## Abstract

The vector wave multiple scattering method is a reliable and efficient technique in treating the photonic band gap problem for photonic crystals composed of spherically scattering objects with metallic components. In this paper, we describe the formalism and its application to the photonic band structures of systems comprising of metallo-dielectric spheres. We show that the photonic band gaps are essentially determined by local short-range order rather than by the translational symmetry if the volume fraction of the metallic core is high.

©2001 Optical Society of America

## 1 Introduction

Popular methods for calculating photonic band structures such as the plane wave method [1] and the finite-difference-time-domain (FDTD) method [2, 3] work well for dielectric photonic crystals but the convergence becomes difficult when the photonic crystals carry metallic components. Dispersion and absorption are difficult to handle with these formalisms. Formalisms such as the multiple-scattering (MS) or Korringa-Kohn-Rostoker (KKR)[4, 5] that take into account of the proper boundary condition of the interface are more desirable. In the MS method [4, 5, 6], the whole crystal is treated as an assembly of scattering centers and its scattering properties are taken as the sum of individual scatterers. Once the scattering properties (the T-matrix) of the individual scatterers are known, the scattered wave of the whole crystal can be constructed from these matrices, so does the band structures of the photonic crystals.

For spherical objects, the scattering T-matrix can be calculated analytically, and thus the effect of the scattering centers can be included in the boundary condition. The extension of the scalar wave KKR method to the case of EM wave was formulated and implemented by several groups [7, 8, 9] from different points of views. We have also implemented the vector wave MS method for the photonic band structure of three dimensional photonic crystals [10].

In this paper, we shall explore the effect of local short-range order on the photonic band gap formation in photonic crystals made of metallo-dielectric spheres. Our study shows that the band gap properties are determined by the local short range order of the metallo-dielectric spheres if the filling ratio is high. Our results remain the same if the metallic spheres are replaced by dielectric spheres with a metallic coating thicker than the skin depth.

## 2 The Vector Wave Multiple Scattering Method

The essence of the multiple scattering method is to assemble the total scattering waves at any space location from those of the individual scatterers. In photonic band structure calculations, we seek for solutions of the Maxwell equations casted as the following second order differential equation for the electric field:

where *κ*=*ω/c* and *ε*(*r*⃗, *ω*) is the position dependent dielectric function. This equation can be put into an integral equation form if one introduces the dyadic Green’s function *d*⃡0(*r*⃗-*r*⃗′)=[*I*⃡+∇⃗∇⃗/*κ*
^{2}]*exp*[*iκ*[*r*⃗-*r*⃗′|]/4*π*|*r*⃗-*r*⃗′|. Then,

This surface integral equation relates the E-field distribution in the photonic crystals to the external incident E-field outside the crystals. The total scattered wave in the photonic crystals can be assembled from those of the individual scatterers, and instead of integrating over the outer surface of the whole crystals, the integration can be carried out as the sum of surface integrations over individual scatterers.

This equation can be further reduced to a linear equation set after one expands locally the electric field using the vector spherical harmonics ${\stackrel{\u20d7}{J}}_{i}^{\mathit{\text{lm}}\sigma}$
(*r*⃗) and *H*
^{⃗lmσ}_{i}(*r*⃗) around each scatterer *i* ((*l,m*) are the angular momentum indices). Within this basis the incident and total E-field near the scatterer *i* are expressed as follows:

where ${\stackrel{\u20d7}{P}}_{i}^{\mathit{\text{lm}}\sigma}$
(*r*⃗)=${\stackrel{\u20d7}{J}}_{i}^{\mathit{\text{lm}}\sigma}$
(*r*⃗)+∑_{l'm'σ'}
${t}_{i}^{lm\mathrm{\sigma};}$
^{l'm''σ'}
${\stackrel{\u20d7}{H}}_{i}^{l\mathit{\prime}m\mathit{\prime}\sigma \mathit{\prime}}$
(*r*⃗′) is the total E-field outside the scattering region of scatterer *i*. After considering the Wronskian-like surface integral, the equation set governing the coefficients *a*_{i}
reads

