Abstract

We describe a theoretical formalism to study the second-harmonic generation in periodically corrugated surfaces illuminated by a plane wave. The incident wave vector is contained in the plane perpendicular to the grating grooves. Our analysis is based on the most general expression for the nonlinear polarization of a homogeneous and isotropic medium. The diffraction problem is solved using a Rayleigh method, and the numerical technique is illustrated by examples for which the nonlinear susceptibilities are calculated with a free-electron model.

© 2011 Optical Society of America

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    [CrossRef]
  39. R. A. Depine and M. L. Gigli, “Conversion between polarization states at the sinusoidal boundary of a uniaxial crystal,” Phys. Rev. B 49, 8437–8445 (1994).
    [CrossRef]
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2009 (4)

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
[CrossRef]

C. I. Valencia, and E. R. Méndez, “Weak localization effects in the second-harmonic light scattered by random systems of particles,” Opt. Commun. 282, 1706–1709 (2009).
[CrossRef]

A. C. Kwan, K. Duff, G. K. Gouras, and W. W. Webb, “Optical visualization of Alzheimer’s pathology via multiphoton-excited intrinsic fluorescence and second harmonic generation,” Opt. Express 17, 3679–3689 (2009).
[CrossRef] [PubMed]

2008 (2)

J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B 78, 205322 (2008).
[CrossRef]

Y. Maeda, T. Iwai, Y. Satake, K. Fujii, S. Miyatake, D. Miyazaki, and G. Mizutani, “Optical second-harmonic spectroscopy of Au(887) and Au(443) surfaces,” Phys. Rev. B 78, 075440 (2008).
[CrossRef]

2007 (1)

D. B. Singh and V. K. Tripathi, “Surface plasmon excitation at second harmonic over a rippled surface,” J. Appl. Phys. 102, 083301 (2007).
[CrossRef]

2005 (1)

K. A. O’Donnell and R. Torre, “Characterization of the second-harmonic response of a silver-air interface,” New J. Phys. 7, 154–164 (2005).
[CrossRef]

2004 (2)

2003 (2)

2001 (2)

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87, 103902 (2001).
[CrossRef] [PubMed]

M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
[CrossRef]

1999 (2)

G. Lüpke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. 35, 75–161 (1999).
[CrossRef]

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

1997 (3)

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

Y. R. Shen, “Wave mixing spectroscopy for surface studies,” Solid State Commun. 102, 221–229 (1997).
[CrossRef]

K. A. O’Donnell and R. Torre, “Second-harmonic generation from a strongly rough metal surface,” Opt. Commun. 138, 341–344 (1997).
[CrossRef]

1996 (3)

1995 (1)

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

1994 (4)

R. M. Corn and D. A. Higgins, “Optical second harmonic generation as a probe of surface chemistry,” Chem. Rev. 94, 107–125(1994).
[CrossRef]

E. Popov and M. Nevière, “Surface-enhanced second-harmonic generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
[CrossRef]

R. A. Depine and M. L. Gigli, “Diffraction from corrugated gratings made with uniaxial crystals: Rayleigh methods,” J. Mod. Opt. 41, 695–715 (1994).
[CrossRef]

R. A. Depine and M. L. Gigli, “Conversion between polarization states at the sinusoidal boundary of a uniaxial crystal,” Phys. Rev. B 49, 8437–8445 (1994).
[CrossRef]

1990 (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

1985 (1)

J. L. Coutaz, M. Nevière, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

1984 (2)

G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3012 (1984).
[CrossRef]

R. T. Deck and R. K. Grygier, “Surface-plasmon enhanced harmonic generation at a rough metal surface,” Appl. Opt. 23, 3202–3213 (1984).
[CrossRef] [PubMed]

1982 (1)

1981 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1971 (1)

J. Rudnick and E. A. Stern, “Second harmonic generation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[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]

1966 (2)

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509(1966).
[CrossRef]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[CrossRef]

1965 (1)

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[CrossRef]

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378(1951).
[CrossRef]

1907 (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. A 79, 399–416 (1907).
[CrossRef]

Angerer, W. E.

