter, we theoretically investigate the transmittivity, reflectivity and absorptance of 1D PCs with one and two complex defects containing SC constituents. Two cases of asymmetric positions of complex SC-containing defects (when both SC-sublayers are located on the righthand, or on the left-hand sides of the dielectric defect sublayers inside the photonic structure) are investigated, as well as the symmetric ones. We also analyze the dependencies of the optical properties of such PCs as functions of SC-sublayer thicknesses and temperature. The chapter is organized as follows. In Sec. 2 we describe the aforementioned photonic structure with one complex defect layer with an analytical approach. In Sec. 3 the numerical calculations of the transmittivity spectra for the photonic structure two superconducting defect layers are presented. In Sec. 4, conclusions, we summarize the obtained results.

Photonic crystal with one defect
Let us consider finite 1D periodic photonic structure composed of two finite size PCs of structure (AB) 5 with the period D=d 1 +d 2 , where the layer A is the strontium titanate SrTiO 3 with thickness d 1 , the layer B is the aluminum oxide Al 2 O 3 with thickness d 2 , and the complex defect layer placed between these PCs, as shown in Fig. 1. We assume the medium surrounding the photonic structure to be a vacuum. The layers of PC are located in the xy-plane and the z-axis to be normal to the interfaces. The defect layer consists of SC sublayer YBa 2 Cu 3 O 7 of thickness d s and dielectric sublayer SrTiO 3 of thickness d 1d , respectively. The selection of these dielectrics is due to the fact that both these materials SrTiO 3 and Al 2 O 3 are widely used as substrates for YBa 2 Cu 3 O 7 SC films. For the SC layers, we consider the case when the crystallographic a-axis (electrically "light" axis) and c-axis (electrically "heavy" axis) of YBa 2 Cu 3 O 7 coincide with the x-and y-axes, respectively. In our study we used the frequency-and temperature-dependent dielectric permittivity tensor for YBa 2 Cu 3 O 7 with nonzero diagonal components ε xx (ω, T ) = ε zz (ω, T ) and ε yy (ω, T ) , introduced in Ref. [19] on the base of a generalized two-fluid model in the following form: Here the dynamic electrical conductivity tensor σ s,v (ω, T ) is defined as , , s w w s w w s w where σ s,v + (ω, T ) are determined as follows:  where T c and f ± (ω, T ) are the critical temperature and the electron distribution functions, respectively, where k B and ℏ are the Boltzmann's and Planck's constants, respectively. In Eq. (4) the pair braking frequency ω s is determined via the temperature-dependent superconductor half-gap 4 (see for details Ref. [19]).
In the optical and near infra-red regimes the electrodynamical properties of YBa 2 Cu 3 O 7 can be described by a dielectric permittivity only. The magnetic permeability of YBa 2 Cu 3 O 7 is assumed to be μ s =1.
It should be noted that SC sublayer exhibits strongly pronounced anisotropy of optical properties. Both components of the permittivity tensor ε s,xx and ε s, yy are complex values, moreover, in considered frequency region (first PGB) both real and imaginary parts of ε s,xx are about two order of magnitude larger then corresponding ones for ε s, yy . As ε s, yy and ε s,xx are responsible for TE-and TM-polarized modes, respectively, it means that decaying of TMpolarized mode inside SC sublayer is much stronger then of TE-polarized one. This difference in values of ε s,xx and ε s, yy leads to drastic contrast in behavior TE-and TM-modes with variation of the SC sublayer thickness d s .
We investigate the case of the normal incidence of light on the right hand surface of the PCs. We assume the incident light to be linearly polarized: x-or y-polarizations (the EMWs with electric field vector E vibrations along the x-or y-axis, respectively).
We calculate the transmittivity and reflectivity spectra of the 1D PC using the four-dimensional transfer matrix method by Berreman [29]. The details of the method are given in our previous papers [17,18,30].
In the case of non-SC defect layer (single dielectric layer with thickness d def =d 1 ) symmetrically embedded between two identical PCs, the dependence of transmittivity for both TE-and TMmodes in 1D PC is characterized by the presence of defect mode in the center of PBGs, as depicted on Fig. 2.
As shown in Ref. [19], a thin SC layer, deposited at one side of dielectric PC, leads to the decrease of transmittivity for the EMW, polarized along the x-axis (TM-mode), whereas TEpolarized EMW propagates through this photonic structure practically without losses.
Below we analyze numerically both intensity and position of the defect mode inside the first PBG as a function of thickness of SC sublayer for the system under consideration (see We calculated the intensities and positions of both TE-and TM-polarized defect modes inside the first PBG as a function of SC sublayer thickness for different temperatures: T=4. 2 The results of our calculations for TE-mode are presented in Figs. 3 -5 and for TM-mode are given in Fig. 6. One can see that the position of TM-polarized defect mode strongly depends on the SC sublayer thickness: the peak of the defect mode shifts to the right edge of the PBG with the increase of d s .  From Fig. 3 one can see, that at the temperature T=4.2 K the defect mode intensity is monotonically increasing with growth of SC sublayer thickness, and the corresponding defect peak approaches the right edge of the PBG. At higher temperatures (T=77 K and T=90 K) the intensity of the defect mode fist is going down to with growth of d s , and then goes up and merge with the right PBG edge.
In contrast of TM-mode, the defect mode of TE-polarization almost does not change its position under growth of temperature and SC sublayer thickness.

