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Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties J. M. Elson, J. P. Rahn, and J. M. Bennett Previously published vector equations describing angle-resolved scattering from single-layer- and multilayer-coated optics have been integrated numerically and analytically over all angles in the reflecting hemi- sphere to obtain numerical results and analytical expressions for total integrated scattering (TIS). The effects of correlation length, polarization, angle of incidence, roughness height distribution, scattered light missed by the collecting hemisphere, and roughness cross-correlation properties of the multilayer stack on the TIS expression are considered. Background material on TIS from optics coated with single opaque re- flecting layers is given for completeness and comparison to corresponding multilayer TIS results. It is shown that errors can occur in calculating the true rms surface roughness from actual TIS measurements; ways to correct these errors are discussed. 1. Introduction Total integrated scattering (TIS) measurements have been used for many years to estimate the rms roughness of optical surfaces covered with single opaque reflecting coatings. Davies1 was an early contributor to a theoretical relationship between rms roughness and TIS. His theory assumed that (1) the surface roughness was small compared with the wavelength of light illuminating the surface, (2) the surface was perfectly conducting (100%reflectance), and (3) the distribution of surface heights was Gaussian. Later Bennett and Porteus 2 and Porteus3 generalized Davies' work. All these authors based their theories on the scalar Fresnel-Kirchhoff diffraction formula and considered only opaque reflecting surfaces. Later Eastman4 applied Airy summations and matrix methods to TIS and other problems involving multilayer optics. His work was also scalar, assumed normal incidence illumination, and invoked the Kirchhoff boundary conditions.5 In all of the above theoretical treatments, it is assumed that the lateral separation of surface irregularities is for the most part greater than a wavelength, so that the scattered light is approximately confined to a diffuse cone centered about the specular beam. With this assumption, small angle approximations can be made, although the assumption is not valid for some common types of surfaces. Additionally, most scalar theories such as those mentioned above do not consider illumination of the surface at a non-normal angle of incidence, the degree of polarization of the incident beam, and the effects of different correlation functions to describe the statistics of the rough surface. In this paper we use angle-resolved scattering (ARS) formulas6 7 to investigate TIS. These formulas have been derived by perturbation methods and assume only that the rms surface roughness is much less than the wavelength of the incident light beam. The angle of incidence, polarization, optical constants, and correlation function for the surface roughness are parameters that may be chosen to fit a specific situation. In the case of multilayer optics, the autocovariance function of each interface and the cross-correlation functions between interfaces may be selected based on available evidence or reasonable assumptions. The TIS is calculated from the ARS formulas by integrating them over the hemisphere containing the incident, reflected, and scattered beams. This approach also gives us the opportunity to study the effects of scattered light which may be lost through the aperture in the collecting hemisphere that admits the incident and specularly reflected beams as well as any large angle scattering that The authors are with U.S. Naval Weapons Center, Physics Division, Michelson Laboratory, China Lake, California 93555. Received 10 November 1982. also misses the collecting hemisphere (Fig. 1). The experimental arrangement for a TIS measurement is shown in Fig. 1. The laser beam is normally 15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS 3207 mirror used to collect the scattered light (Fig. 1), thus explaining why the roughness of diamond-turned mirrors measured on such an instrument may be much less than a profile roughness measurement on the same surface. In Sec. V the relation between TIS and correlation length is discussed for one example of a dielectric multilayer stack, both for normal and nonnormal incidence illumination, and for two types of roughness cross-correlation. All the above results are BEAM INCIDENT BEAM summarized in Sec. VI. Sections II-IV consist primarily of background material designed to establish a base for comparison with TIS from multilayer optics and to discuss some basic concepts of TIS. The material in these sections, and more, has been previously discussed in detail by Church et al.9 Fig. 1. Schematic diagram of apparatus to measure TIS. Most of the normally incident laser light scattered by sample S is collected by the aluminized hemisphere (Coblentz sphere) C and focused onto detector D. Some scattered light (-* ) is lost through the hole along with the specular beam and beyond the large angle limits. incident on the sample through an aperture in the collecting hemispherical mirror (Coblentz sphere). The specular beam also exits through this aperture. Light scattered by the sample is focused by the sphere onto a detector, yielding a measure of the diffuse reflectance. Note that although most of the scattered light is collected, that light scattered close to the incident and specularly reflected beams is lost as well as that scattered nearly parallel to the surface. TIS is defined according to the relations TIS = diffuse reflectance specular + diffuse reflectance (la) where the diffuse reflectance is the fraction of the incident beam which is collected by the Coblentz sphere, and the specular reflectance is the fraction of the incident beam that is reflected into the specular direction. Thus the numerator is the light scattered out of the specular direction, and the denominator is the total reflected light. In Sec. II we discuss the concept of spatial wavelengths of the surface roughness and show how these relate to the autocovariancefunction for the surface and to the so-called correlation length. Then we integrate the ARS expressionsto obtain the TIS for the casewhen the surface correlation length is much longer than the illuminating wavelength (the usual assumption) and, second, when the correlation length is much shorter than the wavelength. A Gaussian autocorrelation function is assumed in the derivations. All the above derivations are for normal incidence illumination on the surface. In Sec. II.B is shown how the results are modified for non-normal incidence illumination. In Sec. III we show that it is not necessary to have a Gaussian distribution of surface heights for the TIS expressions to hold, and, in fact, the surface height distribution need not even be symmetric about the mean surface level. Section IV is important in that it considers the effect on measured surface roughness of light lost through the aperture in the hemispherical 3208 APPLIED OPTICS/ Vol. 22, No. 20 / 15 October 1983 11. TIS from an Opaque Reflecting Coating as a Function of Correlation Length In this section we will (1) introduce the concepts of surface spatial wavelength, autocorrelation function, correlation length, and power spectral density and (2) show how the ARS expressions can be integrated to obtain a relation for the TIS. The case where the correlation length is long compared to the illuminating wavelength is considered first, followed by the other extreme of a correlation length much shorter than the