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The angular dependence and the polarisation of light emitted by surface plasmons on metals due to roughness
Volume 5, number 5
OPTICS COMMUNICATIONS
THE ANGULAR
DEPENDENCE
August 1972
AND THE POLARISATION
O F L I G H T E M I T T E D BY S U R F A C E P L A S M O N S O N M E T A L S D U E T O R O U G H N E S S
E. KRETSCHMANN Institut ffir Angewandte Physik der Universitiit Hamburg, Hamburg, Germany Received 3 May 1972 Surface plasmons on a metal surface excited by light can radiate light via surface roughness or inhomogeneities inside the metal ("interior roughness"). The angular dependence and the polarisation of this surface plasma radiation are calculated. It is shown that by measuring the angular dependence of the radiation it can be decided whether surface roughness or bulk inhomogeneities of the dielectric constant produce this radiation.
1. Introduction It is known that the electromagnetic surface plasma waves (SPW) on the boundary metal-air may radiate light with the help of surface roughness. This is valid for the radiative and the non-radiative SPW. The idea of the electromagnetic scattering theory which describes the interaction of electromagnetic waves with surface roughness is as follows. The electromagnetic wave generates on the rough boundary metal-vacuum polarisation currents additional to those on a smooth surface. These extra currents are the sources of the emitted light. That means that in a first approximation the rough surface is replaced by a smooth one and by an additional layer of dipole currents whose amplitude is proportional to the height of the roughness. It was a characteristic difficulty of the earlier theory to decide into which medium these dipole sources have to be placed, if there are polarisation currents which are directed normal to the flat boundary. The polarisation currents paralleFto the flat boundary do not meet this difficulty. In the original theory [1 ] the currents were placed half into the vacuum half into the metal, in ref. [2] completely into the metal*. It has been shown theoretically [4] that in the case of surface roughness the sources have to be placed into the vacuum at the flat boundary to satisfy the boundary conditions in a first approximation. However it may be that the light emission is due to an interior roughness, i.e., small inhomogeneities of the dielectric constant within the metal. In this case the dipoles oscillating perpendicular to the boundary, have to be placed into the metal. The consequences o f the two different assumptions for which we shall use the abreviations SST (surface scattering theory, current sources in the vacuum) and VST (volume scattering theory, current sources in the metal) shall be discussed in this paper. We shall show that there are significant differences in the angular pattern and polarisation of the emitted light in these two cases. At first we shall give the general formulas for the scattering of plane waves on a thin absorbing film which is covered on both sides with different materials. Then we specialise the formulae for the case of exciting SPW and derive approximate formulae. * A quantum mechanical scattering theory [3] starts from static fluctuations of the density of the electron gas in the metal. The results are quite similar to those that can be received by the classical "current" theory which places the sources into the metal.
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August 1972
2. Scattering on a two boundary system in transmission
2.1. Scattering from surface roughness A thin film with the dielectric constant e 1 = e = e r + ie i is bounded on one side by a non-absorbing material e 2 on the other side by vacuum. Parallel light is incident upon the boundary film-material at the angle 0 0 with the z axis. We calculate the intensity of the light which is scattered from the rough boundary foil-vacuum into the vacuum under the angle 0. We limit ourselves to the case where the incident light is polarised parallel to the plane of incidence. Only in this case normal currents exist and a difference between SST and VST results. The geometry used for the calculation is given in fig. 1. z = O: smooth surface, z = S(x,y): height of the roughness. For the scattered power d l per solid angle element d ~ and per incident power I 0 the SST [4] yields dI 4(e2)1/2 rr4 12 [2 I0d~2 - c o s 0 0 )k4 Itp(00) IWI2 ISk_ko •
(1)
)~ is the wavelength in vacuum. The transmission function tp(00) gives the magnetic field strength of the incident light at the boundary foil-vacuum if the incident field strength is set equal to 1 in the dielectric e 2 (see, e.g., [5]). IW 12 indicates the radiation pattern of a single dipole on the boundary foil-vacuum, generated by the incident light. IWI 2 is given by IWl 2 = l e - I 12 i ( % t cos~ + Wpn ) sin ff + Wst sin~b cos ff 12
(2)
with Wpt = cos 0 (l - e 2 sin 200)1/2 [1 -rp(O)] , Wpn = sin O(e2)]/2sin 00 [1 +rp(O)]/e , Wst
= (1 - e 2 sin 200)~a[1 + rs(O)] .
