Abstract

We study theoretically the absorbed power by a dielectric sphere when it is illuminated with partially coherent light coming from two pinholes. We present a general theory of Mie scattering of partially coherent light (based on the angular spectrum method); this theory is applied to the aforementioned particular scattering problem which is solved analytically. We found that, if the diameter of the sphere is smaller than the skin depth, the absorbed power by the sphere depends complicatedly on the degree of coherence of light between the pinholes. The absorbed power for coherent illumination can be smaller or greater than that for incoherent light between pinholes, depending on the geometrical configuration. Furthermore, there are particular setups in which the absorbed power is independent of the degree of coherence, despite that the intensity distribution of the electric field inside the sphere depends significantly on the spatial coherence. Hence, by tuning the coherence length between the pinholes, the absorbed power by the sphere can be controlled; if a whispering gallery mode is excited, the absorbed power can be varied over a wide range. Our study might have implications in the understanding of light absorption in photovoltaic nano-devices.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. G. Mie, “Beiträge zur Optiktrüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [Crossref]
  2. J.-J. Greffet, M. de la Cruz-Gutierrez, P. V. Ignatovich, and A. Radunsky, “Influence of spatial coherence on scattering by a particle,” J. Opt. Soc. Am. A 20, 2315–2320 (2003).
  3. T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
    [Crossref] [PubMed]
  4. D. G. Fischer, T. van Dijk, T. D. Visser, and E. Wolf, “Coherence effects in Mie scattering,” J. Opt. Soc. Am. A 29, 78–84 (2012).
  5. Y. Wang, S. Yan, D. Kuebel, and T. D. Visser, “Dynamic control of light scattering using spatial coherence,” Phys. Rev. A 92, 013806 (2015).
    [Crossref]
  6. Y. Wang, H. F. Schouten, and T. D. Visser, “Tunable, anomalous Mie scattering using spatial coherence,” Opt. Lett. 40, 4779–4782 (2015).
  7. Y. Wang, H. F. Schouten, and T. D. Visser, “Strong suppression of forward or backward Mie scattering by using spatial coherence,” J. Opt. Soc. Am. A 33, 513–518 (2016).
  8. D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Comm. 150, 239–250 (1998).
    [Crossref]
  9. J. Liu, L. Bi, P. Yang, and G. W. Kattawar, “Scattering of partially coherent electromagnetic beams by water droplets and ice crystals,” J. Quant. Spectrosc. Ra. 134, 74–84 (2014).
    [Crossref]
  10. M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express 18, 6628–6641 (2010).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  14. M. W. Hyde, “Physical optics solution for the scattering of a partially coherent wave from a circular cylinder,” Opt. Comm. 338, 233–239 (2015).
    [Crossref]
  15. Y. Liu and X. Zhang, “Coherent effect in superscattering,” J. Opt. Soc. Am. A 338, 2071–2075 (2016).
  16. J. Tervo and J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Comm. 209, 7–16 (2002).
    [Crossref]
  17. J. A. Gonzaga-Galeana and J. R. Zurita-Sánchez, “Alternative angular spectrum derivation of beam-shape coefficients of generalized Lorenz-Mie theory: scattering of light coming from two pinholes,” J. Electromagnet. Wave. (2018).
    [Crossref]
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    [Crossref]
  20. Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 664 (2012).
    [Crossref] [PubMed]
  21. P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
    [Crossref]
  22. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

2016 (2)

2015 (3)

Y. Wang, S. Yan, D. Kuebel, and T. D. Visser, “Dynamic control of light scattering using spatial coherence,” Phys. Rev. A 92, 013806 (2015).
[Crossref]

Y. Wang, H. F. Schouten, and T. D. Visser, “Tunable, anomalous Mie scattering using spatial coherence,” Opt. Lett. 40, 4779–4782 (2015).

