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Enlargement of absolute photonic band gap in modified 2D anisotropic annular photonic crystals
Optics Communications 284 (2011) 3315–3322
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Enlargement of absolute photonic band gap in modified 2D anisotropic annular photonic crystals T. Fathollahi Khalkhali a,b, B. Rezaei a,⁎, M. Kalafi a,b a b
Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran Physics Department, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 8 December 2010 Received in revised form 27 February 2011 Accepted 1 March 2011 Available online 21 March 2011 Keywords: Photonic crystal Absolute photonic band gap Anisotropic tellurium
a b s t r a c t We analyze the absolute photonic band gap in two dimensional (2D) square, triangular and honeycomb lattices composed of air holes or rings with different geometrical shapes and orientations in anisotropic tellurium background. Using the numerical plane wave expansion method, we engineer the absolute photonic band gap in modified lattices, achieved by addition of circular, elliptical, rectangular, square and hexagonal air hole or ring into the center of each lattice unit cell. We discuss the maximization of absolute photonic band gap width as a function of main and additional air hole or ring parameters with different shapes and orientation. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Since the amazing work of Yablonovitch [1] and John [2], photonic crystals (PCs) have attracted considerable attention. Investigations show that PCs exhibit a region of the frequency spectrum where propagating modes are forbidden, called photonic band gap (PBG). PBG gives rise to unusual physical phenomena such as inhibited spontaneous emission [3] and light localization [4]. Although three-dimensional (3D) PCs suggest the most interesting idea for novel applications, 2D PCs have also been strongly studied, since they can be fabricated more easily than 3D ones and may be employed in optical and electronic devices. For in plane propagation, two types of electromagnetic modes exist according to whether the electric (E-Polarization or TM) or magnetic (H-Polarization or TE) field is parallel to the rod axes. An absolute PBG exists for a 2D PC only when PBG in both polarization modes are present and they overlap each other. A large PBG is very remedial for various applications such as defect cavities [5] optical waveguide [6] defect-mode PC lasers [7]. In order to obtain a large absolute PBG, many attempts like symmetry reduction [8–12] and anisotropy in dielectricity [13–17] have been made to enlarge the PBG in 2D PCs. Recently, it has been reported that the suggested class of the so-called annular PCs [18–21], where dielectric rods are embedded into air holes of larger radius, have a dramatic effect on the absolute band gap. More recently, we have engineered PBG in 2D PCs composed of air rings in anisotropic tellurium background, leading to large absolute PBG [22].
⁎ Corresponding author. Tel.: +98 411 3393027; fax: +98 411 334 7050. E-mail addresses: [email protected], [email protected] (B. Rezaei). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.03.006
In this paper we study the band gap properties of 2D PCs, created by square, triangular and honeycomb lattices of air holes or rings with different geometrical shapes and orientations in anisotropic tellurium background by addition of air holes or rings into the center of each lattice unit cell, named modified structures. Given lattice symmetry, we discuss the maximization of absolute PBG as a function of main and additional air hole or ring parameters with different shapes and orientation. 2. Formulation To determine the PBG in periodic dielectric structures, we study the propagation of electromagnetic waves from Maxwell's equations. In inhomogeneous dielectric materials, Maxwell's equation can be written as the following formula for the magnetic field which is called master equation [23]: " ∇×
# → 1 ω2 → → ∇ × H r = 2 H r ε r c
ð1Þ
where, the light velocity in vacuum, ω is the frequency of light and → c is → → ε r = ε r + R is position-dependent dielectric function which is → periodic with → respect to the real space lattice vector R . Since ε r is periodic, we can use Bloch's theorem to expand its inverse as a sum of plane waves: → iG : r r = ∑η G e
−1 →
ε
→
→
G
→
ð2Þ
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→ → where η G is the Fourier transform of the inverse of ε r and plays a key role in determining photonic band structure. For a given lattice → with a unit cell including ns scatterers, η G is given by Ref. [24]: ns → −1 ð jÞ → −i G : uj η G = εb δG;0 + ∑ η G e →
→
ð3Þ
j=1
→ and ηð jÞ G is the Fourier transform of each scatter of lattice unit cell → at positions uj , which is calculated analytically by the following equation: ð jÞ →
η
G
→
=
1 −1 → −iG : r → ∫ ε ð r Þe dr : Ω cell →
ð4Þ →
Here Ω is the surface of unit cell and G is a 2D reciprocal lattice vector.The orientation of main and additional noncircular air holes or rings relative to the lattice axis is defined by angles Θ and θ, respectively, as shown inFig. 3. For nonzero values of Θ and θ, we → need to replace G in ηð jÞ G by G˜ ′ and G˜ for main and additional noncircular scatterers, respectively [25]: G˜ x ˜y G
!
=
cosðθÞ sinðθÞ − sinðθÞ cosðθÞ
Gx Gy
0 1 G˜ ′ Gx cosðΘÞ sinðΘÞ @ xA= : Gy − sinðΘÞ cosðΘÞ G˜y′
ð5Þ We have considered square, triangular and honeycomb lattices of circular, elliptical, rectangular, square and hexagonal air holes or rings in anisotropic tellurium background. We investigate the modification of band gap spectrum in these lattices by addition of circular, elliptical, rectangular, square and hexagonal air hole or ring into the center of each lattice unit cell. As an example, we have shown some of the modified structures in Fig. 1(a–j), i.e. square lattice of circular air rings and honeycomb lattice of square air rings including air ring with different geometrical shapes at the center of lattice unit cell. The corresponding first Brillouin zone for lattices under consideration is sketched in Fig. 2(a–b). The anisotropic tellurium has two different principle-refractive indices as ordinary-refractive index no = 4.8 and extraordinary-refractive index ne = 6.2 over the wavelength range of 4.5 − 6.25 μm with an absorption coefficient of α ≈ 1 cm− 1 [26]. We assume that the extraordinary axis is parallel to the z axis. In this configuration, the eigenequations for the E- and H-polarization modes are the same as those for the isotropic PCs, except that the refractive indices of anisotropic tellurium are ne and no for E- and Hpolarizations, respectively. 3. Results and discussion In this study we have investigated the band spectrum of 2D PCs with different geometrical air holes or rings in anisotropic tellurium background in three kinds of modified square, triangular and honeycomb lattices, obtained by addition of circular, hexagonal, elliptical, rectangular and square air holes or rings into the center of each lattice unit cell. The photonic band structure is calculated by solving Maxwell equation using the well-known numerical plane wave expansion method [27,28]. A total of 961 plane waves were used for all structures in these calculations, which ensures sufficient convergence for the frequencies of interest. The convergence accuracy for most of the lowest photonic bands is better than 1%. Our main purpose is to study the modification of the band gap spectrum and the value of the absolute PBG when the orientation of noncircular and geometrical parameters of air holes or rings varies. In these structures the geometrical parameters and orientation of air holes or rings are treated as adjustable parameters to obtain the maximum absolute PBG.
