Abstract

A Mueller matrix for scattering by a rough plane surface of a glass hemisphere was simulated by using a micro-facet model. The algorithms are formulated in vector representation in terms of the input and output directions. The single-facet scattering simulation used the results of the Kirchhoff integral for medium rough surfaces with exponential height distribution. Scatterings by two or more facets were also simulated. For a fixed angle between the incident and the detection directions, the transmission scattering and its polarization properties were symmetric when plotted against the off-specular incident angle. The single-facet model generated no depolarization or polarization change. When double-facet scattering was included, polarizations were changed appreciably while depolarization was still very small. Depolarization increased appreciably when scattering by higher orders was included. The simulated results that include all orders of scattering fit excellently to the measured scattering transmittance and its polarization and depolarization.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. G. Videen, J.-Y. Hsu, W. S. Bickel, W. Wolfe, “Polarized light scattered from rough surfaces,” J. Opt. Soc. Am. A 9, 1111–1118 (1992).
    [CrossRef]
  4. T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
    [CrossRef]
  5. K. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparison of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
    [CrossRef]
  6. M. E. Knotts, K. A. O’Donnell, “Measurements of light scattering by a series of conducting surfaces with one-dimensional roughness,” J. Opt. Soc. Am. A 11, 697–710 (1994).
    [CrossRef]
  7. E. R. Mendez, 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).
  8. T. A. Germer, C. C. Asmail, “Polarization of light scat-tered by microrough surfaces and subsurface defects,” J. Opt. Soc. Am. A 16, 1326–1332 (1999).
  9. S. F. Nee, R. V. Dewees, T. W. Nee, L. F. Johnson, M. B. Moran, “Slope distribution of a rough surface measured by transmission scattering and polarization,” Appl. Opt. 39, 1561–1569 (2000).
  10. S. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).
  11. S. F. Nee, T. W. Nee, “Principal Mueller matrix of reflection and scattering measured for a one-dimensional rough surface,” Opt. Eng. (Bellingham) 41, 994–1001 (2002).
    [CrossRef]
  12. S. F. Nee, T. W. Nee, “Polarization of scattering by a rough paint surface,” in Surface Scattering and Diffraction for Advanced Metrology II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE4780, 88–98 (2002).
    [CrossRef]
  13. S. F. Nee, T. W. Nee, “Reflection and transmission scattering by rough plane surface of glass hemisphere,” in Surface Scattering and Diffraction for Advanced Metrology II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE5189, 1–10 (2003).
    [CrossRef]
  14. G. D. Lewis, D. L. Jordan, E. Jakeman, “Backscatter linear and circular polarization analysis of roughened aluminum,” Appl. Opt. 37, 5985–5992 (1998).
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    [CrossRef]
  16. S. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
    [CrossRef]
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    [CrossRef]
  18. S. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
    [CrossRef]
  19. S.-M. F. Nee, “Error analysis of null ellipsometry with depolarization,” Appl. Opt. 38, 5388–5398 (1999).
    [CrossRef]
  20. S.-M. F. Nee, “Depolarization and retardation of a birefringent slab,” J. Opt. Soc. Am. A 17, 2067–2073 (2000).
    [CrossRef]
  21. S.-M. F. Nee, “Depolarization and principal Mueller matrix measured by null ellipsometry,” Appl. Opt. 40, 4933–4939 (2001).
    [CrossRef]
  22. S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 20, 1651–1657 (2003).
    [CrossRef]
  23. R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 22.
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    [CrossRef]
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  26. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.
  27. S. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors Handbook, J. G. Webster, ed. (CRC Press, Boca Raton, Fla., 1999), Secs. 60.3 and 60.6–60.13.

2003

2002

S. F. Nee, T. W. Nee, “Principal Mueller matrix of reflection and scattering measured for a one-dimensional rough surface,” Opt. Eng. (Bellingham) 41, 994–1001 (2002).
[CrossRef]

J. Ellis, P. Caillard, A. Dogariu, “Off-diagonal Mueller matrix elements in backscattering from highly diffusive media,” J. Opt. Soc. Am. A 19, 43–48 (2002).
[CrossRef]

2001

2000

1999

1998

1996

1995

1994

1993

1992

1987

1967

Asmail, C. C.