wavelength. Normal incidence illumination on the surface is the most common situation (Sec. II.A), but non-normal incidence illumination (Sec. II.B) is also considered. The roughness on an optical surface can be considered to be composed of a Fourier series of roughness components of various amplitudes and periods. For an isotropic polished surface, these components will be oriented in random directions; but for a diamondturned or lapped surface, most of the roughness components willbe aligned parallel to the cutting or lapping direction. Each of the Fourier roughness components will have a single periodicity or spatial wavelength, which is the reciprocal of the spatial frequency, and will diffract light in a direction that is determined by the well-known diffraction grating equation, X= d Isin 0o ± sinOI. Here X is the illuminating wavelength, d the spatial wavelength (grating spacing), 00 the angle of incidence (measured from the surface normal), and 0 the diffraction angle (also measured from the surface normal). In this equation it is assumed that the plane of incidence is perpendicular to the direction of the spatial wavelengths (grating grooves). From this equation we can obtain two very important facts about optical scattering: (1) surface spatial wavelengths which are large compared with the wavelength of the illuminating light produce scattering very close to the specular direction; (2) large angle scattering (away from the specular direction) is produced by progressively shorter spatial wavelengths. In the limit when d = X, the scattered light is at a grazing angle along the surface for normal incidence illumination. Surface spatial wavelengths that are shorter than the incident wavelength cannot scatter into the reflecting hemisphere except by other mechanisms such as surface plasmon absorption and subsequent reemission. Most optical surfaces do not have a single spatial wavelength roughness component but contain a large range of roughness spatial wavelength components. The roughness of these surfaces can be conveniently described by a so-called autocovariance function 8 which takes into account both the amplitudes and periods of the various roughness components. This function is a measure of the average correlation between two points separated by a distance x, called the correlation or lag distance. For calculational purposes, it is convenient to assume a Gaussian autocovariance function, which has a value equal to the mean square surface roughness for zero correlation distance. tropic so that G(r) = G(r) and g(k) = g(k), where result J' J0 ticularly for lag distances slightly larger than zero. The distance at which the autocovariance function drops to I/e of its initial value is frequently called the correlation length and will be so defined in this paper. When this correlation length is large compared with the illuminating wavelength, most of the light will be scattered dkkg(k) = 2ir 2 (3) . It should be emphasized that this equation is useful for calculating the mean square roughness 62 only when g(k) is known over the full range of surface wave numbers k. The autocovariance function G (r) is a measure of the average correlation between two points separated by a distance '. A common example of a G(T) function is the Gaussian autocovariance function: Many real surfaces have autocovariance functions closer to exponentials, par- = Jr and k = I k . Since d 2k = kdkd4, we can integrate Eq. (2a) over 0 for the case when r = 0 and obtain the G(T) = 62exp(-r 2 /uy2 ), (4a) where u is the correlation length. From Eq. (2b) we find that a Gaussian autocovariance function also has a Gaussian power spectral density function: g(k = r52U2exp(-k2u2/4). (4b) hemisphere to yield the customaryrelation between TIS The power spectral density function plays a significant role in determining the angular distribution of scattered light. This is the part of the angular scattering expression [Eqs. (5)] which contains the statistical properties of the surface roughness. An important assumption made in deriving Eq. (lb), and rms roughness for a surface covered with an opaque reflecting coating1 0 : long compared with X. This is an assumption that is into a cone about the specular direction; when it is equal to or smaller than the illuminating wavelength, there will be appreciable large angle scattering. The ARS expressions 6 can be integrated over the scattering 147b 62 TIS-, (lb) where 6 is the rms roughness and X the illuminating wavelength. The designation indicates that the correlation length u [defined in Eq. (4a) and discussed later in this section] is long compared with the illuminating wavelength. Equation (lb) is for normal incidence illumination. In practice, 6 would be calculated from Eq. (lb) using the definition of TIS in Eq. (la). To establish the notation we will be using, we will now discuss autocovariance functions, power spectral density functions, and their respective parameters: rms roughness and correlation length. The autocovariance function 8 and power spectral density function form a Fourier transform pair where G(= (12 , fd kg(k) 2 exp-ik *r) (2a) is the autocovariance function of the surface roughness and 2 g(k) = Sd rG(r) exp(ik T) (2b) is the power spectral density of the surface roughness. The autocovariance function may be defined by G(r) = (z (p)z (p + r)), where ( ... ) denotes ensemble average and z (p) describes the height of the surface roughness above and below the mean surface level at point p = (x,y). From this definition, it can be shown that G(0) = 62,11 where 2 is the mean square of the surface roughness. G(T) and g(k) are 2-D functions. In this paper, we will assume that the optical surfaces are iso- as we shall see, is that the autocorrelation length a is not necessarily realistic; in what follows we will see how this assumption can lead to serious errors. We reproduce the ARS expression from Ref. 6: 1 dP ((J/C)4 -= P dQ 72 XI cOO cos 2O1ll- EI2 g (k-ko) IX012 + 1x04' ljq + ql 2 jq + q2} I (5a) where X = (q'q' cosk - kkoe) costk' (w/c)q' sino sino' qo + q0 q0 + qoe () [q' sino C coso' q0 +qoE (w/c) cosk sinl/] qc+q0 (5b) (5c) The left-hand side of Eq. (5a) represents the differential power per unit solid angle dQ = sinOdOdk scattered into direction (0,k) (Fig. 2). The scattered power is normalized to the power P incident on the surface. As shown in Fig. 2, Oois the polar angle of incidence where the plane of incidence is the x - y plane. The angles 0 and 0 are the polar and azimuthal angles of scattering, respectively. The angle ' is the angle of the incident electric field vector relative to the plane of incidence. If O = 0 (r/2), the incident beam is p-polarized (spolarized); i.e., the incident electric field vector is parallel (perpendicular) to the plane of incidence. The dielectric constant of the scattering surface is , which can be a complex quantity. The free-space wave number is /c = 27r/X, where X is the incident wavelength. Other definitions include q = (/c) cosO, q = (WO/c)cos00 k = (/c) sinO, ko = (co/c) sinGo, q' = 15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS [E(W/c) 2 3209 Alternately, Eq. (9a) can be written in terms of k-space with dQ = d 2k/[(cw/c)2 cosO], where d 2k = kdkd5. Integrating over 0 yields DENT BEAM P SCATTERED BEAM TIS = -= (w/c)2 2] [2- (kX/27) 0 2,r/X 7d d O g )X (9b) Po 7 f, ~[I - X201/ where g(k) may be modeled based on experimental OPTICAL SURFACE evidence. Recall that Eqs. (9) are for normal incidence illumination, Ro = 1, and isotropic surface roughness. If the correlation length a >> , g(k) in Eq. (4b) may be approximated by Fig. 2. Schematic diagram showing notation for the ARS formulas. Light is incident