Optical Fabrication

Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles

[+] Author Affiliations
James E. Harvey

The University of Central Florida, The Center for Research and Education in Optics and Lasers (CREOL), P.O. Box 162700, 4000 Central Florida Blvd., Orlando, Florida 32826

Sven Schröder

Fraunhofer Institute for Applied Optics and Precision Engineering, Albert-Einstein-Straße 7, 07745 Jena, Germany

Narak Choi

The University of Central Florida, The Center for Research and Education in Optics and Lasers (CREOL), P.O. Box 162700, 4000 Central Florida Blvd., Orlando, Florida 32826

Angela Duparré

Fraunhofer Institute for Applied Optics and Precision Engineering, Albert-Einstein-Straße 7, 07745 Jena, Germany

Opt. Eng. 51(1), 013402 (Feb 06, 2012). doi:10.1117/1.OE.51.1.013402
History: Received August 3, 2011; Revised October 18, 2011; Accepted November 17, 2011
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Abstract.  Surface scatter effects from residual optical fabrication errors can severely degrade optical performance. The total integrated scatter (TIS) from a given mirror surface is determined by the ratio of the spatial frequency band-limited “relevant” root-mean-square surface roughness to the wavelength of light. For short-wavelength (extreme-ultraviolet/x-ray) applications, even state-of-the-art optical surfaces can scatter a significant fraction of the total reflected light. In this paper we first discuss how to calculate the band-limited relevant roughness from surface metrology data, then present parametric plots of the TIS for optical surfaces with arbitrary roughness, surface correlation widths, and incident angles. Surfaces with both Gaussian and ABC or K-correlation power spectral density functions have been modeled. These parametric TIS predictions provide insight that is useful in determining realistic optical fabrication tolerances necessary to satisfy specific optical performance requirements.

Figures in this Article

Surface scatter phenomena continue to be an important issue in diverse areas of science and engineering in the 21st century. In some applications, the total amount of scattered radiation is of primary concern. In other applications, knowing the angular distribution of the scattered light is crucial.

Recall that the reflectance of a surface is defined as the ratio of the (total) reflected radiant power divided by the incident radiant power. However, for real surfaces (exhibiting some residual surface roughness) the total reflected radiant power consists of two components: one specularly reflected (obeying the law of reflection) and the other diffusely reflected (scattered).

Following the work of Bennett and Porteus,1 which built upon the earlier work of Davies,2 the fraction of the total reflected radiant power remaining in the specular beam after reflection from a single moderately rough surface is given by Display Formula

RsRt=exp[(4πcosθiσ/λ)2],(1)
where Rs is the specular reflectance, Rt is the total reflectance, θi is the incident angle, σ is the root-mean-square (rms) surface roughness, and λ is the wavelength of the incident radiation.

The classical definition of total integrated scatter (TIS) follows directly from Eq. (1) as that fraction of the total reflected radiant power that is scattered out of the specularly reflected beam: Display Formula

TIS=diffuse  reflectancetotal  reflectance=diffuse  reflectancespecular  reflectance+diffuse  reflectance=RdRs+Rd,(2)
or, since Rd=RtRs, we obtain Display Formula
TIS=RtRsRt=1RsRt=1exp[(4πcosθiσ/λ)2].(3)
The above definition of TIS and its paraxial smooth surface approximation (for normal incidence)Display Formula
TIS(4πσ/λ)2(4)
have been discussed extensively in the literature.312 Unfortunately, the widely used commercially available ASAP (Advanced Systems Analysis Program) optical analysis code defines the quantity TIS to be identical with the definition of diffuse reflectance.13 Hence, the above definition of TIS is not always applied uniformly and consistently in the literature, in spite of the fact that there have been international standard procedures written for the measurement of TIS as a means of determining the roughness of a surface.14

An international standard has also been established defining total scattering (TS) as the ratio of the diffusely scattered radiant power to the incident radiant power.15 TS directly expresses the scattered radiant power regardless of the reflectance of the surface. Although using TS has several advantages in practical integrated scatter measurements with respect to robustness and comparability among different metrology instruments, we use TIS in this paper for historical reasons and because all theoretical expressions are independent of Rt. The relationship between the two quantities is given simply by TS=TISRt.

After repeated discussions and admonitions by Church,6,16 Church and Takacs,17,18 Stover,19 Germer and Asmail,20 Dittman,21,22 and others, most of the optical surface metrology community is aware that when we associate surface roughness with scattered light, we must specify the spatial frequency band limits of the effective roughness that is relevant to the particular scattering application. In other words, we must replace the total or intrinsic rms roughness, σ, with the relevant band-limited rms roughness, σrel, in Eqs. (3) and (4); hence TIS is Display Formula

TIS=1exp[(4πcosθiσrel/λ)2],(5)
which for smooth surfaces can be approximated as Display Formula
TIS(4πcosθiσrel/λ)2(6)

In the remainder of this paper we will first discuss the statistical surface characteristics relevant to the scattering process and illustrate precisely how to calculate σrel for arbitrary surface power spectral density (PSD) functions, incident angles, and wavelengths. We will then provide parametric plots of σrel2/σtotal2 for optical surfaces as functions of surface correlation width for both Gaussian and ABC or K-correlation PSD functions.