M. W. Hyde, “Physical optics solution for the scattering of a partially coherent wave from a circular cylinder,” Opt. Comm. 338, 233–239 (2015).
[Crossref]

2014 (1)

J. Liu, L. Bi, P. Yang, and G. W. Kattawar, “Scattering of partially coherent electromagnetic beams by water droplets and ice crystals,” J. Quant. Spectrosc. Ra. 134, 74–84 (2014).
[Crossref]

2013 (2)

M. W. Hyde, A. E. Bogle, and M. J. Havrilla, “Scattering of partially -coherent wave from a material circular cylinder,” Opt. Express 21, 32327–32339 (2013).
[Crossref]

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

2012 (3)

2010 (2)

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[Crossref] [PubMed]

M. L. Marasinghe, M. Premaratne, and D. M. Paganin, “Coherence vortices in Mie scattering of statistically stationary partially coherent fields,” Opt. Express 18, 6628–6641 (2010).
[Crossref] [PubMed]

2006 (1)

2003 (1)

2002 (1)

J. Tervo and J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Comm. 209, 7–16 (2002).
[Crossref]

1998 (1)

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Comm. 150, 239–250 (1998).
[Crossref]

1994 (1)

1974 (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optiktrüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Aagesen, M.

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

Alonso, M. A.

Bi, L.

J. Liu, L. Bi, P. Yang, and G. W. Kattawar, “Scattering of partially coherent electromagnetic beams by water droplets and ice crystals,” J. Quant. Spectrosc. Ra. 134, 74–84 (2014).
[Crossref]

Bogle, A. E.

Brouder, C.

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Comm. 150, 239–250 (1998).
[Crossref]

Cabaret, D.

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Comm. 150, 239–250 (1998).
[Crossref]

Cui, Y.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 664 (2012).
[Crossref] [PubMed]

de la Cruz-Gutierrez, M.

Demichel, O.

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

Devaney, A. J.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

Fan, S.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 664 (2012).
[Crossref] [PubMed]

Fischer, D. G.

D. G. Fischer, T. van Dijk, T. D. Visser, and E. Wolf, “Coherence effects in Mie scattering,” J. Opt. Soc. Am. A 29, 78–84 (2012).

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[Crossref] [PubMed]

Friberg, A. T.

Gonzaga-Galeana, J. A.

J. A. Gonzaga-Galeana and J. R. Zurita-Sánchez, “Alternative angular spectrum derivation of beam-shape coefficients of generalized Lorenz-Mie theory: scattering of light coming from two pinholes,” J. Electromagnet. Wave. (2018).
[Crossref]

Greffet, J.-J.

Havrilla, M. J.

Heiss, M.

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

Holm, J. V.

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

Hyde, M. W.

M. W. Hyde, “Physical optics solution for the scattering of a partially coherent wave from a circular cylinder,” Opt. Comm. 338, 233–239 (2015).
[Crossref]

M. W. Hyde, A. E. Bogle, and M. J. Havrilla, “Scattering of partially -coherent wave from a material circular cylinder,” Opt. Express 21, 32327–32339 (2013).
[Crossref]

Ignatovich, P. V.

Jørgensen, H. I.

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

Kaivola, M.

Kattawar, G. W.

J. Liu, L. Bi, P. Yang, and G. W. Kattawar, “Scattering of partially coherent electromagnetic beams by water droplets and ice crystals,” J. Quant. Spectrosc. Ra. 134, 74–84 (2014).
[Crossref]

Krogstrup, P.

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

Kuebel, D.

Y. Wang, S. Yan, D. Kuebel, and T. D. Visser, “Dynamic control of light scattering using spatial coherence,” Phys. Rev. A 92, 013806 (2015).
[Crossref]

Lange, S.

Lindberg, J.

Liu, J.

J. Liu, L. Bi, P. Yang, and G. W. Kattawar, “Scattering of partially coherent electromagnetic beams by water droplets and ice crystals,” J. Quant. Spectrosc. Ra. 134, 74–84 (2014).
[Crossref]

Liu, Y.

Y. Liu and X. Zhang, “Coherent effect in superscattering,” J. Opt. Soc. Am. A 338, 2071–2075 (2016).

Marasinghe, M. L.

Mie, G.

G. Mie, “Beiträge zur Optiktrüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

Morral, A. F. i

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

Narasimhan, V. K.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 664 (2012).
[Crossref] [PubMed]

Nygard, J.

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

Paganin, D. M.

Premaratne, M.

Radunsky, A.

Rossano, S.

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Comm. 150, 239–250 (1998).
[Crossref]

Ruan, Z.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 664 (2012).
[Crossref] [PubMed]

Schouten, H. F.

