Abstract

In this paper, we consider a layered structure consisting of dielectric, metal, and dielectric media and compute the fields produced in the structure by a finite width beam of incident radiation which couples to a long-range surface-plasmon mode in the metallic layer. The analysis allows for complex dielectric constants in all media and focuses on optical radiation. Curves are presented which show the profiles of the reflected field intensity for different values of the propagation length of the mode relative to the width of the incident beam. We discuss the conditions in which the field reradiated from the long-range surface mode can interfere with the specularly reflected radiation so as to produce an interference zero in the reflected profile. In these conditions our analysis allows for a determination of the separate loss coefficients of the mode associated with energy reradiation and dissipation, respectively

© 1983 Optical Society of America

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References

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  1. L. M. Brekhovshikh, Waves in Layered Media (Academic, New York, 1960);D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  2. P. K. Tien, R. Ulrich, J. Opt. Soc. Am. 60, 1325 (1970).
    [CrossRef]
  3. R. Ulrich, J. Opt. Soc. Am. 60, 1337 (1970).
    [CrossRef]
  4. J. Schoenwald, E. Burstein, J. M. Elson, Solid State Commun. 12, 185 (1973);E. Santamato, F. DeMartini, Nuovo Cimento B 59, 223 (1980).
    [CrossRef]
  5. H. J. Simon, D. E. Mitchell, J. G. Watson, Am. J. Phys. 43, 630 (1975).
    [CrossRef]
  6. M. Fukui, V. So, R. Normandin, Phys. Status Solidi B 91, K61 (1979);D. Sarid, Phys. Rev. Lett. 47, 1927 (1981);A. E. Craig, G. A. Olson, D. Sarid, Opt. Lett. 8, 380 (1983);J. C. Quail, J. G. Rako, H. J. Simon, Opt. Lett. 8, 377 (1983).
    [CrossRef] [PubMed]
  7. J. E. Midwinter, F. Zernike, Appl. Phys. Lett. 16, 198 (1970).
    [CrossRef]
  8. T. Tamir, H. L. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971).
    [CrossRef]
  9. W. P. Chen, G. Ritchie, E. Burstein, Phys. Rev. Lett. 37, 993 (1976).
    [CrossRef]
  10. A. Otto, Z. Phys. 216, 398 (1968).
    [CrossRef]
  11. The same equation can be obtained by solution of the coupled equations (44)′ and (45)′ in region x where B∼3(x) has the constant value A.
  12. For a complete lateral shift of the profile it is necessary that there be complete coupling of the incident radiation into the surface mode so that no specular reflection occurs at the trailing edge of the incident beam.

1979 (1)

M. Fukui, V. So, R. Normandin, Phys. Status Solidi B 91, K61 (1979);D. Sarid, Phys. Rev. Lett. 47, 1927 (1981);A. E. Craig, G. A. Olson, D. Sarid, Opt. Lett. 8, 380 (1983);J. C. Quail, J. G. Rako, H. J. Simon, Opt. Lett. 8, 377 (1983).
[CrossRef] [PubMed]

1976 (1)

W. P. Chen, G. Ritchie, E. Burstein, Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

1975 (1)

H. J. Simon, D. E. Mitchell, J. G. Watson, Am. J. Phys. 43, 630 (1975).
[CrossRef]

1973 (1)

J. Schoenwald, E. Burstein, J. M. Elson, Solid State Commun. 12, 185 (1973);E. Santamato, F. DeMartini, Nuovo Cimento B 59, 223 (1980).
[CrossRef]

1971 (1)

1970 (3)

1968 (1)

A. Otto, Z. Phys. 216, 398 (1968).
[CrossRef]

Bertoni, H. L.

Brekhovshikh, L. M.

L. M. Brekhovshikh, Waves in Layered Media (Academic, New York, 1960);D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Burstein, E.

W. P. Chen, G. Ritchie, E. Burstein, Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

J. Schoenwald, E. Burstein, J. M. Elson, Solid State Commun. 12, 185 (1973);E. Santamato, F. DeMartini, Nuovo Cimento B 59, 223 (1980).
[CrossRef]

Chen, W. P.

W. P. Chen, G. Ritchie, E. Burstein, Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

Elson, J. M.