More recently, we have investigated the photonic band structure of 2D square, triangular and honeycomb lattices of air holes or rings with circular, hexagonal, elliptical, rectangular and square shapes in anisotropic tellurium background [22]. The present work is focused on the engineering of absolute PBG in the aforementioned lattices by addition of air holes or rings into the center of each lattice unit cell. Numerical results show that the size of absolute PBG in modified triangular lattices has been decreased in comparison with the previously obtained results [22]. Therefore, in this letter the photonic band spectra of square and honeycomb lattices will be separately analyzed in the next two subsections. 3.1. Square lattice In this subsection, the results for the square lattice are presented. First of all, it should be mentioned that the safest path for calculating PBG is half side of the first Brillouin zone. However, in some cases depending on the type of photonic crystal lattice and geometric shape of scatterers, by choosing less paths, exact and accurate PBG is obtainable. In this investigation the photonic band structure of the square lattice with circular scatterers was traced along the Γ − X − M − Γ path of the Brillouin zone, while for square, hexagonal, elliptical and rectangular scatterers the half side of the first Brillouin zone were traced (Fig. 2a). These are sufficiently precise paths for calculation PBG in our mentioned structures. We begin our discussion with square lattice of circular air holes in anisotropic tellurium background. Investigations reveal that, this lattice has a complete PBG with maximum normalized width of Δωmax = 0.0271(2πc/a) and gap–midgap ratio of ωr = 6.80% at R = 0.482a, a is being the lattice constant. Here Δωmax is the frequency width of the maximum absolute PBG and ω1 and ω2 represent the corresponding lower and upper edge frequencies in units of 2πc/a, respectively. The gap–midgap ratio is defined as ωr ≡ Δωmax/ωg, where ωg is the center frequency of the PBG. Fig. 4 represents the photonic band structures of this lattice. This figure shows a complete PBG between ω1 = 0.3848 and ω2 = 0.4119 frequencies in unit of (2πc/a). Now, for further investigation of absolute PBG in the above mentioned lattice, we modify its structure by addition of a circular air hole into the center of lattice unit cell. Numerical results show that there is an absolute PBG with maximum normalized width of Δωmax = 0.0599 (2πc/a) and ωr = 11.52% at optimum geometrical parameters R = 0.491a (radius of main air holes) and r = 0.170a (radius of additional air holes). Fig. 5 depicts the dispersion relations of this structure with an absolute PBG resulted from the overlapping of the TM4-5 and TE2-3 band gaps and lies between 0.4903(2πc/a) and 0.5502(2πc/a) frequencies. It can be seen that the width of absolute PBG has been increased by addition of a smaller diameter air hole into the center of lattice unit cell. This absolute PBG is almost two times larger than the previous one. At next step, a hexagonal air hole is located into the center of lattice unit cell. After optimization of all the geometrical parameters an absolute PBG with maximum normalized width of Δωmax = 0.0521(2πc/a) and ωr = 11.75% is obtained at optimum parameters R= 0.484a, r = 0.166a (r is hexagonal side length) and θ = 0∘(rotation angel of additional hexagonal air hole). This PBG is created by overlapping of TM4-5 and TE2-3 band gaps and has been increased compared to the band gap of original square lattice. Similar to the previous cases, we add square air hole into the center of lattice unit cell. The results of numerical calculations show that there is an absolute PBG with maximum normalized width of Δωmax = 0.0503 (2πc/a) and ωr = 9.04% between TM4-5 and TE2-3 polarization band gaps at optimum geometrical parameters R = 0.492a, r = 0.167a, (the half side of square air holes) and θ = 45∘(rotation angle of square air hole). This absolute PBG between 0.5314 and 0.5817 frequencies in units of (2πc/a). Finally, the elliptical and rectangular air holes have been considered as additional air holes into the center of lattice unit cell.
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Fig. 1. Schematic representation of modified 2D square lattices of circular air rings (a–e) and honeycomb lattices of square air rings (f–j) in tellurium background with additional circular, square, hexagonal, elliptical and rectangular air rings at the center of each lattice unit cell.
Numerical results show that the width of absolute PBG has been decreased in these cases and approaches to zero by increasing the size of additional elliptical and rectangular air holes.
Now, we consider the case in which the air rings are included into the center of lattice unit cell instead of air holes. Therefore, the modified square lattices of air holes with additional circular, square,
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Fig. 4. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for square lattice of circular air holes in tellurium background with R = 0.482a.
Fig. 2. Tow-dimensional Brillouin zones for (a) square and (b) triangular and honeycomb lattices. The grey shaded area represents the irreducible Brillouin zone.
rectangular, elliptical and hexagonal air rings at the center of each lattice unit cell are obtained. Comprehensive investigation show that in three cases of hexagonal, circular and square air rings which are inserted at the center of lattice unit cell, the width of absolute PBG has been increased while it has been decreased in the case of additional rectangular and elliptical air rings at the center of lattice unit cell. The maximum normalized width of absolute PBG is 0.0710, 0.0762 and 0.0667 in units of 2πc/a, and gap–midgap ratio is 15.24%, 15.64% and 13.14% for hexagonal, circular and square air rings, respectively. The above mentioned values for PBG width are obtained at optimum geometrical parameters R = 0.485a, r2 = 0.184a (outer side length), r1 = 0.141a (inner side length) and θ = 0∘ for hexagonal air ring and
Fig. 3. Modified square lattice of rectangular air rings in tellurium background with additional square air ring at the center of lattice unit cell. The rotation angle of rectangular and square rings is denoted by Θ, θ, respectively, as the angle between the axis of the air ring cross section and the lattice axis.
R = 0.486a, r2 = 0.175a (outer radius) and r1 = 0.129a (inner radius) for circular air ring and R = 0.488a, r2 = 0.168a (outer half side length), r1 = 0.112a (inner half side length) and θ = 45∘ for square air ring. It can be seen that the largest PBG is obtained in the case of circular air ring. Fig. 6 shows the frequency spectrum of this structure at the above mentioned optimum geometrical parameters. Then we concentrate on the square lattices of circular air rings in tellurium background and investigate the effect of additional air hole or ring on PBG with different geometrical shapes into the center of lattice unit cell. The square lattice of circular air rings with inner and outer radius R1 = 0.08a and R2 = 0.482a, respectively, has an absolute PBG with maximum normalized width of Δωmax = 0.0 390(2πc/a) and gap–midgap ratio of ωr = 9.65%. In order to enlarge the width of PBG, we introduce air holes with different shapes into the center of lattice unit cell. Numerical results show that in the case of hexagonal, circular and square air holes the absolute PBG has been increased while it has been decreased for elliptical and rectangular air holes. For additional hexagonal, circular and square air holes the normalized width of absolute PBG are 0.0822, 0.0919 and 0.0668 in units of 2πc/a, with gap–midgap ratio of 15.79%, 17.71% and 8.35%, respectively. These values of maximum absolute PBG are obtained at optimum parameters R2 = 0.487a, R1 = 0.07a, r = 0.186a and θ = 0∘ for hexagonal air hole and R2 = 0.488a, R1 = 0.07a and r = 0.17a for circular air hole and R2 = 0.490a, R1 = 0.154a, r = 0.153a and θ = 45∘ for square air hole, where R1 and R2 represent inner and outer radius of main circular air ring. It is obvious that in the case of additional circular air hole the width of
Fig. 5. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for square lattice of circular air holes of radius R = 0.491a in tellurium background with additional circular air hole at the center of lattice unit cell with r = 0.170a.
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Table 1 Maximum absolute PBG width for square lattice of hexagonal air holes with optimum side length R and rotation angel Θ, in tellurium background including additional air holes or rings at the center of lattice unit cell. Additional scatterers
Geometrical parameters
Δωmax(2πc/a)
ωr ≡Δωωmax g
Circular hole Square hole Hexagonal hole Square ring Hexagonal ring
r = 0.04a r = 0:15a r = 0:20a r2 = 0:20a r2 = 0:20a
0.0300a 0.0341b 0.0314c 0.0394d 0.0416e
8.78% 10.39% 6.06% 8.24% 7.96%
a b c d e
Fig. 6. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for square lattice of circular air holes of radius R = 0.486a in tellurium background with additional circular air ring at the center of lattice unit cell with r2 = 0.175a and r1 = 0.129a.
absolute PBG has been increased almost two times in comparison with the original square lattice of circular air ring. The photonic band structure of this structure is shown in Fig. 7. At next step, the addition of air rings with different shapes into the center of lattice unit cell has been studied. Numerical results showed that in this case the width of absolute PBG has been decreased and approaches zero by increasing the size of additional air rings. Now we investigate the PBG in the modified square lattice of hexagonal air holes in tellurium background. Numerical results show that the square lattice of hexagonal air holes with side length of R = 0.50a and Θ = 0∘ has an absolute PBG with maximum normalized width of Δωmax = 0.0 220(2πc/a) and ωr = 6.52%. In order to modify this structure, we introduce air holes with different geometrical shapes into the center of lattice unit cell. Therefore, the square lattices of hexagonal air holes with additional circular, square, rectangular, elliptical and hexagonal air holes at the center of each lattice unit cell are obtained. Numerical results show that in the case of additional square, circular and hexagonal air holes, the absolute PBG has been increased while it has been decreased for elliptical and rectangular air holes. Maximum normalized width of absolute PBG and the corresponding optimum geometrical parameters for these structures are listed in Table 1. It is obvious that the width of absolute PBG has been increased in the square lattice of hexagonal air hole with additional square air hole.
θ = 0∘ θ = 0∘ r1 = 0:152a θ = 45∘ r1 = 0:08a θ = 0∘
R = 0.50a, Θ = 0∘. R = 0.49a, Θ = 15∘. R = 0.50a, Θ = 45∘. R = 0.50a , Θ = 45∘. R = 0.50a, Θ = 45∘.
