Surface Science 146 (1984) l-9

North-Holland. Amsterdam

1

SCATTERING OF p-POLARIZED LIGHT FROM Au AND Ag GRATINGS AND THE EFFECT OF SURFACE POLARITONS

G. DIAZ and N. GARCIA

Departamento de Fisica Fundamental, Uniuersidad Autbnomu de Madrid, Cantoblanco, Madrid-34.

Spain

and

H. RAETHER

Institut ftir Angavandte Physik, (inioersrtiit Hamburg, D 2000 Hamburg 36, Fed. Rep. of Germany

Received 2 May 1984; accepted for publication 26 June 1984

Light (p-polarized) reflected at silver and gold gratings shows at certain angles of incidence (resonance angles) a minimum of the reflected intensity, due to the excitation of surface plasmons

by coupling with the corrugation of the surface. The depth of the minimum is a sensitive function

of the amplitude h of a sinusoidal profile and it can become zero which indicates the maximum

excitation of surface plasmons or the maximum electro-magnetic field intensity in the surface.

With increasing h the minimum displaces to higher angles and its width increases, a consequence of

the influence of the corrugation of the surface on the dispersion relation. We present a theoretical

description of experimental observations published in the literature. The calculations are based on

the extinction theorem; it is shown that the details of the experimental results as the position of the

intensity minima, its variation with the amplitude h of the sinusoidal profile, the deformation of

the intensity minimum with increasing h, etc. are well reproduced. The calculations display

quantitative agreement by direct comparison of observed and theoretical results.

1. Introduction

One of the important aspects of Surface Plasmons (SP) is their coupling with light via roughness or via gratings, especially sinusoidal, which allows SP to be excited by light (grating coupler). If p-polarized light is reflected at a grating, this coupling can be observed by a minimum of the reflected light (zero order diffraction), if the wavevector of the SP,

K, = (w/c sin 0 + g),

fulfils the dispersion relation of SP (~9 the angle with the normal of the surface, w frequency of the light and g = n(27r)/a, with a the grating constant, and n

0039-6028/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

2 G. Dim et al. / Scattering of p - polarized light from Au and Ag

any integer number). In case of a sinusoidal grating the profile is described by

h(x) = h cos n(2nx)/a.

The experiments for h/a 5 0.1 have demonstrated [l] that the dispersion relation depends on the amplitude h of the sinus profile: the intensity mini- mum displaces to larger K, values (decrease of the phase velocity: w/K.,) and it broadens with increasing h value. This is due to the scattering of the SP in the surface (Rayleigh scattering), and to the strong damping produced by the emission of light into the different diffraction orders n. Similar results have been obtained on rough surfaces.

In the following we describe the characteristic changes which the minima undergo with varying h demonstrating the influence of the geometry of the surface on the dispersion relation

WI E’ ,1/2 i

K,=--I-----,

C / 1 +e’/

(E’ real part of the dielectric function of the metal), the dispersion relation of a smooth surface. We show that these observed modifications of the shape of the reflection minima can be calculated; one needs a computer program, since it is not possible to get analytical expressions; elsewhere the multiple scattering of the light at the corrugated surface cannot be easily handled.

Earlier calculations of the influence of a second harmonic on the shape of the reflection minimum using the Rayleigh method had reproduced very well the observed changes [2], the same is valid for the change of the position 0, and the width 8;” of the intensity minimum as function of h, the amplitude of the sinusoidal profile [3]. The calculation method has been improved in last time by using the extinction theorem [4,5], so that convergence difficulties have been reduced and calculations can be performed up to higher values of h/u.

This method has been applied in the following to compare experimental curves with calculated curves of the intensity reflected at a grating as function of the reflection angle 0. It appears that the observed details are very well reproduced by the calculations. The applicability of this method has been shown in ref. [S]. In this calculation we have used 40 plane waves for the expansion in the Green function and taken 250 points to perform the corre-

sponding integral in the grating. This is enough to obtain good convergence up to h/a = 0.25.

