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
De-partment, National Chiao Tung University, Taiwan, for their support. We acknowl-edge Ben-Yuan Gu for discussions.
Bibliography
[1] E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[2] S. John, Phys. Rev. Lett. 58, 2486 (1987).
[3] B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, J. Appl. Phys. 87, 603 (2000).
[4] O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I.
Kim, Science 284, 1819 (1999).
[5] K. Busch and S. John, Phys. Rev. Lett. 83, 967 (1999).
[6] C. M. Anderson and K. P. Giapis, Phys. Rev. B 56, 7313 (1997).
[7] Z. Y. Li, B. Y. Gu, and G. Z. Yang, Phys. Rev. Lett. 81, 2574 (1998); Eur. Phys.
J. B 11, 65 (1999).
[8] X. H. Wang, B.Y. Gu, Z. Y. Li, and G. Z. Yang, Phys. Rev. B 60, 11417 (1999).
[9] C. Goffaux and J. P. Vigneron, Phys. Rev. B 64, 075118 (2001).
[10] N. Susa, J. Appl. Phys. 91, 3501 (2002).
[11] R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, J. Opt.
Soc. Am. B 10, 328 (1993).
[12] Min Qiu and Sailing He, J. Opt. Soc. Am. B 17, 1027 (2000).
[13] X. D. Zhang, Z. Q. Zhang, L. M. Li, C. Jin, D. Zhang, B. Man, and B. Cheng, Phys. Rev. B 61, 1892 (2000).
[14] Z. Sun and T. Stirner, Physica B 322, 323 (2002).
[15] C. M. Anderson and K. P. Giapis, Phys. Rev. Lett. 77, 2949 (1996).
[16] K. M. Ho, C. T. Chan and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990) [17] Pierr R. Villeneuve and Michel Pich0e, Phys. Rev. B 46, 4696 (1992).
[18] N. Susa, J. Appl. Phys. 91, 3501 (2002).
[19] J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, J. Appl. Phys.
75, 1896 (1994).
[20] S. Nojima, Jpn. J. Appl. Phys., Part 2 37, L565 (1998).
[21] K. Sakoda, K. Ohtaka, and T. Ueta, 4, 481 (1999).
Figure 4.1: (a) Schematic diagram of proposed photonic crystals. The square di-electric rods with a side-length of l and didi-electric ²a are placed in air background with ²b = 1.0 at the center of a 2D square lattice with a lattice constant, a, in the xy−plane. Another circular rod with ²b = 1.0 and diameter d is drilled into square rod in each unit cell. We denote β = 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 = s(ˆx sin γ + ˆy cos γ), where γ is the span angle of the displacement vector with respect to the y−axis. (b) the Brillouin zone with symmetry points, Γ, X, M, U, M0 and X0.
(a) s=0, γ = 45◦
Figure 4.2: Photonic band structures and the corresponding density of states (DOS) for two structures. The parameters in this figure are chosen as a/l=1.63, β=0.35 (corresponding to filling factor f = 0.34017). The solid and dotted curves correspond to the E- and H-polarizations, respectively. The shadow area marks the complete gap region.
(b) s=0.11, γ = 45◦
Figure 4.2: (con’t)
(a) γ = 0◦
Figure 4.3: The PBG map as the the relative shift s of the drilled rod for three different directions(γ = 0◦, 22.5◦, and 45◦). The other parameters are as those quoted in Figure 4.2(a). The black area denotes the complete band gaps.
(b) γ = 22.5◦
Figure 4.3: (con’t)
(b) γ = 45◦
Figure 4.3: (con’t)
Figure 4.4: The PBG map as a function of the parameter β for filling factor f =0.34017, s = 0.
Chapter 5
Photonic band gaps in a
two-dimensional photonic crystal with veins
This study proposes the double-hybrid-rods structure of two-dimensional (2D) pho-tonic crystals of a square lattice. A square dielectric rod connected with slender rectangular dielectric veins on the middle of each side of dielectric square rod. Some specific modes are found to be sensitive to certain structural parameters, such as the length, width the dielectric constant and the shift of the position of the veins, etc., and giving rise to a new complete PBG at lower index bands. These results could be understood by the use of band structure point of view. In particular, by carefully adjusting the structural parameters, the band structure of the photonic crystal can be substantially engineered to achieve large bandgaps.