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87, 103902 (2001).
[CrossRef] [PubMed]

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]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[CrossRef]

Boardman, A. D.

A. D. Boardman, ed., Electromagnetic Surface Modes(Wiley, 1982).

Brown, F.

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509(1966).
[CrossRef]

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[CrossRef]

Cadilhac, 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]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[CrossRef]

Chen, W.-L.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

Chou, C.-K.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Corbalán, R.

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

Corn, R. M.

R. M. Corn and D. A. Higgins, “Optical second harmonic generation as a probe of surface chemistry,” Chem. Rev. 94, 107–125(1994).
[CrossRef]

Coutaz, J. L.

J. L. Coutaz, M. Nevière, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

Dadap, J. I.

J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B 78, 205322 (2008).
[CrossRef]

de Beer, A. G. F.

A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
[CrossRef]

Deck, R. T.

Depine, R. A.

R. A. Depine and M. L. Gigli, “Diffraction from corrugated gratings made with uniaxial crystals: Rayleigh methods,” J. Mod. Opt. 41, 695–715 (1994).
[CrossRef]

R. A. Depine and M. L. Gigli, “Conversion between polarization states at the sinusoidal boundary of a uniaxial crystal,” Phys. Rev. B 49, 8437–8445 (1994).
[CrossRef]

Dong, C.-Y.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

Downer, M. C.

M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
[CrossRef]

Duff, K.

Farias, G. A.

G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3012 (1984).
[CrossRef]

Fujii, K.

Y. Maeda, T. Iwai, Y. Satake, K. Fujii, S. Miyatake, D. Miyazaki, and G. Mizutani, “Optical second-harmonic spectroscopy of Au(887) and Au(443) surfaces,” Phys. Rev. B 78, 075440 (2008).
[CrossRef]

Fwu, P. T.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

Gavrilenko, V. I.

M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
[CrossRef]

Gigli, M. L.

R. A. Depine and M. L. Gigli, “Diffraction from corrugated gratings made with uniaxial crystals: Rayleigh methods,” J. Mod. Opt. 41, 695–715 (1994).
[CrossRef]

R. A. Depine and M. L. Gigli, “Conversion between polarization states at the sinusoidal boundary of a uniaxial crystal,” Phys. Rev. B 49, 8437–8445 (1994).
[CrossRef]

Gouras, G. K.

Grygier, R. K.

Higgins, D. A.

R. M. Corn and D. A. Higgins, “Optical second harmonic generation as a probe of surface chemistry,” Chem. Rev. 94, 107–125(1994).
[CrossRef]

Hübner, W.

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

Hugonin, J. P.

Iwai, T.

Y. Maeda, T. Iwai, Y. Satake, K. Fujii, S. Miyatake, D. Miyazaki, and G. Mizutani, “Optical second-harmonic spectroscopy of Au(887) and Au(443) surfaces,” Phys. Rev. B 78, 075440 (2008).
[CrossRef]

Jha, S. S.

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]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kim, D.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

Kwan, A. C.

Lee, C. H.

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]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[CrossRef]

Leskova, T. A.

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Q. Lu, “Multiple-scattering effects in the second-harmonic generation of light in reflection from a randomly rough metal surface,” Opt. Lett. 21, 1809–1811(1996).
[CrossRef] [PubMed]

Leyva-Lucero, M. A.

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Q. Lu, “Multiple-scattering effects in the second-harmonic generation of light in reflection from a randomly rough metal surface,” Opt. Lett. 21, 1809–1811(1996).
[CrossRef] [PubMed]

Li, T.-H.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

Lin, S.-J.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

Lu, J. Q.

Lüpke, G.

