Vol. 176
Latest Volume
All Volumes
PIER 176 [2023] PIER 175 [2022] PIER 174 [2022] PIER 173 [2022] PIER 172 [2021] PIER 171 [2021] PIER 170 [2021] PIER 169 [2020] PIER 168 [2020] PIER 167 [2020] PIER 166 [2019] PIER 165 [2019] PIER 164 [2019] PIER 163 [2018] PIER 162 [2018] PIER 161 [2018] PIER 160 [2017] PIER 159 [2017] PIER 158 [2017] PIER 157 [2016] PIER 156 [2016] PIER 155 [2016] PIER 154 [2015] PIER 153 [2015] PIER 152 [2015] PIER 151 [2015] PIER 150 [2015] PIER 149 [2014] PIER 148 [2014] PIER 147 [2014] PIER 146 [2014] PIER 145 [2014] PIER 144 [2014] PIER 143 [2013] PIER 142 [2013] PIER 141 [2013] PIER 140 [2013] PIER 139 [2013] PIER 138 [2013] PIER 137 [2013] PIER 136 [2013] PIER 135 [2013] PIER 134 [2013] PIER 133 [2013] PIER 132 [2012] PIER 131 [2012] PIER 130 [2012] PIER 129 [2012] PIER 128 [2012] PIER 127 [2012] PIER 126 [2012] PIER 125 [2012] PIER 124 [2012] PIER 123 [2012] PIER 122 [2012] PIER 121 [2011] PIER 120 [2011] PIER 119 [2011] PIER 118 [2011] PIER 117 [2011] PIER 116 [2011] PIER 115 [2011] PIER 114 [2011] PIER 113 [2011] PIER 112 [2011] PIER 111 [2011] PIER 110 [2010] PIER 109 [2010] PIER 108 [2010] PIER 107 [2010] PIER 106 [2010] PIER 105 [2010] PIER 104 [2010] PIER 103 [2010] PIER 102 [2010] PIER 101 [2010] PIER 100 [2010] PIER 99 [2009] PIER 98 [2009] PIER 97 [2009] PIER 96 [2009] PIER 95 [2009] PIER 94 [2009] PIER 93 [2009] PIER 92 [2009] PIER 91 [2009] PIER 90 [2009] PIER 89 [2009] PIER 88 [2008] PIER 87 [2008] PIER 86 [2008] PIER 85 [2008] PIER 84 [2008] PIER 83 [2008] PIER 82 [2008] PIER 81 [2008] PIER 80 [2008] PIER 79 [2008] PIER 78 [2008] PIER 77 [2007] PIER 76 [2007] PIER 75 [2007] PIER 74 [2007] PIER 73 [2007] PIER 72 [2007] PIER 71 [2007] PIER 70 [2007] PIER 69 [2007] PIER 68 [2007] PIER 67 [2007] PIER 66 [2006] PIER 65 [2006] PIER 64 [2006] PIER 63 [2006] PIER 62 [2006] PIER 61 [2006] PIER 60 [2006] PIER 59 [2006] PIER 58 [2006] PIER 57 [2006] PIER 56 [2006] PIER 55 [2005] PIER 54 [2005] PIER 53 [2005] PIER 52 [2005] PIER 51 [2005] PIER 50 [2005] PIER 49 [2004] PIER 48 [2004] PIER 47 [2004] PIER 46 [2004] PIER 45 [2004] PIER 44 [2004] PIER 43 [2003] PIER 42 [2003] PIER 41 [2003] PIER 40 [2003] PIER 39 [2003] PIER 38 [2002] PIER 37 [2002] PIER 36 [2002] PIER 35 [2002] PIER 34 [2001] PIER 33 [2001] PIER 32 [2001] PIER 31 [2001] PIER 30 [2001] PIER 29 [2000] PIER 28 [2000] PIER 27 [2000] PIER 26 [2000] PIER 25 [2000] PIER 24 [1999] PIER 23 [1999] PIER 22 [1999] PIER 21 [1999] PIER 20 [1998] PIER 19 [1998] PIER 18 [1998] PIER 17 [1997] PIER 16 [1997] PIER 15 [1997] PIER 14 [1996] PIER 13 [1996] PIER 12 [1996] PIER 11 [1995] PIER 10 [1995] PIER 09 [1994] PIER 08 [1994] PIER 07 [1993] PIER 06 [1992] PIER 05 [1991] PIER 04 [1991] PIER 03 [1990] PIER 02 [1990] PIER 01 [1989]
2022-12-30
A Fast Computation Method of Bands and Band Field Solutions of 3D Periodic Structures Using Broadband Green's Function-Multiple Scattering Theory
By
Progress In Electromagnetics Research, Vol. 176, 67-93, 2023
Abstract
We extended the previous 2D method of BBGF-MST (Broadband Green's function-Multiple Scattering Theory) approach to 3D problems in periodic structures. Band Structures and Band Field Solutions are calculated. A feature of BBGF is that the lattice Green's functions are broadband so that the coefficients of the spherical wave expansions are calculated rapidly for many frequencies. These are then used for speedy calculations of the matrix elements of the KKR (Korringa-Kohn-Rostoker) eigenvalue equation. Using BBGF-MST, a low order matrix eigenvalue equation for the bands is derived. For the first two bands, the dimension of the KKR matrix equation is only 4 by 4. With the use of BBGF, the CPU requirement for the BBGF-MST technique is 0.27 secondson a standard laptop for solving the KKR eigenvalue equation. Numerical results of the band diagrams are illustrated. Higher order spherical waves are next used to calculate the normalized band field solutions for the entire cell.
Citation
Leung Tsang Tien-Hao Liao Shurun Tan , "A Fast Computation Method of Bands and Band Field Solutions of 3D Periodic Structures Using Broadband Green's Function-Multiple Scattering Theory," Progress In Electromagnetics Research, Vol. 176, 67-93, 2023.
doi:10.2528/PIER22080101
http://www.jpier.org/PIER/pier.php?paper=22080101
References