We will then use Eq. (5) to make parametric TIS predictions that provide useful insight for determining realistic optical fabrication tolerances necessary to satisfy specific optical performance requirements. Finally, we will briefly demonstrate the capabilities of the generalized Harvey-Shack surface scatter theory in producing angle resolved scatter (ARS) or bidirectional reflectance distribution function (BRDF) curves for optical surfaces with arbitrary surface roughness (up to at least a few waves, an actual limit has not been established), correlation widths, and incident angles.

The behavior of light scattered from randomly rough surfaces is dictated by the statistical surface characteristics. Consider the surface profile illustrated in Fig. 1. The surface has a zero mean with the surface height, h, illustrated as a function of position along a one-dimensional trace of finite length. Two relevant statistical surface characteristics are the surface height distribution function and the surface autocovariance (ACV) function.9 Fortunately, for many cases of interest, the surface heights are normally distributed (i.e., the surface height distribution function is Gaussian). The rms surface roughness, σs, is the standard deviation of that normal distribution.

Grahic Jump LocationF1 :

Schematic diagram of a surface profile and its relevant statistical parameters.

Although it would be convenient (mathematically) if the surface ACV function were also Gaussian, in most instances that is not the case. Instead, the ACV function is material and process dependent. The ACV length, , is usually defined as the half-width of the ACV function at the 1/e height.

The surface PSD function and the surface ACV function are Fourier transforms of each other. Note in Fig. 1 that the value of the surface ACV function at the origin is equal to the surface variance, σs2. From the central ordinate theorem of Fourier transform theory, we therefore know that the volume under the 2D surface PSD is also equal to the surface variance.

The surface PSD can be thought of as a plot of surface variance as a function of the spatial frequency of the surface irregularities. We can thus talk about several different spatial frequency regimes that have distinctly different effects upon image quality, as illustrated in Fig. 2.

Grahic Jump LocationF2 :

Different spatial frequency regimes and their resulting effects upon image quality.

After decades of concerning themselves with only low spatial frequency “figure” errors and high spatial frequency “finish” errors or “microroughness,” optical manufacturers are finally realizing the significance of “midspatial frequency” optical surface irregularities in the degradation of image quality.2325

The low spatial frequency figure-error regime gives rise to conventional wavefront aberrations. The high spatial frequency finish-error/microroughness regime produces wideangle scattering effects that redistribute radiant energy from the image core into a broad scattered halo without substantially affecting the width of the image core. And the midspatial frequency regime that spans the gap between the traditional figure and finish errors produces small angle scatter that broadens or smears out the image core.2628

Historically, optical fabrication tolerances have been specified by placing a tolerance upon only the figure and finish errors. It has only recently become common practice to also specify and measure the midspatial frequency surface irregularities.

The astronomer’s classical definition of resolution has been the full width at half-max (FWHM) of the point spread function. For bright point sources, this image quality criterion is quite insensitive to wide-angle scatter resulting from high spatial frequency microroughness, since the width of the image core is not significantly broadened. However, for faint point sources, the wide-angle scattered halo causes severe signal-to-noise problems and a substantial loss of image contrast. The small-angle scatter produced by the midspatial frequency surface irregularities does broaden the image core and therefore causes a significant decrease in resolution (larger FWHM). The same considerations hold for deep ultraviolet and especially for extreme ultraviolet (EUV) lithography applications.2932 It is thus imperative that optical fabrication tolerances be specified over the entire range of relevant spatial frequencies.

A uniformly rough surface is one whose roughness is homogeneous and isotropic, i.e., the surface height distribution function and the ACV function do not change with location or orientation of the (finite) measured surface profile. For such a surface, the PSD is a 2D rotationally symmetric function.

It is important to recognize that the relevant (or effective) surface roughness is not an intrinsic surface characteristic, but a band-limited quantity that depends upon the wavelength and incident angle.16,18 For normal incidence, those spatial frequencies greater than 1/λ produce evanescent (imaginary) waves that do not result in radiant power being scattered from the specular beam—i.e., spatial frequencies greater than 1/λ are completely irrelevant with regard to scattered light.6 For an arbitrary incident angle, θi, the 2D boundary of the appropriate bandlimited portion of the surface PSD is illustrated in Fig. 3(a), i.e., a circle of radius 1/λ whose center is shifted to a spatial frequency33 given by Display Formula

fo=sinθoλ,θo=θi.(7)
The corresponding relevant roughness, σrel, is given by the square root of the volume under the relevant portion of the surface PSD illustrated in Fig. 3(b). It is thus calculated by the following integral:33Display Formula
σrel(λ,θi)=1/λ+fo1/λ+fo1/λ2(fxfo)21/λ2(fxfo)2PSD(fx,fy)dfxdfy.(8)
It is the relevant roughness, σrel, that determines the fraction of the total reflected light contained in the specular beam and in the associated scattering function. For normal incidence (and isotropic roughness), the relevant roughness expressed by Eq. (8) simplifies to Display Formula
σrel(λ)=2πf=01/λPSD(f)fdf.(9)
For some applications, there is a nonzero low spatial frequency band limit, 1/L, where L represents an inherent measurement bandwidth limit.17,19,34,35 For example, if surface roughness is being inferred from TIS measurements, the upper and lower angle limits of the TIS instrument determine (through the grating equation) the minimum and maximum spatial frequency band limits of the resulting predicted surface roughness. Thus TIS measurements are meaningful only when the limiting angles are known and reported. The additional contribution to roughness due to spatial frequencies between zero and 1/L can often be ignored.

