CHAPTER 7 SURFACE SCATTERING

E . L . Church*

and

P . Z . Takacs Brookha y en National Laboratory Upton , New York

7 . 1 GLOSSARY

A 0 area a , b polarization parameters 5 ' , i

BRDF bidirectional reflectance distribution function BSDF bidirectional scattering distribution function

f spatial frequency ; f x 5 j , f y 5 h

G power spectral density function including the specular contribution I intensity in power per steradian

P generalized pupil function R Fresnel intensity reflection coef ficient S 1 one-dimensional or profile power spectral density S 2 two-dimensional or area power spectral density

θ , w angles s b a bistatic radar cross section F b a polarization factor

7 . 2 INTRODUCTION

The theory of wave scattering by imperfect surfaces has been highly developed in the fields of radiophysics , acoustics , and optics . It is the subject of a number of books , 1 – 3 book sections , 4 – 9 and review articles , 10–16 and its practical application to scattering from optical

* Work performed in part while the Author was with the U . S . Army Armament Research , Development and Engineering Center , Picatinny , N . J .

7 .1

7 .2 PHYSICAL OPTICS

surfaces has been discussed in the readable and informative texts of Bennett 1 7 and Stover . 1 8 Bennett has also published a selection of reprints from the extensive literature in the related areas of surface finish and its measurement . 1 9

The purpose of this article is to provide a frame of reference for interpreting the existing literature and understanding more recent developments in the description , characterization , and specification of scattering from optical-quality surfaces .

Discussions of related subjects can be found in the articles on ‘‘Measurement of Scatter , ’’ ‘‘Control of Stray Light , ’’ ‘‘X-Ray Optics , ’’ and ‘‘Dif fraction’’ elsewhere in this Handbook .

7 . 3 NOTATION

The most important quantity in the discussion of surface scattering is the bidirectional reflectance distribution function (BRDF) , which is defined precisely in Stover’s article on ‘‘Measurement of Scatter . ’’ The BRDF is a multidimensional function that describes the angular and polarization dependence of the intensity of light reflected and scattered from a surface , and is related to the better-known angular-resolved or dif ferential scattered intensity dI / d Ω and the bistatic radar cross section s b a , according to

BRDF b a ( θ i , θ s , w s ) 5 1

cos θ s ?

1 I i S dI s

d Ω D ba

5 s b a

4 π A 0 cos θ i cos θ s (1)

Here I i is the incident intensity (power per steradian) , I s is the far-field reflected and scattered intensity , A 0 is the illuminated surface area , and the subscripts a and b denote the states of linear polarization of the incident and scattered radiation (note the right-to-left ordering) . When the electric vector is perpendicular (or parallel) to the plane defined by the propagation direction and the surface normal , a , b 5 s 5 ' (or p 5 i ) in optics , and TE (or TM) in radiophysics . Polarization aficionados will note that the quantities defined in Eq . (1) are the square magnitudes of the components of the 2 3 2 Jones scattering matrix , and correspond to the upper-left quadrant of the 4 3 4 Stokes scattering matrix . 2 0

Acoustic (scalar) scattering is described in terms of the analogous acoustic cross section , but without the polarization notation . Instead , there are two extreme forms of the acoustic boundary conditions , which correspond to the scattering of ' - and i -polarized electro- magnetic radiation from a perfectly reflecting (i . e ., infinitely conducting) surface . The first is called a ‘‘soft’’ surface , which uses the Dirichlet boundary condition , and the second is a ‘‘hard’’ surface , which uses the Neumann boundary condition .

As explained later , the angular distribution of the scattered intensity is essentially a Fourier spectrogram of the surface errors , so that it is convenient to discuss scattering in the frequency domain , that is , in terms of the spatial frequency vector f lying in the surface plane . The components of this vector are related to the scattering angles by

f 5 S f x f y D 5

1 l S sin θ s cos w s 2 sin θ i

sin θ s sin w s D (2)

which can be viewed as a generalization of the grating equation for first-order dif fraction from a grating with the spatial wavelength d 5 1 / f . f x and f y are frequently denoted by j and h in the Fourier-optics literature .

