Abstract

We present a new theoretical analysis for the light scattering at sub-wavelength dielectric structures. This analysis can provide new intuitive insights into the phase shift of the scattered light that cannot be obtained from the existing approaches. Unlike the traditional analytical (e.g. Mie formalism) and numerical (e.g. FDTD) approaches, which simulate light scattering by rigorously matching electromagnetic fields at boundaries, we consider sub-wavelength dielectric structures as leaky resonators and evaluate the light scattering as a coupling process between incident light and leaky modes of the structure. Our analysis indicates that the light scattering is fundamentally dictated by the eigenvalue of the leaky modes. It indicates that the upper limit for the scattering efficiency of a cylindrical cylinder in free space is 4n, where n is the refractive index. It also indicates that the phase shift of the forward scattered light at dielectric structures can only cover half of the phase space [0, 2π], but backward scattering can provide a full phase coverage.

© 2013 OSA

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    [Crossref] [PubMed]
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  10. A. Alù and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102(23), 233901 (2009).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  19. Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
    [Crossref]
  20. Z. Ruan and S. Fan, “Super-scattering of light from subwavelength nano-structures,” Phys. Rev. Lett. 105(1), 013901 (2010).
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2012 (2)

Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Y. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett. 12(7), 3674–3681 (2012).
[Crossref] [PubMed]

Y. Yu and L. Y. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express 20(13), 13847–13856 (2012).
[Crossref] [PubMed]

2011 (1)

Y. M. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref] [PubMed]

2010 (2)

Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
[Crossref]

Z. Ruan and S. Fan, “Super-scattering of light from subwavelength nano-structures,” Phys. Rev. Lett. 105(1), 013901 (2010).
[Crossref] [PubMed]

2009 (5)

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[Crossref] [PubMed]

O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. 9(3), 930–934 (2009).
[Crossref] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3(7), 388–394 (2009).
[Crossref]

A. Alù and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102(23), 233901 (2009).
[Crossref] [PubMed]

2008 (1)

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

2007 (3)

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
[Crossref]

Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686–1686 (2007).
[Crossref] [PubMed]

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[Crossref]

2005 (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Algra, R. E.

O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. 9(3), 930–934 (2009).
[Crossref] [PubMed]

Alivisatos, A. P.

Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Y. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett. 12(7), 3674–3681 (2012).
[Crossref] [PubMed]

Alù, A.

A. Alù and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102(23), 233901 (2009).
[Crossref] [PubMed]

Bakkers, E. P. A. M.

O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. 9(3), 930–934 (2009).
[Crossref] [PubMed]

Brongersma, M. L.

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[Crossref] [PubMed]

Cai, W. S.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
[Crossref]

Cao, L. Y.

Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Y. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett. 12(7), 3674–3681 (2012).
[Crossref] [PubMed]

Y. Yu and L. Y. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express 20(13), 13847–13856 (2012).
[Crossref] [PubMed]

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[Crossref] [PubMed]

Chettiar, U. K.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
[Crossref]

Clemens, B. M.

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[Crossref] [PubMed]

Diedenhofen, S. L.

O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. 9(3), 930–934 (2009).
[Crossref] [PubMed]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Engheta, N.

A. Alù and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102(23), 233901 (2009).
[Crossref] [PubMed]

Fan, S.

Z. Ruan and S. Fan, “Super-scattering of light from subwavelength nano-structures,” Phys. Rev. Lett. 105(1), 013901 (2010).
[Crossref] [PubMed]

Fan, S. H.

Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
[Crossref]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Ferry, V. E.

Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Y. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett. 12(7), 3674–3681 (2012).
[Crossref] [PubMed]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Genov, D. A.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Giessen, H.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Hamam, R. E.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[Crossref]

Inouye, Y.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3(7), 388–394 (2009).
[Crossref]

Joannopoulos, J. D.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[Crossref]

Kaas, B. C.

O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. 9(3), 930–934 (2009).
[Crossref] [PubMed]

Karalis, A.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[Crossref]

Kästel, J.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Kawata, S.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3(7), 388–394 (2009).
[Crossref]

Kildishev, A. V.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
[Crossref]

Lagendijk, A.

O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. 9(3), 930–934 (2009).
[Crossref] [PubMed]

Langguth, L.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Lee, H.

Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686–1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Liu, M.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

Liu, N.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Liu, Y. M.