Schweiger, G.

Setälä, T.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tervo, J.

J. Tervo and J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Comm. 209, 7–16 (2002).
[Crossref]

Turunen, J.

J. Tervo and J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Comm. 209, 7–16 (2002).
[Crossref]

van Dijk, T.

D. G. Fischer, T. van Dijk, T. D. Visser, and E. Wolf, “Coherence effects in Mie scattering,” J. Opt. Soc. Am. A 29, 78–84 (2012).

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[Crossref] [PubMed]

Visser, T. D.

Y. Wang, H. F. Schouten, and T. D. Visser, “Strong suppression of forward or backward Mie scattering by using spatial coherence,” J. Opt. Soc. Am. A 33, 513–518 (2016).

Y. Wang, H. F. Schouten, and T. D. Visser, “Tunable, anomalous Mie scattering using spatial coherence,” Opt. Lett. 40, 4779–4782 (2015).

Y. Wang, S. Yan, D. Kuebel, and T. D. Visser, “Dynamic control of light scattering using spatial coherence,” Phys. Rev. A 92, 013806 (2015).
[Crossref]

D. G. Fischer, T. van Dijk, T. D. Visser, and E. Wolf, “Coherence effects in Mie scattering,” J. Opt. Soc. Am. A 29, 78–84 (2012).

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[Crossref] [PubMed]

Wang, Y.

Wolf, E.

D. G. Fischer, T. van Dijk, T. D. Visser, and E. Wolf, “Coherence effects in Mie scattering,” J. Opt. Soc. Am. A 29, 78–84 (2012).

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[Crossref] [PubMed]

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

Xie, C.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 664 (2012).
[Crossref] [PubMed]

Yan, S.

Y. Wang, S. Yan, D. Kuebel, and T. D. Visser, “Dynamic control of light scattering using spatial coherence,” Phys. Rev. A 92, 013806 (2015).
[Crossref]

Yang, P.

J. Liu, L. Bi, P. Yang, and G. W. Kattawar, “Scattering of partially coherent electromagnetic beams by water droplets and ice crystals,” J. Quant. Spectrosc. Ra. 134, 74–84 (2014).
[Crossref]

Yao, J.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 664 (2012).
[Crossref] [PubMed]

Yao, Y.

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 664 (2012).
[Crossref] [PubMed]

Zhang, X.

Y. Liu and X. Zhang, “Coherent effect in superscattering,” J. Opt. Soc. Am. A 338, 2071–2075 (2016).

Zurita-Sánchez, J. R.

J. A. Gonzaga-Galeana and J. R. Zurita-Sánchez, “Alternative angular spectrum derivation of beam-shape coefficients of generalized Lorenz-Mie theory: scattering of light coming from two pinholes,” J. Electromagnet. Wave. (2018).
[Crossref]

Ann. Phys. (1)

G. Mie, “Beiträge zur Optiktrüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[Crossref]

J. Math. Phys. (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (1)

J. Quant. Spectrosc. Ra. (1)

J. Liu, L. Bi, P. Yang, and G. W. Kattawar, “Scattering of partially coherent electromagnetic beams by water droplets and ice crystals,” J. Quant. Spectrosc. Ra. 134, 74–84 (2014).
[Crossref]

Nat. Commun. (1)

Y. Yao, J. Yao, V. K. Narasimhan, Z. Ruan, C. Xie, S. Fan, and Y. Cui, “Broadband light management using low-Q whispering gallery modes in spherical nanoshells,” Nat. Commun. 3, 664 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

P. Krogstrup, H. I. Jørgensen, M. Heiss, O. Demichel, J. V. Holm, M. Aagesen, J. Nygard, and A. F. i Morral, “Single nanowire solar cells beyond the Shockley-Queisser limit,” Nat. Photonics 7, 306–310 (2013).
[Crossref]

Opt. Comm. (3)

M. W. Hyde, “Physical optics solution for the scattering of a partially coherent wave from a circular cylinder,” Opt. Comm. 338, 233–239 (2015).
[Crossref]

D. Cabaret, S. Rossano, and C. Brouder, “Mie scattering of a partially coherent beam,” Opt. Comm. 150, 239–250 (1998).
[Crossref]