SURFACE SCATTERING 7 .3

7 . 4 SCATTERING THEORY

Surface scattering arises from any deviation from the ideal interface that upsets the delicate interference ef fects required for rectilinear propagation . In principle , relating the BRDF to specific scattering structures is a straightforward application of Maxwell’s equations but , more often than not , approximations are necessary to get practically useful results . The best known of these approximations are the physical-optics (Kirchhof f) and the perturbation (Rayleigh-Rice) theories , which despite their advanced age are still subjects of lively investigation . 2 1

In the important case of small-angle scattering from weakly scattering surfaces , however , the predictions of all such theories are the same since in those limits the scattering is determined by simple linear-momentum considerations . For the present purposes we consider the most elementary scattering theory that captures that physics— the first-order Born approximation . This approximation has the added advantages that it includes the ef fects of the measurement system and surface ef fects on an equal footing , and can be readily generalized to include both topographic and nontopographic scattering mechanisms .

In general , scattering problems require the solution of an integral equation that relates the incident and scattered fields . The first-order Born approximation is the first term in an iterative solution of that equation , obtained by setting the total field in the integrand equal to the incident field . 3–5 , 9 The key quantity in the first-order Born approximation for the BRDF is the square magnitude of an integral containing three factors—an incident plane wave , a scattered plane wave , and an interaction term that couples them—integrated over the illuminated area of the surface . In particular

BRDF b a 5 1

l 2 F b a ( w s ) R a ( θ i ) G ( f ) (3)

where

G ( f ) 5 1

A 0 U E

A 0

d r e i 2 π f ? r P ( r ) U 2

(4)

and r is the position vector in the surface plane . Note that the single exponential factor appearing here , which causes the integral to appear in the form of a Fourier transform , is the product of the input and scattered plane waves , and P is the interaction that couples them . The remaining factors in Eq . (3) also have a simple physical interpretation : F b a is a polarization factor which equals cos 2 w s when a 5 b and sin 2 w s when a ? b , and R a is the Fresnel intensity reflection coef ficient of the smooth surface .

The interaction , P can be factored into system and surface terms

P ( r ) 5 P syst ( r ) ? P surf ( r ) (5)

where P s y s t is the pupil function of the measuring system and P s u r f accounts for surface imperfections . It is convenient to write P s u r f 5 1 1 p ( r ) , where the 1 represents the ideal surface and p is the ‘‘perturbation’’ that describes deviations from a perfectly smooth surface .

For a perfect surface , p 5 0 , and the BRDF given by Eqs . (3) to (5) is just the response function of the measuring system—a sharp spike in the specular direction whose shape and width are determined by the system pupil function . In the case of an imperfect surface , p ? 0 , Eqs . (3) to (5) predict a modification of shape and intensity of this specular core , plus scattering out of the core into larger angles . In the following discussion we concentrate on

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the scattering distribution and return to core ef fects under ‘‘Finish Specification . ’’ To make the distinction clear we call the scattering part of the BRDF the bidirectional scattering distribution function , or BSDF .

In general , p ( r ) , the perturbation responsible for scattering , has an irregular or random dependence on surface position , which leads to a scattered intensity distribution in the form of a highly irregular speckle pattern . In general , we are not interested in the fine-scale structure of that pattern , but only its average or smoothed value as a function of scattering angle . In order to relate individual scattering measurements to surface statistics , the locally smoothed values of the scattered intensity are taken to equal the average scattering pattern of an ensemble of statistically equivalent surfaces measured under the same conditions .

That average takes the form

k BSDF b a l 5 1

l 2 F b a ( w s ) R a ( θ i ) S p ( f ) (6)

where k l denotes the ensemble average and

S p ( f ) 5 lim A 0 5

K 1 A 0

U E A 0

d r e i 2 π f ? r p ( r ) U 2 L (7)

is the two-dimensional power spectral density (PSD) of the perturbation p . The power spectrum is the most important factor in the expression for the BSDF since it is a purely surface quantity and contains all of the information about the surface responsible for scattering .

In the case of polished surfaces the power spectrum of the surface errors is generally a smooth and broad function of spatial frequency , and the smoothed value of the BSDF is independent of the system pupil function . On the other hand , if the surface roughness contains periodic components , such as tool marks in precision-machined surfaces , the BSDF will contain sharp dif fraction lines . The positions and intensities of those lines are related to the feed rates and amplitudes of the tool marks , while their widths are determined by the system pupil function . 22–24

If the surface roughness is spatially isotropic , as is usually assumed for randomly polished surfaces , the PSD depends only on the magnitude of the surface spatial frequency and is independent of its direction in the surface plane . This means that the BSDF depends only on the magnitude of f in Eq . (2) , so that aside from angular-limit (band-limit) ef fects , all of the information about the surface spectrum can be obtained from scattering measurements made in the plane of incidence ( w s 5 0) .