r_ and r s are the reflexion coefficients [5] for TM (p-polarised) and TE (s-polarised) plane waves which fall from tee vacuum side on the film. As can be seen in eq. (2) IW 12 results from the interference of three terms, the radiation of a normal (Wpn), a tangential p-polarised (Wpt) and a tangential s-polarised (Wst) dipole on the surface of the foil. This would correspond in fig. 1 to a dipole parallel to the z, x, a n d y axis. ISk_ k h2, the "roughness spectrum", is the absolute square of the Fourier transform of the roughness function S(x,y~
E incident
di~ic
o
~
Y x'
x
ligh
1 $..~'7
~z= Six, y)
foil
332
t
vocuum
scattered
light
Fig. l. Geometry of the scattering experiment. Parallel light is incident upon a two boundary system (x, y plane) at an angle 0o relative to the surface normal (z axis). The light is polarised parallel to the plane of incidence (x', z plane). The scattered light from the rough surface foil--vacuum [z = S(x, y)] is observed in the plane x, z at an angle 0 with the z axis and at a variable polarisation angle qJ. ~ = 0 means s-polarisation, = 90° p-polarisation. The angle between the plane of incidence and the plane of observation is q~.ko and k are the components of the k-vectors of the incident and the scattered light along the surface.
Volume 5, number 5
OPTICS COMMUNICATIONS
fs(x,y) exp [ - ( k -
1 IF ISk-ko 12- F1 (2~.)2
August 1972
k0) (x, y)] dx dy 12 .
(3)
In SST, S(x, y) indicates the height of the rough surface. On an isotropic rough surface Is 12 is a function of I k - k 0 I = (27r/X) (e 2 sin 20 0 + sin 20 - 2e 2 sin 0 0 sin 0 cos q~)v2 .
(4)
At a statistically rough surface Is 12 is unknown. Often a gaussian type of the function Is 12 is assumed [1,4]. Other forms are discussed in [4]. 2.2. Scattering from interior roughness If both boundaries are smooth but there is a small statistical variation of the optical constants within the foil
e(x,y,z)=e+6e(x,y,z);
8 e ' ~ lel
one can derive a scattering formula which is very close to eq. (1) [4]. However two differences appear: The dipole pattern IW 12 is changed into IW' 12 by the fact that the current sources are placed inside the metal on the boundary foil-vacuum. Therefore Wpn which is caused by a dipole directed normal to the foil, is reduced by a factor 1/e. Wpt and Wst remain unchanged:
W'pn =(I/6) Wpn ,
t
Wpt = Wpt ,
~ r
st Wst •
(5)
The roughness function S(x,y) becomes the integral d
S(x,y) = f 0
dz
5e(x,y,z) e ~ ~ ~o(0O, O, z),
(6)
-
with a phase factor ~0which is rather complicated in the general case assumed in eq. (1). In the case of a thick metal foil in eq. (12) an approximate formula for S(x,y) is given.
3. Scattering of light if surface plasmons are excited Non-radiative SPW are excited in the boundary foil-vacuum, if the foil contains a plasma (Re e < 1, Im e '~ IRe e I) and the condition . (Ree(~,) 1)v2 00 = 00n = arcsm ~ e ( - ' ~ 1 e2 ' is fulfilled. The foil thickness has to be, in the case of silver, 6 0 0 - 1 0 0 0 A in order to observe a pronounced resonance peak in the emitted light [6, 7]. It may be of interest that the coefficient ] tp(0o) 12 which indicates the generation of SPW becomes ~ 80 in the resonance case (Ag, d = 600 A, ~. = 6000A, e 2 = 2.15). In order to discuss the angular and polarisation dependences of the scattered light in VST and SST, we shall treat approximate formulae for IW 12 and IW' 12. We shall neglect the influence of damping and the t'mite foil thickness and we assume that surface plasmons are excited. That means the formulae below are valid for Ime~
IRe el,
d~X/27r(lel) ~/2 ,
Ree<-I
,
00=0 ~ .
