Abstract

A full-wave method is used to evaluate the Mueller matrix elements for scattering from layered structures with random rough surfaces. These provide a database for applications in optical detection over a broad range of rough surface statistical parameters. They can be used to determine the optimal frequencies and incident angles that provide most reliable measurements for optical detection. The elements of the Mueller matrix that are most sensitive to medium parameters of the layered structures can also be identified. Contributions from individual terms of the full-wave solutions are shown to have distinct physical interpretations.

© 1997 Optical Society of America

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References

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  1. E. Bahar, “Depolarization of electromagnetic waves excited by distribution of electric and magnetic sources in inhomogeneous multi-layered structures of arbitrarily varying thickness—full wave solutions,” J. Math. Phys. 14, 1510–1515 (1973).
    [CrossRef]
  2. E. Bahar, “Depolarization in nonuniform multi-layered structures—full wave solutions,” J. Math. Phys. 15, 202–208 (1974).
    [CrossRef]
  3. E. Bahar, R. D. Kubik, “Scattering by layered structures with rough surfaces: comparison of polarimetric optical scatterometer measurements with theory,” Appl. Opt. 36, 1–7 (1997).
  4. E. Bahar, S. M. Haugland, “Multiple scattering of electromagnetic waves from coated rough surfaces,” in Proceedings of the 1990 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., 1991), pp. 379–384.
  5. 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]
  6. 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).
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.
  8. 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]
  9. P. Beckman, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973).
  10. 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]
  11. E. Bahar, M. A. Fitzwater, “Shadowing by non-Gaussian rough surfaces for which decorrelation implies statistical independence,” Radio Sci. 18, 566–572 (1983).
    [CrossRef]
  12. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Tables, Applied Math Series55 (National Bureau of Standards, Washington, D.C., 1964).
  13. For the convenience of the reader, Beckman’s principal results are summarized here. There are errors in the published results, corrected equations are Eqs. (21) and (25).
  14. M. I. Sancer, “Shadow corrected electromagnetic scattering from a randomly rough surface,” IEEE Trans. Antennas Propag. AP-17, 577–585 (1968).
  15. 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.
  16. 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, “Scattering by layered structures with rough surfaces: comparison of polarimetric optical scatterometer measurements with theory,” Appl. Opt. 36, 1–7 (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]

1974 (1)

E. Bahar, “Depolarization in nonuniform multi-layered structures—full wave solutions,” J. Math. Phys. 15, 202–208 (1974).
[CrossRef]

1973 (1)

E. Bahar, “Depolarization of electromagnetic waves excited by distribution of electric and magnetic sources in inhomogeneous multi-layered structures of arbitrarily varying thickness—full wave solutions,” J. Math. Phys. 14, 1510–1515 (1973).
[CrossRef]

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).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Tables, Applied Math Series55 (National Bureau of Standards, Washington, D.C., 1964).

Bahar, E.

E. Bahar, R. D. Kubik, “Scattering by layered structures with rough surfaces: comparison of polarimetric optical scatterometer measurements with theory,” Appl. Opt. 36, 1–7 (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, “Depolarization in nonuniform multi-layered structures—full wave solutions,” J. Math. Phys. 15, 202–208 (1974).
[CrossRef]

E. Bahar, “Depolarization of electromagnetic waves excited by distribution of electric and magnetic sources in inhomogeneous multi-layered structures of arbitrarily varying thickness—full wave solutions,” J. Math. Phys. 14, 1510–1515 (1973).
[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).

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, S. M. Haugland, “Multiple scattering of electromagnetic waves from coated rough surfaces,” in Proceedings of the 1990 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., 1991), pp. 379–384.

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.

E. Bahar, S. M. Haugland, “Multiple scattering of electromagnetic waves from coated rough surfaces,” in Proceedings of the 1990 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., 1991), pp. 379–384.

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.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

Kubik, R. D.

E. Bahar, R. D. Kubik, “Scattering by layered structures with rough surfaces: comparison of polarimetric optical scatterometer measurements with theory,” Appl. Opt. 36, 1–7 (1997).

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]

Sancer, M. I.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Tables, Applied Math Series55 (National Bureau of Standards, Washington, D.C., 1964).

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).

Appl. Opt. (1)

E. Bahar, R. D. Kubik, “Scattering by layered structures with rough surfaces: comparison of polarimetric optical scatterometer measurements with theory,” Appl. Opt. 36, 1–7 (1997).

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]

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]

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

J. Math. Phys. (2)

E. Bahar, “Depolarization of electromagnetic waves excited by distribution of electric and magnetic sources in inhomogeneous multi-layered structures of arbitrarily varying thickness—full wave solutions,” J. Math. Phys. 14, 1510–1515 (1973).
[CrossRef]

E. Bahar, “Depolarization in nonuniform multi-layered structures—full wave solutions,” J. Math. Phys. 15, 202–208 (1974).
[CrossRef]

Radio Sci. (2)

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

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.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Tables, Applied Math Series55 (National Bureau of Standards, Washington, D.C., 1964).

For the convenience of the reader, Beckman’s principal results are summarized here. There are errors in the published results, corrected equations are Eqs. (21) and (25).