There are two other cases where scanning in the plane of incidence is suf ficient : first , when the surface is ‘‘one-dimensional’’—i . e ., gratinglike—with its grooves oriented perpendicularly to the plane of incidence , and second , when an arbitrary surface is illuminated at an extreme glancing angle of incidence and the scattered intensity is measured by a slit detector perpendicular to the plane of incidence . In both of these cases the Born formalism described above leads to the one-dimensional form for the BSDF :

k BSDF b a l 5 1 l

d b , a R a ( θ i ) S p ( f x ) (8)

This expression dif fers from the two-dimension form in Eq . (6) in two ways : it is proportional to 1 / l rather than 1 / l 2 and involves the one-dimensional or profile PSD instead of the two-dimensional or area PSD . 2 4 The definition of the one-dimensional power spectrum and its connection with the two-dimensional form are given later , under ‘‘Profile Measurements . ’’

There is a final feature of the Born-approximation expression of the BSDF that is

SURFACE SCATTERING 7 .5

worth noting . In the plane of incidence both the one- and two-dimensional forms of the BSDF , although dif ferent , are functions of the quantity

u l f x u 5 u sin θ s 2 sin θ i u (9)

rather than θ i and θ s individually . This is sometimes written as b 2 b 0 and the reduced dependency is described as ‘‘shift invariance . ’’ 2 5 The Born formalism , however , shows that it is a manifestation of the grating equation or , more fundamentally , linear momentum conservation in first-order scattering .

7 . 5 SURFACE MODELS

Surface models introduce new features into the BSDF which can be used to identify specific scattering mechanisms and to determine values of their characteristic physical parameters . One of the nice features of the Born approximation is that it permits a variety of scattering mechanisms to be discussed in a common language . Three such scattering mechanisms are discussed below .

Topographic Scattering

Surface roughness is the principal source of scattering from most optical surfaces at visible wavelengths . It comes from the phase fluctuations impressed on the reflected wavefront by the surface height fluctuations Z :

P surf ( r ) 5 e i (4 π / l ) cos θ i ? Z ( r ) < 1 1 i 4 π l

cos θ i ? Z ( r ) (10)

Optical-quality surfaces are usually ‘‘smooth’’ in the sense that the exponent above is so small that the two-term expansion on the right is suf ficient for describing the scattering outside the specular core . The second term on the right is then p z , the perturbation responsible for topographic scattering . In that case Eqs . (6) and (7) give

k BSDF b a l z 5 16 π 2

l 4 cos 2 θ i F b a ( w s ) R a ( θ i ) S z ( f ) (11)

where the factor S z is the two-dimensional PSD of the height fluctuations , Z . More refined scalar and vector scattering theories give similar results in the smooth-surface limit . For example , the widely used first-order Rayleigh-Rice vector perturbation theory 4 , 26 gives

k BSDF s s l z 5 16 π 2

l 4 cos θ i cos θ s cos 2 w s 4 R s ( θ i ) R s ( θ s ) S z ( f ) (12)

for s to s (i . e ., ' to ' ) scatter . The simplicity of this expression has earned it the name

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of ‘‘the golden rule’’ of smooth-surface scattering . The corresponding expressions for p - s , s - p , and p - p scattering involve more complicated polarization-reflectivity factors and are given in the literature . 4 , 18 , 23 , 24

In the case of rougher surfaces , higher terms in the expansion of the exponential in Eq . (10) must be taken into account . The resulting expression for the BSDF involves a more complicated dependence on the PSD of the height fluctuations , as well as a dependence on their height distribution function , which is irrelevant in the smooth-surface limit . 1 – 7

Material Scattering

A perfectly smooth surface ( Z 5 0) may still scatter because of fluctuations in the composition or density of the surface material . For the lack of a better term we call this material scattering . The simplest way of describing such ef fects is to write the Fresnel- amplitude reflection coef ficient of the surface in the form

P s u r f ( r ) 5 1 1 [ z a ( r ) 2 z # a ] / z # a (13)

where z # a is the average reflection coef ficient . The unity on the right corresponds to a perfect surface and the second term represents the fluctuations about that average , and is the perturbation , p m , responsible for scattering . The corresponding BSDF is then

k BSDF b a l m 5 1

l 2 F b a ( w s ) R a ( θ i ) S m ( f ) (14)

where S m is the PSD of p m given by Eqs . (7) and (13) . This PSD can , in principle , be related to specific models of the material inhomogeneities , such as the magnitudes and spatial distribution of variations in composition .