(7)
333
Volume 5, number 5
OPTICS COMMUNICATIONS
August 1972
In the case of a silver foil the conditions (7) are fulfilled if X > 4000 A and d > 500 A. Then the a p p r o x i m a t e formulae given below are very close to the exact ones, eq. (2). We shall treat them for two cases: (1) plane of incidence equal to plane o f observation (0 = 0 °) and (2) plane of incidence perpendicular to plane o f observation (~ = 90°). (1) Plane of incidence equal to plane of observation, 0 = 0 °. I W 12
=A(O, l e I) sin 2ff
I W' 12 =
A(O, l e I) sin 2ff A(O,le
[(1 + sin 2 0 / l e l ) 1 / : - sin 0] 2
(8)
[( 1 + sin 2 0 / l e l ) l / 2 + sin 0 l i e I] 2 ,
(9)
I)- lel+ 1 iel-1
4 1 +tan20/
el
The values with a dash signify volume scattering. Here I W 12 and I W'I 2 result from the interference o f the radiation of a tangential p-polarised (first term) and a n o r m a l dipole on the b o u n d a r y . Radiation is only observed for ff > 0 °. The factor A (0, l e I) reduces the e m i t t e d intensity for great angles of observation. In fig. 2a the functions IWI 2 and IW' 12 are given for a 700 A Ag film at X = 5000 A (e = 9.4 + i 0.4) calculated from eq. (2) and (for comparison) from eqs. (8) and (9). The " e x a c t " formulas and the a p p r o x i m a t e ones give nearly the same values. The angular pattern is quite different in SST and VST. This is a consequence of the fact that in a metal (Re e < - 1) Won and W" n have different magnitudes and different signs due to the factor 1/e. Due to the inter. V • O ference m e n t i o n e d above I WI2 zs much smaller for 0 > 0 (forward direction) than for 0 < 0 °. (I W ( - 6 0 ° ) l 2 ~ 120 I W(+60°)l 2 at e = - 10.) I W'(0)I 2 is nearly symmetrical to 0 = 0 °. (2) Plane of incidence perpendicular to plane o f observation, ~b = 90 °. :go ° ~--90"
S
........ ~
(p-pot)
z ~e,o"
.,°.\ SST
zle:O°
/ .='°° q) = 0 ° (s-poll
i:;
_
(a) (b) Fig. 2. Polar diagram of 114/(0)12 and l ltg:(0)f 2, the radiation pattern of a dipole on the boundary silver foil-vacuum, generated by SPW. A 700 A Ag foil and h = 5000 A is assumed. - - (SST) and - - - (VST) is calculated fromeqs. (2) and (5) . . . . from the approximate eqs. (8) and (9). (a) The k o vector Of SPW lies in the - x direction (in the plane of observation). The inteference of the radiation pattern of a normal and a tangential p-polarised dipole becomes effective. (b)/co lies perpendicular to the plane of observation x, z. The radiation pattern of the tangential s-polarised (ff = 0 °) and the radiation pattern of a normal dipole (qJ = 90 °) become effective.At ~ = 0°,1WI2 and IW'[2 are equal; at qJ = 90°,lW'12=(1/lel~)lWIz = (1/88)1WI z.
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August 1972
IWI 2 =A(O, lel)lsin 0 sin ff + c o s 0 cos ff [(1 +sin20/lel) 1/2- isin0 t a n 0 / l e l ] 12 ,
(10)
[W' [2 = A(O, le I) I(-sin 0/le l) sin ff + cos 0 cos ff [(1 +sin 20/le 1)1/2-i sin 0 tan 0/le I] [2 .
