PHYSICAL REVIEW B VOLUME 59, NUMBER 3 15 JANUARY 1999-I Angular correlation functions of light scattered from weakly rough metal surfaces C. S. West* and K. A. O'Donnell The School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 Received 23 July 1998 The angular correlation functions of light diffusely scattered from weakly rough metal surfaces are studied. In experimental work, surfaces are well characterized, have highly one-dimensional roughness, and are studied in the case of p polarization. If there is significant plasmon polariton excitation, one type of intensity correla- tion function indicates that the diffuse intensity decorrelates rapidly as the angle of incidence is varied. It exhibits peaks that arise from an autocorrelation of identical intensities, or from the correlation of intensities related by the reciprocity principle. A second intensity correlation function expresses the symmetry of the diffuse intensity about the specular direction and is significant without plasmon polariton excitation. It is shown that the intensity correlation functions are directly related to two fundamental amplitude correlation functions. The latter are studied with perturbation theory, including all terms to fourth order in the surface profile function. The relation of the observed effects to backscattering enhancement is thus established, and favorable comparisons are made with the experimental results. S0163-1829 99 05103-6 I. INTRODUCTION Despite this theoretical interest, there has been a complete lack of experimental studies of these important correlation The scattering of light from randomly rough surfaces has functions in the case of weak roughness. Because the scat- attracted attention over the years. Of considerable interest is tered intensity is readily detected while the amplitude is not, the case of a weakly rough metal surface, for which the our first purpose here is thus to present measurements of the standard deviation of surface height is much less than the intensity correlation functions. We show results for two gold illumination wavelength . Here, not only may the random surfaces for which is of the order of 10 nm, but the sur- roughness allow an incident light wave to launch surface faces have quite different power spectra. Thus, one surface plasmon polaritons, but it also may scatter the excited plas- produces little plasmon polariton excitation, but the second mon polaritons, thus transforming them into diffuse light es- exhibits strong scattering contributions associated with these caping from the surface. If these scattering processes are surface waves. These two surfaces produce intensity correla- significant, it is well known that backscattering enhancement tion functions of quite diverse character and serve to illus- may appear.1 The effect is apparent, in the mean diffusely trate a broad range of behavior. We then draw comparisons scattered intensity, as a peak in the direction of retroreflec- with previous theoretical works by considering the various tion. The peak arises from constructive interference and has predicted contributions to the intensity correlation functions. seen a sustained level of interest, with many theoretical stud- Our second purpose is to present a theoretical interpreta- ies applying either perturbation theory2­10 or numerical tion of the results. In the limit of large illuminated surface methods.11 This type of backscattering peak was first ob- area, we first discuss that a Gaussian moment theorem allows served recently;12,13 the delay is due largely to the difficulties the intensity correlation functions to be expressed in terms of encountered in the microlithographic fabrication of rough two amplitude correlation functions. Because the amplitude surfaces producing sufficient plasmon polariton excitation. correlations are of lower statistical order, we consider them In addition to the mean intensity, the correlation functions to be more fundamental. These are evaluated using perturba- associated with the diffuse scatter have also been of interest. tion theory based on the reduced Rayleigh equations, includ- We ask the following: when will the amplitudes or intensi- ing all terms to fourth order in . It is shown that the calcu- ties scattered into the far field for different incident and lations are qualitatively and, in some cases, quantitatively scattering angles be correlated with one another, and what consistent with the experimental measurements. form will these correlation functions take? For weakly rough This introduction would be incomplete without recogniz- surfaces, this question has been addressed through theoreti- ing parallel studies of angular correlation functions made in cal study of the angular correlation functions of scattered two other fields. First, in the case of a strongly rough surface amplitude,14 where it was claimed that the same constructive with comparable to , the interest in angular correlations interference producing backscattering enhancement also ap- goes back many years18­20 see Fig. 10 of Ref. 18 for early pears in these correlation functions. Other theoretical works examples of the memory effect discussed below . Much of have considered the correlation functions of intensity,15­17 this original interest was stimulated by the possibility of de- which were found to contain contributions arising from a termining directly from measured angular correlations.20 wide variety of plasmon polariton-related scattering pro- More recently, other papers have investigated the correlation cesses. Generally speaking, it was shown that these correla- functions when multiple scattering occurs within deep sur- tion functions contain much more detailed information about face valleys.21­24 Even though backscattering enhancement the dominant scattering processes than does the mean diffuse may arise from this type of multiple scattering, it is unrelated intensity. to polariton excitation; such works should thus be considered 0163-1829/99/59 3 /2393 14 /$15.00 PRB 59 2393 ©1999 The American Physical Society 2394 C. S. WEST AND K. A. O'DONNELL PRB 59 as quite distinct from polariton-related studies with . There has also been much research on the light diffusely scattered from small particles. If multiple scattering is sig- nificant, the diffuse light may exhibit a backscattering peak.25 The peak arises from constructive interference as do the surface effects and has been related to weak Anderson localization.26 It was predicted that three different types of intensity correlations should exist,27 which stimulated a num- ber of theoretical28 and experimental29 investigations. Some of the correlations have been dubbed ``memory effects,'' for they reflect the manner in which the structure of the scattered field is remembered as the angle of incidence is changed. FIG. 1. For positive k, the measured power spectrum G(k) of the These correlation