Abstract

In this paper, we compute the second harmonic radiation generated at a roughened metal surface which is describable in terms of randomly distributed height and spacing roughness variables. The roughness of the surface allows incident radiation to be coupled to a surface-plasmon mode in the metal at frequency ω that beats with the fundamental field at the same frequency to produce a 2ω field of frequency which radiates from the metal in a narrowly defined direction. Given appropriate rough surface parameters, the latter radiation can in principle serve as a well-collimated coherent source of radiation at the second harmonic frequency. We compare the collimated intensity of the harmonic produced by the roughness-induced plasmon coupling with that produced by specular reflection from the metal surface. The former intensity is predicted to exceed the latter by as much as 2 orders of magnitude.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  2. S. S. Jha, “Theory of Optical Harmonic Generation at a Metal Surface,” Phy. Rev. A 140, 2020 (1965); S. S. Jha, C. S. Warke, “Interband Contributions to Optical Harmonic Generation at a Metal Surface,” Phys. Rev. 153, 751 (1967).
    [CrossRef]
  3. N. Bloembergen, R. K. Chang, S. S. Jha, C. M. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); “Errata,” Phys. Rev. E 178, 1528 (1969).
    [CrossRef]
  4. H. J. Simon, D. E. Mitchell, J. G. Watson, “Optical Second Harmonic Generation with Surface Plasmons in Silver Films,” Phys. Rev. Lett. 33, 1531 (1974).
    [CrossRef]
  5. R. A. Ferrell, “Predicted Radiation of Plasma Oscillations in Metal Films,” Phys. Rev. 111, 1214 (1958).
    [CrossRef]
  6. E. Kröger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emissions,” Z. Physi. 237, 1 (1970).
    [CrossRef]
  7. D. Beaglehole, O. Hunderi, “Interaction of Light with Rough Metal Surfaces,” Opt. Commun. 1, 101 (1969); “Study of the Interaction of Light with Rough Metal Surfaces I. Experiment, II. Theory,” Phys. Rev. B 2, 309 (1970).
  8. J. M. Elson, R. H. Ritchie, “Photon Interactions at a Rough Metal Surface,” Phys. Rev. B 4, 4129 (1971); “Diffuse Scattering and Surface-Plasmon Generation by Photons at a Rough Dielectric Surface,” Phys. Status Solid B 62, 461 (1974).
    [CrossRef]
  9. A review is given in Advances in Chemical Physics, Vol. 27, I. Prigogine, S. A. Rice, Eds. (Wiley, New York, 1974).
    [CrossRef]
  10. V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
    [CrossRef]
  11. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. B 15, 5618 (1977).
    [CrossRef]
  12. A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
    [CrossRef]
  13. A. Maradudin, W. Zierau, “Effects of Surface Roughness on the Surface-Polariton Dispersion Relation,” Phys. Rev. B 14, 484 (1976).
    [CrossRef]
  14. See, for example, S. O. Sari, D. K. Cohen, K. D. Scherkoskie, “Study of Surface Plasma-Wave Reflectance and Roughness-Induced Scattering in Silver Foils,” Phys. Rev. B 21, 2162 (1980).
    [CrossRef]
  15. G. S. Agarwal, S. S. Jha, “Surface-Enhanced Second-Harmonic Generation at a Metallic Grating,” Phys. Rev. B 26, 482 (1982).
    [CrossRef]
  16. C. K. Chen, A. R. B. de Castro, Y. R. Shen, “Surface-Enhanced Second-Harmonic Generation,” Phys. Rev. Lett. 46, 145 (1981).
    [CrossRef]
  17. The z dependence of E(1)(r) occurs only in an exponential factor exp(iβω/cz), where β is a number of order 1, and it follows that the difference between E(1)(r) at z = ζ and at z = 0 is proportional to (ω/c)ζE(1) and, therefore, of order (ω/cζ)2.
  18. We compute the Green’s function integral over the Fourier transform of the source term Ps by evaluation of the integrand at the discontinuity of E(0)(r) for z = 0 via the prescription quoted in Ref. 14.
  19. N. Bloembergen, R. K. Chang, S. S. Jha, C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); C. S. Wang, J. M. Chen, J. R. Bower, “Second Harmonic Generation from Alkali Metals,” Opt. Commun. 8, 275 (1973).
    [CrossRef]
  20. Ref. 12, Eq. (2.67).
  21. Although the plots extend over an angular range including the angle θR(0), the plotted ratio is not accurate in the vicinity of this angle since certain scattering terms which contribute at this angle have been neglected.