(11)
I W 12 and I W' 12 result from the interference of a normal and a tangential s-polarised dipole. If ~b = 0 ° (the scattered light is s-polarised) only the radiation of the tangential dipole is effective, if ff = 90 ° (the scattered light is p-polarised) only the radiation of the normal dipole is effective. The angular pattern from the tangential dipole is seen to be equal in SST and VST; however the emitted intensity from normal dipoles is reduced by the factor [ 1/e 12 in VST with respect to SST. In fig. 2b the 0-dependence of IWI 2 and IW' [2 for ff = 0 ° and ~ = 90 ° is given for a 700 )k Ag film. With the assumptions (7) we can simplify the roughness function in VST, if we assume fie(x, y, z) to be nearly independent o f z for z < ~/4rr(lel) 1/2 (penetration depth). One gets
X 1 fie(x,y, O) ( S(x'Y)-27r2(lel)'/2 lel+l 1
cos 2_0] 81el ] '
(12)
Now S(x, y) is proportional to the local variation of fie.
4. D i s c u s s i o n
We shall now discuss the polarisation I(~0) and angular I(0) dependences of the scattered light intensity in SST and VST if SPW are excited. (a) The polarisation dependence I(q/) is given by the dipole pattern IW 12 or IN' 12. The roughness spectrum ISk_ko 12 is independent of qJ. At ~ = 0 ° in VST and in SST eq. (8) and eq. (9) yield I ( ~ ) "~ sin 2~. At qJ = 0 ° (s-pol) the scattered intensity is zero. At ~b= 90 ° the polarisation dependence in SST and in VST is quite different. From eq. (10) and eq. (11) one derives ( le I >> 1): l(q/) ~ cos 2(ff - 0 ) (SST) and I(ff) ~ cos 2 q/(VST). So by comparing the measured ~k-dependence of the scattered light at 0 4:0 ° with the theoretical one, one should be able to decide if SST or VST is applicable. (b) The angular dependence 1(0) isgiven by the product of the dipole pattern function IW 12 and the roughness spectrum. At ¢ = 0 °, Ik - k 0 I varies strongly with the observation angle 0 [see eq. (4)]. That means a strong variation of , k ~ I2 with 0 is probable too. Consequently the 0-dependence of the scattered light can be quite different from ~s IW 12 given in eqs. (8) and (9) or fig. 2a. If one wants to determine the roughness spectrum Is 12 from the measured angular dependence it is necessary to know if VST or SST is applicable*. At ~ = 90 ° the variation of I k - k 01 with 0 is not very strong. Therefore the 0-dependence of the scattered light should be similar to that of IW 12 eqs. (10), (11) and fig. 2b. When the roughness spectrum is unknown we cannot decide from the measured 0 dependence whether VST or SST is correct. But the ratio of s- and p-polarised scattering Ip/I s is independent of Is 12 and should be another good test function for SST and VST. From eqs. (10) and (11) one derives Ip/I s = tan 20/(1 +tan 20/le I) (SST) and Ip/I~ = (1/le 12) Ip/I s (VST). For metals at long wavelength le l is often very large and the difference oflp/I s in SST and VST should be considerable. Recently we have made scattering experiments with SPW on thin silver films evaporated on quartz glass. The angular and polarisation dependence of the scattered pattern have been found to be in good agreement with the * We think this is not taken into consideration in ref. [8]. 335
Volume 5, number 5
OPTICS COMMUNICATIONS
August 1972
theoretical curves of SST. The dependences given by VST are never found. The experiments and the comparison with our theories will be published in a second paper.
Acknowledgement I am indebted to Professor Dr. H. Raether for his support of this work and for many discussions.
References [1] E.A. Stern, Phys. Rev. Letters 19 (1967) 1321; E. Kretschmann and H. Raether, Z. Naturforsch. 22a (1967) 1623. [2] O. Hunderi and D. Beaglehole, Phys. Rev. Letters 2 (1970) 321. [3] R.H. Ritchie, Phys. Stat. Sol. 39 (1970) 297; R.E. Wilemsand R.H. Ritchie, Phys. Rev. Letters 19 (1967) 1325. [4] E. Kroeger and E. Kretschmann, Z. Physik 237 (1970) 1. [5] H. Wolter, Handbuch der Physik, Vol. 24 (Springer, Berlin, 1956). [6] E. Kretschmann and H. Raether, Z. Naturforsch. 23a (1968) 2135; R. Bruns and H. Raether, Z. Physik 237 (1970) 98. [7] E. Kretschmann, Z. Physik 241 (1971) 313. 18] P. Dobberstein, Phys. Letters 31A (1970) 307.
