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Effects of structurally deformed sub-lattice points on the dispersion properties of 2D hybrid triangular–graphite photonic crystal
Optics Communications 285 (2012) 1486–1493
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Effects of structurally deformed sub-lattice points on the dispersion properties of 2D hybrid triangular–graphite photonic crystal Fulya Bagci, Baris Akaoglu ⁎ Department of Engineering Physics, Faculty of Engineering, Ankara University, 06100 Besevler, Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 28 March 2011 Received in revised form 6 October 2011 Accepted 12 October 2011 Available online 29 October 2011 Keywords: Photonic crystal Photonic band gap Symmetry reduction Plane-wave expansion method
a b s t r a c t We investigate a hybrid two dimensional (2D) photonic crystal which is constructed by merging triangular and graphite lattices in air background. Different geometries of scatterers such as circular rods, circular rods with elliptical holes, circular hollow rods and elliptical rods with circular holes in the triangular sublattice are considered and effects of shape on their dispersion properties are discussed. Photonic band gaps (PBGs) for TM- and TE-polarized modes are found to exist for these symmetry-reduced graphite structures. The TE PBGs are found to contribute to produce complete photonic band gaps (CBGs) up to high values of filling fraction and rod diameter ratios. We show that TM and TE PBGs display opposite behaviors as the structure is transformed from graphite to symmetry-reduced graphite (hybrid triangular–graphite lattice) and from symmetry-reduced graphite to hollow centered symmetry-reduced graphite. Concerning the PBGs, inner structural deformation is found to be more beneficial in comparison to outer structural deformation. Small group velocities are observed at the same frequency for both polarizations near the edges of PBGs for the configurations of circular rods, circular rods with elliptical holes and circular hollow rods. These observations seem valuable for optical gain enhancement and low-threshold lasing. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals/solids (PhC) are composite optical materials, consisting of spatially modulated dielectric and metallic structures, and have unusual optical properties and innovative applications [1–4]. The most prominent feature of PhCs is the possibility of customizing their design to application needs. Moreover, since their discovery various extraordinary optical properties of PhCs have been predicted and some of them have been confirmed experimentally such as existence of a photonic band gap (PBG), possibility of creating localized defect modes in the PBG, suppression of spontaneous emission and generation of slow light in photonic crystal waveguides [5–8]. In such materials, variation of refractive index manipulates light propagation through multiple scattering from the arranged scatterers (optical “atoms”), which are analogous to atoms, diffracting X-rays in a lattice of a crystal. These diffracted light waves give rise to stop bands for the waves within a certain frequency range in all directions through multiple interferences. In other words, formation of standing waves leads to PBGs in the same fashion as electrons in semiconductors. In general, a scattering element with a size of the order of λ/2 roughly leads to a PBG at a wavelength λ [9]. Since pioneering works, independently introduced by Yablonovitch [6] and John [10],
⁎ Corresponding author. E-mail address: [email protected] (B. Akaoglu). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.10.035
increasing interest in this field is directed mainly toward realization of a complete PBG (CBG) which can occur in all directions for all polarizations only in three dimensional (3D) PhCs [11,12]. However, fabrication of 3D PhCs still remains a difficult task when the periodicity of the structure has the length on the order of visible to near-infrared wavelengths. Fortunately, since 2D PhCs exhibit most of the essential properties of their 3D counterparts and are much easier to fabricate, many research efforts have been devoted to 2D PhCs. It is possible to simulate experiments for crystals with lattice constants on the order of visible wavelengths, for example, using specimens with larger lattice constants since there is no fundamental length scale for photonic crystals, that is, the wave equation in a periodically modulated dielectric medium is scale invariant. Modes of a 2D PhC, periodic in the xy-plane and having homogenous dielectric function in the z-direction, can be separated into two sub-modes since the (x,y) plane of periodicity of the PhC is the mirror plane of the system. The modes for which the electric field and magnetic field are parallel to z-axis are referred to as transverse-magnetic (TM) and transverse-electric (TE) modes, respectively. Since the direction of electric field is different relative to dielectric interfaces for TM and TE polarizations, corresponding band structures for the two polarizations are usually different. PBGs can exist both for TM and TE polarizations but the positions of the PBGs can be very different. A CBG exists if PBGs for both polarization modes overlap. In order to get a better control of light confinement it is essential to obtain a CBG and enhance its width by properly designing the crystal structures.
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The general approach to controlling the dispersion properties of the PhC is mainly based on appropriate choice of the refractive index ratio of the constituent materials, the lattice type and adjustment of the filling fraction. On the other hand, various types of PhCs with high degrees of freedom have been introduced and these PhCs have additional parameters, defining the structure, which can be utilized to control the dispersion properties of the crystal further to the maximum extent possible. Mostly, the constraints on PBG formation, enlargement and observing low-dispersive bands near gap-edges stem from the degeneracy of the modes at high symmetry points in the Brillouin zone. Reducing the symmetry of the structure by a number of different approaches leads to separation of degenerate adjacent modes, resulting in the formation of a new PBG or enhancement of the width of an already existing PBG. The primary utilized methods for symmetry reduction can be classified as follows: introducing additional lattice scatterers at the center of the unit cell [9,13–15], changing the shape of lattice points [16–18], utilizing birefringent dielectrics [19,20], introducing interfacial layers [20,21], creating structural and rotational deformation in annular PhC structures [22] and combining two lattices belonging the same Bravais lattice to create hybrid lattice geometries [23,24]. In hybrid lattice geometries, the different inner/outer shape and size of the scatterers in each sublattice provide extra degrees of freedom that can be utilized to alter the optical properties of the crystal. Furthermore, the shape of the scatterers in one sub-lattice of the hybrid structure may not be totally different from the shape of the scatterers in the other sub-lattice but may be just structurally and/or rotationally deformed. Such extra degrees of freedom that hybrid PhC geometries offer open many options to alter the dispersion properties of the PhC in the desired fashion. In this work, we have considered a 2D hybrid triangular–graphite lattice in air background in order to benefit from its high degree of freedom. This hybrid structure can also be envisaged as a graphite structure with reduced symmetry when the central triangular sublattice points are kept at different sizes and/or shapes in comparison to graphite ones. In this structure, to obtain a large CBG, one can take advantage of the large TE PBGs of graphite lattice [24] and large TM PBGs of triangular lattice. Graphite lattices in air background have larger TE PBGs than in dielectric background [25]. On the other hand, triangular lattice PhCs in air background are known to possess large TM PBGs [25]. Merging these two types of lattices of the same Bravais form constitute hybrid triangular–graphite lattices. The hybrid triangular–graphite lattice in dielectric background has large CBGs mostly near closed-packed conditions. Most of the gaps in hybrid triangular–graphite lattice of rods in air background belong
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to TM PBGs but there is also an important number of TE PBGs with considerable widths that constitute scattered CBGs at different rod radii far from closed-packed conditions [23]. Laser emission from low-dispersive bands close to the Г3-point in a wide-range of inplane k values with high quality factors has been reported for the hybrid triangular–graphite configuration in the InP slab, including four InAsP quantum wells on top of a Si substrate SiO2 wafer [26]. In this study, not only the size of the scatterers embedded in the triangular sub-lattice is investigated in point of dispersion properties but also we focused on the effects of the existence and shape of the holes at the triangular sub-lattice points where structural deformations are additionally introduced. While taking the radius of the solid circular rods at the graphite sub-lattice constant, different geometries of scatterers such as circular rods, circular rods with elliptical holes, circular hollow rods and elliptical rods with circular holes in the triangular sub-lattice are considered. In order to unambiguously justify the consequences of the shape effect in hollow rods, the air fractions of the holes are kept constant. So far, to the best of our knowledge, the effects of both the introduction of holes in scatterers and independent deformation of inner and outer shapes of hollow rods embedded in one of the sub-lattices of a 2D hybrid PhC lattice have not been reported yet. Moreover, for this hybrid structure, there exist flat bands at the same frequency for both polarizations near gap edges, which is an extraordinary result for 2D PhCs. The importance of hollow PhCs has been recently highlighted in terms of diverse device applications. They are more sensitive to external refractive index change which is attractive for sensor applications [27]. It has been demonstrated earlier that they have polarization independent PBGs [22]. Also some hollow PhC waveguides have been shown to exhibit low group velocity which is an attractive feature for slow light waveguides [27,28]. Besides, inner and outer shape deformations of them introduce shape anisotropy which lead to strong polarization effects leading to birefringence which find applications in PhC fibers [29–31]. 2. The structure and numerical method Since 2D PhCs have different periodicities in different directions, the gaps in different directions do not necessarily overlap unless they are large enough. In principle, the more equal the periodicity in different directions, which corresponds to a more circular Brillouin zone, the easier it generally becomes to create a PBG [1]. For example, triangular PhCs in air background display large TM PBGs due to their isolated patches of high-ε regions and nearly circular Brillouin-zones.