at angle 0o; the polar and azimuthal scattering angles are 0 and 0, respectively. k2]1 2 , q' = [(w/c) - ko]l/2 . For the special case of normal incidence (00 = 0, ko = 0), Eqs. (5) reduce to 4 1 dP (wO/C) - 12 = ( cos20 g(k) P dQ 712 I 1+ ~-J2 X [I q'I2 CoS20+ (W/C)2 in 2 k. (5d) X~j q -,I2 + q 2Sd) - 2 .' According to the definition of TIS in Eq. (la), we may relate Eqs. (5) to TIS by the expression TIS= P/P P Ro +P/Po (6a) RPo d (6b) is the scattered light integrated over the entire scattering hemisphere (SH) and Ro is the specular reflectance. The approximation in Eq. (6a) is consistent with first-order theory in that the diffuse reflectance (scattering) is much less than the specular reflectance. A. Normal Incidence Illumination For the moment we will confine ourselves to normal incidence illumination and consider only Eq. (5d). To calculate TIS as defined in Eq. (6a), we must integrate (1/Po)dP/d over 0 and 0, where 0 ranges over the angular region 0 - 7r/2 and 0 ranges over 0 - 27r, and divide by Ro. We first assume that the surface has a reflectance of 100% (Ro = 1), so that e I 1 and 0' = 0 (electric vector defining the 1 = 0 direction). (5d) then becomes 1 d= PodQ d Po PO SH dQ where the integration is performed over the entire scattering hemisphere. Also, since g[(w/c sinO] is assumed to be independent of 1,the integration over 0 can be performed with the result that TIS = 3210 J (11) The subscript in Eq. (11) again refers to u/X being large compared with unity. Although a Gaussian form of autocovariance function [Eq. (4a)] was used in the derivation, any other form of the autocovariance function would also be valid as long as /X >> 1. The point we are making here is that when a TIS measurement is made on a surface for which /X >> 1 and the reflectance is high, the resulting value of 62obtained from Eqs. (lb) It is also possible to omit the assumption of 100% specular reflectance. To do this, we again assume that /X >> 1 but impose no restrictions on I 1l. We let N 2 = , where N is the complex refractive index. Since a/X >> 1, most of the scattered light will be concentrated in a cone about the specular direction and the significant range of integration of Eq. (5d) will occur for 0 << 7r/2. We can then approximate q' n (co/c)-v'Tand q (co/c), since Ie >> sin 20 for 0 << r/2. Equation (5d) then becomes 1 dP dO sin0(1 + cos 2 O)g[(co/c) sinO]. (W/C)4 1 Po dQ 71.2 - 1 N12 +N Thus, when /X >> 1, the ARS is proportional to the normal incidence specular reflectance Ro = (1 - N)/(1 + N) 2. Using Eq. (4b) forg(k) and integrating Eq. (12) over the reflectance hemisphere yields 1 dP P 167r262 PO 0 (13a) X so that (7) where k = IkI has been replaced by (co/c) sinO. Since Ro = 1, the TIS in Eq. (6a) is given by 1 TIS. = (161r262)/X2 POf dQ 712 P where 6(k)is the Dirac 6-function,not to be confused with the rms roughness 6. Using this result in Eq. (9b) yields Equation 2 (COS20+ sin 2o COS 0)g[(W/C) sinO], () (10) or (11) is accurate. where JSH d 27rb2 [6(k)/k], g(k) - (9a) APPLIED OPTICS / Vol. 22, No. 20 / 15 October 1983 TIS = P RoPo 167r6 A2 (13b) in agreement with Eq. (11). Equation (13b) is valid for surfaces of any reflectance and /X >> 1. The approximation aIX- - for g(k) in Eq. (10) need not be the only approach to deriving Eq. (lb). An alternate method is to recognizethat the argument of the exponential term of Eq. (4b) is 7r2 (af/X)2 sin 2 G [since k = (co/c)sinG],and for a>>X the major contribution to the integral over 0 in Eq. (9a) is for 0 << 7r/2, as mentioned above. Thus we replace sinG - 0, cosO 1 - 02/2, and extend the range of integration over 0 to 0 -. The result is again Eq. (11). At this point, we call attention to the fact that the TIS in Eq. (11) varies as X- 2 , since for /X << 1, considered in the next paragraph, the wavelength dependence will be as X-4 . We now consider the other extreme case when the correlation length is much smaller than the illuminating wavelength. Now the argument of the exponential term in Eq. (4b) is small for k values from 0 - 27r/X (or 0 values from 0 - 7r/2), and the exponential term in this equation may be approximated by unity: 2 2 7r6 t U. g(k) (14) From Eqs. (9) we now obtain the result that 64 r4 62o-2 TISo =- 3 (15) 4 X where the subscript 0 refers to the case a/X <<1. The TIS in Eq. (15) varies as \- 4 and also depends ol the correlation length . Since the exponential term in Eq. (4b) was approximated by unity, any power spectral density function which is nearly constant over the range with Eqs. (11) and (15). However, the quadratic region is only in effect for a/X $S 0.1, which is a shorter corre- lation length than can be measured with a surface profiling instrument (for X = 0.6328 ,m, a 0.06 Am), 12 although correlation lengths considerably smaller than 0.1 um have been calculated from surface plasmon measurements on rough silver films.1 3 On the other hand, for /X ' 0.6, the plot levels off at unity and remains there for larger values of /X. This yields a- > 0.38 Am for X = 0.6328 Am. Thus, it would appear that a/Xfrequently lies in the transition region between the quadratic range and the constant value of unity. This transition region can be approximated by a straight line, as TIS/TIS. = 2.58(a/X) - 0.161 for 0.1 < a/X < 0.4. For this range of /X, 6 _ (X/47r)(TIS)l/2 [2.58(a/X) - 0.161]1/2. Note also that for /X > 0.6, 6 _ (X/4r) (TIS) 1/2 ; for /X < 0.1, 6 (/8)(X2/72af)(TIS)l/2. In summary, whenever a-/X< 0.6, 6 calculated from TIS_ [(Eq. (11)] will be too small. of integration can be used in the deviation. The result B. as in Eq. (15), however, will depend on the value of the In all the preceding derivations it was assumed that the surface was illuminated at normal incidence. For non-normal incidence illumination, it will be shown in Sec. III that for 6 <<X and >>X power spectral density function at the origin. The results obtained from the two limiting cases in Eqs. (11) and (15) do not depend on the overall shape of the power spectral density function. Obviously, a value of a/X not within the range of approximations used in Eqs. (11) and (15) would mean that we would need to know the detailed shape of g(k) over the range of wave numbers k applicable to the TIS scattering hemisphere. When 2 is calculated from Eq. (11) or (15), the results will generally be different. As an example, let = 0.6328 m and a = 0.1 um, typical values for many polished glass surfaces. For a given TIS measurement, calculation of the rms roughness from either Eq. (11) or (15) yields the ratio (bo2 I'DJ 3 4wr2 (cr/X)2 '(16) which is 3.02 for these parameters. The 60and 65. are the rms roughnesses calculated from Eqs. (15) and (11), respectively, and their ratio is thus 1.74. The question arises as to which formula is more appropriate to use with state-of-the-art optical components. Since correlation lengths are typically 0.1-0.2 m for supersmooth polished fused quartz surfaces coated with silver or aluminum, it would seem that Eq. (15) would be more appropriate for such surfaces in light of the a/X << 1 approximation. Non-normal Incidence Illumination TIS show TIS/TIS 0 vs /X for = 30 and 45°, respectively. The rough scattering surface is assumed to be silver with E = (-16.4, 0.53) at X = 0.6328 gm. In both cases, the ~I C : w To answer this question more origin and levels off to unity as a increases in agreement (17) where 00is the angle of incidence of the illuminating beam. There is no restriction on the reflectance of the surface. Equation (17) is thus analogous to Eqs. (lb) and (11) for normal incidence illumination. By comparing Eq. (17) to Eq. (11),it is seen that the TIS values are in the ratio of cos 2 00. Thus 6, calculated from