J. Tervo and J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Comm. 209, 7–16 (2002).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (1)

Y. Wang, S. Yan, D. Kuebel, and T. D. Visser, “Dynamic control of light scattering using spatial coherence,” Phys. Rev. A 92, 013806 (2015).
[Crossref]

Phys. Rev. Lett. (1)

T. van Dijk, D. G. Fischer, T. D. Visser, and E. Wolf, “Effects of spatial coherence on the angular distribution of radiant intensity generated by scattering on a sphere,” Phys. Rev. Lett. 104, 173902 (2010).
[Crossref] [PubMed]

Other (2)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

J. A. Gonzaga-Galeana and J. R. Zurita-Sánchez, “Alternative angular spectrum derivation of beam-shape coefficients of generalized Lorenz-Mie theory: scattering of light coming from two pinholes,” J. Electromagnet. Wave. (2018).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Scattering of an arbitrary partially coherent beam. (b) Stochastic light coming out from two pinholes at plane z′ = 0.
Fig. 2
Fig. 2 Normalized absorbed power as a function of the normalized radius k1a for a source-sphere distance k1d = 50 and a half inter-pinhole spacing k1x0 = 25 when the the light correlation between the pinholes is coherent and incoherent. The insets show a close up of the indicated regions.
Fig. 3
Fig. 3 On-resonance case (TMl=4 WGM). (a) Normalized absorbed power as a function of the normalized half inter-pinhole separation k1x0. Normalized spectral electric intensity S ¯ ( r , ω ) = 2 n inside the sphere when rxy-, xz- and yz-planes for: (b) k1x0 = 22; (c) k1x0 = 34.572; (d) k1x0 = 44. For all cases: three states of source coherence are considered (coherent, partially coherent, and incoherent), the source-sphere distance is k1d = 50, and the normalized radius is k1a = 2.8569.
Fig. 4
Fig. 4 Off-resonance case. (a) Normalized absorbed power as a function of the normalized half inter-pinhole separation k1x0. Normalized spectral electric intensity S ¯ ( r , ω ) = 2 n inside the sphere when rxy-, xz- and yz-planes for: (b) k1x0 = 33; (c) k1x0 = 51.253; (d) k1x0 = 83. For all cases: three states of source coherence are considered (coherent, partially coherent, and incoherent), the source-sphere distance is k1d = 50, and the normalized radius is k1a = 3.275.
Fig. 5
Fig. 5 Case 2aδs. Normalized electric spectral intensity S ¯ ( r , ω ) = 2 n. Three states of source coherence are considered (coherent, partially coherent, and incoherent), the source-sphere distance is k1d = 50, half inter-pinhole separation k1x0 = 25 and the normalized radius is k1a = 20.185.

Equations (59)

Equations on this page are rendered with MathJax. Learn more.