1982 (1)

G. S. Agarwal, S. S. Jha, “Surface-Enhanced Second-Harmonic Generation at a Metallic Grating,” Phys. Rev. B 26, 482 (1982).
[CrossRef]

1981 (1)

C. K. Chen, A. R. B. de Castro, Y. R. Shen, “Surface-Enhanced Second-Harmonic Generation,” Phys. Rev. Lett. 46, 145 (1981).
[CrossRef]

1980 (1)

See, for example, S. O. Sari, D. K. Cohen, K. D. Scherkoskie, “Study of Surface Plasma-Wave Reflectance and Roughness-Induced Scattering in Silver Foils,” Phys. Rev. B 21, 2162 (1980).
[CrossRef]

1977 (1)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. B 15, 5618 (1977).
[CrossRef]

1976 (1)

A. Maradudin, W. Zierau, “Effects of Surface Roughness on the Surface-Polariton Dispersion Relation,” Phys. Rev. B 14, 484 (1976).
[CrossRef]

1975 (2)

A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
[CrossRef]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[CrossRef]

1974 (1)

H. J. Simon, D. E. Mitchell, J. G. Watson, “Optical Second Harmonic Generation with Surface Plasmons in Silver Films,” Phys. Rev. Lett. 33, 1531 (1974).
[CrossRef]

1971 (1)

J. M. Elson, R. H. Ritchie, “Photon Interactions at a Rough Metal Surface,” Phys. Rev. B 4, 4129 (1971); “Diffuse Scattering and Surface-Plasmon Generation by Photons at a Rough Dielectric Surface,” Phys. Status Solid B 62, 461 (1974).
[CrossRef]

1970 (1)

E. Kröger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emissions,” Z. Physi. 237, 1 (1970).
[CrossRef]

1969 (1)

D. Beaglehole, O. Hunderi, “Interaction of Light with Rough Metal Surfaces,” Opt. Commun. 1, 101 (1969); “Study of the Interaction of Light with Rough Metal Surfaces I. Experiment, II. Theory,” Phys. Rev. B 2, 309 (1970).

1968 (2)

N. Bloembergen, R. K. Chang, S. S. Jha, C. M. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); “Errata,” Phys. Rev. E 178, 1528 (1969).
[CrossRef]

N. Bloembergen, R. K. Chang, S. S. Jha, C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); C. S. Wang, J. M. Chen, J. R. Bower, “Second Harmonic Generation from Alkali Metals,” Opt. Commun. 8, 275 (1973).
[CrossRef]

1965 (1)

S. S. Jha, “Theory of Optical Harmonic Generation at a Metal Surface,” Phy. Rev. A 140, 2020 (1965); S. S. Jha, C. S. Warke, “Interband Contributions to Optical Harmonic Generation at a Metal Surface,” Phys. Rev. 153, 751 (1967).
[CrossRef]

1958 (1)

R. A. Ferrell, “Predicted Radiation of Plasma Oscillations in Metal Films,” Phys. Rev. 111, 1214 (1958).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, S. S. Jha, “Surface-Enhanced Second-Harmonic Generation at a Metallic Grating,” Phys. Rev. B 26, 482 (1982).
[CrossRef]

Beaglehole, D.

D. Beaglehole, O. Hunderi, “Interaction of Light with Rough Metal Surfaces,” Opt. Commun. 1, 101 (1969); “Study of the Interaction of Light with Rough Metal Surfaces I. Experiment, II. Theory,” Phys. Rev. B 2, 309 (1970).

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, C. M. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); “Errata,” Phys. Rev. E 178, 1528 (1969).
[CrossRef]

N. Bloembergen, R. K. Chang, S. S. Jha, C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); C. S. Wang, J. M. Chen, J. R. Bower, “Second Harmonic Generation from Alkali Metals,” Opt. Commun. 8, 275 (1973).
[CrossRef]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Celli, V.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. B 15, 5618 (1977).
[CrossRef]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[CrossRef]

Chang, R. K.

N. Bloembergen, R. K. Chang, S. S. Jha, C. M. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); “Errata,” Phys. Rev. E 178, 1528 (1969).
[CrossRef]

N. Bloembergen, R. K. Chang, S. S. Jha, C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); C. S. Wang, J. M. Chen, J. R. Bower, “Second Harmonic Generation from Alkali Metals,” Opt. Commun. 8, 275 (1973).
[CrossRef]

Chen, C. K.

