Abstract

We studied the second harmonic generation (SHG) by two-dimensional dielectric particles made of a centrosymmetric high-index material. The calculated scattered fields at the fundamental and harmonic frequencies are decomposed on a multipolar basis, allowing the evaluation of the relative strengths of the multipolar resonances excited at the particle. With these tools, we studied the strength of the multipoles that produce the second harmonic field and the role played by those excited at the fundamental frequency.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (2)

D. Smirnova, A. I. Smirnov, and Y. S. Kivshar, “Multipolar second-harmonic generation by Mie-resonant dielectric nanoparticles,” Phys. Rev. A 97, 013807 (2018).
[Crossref]

D. Timbrell, J. W. You, Y. S. Kivshar, and N. C. Panoiu, “A comparative analysis of surface and bulk contributions to second-harmonic generation in centrosymmetric nanoparticles,” Sci. Rep. 8, 3586 (2018).
[Crossref] [PubMed]

2017 (2)

M. E. Inchaussandague, M. L. Gigli, K. A. O’Donnell, E. R. Méndez, R. Torre, and C. I. Valencia, “Second-harmonic generation from plasmon polariton excitation on silver diffraction gratings: comparisons of theory and experiment,” J. Opt. Soc. Am. B 34, 27–37 (2017).
[Crossref]

S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, D. A. Smirnova, L. Wang, H. H. Tan, C. Jagadish, D. N. Neshev, and Y. S. Kivshar, “Nonlinear optical magnetism revealed by second-harmonic generation in nanoantennas,” Nano Lett. 17, 3914–3918 (2017).
[Crossref]

2016 (4)

S. Liu, M. B. Sinclair, S. Saravi, G. A. Keeler, Y. Yang, J. Reno, G. M. Peake, F. Setzpfandt, I. Staude, T. Pertsch, and I. Brener, “Resonantly enhanced second-harmonic generation using III-V semiconductor all-dielectric metasurfaces,” Nano Lett. 16, 5426–5432 (2016).
[Crossref]

V. F. Gili, L. Carletti, A. Locatelli, D. Rocco, M. Finazzi, L. Ghirardini, I. Favero, C. Gomez, A. Lemaître, M. Celebrano, C. De Angelis, and G. Leo, “Monolithic AlGaAs second-harmonic nanoantennas,” Opt. Express 2415965–15971 (2016).
[Crossref] [PubMed]

R. Camacho-Morales, M. Rahmani, S. Kruk, L. Wang, L. Xu, D. A. Smirnova, A. Solntsev, A. E. Miroshnichenko, H. H. Tan, F. Karouta, S. Naureen, K. Vora, L. Carletti, C. de Angelis, C. Jagadish, Y. S. Kivshar, and D. N. Neshev, “Nonlinear generation of vector beams from AlGaAs nanoantennas,” Nano Lett. 16, 7191–7197 (2016).
[Crossref]

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016).
[Crossref]

2015 (1)

2013 (1)

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref]

2012 (3)

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. B. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012).
[Crossref]

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Origin of second-harmonic generation enhancement in optical split-ring resonators,” Phys. Rev. B 85, 201403 (2012).
[Crossref]

D. Macías, P.-M. Adam, V. Ruiz-Cortés, R. Rodríguez-Oliveros, and J. A. Sánchez-Gil, “Heuristic optimization for the design of plasmonic nanowires with specific resonant and scattering properties,” Opt. Express 20, 13146–13163 (2012).
[Crossref]

2011 (1)

2010 (2)

2009 (2)

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mat. Today 12, 60–69 (2009).
[Crossref]

J. A. Schuller and M. L. Brongersma, “General properties of dielectric optical antennas,” Opt. Express 17, 24084–24095 (2009).
[Crossref]

2008 (1)

2006 (1)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006)
[Crossref] [PubMed]

2005 (1)

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005).
[Crossref]

2004 (3)

2003 (2)