Now, in above mentioned lattice the air rings with circular, square, hexagonal, elliptical and rectangular shapes are located at the center of lattice unit cell instead of air holes. Comprehensive calculations reveal that in two cases of additional hexagonal and square air rings the width of absolute PBG has been increased while it has been decreased in the case of additional circular, rectangular and elliptical air rings. Maximum absolute PBG for hexagonal and square air rings and the corresponding optimum parameters are also summarized in Table 1. It can be seen that the largest absolute PBG is obtained in the case of additional hexagonal air ring with normalized width of 0.0 416 (2πc/a). This absolute PBG is nearly two times larger than that of simple square lattice with hexagonal air holes. Now we concentrate on the square lattices of hexagonal air rings in tellurium background and engineer the PBG by introducing air holes or rings with different geometrical shapes into the center of lattice unit cell. Numerical calculations demonstrate that the square lattice of hexagonal air rings in tellurium background represents an absolute PBG with normalized width of Δωmax = 0.0 199(2πc/a) at inner and outer side length of R1 = 0.18a, R2 = 0.50a and Θ = 0∘. Similar to the previous cases, we introduce air holes with different geometrical shapes into the center of lattice unit cell to study the enlargement possibility of PBG width. Numerical results show that in the case of additional hexagonal, circular and square air holes, the size of absolute PBG has been increased while it has been decreased for elliptical and rectangular air holes. The maximum normalized width of absolute PBG is 0.0779, 0.0714 and 0.0367 in units of 2πc/a, and gap–midgap ratio is 12.15%, 11.01% and 8.12% for additional square, circular and hexagonal air holes, respectively. These values of absolute PBG are obtained at optimum parameters R2 = 0.50a, R1 = 0.218a, Θ = 0∘, r = 0.20a and θ = 45∘ for additional square air hole and R2 = 0.50a, R1 = 0.20a, Θ = 0∘ and r = 0.209a for additional circular air hole and R2 = 0.50a, R1 = 0.19a, Θ = 0∘, r = 0.22a and θ = 0∘ for additional hexagonal air hole. Then the addition of air rings with different shapes into the center of lattice unit cell has been studied. Numerical results show that in these structures the width of absolute PBG has been decreased and approaches to zero by increasing the size of additional air rings. Finally, we investigate the modification of PBG in square lattices of square, rectangular and elliptical air holes or rings in tellurium background by addition of air hole or rings into the center of lattice unit cell. Extensive calculations reveal that the width of absolute PBG has been decreased in all above mentioned modified lattices. 3.2. Honeycomb lattice
Fig. 7. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for square lattice of circular air rings with R2 = 0.488a and R1 = 0.07a in tellurium background with additional circular air hole with r = 0.17a.
In this subsection the effect of additional scatterers on PBG in honeycomb lattices of air holes or rings with different geometrical shapes in tellurium background is presented. As discussed in the previous subsection, it should be mentioned that the most reliable
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path for calculating PBG is half side of first Brillouin zone. However, in this study the photonic band structures of honeycomb lattice with circular scatterers were traced along the Γ − K − M − Γ path of the Brillouin zone which named irreducible Brillouin zone (Fig. 2b), while for square, hexagonal, elliptical and rectangular scatterers the half side of the first Brillouin zone were traced. These are completely true and accurate paths for calculation PBG in this paper for honeycomb lattice, according to mentioned geometric shape of holes or rings and background material. We start our discussion with honeycomb lattice of circular air holes in anisotropic tellurium background. Results of numerical calculations show that this structure has an absolute PBG with normalized width of Δωmax = 0.0 234(2πc/a) and gap–midgap ratio of ωr = 14.52% at optimum radius R = 0.50a. Fig. 8 represents the photonic band structures of this lattice. This figure shows a complete PBG between ω1 = 0.1492 and ω2 = 0.1726 frequencies in unit of (2πc/a). Now to study the effect of additional scatterers on PBG in this structure, we introduce circular, square, hexagonal, elliptical and rectangular air holes into the center of lattice unit cell. The obtained results show that in the case of additional hexagonal, circular, rectangular and square air holes, the size of absolute PBG has been increased while it has been decreased for elliptical air hole. For brevity, only the maximum normalized width of absolute PBG and the corresponding optimum geometrical parameters are summarized in Table 2 for those structures in which the absolute PBG has been increased by addition of scatterers. It is obvious that the largest absolute PBG is obtained for honeycomb lattice of circular air holes with additional hexagonal air hole. The frequency spectrum of this structure is shown in Fig. 9. This figure represents an absolute PBG with normalized width of Δωmax = 0.0867(2πc/a) at optimum parameters R = 0.47a, r = 0.53a and θ = 30∘ which is nearly four times larger than that of simple honeycomb lattice of circular air holes. Now, instead of air holes the air rings are included into the center of lattice unit cell. Hence, the modified honeycomb lattices of air holes with additional circular, square, rectangular, elliptical and hexagonal air rings at the center of each lattice unit cell are obtained. The numerical results show that the width of absolute PBG has been decreased in all these structures and it approaches to zero by increasing the size of additional air rings. At next step, the honeycomb lattice of circular air ring in tellurium background is considered. Numerical results show that there is no PBG in such structure. Now similar to the previous cases, at first we introduce air holes with different shapes into the center of lattice unit cell and investigate its effect on photonic band structure. The obtained results show that in the case of additional circular and square air holes, the frequency spectrum of these structures exhibit relatively large absolute PBG. While the addition of rectangular, elliptical and hexagonal air holes at the center of lattice unit cell create no PBG. The maximum normalized width of absolute PBG is 0.0627 and 0.0438 in units of 2πc/a, and gap–midgap ratio is 20.05% and 14.79% for honeycomb lattices of circular air rings with additional circular and square air holes at the center of lattice unit cell, respectively. These values of maximum absolute PBG are obtained at optimum parameters R2 = 0.475a, R1 = 0.27a and r = 0.435a for additional circular air hole and R2 = 0.48a, R1 = 0.29a, r = 0.366a and θ = 0∘ for additional square air hole. Then the air rings with different shapes are introduced into the center of lattice unit cell of honeycomb lattice with circular air ring in tellurium background. After extensive calculation we have found that the additional scatterers have no effect on creating PBG in these photonic structures. Now we investigate the effect of additional scatterers on PBG in honeycomb lattice of hexagonal air holes in tellurium background. Investigating all possible structural parameters show that the honeycomb lattice of hexagonal air holes with side length of R = 0.50a and rotation angle of Θ = 45∘ has an absolute PBG with
Fig. 8. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for honeycomb lattice of circular air holes of radius R = 0.50a in tellurium background.
maximum normalized width of Δω max = 0.0114(2πc/a) and ωr = 3.35%. Similar to the previous cases we modify the structure of this photonic crystal by introducing circular, square, rectangular, elliptical and hexagonal air holes at the center of each lattice unit cell. Comprehensive investigations show that the size of absolute PBG has been increased for additional square, circular, rectangular and hexagonal air holes while it has been decreased for elliptical air holes. Table 3 shows the maximum normalized width of absolute PBG and the corresponding optimum geometrical parameters for honeycomb lattice of hexagonal air holes with additional square, circular, rectangular and hexagonal air holes at the center of lattice unit cell. It is obvious from Table 3 that the size of absolute PBG in honeycomb lattice of hexagonal air hole with additional circular air hole is nearly five times larger than that of the simple honeycomb lattice of hexagonal air holes. Next, instead of air holes, the air rings with circular, square, hexagonal, elliptical and rectangular shapes are inserted at the center of unit cell of honeycomb lattice composed of hexagonal air holes. It has been found that the size of absolute PBG in honeycomb lattice of hexagonal air holes has been increased by addition of hexagonal and circular air rings at the center of lattice unit cell and it has been decreased in the case of additional square, rectangular and elliptical air rings. The obtained results for maximum absolute PBG and the corresponding optimum parameters for additional hexagonal and circular air rings are summarized in Table 3. It can be seen that the largest absolute PBG is obtained for honeycomb lattice of hexagonal air holes with additional hexagonal air ring at the center of lattice unit cell with normalized width of 0.0 635(2πc/a). This absolute PBG is almost six times larger than that of simple honeycomb lattice with hexagonal air holes. Now the honeycomb lattice of hexagonal air rings in tellurium background is considered. The numerical calculations show that this photonic structure represents a maximum absolute PBG with
Table 2 Maximum absolute PBG width for honeycomb lattice of circular air holes with optimum radius R, in tellurium background including additional air holes at the center of lattice unit cell. Additional scatterers
Geometrical parameters
Δωmax(2πc/a)
ωr ≡ Δωωmax g
Square hole Circular hole Rectangular hole Hexagonal hole
r = 0.37a, θ = 30∘ r = 0.462a rx = 0.37a, ry = 0.475a, θ = 0∘ r = 0.53a , θ = 30∘
0.0256a 0.0671b 0.0602c 0.0867d
5.51% 20.19% 14.96% 22.32%
a b c d
R = 0.48a. R = 0.463a. R = 0.463a. R = 0.47a.