x = x , y = y , z = z d .
n x = n x , n y = n y , n z = n z .
E 0 i * ( r 1 , ω ) E 0 j ( r 1 , ω ) = W i j ( 00 ) ( r 1 , r 2 , ω ) δ ( ω ω ) ,
W i j ( 00 ) ( r 1 , r 2 , ω ) = W i j ( κ 1 , κ 2 , ω ) e i ( k 1 * r 1 + k 2 r 1 ) d 2 κ 1 d 2 κ 2 .
W i j ( κ 1 , κ 2 , ω ) = 1 ( 2 π ) 4 W i j ( 00 ) ( ρ 1 , ρ 2 , ω ) e i ( κ 1 ρ 1 κ 2 ρ 2 ) d 2 ρ 1 d 2 ρ 2 .
k z n = k 1 2 ( k x n 2 + k y n 2 )
E γ i * ( r 1 , ω ) E γ j ( r 2 , ω ) = W i j ( γ γ ) ( r 1 , r 2 , ω ) δ ( ω ω ) ,
W i j ( 00 ) ( r 1 , r 2 , ω ) = η η [ A η η ( MM ) ( ω ) M η * ( i ) ( r 1 , k 1 ) M η ( j ) ( r 2 , k 1 ) + A η η ( NN ) ( ω ) N η η * ( i ) ( r 1 , k 1 ) × N η ( j ) ( r 2 , k 1 ) + A η η ( MN ) ( ω ) M η * ( i ) ( r 1 , k 1 ) N η ( j ) ( r 2 , k 1 ) + A η η ( NM ) ( ω ) N η * ( i ) ( r 1 , k 1 ) M η ( j ) ( r 2 , k 1 ) ] ,
M σ l m ( r , k ) = × [ r j l ( k r ) y σ l m ( θ , ϕ ) ] ,
N σ l m ( r , k ) = 1 k × M σ l m ( r , k ) ,
y   o e l m ( θ , ϕ ) P l m ( cos θ ) cos ( m ϕ ) sin ( m ϕ )
W i j ( 00 ) ( r 1 , r 2 , ω ) = W i j ( κ 1 , κ 2 , ω ) e i ( k 1 * r 1 + k 2 r 2 ) e i ( k z 1 * d + k z 2 d ) d 2 κ 1 d 2 κ 2 .
W i j ( 00 ) ( r 1 , r 2 , ω ) = k 1 4 R 2 R 1 W i j ( κ 1 , κ 2 , ω ) e i ( k 1 * r 1 + k 2 r 2 ) e i k 1 d ( cos * α 1 + cos α 2 ) (d 2 ζ 1 ) * d 2 ζ 2 .
W i j ( κ 1 , κ 2 , ω ) e i ( k 1 * r 1 + k 2 r 2 ) = η η [ A η η ( MM ) ( κ 1 , κ 2 , ω ) M η * ( i ) ( r 1 , k 1 ) M η ( j ) ( r 2 , k 1 ) + A η η ( NN ) ( κ 1 , κ 2 , ω ) N η * ( i ) ( r 1 , k 1 ) N η ( j ) ( r 2 , k 1 ) × A η η ( MN ) ( κ 1 , κ 2 , ω ) M η * ( i ) ( r 1 , k 1 ) N η ( j ) ( r 2 , k 1 ) + A η η ( NM ) ( κ 1 , κ 2 , ω ) N η * ( i ) ( r 1 , k 1 ) M η ( j ) ( r 2 , k 1 ) ] ,
A η η ( p q ) ( κ 1 , κ 2 , ω ) = f 1 η η ( p q ) ( ζ 1 , ζ 2 ) W x x ( κ 1 , κ 2 , ω ) + f 2 η η ( p q ) ( ζ 1 , ζ 2 ) W x y ( κ 1 , κ 2 , ω ) + f 3 η η ( p q ) ( ζ 1 , ζ 2 ) W y x ( κ 1 , κ 2 , ω ) + f 4 η η ( p q ) ( ζ 1 , ζ 2 ) W y y ( κ 1 , κ 2 , ω ) ,
A η η ( p q ) ( ω ) = k 1 4 R 2 R 1 A η η ( p q ) ( κ 1 , κ 2 , ω ) e i k 1 d ( cos * α 1 + cos α 2 ) (d 2 ζ 1 ) * d 2 ζ 2 , p , q = M , N .
