1. Introduction

Three-dimensionally periodic dielectric structures, often referred to as photonic crystals, have been proposed as a means to modify the electromagnetic density of states. This concept has spawned activities devoted to uncovering new and novel aspects of these artificial crystals. The progress in photonic crystal research has been documented in special issues[1, 2], a book[3] and reviews[4–6], where several promising directions are explored. The field continues to expand as new ideas are explored and developed.

The calculation of photonic band structure is insufficient to provide a qualitative explanation of the transmission spectra. Boundary conditions and finite-size effects are important factors and gaps in the transmission spectra are found where eigenmodes are present in the spectrum. However, the modes are uncoupled from the incident wave due to a symmetry mismatch between the incoming wave and the eigenmode of the photonic crystal; that is, the eigenmodes of the photonic crystal are either symmetric or anti-symmetric with respect to operations of the group that leave the crystal unchanged and the incoming plane wave also has a definite symmetry with respect to the same operations, when the plane wave and the eigenmode have opposite symmetry, then they do not couple. Group theoretic methods can be applied to tag each band by its eigenfunction’s symmetry without resorting to the numerical computation of the eigenfunctions.

In two-dimensional lattices, where Maxwell’s equations can be reduced to a scalar form, uncoupled modes were initially experimentally identified[7, 8]. The group theoretical analysis[9–11] of two-dimensional structures was confirmed by experiments on triangular and square lattice structures[12, 13]. Uncoupled modes add a new aspect to finding novel applications for photonic crystals; they are not a property of one-dimensional photonic crystals. Two-dimensional crystals are also fabricated with micron or sub-micron lattice constants[14–17], which makes them candidates for further study to develop a number of applications using these new concepts.

By contrast three-dimensional photonic lattices are expensive and difficult to fabricate; a sub-micron lattice constant has only been realized for thin samples[18]. The theoretical calculations are somewhat more complicated because Maxwell’s equations can no longer be simplified as a set of scalar equations, but numerical calculations are well developed to cover this situation. More importantly though, the group theoretic analysis can be extended to cover this important case. Group theory provides important insights into the functionality of a particular photonic crystal design and predicts the symmetry properties of each band’s eigenfunction. It can be employed to optimize desired features and guide the development of applications. The group theoretical analysis for the vector wave equation was preformed by several authors[19, 20].

This paper is devoted examining Sakoda’s approach[9, 20] for several cases of the simple cubic (SC) lattice. This lattice has potential for applications, since a full band gap was found[21]. Here we do not cover the cases where a full gap exists; instead, we concentrate our attention to the evaluation of the band’s eigenfunction symmetry and compare our results with the transmittance obtained by the transfer matrix method. We note that other lattices, sometimes called woodpile structures, using the SC symmetry, have also been the object of theoretical[21–23] and experimental studies[18, 24] and are candidates for future experimental development of structures with sub-micron lattice constants. The analysis of Ref. [20] can also be adapted to cover these important structures.

2. Results

The simple cubic lattice of spheres is used to examine the details of the symmetries of the band eigenfunctions. In our computations the band structure was calculated when the radii of the spheres was smaller than close packed. This is not necessary for the group theory assignments, but it simplifies our numerical computations. The lengths are scaled to the lattice constant. We consider two cases: dielectric spheres embedded in a host and air-holes (i..e. spherical voids) cut out of a dielectric medium. In all cases the ratio of high dielectric constant to low dielectric constant was 13:1.

2.1 Group Theory

Assigned colors are assigned to the first seven bands in Fig. 1 based on the group theory symmetry assignments. The angular frequency is scaled by the lattice constant a and a numerical factor including the speed of light, i.e. a/2πc. We devote our attention to waves propagating in the Γ - M direction, where the symmetry is reduced and the lowest bands for the two polarizations are not degenerate. The wavevector is along the (1,1,0) axis. This is a two-fold symmetry direction, C _2v The irreducible representations are : A ₁, A ₂, B ₁, B₂, which are explained below. At the M-point the irreducible representation of the D _4h symmetry is A _1g, A _1u, B _1g, B _1u, A _2g, A _2u, B _2g, B _2u, E_g, E_u. The corresponding symmetry of the H-field vector is discussed in Ref. [20].

 

Figure 1. Photonic band structure of the infinite simple cubic lattice of air-holes. The E-method calculations use 729 plane waves for each polarization and the cubic symmetry was deformed by 1 % for direct comparison with the transmittance. The scaled radius of the air-holes is 0.495 and the dielectric constant is 13. The color and corresponding line-type scheme is discussed in the text.