G. Lüpke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. 35, 75–161 (1999).
[CrossRef]

Maeda, Y.

Y. Maeda, T. Iwai, Y. Satake, K. Fujii, S. Miyatake, D. Miyazaki, and G. Mizutani, “Optical second-harmonic spectroscopy of Au(887) and Au(443) surfaces,” Phys. Rev. B 78, 075440 (2008).
[CrossRef]

Maradudin, A. A.

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, A. A. Maradudin, and J. Q. Lu, “Multiple-scattering effects in the second-harmonic generation of light in reflection from a randomly rough metal surface,” Opt. Lett. 21, 1809–1811(1996).
[CrossRef] [PubMed]

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3012 (1984).
[CrossRef]

Martorell, J.

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

McGilp, J. F.

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

McGurn, A. R.

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Méndez, E. R.

Mendoza, B. S.

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Miyatake, S.

Y. Maeda, T. Iwai, Y. Satake, K. Fujii, S. Miyatake, D. Miyazaki, and G. Mizutani, “Optical second-harmonic spectroscopy of Au(887) and Au(443) surfaces,” Phys. Rev. B 78, 075440 (2008).
[CrossRef]

Miyazaki, D.

Y. Maeda, T. Iwai, Y. Satake, K. Fujii, S. Miyatake, D. Miyazaki, and G. Mizutani, “Optical second-harmonic spectroscopy of Au(887) and Au(443) surfaces,” Phys. Rev. B 78, 075440 (2008).
[CrossRef]

Mizutani, G.

Y. Maeda, T. Iwai, Y. Satake, K. Fujii, S. Miyatake, D. Miyazaki, and G. Mizutani, “Optical second-harmonic spectroscopy of Au(887) and Au(443) surfaces,” Phys. Rev. B 78, 075440 (2008).
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E. Popov and M. Nevière, “Surface-enhanced second-harmonic generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
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K. A. O’Donnell and R. Torre, “Characterization of the second-harmonic response of a silver-air interface,” New J. Phys. 7, 154–164 (2005).
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K. A. O’Donnell and E. R. Méndez, “Enhanced specular peaks in diffuse light scattering from weakly rough metal surfaces,” J. Opt. Soc. Am. A 20, 2338–2346 (2003).
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K. A. O’Donnell and R. Torre, “Second-harmonic generation from a strongly rough metal surface,” Opt. Commun. 138, 341–344 (1997).
[CrossRef]

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of backscattering effects in second-harmonic generation from a weakly rough metal surface,” Opt. Lett. 21, 1738–1740(1996).
[CrossRef] [PubMed]

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E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

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F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509(1966).
[CrossRef]

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
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Pavlyukh, Y.

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

J. L. Coutaz, M. Nevière, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

Pipino, A. C. R.

A. C. R. Pipino, R. P. Van Duyne, and G. C. Schatz, “Surface-enhanced second-harmonic diffraction: experimental investigation of selective enhancement,” Phys. Rev. B 53, 4162–4169(1996).
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[CrossRef]

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J. L. Coutaz, M. Nevière, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378(1951).
[CrossRef]

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A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
[CrossRef]

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J. Rudnick and E. A. Stern, “Second harmonic generation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

Satake, Y.

Y. Maeda, T. Iwai, Y. Satake, K. Fujii, S. Miyatake, D. Miyazaki, and G. Mizutani, “Optical second-harmonic spectroscopy of Au(887) and Au(443) surfaces,” Phys. Rev. B 78, 075440 (2008).
[CrossRef]

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A. C. R. Pipino, R. P. Van Duyne, and G. C. Schatz, “Surface-enhanced second-harmonic diffraction: experimental investigation of selective enhancement,” Phys. Rev. B 53, 4162–4169(1996).
[CrossRef]

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Y. R. Shen, “Wave mixing spectroscopy for surface studies,” Solid State Commun. 102, 221–229 (1997).
[CrossRef]

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D. B. Singh and V. K. Tripathi, “Surface plasmon excitation at second harmonic over a rippled surface,” J. Appl. Phys. 102, 083301 (2007).
[CrossRef]

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J. E. Sipe and G. I. Stegeman, “Nonlinear optical response of metal surfaces,” in Surface Polaritons, V.M.Agranovich and D.L.Mills, eds. (North-Holland, 1982), pp. 661–701.