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electomagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3 and Chap. 5, Sec. 12.4.1.

Bickel, W. S.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.

Burge, D.

Caillard, P.

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 22.

Cole, T.

S. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 54–64 (1998).
[CrossRef]

Dewees, R. V.

Dogariu, A.

Ellis, J.

Germer, T. A.

Hsu, J.-Y.

Iafelice, V. J.

Jakeman, E.

Johnson, L. F.

Jordan, D. L.

Knotts, K. E.

Knotts, M. E.

Lewis, G. D.

Luna, R. E.

Mendez, E. R.

Michel, T. R.

Moran, M. B.

Navarrete, A. G.

Nee, S. F.

S. F. Nee, T. W. Nee, “Principal Mueller matrix of reflection and scattering measured for a one-dimensional rough surface,” Opt. Eng. (Bellingham) 41, 994–1001 (2002).
[CrossRef]

S. F. Nee, R. V. Dewees, T. W. Nee, L. F. Johnson, M. B. Moran, “Slope distribution of a rough surface measured by transmission scattering and polarization,” Appl. Opt. 39, 1561–1569 (2000).

S. F. Nee, C. Yoo, T. Cole, D. Burge, “Characterization of imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 54–64 (1998).
[CrossRef]

S. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

S. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
[CrossRef]

S. F. Nee, T. W. Nee, “Reflection and transmission scattering by rough plane surface of glass hemisphere,” in Surface Scattering and Diffraction for Advanced Metrology II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE5189, 1–10 (2003).
[CrossRef]

S. F. Nee, T. W. Nee, “Polarization of scattering by a rough paint surface,” in Surface Scattering and Diffraction for Advanced Metrology II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE4780, 88–98 (2002).
[CrossRef]

S. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors Handbook, J. G. Webster, ed. (CRC Press, Boca Raton, Fla., 1999), Secs. 60.3 and 60.6–60.13.

S. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).

Nee, S.-M. F.

Nee, T. W.

S. F. Nee, T. W. Nee, “Principal Mueller matrix of reflection and scattering measured for a one-dimensional rough surface,” Opt. Eng. (Bellingham) 41, 994–1001 (2002).
[CrossRef]

S. F. Nee, R. V. Dewees, T. W. Nee, L. F. Johnson, M. B. Moran, “Slope distribution of a rough surface measured by transmission scattering and polarization,” Appl. Opt. 39, 1561–1569 (2000).

S. F. Nee, T. W. Nee, “Reflection and transmission scattering by rough plane surface of glass hemisphere,” in Surface Scattering and Diffraction for Advanced Metrology II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE5189, 1–10 (2003).
[CrossRef]

S. F. Nee, T. W. Nee, “Polarization of scattering by a rough paint surface,” in Surface Scattering and Diffraction for Advanced Metrology II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE4780, 88–98 (2002).
[CrossRef]

S. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).

O’Donnell, K. A.

Sparrow, E. M.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electomagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3 and Chap. 5, Sec. 12.4.1.

Torrance, K. E.

Videen, G.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.

Wolfe, W.

Yoo, C.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Ellis, P. Caillard, A. Dogariu, “Off-diagonal Mueller matrix elements in backscattering from highly diffusive media,” J. Opt. Soc. Am. A 19, 43–48 (2002).
[CrossRef]

S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 20, 1651–1657 (2003).
[CrossRef]

K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
[CrossRef]

S.-M. F. Nee, “Depolarization and retardation of a birefringent slab,” J. Opt. Soc. Am. A 17, 2067–2073 (2000).
[CrossRef]

E. R. Mendez, 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).

M. E. Knotts, K. A. O’Donnell, “Measurements of light scattering by a series of conducting surfaces with one-dimensional roughness,” J. Opt. Soc. Am. A 11, 697–710 (1994).
[CrossRef]

T. A. Germer, C. C. Asmail, “Polarization of light scat-tered by microrough surfaces and subsurface defects,” J. Opt. Soc. Am. A 16, 1326–1332 (1999).