C. K. Chen, A. R. B. de Castro, Y. R. Shen, “Surface-Enhanced Second-Harmonic Generation,” Phys. Rev. Lett. 46, 145 (1981).
[CrossRef]

Cohen, D. K.

See, for example, S. O. Sari, D. K. Cohen, K. D. Scherkoskie, “Study of Surface Plasma-Wave Reflectance and Roughness-Induced Scattering in Silver Foils,” Phys. Rev. B 21, 2162 (1980).
[CrossRef]

de Castro, A. R. B.

C. K. Chen, A. R. B. de Castro, Y. R. Shen, “Surface-Enhanced Second-Harmonic Generation,” Phys. Rev. Lett. 46, 145 (1981).
[CrossRef]

Elson, J. M.

J. M. Elson, R. H. Ritchie, “Photon Interactions at a Rough Metal Surface,” Phys. Rev. B 4, 4129 (1971); “Diffuse Scattering and Surface-Plasmon Generation by Photons at a Rough Dielectric Surface,” Phys. Status Solid B 62, 461 (1974).
[CrossRef]

Ferrell, R. A.

R. A. Ferrell, “Predicted Radiation of Plasma Oscillations in Metal Films,” Phys. Rev. 111, 1214 (1958).
[CrossRef]

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. B 15, 5618 (1977).
[CrossRef]

Hunderi, O.

D. Beaglehole, O. Hunderi, “Interaction of Light with Rough Metal Surfaces,” Opt. Commun. 1, 101 (1969); “Study of the Interaction of Light with Rough Metal Surfaces I. Experiment, II. Theory,” Phys. Rev. B 2, 309 (1970).

Jha, S. S.

G. S. Agarwal, S. S. Jha, “Surface-Enhanced Second-Harmonic Generation at a Metallic Grating,” Phys. Rev. B 26, 482 (1982).
[CrossRef]

N. Bloembergen, R. K. Chang, S. S. Jha, C. M. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); “Errata,” Phys. Rev. E 178, 1528 (1969).
[CrossRef]

N. Bloembergen, R. K. Chang, S. S. Jha, C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); C. S. Wang, J. M. Chen, J. R. Bower, “Second Harmonic Generation from Alkali Metals,” Opt. Commun. 8, 275 (1973).
[CrossRef]

S. S. Jha, “Theory of Optical Harmonic Generation at a Metal Surface,” Phy. Rev. A 140, 2020 (1965); S. S. Jha, C. S. Warke, “Interband Contributions to Optical Harmonic Generation at a Metal Surface,” Phys. Rev. 153, 751 (1967).
[CrossRef]

Kretschmann, E.

E. Kröger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emissions,” Z. Physi. 237, 1 (1970).
[CrossRef]

Kröger, E.

E. Kröger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emissions,” Z. Physi. 237, 1 (1970).
[CrossRef]

Lee, C. H.

N. Bloembergen, R. K. Chang, S. S. Jha, C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); C. S. Wang, J. M. Chen, J. R. Bower, “Second Harmonic Generation from Alkali Metals,” Opt. Commun. 8, 275 (1973).
[CrossRef]

Lee, C. M.

N. Bloembergen, R. K. Chang, S. S. Jha, C. M. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); “Errata,” Phys. Rev. E 178, 1528 (1969).
[CrossRef]

Maradudin, A.

A. Maradudin, W. Zierau, “Effects of Surface Roughness on the Surface-Polariton Dispersion Relation,” Phys. Rev. B 14, 484 (1976).
[CrossRef]

A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
[CrossRef]

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. B 15, 5618 (1977).
[CrossRef]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[CrossRef]

Mills, D. L.

A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
[CrossRef]

Mitchell, D. E.

H. J. Simon, D. E. Mitchell, J. G. Watson, “Optical Second Harmonic Generation with Surface Plasmons in Silver Films,” Phys. Rev. Lett. 33, 1531 (1974).
[CrossRef]

Ritchie, R. H.

J. M. Elson, R. H. Ritchie, “Photon Interactions at a Rough Metal Surface,” Phys. Rev. B 4, 4129 (1971); “Diffuse Scattering and Surface-Plasmon Generation by Photons at a Rough Dielectric Surface,” Phys. Status Solid B 62, 461 (1974).
[CrossRef]

Sari, S. O.

