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Radiation from surface plasmons propagating on rough metal surfaces
Volume
1, number
July/August
OPTICS COMMUNICATIONS
3
RADIATION
FROM ON
SURFACE
ROUGH
PLASMONS
METAL
1969
PROPAGATING
SURFACES
0. HUNDERI and D. BEAGLEHOLE Department
of Physics College
Revised
and Astronomy, Park.
Maryland
Uniawrsity 20742.
of Maryland
USA
Received 9 June 1969 version received 15 July 1969
Radiation from surface plasmons excited by light normally incident onto thick rough metal samples is reported. Its intensity peaks at about the same wavelength as the extra absorbtivity induced by the surface plasmons. Its polarization properties indicate that it is most likely incoherent rather than resonance radiation. We present a simple theory which gives good agreement with the experimental results.
Surface plasmons are evanescent electromagnetic waves which can propagate along the surface of a metal whose dielectric constant E is less than - I. Their n field is transverse. Their E field has both longiqdinal and normal components, in the ratio 1E 1P. They are dominantly longitudinal when E << - 1. Their wave vector in the surface is greater than w/c, and so they cannot be excited by light falling onto a smooth surface. However, an anomalous decrease in the reflectivity of rough metal surfaces has been observed, and this has been assigned to the excitation of surface plasmons [I]. We have observed radiation from the decay of these surface plasmons into photons. The roughness provides the necessary momentum both for the coupling of the incident light into the plasmon mode, and for its subsequent decay. The polarization properties of the radiation we observe suggest that it is incoherent rather than resonance radiation. It is thus different from the coherent resonance radiation reported from plasmons on thin films [2], but similar to the incoherent radiation which has recently been observed [3] from surface plasmons propagating on a metal grating. Our experiment has been performed by shining polarized light normally onto rough metal surfaces, and measuring the distribution of light coming from the sample as a function of the angle .Ofrom the normal. The incident light was either p or s polarized with respect to the observation plane. Rough samples were produced by the technique of Stanford et al. [l] by deposition of a thin CaF2 layer onto a quartz substrate,
evaporating the metal layer on top. Several metals were studied, particularly Ag and +u. Their thicknesses were greater than 1000 A. Most of the light detected at off-normal angles was scattered light, whose distribution followed approximately that of the scalar scattering theory [4]. In fact the distribution was found to depend somewhat upon the incident polarization, the p scattering being greater than the s. However, this difference was also found above the surface plasmon cutoff frequency (E > - l), and with smooth samples, and so was not associated with the surface plasmons. Examples are shown in fig. 1 for a gold sample. When we placed an analyzer in front of the detector and measured only the light with polarization opposite to that incident, we found a significant anomaly associated with the surface plasmons. This is shown in fig. 1. The peak in the emitted p polarized light excited by incident polarized light disappeared above the maximum surface plasmon frequency, and was not found with smooth films. There is no peak in the emitted s light excited by incident p light. The anomalous light is thus always p polarized. Taking the difference between s and p crossed analyzerpolarizer curves, integrating over 0 and assuming it to be isotropic *, we find the strength of the light associated with the surface plasmons. This is shown in fig. 2 for a silver sample. As a func* The theory we have outlined in this article an extra factor
will give of about 2 in the $-integration.
101
Volume 1, number
OPTICS COMMUNICATIONS
3
-
July/August
1969
1.0 0.5
-0.5 2:
iI, 20
Fig. 1. Intensity
J..yYj& 40
30
of scatte:ed
50
60
radiation
70
80
90
versus
angle from the normal for a gold sample at two Lvavelengths. The plasmon excitation, the short wavelength R (4060 A) above the polarizer-analyzer - - - , crossed polarizer-analyzer.---- , calculated
large wavelength A (6328 A) is in a region of surface surface
plasmon
cutoff.
