Abstract

A laboratory model of a layered structure with a rough upper surface (a glass microscope slide cut with a diamond saw) is used to obtain optical polarimetric data. Scatterometer measurements were made of all the Mueller matrix elements associated with light scattered in arbitrary directions. (A preliminary measurement of scattering from a smooth opaque gold film on a silicon wafer was used to validate the calculation of the Mueller matrix elements.) These measurements are compared with corresponding analytical solutions based on the full-wave approach. Physical interpretations of the analytical solutions that account for scattering upon reflection and transmission across rough interfaces are given in a companion paper. The agreement between calculations and measurements suggests that the full wave, polarimetric solutions can provide a reliable database for electromagnetic detection of rough surfaces in remote-sensing applications.

© 1997 Optical Society of America

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References

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  1. E. Bahar, R. D. Kubik, “Measurements from polarimetric optical bistatic scatterometer,” in Conference Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (IEEE Service Center, Piscataway, N.J., 1993), pp. 173–176.
  2. E. Bahar, Y. Zhang, “Measurements of the Mueller matrix elements using a fully polarimetric optical scatterometer,” in Conference Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (IEEE Service Center, Piscataway, N.J., 1995), pp. 208–210.
    [CrossRef]
  3. R. D. Kubik, “Scattering and depolarization upon transmission across and multiple reflections from irregular multi-layered structures,” Ph.D. dissertation (University of Nebraska–Lincoln, Lincoln, Neb., 1993).
  4. E. Bahar, B. S. Lee, “Full wave solutions for rough surface bistatic radar cross sections—comparison with small perturbation, physical optics solutions numerical and experimental results,” Radio Sci. 29, 407–429 (1994).
    [CrossRef]
  5. E. Bahar, R. D. Kubik, “Computations of the Mueller matrix elements for scattering from layered structures with rough surfaces, with applications to optical detection,” Appl. Opt. 36, 1–9 (1997).
  6. M. I. Sancer, “Shadow corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1968).
  7. E. Bahar, M. A. Fitzwater, “Shadowing by non-Gaussian rough surfaces for which decorrelation implies statistical independence,” Radio Sci. 18, 566–572 (1983).
    [CrossRef]
  8. E. D. Palik, ed., Handbook of Optical Constants of Solids, Vol. 1 (Academic, Orlando, Fla., 1985).
  9. F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emmissivity,” Appl. Opt. 9, 1747–1475 (1970).
    [CrossRef]
  10. J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990).
  11. P. Beckman, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973).
  12. E. Bahar, Y. F. Lee, “Scattering cross sections of non-Gaussian rough surfaces: unified full wave approach,” IEEE Trans. Antennas Propag. 39, 1777–1781 (1991).
    [CrossRef]
  13. S. M. Haugland, E. Bahar, A. H. Carrieri, “Polarized IR scattering used to identify contaminant coatings over rough surfaces,” in Proceedings of the 1991 Scientific Conference on Obscuration and Aerosol Research (Chemical Research, Development and Engineering Center, U.S. Army Armament, Munitions and Chemical Command, Aberdeen Proving Ground, Md., 1992), pp. 143–155.
  14. E. Bahar, M. A. Fitzwater, “Full wave physical models of nonspecular scattering in irregular stratified media,” IEEE Trans. Antennas Propag. 37, 1609–1615 (1989).
    [CrossRef]
  15. E. Bahar, J. R. Wait, “Propagation in a model terrestrial waveguide of nonuniform height, theory and experiment,” Radio Sci. J. Res. 69D, 1445–1463 (1965).

1997 (1)

E. Bahar, R. D. Kubik, “Computations of the Mueller matrix elements for scattering from layered structures with rough surfaces, with applications to optical detection,” Appl. Opt. 36, 1–9 (1997).