Here we have made a Fourier transformation to the momentum space, *k*⃗ is the Bloch wavevector and *s* is the index of scatterer in primary cell. *G*^{ss}
′
_{lmσ}
;
_{l}
′
_{σ}
′(*k*⃗) is the Fourier transformation of structure factor which can be expressed as

Here, (*e*
_{1},*e*
_{2}) denote the two polarizations, *C* is the Clebsch-Gordon coefficient between the angular momenta 1 and *l* which combines the vector nature of EM wave and spatial dependence of the EM wave. The scalar structure factor *g* is given by *glml*′*m*′=4*π*∑_{l″m″}
*i*
^{l-l′-l″,}
*C*
_{lm;l′m′;l″m″}
*h*
_{l″}
(*kR*)*Y**_{l″m}″(-*R̂*), and *C*
_{lml′m′l″m″} are the Gaunt coefficients which determine the overlap coefficients among three spherical harmonics (*l,m*), (*l*′,*m*′), and (*I*″,*m*″); and *h*_{l}
and *Y*_{lm}
are the Hankel function of the first kind and spherical harmonics, respectively.

The physical solution of the above equation set can be picked up by counting the number of positive pseudo-eigenvalues for each frequency. We search for the change of the number of positive pseudo-eigenvalues as function of frequency and the degeneracy can determined by the counter change of neighboring pseudo-eigenvalues. The accuracy of the solution is set by the resolution of frequency mesh points and value of *l*, and we generally require the eigenfrequencies to be accurate to the third digits.

## 3 Results and Discussions

An important property of photonic crystals composed of metallo-dielectric spheres is that *any* periodic structures support complete photonic band gaps [10]. Here, we focus on the correlation between the band gap formation and local structural configurations. We will consider two sets of structures which have the same local nearest neighbor environment but differ in their translational symmetries. One set consists of the HCP and FCC structures, and the other set consists of the hexagonal and cubic diamond structures.

We first compare the photonic band structures of an HCP crystal with that of the FCC crystal. The metallo-dielectric spheres have metal cores modeled by dielectric constant *ε*=-200 and an encapsulating *ε*=12 dielectric coating with thickness equal to 5% of the sphere radius. Both HCP and FCC have the same local configuration with 12 nearest neighbors, but different stacking sequences of close-packed planes. The FCC crystal has the ABC stacking sequence while the HCP has the ABAB stacking sequence. Fig. 1a shows the photonic band structures for the metallo-dielectric spheres arranged in the HCP structures. The filling ratio is *f*=0.74 so that the spheres are in close packing form. The background is assumed to be air. The metallic core occupies about 63.5 % of volume. If the metal forms a percolating network, the system will behave like a good mirror. Here, we are interested in what happens when the metallic component is not percolating. Photonic bands for dielectric photonic crystals are usually normalized to the lattice constants because global periodicity governs the Bragg scattering that leads to the photonic gap. For our system, we will see that the local order is more important than the global structure or the shape of the Brillouin zone. For this reason, the frequencies are normalized to *R*/2*πc*, where *R* is the radius of the spheres. There is a gap around *ωR*/2*πc*=0.305 and the gap/mid-gap frequency ratio reaches a large value of 0.39. The photonic bands of the FCC structure is shown in Fig. 1b, and we see that an absolute gap appears at 0.295 and with a gap/mid-gap ratio of 0.40. Therefore, the characteristic properties of the photonic band gaps in these two crystals are almost the same. Since the HCP and FCC crystals have the same nearest neighbor configuration but different translational symmetry, the above results suggest that it is the local configuration rather than the overall symmetry that dictates photonic gaps for metallo-dielectric sphere systems.

We next consider the hexagonal and cubic diamond structures that have a local tetra-hedral (4-fold) coordination, but again different translational symmetries and different Brillouin Zones. The spheres have exactly the same make-up as previously described. We show in Fig. 2a the photonic band structures of the hexagonal diamond structure with the maximum sphere filling ratio of *f*=0.34. One observes that an absolute photonic band gap appears around *ωR*/2*πc*=0.135, with a huge gap/mid-gap ratio of 0.596. The corresponding photonic band structures for the cubic diamond structure is plotted in Fig. 2b for comparison. The gap and gap/mid-gap frequency ratio are *ωR*/2*πc*=0.141 and 0.734, respectively. The gap positions for both the hexagonal and cubic diamonds occur at roughly the same frequency, and there is a modest 15% difference between gap/mid-gap ratios. The two diamond structures with different symmetries have fairly similar photonic band gap properties, but the global symmetry has some influence.

