## Abstract

The probability to realize a full photonic band gap in two-dimensional birefringent photonic crystals can be readily manipulated by introducing symmetry reduction or air holes in the crystal elements. The results lie in either creation of new band gaps or enlargement of existing band gaps. In particular, a combination of the two processes produces an effect much stronger than a simple summation of their individual contributions. Materials with both relatively low refractive index (rutile) and high refractive index (tellurium) were considered. The combined effect of introduction of symmetry reduction and air holes resulted in a maximum enlargement of the band gaps by 8.4% and 20.2%, respectively, for the two materials.

©2008 Optical Society of America

Photonic crystals have been attracting a lot of attention in recent years because of their high potential in the realization of novel nano-optical devices. A variety of applications have been proposed by making use of, for example, photonic crystal cavities [1] and waveguides [2]. Due to the comparatively easy fabrication process, two-dimensional (2D) photonic crystal structures have found applications in diverse situations such as photonic crystal fibers [3, 4], cavity lasers [5], multi channel drop-filters [6] and feed back mirrors in laser diodes [7]. In most cases, the devices are assumed to work only for one of the two possible polarization modes. However, realization of photonic band gaps for both transverse electric (TE) and transverse magnetic (TM) modes can improve the device performances by reducing the optical loss inside defect cavities and defect waveguides, since the leakage of both the polarizations can be avoided [8]. This requires a sufficient overlap between both TE and TM gaps.

The general approach to realize a full photonic band gap (for both TE and TM modes) is to rely on the manipulations of device characteristics such as refractive index ratio and crystal geometry [9–19]. A choice among the possible approaches is to utilize birefringent dielectrics [10,11]. In 1998, Li *et al*. made use of tellurium as anisotropic material for the realization of 2D photonic crystals. Their analysis shows that uniaxial materials can indeed realize a full photonic band gap when arranged in a periodic configuration [10]. However, as we will show later, this is not obvious for a birefringent materials with relatively low refractive index such as rutile, which is more suitable for optical devices. Another idea to increase the possibility of having a full band gap makes use of symmetry reduction. In 1996, Anderson *et al*. introduced this idea for photonic crystals and they succeeded in realizing a full band gap [12]. More recently, Trifonov *et al*. [13] showed that symmetry reduction can affect the photonic band gap. Furthermore, it was also noticed recently that a coating of higher refractive index material can increase the full band gap because of the modification of the effective refractive index [17].

Here we want to identify the relative roles played by the three approaches mentioned above and their limitations in the realization of a full photonic band gap. Besides, we would then show how their combination can maximize the band gap. The materials chosen in our investigation are tellurium and rutile. The reason is both, to perform a comparison between two different birefringent materials, and to verify if a relatively low refractive index material such as rutile can develop a full band gap.

We have utilized the PlaneWave Expansion method to simulate a hexagonal configuration of birefringent rods in air. The number of plane wave utilized depends on the simulated structure, from a minimum of 16384 to a maximum of 262144 plane waves. The convergence accuracies of the results are better than 1%. Three different configurations, shown in Fig. 1, are considered to exploit the full band gap. The symmetry reduction was realized by considering a hexagonal structure of rods with two different radii *r*
_{1} and *r*
_{2}, as illustrated in Fig. 1(a). The radius ratio, defined as *β* = *r*
_{2}/*r*
_{1}, refers to the amount of symmetry reduction, where *β* = 1 means no reduction of symmetry. The second configuration is where there is no reduction of symmetry, but air holes are drilled in the rods to create refractive index variation, as illustrated Fig. 1(b). Here *r*
_{1} and *r*
_{2} define the outer and the inner radii, respectively, of the holed rod. The radius ratio in this case, defined as *β*=(*r*
_{1}-*r*
_{2})/*r*
_{1}, gives a parameter for the measure of variation in refractive index, where *β* = 1 means no air holes. The third configuration is a combination of the two, shown in Fig. 1(c), where both symmetry reduction and air holes are considered together. In this case, apart from the radius ratio *β* = *r*
_{2}/*r*
_{1}, we also define two more radius ratios *β*
_{1}=(*r*
_{1}-*r*
_{3})/*r*
_{1}, and *β*
_{2}=(*r*
_{2}-*r*
_{4})/*r*
_{2}. Here, *β*
_{1} = 1 means no holes in one set of rods, while *β*
_{2} = 1 means no holes in the other set of rods. Beside the radius ratios *β*, *β*
_{1} and *β*
_{2}, the refractive index and the filling ratio *f* are also needed to obtain the band gaps in each configuration.