functions have some analogs in results to surface roughness for surface A solid curve and surface B dotted be presented here; it could be said that our results represent curve . Circles denote the spectral model of Eq. 28 for surface A. memory effects for weakly rough metal surfaces. Normalization is such that 2 2 is G(k)dk. II. EXPERIMENTAL TECHNIQUES the manner of sample A produce a mean diffuse intensity closely consistent with lowest-order perturbation theory. The One of the most challenging aspects of the experimental lowest-order term depends directly on the surface power work was the fabrication of suitable rough surfaces. Care spectrum;4 we have thus inverted the relation to determine was taken to ensure that the roughness was highly one- the spectrum from a diffuse intensity measurement. Surface dimensional, as is often assumed in calculations. Briefly, two B, however, produces scatter quite different from lowest- 50 50-mm glass plates were coated with a 1.5- m layer of order theory, so that the surface was characterized directly Shipley S1400-27 photoresist. The first plate plate A was with a Talystep stylus profilometer. The roughness spectra processed in a manner so as to produce a Gaussianlike power G(k) are shown in Fig. 1. It can be seen that surface A has a spectrum centered on zero wave number. In particular, a Gaussianlike spectrum while, apart from low levels of spec- beam from a HeCd laser of wavelength 442 nm was ex- tral power for small k, the spectrum of surface B is well- panded to approximately 150 mm diameter and was then constrained between kmin and kmax . From the area of G(k), incident on an opal glass. In the transmitted scatter, a photo- we determine to be 12.1 and 15.5 nm for surfaces A and B, resist plate was mounted in a plane parallel to the opal glass respectively. where finely-scaled speckle was present. As the plate was The high degree of roughness one dimensionality was ap- exposed, a motorized translation stage scanned the plate by parent upon reflection of a laser beam from the samples. 0.25 mm in a direction perpendicular to the normal of the With the incident beam directed perpendicular to the grooves opal glass. Thus, the time-integrated exposure was highly of the rough surface, the scattered light presented a one- one-dimensional. Because any point of the plate received dimensional speckled structure that was well-confined to the exposure from a large number of independent speckles, the plane of incidence. Further, for an incident p or s polariza- statistics of the exposure should be consistent with a Gauss- tion state, the diffuse scatter was identically polarized; we ian random process. restrict all discussions here to p polarization because the The second plate plate B was processed in a manner so scatter may then exhibit polariton-related effects. as to produce a roughness power spectrum of a displaced We now briefly present typical measurements of these rectangular form.12 It was exposed to 500 sinusoidal inten- one-dimensional scattering distributions. Results are ex- sity distributions arising at the intersection of two light pressed as a mean diffuse intensity I(q k) that represents beams from the HeCd laser. Each sinusoidal pattern had a scattered power per radian for unit incident power. Here k different spatial wave number k in the direction along the ( /c)sin i and q ( /c)sin s , where is the frequency, plate and was randomly phased with respect to other expo- and i and s are, respectively, the angles of incidence and sures. The minimum and maximum wave numbers kmin scattering. This notation is used for consistency with theoret- 8.56 10 3 nm 1 and kmax 1.33 10 2 nm 1, respec- ical development of Secs. IV and V. The angle brackets in- tively were well known from the exposing geometry. With dicate an ensemble average to remove speckle noise that, in the pattern wave numbers evenly spaced between kmin and the experiments, was approximated by integrating the detec- kmax the net exposure behaves as a Fourier series that, for a tor signal as the surface was spatially scanned. The sources large number of exposures, becomes consistent with a used were an orange HeNe laser 612 nm or a semicon- Gaussian random process.12 The desired one-dimensionality ductor laser 674 nm . The normalization of I(q k) was follows from alignment of the sinusoidal exposing intensi- obtained directly by measuring the incident beam with the ties. detector. Both plates were developed in a manner found to produce Figure 2 shows I(q k) in p polarization for several a linear relation between exposure and resulting surface cases. Surface A produces a broad distribution that, sur- height 30 sec in Shipley 352 developer . Using standard rounding the specular angle ( s i), resembles the shape of vacuum evaporative coating techniques, the samples were G(k) in Fig. 1. Indeed, G(k) was determined from this data then coated with a thick layer 200 nm of gold at a pressure as discussed earlier. In the case of surface B for 612 nm, less than 10 6 Torr. Efforts were also made to characterize the scatter for small s arises almost entirely from plasmon the samples. We have found previously that samples made in polariton coupling; the narrow bandwidth of G(k) produces PRB 59 ANGULAR CORRELATION FUNCTIONS OF LIGHT . . . 2395 FIG. 3. Interpretation of the correlation conditions; solid outgo- FIG. 2. Mean-diffuse intensities I(q k) of surface A for ing rays denote the specular direction. An intensity for q,k may be 612 nm and i 10° solid curve, circles , surface B for 612 correlated with an intensity for (q ,k ) that has deviation of mag- nm and i 10° dotted curve , and surface B with 674 nm and nitude qk (q k) to the right or left of the specular reflection. i 4° solid curve, triangles . Specular reflections are not shown; the total diffuse powers range from 0.0575 surface A to 0.112 tended by the slit was much smaller than the speckle width surface B, 612 nm . /w, so that integration effects should be negligible. We estimate that any systematic alignment errors in strong lowest-order scatter only for i and s s 42° in Fig. 2. In were of the order of 0.01° and were also small compared to particular, G(k) has been designed so that for i max /w. 13°, one may excite counterpropagating plasmon polari- To take a data set, a computer rotated the sample and tons as detector stages to coordinates ( i , s) and recorded the in- k tensity detected. In a direction transverse to the incident sp /c sin i kmin 1a beam, the sample was moved just sufficiently to cause the and intensity to decorrelate, and the new intensity was recorded. This process was repeated until the uniform area of the rough ksp /c sin i kmax , 1b surface had been fully utilized. After saving the data set to a where k disk file, further such data were taken for all other desired sp ( /c) 1 /( 1 1) is the wave number of a plasmon polariton