Grahic Jump LocationF3 :

(a) Illustration of the 2D boundary of the appropriate band-limited portion of the surface PSD for an arbitrary incident angle, θi. (b) Illustration of the relevant portion of the surface PSD, whose integral yields the square of the relevant rms surface roughness.

For a surface with a Gaussian ACV function Display Formula

ACV(r)=σtotal2exp[(σtotal2/2)],(10)
the surface PSD is also a Gaussian function: Display Formula
PSD(f)=π2σtotal2exp[(πf)2].(11)
It can also be readily shown that the cumulative radial integral of a 2D rotationally symmetric Gaussian function is proportional to one minus that Gaussian, and the proportionality constant is the total volume of the Gaussian function. Thus, integrating Eq. (11) we obtain Display Formula
ϕ=02πf=0fPSD(f)fdfdϕ=σtotal2[1exp((πf)2)].(12)
But if the upper limit is set to 1/λ, the integral on the left-hand side of Eq. (12) is just the bandlimited surface variance, σrel2, for normal incidence. We thus have Display Formula
σrel2σtotal2=1exp[(π/λ)2].(13)
Figure 4 shows a family of parametric curves illustrating the ratio of σrel2 to σtotal2 for different incident angles as a function of normalized correlation width. These curves were obtained by numerically integrating the relevant portion of the surface PSD as indicated in Eq. (8). The analytic solution for normal incidence is also included as a check on our numerical model.

Grahic Jump LocationF4 :

Parametric curves illustrating the variation of relevant roughness with incident angle and surface correlation width (Gaussian ACV function).

However, optical surfaces fabricated by conventional grinding and polishing techniques on ordinary amorphous glassy materials and thin film coatings seldom exhibit Gaussian surface ACV functions. Church36,37 has reported upon the fractal nature of many surface finishes, thus suggesting that the surface PSD can be modeled as exhibiting an inverse power law behavior at high spatial frequencies that can conveniently be fit by the following ABC, or K correlation, function of the form Display Formula

PSD(f)1D=A[1+(Bf)2]C/2.(14)
Here A is the height of the low spatial frequency plateau of the 1D surface PSD and 1/B is the location of the knee in the log–log plot of the 1D PSD (B can be considered the correlation width of the surface irregularities).

It has also been demonstrated that thin film coatings exhibit ABC (or combinations of ABC) PSD’s describing the substrate and the intrinsic roughness of the coating.30,38

Assuming isotropic roughness, this 1D measured surface PSD can be converted into the following 2D surface PSD that relates more directly to the surface scatter behavior and hence to the resulting image degradation: Display Formula

PSD(f)2D=KAB[1+(Bf)2](C+1)/2,K=12πΓ((C+1)/2)Γ(C/2)(15)
There is also a convenient analytic expression for the total volume under the 2D surface PSD: Display Formula
σtotal2=2πKAB[(C1)B2],(16)
and even an analytic expression for the 2D Fourier transform of the above 2D surface PSD. This surface ACV function is given by Display Formula
ACV(r)=(2π)1/2AB2C/2Γ(C/2)(2πrB)(C1)/2K(C1)/2(2πrB).(17)

Although surface scatter effects can also be important at visible and infrared wavelengths, we will consider an ultraviolet example with a wavelength of 100 nm at normal incidence to a surface with a PSD given by Eq. (15). Figure 5 illustrates the ratio of σrel2 to σtotal2 as a function of the surface correlation width B for several different values of the parameter C when the parameter A=6.10nm2mm. As for the case of the Gaussian PSD, the relevant roughness decreases with decreasing surface correlation width. And, of course, the relevant roughness increases with increasing C as a larger portion of the total roughness is contained within the circular bandlimited boundary for an inverse power law with a steeper slope. Note that a small percent change in the parameter C caused a greater change in the ratio of σrel2 to σtotal2 than did five decades of variation in the parameter B.

Grahic Jump LocationF5 :

Parametric curves illustrating the variation of relevant roughness with the parameters B and C.

To provide even more insight into the nature of band-limited roughness for practical optical surfaces, Fig. 6 illustrates the ratio of σrel2 to σtotal2 as a function of the parameter C for fixed values of A and B (A=6.10nm2mm and B=120mm), but for different incident angles and wavelengths. Recall that the total surface variance, σtotal2, is infinite for C<1.0 (an inverse power law slope with a magnitude less than 2). The band-limited surface variance, σrel2, thus initially increases rapidly from C=0, then asymptotically approaches σtotal2 for C>1.35 for an EUV wavelength of 30 nm. Increasing the wavelength by a factor of 333 to 10 μm only moves this asymptotic behavior out to a C-value of 1.75. It is thus apparent from Fig. 6 that the ratio of σrel2 to σtotal2 is quite insensitive to both incident angle and wavelength for surfaces with an ABC function PSD.