J. Schoenwald, E. Burstein, J. M. Elson, Solid State Commun. 12, 185 (1973);E. Santamato, F. DeMartini, Nuovo Cimento B 59, 223 (1980).
[CrossRef]

Fukui, M.

M. Fukui, V. So, R. Normandin, Phys. Status Solidi B 91, K61 (1979);D. Sarid, Phys. Rev. Lett. 47, 1927 (1981);A. E. Craig, G. A. Olson, D. Sarid, Opt. Lett. 8, 380 (1983);J. C. Quail, J. G. Rako, H. J. Simon, Opt. Lett. 8, 377 (1983).
[CrossRef] [PubMed]

Midwinter, J. E.

J. E. Midwinter, F. Zernike, Appl. Phys. Lett. 16, 198 (1970).
[CrossRef]

Mitchell, D. E.

H. J. Simon, D. E. Mitchell, J. G. Watson, Am. J. Phys. 43, 630 (1975).
[CrossRef]

Normandin, R.

M. Fukui, V. So, R. Normandin, Phys. Status Solidi B 91, K61 (1979);D. Sarid, Phys. Rev. Lett. 47, 1927 (1981);A. E. Craig, G. A. Olson, D. Sarid, Opt. Lett. 8, 380 (1983);J. C. Quail, J. G. Rako, H. J. Simon, Opt. Lett. 8, 377 (1983).
[CrossRef] [PubMed]

Otto, A.

A. Otto, Z. Phys. 216, 398 (1968).
[CrossRef]

Ritchie, G.

W. P. Chen, G. Ritchie, E. Burstein, Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

Schoenwald, J.

J. Schoenwald, E. Burstein, J. M. Elson, Solid State Commun. 12, 185 (1973);E. Santamato, F. DeMartini, Nuovo Cimento B 59, 223 (1980).
[CrossRef]

Simon, H. J.

H. J. Simon, D. E. Mitchell, J. G. Watson, Am. J. Phys. 43, 630 (1975).
[CrossRef]

So, V.

M. Fukui, V. So, R. Normandin, Phys. Status Solidi B 91, K61 (1979);D. Sarid, Phys. Rev. Lett. 47, 1927 (1981);A. E. Craig, G. A. Olson, D. Sarid, Opt. Lett. 8, 380 (1983);J. C. Quail, J. G. Rako, H. J. Simon, Opt. Lett. 8, 377 (1983).
[CrossRef] [PubMed]

Tamir, T.

Tien, P. K.

Ulrich, R.

Watson, J. G.

H. J. Simon, D. E. Mitchell, J. G. Watson, Am. J. Phys. 43, 630 (1975).
[CrossRef]

Zernike, F.

J. E. Midwinter, F. Zernike, Appl. Phys. Lett. 16, 198 (1970).
[CrossRef]

Am. J. Phys. (1)

H. J. Simon, D. E. Mitchell, J. G. Watson, Am. J. Phys. 43, 630 (1975).
[CrossRef]

Appl. Phys. Lett. (1)

J. E. Midwinter, F. Zernike, Appl. Phys. Lett. 16, 198 (1970).
[CrossRef]

J. Opt. Soc. Am. (3)

Phys. Rev. Lett. (1)

W. P. Chen, G. Ritchie, E. Burstein, Phys. Rev. Lett. 37, 993 (1976).
[CrossRef]

Phys. Status Solidi B (1)

M. Fukui, V. So, R. Normandin, Phys. Status Solidi B 91, K61 (1979);D. Sarid, Phys. Rev. Lett. 47, 1927 (1981);A. E. Craig, G. A. Olson, D. Sarid, Opt. Lett. 8, 380 (1983);J. C. Quail, J. G. Rako, H. J. Simon, Opt. Lett. 8, 377 (1983).
[CrossRef] [PubMed]

Solid State Commun. (1)

J. Schoenwald, E. Burstein, J. M. Elson, Solid State Commun. 12, 185 (1973);E. Santamato, F. DeMartini, Nuovo Cimento B 59, 223 (1980).
[CrossRef]

Z. Phys. (1)

A. Otto, Z. Phys. 216, 398 (1968).
[CrossRef]

Other (3)

The same equation can be obtained by solution of the coupled equations (44)′ and (45)′ in region x where B∼3(x) has the constant value A.