Elson , Bennett , and Stover have recently appplied this model to the study of nontopographic scatter from composite beryllium mirrors . 2 7 A more fundamental source of material scattering from such mirrors is the fact that beryllium metal is optically anisotropic , and the randomly oriented crystallites in the surface lead to significant nontopographic scattering even for pure materials . 2 8

If the fluctuations in material properties that are responsible for scattering are distributed throughout the volume of the material , the expression for the BSDF involves the three-dimensional PSD of those fluctuations rather than the two-dimensional form appearing in surface scattering . However , in the limit where the skin depth of the reflecting material is less than the characteristic size of the inhomogeneities—that is , the in- homogeneities can be considered to be columnar—the three-dimensional formalism reduces to the two-dimensional form in Eq . (14) . 29–31

Defect Scattering

The surface models considered above take the perturbations responsible for scattering to be distributed broadly and continuously over the surface . An alternative is to have them localized at isolated regions of the surface , such as pits or bumps in the topographic case and patches of dif ferent reflectivity in the case of material scattering .

As an illustration , consider a sparse distribution of identical pits or bumps . In that case the surface height deviations can be written

Z ( r ) 5 O j

Z ( j ) d ( r 2 r j ) (15)

SURFACE SCATTERING 7 .7

where r j is the location of the j th defect and Z ( j ) d is its topographic shape . If the defects are

randomly distributed ,

k BSDF b a l d 5 1

l 2 F b a ( w s ) R a ( θ i ) S d ( f ) (16)

where S d is the PSD of the collection of defects . 2 4 For a collection of cylindrical pits or protrusions , for example ,

S d ( f ) 5 1

A 0 O

j F 2 π r 2

j ? J 1 (2 π fr j )

π fr j ? sin S 2 π

l h j cos θ i D G 2

(17)

where r j and h j are the radius and height of the j th defect , and the sum is taken over all defects within the illuminated surface area , A 0 . This result is instructive but would be quantitative unrealistic for large values of h j because of the omission of shadowing and other ef fects in this elementary calculation .

A number of exact calculations of the scattering from particular defects and structures can be found in the literature which may be more useful in modeling real ‘‘scratch-and- dig’’ defects in surfaces than the analysis given above . 4 , 32–35 Another useful calculation is that for the scattering of dust on smooth surfaces by Videen , Wolfe , and Bickel . 36–37

The measured BSDFs of real surfaces may involve contributions from one or more scattering mechanisms , such as those discussed above . The next two sections discuss methods of identifying the dominant mechanisms involved through their dif ferent characteristics , such as their dif ferent dependencies on the radiation wavelength .

7 . 6 WAVELENGTH SCALING

All first-order expressions for the BSDF appear as the product of factors , each corresponding to a dif ferent feature of the scattering process . Equation (12) , for example , involves an inverse power of the radiation wavelength , obliquity factors , a reflectivity factor , and finally , the power spectral density of the perturbation responsible for scattering .

The wavelength factor comes from two sources : an inherent factor of 1 / l 2 from the dimensionality of the problem [1 / l in the one-dimensional case , Eq . (8)] , and an additional factor of 1 / l 2 for topographic scatter since the radiation ‘‘sees’’ the height fluctuations in terms of its wavelength . 3 1 This additional factor means that , although the topographic component may dominate scattering at visible wavelengths , its contribution to the total BSDF can be overwhelmed by nontopographic or ‘‘anomalous’’ scattering mechanisms at infrared wavelengths . 3 8

The reflectivity factor in the BSDF depends on the radiation wavelength through the index of refraction or permittivity of the surface material , although the reflection coef ficient of highly reflective mirrors may be ef fectively constant over a wide range of wavelengths .