See, for example, S. O. Sari, D. K. Cohen, K. D. Scherkoskie, “Study of Surface Plasma-Wave Reflectance and Roughness-Induced Scattering in Silver Foils,” Phys. Rev. B 21, 2162 (1980).
[CrossRef]

Scherkoskie, K. D.

See, for example, S. O. Sari, D. K. Cohen, K. D. Scherkoskie, “Study of Surface Plasma-Wave Reflectance and Roughness-Induced Scattering in Silver Foils,” Phys. Rev. B 21, 2162 (1980).
[CrossRef]

Shen, Y. R.

C. K. Chen, A. R. B. de Castro, Y. R. Shen, “Surface-Enhanced Second-Harmonic Generation,” Phys. Rev. Lett. 46, 145 (1981).
[CrossRef]

Simon, H. J.

H. J. Simon, D. E. Mitchell, J. G. Watson, “Optical Second Harmonic Generation with Surface Plasmons in Silver Films,” Phys. Rev. Lett. 33, 1531 (1974).
[CrossRef]

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. B 15, 5618 (1977).
[CrossRef]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[CrossRef]

Watson, J. G.

H. J. Simon, D. E. Mitchell, J. G. Watson, “Optical Second Harmonic Generation with Surface Plasmons in Silver Films,” Phys. Rev. Lett. 33, 1531 (1974).
[CrossRef]

Zierau, W.

A. Maradudin, W. Zierau, “Effects of Surface Roughness on the Surface-Polariton Dispersion Relation,” Phys. Rev. B 14, 484 (1976).
[CrossRef]

Opt. Commun. (1)

D. Beaglehole, O. Hunderi, “Interaction of Light with Rough Metal Surfaces,” Opt. Commun. 1, 101 (1969); “Study of the Interaction of Light with Rough Metal Surfaces I. Experiment, II. Theory,” Phys. Rev. B 2, 309 (1970).

Phy. Rev. A (1)

S. S. Jha, “Theory of Optical Harmonic Generation at a Metal Surface,” Phy. Rev. A 140, 2020 (1965); S. S. Jha, C. S. Warke, “Interband Contributions to Optical Harmonic Generation at a Metal Surface,” Phys. Rev. 153, 751 (1967).
[CrossRef]

Phys. Rev. (3)

N. Bloembergen, R. K. Chang, S. S. Jha, C. M. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); “Errata,” Phys. Rev. E 178, 1528 (1969).
[CrossRef]

R. A. Ferrell, “Predicted Radiation of Plasma Oscillations in Metal Films,” Phys. Rev. 111, 1214 (1958).
[CrossRef]

N. Bloembergen, R. K. Chang, S. S. Jha, C. H. Lee, “Optical Second-Harmonic Generation in Reflection from Media with Inversion Symmetry,” Phys. Rev. 174, 813 (1968); C. S. Wang, J. M. Chen, J. R. Bower, “Second Harmonic Generation from Alkali Metals,” Opt. Commun. 8, 275 (1973).
[CrossRef]

Phys. Rev. B (7)

J. M. Elson, R. H. Ritchie, “Photon Interactions at a Rough Metal Surface,” Phys. Rev. B 4, 4129 (1971); “Diffuse Scattering and Surface-Plasmon Generation by Photons at a Rough Dielectric Surface,” Phys. Status Solid B 62, 461 (1974).
[CrossRef]

V. Celli, A. Marvin, F. Toigo, “Light Scattering from Rough Surfaces,” Phys. Rev. B 11, 1779 (1975).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical Properties of Rough Surfaces: General Theory and the Small Roughness Limit,” Phys. Rev. B 15, 5618 (1977).
[CrossRef]

A. Maradudin, D. L. Mills, “Scattering and Absorption of Electromagnetic Radiation by a Semi-Infinite Medium in the Presence of Surface Roughness,” Phys. Rev. B 11, 1392 (1975).
[CrossRef]

A. Maradudin, W. Zierau, “Effects of Surface Roughness on the Surface-Polariton Dispersion Relation,” Phys. Rev. B 14, 484 (1976).
[CrossRef]

See, for example, S. O. Sari, D. K. Cohen, K. D. Scherkoskie, “Study of Surface Plasma-Wave Reflectance and Roughness-Induced Scattering in Silver Foils,” Phys. Rev. B 21, 2162 (1980).
[CrossRef]

G. S. Agarwal, S. S. Jha, “Surface-Enhanced Second-Harmonic Generation at a Metallic Grating,” Phys. Rev. B 26, 482 (1982).
[CrossRef]