336
OPTICS COMMUNICATIONS
THE ANGULAR
DEPENDENCE
August 1972
AND THE POLARISATION
O F L I G H T E M I T T E D BY S U R F A C E P L A S M O N S O N M E T A L S D U E T O R O U G H N E S S
E. KRETSCHMANN Institut ffir Angewandte Physik der Universitiit Hamburg, Hamburg, Germany Received 3 May 1972 Surface plasmons on a metal surface excited by light can radiate light via surface roughness or inhomogeneities inside the metal ("interior roughness"). The angular dependence and the polarisation of this surface plasma radiation are calculated. It is shown that by measuring the angular dependence of the radiation it can be decided whether surface roughness or bulk inhomogeneities of the dielectric constant produce this radiation.
1. Introduction It is known that the electromagnetic surface plasma waves (SPW) on the boundary metal-air may radiate light with the help of surface roughness. This is valid for the radiative and the non-radiative SPW. The idea of the electromagnetic scattering theory which describes the interaction of electromagnetic waves with surface roughness is as follows. The electromagnetic wave generates on the rough boundary metal-vacuum polarisation currents additional to those on a smooth surface. These extra currents are the sources of the emitted light. That means that in a first approximation the rough surface is replaced by a smooth one and by an additional layer of dipole currents whose amplitude is proportional to the height of the roughness. It was a characteristic difficulty of the earlier theory to decide into which medium these dipole sources have to be placed, if there are polarisation currents which are directed normal to the flat boundary. The polarisation currents paralleFto the flat boundary do not meet this difficulty. In the original theory [1 ] the currents were placed half into the vacuum half into the metal, in ref. [2] completely into the metal*. It has been shown theoretically [4] that in the case of surface roughness the sources have to be placed into the vacuum at the flat boundary to satisfy the boundary conditions in a first approximation. However it may be that the light emission is due to an interior roughness, i.e., small inhomogeneities of the dielectric constant within the metal. In this case the dipoles oscillating perpendicular to the boundary, have to be placed into the metal. The consequences o f the two different assumptions for which we shall use the abreviations SST (surface scattering theory, current sources in the vacuum) and VST (volume scattering theory, current sources in the metal) shall be discussed in this paper. We shall show that there are significant differences in the angular pattern and polarisation of the emitted light in these two cases. At first we shall give the general formulas for the scattering of plane waves on a thin absorbing film which is covered on both sides with different materials. Then we specialise the formulae for the case of exciting SPW and derive approximate formulae. * A quantum mechanical scattering theory [3] starts from static fluctuations of the density of the electron gas in the metal. The results are quite similar to those that can be received by the classical "current" theory which places the sources into the metal.
331
Volume 5, number 5
OPTICS COMMUNICATIONS
August 1972
2. Scattering on a two boundary system in transmission
2.1. Scattering from surface roughness A thin film with the dielectric constant e 1 = e = e r + ie i is bounded on one side by a non-absorbing material e 2 on the other side by vacuum. Parallel light is incident upon the boundary film-material at the angle 0 0 with the z axis. We calculate the intensity of the light which is scattered from the rough boundary foil-vacuum into the vacuum under the angle 0. We limit ourselves to the case where the incident light is polarised parallel to the plane of incidence. Only in this case normal currents exist and a difference between SST and VST results. The geometry used for the calculation is given in fig. 1. z = O: smooth surface, z = S(x,y): height of the roughness. For the scattered power d l per solid angle element d ~ and per incident power I 0 the SST [4] yields dI 4(e2)1/2 rr4 12 [2 I0d~2 - c o s 0 0 )k4 Itp(00) IWI2 ISk_ko •
(1)
)~ is the wavelength in vacuum. The transmission function tp(00) gives the magnetic field strength of the incident light at the boundary foil-vacuum if the incident field strength is set equal to 1 in the dielectric e 2 (see, e.g., [5]). IW 12 indicates the radiation pattern of a single dipole on the boundary foil-vacuum, generated by the incident light. IWI 2 is given by IWl 2 = l e - I 12 i ( % t cos~ + Wpn ) sin ff + Wst sin~b cos ff 12
(2)
with Wpt = cos 0 (l - e 2 sin 200)1/2 [1 -rp(O)] , Wpn = sin O(e2)]/2sin 00 [1 +rp(O)]/e , Wst
= (1 - e 2 sin 200)~a[1 + rs(O)] .