1994 (1)

E. Bahar, B. S. Lee, “Full wave solutions for rough surface bistatic radar cross sections—comparison with small perturbation, physical optics solutions numerical and experimental results,” Radio Sci. 29, 407–429 (1994).
[CrossRef]

1991 (1)

E. Bahar, Y. F. Lee, “Scattering cross sections of non-Gaussian rough surfaces: unified full wave approach,” IEEE Trans. Antennas Propag. 39, 1777–1781 (1991).
[CrossRef]

1989 (1)

E. Bahar, M. A. Fitzwater, “Full wave physical models of nonspecular scattering in irregular stratified media,” IEEE Trans. Antennas Propag. 37, 1609–1615 (1989).
[CrossRef]

1983 (1)

E. Bahar, M. A. Fitzwater, “Shadowing by non-Gaussian rough surfaces for which decorrelation implies statistical independence,” Radio Sci. 18, 566–572 (1983).
[CrossRef]

1970 (1)

1968 (1)

M. I. Sancer, “Shadow corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1968).

1965 (1)

E. Bahar, J. R. Wait, “Propagation in a model terrestrial waveguide of nonuniform height, theory and experiment,” Radio Sci. J. Res. 69D, 1445–1463 (1965).

Bahar, E.

E. Bahar, R. D. Kubik, “Computations of the Mueller matrix elements for scattering from layered structures with rough surfaces, with applications to optical detection,” Appl. Opt. 36, 1–9 (1997).

E. Bahar, B. S. Lee, “Full wave solutions for rough surface bistatic radar cross sections—comparison with small perturbation, physical optics solutions numerical and experimental results,” Radio Sci. 29, 407–429 (1994).
[CrossRef]

E. Bahar, Y. F. Lee, “Scattering cross sections of non-Gaussian rough surfaces: unified full wave approach,” IEEE Trans. Antennas Propag. 39, 1777–1781 (1991).
[CrossRef]

E. Bahar, M. A. Fitzwater, “Full wave physical models of nonspecular scattering in irregular stratified media,” IEEE Trans. Antennas Propag. 37, 1609–1615 (1989).
[CrossRef]

E. Bahar, M. A. Fitzwater, “Shadowing by non-Gaussian rough surfaces for which decorrelation implies statistical independence,” Radio Sci. 18, 566–572 (1983).
[CrossRef]

E. Bahar, J. R. Wait, “Propagation in a model terrestrial waveguide of nonuniform height, theory and experiment,” Radio Sci. J. Res. 69D, 1445–1463 (1965).

E. Bahar, Y. Zhang, “Measurements of the Mueller matrix elements using a fully polarimetric optical scatterometer,” in Conference Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (IEEE Service Center, Piscataway, N.J., 1995), pp. 208–210.
[CrossRef]

E. Bahar, R. D. Kubik, “Measurements from polarimetric optical bistatic scatterometer,” in Conference Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (IEEE Service Center, Piscataway, N.J., 1993), pp. 173–176.

S. M. Haugland, E. Bahar, A. H. Carrieri, “Polarized IR scattering used to identify contaminant coatings over rough surfaces,” in Proceedings of the 1991 Scientific Conference on Obscuration and Aerosol Research (Chemical Research, Development and Engineering Center, U.S. Army Armament, Munitions and Chemical Command, Aberdeen Proving Ground, Md., 1992), pp. 143–155.

Beckman, P.

P. Beckman, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973).

Carrieri, A. H.

S. M. Haugland, E. Bahar, A. H. Carrieri, “Polarized IR scattering used to identify contaminant coatings over rough surfaces,” in Proceedings of the 1991 Scientific Conference on Obscuration and Aerosol Research (Chemical Research, Development and Engineering Center, U.S. Army Armament, Munitions and Chemical Command, Aberdeen Proving Ground, Md., 1992), pp. 143–155.

Fitzwater, M. A.

E. Bahar, M. A. Fitzwater, “Full wave physical models of nonspecular scattering in irregular stratified media,” IEEE Trans. Antennas Propag. 37, 1609–1615 (1989).
[CrossRef]

E. Bahar, M. A. Fitzwater, “Shadowing by non-Gaussian rough surfaces for which decorrelation implies statistical independence,” Radio Sci. 18, 566–572 (1983).
[CrossRef]

Haugland, S. M.

