## Modeling of light scattering in different regimes of surface roughness |

Optics Express, Vol. 19, Issue 10, pp. 9820-9835 (2011)

http://dx.doi.org/10.1364/OE.19.009820

Acrobat PDF (1127 KB)

### Abstract

The light scattering of rough metallic surfaces with roughness levels ranging from a few to several hundred nanometers is modeled and compared to experimental data. Different modeling approaches such as the classical Rayleigh-Rice vector perturbation theory and the new Generalized Harvey-Shack theory are used and critically assessed with respect to ranges of validity, accuracy, and practicability. Based on theoretical calculations and comparisons with Rigorous Coupled Wave Analysis for sinusoidal phase gratings, it is demonstrated that the approximate scatter models yield surprisingly accurate results and at the same time provide insight into light scattering phenomena. For stochastically rough metal surfaces, the predicted angles resolved scattering is compared to experimental results at 325 nm, 532 nm, and 1064 nm. In addition, the possibilities of retrieving roughness information from measured scattering data for different roughness regimes are discussed.

© 2011 OSA

## 1. Introduction

6. M. Flemming, L. Coriand, and A. Duparré, “Ultra-hydrophobicity through stochastic surface roughness,” J. Adhes. Sci. Technol. **23**(3), 381–400 (2009). [CrossRef]

8. T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media **14**(4), R1–R40 (2004). [CrossRef]

10. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. **4**(2-3), 351–378 (1951). [CrossRef]

15. T. V. Vorburger, E. Marx, and T. R. Lettieri, “Regimes of surface roughness measurable with light scattering,” Appl. Opt. **32**(19), 3401–3408 (1993). [CrossRef] [PubMed]

16. J. E. Harvey, N. Choi, and A. Krywonos, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE **7794**, 77940V, 77940V-11 (2010). [CrossRef]

17. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. **73**(4), 451–455 (1983). [CrossRef]

## 2. Definitions

### 2.1 Surface roughness

1. E. L. Church and P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. **32**(19), 3344–3353 (1993). [CrossRef] [PubMed]

20. A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. **41**(1), 154–171 (2002). [CrossRef] [PubMed]

*z*(

*x*,

*y*) has zero mean.

*L*is the length of the scanned profile.

*f*

_{x}and

*f*

_{y}. It contains all statistical information of random-rough (stochastic) surfaces such as generated by grinding, polishing, etching, as well as thin film growth or erosion.

*z*(

*x*,

*y*) from its mean value can be calculated by integrating the PSD:The integration limits depend on the application at hand or measurement technique used. In the extreme case

*f*

_{min}= 0 and

*f*

_{min}= ∞, σ is called the total roughness.

*τ*

_{c}is defined as the lateral spacing at which the ACV drops to 1/e of its maximum value. The ACV and PSD contain both vertical and lateral information about surface roughness (e.g. rms roughness and correlation length). They form a Fourier transform pair and are thus equivalent. However, only PSDs provide a direct representation of bandwidth limits. Moreover, using PSDs we are able to combine roughness information from different roughness measurement techniques to retrieve multi scale roughness information.

### 2.2 Light scattering

*θ*

_{s}only.

_{b}) is defined as the power

*P*

_{s}scattered from a surface into the backward hemisphere normalized to the incident power

*P*

_{i.}. For normal incidence, the standard requires at least the light scattered within the angular range between 2° and 85° to be collected. From Eq. (4) it follows that:The starting point of scattering theories that link the surface structural properties and the light scattering properties are always Maxwell’s equations. However certain approximations are used to find a solution to the rather complex problem of scattering from certain structures.

## 3. Surface scattering models

### 3.1 Beckmann-Kirchhoff (BK) theory

22. J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. **46**(7), 078002 (2007). [CrossRef]

### 3.2 Rayleigh-Rice (RR) theory

10. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. **4**(2-3), 351–378 (1951). [CrossRef]

*Q*is the angle dependent polarization reflectance, an optical factor that contains information about the dielectric function as well as about the conditions of illumination and detection (angles of incidence and scattering, polarization, etc.).