5.1 Introduction
Since the pioneering works of Yablonovitch and John in 1987 [1, 2], 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 propagation of electromagnetic (EM) waves in any propagating direction and polarization state is inhibited. A PBG can lead to various peculiar physical phenomena [3] and providing potential applications [4, 5, 6]. 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 is an important issue.
Various methods for creating large PBGs or in increasing an existing PBG by altering the dielectric constant ²(r) within a unit cell, have been proposed. These methods include rotating the lattices [7], using anisotropic dielectric materials [8], rotating the noncircular rods [9, 10, 11], and modifying the permittivity distribution in a unit cell [12, 13]. Some research groups have successfully fabricated PCs by holographic lithography [14] that can yield two- and three-dimensional (3D) complete PBGs [15, 16]. Several PC structures consisting of rods, spheres or cubes linked by dielectric veins as a completely closed 2D or 3D structures would give a large complete band gap [17, 18, 19]. In addition, the search for 3D PBG structures based on a non-close-packed face-centered cubic lattice of spherical shells connected by thin cylindrical tubes was proposed [20].
This study proposes the double-hybrid-rods structure of 2D PCs by placing slen-der rectangular dielectric veins on the middle of each side of square rod in each unit cell. There exists one complete photonic band gap (PBG) in higher frequency band of the prototype square lattices with only square rods [21]. When extending the dielectric veins, some specific modes are found to be sensitive to certain structural parameters, such as the length, the dielectric constant and the shift of the position of the veins, etc. Then, this PBG disappears and for a proper value of vein length another complete PBG at lower index bands opens. The variation of bands near the PBG’s boundaries can be interpreted by considering the effects of Mie scatter-ing and interference of EM waves to be significantly modified and enhanced when
introducing the extra dielectric veins into each unit cell. In this study, we want to understand these effects by the use of band structure point of view. When the length of veins increases, the wavelength of resonance mode would increase too. That is the resonance frequency would decrease. Generally, the EM-field distributions bear strong resemblances to electronic orbitals and, like their electronic counterparts, could lead to bonding and anti-bonding interactions between neighboring rods [22].
The relevance (the strength) of the interactions among scattering rods is attributed to the field distribution characteristics. In terms of band structure terminology, the band center of the band reflects the resonance frequency and the band width reflects the relevance (the strength) of the interactions or the EM-field distribution charac-teristics among scattering rods. In this PC, the structural parameters (the length, width, dielectric constant and the shift of the position of the veins) are properly chosen so that the photonic band structure can be optimized. Thus, it will prove useful in designing PBGs of a variety of photonic crystals.
5.2 Theory
Figure 5.1 displays the schematic diagram of the proposed PC structure. The square dielectric rod with a side-length of l and dielectric ²a is placed in air background with ²b = 1.0 at the center of a 2D square lattice with a lattice constant, a, in the xy−plane. Another dielectric vein with ² = ²v, length h and width d is placed in each unit cell on the middle of each side of the dielectric square rod, forming composite lattices. The term δ is the crevice between the edges of square rod and vein. The shift length s is thus given by s = δ + h.