r_ and r s are the reflexion coefficients [5] for TM (p-polarised) and TE (s-polarised) plane waves which fall from tee vacuum side on the film. As can be seen in eq. (2) IW 12 results from the interference of three terms, the radiation of a normal (Wpn), a tangential p-polarised (Wpt) and a tangential s-polarised (Wst) dipole on the surface of the foil. This would correspond in fig. 1 to a dipole parallel to the z, x, a n d y axis. ISk_ k h2, the "roughness spectrum", is the absolute square of the Fourier transform of the roughness function S(x,y~
E incident
di~ic
o
~
Y x'
x
ligh
1 $..~'7
~z= Six, y)
foil
332
t
vocuum
scattered
light
Fig. l. Geometry of the scattering experiment. Parallel light is incident upon a two boundary system (x, y plane) at an angle 0o relative to the surface normal (z axis). The light is polarised parallel to the plane of incidence (x', z plane). The scattered light from the rough surface foil--vacuum [z = S(x, y)] is observed in the plane x, z at an angle 0 with the z axis and at a variable polarisation angle qJ. ~ = 0 means s-polarisation, = 90° p-polarisation. The angle between the plane of incidence and the plane of observation is q~.ko and k are the components of the k-vectors of the incident and the scattered light along the surface.
Volume 5, number 5
OPTICS COMMUNICATIONS
fs(x,y) exp [ - ( k -
1 IF ISk-ko 12- F1 (2~.)2
August 1972
k0) (x, y)] dx dy 12 .
(3)
In SST, S(x, y) indicates the height of the rough surface. On an isotropic rough surface Is 12 is a function of I k - k 0 I = (27r/X) (e 2 sin 20 0 + sin 20 - 2e 2 sin 0 0 sin 0 cos q~)v2 .
(4)
At a statistically rough surface Is 12 is unknown. Often a gaussian type of the function Is 12 is assumed [1,4]. Other forms are discussed in [4]. 2.2. Scattering from interior roughness If both boundaries are smooth but there is a small statistical variation of the optical constants within the foil
e(x,y,z)=e+6e(x,y,z);
8 e ' ~ lel
one can derive a scattering formula which is very close to eq. (1) [4]. However two differences appear: The dipole pattern IW 12 is changed into IW' 12 by the fact that the current sources are placed inside the metal on the boundary foil-vacuum. Therefore Wpn which is caused by a dipole directed normal to the foil, is reduced by a factor 1/e. Wpt and Wst remain unchanged:
W'pn =(I/6) Wpn ,
t
Wpt = Wpt ,
~ r
st Wst •
(5)
The roughness function S(x,y) becomes the integral d
S(x,y) = f 0
dz
5e(x,y,z) e ~ ~ ~o(0O, O, z),
(6)
-
with a phase factor ~0which is rather complicated in the general case assumed in eq. (1). In the case of a thick metal foil in eq. (12) an approximate formula for S(x,y) is given.
3. Scattering of light if surface plasmons are excited Non-radiative SPW are excited in the boundary foil-vacuum, if the foil contains a plasma (Re e < 1, Im e '~ IRe e I) and the condition . (Ree(~,) 1)v2 00 = 00n = arcsm ~ e ( - ' ~ 1 e2 ' is fulfilled. The foil thickness has to be, in the case of silver, 6 0 0 - 1 0 0 0 A in order to observe a pronounced resonance peak in the emitted light [6, 7]. It may be of interest that the coefficient ] tp(0o) 12 which indicates the generation of SPW becomes ~ 80 in the resonance case (Ag, d = 600 A, ~. = 6000A, e 2 = 2.15). In order to discuss the angular and polarisation dependences of the scattered light in VST and SST, we shall treat approximate formulae for IW 12 and IW' 12. We shall neglect the influence of damping and the t'mite foil thickness and we assume that surface plasmons are excited. That means the formulae below are valid for Ime~
IRe el,
d~X/27r(lel) ~/2 ,
Ree<-I
,
00=0 ~ .
