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Exact calculations of p-polarized electromagnetic fields incident on grating surfaces: Surface polariton resonances
Volume 45, number 5
OPTICS COMMUNICATIONS
1 May 1983
EXACT CALCULATIONS OF P-POLARIZED ELECTROMAGNETIC FIELDS INCIDENT ON GRATING SURFACES: SURFACE POLARITON RESONANCES N. GARCIA Departamento de ~'sica Fundamental, UniversidadAutonoma de Madrid, Canto Blanco, Madrid-34, Spain Received 24 January 1983
Exact calculations of the interaction of p-polarized light from grating surfaces are presented by using the theory of Toigo, Marvin, Celli and Hill. New features for the scattered field are found for light resonating with surface plasmon polaritons; the reflectivity present maxima for large amplitude gratings. The electromagnetic field is calculated on a fine grid near the grating. We f'md enhancements of the order of 200 to 300 for the square of the field that could partly explain SERS.
Excitation of surface plasmons [1] by incident light has received recent attention because it has been suggested by Moskowits [2] and Chen et al. [3] that under these resonance conditions the electric field as well as the magnetic field could be enhanced in the vicinity of the grating by a large factor. This could then be the explanation for the observed surface enhanced Raman scattering (SERS) when molecules are adsorbed on Ag-island [ 3 - 5 ] , or on rough surfaces or gratings that couple p-polarized light to surface plasmons. Another proposal for explaining SERS depends basically on the fields created by small irregularities at the surface as for example adatoms [6]. There are then very interesting questions to answer: Is there enhancement of the electric field when the incident conditions are such that the light can excite a surface polariton? If yes, how is this related to the observed reflectivity?, what is the value of the enhanced field near to the grating?, and how is that related to its shape and amplitude?. In this letter we answer these questions with exact theoretical calculations of tlte behaviour of TM polarized light interacting with a grating surface. The most adequate theory to perform the above mentioned calculations is, in our opinion, that developed by Toigo et al. [7]. This theory is based on the application of the extinction theorem to Green's theorem and will be shown to work very well for val0 030-4018/83/0000-0000/$ 03.00 © 1983 North-Holland
ues of the roughness or grating amplitudes much larger than those for which the Rayleigh expansion can be applied [7,8]. From our experience in these calculations, as well as those performed for TE polarization [8], it can be shown that the theory gives convergent results up to values o f h = d/a = 0.3, where d and a are the grating amplitude and period. Our problem is to calculate the electric and magnetic fields scattered from a one-dimensional grating defined by the function DO') = a ~
h n cos(2nny/a).
(1)
n=l
We are interested in the elastic scattering of a p-polar. ized electromagnetic field impinging on the surface, the magnetic field has only one component (7) in the x-direction and behaves as a scalar field. If the incident wave vector has components K 0 and P0, parallel and perpendicular to the surface, with K 0 + P0 = k, and k = w/c, then the incident field is exp[i(K0Y - p0z)] ,
(2)
and the reflected field for z ~ ~ is B n exp[i(KnY +pnZ)] ,
(3)
n~--aa
with K n = K 0 + Gn ( Gn = 2nn/a, are reciprocal vec-
tors) and Kn2 + p2 = k 2. Similarly, the refracted field 307
Volume 45, number 5
OPTICS COMMUNICATIONS
in the metal is oo
qnZ)] ,
Cn e x p [ i ( K n Y -
(4)
w i t h K 2 + q2 = ek 2, where e ( ~ ) is the dielectric con-
stant. From the theory of ref. [7] it can be shown that for a flat surface there is a resonance between the incident light and the surface plasmon polariton when (5)
6n = e(W)Pn + qn "~ O.
