Abstract

We present a detailed study of the optical second harmonic (SH) generated in samples of centrosymmetric muscovite mica. Samples with thicknesses ranging from 30 to 300μm have been investigated in a transmission normal-incidence geometry. We found a strong dependence of the polarization-dependent SH signal on the sample thickness. In particular, in some of the thickest samples, the SH signal is strongly enhanced. This signal amplification is not monotonically increased with thickness, but it varies erratically. These findings show that under the present experimental conditions, quadrupolar bulk second harmonic generation in mica becomes dominant on the SH generated from the surfaces. The large variability of the SH signal with the variation in thickness is then ascribed to partial optical phase-matching effects, controlled by the mica birefringence. In order to corroborate this hypothesis, a detailed theoretical model accounting for the nonlocal response and anisotropy of a generic birefringent crystal has been developed. The predictions of our model are found to be in excellent agreement with the experimental data.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Yang, A. Kuperman, N. Coombs, S. Mamiche-Afara, and G. A. Ozin, “Synthesis of oriented films of mesoporous silica on mica,” Nature 379, 703-705 (1996).
    [CrossRef]
  2. W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivray, and J. Klein, “Boundary lubrication under water,” Nature 444, 191-194 (2006).
    [CrossRef] [PubMed]
  3. M. Qin, L. K. Wang, and X. Z. Feng, “Bioactive surface modification of mica and poly(dimethylsiloxane) with hydrophobins for protein immobilization,” Langmuir 23, 4465-4471 (2007).
    [CrossRef] [PubMed]
  4. B. Zappone, M. Ruths, G. W. Greene, G. D. Jay, and J. N. Israelachvili, “Adsorption lubrication, and wear of lubricin on model surfaces: polymer brush-like behavior of a glycoprotein,” Biophys. J. 92, 1693-1708 (2007).
    [CrossRef]
  5. O. Agam, “Viscous fingering in volatile thin films,” Phys. Rev. E 79, 021603 (2009).
    [CrossRef]
  6. T. Fukuma, Y. Ueda, S. Yoshioka, and H. Asakawa, “Atomic-scale distribution of water molecules at the mica-water interface visualized by three-dimensional scanning force microscopy,” Phys. Rev. Lett. 104, 016101 (2010).
    [CrossRef] [PubMed]
  7. Y. R. Shen, “Surface-properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519-525 (1989).
    [CrossRef]
  8. Y. R. Shen, “Surfaces probed by nonlinear optics,” Surf. Sci. 299, 551-562 (1994).
    [CrossRef]
  9. X. Zhuang, L. Marrucci, and Y. R. Shen, “Surface-monolayer-induced bulk alignment of liquid crystals,” Phys. Rev. Lett. 73, 1513-1516 (1994).
    [CrossRef] [PubMed]
  10. X. Zhuang, D. Wilk, L. Marrucci, and Y. R. Shen, “Orientation of amphiphilic molecules on polar substrates,” Phys. Rev. Lett. 75, 2144-2147 (1995).
    [CrossRef] [PubMed]
  11. L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
    [CrossRef]
  12. A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
    [CrossRef]
  13. P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254-8263 (1986).
    [CrossRef]
  14. J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129-1141(1987).
    [CrossRef]
  15. X. Wei, S.-C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B 104, 3349-3354 (2000).
    [CrossRef]
  16. B. Jérôme and Y. R. Shen, “Anchoring of nematic liquid-crystals on mica in the presence of volatile molecules,” Phys. Rev. E 48, 4556-4574 (1993).
    [CrossRef]
  17. G. Berkovic, “New studies of liquid and solid surfaces using second harmonic generation,” Phys. A 168, 140-148 (1990).
    [CrossRef]
  18. R. Yerushalmi-Rozen, J. Klein, and G. Berkovic, “In situ probing of polymer grafting from solution onto solid substrates by nonlinear optics,” Langmuir 8, 1392-1397 (1992).
    [CrossRef]
  19. M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
    [CrossRef]
  20. A. Yariv, Quantum Electronics (Wiley, 1989).
  21. M. Medhat and S. Y. El-Zaiat, “Interferometric determination of the birefringence dispersion of anisotropic materials,” Opt. Commun. 141, 145-149 (1997).
    [CrossRef]
  22. B. Gauthier-Manuel, “Simultaneous determination of the thickness and optical constants of weakly absorbing thin films,” Meas. Sci. Technol. 9, 485-487 (1998).
    [CrossRef]
  23. A. I. Bailey and S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39-46 (1965).
    [CrossRef]
  24. In crystallography, the glide plane indicates a symmetry operation describing how a reflection in a plane, followed by a translation parallel to that plane, may leave the crystal unchanged . Note that the tensors characterizing the optical response are not dependent on the translations.
  25. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
    [CrossRef]
  26. W. Borchardt-Ott, Crystallography (Springer-Verlag, 1995).
    [CrossRef]

2010 (1)