Fig. 1. (a) Schematic configuration of the symmetry-reduced graphite lattice with hollow triangular sub-lattice points in air background. The lattice constant is a and the basic vectors of the scatterers are defined by u0 = 0, u1 = (a1 + a2)/3 and u2 = −(a1 + a2)/3 relative to the center of the unit cell. The unit cell is restricted by the thick lines and contains three scatterers. (b) Brillouin zone for the symmetry-reduced graphite lattice where shaded region corresponds to their reducible zone.
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Fig. 2. (a) The graph of the ratio of gap-width to gap-center of the TM and TE PBGs as the filling fraction is changed for the symmetry-reduced graphite structure when β = 0.1 (b) Gap map of symmetry-reduced graphite structure of dielectric rods (ε = 12.09) in air at f = 0.45. TM PBGs are remarked as blue, TE PBGs are remarked as red and CBGs are remarked as yellow in color.
On the other hand, graphite structure of dielectric rods in air exhibits large CBGs and consequently, it has a considerable potential for the realization of the PBG materials [32]. If these two lattices are merged while keeping the Brillouin zone symmetry same, a hybrid triangular–graphite lattice, taking advantage of both these sub-lattices, can be obtained and then the effect of symmetry reduction applied to one of these sub-lattices can be investigated. Within this frame, we have placed different sized and shaped dielectric scatterers in the triangular sub-lattice which corresponds to the central points of graphite lattice. The structure and first Brillouin zone of this hybrid lattice are shown in Fig. 1(a) and (b), respectively. The hybrid triangular graphite lattice is constructed byp defining triangular ffiffiffi Bravais sub-lattice with primitive vectors a ¼ a 3 =2; 1=2 and a2 ¼ 1 pffiffiffi a 3=2; −1=2 and introducing three rods in a unit cell at positions u0 = 0, u1 = (a1 + a2)/3 and u2 = −(a1 + a2)/3 with respect to the center of the unit cell as shown in Fig. 1(a). When radius of circular scatterers in the triangular sub-lattice becomes zero (rt = 0), the structure degenerates the basic graphite structure whereas when it becomes equal to the radius of scatterers in graphite sub-lattice (rt = rg), hybrid triangular–graphite lattice pffiffiffican be regarded as triangular lattice with a lattice parameter a= 3. This hybrid triangular– graphite lattice resembles Suzuki-phase [33] and the Archimedean [34] lattices in point that all these lattice types hold several lattice points in the unit cell which support several low-dispersive bands [26]. We have chosen silicon as dielectric material of the rods which has a dielectric constant of ε = 12.09 at λ = 1.55 μm. We have utilized MIT
photonic-bands package [35] which uses plane wave expansion method with periodic boundary conditions to calculate the band structures and field patterns. The computational error was estimated to be less than 0.2%. 3. Results and discussions In order to determine a radius at which the effect of structural deformation on PBGs might be more noticeable, first we examined the dispersion properties of the symmetry-reduced graphite lattice for all rod diameter ratio (β = rt/rg) and filling fraction (f) values. For this purpose, we modified the size of the scatterers by changing either the β or f. f is changed between 0.1 and 0.6 in steps of 0.05 for all values of β and rod diameter ratio is changed between 0 and 1 in steps of 0.05 for all values of f. For all values of β, increase in the filling fraction leads to contrary results on the first PBGs for TM and TE polarizations. The results of numerical calculations revealed that as f increases, the width of the PBG between TM2 and TM3 modes (TM2–3 PBG) decreases whereas the width of TE1–2 PBG, which exists until β > 0.4, increases. However, TM7–8 and TE3–4 PBGs display similar behaviors (i.e. Fig. 2(a)). As f increases, their widths increase up to a critical value of β and then begin to decrease. Very small f values (f around 0.1) lead to high CBG ratio at frequencies approaching (2πc)/a. It has been found earlier that the largest CBG occurs for f = 0.14 and β = 0.11 [13]. Due to the effect of two additional rods in the unit cell, the mid-gap frequency of the lowest CBG decreases as f increases.
Fig. 3. The geometry and relative parameters utilized in the simulations: (a) D1 configuration of graphite lattice, (b) D2 configuration of symmetry-reduced graphite lattice, (c) D3 configuration of circular rods with elliptical holes, (d) D4 configuration of circular hollow rods, and (e) D5 configuration of elliptical rods with circular holes.
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For particular values of f, as β increases TM2–3 PBG decreases while TM3–4 PBG increases. A lattice with β = 1 strongly favors large TM PBGs like triangular lattice and has a gap-midgap ratio of 49% at f = 0.1. PBG map of the symmetry-reduced graphite lattice at a filling fraction of 0.45 is given in Fig. 2(b). At f = 0.45, the graphite structure has a CBG ratio Δω/ω0 of 2.7%. However, the addition of a
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small rod into the graphite cell center increases the CBG ratio to a maximum value of 6.9% at β = 0.3. It is clearly seen that there is a CBG for almost all values of β. Besides, the symmetry-reduced graphite lattice favors TM PBGs more than TE PBGs as can also be seen in Fig. 2(b). It is interesting that the second TE PBG is nearly not affected by the rod diameter ratio variation. Since the field distribution
Fig. 4. Dispersion diagrams of configuration D1 for (a) TM and (b) TE polarizations. Dispersion diagrams of configuration D2 for (c) TM and (d) TE polarizations.
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simulations show that the magnetic field of the modes of upper and lower band edges has nodes on triangular sub-lattice points, this observation sounds reasonable. It should be noted that as β goes from 0 to 1, the radii of graphite sub-lattice points decrease by only 18%. In order to study the effects of deformation of the rods on the PBGs of the hybrid structure, triangular sub-lattice points at the center of graphite sub-lattice are considered. For this symmetry-reduced graphite structure, we choose a high β value, β = 1, to exhibit the influence of the deformations on the dispersion diagrams stronger. At the filling ratio of 0.45 and β = 1 (corresponding to r = 0.2a) the width of the TE PBG and CBG is very narrow (0.4% at ω = 0.7(2πc/a), illustrated in Fig. 2(b)), thus the enlargement of TE PBG and CBG by opening air holes would be more prominent. The graphite lattice and its symmetry-reduced configurations utilized for the structural deformation analysis are illustrated with their structural values in Fig. 3. Fig. 4(a) and (b) shows the dispersion diagrams of the graphite lattice, given in Fig. 3(a), with circular rods having r = 0.2a for TM and TE polarizations, respectively. Dotted and solid lines denote the odd and even numbered bands, introduced for easy guiding of the eye. For this relative radius value, calculation results revealed that three PBGs appear for TM polarization whereas two considerable PBGs appear for TE polarization within the range considered. The
second PBG for TM polarization and first PBG for TE polarization overlap between normalized frequencies of 0.63 and 0.68 for which Δω/ω0 = 8.2%. Fig. 4(c) and (d) shows the dispersion diagrams of the hybrid triangular–graphite lattice (f = 0.45, β = 1), given in Fig. 3(b), with circular rods having r = 0.2a for TM and TE polarizations, respectively. The number of TM and TE bands is considerably increased since the unit cell became denser by the rods embedded in the center. This structure favors large TM PBGs in comparison to the PBGs of the graphite lattice given in Fig. 4(a). In contrast, TE PBGs are observed to be reduced. This trend is expected since high dielectric constant material is now located in the center of the unit cell. A CBG exits for only a very small frequency interval with Δω/ω0 = 0.4%. It is also found that the gap edges shift to lower frequencies. Structural deformation (symmetry reduction) can be created by opening air holes in the scatterers belonging to one of the sublattice of the hybrid lattice. In addition, deformation of the inner or outer shapes of the hollow rods leads to further structural deformation. First, inner shape deformation is investigated for symmetryreduced graphite lattice where elliptical air holes [with eccentricity (γ) of 2, f = 0.4, β = 1] are introduced at the triangular sub-lattice points as shown in Fig. 3(c). Thus, the structural symmetry of the
Fig. 5. Dispersion diagrams of configuration D3 for (a) TM and (b) TE polarizations. Dispersion diagrams of configuration D4 for (c) TM and (d) TE polarizations.