Y. M. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref] [PubMed]

Liu, Z. W.

Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686–1686 (2007).
[Crossref] [PubMed]

Muskens, O. L.

O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. 9(3), 930–934 (2009).
[Crossref] [PubMed]

Park, J. S.

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[Crossref] [PubMed]

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

Rivas, J. G.

O. L. Muskens, S. L. Diedenhofen, B. C. Kaas, R. E. Algra, E. P. A. M. Bakkers, J. G. Rivas, and A. Lagendijk, “Large photonic strength of highly tunable resonant nanowire materials,” Nano Lett. 9(3), 930–934 (2009).
[Crossref] [PubMed]

Ruan, Z.

Z. Ruan and S. Fan, “Super-scattering of light from subwavelength nano-structures,” Phys. Rev. Lett. 105(1), 013901 (2010).
[Crossref] [PubMed]

Ruan, Z. C.

Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
[Crossref]

Schuller, J. A.

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[Crossref] [PubMed]

Shalaev, V. M.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
[Crossref]

Soljacic, M.

R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75(5), 053801 (2007).
[Crossref]

Sun, C.

Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686–1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Verma, P.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3(7), 388–394 (2009).
[Crossref]

Wang, Y.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

Weiss, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref] [PubMed]

White, J. S.

L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. 8(8), 643–647 (2009).
[Crossref] [PubMed]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[Crossref]

Xiong, Y.

Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686–1686 (2007).
[Crossref] [PubMed]

Yariv, A.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).

Yu, Y.

Y. Yu and L. Y. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express 20(13), 13847–13856 (2012).
[Crossref] [PubMed]

Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Y. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett. 12(7), 3674–3681 (2012).
[Crossref] [PubMed]

Zhang, S.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

Zhang, X.

Y. M. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref] [PubMed]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686–1686 (2007).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Chem. Soc. Rev. (1)

Y. M. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref] [PubMed]

Electron. Lett. (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).

J. Phys. Chem. C (1)

Z. C. Ruan and S. H. Fan, “Temporal coupled-mode theory for Fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C 114(16), 7324–7329 (2010).
[Crossref]

Nano Lett. (2)

Y. Yu, V. E. Ferry, A. P. Alivisatos, and L. Y. Cao, “Dielectric core-shell optical antennas for strong solar absorption enhancement,” Nano Lett. 12(7), 3674–3681 (2012).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Schematic illustration for a circular cylinder.

Fig. 2
Fig. 2

Schematic illustration for a spherical structure.

Fig. 3
Fig. 3

Schematic illustration for the coupling of external waves with leaky modes. The dashed line indicates an arbitrary dielectric structure inside.

Fig. 4
Fig. 4

The coupling of an arbitrary incident frequency with multiple leaky modes. The dots are TMml leaky modes plotted as function of the real part (Nreal) of the eigenvalue and the mode number m. The order number l is also noted in the figure. The solid line indicates the normalized parameter for the incident frequency, and the two dashed lines encircle the modes that can couple with the incident frequency.

Fig. 5
Fig. 5

Calculated scattering coefficient and efficiency using the CLMT model (solid red lines) and Mie theory (blue dashed lines) as a function of the normalized parameter nkr0 for 1D cylinders (the refractive index is assumed to be 4). (a) The calculated real and (b) imaginary parts of a typical scattering coefficient, b0. The calculation with the CLMT model involves eight leaky modes TM01, TM02…TM07. (c) Calculated scattering efficiency Qsca. The calculation using the CLMT model involves all the 56 leaky modes of TMml with m = 0-7 and l = 1-7.

Fig. 6
Fig. 6

Calculated scattering coefficient using the CLMT model (solid red lines) and Mie theory (blue dashed lines) as a function of the normalized parameter nkr0 for 0D spheres (the refractive index is assumed to be 4). (a) The calculated real and (b) imaginary parts of a typical scattering coefficient, b1. The calculation using the CLMT model involves seven leaky modes TE11, TE12…TE17.

