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Variation in the Absolute Photonic Band Gap of Rods Ranging from Square to Octagonal in
Square Lattices
View the table of contents for this issue, or go to the journal homepage for more 2011 Jpn. J. Appl. Phys. 50 072002
(http://iopscience.iop.org/1347-4065/50/7R/072002)
Variation in the Absolute Photonic Band Gap of Rods Ranging
from Square to Octagonal in Square Lattices
Chung-An Hu1, Kuo-Pin Chang1, Su-Lin Yang1, Lin Fang Shen2, Jin-Jei Wu3, and Tzong-Jer Yang1;3
1Department of Electrophysics, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China 2Electromagnetic Academy, Zijingang Campus, Zhejiang University, Hangzhou 310058, People’s Republic of China 3Department of Electrical Engineering, Chung Hua University, Hsinchu 30012, Taiwan, Republic of China
Received March 25, 2011; revised April 30, 2011; accepted May 10, 2011; published online July 20, 2011
The band structures and field patterns of dielectric rods in square lattices are calculated using the plane-wave method. The rods with various cross-sectional shapes from square to octagonal at a fixed filling-factor are constructed to assess the geometry effect of photonic crystals to their band gap properties. Analytical results indicate that the corner profiles of rods significantly affect theE- and H-polarization bands in resonance frequency and field distribution. The absolute photonic band gap is closed in the square lattice when square dielectric rods are replaced with octagonal dielectric rods. # 2011 The Japan Society of Applied Physics
1. Introduction
Periodic dielectric structures (photonic crystals) have garnered considerable interest in the recent decade owing to their ability to prohibit certain frequency-ranged electro-magnetic (EM) waves from propagating in such structured dielectric media.1–3) These frequency regions are called
photonic band gaps (PBGs), and are analogous to electronic band gaps resulting from periodic electrostatic potentials in crystals. The polarization-independent PBGs, called absolute PBGs, occur when the band gaps of both H- and E-polarization modes overlap. Absolute PBGs can control spontaneous light emissions for non-threshold semiconduc-tor lasers.4,5)
The fabrication and simulation of photonic crystals in the infrared (IR), ultraviolet (UV), and visible regions have received considerable interest.6–11) However, the greatest constraint to large PBG width is the degeneration of photonic bands at the highly symmetrical points in the Brillouin zone. Several approaches have been developed to reduce band degradation to obtain large absolute PBGs, which requires varying the contrast of the dielectric contrast ratio, designing the lattice element, tuning the filling ratio and reducing structural symmetries.12–17)As a rule of thumb, the formation of band gaps in a photonic crystal is attributed to the arrangement of connected veins and the isolated dielectric material.18–20) This observation suggests that isolated high-dielectric regions are favorable for large E-polarized band gaps, while connected veins are favorable for large H-polarized band gaps. Thus, an absolute PBG should be favored in photonic crystals with highly dielectric regions that are both isolated and connected.
As is well known, although appearing in a square lattice when square dielectric rods are used, the absolute PBG vanishes when these square rods are replaced with circular or hexagonal dielectric rods.21,22) However, how this
variation occurs has not been examined. This study first demonstrates that using octagonal rods in a square lattice cannot open absolute PBGs. Why such distinct outcomes exist from square and octagonal rods in a square lattice is an interesting question. The shape and boundary of rods may
markedly influence band structure. To elucidate how rod shape and boundary affect PBG formation in a square lattice, this study reshapes the corners of initial square rods gradually, forming all the way to octagonal rods with a fixed filling factor. Band structures and field patterns are estimated using the plane-wave method. The effects of rod shape on E- and H-polarization modes are then determined. Additionally, the center and width of bands associated with the absolute PBG of square dielectric rods in a square lattice are calculated.