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outer shape is retained, whereas that of the inner shape is broken. Numerical calculations indicate that the displacement field of the lower edge of the first TM PBG for all symmetry-reduced graphite structure configurations, given in Fig. 3, concentrates on lattice points at Γ symmetry points. In this structure, the localization of the displacement field on the lattice points, which are hollow for triangular sub-lattice points in this case, shifts the lower gap-edge to higher frequencies and narrows the first TM PBG as seen in Fig. 5(a). However, the expansion of the second TE PBG increases the width of the CBG for which Δω/ω0 = 5% between normalized frequencies of 0.71 and 0.75. The degeneracy between TE10 and TE11 bands has lifted and this led to opening of the third TE PBG as shown in Fig. 5(b). This new PBG arises from resonant scattering of short wavelength light from dielectric walls of hollow scatterers. This third TE PBG appears for the D3, D4 and D5 structures. Moreover, appearance of this additional PBG shifts high frequency modes further to higher frequencies which is a consequence of the decrease of the effective (“average”) refractive index. Second, circular air holes (Fig. 3(d)) are introduced in the center of the rods forming the triangular sub-lattice while keeping the filling fraction the same as the previous case (Fig. 3(c)). Thus, apart from the combined effect of filling fraction and shape [22], only the effect of shape is considered in this case. The inner hole radius corresponds to r = 0.1a. Extensive calculations in the literature showed that the CBG width has been increased in 2D triangular lattice with circular and hexagonal air rings and in graphite lattice with elliptical and hexagonal air rings [36]. The size of the TE PBGs and CBG is found to be larger for circular inner rods as seen in Fig. 5(d), in agreement with the previous studies [16]. The CBG occurs between normalized frequencies of 0.71 and 0.75 for which Δω/ω0 = 5.7%. Since the air space size among the rods are invariable (which is also valid for configurations D2 and D3), the size of the first TE PBG is retained. Finally, configuration D5 with circular inner and vertically deformed outer elliptical rods (γ = 2, f = 0.4) are considered for triangular sub-lattice points. In contrast to configuration D3, the structural symmetry of the inner shape is retained, whereas that of the outer is broken. TM PBGs split into smaller pieces narrowing their band gaps as shown in Fig. 6(a), in agreement with the observations in the
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study of broken structural symmetry of triangular lattice of hollow rods [22]. The ninth TM mode is positioned between two TM PBGs at higher frequencies, whereas the lowest edges of other modes seem preserving their positions. Due to the reduction of air size among the rods, the frequency interval between second and third TE modes becomes larger, shrinking the first TE PBG and vanishing the second TE PBG. However, the third TE PBG is not affected due to the resonant scattering, similar to the third TE PBG as given in Fig. 5 (b). This structure does not support an appreciable CBG. This result is reasonable since PBG width is known to be reduced when the symmetry of the scatterer and the lattices are not the same [17]. One remarkable feature of hybrid triangular–graphite lattice configurations (D2, D3, D4, D5) is that TE PBG formation dominates the CBG width. Another remarkable feature is the flatness of ninth TM and sixth TE bands at the same frequency (≈0.7) below the second TM PBG and second TE PBG, respectively at a very broad k-region (Figs. 4(c, d) and 5(a–d)). For graphite lattice such a flat band does not exist (Fig. 4(a, b)). It should also be noted that for the inverse hybrid structure of air holes in silicon background, such flat polarization independent bands near gap edges are not observed. For the D2 and D4 configurations, flat ninth TM band coincides with eighth TM band and sixth TE band coincides with fifth TE band between M–K points, increasing the density of states further. Besides, for the configurations D2 and D4 with circular rods (holes), the ends of the fourth TE band intersect with the sixth TE band at k-points (0 0.175(2π/a) 0) and (−0.1166(2π/a) 0.1166(2π/a) 0) making the interested frequency region almost flat. The frequency difference between the maximum and minimum points in the almost flat k-region can be regarded as a measure to indicate the flatness of the bands. In this respect, if the D2, D3 and D4 configurations are compared, the flatness of the TM and TE bands can be sorted descending as D3, D4 and D2. Especially for the hybrid configuration D4, the ninth TM band and the sixth TE band show small group velocity in a broad region of k-space, resulting to an increase in density of states (Fig. 5(c,d)). To fabricate such a lattice geometry with flat bands centered at λ = 1.55 μm would require a lattice constant of about 1 μm. However, the fabrication of D3 and D5 structures with inner (or outer) elliptical
Fig. 6. Dispersion diagrams of configuration D5 for (a) TM and (b) TE polarizations.