23. M. Trost, S. Schröder, T. Feigl, A. Duparré, and A. Tünnermann, “Influence of the substrate finish and thin film roughness on the optical performance of Mo/Si multilayers,” Appl. Opt. **50**(9), C148–C153 (2011). [CrossRef] [PubMed]

_{rel}cos

*θ*

_{i}/ λ)

^{2}<< 1. Unfortunately, there is no distinct number to decide whether the criterion is fulfilled or not. In [13] it is stated that

15. T. V. Vorburger, E. Marx, and T. R. Lettieri, “Regimes of surface roughness measurable with light scattering,” Appl. Opt. **32**(19), 3401–3408 (1993). [CrossRef] [PubMed]

### 3.3 Generalized Harvey-Shack (GHS) theory

*f*= 0 to 1/λ [Eq. (3)].

*Q*from the Rayleigh-Rice result to take into account polarization properties. Harvey refers to this procedure as “borrowing” the Rayleigh-Rice optical factor. For small incident and scatter angles it can be shown that

*Q*is equal to the total reflectance [13]. This intuitive quasi-vectorization therefore provides results that are at least as accurate as the purely scalar approach in general. The final GHS result therefore becomes:For smooth surfaces the exponential function in the surface transfer function [Eq. (7)] can be approximated and Eq. (9) transforms into the Rayleigh-Rice result except for a slightly different obliquity factor [27

27. J. C. Stover, “Experimental confirmation of the Rayleigh-Rice obliquity factor,” Proc. SPIE **7792**, 77920J, 77920J-5 (2010). [CrossRef]

25. J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE **7426**, 74260I, 74260I-9 (2009). [CrossRef]

1. E. L. Church and P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. **32**(19), 3344–3353 (1993). [CrossRef] [PubMed]

### 3.4 Rigorous treatments

17. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. **73**(4), 451–455 (1983). [CrossRef]

4. C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, “Comparison and optimization of randomly textured surfaces in thin-film solar cells,” Opt. Express **18**(S3Suppl 3), A335–A341 (2010). [CrossRef] [PubMed]

## 4. Experimental

### 4.1 Sample generation

27. J. C. Stover, “Experimental confirmation of the Rayleigh-Rice obliquity factor,” Proc. SPIE **7792**, 77920J, 77920J-5 (2010). [CrossRef]

### 4.2 Roughness measurements

*f*= 1/D and

*f*= 1/λ are directly relevant for the light scattering, where D denotes the diameter of the illuminated portion of the scattering surface. For typical applications, the low spatial frequency limit is in the order of 1/1000 µm

^{−1}. The shortest wavelength employed for scatter measurements in this paper is 325 nm which corresponds to a high spatial frequency limit of 3 µm

^{−1}. In order to sufficiently cover this large spatial frequency range, different roughness measurement techniques and measurement parameters were used.

^{2}and 141x105 µm

^{2}. While WLI is of great use to measure roughness components in the low- and mid-spatial frequency range, the information at high spatial frequencies is always limited by the transfer function of the objective used.

^{2}and 10x10 µm

^{2}, each scan containing 512x512 data points, were analyzed.

20. A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. **41**(1), 154–171 (2002). [CrossRef] [PubMed]

*n*and

*k*, are required to calculate the optical factor,

*Q*, in Eqs. (6) and (8). For this purpose, specular reflectance measurements were performed on the smoothest surface (sample A). The spectral reflectance was measured using a commercial spectrometer Perkin Elmer Lambda 900. In addition, the reflectance as a function of the angle of incidence for both s- and p-polarized light at the wavelengths of interest were measured using the light scattering instrumentation described in the next section. Starting from the optical properties of iron, the optical constants were then adjusted to fit the measured reflectance data. The measured normal incidence reflectance

*R*and optical constants at the wavelengths of interest are: (i) at 325 nm:

*R*= 48%,

*n*= 1.8,

*k*= 2.5, (ii) at 532 nm:

*R*= 59%,

*n*= 2.04,

*k*= 3.3, (iii) at 1064 nm:

*R*= 67%,

*n*= 2.06,

*k*= 3.9. Considering both the non-zero surface roughness and the uncertainty of the measurements, the relative uncertainty of the reflectance values is estimated to be below 3% at all wavelengths.

### 4.3 Light scattering measurements

28. S. Schröder, T. Herffurth, H. Blaschke, and A. Duparré, “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt. **50**(9), C164–C171 (2011). [CrossRef] [PubMed]

_{s}of 3x10

^{−5}sr. The instrument exhibits a dynamic range of up to 15 orders of magnitude and a noise equivalent ARS level of down to 10

^{−7}sr

^{−1}.