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F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[CrossRef]

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W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

Stegeman, G. I.

J. E. Sipe and G. I. Stegeman, “Nonlinear optical response of metal surfaces,” in Surface Polaritons, V.M.Agranovich and D.L.Mills, eds. (North-Holland, 1982), pp. 661–701.

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J. Rudnick and E. A. Stern, “Second harmonic generation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
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W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

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K. A. O’Donnell and R. Torre, “Characterization of the second-harmonic response of a silver-air interface,” New J. Phys. 7, 154–164 (2005).
[CrossRef]

K. A. O’Donnell and R. Torre, “Second-harmonic generation from a strongly rough metal surface,” Opt. Commun. 138, 341–344 (1997).
[CrossRef]

K. A. O’Donnell, R. Torre, and C. S. West, “Observations of backscattering effects in second-harmonic generation from a weakly rough metal surface,” Opt. Lett. 21, 1738–1740(1996).
[CrossRef] [PubMed]

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D. B. Singh and V. K. Tripathi, “Surface plasmon excitation at second harmonic over a rippled surface,” J. Appl. Phys. 102, 083301 (2007).
[CrossRef]

Valencia, C. I.

Van Duyne, R. P.

A. C. R. Pipino, R. P. Van Duyne, and G. C. Schatz, “Surface-enhanced second-harmonic diffraction: experimental investigation of selective enhancement,” Phys. Rev. B 53, 4162–4169(1996).
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J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
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West, C. S.

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N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87, 103902 (2001).
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N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87, 103902 (2001).
[CrossRef] [PubMed]

Ann. Phys. (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, S.-J. Lin, D. Kim, P. T. C. So, and C.-Y. Dong, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94, 183902 (2009).
[CrossRef]

Chem. Rev. (1)

R. M. Corn and D. A. Higgins, “Optical second harmonic generation as a probe of surface chemistry,” Chem. Rev. 94, 107–125(1994).
[CrossRef]

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378(1951).
[CrossRef]

J. Appl. Phys. (1)

D. B. Singh and V. K. Tripathi, “Surface plasmon excitation at second harmonic over a rippled surface,” J. Appl. Phys. 102, 083301 (2007).
[CrossRef]

J. Mod. Opt. (1)

R. A. Depine and M. L. Gigli, “Diffraction from corrugated gratings made with uniaxial crystals: Rayleigh methods,” J. Mod. Opt. 41, 695–715 (1994).
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J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

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K. A. O’Donnell and R. Torre, “Characterization of the second-harmonic response of a silver-air interface,” New J. Phys. 7, 154–164 (2005).
[CrossRef]

Opt. Commun. (3)

C. I. Valencia, and E. R. Méndez, “Weak localization effects in the second-harmonic light scattered by random systems of particles,” Opt. Commun. 282, 1706–1709 (2009).
[CrossRef]

K. A. O’Donnell and R. Torre, “Second-harmonic generation from a strongly rough metal surface,” Opt. Commun. 138, 341–344 (1997).
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M. A. Leyva-Lucero, E. R. Méndez, T. A. Leskova, and A. A. Maradudin, “Destructive interference effects in the second harmonic light generated at randomly rough metal surfaces,” Opt. Commun. 161, 79–94 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55, 4520–4525 (1997).
[CrossRef]

Phys. Rev. B (11)

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[CrossRef]

J. Rudnick and E. A. Stern, “Second harmonic generation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3012 (1984).
[CrossRef]