1. Wang, Z., Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-free one-way edge modes in a gyromagnetic photonic crystal," Phys. Rev. Lett., Vol. 100, 013905, 2008.
doi:10.1103/PhysRevLett.100.013905

2. Yang, Z., F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, "Topological acoustics," Phys. Rev. Lett., Vol. 114, 114301, 2015.
doi:10.1103/PhysRevLett.114.114301

3. Xue, H., Y. Yang, G. Liu, F. Gao, Y. Chong, and B. Zhang, "Realization of an acoustic third-order topological insulator," Phys. Rev. Lett., Vol. 122, 244301, Jun. 2019.
doi:10.1103/PhysRevLett.122.244301

4. Ao, X., Z. Lin, and C. T. Chan, "One way edge modes in a magneto-optical honeycomb photonic crystal," Phys. Rev. B, Vol. 80, 033105, 2009.
doi:10.1103/PhysRevB.80.033105

5. Tsaolamprou, A. C., M. Kafesaki, C. M. Soukoulis, E. N. Economou, and T. Koschny, "Chiral topological surface states on a finite square photonic crystal bounded by air," Physical Review Applied, Vol. 16, 044011, 2021.
doi:10.1103/PhysRevApplied.16.044011

6. Zhao, R., G. D. Xie, M. L. N. Chen, Z. Lan, Z. Huang, and W. E. I. Sha, "First-principle calculation of Chern number in gyrotropic photonic crystals," Optics Express, Vol. 28, 4638, 2020.
doi:10.1364/OE.380077

7. Feng, Z., S. Tan, L. Tsang, and E. Li, "Band characterization of topological photonic crystals using the broadband Green's function technique," Optics Express, Vol. 28, No. 19, 27223, 2020.
doi:10.1364/OE.400205

8. Tsang, L., T.-H. Liao, and S. Tan, "Calculation of bands and band field solutions in topological acoustics using the broadband Green's function-KKR-multiple scattering method," Progress In Electromagnetic Research, Vol. 171, 137-158, 2021.
doi:10.2528/PIER21081706

9. Ho, K. M., C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett., Vol. 65, 3152-3155, 1990.
doi:10.1103/PhysRevLett.65.3152

10. Leung, K. M. and Y. F. Liu, "Full vector wave calculation of photonic band structures in face- centered-cubic dielectric media," Phys. Rev. Lett., Vol. 65, 2646-2649, 1990.
doi:10.1103/PhysRevLett.65.2646

11. Plihal, M. and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B, Vol. 44, 8565-8571, 1991.
doi:10.1103/PhysRevB.44.8565

12. Joannopoulos, J. D., S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 2011.
doi:10.2307/j.ctvcm4gz9

13. Nicolet, A., S. Guenneau, C. Geuzainec, and F. Zollaa, "Modelling of electromagnetic waves in periodic media with finite elements," Journal of Comp. and Appl. Math., Vol. 168, 321-329, 2004.
doi:10.1016/j.cam.2003.07.002

14. Jin, J. M., Finite Element Method in Electromagnetics, 3rd Ed., Wiley, 2014.

15. Tsang, L., "Broadband calculations of band diagrams in periodic structures using the broadband Green's function with low wavenumber extraction (BBGFL)," Progress In Electromagnetics Research, Vol. 153, 57-68, 2015.
doi:10.2528/PIER15082901