T. Fukuma, Y. Ueda, S. Yoshioka, and H. Asakawa, “Atomic-scale distribution of water molecules at the mica-water interface visualized by three-dimensional scanning force microscopy,” Phys. Rev. Lett. 104, 016101 (2010).
[CrossRef] [PubMed]

2009 (2)

O. Agam, “Viscous fingering in volatile thin films,” Phys. Rev. E 79, 021603 (2009).
[CrossRef]

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

2007 (2)

M. Qin, L. K. Wang, and X. Z. Feng, “Bioactive surface modification of mica and poly(dimethylsiloxane) with hydrophobins for protein immobilization,” Langmuir 23, 4465-4471 (2007).
[CrossRef] [PubMed]

B. Zappone, M. Ruths, G. W. Greene, G. D. Jay, and J. N. Israelachvili, “Adsorption lubrication, and wear of lubricin on model surfaces: polymer brush-like behavior of a glycoprotein,” Biophys. J. 92, 1693-1708 (2007).
[CrossRef]

2006 (1)

W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivray, and J. Klein, “Boundary lubrication under water,” Nature 444, 191-194 (2006).
[CrossRef] [PubMed]

2005 (1)

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

2002 (1)

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

2000 (1)

X. Wei, S.-C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B 104, 3349-3354 (2000).
[CrossRef]

1998 (1)

B. Gauthier-Manuel, “Simultaneous determination of the thickness and optical constants of weakly absorbing thin films,” Meas. Sci. Technol. 9, 485-487 (1998).
[CrossRef]

1997 (1)

M. Medhat and S. Y. El-Zaiat, “Interferometric determination of the birefringence dispersion of anisotropic materials,” Opt. Commun. 141, 145-149 (1997).
[CrossRef]

1996 (1)

H. Yang, A. Kuperman, N. Coombs, S. Mamiche-Afara, and G. A. Ozin, “Synthesis of oriented films of mesoporous silica on mica,” Nature 379, 703-705 (1996).
[CrossRef]

1995 (1)

X. Zhuang, D. Wilk, L. Marrucci, and Y. R. Shen, “Orientation of amphiphilic molecules on polar substrates,” Phys. Rev. Lett. 75, 2144-2147 (1995).
[CrossRef] [PubMed]

1994 (2)

Y. R. Shen, “Surfaces probed by nonlinear optics,” Surf. Sci. 299, 551-562 (1994).
[CrossRef]

X. Zhuang, L. Marrucci, and Y. R. Shen, “Surface-monolayer-induced bulk alignment of liquid crystals,” Phys. Rev. Lett. 73, 1513-1516 (1994).
[CrossRef] [PubMed]

1993 (1)

B. Jérôme and Y. R. Shen, “Anchoring of nematic liquid-crystals on mica in the presence of volatile molecules,” Phys. Rev. E 48, 4556-4574 (1993).
[CrossRef]

1992 (1)

R. Yerushalmi-Rozen, J. Klein, and G. Berkovic, “In situ probing of polymer grafting from solution onto solid substrates by nonlinear optics,” Langmuir 8, 1392-1397 (1992).
[CrossRef]

1990 (1)

G. Berkovic, “New studies of liquid and solid surfaces using second harmonic generation,” Phys. A 168, 140-148 (1990).
[CrossRef]

1989 (1)

Y. R. Shen, “Surface-properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519-525 (1989).
[CrossRef]

1987 (1)

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129-1141(1987).
[CrossRef]

1986 (1)

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254-8263 (1986).
[CrossRef]

1965 (1)

A. I. Bailey and S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39-46 (1965).
[CrossRef]

1962 (1)

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Agam, O.

O. Agam, “Viscous fingering in volatile thin films,” Phys. Rev. E 79, 021603 (2009).
[CrossRef]

Aratake, K.

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Ardizzone, S.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

Asakawa, H.

T. Fukuma, Y. Ueda, S. Yoshioka, and H. Asakawa, “Atomic-scale distribution of water molecules at the mica-water interface visualized by three-dimensional scanning force microscopy,” Phys. Rev. Lett. 104, 016101 (2010).
[CrossRef] [PubMed]

Bailey, A. I.

A. I. Bailey and S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39-46 (1965).
[CrossRef]

Berkovic, G.

R. Yerushalmi-Rozen, J. Klein, and G. Berkovic, “In situ probing of polymer grafting from solution onto solid substrates by nonlinear optics,” Langmuir 8, 1392-1397 (1992).
[CrossRef]

G. Berkovic, “New studies of liquid and solid surfaces using second harmonic generation,” Phys. A 168, 140-148 (1990).
[CrossRef]

Borchardt-Ott, W.

W. Borchardt-Ott, Crystallography (Springer-Verlag, 1995).
[CrossRef]

Briscoe, W. H.

W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivray, and J. Klein, “Boundary lubrication under water,” Nature 444, 191-194 (2006).
[CrossRef] [PubMed]

Cerrone, G.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

Chen, W.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254-8263 (1986).
[CrossRef]

Coombs, N.