## 5. Diffraction efficiency of sinusoidal phase gratings

27. J. C. Stover, “Experimental confirmation of the Rayleigh-Rice obliquity factor,” Proc. SPIE **7792**, 77920J, 77920J-5 (2010). [CrossRef]

*a*is merely a delta function at a spatial frequency corresponding to the inverse period of the grating. It is straightforward to show that the Rayleigh-Rice result [Eq. (6)] leads to the following expression for the diffraction efficiency of the first order [13,27

**7792**, 77920J, 77920J-5 (2010). [CrossRef]

30. J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A **23**(4), 858–865 (2006). [CrossRef]

*J*

_{m}is the Bessel function of the first kind, order

*m*. The summation limits in the denominator of Eq. (4) are the minimum and maximum propagating order of diffraction. The sum therefore corresponds to a renormalization of the propagating orders when higher orders go evanescent. This renormalization to take into account the effect of evanescent orders is also a fundamental feature of the non-paraxial diffraction theory.

*p*and amplitude

*a*, the angle of incidence was adjusted such that the first orders appear at (scatter) angles between 0 degrees and 90 degrees. The diffraction efficiencies were then calculated using the Rayleigh-Rice theory (RR) according to Eq. (9), the non-paraxial diffraction theory (NP) according to Eq. (10), as well as the Rigorous Coupled Wave Analysis (RCWA). The results for different grating parameters at a wavelength of 532 nm are shown in Fig. 1 .

*a*/λ = 0.01 and below. Also the NP prediction is in good agreement with the RCWA result, although there is a slight disagreement at near grazing angles. This effect is a topic of ongoing discussion [27

**7792**, 77920J, 77920J-5 (2010). [CrossRef]

*a*= 20 nm (Fig. 1(b)) indicate that for larger amplitudes, the RR theory starts to predict efficiencies too high at small angles. Yet the corresponding error is below 5% for

*a*/λ ≤ 0.04. This quantitative smooth-surface limit is in good agreement with former statements for stochastically rough surfaces [13,15

15. T. V. Vorburger, E. Marx, and T. R. Lettieri, “Regimes of surface roughness measurable with light scattering,” Appl. Opt. **32**(19), 3401–3408 (1993). [CrossRef] [PubMed]

*p*= 10 µm (Fig. 1(c)) reveal that this effect almost vanishes for larger periods corresponding to diffraction angles closer to the specular direction.

## 6. Scattering of stochastically rough surfaces

^{−1}in the PSD of samples C suggests that residual turning marks of the raw samples that have not been removed by the grinding process are present for this sample. Therefore, although care has been taken to generate isotropic roughness structures, the effects of anisotropic roughness might have to be taken into account when comparing light scattering measurements and predictions based on PSD data.

*f*

_{min}= 1e-3 and

*f*

_{max}= 1/λ for wavelengths of 325 nm, 532 nm and 1064 nm. The resulting upper bandwidth limits are indicated by arrows in Fig. 2.

**32**(19), 3401–3408 (1993). [CrossRef] [PubMed]

23. M. Trost, S. Schröder, T. Feigl, A. Duparré, and A. Tünnermann, “Influence of the substrate finish and thin film roughness on the optical performance of Mo/Si multilayers,” Appl. Opt. **50**(9), C148–C153 (2011). [CrossRef] [PubMed]

_{rel}cos

*θ*

_{i}/ λ ≤ 0.02. This is in reasonable agreement with the results of Sec. 5 and former statements [13,15

**32**(19), 3401–3408 (1993). [CrossRef] [PubMed]

_{b}/

*R*≤ (4πσ

_{rel}cos

*θ*

_{i}/ λ)

^{2}[3,13], where

*R*denotes the specular reflectance, an alternative criterion for the total scattering instead of the roughness allowed would be TS

_{b}/

*R*≤ 0.07. Based on the results of Sec. 5 we believe that the smooth surface approximation might still be valid in particular at large angles even if these conditions are violated to some extent. If a surface does not meet the smooth surface requirement at a short wavelength or small incident angles, a larger wavelength or incident angle may be used in order to determine the surface PSD from scatter measurements. However, measurements at short wavelengths might still be required to obtain information about high spatial frequency roughness components.

^{4}wavelength relationship predicted by the RR theory.