T. R. Michel, M. E. Knotts, K. A. O’Donnell, “Stokes matrix of a one-dimensional perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 585–596 (1992).
[CrossRef]

G. Videen, J.-Y. Hsu, W. S. Bickel, W. Wolfe, “Polarized light scattered from rough surfaces,” J. Opt. Soc. Am. A 9, 1111–1118 (1992).
[CrossRef]

K. E. Knotts, T. R. Michel, K. A. O’Donnell, “Comparison of theory and experiment in light scattering from a randomly rough surface,” J. Opt. Soc. Am. A 10, 928–941 (1993).
[CrossRef]

Opt. Eng. (Bellingham)

S. F. Nee, T. W. Nee, “Principal Mueller matrix of reflection and scattering measured for a one-dimensional rough surface,” Opt. Eng. (Bellingham) 41, 994–1001 (2002).
[CrossRef]

Thin Solid Films

S. F. Nee, T. Cole, “Effects of depolarization of optical components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

Other

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Chap. 22.

P. Beckmann, A. Spizzichino, The Scattering of Electomagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 3 and Chap. 5, Sec. 12.4.1.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.

S. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation, and Sensors Handbook, J. G. Webster, ed. (CRC Press, Boca Raton, Fla., 1999), Secs. 60.3 and 60.6–60.13.

S. F. Nee, T. W. Nee, “Polarization of scattering by a rough paint surface,” in Surface Scattering and Diffraction for Advanced Metrology II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE4780, 88–98 (2002).
[CrossRef]

S. F. Nee, T. W. Nee, “Reflection and transmission scattering by rough plane surface of glass hemisphere,” in Surface Scattering and Diffraction for Advanced Metrology II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE5189, 1–10 (2003).
[CrossRef]

S. F. Nee, T. W. Nee, “Polarization of scattering by rough surfaces,” in Scattering and Surface Roughness II, Z. H. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 169–180 (1998).

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Figures (11)

Fig. 1
Fig. 1

Geometry of transmission scattering through a hemispheric glass sample.

Fig. 2
Fig. 2

Measured slope distribution by a stylus profilometer and by the scattering BTDF.

Fig. 3
Fig. 3

Input and output wave vectors and their associated polarizations for facet reflection (left) and transmission from inside (right).

Fig. 4
Fig. 4

Multiple-facets scattering. k^c is expressed in terms of three orthogonal unit vectors: k^o, βˆ, and αˆ. The path of double scattering is from k^i to k^c to -k^c to k^o. The path of triple scattering is from k^i to k^b to -k^b and then to k^c to -k^c to k^o.

Fig. 5
Fig. 5

Simulated BTDF times a solid angle with a half-apex angle of 1° versus OSI angle. The BTDF for the two-facets model with so=5° overlaps that for the many-facets model, with so=4.95°.

Fig. 6
Fig. 6

Simulated and measured BTDF times a solid angle (=0.00036 sr) versus OSI angle. The simulated curve was obtained by using the many-facets model with a best-fit slope angle of 4.95°.

Fig. 7
Fig. 7

Simulated BTDF or effective transmittance for different orders of scattering for the simulated curve of Fig. 6.

Fig. 8
Fig. 8

Simulated and measured ψ versus OSI angle. The many-facets model fits best to the measured data.

Fig. 9
Fig. 9

Simulated and measured depolarization versus OSI angle. The many-facets model fits best to the measured data.

Fig. 10
Fig. 10

Simulated and measured linear polarization (Px) versus OSI angle. The many-facets model fits best to the measured data.

Fig. 11
Fig. 11

Simulated and measured preserved polarization (Py) versus OSI angle. The many-facets model fits best to the measured data.

Tables (1)

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Table 1 Parameters Used for the Simulated Curves in Figs. 511

Equations (62)