J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot. 90, 333–338 (2003).
[Crossref] [PubMed]

C. I. Valencia, E. R. Méndez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by two-dimensional particles,” J. Opt. Soc. Am. B 20, 2150–2161 (2003).
[Crossref]

2000 (1)

V. L. Brudny, B. S. Mendoza, and W. L. Mochan, “Second-harmonic generation from spherical particles,” Phys. Rev. B 62, 11152 (2000).
[Crossref]

1999 (2)

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[Crossref]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[Crossref]

1996 (1)

B. S. Mendoza and W. L. Mochán, “Exactly solvable model of surface second-harmonic generation,” Phys. Rev. B 53, 4999–5006 (1996).
[Crossref]

1968 (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. B 174, 813–822 (1968).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur optik truüber medien, speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[Crossref]

Adam, P.-M.

Aizpurua, J.

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. B 174, 813–822 (1968).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles(Wiley, 1998).
[Crossref]

Brener, I.

S. Liu, M. B. Sinclair, S. Saravi, G. A. Keeler, Y. Yang, J. Reno, G. M. Peake, F. Setzpfandt, I. Staude, T. Pertsch, and I. Brener, “Resonantly enhanced second-harmonic generation using III-V semiconductor all-dielectric metasurfaces,” Nano Lett. 16, 5426–5432 (2016).
[Crossref]

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref]

Brongersma, M. L.

A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016).
[Crossref]

J. A. Schuller and M. L. Brongersma, “General properties of dielectric optical antennas,” Opt. Express 17, 24084–24095 (2009).
[Crossref]

Brudny, V. L.

V. L. Brudny, B. S. Mendoza, and W. L. Mochan, “Second-harmonic generation from spherical particles,” Phys. Rev. B 62, 11152 (2000).
[Crossref]

Cai, W.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications(Springer, 2010).
[Crossref]

Camacho-Morales, R.

S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, D. A. Smirnova, L. Wang, H. H. Tan, C. Jagadish, D. N. Neshev, and Y. S. Kivshar, “Nonlinear optical magnetism revealed by second-harmonic generation in nanoantennas,” Nano Lett. 17, 3914–3918 (2017).
[Crossref]

R. Camacho-Morales, M. Rahmani, S. Kruk, L. Wang, L. Xu, D. A. Smirnova, A. Solntsev, A. E. Miroshnichenko, H. H. Tan, F. Karouta, S. Naureen, K. Vora, L. Carletti, C. de Angelis, C. Jagadish, Y. S. Kivshar, and D. N. Neshev, “Nonlinear generation of vector beams from AlGaAs nanoantennas,” Nano Lett. 16, 7191–7197 (2016).
[Crossref]

Carletti, L.

R. Camacho-Morales, M. Rahmani, S. Kruk, L. Wang, L. Xu, D. A. Smirnova, A. Solntsev, A. E. Miroshnichenko, H. H. Tan, F. Karouta, S. Naureen, K. Vora, L. Carletti, C. de Angelis, C. Jagadish, Y. S. Kivshar, and D. N. Neshev, “Nonlinear generation of vector beams from AlGaAs nanoantennas,” Nano Lett. 16, 7191–7197 (2016).
[Crossref]

V. F. Gili, L. Carletti, A. Locatelli, D. Rocco, M. Finazzi, L. Ghirardini, I. Favero, C. Gomez, A. Lemaître, M. Celebrano, C. De Angelis, and G. Leo, “Monolithic AlGaAs second-harmonic nanoantennas,” Opt. Express 2415965–15971 (2016).
[Crossref] [PubMed]

L. Carletti, A. Locatelli, O. Stepanenko, G. Leo, and C. De Angelis, “Enhanced second-harmonic generation from magnetic resonance in AlGaAs nanoantennas,” Opt. Express 23, 26544–26550 (2015).
[Crossref]

Celebrano, M.