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Table 1 Frequency interval (Δω) and central frequency (ωc) of the flat ninth TM band along the whole Г–M–K–Г path and frequency interval (Δω) and central frequency (ωc) of the flat sixth TE band between the (0 0.225(2π/a) 0) and (− 0.1166(2π/a) 0.1166(2π/a) 0) k-points. TM
TE
f
Δω(2πc/a)
ωc(2πc/a)
f
Δω(2πc/a)
ωc(2πc/a)
0.1 0.2 0.3 0.4 0.5 0.6
0.0112 0.0098 0.0158 0.0234 0.0353 0.0472
1.098 0.835 0.714 0.648 0.603 0.572
0.1 0.2 0.3 0.4 0.5 0.6
0.0322 0.0092 0.0204 0.0191 0.0342 0.0529
1.083 0.837 0.718 0.649 0.598 0.561
holes (rods) would not be easy. Fortunately, the D4 configuration also shows the same flat bands and would require a relatively less effort for fabrication. If one wants to utilize this PhC at λ = 1.55 μm, the ring width would be about 100 nm, which is sufficiently thick to be fabricated using standard technologies for dry etching. Since the fabrication of D4 structure is relatively easy, we further analyzed the flatness of its polarization independent gap-edge bands as a function of filling fraction and the ratio of inner to outer rings of triangular sub-lattice points. The filling fraction for the D4 configuration with circular hollow rods in the hybrid triangular– graphite lattice is calculated as h i pffiffiffi 2 2 2 f ¼ 2πr g þ π r t −r in = 3=2 : The frequency width of the flat ninth TM band (Δω) in the whole k-region and its frequency value in the middle of M–K region (ωc) are listed as a function of filling fraction in Table 1. Since the ends of flat sixth TE bands are curved at Г points, the frequency width for TE polarization (Δω) is calculated between (0 0.225(2π/a) 0) and (−0.1166(2π/a) 0.1166(2π/a) 0), i.e. between 11th and 55th kpoints in our calculated k-region with 60 points. The frequency width of the flat sixth TE band (Δω) and its frequency value in the middle of M–K region (ωc) are also listed as a function of filling fraction in Table 1. The flatness of the ninth TM band increases up to f = 0.2 and then decreases as the filling fraction increases. The same behavior can also be found on account of the flatness of the sixth TE band. The greater value of the frequency of the bands for the filling fraction of 0.2 creates PhC lattices far from closed-packed conditions and in larger lattice dimensions, providing convenience in fabrication. The polarization independent bands can be regarded almost flat up to the filling fraction of 0.45. We also investigated the band flatness of configuration D4 with respect to the ratio of inner ring of the triangular sub-lattice points to that of the outer ring (δ) at all filling fractions in 0.1 steps. Polarization independent flat bands are found to be mutually more flat
when 0.5 ≤ δ ≤ 0.7. The flattest polarization independent flat bands for configuration D4 are obtained at f = 0.2 and δ = 0.6 with Δω = 0.0038(2πc/a) for the TE and Δω = 0.0055(2πc/a) for the TM band. For these flat bands centered at wavelength λ = 1.55 μm, the dielectric rods should have a radius of 190 nm. Since flatness of a band and its group velocity are directly related, maximum group velocity can also be addressed to indicate the flatness of the band. The maximum group velocities for the most (f = 0.2, δ = 0.6) and least flat ninth TM band (f = 0.45, δ = 0.4) are found as 0.012c and 0.12c, respectively. Similar polarization independent flat bands have also been noticed for 2D square lattices of square cross-section dielectric rods with air holes drilled inside [37]. Most of the studies of PhC variations with flat dispersion curves have their flat dispersion positions in the vicinity of the band edges [26,38–41]. In this study, the gap-edge bands are found to be flat through almost all the k-regions. Although dielectric backgrounds are frequently utilized [26,38–40], a surface emitting microlaser based on a 2D square lattice of InP rods lying on a silica layer is also demonstrated by using standard fabrication processes [41]. In addition, for the D5 configuration, when light is propagated through M–K direction, coupling of almost flat TM and TE modes with other modes is less probable because tenth TM band is restricted to a narrow spectral region and the sixth TE band is farther in frequency than other TE bands. These results indicate that the PhC configuration proposed in this study might be considered for optical gain enhancement or low-threshold lasing [26,37–41]. Since the shift of the ninth TM mode in the configuration D5 is remarkable, we further examined its displacement field distribution for contrary configurations D3 (Fig.7(a)) and D5 (Fig. 7(b)). As shown in Fig. 7, a sharp difference stands out between the field distributions of D3 and D5. Reminding that air area in the triangular sub-lattice points are same in both configurations D3 and D5, this variation is solely a result of the shape effect. For the configuration D3, the uniform variation of the displacement fields and resonant scattering of the electromagnetic waves from thin dielectric walls of hollow rods play key roles in the existence of large second TM PBG. However, for the configuration D5 the displacement fields are distributed nonuniformly and the scattered displacement fields from the thin dielectric walls cannot efficiently interfere. Therefore, TM PBGs split and the widths of the TM PBGs vary more than those of the configuration D3, relative to the TM PBGs of configuration D2. Another point to note is that the displacement field of configuration D5 is concentrated much more in the air region among the rods than configuration D3, causing a shift of the mode to higher frequencies. 4. Conclusions In this work, we have investigated the effects of both the introduction of holes and independent deformation of inner and outer shapes of the
Fig. 7. Displacement field distribution of ninth TM mode at Γ symmetry point for (a) D3 and (b) D5 lattice configurations.
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rods embedded in triangular sub-lattice of 2D hybrid triangular–graphite structure in air background. For the symmetry-reduced graphite structure, wide ranges of PBGs at different frequencies with significant sizes have been obtained and a regular behavior of PBGs relating to different modes has been noticed against rod diameter ratio variation. Different geometries of scatterers such as circular rods, circular rods with elliptical holes, circular hollow rods and elliptical rods with circular holes in the triangular sub-lattice are considered for the selected filling fraction and rod diameter ratio values. As the structure is transformed from graphite to a symmetry-reduced graphite structure (hybrid triangular–graphite lattice), TM and TE PBGs have been found to be increasing and decreasing, respectively. Opening air holes in the scatterers leads to an increase in TE PBGs whereas holes produce a negative effect on the width of the first TM PBG. Outer structural deformations have much stronger effect on dispersion properties than the inner structural deformations, leading to splitting of PBGs. Among the more or less flat bands appearing in the band diagrams along the M–K directions, the ones appeared for the configuration with circular rods with elliptical and circular holes are found to be remarkable. Not only these bands have been observed to be quite flat in a broad region of k-space but they are also independent of polarization at a wide filling ratio interval. For the configuration of elliptical rods with circular holes, the absence of modes in the vicinity of observed constant group-velocity bands along M–K direction can reduce propagation loss. These polarization-independent flat bands of the different hybrid triangular–graphite PhC lattice configurations proposed here can be utilized for optical gain enhancement or low-threshold lasing. Symmetry reduction has been mainly utilized in this study to design new PhCs with promising properties. Reminding the absence of a wellestablished physical rule explaining the relation between the degree of symmetry and dispersion properties, this work rather elucidates the dispersion properties of particular geometrical arrangements in a hybrid 2D PhC. Nevertheless, further study is indispensible to have a solid understanding of this kind of hybrid lattices with various deformations by employing all possible values of the variational parameters present in the model configurations. Acknowledgment Part of the numerical calculations reported in this paper were performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRGrid e-Infrastructure). References [1] S.G. Johnson, J.D. Joannopoulos, R.D. Maede, J.N. Winn, Photonic Crystals: Molding the Flow of Light, second ed. Princeton University Press, Princeton, NJ, 2008.