Photonic gaps in 2D materials can correspond to both TE and TM groups of solutions but only when they overlap with each other one can talk about a full band gap. It is interesting to notice that the TM gap is favored in geometrical configurations showing isolated regions of high refractive index (*e.g.*, rods in air) whereas the TE gap is favored in connected regions of high refractive index. Therefore, it is not obvious to construct an optimal geometry to realize an overlapping of both TE and TM gaps. In fact, irrespective of the radius, the refractive index and the Bravais lattice, a 2D configuration of dielectric rods in air cannot produce a full photonic band gap. Therefore, it is necessary to develop some alternative approach to realize full band gaps.

Owing to the two different refractive indices shown by the birefringent materials for two different polarizations of light, photonic crystals made of these materials may realize a full band gap [10, 11]. When the electric field is oriented along the extraordinary axis of a birefringent material, the refractive index would be *n*
* _{e}*, and it would be

*n*

*for perpendicular directions. We can then choose a 2D photonic crystal made of anisotropic dielectric rods having the axis parallel to the extraordinary axis of the material. This choice will make TE and TM modes feel*

_{o}*n*

*and*

_{o}*n*

*, respectively. The main consequence would be the relative shift of the gaps associated with the two polarizations so that we can expect a higher chance to obtain an overlap of the bands and hence a full band gap. In our investigation we have chosen to concentrate on two birefringent materials, namely tellurium and rutile. The former shows high refractive indices of*

_{e}*n*

*= 6.2 and*

_{e}*n*

*= 4.8 at about 5*

_{o}*μm*[20], while the latter takes the values

*n*

*= 2.903 and*

_{e}*n*

*= 2.616 at 595*

_{o}*nm*[21]. The corresponding imaginary parts of the complex refractive indices are negligible. This choice of tellurium and rutile is motivated by the high refractive index of the former versus the relatively low refractive index of the latter. This will allow us to explore the possibility of opening a full band gap for remarkably different situations. Moreover, rutile can represent a fairly good choice for the fabrication of optical devices because of its relatively low refractive index. Even though the average refractive index of rutile is still larger than the typical refractive index of optical fibers, it is still one of the lowest values among birefringent materials which can open a full band gap.

Symmetries often play very important roles in nature [22], and in photonic crystals they decide the properties of the light. For instance, if we consider an isotropic bulk material, which is a structure with the highest possible symmetry, it would show no structural change after going through either rotational, translational or inversion operation. It explains why light inside such structures has no preferential configuration. However, if we slightly modify this bulk by lowering its symmetry, for example by introducing a translational symmetry, the solutions (modes) will have to respect this new translational invariance (each mode will be described as a Bloch function), which would mean that some of the degenerated modes will no longer remain at the same energy and would split creating a band gap. Energetically speaking, it means that solutions which were degenerate (same energy) inside the bulk can no longer correspond to the same eigenvalue in a periodic structure. Hence, they split and in a band picture it means that bands decouple from one another and a band gap can be created. This phenomenon can be easily understood if we consider the variational principle applied to the Maxwell equations, which shows how light tends to concentrate in a region with higher refractive index in order to minimize the energy. Furthermore, if we add the fact that eigenmodes associated to different bands (but same wave vector *k*) must be orthogonal to one another, we can see that solutions associated to different eigenmodes cannot occupy the same space inside the lattice, which leads to a breaking of the degeneration and consequently to an opening of a band gap. We must however be aware that there is an intrinsic limit in the symmetry reduction, which is given by the Bravais lattice configuration [23]. This is because excess of symmetry reduction would result in changing its Bravais form. In particular, in three dimensions a cube represents the Bravais lattice with the highest symmetry whereas a triclinic possesses the lowest symmetry. All the remaining Bravais lattices are located in between the two. We can move from the cubic structure to the triclinic structure by simply reducing the symmetry by *enough* amount. Since the Brillouin zone is intimately related to the crystal structure in the real space, it is a trade between reducing the symmetry and keeping the Brillouin zone intact for achieving an optimum band gap. It is then a delicate game which must be played carefully, as also shown by Anderson *et al*. [15]. In this manuscript symmetry reduction is achieved by modifying the radius of some of the rods, as shown in Fig. 1(a). It guarantees that the Brillouin zone of the structure with reduced symmetry will be the same as the lattice where symmetry reduction was not applied (hexagonal structure). Furthermore, by analyzing the conventional unit cell of the lattice showing symmetry reduction, it is quite straightforward to recognize that its symmetries are described by the group C_{6}
* _{v}* (6mm). Noting that C