Our aim in this paper is to study the behaviour of the field under approximately resonance conditions. For this we choose a case where SERS has been observed from nitrobenzoic acid adsorbed on an Ag grating surface [4] w i t h a = 8000 A and a wavelength of the incident light ~ = 5145 A. For the above conditions the resonance occurs for an angle o f incidence with the normal to the surface 0 i = 24 °, and is a resonance produced for n = 1 in eq. (5). We like this case because the resonating beam is the first order one and we expect therefore stronger coupling with the grating 1.0
h 1 = OJIl(~, h 1 =0.02
h 1 = 0.03
lo,
0.8
1 May 1983
and then stronger resonance effects. The value of the dielectric constant for the incident wavelength is e ( ~ ) = - 1 1 . 0 + 0.33i [9]. In fig. 1 we present the values o f the reflectivity (specular peak (0)), and diffraction peaks (n) for a sinusoidal grating as a function o f the angle 0 i for different amplitude h 1 . Also we show the enhancement of the magnetic field (the ratio between the modulus squared of the scattered field to the modulus squared of the incident field) (ell), in thick lines. The value o f e H is calculated at the maximum height o f the grating and the number, for exam'ple 172, in the case h 1 = 0.02 indicates the maximum enhancement. The enhancement o f the electric field (eE) is "--30% higher (fig. 3). The line indicated by R is the position of the surface plasmon for a flat surface and the resonance is produced by the beam (1) that disappears at 20.8 °. Notice the structure produced in the ( - 1 ) beam at threshold T in the case of h 1 = 0.0775 in agreement with other recent analytical and numerical calculations [8,10]. We also notice that the reflectivity as well as the diffraction beams are minimum at resonance
h 1 = 0.05
167
186/
0.8
0.6
h 4 = 0.0075
0.6 0.4
0.2
eH
0.4
40 30
0.2
20 10
23 2'4
~3 2'4 25
23 24
h 1 = 0.0375
hi=0.0775 ~
0.8
•
a
i
0.4 0.2
22
24
26
22
24
26
Oi(°) Fig. 1. V a l u e s o f t h e s c a t t e r e d i n t e n s i f i e s a n d e H as the incid e n t angle varies through resonance for several values of hi on
a sinusoidal grating. Note that the specular beam is maximum for h I = 0.0775. 308
h = 0.03
_
/
0.6
20
23 :';25 (~0) Saw T°°th
0.8
eH 160
0.6
120
0.4
80
0.2
40
22
24
26
28 Oi(°)
Fig. 2. The same as in fig. 1 for a profile with several Fourier components (a) and a saw-tooth (b). Notice the non-symmetric shape of the resonance as well as the maximum in the (-1) beam in (a).
Volume 45, number 5
OPTICS COMMUNICATIONS
but that the reflectivity becomes a maximum for h I = 0.0775. This effect has been observed [11]. In fact, the reflectivity is maximum for h ~> 0.065. Also the minimum in the reflectivity is smaller for a critical amplitude h 1 = 0.02 which is when the maximum enhancement occurs. In fig. 2 we present results for cases similar to those o f fig. 1, but for different grating shapes, to prove the versatility o f our numerical method, and new interesting results appear. First in the case of several fourier components the resonance is not symmetric as it was before. Also the beam ( - 1 ) now displays a maximum while for the sinusoidal gratings it always showed a minimum. At the same time the values o f e H ,E are reduced even if the reflectivity is very small, but now ~ 6 0 % o f the incident field is p u m p e d into the first order beam. The sawtooth profile has also been studied and the values o f eH,E are reduced by ~25%. Notice that from fig. 1 it is possible to £md the resonance width and plasmon dispersion relation (shift in the resonance position) as function o f the grating amplitude. For large amplitudes the plasmon resonance spreads out to the whole range o f 0 i and e H,E tend to unity. In fig. 3 the values o f ell, E are given at the p o i n t s of a fine grid near the grating for the two surfaces profiles indicated in the figure. We find enhancement of the field o f the order o f ~ 2 0 0 to ~ 0 under condi. tions o f maximum absorption of light. This happens at resonance for h 1 = 0.02 because almost all the indident energy is pumped into the