traveling to the right or left , angle pairs ( i , s). The process required considerable time to complete 3­5 days and all correlations between files 1 i 2 is the dielectric constant, and we estimate that k were then computed; three such data sets were taken. sp 1.06 /c. It is readily verified that for i max as in Fig. 2, counterpropagating plasmon polaritons are launched Because it was impractical to do exhaustive searches for because Eqs. 1 are satisfied for the roughness wave num- correlations, the following considerations dictated our ap- bers in G(k) between k proach. For one-dimensional roughness the lowest-order min and kmax . Now, ksp and ksp may themselves be scattered by the roughness, to be out- scattered amplitude for q,k is proportional to (q k), wardly coupled as in where is the Fourier transform of the surface profile function.4 First, one could expect a correlation between in- /c sin s ksp kr 2a tensities at q,k and (q ,k ) if (q k ) qk is identical and to (q k) qk , because is evaluated at the same coordi- nate to produce both amplitudes. Second, one may also ex- /c sin s ksp kr , 2b pect an intensity correlation for qk qk ; the Fourier transform of the real surface profile is Hermitian, and the two where kr and kr are roughness wave numbers available in G(k). It is easily shown that the rectangular part of G(k) scattered amplitudes are related because (q k) constrains the outward coupling to *(q k ). These conditions are illustrated in Fig. 3. For s max , thus produc- ing the central distribution of Fig. 2. It is the interference the conditions q,k , qk is the deviation from specularity between the distinct processes involving k (q k) and we employ a small value away from specular sp and ksp that produces the backscattering peak at glare. Upon changing the illumination to condition k , the s i . For the other case of Fig. 2 with 674 nm, the distributions associated above arguments imply that we may find a correlated inten- with k sity either at position q with identical deviation from specu- sp and ksp shear apart from one another,13 but the backscattering peak still appears for lar i.e., q k i 4°, as shown. qk , or else at a second q equally dis- The following procedure was used to measure the angular placed to the other side of specular at q k qk . In Sec. correlation functions. The p-polarized laser beam was spa- III, results may be visualized by choosing q,k in Fig. 3 a tially filtered and focused to a waist on the surface. The and then plotting the correlation as k varies, with it being illumination width w 1/e diameter of the intensity was 67 understood that q follows one of the two correlated direc- or 53 m for 612 or 674 nm, respectively. The sample tions in Fig. 3 b . was mounted on a rotation stage to set The general correlation function may be expressed as i ; an arm mounted on a concentric rotation stage was used to record the inten- sity present at scattering angle s . On this arm, in the far C I q,k q ,k I q k I q k , 3 field 70 cm from the rough surface, was a slit of width 130 m. All light entering the slit was collected by a field lens where I I I . The first type of correlation function dis- and was focused onto a silicon photodetector. The angle sub- cussed is represented by 2396 C. S. WEST AND K. A. O'DONNELL PRB 59 FIG. 4. Mean-diffuse intensities I(k qk k) computed from averages of the data. Cases shown are surface A with 612 nm circles , surface B with 612 nm squares , and surface B for 674 nm triangles . C I k,k , qk C I k qk ,k k qk ,k , 4 while the second type is given by C I k,k , qk C I k qk ,k k qk ,k . 5 We also find it useful to present the statistically normalized correlation functions C k,k , FIG. 5. Correlation functions C (k,k , I qk I qk) solid lines and I k,k , qk , 6 I2 q k I2 q k C I(k,k , qk) dashed lines for surface A, 612 nm, and qk 0.12 /c. The autocorrelation A and reciprocal R points are where 0 indicated. I 1. III. EXPERIMENTAL RESULTS Data sets were taken for surface A with qk 0.12 /c and 612 nm, and for surface B with qk 0.04 /c, with wavelengths either 612 or 674 nm. For each pair ( i , s), 1.2 104 and 5.1 103 nearly independent intensi- ties were measured for surfaces A and B, respectively, to compute the correlation functions. The number of angle pairs was 26 for surface A, and 170 and 262 for surface B with 612 and 674 nm, respectively. In the latter two cases, data are too dense to be plotted distinctly in the forthcoming fig- ures. Throughout results the intensity is normalized as in Fig. 2; the resulting correlation units are rad 2. All statistical er- ror bars shown are computed directly from moments of the data and extend to plus or minus one standard deviation. The mean intensities I(q k) computed by averaging data sets for positive qk are shown in Fig. 4. As had been noted in Fig. 2, surface A produces a broad distribution with- out distinct features, while surface B produces a more com- pact distribution having a backscattering peak at both wave- lengths. The peak falls at q k; because of the relation between q and k this occurs for k qk/2, or thus at i 1.1° for surface B. The steep slopes from polariton ex- citation are obvious for surface B and, as plotted in Fig. 4, they appear symmetrically about the backscattering peak. Correlation functions for surface A are shown in Figs. 5 FIG. 6. Correlation functions (k,k , and 6 with 612 nm. In Fig. 5, it is seen that I qk) solid lines and (k,k , C I qk) dashed lines for surface A, 612 nm, and qk I(k,k , qk) and C I(k,k , qk) present broad curves cen- 0.12 /c. The autocorrelation A and reciprocal R points are tered near i 0°. The slow decay arises largely from the indicated. PRB 59 ANGULAR CORRELATION FUNCTIONS OF LIGHT . . . 2397 FIG. 7. Correlation functions C I(k,k , qk) solid lines and C I(k,k , qk) dashed lines for surface B, 612 nm, and qk 0.04 /c. The autocorrelation A and reciprocal R peaks are apparent in b ­ d . FIG. 8. Correlation functions I(k,k , qk) solid lines and fall of the mean intensity; this is clear because I(k,k , qk) dashed lines for surface B, 612 nm, and qk 0.04 /c. The autocorrelation A and reciprocal R points show I(k,k , qk) and I(k,k , qk) as shown in Fig. 6 remain strong, indicating little actual decorrelation. Throughout Fig. essentially perfect correlation. 