Eq. (11) but using non-normal incidence illumination, will be too small. The error will increase as 0 increases. When a-/Xis no longer large compared to unity, it is most convenient to integrate numerically Eq. (5a) to obtain the dependence of TIS on /X. Figures 4 and 5 I I I I I 1.0 I I I / 0.8 / t 0.6- precisely, we have numerically integrated Eq. (5d) over the reflectance hemisphere and plotted the results in Fig. 3, using Eq. (4b) for g(k). The y axis is the TIS normalized with respect to TISc-, Eq. (11), while the x axis is the ratio of correlation length to wavelength. The scattering surface is assumed to be silver with = (-16.4, 0.53) at X = 0.6328 m [ = El + iE2]. In fact, very nearly the same curve is obtained for uncoated glass as long as the scattered light is divided by the specular reflectance [see Eq. (la) for the definition of TIS]. As seen in Fig. 3 the plot is quadratic near the = (47rbcosOo)2 o 0.4- n 0.2- 0 Z 0 0 0.2 0.4 0.6 0.8 1.0 CORRELATION LENGTH/WAVELENGTH, a/X Fig. 3. Normalized TIS/TIS_ vs o-/Xfor an opaque highly reflecting surface. The light is assumed to be normally incident on the surface, and the scattered light is collected for all angles between 0 and 90°. TIS_ is for the same surface when >> X. 15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS 3211 4 Although in most analytical treatments of scattering 1.0 the heights of surface irregularities are assumed to have C-) a Gaussian distribution about the mean surface level, this is not necessary for the TIS relations to hold. In fact, the height distribution function need not even be @ 0.8 ZT t 0.6 symmetric, as will be shown in this section. Since en- ergy is conserved, the TIS for an opaque highly reflecting surface must be equal to 1 - R 0 , where Ro is the 04 I-_ 0 N 4 specular or coherent reflectance. We assume that the ratio of rms roughness to wavelength is <<1and that 0.2 there are no correlation length effects. Thus we assume 0z n 1,s -0 0.5 1.0 1.5 2.0 CORRELATION LENGTH/WAVELENGTH,a/X Fig. 4. Same as Fig. 3 except that the angle of incidence o is 30°. TIS_ is for normal incidence and a >>X. The solid (dashed) curve is for p-polarized (s-polarized) incident light. For aIX > 2, the TIS/TIS. curve is nearly constant at cos 230' = 0.75. Qir us that the exiting wave fronts have height variations that are in 2:1 correspondence to the respective surface height variations independent of the transverse separation of the scatterers. With these assumptions, we use the Kirchhoff diffraction integral to compute the specular fields ER and ES for a rough and smooth surface, respectively. The ratio Ro = (ER/ES)2 , which for 6/X << 1 will be slightly less than unity, and, therefore, TIS = 1 - (ER/Es) 2 . The form of the Kirchhoff diffraction integral for the specular field produced by the reflection of a plane wave 0 from a slightly rough surface A' at a distance r >> 8 0 is V) ~_ T E(r) = cosoo da' E(x'y) exp 27- 1 rO+ 2(x',y') cosOol (18) N where 0 0 is the angle of incidence. .: To Z0 0 IC I 0.5 CORRELATION' 1.0 1.5 LENGTH/WAVELENGTH, 2.0 a/\ Fig. 5. Same as Figs. 3 and 4, except that the angle of incidence 0 0 is 45°. The solid (dashed) curve is for p-polarized (s-polarized) in- cident light. For a/ > 2, the TIS/TIS- curve is nearly constant at cos245 0.50. results are significantly different than for 00= 0, and also the curves are different for s- or p-polarized incident light. The asymptotic values of the curves in Figs. 3-5 are proportional to cos2 00 , as predicted in Eq. (17). All the theoretical calculations in Figs. 3-5 have assumed a Gaussian autocovariance function for the rough surface. This is not necessary since the formulas could be numerically integrated with any analytical autocovariance function that fits the experimental surface. Also, the assumption of a Gaussian distribution of surface heights is not necessary, as has been discussed by Porteus 3 and is further considered in the following section. Ill. TIS Independent of the Form of the Surface Roughness Height DistributionFunction In this section wepresent a development based on the Kirchhoff diffraction integral. The primary reason for this change in formalism is that the surface roughness height distribution function is relatively simple to incorporate. Conclusions similar to those of this section have been obtained by Church et al.9 by verbal and analytical arguments. 3212 APPLIED OPTICS / Vol. 22, No. 20 / 15 October 1983 A point on the sur- face is defined by its position (x',y') and height (in the z direction) ¢(x',y'). The parameter r = {(x - XI)2 + (y - y') 2 + [ - (x',y')]2}1 /2 is the distance from the point on the surface [x',y',(x',y')] to the observation y') 2 + z 2] 1/2 is the distance from the point on the mean surface plane (x', point (x,y,z); ro = [(x - x') 2 + (y - y',O) to the observation point. Since the observation point is many wavelengths removed from the surface, we may write r n ro, and then the only dependence on the roughness t(x',y') is in the exponential term. The average value of the electric field for an ensemble of surfaces may be computed by using the height probability density function D(P). This function has the propertiesthat SJD(O)dt = 1, (19a) or that the area under the probability density is unity, and S D(g)tdv = A (19b) which is taken to be zero ( = 0) by definition of the mean surface level. Also X D()? 2d!= a, (19c) which is the mean square roughness value. More generally, E. D()f()d = (f()), (19d) wql where (f (g)) is the ensemble averaged value of a function f(A). Note that wehave made no assumptions that D (P)is symmetric about the origin D= 0 (mean surface level). Since the surface roughness is nondeterministic, we cannot calculate Eq. (18) exactly. Instead we must use ensemble averaging techniques as in Eq. (19d). The ensemble average of E(r) is (E(r)) = dA' E Ai 2Xi A' i X exp o IV. Surface Autocovariance Functions and TIS Angular Measurement Limitations In previous sections we have considered only the TIS C d D E [ro + 2t(x',y') The primary conclusion from this section is that the standard TIS formula does not depend on the surface height distribution function being Gaussian or even symmetric about the mean surface level if 6/X <<1. arising from the spatial wavelengths on a rough surface coso] (20a) and have not considered the limitations introduced by the system that measures the TIS. A hemispherical collector of the type shown in Fig. 1 must have a hole to which may be rewritten as (E(r)) = El(r) dgD(P) exp X cosOo), Ar (20b) where El(r) = E cos~o r dAE(x',y') 10 dA' 2Xi fA' ro (27riro( exp t (21) For a perfectly smooth surface t(x',y) = 0 and D(P) = W(r),which is the Dirac -function. This yields the simple result that (E(r)) = E1 = Es. For more general cases where (22) /X << 1 and 62 = ( t2(x',y') ), we may expand the exponential term in the integrand in Eq. (20a) to yield ER = (E(r) ) X [1 + 4-i E1 pass the incident and specularly reflected beams; it also cannot collect all the light scattered at grazing angles to the surface. Thus very near angle scattering and grazing angle scattering will be missed, and the roughnesses of the surface spatial wavelengths that produce this scattering will not be measured. [Any roughness features with surface spatial wavelengths that are shorter than the illuminating wavelength (for normal incidence illumination) cannot be measured even with a perfect collecting hemisphere since they would produce scattering at larger than grazing angles to the surface.] In this section we will introduce the concept of an effective roughness 6e which is that value that would be measured if the polar angle limits on the TIS measurements and show how d gD() cosSo + I (4ri COSo) (23) 2]. By the conditions given in Eqs. (19), the first two integrals are unity and zero, respectively. Since TIS = 1 - (ER/ES)2 , we obtain from Eqs. (23) and (22) were >0 and <7r/2. We will derive general relations with variable collection angle limits 6 e differs from as a function of these limits and of the correlation length of the surface. Finally, we will give examples of surfaces having different correlation lengths including one that has both long and short correlation lengths to show how the effective roughness measured by TIS differs from the true roughness. We will assume that g(k), the power spectral density of the true surface roughness, is known. Then, if the surface is isotropic so that there is azimuthal symmetry, TIS = 1 - [1 - 2 2 87r252 cos 00/X2] 16r22 cos2 0 4) Note the dependence of the TIS on angle of incidence 00 and also that, for normal incidence ( 0 Eq. (3) yields the true mean square roughness surface is less than a wavelength. In terms of the spatial wavelength components on the surface, no wave front height variation may occur when the spatial wavelengths are less than the illuminating wavelength and the surface is being illuminated at normal incidence. In other words, the light is not scattered by more than 90° from the mean surface normal. 6 e by the relation = 0), the result reduces to that of Eqs. (lb) and (11). Note also that this result does not depend on D(t) being Gaussian or even symmetrical. This is by virtue of the assumption b/X <<1, which means that the light is not sensitive to the roughness peak-to-valley ratio because it cannot resolve this ratio. The result for TIS given in Eq. (24) assumed that all the roughness frequency spectrum contributed to the scattering, which is equivalent to the assumption o/X >> 1. It does not take into account the reduction in exiting wave front height variation which occurs when the spacing between adjacent peaks and valleys on the By 62. analogy we can define an effective rms roughness 52= f'dkkg(k), e 27r (25) where the limits of integration are variable to include the nonideal collection limits of the hemispherical collecting mirror. In k-space, these limits are ca= (27r/X) sinG1 , ( > 0), and 3 = (2ir/X) sinG2 , ( < 27r/X), where G1 and 02 are the polar angles subtended by the limits of the specular exit hole and the rim of the hemisphere. In other words, some scattered light is lost through the specular exit hole and also beyond the large angle limit of the Coblentz sphere, as shown in Fig. 1. These losses can affect the accuracy of 62 obtained from actual TIS measurements. Equation (25) can be related to the autocovariance function using Eq. (2b) for (ko = 0), which yields eS= J dk k f d T G()Jo(k,), (26) where G(r) = G (T) (azimuthally symmetric), and the 15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS 3213 wI azimuth integration of r has been done yielding the zero-order Bessel function JO. If we choose a Gaussian autocovariance function G(T) = 62 exp(-r 2 /o 2 ) and an exponential autocovariance function G (r) = 62 exp(-IT I/o), we may derive from Eq. (26) (1e)2 1 22d 1 P2a2\ 2 - exp (- l(^)2= exp (- ) (27a) Equation (32a) agrees with Eq. (11). However, Eq. (15) is larger than Eq. (32b) by a factor of 4/3. The difference is attributed to the lack of information in the development of this section of the angular dependence of dipole scattering, which is inherent in the ARS formulas. To a first approximation and for normally incident light, the dipole scattering currents on a slightly rough surface are oriented in the same direction as the incident for the Gaussian case, and electric field vector. Thus the dipole scattering cur(27b) rents are parallel to the mean surface plane. A Rayleigh for the exponential case. Equations (27) are in agreement with Church et al.9 In the indicated asymptotic angles. Hence it is clear why Eq. (32b) (which does not lf = 1 1+ I cases we find (Se2 I ) fJr 2 ( 2 - ae2)/4 _ exp(-a2a2/4) a <<A, (28a) a >> A, (28b) a <<A, a»> A, (28c) (28d) for Eq. (27a) and re2 0a2(02- a 2 )/2 l1/ar for Eq. (27b). The result for >>Xindicates that most of the scattered light is passing through the specular exit hole, and thus the amount of light collected approaches zero. In this case, the effective roughness be obviously approaches zero. Note, however, that the results of Eqs. (28b) and (28d) are very different in their asymptotic behavior. Because of this, it is possible to realize large numerical differences with these asymptotic formulas. In this case, the choice of the autocovariance function becomes very important. On the other extreme, when /X << 1, the TIS also approaches zero. This is because the surface spatial wavelengths are, for the most part, shorter than the incident wavelengthand thus cannot produce direct scattering into the hemisphere (for normal incidence illumination). In the perfect case when a = 0 and 1 = 27r/X,we have (5e)2 )l 72oa2 /X2 I A, a >> , a (29a) (29b) for the Gaussian case and lbeU (b )/ f 2ir2 a 2 /A2 1 a << A, (29c) a >> A, (29d) for the exponential case. To relate Eqs. (29a) and (29b)to previous TIS results [Eqs. (11) and (15) for Gaussian autocovariance functions, a = 0 and 13 27r/A]in the same limiting cases, we write TIS = (167r2 52)/X 2, (30) where e has replaced 6 in Eq. (11). From Eq. (27a) for a = 0 and 13= 2r/X and Eq. (30), we have dipole exhibits a cos2 0 dependence [seeEq. (12)], and thus the scattering intensity falls off for larger scattering take into account dipole scattering properties) is slightly larger than Eq. (15) (which does contain the dipole angular scattering properties). The result of Porteus3 was limited to small angle approximations and thus did not contain the larger angle dipole scattering properties. This explains the agreement of Eq. (31) with the Porteus results.3 If Eq. (31) is plotted as TIS/TIS., vs the ratio of correlation length to wavelength o/X, the curve is essentially identical to that in Fig. 3. Thus, it follows that in principle Eqs. (27) may be used to correct 6e for general values of a and for lost scattered light provided that the autocovariance function is Gaussian. The values of a and will be known from the geometry of the collecting hemisphere, but the correlation length a may not be accurately known for a given sample. If there is a reason to believe that much of the shape of the autocovariance function is exponential, perhaps Eqs. (28) can be used to correct the TIS obtained value 62to the true value 62. Strictly speaking, exponential autocovariance functions are not physically realistic when the origin ( = 0) is included. However, many experimentally determined autocovariance functions have an exponential shape away from the origin. We will now show how the angular limits on the col- lecting hemisphere affect the measured TIS and thus be for surfaces having short ( = 0.2-,m), medium (a = 2m), and long ( = 10-,m) correlation lengths. The short correlation length surface might be a conventionally polished glass surface, while the longer correlation lengths might be associated with chemically polished, electropolished, or diamond-turned surfaces. All surfaces are assumed to have true rms roughnesses 6 of 10 A. We assume that the hole in the collecting hemisphere that passes the incident and specularly reflected beams subtends a polar angle 1 = 2.85° (normal incidence illumination) and that the large angle limit 0 2 is 79.1° (the values for the China Lake instrument). If the measuring wavelengthX= 0.6328,um, / = 0.316, 3.16, and 15.8, respectively, for the three surfaces. Figure 3 shows that for a/X = 0.316, TIS/TIS_ (31) = 0.66; for the larger values of o/X, the ratio is unity. Thus, for a perfect collectinghemisphere, the measured which is in agreement with the result of Porteus. 