polariton that enhances the field. We should notice that the field enhancement e H produced by the surface polariton wave are " 1 7 3 and ~ 5 3 for the cases o f fig. 3a and 3b respectively. Therefore the deviation o f these numbers to those given in fig. 3 are because the total field is a summation o f many partial waves plus the incident field. I f the Raman cross section is approximately proportional to the "fourth power" o f the scattered electric field, then we have an increase o f the cross section o f the order of 1 0 - 4 to 1 0 - 5 which could explain partly SERS [1 ~ 5 ] . Recently some calculations o f e E at the maximum point o f the grating for a saw-tooth profile [12] have reported a value o f e E = 25. The difference o f this value to those presented here m a y be because the conditions in the two calculations were not the same, i.e. in ref. [12] the n vector associated with the polariton resonance was n = 3, while in our results n = 1. This makes a great
1 May 1983 Vacuum
eE
IZ,4
11,0
153
179
11,7
11,1,
158
185
(a)
aH 11,1
115
f10
116
2071I 175 11,5 151 148 f63 192 215 / 181 150 I 155153169199221,1'86151,128120129
1991 170
118
114
120
122
117
125
129
]2,
131,
133
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138
\137
,32
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~
137
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159
157
171,
163
161
180/
207~
_ _
,57
,++
_
,71
,7o
~++/ ,+~/
175
171,
179
179 /
,83 'eL./,
/
~58 X
/
-a 2
\
/
+'2": '.7'
X
' -- z~'-i--
-3a 8
,
,
-a 4
-a 8
X
-O.O',a , +,
153
146
158
\
0
a 8
'+3
,
,
a 4
3a 8
Vacuum
eE
11,1
~',++
eH
a 2 (b)
50
28
50
41,
95.[.168
60
22
25
27
52
29
52
44
62
23
25
28
++
30
+3
4,/
/ _ \ / "t
+4
23
2+
2+
58
30
55
44/
65
24
27
29
+,
3,
+7
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.
32
5+
++
32
71,
33 33
80
y33 /
-a_ 2
1,3/
_ \
\
21,
27
_
,~0
2+
2+
3,
51/
_
\
2+
.
32
63 / 65/
Metal (Aa)
30 31
33 34
1,y
_
\58
X
-
0i!24 T
"~ 32y
w,- . . . .- +. . -3a 8
-a 4
-a 8
0
26 ~27
a 8
a 4
2, 3a 8
a 2
Fig. 3. Values ofe H and e E for a sinusoidal grating (a) and grating (b) of fig. 2(a). The positions of the numbers define the grid points where the fields are calculated. The numbers are the values of ell,E rounded to the nearest integer. The enhancement is larger for the sinusoidal case. The values for a saw-tooth grating are ~25% smaller, in general, than for the sinusoidal one. 0 i = 24 ° . h = 5145 A. The results are for the illuminated region of the surface x < 0, but those of x> 0 which correspond to the shaded surface are practically the same.
difference because for h "" 0.02, n = 1 has stronger interaction than n = 3 and favours our value e. Also our angle is 0 i = 24 ° while in ref. [12] it is 45 °. In conclusion we have presented exact numerical calculations for the interaction o f TM polarized radiation with grating surfaces that show new features as functions o f the incident angle, grating amplitudes 309
Volume 45, number 5
OPTICS COMMUNICATIONS
and shapes. Also field enhancements calculated on a fine grid show that SERS could be explained by surface plasmon resonances. While this letter was written, recent experirnental data by Raether [ 13] appeared that support the shape and behaviour o f the calculated intensities. We hope that this work will stimulate further experimental work to see how precisely the calculations agree with experiments because then one can obtain much information from fitting experimental resonant shapes to the calculated ones. Also calculations up to h = 0.25 show that the factor e H is still decreasing smoothly having a value e H "" 5. It is a pleasure to acknowledge A.A. Maradudin for continuing discussions, suggestions and reading the manuscript. I appreciate also suggestions by V. Celli, T. Lopez-Rios, M.R. Philpott, I. Pockrand and H. Seki. I am also very grateful to E. Burstein for indicating me some numerical errors and unphysical results presented in a first version o f this paper. Finally I thank IBM Research Laboratory San Jos6 for providing the computer facilities and financial support for performing these calculations.
