Abstract

The differential formalism previously developed to study the diffraction by gratings in linear optics is generalized to metallic gratings in nonlinear optics for TM polarization. It is applied to the study of the enhancement of second-harmonic generation at a silver grating, linked with the resonant excitation of surface plasmons.

© 1988 Optical Society of America

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References

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  1. R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
    [Crossref]
  2. D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
    [Crossref]
  3. 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. 174, 813–822 (1968).
    [Crossref]
  4. R. Reinisch, E. Pic, D. Maystre, and M. Nevière, “Nonlinear polarization and boundary conditions for second-harmonic generation and metal surfaces: a critical study,” submitted to J. Opt. Soc. Am. B.
  5. D. Maystre, M. Nevière, R. Reinisch, and J. L. Coutaz, “Integral theory for metallic gratings in nonlinear optics and comparison with experimental results on second-harmonic generation,” J. Opt. Soc. Am. B 5, 338–346 (1988).
    [Crossref]
  6. M. Nevière, P. Vincent, and R. Petit, “Sur la théorie due réseau conducteur et ses applications à l’optique,” Nouv. Rev. Optique 5, 65–77 (1974).
    [Crossref]
  7. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, Vol. XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
    [Crossref]
  8. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).
  9. J. E. Sipe and G. I. Stegeman, in Surface Polaritons, V. M. Agranovitch and D. L. Mills, eds. (North-Holland, Amsterdam, 1982).
  10. J. Rudnick and E. Stern, “Second harmonic radiation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
    [Crossref]
  11. M. Nevière, D. Maystre, J. L. Coutaz, E. Pic, and R. Reinisch, “Surface-enhanced second harmonic generation on silver gratings: theory and experiment,” in Digest of the XIV International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1986), p. 68.
  12. 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]
  13. M. Nevière, R. Reinisch, and D. Maystre, “Surface-enhanced second-harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
    [Crossref]

1988 (1)

1986 (1)

D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[Crossref]

1985 (2)

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]

M. Nevière, R. Reinisch, and D. Maystre, “Surface-enhanced second-harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
[Crossref]

1983 (1)

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
[Crossref]

1974 (1)

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie due réseau conducteur et ses applications à l’optique,” Nouv. Rev. Optique 5, 65–77 (1974).
[Crossref]

1971 (1)

J. Rudnick and E. Stern, “Second harmonic radiation 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. 174, 813–822 (1968).
[Crossref]

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. 174, 813–822 (1968).
[Crossref]

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. 174, 813–822 (1968).
[Crossref]

Coutaz, J. L.

D. Maystre, M. Nevière, R. Reinisch, and J. L. Coutaz, “Integral theory for metallic gratings in nonlinear optics and comparison with experimental results on second-harmonic generation,” J. Opt. Soc. Am. B 5, 338–346 (1988).
[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]

M. Nevière, D. Maystre, J. L. Coutaz, E. Pic, and R. Reinisch, “Surface-enhanced second harmonic generation on silver gratings: theory and experiment,” in Digest of the XIV International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1986), p. 68.

Henrici, P.

P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).

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. 174, 813–822 (1968).
[Crossref]

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. 174, 813–822 (1968).
[Crossref]

Maystre, D.

D. Maystre, M. Nevière, R. Reinisch, and J. L. Coutaz, “Integral theory for metallic gratings in nonlinear optics and comparison with experimental results on second-harmonic generation,” J. Opt. Soc. Am. B 5, 338–346 (1988).
[Crossref]

D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[Crossref]

M. Nevière, R. Reinisch, and D. Maystre, “Surface-enhanced second-harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
[Crossref]

M. Nevière, D. Maystre, J. L. Coutaz, E. Pic, and R. Reinisch, “Surface-enhanced second harmonic generation on silver gratings: theory and experiment,” in Digest of the XIV International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1986), p. 68.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, Vol. XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
[Crossref]

R. Reinisch, E. Pic, D. Maystre, and M. Nevière, “Nonlinear polarization and boundary conditions for second-harmonic generation and metal surfaces: a critical study,” submitted to J. Opt. Soc. Am. B.