H. Yang, A. Kuperman, N. Coombs, S. Mamiche-Afara, and G. A. Ozin, “Synthesis of oriented films of mesoporous silica on mica,” Nature 379, 703-705 (1996).
[CrossRef]

de Lisio, C.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

di Uccio, U. Scotti

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

El-Zaiat, S. Y.

M. Medhat and S. Y. El-Zaiat, “Interferometric determination of the birefringence dispersion of anisotropic materials,” Opt. Commun. 141, 145-149 (1997).
[CrossRef]

Feng, X. Z.

M. Qin, L. K. Wang, and X. Z. Feng, “Bioactive surface modification of mica and poly(dimethylsiloxane) with hydrophobins for protein immobilization,” Langmuir 23, 4465-4471 (2007).
[CrossRef] [PubMed]

Fukuma, T.

T. Fukuma, Y. Ueda, S. Yoshioka, and H. Asakawa, “Atomic-scale distribution of water molecules at the mica-water interface visualized by three-dimensional scanning force microscopy,” Phys. Rev. Lett. 104, 016101 (2010).
[CrossRef] [PubMed]

Gauthier-Manuel, B.

B. Gauthier-Manuel, “Simultaneous determination of the thickness and optical constants of weakly absorbing thin films,” Meas. Sci. Technol. 9, 485-487 (1998).
[CrossRef]

Granozio, F. Miletto

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

Greene, G. W.

B. Zappone, M. Ruths, G. W. Greene, G. D. Jay, and J. N. Israelachvili, “Adsorption lubrication, and wear of lubricin on model surfaces: polymer brush-like behavior of a glycoprotein,” Biophys. J. 92, 1693-1708 (2007).
[CrossRef]

Guyot-Sionnest, P.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254-8263 (1986).
[CrossRef]

Held, H.

X. Wei, S.-C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B 104, 3349-3354 (2000).
[CrossRef]

Hong, S.-C.

X. Wei, S.-C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B 104, 3349-3354 (2000).
[CrossRef]

Israelachvili, J. N.

B. Zappone, M. Ruths, G. W. Greene, G. D. Jay, and J. N. Israelachvili, “Adsorption lubrication, and wear of lubricin on model surfaces: polymer brush-like behavior of a glycoprotein,” Biophys. J. 92, 1693-1708 (2007).
[CrossRef]

Jay, G. D.

B. Zappone, M. Ruths, G. W. Greene, G. D. Jay, and J. N. Israelachvili, “Adsorption lubrication, and wear of lubricin on model surfaces: polymer brush-like behavior of a glycoprotein,” Biophys. J. 92, 1693-1708 (2007).
[CrossRef]

Jérôme, B.

B. Jérôme and Y. R. Shen, “Anchoring of nematic liquid-crystals on mica in the presence of volatile molecules,” Phys. Rev. E 48, 4556-4574 (1993).
[CrossRef]

Kay, S. M.

A. I. Bailey and S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39-46 (1965).
[CrossRef]

Kitaoka, H.

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Klein, J.

W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivray, and J. Klein, “Boundary lubrication under water,” Nature 444, 191-194 (2006).
[CrossRef] [PubMed]

R. Yerushalmi-Rozen, J. Klein, and G. Berkovic, “In situ probing of polymer grafting from solution onto solid substrates by nonlinear optics,” Langmuir 8, 1392-1397 (1992).
[CrossRef]

Kobayashi, E.

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Kuperman, A.

H. Yang, A. Kuperman, N. Coombs, S. Mamiche-Afara, and G. A. Ozin, “Synthesis of oriented films of mesoporous silica on mica,” Nature 379, 703-705 (1996).
[CrossRef]

Lvovsky, A. I.

X. Wei, S.-C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B 104, 3349-3354 (2000).
[CrossRef]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Mamiche-Afara, S.

H. Yang, A. Kuperman, N. Coombs, S. Mamiche-Afara, and G. A. Ozin, “Synthesis of oriented films of mesoporous silica on mica,” Nature 379, 703-705 (1996).
[CrossRef]

Mannhart, J.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

Marrucci, L.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

X. Zhuang, D. Wilk, L. Marrucci, and Y. R. Shen, “Orientation of amphiphilic molecules on polar substrates,” Phys. Rev. Lett. 75, 2144-2147 (1995).
[CrossRef] [PubMed]

X. Zhuang, L. Marrucci, and Y. R. Shen, “Surface-monolayer-induced bulk alignment of liquid crystals,” Phys. Rev. Lett. 73, 1513-1516 (1994).
[CrossRef] [PubMed]

McGillivray, D. J.

W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivray, and J. Klein, “Boundary lubrication under water,” Nature 444, 191-194 (2006).
[CrossRef] [PubMed]

Medhat, M.

M. Medhat and S. Y. El-Zaiat, “Interferometric determination of the birefringence dispersion of anisotropic materials,” Opt. Commun. 141, 145-149 (1997).
[CrossRef]

Mizutani, G.

