Abstract

Second harmonic generation of long-range surface plasmon polaritons excited by a finite 1-D beam are discussed. To simplify the analysis of second harmonic generation, we adopt the “pole cancellation” method originally developed by Sipe. A numerical calculation of second harmonic intensity is presented. It is found that the effect of beam size on second harmonic intensity should be taken into account in actual experiments.

© 1985 Optical Society of America

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References

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  1. M. Fukui, V. So, R. Normandin, “Lifetimes of Surface Plasmons in Thin Silver Films,” Phys. Status Solidi B 91, K61 (1979).
    [CrossRef]
  2. D. Sarid, “Long-Range Surface-Plasma Waves on Very Thin Metal Films,” Phys. Rev. Lett. 47, 1927 (1981).
    [CrossRef]
  3. G. I. Stegeman, J. J. Burke, D. G. Hall, “Nonlinear Optics of Long-Range Surface Plasmons,” Appl. Phys. Lett. 41, 906 (1982).
    [CrossRef]
  4. G. I. Stegeman, C. Liao, C. Karaguleff, “Second Harmonic Generation by Oppositely Traveling Long Range Surface Polaritons,” Opt. Commun. 46, 253 (1983).
    [CrossRef]
  5. R. T. Deck, D. Sarid, “Enhancement of Second-Harmonic Generation by Coupling to Long-Range Surface Plasmons,” J. Opt. Soc. Am. 72, 1613 (1982).
    [CrossRef]
  6. J. C. Quail, J. G. Rako, H. J. Simon, R. T. Deck, “Optical Second-Harmonic Generation with Long-Range Surface Plasmons,” Phys. Rev. Lett. 50, 1987 (1983);J. C. Quail, H. J. Simon, “Second-Harmonic Generation with Phase-Matched Long-Range and Short-Range Surface Plasmons,” J. Appl. Phys. 56, 2589 (1984).
    [CrossRef]
  7. R. Ulrich, “Plane-Wave Analysis of Prism-Film Coupler,” J. Opt. Soc. Am. 60, 1337 (1970).
    [CrossRef]
  8. W. P. Chen, G. Ritchie, E. Burstein, “Excitation of Surface Electromagnetic Waves in Attenuated Total-Reflection Prism Configurations,” Phys. Rev. Lett. 37, 993 (1976).
    [CrossRef]
  9. R. T. Deck, D. Sarid, G. A. Olson, J. M. Elson, “Coupling Between Finite Electromagnetic Beam and Long-Range Surface-Plasmon Mode,” Appl. Opt. 22, 3397 (1983).
    [CrossRef] [PubMed]
  10. J. E. Sipe, “A New Treatment of the Growing Wave Problem in Surface Optics,” Solid State Commun. 39, 493 (1981).
    [CrossRef]
  11. W. N. Hansen, “Electric Fields in Stratified Medium,” J. Opt. Soc. Am. 58, 380 (1968).
    [CrossRef]
  12. M. Fukui, V. So, G. I. Stegeman, “Harmonic Generation of a Surface-Plasmon Nature in Thin-Film Geometries: Off-Synchronous Treatment,” Phys. Rev. B 18, 2484 (1978).
    [CrossRef]
  13. J. E. Sipe, “The ATR Spectra of Multiple Surface Plasmons,” Surf. Sci. 84, 75 (1979).
    [CrossRef]

1983 (3)

G. I. Stegeman, C. Liao, C. Karaguleff, “Second Harmonic Generation by Oppositely Traveling Long Range Surface Polaritons,” Opt. Commun. 46, 253 (1983).
[CrossRef]

J. C. Quail, J. G. Rako, H. J. Simon, R. T. Deck, “Optical Second-Harmonic Generation with Long-Range Surface Plasmons,” Phys. Rev. Lett. 50, 1987 (1983);J. C. Quail, H. J. Simon, “Second-Harmonic Generation with Phase-Matched Long-Range and Short-Range Surface Plasmons,” J. Appl. Phys. 56, 2589 (1984).
[CrossRef]