Photonic crystal with two defects
Let us consider a finite size 1D PC of the structure (BA) N Def 2 (BA) M Def 1 (BA) N consisting of 2N +M regular unit cells (BA) with two combined defect layers Def 1 and Def 2 embedded into the PC symmetrically with respect to the PC's edges, as shown in Fig. 1  Below we present the numerical calculations of the transmittivity spectra of the PCs under consideration as function of the SC defect sublayer thickness d s , as well as of the dielectric defect sublayer thickness d 1 . Also, we numerically investigate the temperature dependence of the corresponding spectra. We restrict our considerations to the frequency range within the first PBG and its vicinity. For the numerical calculations presented below we chose the same parameters of the PC with unit cell numbers of N=5 and M=2, as in Sec. 2.

PC with two RH defects
First, we start our investigation with a PC containing two RH defects (see Fig. 7(a)). In Figs In Fig. 9(a) and 9(b) we present the transmitivity spectrum dependence on d s for y-polarized DM, calculated for T=4.2 K (the top-view and profiles, respectively).
In contrast to the case of the x-polarized DMs given in Figs Above we investigated the PC with two identical complex (SC + dielectric) defects but with different thicknesses of SC sublayers. We can fix one of the SC sublayer thickness d sR or d sL (in the left or right combined defect, respectively) and vary the other SC sublayer thickness to obtain a possible changing of the PBG spectra. In Figs. 11(a) -11(c) we present top views of  The corresponding spectra for the opposite case (with fixed thickness of the right-side SC defect d sR =10 nm, 20 nm, and 30 nm) are given in panels (d), (e), and (f), respectively. Comparing Figs. 8(a) and 11(a) for symmetrical and asymmetrical changes of the SC defect layers thickness, one can see that the behavior of the DMs changed. In contrast to the case of identical SC sublayers given in Figs. 8(a), for the case of fixed d sL =10 nm the LF DM is more narrow and it decays fast with increasing of d sL , not merging with the HF PBG edge as shown in Fig. 8(a). With the increase of the fixed value of d sL to 20 nm (Fig. 11(b)   one can see that the behavior of the DMs changed. In contrast to the case of identical SC sublayers given in Figs. 8(a), for the case of fixed d sL =10 nm the LF DM is more narrow and it decays fast with increasing of d sL , not merging with the HF PBG edge as shown in Fig. 8(a). With the increase of the fixed value of d sL to 20 nm (Fig. 11(b)) the HF DM comes closer to the HF PBG edge and merges with it for d sL =30 nm for all values of d sR . For d sL =20 nm and d sL =30 nm the LF DM grows with increasing of d sR (Figs. 11(b) and 11(d)) and merges with the HF PBG edge for d sL =30 nm at about d sR =40 nm. Varying the left-side SC defect layer thickness one can obtain another dependence of the spectra on d sL (Figs. 11(d) -11(f)). For this situation the LF DM becomes visible for larger values of d sL . The tendency of the shift of DM to higher frequencies with the increase of both d sL and d sR remains.

PC with two LH defects
In this subsection we present the numerical results for the PC with two LH defect layers depicted schematically in Fig. 7(b). Similar to the previous case, we study the EMW's incidence on the right-hand surface of the PC, and all the parameters of the PC are the same, as described in detail above in section 3.1, except for the replacing of the RH combined defects to LH ones.