For a complete lateral shift of the profile it is necessary that there be complete coupling of the incident radiation into the surface mode so that no specular reflection occurs at the trailing edge of the incident beam.

L. M. Brekhovshikh, Waves in Layered Media (Academic, New York, 1960);D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

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Figures (4)

Fig. 1
Fig. 1

(a) Geometry of boundaries between three media which define layered structure, (b) Geometry used to excite boundary layer mode on layered structure 2−1−0.

Fig. 2
Fig. 2

Ratio of reflected power to incident peak power plotted as function of transverse distance along interface measured in microns. Position zero marks center of incident beam which extends from −500 to +500 μm. Parameters are as listed in text: (a) d2 = 1.25 μm, L/W = 0.35; (b) d2 = 1 μm, L/W = 0.23; (c) d2 = 0.9, μm, L/W = 0.174.

Fig. 3
Fig. 3

Same as Fig. 2. (d) d2 = 0.75 μm, L/W = 0.106; (e) d2 = 0.6 μm, L/W = 0.059.

Fig. 4
Fig. 4

Propagation vectors associated with boundary layer mode on layered structure 2–1–0.

Equations (80)

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ψ 0 ( β ) 2 k 1 z d 1 2 ϕ 12 2 ϕ 10 = 2 m π , m = 0 , 1 , 2 , .
k | | = ω c β , k lz = ω c l β 2 , ϕ jl = i 2 ln r jl , j , l = 0 , 1 , 2 , 3 ,
r jl = exp ( 2 i ϕ jl ) = l k jz j k lz l k jz + j k lz = r lj ,
ϕ jl = tan 1 ( j k lz i l k jz ) .
r 210 = r 21 + r 10 exp ( 2 i k 1 z d 1 ) 1 + r 21 r 10 exp ( 2 i k 1 z d 1 ) ,
R 0 ( β ) = r 12 r 10 exp ( 2 i k 1 z d 1 ) .
R 0 ( β m ) = exp [ 2 i ( k 1 z d 1 ϕ 12 ϕ 10 ) ] | β m = exp [ i ψ 0 ( β m ) ] = 1 .
h exp ( i k 2 z d 2 ) = exp [ ( k | | 2 2 ω 2 c 2 ) 1 / 2 d 2 ]
exp ( 2 i k 1 z d 1 ) = exp [ 2 ω c d 1 β 2 1 ]
1 r 10 = 0 , 0 k 1 z = 1 k 0 z ,
1 r 12 = 0 , 1 k 2 z = 2 k 1 z ,
r ijlm = r ij + r jlm exp ( 2 i k jz d j ) 1 + r ij r jlm exp ( 2 i k jz d j ) .
u ( β ) [ 1 r 123 r 10 exp ( 2 i k 1 z d 1 ) ] 1 .
r 123 = r 12 + r 23 exp ( 2 i k 2 z d 2 ) 1 + r 12 r 23 exp ( 2 i k 2 z d 2 ) ,
r 3210 = υ ( β ) u ( β ) ,
υ ( β ) r 32 + r 32 r 21 r 10 exp ( 2 i k 1 z d 1 ) + h 2 [ r 21 + r 10 exp ( 2 i k 1 z d 1 ) ] ( 1 + h 2 r 12 r 23 ) .
r 123 = r 12 ( 1 + 2 i h 2 r 23 sin 2 ϕ 12 ) ,
u ( β ) = [ 1 R ( β ) ] 1 ,