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Moss, D. J.

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129-1141(1987).
[CrossRef]

Nisenoff, M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Omote, M.

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Ozin, G. A.

H. Yang, A. Kuperman, N. Coombs, S. Mamiche-Afara, and G. A. Ozin, “Synthesis of oriented films of mesoporous silica on mica,” Nature 379, 703-705 (1996).
[CrossRef]

Paparo, D.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

Perna, P.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

Podloucky, R.

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Qin, M.

M. Qin, L. K. Wang, and X. Z. Feng, “Bioactive surface modification of mica and poly(dimethylsiloxane) with hydrophobins for protein immobilization,” Langmuir 23, 4465-4471 (2007).
[CrossRef] [PubMed]

Quagliotto, P. G.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

Richter, C.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

Ristic, Z.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

Ruths, M.

B. Zappone, M. Ruths, G. W. Greene, G. D. Jay, and J. N. Israelachvili, “Adsorption lubrication, and wear of lubricin on model surfaces: polymer brush-like behavior of a glycoprotein,” Biophys. J. 92, 1693-1708 (2007).
[CrossRef]

Salluzzo, M.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

Sano, H.

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Santamato, E.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Savoia, A.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

Shen, Y. R.

X. Wei, S.-C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B 104, 3349-3354 (2000).
[CrossRef]

X. Zhuang, D. Wilk, L. Marrucci, and Y. R. Shen, “Orientation of amphiphilic molecules on polar substrates,” Phys. Rev. Lett. 75, 2144-2147 (1995).
[CrossRef] [PubMed]

X. Zhuang, L. Marrucci, and Y. R. Shen, “Surface-monolayer-induced bulk alignment of liquid crystals,” Phys. Rev. Lett. 73, 1513-1516 (1994).
[CrossRef] [PubMed]

Y. R. Shen, “Surfaces probed by nonlinear optics,” Surf. Sci. 299, 551-562 (1994).
[CrossRef]

B. Jérôme and Y. R. Shen, “Anchoring of nematic liquid-crystals on mica in the presence of volatile molecules,” Phys. Rev. E 48, 4556-4574 (1993).
[CrossRef]

Y. R. Shen, “Surface-properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519-525 (1989).
[CrossRef]

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254-8263 (1986).
[CrossRef]

Sipe, J. E.

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129-1141(1987).
[CrossRef]

Solimeno, S.

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

Suzuki, O.

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Thiel, S.

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

Thomas, R. K.

W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivray, and J. Klein, “Boundary lubrication under water,” Nature 444, 191-194 (2006).
[CrossRef] [PubMed]

Tiberg, F.

W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivray, and J. Klein, “Boundary lubrication under water,” Nature 444, 191-194 (2006).
[CrossRef] [PubMed]

Timtuss, S.

W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivray, and J. Klein, “Boundary lubrication under water,” Nature 444, 191-194 (2006).
[CrossRef] [PubMed]

Ueda, Y.

T. Fukuma, Y. Ueda, S. Yoshioka, and H. Asakawa, “Atomic-scale distribution of water molecules at the mica-water interface visualized by three-dimensional scanning force microscopy,” Phys. Rev. Lett. 104, 016101 (2010).
[CrossRef] [PubMed]

van Driel, H. M.

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129-1141(1987).
[CrossRef]

Wang, L. K.

M. Qin, L. K. Wang, and X. Z. Feng, “Bioactive surface modification of mica and poly(dimethylsiloxane) with hydrophobins for protein immobilization,” Langmuir 23, 4465-4471 (2007).
[CrossRef] [PubMed]

Wei, X.

X. Wei, S.-C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B 104, 3349-3354 (2000).
[CrossRef]

Wilk, D.

X. Zhuang, D. Wilk, L. Marrucci, and Y. R. Shen, “Orientation of amphiphilic molecules on polar substrates,” Phys. Rev. Lett. 75, 2144-2147 (1995).
[CrossRef] [PubMed]

Wolf, W.

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Yang, H.

H. Yang, A. Kuperman, N. Coombs, S. Mamiche-Afara, and G. A. Ozin, “Synthesis of oriented films of mesoporous silica on mica,” Nature 379, 703-705 (1996).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, 1989).

Yerushalmi-Rozen, R.

R. Yerushalmi-Rozen, J. Klein, and G. Berkovic, “In situ probing of polymer grafting from solution onto solid substrates by nonlinear optics,” Langmuir 8, 1392-1397 (1992).
[CrossRef]

Yoshioka, S.

T. Fukuma, Y. Ueda, S. Yoshioka, and H. Asakawa, “Atomic-scale distribution of water molecules at the mica-water interface visualized by three-dimensional scanning force microscopy,” Phys. Rev. Lett. 104, 016101 (2010).
[CrossRef] [PubMed]

Zappone, B.