In Figs. 12(a) -12(d) we present the resulting transmittivities T (x) for the PC with two LH defects, analogous to that of Figs. 8(a) -8(d).
In Figs. 12(a) and 12(b) one can see the top view of the transmittivities vs the normalized frequency and the defect SC sublayer thickness d s for the temperatures T=4.2 K and T=77 K, respectively. The SC sublayer thickness d s varies in the range (0 ÷ 70) nm, while the dielectric defect layer thickness is fixed to be d 1d =d 1 =2.1 µm. In Fig. 12(c) we demonstrate the evolution of the transmittivity profiles T (x) with the increase of d s .
Comparing the corresponding parts of Figs. 12 and 8, one can see that the behavior of the transmittivity T (x) has drastically changed. Though analogously to the case of the PC with two RH defects, both DMs have a tendency to shift to higher frequencies with the increase of d s , now this shift is substantially smaller. The solid lines in Fig. 12 As for the behavior of the transmittivity of the y-polarized light T (y) , it practically does not depend on the type of defects (RH or LH) in the PC, and its variation with increasing temperature and d s remains similar as depicted in Fig. 9(a) for the PC with two RH defects. Examining the transmittivity spectra dependence on the normalized thickness of the dielectric sublayer d 1d /D, analogously to the case investigated above for the 1D PC with two RH defects, we obtained the periodically repeating structures in the top views of T (x) vs ω and d 1d (Figs. 13(a) and 13(b)). As before, the spectra variations with d 1d are calculated for the SC defect sublayer fixed to be d s =10 nm (Fig. 13(a)) and d s =20 nm (Fig. 13(b)). The thickness increase of both dielectric defect sublayers results in a shift of the DMs to the LF PBG edge, as it was obtained above for a 1D PC with two RH defects. But for a 1D PC with two LH defects, one can see the distortion of the DMs on the LF PBG edge with the increase of d s to 20 nm. A further enlargement of d s leads to the DM's lines breaking. Figure 13. The same as for Fig. 11 Fig. 12(a), we see that in the case of asymmetrical SC defects the LF PBG edge does not merge with the LF DM when we change the right-sided SC defect sublayer thickness d sR . Varying d sL while d sR is fixed we obtain a mergence of the decreasing LF DM with the LF PBG edge for d sR =30 nm, similar to that in Fig. 12(a) for d sL ≈ 60 nm (Fig. 14(f)).
Considering asymmetrical changes of the SC defect sublayer thicknesses, as it was done analogously for the case of two RH defects in section 3.1, for the case of a 1D PC with two LH we obtain different spectra when changing one of the SC defect thicknesses while the other one is fixed.

PC with RH -LH and LH -RH defects
In this subsection we present the numerical results for the PCs with two defect layers depicted schematically in Fig. 7(c) and (d). In Figs. 15(a) and 15(b) we present the top view of the transmittivities T (x) for the x-polarized EMWs vs the normalized frequency  and the defect SC sublayer thickness d s for RH -LH and LH -RH geometries, respectively.

PC with RH -LH and LH -RH defects
In this subsection we present the numerical results for the PCs with two defect layers depicted schematically in Fig. 7(c) and (d). In Figs. 15(a) and 15(b) we present the top view of the transmittivities T (x) for the x-polarized EMWs vs the normalized frequency ω and the defect SC sublayer thickness d s for RH -LH and LH -RH geometries, respectively.
The SC sublayer thickness d s varies within the range (0 ÷ 70) nm, while the dielectric defect layer thickness is fixed to be d 1d =d 1 =2.1 µm. The color in Figs. 2(a) and 2(b) denotes the value of the transmittivity, as shown on the panels. The results are obtained for temperature T=4.2 K. The presence of two combined defect layers in the PC leads to appearance of two narrow DMs inside the PBG (see Figs. 15(a), 15(b)) which further are referred to as low-frequency (LF) and high-frequency (HF) DMs.
As one can see from Figs. 15(a) and 15(b), for both geometries (RH -LH and LH -RH), the increase of d s from 0 to 70 nm leads to a shift of both LH and HF DMs to higher frequencies, but the shifts of HF DMs are larger than of the LF ones.
In the case of RH -LH geometry, first the LF defect mode slightly deviates to higher frequencies with increasing d s and further, after d s ≈20 nm, the LF DM gets thin and its position does not change with further increase of d s , while the HF DM peak merges with the HF PBG edge and the transmittivity values of both HF PBG edge and HF DM decrease. For all values of d s from the considered interval (except d s =0 nm) the LF defect mode is more pronounced then the HF one.
For the case of LH -RH geometry, the behavior of the defect modes is opposite: the LF defect mode is less pronounced than the HF one. For d s =10 nm the LF DM is about 8 times smaller in magnitude then the HF DM, and for d s >20 nm the LF DM is practically suppressed, while the HF DM still exists up to d s =30 nm.
In Figs. 15(c) we give the transmittivity T (x) spectra for d s =0 (solid lines), d s =10 nm (dashed lines), d s =20 nm (dash-dotted lines), d s =30 nm (dotted lines), d s =40 nm (dash-dot-doted lines), and d s =50 nm (long dash-dotted lines) for the case of RH -LH geometry. In the case of LH -RH geometry, for d s increasing 30 nm the both DMs are suppressed and in Fig. 15(d) we show the T (x) spectra only for d s =0 (solid lines), d s =10 nm (dashed lines), d s =20 nm (dash-dotted lines), and d s =30 nm (dotted lines). Obviously, in the case of pure dielectric defects (d s =0) the spectra for RH -LH and LH -RH geometries are identical (the solid lines in Figs. 15(c) and 15(d)). In this case two DM peaks of equal values T LF (x) = T HF (x) ≈0.57 are localized symmetrically inside the PBG at ω LF peak =0.2307 and ω HF peak =0.2577 which correspond to ω LF peak =86.91 THz and ω HF peak =97.08   For the case of RH -LH geometry ( Fig. 16(a) Fig. 16(a) one can see an essential lowering of the LF PBG edge in contrast to the HF PBG edge, which practically does not change with temperature. In the case of LH -RH geometry ( Fig. 16(b)) the temperature effects substantially on the HF defect mode: its magnitude drops more than four times from Our numerical results demonstrate that the PBG spectra of the y-polarized waves practically do not change when d s is about several tens of nanometers, in contrast to the case of the xpolarized EMWs. The analogous result was obtained in our previous calculations for the PCs with two RH and two LH combined SC defects in the previous subsection. In Fig. 17 we show the top-view of the transmittivity T (y) of y-polarized EMW as function of normalized frequency ω and the SC defect sublayer thickness d s for RH -LH geometry. The corresponding dependence for LH -RH geometry looks similarly. As one can see form   Analogously, in the case of the PC of the LH -RH geometry, the asymmetric changing of the SC defect sublayer thickness modifies the dependence of the transmittivity specter given in Fig. 15 Figure 19. The same as for Fig. 9, except for LH -RH geometry.

