Variation of group velocity and complete bandgaps in two-dimensional photonic crystals with drilling holes into the dielectric rods

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Physica B 368 (2005) 151–156

Variation of group velocity and complete bandgaps

in two-dimensional photonic crystals with drilling holes

into the dielectric rods

Wen-Long Liu

, Tzong-Jer Yang

Department of Electrophysics, National Chiao Tung University, Hsinchu 30050, Taiwan, ROC Received 22 June 2005; received in revised form12 July 2005; accepted 12 July 2005

Abstract

Two-dimensional square lattices of square cross-section dielectric rods in air, designed with an air hole drilled into each square rod, are studied theoretically. By adjusting the shift of the hole position in the square rod in each unit cell, the dielectric distribution of the square rod will be modiﬁed. A sizable complete band gap occurs for certain structural parameters and exhibits very ﬂat photonic bands near such gap edge, which resulting in a sharp peak of density of states. In addition, the zero or small group velocities are observed in a broad region of k-space. This structure can be fabricated with materials widely used today and opens a fascinating area for applications in optoelectric devices. r2005 Elsevier B.V. All rights reserved.

PACS: 42.70.Qs; 42.25.Bs; 41.20.q

Keywords: Photonic band gap; Photonic crystals; Density of states; Group velocity

1. Introduction

First introduced by Yablonovitch[1] and John

[2] in 1987, photonic crystals (PCs) are now a fascinating issue of research. PCs are of artiﬁcial materials having the periodical modulation of dielectric structures in space and there exist photonic band gaps (PBGs) in which the

propaga-tion of electromagnetic (EM) waves in any propagating direction and polarization state is inhibited. In the PBG the spontaneous emission from the atoms or molecules can be rigorously forbidden [1]. The absence of normal modes of EM waves along certain directions provides the potential for application to various optical devices, such as resonant antennas [3], microscopic lasers

[4], and optical switches[5], etc.

The wider a PBG is, the greater the forbidden region of the frequency spectrum. Thus, the search for photonic crystals that possess wider band gaps

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is an important issue. Various methods for creating large PBGs or in increasing an existing PBG by altering the dielectric constant ðrÞ within a unit cell, have been proposed. These methods include rotating the lattices[6], using anisotropic dielectric materials [7], rotating the noncircular rods [8–10], and modifying the permittivity dis-tribution in a unit cell [11–13]. In such cases, an EM wave can be decomposed into the E- and H-polarization modes for two-dimensional (2D) photonic crystal. A complete PBG exists for 2D PBG crystal only when band gaps in both E- and H-polarization modes are present and they overlap each other. That is a PBG independent of the polarization of the EM waves. Many crystals generating band gaps for some light polarizations, but these may not overlap to produce a complete PBG [14]. It was reported that the symmetry reduction of atomconﬁguration by introducing a two-point basis set in simple 2D lattice can remarkably increase complete PBG [15], quite similar to the 3D case for diamond structure[16]. In contrast, symmetry breaking in a square lattice by changing the shape of square air rods to rectangular[17]or cylinder[18]reduces the width of complete PBGs.

In this work we study a type of square photonic lattice in two dimensions, which is formed by an air hole drilled into each square dielectric rod in air. We shift the air hole to modify dielectric distribution without changing the shape and orientation of dielectric scatterers. A sizable complete PBG occurs for certain structural para-meters and exhibits very ﬂat photonic bands near such gap edge, which results in a sharp peak of density of states. In addition, the zero or small group velocities are observed in a broad region of k-space. These small group velocities of the eigenmode cause a long optical path in this structure [19]. It brings about the optical gain enhancement or low-threshold lasing[20,21].

2. Theory

Fig. 1 displays the schematic diagram of our proposed 2D photonic band structure. The square dielectric rods with a side-length of l and dielectric

aare placed in air background with b¼1:0 at the

center of a 2D square lattice with a lattice constant, a, in the xy-plane. Another circular rod with b¼1:0 and diameter d is drilled into square

rod in each unit cell. We denote b ¼ d=l for convenience.

The electromagnetic (EM) ﬁelds with the E=H-polarization (in-pane magnetic/electric ﬁelds) in the 2D PC are governed by Maxwell’s equations:

r 1 ðrÞr HðrÞ ¼o 2 c2HðrÞ, (1)

Fig. 1. (a) Schematic diagram of proposed photonic crystals. The square dielectric rods with a side-length of l and dielectric aare placed in air background with b¼1:0 at the center of a

2D square lattice with a lattice constant, a, in the xy-plane. Another circular rod with b¼1:0 and diameter d is drilled into

square rod in each unit cell. We denote b ¼ d=l for convenience. We assume that there is a shift s of the drilled circular rod with respect to the center of the unit cell, that is s ¼ sð ^x sin g þ ^y cos gÞ, where g is the span angle of the displacement vector with respect to the y-axis, (b) the Brillouin zone with symmetry points, C, X, M, U, M0_{and X}0_.