Chang, R. K.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. B 174, 813–822 (1968).
[Crossref]

Chantada, L.

Chichkov, B. N.

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[Crossref]

Chipouline, A.

Ciracì, C.

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Origin of second-harmonic generation enhancement in optical split-ring resonators,” Phys. Rev. B 85, 201403 (2012).
[Crossref]

Dadap, J. I.

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21, 1328–1347 (2004).
[Crossref]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[Crossref]

de Angelis, C.

R. Camacho-Morales, M. Rahmani, S. Kruk, L. Wang, L. Xu, D. A. Smirnova, A. Solntsev, A. E. Miroshnichenko, H. H. Tan, F. Karouta, S. Naureen, K. Vora, L. Carletti, C. de Angelis, C. Jagadish, Y. S. Kivshar, and D. N. Neshev, “Nonlinear generation of vector beams from AlGaAs nanoantennas,” Nano Lett. 16, 7191–7197 (2016).
[Crossref]

V. F. Gili, L. Carletti, A. Locatelli, D. Rocco, M. Finazzi, L. Ghirardini, I. Favero, C. Gomez, A. Lemaître, M. Celebrano, C. De Angelis, and G. Leo, “Monolithic AlGaAs second-harmonic nanoantennas,” Opt. Express 2415965–15971 (2016).
[Crossref] [PubMed]

L. Carletti, A. Locatelli, O. Stepanenko, G. Leo, and C. De Angelis, “Enhanced second-harmonic generation from magnetic resonance in AlGaAs nanoantennas,” Opt. Express 23, 26544–26550 (2015).
[Crossref]

Decker, M.

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref]

N. Feth, S. Linden, M. W. Klein, M. Decker, F. B. P. Niesler, Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, J. V. Moloney, and M. Wegener, “Second-harmonic generation from complementary split-ring resonators,” Opt. Lett. 33, 1975–1977 (2008).
[Crossref]

Dominguez, J.

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref]

Eisenthal, K. B.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[Crossref]

Enkrich, C.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006)
[Crossref] [PubMed]

Evlyukhin, A. B.

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82, 045404 (2010).
[Crossref]

Favero, I.

Feth, N.

Finazzi, M.

Fofang, N. T.

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref]

Froufe-Pérez, L. S.

Fu, Y. H.

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. B. Zhang, and B. Luk’yanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012).
[Crossref]

García-Etxarri, A.

Ghirardini, L.

Gielis, J.

J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot. 90, 333–338 (2003).
[Crossref] [PubMed]

Gigli, M. L.

Gili, V. F.

Gomez, C.

Gómez-Medina, R.

Gonzales, E.

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref]

Heinz, T. F.

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21, 1328–1347 (2004).
[Crossref]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[Crossref]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Hoyer, W.

Hübner, W.

Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245434 (2004).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Schematic diagram of the geometry considered.
Fig. 2
Fig. 2 Illustration of the lowest order functions of the symmetric and antisymmetric bases. In the symmetric base the lowest order terms represent the magnetic dipole (MD), the electric dipole (ED) and the electric quadrupole (EQ) contributions. In the antisymmetric base they are the electric dipole (ED) and the electric quadrupole (EQ) contributions.
Fig. 3
Fig. 3 Illustration of situations in which the second harmonic scattered field is assumed to be symmetric (a) and antisymmetric (b).
Fig. 4
Fig. 4 Scattering efficiency at the fundamental frequency ω for an infinite cylinder (2D particle with a circular cross section) of radius a = 150 nm. The first three multipolar contributions are also shown. Note that due to the symmetry of the system, only multipoles of the base ψ m + are excited in the linear case.
Fig. 5
Fig. 5 Nonlinear and linear scattering efficiency of the cylinder considered in Fig. 4. (a) Generated second harmonic and the first three multipolar contributions when the fundamental wavelength λ spans the range shown in Fig. 4. Note that due to the symmetry of the system only multipoles of the base ψ m are excited. (b) Linear scattering efficiency in the same spectral region as in (a). In this case, only multipoles of the base ψ m + are excited. In both cases, λ denotes the wavelength of incidence.
Fig. 6
Fig. 6 Modulus of the magnetic field in the near field of the cylinder considered in Fig. 4 at λ/2 = 977 nm, which coincides with the maximum of the electric dipole resonance in the second harmonic field. Note that even at the peak of the dipolar resonance, the quadrupole contribution is predominant.
Fig. 7
Fig. 7 Scattering efficiency in the SHG for the cylinder considered in Fig. 4, considering the effect of the fundamental multipolar excitations. The full curve includes all the multipolar contributions at the fundamental. The green dashed curve considers only the dipolar electric contribution, the gray continuous line considers all the contributions but the dipolar electric one, and the short-dashed blue curve considers the dipolar and quadrupolar electric contributions.
Fig. 8
Fig. 8 Second harmonic scattering efficiency for a dielectric particle with a 300 nm per side square cross section. Also shown are the first three multipolar contributions to the efficiency. The insets illustrate the form of the far field scattering distributions at the wavelengths of the three multipolar resonances.
Fig. 9
Fig. 9 Modulus of the magnetic field in the near field of the particle with square cross section considered in Fig. 8 at λ/2 = 1052 nm, which coincides with the maximum of the electric dipole resonance at the second harmonic wavelength.
Fig. 10
Fig. 10 Modulus of the magnetic field in the near field of the particle with triangular cross section with different orientations at λ/2 = 626 nm, which coincides with the maximum of the electric quadrupole resonance at the second harmonic wavelength. The symmetric orientation of the particle is shown in (a), while (b), (c) and (d) correspond to rotations by α = 10°, 20° and 30°.

Tables (1)

Tables Icon

Table 1 Values of the multipolar expansion coefficients of the second harmonic field for different orientations of the particle of triangular cross section (see Fig. 10) at λ/2 = 626 nm. For simplicity, the common factor 1/(nBe) has been omitted from the values shown.

Equations (35)