6, I contains a peak of unit height at k k that represents an autocorrelation between identical intensities. In addition, C I(k,k , qk) and demonstrates nearly perfect correlation at a second peak arises from reciprocity; the reciprocity points where the two intensities are related by the reciprocity principle30 states that reversing the incident and scattered principle. Both I and I exhibit modest values outside of wave vectors produces an identical scattered amplitude. the polariton coupling region, but generally I and C I re- Here, an amplitude for q,k must then be identical to another main small. with (q ,k ) ( k, q).4 The two intensities correlated in The results of Figs. 5­8 do little justice to the quantity of I are such a reciprocal pair for k k qk , producing data taken, and, to accentuate the differences between sur- the second peak Figs. 6 a and 6 b . Further, the case of Fig. faces A and B, we provide more complete plots of 6 c shows but a single peak, which appears where the auto- C I(k,k , qk) in Fig. 9. For C I in the case of surface A, correlation and reciprocal conditions are simultaneously sat- Fig. 9 a shows a broad envelope and thus indicates that the isfied at backscattering with k k qk/2. In the case of speckles generally decorrelate slowly. In Fig. 9 c , C I for I no distinct peaks appear, but I remains near 0.8 and surface B exhibits essentially identical autocorrelation and thus represents a significant and persistent correlation. reciprocal peaks, which overlap to produce a high central The correlation functions of surface B take different forms peak at the point for which both correlated intensities are at as shown in Figs. 7 and 8 with 612 nm. In Fig. 7 a , k is backscattering. Otherwise the decorrelation is rapid and outside of the polariton coupling region of Fig. 4 and strong; it appears that C I only rises where it must to pro- C I(k,k , qk) remains small for all i . However, the low duce the autocorrelation and reciprocal peaks. In the case of levels of intensity nonetheless exhibit correlations; Fig. 8 a C I , Fig. 9 b shows that it is generally comparable to C I shows that I is strong outside of the coupling region, but for surface A but, in Fig. 9 d , it is seen to be almost insig- decays within it, and decays further when the intensity at k nificant for surface B. falls within the backscattering peak. In the other three cases The decorrelation seen in C I for surface B may be made I(q/k) has strong polariton coupling; C I in Figs. 7 b and even more rapid by using a longer wave length, as is appar- 7 c exhibits narrow peaks surrounding the autocorrelation ent with 674 nm in Fig. 10. In Fig. 10 d , for example, and reciprocal points, with low levels elsewhere. In Fig. 7 c the width of the correlation peak of C I(k,k , qk) has nar- the two peaks are interacting with one another and, in Fig. rowed to approximately 60% of the width of the result of 7 d , they coincide when I(q k) is at backscattering. In Figs. Fig. 7 d . This observation will be discussed in Sec. V. An- 8 b ­8 d , I exhibits peaks related to those of other notable difference is that the polariton coupling region 2398 C. S. WEST AND K. A. O'DONNELL PRB 59 FIG. 9. Experimental results for C I(k,k , qk) of surface A a and b and surface B c and d . In C I , solid- and dashed-diagonal lines follow the autocorrelation and reciprocal peaks, respectively. Units of C I are rad 2, and i and i are expressed in degrees. has become wider in Fig. 10, but this had been seen previ- qk have been widely studied.19­24 In the case of volume ously in Fig. 4. Otherwise, Fig. 10 demonstrates that C I scattering,27­29 the analogous intensity correlation is known maintains much qualitative similarity to Fig. 7; the similari- as the memory effect; the reciprocal peak in this correlation ties also extend to was termed the time-reversed memory effect. For weakly I , although we do not show the latter measurements here. rough surfaces,15,16 C I has been called the C(1) correlation A measure of the quality of the data is the degree of elsewhere, and C I has been termed the C(10) correlation. correlation at a reciprocal configuration expressed by The above measurements are the first experimental observa- I(k, k qk , qk). It is expected that angular positioning tions of the latter correlation function. errors could greatly reduce the reciprocal correlation. The It is equally important to note what correlation functions average value of have not been observed in our experiments; these have been I(k, k qk , qk) is 0.99 and 0.98 for surface B at with 612 and 674 nm, respectively, and is termed long-range correlations. To our knowledge, none of 0.97 for surface A. These values indicate that such errors the following effects have ever been observed in surface were of little consequence. scattering. One such correlation has been predicted to occur Following the conventions of laser speckle theory,31 we when (q k) (q k ).23 We have tried but failed to find define the intensity contrast C as statistically significant correlations in such cases. Another possibility is the C(1.5) correlation15,16 that implies a correla- I2 I 2 tion between the intensity of the single scatter and of the C I . 7 plasmon-polariton related scatter of surface B. We regret that the single scatter of surface B appears artificially non- A contrast of unity is consistent with the complex circular Gaussian due to the discrete nature of the 500 exposures used Gaussian amplitude statistics commonly assumed for well- in the fabrication; this sample is unsuitable for such measure- developed speckle.31 We find that the average C for our data ments. Other polariton-related possibilities include the C(2) sets is 1.00 and 1.01 for surface B with 612 and 674 nm, correlation,15,16 which would, for example, predict six ridges respectively, which is consistent with these amplitude statis- at and near the lines k k in Fig. 9 d , but these are not tics. However, for surface A the average contrast is only present. Another is C(3), which predicts a correlation distrib- 0.95, but we do not attribute this to other amplitude statistics. uted throughout the entire speckle pattern,15,16 yet we find no Instead, the roughness of surface B is somewhat less one- such effects. Using a narrower illumination region, which dimensional than A; we observe that its speckle has a slight may make such correlations stronger, is impractical for us two-dimensional structure. Integrating this two-dimensional due to alignment difficulties. speckle over the length of the detection slit inevitably re- duces C, as generally occurs for integrated speckle.31 We now relate these observations to previous work. A IV. DISCUSSION correlation function like C I(k,k , qk) has appeared previ- We now consider a fundamental question. Let us assume ously. For strongly rough surfaces, similar correlation func- that the surface roughness is a statistically stationary random tions of both amplitude and intensity that occur for qk process and that the illuminated region is large. From these PRB 59 ANGULAR CORRELATION FUNCTIONS OF LIGHT . . . 