W i j ( r 1 , r 2 , ω ) = [ W i j ( 00 ) ( r 1 , r 2 , ω ) W i j ( 01 ) ( r 1 , r 2 , ω ) W i j ( 02 ) ( r 1 , r 2 , ω ) W i j ( 10 ) ( r 1 , r 2 , ω ) W i j ( 11 ) ( r 1 , r 2 , ω ) W i j ( 12 ) ( r 1 , r 2 , ω ) W i j ( 20 ) ( r 1 , r 2 , ω ) W i j ( 21 ) ( r 1 , r 2 , ω ) W i j ( 22 ) ( r 1 , r 2 , ω ) ] .
W i j ( r 1 , r 2 , ω ) = η η [ A η η ( MM ) ( ω ) u η * ( i ) ( r 1 , ω ) u η T ( j ) ( r 2 , ω ) + A η η ( MN ) ( ω ) u η * ( i ) ( r 1 , ω ) v η T ( j ) ( r 2 , ω ) + A η η ( NM ) ( ω ) v η * ( i ) ( r 1 , ω ) u η T ( j ) ( r 2 , ω ) + A η η ( NM ) ( ω ) v η * ( i ) ( r 1 , ω ) v η T ( j ) ( r 2 , ω ) ] ,
u σ l m ( i ) ( r , ω ) = [ M σ l m ( i ) ( r , k 1 ) a ˜ l ( ω ) M ¯ σ l m ( i ) ( r , k 1 ) c ˜ l ( ω ) M σ l m ( i ) ( r , k 2 ) ] , v σ l m ( i ) ( r , ω ) = [ N σ l m ( i ) ( r , k 1 ) b ˜ l ( ω ) N ¯ σ l m ( i ) ( r , k 1 ) d ˜ l ( ω ) N σ l m ( i ) ( r , k 2 ) ] .
S ( r , ω ) = i W i i ( 22 ) ( r , r , ω ) .
P ¯ T = 0 P ( ω ) d ω
P ( ω ) = 2 ε o Im [ ( ϵ 2 ω ) ] ω V S ( r , ω ) d 3 r
M η * ( r , k 2 ) N η * ( r , k 2 ) M η ( r , k 2 ) N η ( r , k 2 ) .
0 a r 2 | j l ( k r ) | 2 d r = a 2 Re [ k ] Im [ k ] Im [ k a j l * ( k a ) j l + 1 ( k a ) ] .
V S ( r , ω ) d 3 r = π a Re [ k 2 ] Im [ k 2 ] σ l m ( 1 + δ 0 m ) l ( l + 1 ) 2 l + 1 ( l + m ) ! ( l m ) ! { A σ l m σ l m ( MM ) ( ω ) | c ˜ l ( ω ) | 2 × Im [ k 2 a j l * ( k 2 a ) j l + 1 ( k 2 a ) ] + A σ l m σ l m ( NN ) ( ω ) | d ˜ l ( ω ) | 2 2 l + 1 [ ( l + 1 ) × Im [ k 2 a j l 1 * ( k 2 a ) j l ( k 2 a ) ] + l Im [ k 2 a j l + 1 * ( k 2 a ) j l + 2 ( k 2 a ) ] ] } .
W ξ ζ ( 00 ) ( ρ 1 , ρ 1 , ω ) = Q i j ( ρ 01 , ρ 01 , ω ) δ ( 2 ) ( ρ 1 ρ 01 ) δ ( 2 ) ( ρ 2 ρ 01 ) + Q i j ( ρ 02 , ρ 02 , ω ) × δ ( 2 ) ( ρ 1 ρ 02 ) δ ( 2 ) ( ρ 2 ρ 02 ) + Q i j ( ρ 01 , ρ 02 , ω ) δ ( 2 ) ( ρ 1 ρ 01 ) × δ ( 2 ) ( ρ 2 ρ 02 ) + Q i j ( ρ 02 , ρ 01 , ω ) δ ( 2 ) ( ρ 1 ρ 02 ) δ ( 2 ) ( ρ 2 ρ 01 ) .
W ξ ζ ( κ 1 , κ 2 , ω ) = 1 ( 2 π ) 4 n , s = 1 2 Q ξ ζ ( ρ 0 n , ρ 0 s , ω ) exp [ i ( k x 1 x 0 n + k y 1 y 0 n ) ] × exp [ i ( k x 2 x 0 s + k y 2 y 0 s ) ] .
h l ( k r ) y σ l m ( θ , ϕ ) = ( 1 ) l 2 π R sin α y σ l m ( α , β ) exp [ i k r ] d β d α