B. Zappone, M. Ruths, G. W. Greene, G. D. Jay, and J. N. Israelachvili, “Adsorption lubrication, and wear of lubricin on model surfaces: polymer brush-like behavior of a glycoprotein,” Biophys. J. 92, 1693-1708 (2007).
[CrossRef]

Zhuang, X.

X. Zhuang, D. Wilk, L. Marrucci, and Y. R. Shen, “Orientation of amphiphilic molecules on polar substrates,” Phys. Rev. Lett. 75, 2144-2147 (1995).
[CrossRef] [PubMed]

X. Zhuang, L. Marrucci, and Y. R. Shen, “Surface-monolayer-induced bulk alignment of liquid crystals,” Phys. Rev. Lett. 73, 1513-1516 (1994).
[CrossRef] [PubMed]

Biophys. J. (1)

B. Zappone, M. Ruths, G. W. Greene, G. D. Jay, and J. N. Israelachvili, “Adsorption lubrication, and wear of lubricin on model surfaces: polymer brush-like behavior of a glycoprotein,” Biophys. J. 92, 1693-1708 (2007).
[CrossRef]

Br. J. Appl. Phys. (1)

A. I. Bailey and S. M. Kay, “Measurement of refractive index and dispersion of mica, employing multiple beam interference techniques,” Br. J. Appl. Phys. 16, 39-46 (1965).
[CrossRef]

J. Phys. Chem. B (1)

X. Wei, S.-C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B 104, 3349-3354 (2000).
[CrossRef]

J. Phys. Condens. Matter (1)

M. Omote, H. Kitaoka, E. Kobayashi, O. Suzuki, K. Aratake, H. Sano, G. Mizutani, W. Wolf, and R. Podloucky, “Spectral, tensor, and ab initio theoretical analysis of optical second harmonic generation from the rutile TiO2 (110) and (001) faces,” J. Phys. Condens. Matter 17, S175-S200 (2005).
[CrossRef]

Langmuir (2)

R. Yerushalmi-Rozen, J. Klein, and G. Berkovic, “In situ probing of polymer grafting from solution onto solid substrates by nonlinear optics,” Langmuir 8, 1392-1397 (1992).
[CrossRef]

M. Qin, L. K. Wang, and X. Z. Feng, “Bioactive surface modification of mica and poly(dimethylsiloxane) with hydrophobins for protein immobilization,” Langmuir 23, 4465-4471 (2007).
[CrossRef] [PubMed]

Meas. Sci. Technol. (1)

B. Gauthier-Manuel, “Simultaneous determination of the thickness and optical constants of weakly absorbing thin films,” Meas. Sci. Technol. 9, 485-487 (1998).
[CrossRef]

Nature (3)

H. Yang, A. Kuperman, N. Coombs, S. Mamiche-Afara, and G. A. Ozin, “Synthesis of oriented films of mesoporous silica on mica,” Nature 379, 703-705 (1996).
[CrossRef]

W. H. Briscoe, S. Timtuss, F. Tiberg, R. K. Thomas, D. J. McGillivray, and J. Klein, “Boundary lubrication under water,” Nature 444, 191-194 (2006).
[CrossRef] [PubMed]

Y. R. Shen, “Surface-properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519-525 (1989).
[CrossRef]

Opt. Commun. (1)

M. Medhat and S. Y. El-Zaiat, “Interferometric determination of the birefringence dispersion of anisotropic materials,” Opt. Commun. 141, 145-149 (1997).
[CrossRef]

Opt. Lasers Eng. (1)

L. Marrucci, D. Paparo, G. Cerrone, C. de Lisio, E. Santamato, S. Solimeno, S. Ardizzone, and P. G. Quagliotto, “Probing interfacial properties by optical second-harmonic generation,” Opt. Lasers Eng. 37, 601-610 (2002).
[CrossRef]

Phys. A (1)

G. Berkovic, “New studies of liquid and solid surfaces using second harmonic generation,” Phys. A 168, 140-148 (1990).
[CrossRef]

Phys. Rev. B (3)

A. Savoia, D. Paparo, P. Perna, Z. Ristic, M. Salluzzo, F. Miletto Granozio, U. Scotti di Uccio, C. Richter, S. Thiel, J. Mannhart, and L. Marrucci, “Polar catastrophe and electronic reconstructions at the LaAlO3/SrTiO3 interface: evidence from optical second harmonic generation,” Phys. Rev. B 80, 075110 (2009).
[CrossRef]

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254-8263 (1986).
[CrossRef]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129-1141(1987).
[CrossRef]

Phys. Rev. E (2)

B. Jérôme and Y. R. Shen, “Anchoring of nematic liquid-crystals on mica in the presence of volatile molecules,” Phys. Rev. E 48, 4556-4574 (1993).
[CrossRef]

O. Agam, “Viscous fingering in volatile thin films,” Phys. Rev. E 79, 021603 (2009).
[CrossRef]

Phys. Rev. Lett. (4)