16. Tsang, L. and S. Tan, "Calculations of band diagrams and low frequency dispersion relations of 2D periodic dielectric scattering using broadband Green's function with low wavenumber extraction (BBGFL)," Optics Express, Vol. 24, 945-965, 2016.
doi:10.1364/OE.24.000945

17. Gao, R., L. Tsang, S. Tan, and T.-H. Liao, "Band calculations using broadband Green's functions and the KKR method with applications to magneto-optics and photonic crystals," Journal of Optical Society of America B, Vol. 37, 3896-3907, 2020.
doi:10.1364/JOSAB.400824

18. Gao, R., L. Tsang, S. Tan, and T.-H. Liao, "Broadband Green's function-KKR-multiple scattering method for calculations of normalized band field solutions in magnetic-optics crystals," Journal of Optical Society of America B, Vol. 38, 3159-3171, 2021.
doi:10.1364/JOSAB.422574

19. Tan, S. and L. Tsang, "Efficient broadband evaluations of lattice Green's functions via imaginary wavenumber components extractions," Progress In Electromagnetics Research, Vol. 164, 63-74, 2019.

20. Sanamzadeh, M. and L. Tsang, "Fast and broad band calculation of the dyadic Green's function in the rectangular cavity; An imaginary wave number extraction technique," Progress In Electromagnetic Research C, Vol. 96, 243-258, 2019.
doi:10.2528/PIERC19090301

21. Liao, T.-H., L. Tsang, and W. Kwek, "Broadband Green's Funtion (BBGFL) method with imaginary wavenumber extractions for simulations of radiated emissions from irregular shaped printed circuit board," IEEE Transactions on Electromagnetic Compatibility, Vol. 62, No. 5, 2209-2216, Oct. 2020.
doi:10.1109/TEMC.2019.2939136

22. Korringa, J., "On the calculation of the energy of a Bloch wave in a metal," Physica, Vol. 13, 392-400, 1947.
doi:10.1016/0031-8914(47)90013-X

23. Kohn, W. and N. Rostoker, "Solution of the Schrodinger equation in periodic lattices with an application to metallic lithium," Phys. Rev., Vol. 94, 1111-1120, 1954.
doi:10.1103/PhysRev.94.1111

24. Ham, F. S. and B. Segall, "Energy bands in periodic lattices --- Green's function method," Phys. Rev., Vol. 124, No. 6, 1786-1796, 1961.
doi:10.1103/PhysRev.124.1786

25. Wang, X., X. G. Zhang, Q. Yu, and B. N. Harmon, "Multiple scattering theory for electromagnetic waves," Phys. Rev. B, Vol. 47, No. 8, 4161-4167, 1993.
doi:10.1103/PhysRevB.47.4161

26. Leung, K. M. and Y. Qiu, "Multiple-scattering calculation of the two-dimensional photonic band structure," Phys. Rev. B, Vol. 48, 7767-7771, 1993.
doi:10.1103/PhysRevB.48.7767

27. Kafesaki, M. and F. Economou, "Multiple-scattering theory for three-dimensional periodic acoustic composites," Phys. Rev. B, Vol. 60, 11993-12001, 1999.
doi:10.1103/PhysRevB.60.11993

28. Liu, Z., C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: Theory and experiment," Phys. Rev. B, Vol. 62, 2446-2457, 2000.
doi:10.1103/PhysRevB.62.2446

29. Kambe, K., "Theory of low-energy electron diffraction," I. Application of the Cellular Method to Monatomic Layers, Vol. 22, No. 3, 322-330, 1967.

30. Kambe, K., "Theory of electron diffraction by crystals," I. Green's Function and Integral Equation, Vol. 22, No. 4, 422-431, 1967.

31. Kambe, K., "Theory of low-energy electron diffraction," II. Cellular Method for Complex Monolayers and Multilayers, Vol. 23, No. 9, 1280-1294, 1968.

32. Jia, P.-H., et al., "Two fold domain decomposition method for the analysis of multiscale composite structures," IEEE Transactions on Antennas and Propagation, Vol. 67, No. 9, 6090-6103, Sep. 2019.
doi:10.1109/TAP.2019.2925120

33. Tsang, L., J. A. Kong, and K. H. Ding, Scattering of Electromagnetic Waves, Vol. 1: Theory and Applications, 426 pages, Wiley Interscience, 2000.
doi:10.1002/0471224286

34. Foldy, L. L., "The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers," Phys. Rev., Vol. 67, 107-119, 1945.
doi:10.1103/PhysRev.67.107

35. Lax, M., "Multiple scattering of waves," Rev. Mod. Phys., Vol. 23, No. 4, 287-310, 1951.
doi:10.1103/RevModPhys.23.287