A σ l m σ l m ( p q ) ( ω ) = V l m ( k 1 ) V l m ( k 1 ) n , s = 1 2 p σ l m * ( p ) T ( r 0 n , k 1 ) ( ρ 0 n , ρ 0 , s , ω ) p σ l m ( q ) ( r 0 s , k 1 ) ,
V l m ( k 1 ) = k 1 2 π ( 1 ) l 1 + δ ˜ 0 m 2 l + 1 l ( l + 1 ) ( l m ) ! ( l + m ) ! ,
p σ l m ( M ) ( r , k 1 ) = [ N ¯ σ l m ( y ) ( r , k 1 ) N ¯ σ l m ( x ) ( r , k 1 ) ] , p σ l m ( N ) ( r , k 1 ) = [ M ¯ σ l m ( y ) ( r , k 1 ) M ¯ σ l m ( x ) ( r , k 1 ) ] ,
( ρ 0 n , ρ 0 s , ω ) = [ Q x x ( ρ 0 n , ρ 0 s , ω ) Q x y ( ρ 0 n , ρ 0 s , ω ) Q y x ( ρ 0 n , ρ 0 s , ω ) Q y y ( ρ 0 n , ρ 0 s , ω ) ] , n , s = 1 , 2 .
Q ξ ζ ( ρ 0 n , ρ 0 s , ω ) = Q ˜ ξ ζ ( ω ) exp [ ( | ρ 0 n | 2 + | ρ 0 s | 2 ) / w 0 2 ] exp [ ( | ρ 0 n ρ 0 s | 2 ) / ( 2 σ 0 2 ) ] ,
P ¯ = a 2 P ( ω ) / [ ε 0 c Q ˜ 0 ( ω ) ] .
S ¯ ( r , ω ) = S ( r , ω ) k 1 4 Q ˜ 0 ( ω ) .
f 1 η η ( p q ) ( ζ 1 , ζ 2 ) = Ψ η * ( p ) ( ζ 1 ) Ψ η ( q ) ( ζ 2 ) ,
f 2 η η ( p q ) ( ζ 1 , ζ 2 ) = Ψ η * ( p ) ( ζ 1 ) Γ η ( q ) ( ζ 2 ) ,
f 3 η η ( p q ) ( ζ 1 , ζ 2 ) = Γ η * ( p ) ( ζ 1 ) Ψ η ( q ) ( ζ 2 ) ,
f 4 η η ( p q ) ( ζ 1 , ζ 2 ) = Γ η * ( p ) ( ζ 1 ) Γ η ( q ) ( ζ 2 ) ,
Ψ   o e l m ( M ) ( ζ ) = ± P l m cos α [ Λ   e o l m ( 1 ) ( α , β ) + Λ   e o l m ( 2 ) ( α , β ) ] ,
Ψ   o e l m ( N ) ( ζ ) = i ( 2 l + 1 ) P l m cos α [ Λ   o e l m ( 3 ) ( α , β ) + Λ   o e l m ( 4 ) ( α , β ) ] ,
Γ   o e l m ( M ) ( ζ ) = P l m cos α [ Λ   o e l m ( 1 ) ( α , β ) + Λ   o e l m ( 2 ) ( α , β ) ] ,
Γ   o e l m ( M ) ( ζ ) = i ( 2 l + 1 ) P l m cos α [ Λ   e o l m ( 3 ) ( α , β ) + Λ   e o l m ( 4 ) ( α , β ) ] .
P l m = i l l ( l + 1 ) ( l m ) ! ( l + m ) ! ,
Λ   e o l m ( 1 ) ( α , β ) = l y   e o l + 1 m + 1 ( α , β ) + ( l + 1 ) y   e o l 1 m + 1 ( α , β ) ,
Λ   e o l m ( 2 ) ( α , β ) = ( 1 δ ˜ 0 m ) [ l ( l m + 1 ) ( l m + 2 ) y   e o l + 1 m 1 ( α , β ) + ( l + 1 ) ( l + m ) ( l + m 1 ) y   e o l 1 m 1 ( α , β ) ] ,
Λ   e o l m ( 3 ) ( α , β ) = y   e o l m + 1 ( α , β ) ,
Λ   e o l m ( 4 ) ( α , β ) = ( 1 δ ˜ 0 m ) ( l + m ) ( l m + 1 ) y   e o l m 1 ( α , β )