The final factor in the BSDF , the PSD , depends on the radiation wavelength principally through its argument , the spatial frequency , although its magnitude can also involve inherent wavelength dependencies . In the case of topographic scattering , the PSD is a purely geometric quantity , independent of the radiation wavelength , while the PSD of material fluctuations can depend on the wavelength .

The radiation-wavelength dependence of the argument of the power spectrum comes from purely geometric considerations . According to Eq . (2) , the spatial frequency corresponding to a given scattering direction is inversely proportional to the radiation

7 .8 PHYSICAL OPTICS

wavelength . This predictable dependence can be eliminated in data analysis by comparing BSDF measurements made at dif ferent radiation wavelengths as a function of spatial frequency rather than scattering angle .

Recent results indicate that mirrors made of some materials , such as molybdenum and silicon , behave as purely topographic scatterers from visible to 10-micron radiation wavelengths , while others show significant nontopographic contributions . 38–40

If dif ferent scattering mechanisms are present , and they are statistically independent , their BSDFs add . If they are not independent , there may be important interference ef fects which have an intermediate wavelength dependence . 31 , 41

7 . 7 PROFILE MEASUREMENTS

The models described above relate the PSD of surface imperfections to more fundamental physical quantities that are known a priori or can be measured independently . Surface topography , for example , can be measured directly using optical or mechanical profiling techniques , which raises the question of how to use profile data to determine the two-dimensional PSD appearing in Eq . (12) . That is , how can BSDFs be predicted from profile measurements?

Algorithms for estimating two-dimensional spectra from two-dimensional arrays of profile data are readily available . 4 2 The usual procedure , though , is to use the PSDs estimated from profiles taken by ‘‘one-dimensional’’ profiling instruments , or profiles stripped from rectangular arrays of ‘‘two-dimensional’’ measurements . In either case , one gets estimates of the one - dimensional power spectral density of the surface :

S z ( f x ) 5 lim L 5

K 2 L U E 1 L /2

2 L /2 dxe i 2 π f x x Z ( x ) U 2 L (18)

which contains information over the range of surface frequencies

1 L

, f x , N

2 L (19)

where L is the profile length and N is the number of data points in the profile . The dynamic range of such measurements—the ratio of the highest to lowest frequency included—is then N / 2 , which can be 1000 or more in practice . In principle , the estimation of power spectra from limited data sets is straightforward , but it involves some subtleties and must be done with care . 42–45 Procedures for doing this are presently being codified in an ASTM standard . 4 6

It is important to recognize , though , that the profile spectrum is not just a special form of the area spectrum , but is related to it in a more complicated way than might have been supposed . In particular , they are connected by the integral transform

S 1 ( f x ) 5 2 E 1

2

df y S 2 ( f x , f y ) (20)

where f x . 0 . This accounts for their dif ferent units—for example , S 1 ( f x ) for topographic errors is usually expressed in units of micrometers to the third power , while S 2 ( f x , f y ) is in

SURFACE SCATTERING 7 .9

micrometers to the fourth power . 24 , 45 (We have added subscripts 1 and 2 in Eqs . (20) and (21) to emphasize the distinction between one- and two-dimensional spectra .

In general , Eq . (20) cannot be solved to give S 5 S 2 (the quantity that appears in the expression for the BSDF) in terms of S 1 determined from profile measurements . However , in the special case of an isotropically rough surface , it can , and S 2 can be written as a dif ferent but related integral transform of S 1 , namely : 24 , 45

S 2 ( f ) 5 2 1

2 π E

f

df x

4 f 2 x 2 f 2 ?

d df x

S 1 ( f x ) (21)

Unfortunately , the integral on the right cannot be evaluated directly from measured (estimated) profile spectra since real data sets involve strong speckle fluctuations and do not include the high-frequency information needed to extend the upper limit of the integration to infinity . The usual way of handling these dif ficulties is fit the measured profile spectrum to a physically reasonable analytic model before evaluating Eq . (21) . This does three things : it smooths the fluctuations , adds a priori physical information required to extrapolate the data outside the measurement bandwidth , and condenses the data into a few finish parameters . 4 5

A useful model for this purpose is the ABC or K-correlation model ,

S 1 ( f x ) 5 A [1 1 ( Bf x ) 2 ] 2 C /2 (22)

where A , B , and C are the adjustable model parameters . A is the value of the profile spectrum in the low-frequency limit , B / 2 π is a ‘‘correlation length’’ which determines the location of the transition between the low- and high-frequency behavior , and C is the exponent of the power-law fall-of f at high frequencies . The corresponding two-dimensional form of the ABC spectrum , determined from the above transformations , is