2. Theory
The photonic band gap can be obtained by solving Maxwell’s equations. Assume a source free, time-invariant, and nonpermeable ( ¼ 0) space; thus, Maxwell’s equa-tions can be written in terms of magnetic fieldH as a master equation, r 1 "ðrÞr HðrÞ ¼!2 c2 HðrÞ; ð1Þ
whereHðrÞ is the magnetic field, "ðrÞ is a position-dependent dielectric constant, ! is angular frequency, and c is the speed of light in a vacuum. For a periodic system, the magnetic field,HðrÞ, and dielectric function, "ðrÞ, can be expressed as sums of plane waves:
HðrÞ ¼X G X ¼1;2 hG;^eeiðkþGÞr; ð2Þ "ðrÞ ¼X G "ðGÞeiGr; ð3Þ
where hG;is a coefficient of theH component, k is the wave vector in the Brillouin zone, and G is the reciprocal-lattice vector. Two independent polarizations characterized by unit vectors ^eð ¼ 1; 2Þ are perpendicular to the propagation vector (k þ G). Under the Fourier transform, the coefficient of "ðGÞ is defined as "ðGÞ ¼ 1 Acell Z cell "ðrÞeiGrdr; ð4Þ where Acell is the area of a primitive lattice cell. Thus, eq. (1) can be expressed in a matrix form as
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Japanese Journal of Applied Physics 50 (2011) 072002 REGULAR PAPER
X G0 jk þ Gjjk þ G0j ^e2 "1G;G0 ^e20 ^e2 "1G;G0 ^e10 ^e1 "1G;G0 ^e20 ^e1 "1G;G0 ^e10 " # h1;G0 h2;G0 ¼!2 c2 h1;G h2;G ; ð5Þ
where "1G;G0 ¼ "1ðG G0Þ is the inverse of matrix
"ðG G0Þ. The eigenvalue equation, eq. (5), can then be solved using the matrix diagonalization method.
For a two-dimensional (2D) photonic crystal, an electro-magnetic (EM) wave can be decomposed into E-polarized (the electrical field is parallel to the z-axis) and H-polarized (the magnetic field is parallel to the z-axis) modes. Calculations are restricted to the case in which wave vectors of eigenmodes are on the 2D x–y plane and are uniform in the z-direction. The dielectric constant is given by
"ðGÞ ¼ "a f þ "b ð1 f Þ for G ¼ 0 ð"a "bÞ SðGÞ forG 6¼ 0
; ð6Þ
where "a and "b are the dielectric constants of rods and the background medium, respectively. Filling factor f is the cross-sectional area fraction of rods in a primitive unit cell. The structure factor is denoted by
SðGÞ ¼ 1
Acell Z
Rod
eiGrdr: ð7Þ
Here, the integral is over the cross section of rods in a unit cell of the lattice. In this study, the structural factor for each corner-cutting structure can be derived from polygonal rod concepts. For a polygonal rod, this study considers a polygonal rod with N sides; the coordinate of the jth vertex is denoted byPj¼ ðxj; yjÞ. According to the Stokes theorem, the structural factor for a polygonal rod can be written as
SðGÞ ¼ XN j¼1 iyjeiGCj Gx sinðG SjÞ G Sj for Gy¼ 0 XN j¼1 ixjeiGCj Gy sinðG SjÞ G Sj for Gx¼ 0 XN j¼1
2i^ezðG SjÞeiGCj G2 sinðG SjÞ G Sj for Gx6¼ 0; Gy6¼ 0 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : ; ð8Þ with Cj Pjþ1þ Pj 2 ; Sj Pjþ12 Pj ; x xjþ1 xj; y yjþ1 yj:
The band structures for a photonic crystal are calculated using the standard plane-wave method. In this study, 841 plane waves were adopted, and computational errors in the E- and H-polarized modes for each structure were estimated at <1%.