R ( β ) = r 12 r 10 exp ( 2 i k 1 z d 1 ) ( 1 + 2 i h 2 r 23 sin 2 ϕ 12 ) = R 0 ( β ) [ 1 + δ R ( β ) R 0 ( β ) ] = exp [ i ψ 0 ( β ) + i δ ψ ( β ) ] δ ψ ( β ) = 2 h 2 r 23 sin 2 ϕ 12 . }
ψ 0 ( β ̂ m ) + δ ψ ( β ̂ m ) = 2 m π , m = 0 , 1 , 2 , ,
β ̂ m = β m + δ β m .
ψ 0 ( β m ) + d ψ d β | β m δ β m + δ ψ ( β m ) 2 m π ,
δ β m δ ψ ( β m ) / d ψ d β | β m = 2 h 2 r 23 sin 2 ϕ 12 ( d ψ d β ) | β m | β m .
d ψ d β | β m .
u ( β ) i d ψ d β | β m ( β β ̂ m ) .
υ ( β ) i d ψ d β | β m r 32 ( β ) [ β β m δ β m r 32 2 ( β m ) ] ,
r 3210 ( β ) r 32 ( β ) [ β β m δ β m r 32 2 ( β m ) ] ( β β ̂ m ) ,
Δ ψ ( β m ) = d ψ d β | β m Δ β m = 2 π ,
Δ β m = 2 π / d ψ d β | β m .
r 3210 ( β ) = r 32 ( β m ) { 1 + δ β m [ 1 1 r 32 2 ( β m ) ] β Re ( β m + δ β m ) + ilm ( β m + δ β m ) } .
β = c ω k | | = c ω k sin θ i = 3 sin θ i ,
a ( β , z ) = { exp ( i k 1 z z ) + r 10 exp [ i k 1 z z + 2 i k 1 z ( d 1 + d 2 ) ] } ,
a ( β , z ) t 321 ( β ) ,
t 321 ( β ) = ( 1 + r 32 ) ( 1 + r 21 ) 1 + r 32 r 21 exp ( 2 i k 2 z d 2 ) u ( β ) exp [ i ( k 2 z k 1 z ) d 2 ] .
a ( β , z ) t 321 ( β ) i t ¯ m ( β m , z ) ( β β ̂ m ) ,
t ¯ m ( β m , z ) = 2 i h ( 1 + r 32 ) d ψ d β | β m sin ϕ 12 × { exp [ i k 1 z ( z d 2 ) i ϕ 12 ] + exp [ i k 1 z ( z d 2 ) + i ϕ 12 ] } | β m .
B 3 ( x , t ) = B 3 y ( x , z , t ) | z = 0 B 3 ( x ) exp ( i ω t ) , B 3 ( x ) = F ( x ) exp ( i k | | 0 x ) ,
B 3 ( x ) = d β b 3 ( β ) exp ( i ω c β x )
b 3 ( β ) = ω / c 2 π d x F ( x ) exp [ i ( k | | 0 ω c β ) x ] .
B 3 ( x ) = d β r 3210 ( β ) b 3 ( β ) exp ( i ω c β x ) ,
B 1 ( x , y ) = d β a ( β , z ) t 321 ( β ) b 3 ( β ) exp ( i ω c β x ) .
B 3 ( x ) = ω / c 2 π d x R 3210 ( x x ) B 3 ( x ) ,
B 1 ( x , z ) = ω / c 2 π d x T 321 ( x x , z ) B 3 ( x ) ,
R 3210 ( x ) d β r 3210 ( β ) exp ( i ω c β x ) ,
T 321 ( x , y ) d β a ( β , z ) t 321 ( β ) exp ( i ω c β x ) .
Δ β 2 π / ( ω / c ) w .
ω c w d ψ d β | β m ,
R 3210 ( x x ) = r 32 ( β m ) { 2 π ω / c δ ( x x ) + δ β m [ 1 1 r 32 2 ( β m ) ] ζ m ( x x ) } ,
T 321 ( x x , z ) = t ¯ m ( β m , z ) ζ m ( x x ) ,
ζ m ( x x ) { 2 π exp [ i ω c β ̂ m ( x x ) ] , x x , 0 , x > x .
B 3 ( x ) = r 32 ( β m ) B 3 ( x ) + 2 ω c δ β m sin 2 ϕ 32 ( β m ) × x d x B 3 ( x ) exp [ i ω c β ̂ m ( x x ) ]
B 1 ( x , z ) = ω c t ¯ m ( β m , z ) x d x B 3 ( x ) exp [ i ω c β ̂ m ( x x ) ] ,