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[2] C.M. Soukolis, Photonic Crystals and Light Localization in the 21st Century, second ed. Kluwer, Dortrecht, 2000. [3] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, Physical Review B 58 (1998) R10096. [4] E. Ozbay, I. Bulu, K. Aydin, H. Caglayan, K. Guven, Photonics and Nanostructures — Fundamentals and Applications 2 (2004) 87. [5] K. Sakoda, Journal of Applied Physics 84 (1998) 1210. [6] E. Yablonovitch, Physical Review Letters 58 (1987) 2059. [7] M. Boroditsky, R.B. Vrijen, T.F. Krauss, R. Coccioli, R.J. Bhat, E. Yablonovitch, LightEmitting Diodes: Research, Manufacturing and Applications III, 3621, 1999, p. 190. [8] Y. Hamachi, S. Kubo, T. Baba, Optics Letters 34 (2009) 71072. [9] C.M. Anderson, K.P. Giapis, Physical Review B 56 (1997) 7313. [10] S. John, Physical Review Letters 58 (1987) 2486. [11] E. Yablonovitch, T.J. Gmitter, K.M. Leung, Physical Review Letters 67 (1991) 2295. [12] S.Y. Lin, J.G. Fleming, D.L. Hetherington, B.K. Smith, R. Biswas, K.M. Ho, M.M. Sigalas, W. Zubrzycki, S.R. Kurtz, J. Bur, Nature 394 (1998) 251. [13] C.M. Anderson, K.P. Giapis, Physical Review Letters 77 (14) (1996) 2949. [14] X.-H. Wang, B.-Y. Gu, Z.Y. Li, G.-Z. Yang, Physical Review B 60 (1999) 11417. [15] T. Trifonov, L.F. Marsal, A. Rodriguez, J. Palleres, R. Alcubilla, Physical Review B 69 (2004) 235112. [16] N. Susa, Journal of Applied Physics Letters 91 (2002) 3501. [17] R. Wang, X.-H. Wang, B.-Y. Gu, G.-Z. Yang, Journal of Applied Physics 90 (2001) 4307. [18] S. Zarei, M. Shahabadi, S. Mohajerzadeh, Journal of Modern Optics 55 (2008) 2971. [19] Z.Y. Li, B.Y. Gu, G.Z. Yang, Physical Review Letters 81 (1998) 2574. [20] R.P. Zaccaria, P. Verma, S. Kawaguchi, S. Shoji, S. Kawata, Optics Express 16 (2008) 14812. [21] H. Kurt, D.S. Citrin, Optics Express 13 (2005) 10316. [22] K.P. Chang, S.L. Yang, Journal of Applied Physics 100 (2006) 073104. [23] L.J. Martinez, A. Garcia-Martin, P.A. Postigo, Optics Express 12 (2004) 5684. [24] B. Wu, R. Wei, H. Zhao, C. He, Acta Mechanica Solida Sinica 23 (2010) 255. [25] D. Cassagne, C. Jouanin, D. Bertho, Physical Review B 53 (1996) 7134. [26] L.J. Martinez, B. Alen, I. Prieto, J.F. Galisteo-Lopez, M. Galli, L.C. Andreani, C. Seassal, P. Viktorovitch, P.A. Postigo, Optics Express 17 (2009) 15043. [27] A. Saynatjoki, M. Mulot, K. Vynck, D. Cassagne, J. Ahopelto, H. Lipsanen, Photonics and Nanostructures — Fundamentals and Applications 6 (2008) 42. [28] M. Mulot, A. Saynatjoki, H. Lipsanen, J. Ahopelto, Journal of Optics A: Pure and Applied Optics 9 (2007) S415. [29] M.J. Steel, R.M. Jr Osgood, Optics Letters 15 (2001) 229. [30] T.-L. Wu, C.-H. Chao, IEEE Photonics Technology Letters 16 (2004) 126. [31] M.J. Steel, R.M. Osgood, IEEE Journal of Lightwave Technology 19 (2001) 495. [32] D. Cassagne, C. Jouanin, D. Bertho, Physical Review B 52 (1995) R2217. [33] A.R. Alija, L.J. Marınez, P.A. Postigo, J. Sanchez-Dehesa, M. Galli, A. Politi, M. Patrini, L.C. Andreani, C. Seassal, P. Viktorovitch, Optics Express 15 (2007) 704. [34] S. David, A. Chelnokov, J.-M. Lourtioz, IEEE Journal of Quantum Electronics 37 (2001) 1427. [35] S.G. Johnson, J.D. Joannopoulos, Optics Express 8 (2001) 173. [36] B. Rezaei, T.F. Khalkhali, A.S. Vala, M. Kalafi, Optics Communication 282 (2009) 2861. [37] W.-L. Liu, T.-J. Yang, Physica B: Condensed Matter 368 (2005) 151. [38] S. Nojima, Japanese Journal of Applied Physics 37 (1998) L565. [39] S. Nojima, Journal of Applied Physics 90 (2) (2001) 545. [40] S.-H. Kwon, H.-Y. Ryu, G.-H. Kim, Y.-H. Lee, S.-B. Kim, Applied Physics Letters 83 (19) (2003) 3870. [41] L. Ferrier, O. El Daif, X. Letartre, P.R. Romeo, C. Seassal, R. Mazurczyk, P. Viktorovitch, Optics Express 17 (2009) 9780.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Effects of structurally deformed sub-lattice points on the dispersion properties of 2D hybrid triangular–graphite photonic crystal Fulya Bagci, Baris Akaoglu ⁎ Department of Engineering Physics, Faculty of Engineering, Ankara University, 06100 Besevler, Ankara, Turkey
a r t i c l e
i n f o
Article history: Received 28 March 2011 Received in revised form 6 October 2011 Accepted 12 October 2011 Available online 29 October 2011 Keywords: Photonic crystal Photonic band gap Symmetry reduction Plane-wave expansion method
a b s t r a c t We investigate a hybrid two dimensional (2D) photonic crystal which is constructed by merging triangular and graphite lattices in air background. Different geometries of scatterers such as circular rods, circular rods with elliptical holes, circular hollow rods and elliptical rods with circular holes in the triangular sublattice are considered and effects of shape on their dispersion properties are discussed. Photonic band gaps (PBGs) for TM- and TE-polarized modes are found to exist for these symmetry-reduced graphite structures. The TE PBGs are found to contribute to produce complete photonic band gaps (CBGs) up to high values of filling fraction and rod diameter ratios. We show that TM and TE PBGs display opposite behaviors as the structure is transformed from graphite to symmetry-reduced graphite (hybrid triangular–graphite lattice) and from symmetry-reduced graphite to hollow centered symmetry-reduced graphite. Concerning the PBGs, inner structural deformation is found to be more beneficial in comparison to outer structural deformation. Small group velocities are observed at the same frequency for both polarizations near the edges of PBGs for the configurations of circular rods, circular rods with elliptical holes and circular hollow rods. These observations seem valuable for optical gain enhancement and low-threshold lasing. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals/solids (PhC) are composite optical materials, consisting of spatially modulated dielectric and metallic structures, and have unusual optical properties and innovative applications [1–4]. The most prominent feature of PhCs is the possibility of customizing their design to application needs. Moreover, since their discovery various extraordinary optical properties of PhCs have been predicted and some of them have been confirmed experimentally such as existence of a photonic band gap (PBG), possibility of creating localized defect modes in the PBG, suppression of spontaneous emission and generation of slow light in photonic crystal waveguides [5–8]. In such materials, variation of refractive index manipulates light propagation through multiple scattering from the arranged scatterers (optical “atoms”), which are analogous to atoms, diffracting X-rays in a lattice of a crystal. These diffracted light waves give rise to stop bands for the waves within a certain frequency range in all directions through multiple interferences. In other words, formation of standing waves leads to PBGs in the same fashion as electrons in semiconductors. In general, a scattering element with a size of the order of λ/2 roughly leads to a PBG at a wavelength λ [9]. Since pioneering works, independently introduced by Yablonovitch [6] and John [10],
⁎ Corresponding author. E-mail address: [email protected] (B. Akaoglu). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.10.035
increasing interest in this field is directed mainly toward realization of a complete PBG (CBG) which can occur in all directions for all polarizations only in three dimensional (3D) PhCs [11,12]. However, fabrication of 3D PhCs still remains a difficult task when the periodicity of the structure has the length on the order of visible to near-infrared wavelengths. Fortunately, since 2D PhCs exhibit most of the essential properties of their 3D counterparts and are much easier to fabricate, many research efforts have been devoted to 2D PhCs. It is possible to simulate experiments for crystals with lattice constants on the order of visible wavelengths, for example, using specimens with larger lattice constants since there is no fundamental length scale for photonic crystals, that is, the wave equation in a periodically modulated dielectric medium is scale invariant. Modes of a 2D PhC, periodic in the xy-plane and having homogenous dielectric function in the z-direction, can be separated into two sub-modes since the (x,y) plane of periodicity of the PhC is the mirror plane of the system. The modes for which the electric field and magnetic field are parallel to z-axis are referred to as transverse-magnetic (TM) and transverse-electric (TE) modes, respectively. Since the direction of electric field is different relative to dielectric interfaces for TM and TE polarizations, corresponding band structures for the two polarizations are usually different. PBGs can exist both for TM and TE polarizations but the positions of the PBGs can be very different. A CBG exists if PBGs for both polarization modes overlap. In order to get a better control of light confinement it is essential to obtain a CBG and enhance its width by properly designing the crystal structures.