Phys. Rev. Lett. (2)

C. K. Chen, A. R. B. de Castro, Y. R. Shen, “Surface-Enhanced Second-Harmonic Generation,” Phys. Rev. Lett. 46, 145 (1981).
[CrossRef]

H. J. Simon, D. E. Mitchell, J. G. Watson, “Optical Second Harmonic Generation with Surface Plasmons in Silver Films,” Phys. Rev. Lett. 33, 1531 (1974).
[CrossRef]

Z. Physi. (1)

E. Kröger, E. Kretschmann, “Scattering of Light by Slightly Rough Surfaces or Thin Films Including Plasma Resonance Emissions,” Z. Physi. 237, 1 (1970).
[CrossRef]

Other (6)

A review is given in Advances in Chemical Physics, Vol. 27, I. Prigogine, S. A. Rice, Eds. (Wiley, New York, 1974).
[CrossRef]

Ref. 12, Eq. (2.67).

Although the plots extend over an angular range including the angle θR(0), the plotted ratio is not accurate in the vicinity of this angle since certain scattering terms which contribute at this angle have been neglected.

The z dependence of E(1)(r) occurs only in an exponential factor exp(iβω/cz), where β is a number of order 1, and it follows that the difference between E(1)(r) at z = ζ and at z = 0 is proportional to (ω/c)ζE(1) and, therefore, of order (ω/cζ)2.

We compute the Green’s function integral over the Fourier transform of the source term Ps by evaluation of the integrand at the discontinuity of E(0)(r) for z = 0 via the prescription quoted in Ref. 14.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

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 (3)

Fig. 1
Fig. 1

Geometry at boundary of rough metal surface.

Fig. 2
Fig. 2

Ratio R ( θ R ) / S R ( 0 ) plotted as a function of the angle of emission of the harmonic radiation θR in different angular intervals (a)–(d) with the angle of incidence of the fundamental radiation taken to be 26°. Ratio R ( θ R ) / S R ( 0 ) integrated over a detector solid angle represents the ratio of the Poynting vector for harmonic radiation emitted at angle θR to that emitted at the specular angle θ R ( 0 ). The roughness parameters have values a = 500 Å, δ = 100 Å. Note scale factor E3 = 103 in Figs. (b) and (d).

Fig. 3
Fig. 3

Same as Fig. 2 with the angle of incidence of the fundamental radiation taken to be 45°.

Equations (70)