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Table 4 Maximum absolute PBG width for honeycomb lattice of square air holes with half side length R and rotation angel Θ, in tellurium background, including air holes or rings at the center of lattice unit cell. Additional scatterers Circular hole Hexagonal hole Square hole Elliptical hole Rectangular hole Circular ring a b c d e f
Fig. 9. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for honeycomb lattice of circular air holes of radius R = 0.47a in tellurium background with additional hexagonal air hole of side length r = 0.53a and rotation angle θ = 30∘ at the center of lattice unit cell.
normalized width of Δωmax = 0.0 170(2πc/a) and ωγ = 4.88% at inner and outer side lengths of R1 = 0.115a, R2 = 0.50a and rotation angle of Θ = 56∘. Now in the same process, air holes with shapes of circular, square, hexagonal, elliptical and rectangular are located into the center of lattice unit cell of this structure. Numerical results show that in these modified structures the size of absolute PBG has been increased for additional circular and hexagonal air holes and decreased for additional rectangular, elliptical and square air holes. The maximum absolute PBG with normalized width of 0.0 627(2πc/a) and gap– midgap ratio of 21.89% is obtained at optimum parameters R2 = 0.50a, R1 = 0.150a, Θ = 41∘ and r = 0.448a for additional circular air hole in honeycomb lattice of hexagonal air rings. The maximum absolute PBG and gap–midgap ratio are 0.0 670(2πc/a) and 22.65%, respectively, for additional hexagonal air holes which is took placed at optimum parameters R2 = 0.50a, R1 = 0.15a, Θ = 43∘, r = 0.495a and θ = 4∘. At next step, the air holes are replaced with air rings at the center of lattice unit cell. Numerical results show that in these cases the width of absolute PBG has been decreased and approaches to zero by increasing the size of additional air rings. At this step we consider another structure, i.e. honeycomb lattice of square air holes, and repeat the same processes as in previous cases. The honeycomb lattice of square air holes with half side length of R = 0.42a and Θ = 0∘ represents an absolute PBG with maximum normalized width of Δωmax = 0.0 127(2πc/a) and gap–midgap ratio of ωr = 4.99%. By introducing air holes with different shapes of circular, square, rectangular, elliptical and hexagonal at the center of lattice unit cell, Table 3 Maximum absolute PBG width for honeycomb lattice of hexagonal air holes with optimum side length R and rotation angel Θ, in tellurium background including additional air holes or rings at the center of lattice unit cell. Additional scatterers Square hole Rectangular hole Circular hole Hexagonal hole Circular ring Hexagonal ring a b c d e f
R = 0.50a, Θ = 10∘. R = 0.494a , Θ = 0∘. R = 0.50a, Θ = 43∘. R = 0.50a, Θ = 44∘. R = 0.50a, Θ = 41∘. R = 0.50a, Θ = 43∘.
Geometrical parameters ∘
r = 0.22a, θ = 0 rx = 0.365a, ry = 0.49a, θ = 0∘ r = 0.45a r = 0.496a , θ = 4∘ r2 = 0.46a, r1 = 0.13a r2 = 0.50a, r1 = 0.20a, θ = 4∘
Δωmax(2πc/a) a
0.0326 0.0502b 0.0528c 0.0527d 0.0570e 0.0635f
ωr ≡ Δωωmax g 11.06% 13.41% 18.31% 17.86% 19.00% 21.30%
Geometrical parameters r = 0.46a r = 0.456a, θ = 0∘ r = 0.396a, θ = 0∘ rx = 0.39a , ry = 0.57 , θ = 0∘ rx = 0.39a, ry = 0.45a, θ = 0∘ r2 = 0.45a , r1 = 0.091
Δωmax(2πc/a) a
0.0331 0.0383b 0.0549c 0.0295d 0.0738e 0.0366f
ωr ≡ Δωωmax g 8.71% 11.87% 15.90% 5.91% 15.50% 9.11%
R = 0.39a, Θ = 0∘. R = 0.404a, Θ = 0∘. R = 0.408a, Θ = 0∘. R = 0.43a , Θ = 0∘. R = 0.43a , Θ = 0∘. R = 0.41a , Θ = 0∘.
the modified honeycomb lattices of square air holes in tellurium background is obtained. Numerical results show that the width of absolute PBG has been increased for all shapes of additional air holes. Maximum normalized width of absolute PBG and the corresponding optimum geometrical parameters are listed in Table 4 for these structures. It is clear from Table 4 that the width of absolute PBG in honeycomb lattice of square air holes with additional rectangular air hole is almost six times larger than that of the simple honeycomb lattice of square air holes. Next, the air rings with circular, square, hexagonal, elliptical and rectangular shapes are located at the center of lattice unit cell of honeycomb lattice of square air holes. Numerical results show that the width of absolute PBG has been decreased for all additional air rings except the circular air ring. The size of maximum absolute PBG and the corresponding optimum parameters are summarized in Table 4 for additional circular air rings. It can be seen that the size of absolute PBG in the honeycomb lattice of square air holes with additional circular air ring is nearly three times larger than that of the simple honeycomb lattice of square air holes. Now the honeycomb lattice of square air rings in tellurium background is considered. Investigate all possible geometrical parameters demonstrate that there exist no PBG in this structure. Thus by adding air holes with different shapes of circular, square, hexagonal, elliptical and rectangular into the center of lattice unit cell, we study the properties of PBG in these structures. Numerical results show that the size of absolute PBG has been increased only in the case of additional elliptical air hole, while it has been decreased for other shapes of air holes. The honeycomb lattice of square air rings with additional elliptical air holes represents an absolute PBG with maximum normalized width of 0.0213(2πc/a) and ωr = 5.45% at optimum parameters R2 = 0.43a, R1 = 0.15a (R2 and R1 are outer and inner half sides of square air rings, respectively) Θ = 0∘, rx = 0.39a, ry = 0.563a (half diameter of elliptical air holes along x and y axes) and θ = 0∘. At the end, the air rings with circular, square, hexagonal, elliptical and rectangular shapes are included into the center of lattice unit cell. Numerical results show that there is no increment in the size of absolute PBG and approaches to zero by increasing the size of additional air rings. Finally, the effect of additional scatterers has been studied in honeycomb lattices of rectangular and elliptical air holes or rings in tellurium background. After extensive calculations we have found that the width of absolute PBG has been decreased in all of these modified structures. 4. Conclusion Using the numerical plane wave method, we have performed a detailed numerical analysis on photonic band structures of modified
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2D square, triangular and honeycomb lattice PCs, composed of air holes or rings with different shapes and orientations in anisotropic tellurium background. The modified structures are obtained by introducing air holes or rings with square, circular, elliptical, rectangular and hexagonal shapes into the center of each lattice unit cell. Extensive calculations reveal that the width of absolute PBG has been increased for some shapes of additional air hole or ring in the studied 2D PCs. Among all types of lattices considered here, the largest absolute PBG has been appeared in the case of square lattice of circular air rings in tellurium background with additional circular air hole at the center of lattice unit cell. These results should be helpful in designing 2D PCs with large absolute PBGs. References [1] [2] [3] [4] [5]
E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. S. John, Phys. Rev. Lett. 58 (1987) 2486. D.L. Bullock, C. Shih, R.S. Margulies, J. Opt. Soc. Am. B10 (1993) 399. J.C. Knight, J. Broeng, T.A. Birks, P.St.J. Russell, Science 282 (1998) 1476. O. Painter, R.K. Lee, A. Scherer, A. Yariv, J.D. O'Brien, P.D. Apkus, I. Kim, Science 284 (1999) 1819. [6] E. Miyai, M. Okano, M. Mochizuki, S. Noda, Appl. Phys. Lett. 81 (No. 20) (2002) 3729. [7] T.D. Happ, M. Kamp, A. Forchel, J.-L. Gentner, L. Goldstein, Appl. Phys. Lett. 82 (No. 1) (2003) 4.