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where HðrÞ denotes the magnetic fields; o the angular frequency; c the speed of light in vacuum, and ðrÞ the periodically modulated dielectric function. The magnetic fields and the dielectric function can be expanded in terms of Fourier series as HðrÞ ¼X G X2 l¼1 hG;lêleiðkþGÞ r, (2) ðrÞ ¼X G ðGÞeiG r, (3)

where k is the Bloch wave vector within the ﬁrst Brillouin zone, and G the 2D reciprocal lattice vector. The polarization unit vectors ^el with l ¼

1; 2 are perpendicular to ðk þ GÞ, and hG;l is the

Fourier expansion component of the magnetic ﬁelds.

The Fourier coefﬁcient ðGÞ is given by

ðGÞ ¼ 1 Acell

Z

cell

ðrÞeiG rdr, (4)

where the integration is performed over the unit cell. Here, the ﬁlling factor f, which is the ratio of the areas Ascat of dielectric scatterers in a unit cell

to the area Acellof a unit cell of square lattice, is

f ¼ l 2 a2 1 pb2 4 . (5)

For the proposed PC, ðGÞ is evaluated by

ðGÞ ¼ f aþ ð1 f Þb for G ¼ 0; ðabÞSðGÞ for Ga0:

(

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We assume that there is a shift s of the drilled circular rod with respect to the center of the unit cell, that is s ¼ sð ^x sin g þ ^y cos gÞ, where g is the span angle of the displacement vector with respect to the y-axis. The structural factor SðGÞ is then given by SðGÞ ¼ S1ðGÞ eiG sS2ðGÞ, (7) where S1ðGÞ ¼ l2 a2 Sinc Gxl 2 Sinc Gyl 2 (8)

with SincðxÞ ¼ sin x=x and

S2ðGÞ ¼ l2 a2 pb2 2 J1ðGaÞ Ga , (9)

where J1ðxÞ is the Bessel function of the ﬁrst kind,

and G ¼ jGj.

The band structures are then determined from solving the following equation:

X G0 Aðk þ G; k þ G0ÞHðG0Þ ¼o2HðGÞ (10) with AðK; K0 Þ ¼ jKjjK0j1_{ðK K}0 Þ for the E-polarization state; K K01_{ðK K}0 Þ for the H-polarization state; 8 > > > > < > > > > : (11) where K ¼ k þ G, K0¼k þ G0. 1_{ðK K}0_{Þ ¼}

1_{ðG G}0_Þ _{can be computed from solving the}

following equation: X

G00

1ðG G00ÞðG00G0Þ ¼dGG0. (12)

3. Results and discussion

All our calculations have been performed for a¼ ¼ 13:6 appropriate for galliumarsenide

(GaAs), and b¼1:0 in air. GaAs has been used

because this material exhibits fascinating optical properties in the infrared region and is representa-tive of many semiconductors. The design of this structure has many degrees of freedom which can be used to optimize the size of the gap, depending on the materials used in the fabrication. Although GaAs is used in this example, they can be replaced by other material with a different index contrast. To calculate PBGs for the EðHÞ-polarization 1521 plane waves in the Fourier expansion are used. First, the PBG structures and the corresponding density of states (DOS) of an air hole drilled into the center of each square dielectric rod in each unit cell are calculated, as shown in Fig. 2(a). The parameters in this ﬁgure are chosen as a=l ¼ 1:63, b ¼ 0:35 (corresponding to ﬁlling factor f ¼ 0:34017) and s ¼ 0. The solid (dotted)

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curves correspond to the EðHÞ-polarization. It is shown that there are four PBGs for the E-polarization and two PBGs for the H-E-polarization. However, overlap of PBG for E- and H-polariza-tion does not exist. Along the left-hand and right-hand margins of this ﬁgure the density of photonic states in arbitrary units were plotted. The eigen-frequencies for 6400 uniformly spaced values of k vectors inside the ﬁrst Brillouin zone were

calculated. In Fig. 2(b), the calculated result for s ¼ 0:11a and g ¼ 45 is illustrated. The other parameters are the same as those quoted in Fig. 2(a). A complete PBG with a gapwidth of Do ¼ 0:0415ð2pc=aÞ, and a central value, og¼0:66335ð2pc=aÞ, which is in the region of

overlap of E8 and H6 band gaps, is found. Eiand

Hi denotes the gaps that appear between the ith

and ði þ 1Þth bands, for the corresponding polar-ization. In the spectral range of this complete bandgap neither E-polarized nor H-polarized photonic states exist (DOS ¼ 0). Notably, modify-ing the position of circular hole in the square dielectric rod in air seems to lower the frequency of the ninth E-polarization band and the ﬁfth and sixth H-polarization bands at the MðM0_Þ_{point of}

the Brillouin zone that depicted in Fig. 2(a), then the overlap of E8 and H6 band gaps occurs. This