R. T. Deck, D. Sarid, G. A. Olson, J. M. Elson, “Coupling Between Finite Electromagnetic Beam and Long-Range Surface-Plasmon Mode,” Appl. Opt. 22, 3397 (1983).
[CrossRef] [PubMed]

1982 (2)

R. T. Deck, D. Sarid, “Enhancement of Second-Harmonic Generation by Coupling to Long-Range Surface Plasmons,” J. Opt. Soc. Am. 72, 1613 (1982).
[CrossRef]

G. I. Stegeman, J. J. Burke, D. G. Hall, “Nonlinear Optics of Long-Range Surface Plasmons,” Appl. Phys. Lett. 41, 906 (1982).
[CrossRef]

1981 (2)

J. E. Sipe, “A New Treatment of the Growing Wave Problem in Surface Optics,” Solid State Commun. 39, 493 (1981).
[CrossRef]

D. Sarid, “Long-Range Surface-Plasma Waves on Very Thin Metal Films,” Phys. Rev. Lett. 47, 1927 (1981).
[CrossRef]

1979 (2)

M. Fukui, V. So, R. Normandin, “Lifetimes of Surface Plasmons in Thin Silver Films,” Phys. Status Solidi B 91, K61 (1979).
[CrossRef]

J. E. Sipe, “The ATR Spectra of Multiple Surface Plasmons,” Surf. Sci. 84, 75 (1979).
[CrossRef]

1978 (1)

M. Fukui, V. So, G. I. Stegeman, “Harmonic Generation of a Surface-Plasmon Nature in Thin-Film Geometries: Off-Synchronous Treatment,” Phys. Rev. B 18, 2484 (1978).
[CrossRef]

1976 (1)

W. P. Chen, G. Ritchie, E. Burstein, “Excitation of Surface Electromagnetic Waves in Attenuated Total-Reflection Prism Configurations,” Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

1970 (1)

1968 (1)

Burke, J. J.

G. I. Stegeman, J. J. Burke, D. G. Hall, “Nonlinear Optics of Long-Range Surface Plasmons,” Appl. Phys. Lett. 41, 906 (1982).
[CrossRef]

Burstein, E.

W. P. Chen, G. Ritchie, E. Burstein, “Excitation of Surface Electromagnetic Waves in Attenuated Total-Reflection Prism Configurations,” Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

Chen, W. P.

W. P. Chen, G. Ritchie, E. Burstein, “Excitation of Surface Electromagnetic Waves in Attenuated Total-Reflection Prism Configurations,” Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

Deck, R. T.

R. T. Deck, D. Sarid, G. A. Olson, J. M. Elson, “Coupling Between Finite Electromagnetic Beam and Long-Range Surface-Plasmon Mode,” Appl. Opt. 22, 3397 (1983).
[CrossRef] [PubMed]

J. C. Quail, J. G. Rako, H. J. Simon, R. T. Deck, “Optical Second-Harmonic Generation with Long-Range Surface Plasmons,” Phys. Rev. Lett. 50, 1987 (1983);J. C. Quail, H. J. Simon, “Second-Harmonic Generation with Phase-Matched Long-Range and Short-Range Surface Plasmons,” J. Appl. Phys. 56, 2589 (1984).
[CrossRef]

R. T. Deck, D. Sarid, “Enhancement of Second-Harmonic Generation by Coupling to Long-Range Surface Plasmons,” J. Opt. Soc. Am. 72, 1613 (1982).
[CrossRef]

Elson, J. M.

Fukui, M.

M. Fukui, V. So, R. Normandin, “Lifetimes of Surface Plasmons in Thin Silver Films,” Phys. Status Solidi B 91, K61 (1979).
[CrossRef]

M. Fukui, V. So, G. I. Stegeman, “Harmonic Generation of a Surface-Plasmon Nature in Thin-Film Geometries: Off-Synchronous Treatment,” Phys. Rev. B 18, 2484 (1978).
[CrossRef]

Hall, D. G.