3. Results and Discussion
Figure 1 shows the cross-sectional geometry of the octago-nal rod formed by cutting the corners of a square rod. The lengths of the cut-corner edge and rod central side are
designated as b and l, respectively. Each photonic crystal with exact, inexact octagonal, or square rods has the same optimal filling factor of f ¼ 0:45. The difference in the size of the cut corner edge between the exact octagonal rod and square rod is l8=pffiffiffi2, where l8 is the side length of the exact octagonal rod. The dielectric constant of the rod is "a¼ 12:96, corresponding to that of gallium arsenide (GaAs) or silicon at 1.55m, and the dielectric constant for the background material is taken as "b¼ 1 for air or the vacuum. In this study, the corners are cut from the square rod to generate an octagon by 15 steps with an increment of ¼ l8=ð15 pffiffiffi2Þ. For convenience, the size of cut corner edge is denoted as b ¼ N , for N ¼ 0; 1; . . . ; 14; 15.
Figures 2(a) and 2(b) show the photonic band structures associated with the square rods (b ¼ 0 ) and octagonal rods (b ¼ 15 ) in square lattices, respectively, with a filling factor f ¼ 0:45. In Fig. 2, the solid curves represent E-polarized modes and dotted curves represent H-polarized modes. The dispersion curves are traced along the T –X–M–T path in the first Brillouin zone of square lattices. The square dielectric rods in square lattices generate an absolute PBG occurring where the E8and H6 gaps overlap, where Eior Hiis the gap between the ith and (i þ 1)th bands for the corresponding polarizations [Fig. 2(a)]. The absolute gap center is at 0:628ð!a=2cÞ and the absolute PBG width is 0:036ð!a=2cÞ. Conversely, the absolute PBG closes when the square dielectric rods are replaced with octagonal dielectric rods [Fig. 2(b)], suggesting that the four corners of square rods may markedly influence band structure.
To determine why such a difference exists in appearance of the absolute PBG between square rods and octagonal rods in the square lattice, the corners of square rods were cut to form an octagonal rod gradually while the filling factor remained constant. Figure 3 shows the dependence of the gap map on the cutting step N. Here this study only considers the frequency range of interest of 0:55 0:7ð!a=2cÞ. Analytical results indicate that the widths of E8and H6 gaps decline simultaneously as N increases. The variation in frequencies of air bands (E9 and H7 bands) is larger than that of dielectric bands (E8 and H6 bands), and thus dramatically reduces absolute band gap width.
Fig. 1. Representation and cross-sectional geometry of rods.
C.-A. Hu et al. Jpn. J. Appl. Phys. 50 (2011) 072002
To examine the effect of cutting the edges on E-polariza-tion modes, Figs. 4(a)–4(d) plot the field patterns for structures of N ¼ 0, 8, 11, and 15, respectively. At the M-symmetrical point, field patterns of each figure are displayed for the E8 (left) and E9 (right) bands. The field distributions of the E8band are generally independent of N; however, the amounts of field energy inside rods increase as N increases. For field patterns of the E9 band, the shape of rods strongly affects the field distribution and the amount of field energy. When N increases, the fields near the boundary of a rod are expelled from dielectric regions and the amount of field energy decreases. Because the amount of field energy inside dielectric rods is closely related to band frequency, the difference in the amount of field energy between E8 and E9 bands may lead to the absence of the E-polarized gap.