2399 A moment theorem of the form of Eq. 8 has rarely been used elsewhere; it has been applied to the spatial speckle statistics of imaged phase screens.33 In the vast majority of speckle work, the second term of Eq. 8 did not appear because of the additional assumption of circular statistics.31,34,35 For the angular correlations, both terms must be retained. Further, we may make a stronger statement: The second-order moments A(q k)A*(q k ) and A(q k)A(q k ) in Eq. 8 are fundamental because, for zero-mean Gaussian statistics, they fully specify the ampli- tude's statistical properties. Thus, all joint probability densi- ties of amplitude immediately follow, and joint moments of arbitrary statistical order are guaranteed to follow from these amplitude moments.34,35 Equation 8 is but a special case of the latter statement. Our original question may thus be answered completely through accurate determination of the two amplitude mo- ments. For this purpose, we employ the approach of Maradu- din and MeŽndez4 that, in each perturbation order, allows terms to be evaluated exactly. In this formulation, a p-polarized plane wave illuminates a long length L1 of a surface with one-dimensional roughness. The diffuse ampli- tude is given by A q k i 2 1/2 L N i , s G0 q T q k G0 k , 1 FIG. 10. Correlation functions C (k,k , 9 I qk) solid lines and C I(k,k , qk) dashed lines for surface B, 674 nm, and qk where T(q k) is the transition matrix, T T T , and 0.04 /c. The autocorrelation A and reciprocal R peaks of G0(k) is the plasmon polariton Green's function for a flat C I have narrowed as compared to the case for 612 nm in metal surface Fig. 7. i considerations alone, is it possible to determine what types G0 k , 10 of correlation functions of amplitude or intensity follow? De- 0 k k spite the wide variety of correlation functions discussed in where Sec. III, the issue has not been adequately addressed previ- 0(k) ( /c)2 k2 and (k) ( /c)2 k2. The factor ously. Nonetheless, our main conclusions are related to those of a number of previous works,14­17 but this will be dis- cos s cussed in detail later. In our arguments that follow there are N 3/2 i , s c cos i 11 many subtleties; the discussion will stress plausibility at the expense of mathematical rigor. guarantees that the mean diffuse intensity I(q k) Under these physical conditions, the far-field scattered A(q k) 2 is normalized as in Figs. 2 and 4, as is seen amplitude arises from the sum of many independent contri- from Eq. 3.4 of Ref. 4. The transition matrix is expanded butions from different regions of the rough surface. As in into the perturbation series speckle theory, we heuristically invoke the central limit theo- rem to conclude that the amplitude follows complex Gauss- i n ian statistics.31 It has also been shown that, for a perfectly T q k n 1 n! T n q k 12 conducting rough surface, the scattered amplitude may be of nonvanishing mean only at the specular angle.32 The proof where T(n)(q k) is of order n in the surface profile function relies only on the statistical stationarity of the surface source (x). These terms may be cast as functions; we also expect this to be the case for the pen- etrable surface of interest here. We thus denote the diffusely T 1 q k A 1 q k q k , 13 scattered amplitude by A(q k), with it being understood that it has zero mean and that any specular reflection has been dp subtracted. T 2 q k 2 A 2 q p k q p p k , Consider now, for example, the intensity I(q k) 14 A(q k) 2. Making only the assumptions of the previous paragraph, it is readily shown that dp dr T 3 q k I q k I q k A q k A* q k 2 2 2 A 3 q p r k A q k A q k 2. 8 q p p r r k , 15 2400 C. S. WEST AND K. A. O'DONNELL PRB 59 and so on, where (k) is the Fourier transform of the surface T 1 q k T 3 * q k 2 2 2 q k q k profile A 1 q k H* q k g q k , k dx x exp ikx . 16 20c while those of Eq. 19 are given by The functions A(n) are derived from the reduced Rayleigh T 1 q k T 1 q k 2 2 q k q k equations and follow from the results of Ref. 4. For example, A 1 q k A 1 q k g q k , 1 A 1 q k i 21a 2 qk q k , 17 T 2 q k T 2 q k and an expression for A(3) has appeared elsewhere.10 It is only necessary to insert Eqs. 12 ­ 15 into Eq. 9 dp and compute the two desired amplitude correlation functions. 2 2 2 q k q k 2 2 For convenience, we now assume that the surface profile (x) is a real Gaussian process. Retaining terms to fourth A 2 q p k S q,q ,p,k g q p g p k , order in , which is sufficient to obtain polariton-related effects,4 we average over the statistics of (x) to obtain 21b and A q k A* q k 2 L N i , s N i , s G0 q T 1 q k T 3 q k 2 2 2 q k q k 1 G A 1 q k H q k g q k . 0* q T 1 q k T 1 * q k 21c 14 T 2 q k T 2 * q k Here denotes the Dirac delta function, G(k) 2g(k), 16 T 1 q k T 3 * q k and we have defined 16 T 1 * q k T 3 q k S q,q ,p,k A 2 q q q p k G0 k G0* k , 18 A 2 q k q p k , 22 and and dp dp A q k A q k 2 H q k L N i , s N i , s G0 q G0 q 1 2 2 A 3 q p q k g q p 2 2 T 1 q k T 1 q k A 3 q k p k A 3 q q k p p k 14 T 2 q k T 2 q k g p k . 23 16 T 1 q k T 3 q k In Eqs. 20 all contributions to A(q k)A*(q k ) con- 1 tain delta functions with argument (q k) (q k ), while 6 T 1 q k T 3 q k Eqs. 21 for A(q k)A(q k ) contain delta functions in G0 k G0 k . 19 (q k) (q k ). This is not entirely surprising. It has been shown elsewhere that, for a perfectly conducting rough The moments of Eq. 18 may be expressed as surface, the statistical stationarity of the surface source func- tion restricts the correlation A(q k)A*(q k ) to the con- T 1 q k T 1 * q k 2 2 q k q k dition noted.21 We expect that similar stationarity arguments A 1 q k A 1 * q k g q k , should apply to penetrable surfaces. Further, in extending the approach of Ref. 21 to the unconjugated correlation 20a A(q k)A(q k ) , it is straightforward to show that the re- striction to (q k) (q k ) follows directly. It is thus T 2 q k T 2 * q k highly plausible that, for any statistically stationary rough- ness, the delta functions present in either Eqs. 20 or 21 dp 2 2 2 q k q k persist throughout terms of arbitrary order. 2 2 Thus, A(q k)A*(q k ) and A(q k)A(q k ) usually appear distinctly; in particular, when (q k) (q k ) the A 2 q p k S * q,q ,p,k g q p g p k , former moment is nonzero but the latter vanishes, and the 20b situation is reversed for (q k) (q k ). Both of these correlations are of the short-range type, arising as delta func- and tions in q and k. Indeed, the approach taken here has appar- PRB 59 ANGULAR CORRELATION FUNCTIONS OF LIGHT . . . 