G. I. Stegeman, J. J. Burke, D. G. Hall, “Nonlinear Optics of Long-Range Surface Plasmons,” Appl. Phys. Lett. 41, 906 (1982).
[CrossRef]

Hansen, W. N.

Karaguleff, C.

G. I. Stegeman, C. Liao, C. Karaguleff, “Second Harmonic Generation by Oppositely Traveling Long Range Surface Polaritons,” Opt. Commun. 46, 253 (1983).
[CrossRef]

Liao, C.

G. I. Stegeman, C. Liao, C. Karaguleff, “Second Harmonic Generation by Oppositely Traveling Long Range Surface Polaritons,” Opt. Commun. 46, 253 (1983).
[CrossRef]

Normandin, R.

M. Fukui, V. So, R. Normandin, “Lifetimes of Surface Plasmons in Thin Silver Films,” Phys. Status Solidi B 91, K61 (1979).
[CrossRef]

Olson, G. A.

Quail, J. C.

J. C. Quail, J. G. Rako, H. J. Simon, R. T. Deck, “Optical Second-Harmonic Generation with Long-Range Surface Plasmons,” Phys. Rev. Lett. 50, 1987 (1983);J. C. Quail, H. J. Simon, “Second-Harmonic Generation with Phase-Matched Long-Range and Short-Range Surface Plasmons,” J. Appl. Phys. 56, 2589 (1984).
[CrossRef]

Rako, J. G.

J. C. Quail, J. G. Rako, H. J. Simon, R. T. Deck, “Optical Second-Harmonic Generation with Long-Range Surface Plasmons,” Phys. Rev. Lett. 50, 1987 (1983);J. C. Quail, H. J. Simon, “Second-Harmonic Generation with Phase-Matched Long-Range and Short-Range Surface Plasmons,” J. Appl. Phys. 56, 2589 (1984).
[CrossRef]

Ritchie, G.

W. P. Chen, G. Ritchie, E. Burstein, “Excitation of Surface Electromagnetic Waves in Attenuated Total-Reflection Prism Configurations,” Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

Sarid, D.

Simon, H. J.

J. C. Quail, J. G. Rako, H. J. Simon, R. T. Deck, “Optical Second-Harmonic Generation with Long-Range Surface Plasmons,” Phys. Rev. Lett. 50, 1987 (1983);J. C. Quail, H. J. Simon, “Second-Harmonic Generation with Phase-Matched Long-Range and Short-Range Surface Plasmons,” J. Appl. Phys. 56, 2589 (1984).
[CrossRef]

Sipe, J. E.

J. E. Sipe, “A New Treatment of the Growing Wave Problem in Surface Optics,” Solid State Commun. 39, 493 (1981).
[CrossRef]

J. E. Sipe, “The ATR Spectra of Multiple Surface Plasmons,” Surf. Sci. 84, 75 (1979).
[CrossRef]

So, V.

M. Fukui, V. So, R. Normandin, “Lifetimes of Surface Plasmons in Thin Silver Films,” Phys. Status Solidi B 91, K61 (1979).
[CrossRef]

M. Fukui, V. So, G. I. Stegeman, “Harmonic Generation of a Surface-Plasmon Nature in Thin-Film Geometries: Off-Synchronous Treatment,” Phys. Rev. B 18, 2484 (1978).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, C. Liao, C. Karaguleff, “Second Harmonic Generation by Oppositely Traveling Long Range Surface Polaritons,” Opt. Commun. 46, 253 (1983).
[CrossRef]

G. I. Stegeman, J. J. Burke, D. G. Hall, “Nonlinear Optics of Long-Range Surface Plasmons,” Appl. Phys. Lett. 41, 906 (1982).
[CrossRef]

M. Fukui, V. So, G. I. Stegeman, “Harmonic Generation of a Surface-Plasmon Nature in Thin-Film Geometries: Off-Synchronous Treatment,” Phys. Rev. B 18, 2484 (1978).
[CrossRef]