In our calculations, the band structures of the PCs were calculated using the plane-wave expansion method, described in detail in the literature [23, 24, 25]. The electromagnetic (EM) fields with the E/H−polarization ( in-pane magnetic/electric
fields ) in the 2D PC are governed by Maxwell’s equations
where H(r) denotes the magnetic fields; ω the angular frequency; c the speed of light in vacuum, and ²(r) the periodically modulated dielectric function. The magnetic fields and the dielectric function can be expanded in terms of Fourier series as
H(r) =∑
where k is the Bloch wave vector in the first Brillouin zone (FBZ), and G is the 2D reciprocal lattice vector. The polarization unit vectors ˆeλ with λ = 1, 2 are per-pendicular to (k+G) and hG,λ is the Fourier expansion component of the magnetic fields. The Fourier coefficient ε(G) is given by
²(G) = 1 Acell
∫
cell
²(r)e−iG·rdr, (5.4)
where the integration is performed over the unit cell. For structures with a unit cell including veins centered at ui, the corresponding dielectric constant is expressed as
²−1(r) = ²−1b + (²−1a −²−1b )∑ where Psq and Pv describe the probability of the square rod and veins, respectively, and R denotes the translation vector of the Bravais lattice, and
Psq(r) =
where Rsq and R(i)v are the region in the xy-plane defined by the cross-section of the square rod and the ith vein, respectively. The Fourier transforms of ²−1(r) are given by
The band structures are then determined from solving the following equation
∑
5.3 Results and discussion
The following parameters were used in the calculations: ²a = 11.4 appropriate for gallium arsenide (GaAs) at wavelength λ ≈ 1.5µm and ²b = 1.0 in air. The Fourier expansion with 625 plane waves was used to calculate the PBGs for the E/H-polarization (in-pane magnetic/electric fields) and the convergence accuracy for the several lowest photonic bands was better than 1%. This study explored the influence of the slender dielectric veins on the 2D complete PBG. As an example, three cases of the dielectric constant of veins ²v were considered, with ²v = 6, 11.4 and 16. First, the PBG structures of the prototype square lattices with only square rods were calculated, as shown in Fig. 5.2(a), the side-length of square rod fixed at l = 0.57a. The solid (dotted ) curves correspond to the E(H)-polarization. The diagram clearly shows that a complete PBG exists at higher index bands resulting from the superposition of the E8− 9 and H6 − 7 gaps. If the square rods are linked with dielectric veins at each middle side of the square rods, the influences of the length h of the dielectric vein on the PBGs is now investigated. The calculated band structures for three choices of the dielectric constant of veins are demonstrated in Fig. 5.2(b) as ²v = 6, (c) ²v = 11.4 and (d) ²v = 16, respectively. The dielectric vein has a width of d = 0.08a, and a crevice of δ=0 between the edges of vein and square rod.
We find that the higher complete PBG shown in Fig. 5.2(a) disappears in Figs.
5.2(b)–(d) and another complete PBG at lower index bands opens while the length of veins continues to increase. For ²v=6 the overlap of the H2− 3 band gap and the far wider E3−4 band gaps creates a complete PBG, with the band edges lying at the M symmetry point. On increasing the vein length substantially lowers the frequency of M symmetry point of H2 band. The same happens when ²v increases from 6 to
16. Consequently, the M point is lower than the Γ point somewhere for the case of
²v = 11.4 and ²v = 16, and then this complete PBG is bounded on the lower side by the Γ point of H2 band . On the other hand, the top edge of this complete PBG remains unchanged and lies at M point of H3 band. When it reaches ²v=16 and h = 0.19a, the Γ point of E4 band is below than the M point of H3 band; thus the complete PBG is bounded on the upper side by the Γ point of E4 band. Comparison with the variation of complete PBG boundaries of Figs. 5.2 (b)–(d) shows that the lower boundary first shifts downwards, then remains unmodified. While the upper boundary first remains unchanged, then moves downwards. It is clearly seen that the largest complete PBG occurs at h = 0.215a, namely, the veins are fully connected at the lattice unit cell boundary for the case ²v = 6. However the complete PBG reaches its maximum width with midgap frequency ωg = 0.42385(2πc/a) and the gap size ∆ω = 0.0557(2πc/a) at the intermediate value of h = 0.19a for the case
²v=11.4, and then remains unchanged where h = 0.215a. Here c is the light speed in vacuum. In particular, for ²v = 16 increasing h from 0.155a to 0.19a, we find that both the lower and upper boundaries shift towards lower frequencies. The lower boundary of this complete PBG (i.e., the lower band edge of the H-polarized gap) moves a bit faster than the upper boundary (i.e., the upper band edge of the E-polarized gap); therefore, this complete PBG becomes wider. If h continues to increase and reaches 0.215a, the complete PBG shrinks again.