To better understand the systematics, we have calculated the photonic band structures for a series of filling ratios for the above two sets of crystal types. The filling ratio is varied by changing the lattice constant at fixed radius of the spheres. As the dominant effect is due to the field exclusion effect of metal core and the high dielectric coating layer is mainly used to avoid the metal contact between the neighboring spheres, here we consider only the photonic band structures for crystals made of pure metal spheres. The gap/mid-gap ratio is plotted in Fig. 3 as a function of the filling ratio of the spheres for both HCP and FCC structures. The overall feature suggests that the gap/mid-gap ratio is almost the same for FCC and HCP structures at high filling fractions, but starts to deviate at lower filling ratios so that the threshold filling ratio for the gap to appear is slightly smaller for HCP than FCC. In Fig. 4, we plot the gap/mid-gap ratio for hexagonal and cubic diamonds. Although the gap/mid-gap frequency near the maximum filling ratio has similar magnitude, their dependence with filling ratio has more conspicuous differences, and the gap diminishes faster for hexagonal diamond(*f*≈0.2) than for cubic diamond(*f*≈0.1).

These results show that the photonic gap properties near maximum filling is determined essentially by the local configuration, but the translational symmetry starts to influence the gap features at lower filling ratio of the spheres. We note that EM waves can only propagate in the void regions not occupied by the metallic cores. Here, the term “metallic core” can also be taken to mean the volume occupied by a dielectric core coated by a metallic coating. A solid metal core and a dielectric core with a metal coating have the same effect for external EM waves as long as the EM wave cannot penetrate the coating. As the filling ratio of the metallic core increases, EM wave becomes increasingly squeezed into the void regions. A high filling ratio of spheres has two direct effects: (i) it reduces the void volume between the metal spheres and thus increases the frequency separation between the resonance modes in the voids; (ii) it reduces the inter-void coupling, and thus reduces the dispersion of the resonance mode induced bands. These two factors combine to yield photonic band gap obtained in the spectra. When the filling ratio is high enough, the photonic band structure may be regarded as the result of the hopping of resonance modes in the voids. In this high filling limit, a tight-binding picture is a good description and the gap features are dictated by the local coordination. Translational symmetry and even orderness have only secondary effects in this regime. As the filling ratio is reduced, the voids increase in volume and are better interconnected. Eventually, in the low filling limit, the propagation is better described by a freely propagating wave interrupted by an periodic array of scatterers, and the shape of the Brillouin zone and translational symmetry plays an important role. The critical filling ratio for a vanishing photonic gap is usually at an intermediate filling ratio and in that regime both the local configuration as well as the global translational symmetry matters. We also note that the FCC and HCP polytypes have the highest possible filling fraction (74 %) for all known periodic structures of touching mono-sized spheres. Thus, the void regions not occupied by the metallic cores in the FCC/HCP family of structures are the most isolated and hence the photonic gap properties are mainly governed by the local configuration and the effect of the translational symmetry becomes apparent only when the metallic cores are reduced significantly in size. For the diamond structure, even touching spheres occupy only about 34 % of volume and the voids are well interconnected even in the touching sphere limit. In this case, the effect of the translational symmetry should be more apparent, and indeed it is what we obtained numerically.

The importance of the local configuration in governing the photonic band gap properties is actually valid for all the crystal structures when the filling ratio is near its maximum.

## 4 Conclusion

In summary, we have studied the photonic band structures of photonic crystals comprising of metallo-dielectric spheres, and showed that the local configuration determines the photonic band gap properties at high filling fractions of the metallic core. These results are obtained with the multiple scattering method that offers a reliable and efficient way to calculate the photonic band structures for both dielectric and metallic systems.

## 5 Acknowledgment

We thank Dr. Z.L. Wang for many discussions. This work is supported by RGC Hong Kong through HKUST6145/99P. WYZ is also supported in part by the ‘Climbing Program’ by NSTC, and NSFC of China under ‘Excellent Youth Foundation”.

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