Engineering the bandgap of a two-dimensional photonic crystal
with slender dielectric veins
Wen-Long Liu
∗, Tzong-Jer Yang
Department of Electrophysics, National Chiao Tung University, Hsinchu 30050, Taiwan, ROC Received 21 December 2006; received in revised form 22 March 2007; accepted 7 May 2007
Available online 23 May 2007 Communicated by J. Flouquet
Abstract
This study proposes the double-hybrid-rods structure of two-dimensional (2D) photonic 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, 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. ©2007 Elsevier B.V. All rights reserved.
PACS: 42.70.Qs; 42.25.Bs; 41.20.-q
Keywords: Photonic band gap; Photonic crystals; Band structure
Since the pioneering works of Yablonovitch and John in 1987 [1,2], photonic crystals (PCs) are now a fascinating is-sue of research. PCs are of artificial materials having the pe-riodical 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 pe-culiar physical phenomena [3] and providing potential appli-cations [4–6]. 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 (r) within a unit cell, have been proposed. These methods include rotat-ing the lattices [7], using anisotropic dielectric materials [8], rotating the noncircular rods[9–11], and modifying the permit-tivity distribution in a unit cell[12,13]. Some research groups have successfully fabricated PCs by holographic lithography
* Corresponding author.
E-mail address:[email protected](W.-L. Liu).
[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– 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]. More re-cently, it was demonstrated that a 2D PBG structures with open veins can also remarkably increase complete PBG[21].
This study proposes the double-hybrid-rods structure of 2D PCs by placing slender rectangular dielectric veins on the mid-dle 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[22]. 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 scattering and interference of EM waves to be significantly modified and enhanced when
intro-0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
Fig. 1. Schematic diagram of the proposed photonic crystals. The square dielec-tric rod with a side-length l and dielecdielec-tric ais placed in air with b= 1.0 at
the center of a 2D square lattice with a lattice constant, a, in the xy-plane. An-other dielectric vein with = vand length h and width d is inserted in each
unit cell on the middle of each side of dielectric square rod, forming compos-ite lattices. The shift length s of the inserted vein is defined with respect to the edge of the square rod, where δ is the crevice between the edges of the square rod and the vein, thus s is denoted by s= δ + h.
ducing 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[23]. The relevance (the strength) of the interactions among scatter-ing 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 dis-tribution characteristics among scattering rods. In this PC, the structural parameters are properly chosen so that the photonic band structure can be optimized. Although this proposal is sim-ilar to Ref.[21]mentioned in the preceding paragraph, it has an advantage that enables us to more precisely design the band structures by utilizing additional arbitrary parameters (the di-electric constant and the shift of the position of the veins), and hence it will prove useful in designing PBGs of a variety of photonic crystals.
Fig. 1 displays the schematic diagram of the proposed PC structure. The square dielectric rod with a side-length of l and dielectric a is placed in air background with b= 1.0 at the
center of a 2D square lattice with a lattice constant, a, in the xy-plane. Another dielectric vein with = v, 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. The following pa-rameters were used in the calculations: a= 11.4 appropriate
for gallium arsenide (GaAs) at wavelength λ≈ 1.5 µm and b= 1.0 in air. In our calculations, the band structures of the
PCs were calculated using the plane-wave expansion method, described in detail in the literature [24–26]. The Fourier ex-pansion 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 ex-ample, three cases of the dielectric constant of veins v were
considered, with v= 6, 11.4 and 16. First, the PBG structures
of the prototype square lattices with only square rods were cal-culated, as shown in Fig. 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 superposi-tion of the E8–9 and H 6–7 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. 2(b) as v= 6, (c) v= 11.4 and (d) v= 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 inFig. 2(a) dis-appears in Figs. 2(b)–(d) and another complete PBG at lower index bands opens while the length of veins continues to in-crease. For v= 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 symme-try point of H 2 band. The same happens when v increases
from 6 to 16. Consequently, the M point is lower than the point somewhere for the case of v= 11.4 and v= 16, and
then this complete PBG is bounded on the lower side by the point of H 2 band. On the other hand, the top edge of this complete PBG remains unchanged and lies at M point of H 3 band. When it reaches v=16 and h= 0.19a, the point of E4
band is below than the M point of H 3 band; thus the complete PBG is bounded on the upper side by the point of E4 band. Comparison with the variation of complete PBG boundaries of
Figs. 2(b)–(d) shows that the lower boundary first shifts down-wards, 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 v= 6. However the complete PBG reaches its
maxi-mum 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 v= 11.4, and then remains unchanged
where h= 0.215a. Here c is the light speed in vacuum. In par-ticular, for v= 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 hcontinues to increase and reaches 0.215a, the complete PBG shrinks again.