S 2 ( f ) 5 A 9 [1 1 ( Bf ) 2 ] 2 ( C 1 1)/2 (23)

where A 9 5 AB G (( C 1 1) / 2) / (2 4 π G ( C / 2)) for C . 0 . 40 , 45

The ABC model is useful since it contains a minimal set of physical parameters , is mathematically tractable , and reduces to a number of more familiar forms as special cases . For example , C 5 2 gives a lorentzian spectrum , and large B gives the fractal-like forms :

S 1 ( f x ) 5 K n f 2 n x (24)

and

S 2 ( f ) 5 G (( n 1 1) / 2) 2 4 π G ( n / 2)

? K n f 2 ( n 1 1) (25)

Equation (25) corresponds to an inverse-power-law form for the BSDF . Isolated examples of such scattering distributions have been reported for a number of years 24 , 25 , 47 but their ubiquity and importance is only now being recognized . 4 8

Equations (22) to (25) provide a mechanism for comparing profile and BSDF measurements of a given surface , and this—along with wavelength scaling—of fers another way of distinguishing topographic and nontopographic scattering mechanisms in real surfaces . Such comparisons have been reported in several recent papers . 27 , 40 , –49

A related procedure , that does not involve scattering as such , is to make mechanical and optical profile measurements of a given surface and to compare spectra deduced from them over a common frequency range . Dif ferences could come from the fact that mechanical measurements are sensitive to topographic ef fects , while optical measurements are sensitive to material ef fects as well . Preliminary comparisons of this type indicate that machined silicon is essentially a pure topographic scatterer at HeNe wavelengths . 50 , 51

7 .10 PHYSICAL OPTICS

7 . 8 FINISH PARAMETERS

Optical finish measurements are frequently reported in terms of one or more classical topographic finish parameters—the rms (root mean square) surface roughness s , the rms profile slope m , and the surface correlation length l . These are appealing parameters since they are ‘‘obvious’’ measures of quality and numerical values can be estimated directly from raw profile data using simple , intuitive algebraic expressions (estimators) .

On the other hand , they suf fer from the fact that their values can depend significantly on the measuring instrument used , and do not necessarily represent intrinsic properties of the surfaces being measured . In the early days , when there were only one or two instruments that could measure optical surface finish at all , and measurements were used mainly for comparative purposes , instrumental ef fects were not an important concern . But nowadays , when many dif ferent measurement techniques are available and there is a need for a more quantitative understanding of the measured data , these instrumental ef fects must be recognized and understood .

The royal road to understanding the ef fects of the measurement process lies in the frequency domain , which has already appeared naturally in the discussion of surface scattering . For example , it is easy to see that for linear measurement systems the measured value of the mean-square profile roughness and slope are given by weighted integrals or moments of the profile PSD :

S s

m D 2

meas

5 E

0 df x S 1 ( f x ) T ( f x ) S 1

2 π f x D 2

(26)

The factor T ( f x ) in the integrand is the square magnitude of the transfer function of the particular measurement process involved , and accounts for the fact that real measurements are sensitive to only a limited range of surface spatial frequencies . In contrast , the intrinsic values of these parameters are defined by Eq . (26) where T ( f x ) 5 1 .

In the case of mechanical profile measurements T ( f x ) is usually taken to be a unit rectangle ; that is , a constant equal to unity within the measurement bandpass and zero outside . In the case of optical profile measurements it is usually taken to be a triangle function that falls from unity at zero frequency to zero at the incoherent cutof f of the optical system involved . 43 , 44 , 51 , 52 Although Eq . (26) applies to profile measurements , it is readily generalized to two-dimensional measurement processes , 44 , 53 where the area in the frequency plane defined by the nonvanishing transfer function is called the frequency footprint of the measurement process . 5 3

The literature also describes topographic finish in terms of the ‘‘correlation length , ’’ a length parameter characterizing its transverse properties . Unfortunately , though , most of the values in the literature have been determined without taking the possibly large ef fects of the measurement bandwidth into account , and must be interpreted with considerable care . 43 , 48 , 54