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

ω c ζ = 2 π λ ζ 1.
ɛ ( z ) = ɛ o ( z ) + ɛ s ( z ) ,
ɛ s ( z ) = ( ɛ - ɛ i ) { θ ( z ) - θ [ z - ζ ( x , y ) ] } ,
θ ( z ) { 1 , z > 0 0 , z < 0 , ɛ o ( z ) { ɛ i , z > 0 ɛ , z < 0.
D ( r ) = E ( r ) + 4 π P ( r ) = ɛ ( z ) E ( r ) ,
D ( r ) = ɛ o ( z ) E ( r ) + 4 π P s ( r ) ,
P s ( r ) = 1 4 π ɛ s ( z ) E ( r ) .
P s ( r ) = 1 4 π ( ɛ - ɛ i ) δ ( z ) ζ ( x , y ) E ( r ) .
ω c P s = 2 π λ P s
ω c ζ = 2 π λ ζ .
E ( r ) = E ( 0 ) ( r ) + E ( 1 ) ( r ) ,
P s ( r ) = 1 4 π ( ɛ - ɛ i ) δ ( z ) ζ ( x , y ) E ( 0 ) ( r ) .
E ( 0 ) ( r ) = { E i ( 0 ) ( r ) + E r ( 0 ) ( r ) , z > 0 , E t ( 0 ) ( r ) , z < 0.
E i ( 0 ) ( r ) = E i ( 0 ) exp ( i k i · r ) = ( x ^ k i z k i + z ^ k k i ) E i ( 0 ) exp [ i ( k x - k i z z ) , E r ( 0 ) ( r ) = E r ( 0 ) exp ( i k r · r ) = - ( x ^ k i z k i - z ^ k k i ) r E i ( 0 ) × exp [ i ( k x + k i z z ) ] , E t ( 0 ) ( r ) = E t ( 0 ) exp ( i k · r ) = ( x ^ k z k + z ^ k k ) t E i ( 0 ) exp [ i ( k x - k z z ) ] ,
k i = k r = ɛ i ω c = ( k 2 + k i z 2 ) 1 / 2 , k = ɛ ω c = ( k 2 + k z 2 ) 1 / 2 ,
r = ɛ k i z - ɛ i k z ɛ k i z + ɛ i k z ,             t = 2 ɛ ɛ i k i z ɛ k i z + ɛ i k z .
k i x = k r x = k x k .
E ( 1 ) ( r ) = d q E ( 1 ) ( q , z ) exp ( i q · R ) ,
ζ ( x , y ) exp ( i k x ) = d q ζ ( q - k x ^ ) exp ( i q · R ) , ζ ( q - k x ^ ) = 1 ( 2 π ) 2 d R ζ ( x , y ) exp [ - i ( q - k x ^ ) · R ] ,
R x x ^ + y y ^ ,             q q x x ^ + q y y ^ .
E ( r ) = { E i ( 0 ) ( r ) + E r ( 0 ) ( r ) + E r ( 1 ) ( r ) , z > 0 E t ( 0 ) ( r ) + E t ( 1 ) ( r ) , z < 0
E r ( 1 ) ( r ) d q E r ( 1 ) ( q , z ) exp ( i q · R ) , E t ( 1 ) ( r ) d q E t ( 1 ) ( q , z ) exp ( i q · R ) .
E t ( 1 ) ( q , z ) = E ( 1 ) ( q , z < 0 ) = i ζ ( q - k x ^ ) ( ɛ - 1 ) ( ɛ α + γ ) [ α q x q k z k + q k k ] t E i ( 0 ) ( γ x ^ + q z ^ ) × exp ( - i γ z ) - i ζ ( q - k x ^ ) ( ɛ - 1 ) ( α + γ ) ( ω c ) 2 q y q k z k × t E i ( 0 ) y ^ exp ( - i γ z ) ,
α = ω 2 c 2 - q 2 , γ = { ɛ ω 2 c 2 - q 2 , Re ɛ > 0 i - ɛ ω 2 c 2 + q 2 , Re ɛ < 0 ,
P N L ( r , 2 ω ) = b 1 [ E t 2 ( r , ω ) ] + b 2 E t ( r , ω ) · E t ( r , ω ) ,
P N L ( r , 2 ω ) = b 1 [ E t 2 ( r , ω ) ] ,
P N L ( r ) = b 1 ( [ E t ( 0 ) ( r ) ] 2 + 2 E t ( 0 ) ( r ) · E t ( 1 ) ( r ) + [ E t ( 1 ) ( r ) ] 2 ) .
P N L ( r ) = P 0 N L ( r ) + P 1 N L ( r ) + P 2 N L ( r ) ,