S. M. Haugland, E. Bahar, A. H. Carrieri, “Polarized IR scattering used to identify contaminant coatings over rough surfaces,” in Proceedings of the 1991 Scientific Conference on Obscuration and Aerosol Research (Chemical Research, Development and Engineering Center, U.S. Army Armament, Munitions and Chemical Command, Aberdeen Proving Ground, Md., 1992), pp. 143–155.

Kubik, R. D.

E. Bahar, R. D. Kubik, “Computations of the Mueller matrix elements for scattering from layered structures with rough surfaces, with applications to optical detection,” Appl. Opt. 36, 1–9 (1997).

E. Bahar, R. D. Kubik, “Measurements from polarimetric optical bistatic scatterometer,” in Conference Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (IEEE Service Center, Piscataway, N.J., 1993), pp. 173–176.

R. D. Kubik, “Scattering and depolarization upon transmission across and multiple reflections from irregular multi-layered structures,” Ph.D. dissertation (University of Nebraska–Lincoln, Lincoln, Neb., 1993).

Lee, B. S.

E. Bahar, B. S. Lee, “Full wave solutions for rough surface bistatic radar cross sections—comparison with small perturbation, physical optics solutions numerical and experimental results,” Radio Sci. 29, 407–429 (1994).
[CrossRef]

Lee, Y. F.

E. Bahar, Y. F. Lee, “Scattering cross sections of non-Gaussian rough surfaces: unified full wave approach,” IEEE Trans. Antennas Propag. 39, 1777–1781 (1991).
[CrossRef]

Nicodemus, F. E.

Sancer, M. I.

M. I. Sancer, “Shadow corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1968).

Stover, J. C.

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990).

Wait, J. R.

E. Bahar, J. R. Wait, “Propagation in a model terrestrial waveguide of nonuniform height, theory and experiment,” Radio Sci. J. Res. 69D, 1445–1463 (1965).

Zhang, Y.

E. Bahar, Y. Zhang, “Measurements of the Mueller matrix elements using a fully polarimetric optical scatterometer,” in Conference Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (IEEE Service Center, Piscataway, N.J., 1995), pp. 208–210.
[CrossRef]

Appl. Opt. (2)

E. Bahar, R. D. Kubik, “Computations of the Mueller matrix elements for scattering from layered structures with rough surfaces, with applications to optical detection,” Appl. Opt. 36, 1–9 (1997).

F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emmissivity,” Appl. Opt. 9, 1747–1475 (1970).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

E. Bahar, M. A. Fitzwater, “Full wave physical models of nonspecular scattering in irregular stratified media,” IEEE Trans. Antennas Propag. 37, 1609–1615 (1989).
[CrossRef]

M. I. Sancer, “Shadow corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1968).

E. Bahar, Y. F. Lee, “Scattering cross sections of non-Gaussian rough surfaces: unified full wave approach,” IEEE Trans. Antennas Propag. 39, 1777–1781 (1991).
[CrossRef]

Radio Sci. (2)

E. Bahar, M. A. Fitzwater, “Shadowing by non-Gaussian rough surfaces for which decorrelation implies statistical independence,” Radio Sci. 18, 566–572 (1983).
[CrossRef]

E. Bahar, B. S. Lee, “Full wave solutions for rough surface bistatic radar cross sections—comparison with small perturbation, physical optics solutions numerical and experimental results,” Radio Sci. 29, 407–429 (1994).
[CrossRef]

Radio Sci. J. Res. (1)

E. Bahar, J. R. Wait, “Propagation in a model terrestrial waveguide of nonuniform height, theory and experiment,” Radio Sci. J. Res. 69D, 1445–1463 (1965).

Other (7)

E. D. Palik, ed., Handbook of Optical Constants of Solids, Vol. 1 (Academic, Orlando, Fla., 1985).

J. C. Stover, Optical Scattering: Measurement and Analysis (McGraw-Hill, New York, 1990).

P. Beckman, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973).

S. M. Haugland, E. Bahar, A. H. Carrieri, “Polarized IR scattering used to identify contaminant coatings over rough surfaces,” in Proceedings of the 1991 Scientific Conference on Obscuration and Aerosol Research (Chemical Research, Development and Engineering Center, U.S. Army Armament, Munitions and Chemical Command, Aberdeen Proving Ground, Md., 1992), pp. 143–155.