J. L. Coutaz, M. Nevière, E. Pic, and R. Reinisch, “Experimental study of surface-enhanced second-harmonic generation on silver gratings,” Phys. Rev. B 32, 2227–2232 (1985).
[CrossRef]

A. C. R. Pipino, R. P. Van Duyne, and G. C. Schatz, “Surface-enhanced second-harmonic diffraction: experimental investigation of selective enhancement,” Phys. Rev. B 53, 4162–4169(1996).
[CrossRef]

R. A. Depine and M. L. Gigli, “Conversion between polarization states at the sinusoidal boundary of a uniaxial crystal,” Phys. Rev. B 49, 8437–8445 (1994).
[CrossRef]

Y. Maeda, T. Iwai, Y. Satake, K. Fujii, S. Miyatake, D. Miyazaki, and G. Mizutani, “Optical second-harmonic spectroscopy of Au(887) and Au(443) surfaces,” Phys. Rev. B 78, 075440 (2008).
[CrossRef]

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

J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B 78, 205322 (2008).
[CrossRef]

A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79, 155420 (2009).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett. (4)

N. Yang, W. E. Angerer, and A. G. Yodh, “Angle-resolved second-harmonic light scattering from colloidal particles,” Phys. Rev. Lett. 87, 103902 (2001).
[CrossRef] [PubMed]

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[CrossRef]

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509(1966).
[CrossRef]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[CrossRef]

Proc. R. Soc. A (1)

L. Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. A 79, 399–416 (1907).
[CrossRef]

Prog. Surf. Sci. (1)

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

Solid State Commun. (1)

Y. R. Shen, “Wave mixing spectroscopy for surface studies,” Solid State Commun. 102, 221–229 (1997).
[CrossRef]

Surf. Interface Anal. (1)

M. C. Downer, B. S. Mendoza, and V. I. Gavrilenko, “Optical second harmonic spectroscopy of semiconductor surfaces: advances in microscopic understanding,” Surf. Interface Anal. 31, 966–986 (2001).
[CrossRef]

Surf. Sci. Rep. (1)

G. Lüpke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. 35, 75–161 (1999).
[CrossRef]

Other (4)

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

A. D. Boardman, ed., Electromagnetic Surface Modes(Wiley, 1982).

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

J. E. Sipe and G. I. Stegeman, “Nonlinear optical response of metal surfaces,” in Surface Polaritons, V.M.Agranovich and D.L.Mills, eds. (North-Holland, 1982), pp. 661–701.

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Figures (8)

Fig. 1
Fig. 1

View of the principal section of the grating.

Fig. 2
Fig. 2

Computed values of p 1 2 ω as a function of the angle of incidence θ i , for p incidence with λ = 1.064 μm from the vacuum onto a sinusoidal silver grating with period d = 0.556 μm and groove depths h = 230 Å , h = 350 Å , and h = 460 Å (gratings 1, 2, and 3, respectively). These parameters are those used in Fig. 4 of [15]. (a) For the permittivities of silver, we used ϵ I I ( ω ) = 59.55 + 1.15 i and ϵ I I ( 2 ω ) = 14.14 + 0.14 i [42]. (b) For the permittivities of silver, we used ϵ I I ( ω ) = 51.99 + 3.38 i and ϵ I I ( 2 ω ) = 10.18 + 0.83 i [45].

Fig. 3
Fig. 3

Angular behavior of the (a) linear and (b) second-harmonic diffraction orders. The curves illustrate the angular position of the various orders as functions of the angle of incidence; the diffraction order is indicated. The vertical lines indicate the angles of incidence for which an order coincides with the surface plasmons propagation constant. The number next to each vertical line indicates the diffraction order involved in the coupling. The wavelength to period ratio is λ / d = 1 , and we have used the dielectric constants ϵ I I ( ω ) 59.55 + 1.15 i and ϵ I I ( 2 ω ) 14.14 + 0.14 i .