P 0 N L ( r ) P 0 k s 0 exp ( i k s 0 · r ) = i b 1 t 2 [ E i ( 0 ) ] 2 k s 0 exp ( i k s 0 · r ) , P 1 N L ( r ) d q P 1 ( q ) k s 1 ( q ) exp [ i k s 1 ( q ) · r ] = - 2 b 1 t 2 [ E i ( 0 ) ] 2 d q ζ ( q - k x ^ ) ( ɛ - 1 ) ( ɛ α + γ ) [ α q x k z q k + q k k ) × ( γ k z k + q k k ) k s 1 ( q ) exp [ i k s 1 ( q ) · r ]
P 2 N L ( r ) d q d q P 2 ( q , q ) k s 2 ( q , q ) exp [ i k s 2 ( q , q ) · r ] = - i b 1 t 2 [ E i ( 0 ) ] 2 d q d q ( q , q ) k s 2 ( q , q ) exp [ i k s 2 ( q , q ) · r ] , ( q , q ) ζ ( q - k x ^ ) ζ ( q - k x ^ ) [ ( ɛ - 1 ) 2 ( ɛ α + γ ) ( ɛ α + γ ) ( α q x q k z k + q k k ) ( α q x q k z k + q k k ) ( γ γ + q q ) + ( ω c ) 4 ( ɛ - 1 ) 2 ( α + γ ) ( α + γ ) q y q q y q ( k z k ) 2 ] ,
k s 0 = 2 k x ^ - 2 k z z ^ , k s 1 ( q ) = ( k + q x ) x ^ + q y y ^ - ( k z + γ ) z ^ , k s 2 ( q , q ) = ( q x + q x ) x ^ + ( q y + q y ) y ^ - ( γ + γ ) z ^ .
- 4 π c 2 2 t 2 [ P N L ( r ) exp ( - 2 i ω t ) ] = 4 π ( 2 ω c ) 2 P N L ( r ) exp ( - 2 i ω t ) ,
× × E ¯ - ɛ ( 2 ω ) ( 2 ω c ) 2 E ¯ = 4 π ( 2 ω c ) 2 P N L .
E ¯ ( r ) = Σ s exp ( i k s · r ) Σ exp ( i K · r ) ,
K · Σ = 0 ,
( k s 2 - K 2 ) Σ s - k s ( k s · Σ s ) = 4 π ( 2 ω c ) 2 P .
Σ s = 4 π ( 2 ω c ) 2 k s 2 - K 2 [ P - k s ( k s · P ) K 2 ] ,
Σ s = - 4 π ( 2 ω c ) 2 K 2 P = - 4 π ɛ ( 2 ω ) P .
E ¯ ( r ) = E ¯ 0 ( r ) + E ¯ 1 ( r ) + E ¯ 2 ( r ) ,
E ¯ 0 ( r ) = - 4 π ɛ ( 2 ω ) P 0 k s 0 exp ( i k s 0 · r ) + Σ 0 exp ( i K 0 · r ) ,
E ¯ 1 ( r ) = d q [ - 4 π ɛ ( 2 ω ) P 1 ( q ) k s 1 ( q ) exp [ i k s 1 ( q ) · r + Σ 1 ( q ) exp [ i K 1 ( q ) · r ] ,
E ¯ 2 ( r ) = d q d q { - 4 π ɛ ( 2 ω ) P 2 ( q , q ) k s 2 ( q , q ) × exp [ i k s 2 ( q , q ) · r ] + Σ 2 ( q , q ) exp [ i K 2 ( q , q ) · r ] }
K n · Σ n = 0 ,             ( n = 0 , 1 , 2 ) ,
E ¯ ( r ) = E ¯ ( 0 ) ( r ) + E ¯ ( 1 ) ( r ) .
E ¯ ( 0 ) ( r ) = { E ¯ R 0 ( 0 ) ( r ) + E ¯ R 1 ( 0 ) ( r ) + E ¯ R 2 ( 0 ) ( r ) , z > 0 E ¯ 0 ( r ) + E ¯ 1 ( r ) + E ¯ 2 ( r ) , z < 0 ,
E ¯ R 0 ( 0 ) ( r ) = Σ R 0 ( 0 ) exp ( i K R 0 · r ) , E ¯ R 1 ( 0 ) ( r ) = d q Σ R 1 ( 0 ) ( q ) exp [ i K R 1 ( q ) · r ] , E ¯ R 2 ( 0 ) ( r ) = d q d q Σ R 2 ( 0 ) ( q , q ) exp [ i K R 2 ( q , q ) · r ]
Σ R n ( 0 ) = 4 π ( 2 ω c ) ( K R n ) P n ɛ ( 2 ω ) ( 2 ω c ) 2 - ( K R n ) 2 + ɛ ( 2 ω ) ( 2 ω c ) 2 - ( K R n ) 2 , [ Σ R n ( 0 ) ] z = ( K R n ) K R n Σ R n ( 0 ) ,             [ Σ R n ( 0 ) ] = - ( K R n ) z K R n Σ R n ( 0 ) , K R n = 2 ω c , ( K R n ) = ( k s n ) = ( K n ) ,             ( n = 0 , 1 , 2 ) ,
E ¯ R ( r ) = E ¯ R 0 ( 0 ) ( r ) + E ¯ R 0 ( 1 ) ( r ) + E ¯ R 1 ( 0 ) ( r ) + E ¯ R 2 ( 0 ) ( r ) .