Ulrich, R.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. I. Stegeman, J. J. Burke, D. G. Hall, “Nonlinear Optics of Long-Range Surface Plasmons,” Appl. Phys. Lett. 41, 906 (1982).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

G. I. Stegeman, C. Liao, C. Karaguleff, “Second Harmonic Generation by Oppositely Traveling Long Range Surface Polaritons,” Opt. Commun. 46, 253 (1983).
[CrossRef]

Phys. Rev. B (1)

M. Fukui, V. So, G. I. Stegeman, “Harmonic Generation of a Surface-Plasmon Nature in Thin-Film Geometries: Off-Synchronous Treatment,” Phys. Rev. B 18, 2484 (1978).
[CrossRef]

Phys. Rev. Lett. (3)

D. Sarid, “Long-Range Surface-Plasma Waves on Very Thin Metal Films,” Phys. Rev. Lett. 47, 1927 (1981).
[CrossRef]

J. C. Quail, J. G. Rako, H. J. Simon, R. T. Deck, “Optical Second-Harmonic Generation with Long-Range Surface Plasmons,” Phys. Rev. Lett. 50, 1987 (1983);J. C. Quail, H. J. Simon, “Second-Harmonic Generation with Phase-Matched Long-Range and Short-Range Surface Plasmons,” J. Appl. Phys. 56, 2589 (1984).
[CrossRef]

W. P. Chen, G. Ritchie, E. Burstein, “Excitation of Surface Electromagnetic Waves in Attenuated Total-Reflection Prism Configurations,” Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

Phys. Status Solidi B (1)

M. Fukui, V. So, R. Normandin, “Lifetimes of Surface Plasmons in Thin Silver Films,” Phys. Status Solidi B 91, K61 (1979).
[CrossRef]

Solid State Commun. (1)

J. E. Sipe, “A New Treatment of the Growing Wave Problem in Surface Optics,” Solid State Commun. 39, 493 (1981).
[CrossRef]

Surf. Sci. (1)

J. E. Sipe, “The ATR Spectra of Multiple Surface Plasmons,” Surf. Sci. 84, 75 (1979).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Curves of R (x) vs x for W = 2 mm. The lateral form of incident beam is symmetrical with its center at x = 0.

Tables (2)

Tables Icon

Table I Wave Vectors of Normal Mode of L-SPs and Various Parameters for Different Thicknesses of Ag Film

Tables Icon

Table II Final Values of R(W → ∞, x → ∞) for Different Thicknesses of Ag Film

Equations (51)