310
1 May 1983
References [ 1] H. Raether, Surface plasmons and roughness, in: Surface polaritons, eds. D.L. Mills and V.M. Agranovich (NorthHolland, Amsterdam, 1982) p. 331. [2] M. Moskowits, J. Chem. Phys. 69 (1978) 4159. [3] E. Burstein, C.Y. Chen and S. Lundqvist, Solid State Commun. 32 (1979) 63; C.Y. Chen and E. Burstein, Phys. Rev. Lett. 45 (1980) 1287. [4] J.C. Tsang, J.R. Kirtley and T.N. Theis, Solid State Commun. 35 (1980) 667. [5] H. Seki and M.R. Philpott, J. Chem. Phys. 72 (1980) 2166. [6] A. Otto, J. Timper, J. Billmann and I. Pockrang, Phys. Rev. Lett. 45 (1980) 46; A. Otto, Appl. Surf. Science 6 (1980) 309. [7] F. Toigo, A. Marvin, C. Celli and N.R. Hill, Phys. Rev. B15 (1977) 5618. [8] N. Garcia and A.A. Maradudin, OpticsComm. 45 (1983) 301. [9] P.B. Johnson and R.W. Christy, Phys. Rev. B6 (1972) 4370. [10] N. Garcia and W.A. Schlup, Surface Sci. 122 (1982) 1.657 L657. [11] E.H. Rosengart and I. Pockrand, Optics Lett. 1 (1977) 194. [12] D.L. Mills and M. Weber, Phys. Rev. B26 (1982) 1075. [13] H. Raether, Optics Comm. 42 (1982) 217.
OPTICS COMMUNICATIONS
1 May 1983
EXACT CALCULATIONS OF P-POLARIZED ELECTROMAGNETIC FIELDS INCIDENT ON GRATING SURFACES: SURFACE POLARITON RESONANCES N. GARCIA Departamento de ~'sica Fundamental, UniversidadAutonoma de Madrid, Canto Blanco, Madrid-34, Spain Received 24 January 1983
Exact calculations of the interaction of p-polarized light from grating surfaces are presented by using the theory of Toigo, Marvin, Celli and Hill. New features for the scattered field are found for light resonating with surface plasmon polaritons; the reflectivity present maxima for large amplitude gratings. The electromagnetic field is calculated on a fine grid near the grating. We f'md enhancements of the order of 200 to 300 for the square of the field that could partly explain SERS.
Excitation of surface plasmons [1] by incident light has received recent attention because it has been suggested by Moskowits [2] and Chen et al. [3] that under these resonance conditions the electric field as well as the magnetic field could be enhanced in the vicinity of the grating by a large factor. This could then be the explanation for the observed surface enhanced Raman scattering (SERS) when molecules are adsorbed on Ag-island [ 3 - 5 ] , or on rough surfaces or gratings that couple p-polarized light to surface plasmons. Another proposal for explaining SERS depends basically on the fields created by small irregularities at the surface as for example adatoms [6]. There are then very interesting questions to answer: Is there enhancement of the electric field when the incident conditions are such that the light can excite a surface polariton? If yes, how is this related to the observed reflectivity?, what is the value of the enhanced field near to the grating?, and how is that related to its shape and amplitude?. In this letter we answer these questions with exact theoretical calculations of tlte behaviour of TM polarized light interacting with a grating surface. The most adequate theory to perform the above mentioned calculations is, in our opinion, that developed by Toigo et al. [7]. This theory is based on the application of the extinction theorem to Green's theorem and will be shown to work very well for val0 030-4018/83/0000-0000/$ 03.00 © 1983 North-Holland
ues of the roughness or grating amplitudes much larger than those for which the Rayleigh expansion can be applied [7,8]. From our experience in these calculations, as well as those performed for TE polarization [8], it can be shown that the theory gives convergent results up to values o f h = d/a = 0.3, where d and a are the grating amplitude and period. Our problem is to calculate the electric and magnetic fields scattered from a one-dimensional grating defined by the function DO') = a ~
h n cos(2nny/a).