Tables (2)

Tables Icon

Table 1 Calculated Intrinsic Phase of TM Leaky Modes in 1D Cylinders (π)

Tables Icon

Table 2 Calculated Intrinsic Phase of TE Leaky Modes in 0D Spheres (π)

Equations (35)

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Inside, E z (r,ϕ,z)=C J m (κr) J m (κ r 0 ) e iβz e imϕ
Outside, E z (r,ϕ,z)=C H m (γr) H m (γ r 0 ) e iβz e imϕ
κ 2 = n 2 k 2 β 2
γ 2 = k 2 β 2
Inside, E z (r,ϕ,z)=C J m (κr) J m (κ r 0 ) e iβz sin(mϕ)
Outside, E z (r,ϕ,z)=C H m (γr) H m (γ r 0 ) e iβz sin(mϕ)
Inside, E z (r,ϕ,z)=C J m (κr) J m (κ r 0 ) e iβz cos(mϕ)
Outside, E z (r,ϕ,z)=C H m (γr) H m (γ r 0 ) e iβz cos(mϕ)
Inside, H z (r,ϕ,z)= C ' J m (κr) J m (κ r 0 ) e iβz cos(mϕ)
Outside, H z (r,ϕ,z)= C ' H m (γr) H m (γ r 0 ) e iβz cos(mϕ)
Inside, H z (r,ϕ,z)= C ' J m (κr) J m (κ r 0 ) e iβz sin(mϕ)
Outside, H z (r,ϕ,z)= C ' H m (γr) H m (γ r 0 ) e iβz sin(mϕ)
E ϕ = i α 2 ( β r E z ϕ ωμ H z r )
H ϕ = i α 2 (ωε E z ϕ + β r H z r )
( 1 κ 2 1 γ 2 ) βm r 0 C ' =Cω( n 2 J m ' (κ r 0 ) κ J m (κ r 0 ) H m ' (γ r 0 ) γ H m (γ r 0 ) )
( 1 κ 2 1 γ 2 ) βm r 0 C= C ' ω( μ J m ' (κ r 0 ) κ J m (κ r 0 ) μ H m ' (γ r 0 ) γ H m (γ r 0 ) )
J m ' (nk r 0 ) n J m (nk r 0 ) H m ' (k r 0 ) H m (k r 0 ) =0
n J m ' (nk r 0 ) J m (nk r 0 ) H m ' (k r 0 ) H m (k r 0 ) =0
Inside: E t =D F e,olm (θ,ϕ) j m (nkr)
Outside: E t = D ' F e,olm (θ,ϕ) h m (kr)
Inside: H t =D nk iωμ G e,olm (θ,ϕ) [ nkr j m (nkr) ] nkr
Outside: H t = D ' k iωμ G e,olm (θ,ϕ) [ kr h m (kr) ] kr
D F e,olm (θ,ϕ) j m (nk r 0 )= D ' F e,olm (θ,ϕ) h m (k r 0 )
D ' nk iωμ G e,olm (θ,ϕ) [ nk r 0 j m (nk r 0 ) ] ' nk r 0 = D ' k iωμ G e,olm (θ,ϕ) [ k r 0 h m (k r 0 ) ] ' k r 0
[nk r 0 j m (nk r 0 )] ' j m (nk r 0 ) = [k r 0 h m (k r 0 )] ' h m (k r 0 )
[nk r 0 j m (nk r 0 )] ' n 2 j m (nk r 0 ) = [k r 0 h m (k r 0 )] ' h m (k r 0 )
TE modes: ψ m (nk r 0 ) ψ m ' (nk r 0 ) =n ξ m (k r 0 ) ξ m ' (k r 0 )
TM modes: n ψ m (nk r 0 ) ψ m ' (nk r 0 ) = ξ m (k r 0 ) ξ m ' (k r 0 )
d a ml dt =(iω γ ml ) a ml + κ ml W ml +
W ml = C a,ml a ml + C W,ml W ml +
R ml = W ml W ml + = γ ml i( ω ml ω) γ ml +i( ω ml ω) e i2 θ ml = e i(2 θ ml + Δ ml )
b ml = R ml 1 2 = e i(2 θ ml + Δ ml ) 1 2 =sin( θ ml + Δ ml /2) e i( θ ml + Δ ml /2+π/2)
e i Δ ml = γ ml i( ω ml ω) γ ml +i( ω ml ω) = 1iβ 1+iβ
e i θ ml = W ml 2 γ ml a ml = J m (nk r 0 )/ H m (1) (k r 0 ) | J m (nk r 0 )/ H m (1) (k r 0 )|
e i θ ml = j m (nk r 0 )/ h m (k r 0 ) | j m (nk r 0 )/ h m (k r 0 )|

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