Optimal higher-lying band gaps for
photonic crystals with large dielectric
contrast
Ruey-Lin Chern∗and Sheng D. Chao
Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, Republic of China
∗Corresponding author:[email protected]
Abstract: We investigate the characteristics of higher-lying band gaps for two-dimensional photonic crystals with large dielectric contrast. An optimal common band gap is attained on a hexagonal lattice of circular dielectric cylinders at relatively higher bands. The corresponding TM and TE modes exhibit simultaneous band edges, around which the frequency branches tend to be dispersionless. Unlike the fundamental band gap which usually appears between the dielectric and air bands, the optimal higher-lying gap in the present study occurs between two consecutive dielectric-like bands with high energy fill factors. The underlying mechanism is illustrated with the apparent change of eigenmode patterns inside the dielectric regions for both polarizations. In particular, the common gap region is bounded by two successive orders of Mie resonance frequencies on a single dielectric cylinder with the same geometry and material, where the Mie resonance modes show similar internal fields with the respective eigenmodes for the photonic crystal.
© 2008 Optical Society of America
OCIS codes: (160.5298) Photonic crystals; (260.5740) Resonance; (290.4020) Mie theory.
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1. Introduction
Photonic crystals have been the subject of intensive research in the past two decades [1, 2, 3]. One of the most distinguished features is the existence of photonic band gap. Formation of band gap is considered as a result of hybridization of individual Mie resonance due to single scatterers and the Bragg-like multiple scattering due to periodicity [4]. The former corresponds to a strongly localized photon state, and the latter to a nearly free photon state [5]. Nature of photonic band gap was elucidated with the variation of electrical energy inside the dielectric regions, through the use of a fill factor [6]. As two consecutive bands exhibit markedly different fill factors, the discrepancy in frequency will be large and the band gap would be significant.
Due to the vector nature of electromagnetic fields, band gaps for two-dimensional structures behave in different manners for different polarizations. The common band gap for TM and TE modes (with respect to the normal of lattice plane) is an important issue for confining the light from arbitrary orientation. A general rule of thumb was proposed to characterize the band gap features for two-dimensional crystals: TM band gaps are favoured in a lattice of isolated high dielectric region, and TE gaps are favoured in a connected lattice [7]. This rule is useful for
describing the basic features of a fundamental band gap. A compromise between an isolated and a connected lattice then leads to a common band gap for both polarizations. One typical example is the triangular lattice of air columns [8, 9]. By incorporating two geometric parameters, the band gap size can be enlarged. The connected hexagonal lattice has an optimal common band gap (over 24%) for the silicon-air structure (with the dielectric contrast 13) [10, 11]. As the optimal condition is reached, the corresponding TM and TE modes exhibit simultaneous band edges. Accordingly, the common gap width is not trimmed off for either polarization [10, 12].
The higher-lying band gap (which occurs at relatively higher frequency branches) shows a different characteristic. It is not necessary for the structure to be connected [13, 14, 15]. As the mid-gap frequency is higher, the minimum feature size is larger. This can be a very important issue in fabrication [7]. However, higher-lying band gaps are more difficult to appear [16] and receive much less attention in the past [17]. This may be attributed to a more complicated behavior at higher-order Bragg scattering. Although a genetic algorithm based on the selection of dielectric pixels can be utilized to obtain a large higher-lying band gap [18], the rule or mechanism for opening optimal gaps at higher bands still demands a further study.
In this article, we investigate the features of higher-lying band gaps for two-dimensional pho-tonic crystals. The hexagonal lattice of circular cylinders, consisting of two dielectric atoms in one unit cell, is used to seek for a large higher-lying band gap. Experimental and numerical studies of the hexagonal structures fabricated in silicon have demonstrated the pronounced band gap effect [19, 20]. In the present study, an optimal gap ratio is attained at a large di-electric contrast. The corresponding TM and TE modes exhibit simultaneous band edges, as in the case of fundamental optimal band gap. Two features associated with the higher-lying band gaps are different from the low-order ones. First, the band gap does not appear between the dielectric and air bands; instead, it occurs between two consecutive dielectric-like bands, where the field energy is strongly concentrated in the dielectric regions. Second, the frequency branches around the band edges become dispersionless or insensitive to the change of wave vectors. The corresponding modes are represented as flattened bands within small band widths for both polarizations in the whole wave vector space. The underlying mechanism for opening large higher-lying band gaps can be understood by the apparent change of eigenmode patterns between the upper and lower band edges. This feature is closely related to Mie resonances and waveguide modes on a single cylinder with the same geometry and material. In particular, the common band gap is bounded by two successive orders of Mie resonance frequencies, and the Mie resonance modes show similar internal field patterns with the respective eigenmodes for the photonic crystal.