What is the key factor that leads to lower band edges at cer-tain symmetry points and hence create a gap when the square dielectric rod is connected with slender dielectric veins? In or-der to clarify this issue, we calculate the spatial energy dis-tribution for the corresponding states. Figs. 3(a)–(c) plot the
(a) (b)
(c) (d)
Fig. 2. Photonic band structures for two structures: the prototype structure without the inserted veins for fixing the side-length of square rod at l= 0.57a, a= 11.4,
appropriate for GaAs material; b= 1.0 in air as shown in (a) and three choices of the dielectric constant of veins are demonstrated in (b) v= 6.0, (c) v= 11.4,
(d) v= 16.0. The other parameters are the same as those in (a) except for parameters of veins: δ = 0, d = 0.08a and h: 0.155a (left panel), 0.19a (middle panel),
0.215a (right panel). The solid and dotted curves correspond to the E- and H -polarizations, respectively. The gray area marks the complete gap region.
spatial distributions of the electric field intensity |E2| within a unit cell of the PC at the M point of (a) H 2 band (or de-noted by H(2,M)) for h= 0, (b) H 4 band (H(4,M)) for h= 0, and (c) H 2 band (H(4,M)) for v= 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 in-side 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.Fig. 3(c) plots the spatial|E2| dis-tribution of H(4,M)mode (corresponding to the M point of H 2 band shown in the right panel ofFig. 2(b)). The energy distribu-tions 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.
Figs. 3(d) and 3(e)plot the band center (BC) and band width (BW) versus vein length for H 2 band with v= 6, 11.4, 16
and E4 band with v= 6; the other parameters are as those
in Fig. 2. The curves of BC for H 2 band in Fig. 3(d) ex-hibit 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(2π c/a) in turn at a certain h, depend-ing on v. The BC of H 2 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 )), ex-cept for the solid one with v= 6. In the same figure, the
BC curves of E4 band for v= 16 is shown. Notably, the BC
curve of E4 band decline monotonically to its minimum value around 0.456(2π c/a) when dielectric veins are fully connected (h= 0.215a). The curves of BW for H 2 band of v= 6, 11.4
(a) (b)
(c) (d)
(e)
Fig. 3. The spatial distributions of the electric field intensity|E2| at the M point of (a) H 2 band (or denoted by H(2,M)) for h= 0, (b) H 4 band (H(4,M)) for h= 0, and (c) H 2 band (H(4,M)) for v= 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. The band center (d) and the band width (e) as functions of h for H 2 band with v= 6
(solid line), 11.4 (dotted line), 16 (dashed line) and E4 band with v= 16. and 16 versus h are shown inFig. 3(e). Apparently, all of the BW curves of H 2 band exhibit a similar profile to the corre-sponding BC curves. However, they have same value of BW in the flat region at around 0.135(2π c/a) and 0.08(2π c/a), re-spectively. 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 H 2 band lying at M point) is sensitive to the ex-tension of dielectric veins. Besides, this means that field energy is more spreading out from the square rod. The BW curves of
(a)
(b)
Fig. 4. The positions of the edge states of the lower complete PBG (CPBG): (a) the evolution of the edge states of the lower complete PBG as a function of vein length, h, for v= a= 11.4. The other parameters are: a= 11.4,
l= 0.57a, δ = 0 and d = 0.08a. The dark region indicates the complete PBG. H(n,Γ )(E(n,Γ )) denote the nth band for the H (E)-polarization modes at point. (b) The evolution of the edge states of the lower complete PBG as func-tions of the vein refractive index for three different vein lengths (h= 0.155a, 0.19a, and 0.215a). The vein refractive index of n= 2.45, 3.376 and 4.0 (i.e., v= 6.0, 11.4 and 16.0) are indicated by the vertical dotted lines. The
dashed–dotted lines represent the curves of H2,M(H3,M) or E2,Γmodes for different h values.
E4 band (v= 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(2π c/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) down-shifts to the lower frequency, and thus the field energy is more spreading out from the square rod.
To get better insight into superior features of the hybrid structure, we investigate in detail the edge states of the com-plete PBGs. Here we will address the lower comcom-plete PBG that form the structure described above. In our case slender
dielectric veins play a crucial role in opening the lower com-plete 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.Fig. 4(a) plots the evolution of edge states of the lower complete PBG as functions of the vein length, h, for a= v= 11.4. The other parameters are as those quoted
inFig. 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 degen-erate 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(2π c/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 re-gion 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.