(7)
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Volume 5, number 5
OPTICS COMMUNICATIONS
August 1972
In the case of a silver foil the conditions (7) are fulfilled if X > 4000 A and d > 500 A. Then the a p p r o x i m a t e formulae given below are very close to the exact ones, eq. (2). We shall treat them for two cases: (1) plane of incidence equal to plane o f observation (0 = 0 °) and (2) plane of incidence perpendicular to plane o f observation (~ = 90°). (1) Plane of incidence equal to plane of observation, 0 = 0 °. I W 12
=A(O, l e I) sin 2ff
I W' 12 =
A(O, l e I) sin 2ff A(O,le
[(1 + sin 2 0 / l e l ) 1 / : - sin 0] 2
(8)
[( 1 + sin 2 0 / l e l ) l / 2 + sin 0 l i e I] 2 ,
(9)
I)- lel+ 1 iel-1
4 1 +tan20/
el
The values with a dash signify volume scattering. Here I W 12 and I W'I 2 result from the interference o f the radiation of a tangential p-polarised (first term) and a n o r m a l dipole on the b o u n d a r y . Radiation is only observed for ff > 0 °. The factor A (0, l e I) reduces the e m i t t e d intensity for great angles of observation. In fig. 2a the functions IWI 2 and IW' 12 are given for a 700 A Ag film at X = 5000 A (e = 9.4 + i 0.4) calculated from eq. (2) and (for comparison) from eqs. (8) and (9). The " e x a c t " formulas and the a p p r o x i m a t e ones give nearly the same values. The angular pattern is quite different in SST and VST. This is a consequence of the fact that in a metal (Re e < - 1) Won and W" n have different magnitudes and different signs due to the factor 1/e. Due to the inter. V • O ference m e n t i o n e d above I WI2 zs much smaller for 0 > 0 (forward direction) than for 0 < 0 °. (I W ( - 6 0 ° ) l 2 ~ 120 I W(+60°)l 2 at e = - 10.) I W'(0)I 2 is nearly symmetrical to 0 = 0 °. (2) Plane of incidence perpendicular to plane o f observation, ~b = 90 °. :go ° ~--90"
S
........ ~
(p-pot)
z ~e,o"
.,°.\ SST
zle:O°
/ .='°° q) = 0 ° (s-poll
i:;
_
(a) (b) Fig. 2. Polar diagram of 114/(0)12 and l ltg:(0)f 2, the radiation pattern of a dipole on the boundary silver foil-vacuum, generated by SPW. A 700 A Ag foil and h = 5000 A is assumed. - - (SST) and - - - (VST) is calculated fromeqs. (2) and (5) . . . . from the approximate eqs. (8) and (9). (a) The k o vector Of SPW lies in the - x direction (in the plane of observation). The inteference of the radiation pattern of a normal and a tangential p-polarised dipole becomes effective. (b)/co lies perpendicular to the plane of observation x, z. The radiation pattern of the tangential s-polarised (ff = 0 °) and the radiation pattern of a normal dipole (qJ = 90 °) become effective.At ~ = 0°,1WI2 and IW'[2 are equal; at qJ = 90°,lW'12=(1/lel~)lWIz = (1/88)1WI z.
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August 1972
IWI 2 =A(O, lel)lsin 0 sin ff + c o s 0 cos ff [(1 +sin20/lel) 1/2- isin0 t a n 0 / l e l ] 12 ,
(10)
[W' [2 = A(O, le I) I(-sin 0/le l) sin ff + cos 0 cos ff [(1 +sin 20/le 1)1/2-i sin 0 tan 0/le I] [2 .