**32**(19), 3401–3408 (1993). [CrossRef] [PubMed]

## 7. Summary and conclusions

_{rel}cos

*θ*

_{i}/ λ)

^{2}<< 1 is met. As a quantitative criterion, the surface can be considered optically smooth if: σ

_{rel}cos

*θ*

_{i}/ λ ≤ 0.02 or, equivalently, if TS

_{b}/R ≤ 0.07. In this regime, only the first diffracted orders of roughness components contain significant power. PSD and ARS are directly proportional and either of the functions can be determined by measuring the other.

28. S. Schröder, T. Herffurth, H. Blaschke, and A. Duparré, “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt. **50**(9), C164–C171 (2011). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | E. L. Church and P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. |

2. | J. E. Harvey, K. L. Lewotsky, and A. Kotha, “Effects of surface scatter on the optical performance of x-ray synchrotron beam-line mirrors,” Appl. Opt. |

3. | A. Duparré, “Scattering from surfaces and thin films,” in |

4. | C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, “Comparison and optimization of randomly textured surfaces in thin-film solar cells,” Opt. Express |

5. | S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of Roughened TCO Films - Modeling and Measurement,” in Optical Interference Coatings, OSA Technical Digest (Optical Society of America, 2010), paper ThD3. |

6. | M. Flemming, L. Coriand, and A. Duparré, “Ultra-hydrophobicity through stochastic surface roughness,” J. Adhes. Sci. Technol. |

7. | S. Schröder, A. Duparré, “Finish assessment of complex surfaces by advanced light scattering techniques,” Proc. SPIE 7102, paper 0F (2008). |

8. | T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media |

9. | P. Beckmann and A. Spizzichino, |

10. | S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. |

11. | E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship #res,” Opt. Eng. |

12. | J. M. Elson and J. M. Bennett, “Vector scattering theory,” Opt. Eng. |

13. | J. C. Stover, |

14. | J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. Dissertation, University of Arizona (1976). |

15. | T. V. Vorburger, E. Marx, and T. R. Lettieri, “Regimes of surface roughness measurable with light scattering,” Appl. Opt. |

16. | J. E. Harvey, N. Choi, and A. Krywonos, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE |

17. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. |

18. | A. Taflove and S. C. Hagness, |

19. | E. L. Church, “Statistical effects in the measurement and characterization of smooth scattering surfaces,” Proc. SPIE |

20. | A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. |

21. | “Optics and optical instruments-test methods for radiation scattered by optical components,” ISO 13696:2002 (International Organization for Standardization, 2002). |

22. | J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. |

23. | M. Trost, S. Schröder, T. Feigl, A. Duparré, and A. Tünnermann, “Influence of the substrate finish and thin film roughness on the optical performance of Mo/Si multilayers,” Appl. Opt. |

24. | A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. Dissertation, College of Optics and Photonics, University of Central Florida (2006). |

25. | J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE |

26. | A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” Accepted for publication in J. Opt. Soc. Am. A (March 25, 2011) |

27. | J. C. Stover, “Experimental confirmation of the Rayleigh-Rice obliquity factor,” Proc. SPIE |

28. | S. Schröder, T. Herffurth, H. Blaschke, and A. Duparré, “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt. |

29. | A. von Finck, M. Hauptvogel, and A. Duparré, “Instrument for close-to-process light scatter measurements of thin film coatings and substrates,” in Optical Interference Coatings, OSA Technical Digest (Optical Society of America, 2010), paper ThD4. |

30. | J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A |

**OCIS Codes**

(240.5770) Optics at surfaces : Roughness

(290.0290) Scattering : Scattering

(290.5835) Scattering : Scattering, Harvey

**ToC Category:**

Scattering

**History**

Original Manuscript: February 14, 2011

Revised Manuscript: April 7, 2011

Manuscript Accepted: May 1, 2011

Published: May 5, 2011

**Citation**

Sven Schröder, Angela Duparré, Luisa Coriand, Andreas Tünnermann, Dayana H. Penalver, and James E. Harvey, "Modeling of light scattering in different regimes of surface roughness," Opt. Express **19**, 9820-9835 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9820