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The general approach to controlling the dispersion properties of the PhC is mainly based on appropriate choice of the refractive index ratio of the constituent materials, the lattice type and adjustment of the filling fraction. On the other hand, various types of PhCs with high degrees of freedom have been introduced and these PhCs have additional parameters, defining the structure, which can be utilized to control the dispersion properties of the crystal further to the maximum extent possible. Mostly, the constraints on PBG formation, enlargement and observing low-dispersive bands near gap-edges stem from the degeneracy of the modes at high symmetry points in the Brillouin zone. Reducing the symmetry of the structure by a number of different approaches leads to separation of degenerate adjacent modes, resulting in the formation of a new PBG or enhancement of the width of an already existing PBG. The primary utilized methods for symmetry reduction can be classified as follows: introducing additional lattice scatterers at the center of the unit cell [9,13–15], changing the shape of lattice points [16–18], utilizing birefringent dielectrics [19,20], introducing interfacial layers [20,21], creating structural and rotational deformation in annular PhC structures [22] and combining two lattices belonging the same Bravais lattice to create hybrid lattice geometries [23,24]. In hybrid lattice geometries, the different inner/outer shape and size of the scatterers in each sublattice provide extra degrees of freedom that can be utilized to alter the optical properties of the crystal. Furthermore, the shape of the scatterers in one sub-lattice of the hybrid structure may not be totally different from the shape of the scatterers in the other sub-lattice but may be just structurally and/or rotationally deformed. Such extra degrees of freedom that hybrid PhC geometries offer open many options to alter the dispersion properties of the PhC in the desired fashion. In this work, we have considered a 2D hybrid triangular–graphite lattice in air background in order to benefit from its high degree of freedom. This hybrid structure can also be envisaged as a graphite structure with reduced symmetry when the central triangular sublattice points are kept at different sizes and/or shapes in comparison to graphite ones. In this structure, to obtain a large CBG, one can take advantage of the large TE PBGs of graphite lattice [24] and large TM PBGs of triangular lattice. Graphite lattices in air background have larger TE PBGs than in dielectric background [25]. On the other hand, triangular lattice PhCs in air background are known to possess large TM PBGs [25]. Merging these two types of lattices of the same Bravais form constitute hybrid triangular–graphite lattices. The hybrid triangular–graphite lattice in dielectric background has large CBGs mostly near closed-packed conditions. Most of the gaps in hybrid triangular–graphite lattice of rods in air background belong
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to TM PBGs but there is also an important number of TE PBGs with considerable widths that constitute scattered CBGs at different rod radii far from closed-packed conditions [23]. Laser emission from low-dispersive bands close to the Г3-point in a wide-range of inplane k values with high quality factors has been reported for the hybrid triangular–graphite configuration in the InP slab, including four InAsP quantum wells on top of a Si substrate SiO2 wafer [26]. In this study, not only the size of the scatterers embedded in the triangular sub-lattice is investigated in point of dispersion properties but also we focused on the effects of the existence and shape of the holes at the triangular sub-lattice points where structural deformations are additionally introduced. While taking the radius of the solid circular rods at the graphite sub-lattice constant, different geometries of scatterers such as circular rods, circular rods with elliptical holes, circular hollow rods and elliptical rods with circular holes in the triangular sub-lattice are considered. In order to unambiguously justify the consequences of the shape effect in hollow rods, the air fractions of the holes are kept constant. So far, to the best of our knowledge, the effects of both the introduction of holes in scatterers and independent deformation of inner and outer shapes of hollow rods embedded in one of the sub-lattices of a 2D hybrid PhC lattice have not been reported yet. Moreover, for this hybrid structure, there exist flat bands at the same frequency for both polarizations near gap edges, which is an extraordinary result for 2D PhCs. The importance of hollow PhCs has been recently highlighted in terms of diverse device applications. They are more sensitive to external refractive index change which is attractive for sensor applications [27]. It has been demonstrated earlier that they have polarization independent PBGs [22]. Also some hollow PhC waveguides have been shown to exhibit low group velocity which is an attractive feature for slow light waveguides [27,28]. Besides, inner and outer shape deformations of them introduce shape anisotropy which lead to strong polarization effects leading to birefringence which find applications in PhC fibers [29–31]. 2. The structure and numerical method Since 2D PhCs have different periodicities in different directions, the gaps in different directions do not necessarily overlap unless they are large enough. In principle, the more equal the periodicity in different directions, which corresponds to a more circular Brillouin zone, the easier it generally becomes to create a PBG [1]. For example, triangular PhCs in air background display large TM PBGs due to their isolated patches of high-ε regions and nearly circular Brillouin-zones.
Fig. 1. (a) Schematic configuration of the symmetry-reduced graphite lattice with hollow triangular sub-lattice points in air background. The lattice constant is a and the basic vectors of the scatterers are defined by u0 = 0, u1 = (a1 + a2)/3 and u2 = −(a1 + a2)/3 relative to the center of the unit cell. The unit cell is restricted by the thick lines and contains three scatterers. (b) Brillouin zone for the symmetry-reduced graphite lattice where shaded region corresponds to their reducible zone.
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Fig. 2. (a) The graph of the ratio of gap-width to gap-center of the TM and TE PBGs as the filling fraction is changed for the symmetry-reduced graphite structure when β = 0.1 (b) Gap map of symmetry-reduced graphite structure of dielectric rods (ε = 12.09) in air at f = 0.45. TM PBGs are remarked as blue, TE PBGs are remarked as red and CBGs are remarked as yellow in color.
On the other hand, graphite structure of dielectric rods in air exhibits large CBGs and consequently, it has a considerable potential for the realization of the PBG materials [32]. If these two lattices are merged while keeping the Brillouin zone symmetry same, a hybrid triangular–graphite lattice, taking advantage of both these sub-lattices, can be obtained and then the effect of symmetry reduction applied to one of these sub-lattices can be investigated. Within this frame, we have placed different sized and shaped dielectric scatterers in the triangular sub-lattice which corresponds to the central points of graphite lattice. The structure and first Brillouin zone of this hybrid lattice are shown in Fig. 1(a) and (b), respectively. The hybrid triangular graphite lattice is constructed byp defining triangular ffiffiffi Bravais sub-lattice with primitive vectors a ¼ a 3 =2; 1=2 and a2 ¼ 1 pffiffiffi a 3=2; −1=2 and introducing three rods in a unit cell at positions u0 = 0, u1 = (a1 + a2)/3 and u2 = −(a1 + a2)/3 with respect to the center of the unit cell as shown in Fig. 1(a). When radius of circular scatterers in the triangular sub-lattice becomes zero (rt = 0), the structure degenerates the basic graphite structure whereas when it becomes equal to the radius of scatterers in graphite sub-lattice (rt = rg), hybrid triangular–graphite lattice pffiffiffican be regarded as triangular lattice with a lattice parameter a= 3. This hybrid triangular– graphite lattice resembles Suzuki-phase [33] and the Archimedean [34] lattices in point that all these lattice types hold several lattice points in the unit cell which support several low-dispersive bands [26]. We have chosen silicon as dielectric material of the rods which has a dielectric constant of ε = 12.09 at λ = 1.55 μm. We have utilized MIT
photonic-bands package [35] which uses plane wave expansion method with periodic boundary conditions to calculate the band structures and field patterns. The computational error was estimated to be less than 0.2%. 3. Results and discussions In order to determine a radius at which the effect of structural deformation on PBGs might be more noticeable, first we examined the dispersion properties of the symmetry-reduced graphite lattice for all rod diameter ratio (β = rt/rg) and filling fraction (f) values. For this purpose, we modified the size of the scatterers by changing either the β or f. f is changed between 0.1 and 0.6 in steps of 0.05 for all values of β and rod diameter ratio is changed between 0 and 1 in steps of 0.05 for all values of f. For all values of β, increase in the filling fraction leads to contrary results on the first PBGs for TM and TE polarizations. The results of numerical calculations revealed that as f increases, the width of the PBG between TM2 and TM3 modes (TM2–3 PBG) decreases whereas the width of TE1–2 PBG, which exists until β > 0.4, increases. However, TM7–8 and TE3–4 PBGs display similar behaviors (i.e. Fig. 2(a)). As f increases, their widths increase up to a critical value of β and then begin to decrease. Very small f values (f around 0.1) lead to high CBG ratio at frequencies approaching (2πc)/a. It has been found earlier that the largest CBG occurs for f = 0.14 and β = 0.11 [13]. Due to the effect of two additional rods in the unit cell, the mid-gap frequency of the lowest CBG decreases as f increases.
Fig. 3. The geometry and relative parameters utilized in the simulations: (a) D1 configuration of graphite lattice, (b) D2 configuration of symmetry-reduced graphite lattice, (c) D3 configuration of circular rods with elliptical holes, (d) D4 configuration of circular hollow rods, and (e) D5 configuration of elliptical rods with circular holes.
