Abstract

We confirm experimentally that the degree and state of polarization of a random, partially polarized electromagnetic beam can be obtained by probing the field with a nanoscatterer. We use a gold nanocube on silicon substrate as a local scatterer and detect the polarization characteristics of the scattered far field, which enables us to deduce the state of partial polarization of the field at the nanoprobe site. In contrast to previous beam characterization methods where spatial resolution is limited by the pixel size of the detector, the accuracy of the current technique is specified by the particle size. Our work is the first step towards polarization-state detection of random optical near fields for which the use of nanoprobes is required.

© 2016 Optical Society of America

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References

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  1. B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, 2007).
  2. E. Collet, Polarized Light in Fiber Optics (The PolaWave Group, 2003).
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  4. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
    [Crossref]
  5. T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
    [Crossref] [PubMed]
  6. A. Madrazo, R. Carminati, M. Nieto-Vesperinas, and J.-J. Greffet, “Polarization effects in the optical interaction between a nanoparticle and a corrugated surface: implications for apertureless near-field microscopy,” J. Opt. Soc. Am. A 15, 109–119 (1998).
    [Crossref]
  7. R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. A: Pure Appl. Opt. 6, S18–S23 (2004).
    [Crossref]
  8. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [Crossref] [PubMed]
  9. K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
    [Crossref]
  10. T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
    [Crossref] [PubMed]
  11. L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635, (2014).
    [Crossref]
  12. E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
    [Crossref]
  13. T. Hakkarainen, T. Setälä, and A. T. Friberg, “Electromagnetic near-field interactions of a dipolar emitter with metal and metamaterial nanoslabs,” Phys. Rev. A 84, 033849 (2011).
    [Crossref]
  14. C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, 1971).
  15. L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
    [Crossref]
  16. L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young’s interferometer,” Opt. Lett. 40, 2898–2901 (2015).
    [Crossref]

2015 (2)

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young’s interferometer,” Opt. Lett. 40, 2898–2901 (2015).
[Crossref]

2014 (2)

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635, (2014).
[Crossref]

2013 (1)

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[Crossref]

2011 (1)

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Electromagnetic near-field interactions of a dipolar emitter with metal and metamaterial nanoslabs,” Phys. Rev. A 84, 033849 (2011).
[Crossref]

2007 (1)

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

2004 (1)

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. A: Pure Appl. Opt. 6, S18–S23 (2004).
[Crossref]

2001 (2)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[Crossref] [PubMed]

1998 (1)

Banzer, P.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Bauer, T.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Boyd, R. W.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Carminati, R.

Collet, E.

E. Collet, Polarized Light in Fiber Optics (The PolaWave Group, 2003).

Dändliker, R.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. A: Pure Appl. Opt. 6, S18–S23 (2004).
[Crossref]

Friberg, A. T.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young’s interferometer,” Opt. Lett. 40, 2898–2901 (2015).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635, (2014).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Electromagnetic near-field interactions of a dipolar emitter with metal and metamaterial nanoslabs,” Phys. Rev. A 84, 033849 (2011).
[Crossref]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

Giannini, V.

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[Crossref]

Greffet, J.-J.

Hakkarainen, T.

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Electromagnetic near-field interactions of a dipolar emitter with metal and metamaterial nanoslabs,” Phys. Rev. A 84, 033849 (2011).
[Crossref]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Kaivola, M.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

Kalkbrenner, T.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[Crossref] [PubMed]

Karimi, E.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Leppänen, L.-P.

Leuchs, G.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Lindfors, K.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

Madrazo, A.

Maier, S. A.

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Marrucci, L.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Massa, E.

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[Crossref]

Mlynek, J.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[Crossref] [PubMed]

Nesci, A.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. A: Pure Appl. Opt. 6, S18–S23 (2004).
[Crossref]

Nieto-Vesperinas, M.

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Orlov, S.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Priimagi, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

Ramstein, M.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[Crossref] [PubMed]

Rubano, A.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Saastamoinen, K.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young’s interferometer,” Opt. Lett. 40, 2898–2901 (2015).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

Saleh, B.

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, 2007).

Sandoghdar, V.

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[Crossref] [PubMed]

Santamato, E.

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Setälä, T.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young’s interferometer,” Opt. Lett. 40, 2898–2901 (2015).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

L.-P. Leppänen, A. T. Friberg, and T. Setälä, “Partial polarization of optical beams and near fields probed with a nanoscatterer,” J. Opt. Soc. Am. A 31, 1627–1635, (2014).
[Crossref]

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Electromagnetic near-field interactions of a dipolar emitter with metal and metamaterial nanoslabs,” Phys. Rev. A 84, 033849 (2011).
[Crossref]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

Shevchenko, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

Tai, C.-T.

C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, 1971).

Teich, M.

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, 2007).

Tortora, P.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. A: Pure Appl. Opt. 6, S18–S23 (2004).
[Crossref]

Vaccaro, L.