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

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).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

E. Bahar, S. M. Haugland, “Multiple scattering of electromagnetic waves from coated rough surfaces,” in Proceedings of the 1990 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., 1991), pp. 379–384.

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Figures (6)

Fig. 1
Fig. 1

Three-media irregular structure with a random rough uppermost interface.

Fig. 2
Fig. 2

Scatter from the uppermost interface, corresponding to n x = 0 (zero round trips).

Fig. 3
Fig. 3

Scatter upon transmission across a rough interface, n x = 1, from (a) medium 1 to medium 0, (b) medium 0 to medium 1.

Fig. 4
Fig. 4

Plots of total (incoherent and coherent) modified Mueller matrix elements M ijm as functions of incident (and specularly reflected) angles (see Tables 2 and 3): (a) M 11m , (b) M 22m , (c) M 12m , (d) M 33m , (e) M 34m .

Fig. 5
Fig. 5

Contributions to the total modified Mueller matrix elements M 34m from waves that make n x = 0 and n x = 1 round trips in medium 1 as functions of incident (and specularly reflected) angles (see Tables 2 and 3).

Fig. 6
Fig. 6

Plots of incoherent backscatter modified Mueller matrix elements M ijm as functions of incident angles (see Tables 4 and 5): (a) M 11m , (b) M 22m , (c) M 12m , (d) M 33m , (e) M 34m .

Tables (4)

Tables Icon

Table 1 Number of Terms in the Truncated Series Corresponding to n x = 0, 1, 2, 3, … (Number or Round Trips)

Tables Icon

Table 2 Input Parameters for the Data Shown in Fig. 4

Tables Icon

Table 3 Input Parameters for the Data Shown in Fig. 4

Tables Icon

Table 4 Input Parameters for the Data Shown in Fig. 6

Equations (34)

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

y=hsxs, zs; xs, zs-l, l, y=h01=constant elsewhere.
medium 0, y > hsxs, zs free space 0, μ0,medium 1, hsxs, zs > y >h121, μ1,medium 2, h12 >y 2, μ2.
E0i=E0ViE0Hi.
E0f=E0VfE0Hf=DE0iG0,
G0=k0 exp-ik0rf/2πirf,
D=DVVDVHDHVDHH=D1-D2, Vvertical, Hhorizontal,
D1=-LL-llD00+q=1D01 exp-i2qv1ih+p=1D10 exp-i2pv1fh+p=1q=1D11 exp-i2hqv1i+pv1f×expik0f-k0irsdxsdzs,
k0i=k0n0i=k0S0iCϕiax-C0iay+S0iSϕiaz=uiax-v0iay+wiaz,
k1i=k1n1i=k1S1iCϕiax+C1iay+S1iSϕiaz=uiax+v1iay+wiaz.
k0f=k0n0f=k0S0fCϕfax+C0fay+S0fSϕfaz=ufax+v0fay+wfaz,
k1f=k1n1f=k1S1fCϕfax-C1fay+S1fSϕfaz=ufax-v1fay+wfaz,
If=MIi.
σRSPQ=ESPQESRS*-ESPQESRS*E0QiE0Si4πrf2Ay,
σRSPQ=G024πrf2AyDPQDRS*-DPQDRS*=k02πAyDPQDRS*-DPQDRS* ρDPQDRS*.
fp, q, r, s =cdabIRSPQQf, gexpi2pv1f+qv1i-rv1f*-sv1i*h12,
cdabIRSPQ= DabPQDcdRS*pn×P2naf, nbi|nP2ncf, ndi|n¯dn×a, b and c, d=medium 0, 1,
Qf, g=2k020χ2f, g; rd-χfχgJ0vxzrdrddrd,
f=vy-2pv1f+qv1i, g=-vy+2rv1f+sv1i*, vy=k0f-k0iay.
χ2f, g; rd=-- phs, hs; rdexp×ifhs+ghsdhsdhs,
χf=-phsexpifhsdhs, χg=-phsexpighsdhs,
phs=νKhsK-1exp-νhsK-1!hs  00hs < 0,
phs, hs; rd=phsphsn=0Rnrdqn2 LnK-1νhs×LnK-1νhs,
qn2=K+n-1!K-1!n!.
χf=11-if/νK.
χ2f, g; rd=n=0K+n-1!RnrdK-1!n!×-fg/ν2n1-fg/ν2-if+g/νK+n.
Rrd=exp-rd/lc2,
phx=22K exp-2Khx/σxσx22KK-1!j=0K-1×2K-j-2!j!K-j-1!22Khxσxj,
pn=phx, hz=phxphz,
P2naf, nai|n=U-nai·n1+Γai/uaiUnaf·n1+Γaf/uaf.
P2nbf, nai|n=U-nai·n1+Γai/uaiUnbf·n1+Γbf/ubf,
Γaj Γuaj=uajhx-uajphxdhx.
Γuajuaj=exp-2Kuaj/σx22K-1K-1!j=0K-12K-j-2!2jK-j-1!×r=1j+1r2Kuaj/σxj-rj+1-r!.
cdabIRRPP=DabPPDcdRR*nay
Qf, g=Ayk02π χfχg.

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