Grahic Jump LocationF6 :

Variation of relevant roughness to parameter C. Note insensitivity to incident angle.

Note that the surface correlation width does not appear explicitly in Eq. (5) or Eq. (6) for TIS. Yet Elson7 was aware in 1983 that the derivation of Eq. (4) involved an assumption that the correlation width was long compared with the wavelength (λ). He also recognized that any surface spatial wavelengths shorter than the wavelength of the incident radiation would not contribute to the normal incidence TIS. Equation (4) is thus not valid in general, even for smooth surfaces. He then calculated that for λ, the value of TIS varies inversely as the fourth power of the wavelength for a smooth surface with a Gaussian ACV function illuminated at normal incidence,7 so that TIS is Display Formula

TIS=(643)(π4σ22λ4),λ.(18)

Elson continued his analysis of the variation of TIS with correlation width by plotting the ratio of the actual TIS for arbitrary correlation widths to TIS, which is given by Eq. (4) when λ. He calculated the actual TIS by performing numerical integrations of the ARS predicted by the Rayleigh-Rice surface scatter theory. The result of these calculations is illustrated by the discrete data points in Fig. 7. Elson’s quantity TIS/TIS (plotted as the solid line in Fig. 7) can also be calculated by merely dividing Eq. (5) by Eq. (3): Display Formula

TISTIS=1exp[(4πcosθiσrel/λ)2]1exp[(4πcosθiσtotal/λ)2].(19)

Grahic Jump LocationF7 :

TIS/TIS versus /λ for a smooth surface with normally incident light. The solid line is in excellent agreement with Elson’s original analysis.

Both the numerator and the denominator of this ratio are equal for large correlation widths, yielding a value of unity for the ratio. There are thus two asymptotic regions in Fig. 7 with analytic solutions, illustrated by the dashed line given by Eq. (18) for λ, and unity as approaches λ. Equation (18) for small correlation widths has been quite useful in predicting surface scatter from optical thin films exhibiting columnar growth.39,40

Elson performed the above analysis, which provides valuable insight into surface scatter behavior, without ever mentioning or acknowledging the concept of band-limited roughness. He also determined that Eq. (4) is not limited to surfaces with a Gaussian surface height distribution function or a Gaussian ACV function.7,19

The excellent agreement between Elson’s calculations and the predictions from Eq. (19) dramatically illustrates that Eqs. (3) and (4) are ambiguous and incorrect for surfaces with correlation widths less than the wavelength of the operational wavelength due to their failure to identify the relevant spatial frequency bandwidth limits, as does Eq. (5). Furthermore, by merely performing the 2D integral of the surface PSD over the shifted circular boundary discussed in section 2, we can readily calculate σrel, and therefore TIS, for arbitrary surface ACV functions without the necessity of implementing a given surface scatter theory to predict the ARS or the BRDF.

Since Eq. (19) is valid for surfaces with arbitrary roughness, correlation widths, and incident angles, a more thorough parametric analysis, providing even more insight into surface scatter phenomena, can now be readily performed. Figure 8 illustrates a set of parametric curves of TIS/TIS versus correlation width for different surface roughnesses (0.02<σ/λ<0.50) for a surface with a Gaussian ACV function. Again, the smooth surface curve agrees very well with Elson’s original data.

Grahic Jump LocationF8 :

Parametric curves of TIS/TIS versus /λ for surfaces with a Gaussian ACV and different roughnesses for normally incident light.

Figure 9 illustrates a similar set of parametric curves of TIS/TIS versus incident angle for a fixed correlation width of /λ=1.0. This is again for surfaces exhibiting a Gaussian ACV function. Note that TIS/TIS is equal to unity, as expected for small incident angles. At an incident angle of about 30 deg, the curves diverge until about 65 deg. They then asymptotically converge to a common value of 0.455 at grazing incidence.

This behavior becomes intuitive when one realizes that even moderately rough surfaces become specular at grazing incidence. Both the numerator and the denominator thus become zero, and Eq. (19) becomes indeterminant for θi=90deg. Applying L’Hospital’s Rule yields TIS/TIS equal to Display Formula

TISTIS=1exp[(4πcosθiσrel/λ)2]1exp[(4πcosθiσtotal/λ)2]σrel2σtotal2asθi90°(20)
for all roughness values. Clearly, this asymptotic value of TIS/TIS will vary for different correlation widths. For example, for a correlation width of 2.0λ, we obtain the set of parametric curves illustrated in Fig. 10. For this case, the values of TIS/TIS are drastically different at normal incidence for different roughnesses, having substantially lower values for the smoother surfaces. And, indeed, the asymptotic value for grazing incidence has been reduced to a value of 0.236.

Grahic Jump LocationF9 :

Parametric curves of TIS/TIS versus incident angle for surfaces of different roughnesses with a Gaussian ACV and /λ=1.0.