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### References

- E. L. Church and P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. 32(19), 3344–3353 (1993). [CrossRef] [PubMed]
- J. E. Harvey, K. L. Lewotsky, and A. Kotha, “Effects of surface scatter on the optical performance of x-ray synchrotron beam-line mirrors,” Appl. Opt. 34(16), 3024–3032 (1995). [CrossRef] [PubMed]
- A. Duparré, “Scattering from surfaces and thin films,” in Encyclopedia of Modern Optics, B. D. Guenther, D. G. Steel, and L. Bayvel, eds. (Elsevier, 2004).
- C. Rockstuhl, S. Fahr, K. Bittkau, T. Beckers, R. Carius, F.-J. Haug, T. Söderström, C. Ballif, and F. Lederer, “Comparison and optimization of randomly textured surfaces in thin-film solar cells,” Opt. Express 18(S3Suppl 3), A335–A341 (2010). [CrossRef] [PubMed]
- S. Schröder, A. Duparré, K. Füchsel, N. Kaiser, A. Tünnermann, and J. E. Harvey, “Scattering of Roughened TCO Films - Modeling and Measurement,” in Optical Interference Coatings, OSA Technical Digest (Optical Society of America, 2010), paper ThD3.
- M. Flemming, L. Coriand, and A. Duparré, “Ultra-hydrophobicity through stochastic surface roughness,” J. Adhes. Sci. Technol. 23(3), 381–400 (2009). [CrossRef]
- S. Schröder, A. Duparré, “Finish assessment of complex surfaces by advanced light scattering techniques,” Proc. SPIE 7102, paper 0F (2008).
- T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14(4), R1–R40 (2004). [CrossRef]
- P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, (Pergamon Press, 1963).
- S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4(2-3), 351–378 (1951). [CrossRef]
- E. L. Church, H. A. Jenkinson, and J. M. Zavada, “Relationship #res,” Opt. Eng. 18, 125 (1979).
- J. M. Elson and J. M. Bennett, “Vector scattering theory,” Opt. Eng. 18, 116–124 (1979).
- J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE Press, 1995).
- J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. Dissertation, University of Arizona (1976).
- T. V. Vorburger, E. Marx, and T. R. Lettieri, “Regimes of surface roughness measurable with light scattering,” Appl. Opt. 32(19), 3401–3408 (1993). [CrossRef] [PubMed]
- J. E. Harvey, N. Choi, and A. Krywonos, “Scattering from moderately rough interfaces between two arbitrary media,” Proc. SPIE 7794, 77940V, 77940V-11 (2010). [CrossRef]
- M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 73(4), 451–455 (1983). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).
- E. L. Church, “Statistical effects in the measurement and characterization of smooth scattering surfaces,” Proc. SPIE 511, 18 (1984).
- A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, and J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. 41(1), 154–171 (2002). [CrossRef] [PubMed]
- “Optics and optical instruments-test methods for radiation scattered by optical components,” ISO 13696:2002 (International Organization for Standardization, 2002).
- J. E. Harvey, A. Krywonos, and C. L. Vernold, “Modified Beckmann-Kirchhoff scattering model for rough surfaces with large incident and scattering angles,” Opt. Eng. 46(7), 078002 (2007). [CrossRef]
- M. Trost, S. Schröder, T. Feigl, A. Duparré, and A. Tünnermann, “Influence of the substrate finish and thin film roughness on the optical performance of Mo/Si multilayers,” Appl. Opt. 50(9), C148–C153 (2011). [CrossRef] [PubMed]
- A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. Dissertation, College of Optics and Photonics, University of Central Florida (2006).
- J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 74260I, 74260I-9 (2009). [CrossRef]
- A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” Accepted for publication in J. Opt. Soc. Am. A (March 25, 2011)
- J. C. Stover, “Experimental confirmation of the Rayleigh-Rice obliquity factor,” Proc. SPIE 7792, 77920J, 77920J-5 (2010). [CrossRef]
- S. Schröder, T. Herffurth, H. Blaschke, and A. Duparré, “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt. 50(9), C164–C171 (2011). [CrossRef] [PubMed]
- A. von Finck, M. Hauptvogel, and A. Duparré, “Instrument for close-to-process light scatter measurements of thin film coatings and substrates,” in Optical Interference Coatings, OSA Technical Digest (Optical Society of America, 2010), paper ThD4.
- J. E. Harvey, A. Krywonos, and D. Bogunovic, “Nonparaxial scalar treatment of sinusoidal phase gratings,” J. Opt. Soc. Am. A 23(4), 858–865 (2006). [CrossRef]

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