sin 2 ϕ 32 ( β m ) = i 2 [ r 32 ( β m ) 1 r 32 ( β m ) ] .
B 3 ( x ) = B 3 ( x ) exp ( i k | | 0 x ) , B 3 ( x ) = B 3 ( x ) exp ( i k | | 0 x ) , B 1 ( x , z ) = B 1 ( x , z ) exp ( i k | | 0 x ) .
d B 1 d x = i ω c ( β ̂ m ҟ | | 0 ) B 1 + ω c t ¯ m B 3 ,
B 3 = r 32 B 3 + 2 δ β m t ¯ m sin ( 2 ϕ 32 ) B 1 ,
β m β + i β α i ω c ( β ̂ m ҟ | | 0 ) = ω c ( β + l m δ β m ) + i ω c ( Re β ̂ m ҟ | | 0 ) , α 1 ω c δ β m sin 2 ϕ 32 ,
d B 1 d x = α B 1 + ω c t ¯ m B 3 ,
B 3 = r 32 B 3 2 α 1 ( ω / c ) t ¯ m B 1 .
F ( x ) = { 0 , x < w 2 , A , w 2 x w 2 , 0 , x > w 2 ,
B 3 ( x ) = r 32 B 3 ( x ) 2 α 1 α A { } ,
B 1 ( x ) = ( ω / c ) t ¯ m α A { } ,
{ } { 0 , x < w 2 , { 1 exp [ α ( x + w 2 ) ] } , w 2 x w 2 , 2 i exp ( α x ) sin ( i α w 2 ) , x > w 2 ,
B 3 ( x ) B 3 ( 0 ) = r 32 2 α 1 α { 1 exp [ α ( x + w 2 ) ] } .
2 α 1 α n ( x 1 ) = r 32 ,
n ( x ) { 1 exp [ α ( x + w c ) ] } .
ω c lm δ β m = α 2 n ( x 1 ) lm [ r 32 ( β m ) sin 2 ϕ 32 ( β m ) ] ,
ω c Re δ β m = α 2 n ( x 1 ) Re [ r 32 ( β m ) sin 2 ϕ 32 ( β m ) ] ,
1 2 [ r 32 ( β m ) sin 2 ϕ 32 ( β m ) ] = i [ 1 1 / r 32 2 ( β m ) ] 1 ,
r 32 ( β m ) = 2 3 β m 2 3 2 β m 2 2 3 β m 2 + 3 2 β m 2 .
β ̂ m = Re β ̂ m + i lm β ̂ m = 3 sin θ l + i c ω α .
P r P i = | B 3 ( x ) | 2 | B 3 ( 0 ) | 2 ,
L = 1 ω | c lm β ̂ m = [ ω c ( β + lm δ β m ) ] 1 ,
d ψ d β = 2 ( ω c ) 2 β k 1 z D 1 ( β ) ,
D 1 ( β ) = ( d 1 + i 1 2 k 2 z k 1 z 2 / k 2 z 1 2 k 2 z 2 2 2 k 2 z 2 + i 1 0 k 0 z k 1 z 2 / k 0 2 1 2 k 0 z 2 0 2 k 1 z 2 ) .
P = d z S x ̂ = c 8 π Re d z E z B y * ,
P = c 8 π | a 1 | 2 ( Re β ) exp ( 2 lm ϕ 12 2 lm k | | x ) { } ,
{ } = | 1 / 2 | ( lm k 2 z ) [ cosh ( 2 lm ϕ 12 ) + cos ( 2 Re ϕ 12 ) ] + | 1 / 0 | ( lm k 0 z ) [ cosh ( 2 lm ϕ 10 ) + cos ( 2 Re ϕ 10 ) ] + 1 ( Re k 1 z ) [ sin ( 2 Re ϕ 12 ) + sin ( 2 Re ϕ 10 ) ] + 1 lm k 1 z [ sinh ( 2 lm ϕ 12 ) + sinh ( 2 lm ϕ 10 ) ]
{ } = 1 2 exp ( 2 lm ϕ 12 [ | 1 / 2 | ( lm k 2 z ) | 1 + 1 r 12 | 2 + | 1 / 2 | lm k 0 z | 1 + r 10 | 2 exp ( 2 lm k 1 z d 1 ) + 1 i Re k 1 z ( 1 r 12 1 r 12 * ) + 1 i Re k 1 z ( r 10 * r 10 ) exp ( 2 lm k 1 z d 1 ) + 1 lm k 1 z ( 1 1 | r 12 | 2 ) 1 lm k 1 z ( 1 | r 10 | 2 ) exp ( 2 lm k 1 z d 1 ) ] .
P = { c 4 π | a 1 | 2 exp ( 2 lm ϕ 12 ) β D 1 ( β ) surface plasmon , c 4 π | a 1 | 2 β D 1 ( β ) guided wave ,

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