The perspective of the band structure is adopted to understand why the absolute PBG appears in square rods, but is disappear in octagonal rods. For a given band, the band center (BC) reflects the resonant frequency of modes, while band width (BW) reflects the strength of the interaction of EM waves among rods. These band-structure concepts are similar to those in the A linear combination of atomic orbitals (LCAO) method.23–25)Here, the BC and BW are defined as
FiðBCÞ ¼Fi;maxþ Fi;min2 ;
FiðBWÞ ¼ Fi;max Fi;min ð9Þ
where Fi;maxand Fi;mindenote the maximum and minimum frequencies of the ith band in the first Brillouin zone,
respectively. Here, the E- and H-polarized bands associated with the absolute PBG in the photonic crystal are utilized. Figures 5(a) and 5(b) show the BC and BW of the E8and E9 bands as a function of N, respectively. The BC of the E9 band varies faster than that of E8 band when N exceeds 4. These analytical results demonstrate that the effect of the cutting edge on the resonance frequency of the E9 band is stronger than that on the E8 band, and the resonance frequencies for both bands gradually approach each other and move closer as N increases. Furthermore, this study examines BW to investigate how field energy concentrates inside dielectric regions and influences band structures. The BW of the E9 band when N increases is markedly different than that of the E8band. The BW for the E8band is almost independent of N, implying that field energies are localized in the dielectric rods. However, the BW of the E9 band exhibits a large variation and increases to 137% at N ¼ 11. These analytical results likely indicate that the partial fields of the E9band are expelled from the dielectric rods when the cutting edge of the rod changes. Although the BC and BW of the E8 band are insensitive to the shape or boundary of rods, the BC decreases and BW increases for the E9 band close the E-polarized gap when N exceeds 11. Figures 6(a) and 6(b) show the BC and BW associated with the H6and H7bands varying with N. The BC of the H6 band varies slightly with N; however, the BC of the H7band varies markedly, indicating that the resonance frequency of H7band depends strongly on rod shape. Moreover, the BWs for both H6 and H7 bands have large variations with N and both increase by roughly 0.017 when N ¼ 11. Thus, the fields propagated in the x–y plane cannot be localized well inside the dielectric region when the cut edge of the rod changes. Notably, the fields of the H-polarization modes concentrated in the rods are closely related to the air-space size among the rods.13) This study fixes the filling factor
throughout calculations, and the air-space size among rods consequently varies with N. Therefore, both bands have the same BW increment is acceptable. Because of the decrease in resonance frequency of the H7band and increase in BWs for both the H6and H7 bands, the H-polarized gap may close.
4. Conclusions
Exactly why the absolute PBG appears in square rods but closes in octagonal rods is investigated from the aspect of
Fig. 3. The gap map as a function of N. The corresponding size of cutting edges for each structure is N .
(a)
(b)
Fig. 2. Photonic band structures for (a) square rods (b ¼ 0) and (b) octagonal rods (15 ) in the square lattice with a filling factor of 0.45.
band structure and field patterns. The perspective of band structure is helpful in understanding the formation of the PBG in photonic crystals. This study employed the plane-wave method to calculate band structures and field patterns. The corners of square rods were changed step by step to fabricate octagonal rods with a fixed filling factor throughout calculations. Analytical results for the E-polarized gap (E8 gap) indicate that the cutting edges of rods strongly affect the E9band in the resonance frequency and field distribution
inside rods. The decrease in resonance frequency and increase in band width caused the E-polarized gap to close. Moreover, the H-polarized gap (H6gap) was closely related to the cutting edges of rods. The decrease in resonance frequency of the H7 band and increase in bandwidths of both H6 and H7bands caused the H-polarized gap to close. Accordingly, the absolute PBG is closed in the square lattice when square dielectric rods are replaced with octagonal dielectric rods.
(a)
(b)
Fig. 5. (a) Band center and (b) band width associated with the E8and E9
bands as a function of N.
(a)
(b)
Fig. 6. (a) Band center and (b) band width associated with the H6and H7
bands as a function of N.
(a) (b)
(c) (d)
Fig. 4. Field patterns of E-polarized modes inside dielectric rods when (a) N ¼ 0, (b) N ¼ 8, (c) N ¼ 10, and (d) N ¼ 15, for the E8band and at the M-symmetrical point.
C.-A. Hu et al. Jpn. J. Appl. Phys. 50 (2011) 072002
Acknowledgments
The authors would like to thank the National Science Council of the Republic of China, Taiwan (Contract No. 009-019-MY2 and NSC-98-2112-M-216-001), and the National Science Foundation of China (Contract No. 60531020) for financially supporting this research.
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