Parallel
of wavelength it has approximately the form of the extra roughness-induced absorbtivity. A simple theory which assumes that the plasmons are excited propagating in all directions, and are then scattered into the p direction to radiate, predicts the observed angular variation of the anomalous radiation very well. Details of the theory are described elsewhere [5]. With normally incident light, a plasmon with wave vector k will be excited with a strength depending only upon e. ek, where c and ekare the electric field polarization vectors for the incident light and the plasmon respectively. Let K be one of the wave vectors associated with the surface roughness (these are assumed to be isotropically distributed). Plasmons propagating on a rough surface may then be represented by a sum over the unperturbed states of plasmons on a smooth surface
tion
+k = $ CZ~Tekl exp (ik’ .r) with k’ = k +K, where degenerate perturbation 102
2
/ uk7 1 , using first order theory,
follows
a
Lorentzian
distribution
F(K)(e’ek)2(ek. In , $2 CL~_. _____~.____ k ( Ldk-tik,)2
ek’)2 _. + r2
peaking at /k’ j = /k ’ with a width I. Wk and tik 1 are the unperturbed plasmon energies. If we take the scattering strength proportional to the amplitude of the surface roughness, F(hf) will be proportional to the Fourier transform of the autocorrelation coefficient of the height variation of the surface. The intensity of the radiation in the observation plane at an angle B to the normal Will be proportional to :a& I2 with k’ in the p direction of magnitude (w/c) iin 0. If we assume that each component of the plasmon has the same ratio of longitudinal to normal electric fields as that of a plasmon on a smooth surface, then the angular variation of the radiation in the p direction will vary as
OPTICS COMMUNICATIONS
Volume 1, number 3
co.52 e
dZ(Q) ~ cc
da
1(c -
sin2 Q)+ + E ~0s 0 I2
x{@~jE-sin2QI - 20,0,,
sin 0
sin’
+ (C!/IEI)
Q
1(E - sin2 0)/e 1”) p
where ?r/2 P=
J
(sin $ + 1E (wk). E(W~T)I-+~ F(K) cos2qJ
-n/2
(wk-wk’)
2
+r
do
2
and K2 =
(:)’[(2s
+sin’Q
-2&sin6sin@
I
w/c
.
This expression is the sum of radiation from longitudinal and normal currents. c,, and CJ~are the conductivities parallel and normal to the surface (we allow for the possibility that these may be different). To evaluate this expression it is necessary to know F(K) in the region from about (10 11)-l to 2w/c. Information on the form of F(K) can be found from the scattered light, because it too is determined by F(K) in the region up to 1K 1 = W/C [4]. A Gaussian distribution for F(K) fits the scattered light reasonably well, the width of the Gaussian corresponding to a widthOof the autocorrelation function of about 1650 A. F(K) is thus rather flat. dI/dS1 has been evaluated using the same Gaussian, treating r as a parameter. The calculated curve is shown in fig. 1. It has been normalized to the experimental curve at
I
z t .75
I .50
Api
.25
Fig. 2. Variation of the anomalous radiation with wavelength for a silver sample (curve 1) and the difference in absorbtivity between rough and smooth samples (curve 2). &.,, is the surface plasmon cutoff wavelength.
July/August 1969
0 = 60°. We have taken oL/c,, = 1.25 although the shape is not sensitive to this ratio. We obtain a good fit to the experiment with r/wk = 0.37. The theory we have outlined is different from those which Stern [6] and Wilems and Ritchie [7] have developed to explain the radiation by transverse plasmons on thin films. Stern has considered only coherent (resonance) radiation while Wilems and Ritchie have given a quantum mechanical description of the radiation by the normal plasmons currents. We must consider the possibility that our observed anomaly might be due to coherent rather than incoherent radiation. Resonance radiation from plasmons on thick films will be purely p polarized for p polarized incident light. For s polarized incident light there will be a small coherent p component due to the normal currents, having much the same angular variation as we observe [6]. This component will be roughly l/ 1E 1 smaller than that emitted in the s direction by the longitudinal currents. We have found the ratio of intensities for the two directions of the analyzer to be in fact about 1E 1. Thus if the observed anomaly were all coherent radiation, this would imply that all the light found with the polarizer and analyzer parallel would be coherent, and so the true scattered light would be very small. This can be ruled out, since we have measured the scattered intensity above the maximum plasmon frequency where there are no anomalies [5], and an extrapolation into the surface plasmon region shows that at least 2/3 of the parallel polarizer-analyzer light is not associated with the plasmons. It is also important to distinguish this surface plasmon radiation from the effects recently considered by Berreman [a]. Berreman has shown that pits and bumps much smaller than the wavelength of light produce a coherent distribution to the specularly reflected light, through dipoles induced across the pits and bumps by the incident electric field. These presumably will also produce coherent and incoherent contributions at other angles. Radiation from modes of this type will have the same polarization as that of the incident light, and will not be seen in the opposite polarization. Otto [9], in a different experiment involving rough surfaces, has also recently observed p polarized surface plasmon radiation using s incident light, this radiation being about half as strong as found using p incident light (roughly the ratio we observe). He has proposed another mechanism for the cross polarization coupling. He suggests that plasmons in the p direction are 103
Volume
1, number 3
OPTICS COMMUNICATIONS
excited by the component of the scattered light which has its polarization rotated from the s to the p direction in the scattering process. However, we have found the depolarization of the scattered light to be typically only lOTo, suggesting a much smaller ratio than 1:2 between the surface plasmon radiation with the two incident polarizations. We suggest that our mechanism provides a more satisfactory explanation for his results. Schrijder [lo] also has observed a scattering of light from very rough surfaces, 400 A rms compared with the 50 A rms of our experiments, which he has assigned to surface plasmons, though he finds no polarization dependence. With surfaces as rough as these, Schrijder may well be seeing dipole field effects.