2. Basic equations
Consider a periodic lattice of dielectric cylinders whose geometry is constant along the cylinder axis. For propagation of waves parallel to the lattice plane, the time-harmonic electromagnetic modes (with time dependence e−iωt) are described as
−∇2E=εω c 2 E, (1) −∇· 1ε∇H =ω c 2 H, (2)
for transverse magnetic (TM) and transverse electric (TE) polarizations, respectively, where E and H are field components along the cylinder axis. For periodic structures with infinite extent, it is sufficient to solve the problem in one unit cell, along with the Bloch condition
applying at the unit cell boundary, whereφ is either E or H field, k is the Bloch wave vector, and ai(i = 1, 2) is the lattice translation vector. For convenience in computation, the primitive unit cell (a hexagon) is replaced by a rectangle with the same area (cf. the right of Fig. 2). Accordingly, the lattice vectors are changed to a1= (a, 0) and a2= (a/2,
√
3a/2), where a is the lattice period.
The eigensystems (1) and (2) are solved by the inverse iteration method [10, 21], in which the
Hermitian property of the differential operators is of full use. An arbitrary distribution of fields
over the unit cell is given as the initial guess of the eigenfunction, and the Rayleigh quotients
QE= R |∇E|2dτ Rε |E|2dτ, QH= R 1 ε|∇H|2dτ R |H|2dτ , (4)
are employed to calculate the eigenfrequencies for TM and TE modes, respectively. By repeat-edly solving a matrix inversion, the solution is refined through iterations until it is converged. The Rayleigh quotients (4) are utilized not only for obtaining the solutions, but also in the anal-ysis of band gap features for different polarizations [22]. Details of the inverse iteration method can be found in Ref. [10].
r/a 0.15 0.2 0.25 εd 20 25 30 35 G a p ra tio 0 0.1 0.2 0.3
Fig. 1. Contours of the higher-lying common gap ratio for a hexagonal lattice of circular cylinders with varying the dielectric constantεdand cylinder radius r/a.
3. Results and discussion
3.1. Band gap features and localized modes
The hexagonal lattice of circular cylinders, where there are two dielectric atoms in one unit cell, is useful for studying higher-lying band gaps [7]. In order to seek for an optimal condition, two parameters have been incorporated to arrange the structures [10]. In the present study, the optimal structure is attained by increasing the dielectric contrast on one hand, and varying the cylinder radius on the other. Let the dielectric constant of the cylinder beεd, and that of the surrounding medium be unity. Figure 1 shows the contours of higher-lying common gap ratio with respect toεdand the cylinder radius r/a. The optimal condition is reached at r/a = 0.19 andεd= 26, and the corresponding band structure is plotted in Fig. 2. A common band gap is opened between the third and fourth bands for TE polarization, and between the sixth and seventh bands for TM polarization. The lower and upper band edges occur atω= 0.446(2πc/a) and 0.575(2πc/a), respectively, with a band gap width 0.129(2πc/a) and the gap to mid-gap ratio 25.3%.
TM modes TE modes F re q u e n c y ( ω a /2 π c ) 0 0.2 0.4 0.6 0.8 K Γ M Γ 1 a 2 a r
Fig. 2. Band structure for a hexagonal lattice of circular cylinders with radius r/a = 0.19
and dielectric constantεd= 26. Shaded area is the common band gap for both polarizations;
the band gap width is 0.129(2πc/a) and the gap to mid-gap ratio is 25.3%. The unit cell
and geometric parameters are shown on the right.
