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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics


  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 8 — Sep. 4, 2013

Three-dimensional integral equation approach to light scattering, extinction cross sections, local density of states, and quasi-normal modes

Jakob Rosenkrantz de Lasson, Jesper Mørk, and Philip Trøst Kristensen  »View Author Affiliations

JOSA B, Vol. 30, Issue 7, pp. 1996-2007 (2013)

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We present a numerical formalism for solving the Lippmann–Schwinger equation for the electric field in three dimensions. The formalism may be applied to scatterers of different shapes and embedded in different background media, and we develop it in detail for the specific case of spherical scatterers in a homogeneous background medium. In addition, we show how several physically important quantities may readily be calculated with the formalism. These quantities include the extinction cross section, the total Green’s tensor, the projected local density of states, and the Purcell factor as well as the quasi-normal modes of leaky resonators with the associated resonance frequencies and quality factors. We demonstrate the calculations for the well-known plasmonic dimer consisting of two silver nanoparticles and thus illustrate the versatility of the formalism for use in modeling of advanced nanophotonic devices.

© 2013 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(240.6680) Optics at surfaces : Surface plasmons
(290.4210) Scattering : Multiple scattering
(050.1755) Diffraction and gratings : Computational electromagnetic methods

ToC Category:

Original Manuscript: March 12, 2013
Revised Manuscript: April 23, 2013
Manuscript Accepted: April 24, 2013
Published: June 27, 2013

Virtual Issues
Vol. 8, Iss. 8 Virtual Journal for Biomedical Optics

Jakob Rosenkrantz de Lasson, Jesper Mørk, and Philip Trøst Kristensen, "Three-dimensional integral equation approach to light scattering, extinction cross sections, local density of states, and quasi-normal modes," J. Opt. Soc. Am. B 30, 1996-2007 (2013)