3 From TIS for the two longer correlation length surfaces would Eq. (31) we find that for the indicated limiting cases give the correct rms roughness, but the measured TIS for the 0.2-,m correlation length surface would be too TIS = 2 [1 - exp(-7r 2a2 /X2 )], ),2 TIS = 167r22/A2 167r462a 2 /X4 3214 a>> , a << . APPLIED OPTICS/ Vol. 22, No. 20 (32a) (32b) / 15 October 1983 small, and the value of 6 calculated from Eq. (11) would be \ X 10 A, or 8.1 A rms. 1.00 0. Z 061 4 0.2 0 5.00 10.00 15.00 20.00 25.00 CORRELATION LENGTH/WAVELENGTH, U/\ Fig. 6. TIS/TIS- vs u1Xfor a surface having a two-Gaussian autocorrelation function and measured using an apparatus similar to that shown in Fig. 1. The short-range correlation length ars is held constant at 0.35 Aum,while the long-range correlation length varies from to 16.0 m. The short- and long-range rms roughness values are 33.6 and 47.5 A, respectively. For this type of surface (similar to an actual diamond-turned surface), significant light is lost through the specular beam exit hole. Now we will calculate what the restricted collection angles of the hemisphere do to the TIS measurements. We will assume that the autocovariance functions for all three surfaces are Gaussian so that Eq. (27a) is applicable. The collection angles 1 = 2.85° and 2 = 79.1° yield a = 0.494 gm- 1 and 13= 9.75 gm- 1 , respectively. Equation (27a) predicts be/6 values of 0.78,0.88, and 0.047 for the o-values of 0.2, 2, and 10 gm, respec- tively. Physically, we see that for the o-= 0.2-gm surface the limiting inner aperture introduces essentially no error, but that appreciable light is scattered beyond 79.10. For the o = 2-gm surface, the inner aperture is cutting out some of the scattered light, but the large angle limit has no adverse effect. For the o-= 10-gm surface, most of the light is scattered into angles of <2.85° so that only 0.0472 or 0.0022 of the correct TIS amount is collected by the hemisphere. This last result is somewhat surprising since if only a single spatial wavelength of 10 gm were present on the surface it would be scattered (diffracted) at an angle of 3.60 from the specular direction and would thus easily be collected by the hemisphere. This result is particularly relevant to diamond-turned surfaces and explains why TIS measurements on some diamond-turned surfaces give much smaller roughness values than those measured by stylus instruments or interferometry.1 4 To complete the calculation that was proposed, by combining the roughnesses are isotropic so that the previously obtained expressions are valid. As shown above, when a surface has a long correlation length, most of the light will be scattered at angles close to the specular direction and thus will be lost through the specular exit hole in the collecting hemisphere. Experimentally, it has been found that for six diamond-turned samples tested the light lost through the central aperture ranged from 63 to 97%of the total measured scattered light.14 Thus the TIS measures primarily the short-range roughness of these surfaces. To show how the TIS from short- and long-range roughnesses combine, we can model the surface with a two-Gaussian autocovariance function. Comparisons between theory and experiment have shown that generally two-part autocovariance functions are needed.1 5 We can write the two-Gaussian autocovariance function as 15 G(T)= 6' exp(-T 2 /U2) + t2 exp(-- 2 /aj2). (33a) The parameters 6s, L, s, and L are the short-range rms roughness, long-range rms roughness, short-range correlation length, and long-range correlation length, respectively. We note that the long-range parameters direct scattered light in the near-specular direction, whereas the short-range parameters yield wide-angle scattering. The corresponding g(k) is effects of small aiX with the restricted collection angles g(k) = 7r sas exp(-k 2 aS/4) + a of the hemisphere, the three original 10-A rms rough- exp(-k 2 L/4)]. (33b) ness surfaces would have roughnesses of 6.3, 8.8, and 0.5 A for o-values of 0.2, 2, and 10 gm, respectively, as cal- In Eq. (33a) the long-range culated from uncorrected measured values of TIS. We carry the TIS and effective surface roughness calculations one step further by considering a surface somewhat similar to a diamond-turned surface which contains both short-range and long-range roughness components. Diamond-turned surfaces can have short-range roughness caused by tool chatter, interactions between the chip and the surface, material im- diamond tool. The short-range as is intended to include the effects of residual random roughness, which typically has a much shorter correlation length. We L is intended to simulate the effects of the long-range correlation produced by the realize that a diamond machined surface has anisotropic the form of grooves cut by the diamond tool. For this surface topography and that this is contrary to Eqs. (33). However, similar effects will be seen in TIS measurements from diamond turned surfaces as are predicted from Eqs. (33). The important condition is that both long and short range parameters are used, which is consistent with the situation for diamond turned sur- example, we will assume that both short- and long-range faces. perfections, etc., as well as the long-range roughness in 15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS 3215 W 1.4 r In 1.2 a < 1.0 cr 0 0.8 E 0.6 4-1 T 0a 0.4 0.2 z0 Fig. 7. 0 0 0.5 1.0 1.5 2.0 CORRELATION LENGTH/WAVELENGTH,a/X TIS/TIS_ vs a/A for light normally incident on a 23-layer dielectric stack, with the scattered light collected from all angles between 0and 90°. The thin films are quarterwave optical thickness at normal incidence. The solid (dashed) curve is for a correlated (uncorrelated) multilayer stack with Gaussian autocorrelation functions assumed for the film interfaces. TIS_ is for an opaque highly reflecting surface with a >>A. For a/M > 2, the solid curve remains nearly constant at unity. In the illustrative example, the 6s and 6 L values are chosen to be 33.6 and 47.5 A, respectively, which yields 6 = vAJsTWL = 58.2 A. These values are consistent with experimentally measured quantities. We have integrated Eq. (5d) over the scattering hemisphere while limiting the collection angles to the 2.85-79.1° range, as discussed above. The short-range correlation length us = 0.35 gm is held constant, while aoLvaries from 0 to 16 gim. Since as is held constant, the TIS for this example will never vanish in small or large limits of aL because there will always be a background (in this case, a constant background) of scattered light from the short-range roughness. The results of the calculations are plotted in Fig. 6 as TIS/TIS_ vs CL/X. We can learn several interesting points from Fig. 6. As the long-range uL approaches the small and large limits, the TIS/TISvalues approach the residual value produced by the short-range roughness background scattering. When cYL/X- 0, the long-range roughness scattering ceases where 62 = 6L + 6S. A plot of this equation for the same parameters associated with Fig. 6 again yields a plot which is very nearly identical to Fig. 6. Thus it is seen that for surfaces having significant long-range spatial wavelength components, there can be large errors made in calculating the total rms roughness because of scattered light lost through the specular exit hole. It should be emphasized that the discussion in this section has been based on the assumption of a known power spectral density function g(k). Thus the usefulness and validity of the formulas in this section depend on the degree to which