Abstract

The polarization of light scattered into directions out of the plane of incidence for 532-nm light incident at 45° with p polarization was measured from rough silicon, rough titanium nitride, polished fused silica and glass ceramic, and ground and incompletely polished black glass. Models for polarized light scattering from microroughness, subsurface defects, and facets are reviewed. The measurements demonstrate the validity of the models and the utility of polarized light scattering measurements for distinguishing between roughness and defects.

© 1999 Optical Society of America

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References

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  1. J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE Press, Bellingham, Wash., 1995).
  2. J. C. Stover, ed., Optical Scattering in the Optics, Semiconductor, and Computer Disk Industries, Proc. SPIE2541, (1995).
  3. J. C. Stover, ed., Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays, Proc. SPIE2862, (1996).
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  5. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  6. R. E. Luna, “Scattering by one-dimensional random rough metallic surfaces in a conical configuration: several polarizations,” Opt. Lett. 21, 1418–1420 (1996).
    [CrossRef] [PubMed]
  7. J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-turned mirror,” Opt. Eng. (Bellingham) 34, 1593–1598 (1995).
    [CrossRef]
  8. E. L. Church, J. C. Stover, “Surface haze in the Stokes–Mueller representation,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 54–68 (1996).
    [CrossRef]
  9. E. R. Méndez, A. G. Navarrete, R. E. Luna, “Statistics of the polarization properties of one-dimensional randomly rough surfaces,” J. Opt. Soc. Am. A 12, 2507–2516 (1995).
    [CrossRef]
  10. S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
    [CrossRef]
  11. T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
    [CrossRef]
  12. T. A. Germer, “Angular dependence and polarization of out-of-plane optical scattering from particulate contamination, subsurface defects, and surface microroughness,” Appl. Opt. 36, 8798–8805 (1997).
    [CrossRef]
  13. D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
    [CrossRef]
  14. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  15. G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
    [CrossRef]
  16. D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970).
  17. E. Kröger, E. Kretschmann, “Scattering of light by slightly rough surfaces or thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
    [CrossRef]
  18. C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
    [CrossRef]
  19. T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997).
    [CrossRef]
  20. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  21. R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. II, pp. 22.1–37.
  22. E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).
  23. B. W. Scheer, “Development of a physical haze and microroughness standard,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 78–95 (1996).
    [CrossRef]
  24. E. D. Palik, Handbook of Optical Constants of Solids II (Academic, San Diego, Calif., 1991).
  25. Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.
  26. C. Asmail, J. Fuller, R. Parks, “Status of bidirectional reflectance distribution function (BRDF) calibration standards development,” in Quality and Reliability for Optical Systems, J. W. Bilbro, R. E. Parks, eds., Proc. SPIE1993, 44–53 (1993).
    [CrossRef]
  27. B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Tech. Note1297 (National Institute of Standards and Technology, 1994).
  28. J. M. Elson, “Light scattering from surfaces with a single dielectric overlayer,” J. Opt. Soc. Am. 66, 682–694 (1976).
    [CrossRef]
  29. C. Asmail, J. Hsia, A. Parr, J. Hoeft, “Rayleigh scattering limits for low-level bidirectional reflectance distribution function measurements,” Appl. Opt. 33, 6084–6091 (1994).
    [CrossRef] [PubMed]
  30. T. A. Germer, C. C. Asmail, “Microroughness-blind hemispherical optical scatter instrument,” U.S. patent application09/058,182 (April10, 1998).
  31. J. Ryuta, E. Morita, T. Tanaka, Y. Shimanuki, “Crystal-originated singularities on Si wafer surface after SC1 cleaning,” Jpn. J. Appl. Phys., Part 2 29, L1947–L1949 (1990).
    [CrossRef]
  32. L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light scattering measurements of dielectric spheres on a silicon surface,” Opt. Lett. (to be published).
  33. T. A. Germer, “Application of bidirectional ellipsometry to the characterization of roughness and defects in dielectric layers,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 121–131 (1998).
    [CrossRef]

1997 (2)

1996 (2)

1995 (2)

E. R. Méndez, A. G. Navarrete, R. E. Luna, “Statistics of the polarization properties of one-dimensional randomly rough surfaces,” J. Opt. Soc. Am. A 12, 2507–2516 (1995).
[CrossRef]