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For particular values of f, as β increases TM2–3 PBG decreases while TM3–4 PBG increases. A lattice with β = 1 strongly favors large TM PBGs like triangular lattice and has a gap-midgap ratio of 49% at f = 0.1. PBG map of the symmetry-reduced graphite lattice at a filling fraction of 0.45 is given in Fig. 2(b). At f = 0.45, the graphite structure has a CBG ratio Δω/ω0 of 2.7%. However, the addition of a
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small rod into the graphite cell center increases the CBG ratio to a maximum value of 6.9% at β = 0.3. It is clearly seen that there is a CBG for almost all values of β. Besides, the symmetry-reduced graphite lattice favors TM PBGs more than TE PBGs as can also be seen in Fig. 2(b). It is interesting that the second TE PBG is nearly not affected by the rod diameter ratio variation. Since the field distribution
Fig. 4. Dispersion diagrams of configuration D1 for (a) TM and (b) TE polarizations. Dispersion diagrams of configuration D2 for (c) TM and (d) TE polarizations.
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simulations show that the magnetic field of the modes of upper and lower band edges has nodes on triangular sub-lattice points, this observation sounds reasonable. It should be noted that as β goes from 0 to 1, the radii of graphite sub-lattice points decrease by only 18%. In order to study the effects of deformation of the rods on the PBGs of the hybrid structure, triangular sub-lattice points at the center of graphite sub-lattice are considered. For this symmetry-reduced graphite structure, we choose a high β value, β = 1, to exhibit the influence of the deformations on the dispersion diagrams stronger. At the filling ratio of 0.45 and β = 1 (corresponding to r = 0.2a) the width of the TE PBG and CBG is very narrow (0.4% at ω = 0.7(2πc/a), illustrated in Fig. 2(b)), thus the enlargement of TE PBG and CBG by opening air holes would be more prominent. The graphite lattice and its symmetry-reduced configurations utilized for the structural deformation analysis are illustrated with their structural values in Fig. 3. Fig. 4(a) and (b) shows the dispersion diagrams of the graphite lattice, given in Fig. 3(a), with circular rods having r = 0.2a for TM and TE polarizations, respectively. Dotted and solid lines denote the odd and even numbered bands, introduced for easy guiding of the eye. For this relative radius value, calculation results revealed that three PBGs appear for TM polarization whereas two considerable PBGs appear for TE polarization within the range considered. The
second PBG for TM polarization and first PBG for TE polarization overlap between normalized frequencies of 0.63 and 0.68 for which Δω/ω0 = 8.2%. Fig. 4(c) and (d) shows the dispersion diagrams of the hybrid triangular–graphite lattice (f = 0.45, β = 1), given in Fig. 3(b), with circular rods having r = 0.2a for TM and TE polarizations, respectively. The number of TM and TE bands is considerably increased since the unit cell became denser by the rods embedded in the center. This structure favors large TM PBGs in comparison to the PBGs of the graphite lattice given in Fig. 4(a). In contrast, TE PBGs are observed to be reduced. This trend is expected since high dielectric constant material is now located in the center of the unit cell. A CBG exits for only a very small frequency interval with Δω/ω0 = 0.4%. It is also found that the gap edges shift to lower frequencies. Structural deformation (symmetry reduction) can be created by opening air holes in the scatterers belonging to one of the sublattice of the hybrid lattice. In addition, deformation of the inner or outer shapes of the hollow rods leads to further structural deformation. First, inner shape deformation is investigated for symmetryreduced graphite lattice where elliptical air holes [with eccentricity (γ) of 2, f = 0.4, β = 1] are introduced at the triangular sub-lattice points as shown in Fig. 3(c). Thus, the structural symmetry of the
Fig. 5. Dispersion diagrams of configuration D3 for (a) TM and (b) TE polarizations. Dispersion diagrams of configuration D4 for (c) TM and (d) TE polarizations.
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outer shape is retained, whereas that of the inner shape is broken. Numerical calculations indicate that the displacement field of the lower edge of the first TM PBG for all symmetry-reduced graphite structure configurations, given in Fig. 3, concentrates on lattice points at Γ symmetry points. In this structure, the localization of the displacement field on the lattice points, which are hollow for triangular sub-lattice points in this case, shifts the lower gap-edge to higher frequencies and narrows the first TM PBG as seen in Fig. 5(a). However, the expansion of the second TE PBG increases the width of the CBG for which Δω/ω0 = 5% between normalized frequencies of 0.71 and 0.75. The degeneracy between TE10 and TE11 bands has lifted and this led to opening of the third TE PBG as shown in Fig. 5(b). This new PBG arises from resonant scattering of short wavelength light from dielectric walls of hollow scatterers. This third TE PBG appears for the D3, D4 and D5 structures. Moreover, appearance of this additional PBG shifts high frequency modes further to higher frequencies which is a consequence of the decrease of the effective (“average”) refractive index. Second, circular air holes (Fig. 3(d)) are introduced in the center of the rods forming the triangular sub-lattice while keeping the filling fraction the same as the previous case (Fig. 3(c)). Thus, apart from the combined effect of filling fraction and shape [22], only the effect of shape is considered in this case. The inner hole radius corresponds to r = 0.1a. Extensive calculations in the literature showed that the CBG width has been increased in 2D triangular lattice with circular and hexagonal air rings and in graphite lattice with elliptical and hexagonal air rings [36]. The size of the TE PBGs and CBG is found to be larger for circular inner rods as seen in Fig. 5(d), in agreement with the previous studies [16]. The CBG occurs between normalized frequencies of 0.71 and 0.75 for which Δω/ω0 = 5.7%. Since the air space size among the rods are invariable (which is also valid for configurations D2 and D3), the size of the first TE PBG is retained. Finally, configuration D5 with circular inner and vertically deformed outer elliptical rods (γ = 2, f = 0.4) are considered for triangular sub-lattice points. In contrast to configuration D3, the structural symmetry of the inner shape is retained, whereas that of the outer is broken. TM PBGs split into smaller pieces narrowing their band gaps as shown in Fig. 6(a), in agreement with the observations in the
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study of broken structural symmetry of triangular lattice of hollow rods [22]. The ninth TM mode is positioned between two TM PBGs at higher frequencies, whereas the lowest edges of other modes seem preserving their positions. Due to the reduction of air size among the rods, the frequency interval between second and third TE modes becomes larger, shrinking the first TE PBG and vanishing the second TE PBG. However, the third TE PBG is not affected due to the resonant scattering, similar to the third TE PBG as given in Fig. 5 (b). This structure does not support an appreciable CBG. This result is reasonable since PBG width is known to be reduced when the symmetry of the scatterer and the lattices are not the same [17]. One remarkable feature of hybrid triangular–graphite lattice configurations (D2, D3, D4, D5) is that TE PBG formation dominates the CBG width. Another remarkable feature is the flatness of ninth TM and sixth TE bands at the same frequency (≈0.7) below the second TM PBG and second TE PBG, respectively at a very broad k-region (Figs. 4(c, d) and 5(a–d)). For graphite lattice such a flat band does not exist (Fig. 4(a, b)). It should also be noted that for the inverse hybrid structure of air holes in silicon background, such flat polarization independent bands near gap edges are not observed. For the D2 and D4 configurations, flat ninth TM band coincides with eighth TM band and sixth TE band coincides with fifth TE band between M–K points, increasing the density of states further. Besides, for the configurations D2 and D4 with circular rods (holes), the ends of the fourth TE band intersect with the sixth TE band at k-points (0 0.175(2π/a) 0) and (−0.1166(2π/a) 0.1166(2π/a) 0) making the interested frequency region almost flat. The frequency difference between the maximum and minimum points in the almost flat k-region can be regarded as a measure to indicate the flatness of the bands. In this respect, if the D2, D3 and D4 configurations are compared, the flatness of the TM and TE bands can be sorted descending as D3, D4 and D2. Especially for the hybrid configuration D4, the ninth TM band and the sixth TE band show small group velocity in a broad region of k-space, resulting to an increase in density of states (Fig. 5(c,d)). To fabricate such a lattice geometry with flat bands centered at λ = 1.55 μm would require a lattice constant of about 1 μm. However, the fabrication of D3 and D5 structures with inner (or outer) elliptical
Fig. 6. Dispersion diagrams of configuration D5 for (a) TM and (b) TE polarizations.