g (k) for the unknown surface is Gaussian or Lorentzian. In reality, only a portion of g(k) or G(T) is known or can be measured, and making assumptions about g(k) beyond the measurable region can lead to errors. However, it is felt that the methods outlined in this section can provide reasonable estimates for corrections to the mean square roughness to exist because when aL/X << 1 very little scattering can occur within the hemisphere; also, that which does occur is not collected because of the large angle limits. When cJL/X >> 0, the long-range roughness is not collected for most state-of-the-art optical components, excluding because the scattered light escapes through the specular The previous sections have provided background material for scattering from a surface covered with a single opaque reflecting coating. The interpretation of the TIS from such a surface is much simpler than for a surface covered with a multilayer dielectric stack. There are two primary reasons for this: (1) thin films exit hole. The maximum value of TIS/TIS.. occurs for cJL/X= 0.83, which corresponds to CL = 0.53 gm. The TIS/TIS., curve passes through a maximum when the optimum L is reached so that losses through the specular exit hole and beyond the large angle limits are minimized. We can obtain the results of Fig. 6 in another way by using a two-Gaussian autocovariance function in an equation analogous to Eq. (27a): (t)2 = ()2 + 3216 [exp(-a 2 2L/4)- exp(_02ai/4) 2 [exp(-a 2a2/4) - exp(-2 2a /4), ( APPLIED OPTICS/ Vol. 22, No. 20 / 15 October 1983 diamond-turned optics. V. TIS from Multilayer-Coated Optics with parallel rough boundaries can produce interference effects in the scattered light and (2) the statistical relationships between the roughness at a given interface relative to the other interfaces can significantly affect the TIS. This latter effect is caused by roughnessinduced phase relationships in the light scattered from each interface. The angle-resolved vector scattering theory8 used to predict scattering from multilayer stacks is not reproduced here. However, its validity is the same as for the ARS theory presented in previous sections that applies to a single opaque metal coating. In X 1.4 < 1.2 F 1.0 w 0.8 0 t a: 0 I- CI - - t 0 5- 0.4 tI 0R J1 0 d 0.5 1.0 1.5 2.0 CORRELATIONLENGTH/WAVELENGTH,a/X Fig. 8. TIS/TIS vs a/X for p-polarized light incident at 0o = I I I l 10 O 0o 30° I I I I I I I I I 2. 3 E0 4.0 LENGTH/WAVELENGTH, CORRELATION on the same dielectric stack as in Fig. 7, except that the thin films have a quarterwave optical thickness at 300 incidence. The solid (dashed) curve is for a correlated (uncorrelated) multilayer stack with Gaussian I 0.1 Z0 MI I I E o I - I O., rzj 4 0.2 I I 0.6 N I // - t 0 z I 2- U/X Fig. 10. Same as Fig. 8 (p-polarized incident light), except that the angle of incidence is 450, and the films are quarterwave optical thickness at 450 incidence. For a/X > 4.5, the TIS/TIS_ curve for the correlated case is nearly constant at cos 2 450 = 0.50. autocorrelation functions assumed for the film interfaces. TIS, is for an opaque highly reflecting surface at 00 = 00 and a >>X. For a/X > 2, the TIS/TIS_ curve for the correlated case is nearly constant at cos2 30' = 0.75. .. . 1.4 4 I I I I I I I I I 1.2 - X r 0 1.4 4 _ (J 2 Sc0.8 - 1.0 a: - 4 0 < 1.0 cc 1.2 In ziG 0.6 0 Z g0.8 0.4 0 D _ 0.6 N 0 E Or - < n 0 0.2 0 Z 0.5 1.0 1.5 2.0 CORRELATION LENGTH/WAVELENGTH, X Fig. 9. 0.2 2 0.4 L4 Fig. 11. I I I I I I I I I 1.0 2.0 3.0 4.0 CORRELATION LENGTH/WAVELENGTH. Same as Fig. 10, except that the incident polarized. light is s- Same as Fig. 8, except that the incident light is s-polarized. this section, we will give examples of scattering from surfaces covered with multilayer dielectric films and compare them to previous results for scattering from a surface covered with a single opaque metal layer. In this section, we consider a 23-layer dielectric stack of a SH(LH)1 1A design, where S, H, L, and A represent the substrate (s = 2.25, 0.0), high index film (H = 5.29, 0.0), low index film (EL = 1.9, 0.0), and air (A = 1.0, 0.0), respectively, at a wavelength X = 0.6328 gim. The op- tical thicknesses of the films are assumed to be X/4 at the angle at which the stack is illuminated. Each interface is assumed to be rough and to have the same statistical properties of the roughness. However, two limiting cases are considered: (1) correlated roughness and (2) uncorrelated roughness. In the case of correlated roughness, each interface in the stack is identical in shape so that the roughnesses at all interfaces are correlated. Naturally, each interface has the same autocovariance function. Also, all interface pairs have however, that all interfaces have the same autocovariance function (which is the same as that of the correlated case). However, because of the assumption of independence between interfaces, all cross-correlation functions vanish. The TIS from dielectric stacks having correlated and uncorrelated roughness is quite different, as we will illustrate by the following examples. In Fig. 7 we show a plot similar to that in Fig. 3 except that the TIS is for the multilayer stack (the optical thickness of the layers is X/4 for normal incidence illumination), and TIS is the scattering that would be obtained from a highly reflecting opaque surface of the same roughness but having a large correlation length/wavelength ratio. Note that even though the illumination is at normal incidence, the correlation properties of the surface greatly affect the amount of TIS. The curve for the correlated case is very much like that in Fig. 3 for an opaque highly reflecting surface, but the TIS for the the same cross-correlation functions which are identical uncorrelated to the autocovariance function. In the case of uncorrelated roughness, the roughness shapes between different interfaces are independent. It is assumed, Since the illumination is at normal incidence, there is no difference between p-polarized and s-polarized incidence. case is approximately a factor of 2 lower. 15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS 3217 xxxx a 0 z~KzS 4LI :f 2 2 <t 11 2 Z Fig. 12. Plot of the electric field intensity for p-polarized (solid curves) and s-polarized (dashed curves) light incident on a 23-layer dielectric stack. Angles of incidence are (a) O (b) 30, and (c) 45°. The optical thicknesses of the layers are quarterwave at the given angle of incidence. At normal incidence there is no difference between s- and p-polarized incident light. Figures 8-11 show curves similar to those in Figs. 4 and 5 for multilayer stacks illuminated at 30 and 450 angles of incidence, respectively. The optical thicknesses of the layers have been adjusted to be X/4 at the appropriate illumination angle. Also TIS_ is again the scattering that would be obtained from a highly reflecting opaque surface of the same roughness, with a large a/X ratio, and illuminated at normal incidence. With regard to Figs. 8-11, first consider the curves for correlated roughness and s- and p-polarized incident light (solid curves in Figs. 8-11). These curves are quite similar to the corresponding curves in Figs. 4 and 5 in that the long-correlation length limits for the TISMtIS_ ratio tend toward the ratios that would be calculated from the cos20 value for the illumination angle: 0.75 (0.50) for 30° (45°) incident illumination. The interpretation is straightforward, since the roughness at each interface is identical, and, therefore, the scattering currents are correlated in phase throughout the multilayer stack. Also the layer pairs are halfwave optical thickness. Thus considering the optical thicknesses and phase relationships