T. Fukuma, Y. Ueda, S. Yoshioka, and H. Asakawa, “Atomic-scale distribution of water molecules at the mica-water interface visualized by three-dimensional scanning force microscopy,” Phys. Rev. Lett. 104, 016101 (2010).
[CrossRef] [PubMed]

X. Zhuang, L. Marrucci, and Y. R. Shen, “Surface-monolayer-induced bulk alignment of liquid crystals,” Phys. Rev. Lett. 73, 1513-1516 (1994).
[CrossRef] [PubMed]

X. Zhuang, D. Wilk, L. Marrucci, and Y. R. Shen, “Orientation of amphiphilic molecules on polar substrates,” Phys. Rev. Lett. 75, 2144-2147 (1995).
[CrossRef] [PubMed]

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Surf. Sci. (1)

Y. R. Shen, “Surfaces probed by nonlinear optics,” Surf. Sci. 299, 551-562 (1994).
[CrossRef]

Other (3)

W. Borchardt-Ott, Crystallography (Springer-Verlag, 1995).
[CrossRef]

In crystallography, the glide plane indicates a symmetry operation describing how a reflection in a plane, followed by a translation parallel to that plane, may leave the crystal unchanged . Note that the tensors characterizing the optical response are not dependent on the translations.

A. Yariv, Quantum Electronics (Wiley, 1989).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Axis orientation for the laboratory reference frame x y ^ z and crystal reference frame i j ^ k . Axes a ^ and b ^ represent the projection of the dielectric axis ı ^ and ĵ on the cleavage plane parallel to the x y ^ plane. The tensor ϵ ¯ q j is diagonal respect to these axes (see text). ϑ is the angle formed between axis x ^ and a ^ . For mica, the k ^ axis forms a small 5 ° angle with the z ^ axis (this angle has been exaggerated in figure for clarity) and the shadowed rectangle represents the intersection between the glide plane and crystal cell itself, whereas the r ^ axis is the twofold axis.

Fig. 2
Fig. 2

Results of the simplified model developed in Appendix A. The SH is plotted as a function of the azimuthal angle ϑ and sample thickness d ˜ (in units of wavelength). Note the strong variability of the maximum SH intensity as a function of d ˜ .

Fig. 3
Fig. 3

Layout of the experimental setup: P, Glan–Laser polarizer; λ / 2 , half-wave plate; L, lens; F, filter; MC, monochromator; and PMT, photomultiplier tube. For clarity, the fundamental and SH beams have been displaced, but in reality, they are collinear.

Fig. 4
Fig. 4

SH signal of sample M1 for all polarization combinations. Error bars are given at a 90% level of confidence. The solid lines are the results of a best-fit procedure based on Eqs. (35, 36). The four panels have the same units for the signal S.

Fig. 5
Fig. 5

Comparison of the SH signal of all the samples for the parallel polarization combinations: (a) M1, (b) M2, (c) M3, and (d) M4. Error bars and solid lines are the same as in the caption of Fig. 4. The four panels have the same units for the signal S.

Fig. 6
Fig. 6

Same comparison as in Fig. 5 but for the perpendicular polarization combinations.

Tables (1)

Tables Icon

Table 1 Values of the Parameters Used for the Fitting Curves of Figs. 4, 5, 6 a

Equations (59)