36. Waterman, P. C. and R. Truell, "Multiple scattering of waves," Journal of Mathematical Physics, Vol. 2, No. 4, 512-537, 1961.
doi:10.1063/1.1703737

37. Ishimaru, A., Wave Propagation and Scattering in Random Media, Academic Press, 1978.

38. Tsang, L., J. A. Kong, and T. Habashy, "Multiple scattering of acoustic waves by random distribution of discrete spherical scatterers with the quasicrystalline and Percus-Yevick approximation," Journal of the Acoustical Society of America, Vol. 71, No. 3, 552-558, Mar. 1982.
doi:10.1121/1.387524

39. Tsang, L., J. A. Kong, and R. Shin, Theory of Microwave Remote Sensing, Wiley-Interscience, New York, 1985.

40. Tsang, L., J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol. 2: Numerical Simulations, 705 pages, Wiley Interscience, 2001.
doi:10.1002/0471224308

41. Mishchenko, M. I., L. D. Travis, and A. A. Laci's, Multiple Scattering of Light by Particles, Radiative Transfer and Coherent Backscattering, Cambridge University Press, 2006.

42. Stein, S., "Addition theorem for spherical wave functions," Quarterly of Applied Mathematics, Vol. 19, No. 1, 15-24, 1961.
doi:10.1090/qam/120407

43. Edmonds, A. R., Angular Momentum in Quantum Mechanics, Princeton University Press, 1960.

44. Cruzan, O. R., "Translational addition theorem for spherical vector wave functions," Q. Appl. Math, Vol. 20, 33-40, 1962.
doi:10.1090/qam/132851

45. Chew, W. C., Waves and Fields in Inhomogeneous Media, IEEE Press, 1995.

46. Abramowitz, M. and I. Stegun, Handbook of Mathematical Functions, Dover, 1956.

47. Waterman, P. C., "Matrix formulation of electromagnetic scattering," Proceedings of IEEE, Vol. 53, 805-812, 1965.
doi:10.1109/PROC.1965.4058

48. Gradshteyn, I. S. and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 2007.

49. Huang, H., L. Tsang, A. Colliander, R. Shah, X. Xu, and S. H. Yueh, "Multiple scattering of waves by complex objects using hybrid method of T-matrix and foldy-lax equations using vector spherical waves and vector spheroidal waves," Progress In Electromagnetic Research, Vol. 168, 87-111, 2020.
doi:10.2528/PIER20080409

50. Gu, W., L. Tsang, A. Colliander, and S. Yueh, "Propagation of waves in vegetations using a hybrid method," IEEE Transactions on Antennas and Propagation, Vol. 69, No. 10, 6752-6761, Oct. 2021.
doi:10.1109/TAP.2021.3069487

51. Gu, W., L. Tsang, A. Colliander, and S. Yueh, "Multifrequency full-wave simulations of vegetation using a hybrid method," IEEE Transactions on Microwave Theory and Techniques, Vol. 70, No. 1, 275-285, Jan. 2022.
doi:10.1109/TMTT.2021.3107313

52. Tsang, L., T.-H. Liao, R. Gao, H. Xu, W. Gu, and J. Zhu, "Theory of microwave remote sensing of vegetation effects, soop and rough soil surface backscattering," Remote Sensing, Vol. 14, No. 15, 2022.
doi:10.3390/rs14153640

53. Tan, S. and L. Tsang, "Green functions, including scatterers, for photonic crystals and metamaterials," Journal of Optical Society of America B, Vol. 34, 1450-1458, 2017.
doi:10.1364/JOSAB.34.001450

54. Tan, S. and L. Tsang, "Scattering of waves by a half-space of periodic scatterers using broadband Green's function," Opt. Lett., Vol. 42, No. 22, 4667-4670, Nov. 2017.
doi:10.1364/OL.42.004667

55. Tsang, L., K.-H. Ding, and S. Tan, "Broadband point source Green's function in a one-dimensional infinite periodic lossless medium based on BBGFL with modal method," Progress In Electromagnetics Research, Vol. 163, 51-77, 2018.
doi:10.2528/PIER18071802

56. Tsang, L. and S. Tan, "Full wave simulations of photonic crystals and metamaterials using the broadband Green's functions,", US patent number 11,087,043, Aug. 10, 2021.

57. Gu, W., L. Tsang, A. Colliander, and S. Yueh, "Hybrid method for full-wave simulations of forests at L-band," IEEE Access, Vol. 10, 105898-105909, 2022.
doi:10.1109/ACCESS.2022.3211323

58. Harrington, R. F., Time-harmonic Electromagnetic Fields, McGraw-Hill, 1961.

59. Sarabandi, K., Foundations of Applied Electromagnetics, Michigan Publishing, 2022.