Grahic Jump LocationF10 :

Parametric curves of TIS/TIS versus incident angle for surfaces of different roughnesses with a Gaussian ACV and /λ=2.0.

Figure 11 illustrates cumulative surface roughness as a function of spatial frequency for a state-of-the-art EUV telescope mirror characterized by an ABC function PSD.41 The relevant roughness (determined by a wavelength of 303.8 Å) is indicated, as is the total intrinsic roughness. The values of the ABC parameters are indicated in the figure, and several different metrology regions are shown over which band-limited optical fabrication tolerances are specified. The maximum relevant spatial frequency, relevant surface roughness, and resulting TIS as calculated from Eq. (5) are tabulated for each of six specific EUV wavelengths of interest. Note that at the longest wavelength of interest, only 7% of the reflected radiant power is scattered, whereas for the shortest wavelength of interest, over 56% of the reflected radiant power is scattered.

Grahic Jump LocationF11 :

Plot of cumulative surface roughness versus spatial frequency illustrates the difference between the relative roughness and the total roughness of an EUV telescope mirror characterized by an ABC function surface PSD. The TIS is tabulated for each of six EUV wavelengths of interest.

The TIS of an optical surface can be a very useful metric for evaluating different optical materials and optical fabrication processes, particularly for short-wavelength imaging systems. However, when making image quality predictions from optical metrology data, or when deriving practical optical fabrication tolerances necessary to satisfy specific image quality requirements, it is frequently not sufficient to merely know the TIS. It is often necessary to also know the angular distribution of the scattered radiation, i.e., the ARS or the BRDF for different incident angles and wavelengths. Rayleigh-Rice,42,6 Beckmann–Kirchhoff,43 or Harvey–Shack44,45 surface scatter theory is commonly used to predict surface scatter effects.

For short-wavelength imaging systems, where even state-of-the-art surfaces are moderately rough, this is a complicated problem that requires more than knowledge of the relevant bandlimited roughness of the optical surfaces making up the imaging system. The smooth-surface limitation of the classical Rayleigh–Rice surface scatter theory and the paraxial limitation of the Beckmann–Kirchhoff and the original Harvey–Shack theories have inhibited the widespread analysis of image degradation due to surface scatter phenomena. However, recent advances in surface scatter theory have resulted in a unified theory4648 that appears to combine the advantages of the Rayleigh–Rice theory and the Beckmann–Kirchhoff theories without the disadvantages of either. This generalized Harvey–Shack surface scatter theory has been demonstrated to be in good agreement with rigorous calculations and experimental results, even for moderately rough surfaces with arbitrary incident and scattered angles.48

As an example of the capabilities of the generalized Harvey-Shack surface scatter theory, Fig. 12 illustrates the previous TIS curves from Fig. 8 with a variety of inserts depicting the ARS curves corresponding to specific surface roughness and correlation width values. The ARS curves are all plotted on the same scale so one can readily see that the peak scattered intensity is (i) small in insert 1 due to the low TIS for this smooth surface, (ii) large in insert 3 due to the large TIS and correlation width, and (iii) small in insert 6 due to the small correlation width that causes σrel2 to be a small fraction of σtotal2, thus reducing the TIS in spite of the fact that the surface is quite rough. Additional insight can be gained by studying the values of TIS, the ratio of σrel2 to σtotal2, and TIS/TIS for each of the six points represented by the ARS inserts. Table 1 provides these tabulated data.

Grahic Jump LocationF12 :

Inserts added to Fig. 8 depict ARS curves calculated with the generalized Harvey-Shack surface scatter theory, which has been demonstrated to be valid for moderately rough surfaces with arbitrary correlation widths and incident angles.

Table Grahic Jump Location
Table 1Tabulated data for each point represented by ARS inserts.

The ARS data represented by the inserts in Fig. 12 can be input into several commercially available image analysis codes for predicting image quality as degraded by not only diffraction effects and geometrical aberrations, but surface scatter effects resulting from residual optical fabrication errors.49 Finally, it should again be mentioned that many deep ultraviolet and EUV components involve multilayer coatings that require multilayer scattering theories, or scatter measurements at the operational wavelength.2932

We first reviewed the historical (50-year-old) expression for TIS as a function of rms surface roughness, and its widely used smooth surface approximation. This was followed by a thorough discussion of the spatial frequency bandlimited roughness of an optical surface that is relevant to surface scatter phenomena. A simple procedure for calculating that relevant roughness for arbitrary surface PSDs, wavelengths, and incident angles was presented.

The classical equation for calculating TIS was then updated to be explicitly expressed in terms of this relevant band-limited roughness. Only then does it properly account for the effects upon the TIS caused by variations in surface correlation width, wavelength, and incident angle. The resulting Eq. (5) incorporates the concept of the relevant bandlimited roughness into the definition of TIS and renders the classical ambiguous expressions for TIS, Eqs. (3) and (4), obsolete and inaccurate for many applications involving short surface correlation widths and large incident angles.

Extensive parametric predictions were then presented of the TIS for optical surfaces of arbitrary roughness, correlation widths, and incident angles. This parametric analysis provides valuable insight and understanding to optical fabrication and metrology engineers that is not readily available from the previously existing literature.