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

P j ( k , ω ) = P j , ω ( 1 ) ( k ) + P j , 2 ω ( 2 ) ( k ) = ϵ 0 χ ˜ j h ( 1 ) ( k , ω ) E h ( k , ω ) + χ ˜ j h l ( 2 ) ( k , k , ω , ω ) × E h ( k , ω ) E l ( k , ω ) δ ( ω ω ω ) δ ( k k k ) d 3 k d 3 k d ω d ω ,
χ ˜ j h ( 1 ) ( k , ω ) = χ j h ( 1 ) ( ω ) + g j h l ( ω ) k l ,
χ ˜ j h l ( 2 ) ( k , k , ω , ω ) = χ j h l ( 2 ) ( ω , ω ) + γ j h l q ( ω , ω ) k q + γ ¯ j h l q ( ω , ω ) k q ,
χ j h ( 1 ) ( ω ) = χ ˜ j h ( 1 ) ( 0 , ω ) ,
χ j h l ( 2 ) ( ω , ω ) = χ ˜ j h l ( 2 ) ( 0 , 0 , ω , ω ) ,
g j h l ( ω ) = χ ˜ j h ( 1 ) / k l | k = 0 ,
γ j h l q ( ω , ω ) = χ ˜ j h l ( 2 ) / k q | k = 0 , k = 0 ,
γ ¯ j h l q ( ω , ω ) = χ ˜ j h l ( 2 ) / k q | k = 0 , k = 0 .
χ j h l ( 2 ) ( ω , ω ) = χ j l h ( 2 ) ( ω , ω ) ,
γ j h l q ( ω , ω ) = γ ¯ j l h q ( ω , ω ) .
P j ( r , ω ) = ϵ 0 χ j h ( 1 ) ( ω ) E h ( r , ω ) + d ω d ω χ j h l ( 2 ) ( ω , ω ) E h ( r , ω ) E l ( r , ω ) δ ( ω ω ω ) i d ω d ω γ j h l q ( ω , ω ) E l ( r , ω ) x q E h ( r , ω ) δ ( ω ω ω ) i d ω d ω γ ¯ j h l q ( ω , ω ) E h ( r , ω ) x q E l ( r , ω ) δ ( ω ω ω ) .
E j ( r , ω ) = E j , ω ¯ ( r ) δ ( ω ω ¯ ) + E j , ω ¯ * ( r ) δ ( ω + ω ¯ ) + E j , 2 ω ¯ ( r ) δ ( ω 2 ω ¯ ) + E j , 2 ω ¯ * ( r ) δ ( ω + 2 ω ¯ ) ,
P j ( r , ω ) = P j , ω ¯ ( r ) δ ( ω ω ¯ ) + P j , ω ¯ * ( r ) δ ( ω + ω ¯ ) + P j , 2 ω ¯ ( r ) δ ( ω 2 ω ¯ ) + P j , 2 ω ¯ * ( r ) δ ( ω + 2 ω ¯ ) ,
P j , ω ¯ ( r ) = ϵ 0 χ j h ( 1 ) ( ω ¯ ) E h , ω ¯ ( r ) ,
P j , 2 ω ¯ ( r ) = ϵ 0 χ j h ( 1 ) ( 2 ω ¯ ) E h , 2 ω ¯ ( r ) + χ j h l ( 2 ) ( ω ¯ , ω ¯ ) E h , ω ¯ ( r ) E l , ω ¯ ( r ) i γ j h l q ( ω ¯ , ω ¯ ) E l , ω ¯ ( r ) x q E h , ω ¯ ( r ) i γ ¯ j h l q ( ω ¯ , ω ¯ ) E h , ω ¯ ( r ) x q E l , ω ¯ ( r ) .
P j , 2 ω ¯ ( r ) = ϵ 0 χ j h ( 1 ) ( 2 ω ¯ ) E h , 2 ω ¯ ( r ) + P j , 2 ω ¯ NL ( r ) ,
P j , 2 ω ¯ NL ( r ) = Γ j h l q ( ω ¯ , ω ¯ ) E h , ω ¯ ( r ) x q E l , ω ¯ ( r ) ,
Γ j h l q ( ω ¯ , ω ¯ ) = 2 i γ j l h q ( ω ¯ , ω ¯ ) = 2 i γ ¯ j h l q ( ω ¯ , ω ¯ ) .
d 2 d z 2 E j , ω ( z ) δ j z d 2 d z 2 E z , ω ( z ) + μ 0 ω 2 ϵ j h ( ω ) E h , ω ( z ) = 0 ,
d 2 d z 2 E j , 2 ω ( z ) δ j z d 2 d z 2 E z , 2 ω ( z ) + 4 μ 0 ω 2 ϵ j h ( 2 ω ) E j , 2 ω ( z ) = 4 μ 0 ω 2 P j , 2 ω NL ( z ) ,
E z , ω = j = x , y ϵ z j ϵ z z E j , ω ,
E z , 2 ω = j = x , y ϵ z j ϵ z z E j , 2 ω 1 ϵ z z P z , 2 ω NL .
d 2 d z 2 E j , ω ( z ) + μ 0 ω 2 ϵ ¯ j h ( ω ) E h , ω ( z ) = 0 ,
d 2 d z 2 E j , 2 ω ( z ) + 4 μ 0 ω 2 ϵ ¯ j h ( 2 ω ) E j , 2 ω ( z ) = 4 μ 0 ω 2 Γ ¯ j h l E h , ω d d z E l , ω ,
ϵ ¯ j h = ϵ j h ϵ j z ϵ z h ϵ z z ,
Γ ¯ j h l = Γ j h l z ϵ j z ϵ z z Γ z h l z ϵ z l ϵ z z Γ j h z z ϵ z h ϵ z z Γ j z l z + ϵ j z ϵ z l ϵ z z 2 Γ z h z z + ϵ j z ϵ z h ϵ z z 2 Γ z z l z + ϵ z l ϵ z h ϵ z z 2 Γ j z z z ϵ j z ϵ z h ϵ z l ϵ z z 3 Γ z z z z .