[8] Cheryl M. Anderson, Konstantinos P. Giapis, Phys. Rev. Lett. 77 (1996) 2949. [9] Cheryl M. Anderson, Konstantinos P. Giapis, Phys. Rev. B 56 (1997) 7313. [10] L.F. Marsal, T. Trifonov, A. Rodriguez, J. Pallares, R. Alcubilla, Phys. E. 16 (2003) 580. [11] N. Malkova, S. Kim, T. DiLazaro, V. Gopalan, Phys. Rev. B 67 (2003) 125203. [12] T. Trifonov, L.F. Marsal, A. Rodriguez, J. Pallares, R. Alcubilla, Phys. Rev. B 69 (2004) 235112. [13] Z.Y. Li, B.Y. Gu, G.Z. Yang, Phys. Rev. Lett. 81 (1998) 2574. [14] Z.Y. Li, J. Wang, B.Y. Gu, Phys. Rev. B 58 (1998) 3721. [15] B. Rezaei, M. Kalafi, Opt. Commun. 266 (2006) 159. [16] B. Rezaei, M. Kalafi, Mater. Sci. Semicond. Process. 10 (2007) 159. [17] B. Rezaei, T. Fathollahi Khalkhali, M. Kalafi, Opt. Commun. 284 (2011) 813. [18] T. Pan, F. Zhuang, Z.-Y. Li, Solid State Commun. 129 (2004) 501. [19] T. Trifonov, L.F. Marsal, A. Rodriguez, J. Pallares, R. Alcubilla, Phys. Rev. B 70 (2004) 195108. [20] H. Kurt, D.S. Citrin, Opt. Express 13 (2005) 10316. [21] K.P. Chang, S.L. Yang, J. Appl. Phys. 100 (2006) 073104. [22] B. Rezaei, T. Fathollahi Khalkhali, A. Soltani Vala, M. Kalafi, Opt. Commun. 282 (2009) 2861. [23] K. Sakoda, Optical Properties of Photonic Crystals, Springer-Verlag, Germany, 2001. [24] D. Cassagne, C. Jouanin, D. Bertho, Phys. Rev. B 53 (1996) 7134. [25] R. Wang, X.-H. Wang, B.-Y. Gu, G.-Z. Yang, J. Appl. Phys. 90 (2001) 4307. [26] E.D. Palik, Hand Book of Optical Constants of Solids, Vol. 2, Academic Press, Boston, USA, 1998. [27] K.M. Ho, C.T. Chan, C.M. Soukoulis, Phys. Rev. Lett. 65 (1990) 3152. [28] K. Busch, S. John, Phys. Rev. E 58 (1998) 3896.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Enlargement of absolute photonic band gap in modified 2D anisotropic annular photonic crystals T. Fathollahi Khalkhali a,b, B. Rezaei a,⁎, M. Kalafi a,b a b
Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran Physics Department, University of Tabriz, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 8 December 2010 Received in revised form 27 February 2011 Accepted 1 March 2011 Available online 21 March 2011 Keywords: Photonic crystal Absolute photonic band gap Anisotropic tellurium
a b s t r a c t We analyze the absolute photonic band gap in two dimensional (2D) square, triangular and honeycomb lattices composed of air holes or rings with different geometrical shapes and orientations in anisotropic tellurium background. Using the numerical plane wave expansion method, we engineer the absolute photonic band gap in modified lattices, achieved by addition of circular, elliptical, rectangular, square and hexagonal air hole or ring into the center of each lattice unit cell. We discuss the maximization of absolute photonic band gap width as a function of main and additional air hole or ring parameters with different shapes and orientation. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Since the amazing work of Yablonovitch [1] and John [2], photonic crystals (PCs) have attracted considerable attention. Investigations show that PCs exhibit a region of the frequency spectrum where propagating modes are forbidden, called photonic band gap (PBG). PBG gives rise to unusual physical phenomena such as inhibited spontaneous emission [3] and light localization [4]. Although three-dimensional (3D) PCs suggest the most interesting idea for novel applications, 2D PCs have also been strongly studied, since they can be fabricated more easily than 3D ones and may be employed in optical and electronic devices. For in plane propagation, two types of electromagnetic modes exist according to whether the electric (E-Polarization or TM) or magnetic (H-Polarization or TE) field is parallel to the rod axes. An absolute PBG exists for a 2D PC only when PBG in both polarization modes are present and they overlap each other. A large PBG is very remedial for various applications such as defect cavities [5] optical waveguide [6] defect-mode PC lasers [7]. In order to obtain a large absolute PBG, many attempts like symmetry reduction [8–12] and anisotropy in dielectricity [13–17] have been made to enlarge the PBG in 2D PCs. Recently, it has been reported that the suggested class of the so-called annular PCs [18–21], where dielectric rods are embedded into air holes of larger radius, have a dramatic effect on the absolute band gap. More recently, we have engineered PBG in 2D PCs composed of air rings in anisotropic tellurium background, leading to large absolute PBG [22].
⁎ Corresponding author. Tel.: +98 411 3393027; fax: +98 411 334 7050. E-mail addresses: [email protected], [email protected] (B. Rezaei). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.03.006
In this paper we study the band gap properties of 2D PCs, created by square, triangular and honeycomb lattices of air holes or rings with different geometrical shapes and orientations in anisotropic tellurium background by addition of air holes or rings into the center of each lattice unit cell, named modified structures. Given lattice symmetry, we discuss the maximization of absolute PBG as a function of main and additional air hole or ring parameters with different shapes and orientation. 2. Formulation To determine the PBG in periodic dielectric structures, we study the propagation of electromagnetic waves from Maxwell's equations. In inhomogeneous dielectric materials, Maxwell's equation can be written as the following formula for the magnetic field which is called master equation [23]: " ∇×
# → 1 ω2 → → ∇ × H r = 2 H r ε r c
ð1Þ
where, the light velocity in vacuum, ω is the frequency of light and → c is → → ε r = ε r + R is position-dependent dielectric function which is → periodic with → respect to the real space lattice vector R . Since ε r is periodic, we can use Bloch's theorem to expand its inverse as a sum of plane waves: → iG : r r = ∑η G e
−1 →
ε
→
→
G
→
ð2Þ
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→ → where η G is the Fourier transform of the inverse of ε r and plays a key role in determining photonic band structure. For a given lattice → with a unit cell including ns scatterers, η G is given by Ref. [24]: ns → −1 ð jÞ → −i G : uj η G = εb δG;0 + ∑ η G e →
→
ð3Þ
j=1
→ and ηð jÞ G is the Fourier transform of each scatter of lattice unit cell → at positions uj , which is calculated analytically by the following equation: ð jÞ →
η
G
→
=
1 −1 → −iG : r → ∫ ε ð r Þe dr : Ω cell →
ð4Þ →
Here Ω is the surface of unit cell and G is a 2D reciprocal lattice vector.The orientation of main and additional noncircular air holes or rings relative to the lattice axis is defined by angles Θ and θ, respectively, as shown inFig. 3. For nonzero values of Θ and θ, we → need to replace G in ηð jÞ G by G˜ ′ and G˜ for main and additional noncircular scatterers, respectively [25]: G˜ x ˜y G
!
=
cosðθÞ sinðθÞ − sinðθÞ cosðθÞ
Gx Gy
0 1 G˜ ′ Gx cosðΘÞ sinðΘÞ @ xA= : Gy − sinðΘÞ cosðΘÞ G˜y′
ð5Þ We have considered square, triangular and honeycomb lattices of circular, elliptical, rectangular, square and hexagonal air holes or rings in anisotropic tellurium background. We investigate the modification of band gap spectrum in these lattices by addition of circular, elliptical, rectangular, square and hexagonal air hole or ring into the center of each lattice unit cell. As an example, we have shown some of the modified structures in Fig. 1(a–j), i.e. square lattice of circular air rings and honeycomb lattice of square air rings including air ring with different geometrical shapes at the center of lattice unit cell. The corresponding first Brillouin zone for lattices under consideration is sketched in Fig. 2(a–b). The anisotropic tellurium has two different principle-refractive indices as ordinary-refractive index no = 4.8 and extraordinary-refractive index ne = 6.2 over the wavelength range of 4.5 − 6.25 μm with an absorption coefficient of α ≈ 1 cm− 1 [26]. We assume that the extraordinary axis is parallel to the z axis. In this configuration, the eigenequations for the E- and H-polarization modes are the same as those for the isotropic PCs, except that the refractive indices of anisotropic tellurium are ne and no for E- and Hpolarizations, respectively. 3. Results and discussion In this study we have investigated the band spectrum of 2D PCs with different geometrical air holes or rings in anisotropic tellurium background in three kinds of modified square, triangular and honeycomb lattices, obtained by addition of circular, hexagonal, elliptical, rectangular and square air holes or rings into the center of each lattice unit cell. The photonic band structure is calculated by solving Maxwell equation using the well-known numerical plane wave expansion method [27,28]. A total of 961 plane waves were used for all structures in these calculations, which ensures sufficient convergence for the frequencies of interest. The convergence accuracy for most of the lowest photonic bands is better than 1%. Our main purpose is to study the modification of the band gap spectrum and the value of the absolute PBG when the orientation of noncircular and geometrical parameters of air holes or rings varies. In these structures the geometrical parameters and orientation of air holes or rings are treated as adjustable parameters to obtain the maximum absolute PBG.