Conclusions
In conclusion, we have investigated the behavior of DMs in a 1D dielectric PC with two complex bilayer defects composed of SC and dielectric constituents. We have considered the case of a fixed distance between the two defects embedded into the PC asymmetrically, i.e., when both SC sublayers are located on the right-hand side (RH -RH geometry) or on the lefthand side (LH -LH geometry) with respect to the dielectric parts of the complex defects. The normal incidence of linearly polarized EMWs (with electric field vector E perpendicular to the axis of the PC's growth direction) on the PC is investigated.
The positions of the transmittivity peaks at the DM frequencies inside the first PBG are studied both analytically and numerically, for different temperatures and thicknesses of SC and dielectric sublayers. We have shown that the increase of temperature from liquid helium temperature (T=4.2 K) to the critical temperature of the SC sublayer (T C =90 K for YBa 2 Cu 3 O 7 ) leads to significant changes of the DM's transmittivity peaks in the PBG spectra of x-polarized EMWs, up to a practically complete suppression of one of them (the low-frequency DM for RH -RH geometry), while the spectra of y-polarized EMWs are practically unchanged. The pronounced contrast in behavior of x-and y-polarized modes is based on the in-plane anisotropy of the dielectric tensor components of the SC sublayer. The positions of PBG edges and DM's remain invariable with temperature for both EMW's polarizations. We have demonstrated a high sensitivity of the PBG spectra of x-polarized EMWs to variations of the SC defect sublayers thicknesses in both RH -RH and LH -LH geometries. The increase of the SC defect sublayer thickness leads to a shift of both DM peaks towards the high-frequency PBG range up to their mergence with it, as well as to a decrease of the distance between these peaks (for RH -RH geometry), or a substantial shift of the position of the low-frequency PBG edge to higher frequencies (LH -LH geometry).
We have also considered the case of a fixed distance between the two defects symmetrically embedded into the PC, i.e., when one of the SC sublayers is located on the right-hand side and the another one on the left-hand side with respect to the dielectric parts of the complex defects (the cases of RH -LH and LH -RH geometries). The increase of the SC defect sublayer thickness leads to a shift of both DM peaks towards the high-frequency PBG range up to mergence of the one of them with it (in the RH -LH geometry), as well as to an increase of the distance between these peaks, or a substantial shift of the position of the low-frequency PBG edge to higher frequencies (in both RH -LH geometries). The variation of the dielectric defect sublayer's thicknesses allows shifting the DM peaks in spectra of both x-and y-polarized EMWs towards the low-frequency PBG edge for each geometry (RH -RH, LH -LH, RH -LH, and LH -RH).
Changing the SC defect sublayer thicknesses asymmetrically (when the right or the left SC defect sublayer's thickness is fixed while the another one is varied), one can modify the PBG spectra of x-polarized EMWs, changing the number of the defect modes in the PBG with chosen thicknesses of the SC defect sublayers.
As follows from our numerical results, the dielectric PCs with SC defects can be used as constituents of polarization-selective narrow-band filters for THz radiation [31,32]. The optimal parameters of these filters can be obtained choosing the proper SC and dielectric defect sublayer thicknesses. The high sensitivity to temperature of the PCs with SC defects opens possibilities to use such a structures as the basis of temperature tunable electromagnetic filters in the THz regime [33].

Author details
N. N. Dadoenkova