What is the the key factor that leads to lower band edges at certain symme-try points and hence create a gap when the square dielectric rod is connected with slender dielectric veins? In order to clarify this issue, we calculate the spatial en-ergy distribution for the corresponding states. Figures 5.3 (a)–(c) plot the spatial distributions of the electric field intensity |E2| within a unit cell of the PC at the M point of (a) H2 band (or denoted by H(2,M )) for h=0, (b) H4 band (H(4,M ))
for h = 0, and (c) H2 band (H(4,M )) for ²v = 6 and h = 0.215a. Here we mark the states in accordance with their ordering in frequency for the prophase, namely, the initial mode H(n,M) denotes the nth band for the H-polarization mode at M point. When h = 0 it is apparent that the spatial |E2| distributions of the H(2,M ) mode always concentrates inside the square rod, exhibiting a single parallelogram-like spot; on the contrary, the high index mode (H(4,M )) possesses four spots close to the edges of square rod. While the square dielectric rod is connected with slender dielectric veins, the|E2| distribution of H(4,M ) mode in the unit cell will spread out from square rod and concentrate inside the dielectric veins, then leads to the shift of frequencies of H(4,M ) mode; consequently, the H(4,M ) mode is below the H(2,M ) and H(3,M ) modes. Figure 5.3(c) plots the spatial |E2| distribution of H(4,M ) mode (corresponding to the M point of H2 band shown in the right panel of Fig. 5.2(b)).
The energy distributions for other modes of certain symmetry points at band edges are also investigated, and the same phenomenon is observed. It is worth pointing out that field is more spreading out from the square rod owing to extending the vein length.
Figures 5.3(d) and 5.3(e) plot the band center (BC) and band width (BW) versus vein length for H2 band with ²v=6, 11.4, 16 and E4 band with ²v=6; the other parameters are as those in Fig. 5.2. The curves of BC for H2 band in Fig. 5.3(d) exhibit a plateau profile with slightly sloping at the beginning of curves, and then decline rapidly to their minimum values with 0.372, 0.356 and 0.354(2πc/a) in turn at a certain h, depending on ²v. The BC of H2 band falls off and so the complete PBG occurs (owing to H(4,M ) < H(2,M ) (or H(3,M ))). It can be understood here that the resonance frequency would decrease. Clearly, it is also seen that the curves exhibit another plateau profile at the end of curves (owing to H(4,M ) < H(2,Γ)), except for the solid one with ²v = 6. In the same figure, the BC curves of E4 band
for ²v = 16 is shown. Notably, the BC curve of E4 band decline monotonically to its minimum value around 0.456(2πc/a) when dielectric veins are fully connected (h = 0.215a). The curves of BW for H2 band of ²v=6, 11.4 and 16 versus h are shown in Fig. 5.3(e). Apparently, all of the BW curves of H2 band exhibit a similar profile to the corresponding BC curves. However, they have same value of BW in the flat region at around 0.135(2πc/a) and 0.08(2πc/a),respectively. Its existence shows that the BW is insensitive to the extension of dielectric veins. However, the curves of BW with a sharp slant because the H(4,M ) mode (i.e., the mode of top band edge of H2 band lying at M point) is sensitive to the extension of dielectric veins. Besides, this means that field energy is more spreading out from the square rod. The BW curves of E4 band (²v = 16) is also shown in this figure. Here, we have mostly paid attention to the curve of E4 band for which the complete PBG is bounded when h > 0.155a. It is clearly seen that the BW curve mount up soon from 0.046 to 0.076(2πc/a) in this region (h > 0.155a) because the E(6,Γ) mode (i.e., the mode of bottom band edge of E4 band lying at Γ point) downshifts to the lower frequency, and thus the field energy is more spreading out from the square rod.
The complete PBG is interesting issue for the extra veins formed on rods. An additional plot in Fig. 5.4 presents the PBG map as a function of length of veins, h.