_{6}

*also describes the symmetries of any standard hexagonal lattice (with no symmetry reduction) and that the lines of high symmetry dictate the k-space on which the band structure can be calculated, one can conclude that the G-M-K-G pathway correctly represents the optical behavior of the structure with symmetry reduction.*

_{v}Another method to produce a full band gap is by introducing an interfacial layer on the surface of dielectric rods, which can be explained by an effective refractive index description. Such idea can be extended by considering air holes inside dielectric rods. What we expect is an increase of the band gap because of the different distribution pattern of the modes compared to the case where the photonic crystal is made of solid rods. Indeed, because the regions with high refractive index becomes limited after drilling the holes, and because light tends to concentrate in the region with higher refractive index, the energy associated to the lower and higher bands of a gap would experience shifts to lower and higher energies, respectively, resulting in an increase of the band gap.

Out of the five different Bravais lattices that are possible in 2D structure, we have chosen the hexagonal lattice in our simulation, because the high symmetry of this lattice makes it the most isotropic among the 2D Bravais structures. As already mentioned, if the rods forming the hexagonal photonic crystal are made of isotropic dielectric, no full band gap can be generated. However, in uniaxial materials, the situation can be different. In the specific case of rutile (low refractive index), we have performed simulations by varying the filling ratio *f* of the photonic crystal from 0.0 to 0.6 (over which the rods start to overlap) at the steps of 0.1. In this case, no full band gap was observed. On the other hand, if the rods are made of tellurium (high refractive index) a full band gap exists in all the simulated range, as shown by green empty triangles in Fig. 2. For a comparison, simulation results for two isotropic materials with *n* = 5.5 and *n* = 2.75 are also shown by the black empty circles and dotted lines in Fig. 2. The two chosen values of refractive index correspond to the average of the refractive indices of tellurium and rutile, respectively.

In case of a rutile photonic crystal, the shift is not enough to allow an overlap between TE and TM gaps so that no full band gap is realized. Hence it is necessary to introduce a new approach able to generate new families of band gap to actually enhance the probability of obtaining an overlap between the two polarizations. We would show that symmetry reduction is suitable for such a purpose.

When TE and TM gaps cannot overlap with each other or, even worse, when such gaps are absent, the introduction of a method that could be capable of generating a new band gap is of crucial importance. Here we discuss the calculations of hexagonal birefringent photonic crystal on which symmetry reduction operations were applied by considering rods of two different radii in the unit cell of the crystal, illustrated in Fig. 1(a). The filling ratio *f* was varied from 0.0 to 0.6 at the steps of 0.1, and for each value of *f*, the radius ratio *β* was scanned from 0.0 to 1.0 at the steps of 0.2. The results, both for tellurium and rutile, are shown by blue full circles in Fig. 2.

This is evident by comparing the green empty triangles with blue full circles in Fig. 2 that the reduction of symmetry can actually increase the size of band gap. In particular, rutile, which did not show any band gap with full symmetry, is now able to present an optical gap. Also, the band gap for tellurium is increased for *f* < 0.3. It can be noted here that the value of *β* in Fig. 2 is chosen to maximize the band gap for each value of *f*. For this reason, the value of *β* in Fig. 2(b) for *f* > 0.3 is 1, which means no symmetry reduction. In fact, when symmetry reduction is considered (*i.e.*, *β* 6≠1), the band gap reduces, indicating a negative effect produced by symmetry reduction. It can be concluded from the above simulation that symmetry reduction plays a positive role for materials with low refractive index, whereas it can even give negative contribution for materials with high refractive index. The explanation is quite straightforward – the reduction of the symmetry in materials with high refractive index can produce new band gaps, but at the cost of reducing the already existing wide gaps. The results therefore may not be positive, exactly because some gaps can be reduced to the extent that the TE and TM modes do not overlap anymore. On the other hand, such a problem does not exist in materials with low refractive index, because these materials do not originally have any full band gap (or, at the most, their number is very limited). To further confirm this hypothesis, we would discuss what happens with the TE and TM modes when symmetry reduction is imposed for both rutile and tellurium. Calculations of full energy dispersions for rutile and tellurium are shown in Figs. 3 and 4, respectively.