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t pgms = t p g t gms exp ( β 0 d ) / [ 1 + r p g r gms exp ( 2 β 0 d ) ] ,
r gms = [ r g m + r m s exp ( 2 α 0 h ) ] / [ 1 + r g m r m s exp ( 2 α 0 h ) ] ,
t gms = t g m t m s exp ( α 0 h ) / [ 1 + r g m r m s exp ( 2 α 0 h ) ] , α 0 = [ k 2 ε m ( ω ) ω 2 / c 2 ] 1 / 2 , β 0 = [ k 2 ε g ( ω ) ω 2 / c 2 ] 1 / 2 ,
t pgms T m / ( k k s p ) ,
T m = [ 1 + r g m r m s exp ( 2 α 0 h ) ] t p g t gms × exp ( β 0 d ) / ( [ d r 0 d k ] k = k s p ) ,
r 0 = r g m r m s exp ( 2 α 0 h ) + r p g [ r g m + r m s exp ( 2 α 0 h ) ] exp ( 2 β 0 d ) ,
H s ( x ) = i T m x d x H p ( x ) exp [ i k s p ( x x ) ] .
x < W / 2 , H p = 0 ,
W / 2 x W / 2 , H p = H 0 exp [ i ( k x ω t ) ] ,
x > W / 2 , H p = 0 .
x < W / 2 , H s ( x ) = 0 ,
W / 2 x W / 2 , H s ( x ) = i T m H 0 exp [ i ( k x ω t ) ] × { 1 exp [ ξ ( x + W / 2 ) ] } / ξ ,
x > W / 2 H s ( x ) = 2 T m H 0 sin ( i ξ W / 2 ) × exp [ i ( k x ω t ) ] × exp ( ξ x ) / ξ ,
E s ( x , z ) = [ i c / ω ε s ( ω ) ] H s ( x , z ) z .
P N L = P ¯ exp [ i ( 2 k x 2 ω t ) + 2 γ 0 ( z + d + h ) ] ,
P ̅ x = 4 i d 11 k E s 2 / γ 0 ,
P ̅ z = 2 d 11 ( γ 0 2 + k 2 ) E s 2 / γ 0 2 ,
P ¯ x , z = P x , z 01 { 1 exp [ ξ ( x + W / 2 ) ] } 2 / ξ 2 , P x 01 = i 4 c 2 d 11 T m 2 H 0 2 γ 0 k / ω 2 ε s 2 ( ω ) , P z 01 = 2 c 2 d 11 T m 2 H 0 2 ( γ 0 2 + k 2 ) / ω 2 ε s 2 ( ω ) , for | x | W / 2 , }
P ¯ x , z = P x , z 02 exp ( 2 ξ x ) / ξ 2 , P x 02 = i 16 c 2 d 11 T m 2 H 0 2 γ 0 k sin 2 ( i ξ W / 2 ) / ω 2 ε s 2 ( ω ) , P z 02 = 8 c 2 d 11 T m 2 H 0 2 ( γ 0 2 + k 2 ) sin 2 ( i ξ W / 2 ) / ω 2 ε s 2 ( ω ) , for x > W / 2 . }
E p = E ¯ p exp [ i ( 2 k x 2 ω t ) ] ,
d E ¯ p d x + i [ 2 k k s p ( 2 ω ) ] E ¯ P = i d h K s ( z ; z ) P ¯ ( x ) d z ,
I s = c { [ Re ( E ¯ p z H ̅ * p y ) ] 2 + [ Re ( E ¯ p x H ̅ p y ) ] 2 } 1 / 2 / 8 π = R ( x ) I i 2 ,
I i = c H 0 2 / 8 π [ ε p ( ω ) ] 1 / 2 ,
e p ( z ) = M p ( z ) M p g M g ( d ) M g m M m ( h ) M m s e s ( d h ) , e p ( z ) = [ E p ( z ) 0 ] , e s ( d h ) = [ E s + E s ] ,
E p ( z ) = A S E s + exp [ i k p z ( 2 ω ) z ] ,
A = a 1 S + a 2 { r s m + r m g exp ( 2 α h ) + r g p × [ exp ( 2 α h ) + r g m r m s ] exp ( 2 β d ) } ,
a 1 = exp ( α h β d ) + r g m r m s exp ( α h β d ) + r p g [ r g m exp ( α h + β d ) + r m s exp ( α h + β d ) ] ,
a 2 = r m s exp ( α h β d ) + r g m exp ( α h β d ) + r p g [ r g m r m s exp ( α h + β d ) + exp ( α h + β d ) ] ,