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. A: Pure Appl. Opt. 6, S18–S23 (2004).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

J. Microsc. (1)

T. Kalkbrenner, M. Ramstein, J. Mlynek, and V. Sandoghdar, “A single gold particle as a probe for apertureless scanning near-field optical microscopy,” J. Microsc. 202, 72–76 (2001).
[Crossref] [PubMed]

J. Opt. A: Pure Appl. Opt. (1)

R. Dändliker, P. Tortora, L. Vaccaro, and A. Nesci, “Measuring three-dimensional polarization with scanning optical probes,” J. Opt. A: Pure Appl. Opt. 6, S18–S23 (2004).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Photonics (1)

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

New J. Phys. (2)

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Electromagnetic near-field interactions of a dipolar emitter with metal and metamaterial nanoslabs,” Phys. Rev. A 84, 033849 (2011).
[Crossref]

Phys. Rev. Lett. (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Science (1)

T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization Möbius strips,” Science 347, 964–966 (2015).
[Crossref] [PubMed]

Other (5)

C.-T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, 1971).

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, 2007).

E. Collet, Polarized Light in Fiber Optics (The PolaWave Group, 2003).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Illustration of the polarization-state measurement of a partially polarized electromagnetic beam field E(r,ω) with a nanoscatterer. A cubic nanoparticle deposited on a substrate is placed in the beam and the polarization properties of the field scattered by the particle are measured in the far zone with a conventional quarter-wave plate, polarizer, and detector combination. The observation direction is in the yz plane and makes an angle of 45° with respect to the y axis.
Fig. 2
Fig. 2 Setup for the measurement of the polarization properties of beams with a nanoscatterer. A linearly polarized He-Ne laser beam (wavelength 632.8 nm, power 5 mW) is prepared with a linear polarizer (LP), a neutral density filter (ND), and a half-wave plate (HWP), and directed to a convex lens LA (focal length 7.5 cm) that focuses the beam onto a 100 nm gold nanocube (the inset shows a scanning electron microscope image of the particle). The field scattered from the nanoparticle is analyzed in the far zone with a quarter-wave plate (QWP) and a linear polarizer (LP) that are located at the distance of approximately 6 cm from the cube. Due to low scattered power, light emerging from the waveplate-polarizer system is focused with a lens LB (focal length 3.5 cm, located at approximately 8.5 cm from the particle) onto a detector with picowatt-range sensitivity.
Fig. 3
Fig. 3 Degree of polarization (P) and the normalized Stokes parameters (s 1,s 2,s 3) of a light beam as a function of the intensity difference δ = (Iy − Ix )/Iy of the x and y field components. Values of P obtained by probing with a nanocube are shown with red circles, while the Stokes parameters are presented with triangles: blue (s 1), magenta (s 2), and green (s 3) (note the negative axis). The red dashed line represents the theoretical curve of s 1. The black curve depicts the degree of polarization obtained with a traditional polarimetric measurement.
Fig. 4
Fig. 4 Illustration of the coordinate system and the unit vectors in treating the reflection of the field emitted into the far zone by the point dipole. The particle center is at the origin and the substrate surface is marked with a vertical line. The figure is drawn for a 45° reflection corresponding to the actual measurements.

Equations (10)

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α eff ( ω ) = 32 π h 3 c 2 μ 0 r p [ 1 0 0 0 1 0 0 0 1 2 ] .
E f ( r , ω ) = μ 0 ω 2 e i k | r r 0 | 4 π | r r 0 | [ A u ^ φ u ^ φ T + u ^ θ u ^ θ T + B u ^ θ u ^ θ T ] α eff ( ω ) E ( r 0 , ω ) ,
ϕ ( r , ω ) = μ 0 2 ω 4 16 π 2 | r r 0 | 2 ( u ^ θ T + B * u ^ θ T A * u ^ φ T ) α eff * ( ω ) Φ ( r 0 , ω ) α eff T ( ω ) ( u ^ θ T + B u ^ θ T A u ^ φ T ) T .
P = | I x I y | I x + I y .
P = s 1 2 + s 2 2 + s 3 2
E d ( r , ω ) = μ 0 ω 2 e i k | r r 0 | 4 π | r r 0 | ( I u ^ r u ^ r T ) α eff ( ω ) E ( r 0 , ω ) ,
E i ( r , ω ) = μ 0 ω 2 e i k | r r 0 | 4 π | r r 0 | ( I u ^ r + u ^ r + T ) α eff ( ω ) E ( r 0 , ω ) .
E i ( r , ω ) = μ 0 ω 2 e i k | r r 0 | 4 π | r r 0 | [ u ^ φ + u ^ φ + T + u ^ θ + u ^ θ + T ] α eff ( ω ) E ( r 0 , ω ) .
E i ( r , ω ) = μ 0 ω 2 e i k ( | r r 0 | + Δ r ) 4 π ( | r r 0 | + Δ r ) [ R s u ^ φ + u ^ φ + T R p u ^ θ u ^ θ T ] α eff ( ω ) E ( r 0 , ω ) .
E f ( r , ω ) = E d ( r , ω ) + E i ( r , ω ) = μ 0 ω 2 e i k | r r 0 | 4 π | r r 0 | [ A u ^ φ u ^ φ T + u ^ θ u ^ θ T + B u ^ θ u ^ θ T ] α eff ( ω ) E ( r 0 , ω ) ,

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