As one can see from the figures, symmetry reduction opens new band gap as well as reduces the existing band gaps. For rutile, this means overlap of TE and TMmodes providing a full band gap (Fig. 3), while for tellurium, this means narrowing down of already overlapped part of TE and TM modes, resulting in a negative effect (Fig. 4). Indeed, the width of the band gaps in the case of tellurium tends to shrink when band degeneration is broken, hence the probability to have a full band gap is reduced. Such phenomenon can be explained if we notice that the change in geometry can affect the dispersion relation in two different ways. If the geometry is changed in a manner that does not affect the symmetries of the photonic crystal, the modification of the band structure will be “proportional” to the introduced geometrical changes. On the other hand, if the geometry is changed in a manner that affects the symmetries of the photonic crystal, then even a minor modification in geometry can introduce new gaps without strongly modifying the band structure. In order to accommodate the newly created gaps by the symmetry reduction, the already existing gaps must compress. This is the situation for tellurium, which already has some existing band gaps.

In order to understand the effect of higher refractive index at the outer part of the rods, we considered rods with holes, as illustrated in Fig. 1(b). Simulations were performed for three materials, silicon, tellurium, and rutile. Silicon was examined for the purpose of comparing birefringent materials with a commonly used isotropic material. The filling ratio *f* was varied from 0.1 to 0.6 in the steps of 0.1 and the radius ratio *β* was varied from 0.10 to 1.00 (no holes) in the steps of 0.01. In every case, the simulations resulted in positive contribution of air holes in increasing the band gap. For silicon and rutile, no full band gaps were generated but it was possible to notice that TE and TM gaps tend to approach each other. For tellurium, a small increase of the existing band gap was observed. In particular, at *f* = 0.40 and *β* = 0.78, which corresponds to the maximum value of band gap in the absence of air holes, the ratio between the bandwidth and the band-center was increased from 17.8% to 18.4% due to the presence of air holes. The accuracy of this calculation, as estimated from the calculation parameters, was better than 0.2%.

Finally, we consider the combined effect of symmetry reduction and air holes in birefringent photonic crystals. The geometry of the structure as well as the geometrical parameters are illustrated in Fig. 1(c). The simulation was performed for the filling ratio *f* ranging from 0.1 to 0.6 in the steps of 0.1, and the radius ratios *β*, *β*
_{1} and *β*
_{2} were all in the range from 0.2 to 1.0 with steps of 0.2, for each value of *f*. The red full triangles in Fig. 2 summarize the simulation results for both rutile and tellurium, where the highest values of the ratio between the bandwidth and band-center are plotted.

The results confirm that air holes can increase an already existing band gap. Indeed, by looking at rutile, one realizes that adding air holes in the band-gap-free-regions does not realize any frequency range at zero transmission. Further, the combination of symmetry reduction and air holes gives a higher contribution to the enlargement of the gap than just a simple summation of their individual contributions. On the other hand, by looking at tellurium at *f* = 0.3, one can realize that there is no effect due to symmetry reduction, indicating that only air holes can contribute to the enlargement of the gap. However, it is interesting to note that the combination of air hole and symmetry reduction increases the gap by 6%, while air holes alone could increase the gap by only 2.6%, even though there is no individual contribution of symmetry reduction at this point. This emphasizes the effect of combination, which is larger than the sum of their individual contributions. The maximum values for the gaps are also increased compared to all the previous situations. The maximum gap for rutile is now 8.4% while tellurium has passed 20%.

In summary, we have discussed the optical behavior of birefringent photonic crystals with relatively low and high refractive indices (rutile and tellurium, respectively). In particular, we have analyzed the full photonic band gap in order to understand how it is affected by symmetry reduction in the lattice and by the presence of air holes in such structures. We have demonstrated that both the symmetry reduction and the presence of air holes play important roles in increasing or modifying the band gap of the photonic crystal. Symmetry reduction, due to the breaking of degeneration, is able to create photonic gaps. For example, it can create a gap in the gap-free rutile photonic crystals. However, as a drawback, it tends to reduce the gap if the refractive index of the dielectric is too high. On the other hand, air holes can only enlarge an already existing gap, but they cannot generate new gaps. More importantly, the effects of symmetry reduction and air holes are much stronger when taken together than in configurations where they are treated individually. A combination of the two effects, properly weighted by the geometrical parameters of the photonic crystal, can help to maximize the value of the full photonic band gap.

The Authors gratefully acknowledge support from CREST of Japan Science and Technology Agency and Photonics Advanced Research Center of Osaka University.

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**23. **
In fact, it is more appropriate to talk about Crystal Systems instead of Bravais Lattices. Indeed, the symmetry level of a lattice with translational invariance is dictated by the corresponding crystal system. However we have chosen to use the Bravais lattice terminology to avoid confusion in the text. Besides, this approximation can work well if we consider that the fourteen Bravais lattices are intimately related to the seven crystal systems that are defined in three dimensions.