Finally, the capabilities of a new unified surface scatter theory combining the advantages of both the classical Rayleigh-Rice and Beckmann-Kirchhoff theories was demonstrated by calculating ARS curves for surfaces with arbitrary roughness, correlation widths, and incident angles.

We have not proven, either by experimental verification or by rigorous numerical electromagnetic theory, that Eq. (5) is accurate for roughnesses and incident angles that result in arbitrarily high TIS values; however, there have been numerical validations by the optical design community50 that the complementary expression for the fraction of the energy remaining in the image core (when degraded by a combination of various aberrations, or figure errors, rather than microroughness) is accurate for values of Strehl 0.1. This would correspond to TIS 0.9. It is the authors’ hope that the publication of this paper will not only benefit metrology engineers and image analysts, but also stimulate the more theoretically inclined readers to help determine the limit of the validity of Eq. (5).

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Noll  R. J., “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng.. 18, (2 ), 137 –142 (1979). 0091-3286 
Harvey  J. E., Kotha  A., “Scattering effects from residual optical fabrication errors,” Proc. SPIE. 2576, , 155 –174 (1995). 0277-786X doi:10.1117/12.215588
Harvey  J. E., “Bridging the gap between ‘figure’ and ‘finish’,” presented at the OSA Optical Fabrication and Testing Meeting. ,  Boston, MA  ( May 3 1996).
Schröder  S., Gliech  S., Duparré  A., “Measurement system to determine the total and angle-resolved light scattering of optical components in the deep-ultraviolet and vacuum-ultraviolet spectral regions,” Appl. Opt.. 44, (29 ), 6093 –6107 ( October 2005). 0003-6935 CrossRef
Schröder  S. et al., “Angle-resolved scattering and reflectance of extreme-ultraviolet multilayer coatings: measurement and analysis,” Appl. Opt.. 49, (9 ), 1503 –1512 (20  March 2010). 0003-6935 CrossRef
Schröder  S. et al., “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt.. 50, (9 ), C164 –C171 (20  March 2011). 0003-6935 CrossRef
Trost  M. et al., “Influence of the substrate finish and thin film roughness on the optical performance of Mo/Si multilayers,” Appl. Opt.. 50, (9 ), C148 –C153 (20  March 2011). 0003-6935 CrossRef
Harvey  J. E. et al., “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE. 7794, 77940V  (2010).doi:10.1117/12.863995
Church  E. L., Takacs  P. Z., “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE. 1164, , 46 –59 (1989). 0277-786X 
Duparré  A. et al., “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt.. 41, (1 ), 154 –171 (2002). 0003-6935 CrossRef
Church  E. L., “Fractal surface finish,” Appl. Opt.. 27, (8 ), 1518 –1526 (15  April 1988). 0003-6935 CrossRef
Church  E. L., Takacs  P. Z., Leonard  T. A., “The prediction of BRDFs from surface profile measurements,” Proc. SPIE. 1165, , 136 –150 (1989). 0277-786X 
Ferré-Borrull  J., Duparré  A., Quesnel  E., “Procedure to characterize microroughness of optical thin films: application to ion-beam-sputtered vacuum-ultraviolet coatings,” Appl. Opt.. 40, (13 ), 2190 –2199 (1  May 2001). 0003-6935 CrossRef
Duparré  A., Kassam  S., “Relation between light scattering and the microstructure of optical thin films,” Appl. Opt.. 32, (28 ), 5475 –5480 (1993). 0003-6935 CrossRef
Duparré  A., “Scattering from surfaces and thin films,” in Encyclopedia of Modern Optics. , Guenther  B. D., Steel  D. G., Bayvel  L., eds.,  Elsevier ,  Amsterdam  (2004).
Martinez-Galarce  D. et al., “A novel forward-model technique for estimating EUV imaging performance—design and analysis of the SUVI telescope,” Proc. SPIE. 7732, , 773237  (2010). 0277-786X CrossRef
Rice  S. O., “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math.. 4, (2–3 ), 351 –378 (1951). 0010-3640 CrossRef
Beckmann  P., Spizzichino  A., The Scattering of Electromagnetic Waves from Rough Surfaces. ,  Pergamon Press ,  New York  (1963).
Harvey  J. E., “Light-scattering characteristics of optical surfaces,” Ph.D dissertation, University of Arizona (1976).
Harvey  J. E., “Surface scatter phenomena: a linear, shift-invariant process,” in Scatter from Optical Components. , Stover  J. C., ed., Proc. SPIE. 1165, , 87 –99 (1989). 0277-786X 
Harvey  J. E., Krywonos  A., Stover  J. C., “Unified scatter model for rough surfaces at large incident and scattered angles,” SPIE International Symposium on Optics and Photonics, Proc. SPIE. 6672, , 66720C  (2007).doi:10.1117/12.739139
Harvey  J. E. et al., “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE. 7426, , 74260I  (2009).doi:10.1117/12.831302
Krywonos  A., Harvey  J. E., Choi  N., “Linear systems formulation of surface scatter theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A. 28, (6 ), 1121 –1138 ( June 2011). 0740-3232 CrossRef
Harvey  J. E. et al., “Image degradation due to scattering effects in two-mirror telescopes,” Opt. Eng.. 49, (6 ), 063202  (2010). 0091-3286 CrossRef
Zemax Development Corporation, Zemax Optical Design Program User’s Guide. , p. 182  ( August 2007).