It is noticed that the corresponding TM and TE modes exhibit simultaneous band edges, as has been observed in the fundamental (low-order) optimal band gap for a connected hexagonal structure [10]. The common gap width is therefore not trimmed off for either polarization. Around the band edges, the frequency branches becomes flattened over the whole wave vector space, which means that the resonant modes are dispersionless or insensitive to the change of Bloch wave vector. This is a typical feature of strong resonance and has been termed as the heavy photon state [23]. In the present study, the band flattening is a consequence of large dielectric contrast, as the energy has been strongly confined within the dielectric material and the frequency becomes insensitive to the change of wave vector. This phenomenon appears in other structures as well, such as the textured metallic microcavities [24], bi-dimensional metallic mesa gratings [25], and periodic arrays of plasmonic cylinders [26].
(a) (b)
Fig. 3. Magnetic field contours of the TE eigenmodes at the pointΓfor the photonic crystal in Fig. 2. (a) eigenmode near the lower edge withω= 0.429(2πc/a), (b) eigenmode near
Figure 3 shows the TE eigenmode patterns near the band edges at the pointΓ. The fields are strongly localized within the dielectric regions, where a lump of inphase oscillation is associ-ated with the lower band edge [Fig. 3(a)], and a pair of antiphase oscillations is with the upper band edge [Fig. 3(b)]. Unlike the fundamental band gap which are likely to appear between the dielectric and air bands, the higher-lying gap in the present problem occurs between two con-secutive dielectric-like bands with high energy fill factors. The fill factor f ≡R
ε=εduda/
R
uda
(u=εε0E2+µ0H2) for measuring the energy concentration within the dielectric regions [6] does not exhibit a marked difference. In Figs. 3(a) and 3(b), f= 0.9 and 0.88, respectively.
(a) (b)
Fig. 4. Electric field contours of the TM eigenmodes at the pointΓfor the photonic crystal in Fig. 2. (a) eigenmode near the lower edge withω= 0.425(2πc/a), (b) eigenmode near
the upper edge withω= 0.618(2πc/a).
Similar localized field patterns of the respective eigenmodes are observed for TM polariza-tion as well, as shown in Fig. 4. The fill factors are still large for both modes; f= 0.79 and 0.87 for Figs. 4(a) and 4(b), respectively. However, the field localization within the dielectric regions are not so strong. This is due to the boundary condition for tangential E field, which is required to be smooth across the boundary(∂E
∂n = 0). In contrast, the tangential H field is constrained in a different manner(1
ε∂∂Hn = 0). Consequently, the H field could be more localized within the high dielectric region than the E field. In addition, the respective eigenmodes display one higher order of oscillations than those in TE polarization; one pair of oscillations is associated with the lower edge [Fig. 4(a)] and two pairs are with the upper edge [Fig. 4(b)].
The drastic change of field patterns between Figs. 3(a) and 3(b) [or Figs. 4(a) and 4(b)] ac-companies a notable difference of resonance frequency between the lower and the upper edges, which makes up the band gap width. This distinction is also identified in terms of multipole-like fields: the magnetic dipole [Fig. 3(a)] and quadrupole [Fig. 3(b)] for TE modes, and the electric quadrupole [Fig. 4(a)] and octupole [Fig. 4(b)] for TM modes. This will be more realized when the band gap features are correlated to Mie resonances and waveguide modes, as discussed in the next two subsections.
3.2. Connection with Mie resonances
The higher-lying band gap features reported in Figs. 2−4 are closely related to Mie resonance in two aspects. First, the upper and lower band edges are approximately bounded by the Mie resonance frequencies of successive two orders. The TE band edges occur near the first and second Mie resonances with the same polarization, while the TM band edges locate near the second and third Mie resonances. Second, the respective eigenmodes exhibit similar internal field patterns with the Mie resonance modes. This means that resonances on single scatterers dominate the dispersion characteristics (around the band edges) in the underlying problem. This
feature also responds to the dispersionless behavior (insensitive to the change of Bloch’s wave vector) for the corresponding frequency branches.