E. Bahar, R. D. Kubik, “Measurements from polarimetric optical bistatic scatterometer,” in Conference Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (IEEE Service Center, Piscataway, N.J., 1993), pp. 173–176.

E. Bahar, Y. Zhang, “Measurements of the Mueller matrix elements using a fully polarimetric optical scatterometer,” in Conference Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (IEEE Service Center, Piscataway, N.J., 1995), pp. 208–210.
[CrossRef]

R. D. Kubik, “Scattering and depolarization upon transmission across and multiple reflections from irregular multi-layered structures,” Ph.D. dissertation (University of Nebraska–Lincoln, Lincoln, Neb., 1993).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of scattering from a single rough interface, with notation used in this paper.

Fig. 2
Fig. 2

Schematic diagram of scatterometer. The optical table rotates about the z axis to vary the incident elevation angle θ i . The detector base rotates about the y axis to vary the detector azimuth angle ϕ f (scatter plane). The detector mount rotates around a semicircular arc in the scatter plane to vary the scatter elevation angle θ f . The normal to the mean plane of the sample–air interface is the y axis. The sample holder can be rotated around the y axis to vary the incident azimuth angle (ϕ i ). The sample is 50 cm from the detector.

Fig. 3
Fig. 3

Specular scatter measurements for an opaque gold film on a silicon wafer substrate and full-wave computations (theory) of the Mueller matrix elements. Rough surface rms height h = 0.5 nm: (a) M 11; (b) M 22; (c) M 12, M 21; (d) M 34 = - M 43; (e) M 33; (f) M 44. All the Mueller matrix elements (except M 11) are plotted relative to M 11.

Fig. 4
Fig. 4

Glass slide, cut by a circular diamond saw.

Fig. 5
Fig. 5

AFM surface data for a rough glass plate: (a) Image of the 100 µm × 100 µm rough surface area, (b) surface height autocorrelation function, (c) histogram of surface heights (scan 1). The shaded area is normalized to unity.

Fig. 6
Fig. 6

AFM surface data for a rough glass plate: (a) Image of the 100 × 100 µm rough surface area, (b) surface height autocorrelation function, (c) histogram of surface heights (scan 2). The shaded area is normalized to unity.

Fig. 7
Fig. 7

Cut glass slab: (a) (- n i · a y) > (- n i · n l), (b) (- n i · a y) < (- n i · n l).

Fig. 8
Fig. 8

Measured and computed (theory) Mueller matrix data for the condition shown in Fig. 7(a): (a) M 12, (b) M 22, (c) M 33, (d) M 34, (e) M 44, for the specularly scattered field.

Fig. 9
Fig. 9

Measured and computed (theory) Mueller matrix data for the condition shown in Fig. 7(b): (a) M 12, (b) M 22, (c) M 33, (d) M 34, (e) M 44, for the specularly scattered field.

Tables (1)

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Table 1 Illuminating Beam and Glass Slide Parameters

Equations (8)

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ρDPQDRS*=Qvy,-vy0000IRSPQ,
0000IRSPQ=D00PQD00RS*pnP2n0f, n0i|ndn,
Qvy, -vy=2k020χ2vy,-vy; rd)-χvyχ-vyJ0vxzrdrddrd,
v=k0f-k0i=vxax+vyay+vzaz, vxz=vx2+vz2, 
ESf=ES1f-ES2f, ES1f=DE0iG0,
E0i=E0ViE0Hi, ES1f=E0VfE0Hf, D=DVVDHVDHVDHH,
G0=-ik0 exp-ik0rf2πrf,
D=-LL-llD00+q=1D01 exp-i2qv1ih+p=1D10 exp-i2pv1fh+p=1q=1D11 exp-i2hqv1i+pv1f×expik0f-k0irsdxsdzs,

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