Nevière, M.

D. Maystre, M. Nevière, R. Reinisch, and J. L. Coutaz, “Integral theory for metallic gratings in nonlinear optics and comparison with experimental results on second-harmonic generation,” J. Opt. Soc. Am. B 5, 338–346 (1988).
[Crossref]

D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[Crossref]

M. Nevière, R. Reinisch, and D. Maystre, “Surface-enhanced second-harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
[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]

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
[Crossref]

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie due réseau conducteur et ses applications à l’optique,” Nouv. Rev. Optique 5, 65–77 (1974).
[Crossref]

R. Reinisch, E. Pic, D. Maystre, and M. Nevière, “Nonlinear polarization and boundary conditions for second-harmonic generation and metal surfaces: a critical study,” submitted to J. Opt. Soc. Am. B.

M. Nevière, D. Maystre, J. L. Coutaz, E. Pic, and R. Reinisch, “Surface-enhanced second harmonic generation on silver gratings: theory and experiment,” in Digest of the XIV International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1986), p. 68.

Petit, R.

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie due réseau conducteur et ses applications à l’optique,” Nouv. Rev. Optique 5, 65–77 (1974).
[Crossref]

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]

M. Nevière, D. Maystre, J. L. Coutaz, E. Pic, and R. Reinisch, “Surface-enhanced second harmonic generation on silver gratings: theory and experiment,” in Digest of the XIV International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1986), p. 68.

R. Reinisch, E. Pic, D. Maystre, and M. Nevière, “Nonlinear polarization and boundary conditions for second-harmonic generation and metal surfaces: a critical study,” submitted to J. Opt. Soc. Am. B.

Reinisch, R.

D. Maystre, M. Nevière, R. Reinisch, and J. L. Coutaz, “Integral theory for metallic gratings in nonlinear optics and comparison with experimental results on second-harmonic generation,” J. Opt. Soc. Am. B 5, 338–346 (1988).
[Crossref]

D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[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]

M. Nevière, R. Reinisch, and D. Maystre, “Surface-enhanced second-harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
[Crossref]

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
[Crossref]

R. Reinisch, E. Pic, D. Maystre, and M. Nevière, “Nonlinear polarization and boundary conditions for second-harmonic generation and metal surfaces: a critical study,” submitted to J. Opt. Soc. Am. B.

M. Nevière, D. Maystre, J. L. Coutaz, E. Pic, and R. Reinisch, “Surface-enhanced second harmonic generation on silver gratings: theory and experiment,” in Digest of the XIV International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1986), p. 68.

Rudnick, J.

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

Sipe, J. E.

J. E. Sipe and G. I. Stegeman, in Surface Polaritons, V. M. Agranovitch and D. L. Mills, eds. (North-Holland, Amsterdam, 1982).

Stegeman, G. I.

J. E. Sipe and G. I. Stegeman, in Surface Polaritons, V. M. Agranovitch and D. L. Mills, eds. (North-Holland, Amsterdam, 1982).

Stern, E.

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

Vincent, P.

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie due réseau conducteur et ses applications à l’optique,” Nouv. Rev. Optique 5, 65–77 (1974).
[Crossref]

Appl. Phys. A (1)

D. Maystre, M. Nevière, and R. Reinisch, “Nonlinear polarization inside metals: a mathematical study of the free electron model,” Appl. Phys. A 39, 115–121 (1986).
[Crossref]

J. Opt. Soc. Am. B (1)

Nouv. Rev. Optique (1)

M. Nevière, P. Vincent, and R. Petit, “Sur la théorie due réseau conducteur et ses applications à l’optique,” Nouv. Rev. Optique 5, 65–77 (1974).
[Crossref]

Phys. Rev. (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. 174, 813–822 (1968).
[Crossref]

Phys. Rev. B (4)

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

M. Nevière, R. Reinisch, and D. Maystre, “Surface-enhanced second-harmonic generation at a silver grating: a numerical study,” Phys. Rev. B 32, 3634–3641 (1985).
[Crossref]

R. Reinisch and M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
[Crossref]

Other (5)

M. Nevière, D. Maystre, J. L. Coutaz, E. Pic, and R. Reinisch, “Surface-enhanced second harmonic generation on silver gratings: theory and experiment,” in Digest of the XIV International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1986), p. 68.