S = 1 + r g m r m s exp ( 2 α h ) + r p g [ r g m + r m s exp ( 2 α h ) ] × exp ( 2 β d ) = 1 + r ( k ) [ 2 k k s p ( 2 ω ) ] × [ d r ( k ) d k ] k = k s p ( 2 ω ) / 2 ,
E s + = 8 π ω 2 γ c 2 p ̂ s + d h P N L ( x , z ) exp [ γ ( z + d + h ) ] d z ,
p ̂ s + = c [ 2 k + i γ x ̂ ] / 2 ω [ ε s ( 2 ω ) ] 1 / 2 ,
E p ( z ) = 8 π ω 2 A c 2 γ S exp [ i k p z ( 2 ω ) z ] p ̂ p + p ̂ s + d h P N L ( x , z ) exp [ γ ( z + d + h ) ] d z ,
p ̂ p + = c [ 2 k k p z ( 2 ω ) x ̂ ] / 2 ω [ ε p ( 2 ω ) ] 1 / 2 .
d E ¯ p d x + i 2 k E ¯ p = i 2 k 8 π ω 2 A c 2 γ S exp [ i k p z ( 2 ω ) z ] p ̂ p + p ̂ s + d h P ¯ ( x ) exp [ ( γ + 2 γ 0 ) ( z + d + h ) ] d z .
d E ¯ p d x + i [ 2 k k s p ( 2 ω ) ] E ¯ P = 8 π i ω 2 A c 2 γ [ dr ( k ) d k ] k = k sp ( 2 ω ) / 2 × exp [ i k p z ( 2 ω ) z ] p ̂ p + p ̂ s + d h P ¯ ( x ) exp [ ( γ + 2 γ 0 ) × ( z + d + h ) ] d z .
K s ( z ; z ) = 8 π ω 2 A c 2 γ [ dr ( k ) d k ] k = k sp ( 2 ω ) / 2 exp [ i k p z ( 2 ω ) z ] × exp [ ( γ + 2 γ 0 ) ( z + d + h ) ] p ̂ p + p ̂ s + .
= K s exp [ i k p z ( 2 ω ) z ] exp ( γ + 2 γ 0 ) ( z + d + h ) p ̂ p + p ̂ s + .
E ¯ p z = c k p z ( 2 ω ) B P 01 f 1 ( ξ , ξ , x ) exp [ i k p z ( 2 ω ) z ] / 2 ω [ ε p ( 2 ω ) ] 1 / 2 ,
E ¯ p z = 2 k E p x / k p z ( 2 ω ) ,
H ̅ p y = c 2 B P 01 [ 2 i ω 2 ε p ( 2 ω ) f 1 ( ξ , ξ , x ) / c 2 + k d f 1 ( ξ , ξ , x ) d x ] exp [ i k p z ( 2 ω ) z ] / 2 i ω 2 [ ε p ( 2 ω ) ] 1 / 2 ,
B = i k s / ( γ + 2 γ 0 ) ,
P 01 = c ( i γ P x 01 + 2 k P z 01 ) / 2 ω [ ε s ( 2 ω ) ] 1 / 2 ,
f 1 ( ξ , ξ , x ) = ( { 1 exp [ ξ ( x + W / 2 ) ] } / ξ 2 { exp [ ξ ( x + W / 2 ) ] exp [ ξ ( x + W / 2 ) ] } / ( ξ ξ ) + { exp [ 2 ξ ( x + W / 2 ) ] exp [ ξ ( x + W / 2 ) ] } / ( ξ 2 ξ ) ) / ξ 2 ,
ξ = i [ 2 k k s p ( 2 ω ) ] .
E ¯ p x = c k p z ( 2 ω ) B F / 2 ω [ ε p ( 2 ω ) ] 1 / 2 ,
E ¯ p z = 2 k B F / 2 ω [ ε p ( 2 ω ) ] 1 / 2 ,
H ¯ p y = [ ε p ( 2 ω ) ] 1 / 2 B F + c 2 k B ( d F / d x ) / 2 i ω 2 [ ε p ( 2 ω ) ] 1 / 2 ,
F = P 01 f 2 ( ξ , ξ , x ) + P 02 g ( ξ , ξ , x ) ,
f 2 ( ξ , ξ , x ) = exp ( ξ x ) ( [ exp ( ξ W / 2 ) exp ( ξ W / 2 ) ] / ξ + 2 { exp ( ξ W / 2 ) exp [ ( ξ 2 ξ ) W / 2 } / ( ξ ξ ) + { exp [ ( ξ 4 ξ ) W / 2 ] exp ( ξ W / 2 } / ( ξ 2 ξ ) ) / ξ 2 ,
g ( ξ , ξ , x ) = exp ( ξ x ) { exp [ ( ξ 2 ξ ) x ] exp [ ( ξ 2 ξ ) W / 2 ] } / ( ξ 2 ξ ) ,
P 02 = P 01 ( P x 01 P x 02 , P z 01 P z 02 ) .

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