More recently, we have investigated the photonic band structure of 2D square, triangular and honeycomb lattices of air holes or rings with circular, hexagonal, elliptical, rectangular and square shapes in anisotropic tellurium background [22]. The present work is focused on the engineering of absolute PBG in the aforementioned lattices by addition of air holes or rings into the center of each lattice unit cell. Numerical results show that the size of absolute PBG in modified triangular lattices has been decreased in comparison with the previously obtained results [22]. Therefore, in this letter the photonic band spectra of square and honeycomb lattices will be separately analyzed in the next two subsections. 3.1. Square lattice In this subsection, the results for the square lattice are presented. First of all, it should be mentioned that the safest path for calculating PBG is half side of the first Brillouin zone. However, in some cases depending on the type of photonic crystal lattice and geometric shape of scatterers, by choosing less paths, exact and accurate PBG is obtainable. In this investigation the photonic band structure of the square lattice with circular scatterers was traced along the Γ − X − M − Γ path of the Brillouin zone, while for square, hexagonal, elliptical and rectangular scatterers the half side of the first Brillouin zone were traced (Fig. 2a). These are sufficiently precise paths for calculation PBG in our mentioned structures. We begin our discussion with square lattice of circular air holes in anisotropic tellurium background. Investigations reveal that, this lattice has a complete PBG with maximum normalized width of Δωmax = 0.0271(2πc/a) and gap–midgap ratio of ωr = 6.80% at R = 0.482a, a is being the lattice constant. Here Δωmax is the frequency width of the maximum absolute PBG and ω1 and ω2 represent the corresponding lower and upper edge frequencies in units of 2πc/a, respectively. The gap–midgap ratio is defined as ωr ≡ Δωmax/ωg, where ωg is the center frequency of the PBG. Fig. 4 represents the photonic band structures of this lattice. This figure shows a complete PBG between ω1 = 0.3848 and ω2 = 0.4119 frequencies in unit of (2πc/a). Now, for further investigation of absolute PBG in the above mentioned lattice, we modify its structure by addition of a circular air hole into the center of lattice unit cell. Numerical results show that there is an absolute PBG with maximum normalized width of Δωmax = 0.0599 (2πc/a) and ωr = 11.52% at optimum geometrical parameters R = 0.491a (radius of main air holes) and r = 0.170a (radius of additional air holes). Fig. 5 depicts the dispersion relations of this structure with an absolute PBG resulted from the overlapping of the TM4-5 and TE2-3 band gaps and lies between 0.4903(2πc/a) and 0.5502(2πc/a) frequencies. It can be seen that the width of absolute PBG has been increased by addition of a smaller diameter air hole into the center of lattice unit cell. This absolute PBG is almost two times larger than the previous one. At next step, a hexagonal air hole is located into the center of lattice unit cell. After optimization of all the geometrical parameters an absolute PBG with maximum normalized width of Δωmax = 0.0521(2πc/a) and ωr = 11.75% is obtained at optimum parameters R= 0.484a, r = 0.166a (r is hexagonal side length) and θ = 0∘(rotation angel of additional hexagonal air hole). This PBG is created by overlapping of TM4-5 and TE2-3 band gaps and has been increased compared to the band gap of original square lattice. Similar to the previous cases, we add square air hole into the center of lattice unit cell. The results of numerical calculations show that there is an absolute PBG with maximum normalized width of Δωmax = 0.0503 (2πc/a) and ωr = 9.04% between TM4-5 and TE2-3 polarization band gaps at optimum geometrical parameters R = 0.492a, r = 0.167a, (the half side of square air holes) and θ = 45∘(rotation angle of square air hole). This absolute PBG between 0.5314 and 0.5817 frequencies in units of (2πc/a). Finally, the elliptical and rectangular air holes have been considered as additional air holes into the center of lattice unit cell.
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Fig. 1. Schematic representation of modified 2D square lattices of circular air rings (a–e) and honeycomb lattices of square air rings (f–j) in tellurium background with additional circular, square, hexagonal, elliptical and rectangular air rings at the center of each lattice unit cell.
Numerical results show that the width of absolute PBG has been decreased in these cases and approaches to zero by increasing the size of additional elliptical and rectangular air holes.
Now, we consider the case in which the air rings are included into the center of lattice unit cell instead of air holes. Therefore, the modified square lattices of air holes with additional circular, square,
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Fig. 4. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for square lattice of circular air holes in tellurium background with R = 0.482a.
Fig. 2. Tow-dimensional Brillouin zones for (a) square and (b) triangular and honeycomb lattices. The grey shaded area represents the irreducible Brillouin zone.
rectangular, elliptical and hexagonal air rings at the center of each lattice unit cell are obtained. Comprehensive investigation show that in three cases of hexagonal, circular and square air rings which are inserted at the center of lattice unit cell, the width of absolute PBG has been increased while it has been decreased in the case of additional rectangular and elliptical air rings at the center of lattice unit cell. The maximum normalized width of absolute PBG is 0.0710, 0.0762 and 0.0667 in units of 2πc/a, and gap–midgap ratio is 15.24%, 15.64% and 13.14% for hexagonal, circular and square air rings, respectively. The above mentioned values for PBG width are obtained at optimum geometrical parameters R = 0.485a, r2 = 0.184a (outer side length), r1 = 0.141a (inner side length) and θ = 0∘ for hexagonal air ring and
Fig. 3. Modified square lattice of rectangular air rings in tellurium background with additional square air ring at the center of lattice unit cell. The rotation angle of rectangular and square rings is denoted by Θ, θ, respectively, as the angle between the axis of the air ring cross section and the lattice axis.
R = 0.486a, r2 = 0.175a (outer radius) and r1 = 0.129a (inner radius) for circular air ring and R = 0.488a, r2 = 0.168a (outer half side length), r1 = 0.112a (inner half side length) and θ = 45∘ for square air ring. It can be seen that the largest PBG is obtained in the case of circular air ring. Fig. 6 shows the frequency spectrum of this structure at the above mentioned optimum geometrical parameters. Then we concentrate on the square lattices of circular air rings in tellurium background and investigate the effect of additional air hole or ring on PBG with different geometrical shapes into the center of lattice unit cell. The square lattice of circular air rings with inner and outer radius R1 = 0.08a and R2 = 0.482a, respectively, has an absolute PBG with maximum normalized width of Δωmax = 0.0 390(2πc/a) and gap–midgap ratio of ωr = 9.65%. In order to enlarge the width of PBG, we introduce air holes with different shapes into the center of lattice unit cell. Numerical results show that in the case of hexagonal, circular and square air holes the absolute PBG has been increased while it has been decreased for elliptical and rectangular air holes. For additional hexagonal, circular and square air holes the normalized width of absolute PBG are 0.0822, 0.0919 and 0.0668 in units of 2πc/a, with gap–midgap ratio of 15.79%, 17.71% and 8.35%, respectively. These values of maximum absolute PBG are obtained at optimum parameters R2 = 0.487a, R1 = 0.07a, r = 0.186a and θ = 0∘ for hexagonal air hole and R2 = 0.488a, R1 = 0.07a and r = 0.17a for circular air hole and R2 = 0.490a, R1 = 0.154a, r = 0.153a and θ = 45∘ for square air hole, where R1 and R2 represent inner and outer radius of main circular air ring. It is obvious that in the case of additional circular air hole the width of
Fig. 5. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for square lattice of circular air holes of radius R = 0.491a in tellurium background with additional circular air hole at the center of lattice unit cell with r = 0.170a.
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Table 1 Maximum absolute PBG width for square lattice of hexagonal air holes with optimum side length R and rotation angel Θ, in tellurium background including additional air holes or rings at the center of lattice unit cell. Additional scatterers
Geometrical parameters
Δωmax(2πc/a)
ωr ≡Δωωmax g
Circular hole Square hole Hexagonal hole Square ring Hexagonal ring
r = 0.04a r = 0:15a r = 0:20a r2 = 0:20a r2 = 0:20a
0.0300a 0.0341b 0.0314c 0.0394d 0.0416e
8.78% 10.39% 6.06% 8.24% 7.96%
a b c d e
Fig. 6. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for square lattice of circular air holes of radius R = 0.486a in tellurium background with additional circular air ring at the center of lattice unit cell with r2 = 0.175a and r1 = 0.129a.
absolute PBG has been increased almost two times in comparison with the original square lattice of circular air ring. The photonic band structure of this structure is shown in Fig. 7. At next step, the addition of air rings with different shapes into the center of lattice unit cell has been studied. Numerical results showed that in this case the width of absolute PBG has been decreased and approaches zero by increasing the size of additional air rings. Now we investigate the PBG in the modified square lattice of hexagonal air holes in tellurium background. Numerical results show that the square lattice of hexagonal air holes with side length of R = 0.50a and Θ = 0∘ has an absolute PBG with maximum normalized width of Δωmax = 0.0 220(2πc/a) and ωr = 6.52%. In order to modify this structure, we introduce air holes with different geometrical shapes into the center of lattice unit cell. Therefore, the square lattices of hexagonal air holes with additional circular, square, rectangular, elliptical and hexagonal air holes at the center of each lattice unit cell are obtained. Numerical results show that in the case of additional square, circular and hexagonal air holes, the absolute PBG has been increased while it has been decreased for elliptical and rectangular air holes. Maximum normalized width of absolute PBG and the corresponding optimum geometrical parameters for these structures are listed in Table 1. It is obvious that the width of absolute PBG has been increased in the square lattice of hexagonal air hole with additional square air hole.