Frequency (ωa/2πc) A m p lit u d e co e ff ici e n t (T E ) 0 0.2 0.4 0.6 0.8 0 1 2 3 4 |a0| |a1| |b0| |b1| (a) Frequency (ωa/2πc) A m p lit u d e co e ff ici e n t (T M ) 0 0.2 0.4 0.6 0.8 0 1 2 3 4 |a1| |a2| |b1| |b2| (b)
Fig. 5. Amplitude coefficients of the scattered and internal fields for a dielectric circular cylinder of radius r= 0.19a and dielectric constantεd= 26. (a) TE polarization, (b) TM
polarization. Shaded areas correspond to the common band gap region for the photonic crystal in Fig. 2. Vertical solid and dashed lines indicate the waveguide mode frequencies related to Mie resonances.
Mie resonance is the scattering excitation of a single scatterer [27], which occurs when the wavelength matches the size of scatterer. At resonance, the scattered and internal fields may be strongly enhanced. For a dielectric circular cylinder of radius r and dielectric constantεd, the scattered and internal fields subject to an incident plane wave are given as [28]
Hscat= ∞
∑
n=−∞ anHn(kρ)e−inφ, an= in Jn(x)Jn′(x1) −√εdJn′(x)Jn(x1) √ε dHn′(x)Jn(x1) − Hn(x)Jn′(x1) , (5) Hint= ∞∑
n=−∞ bnJn(√εdkρ) e−inφ, bn= in 2i/πx √ε dHn′(x)Jn(x1) − Hn(x)Jn′(x1) , (6)for TE polarization, whereρ andφ are cylindrical coordinates with the origin at the cylinder center, x≡ kr =ωr/c, x1≡√εdx, Jn(x) and Hn(x) are the nth order Bessel function and Hankel function of the first kind, respectively, the prime denotes derivative with respect to the argument, and the dielectric constant of surrounding medium is assumed to be unity. Mie resonance cor-responds to divergence of the amplitude coefficients for the scattered and internal fields; that is, the vanishing denominator of anor bn[29]. Let r= 0.19a andεd= 26 be the same as the photonic structure in Fig. 2. The first two orders of anand bnfor TE polarization are plotted in Fig. 5(a), where the Mie resonances (located at the peak positions) occur atω= 0.382(2πc/a) andω = 0.607(2πc/a). For comparison, the common gap region for the photonic crystal in Fig. 2 is overlaid in the same plot (shaded area), showing that the common gap region is ap-proximately bounded by the two Mie resonance frequencies. The corresponding internal field is given by Hint
nm= Jn(√εdxnmρ/r)cos(nφ), where xnm is the mth zero of the denominator of
anor bn. The field patterns for H01intand H11intare shown in Figs. 6(a) and 6(b), respectively. A notable similarity is observed with the respective eigenmodes (in the dielectric region) for the photonic crystal [cf. Figs. 3(a) and 3(b)]. This feature further confirms the correlation of Mie resonances to the higher-lying band gaps in the present problem.
For TM polarization, similar expressions are given for the scattered and internal fields as [28]
Escat= ∞
∑
n=−∞ anHn(kρ)e−inφ, an= in √ε dJn(x)Jn′(x1) − Jn′(x)Jn(x1) H′ n(x)Jn(x1) −√εdHn(x)Jn′(x1) , (7)(a) (b) (c) (d)
Fig. 6. Internal field patterns of Mie resonances for a dielectric circular cylinder of radius
r= 0.19a and dielectric constantεd= 26. (a) H01intfield withω= 0.382(2πc/a), (b) H11int
field withω= 0.607(2πc/a), (c) Eint
11 field withω= 0.382(2πc/a), (d) E21int field with
ω= 0.617(2πc/a). Eint= ∞
∑
n=−∞ bnJn(√εdkρ) e−inφ, bn= in 2i/πx Hn′(x)Jn(x1) −√εdHn(x)Jn′(x1) , (8)where the amplitude coefficients are plotted in Fig. 5(b) for the same r andεd. The peaks of the second and third orders of an and bn occur atω = 0.382(2πc/a) and 0.617(2πc/a), respec-tively. As in the case of TE polarization, the common gap region is approximately bounded by the two Mie resonance frequencies. Note that the peak of a1(b1) in TM modes coincides with that of a0(b0) in TE modes, which can be realized by the relations: J0′= −J1and H0′= −H1. In addition, the connection with the zeroth order coefficients a0and b0is not obvious. This is in accordance with the observation that the band gap locates at relatively higher bands for TM polarization (cf. Fig. 2). The internal field is given by Enmint= Jn(√εdxnmρ/r)cos(nφ), as shown in Figs. 6(c) and 6(d) for E11int and E21int, respectively. As expected, the Mie resonance modes display similar field patterns with the respective eigenmodes (in the dielectric region) for the photonic crystal [cf. Figs. 4(a) and 4(b)].