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-turned mirror,” Opt. Eng. (Bellingham) 34, 1593–1598 (1995).
[CrossRef]

1994 (1)

1990 (1)

J. Ryuta, E. Morita, T. Tanaka, Y. Shimanuki, “Crystal-originated singularities on Si wafer surface after SC1 cleaning,” Jpn. J. Appl. Phys., Part 2 29, L1947–L1949 (1990).
[CrossRef]

1978 (1)

1976 (1)

1970 (1)

E. Kröger, E. Kretschmann, “Scattering of light by slightly rough surfaces or thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[CrossRef]

1968 (1)

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[CrossRef]

1967 (1)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

1951 (1)

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

Asmail, C.

C. Asmail, J. Hsia, A. Parr, J. Hoeft, “Rayleigh scattering limits for low-level bidirectional reflectance distribution function measurements,” Appl. Opt. 33, 6084–6091 (1994).
[CrossRef] [PubMed]

C. Asmail, J. Fuller, R. Parks, “Status of bidirectional reflectance distribution function (BRDF) calibration standards development,” in Quality and Reliability for Optical Systems, J. W. Bilbro, R. E. Parks, eds., Proc. SPIE1993, 44–53 (1993).
[CrossRef]

Asmail, C. C.

T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
[CrossRef]

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

T. A. Germer, C. C. Asmail, “Microroughness-blind hemispherical optical scatter instrument,” U.S. patent application09/058,182 (April10, 1998).

T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997).
[CrossRef]

Azzam, R. M. A.

Barrick, D. E.

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[CrossRef]

D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Chipman, R. A.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-turned mirror,” Opt. Eng. (Bellingham) 34, 1593–1598 (1995).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. II, pp. 22.1–37.

Church, E. L.

E. L. Church, J. C. Stover, “Surface haze in the Stokes–Mueller representation,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 54–68 (1996).
[CrossRef]

Cromer, C. L.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

Elson, J. M.

Fuller, J.

C. Asmail, J. Fuller, R. Parks, “Status of bidirectional reflectance distribution function (BRDF) calibration standards development,” in Quality and Reliability for Optical Systems, J. W. Bilbro, R. E. Parks, eds., Proc. SPIE1993, 44–53 (1993).
[CrossRef]

Germer, T. A.

T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
[CrossRef]

T. A. Germer, “Angular dependence and polarization of out-of-plane optical scattering from particulate contamination, subsurface defects, and surface microroughness,” Appl. Opt. 36, 8798–8805 (1997).
[CrossRef]

T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997).
[CrossRef]

T. A. Germer, “Application of bidirectional ellipsometry to the characterization of roughness and defects in dielectric layers,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 121–131 (1998).
[CrossRef]

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light scattering measurements of dielectric spheres on a silicon surface,” Opt. Lett. (to be published).

T. A. Germer, C. C. Asmail, “Microroughness-blind hemispherical optical scatter instrument,” U.S. patent application09/058,182 (April10, 1998).

Hoeft, J.

Hsia, J.

Hsia, J. J.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kretschmann, E.

E. Kröger, E. Kretschmann, “Scattering of light by slightly rough surfaces or thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[CrossRef]

Kröger, E.

E. Kröger, E. Kretschmann, “Scattering of light by slightly rough surfaces or thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[CrossRef]

Kuyatt, C. E.

B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Tech. Note1297 (National Institute of Standards and Technology, 1994).

Luna, R. E.

Méndez, E. R.

Morita, E.

J. Ryuta, E. Morita, T. Tanaka, Y. Shimanuki, “Crystal-originated singularities on Si wafer surface after SC1 cleaning,” Jpn. J. Appl. Phys., Part 2 29, L1947–L1949 (1990).
[CrossRef]

Mulholland, G. W.

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light scattering measurements of dielectric spheres on a silicon surface,” Opt. Lett. (to be published).

Navarrete, A. G.

Nee, S.-M. F.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids II (Academic, San Diego, Calif., 1991).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

Parks, R.

C. Asmail, J. Fuller, R. Parks, “Status of bidirectional reflectance distribution function (BRDF) calibration standards development,” in Quality and Reliability for Optical Systems, J. W. Bilbro, R. E. Parks, eds., Proc. SPIE1993, 44–53 (1993).
[CrossRef]

Parr, A.