104
July/August 1969
REFERENCES [l] P. Dobberstein
et al., J. L. Stanford et al., and S. E. Schnatterly, reporting at the 2nd Intern. Vacuum Ultraviolet Conf. Gatlinburg, Tennessee (1968). [Zf J. Brambring and H. Raether, Z. Physik 199 (1967) 118; P. E. Schreiber, Z. Physik 211 (1968) 257; E. Kretschmann and H. Raether, 2. Naturforsch, 23 (1968) 2135. [3] D.Beaglehole, Phys. Rev. Letters 22 (1969) 708. 141 H.Davis, Proc. Inst. Elec. Engrs. 101 (1954) 209. [5] D.Beaglehole and 0. Hunderi, Univ. of Maryland. Tech. Rept. no. 999, to be published. [6] E.A.Stern, Phys. Rev. Letters 19 (1967) 1321. [7] R.E.Wilems and R.H.Ritchie, Phys. Rev. Letters 19 (1967) 1325. [S] D. W.Berreman, Phys. Rev. 163 (1967) 855. [9] A.Otto, Z. Physik 224 (1969) 65. [lo] E.Schroder, Opt. Commun. 1 (1969) 13.
1, number
July/August
OPTICS COMMUNICATIONS
3
RADIATION
FROM ON
SURFACE
ROUGH
PLASMONS
METAL
1969
PROPAGATING
SURFACES
0. HUNDERI and D. BEAGLEHOLE Department
of Physics College
Revised
and Astronomy, Park.
Maryland
Uniawrsity 20742.
of Maryland
USA
Received 9 June 1969 version received 15 July 1969
Radiation from surface plasmons excited by light normally incident onto thick rough metal samples is reported. Its intensity peaks at about the same wavelength as the extra absorbtivity induced by the surface plasmons. Its polarization properties indicate that it is most likely incoherent rather than resonance radiation. We present a simple theory which gives good agreement with the experimental results.
Surface plasmons are evanescent electromagnetic waves which can propagate along the surface of a metal whose dielectric constant E is less than - I. Their n field is transverse. Their E field has both longiqdinal and normal components, in the ratio 1E 1P. They are dominantly longitudinal when E << - 1. Their wave vector in the surface is greater than w/c, and so they cannot be excited by light falling onto a smooth surface. However, an anomalous decrease in the reflectivity of rough metal surfaces has been observed, and this has been assigned to the excitation of surface plasmons [I]. We have observed radiation from the decay of these surface plasmons into photons. The roughness provides the necessary momentum both for the coupling of the incident light into the plasmon mode, and for its subsequent decay. The polarization properties of the radiation we observe suggest that it is incoherent rather than resonance radiation. It is thus different from the coherent resonance radiation reported from plasmons on thin films [2], but similar to the incoherent radiation which has recently been observed [3] from surface plasmons propagating on a metal grating. Our experiment has been performed by shining polarized light normally onto rough metal surfaces, and measuring the distribution of light coming from the sample as a function of the angle .Ofrom the normal. The incident light was either p or s polarized with respect to the observation plane. Rough samples were produced by the technique of Stanford et al. [l] by deposition of a thin CaF2 layer onto a quartz substrate,
evaporating the metal layer on top. Several metals were studied, particularly Ag and +u. Their thicknesses were greater than 1000 A. Most of the light detected at off-normal angles was scattered light, whose distribution followed approximately that of the scalar scattering theory [4]. In fact the distribution was found to depend somewhat upon the incident polarization, the p scattering being greater than the s. However, this difference was also found above the surface plasmon cutoff frequency (E > - l), and with smooth samples, and so was not associated with the surface plasmons. Examples are shown in fig. 1 for a gold sample. When we placed an analyzer in front of the detector and measured only the light with polarization opposite to that incident, we found a significant anomaly associated with the surface plasmons. This is shown in fig. 1. The peak in the emitted p polarized light excited by incident polarized light disappeared above the maximum surface plasmon frequency, and was not found with smooth films. There is no peak in the emitted s light excited by incident p light. The anomalous light is thus always p polarized. Taking the difference between s and p crossed analyzerpolarizer curves, integrating over 0 and assuming it to be isotropic *, we find the strength of the light associated with the surface plasmons. This is shown in fig. 2 for a silver sample. As a func* The theory we have outlined in this article an extra factor
will give of about 2 in the $-integration.