2. The reflection minimum dependent on the amplitude of a sinusoidal grating

2. I. Gold grating

The experiments on holographic (interference) gratings have shown that the depths of the minima of the reflected light passes a minimum as function of h

G. Diar et al. / Scattering of p -polarized light from Au and Ag 3

and can reach values equal to zero. This is a consequence of the matching between the interior damping yi (Joule heat) and the light emitted into the different diffraction orders n given by yrad which in the linear approximation (h -C 70 A) is proportional to h2 [6]; this matching is well known from different electromagnetic problems and has already been seen at the ATR minimum of smooth surfaces where the radiation damping is due to the light radiated by the surface plasmons back through the metal film into the prism.

Fig. 1 shows this phenomenon at a gold grating with a = 7965 A of different values of h irradiated with light of h = 6471 A. It displays the calculated data (full lines) in which observed data from ref. [3] are inserted. The amplitude h varies from 89 to 514 A. I0 means the relative reflected intensity or zero

I

0.8 :_

3

0.6

4 0.4

0.6 1

0.4

0.2

0.0

0.8

0.6

0.4

0.2

0.0 I I I

12 14 l6 OIdegl I8

Fig. 1. Observed (points) and calculated (full lines) intensity reflected at a sinusoidal gold (interference) grating with Q = 7965 A and X = 6471 A. I0 and I-’ are the intensities in the zero

and (- 1) diffraction order, I0 + I-t means the sum of both light intensities. These are relative intensities referred to the incoming intensity (I,,). The amplitude h of the profile is given as a parameter: (1) 89 A, (2) 174 A, (3) 308 A, (4) 514 ?L The deepest minimum occurs at h = 300 A, at which an enhancement of es - 59 and eH - 52 is calculated. The displacement of the minimum to higher angles and the increasing width of the minima with larger h display the dependence of the dispersion relation of SP of the amplitude h of the grating.

4 G. Dim et al. / Scattering of p-p&wired light front Au and Ag

diffraction order, I- ’ the ( - 1) diffraction order, and I” + I-’ the total light emitted from the grating. At B = 10.8” a change of the intensity takes place due to the disappearing (+ 1) diffraction order (Woods anomaly); its position is independent of h as expected, but the intensity step increases with h since the diffraction intensity increases with h’ as a first approximation. At larger 4 the reflected intensity goes through a minimum of nearly zero intensity given by yi = yrad at h = 300 A.

Here the resonance is maximum, so that the field intensity reaches high values; the enhancement (the square of the field divided by the square of the incident field), amounts to eE - 59 and 0, - 52 [5], which is higher than on a smooth surface. (The subscripts E and H are for electric and magnetic field, respectively.) At larger h values, the intensity I0 at larger angles reaches no more the intensity left of the minimum. Further the position of the minimum displaces to larger angles 0 (decrease of the phase velocity) and their width Q1j2 increases (increase of radiation damping). These details are very well reproduced by the calculated curves.

A similar behaviour, but less pronounced, is found in I-” (fig. 1). Observed and calculated values agree well, indicating, that the second harmonic in the grating profile is small so that the asymmetry of the intensities right and left of the minimum is rather weak. Such a contribution of a second harmonic can be produced by the fabrication of the photoresist gratings and leads to strong asymmetric curves [2].

If contributions of a second harmonic are present, the intensity minima change into intensity maxima accompanied by a strong asymmetry of the intensity right and left of the resonance position. These results come out in good agreement with the calculated data, using the Rayleigh method [2] as well as the extinction theorem [5].

The total reflected intensity (1’ + I- ‘) represents a very good reproduction of the observed data. Here the curve (1) of 1’ of fig. 1 has been suppressed since the intensity of I-’ is so small (- O.Ol), that it is identical with curve (1) in I*,

The full curves have been calculated with a dielectric function C(W) = r’f w) + it”(w), C’(W)= -9.75 and E”(W)= 1.1 for h = 6471 A. The values of C(W) taken from the literature [7] are somewhat larger: --C’(W)= 11.9-13.x and =E”(w) = 1.1-1.4; the agreement in fig. 1, however, is very good; probably the sensitivity to changes of the value of C(W) is small due to the relatively large l ” value of gold.