E γ , 2 ω ( d ) = i d 2 μ 0 ω 2 k γ [ T a a γ E a , ω 2 + ( T a b γ + T b a γ ) E a , ω E b , ω + T b b γ E b , ω 2 ] e i k γ d ,
T α β γ = i η β Γ ¯ γ α β ( ω , ω ) e i ( η α + η β + k γ ) d 2 × sync [ ( η α + η β k γ ) d 2 ] ,
( a ^ b ^ ) = R ( x ^ y ^ ) = ( cos ϑ sin ϑ sin ϑ cos ϑ ) ( x ^ y ^ ) .
E j , 2 ω ( d ) = γ = a , b R j γ E γ , 2 ω ( d ) e i k γ d .
E j , 2 ω ( h ) ( d ) = i 2 d μ 0 ω 2 h E h , ω 2 γ = a , b k γ 1 R j γ [ T a a γ R a h 1 R a h 1 + ( T a b γ + T b a γ ) R a h 1 R b h 1 + T b b γ R b h 1 R b h 1 ] ,
S x x | E x , ω 2 | 2 S ( ϑ ) = | E x , ω 2 | 2 | T a a a k a cos 3 ϑ + ( T a a b k b + T a b a + T b a a k a ) sin ϑ cos 2 ϑ + ( T a b b + T b a b k b + T b b a k a ) sin 2 ϑ cos ϑ + T b b b k b sin 3 ϑ | 2 ,
S y x | E y , ω 2 | 2 S ( ϑ ) = | E y , ω 2 | 2 | T b b a k a cos 3 ϑ + ( T b b b k b T a b a + T b a a k a ) sin ϑ cos 2 ϑ + ( T a b b + T b a b k b + T a a a k a ) sin 2 ϑ cos ϑ + T a a b k b sin 3 ϑ | 2 ,
S y y | E y , ω 2 | 2 S ( ϑ + π / 2 ) ,
S x y | E x , ω 2 | 2 S ( ϑ + π / 2 ) .
S ( ϑ ) cos 2 ϑ [ ( A cos 2 ϑ + B ) 2 + C ] ,
S ( ϑ ) cos 2 ϑ [ ( A cos 2 ϑ + D ) 2 + F ] ,
L = T a a a T b b a k a T a b b + T b a b k b ,
ϕ a = atan [ Im ( L ) Re ( L ) ] ,
H 1 = Re [ ( T b b a k a + T a b b + T b a b k b ) e i ϕ a ] ,
H 2 = Re [ ( T a b b + T b a b k b T a a a k a ) e i ϕ a ] ,
A = | L | ,
B = Re ( H 1 ) ,
C = [ Im ( H 1 ) ] 2 ,
D = Re ( H 2 ) ,
F = [ Im ( H 2 ) ] 2 .
( A + B ) 2 + C = | T a a a / k a | = | η a k a | | Γ a a a z | sync [ ω δ a d c ] ,
( A + D ) 2 + F = | T b b a / k a | = | η b k a | | Γ a b b z | sync [ ω ( Δ ω δ a ) d c ] ,
( A + B + D ) 2 + ( C ± F ) 2 = | ( T a b b + T b a b ) / k b | = | η b Γ b a b z + η a Γ b b a z | | k b | × sync [ ω ( Δ ω 2 Δ 2 ω δ a ) d c ] .
S x x | [ Γ a a a z f 0 ( d ˜ ) Γ a b b z f 1 ( d ˜ ) ( Γ b a b z + Γ b b a z ) f 2 ( d ˜ ) ] cos 3 ϑ + Γ a b b z f 1 ( d ˜ ) cos ϑ + ( Γ b a b z + Γ b b a z ) f 2 ( d ˜ ) cos ϑ | 2 ,
f 0 ( d ˜ ) = sync ( 2 π δ a d ˜ ) ,
f 1 ( d ˜ ) = sync [ 2 π ( Δ ω δ a ) d ˜ ] ,
f 2 ( d ˜ ) = e i π Δ ω d ˜ × sync [ 2 π ( Δ ω / 2 Δ 2 ω δ a ) d ˜ ] .
S x x 1 16 | ( Γ a a a z Γ a b b z Γ b a b z Γ b b a z ) cos ( 3 ϑ ) + ( 3 Γ a a a z + Γ a b b z + Γ b a b z + Γ b b a z ) cos ϑ | 2 ,
S x x cos 2 ϑ | { [ f 0 ( d ˜ ) + 3 f 2 ( d ˜ ) ] Γ a a a z + [ f 2 ( d ˜ ) f 1 ( d ˜ ) ] Γ a b b z } cos 2 ϑ 3 f 2 ( d ˜ ) Γ a a a z [ f 2 ( d ˜ ) f 1 ( d ˜ ) ] Γ a b b z | 2 .
S x x | [ Γ a a a z Γ a b b z g 0 ( d ˜ ) + g 1 ( d ˜ ) ] cos 2 ϑ Γ a a a z Γ a b b z g 2 ( d ˜ ) g 1 ( d ˜ ) | 2 ,
g 0 ( d ˜ ) = sin ( 2 π δ a d ˜ ) + 3 e i π Δ ω d ˜ sin [ 2 π ( Δ ω / 2 Δ 2 ω δ a ) ] ,
g 1 ( d ˜ ) = e i π Δ ω d ˜ sin [ 2 π ( Δ ω / 2 Δ 2 ω δ a ) ] sin [ 2 π ( Δ ω δ a ) d ˜ ] ,
g 2 ( d ˜ ) = g 0 ( d ˜ ) sin ( 2 π δ a d ˜ ) .

Metrics