101
Volume 1, number
OPTICS COMMUNICATIONS
3
-
July/August
1969
1.0 0.5
-0.5 2:
iI, 20
Fig. 1. Intensity
J..yYj& 40
30
of scatte:ed
50
60
radiation
70
80
90
versus
angle from the normal for a gold sample at two Lvavelengths. The plasmon excitation, the short wavelength R (4060 A) above the polarizer-analyzer - - - , crossed polarizer-analyzer.---- , calculated
large wavelength A (6328 A) is in a region of surface surface
plasmon
cutoff.
Parallel
of wavelength it has approximately the form of the extra roughness-induced absorbtivity. A simple theory which assumes that the plasmons are excited propagating in all directions, and are then scattered into the p direction to radiate, predicts the observed angular variation of the anomalous radiation very well. Details of the theory are described elsewhere [5]. With normally incident light, a plasmon with wave vector k will be excited with a strength depending only upon e. ek, where c and ekare the electric field polarization vectors for the incident light and the plasmon respectively. Let K be one of the wave vectors associated with the surface roughness (these are assumed to be isotropically distributed). Plasmons propagating on a rough surface may then be represented by a sum over the unperturbed states of plasmons on a smooth surface
tion
+k = $ CZ~Tekl exp (ik’ .r) with k’ = k +K, where degenerate perturbation 102
2
/ uk7 1 , using first order theory,
follows
a
Lorentzian
distribution
F(K)(e’ek)2(ek. In , $2 CL~_. _____~.____ k ( Ldk-tik,)2
ek’)2 _. + r2
peaking at /k’ j = /k ’ with a width I. Wk and tik 1 are the unperturbed plasmon energies. If we take the scattering strength proportional to the amplitude of the surface roughness, F(hf) will be proportional to the Fourier transform of the autocorrelation coefficient of the height variation of the surface. The intensity of the radiation in the observation plane at an angle B to the normal Will be proportional to :a& I2 with k’ in the p direction of magnitude (w/c) iin 0. If we assume that each component of the plasmon has the same ratio of longitudinal to normal electric fields as that of a plasmon on a smooth surface, then the angular variation of the radiation in the p direction will vary as
OPTICS COMMUNICATIONS
Volume 1, number 3
co.52 e
dZ(Q) ~ cc
da
1(c -
sin2 Q)+ + E ~0s 0 I2
x{@~jE-sin2QI - 20,0,,
sin 0
sin’
+ (C!/IEI)
Q
1(E - sin2 0)/e 1”) p
where ?r/2 P=
J
(sin $ + 1E (wk). E(W~T)I-+~ F(K) cos2qJ
-n/2
(wk-wk’)
2
+r
do
2
and K2 =
(:)’[(2s
+sin’Q
-2&sin6sin@
I
w/c
.
This expression is the sum of radiation from longitudinal and normal currents. c,, and CJ~are the conductivities parallel and normal to the surface (we allow for the possibility that these may be different). To evaluate this expression it is necessary to know F(K) in the region from about (10 11)-l to 2w/c. Information on the form of F(K) can be found from the scattered light, because it too is determined by F(K) in the region up to 1K 1 = W/C [4]. A Gaussian distribution for F(K) fits the scattered light reasonably well, the width of the Gaussian corresponding to a widthOof the autocorrelation function of about 1650 A. F(K) is thus rather flat. dI/dS1 has been evaluated using the same Gaussian, treating r as a parameter. The calculated curve is shown in fig. 1. It has been normalized to the experimental curve at
I
z t .75
I .50
Api
.25
Fig. 2. Variation of the anomalous radiation with wavelength for a silver sample (curve 1) and the difference in absorbtivity between rough and smooth samples (curve 2). &.,, is the surface plasmon cutoff wavelength.