(1)
n=l
We are interested in the elastic scattering of a p-polar. ized electromagnetic field impinging on the surface, the magnetic field has only one component (7) in the x-direction and behaves as a scalar field. If the incident wave vector has components K 0 and P0, parallel and perpendicular to the surface, with K 0 + P0 = k, and k = w/c, then the incident field is exp[i(K0Y - p0z)] ,
(2)
and the reflected field for z ~ ~ is B n exp[i(KnY +pnZ)] ,
(3)
n~--aa
with K n = K 0 + Gn ( Gn = 2nn/a, are reciprocal vec-
tors) and Kn2 + p2 = k 2. Similarly, the refracted field 307
Volume 45, number 5
OPTICS COMMUNICATIONS
in the metal is oo
qnZ)] ,
Cn e x p [ i ( K n Y -
(4)
w i t h K 2 + q2 = ek 2, where e ( ~ ) is the dielectric con-
stant. From the theory of ref. [7] it can be shown that for a flat surface there is a resonance between the incident light and the surface plasmon polariton when (5)
6n = e(W)Pn + qn "~ O.
Our aim in this paper is to study the behaviour of the field under approximately resonance conditions. For this we choose a case where SERS has been observed from nitrobenzoic acid adsorbed on an Ag grating surface [4] w i t h a = 8000 A and a wavelength of the incident light ~ = 5145 A. For the above conditions the resonance occurs for an angle o f incidence with the normal to the surface 0 i = 24 °, and is a resonance produced for n = 1 in eq. (5). We like this case because the resonating beam is the first order one and we expect therefore stronger coupling with the grating 1.0
h 1 = OJIl(~, h 1 =0.02
h 1 = 0.03
lo,
0.8
1 May 1983
and then stronger resonance effects. The value of the dielectric constant for the incident wavelength is e ( ~ ) = - 1 1 . 0 + 0.33i [9]. In fig. 1 we present the values o f the reflectivity (specular peak (0)), and diffraction peaks (n) for a sinusoidal grating as a function o f the angle 0 i for different amplitude h 1 . Also we show the enhancement of the magnetic field (the ratio between the modulus squared of the scattered field to the modulus squared of the incident field) (ell), in thick lines. The value o f e H is calculated at the maximum height o f the grating and the number, for exam'ple 172, in the case h 1 = 0.02 indicates the maximum enhancement. The enhancement o f the electric field (eE) is "--30% higher (fig. 3). The line indicated by R is the position of the surface plasmon for a flat surface and the resonance is produced by the beam (1) that disappears at 20.8 °. Notice the structure produced in the ( - 1 ) beam at threshold T in the case of h 1 = 0.0775 in agreement with other recent analytical and numerical calculations [8,10]. We also notice that the reflectivity as well as the diffraction beams are minimum at resonance
h 1 = 0.05
167
186/
0.8
0.6
h 4 = 0.0075
0.6 0.4
0.2
eH
0.4
40 30
0.2
20 10
23 2'4
~3 2'4 25
23 24
h 1 = 0.0375
hi=0.0775 ~
0.8
•
a
i
0.4 0.2
22
24
26
22
24
26
Oi(°) Fig. 1. V a l u e s o f t h e s c a t t e r e d i n t e n s i f i e s a n d e H as the incid e n t angle varies through resonance for several values of hi on
a sinusoidal grating. Note that the specular beam is maximum for h I = 0.0775. 308
h = 0.03
_
/
0.6
20
23 :';25 (~0) Saw T°°th
0.8
eH 160
0.6
120
0.4
80
0.2
40
22
24
26
28 Oi(°)
Fig. 2. The same as in fig. 1 for a profile with several Fourier components (a) and a saw-tooth (b). Notice the non-symmetric shape of the resonance as well as the maximum in the (-1) beam in (a).