2401 ently excluded all long-range correlations; these have V. NUMERICAL RESULTS strengths proportional to the inverse of the illumination Here we present results based on numerical integration of width15,16 and should thus be absent in the limit taken here. the expressions from Sec. IV. The divergence from the delta Still, our assumption of plane-wave illumination differs from functions of Eqs. 20 and 21 is artificial and arises from the Gaussian beam of the experiments, but the consequences treating the profile (x) as having infinite length. To evaluate of such differences should be small. In the case of A(q k)A*(q k ) and A(q k)A(q k ) from Eqs. 18 ­ A(q k)A*(q k ) , it has been shown that the amplitude 21 , we have thus transformed factors as correlation function with beam illumination is nearly identi- cal to that with a plane wave, as long as the beam has a 2 diffraction width much narrower than the structure in L q k q k q k , q k 24a 1 A(q k)A*(q k ) ;21 an analogous conclusion has also been reached from direct studies of the intensity correlation and function.16 This condition seems satisfied in the experiments; with 612 nm, for example, the full width at half maxi- 2 q k q k q k , q k , 24b mum of the intensity of the incident beam is 0.28°, which is L1 far less than the 2.4° width of C I in Fig. 7 d . where subscripted deltas are of the Kronecker type; these are We now relate our approach to those taken elsewhere. analogous to Eq. 4.1 of Ref. 4. The amplitude correlations Reference 14 was unaware of the second term of Eq. 8 , and are nonzero but finite as determined by the arguments of the evaluated the first term with a perturbation theory of a dif- Kronecker delta functions, and results may thus be expressed ferent type. In Refs. 15 and 16, perturbation theory was de- as veloped for the full intensity correlation with the surface il- luminated by a beam of Gaussian profile. Here, to leading CA k,k , qk A k qk k A* k qk k 25 order in , the moment theorem of Eq. 8 can be seen to be and valid and the results exhibit appropriate delta functions with arguments (q k) (q k ). However, in higher-order C A k,k , qk A k qk k A k qk k , 26 terms, the long-range correlations contribute and Eq. 8 no longer applies. Nevertheless, in the limit of large illumina- that are similar to Eqs. 4 and 5 . We also define the sta- tion width the long-range effects should vanish and Eq. 8 tistically normalized correlation functions should be recovered, but this is not obvious due to the com- plexity of the theory. C k,k , k,k , A qk , 27 In Ref. 17 the noncircular moment theorem of Eq. 8 was A qk A q k 2 A q k 2 indeed originally proposed for the angular correlation func- tions. From numerical simulations of scattering from a sta- with 0 A 1, where A(q k) 2 follows as a special tistical ensemble of rough surfaces, the intensity correlation case of Eq. 18 . function was calculated by averaging over this ensemble. It In calculations for surface A, we have employed the spec- was thus shown that the two terms of Eq. 8 contributed trum distinctly, in the same manner as discussed above. For a rectangular spectrum G(k), it was found that the scattered 2b exp a2/b2 exp k2a2 g k amplitude was highly non-Gaussian, having speckle contrast 1 erf a/b 1 k2b2 . 28 C that deviated 70% from unity, and that the long-range cor- This model agrees closely with the experimental spectrum as relations were several times stronger than C I(k,k , qk). It is seen in Fig. 1; a least-squares fit has provided the param- is unclear why these claims are different from the results of eters a 89 nm and b 190 nm. We show results for surface our controlled experiments with surface B in Sec. III, where A in Fig. 11, assuming the dielectric constant of gold to be C was essentially unity and the long-range correlations were 9.0 1.29i for 612 nm. The two amplitude correla- absent. It was also claimed that the moment theorem of Eq. tion functions show a broad angular structure and are domi- 8 does not affect the speckle contrast C. In Sec. V, we give nated by the 1-1 terms of Eqs. 18 and 19 ; the 1-3 terms several examples in which there may be significant effects on make contributions of similar shape but are approximately C. While the circular moment theorem i.e., Eq. 8 with the one order of magnitude smaller. It is seen that CA (k,k , qk) second term absent has sometimes been referred to as a is largely real and positive, while CA (k,k , qk) is domi- factorization approximation, this name may be misleading. nated by a negative imaginary part. There are peaks at the Indeed, here we have made one different approximation: that autocorrelation and reciprocal positions in Re CA that are the amplitude follows complex Gaussian statistics. It is then hardly discernible; they arise from a small 2-2 contribution only an exercise to determine, from physical principles, all from Eq. 20b . Most of decay of CA arises from the fall of amplitude moments of first and second statistical order, thus the mean intensity and there is little actual decorrelation; we specifying all parameters of the Gaussian probability distri- find that A 0.93 throughout Fig. 11, although we do not bution. At this point the problem is as good as solved; any show results for A here. required statistical moment follows immediately from the These calculations may be readily compared with the distribution. This approach thus leads directly to Eq. 8 and experimental results of Sec. III. In particular, it follows there is no violation of a factorization approximation. directly from earlier discussions that C I(k,k , qk) 2402 C. S. WEST AND K. A. O'DONNELL PRB 59 FIG. 11. Calculated amplitude correlation functions C A (k,k , qk) circles and CA (k,k , qk) triangles for the spec- trum of Eq. 28 with 612 nm, 9.0 1.29i, qk 0.12 /c, and k 0.18 /c. Solid and dashed lines denote, respec- tively, the real and imaginary parts. Autocorrelation A and recip- rocal R points are indicated. C A (k,k , qk) 2 and I(k,k , qk) A (k,k , qk) 2. We show such comparisons for surface A in Fig. 12. We had FIG. 13. Results for the Gaussian spectrum of Eq. 29 with pointed out in Sec. III that the experimental contrast C was 457.9 nm, 7.5 0.24i, 0.95 because of surface defects; we have crudely corrected qk 0.12 /c, and k 0.18 /c. Top: Calculations of C circles and C triangles ; solid and for this by scaling C A A A 2 by the factor C2. There are other- dotted lines denote, respectively, the real and imaginary parts. Bot- wise no free parameters and the agreement between C I and tom: Calculations of A 2 solid line and A 2 dotted line ; the C A 2 in Fig. 12 a is thus remarkably good. In the compari- autocorrelation A and reciprocal R peaks in A 2 imply perfect son between the calculated correlation. A 2 and the experimental I in Fig. 12 b , both results share peaks at the autocorrelation and reciprocal points, but the latter decays more rapidly for large We next consider the case of the Gaussian spectrum i . It is seen that A 2 and I show somewhat less cor- relation, with the experimental curve being lower. Including g k a exp k2a2/4 . 