θ = 0∘ θ = 0∘ r1 = 0:152a θ = 45∘ r1 = 0:08a θ = 0∘
R = 0.50a, Θ = 0∘. R = 0.49a, Θ = 15∘. R = 0.50a, Θ = 45∘. R = 0.50a , Θ = 45∘. R = 0.50a, Θ = 45∘.
Now, in above mentioned lattice the air rings with circular, square, hexagonal, elliptical and rectangular shapes are located at the center of lattice unit cell instead of air holes. Comprehensive calculations reveal that in two cases of additional hexagonal and square air rings the width of absolute PBG has been increased while it has been decreased in the case of additional circular, rectangular and elliptical air rings. Maximum absolute PBG for hexagonal and square air rings and the corresponding optimum parameters are also summarized in Table 1. It can be seen that the largest absolute PBG is obtained in the case of additional hexagonal air ring with normalized width of 0.0 416 (2πc/a). This absolute PBG is nearly two times larger than that of simple square lattice with hexagonal air holes. Now we concentrate on the square lattices of hexagonal air rings in tellurium background and engineer the PBG by introducing air holes or rings with different geometrical shapes into the center of lattice unit cell. Numerical calculations demonstrate that the square lattice of hexagonal air rings in tellurium background represents an absolute PBG with normalized width of Δωmax = 0.0 199(2πc/a) at inner and outer side length of R1 = 0.18a, R2 = 0.50a and Θ = 0∘. Similar to the previous cases, we introduce air holes with different geometrical shapes into the center of lattice unit cell to study the enlargement possibility of PBG width. Numerical results show that in the case of additional hexagonal, circular and square air holes, the size of absolute PBG has been increased while it has been decreased for elliptical and rectangular air holes. The maximum normalized width of absolute PBG is 0.0779, 0.0714 and 0.0367 in units of 2πc/a, and gap–midgap ratio is 12.15%, 11.01% and 8.12% for additional square, circular and hexagonal air holes, respectively. These values of absolute PBG are obtained at optimum parameters R2 = 0.50a, R1 = 0.218a, Θ = 0∘, r = 0.20a and θ = 45∘ for additional square air hole and R2 = 0.50a, R1 = 0.20a, Θ = 0∘ and r = 0.209a for additional circular air hole and R2 = 0.50a, R1 = 0.19a, Θ = 0∘, r = 0.22a and θ = 0∘ for additional hexagonal air hole. Then the addition of air rings with different shapes into the center of lattice unit cell has been studied. Numerical results show that in these structures the width of absolute PBG has been decreased and approaches to zero by increasing the size of additional air rings. Finally, we investigate the modification of PBG in square lattices of square, rectangular and elliptical air holes or rings in tellurium background by addition of air hole or rings into the center of lattice unit cell. Extensive calculations reveal that the width of absolute PBG has been decreased in all above mentioned modified lattices. 3.2. Honeycomb lattice
Fig. 7. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for square lattice of circular air rings with R2 = 0.488a and R1 = 0.07a in tellurium background with additional circular air hole with r = 0.17a.
In this subsection the effect of additional scatterers on PBG in honeycomb lattices of air holes or rings with different geometrical shapes in tellurium background is presented. As discussed in the previous subsection, it should be mentioned that the most reliable
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path for calculating PBG is half side of first Brillouin zone. However, in this study the photonic band structures of honeycomb lattice with circular scatterers were traced along the Γ − K − M − Γ path of the Brillouin zone which named irreducible Brillouin zone (Fig. 2b), while for square, hexagonal, elliptical and rectangular scatterers the half side of the first Brillouin zone were traced. These are completely true and accurate paths for calculation PBG in this paper for honeycomb lattice, according to mentioned geometric shape of holes or rings and background material. We start our discussion with honeycomb lattice of circular air holes in anisotropic tellurium background. Results of numerical calculations show that this structure has an absolute PBG with normalized width of Δωmax = 0.0 234(2πc/a) and gap–midgap ratio of ωr = 14.52% at optimum radius R = 0.50a. Fig. 8 represents the photonic band structures of this lattice. This figure shows a complete PBG between ω1 = 0.1492 and ω2 = 0.1726 frequencies in unit of (2πc/a). Now to study the effect of additional scatterers on PBG in this structure, we introduce circular, square, hexagonal, elliptical and rectangular air holes into the center of lattice unit cell. The obtained results show that in the case of additional hexagonal, circular, rectangular and square air holes, the size of absolute PBG has been increased while it has been decreased for elliptical air hole. For brevity, only the maximum normalized width of absolute PBG and the corresponding optimum geometrical parameters are summarized in Table 2 for those structures in which the absolute PBG has been increased by addition of scatterers. It is obvious that the largest absolute PBG is obtained for honeycomb lattice of circular air holes with additional hexagonal air hole. The frequency spectrum of this structure is shown in Fig. 9. This figure represents an absolute PBG with normalized width of Δωmax = 0.0867(2πc/a) at optimum parameters R = 0.47a, r = 0.53a and θ = 30∘ which is nearly four times larger than that of simple honeycomb lattice of circular air holes. Now, instead of air holes the air rings are included into the center of lattice unit cell. Hence, the modified honeycomb lattices of air holes with additional circular, square, rectangular, elliptical and hexagonal air rings at the center of each lattice unit cell are obtained. The numerical results show that the width of absolute PBG has been decreased in all these structures and it approaches to zero by increasing the size of additional air rings. At next step, the honeycomb lattice of circular air ring in tellurium background is considered. Numerical results show that there is no PBG in such structure. Now similar to the previous cases, at first we introduce air holes with different shapes into the center of lattice unit cell and investigate its effect on photonic band structure. The obtained results show that in the case of additional circular and square air holes, the frequency spectrum of these structures exhibit relatively large absolute PBG. While the addition of rectangular, elliptical and hexagonal air holes at the center of lattice unit cell create no PBG. The maximum normalized width of absolute PBG is 0.0627 and 0.0438 in units of 2πc/a, and gap–midgap ratio is 20.05% and 14.79% for honeycomb lattices of circular air rings with additional circular and square air holes at the center of lattice unit cell, respectively. These values of maximum absolute PBG are obtained at optimum parameters R2 = 0.475a, R1 = 0.27a and r = 0.435a for additional circular air hole and R2 = 0.48a, R1 = 0.29a, r = 0.366a and θ = 0∘ for additional square air hole. Then the air rings with different shapes are introduced into the center of lattice unit cell of honeycomb lattice with circular air ring in tellurium background. After extensive calculation we have found that the additional scatterers have no effect on creating PBG in these photonic structures. Now we investigate the effect of additional scatterers on PBG in honeycomb lattice of hexagonal air holes in tellurium background. Investigating all possible structural parameters show that the honeycomb lattice of hexagonal air holes with side length of R = 0.50a and rotation angle of Θ = 45∘ has an absolute PBG with
Fig. 8. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for honeycomb lattice of circular air holes of radius R = 0.50a in tellurium background.
