## Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres

Optics Express, Vol. 8, Issue 3, pp. 203-208 (2001)

http://dx.doi.org/10.1364/OE.8.000203

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### 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.

© Optical Society of America

## 1 Introduction

1. See, e.g., K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. **65**, 3152 (1990). [CrossRef] [PubMed]

4. J. Korringa, Physica **13**, 392 (1947). [CrossRef]

5. W. Kohn and N. Rostoker, “Solution of Schrodinger equation in periodic lattice with an application to metallic lithium,” Phys. Rev. **94**, 1111 (1954). [CrossRef]

4. J. Korringa, Physica **13**, 392 (1947). [CrossRef]

5. W. Kohn and N. Rostoker, “Solution of Schrodinger equation in periodic lattice with an application to metallic lithium,” Phys. Rev. **94**, 1111 (1954). [CrossRef]

9. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Computer Phys. Commun. **113**, 49 (1998). [CrossRef]

10. W.Y. Zhang, X.Y. Lei, Z.L. Wang, D.G. Zheng, W.Y. Tam, C.T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. **84**, 2853 (2000). [CrossRef] [PubMed]

## 2 The Vector Wave Multiple Scattering Method

*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,

*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:

*r*⃗)=

*r*⃗)+∑

_{l'm'σ'}

^{l'm''σ'}

*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

*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

*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.

*l*, and we generally require the eigenfrequencies to be accurate to the third digits.

## 3 Results and Discussions

*any*periodic structures support complete photonic band gaps [10

10. W.Y. Zhang, X.Y. Lei, Z.L. Wang, D.G. Zheng, W.Y. Tam, C.T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. **84**, 2853 (2000). [CrossRef] [PubMed]

*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.

*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.

*f*≈0.2) than for cubic diamond(

*f*≈0.1).

## 4 Conclusion

## 5 Acknowledgment

## References and links

1. | See, e.g., K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. |

2. | See, e.g., A. J. Ward and J. B. Pendry, “Calculating photonic Green’s function using a nonorthogonal finite-difference time-domain method,” Phys. Rev. |

3. | S. Fan, P. R. Villeneuve, and Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. |

4. | J. Korringa, Physica |

5. | W. Kohn and N. Rostoker, “Solution of Schrodinger equation in periodic lattice with an application to metallic lithium,” Phys. Rev. |

6. | J. L. Beeby, “The electronic structures of disordered systems,” Proc. R. Soc. |

7. | See, e.g., K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. |

8. | X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple scattering theory for electro-magnetic waves,” Phys. Rev. |

9. | N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Computer Phys. Commun. |

10. | W.Y. Zhang, X.Y. Lei, Z.L. Wang, D.G. Zheng, W.Y. Tam, C.T. Chan, and P. Sheng, “Robust photonic band gap from tunable scatterers,” Phys. Rev. Lett. |

**OCIS Codes**

(160.4670) Materials : Optical materials

(260.2110) Physical optics : Electromagnetic optics

(260.3910) Physical optics : Metal optics

**ToC Category:**

Focus Issue: Photonic bandgap calculations

**History**

Original Manuscript: November 13, 2000

Published: January 29, 2001

**Citation**

Weiyi Zhang, Che Ting Chan, and Ping Sheng, "Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres," Opt. Express **8**, 203-208 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-203

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### References

- See, e.g., K. M. Ho, C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152 (1990). [CrossRef] [PubMed]
- See, e.g., A. J. Ward and J. B. Pendry, "Calculating photonic Green's function using a nonorthogonal finite-difference time-domain method," Phys. Rev. B58, 7252 (1998).
- S. Fan, P. R. Villeneuve, and Joannopoulos, "Large omnidirectional band gaps in metallodielectric photonic crystals," Phys. Rev. B54, 11245 (1996).
- J. Korringa, Physica 13, 392 (1947). [CrossRef]
- W. Kohn and N. Rostoker, "Solution of Schrodinger equation in periodic lattice with an application to metallic lithium," Phys. Rev. 94, 1111 (1954). [CrossRef]
- J. L. Beeby, "The electronic structures of disordered systems," Proc. R. Soc. A279, 82 (1964).
- See, e.g., K. Ohtaka, "Energy band of photons and low-energy photon diffraction," Phys. Rev. B19, 5057 (1979).
- X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, "Multiple scattering theory for electro-magnetic waves," Phys. Rev. B47, 4161 (1993).
- N. Stefanou, V. Yannopapas, and A. Modinos, "Heterostructures of photonic crystals: frequency bands and transmission coefficients," Computer Phys. Commun. 113, 49 (1998). [CrossRef]
- W.Y. Zhang, X.Y. Lei, Z.L. Wang, D.G. Zheng, W.Y. Tam, C.T. Chan, and P. Sheng, "Robust photonic band gap from tunable scatterers," Phys. Rev. Lett. 84, 2853 (2000). [CrossRef] [PubMed]

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