Pezzaniti, J. L.

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-turned mirror,” Opt. Eng. (Bellingham) 34, 1593–1598 (1995).
[CrossRef]

Proctor, J. E.

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

Rice, S. O.

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

Ryuta, J.

J. Ryuta, E. Morita, T. Tanaka, Y. Shimanuki, “Crystal-originated singularities on Si wafer surface after SC1 cleaning,” Jpn. J. Appl. Phys., Part 2 29, L1947–L1949 (1990).
[CrossRef]

Scheer, B. W.

T. A. Germer, C. C. Asmail, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
[CrossRef]

B. W. Scheer, “Development of a physical haze and microroughness standard,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 78–95 (1996).
[CrossRef]

Shimanuki, Y.

J. Ryuta, E. Morita, T. Tanaka, Y. Shimanuki, “Crystal-originated singularities on Si wafer surface after SC1 cleaning,” Jpn. J. Appl. Phys., Part 2 29, L1947–L1949 (1990).
[CrossRef]

Stover, J. C.

E. L. Church, J. C. Stover, “Surface haze in the Stokes–Mueller representation,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 54–68 (1996).
[CrossRef]

J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE Press, Bellingham, Wash., 1995).

Sung, L.

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light scattering measurements of dielectric spheres on a silicon surface,” Opt. Lett. (to be published).

Tanaka, T.

J. Ryuta, E. Morita, T. Tanaka, Y. Shimanuki, “Crystal-originated singularities on Si wafer surface after SC1 cleaning,” Jpn. J. Appl. Phys., Part 2 29, L1947–L1949 (1990).
[CrossRef]

Taylor, B. N.

B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Tech. Note1297 (National Institute of Standards and Technology, 1994).

Valenzuela, G. R.

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Appl. Opt. (3)

Commun. Pure Appl. Math. (1)

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

IEEE Trans. Antennas Propag. (2)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. AP-15, 552–557 (1967).
[CrossRef]

D. E. Barrick, “Rough surface scattering based on the specular point theory,” IEEE Trans. Antennas Propag. AP-16, 449–454 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys., Part 2 (1)

J. Ryuta, E. Morita, T. Tanaka, Y. Shimanuki, “Crystal-originated singularities on Si wafer surface after SC1 cleaning,” Jpn. J. Appl. Phys., Part 2 29, L1947–L1949 (1990).
[CrossRef]

Opt. Eng. (Bellingham) (1)

J. L. Pezzaniti, R. A. Chipman, “Mueller matrix scatter polarimetry of a diamond-turned mirror,” Opt. Eng. (Bellingham) 34, 1593–1598 (1995).
[CrossRef]

Opt. Lett. (3)

Z. Phys. (1)

E. Kröger, E. Kretschmann, “Scattering of light by slightly rough surfaces or thin films including plasma resonance emission,” Z. Phys. 237, 1–15 (1970).
[CrossRef]

Other (19)

C. C. Asmail, C. L. Cromer, J. E. Proctor, J. J. Hsia, “Instrumentation at the National Institute of Standards and Technology for bidirectional reflectance distribution function (BRDF) measurements,” in Stray Radiation in Optical Systems III, R. P. Breault, ed., Proc. SPIE2260, 52–61 (1994).
[CrossRef]

T. A. Germer, C. C. Asmail, “A goniometric optical scatter instrument for bidirectional reflectance distribution function measurements with out-of-plane and polarimetry capabilities,” in Scattering and Surface Roughness, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3141, 220–231 (1997).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. II, pp. 22.1–37.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

B. W. Scheer, “Development of a physical haze and microroughness standard,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 78–95 (1996).
[CrossRef]

E. D. Palik, Handbook of Optical Constants of Solids II (Academic, San Diego, Calif., 1991).

Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

C. Asmail, J. Fuller, R. Parks, “Status of bidirectional reflectance distribution function (BRDF) calibration standards development,” in Quality and Reliability for Optical Systems, J. W. Bilbro, R. E. Parks, eds., Proc. SPIE1993, 44–53 (1993).
[CrossRef]

B. N. Taylor, C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Tech. Note1297 (National Institute of Standards and Technology, 1994).