Volume 45, number 5
OPTICS COMMUNICATIONS
but that the reflectivity becomes a maximum for h I = 0.0775. This effect has been observed [11]. In fact, the reflectivity is maximum for h ~> 0.065. Also the minimum in the reflectivity is smaller for a critical amplitude h 1 = 0.02 which is when the maximum enhancement occurs. In fig. 2 we present results for cases similar to those o f fig. 1, but for different grating shapes, to prove the versatility o f our numerical method, and new interesting results appear. First in the case of several fourier components the resonance is not symmetric as it was before. Also the beam ( - 1 ) now displays a maximum while for the sinusoidal gratings it always showed a minimum. At the same time the values o f e H ,E are reduced even if the reflectivity is very small, but now ~ 6 0 % o f the incident field is p u m p e d into the first order beam. The sawtooth profile has also been studied and the values o f eH,E are reduced by ~25%. Notice that from fig. 1 it is possible to £md the resonance width and plasmon dispersion relation (shift in the resonance position) as function o f the grating amplitude. For large amplitudes the plasmon resonance spreads out to the whole range o f 0 i and e H,E tend to unity. In fig. 3 the values o f ell, E are given at the p o i n t s of a fine grid near the grating for the two surfaces profiles indicated in the figure. We find enhancement of the field o f the order o f ~ 2 0 0 to ~ 0 under condi. tions o f maximum absorption of light. This happens at resonance for h 1 = 0.02 because almost all the indident energy is pumped into the polariton that enhances the field. We should notice that the field enhancement e H produced by the surface polariton wave are " 1 7 3 and ~ 5 3 for the cases o f fig. 3a and 3b respectively. Therefore the deviation o f these numbers to those given in fig. 3 are because the total field is a summation o f many partial waves plus the incident field. I f the Raman cross section is approximately proportional to the "fourth power" o f the scattered electric field, then we have an increase o f the cross section o f the order of 1 0 - 4 to 1 0 - 5 which could explain partly SERS [1 ~ 5 ] . Recently some calculations o f e E at the maximum point o f the grating for a saw-tooth profile [12] have reported a value o f e E = 25. The difference o f this value to those presented here m a y be because the conditions in the two calculations were not the same, i.e. in ref. [12] the n vector associated with the polariton resonance was n = 3, while in our results n = 1. This makes a great
1 May 1983 Vacuum
eE
IZ,4
11,0
153
179
11,7
11,1,
158
185
(a)
aH 11,1
115
f10
116
2071I 175 11,5 151 148 f63 192 215 / 181 150 I 155153169199221,1'86151,128120129
1991 170
118
114
120
122
117
125
129
]2,
131,
133
128
138
\137
,32
,1,3
~
137
,1,8
/
159
157
171,
163
161
180/
207~
_ _
,57
,++
_
,71
,7o
~++/ ,+~/
175
171,
179
179 /
,83 'eL./,
/
~58 X
/
-a 2
\
/
+'2": '.7'
X
' -- z~'-i--
-3a 8
,
,
-a 4
-a 8
X
-O.O',a , +,
153
146
158
\
0
a 8
'+3
,
,
a 4
3a 8
Vacuum
eE
11,1
~',++
eH
a 2 (b)
50
28
50
41,
95.[.168
60
22
25
27
52
29
52
44
62
23
25
28
++
30
+3
4,/
/ _ \ / "t
+4
23
2+
2+
58
30
55
44/
65
24
27
29
+,
3,
+7
30
.