2.2. Silver gratings

A similar comparison can be performed with silver gratings of sinusoidal profile. Fig. 2 (bottom) shows the experimental results for I0 + I-’ with h as the parameter at a grating of a = 6015 A and at a wavelength X = 5145 A [8], whereas fig. 2 displays above the calculations for IO, I- ’ and I0 + I-‘. The

G. Dim et al. / Scattering of p -polarized light from Au and Ag 5

observed curve I0 + I-’ agrees with the calculated I0 + I-’ in all details. Since

the curves lie close together due to the small damping of the SP in silver, it was not possible to insert the measured data into the calculated curve. A character- istic feature at large h values (550 A) is the deformation of the minimum of I0

0.8 -

a 10 12 IL I6 O(degl ‘8

Fig. 2. The observed reflected intensity (I” + I- ‘) of a sinusoidal silver grating ((I = 6015 A) with

h, the amplitude of the sinus profile as a parameter, as a function of the angle of incidence (0); X = 5145 A. At h = 130 A the resonance is strongest leading to an enhancement of the electromag-

netic field intensity of about ef: - 231 and e,, - 184. The calculated intensity IO. I-’ and I”+ I-’ for the same grating and the same parameters h is given above. Values of h (in A): (_.__) 25; (_. _. _ .) 75; (- ) 135; (......) 215; (--.-.) 315; (----_) 550.

6 G. Dim et al. / Scattering of p -polarized light from Au and Ag

which is restored by adding I-r, so that the broad minimum in I0 + I-’ comes out, similarly as in fig. 1; see also ref. [6].

Apparently, the strong intensity of I-’ at larger h values modifies the minimum; the displacement of the resonance position to higher angles and the increase of the width of the resonance minimum is another striking feature of the curves. One obtains this displacement of the resonance only if one uses a

_o-o-’ o--o--

n=O

;r: n-1

0.3 t

n=2

0.0 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’

0.5 - x 10-2 n-L

60 55 60

Fig. 3. Calculated reflectivities (a), full lines. and experimental data (b), full lines. of different diffraction orders u. The grating parameters are h = 1200 k and u = 20140 A: X = $582 ,& Solid and open circles in (a) *I = 0 ate observed data. see text.

G. Dim ei al. / &altering of p -polarized light from Au and Ag 7

calculation which takes into account an approximation of at least second order, since the physical reason for this displacement is the scattering multiple

of SP in the surface [9]. The values of E(W) for X = 5145 A have been chosen as E’(W) = - 11 and

c”(w) = 0.33; the corresponding values of the literature are - 6’ = 10.3-10.5 and c” = 0.38. A slight displacement between the minima of observed and calculated curves of about 0.5” comes out.

Similar calculations applied to the silver grating in ref. [6]. Fig. 3, of a = 8910 A and X = 5500 A, yields the deepest minimum at h = 180 A and not

at 460 A as in ref. [6]; perhaps this discrepancy comes from an error of a factor 2, putting 2h instead of h in the referred figure. At h = 180 A the maximum enhancement of eE = 215 is calculated; see also fig. 2.

3. Reflection intensities of higher order

Some further examples of reflection curves at silver gratings will demon- strate that the calculations reproduce the higher order reflections.

Fig. 3 displays the calculated reflections h = 5682 A for n = 0 up to n = 4 for a sinusoidal grating with h = 1200 A and a = 20140 A. Full lines are calculated curves with c = - 13.2 + i0.49 [7]; for n = 0 the lower curve with a maximum at 8 = 50’ is meant. The solid circles are measurements for n = 0 [lo] and show - except for a discrepancy of about 1” in the position of the minima of the two curves - rather good agreement. The minimum is strongly deformed, similarly to that of I0 in fig. 1 (h = 514 A) and fig. 2 (h = 550 A). Comparing the calculated (fig. 3 left) with the observed (fig. 3 right) curves of the higher orders, one states a general agreement of the shape and the absolute values of the reflectivities. Fig. 3: n = 4 is an exception; here the intensities

differ by a factor of 2, but the intensity is already very small. The sharp peak visible at 8 = 45.9” is produced by the disappearance of the

order n = + 1 under the horizon. Summing up the intensities of all orders one obtains the full line in fig. 3

(n = 0, upper curve) as the calculated curve which agrees rather well with the open circles taken from the experiments; it is again a curve with a well developed minimum.