The standards literature does not yet completely address the issue of measurement bandwidths . The 1985 version of ANSI / ASME B46 , which is principally directed towards mechanical surfaces , lists low-frequency limits but not high-frequency ones , although the updated version will . 5 5 On the other hand , ASTM F1408 , for the total integrated scatter (TIS) method for measuring the ef fective rms surface roughness 5 6 specifies wavelength and angular parameters that translate to a measurement bandwidth of

0 . 07 , f , 1 . 48 m m 2 1 (27)

It is important to note that these limits correspond to a dynamic range of only 21 . 5 , which is much less than that of many current profiling instruments . This means that the ef fective roughness value of a given surface determined by conventional TIS techniques can be significantly smaller than that calculated from profile measurements . 53 , 57 Noll and Glenn ,

SURFACE SCATTERING 7 .11

for example , have reported TIS roughness values that are less than Talystep roughnesses by a factor of 3 or more . 5 8

The most striking example of bandwidth ef fects occurs for fractal surfaces , that is , surfaces with inverse-power-law spectra . Such surfaces do not have finite intrinsic values of the roughness , slope , or correlation length , although they do have finite measured y alues of these quantities , since the low- and high-frequency divergences are cut of f by the finite measurement band-pass . 24 , 48 , 59–61

In the light of this , it is more realistic to use the classical topographic finish parameters measured under standardized conditions for descriptive and comparative purposes only , and to use the parameters of the power spectrum for a more comprehensive and precise description—for example , A , B , and C in the ABC model and n and K n in the fractal model .

The advantages of the power-spectrum description come from the fact that the PSD is directly related to the BSDF , it is an intrinsic surface property and can be used to calculate conventional finish parameters from Eq . (26) . 38 , 40 , 49

7 . 9 FINISH SPECIFICATION

Imperfect surface finish is undesirable since it degrades the performance of an optical system . This raises the question of how to specify surface finish in terms of performance requirements . Although this subject is outside the scope of this article , it is useful to indicate how the ideas discussed above are involved .

The simplest performance requirement is that the distribution of the scattered light intensity is less than some prescribed function of the scattering angle away from the specular direction . This can be translated directly into limitations on error spectra using Eqs . (12) (14) and (16) . This method—using an inverse-power-law spectrum—was proposed a number of years ago for x-ray mirrors . 4 7 and has recently been adopted in an international (ISO) standard for the designation of optical surface finish . 6 2

A variant of this is the requirement that the total integrated intensity scattered out of the image core be less than some fraction of the total reflected intensity . In the case of topographic errors this appears in the form of the famous Strehl ratio :

Core intensity , Z ? 0 Core intensity , Z 5 0

< e 2 (4 π s cos θ i / l ) 2 (28)

where s is the ef fective rms roughness of the surface evaluated by integrating the two-dimensional power spectrum of the equivalent surface roughness using the two- dimensional analog of Eq . (26) , where the frequency limits of T ( f ) are determined from the angular limits through the grating equation , Eq . (2) . The extreme values of these angles are determined by the core size in one limit , and scattering parallel to the surface plane in the other . 5 7

The integrated scattering intensity is important since it represents a loss of the core intensity . In the case of imaging optics , the ef fects of surface errors on the detailed distribution of the core intensity are also of interest : for example , its on-axis image intensity , the image width , and its spatial-frequency distribution . Expressions for these ef fects follow from precisely the same Born-approximation result , Eqs . (3) to (5) , that to this point we have used only to examine the properties of the scattered radiation , that is , the BSDF .

7 .12 PHYSICAL OPTICS

One interesting result of this analysis is the fact that the ef fects on imaging are automatically finite for surfaces with fractal surface errors , even though fractal power spectra diverge at long surface spatial wavelengths . This is because the finite size of the pupil function makes the imaging ef fects insensitive to spatial wavelengths longer than the size of the illuminated area of the mirror . The interested reader is referred to the literature for details of these and related issues . 59–61

7 . 1 0 REFERENCES

1 . P . Beckmann and A . Spizzichino , The Scattering of Electromagnetic Wa y es from Rough Surfaces , Pergamon Press , New York , N . Y ., 1963 .

2 . F . G . Bass and I . M . Fuks , Wa y e Scattering from Statistically Rough Surfaces , Pergamon Press , New York , N . Y ., 1979 .

3 . J . A . Ogilvy , Theory of Wa y e Scattering from Random Rough Surfaces , Adam Hilger , New York , N . Y ., 1991 .