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The Γ-M symmetry contains invariance under two mirror reflection operations. One is the vertical plane defined by the Γ - Z and Γ - M lines; the other is the horizontal plane defined by the Γ - X and Γ - M lines. The eigenfunctions will be symmetric or anti-symmetric with respect to these operations. We define the symmetry with respect to the E-field vector, a complex vector field amplitude.

B ₁ is symmetric with respect to the horizontal plane and anti-symmetric with respect to the vertical plane. It can be coupled to an incident S-polarized wave, which is polarized parallel to the horizontal plane; this mode is colored blue and its a dashed line. By contrast B ₂, denoted by the red colored line or the solid line, is symmetric with respect to the vertical plane and anti-symmetric with respect to the horizontal plane; it can couple with a P-polarized wave. A ₁ (the light-blue colored line or dashed dotted line) is symmetric with respect to both planes and A ₂ (the green colored line, also a dashed-dotted line) is anti-symmetric in both planes. These modes are not activated by incident plane waves. In the following we apply these assignments for the bands.

2.2 Transmission Spectrum

For corroboration of the group theory assignments we have calculated the transmittance spectrum for the SC lattice by using the transfer matrix method[25] with the same parameters. For numerical computations of the transmissivity we use the computer program developed by Pendry’s group [26]; the result for propagation along the Γ - M direction with a crystal that is 32 periods thick is given in Figure 2; the lateral direction is infinite. There is a considerable shift in the width and the depth of the gaps as the sample thickness is increased, but 32 layers provides a clear determination of the gap positions. The oscillations at low frequencies are Fabry-Perot interference due to reflections from opposite surfaces. The interference is strongly affected by the sample thickness; at low frequencies where only one band is found, the number of oscillations is used by us to verify the number of layers. Oscillations occur at the higher frequencies, but they are difficult to interpret becuase of the strong dispersion in the bands and the existence of multiply excited bands with different dispersion.

 

Figure 2. The P- and S-polarization transmittances along the Γ - M direction for the simple cubic lattice of air-holes for a sample that is 32 periods thick. The scaled radius of the air-holes is 0.495 and the dielectric constant is 13.

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To apply Pendry’s method to the Γ - M direction the unit cell is deformed. The separation of the sphere centers is √2 in the propagation direction, but is unity in the transverse directions. We modified the unit cell’s geometry to make the lateral to longitudinal length ratio 10:14. This creates a deformed SC geometry, a contraction of 1% along the longitudinal direction making the lattice parameters 0.99:0.99:1.0. This distortion does not noticably affect the band structure, which is discussed next.

2.3 Band Structure

The band structures are calculated for the 1% deformed SC lattice using the plane-wave method. The convergence was examined by using two-methods and extrapolating the truncated plane-wave expansion to an infinite number[27]. The E-method which expands the dielectric function in a truncated Fourier series, and the H-method which expands the inverse of the dielectric function in a truncated Fourier series. Generally, the extrapolation of the two methods to an infinite number of plane waves yields consistent results, but their rates of convergence differ.

 

Figure 3. The convergence of the E- and H-methods versus N ^-1/3 for bands calculated at the M-point of the Brillouin zone, where N is the number of plane waves in the truncated expansion for each polarization. The radius of the air-holes is 0.495 and the dielectric constant of the host is 13.

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To illustrate the convergence of the two methods we plot the eigenfrequencies versus N ^-1/3 in Figure 3, where N is the number of plane-waves used. The eigenfrequencies from the H-methods (shown on the left side) have a steep slope as they converge. By contrast the E-method eigenfrequencies are nearly flat as a function of N; this is especially true of the lower bands. Throughout the entire Brillouin zone the H-method band structure is qualitatively the same as that given by the E-method. However, even for 1000 plane waves the frequencies are shifted by 10% or more from those of the E-method and the extrapolated values by both methods. Therefore in our analysis we use the E-method band structure calculations. In all cases presented in this paper, the E-method has better convergence. It is interesting to note that the band structure presented for the smallest spheres in this paper exhibited the slowest rate of convergence.

2.4 Air-holes

The band structure presented in Fig. 1 was calculated by the E-method for the deformed SC lattice of air-holes with radii r = 0.495 (i.e. they are nearly touching). the symmetries of the lowest 7 bands in the Γ - M direction assigned. A direct gap opens between the second- and third-bands, but there is no common gap over all directions. The direct gap is found for both the E-and H-methods. The Γ - M direction is distinguished by the broken degeneracy of the bands. By examining the symmetry of each band we determine whether an incoming wave will be coupled to it. The validity of our analysis is checked by the structure of the transmission spectra. From the band structure calculations and group analysis two gaps are identified for the P-polarization and one for the S-polarization. The positions of the gaps are indicated in Fig. 1 and their numerical values appear in Table 1.