29 terms only to order 2, it is straightforward to show that To be consistent with theoretical works studying backscatter- A 1; thus the fourth-order terms are required to produce ing enhancement for this surface model,1,4,11 we assume a decorrelation. It may require yet higher-order perturbation 100 nm, 5 nm, 457.9 nm, and 7.5 0.24i. It is terms to predict further decorrelation, so as to be more simi- expected that the plasmon polariton excitation will be more lar to the experimental results. apparent; g(ksp)/g(0) is eight times higher than for surface A. However, no experimental studies of this surface have appeared because of fabrication difficulties that we also can- not surmount. Results for C A(k,k , qk) are shown for the Gaussian spectrum in Fig. 13 a ; the results are of lower scale than Fig. 11 because of the reduction in . The broad features resemble the results for surface A and arise similarly from the 1-1 and 1-3 terms. However, there are now distinct peaks in Re C A that appear at the autocorrelation and reciprocal points. These arise entirely from the 2-2 term, as do the associated zero crossings in Im C A . No such structures are seen in C A and we find that its 2-2 term makes a small, featureless contribution. In Fig. 13 b , we also show the cor- relation functions A(k,k , qk) 2 that indicate a substantial decorrelation of the speckle intensity. There are distinct minima that appear as the normalization factor A(q k ) 2 in Eq. 27 passes through its backscattering peak. In A 2, there are narrow autocorrelation and reciprocal peaks where the correlation is perfect; these also arise from the 2-2 con- FIG. 12. Results for surface A, 612 nm, qk 0.12 /c, and k 0.18 /c. Top: Measurements of C circles and C squares tribution to C A . I I are compared with calculations of C 2 solid curve and C 2 We now consider the case of an ideal rectangular spec- A A dotted curve . Bottom: Measurements of circles and trum for which g(k) has constant height for k I I min k squares are compared with calculations of A 2 solid curve and kmax , with g(k) 0 for all other k. Using the parameters of A 2 dotted curve . surface B we find that, keeping terms of up to fourth order, PRB 59 ANGULAR CORRELATION FUNCTIONS OF LIGHT . . . 2403 from the radiative damping of plasmon polaritons1 on the strong experimental surface or, as will be seen, from differ- ences between the assumed and actual values of . Finally, we note that our calculations for A(k,k , qk) 2 for the rectangular spectrum bear much qualitative similarity to the experimental results of Fig. 8, even to the point of resem- bling the unusual forms of Fig. 8 a , but we do not show these results here. We claim that the peaks of C A(k,k , qk) seen through- out results arise from processes that are generalizations of those producing backscattering enhancement. This may be shown through approximate evaluation of the 2-2 contribu- tion to C A . For this purpose, we write A 2 q p k 2A 1 q p G0 p A 1 p k , 30 which neglects terms that make small contributions. This simplification is made in Eqs. 20b and 22 , and the prod- ucts of displaced Green's functions may be treated with a pole approximation4 2 iC2 G0 p G0* x p 2i p ksp x 2 iC2 2i p ksp , 31 x where C 1 3/2 2 32 1 1 FIG. 14. Calculations for the case of the ideal rectangular spec- and trum with 5 nm, 612, 9.0 1.29i, and qk 0.04 /c. Shown are C A (k,k , qk) circles and CA (k,k , qk) triangles ; k solid and dashed lines denote, respectively, the real and imaginary sp 12 2 . 33 1 1 1 parts. Upon performing the integral we obtain, to an excellent ap- the predicted diffuse intensity in the polariton-coupling re- proximation, gion is twice that of the experimental results of Figs. 2 and 4. This is evidence that the surface is too rough for direct com- T 2 q k T 2 * q k parison with perturbation theory of this order. We thus present numerical results for a weaker and fictitious sur- 2L 2 2i 1 4C q k , q k face, and confine the discussion to qualitative experimental 2i k k comparisons. In the calculations, we assume the surface to F have an ideal rectangular spectrum with k q k F * q k g q ksp g ksp k min and kmax as in Sec. II, 5.0 nm, 612 nm, and 9.0 1.29i. 2i The results for C F A (k,k , qk) are shown for qk 2i q k F * q k g q ksp q k 0.04 /c in Fig. 14 and exhibit much similarity to the ex- perimental results of Fig. 7. It is notable that C 2i A arises solely from the 2-2 terms; contributions from all other terms of Eqs. g ksp k 2i F q k F * q k 20 and 21 vanish because of the common factor g(q k), which is zero under the conditions of Fig. 14. 2i q k g q k Peaks similar to those in C sp g ksp k 2i I(k,k , qk) of Fig. 7 appear in k k Re C A and related structures are seen in Im CA . As had been the case for C I in Fig. 7, CA remains small throughout F q k F * q k g q ksp g ksp k , 34 Fig. 14; its levels are comparable to those of C A in Fig. 14 a . There are, however, subtle differences in comparisons where F (q k) A(1)(q ksp)A(1)( ksp k) represents the with experimental data. The peak width the full width at coupling of an incident state k to a scattered state q, via the half maximum of, for example, C I in Fig. 7 d is 2.4° plasmon polariton ksp . while, upon taking C A 2 from the result of Fig. 14 d , we For the ideal rectangular spectrum, we employ the ap- find a peak width of only 1.4°. This difference may arise proximate 2-2 term of Eq. 34 to evaluate C A(k,k , qk) in 2404 C. S. WEST AND K. A. O'DONNELL PRB 59 FIG. 15. From the pole approximation, the contributions to FIG. 16. The calculated intensity contrast C at specular for the Re C A(k,k , qk) from Eq. 34 for the ideal rectangular spectrum. model of surface A dot-dashed curve , the Gaussian spectrum The autocorrelation peak right contains contributions from the solid curve , and the ideal rectangular spectrum dotted curve . first term dashed curve and last term dotted curve of Eq. 34 ; the reciprocal peak left shows contributions from the second dot- background when there are significant 1-1 and 1-3 contribu- dashed curve and third terms broadly dashed curve . The solid tions. It had been noted in Sec. III that the peaks were nar- curve indicates the total. rower for 674 nm than 612 nm; this trend is consistent Fig. 15; the 1-1 and 1-3 contributions here vanish. The re- with the behavior of as 1 changes rapidly with .36 sults are quite similar to those obtained in Fig. 14 b from Finally, we consider effects on the speckle contrast C of exact numerical integration. Further, the contributions of the Eq. 7 . The variance of the intensity follows from Eq. 8 four terms of Eq. 34 are plotted distinctly and reveal struc- with (q,k) (q ,k ). Assuming that qk 0, the delta func- tures that may be interpreted as arising from interference tions noted in Sec. IV guarantee that the second term of Eq. between scattering processes. The first and fourth terms of 8 vanishes so that I2 I 2. Thus C is constrained to Eq. 34 represent interference terms between the processes unity, as would have been the case for circular Gaussian k k statistics. This conclusion is consistent with the experimental sp q and k ksp q . In Fig. 15, these pro- cesses interfere constructively to contribute to the autocorre- results of Sec. III where C 1. However, for the special case lation peak at k k. The reciprocal peak arises from the at specular with q k q k , both terms of Eq. 8 con- second and third terms of Eq. 