Fig. 4
Fig. 4

(a)  p 0 ω as a function of the angle of incidence θ i , for p incidence from the vacuum onto a sinusoidal silver grating, λ / d = 1 and several values of the height to period ratio h / d . On the left, we show p 0 ω for 5 ° < θ i < 90 ° ; on the right, the range 0 ° < θ i < 5 ° is plotted. (b)  p 0 2 ω as a function of the angle of incidence θ i . On the left, we show p 0 2 ω for 5 ° < θ i < 90 ° ; on the right, the range 0 ° < θ i < 5 ° is plotted.

Fig. 5
Fig. 5

Schematic diagram of some of the multiple scattering and nonlinear optical processes that can take place in the interaction of the electromagnetic field with the grating. The diagram shows that the specular reflection at the fundamental frequency arises from single and multiple-scattering interactions. Similarly, contributions to the second-harmonic signal can arise from a variety of nonlinear interactions. The figure illustrates the generation of second-harmonic plasmons through the interaction of fundamental plasmons and their subsequent decay into propagating waves.

Fig. 6
Fig. 6

(a)  p 1 ω as a function of the angle of incidence θ i . The parameters are the same as in Fig. 4. On the left, we show p 1 ω for 5 ° < θ i < 90 ° ; on the right, the range 0 ° < θ i < 5 ° is plotted. (b)  p 1 2 ω as a function of the angle of incidence θ i . The parameters are the same as in Fig. 4. On the left, we show p 1 2 ω for 5 ° < θ i < 90 ° ; on the right, the range 0 ° < θ i < 5 ° is plotted.

Fig. 7
Fig. 7

p 2 2 ω as a function of the angle of incidence θ i . The parameters are the same as in Fig. 4. On the left, we show p 2 2 ω for 5 ° < θ i < 90 ° ; on the right, the range 0 ° < θ i < 5 ° is plotted.

Fig. 8
Fig. 8

p + 1 2 ω as a function of the angle of incidence θ i . The parameters are the same as in Fig. 4.

Equations (61)