Grahic Jump LocationImage not available.

James E. Harvey is an associate professor in the College of Optics and Photonics at the University of Central Florida and a senior staff member of the Center for Research and Education in Optics and Lasers (CREOL). He has a PhD in optical sciences from the University of Arizona and is credited with over 195 publications and conference presentations in the areas of diffraction theory, surface scatter phenomena, adaptive optics, wavefront sensing, beam sampling technology, optical properties of materials, phased telescope arrays, and X-ray/EUV imaging systems. He is a member of OSA and a Fellow and past board member of SPIE.

Grahic Jump LocationImage not available.

Sven Schroeder received his PhD degree in physics from the Friedrich Schiller University in Jena, Germany in 2008. His dissertation dealt with light scattering properties of optical surfaces and thin film coatings at 193 nm and 13.5 nm. He has been with the Surface and Thin Film Characterization group at the Fraunhofer Institute for Applied Optics and Precision Engineering (IOF) in Jena since 2004. In 2007, he was cowinner of the Thuringian Research Prize for IOF’s contributions to the development of optical components and characterization techniques for EUV lithography. From July 2010 to July 2011 he was a visiting research scientist at CREOL/UCF in Orlando, Florida, working in the Imaging Group on light scattering models. He has 16 refereed journals to his credit.

Grahic Jump LocationImage not available.

Narak Choi received a bachelor’s and a master’s degree in physics from Seoul National University, South Korea, in 2005 and 2007, respectively. He is currently a PhD student at the College of Optics and Photonics at the University of Central Florida, doing research on surface scatter theory.

Grahic Jump LocationImage not available.

Angela Duparré received her PhD degree (thesis on optical properties of high-power laser mirrors) from the physics faculty of the Friedrich Schiller University of Jena in 1985. In 1992, she joined the Fraunhofer Institute for Applied Optics and Precision Engineering in Jena to become head of the Surface and Thin Film Characterization Group. Her interests have since been directed to the study of optical and non-optical surface and thin film properties such as light scattering, nano-/microstructures, and roughness, as well as to the development of light scattering measurement and modeling techniques. She has published more than 100 papers and is involved in standardization committees and in the organization of conferences on optical metrology, fabrication, and thin film coatings.

© 2012 Society of Photo-Optical Instrumentation Engineers

Citation

James E. Harvey ; Sven Schröder ; Narak Choi and Angela Duparré
"Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles", Opt. Eng. 51(1), 013402 (Feb 06, 2012). ; http://dx.doi.org/10.1117/1.OE.51.1.013402


Figures

Grahic Jump LocationF1 :

Schematic diagram of a surface profile and its relevant statistical parameters.

Grahic Jump LocationF2 :

Different spatial frequency regimes and their resulting effects upon image quality.

Grahic Jump LocationF3 :

(a) Illustration of the 2D boundary of the appropriate band-limited portion of the surface PSD for an arbitrary incident angle, θi. (b) Illustration of the relevant portion of the surface PSD, whose integral yields the square of the relevant rms surface roughness.

Grahic Jump LocationF4 :

Parametric curves illustrating the variation of relevant roughness with incident angle and surface correlation width (Gaussian ACV function).

Grahic Jump LocationF12 :

Inserts added to Fig. 8 depict ARS curves calculated with the generalized Harvey-Shack surface scatter theory, which has been demonstrated to be valid for moderately rough surfaces with arbitrary correlation widths and incident angles.

Grahic Jump LocationF11 :

Plot of cumulative surface roughness versus spatial frequency illustrates the difference between the relative roughness and the total roughness of an EUV telescope mirror characterized by an ABC function surface PSD. The TIS is tabulated for each of six EUV wavelengths of interest.

Grahic Jump LocationF10 :

Parametric curves of TIS/TIS versus incident angle for surfaces of different roughnesses with a Gaussian ACV and /λ=2.0.

Grahic Jump LocationF9 :

Parametric curves of TIS/TIS versus incident angle for surfaces of different roughnesses with a Gaussian ACV and /λ=1.0.

Grahic Jump LocationF8 :

Parametric curves of TIS/TIS versus /λ for surfaces with a Gaussian ACV and different roughnesses for normally incident light.

Grahic Jump LocationF7 :

TIS/TIS versus /λ for a smooth surface with normally incident light. The solid line is in excellent agreement with Elson’s original analysis.

Grahic Jump LocationF6 :

Variation of relevant roughness to parameter C. Note insensitivity to incident angle.

Grahic Jump LocationF5 :

Parametric curves illustrating the variation of relevant roughness with the parameters B and C.

Tables

Table Grahic Jump Location
Table 1Tabulated data for each point represented by ARS inserts.