Another interesting case is represented in the series of measurements at h = 700 A at different orders with the same conditions as in fig. 4 [lo]. The observed data are indicated by open circles for the different orders, whereas the full lines display the calculated values. The curves for n = 0 agree rather well except a displacement of about 0.5” of the minimum. The different shapes compared to the curve n = 0 at h = 1200 A as in fig. 3 are remarkable. It is interesting that we find a well developed minimum still at h = 700 A for a = 20140 A or h/a = 3.5 X lo-* similar to fig. 2 at h = 315 A for a = 6015A

x G. DILIZ et al. / Stuttering of p - polurrred light from Au und Ag

or h/a = 5.2 x 10p2; the minimum is no more well developed at h = 1200 A for a=20140 (h/a=6x10p2) (fig. 3) and at h=550 A for a=6015 P\

(h/a = 9 x 10-l) (fig. 2). The agreement of the curves n > 2 is less satisfying, since the observed

minima are strongly deformed and maxima appear. This behaviour can be reproduced by adding to the sinus profile a second harmonic contribution h"' = - 84 A. A similar influence of the second harmonic on the structure of the reflected intensity has been found at gratings of smaller a values (a - 6000

A PI). In the experiments discussed here, the value of hc2' has not been measured,

since its influence was not yet known in ref. [lo]. The good agreement of calculated and observed curves is in favour of the assumed value of h(I),

because a small hc2' component can appear in the preparation of the grating.

55 0 Ideg)

Fig. 4. Calculated reflectivities of different orders n at a silver grating of h = 700 A. The left curve is calculated with a sinus profile, the right one is modified by adding a second harmonic

component of h”’ = - 84 I\; a = 20140 k and X = 5682 .k Open circles are observed data [IO].

G. Dmz et al. / Scattering of p -polarrred light from Au and Ag 9

Earlier measurements at commercial gratings with a saw tooth profile [II] showed, instead of intensity minima, strong maxima when the dispersion

relation was fulfilled. here also the influence of higher harmonics existing in the saw tooth profile is the reason of observing intensity maxima in the

reflected light instead of minima [2,3,5].

4. Conclusions

We have shown that the extinction theorem can be applied to calculate the light intensity reflected at gratings with a sinusoidal profile (and those with a second harmonic contribution). The shape and depth of the reflection mini- mum produced by surface plasmons at silver and gold surfaces is a function of the amplitude of the Fourier components of the grating. The calculated and observed dependencies are in good agreement. The dielectric function e(w) as known from literature has been used and reproduces the observed data better in silver that in gold.

Acknowledgement

This work has been supported in part by the Comision Asesora de Investi- gacion Cientifica y Tecnica (CAICYT) through contract No. 1426-82.

References

[I] For a review of these observations, see H. Raether, Surface Plasmons and Roughness, in:

Surface Polaritons, Eds. V.M. Agranovich and D.L. Mills (North-Holland, Amsterdam, 1982)

ch. 9, p. 331.

(21 E.H. Rosengart and I. Pockrand. Opt. Letters 1 (1977) 194.

[3] H. Raether. Opt. Commun. 42 (19X2) 217.

[4] F. Toigo, A. Marvin, C. Celli and N.R. Hill, Phys. Rev. B15 (1977) 5618.

[5] N. Garcia, Opt. Commun. 45 (1983) 307; J. Electron Spectrosc. 29 (1983) 421:

N. Garcia. G. Diaz, J.J. Saenz and C. Ocal. Surface Sci., submitted;

B. Laks. D.L. Mills and A.A. Maradudin, Phys. Rev. B23 (1981) 4965.

[6] I. Pockrand. J. Phys. D (Appl. Phys.) 9 (1976) 2423.

Recently the same explanation has been given in:

D.L. Mills and M. Weber, Phys. Rev. B26 (1982) 1075.

[7] P.B. Johnson and R.W. Christy, Phys. Rev. B6 (1972) 4370;

U. Schroder, Surface Sci. 102 (1981) 118, see appendix.

[g] I. Pockrand. Thesis, Hamburg (1976). fig. 19 (unpublished).

[9] I. Pockrand and H. Raether, Opt. Commun. 18 (1976) 395; Appl. Opt. 16 (1977) 1784.

[lo] W. RothbaIler. Opt. Commun. 9 (1977) 429.

[ll] R.H. Ritchie, E.T. Arakawa. J.J. Cowan and R.N. Hamm, Phys. Rev. Letters 21 (1968) 1530.