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× E ( r | 2 ω ) = 2 i ω c H ( r | 2 ω ) ,
× H ( r | 2 ω ) = 2 i ω c D ( r | 2 ω ) ,
· D ( r | 2 ω ) = 0 ,
· H ( r | 2 ω ) = 0 ,
D ( r | 2 ω ) = ϵ II ( 2 ω ) E ( r | 2 ω ) + 4 π P NL ( r | 2 ω ) ,
P NL ( r | 2 ω ) = α [ E ( r | ω ) · ] E ( r | ω ) + β E ( r | ω ) [ · E ( r | ω ) ] + γ [ E ( r | ω ) · E ( r | ω ) ] ,
[ 13 2 + ( 2 ω c ) 2 ϵ II ( 2 ω ) ] ψ p ( II ) ( r | 2 ω ) = i 4 π ( 2 ω c ) ( P 1 NL x 3 P 3 NL x 1 ) ,
E t ( I ) ( r s | 2 ω ) E t ( II ) ( r s | 2 ω ) = 4 π t P z s ( r s | 2 ω ) ,
H t ( I ) ( r s | 2 ω ) H t ( II ) ( r s | 2 ω ) = 4 π ( 2 i ω c ) z ^ × P t s ( r s | 2 ω ) ,
P z s ( r s | 2 ω ) = lim τ 0 τ τ P z NL ( r s , z | 2 ω ) ϵ II ( z | 2 ω ) d z ,
P t s ( r s | 2 ω ) = lim τ 0 τ τ P t NL ( r s , z | 2 ω ) d z .
P z s ( r s | 2 ω ) = χ z z z s [ D z ( r s | ω ) ] 2 + χ z t t s [ E t ( r s | ω ) · E t ( r s | ω ) ] ,
P t s ( r s | 2 ω ) = χ t t z s [ E t ( r s | ω ) D z ( r s | ω ) ] .
r s ( t ) = [ f ( t ) , g ( t ) ] ,
Z = [ g ( t ) , f ( t ) ] , X = [ f ( t ) , g ( t ) ] ,
Z = [ g ( t ) x 1 + f ( t ) x 3 ] ,
X = [ f ( t ) x 1 + g ( t ) x 3 ] .
F ( r ) X | r = r s = d F ( t ) d t ,
ψ p ( R ) ( t | Ω ) = ψ p ( R ) ( r | Ω ) | r = r s ,
ϒ p ( R ) ( t | Ω ) = ψ p ( R ) ( r | Ω ) Z | r = r s .
1 ϵ I ( 2 ω ) ϒ p ( I ) ( t | 2 ω ) 1 ϵ II ( 2 ω ) ϒ p ( II ) ( t | 2 ω ) = 4 π 2 i ω c [ d P z s ( t | 2 ω ) d t + ϕ ( t ) P x NL ( t | 2 ω ) ϵ II ( 2 ω ) ] ,
ψ p ( I ) ( t | 2 ω ) ψ p ( II ) ( t | 2 ω ) = 4 π 2 i ω c P x s ( t | 2 ω ) .
ϕ ( t ) P x NL ( t | 2 ω ) = ( α / 2 + γ ) c 2 ω 2 { 1 ϵ I 2 ( ω ) d d t [ ϒ p ( I ) ( t | ω ) ϕ ( t ) ] 2 + 1 ϵ II 2 ( ω ) d d t [ 1 ϕ ( t ) d ψ p ( I ) ( t | ω ) d t ] 2 } α 2 ϵ II ( ω ) d d t [ ψ p ( I ) ( t | ω ) ] 2 ,
P x s ( t | 2 ω ) = χ t t z s ϵ I ( ω ) ( c ω ) 2 1 ϕ 2 ( t ) ϒ p ( I ) ( t | ω ) d ψ p ( I ) ( t | ω ) d t ,
P z s ( t | 2 ω ) = χ z z z s ( c ω ) 2 [ 1 ϕ ( t ) d ψ p ( I ) ( t | ω ) d t ] 2 χ z t t s 1 ϵ I 2 ( ω ) ( c ω ) 2 [ ϒ p ( I ) ( t | ω ) ϕ ( t ) ] 2 .
H i ( r ) = x ^ 2 ψ 0 exp ( i k i . r ) ,
k i = α 0 x ^ 1 β 0 x ^ 3 ,
ψ p ( I ) ( r | ω ) = n = R n exp [ i ( α n x 1 + β n x 3 ) ] ,
α n = α 0 + n 2 π d ,
α n 2 + β n 2 = ϵ I ( ω ) ( ω c ) 2 .
ψ p ( II ) ( r | ω ) = n = T n exp [ i ( α n x 1 + γ n x 3 ) ] ,
α n 2 + γ n 2 = ϵ II ( ω ) ( ω c ) 2 .
ψ p ( I ) ( r | 2 ω ) = n = S n exp [ i ( α n x 1 + β n x 3 ) ] ,