(11)
I W 12 and I W' 12 result from the interference of a normal and a tangential s-polarised dipole. If ~b = 0 ° (the scattered light is s-polarised) only the radiation of the tangential dipole is effective, if ff = 90 ° (the scattered light is p-polarised) only the radiation of the normal dipole is effective. The angular pattern from the tangential dipole is seen to be equal in SST and VST; however the emitted intensity from normal dipoles is reduced by the factor [ 1/e 12 in VST with respect to SST. In fig. 2b the 0-dependence of IWI 2 and IW' [2 for ff = 0 ° and ~ = 90 ° is given for a 700 )k Ag film. With the assumptions (7) we can simplify the roughness function in VST, if we assume fie(x, y, z) to be nearly independent o f z for z < ~/4rr(lel) 1/2 (penetration depth). One gets
X 1 fie(x,y, O) ( S(x'Y)-27r2(lel)'/2 lel+l 1
cos 2_0] 81el ] '
(12)
Now S(x, y) is proportional to the local variation of fie.
4. D i s c u s s i o n
We shall now discuss the polarisation I(~0) and angular I(0) dependences of the scattered light intensity in SST and VST if SPW are excited. (a) The polarisation dependence I(q/) is given by the dipole pattern IW 12 or IN' 12. The roughness spectrum ISk_ko 12 is independent of qJ. At ~ = 0 ° in VST and in SST eq. (8) and eq. (9) yield I ( ~ ) "~ sin 2~. At qJ = 0 ° (s-pol) the scattered intensity is zero. At ~b= 90 ° the polarisation dependence in SST and in VST is quite different. From eq. (10) and eq. (11) one derives ( le I >> 1): l(q/) ~ cos 2(ff - 0 ) (SST) and I(ff) ~ cos 2 q/(VST). So by comparing the measured ~k-dependence of the scattered light at 0 4:0 ° with the theoretical one, one should be able to decide if SST or VST is applicable. (b) The angular dependence 1(0) isgiven by the product of the dipole pattern function IW 12 and the roughness spectrum. At ¢ = 0 °, Ik - k 0 I varies strongly with the observation angle 0 [see eq. (4)]. That means a strong variation of , k ~ I2 with 0 is probable too. Consequently the 0-dependence of the scattered light can be quite different from ~s IW 12 given in eqs. (8) and (9) or fig. 2a. If one wants to determine the roughness spectrum Is 12 from the measured angular dependence it is necessary to know if VST or SST is applicable*. At ~ = 90 ° the variation of I k - k 01 with 0 is not very strong. Therefore the 0-dependence of the scattered light should be similar to that of IW 12 eqs. (10), (11) and fig. 2b. When the roughness spectrum is unknown we cannot decide from the measured 0 dependence whether VST or SST is correct. But the ratio of s- and p-polarised scattering Ip/I s is independent of Is 12 and should be another good test function for SST and VST. From eqs. (10) and (11) one derives Ip/I s = tan 20/(1 +tan 20/le I) (SST) and Ip/I~ = (1/le 12) Ip/I s (VST). For metals at long wavelength le l is often very large and the difference oflp/I s in SST and VST should be considerable. Recently we have made scattering experiments with SPW on thin silver films evaporated on quartz glass. The angular and polarisation dependence of the scattered pattern have been found to be in good agreement with the * We think this is not taken into consideration in ref. [8]. 335
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OPTICS COMMUNICATIONS
August 1972
theoretical curves of SST. The dependences given by VST are never found. The experiments and the comparison with our theories will be published in a second paper.
Acknowledgement I am indebted to Professor Dr. H. Raether for his support of this work and for many discussions.
References [1] E.A. Stern, Phys. Rev. Letters 19 (1967) 1321; E. Kretschmann and H. Raether, Z. Naturforsch. 22a (1967) 1623. [2] O. Hunderi and D. Beaglehole, Phys. Rev. Letters 2 (1970) 321. [3] R.H. Ritchie, Phys. Stat. Sol. 39 (1970) 297; R.E. Wilemsand R.H. Ritchie, Phys. Rev. Letters 19 (1967) 1325. [4] E. Kroeger and E. Kretschmann, Z. Physik 237 (1970) 1. [5] H. Wolter, Handbuch der Physik, Vol. 24 (Springer, Berlin, 1956). [6] E. Kretschmann and H. Raether, Z. Naturforsch. 23a (1968) 2135; R. Bruns and H. Raether, Z. Physik 237 (1970) 98. [7] E. Kretschmann, Z. Physik 241 (1971) 313. 18] P. Dobberstein, Phys. Letters 31A (1970) 307.
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