The corresponding transmissivity spectra distinguished by incident polarization are plotted in Fig. 2. The gaps extracted from the computations are compared with the band structure results from Fig. 1; they are also found in Table 1. The correspondence of the two is generally close. An exception is found, a small second gap is identified in the S-polarization transmissivity spectrum at around the frequency 0.27, which is not found in the band structure. This could be attributed to an interference or resonance affect. It is not simply a boundary interference affect similar to Fabry-Perot resonances, since it is also present for 8 and 16 layered samples.

2.5 Dielectric spheres

 

Figure 4. Photonic band structure of the infinite simple cubic lattice of dielectric spheres. The E-method calculations use 729 plane waves for each polarization. The scaled radius of the spheres is 0.495 and the dielectric constant is 13. The color and line-type scheme is identical to Fig. 1.

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Fig. 4 is the band structure of the deformed SC lattice composed of dielectric spheres with dielectric constant 13 and sphere radii r = 0.495. Non-intersecting spheres are not constructed in the laboratory, but this case is an interesting example of the relation between band symmetry and transmittance. In this case a direct gap opens between the fifth and sixth bands. We also note that the dispersion of the third band in the Γ - M direction is not monotonic; this is a common feature in waveguide dispersion curves. It is of interest since it yields both a negative and positive value for the group velocity at a single frequency. This feaure is also found in two-dimensional photonic crystals, but is absent from one-dimensional ones.

The transmission spectra in Fig. 5 is calculated for 32 periods. The color coding of the bands is the same as before. The gaps have positions that are in good correspondence with those predicted from the band structure in Fig. 4. Referring to Table 1, the gaps found for both polarizations agree well with one another. Fig. 5 also displays small gap-like features at other frequencies, but they are attributable to minima arising from interference between waves in the crystal.

 

Figure 5. The P- and S-polarization transmittance of the simple cubic lattice of dielectric spheres along the Γ - M direction for a 32 periods thick sample. The scaled radius of the speheres is 0.495 and the dielctric constant is 13.

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Figs. 6 and 7 are the results for a photonic lattice of dielectric spheres with radii r = 0.297. The volume fraction of spheres is low enough that no direct gap is observed, but there is strong dispersion, including the appearance of distinct nonmonotonic bands. The density of states is nonzero over the entire frequency range, which makes this case a good candidate to demonstrate the correspondence between band symmetry and the transmissivity features.

 

Figure 6. Photonic band structure of the infinite simple cubic lattice of dielectric spheres. The E-method calculations use 729 plane waves for each polarization. The scaled radius of the spheres is 0.297 and the dielectric constant is 13.

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Figure 7. The P- and S-polarization transmittance of the simple cubic lattice along the Γ - M direction of dielectric spheres for a 32 periods thick sample. The scaled radius of the speheres is 0.297 and the dielectric constant is 13.

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Fig. 6 displays large, distinct gaps expected in the transmissivity based on group theory arguments. In each case the appearance of the uncoupled A ₁ or A ₂ modes spans a portion of the gap region. The lowest two bands for each polarization are in good quantitative agreement with the transmissivity, see Table 1. Although the volume fraction is small, the appearance of large gaps due entirely to predicted uncoupled modes means that the device design parameters based on these features are not stringent; they appear over a wide range of volume fractions.

Table 1. Gaps identified from the band structure calculations and the corresponding gaps determined by the transfer matrix computations.

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3. Conclusions

Important information about eigenmode symmetry can be gleaned from the group theoretic treatment of the vector-wave band structure. Sakoda’s approach[20] is simple and accurate for assigning the eigenmode symmetries. We have focused our attention on the Γ - M direction here, but other directions can likewise be evaluated. The SC lattice transmissivity provided direct confirmation of the symmetry assignments. Our findings demonstrate excellent correspondence between the gaps determined from the symmetry assigned to the bands and the gaps in the transmittance. There were no cases where a gap due to an uncoupled mode was expected, but not found in the transmissivity. The appearance of gaps in the density of states require high volume fractions of air in the structure and as a consequence the structures are extremely fragile and difficult to fabricate. Uncoupled modes on the other hand appear over a wide range of volume fractions yielding mechanically stronger structures.

By identifying the mode-symmetries we can take advantage of their special properties. The existence of uncoupled modes can be used as a filter for the unwanted polarization or it can relax the constraint of searching for the existence of a complete band gap in applications, which desire to reflect an incident wave of a particular polarization. One potential application would be a mirror designed to bend light in a waveguide around a corner. Another application would be the identification of surface modes on the photonic crystal, whose symmetry prevents them from coupling to external radiation. There is also the tantalizing possibility that uncoupled modes can be further identified by internally exciting a dipole that is placed in the lattice; the radiation could be probed on the ouside by the surface leakage.