July/August 1969
0 = 60°. We have taken oL/c,, = 1.25 although the shape is not sensitive to this ratio. We obtain a good fit to the experiment with r/wk = 0.37. The theory we have outlined is different from those which Stern [6] and Wilems and Ritchie [7] have developed to explain the radiation by transverse plasmons on thin films. Stern has considered only coherent (resonance) radiation while Wilems and Ritchie have given a quantum mechanical description of the radiation by the normal plasmons currents. We must consider the possibility that our observed anomaly might be due to coherent rather than incoherent radiation. Resonance radiation from plasmons on thick films will be purely p polarized for p polarized incident light. For s polarized incident light there will be a small coherent p component due to the normal currents, having much the same angular variation as we observe [6]. This component will be roughly l/ 1E 1 smaller than that emitted in the s direction by the longitudinal currents. We have found the ratio of intensities for the two directions of the analyzer to be in fact about 1E 1. Thus if the observed anomaly were all coherent radiation, this would imply that all the light found with the polarizer and analyzer parallel would be coherent, and so the true scattered light would be very small. This can be ruled out, since we have measured the scattered intensity above the maximum plasmon frequency where there are no anomalies [5], and an extrapolation into the surface plasmon region shows that at least 2/3 of the parallel polarizer-analyzer light is not associated with the plasmons. It is also important to distinguish this surface plasmon radiation from the effects recently considered by Berreman [a]. Berreman has shown that pits and bumps much smaller than the wavelength of light produce a coherent distribution to the specularly reflected light, through dipoles induced across the pits and bumps by the incident electric field. These presumably will also produce coherent and incoherent contributions at other angles. Radiation from modes of this type will have the same polarization as that of the incident light, and will not be seen in the opposite polarization. Otto [9], in a different experiment involving rough surfaces, has also recently observed p polarized surface plasmon radiation using s incident light, this radiation being about half as strong as found using p incident light (roughly the ratio we observe). He has proposed another mechanism for the cross polarization coupling. He suggests that plasmons in the p direction are 103
Volume
1, number 3
OPTICS COMMUNICATIONS
excited by the component of the scattered light which has its polarization rotated from the s to the p direction in the scattering process. However, we have found the depolarization of the scattered light to be typically only lOTo, suggesting a much smaller ratio than 1:2 between the surface plasmon radiation with the two incident polarizations. We suggest that our mechanism provides a more satisfactory explanation for his results. Schrijder [lo] also has observed a scattering of light from very rough surfaces, 400 A rms compared with the 50 A rms of our experiments, which he has assigned to surface plasmons, though he finds no polarization dependence. With surfaces as rough as these, Schrijder may well be seeing dipole field effects.
104
July/August 1969
REFERENCES [l] P. Dobberstein
et al., J. L. Stanford et al., and S. E. Schnatterly, reporting at the 2nd Intern. Vacuum Ultraviolet Conf. Gatlinburg, Tennessee (1968). [Zf J. Brambring and H. Raether, Z. Physik 199 (1967) 118; P. E. Schreiber, Z. Physik 211 (1968) 257; E. Kretschmann and H. Raether, 2. Naturforsch, 23 (1968) 2135. [3] D.Beaglehole, Phys. Rev. Letters 22 (1969) 708. 141 H.Davis, Proc. Inst. Elec. Engrs. 101 (1954) 209. [5] D.Beaglehole and 0. Hunderi, Univ. of Maryland. Tech. Rept. no. 999, to be published. [6] E.A.Stern, Phys. Rev. Letters 19 (1967) 1321. [7] R.E.Wilems and R.H.Ritchie, Phys. Rev. Letters 19 (1967) 1325. [S] D. W.Berreman, Phys. Rev. 163 (1967) 855. [9] A.Otto, Z. Physik 224 (1969) 65. [lo] E.Schroder, Opt. Commun. 1 (1969) 13.