T. A. Germer, C. C. Asmail, “Microroughness-blind hemispherical optical scatter instrument,” U.S. patent application09/058,182 (April10, 1998).

E. L. Church, J. C. Stover, “Surface haze in the Stokes–Mueller representation,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays, J. C. Stover, ed., Proc. SPIE2862, 54–68 (1996).
[CrossRef]

D. E. Barrick, Radar Cross Section Handbook (Plenum, New York, 1970).

J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE Press, Bellingham, Wash., 1995).

J. C. Stover, ed., Optical Scattering in the Optics, Semiconductor, and Computer Disk Industries, Proc. SPIE2541, (1995).

J. C. Stover, ed., Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays, Proc. SPIE2862, (1996).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light scattering measurements of dielectric spheres on a silicon surface,” Opt. Lett. (to be published).

T. A. Germer, “Application of bidirectional ellipsometry to the characterization of roughness and defects in dielectric layers,” in Flatness, Roughness, and Discrete Defect Characterization for Computer Disks, Wafers, and Flat Panel Displays II, J. C. Stover, ed., Proc. SPIE3275, 121–131 (1998).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Sample and beam coordinate systems.

Fig. 2
Fig. 2

Schematic of the intensity distribution measured by a rotating linear polarizer, defining angle η and maximum and minimum signals fmax and fmin, respectively. The axes are defined so that the viewer is looking into the scattered beam.

Fig. 3
Fig. 3

Results of bidirectional ellipsometry measurements for the four silicon samples (Si1–Si4) as functions of azimuthal scattering angle ϕr: (top) degree of linear polarization [PL(p)] and (bottom) principal polarization angle η(p) for p-polarized incident light. The incident and scattering polar angles were both 45°. The curves represent the models for surface microroughness (solid), subsurface defects (dashed), and facets (dotted).

Fig. 4
Fig. 4

Results of bidirectional ellipsometry measurements for the TiN sample as functions of azimuthal scattering angle ϕr: (top) degree of linear polarization [PL(p)] and (bottom) principal polarization angle η(p) for p-polarized incident light. The incident and scattering polar angles were both 45°. The curves represent the models for surface microroughness (solid) and subsurface defects (dashed).

Fig. 5
Fig. 5

Results of bidirectional ellipsometry measurements for the two transparent samples (FS and ZD) as functions of azimuthal scattering angle ϕr: (top) degree of linear polarization [PL(p)] and (bottom) principal polarization angle η(p) for p-polarized incident light. The incident and scattering polar angles were both 45°. The curves represent the models for surface microroughness (solid) and subsurface defects (dashed).

Fig. 6
Fig. 6

Results of bidirectional ellipsometry measurements for the six black-glass samples (BgA, BgB, BgC, BgD, BgE, and BgF) as functions of azimuthal scattering angle ϕr: (top) degree of linear polarization [PL(p)] and (bottom) principal polarization angle η(p) for p-polarized incident light. The incident and scattering polar angles were both 45°. The curves represent the models for surface microroughness (solid) and subsurface defects (dashed).

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

EsscatEpscat=A exp(ikR)RqssqpsqspqppEsincEpinc
qsssub=k cos ϕr/[(kzi+kzi)(kzr+kzr)],
qpssub=-kzi sin ϕr/[(kzi+kzi)(kzr+kzr)],
qspsub=-kzr sin ϕr/[(kzi+kzi)(kzr+kzr)],
qppsub=(kxyikxyr-kzikzr cos ϕr)/[(kzi+kzi)×(kzr+kzr)],
kzj=k cos θj,
kxyj=k sin θj,
kzj=k(-sin2 θj)1/2,
qssrough=k cos ϕr/[(kzi+kzi)(kzr+kzr)],
qpsrough=-kzi sin ϕr/[(kzi+kzi)(kzr+kzr)],
qsprough=-kzr sin ϕr/[(kzi+kzi)(kzr+kzr)],
qpprough=(kxyikxyr-kzikzr cos ϕr)/[(kzi+kzi)×(kzr+kzr)].
η(p)=12arctan(S1, S2),
PL(p)=(fmax-fmin)/(fmax+fmin)=[(S1)2+(S2)2]1/2/S0,

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