3.3. Related waveguide modes
It is noticed that the Mie resonances stated above can be closely related to waveguide modes, but with mutually exchanged polarizations. For sufficiently largeεd, Mie resonance condition for TE polarization is approximated to Jn(√εdx) ≈ 0 [the vanishing denominator of Eq. (5) or (6)], which is the boundary condition for TM waveguides [30]. The frequency of TMnmwaveguide mode is given asωnmTE= xnmc/r√εd, where xnmis the mth zero of Jn(x). For the same geometry and material (r= 0.19a and εd= 26), the lowest two modes occur at ω01TM= 0.395(2πc/a) andω11TM= 0.629(2πc/a) (x01≈ 2.405 and x11≈ 3.832), which are close to Mie resonance frequencies at 0.382(2πc/a) and 0.607(2πc/a), respectively [cf. Figs. 6(a) and 6(b)].
Likewise, Mie resonance condition for TM polarization is approximated (but not as good as the TE case) to Jn′(√εdx) ≈ 0 [the vanishing denominator of Eq. (7) or (8)] for largeεd. This is the boundary condition for TE waveguides [30], and the frequency of TEnmwaveguide mode is given asωTE
nm = x′nmc/r √ε
d, where x′nm is the mth zero of Jn′(x). Accordingly, the second the third modes occur atωTE
11 = 0.302(2πc/a) andω21TE= 0.502(2πc/a) (x′11≈ 1.841 and x21′ ≈ 3.054), respectively. Note that bothωTE
11 andω21TEare not so close to Mie resonances at 0.382(2πc/a) and 0.617(2πc/a), respectively [cf. Figs. 6(c) and 6(d)]. This is due to the
weaker localization of fields (inside the dielectric) in TM polarization (compare Figs. 3 and 4),
so that TM waveguide modes are not indeed very accurate approximations to Mie resonances. Nonetheless, they serve as a convenient way for roughly estimating the Mie resonances and describing the band gap features of the underlying problem.
4. Concluding remarks
In conclusion, the higher-lying band gaps for two-dimensional photonic crystals are investi-gated, with emphasis on the resonance features and related mechanism. An optimal common band gap width [0.129(2πc/a)] with the gap to mid-gap ratio (25.3%) was attained on a hexag-onal lattice of circular cylinders (r/a = 0.19) with large dielectric contrast (εd= 26). The high permittivity can be achieved by using the ceramic as the dielectric material, whose dielectric constant may range from twenties to thirties [31]. The feature of higher band gap holds for lower index contrast as well, but is not optimal; for instance, the band gap ratio for a silicon-based structure with the same geometry is 10.4%. In the present problem, an obvious distinct feature from the fundamental band gap is the opening of gap region between two consecu-tive dielectric-like bands, where the fill factor for measuring the energy concentration does not exhibit a marked difference. The apparent change of eigenmode patterns inside the cylinders, which occurs simultaneously for TM and TE modes, is responsible for a large higher-lying common band gap. At the optimal condition, the band edges for both polarizations are nearly located at the same positions, around which the frequency branches tend to be dispersionless (flattened) due to the strong localization of fields inside the dielectric regions. It follows that the eigenmodes at the two band edges are analogous to Mie resonances and related waveg-uide modes. The field patterns (inside the cylinder) exhibit a sudden change from dipole to quadrupole oscillations (TE polarization), or from quadrupole to octupole ones (TM polar-ization), between two consecutive eigenmodes for the photonic crystal, which occur as well between two successive orders of Mie resonances.
Acknowledgments
This work was supported in part by National Science Council of the Republic of China under Contract No. NSC 96-2221-E-002-190-MY3 and NSC 97-2120-M-002-013.