34 . The second term repre- tribute so that sents the interference between processes k ksp q and k k I 2 A2 2 sp q . The latter process becomes a time-reversed C version of the former at a reciprocal configuration with I , 35 (q ,k ) ( k, q), thus producing the reciprocal peak. A and it is clear that C may exceed unity. At specular, this similar argument applies to the third term of Eq. 34 , which definition of C is slightly different from that often expresses the interference between k ksp q and k employed31 because the mean amplitude was subtracted to ksp q . Previously, related arguments have been ap- define A(q k) in Sec. IV. However, the mean-specular am- plied to multiple scattering from a strongly rough metal plitude is infinite in this theory and its subtraction is neces- surface,22 but the intermediate state is not a plasmon polar- sary to obtain a nonvanishing C. iton. The specular contrast was evaluated from numerical inte- The backscattering peak in the mean intensity of Figs. 2 gration of Eqs. 18 and 19 to determine I and A2 , and 4 is also a consequence of this interference. It arises for respectively. Figure 16 shows the results for the same param- q k q k where all four processes are correlated; eters used earlier for surface A, the Gaussian model, and the the four terms of Eq. 34 produce equal and simultaneous ideal rectangular spectrum. It is seen that surface A produces contributions to this peak. Thus, C A(k,k , qk) has allowed a contrast of nearly 1.4, with a shallow minimum at us to see more distinctly the contributions that produce the i 0° as the intensity passes through backscattering. The Gaussian backscattering peak. Further, it is remarkable that the peaks model has nearly unit contrast for large occur when both q,k and (q ,k ) are far from backscatter- i , but C rises above 1.2 for smaller ing in Fig. 15, but these coherent processes make distinct i , only to fall to 1.15 at backscattering. The contrast for the rectangular spectrum is similar to surface A contributions nonetheless. The results also indicate that the for large amplitude arising via the polariton-related processes decor- i but falls to less than 1.02 for i 11°, and falls further to C 1 at relates rapidly; such rapid decorrelation is not associated i 0°. These results suggest that polariton-related scattering contributions tend to reduce the with the scattering processes contained in the 1-1 or 1-3 contrast; this is most clear from the abrupt fall of contrast, terms. It is readily shown from Eq. 34 that the width full either for the rectangular spectrum for small width at half maximum of either peak in Re i , or for all C A(k,k , qk) or three cases at backscattering. For a Gaussian-like spectrum, a C A(k,k , qk) 2 is given by k 4 . In terms of i , this substantial increase in will produce stronger polariton cou- implies widths of i 4c /( cos i) and i pling and may then drive the specular contrast toward unity. 4c /( cos s) for the autocorrelation and reciprocal However, a thorough investigation of this speculation would peaks, respectively. These arguments apply directly to peak require a more complete theory than the low-order terms widths for surface B, or else to the peak width above the employed here. PRB 59 ANGULAR CORRELATION FUNCTIONS OF LIGHT . . . 2405 VI. CONCLUSIONS This theoretical approach is less sophisticated than others We have presented a detailed investigation of the angular employed to investigate long-range correlations, yet it is correlation functions for p-polarized light scattered from fully capable of reproducing all essential features of our ex- weakly rough metal surfaces. In experimental work, we have perimental results. Further, should the fourth-order theory be used sophisticated fabrication techniques to produce two ran- inadequate, it is only necessary to find another means of domly rough surfaces with different power spectra. For the calculating C A and CA , which may be a far less formidable first surface, there is little plasmon polariton excitation and task than the direct calculation of C I and C I . the correlation function C I(k,k , qk) indicates that the in- It has been discussed that CA and CA both express per- tensity decorrelates slowly as the angle of incidence is var- fect correlation for scatter consistent with lowest-order per- ied. Further, C I(k,k , qk) is strong and demonstrates that turbation theory but, as higher-order terms become signifi- the scattered intensity, although random, possesses some- cant, their behavior differs. When there is significant what symmetric structure about the specular direction. In the plasmon polariton excitation, C A presents peaks that arise case of the second surface, plasmon polariton excitation is from coherent mechanisms identical to those producing significant and the behavior is strikingly different; C I indi- backscattering enhancement. Further, C must remain im- cates that the intensity decorrelates extremely rapidly, and A portant even as the roughness is increased to arbitrary levels; C I remains quite small, so that the symmetry is nearly ab- at the least, it must rise to express perfect correlation for its sent. Under our experimental conditions, these two correla- autocorrelation and reciprocal peaks. In the case of C , tion functions are all that are observed, and long-range cor- A however, we have found that higher-order perturbation terms relations seem to be below statistical noise levels. reduce the degree of correlation implicit throughout C . In It has also been shown that the statistical properties of the A addition, we have not found peaks within C that arise from scattered light are fully specified by the amplitude correla- A coherent mechanisms, even under conditions for which such tion functions C A (k,k , qk) and CA (k,k , qk), and that a peaks are present in C . Although C is, in principle, al- rarely-used Gaussian moment theorem relates these directly A A ways present for any rough surface, its behavior and signifi- to C I and C I . These conclusions are based on plausibility cance for stronger roughness remain open issues. arguments that appear to be valid for statistically stationary roughness in the limit of large illuminated area. Using a per- turbation approach exact to fourth order in the surface pro- file, C ACKNOWLEDGMENT A and CA have been evaluated, and comparisons with experimental results for C I and C I are generally favorable. We are grateful for discussions with E. R. MeŽndez. *Present address: Optical Sciences Division, Naval Research Labo- 15 V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. Maradudin, ratory, 4555 Overlook Ave. SW, Washington, D.C. 20375. and M. Nieto-Vesperinas, Opt. Lett. 22, 946 1997 . Present address: Division de Fisica Aplicada, Centro de Investiga- 16 V. Malyshkin, A. R. McGurn, T. A. Leskova, A. A. 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