Fig. 4(b) shows the positions of edge states of the complete PBGs as functions 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 (indi-cated by light gray region). It is seen that the curves of H(2,M) (or H(3,M)) and H(2,Γ ) modes are flat in the region given by n= [2.0, 4.5]. However, The H(4,M), E(4,Γ )and E(6,Γ )modes decrease monotonously for increasing n. For h= 0.155a the bottom edge state is always H(4,M)as n= [3.43, 4.4], while the top edge states of this complete PBG are H(2,M) (or H(3,M)) and E(6,Γ ) modes as n= [3.43, 4.03] and n = [4.03, 4.4] in turn. The maximum width of this gap occurs at n= 4.03. For h= 0.19a the complete PBG opens for n = 2.4 and closes above n= 4.5. When n is increased from 2.4, the top edge states are H(2,M) (or H(3,M)) modes, while the bottom edge state is H(4,M) mode. As n > 3.33 the H(4,M)mode is below the H(2,Γ ) mode, thus the complete PBG is bounded on the lower side by the H(2,Γ ) mode. Meanwhile, the E(6,Γ ) mode is below the H(2,M) (H(3,M)) mode for n > 3.65, and hence the complete PBG is bounded on the upper side by the E(6,Γ ) mode. For h= 0.215a, the edge states are the same as those
Fig. 5. Variations of ω/ωgwith the shift length s for different vein lengths h:
0.17a, 0.18a and 0.2a for (a) v= 6.0, (b) v= 11.4 and (c) v= 16.0. The
other parameters are a= 11.4, l = 0.57a and d = 0.08a.
for h= 0.19a. However, both left- and right-hand ends of the parallelogram-like outline are shifted toward lower n regime. In this case the complete PBG starts near n= 2.13 and ends at about n= 4.29. Notably, in the range of n = [3.33, 3.65] for h= 0.19a and n = [2.78, 3.43] for h = 0.215a the complete PBGs remain unchanged or vary a little. Moreover, there exists the equal maximum size of the complete PBG in the overlap re-gion of n= [3.33, 3.43] for h = 0.19a and 0.215a. This broad profile with large gap size manifests the large freedom in the choice of the structural parameters, which provide the benefit of the facilitated construction of the PCs with a large allowance of tolerance.
The influence of the shift s outward of the veins on disper-sion spectrum is demonstrated by the plot of the dependence of ω/ωg(the gap width to midgap frequency ratio) as a function
of s for different h values (0.17a, 0.18a and 0.2a) as shown inFig. 5for (a) v= 6.0, (b) v= 11.4 and (c) v= 16.0,
re-spectively. The other parameters were also chosen as a= 11.4,
b= 1, l = 0.57a and d = 0.08a. The shift length s, is given as
the sum of δ and h. The varying region of s is limited, i.e., only from s= h to s = (a − l)/2. All ω/ωg versus s curves
appear to exhibit an asymmetric profile in a finite s range. Moreover, the right-hand end of the curve in a larger vextends
to a wider region of s. For each value of s, (ω/ωg)h=0.2a>
(ω/ωg)h=0.18a> (ω/ωg)h=0.17awhen v= 6; by contrast,
(ω/ωg)h=0.17a > (ω/ωg)h=0.18a > (ω/ωg)h=0.2a when
v= 16. In the lower-v, the vein length of h= 0.2a widens
the PBG while shifting the vein towards the lattice unit cell boundary (seeFig. 5(a)). All the curves inFig. 5(b) exhibit a plateau profile at the right-hand end of the curves spanning a fi-nite region of s in which ω/ωgis insensitive to changes of s.
However, the right-hand end of the curves inFig. 5(c) declines gradually. These results can be understood to be related to the effective dielectric constant of veins in the region of s. In fact, the complete PBGs can be optimized for a right choice of h and
sfor a given v, since the complete PBGs are always governed
by the vein dielectric constant and the vein length.
As a conclusion, we have investigated in detail the photonic band structures of 2D square lattices of a square dielectric rod connected with slender rectangular dielectric veins on the mid-dle of each side of dielectric square rod. Properly adjusting the length, dielectric constant and the shift of the position of veins in the unit cell enables the large complete PBG generated from the composite structure to be achieved. Additionally, the large freedom in the choice of the structural parameters which pro-vide the benefit of the facilitated construction of the PCs with a large allowance of tolerance. The PCs can be easily fabricated and operated in the micro-wave region because a is in the order of microwave wavelengths—several mm or cm, and hence it is anticipated to be encouraged in applications to new microwave devices.
Acknowledgements
The authors would like to thank the National Science Coun-cil of the Republic of China, Taiwan (Contract No. NSC 95-2119-M-009-029) and the Electrophysics Department, National Chiao Tung University, Taiwan, for their support and to Young-Chung Hsue for his useful discussions.
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