4 . G . T . Ruck , D . E . Barrick , W . D . Stuart , and C . K . Krichbaum , Radar Cross - Section Handbook , 2 vols ., Plenum Press , New York , N . Y ., 1970 .

5 . A . Ishimaru , Wa y e Propagation and Scattering in Random Media , 2 vols ., Academic Press , New York , N . Y ., 1978 .

6 . F . T . Ulaby , R . K . Moore , and A . K . Fung , Microwa y e Remote Sensing : Acti y e and Passi y e , Addison-Wesley , Reading , Mass ., 1981 , vols . 1 , 2 , and 3 , Artech House , Dedham , Mass ., 1986 .

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SURFACE SCATTERING 7 .13

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24 . E . L . Church , H . A . Jenkinson , and J . M . Zavada , ‘‘Relationship Between Surface Scattering and Microtopographic Features , ’’ Optical Engineering 18 : 125 – 136 (1979) .

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44 . E . L . Church and P . Z . Takacs , ‘‘Ef fects of the Optical Transfer Function in Surface-Profile Measurements , ’’ Proc . SPIE 1164 : 46 – 59 (1989) .

7 .14 PHYSICAL OPTICS

45 . E . L . Church and P . Z . Takacs , ‘‘The Optimal Estimation of Finish Parameters , ’’ Proc . SPIE 1530 : 71 – 85 (1991) .

46 . ‘‘Standard Practice for Characterizing Roughness of Optical Surfaces from Surface Profile Data , ’’ ASTM Subcommittee E12 . 09 . (This Standard was in the process of being written , 1993 . )

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PSDs Derived from Multiple-Wavelength BRDFs and Wyko Profilometer Data , ’’ Proc . SPIE 1530 : 86 – 103 (1991) .

50 . E . L . Church , T . V . Vorburger , and J . C . Wyant , ‘‘Direct Comparison of Mechanical and Optical Measurements of the Finish of Precision-Machined Optical Surfaces , ’’ Optical Engineering 24 : 388 – 395 (1985) .

51 . E . L . Church , J . C . Dainty , D . M . Gale , and P . Z . Takacs , ‘‘Comparison of Optical and Mechanical Measurements of Surface Finish , ’’ Proc . SPIE 1531 : 234 – 250 (1991) .

52 . E . L . Church and P . Z . Takacs , ’‘Use of an Optical-Profiling Instrument for the Measurement of the Figure and Finish of Optical-Quality Surfaces , ’’ Wear 109 : 241 – 257 (1986) .

53 . E . L . Church , ‘‘The Precision Measurement and Characterization of Surface Finish , ’’ Proc . SPIE 429 : 86 – 95 (1983) .

54 . E . L . Church , ‘‘Comments on the Correlation Length , ’’ Proc . SPIE 680 : 102 – 111 (1987) . 55 . ANSI / ASME B46 . 1 , Surface Texture ( Surface Wa y iness and Lay ) , American Society of

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Components by Total Integrated Scattering , 1987 . 57 . E . L . Church , G . M . Sanger , and P . Z . Takacs , ‘‘Comparison of Wyko and TIS Measurements of

Surface Finish , ’’ Proc . SPIE 740 : 65 – 73 (1987) . 58 . R . J . Noll and P . Glenn , ‘‘Mirror Surface Autocovariance Functions and Their Associated Visible

Scattering , ’’ Applied Optics 21 : 1824 – 1838 (1982) . 59 . E . L . Church and P . Z . Takacs , ‘‘Specification of the Figure and Finish of Optical Elements in

Terms of System Performance , ’’ Proc . SPIE 1781 : 118 – 130 (1992) . 60 . E . L . Church and P . Z . Takacs , ‘‘Specification of Surface Figure and Finish in Terms of System

Performance , ’’ Applied Optics 32 : 3344 – 3353 (1993) . 61 . E . L . Church and P . Z . Takacs , ‘‘Specifying the Surface Finish of X-ray Mirrors , ’’ Proceedings on

Soft X - ray Projection Lithography , vol . 18 , OSA Proceedings Series , Optical Society of America , Washington , D . C ., 1993 .

62 . ISO 10110 Standard , Indications on Optical Drawings , ‘‘Part 8 : Surface Texture , ’’ 1992 .