References

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Aikens  D., “Specification and control of mid-spatial frequency wavefront errors in optical systems,” presented at the Optical Society of America Topical Technical Digest Meeting on Optical Fabrication and Testing. ,  Rochester, NY , Paper OTuA1 (2008).
Murphy  P., “Methods and challenges in quantifying mid-spatial frequencies,” presented at the OSA Technical Digest Topical Meeting on Optical Fabrication and Testing. ,  Rochester, NY , Paper OTuA3 (2008).
Duparré  A., “Characterization of surface and thin-film roughness using PSD functions,” presented at the OSA Topical Meeting on Optical Fabrication and Testing. ,  Rochester, NY  (2008).
Noll  R. J., “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng.. 18, (2 ), 137 –142 (1979). 0091-3286 
Harvey  J. E., Kotha  A., “Scattering effects from residual optical fabrication errors,” Proc. SPIE. 2576, , 155 –174 (1995). 0277-786X doi:10.1117/12.215588
Harvey  J. E., “Bridging the gap between ‘figure’ and ‘finish’,” presented at the OSA Optical Fabrication and Testing Meeting. ,  Boston, MA  ( May 3 1996).
Schröder  S., Gliech  S., Duparré  A., “Measurement system to determine the total and angle-resolved light scattering of optical components in the deep-ultraviolet and vacuum-ultraviolet spectral regions,” Appl. Opt.. 44, (29 ), 6093 –6107 ( October 2005). 0003-6935 CrossRef
Schröder  S. et al., “Angle-resolved scattering and reflectance of extreme-ultraviolet multilayer coatings: measurement and analysis,” Appl. Opt.. 49, (9 ), 1503 –1512 (20  March 2010). 0003-6935 CrossRef
Schröder  S. et al., “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt.. 50, (9 ), C164 –C171 (20  March 2011). 0003-6935 CrossRef
Trost  M. et al., “Influence of the substrate finish and thin film roughness on the optical performance of Mo/Si multilayers,” Appl. Opt.. 50, (9 ), C148 –C153 (20  March 2011). 0003-6935 CrossRef
Harvey  J. E. et al., “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE. 7794, 77940V  (2010).doi:10.1117/12.863995
Church  E. L., Takacs  P. Z., “Effects of the optical transfer function in surface profile measurements,” Proc. SPIE. 1164, , 46 –59 (1989). 0277-786X 
Duparré  A. et al., “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt.. 41, (1 ), 154 –171 (2002). 0003-6935 CrossRef
Church  E. L., “Fractal surface finish,” Appl. Opt.. 27, (8 ), 1518 –1526 (15  April 1988). 0003-6935 CrossRef
Church  E. L., Takacs  P. Z., Leonard  T. A., “The prediction of BRDFs from surface profile measurements,” Proc. SPIE. 1165, , 136 –150 (1989). 0277-786X 
Ferré-Borrull  J., Duparré  A., Quesnel  E., “Procedure to characterize microroughness of optical thin films: application to ion-beam-sputtered vacuum-ultraviolet coatings,” Appl. Opt.. 40, (13 ), 2190 –2199 (1  May 2001). 0003-6935 CrossRef
Duparré  A., Kassam  S., “Relation between light scattering and the microstructure of optical thin films,” Appl. Opt.. 32, (28 ), 5475 –5480 (1993). 0003-6935 CrossRef
Duparré  A., “Scattering from surfaces and thin films,” in Encyclopedia of Modern Optics. , Guenther  B. D., Steel  D. G., Bayvel  L., eds.,  Elsevier ,  Amsterdam  (2004).
Martinez-Galarce  D. et al., “A novel forward-model technique for estimating EUV imaging performance—design and analysis of the SUVI telescope,” Proc. SPIE. 7732, , 773237  (2010). 0277-786X CrossRef
Rice  S. O., “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math.. 4, (2–3 ), 351 –378 (1951). 0010-3640 CrossRef
Beckmann  P., Spizzichino  A., The Scattering of Electromagnetic Waves from Rough Surfaces. ,  Pergamon Press ,  New York  (1963).
Harvey  J. E., “Light-scattering characteristics of optical surfaces,” Ph.D dissertation, University of Arizona (1976).
Harvey  J. E., “Surface scatter phenomena: a linear, shift-invariant process,” in Scatter from Optical Components. , Stover  J. C., ed., Proc. SPIE. 1165, , 87 –99 (1989). 0277-786X 
Harvey  J. E., Krywonos  A., Stover  J. C., “Unified scatter model for rough surfaces at large incident and scattered angles,” SPIE International Symposium on Optics and Photonics, Proc. SPIE. 6672, , 66720C  (2007).doi:10.1117/12.739139
Harvey  J. E. et al., “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE. 7426, , 74260I  (2009).doi:10.1117/12.831302
Krywonos  A., Harvey  J. E., Choi  N., “Linear systems formulation of surface scatter theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A. 28, (6 ), 1121 –1138 ( June 2011). 0740-3232 CrossRef
Harvey  J. E. et al., “Image degradation due to scattering effects in two-mirror telescopes,” Opt. Eng.. 49, (6 ), 063202  (2010). 0091-3286 CrossRef
Zemax Development Corporation, Zemax Optical Design Program User’s Guide. , p. 182  ( August 2007).

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