The parameters were chosen as: ²a = 11.4, l = 0.57a, d = 0.08a, δ=0. Three cases of the dielectric constant of veins were demonstrated in (a) ²v = 6.0, (b) ²v = 11.4 and (c) ²v = 16.0, respectively. The gap map for E-polarization showing in Fig. 5.4(a) exhibits four large gaps. Significantly, the higher gap gradually shrinks as h increase;
nevertheless, three other gaps remain almost unchanged. For H-polarization, two remarkable gaps in this structure. The first gap varies significantly as the vein length h increases toward 0.19a, while the higher frequency gap width remains almost unchanged. As ²v increases, in Fig. 5.4(b) and (c) the higher frequency gaps for
E-and H-polarized mode shrink. Figures 5.4(a), (b) E-and (c) each shows two complete PBGs. The complete PBGs for three different dielectric constants of veins found in the higer index bands are in the range h = [0, 0.155]a for ²v=6; h = [0, 0.079]a for
²v=11.4, and h = [0, 0.059]a for ²v=16. Notably, the complete PBG spans a wider range of h for ²v=6, while the lower complete PBG results from the overlap between the first gap for H-polarized mode and the second gap for E-polarized mode. The figure shows that the lower complete PBGs are opened near: h > 0.19a for ²v=6.0;
h > 0.155a for ²v=11.4, and h > 0.145a for ²v=16.0. In these cases the complete PBG spans a wider range of h for ²v=16 than for ²v = 6, while the larger size of the complete PBG occurs for ²v=11.4 as shown in Fig. 5.4(b). When the vein lengthens up to about h = 0.155a, the bottom edge band of gap drops dramatically, but the top edge band remains almost unmodified, and consequently overlaps with the gap formed by E-polarization. Hence, the complete PBG increases in size rapidly and reaches its maximum value at h = 0.19a, and then remains almost unaltered until the vein is completely closed. Notably, for extending the length of veins substantially shrinks the complete PBG located at higher index bands, and for an appropriate vein length generates the new complete PBG at lower index bands. These results were dominated at higher index bands by E-polarization and at lower index bands by H-polarization through extending veins.
To get better insight into superior features of the hybrid structure, we investigate in detail the edge states of the complete PBGs. Here we will address the lower complete PBG that form the structure described above. In our case slender dielectric veins play a crucial role in opening the lower complete PBG, therefore we have performed two kind of evolutions. First, we have calculated the positions of edge states of the lower complete PBG for a fixed value of dielectric constant of veins. We examined the PBG structures with only square rods (i.e., h=0) to start with and
then varied the value of vein length, h. Second, we have investigated the positions of edge states of the lower complete PBG for three different h values as functions of the index of refraction of the slender dielectric veins. Figure 5.5(a) plots the evolution of edge states of the lower complete PBG as functions of the vein length, h, for ²a = ²v = 11.4. The other parameters are as those quoted in Fig. 5.2(c) (i.e., δ = 0, l = 0.57a, d = 0.08a). According to the calculation of the photonic band structures, the edge states of the lower complete PBG are H(2,M), H(3,M), H(4,M) and H(2,Γ) modes. While, the H(2,M) and H(3,M) modes are degenerate in the region given by h = [0, 0.215]a. The frequencies of these two modes and H(2,Γ) remain almost unmodified at around 0.452 and 0.396(2πc/a) in turn. The frequencies of H(4,M) decrease significantly for increasing h. As the H(4,M) mode is below H(2,M) and H(3,M) modes for h > 0.155a, the lower complete PBG is opened, and its width increases quite sharply. In the region h > 0.155a, the complete PBG is bounded on the lower side by the H(4,M) boundary, and on its upper side by the H(2,M) or H(3,M) boundary. Furthermore, the vein length increases up to about h = 0.19a, the H(4,M) mode is again lower than H(2,Γ) mode. The complete PBG is thus bounded on the lower side by the H(2,Γ) boundary, and on its upper side by the H(2,M) (H(3,M)) boundary in the region h = [0.19, 0.215]a. Notably, this complete PBG tends to increase in size dramatically in the region h = [0.155, 0.19]a, and reaches its maximum value at h = 0.19a. Then, the width of this complete PBG remain unmodified.
Figure 5.5(b) shows the positions of edge states of the complete PBGs as func-tions of the vein refractive index (in the range of 2.0≤ n ≤ 4.5) for three different length of veins (h = 0.155a, 0.19a, and 0.215a). Apparently, the appearance of the complete PBGs exhibits a triangle-like outline for h = 0.155a (indicated by the dark gray region) and two parallelogram-like outlines for h = 0.19a (shaded by vertical
solid lines) and h = 0.215a (indicated by light gray region). It is seen that the curves
solid lines) and h = 0.215a (indicated by light gray region). It is seen that the curves