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Table 1 Frequency interval (Δω) and central frequency (ωc) of the flat ninth TM band along the whole Г–M–K–Г path and frequency interval (Δω) and central frequency (ωc) of the flat sixth TE band between the (0 0.225(2π/a) 0) and (− 0.1166(2π/a) 0.1166(2π/a) 0) k-points. TM
TE
f
Δω(2πc/a)
ωc(2πc/a)
f
Δω(2πc/a)
ωc(2πc/a)
0.1 0.2 0.3 0.4 0.5 0.6
0.0112 0.0098 0.0158 0.0234 0.0353 0.0472
1.098 0.835 0.714 0.648 0.603 0.572
0.1 0.2 0.3 0.4 0.5 0.6
0.0322 0.0092 0.0204 0.0191 0.0342 0.0529
1.083 0.837 0.718 0.649 0.598 0.561
holes (rods) would not be easy. Fortunately, the D4 configuration also shows the same flat bands and would require a relatively less effort for fabrication. If one wants to utilize this PhC at λ = 1.55 μm, the ring width would be about 100 nm, which is sufficiently thick to be fabricated using standard technologies for dry etching. Since the fabrication of D4 structure is relatively easy, we further analyzed the flatness of its polarization independent gap-edge bands as a function of filling fraction and the ratio of inner to outer rings of triangular sub-lattice points. The filling fraction for the D4 configuration with circular hollow rods in the hybrid triangular– graphite lattice is calculated as h i pffiffiffi 2 2 2 f ¼ 2πr g þ π r t −r in = 3=2 : The frequency width of the flat ninth TM band (Δω) in the whole k-region and its frequency value in the middle of M–K region (ωc) are listed as a function of filling fraction in Table 1. Since the ends of flat sixth TE bands are curved at Г points, the frequency width for TE polarization (Δω) is calculated between (0 0.225(2π/a) 0) and (−0.1166(2π/a) 0.1166(2π/a) 0), i.e. between 11th and 55th kpoints in our calculated k-region with 60 points. The frequency width of the flat sixth TE band (Δω) and its frequency value in the middle of M–K region (ωc) are also listed as a function of filling fraction in Table 1. The flatness of the ninth TM band increases up to f = 0.2 and then decreases as the filling fraction increases. The same behavior can also be found on account of the flatness of the sixth TE band. The greater value of the frequency of the bands for the filling fraction of 0.2 creates PhC lattices far from closed-packed conditions and in larger lattice dimensions, providing convenience in fabrication. The polarization independent bands can be regarded almost flat up to the filling fraction of 0.45. We also investigated the band flatness of configuration D4 with respect to the ratio of inner ring of the triangular sub-lattice points to that of the outer ring (δ) at all filling fractions in 0.1 steps. Polarization independent flat bands are found to be mutually more flat
when 0.5 ≤ δ ≤ 0.7. The flattest polarization independent flat bands for configuration D4 are obtained at f = 0.2 and δ = 0.6 with Δω = 0.0038(2πc/a) for the TE and Δω = 0.0055(2πc/a) for the TM band. For these flat bands centered at wavelength λ = 1.55 μm, the dielectric rods should have a radius of 190 nm. Since flatness of a band and its group velocity are directly related, maximum group velocity can also be addressed to indicate the flatness of the band. The maximum group velocities for the most (f = 0.2, δ = 0.6) and least flat ninth TM band (f = 0.45, δ = 0.4) are found as 0.012c and 0.12c, respectively. Similar polarization independent flat bands have also been noticed for 2D square lattices of square cross-section dielectric rods with air holes drilled inside [37]. Most of the studies of PhC variations with flat dispersion curves have their flat dispersion positions in the vicinity of the band edges [26,38–41]. In this study, the gap-edge bands are found to be flat through almost all the k-regions. Although dielectric backgrounds are frequently utilized [26,38–40], a surface emitting microlaser based on a 2D square lattice of InP rods lying on a silica layer is also demonstrated by using standard fabrication processes [41]. In addition, for the D5 configuration, when light is propagated through M–K direction, coupling of almost flat TM and TE modes with other modes is less probable because tenth TM band is restricted to a narrow spectral region and the sixth TE band is farther in frequency than other TE bands. These results indicate that the PhC configuration proposed in this study might be considered for optical gain enhancement or low-threshold lasing [26,37–41]. Since the shift of the ninth TM mode in the configuration D5 is remarkable, we further examined its displacement field distribution for contrary configurations D3 (Fig.7(a)) and D5 (Fig. 7(b)). As shown in Fig. 7, a sharp difference stands out between the field distributions of D3 and D5. Reminding that air area in the triangular sub-lattice points are same in both configurations D3 and D5, this variation is solely a result of the shape effect. For the configuration D3, the uniform variation of the displacement fields and resonant scattering of the electromagnetic waves from thin dielectric walls of hollow rods play key roles in the existence of large second TM PBG. However, for the configuration D5 the displacement fields are distributed nonuniformly and the scattered displacement fields from the thin dielectric walls cannot efficiently interfere. Therefore, TM PBGs split and the widths of the TM PBGs vary more than those of the configuration D3, relative to the TM PBGs of configuration D2. Another point to note is that the displacement field of configuration D5 is concentrated much more in the air region among the rods than configuration D3, causing a shift of the mode to higher frequencies. 4. Conclusions In this work, we have investigated the effects of both the introduction of holes and independent deformation of inner and outer shapes of the
Fig. 7. Displacement field distribution of ninth TM mode at Γ symmetry point for (a) D3 and (b) D5 lattice configurations.
F. Bagci, B. Akaoglu / Optics Communications 285 (2012) 1486–1493
rods embedded in triangular sub-lattice of 2D hybrid triangular–graphite structure in air background. For the symmetry-reduced graphite structure, wide ranges of PBGs at different frequencies with significant sizes have been obtained and a regular behavior of PBGs relating to different modes has been noticed against rod diameter ratio variation. Different geometries of scatterers such as circular rods, circular rods with elliptical holes, circular hollow rods and elliptical rods with circular holes in the triangular sub-lattice are considered for the selected filling fraction and rod diameter ratio values. As the structure is transformed from graphite to a symmetry-reduced graphite structure (hybrid triangular–graphite lattice), TM and TE PBGs have been found to be increasing and decreasing, respectively. Opening air holes in the scatterers leads to an increase in TE PBGs whereas holes produce a negative effect on the width of the first TM PBG. Outer structural deformations have much stronger effect on dispersion properties than the inner structural deformations, leading to splitting of PBGs. Among the more or less flat bands appearing in the band diagrams along the M–K directions, the ones appeared for the configuration with circular rods with elliptical and circular holes are found to be remarkable. Not only these bands have been observed to be quite flat in a broad region of k-space but they are also independent of polarization at a wide filling ratio interval. For the configuration of elliptical rods with circular holes, the absence of modes in the vicinity of observed constant group-velocity bands along M–K direction can reduce propagation loss. These polarization-independent flat bands of the different hybrid triangular–graphite PhC lattice configurations proposed here can be utilized for optical gain enhancement or low-threshold lasing. Symmetry reduction has been mainly utilized in this study to design new PhCs with promising properties. Reminding the absence of a wellestablished physical rule explaining the relation between the degree of symmetry and dispersion properties, this work rather elucidates the dispersion properties of particular geometrical arrangements in a hybrid 2D PhC. Nevertheless, further study is indispensible to have a solid understanding of this kind of hybrid lattices with various deformations by employing all possible values of the variational parameters present in the model configurations. Acknowledgment Part of the numerical calculations reported in this paper were performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRGrid e-Infrastructure). References [1] S.G. Johnson, J.D. Joannopoulos, R.D. Maede, J.N. Winn, Photonic Crystals: Molding the Flow of Light, second ed. Princeton University Press, Princeton, NJ, 2008.
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