R. Reinisch, E. Pic, D. Maystre, and M. Nevière, “Nonlinear polarization and boundary conditions for second-harmonic generation and metal surfaces: a critical study,” submitted to J. Opt. Soc. Am. B.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, Vol. XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984).
[Crossref]

P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).

J. E. Sipe and G. I. Stegeman, in Surface Polaritons, V. M. Agranovitch and D. L. Mills, eds. (North-Holland, Amsterdam, 1982).

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Figures (6)

Fig. 1
Fig. 1

Schematic representation of the grating and notations.

Fig. 2
Fig. 2

Nonlinear reflectivity of a silver plane as a function of incidence.

Fig. 3
Fig. 3

Resonance lines at pump frequency of the +1 Rayleigh coefficient below the grating as a function of incidence due to the excitation of a surface plasmon at circular frequency ω1.

Fig. 4
Fig. 4

Resonance lines at second-harmonic frequency of various Rayleigh coefficients Tn below the grating as functions of incidence, for the grating of Fig. 3.

Fig. 5
Fig. 5

Resonance lines for propagating orders and the same grating as in Figs. 3 and 4.

Fig. 6
Fig. 6

Illustration of the definitions of functions x1(y) and x2(y).

Tables (1)

Tables Icon

Table 1 Contribution of Bulk and Surface Terms in Nonlinear Reflectivity

Equations (46)