of the scattering currents, it follows that for scattering reasonably near the specular direction the scattered light will behave much like the specular beam from a single opaque surface. Since ratios oaI > 1 confine scattered light reasonably near 3218 APPLIED OPTICS/ Vol. 22, No. 20 / 15 October 1983 the specular beam, this explains why the solid curves of Figs. 8-11 approach the corresponding curves of Figs. 4 and 5. Next consider the curves for uncorrelated roughness and s- and p-polarized incident light (dashed curves in Figs. 8-11). In the case of uncorrelated roughness, there is no correlated phase relationship of the roughness-induced scattering currents between interfaces. This lack of correlation can produce significant differences in the ARS1 5,1 6 and, consequently, also the TIS. Note that in Figs.7-11 for s-polarized incident light, the TIS for the uncorrelated case generally decreases with increasing angle of illumination. On the other hand, for p-polarized incident light and uncorrelated roughness, the TIS generally increases with increasing angle of illumination. This is somewhat easier to understand by looking at the electric field distribution in the 23layer stack for angles of incidence of 0, 30, and 45° shown in Fig. 12. In the case of s-polarized incident light, the field intensity consistsof nodes and antinodes. The intensity of the antinodes decreases with increasing angle of incidence. This is consistent with the decrease in TIS, for s-polarized incident light, as the angle of incidence increases. In other words, the dipole scattering currents generated by the fields at the antinodes decrease with increasing angle of incidence. Further- more, for uncorrelated stacks, each interface is an independent sheet of currents, and thus phase relationships due to roughness shape do not yield the interference effects seen in the correlated stacks. It follows that when the electric fields decrease, the TIS will also decrease. In the case of p-polarized incident light, there are nodes and antinodes in the electric field for normal incidence. However, for the 30 and 450 cases, there are no nodes because of the normal component of the incident field. The original antinodes actually decrease very little when going from normal incidence to nonnormal incidence illumination. Thus, considering this fact along with the disappearance of the nodes, the overall distribution of the electric field in the stack increases with increasing angle of incidence for p-polarized incident light. This is consistent with the increase of TIS for increasing angle of incidence with p-polarized incident light and uncorrelated roughness. VI. have been given of surfaces having different roughness correlation lengths to show the extent of the decrease of their TIS from the ideal TIS. Also an example is given of a surface that has both short-range and longrange correlation length roughnesses to simulate a diamond-turned surface. The results for this surface clearly show that the scattering from the long-range roughness component passes through the hole in the collecting hemisphere, so that only the scattering from the short-range roughness is actually collected. Finally (Sec. V), the TIS has been calculated for a 23-layer multilayer stack assuming that the roughness between adjacent layers in the stack is either correlated or uncorrelated. Normal incidence, 300, and 450 incident illumination are considered. The magnitude of the TIS is quite different depending on whether the roughness between adjacent layers is correlated or uncorrelated. It is also shownthat the relative amount of TIS for the correlated and uncorrelated cases depends on the incident polarization. Conclusions We have shown that the previously published vector equations 6 for ARS from an opaque reflecting surface (normal incidence illumination) can be integrated over all angles in the reflecting hemisphere to yield an expression for the TIS (Sec. II). When the correlation length a of the surface roughness is long compared with the illuminating wavelength X, the TIS expression is identical to the previously published expression and shows that TIS is directly proportional to the square of the rms roughness 6 and inversely proportional to the square of the wavelength. When a is small compared to X, the TIS is directly proportional to the product 62a-2 and inversely proportional to 4. The ARS expression has also been numerically integrated to yield TIS for intermediate values of a relative to Xand is plotted in Fig. 3. The ARS expressions for non-normal incidence illumination have also been numerically integrated for in- cident illumination angles of 30 and 450, respectively (Sec. IIB), and the results are plotted in Figs. 4 and 5. The curves show that when /X > 1, the TIS value is identical to that predicted by the customary TIS equation for non-normal incidence [Eq. (17)]. A proof based on the Kirchhoff diffraction integral (Sec. III) shows that when 6 << X the TIS is independent of the form of the function for the distribution of surface heights. Specifically, the height distribution function does not need to be Gaussian or even symmetric about the mean surface level. One type of instrument used to measure TIS (Fig. 1) contains an aluminized hemisphere with a central hole through which the normally incident light beam passes and the specularly reflected beam exits. The hemisphere on one such instrument collects scattered light for angles from 3 to 790 from the specular beam and thus misses the very near angle and very large angle scattered light. In Sec. IV the ideal TIS expressions have been modified to take into account the limited collection angles for the scattered light, and examples The authors would like to thank the referees for helpful comments and suggestions. References 1. H. Davies, Proc. IEE London 101, 209 (1954). 2. H. E. Bennett and J. 0. Porteus, J. Opt. Soc. Am. 51, 123 (1961). 3. J. 0. Porteus, J. Opt. Soc. Am. 53,1394 (1963). 4. J. M. Eastman, "Surface Scattering in Optical Interference Coatings," Dissertation, U. Rochester, Rochester, N.Y. (1974). 5. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 379. 6. J. M. Elson, Phys. Rev. B 12, 2541 (1975); J. M. Elson, Proc. Soc. Photo-Opt. Instrum. Eng. 240, 296 (1981). 7. The angle-resolved scattering equations pertaining to surfaces coated with single opaque reflecting films are given in the present notation in Ref. 6. These equations have been obtained previously by D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970), Chap. 9 and subsequently by numerous other workers using various methods. 8. J. M. Elson and J. M. Bennett, J. Opt. Soc. Am. 69, 31 (1979). 9. E. L. Church, H. A. Jenkinson, and J. M. Zavada, Opt. Eng. 18, 125 (1979). 10. H. E. Bennett, Opt. Eng. 17, 480 (1978). 11. For properties of G () and g(k), see, e.g.,J. S. Bendat and A.G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), p. 18. 12. J. M. Bennett and J. H. Dancy, Appl. Opt. 20, 1785 (1981). 13. H. Raether, "Surface Plasmons and Roughness," in Surface Polaritons, V. M. Agranovich and D. L. Mills, Eds. (North-Hol- land, Amsterdam, 1982), Chap. 9. 14. J. M. Bennett, J. P. Rahn, P. C. Archibald, and D. L. Decker, "Specifying the Surface Finish of Diamond-Turned Optics-A Study of the Relation Between Surface Profiles and Scattering," in Technical Digest, Workshop on Optical Fabrication and Testing (Optical Society of America, Washington, D.C., 1981). 15. A two-Gaussian autocorrelation function was used in earlier work by J. M. Elson,J. P. Rahn, and J. M. Bennett, Appl. Opt. 19,669 (1980). 16. J. M. Elson, J. Opt. Soc. Am. 69, 48 (1979). 15 October 1983 / Vol. 22, No. 20 / APPLIED OPTICS 3219