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r s ( t ) = [ ξ ( t ) , η ( t ) ] .
P N L ( r | 2 ω ) = α [ E ( r | ω ) ] E ( r | ω ) + β E ( r | ω ) [ E ( r | ω ) ] + γ [ E ( r | ω ) E ( r | ω ) ] ,
ψ p ( I ) ( t | 2 ω ) ψ p ( I I ) ( t | 2 ω ) = A p ( t | ω ) ,
1 ϵ I ( 2 ω ) ϒ p ( I ) ( t | 2 ω ) 1 ϵ I I ( 2 ω ) ϒ p ( I I ) ( t | 2 ω ) = B p ( t | ω ) ,
A p ( t | ω ) = 8 π i c ω χ   ϵ I ( ω ) 1 ϕ 2 ( t ) ϒ p ( I ) ( t | ω ) d ψ p ( I ) ( t | ω ) d t ,
B p ( t | ω ) = 8 π i c ω [ χ d d t ( 1 ϕ ( t ) d ψ p ( I ) ( t | ω ) d t ) 2 + χ ϵ I 2 ( ω ) d d t ( ϒ p ( I ) ( t | ω ) ϕ ( t ) ) 2 + χ b d [ ψ p ( I ) ( t | ω ) ] 2 d t η d [ ψ s ( I ) ( t | ω ) ] 2 d t ] ,
χ   = χ t t z s ,
χ = χ z z z s + α / 2 + γ ϵ I I ( 2 ω ) ϵ I I 2 ( ω ) ,
χ = χ z t t s + α / 2 + γ ϵ I I ( 2 ω ) ,
η = χ z t t s + γ ϵ I I ( 2 ω ) ,
χ b = α 2 ϵ I I ( ω ) ϵ I I ( 2 ω ) ( ω c ) 2 ,
χ ( ω ) = e 2 / m ω 0 2 ω 2 i Γ ω ,
α ( ω ) = ϵ ( ω ) 1 32 π 2 n B e [ 4 ϵ ( 2 ω ) ϵ ( ω ) 3 ] ,
β ( ω ) = ϵ ( ω ) 1 32 π 2 n B e ,
γ ( ω ) = ϵ ( ω ) 1 32 π 2 n B e [ ϵ ( 2 ω ) 1 ] ,
χ z z z s ( ω ) = ( ϵ ( ω ) 1 ) 2 32 π 2 n B e ϵ ( ω ) 2 { 2 ϵ ( ω ) ϵ ( 2 ω ) ϵ ( ω ) ϵ ( 2 ω ) [ ϵ ( 2 ω ) ϵ ( ω ) ] + [ 1 ϵ ( 2 ω ) ] [ ϵ ( 2 ω ) ϵ ( ω ) ] 2 ln ( ϵ ( ω ) ϵ ( 2 ω ) ) } ,
χ t t z s ( ω ) = χ t z t s ( ω ) = 1 32 π 2 n B e [ ϵ ( ω ) 1 ] 2 ϵ ( ω ) ,
χ z t t s ( ω ) = 0 .
ψ s c ( I ) ( r , θ | Ω ) = e i π 4 e i k Ω r [ 8 π k Ω r ] 1 / 2 S ( θ | Ω ) ,
S ( θ | Ω ) = S [ i k Ω [ η ( t ) cos θ ξ ( t ) sin θ ] ψ p ( I ) ( t | Ω ) ϒ p ( I ) ( t | Ω ) ] × exp { i k Ω [ ξ ( t ) cos θ + η ( t ) sin θ ] } d t .
ψ s c ( I ) ( r , θ | Ω ) = m = c m i m H m ( 1 ) ( k Ω r ) e i m θ ,
ψ s c ( I ) ( r , θ | Ω ) = m = 0 c m + ψ m + ( r , θ | Ω ) + m = 1 c m ψ m ( r , θ | Ω ) ,
ψ m ± ( r , θ , | Ω ) = i m H m ( 1 ) ( k Ω r ) { cos m θ i sin m θ } ,
c m ± = ( c m ± c m ) ,
ψ 0 + ( r , θ , | Ω ) = H 0 ( 1 ) ( k Ω r ) ,
c 0 + = c 0 .
c m ± = 0 2 π ψ s c ( I ) ( r 0 , θ | Ω ) ψ m ± ( r 0 , θ | Ω ) d θ 0 2 π | ψ m ± ( r 0 , θ | Ω ) | 2 d θ ,
c 2 ± = 1 4 π 0 2 π S ( θ | Ω ) { cos m θ i sin m θ } d θ ,
c 0 = 1 8 π 0 2 π S ( θ | Ω ) d θ .
Q s c ( ω ) = P s c ( ω ) P i n c ( ω ) = 0 2 π q s c ( θ | ω ) d θ ,
q s c ( θ | ω ) = 1 8 π D ( ω / c ) | S ( θ | ω ) | 2 | ψ 0 | 2 ,
Q s c ( 2 ω ) = σ g P s c ( 2 ω ) [ P i n c ( ω ) ] 2 = 0 2 π q s c ( θ | 2 π ) d θ ,
q s c ( θ | 2 ω ) = 1 2 ω D | S ( θ | 2 ω ) | 2 | ψ 0 | 4 .
Q s c ( Ω ) = [ | c 0 | 2 + 1 2 m = 1 [ | c m + | 2 + | c m | 2 ] ] × { 2 k D , Ω = ω , 8 π ω D | ψ 0 | 4 , Ω = 2 ω ,
q s c ( θ | Ω ) = [ c 0 + m = 1 [ c m + cos m θ + i c m sin m θ ] ] 2 × { 1 π k D , Ω = ω , 16 ω D | ψ 0 | 4 , Ω = 2 ω .

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