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P NL = a γ grad ( E · E ) + b β [ E · div ( E ) ] ,
curl H = - i 2 ω 1 0 - i 2 ω 1 P NL ,
curl H = - i 2 ω 1 0 { } - i 2 ω 1 P V NL - i 2 ω 1 P S t NL δ S .
{ curl H } + n · J ( H ) δ S = - 2 i ω 1 { } - 2 i ω 1 P V NL - 2 i ω 1 P S t NL δ S ,
{ curl H e z } = - 2 i ω 1 { } - 2 i ω 1 P V NL
{ } = - 1 2 i ω 1 0 { grad H } × e z - 1 0 P V NL .
curl { } + curl n δ S = 2 i ω 1 μ 0 H .
curl ( { grad H } × e z 2 i ω 1 0 ) + curl P V NL 0 - n × grad t n = - 2 i ω 1 μ 0 H .
e z div ( β { grad H } ) + H e z = 2 i ω 1 curl ( β P V NL ) - 2 i ω 1 0 n × grad t ( n c 2 4 ω 1 2 ) δ S .
div { β [ grad H - n J ( H ) δ S ] } + H = 2 i ω 1 { e z · [ n × J ( β P V NL ) - 0 n × grad t ( n c 2 4 ω 1 2 ) ] δ S } .
div [ β ( grad H + A δ S ) ] + H = B δ S ,
A = - n J ( H )
B = J [ β ( d H d n ) ] .
H = m = - + H m ( y ) exp ( i γ m x ) ,
A x δ S = m A m x ( y ) exp ( i γ m x ) ,
A y δ S = m A m y ( y ) exp ( i γ m x ) ,
B δ S = m B m ( y ) exp ( i γ m x ) ,
γ m = 2 k 1 sin θ 1 + n 2 π d ,
β = m β m ( y ) exp ( i m 2 π d x ) .
x ( β H x + A x δ S ) + y ( β H y + A y δ S ) + H = B δ S .
i γ n m β n - m ( i γ m H m + A m x ) + d d y [ m β n - m ( d H m d y + A m y ) ] + H n = B n ( y ) .
K n ( y ) = m β n - m ( d H m d y + A m y ) .
d H n d y = m ξ n - m K m - A n y ,
d K n d y = - H n - i γ n m β n - m ( i γ m H m A m x ) + B n ( y ) ,
H ( ω 3 , r ) = n = - + R n exp [ i ( α 1 , n y + γ n x ) ]             if y δ ,
H ( ω 3 , r ) = n = - + T n exp [ i ( - α 2 , n y + γ n x ) ]             if y 0 ,
d H n d y = m = - N + N ξ n - m K m - A n y ( y ) ,
d K n d y = - i γ n m = - N + N β n - m [ i γ m H m + A m x ( y ) ] - H n ( y ) + B n ( y ) ,
H ˜ n p ( 0 ) = δ n p
d H ˜ n p d y ( y ) | y = 0 = - i α 2 , n δ n , p = - i α 2 , n H ˜ n p ( 0 ) ,
[ H ( y ) ] = p = - N + N T p [ H ˜ ( y ) ] p + [ H NL ( y ) ] ,
p T p H ˜ n p ( δ ) + H n NL ( δ ) = R n exp ( i α i n δ )
p T p d H ˜ n p d y ( δ ) + d H n NL d y ( δ ) = i α 1 n R n exp ( i α 1 n δ ) .
p T p [ d H ˜ n p ( δ ) d y - i α 1 n H ˜ n p ( δ ) ] = - d H n NL ( δ ) d y + i α 1 n H n NL ( δ ) ,
K ˜ n p ( 0 ) = 1 k 2 2 ( ω 3 ) d H ˜ n p ( 0 ) d y .
J ( H ) = - ( H i ) + ,
J ( 1 d H d n ) = - 1 1 ( ω 1 ) ( d H i d n ) + .
A x = n x ( H i ) + e ( H i ) + 1 + g ( x ) 2 , A y = n y ( H i ) + = - e g ( x ) ( H i ) + 1 + g 2 , B = - 1 ω 1 2 0 μ 0 1 ( ω 1 ) ( d H inc d n ) + = i k 1 ( n y cos θ 1 - n x sin θ 1 ) ω 1 2 0 μ 0 1 ( ω 1 ) × exp [ i k 1 ( x sin θ 1 - y cos θ 1 ) ] .
A m x ( y ) = [ H i ( x 1 ) ] + d exp [ - i γ m x 1 ( y ) ] - [ H i ( x 2 ) ] + d × exp [ - i γ m x 2 ( y ) ] , A m y ( y ) = - g ( x 1 ) [ H i ( x 1 ) ] + d exp [ - i γ m x 1 ( y ) ] + g ( x 2 ) [ H i ( x 2 ) ] + d exp [ - i γ m x 2 ( y ) ] , B n ( y ) = i [ n y cos θ - n x sin θ ] k 1 d { [ H i ( x 1 ) ] + 1 + g ( x 1 ) 2 × exp [ - i γ m x 1 ( y ) ] + [ H i ( x 2 ) ] + 1 + g ( x 2 ) 2 × exp [ - i γ m x 2 ( y ) ] } .
P NL = a γ grad ( E · E ) + b β [ E · div ( E ) ] .
B δ S = m B m ( y ) exp ( i γ m x ) .
B n ( y ) , φ ( y ) = 1 d B ( x , y ) δ S , φ ( y ) exp ( - i γ n x ) = 1 d period B ( M ) φ ( M ) exp ( - i γ n x ) d s ,
d s = 1 + ( d x d y ) 2 d y , = - 1 if x = x 1 , = + 1 if x = x 2 .
B n ( y ) , φ ( y ) = 1 d [ - δ 0 B ( x 1 , y ) φ ( y ) exp ( - i γ n x 1 ) × 1 + ( d x 1 d y ) 2 d y + 0 δ B ( x 2 , y ) φ ( y ) × exp ( - i γ n x 2 ) 1 + ( d x 2 d y ) 2 d y ] .
B n ( y ) , φ ( y ) = 0 δ B n ( y ) φ ( y ) d y ,
B n ( y ) = 1 d [ B ( x 1 , y ) exp ( - i γ n x 1 ) 1 + ( d x 1 d y ) 2 + B ( x 2 , y ) exp ( - i γ n x 2 ) 1 + ( d x 2 d y ) 2 ] .

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