32
5+
++
32
71,
33 33
80
y33 /
-a_ 2
1,3/
_ \
\
21,
27
_
,~0
2+
2+
3,
51/
_
\
2+
.
32
63 / 65/
Metal (Aa)
30 31
33 34
1,y
_
\58
X
-
0i!24 T
"~ 32y
w,- . . . .- +. . -3a 8
-a 4
-a 8
0
26 ~27
a 8
a 4
2, 3a 8
a 2
Fig. 3. Values ofe H and e E for a sinusoidal grating (a) and grating (b) of fig. 2(a). The positions of the numbers define the grid points where the fields are calculated. The numbers are the values of ell,E rounded to the nearest integer. The enhancement is larger for the sinusoidal case. The values for a saw-tooth grating are ~25% smaller, in general, than for the sinusoidal one. 0 i = 24 ° . h = 5145 A. The results are for the illuminated region of the surface x < 0, but those of x> 0 which correspond to the shaded surface are practically the same.
difference because for h "" 0.02, n = 1 has stronger interaction than n = 3 and favours our value e. Also our angle is 0 i = 24 ° while in ref. [12] it is 45 °. In conclusion we have presented exact numerical calculations for the interaction o f TM polarized radiation with grating surfaces that show new features as functions o f the incident angle, grating amplitudes 309
Volume 45, number 5
OPTICS COMMUNICATIONS
and shapes. Also field enhancements calculated on a fine grid show that SERS could be explained by surface plasmon resonances. While this letter was written, recent experirnental data by Raether [ 13] appeared that support the shape and behaviour o f the calculated intensities. We hope that this work will stimulate further experimental work to see how precisely the calculations agree with experiments because then one can obtain much information from fitting experimental resonant shapes to the calculated ones. Also calculations up to h = 0.25 show that the factor e H is still decreasing smoothly having a value e H "" 5. It is a pleasure to acknowledge A.A. Maradudin for continuing discussions, suggestions and reading the manuscript. I appreciate also suggestions by V. Celli, T. Lopez-Rios, M.R. Philpott, I. Pockrand and H. Seki. I am also very grateful to E. Burstein for indicating me some numerical errors and unphysical results presented in a first version o f this paper. Finally I thank IBM Research Laboratory San Jos6 for providing the computer facilities and financial support for performing these calculations.
310
1 May 1983
References [ 1] H. Raether, Surface plasmons and roughness, in: Surface polaritons, eds. D.L. Mills and V.M. Agranovich (NorthHolland, Amsterdam, 1982) p. 331. [2] M. Moskowits, J. Chem. Phys. 69 (1978) 4159. [3] E. Burstein, C.Y. Chen and S. Lundqvist, Solid State Commun. 32 (1979) 63; C.Y. Chen and E. Burstein, Phys. Rev. Lett. 45 (1980) 1287. [4] J.C. Tsang, J.R. Kirtley and T.N. Theis, Solid State Commun. 35 (1980) 667. [5] H. Seki and M.R. Philpott, J. Chem. Phys. 72 (1980) 2166. [6] A. Otto, J. Timper, J. Billmann and I. Pockrang, Phys. Rev. Lett. 45 (1980) 46; A. Otto, Appl. Surf. Science 6 (1980) 309. [7] F. Toigo, A. Marvin, C. Celli and N.R. Hill, Phys. Rev. B15 (1977) 5618. [8] N. Garcia and A.A. Maradudin, OpticsComm. 45 (1983) 301. [9] P.B. Johnson and R.W. Christy, Phys. Rev. B6 (1972) 4370. [10] N. Garcia and W.A. Schlup, Surface Sci. 122 (1982) 1.657 L657. [11] E.H. Rosengart and I. Pockrand, Optics Lett. 1 (1977) 194. [12] D.L. Mills and M. Weber, Phys. Rev. B26 (1982) 1075. [13] H. Raether, Optics Comm. 42 (1982) 217.