result can be understood to be due to the fact that reducing the symmetry of the dielectric distribu-tion in the square rod. Apparently, the band structure in Fig. 2(b) also exhibits very gently sloped bands near the complete PBGs edge. Thus, a sharp peak of density of states can be observed due to the ﬂat band. Since the group velocity vgof

the modes given by the slope of the dispersion curves, qo=qk, is expected to be zero or very small and correspondingly, the optical path is expected to be long. Comparison betweenFig. 2(a) and (b) shows that the zero or small group velocities are observed in a broad region of k-space as the increase of s. The ﬁfth, sixth H-polarization bands and the ninth E-polarization band become more restricted to a narrow spectral region, thus, light waves become more localized as s increased. There are k points between the M–U direction at which the sixth H-polarization and the eighth E-polarization bands are almost ﬂat. That is to say, group velocities of both mode approach to zero. Generally, the zero group velocity appears near photonic band edge only for E- or H-polarization. In this case, the zero group velocity is allowed for both E- and H-polarization simul-taneously.

An additional plot in Fig. 3 provides more information on PCs. The PBG map as the relative shift s of the drilled rod for three different directions of (a) g ¼ 0, (b) g ¼ 22:5 _{and (c)}

Fig. 2. Photonic band structures and the corresponding density of states (DOS) for two structures. The parameters in this ﬁgure are chosen as a=l ¼ 1:63, b ¼ 0:35 (corresponding to ﬁlling factor f ¼ 0:34017). (a) s ¼ 0 and (b) s ¼ 0:11, g ¼ 45_{. The}

solid and dotted curves correspond to the E- and H-polariza-tions, respectively. The shadow area marks the complete gap region.

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g ¼ 45_{. The other parameters are as those in}

Fig. 2(a). Only the ﬁrst ten-bands are involved in this map for both E- and H-polarizations. Notably, for a given g, the varying region of s is limited, i.e., only from zero to a certain value at which the outermost edge of the internal air circular rod just touches the outermost edge of the square dielectric rod at the lattice. The gap map for E-polarization shown inFig. 3(a) exhibits six large gaps. We note that E1 and E3gaps occur

over the range of the shift s within ½0; 0:199 a. Moreover, a remarkable gap H6 occurs in the

same range for H-polarization. Some other gaps only lie in the intermediate range of s. There are three complete PBGs in this conﬁguration due to the overlap of E8 with H5; E7 with H5, and E8

with H6 gaps. Comparison among Fig. 3(a)–(c)

shows that gap widths strongly depend on the shift of the air hole position. The most important result is the appearance of the overlap of E8 with H6

gap, which occurs for s in the region ½0:015; 0:18 a for g ¼ 0, ½0:016; 0:215 a for g ¼ 22:5 and ½0:014; 0:253 a for g ¼ 45, in turn. One would see this complete PBG to get larger width as the g is increased. This complete PBG is always bounded at the top by the upper boundary of the E8 gap. The lower boundary switches from H6 to

E8 gap both inFig. 3(a) and (b). Furthermore, its

bottomside shown inFig. 3(c) is wholly bounded by the lower boundary of the E8gap. In fact, since

E and H polarized modes are decoupled and are governed by different equations for a right choice of s and g.

We have also examined the case of an air hole drilled at the center of each square dielec-tric rod in air. Fig. 4 shows the PBG map as a function of the parameter b for ﬁlling factor f ¼ 0:34017. Several gaps in both E- and H-polarization appear and disappear as b is varied. We should note here that one H-polariza-tion and four E-polarizaH-polariza-tion gaps exhibit near b ¼ 0 when air hole is absent. One large complete PBG occurs due to the overlap of H6 and E8

gaps. This complete PBG starts near b ¼ 0 and ends at about 0.34. The gap size Do reaches the maximum value 0.0427ð2pc=aÞ at about b ¼ 0:19 when the same total ﬁlling factor f ¼ 0:34017 and a=l ¼ 1:69.

Fig. 3. The PBG map as the relative shift s of the drilled rod for three different directions of (a) g ¼ 0_{, (b) g ¼ 22:5} _{and (c)}

g ¼ 45_{. The other parameters are as those quoted in}_{Fig. 2}_(a).

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4. Conclusion

This study proposes two-dimensional square lattices of square cross-section dielectric rods in air, designed with an air hole drilled into each square rod. By adjusting the shift of the hole position in the square rod in each unit cell, the dielectric distribution of the square rod will be modiﬁed. The calculations show that the photonic crystal structure proposed here has a sizable complete band gap and exhibits very gently sloped bands near such gap edge, which resulting in a sharp peak of density of state. In addition, the zero or small group velocities are observed in a broad region of k-space. This property can be utilized for optical gain enhancement or low-threshold lasing.

Acknowledgements

The authors would like to thank the National Science Council of the Republic of China, Taiwan (Contract no. NSC 93-2112-M-009-010) and the Electrophysics Department, National Chiao Tung University, Taiwan, for their support. We ac-knowledge Ben-Yuan Gu for discussions.

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