a ˜ l ( ω ) = μ 2 j l ( ρ 2 ) [ ρ 1 j l ( ρ 1 ) ] μ 1 j l ( ρ 1 ) [ ρ 2 j l ( ρ 2 ) ] μ 1 h l ( ρ 1 ) [ ρ 2 j l ( ρ 2 ) ] μ 2 j l ( ρ 2 ) [ ρ 1 h l ( ρ 1 ) ] ,
b ˜ l ( ω ) = ϵ 2 j l ( ρ 2 ) [ ρ 1 j l ( ρ 1 ) ] ϵ 1 j l ( ρ 1 ) [ ρ 2 j l ( ρ 2 ) ] ϵ 1 h l ( ρ 1 ) [ ρ 2 j l ( ρ 2 ) ] ϵ 2 j l ( ρ 2 ) [ ρ 1 h l ( ρ 1 ) ] ,
c ˜ l ( ω ) = μ 2 h l ( ρ 1 ) [ ρ 1 j l ( ρ 1 ) ] μ 2 j l ( ρ 1 ) [ ρ 1 h l ( ρ 1 ) ] μ 1 h l ( ρ 1 ) [ ρ 2 j l ( ρ 2 ) ] μ 2 j l ( ρ 2 ) [ ρ 1 h l ( ρ 1 ) ] ,
d ˜ l ( ω ) = Z 2 Z 1 ϵ 2 h l ( ρ 1 ) [ ρ 1 j l ( ρ 1 ) ] ϵ 2 j l ( ρ 1 ) [ ρ 1 h l ( ρ 1 ) ] ϵ 1 h l ( ρ 1 ) [ ρ 2 j l ( ρ 2 ) ] ϵ 2 j l ( ρ 2 ) [ ρ 1 h l ( ρ 1 ) ] ,
M η * ( r , k ) M η ( r , k ) d Ω = 2 π ( 1 + δ 0 m ) l ( l + 1 ) ( 2 l + 1 ) ( l + m ) ! ( l m ) ! δ ˜ σ σ δ ˜ m m δ ˜ l l | j l ( k r ) | 2 ,
N η * ( r , k ) N η ( r , k ) d Ω = 2 π ( 1 + δ 0 m ) l ( l + 1 ) ( 2 l + 1 ) 2 ( l + m ) ! ( l m ) ! δ ˜ σ σ δ ˜ m m δ ˜ l l × [ ( l + 1 ) | j l 1 ( k r ) | 2 + l | j l + 1 ( k r ) | 2 ] ,
M η * ( r , k ) N η ( r , k ) d Ω = 0 .
M ¯   o e l m ( x ) ( r , k 1 ) = ± h l ( k 1 r ) 2 [ ( 1 ± δ ˜ 0 m ) y   e o l m + 1 ( θ , ϕ ) + ( 1 δ ˜ 0 m ) ( l + m ) ( l m + 1 ) × y   e o l m + 1 ( θ , ϕ ) ] ,
M ¯   o e l m ( y ) ( r , k 1 ) = h l ( k 1 r ) 2 [ ( 1 ± δ ˜ 0 m ) y   o e l m + 1 ( θ , ϕ ) + ( 1 δ ˜ 0 m ) ( l + m ) ( l m + 1 ) × y   o e l m 1 ( θ , ϕ ) ] ,
N ¯   o e l m ( x ) ( r , k 1 ) = 1 2 ( 2 l + 1 ) { ( 1 ± δ ˜ 0 m ) [ l h l + 1 ( k 1 r ) y   o e l + 1 m + 1 ( θ , ϕ ) + ( l + 1 ) h l 1 ( k 1 r ) × y   o e l 1 m + 1 ( θ , ϕ ) ] + ( 1 δ ˜ 0 m ) [ l ( l m + 1 ) ( l m + 2 ) h l + 1 ( k 1 r ) × y   o e l + 1 m 1 ( θ , ϕ ) ( l + 1 ) ( l + m 1 ) ( l + m ) h l 1 ( k 1 r ) × y   o e l 1 m 1 ( θ , ϕ ) ] } ,
N ¯   o e l m ( y ) ( r , k 1 ) = ± 1 2 ( 2 l + 1 ) { ( 1 ± δ ˜ 0 m ) [ l h l + 1 ( k 1 r ) y   e o l + 1 m + 1 ( θ , ϕ ) + ( l + 1 ) h l 1 ( k 1 r ) × y   e o l 1 m + 1 ( θ , ϕ ) ] + ( 1 δ ˜ 0 m ) [ l ( l m + 1 ) ( l m + 2 ) h l + 1 ( k 1 r ) × y   e o l + 1 m 1 ( θ , ϕ ) + ( l + 1 ) ( l + m 1 ) ( l + m ) h l 1 ( k 1 r ) × y   e o l 1 m 1 ( θ , ϕ ) ] } .

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