maximum normalized width of Δω max = 0.0114(2πc/a) and ωr = 3.35%. Similar to the previous cases we modify the structure of this photonic crystal by introducing circular, square, rectangular, elliptical and hexagonal air holes at the center of each lattice unit cell. Comprehensive investigations show that the size of absolute PBG has been increased for additional square, circular, rectangular and hexagonal air holes while it has been decreased for elliptical air holes. Table 3 shows the maximum normalized width of absolute PBG and the corresponding optimum geometrical parameters for honeycomb lattice of hexagonal air holes with additional square, circular, rectangular and hexagonal air holes at the center of lattice unit cell. It is obvious from Table 3 that the size of absolute PBG in honeycomb lattice of hexagonal air hole with additional circular air hole is nearly five times larger than that of the simple honeycomb lattice of hexagonal air holes. Next, instead of air holes, the air rings with circular, square, hexagonal, elliptical and rectangular shapes are inserted at the center of unit cell of honeycomb lattice composed of hexagonal air holes. It has been found that the size of absolute PBG in honeycomb lattice of hexagonal air holes has been increased by addition of hexagonal and circular air rings at the center of lattice unit cell and it has been decreased in the case of additional square, rectangular and elliptical air rings. The obtained results for maximum absolute PBG and the corresponding optimum parameters for additional hexagonal and circular air rings are summarized in Table 3. It can be seen that the largest absolute PBG is obtained for honeycomb lattice of hexagonal air holes with additional hexagonal air ring at the center of lattice unit cell with normalized width of 0.0 635(2πc/a). This absolute PBG is almost six times larger than that of simple honeycomb lattice with hexagonal air holes. Now the honeycomb lattice of hexagonal air rings in tellurium background is considered. The numerical calculations show that this photonic structure represents a maximum absolute PBG with
Table 2 Maximum absolute PBG width for honeycomb lattice of circular air holes with optimum radius R, in tellurium background including additional air holes at the center of lattice unit cell. Additional scatterers
Geometrical parameters
Δωmax(2πc/a)
ωr ≡ Δωωmax g
Square hole Circular hole Rectangular hole Hexagonal hole
r = 0.37a, θ = 30∘ r = 0.462a rx = 0.37a, ry = 0.475a, θ = 0∘ r = 0.53a , θ = 30∘
0.0256a 0.0671b 0.0602c 0.0867d
5.51% 20.19% 14.96% 22.32%
a b c d
R = 0.48a. R = 0.463a. R = 0.463a. R = 0.47a.
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Table 4 Maximum absolute PBG width for honeycomb lattice of square air holes with half side length R and rotation angel Θ, in tellurium background, including air holes or rings at the center of lattice unit cell. Additional scatterers Circular hole Hexagonal hole Square hole Elliptical hole Rectangular hole Circular ring a b c d e f
Fig. 9. Dispersion relation of E-polarization (dashed line) and H-polarization (solid line) for honeycomb lattice of circular air holes of radius R = 0.47a in tellurium background with additional hexagonal air hole of side length r = 0.53a and rotation angle θ = 30∘ at the center of lattice unit cell.
normalized width of Δωmax = 0.0 170(2πc/a) and ωγ = 4.88% at inner and outer side lengths of R1 = 0.115a, R2 = 0.50a and rotation angle of Θ = 56∘. Now in the same process, air holes with shapes of circular, square, hexagonal, elliptical and rectangular are located into the center of lattice unit cell of this structure. Numerical results show that in these modified structures the size of absolute PBG has been increased for additional circular and hexagonal air holes and decreased for additional rectangular, elliptical and square air holes. The maximum absolute PBG with normalized width of 0.0 627(2πc/a) and gap– midgap ratio of 21.89% is obtained at optimum parameters R2 = 0.50a, R1 = 0.150a, Θ = 41∘ and r = 0.448a for additional circular air hole in honeycomb lattice of hexagonal air rings. The maximum absolute PBG and gap–midgap ratio are 0.0 670(2πc/a) and 22.65%, respectively, for additional hexagonal air holes which is took placed at optimum parameters R2 = 0.50a, R1 = 0.15a, Θ = 43∘, r = 0.495a and θ = 4∘. At next step, the air holes are replaced with air rings at the center of lattice unit cell. Numerical results show that in these cases the width of absolute PBG has been decreased and approaches to zero by increasing the size of additional air rings. At this step we consider another structure, i.e. honeycomb lattice of square air holes, and repeat the same processes as in previous cases. The honeycomb lattice of square air holes with half side length of R = 0.42a and Θ = 0∘ represents an absolute PBG with maximum normalized width of Δωmax = 0.0 127(2πc/a) and gap–midgap ratio of ωr = 4.99%. By introducing air holes with different shapes of circular, square, rectangular, elliptical and hexagonal at the center of lattice unit cell, Table 3 Maximum absolute PBG width for honeycomb lattice of hexagonal air holes with optimum side length R and rotation angel Θ, in tellurium background including additional air holes or rings at the center of lattice unit cell. Additional scatterers Square hole Rectangular hole Circular hole Hexagonal hole Circular ring Hexagonal ring a b c d e f
R = 0.50a, Θ = 10∘. R = 0.494a , Θ = 0∘. R = 0.50a, Θ = 43∘. R = 0.50a, Θ = 44∘. R = 0.50a, Θ = 41∘. R = 0.50a, Θ = 43∘.
Geometrical parameters ∘
r = 0.22a, θ = 0 rx = 0.365a, ry = 0.49a, θ = 0∘ r = 0.45a r = 0.496a , θ = 4∘ r2 = 0.46a, r1 = 0.13a r2 = 0.50a, r1 = 0.20a, θ = 4∘
Δωmax(2πc/a) a
0.0326 0.0502b 0.0528c 0.0527d 0.0570e 0.0635f
ωr ≡ Δωωmax g 11.06% 13.41% 18.31% 17.86% 19.00% 21.30%
Geometrical parameters r = 0.46a r = 0.456a, θ = 0∘ r = 0.396a, θ = 0∘ rx = 0.39a , ry = 0.57 , θ = 0∘ rx = 0.39a, ry = 0.45a, θ = 0∘ r2 = 0.45a , r1 = 0.091
Δωmax(2πc/a) a
0.0331 0.0383b 0.0549c 0.0295d 0.0738e 0.0366f
ωr ≡ Δωωmax g 8.71% 11.87% 15.90% 5.91% 15.50% 9.11%
R = 0.39a, Θ = 0∘. R = 0.404a, Θ = 0∘. R = 0.408a, Θ = 0∘. R = 0.43a , Θ = 0∘. R = 0.43a , Θ = 0∘. R = 0.41a , Θ = 0∘.
the modified honeycomb lattices of square air holes in tellurium background is obtained. Numerical results show that the width of absolute PBG has been increased for all shapes of additional air holes. Maximum normalized width of absolute PBG and the corresponding optimum geometrical parameters are listed in Table 4 for these structures. It is clear from Table 4 that the width of absolute PBG in honeycomb lattice of square air holes with additional rectangular air hole is almost six times larger than that of the simple honeycomb lattice of square air holes. Next, the air rings with circular, square, hexagonal, elliptical and rectangular shapes are located at the center of lattice unit cell of honeycomb lattice of square air holes. Numerical results show that the width of absolute PBG has been decreased for all additional air rings except the circular air ring. The size of maximum absolute PBG and the corresponding optimum parameters are summarized in Table 4 for additional circular air rings. It can be seen that the size of absolute PBG in the honeycomb lattice of square air holes with additional circular air ring is nearly three times larger than that of the simple honeycomb lattice of square air holes. Now the honeycomb lattice of square air rings in tellurium background is considered. Investigate all possible geometrical parameters demonstrate that there exist no PBG in this structure. Thus by adding air holes with different shapes of circular, square, hexagonal, elliptical and rectangular into the center of lattice unit cell, we study the properties of PBG in these structures. Numerical results show that the size of absolute PBG has been increased only in the case of additional elliptical air hole, while it has been decreased for other shapes of air holes. The honeycomb lattice of square air rings with additional elliptical air holes represents an absolute PBG with maximum normalized width of 0.0213(2πc/a) and ωr = 5.45% at optimum parameters R2 = 0.43a, R1 = 0.15a (R2 and R1 are outer and inner half sides of square air rings, respectively) Θ = 0∘, rx = 0.39a, ry = 0.563a (half diameter of elliptical air holes along x and y axes) and θ = 0∘. At the end, the air rings with circular, square, hexagonal, elliptical and rectangular shapes are included into the center of lattice unit cell. Numerical results show that there is no increment in the size of absolute PBG and approaches to zero by increasing the size of additional air rings. Finally, the effect of additional scatterers has been studied in honeycomb lattices of rectangular and elliptical air holes or rings in tellurium background. After extensive calculations we have found that the width of absolute PBG has been decreased in all of these modified structures. 4. Conclusion Using the numerical plane wave method, we have performed a detailed numerical analysis on photonic band structures of modified
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2D square, triangular and honeycomb lattice PCs, composed of air holes or rings with different shapes and orientations in anisotropic tellurium background. The modified structures are obtained by introducing air holes or rings with square, circular, elliptical, rectangular and hexagonal shapes into the center of each lattice unit cell. Extensive calculations reveal that the width of absolute PBG has been increased for some shapes of additional air hole or ring in the studied 2D PCs. Among all types of lattices considered here, the largest absolute PBG has been appeared in the case of square lattice of circular air rings in tellurium background with additional circular air hole at the center of lattice unit cell. These results should be helpful in designing 2D PCs with large absolute PBGs. References [1] [2] [3] [4] [5]
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