( α n ) 2 + ( β n ) 2 = ϵ I ( 2 ω ) ( 2 ω c ) 2 ,
α n = 2 α 0 + n 2 π d .
ψ p ( II ) ( r | 2 ω ) = n = W n exp [ i ( α n x 1 + γ n x 3 ) ] ,
( α n ) 2 + ( γ n ) 2 = ϵ II ( 2 ω ) ( 2 ω c ) 2 .
[ u m v m ] = [ A n m B n m C n m D n m ] [ S n W n ] ,
A m n = 0 1 exp [ i ( α n α m ) f ( t ) ] exp ( i β n g ( t ) ) d t ,
B m n = 0 1 exp [ i ( α n α m ) f ( t ) ] exp ( i γ n g ( t ) ) d t ,
C m n = 0 1 h n ( t ) exp [ i ( α n α m ) f ( t ) ] exp ( i β n g ( t ) ) d t ,
D m n = 1 ϵ II ( 2 ω ) 0 1 k n ( t ) exp [ i ( α n α m ) f ( t ) ] exp ( i γ n g ( t ) ) d t ,
h n ( t ) = i [ g ( t ) α n + β n f ( t ) ] , k n ( t ) = i [ g ( t ) α n + γ n f ( t ) ] ,
u m = 8 π i c χ t t z s ω { ψ 0 2 0 1 1 ϕ 2 BC E 1 ( m ) d t + ψ 0 n R n 0 1 1 ϕ 2 ( E n C F n B ) E 2 ( n , m ) d t + n , l R n R l 0 1 1 ϕ 2 E n F l E 3 ( n , m , l ) d t } ,
v m = 16 π i ω c { ψ 0 2 0 1 [ a 1 ( ϕ ϕ 3 C 2 1 ϕ 2 CG ) + a 2 ( ϕ ϕ 3 B 2 1 ϕ 2 BH ) + i a 3 C ] E 1 ( m ) d t + ψ 0 n R n 0 1 [ a 1 ( ϕ ϕ 3 2 C F n 1 ϕ 2 ( C J n + G F n ) ) a 2 ( ϕ ϕ 3 2 B E n 1 ϕ 2 ( B I n + H E n ) ) + i a 3 ( F n + C ) ] E 2 ( n , m ) d t + n , l R n R l 0 1 [ a 1 ( ϕ ϕ 3 F n F l 1 ϕ 2 F n J l ) + a 2 ( ϕ ϕ 3 E n E l 1 ϕ 2 E n I l ) + i a 3 F l ] E 3 ( n , m , l ) d t } ,
a 1 = c 2 ω 2 [ χ z z z s + α / 2 + γ ϵ II ( 2 ω ) ϵ II 2 ( ω ) ] ,
a 2 = c 2 ω 2 [ χ z t t s + α / 2 + γ ϵ II ( 2 ω ) ] ,
a 3 = α 2 ϵ II ( ω ) ϵ II ( 2 ω ) ,
a 4 = χ z t t s + γ ϵ II ( 2 ω ) ,
E 1 ( m ) = exp [ i ( 2 π d m f ( t ) + 2 β 0 g ( t ) ) ] ,
E 2 ( n , m ) = exp [ i ( 2 π d ( n m ) f ( t ) + ( β n β 0 ) g ( t ) ) ] ,
E 3 ( n , m , l ) = exp [ i ( 2 π d ( n + l m ) f ( t ) + ( β n + β l ) g ( t ) ) ] .
B = α 0 g ( t ) + β 0 f ( t ) , C = α 0 f ( t ) β 0 g ( t ) , G = ( C ) + i ( C ) 2 , H = ( B ) + i BC , E j = α j g ( t ) + β j f ( t ) , F j ( t ) = α j f ( t ) + β j g ( t ) , I j = ( E j ) + i E j F j , J j = ( F j ) + i ( F j ) 2 .
p n 2 ω = | P n 2 ω | | P inc | 2 ,
p n ω = | P n ω | | P inc | ,
χ z z z s = 2 3 β [ ( ϵ II R ( ω ) 1 ) ( ϵ II R ( ω ) 3 ) 2 ϵ II R 2 ( ω ) 2 3 ln ( ϵ II R ( ω ) ϵ II R ( 2 ω ) ) ] ,
χ z t t s = 0 ,
χ t t z s = β ( ϵ II R ( ω ) 1 ϵ II R ( ω ) ) .
p = | P ref | | P inc | 2 = 8 π c | S ( 2 ω ) | 2 | ψ 0 | 4 ,
α n = α 0 + n 2 π d Re ( α p ( ω ) ) ,
α m = 2 α 0 + m 2 π d Re ( α p ( 2 ω ) ) ,

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