E ¯ R 0 ( 1 ) ( r ) = d q x Σ R 0 ( 1 ) ( q x , z ) z > 0 exp ( i q x x ) , E ¯ R 1 ( 1 ) ( r ) = d q x Σ R 1 ( 1 ) ( q x , z ) z > 0 exp ( i q x x ) ,
{ [ Σ R 0 ( 1 ) ] x , [ Σ R 0 ( 1 ) ] z } = i ζ ( q x - 2 k ) ( ɛ s - 1 ) ( ɛ s α s + γ s ) [ { } x , { } z ] , { } x = { { [ Σ R 0 ( 0 ) ] α s γ s - [ Σ R 0 ( 0 ) ] z α s q x + 2 π ɛ s P 0 ( k s 0 ) z α s q x } exp ( i α s z ) ,             z > 0 , { [ Σ R 0 ( 0 ) ] α s γ s + [ Σ R 0 ( 0 ) ] z γ s ɛ s q x - 2 π ɛ s P 0 ( k s 0 ) z γ s ɛ s q x } exp ( - i γ s z ) ,             z < 0 , { } z = { { - [ Σ R 0 ( 0 ) ] γ s q x + [ Σ R 0 ( 0 ) ] z q x 2 - 2 π ɛ s P 0 ( k s 0 ) z q x 2 } exp ( i α s z ) ,             z > 0 , { [ Σ R 0 ( 0 ) ] α s q x + [ Σ R 0 ( 0 ) ] z q x 2 ɛ s - 2 π ɛ s P 0 ( k s 0 ) z q x 2 ɛ s } exp ( - i γ s z ) ,             z < 0 , { [ Σ R 1 ( 1 ) ] x , [ Σ R 1 ( 1 ) ] z } = i d q x ζ [ q x - k s 1 x ( q x ) ] ( ɛ s - 1 ) ( ɛ s α s + γ s ) [ { } x , { } z ] , { } x , z = { } x , z Σ R 0 ( 0 ) Σ R 1 , P 0 ( 0 ) k s 0 P 1 ( q x ) k s 1 ( q x )
ɛ s ɛ ( 2 ω ) , α s = ( 2 ω c ) 2 - q x 2 ,             γ s = ɛ s ( 2 ω c ) 2 - q x 2 .
S R = c 8 π E ¯ R ( r ) × B ¯ R * ( r )
B ¯ R = - i ( c 2 ω ) × E ¯ R .
ζ ( q ) = 0 , ζ ( q ) ζ ( q ) = ( 2 π L ) 2 δ ( q + q ) ζ ( q ) 2 , ζ ( q ) ζ ( q ) ζ ( q ) = 0 ,
c 8 π E ¯ R 0 ( 0 ) × B ¯ R 0 ( 0 ) * ,
S R ( 0 ) c 8 π E ¯ R 0 ( 0 ) × B ¯ R 0 ( 0 ) * ,
S R ( a ) = c 8 π E ¯ R 1 ( 0 ) × B ¯ R 1 ( 0 ) * , S R ( b ) = c 8 π E ¯ R 0 ( 1 ) × B ¯ R 0 ( 1 ) * , S R ( c ) = c 8 π [ E ¯ R 1 ( 0 ) × B ¯ R 0 ( 1 ) * + E ¯ R 0 ( 1 ) × B ¯ R 1 ( 0 ) * ] , S R ( d ) = c 8 π [ E ¯ R 0 ( 0 ) × B ¯ R 1 ( 1 ) * + E ¯ R 1 ( 1 ) × B ¯ R 0 ( 0 ) * ] , S R ( e ) = c 8 π [ E ¯ R 0 ( 0 ) × B ¯ R 2 ( 0 ) * + E ¯ R 2 ( 0 ) × B ¯ R 0 ( 0 ) * ] . }
S R = S R ( 0 ) + S R ( a ) + S R ( b ) + S R ( c ) + S R ( d ) + S R ( e ) .
S R ( 0 ) = c 8 π ( c 2 ω ) Σ R 0 ( 0 ) Σ R 0 ( 0 ) * K R 0 S R ( 0 ) K ^ R 0 ,
d q = q d q d ϕ = K R 2 cos θ R d Ω R = ( 2 ω c ) 2 cos θ R d Ω R ,
R ( θ R ) = R ( a ) ( θ R ) + R ( b ) ( θ R ) + R ( c ) ( θ R )
q x = Re ( k ± ω c ɛ ɛ + 1 ) ,
q x = ± Re ( 2 ω c ɛ s ɛ s + 1 ) .
( 2 π L ) 2 ζ ( q ) 2 = ( 2 π ) 2 π δ 2 a 2 exp ( - a 2 4 q 2 ) ,
a = 500 Å ,             δ = 100 Å .
θ i = θ R ( 0 ) = 26 ° ,
θ i = θ R ( 0 ) = 45 ° .
( θ R ) R e s o n a n c e = { 46.35 ° , - 16.57 ° , θ i = 26 ° 59.08 ° , - 8.67 ° , θ i = 45 ° .
S max 200 ,             θ R 46.35 °
S max 30 ,             θ R 59.08 °

Metrics