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  1. V. P. Bykov, “Spontaneous emission from a medium with a band spectrum,” Sov. J. Quantum Electron. 4, 861–871 (1975). [CrossRef]
  2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef]
  3. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef]
  4. K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–21800 (2008). [CrossRef]
  5. N. Liu, M. Hentschel, T. Weiss, A. P. Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science 332, 1407–1410 (2011). [CrossRef]
  6. J. Z. Zhang and C. Noguez, “Plasmonic optical properties and applications of metal nanostructures,” Plasmonics 3, 127–150 (2008). [CrossRef]
  7. M. Dems, I.-S. Chung, P. Nyakas, S. Bischoff, and K. Panajotov, “Numerical methods for modeling photonic-crystal VCSELs,” Opt. Express 18, 16042–16054 (2010). [CrossRef]
  8. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  9. J. Reddy, An Introduction to the Finite Element Method, 3rd ed. (McGraw-Hill Science/Engineering/Math, 2005).
  10. K. Busch, M. König, and J. Niegemann, “Discontinuous Galerkin methods in nanophotonics,” Laser Photon. Rev. 5, 773–809 (2011). [CrossRef]
  11. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). [CrossRef]
  12. L. C. Botten, R. C. McPhedran, N. A. Nicorovici, A. A. Asatryan, C. M. de Sterke, P. A. Robinson, K. Busch, G. H. Smith, and T. N. Langtry, “Rayleigh multipole methods for photonic crystal calculations,” Prog. Electromagn. Res. 41, 21–60 (2003). [CrossRef]
  13. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys 114, 185–200 (1994). [CrossRef]
  14. L. Novotny and B. Hecht, Principles of Nano-Optics, 1st ed. (Cambridge University, 2006).
  15. A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics, 1st ed. (IEEE, 1998).
  16. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]
  17. H. Levine and J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948). [CrossRef]
  18. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  19. C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1994).
  20. M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000). [CrossRef]
  21. P. A. Martin, Multiple Scattering. Interaction of Time-Harmonic Waves with N Obstacles, 1st ed. (Cambridge University, 2006).
  22. P. T. Kristensen, P. Lodahl, and J. Mørk, “Light propagation in finite-sized photonic crystals: multiple scattering using an electric field integral equation,” J. Opt. Soc. Am. B 27, 228–237 (2010). [CrossRef]
  23. G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. 25, 377–445 (1908). [CrossRef]
  24. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef]
  25. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996). [CrossRef]
  26. F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999). [CrossRef]
  27. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed. (Pergamon, 1981).
  28. O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998). [CrossRef]
  29. R. Sprik, B. A. van Tiggelen, and A. Lagendijk, “Optical emission in periodic dielectrics,” Europhys. Lett. 35, 265–270 (1996). [CrossRef]
  30. E. M. Purcell, “Proceedings of the American Physical Society, b10. Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 674 (1946). [CrossRef]
  31. K. M. Lee, P. T. Leung, and K. M. Pang, “Dyadic formulation of morphology-dependent resonances. I. Completeness relation,” J. Opt. Soc. Am. B 16, 1409–1417 (1999). [CrossRef]
  32. P. T. Kristensen, C. V. Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. 37, 1649–1651 (2012). [CrossRef]
  33. J. R. de Lasson, P. T. Kristensen, and J. Mørk, “Multiple-scattering formalism beyond the quasistatic approximation: analyzing resonances in plasmonic chains,” AIP Conf. Proc. 1475, 158–160 (2012). [CrossRef]
  34. A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980). [CrossRef]
  35. F. Capolino, Theory and Phenomena of Metamaterials, 1st ed. (CRC Press, 2009).
  36. R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007). [CrossRef]
  37. P. A. Martin, “Multiple scattering and the Rehr–Albers–Fritzsche formula for the propagator matrix,” J. Phys. A 31, 8923 (1998). [CrossRef]
  38. V. N. Pustovit and T. V. Shahbazyan, “Plasmon-mediated superradiance near metal nanostructures,” Phys. Rev. B 82, 075429 (2010). [CrossRef]
  39. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998). [CrossRef]
  40. M. R. Spiegel, S. Lipschutz, and J. Liu, Schaum’s Outline of Mathematical Handbook of Formulas and Tables, 3rd ed. (McGraw-Hill, 2008).
  41. S. A. Maier, Plasmonics: Fundamentals and Applications, 1st ed. (Springer, 2007).
  42. V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Marzan, and F. J. García de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. 37, 1792–1805 (2008). [CrossRef]
  43. E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004). [CrossRef]
  44. M. Chen, Y.-F. Chau, and D. Tsai, “Three-dimensional analysis of scattering field interactions and surface plasmon resonance in coupled silver nanospheres,” Plasmonics 3, 157–164 (2008). [CrossRef]
  45. P. K. Jain, W. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7, 2080–2088 (2007). [CrossRef]
  46. N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113, 2784–2791 (2009). [CrossRef]
  47. A. F. Koenderink, “On the use of Purcell factors for plasmon antennas,” Opt. Lett. 35, 4208–4210 (2010). [CrossRef]
  48. Y. Chen, T. R. Nielsen, N. Gregersen, P. Lodahl, and J. Mørk, “Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides,” Phys. Rev. B 81, 125431 (2010). [CrossRef]
  49. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011). [CrossRef]
  50. M. Liu, T.-W. Lee, S. K. Gray, P. Guyot-Sionnest, and M. Pelton, “Excitation of dark plasmons in metal nanoparticles by a localized emitter,” Phys. Rev. Lett. 102, 107401 (2009). [CrossRef]
  51. F. J. García de Abajo, “Relativistic energy loss and induced photon emission in the interaction of a dielectric sphere with an external electron beam,” Phys. Rev. B 59, 3095–3107 (1999). [CrossRef]
  52. J. Kellendonk and S. Richard, “Weber–Schafheitlin-type integrals with exponent 1,” Integral Transforms Spec. Funct. 20, 147–153 (2009). [CrossRef]

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