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n^r(k^i, k^o, rn)=sign(1-rn)(k^i-k^o)/[2(1-k^i  k^o)]1/2 forreflection,
n^t(k^i, k^o, rn)=(k^i-rnk^o)/(1+rn2-2rnk^i  k^o)1/2 fortransmission;
sˆ=k^i×k^o/[1-(k^i  k^o)2]1/2,
pˆ=sˆ×kˆ,
(n^t  k^i)(n^t  k^o)0,
zˆ  n^t0.
k^i  k^ona/nb.
T0=Tχ2(zˆ  v)sinc2(Lxˆ  v)sinc2(Lyˆ  v),
v=2π(nik^i-nok^o)/λfortransmission2π(k^o-k^i)/λforreflection,
χ2(vz)=(1+vz2σ2)-2,
T1=TΔΩf(s)(zˆ  k^o)-1(zˆ  nˆ)-4,
f(s)=(2πso2)-1exp(-s/so),
s2=(zˆ  nˆ)-2-1.
ts=2[rn(1-rnk^i  k^o)(k^i  k^o-rn)]1/2/|1-rn2|,
tp=ts/k^i  k^o
rs=(1-k^i  k^o)1/2-(2rn2-1-k^i  k^o)1/2(1-k^i  k^o)1/2+(2rn2-1-k^i  k^o)1/2,
rp=rn2(1-k^i  k^o)1/2-(2rn2-1-k^i  k^o)1/2rn2(1-k^i  k^o)1/2+(2rn2-1-k^i  k^o)1/2.
T(tp, ts)=TX00XT0000YZ00-Z Y,
T=(tptp*+tsts*)/2,X=(tptp*-tsts*)/2,
Y=Re(tpts*),Z=Im(tpts*),
Tt=2rn(1-rnk^i  k^o)(k^i  k^o-rn)[1+(k^i  k^o)2]/[(1-rn2)(k^i  k^o)]2,
Xt=T[1-(k^i  k^o)2]/[1+(k^i  k^o)2],
Yt=2T(k^i  k^o)/[1+(k^i  k^o)2],
Zt=0.
s^i=k^i×zˆ/[1-(k^i  zˆ)2]1/2.
M1i(s^1; p^i, s^i)=10000(s^1  s^i)2-(s^1  p^i)2-2(s^1  s^i)(s^1  p^i)002(s^1  s^i)(s^1  p^i)(s^1  s^i)2-(s^1  p^i)200001,
Mo1(p^o, s^o; s^1)=10000(s^1  s^o)2-(s^1  p^o)22(s^1  s^o)(s^1  p^o)00-2(s^1  s^o)(s^1  p^o)(s^1  s^o)2-(s^1  p^o)200001.
MI=Mo1T1M1i.
MI=[(T0+T1)/T]Mo1T1M1i.
k^c=-k^ocos β+βˆ sin β cos α+αˆ sin β sin α,
βˆ=[-zˆ+zˆ  k^ok^o]/(1-(zˆ  k^o)2)1/2,
αˆ=-k^o×βˆ=-zˆ×k^o/[1-(zˆ  k^o)2]1/2,
rn=na/nbcos β1.
n^1=(k^i-k^c)/[2(1-k^i  k^c)]1/2,
n^2=-(k^c+rnk^o)/(1+rn2+2rnk^c  k^o)1/2,
MII=ΔΩzˆ  k^o02πdαrn1dcos β f(s2)f(s1)(zˆ  n^2)4(zˆ  n^1)4×Mo2T2M2cRnMc1R1M1i,
MIII=ΔΩzˆ  k^o02πdαrn1dcos β f(s3)f(s2)(zˆ  n^3)4(zˆ  n^2)4×Mo3T3M3cRnMc2R2M2b×02πdφ-10dcos θ f(s1)(zˆ  n^1)4RnMb1R1M1i,
M=MI+C2MII+C3MIII.
MIII=ΔΩzˆ  k^o02πdαrn1dcos β f(s3)f(s2)(zˆ  n^3)4(zˆ  n^2)4×Mo3T3M3cRnMc2R2M2b×02πdφ-1-cos 49°dcos θ f(s1)f(si)(zˆ  n^1)4(zˆ  k^i)4×RnMb1R1M1i,
si2=(zˆ  k^i)-2-1.
Mo=T1Px00Px1-2Dv0000PyPz00-PzPy,
Px=-Pcos 2ψ,
Py=Psin 2ψ cos Δ,
Pz=Psin 2ψ sin Δ.
P=(Px2+Py2+Pz2)1/21,
D1-P=Du+Dv,
v=2π(nik^i-nok^o)/λfortransmission2π(k^o-k^i)/λforreflection.
w(z)=exp(-|z|/σ)/2σ,
χ(vz)=(1+vz2σ2)-1=(1+g)-1,
g=vz2σ2.
T0=Tχ2(vz)sinc2(Lxˆ  v)sinc2(Lyˆ  v).
w2(z1, z2)=(2σ)-2exp[-(|z1|+|z2|)/σ],
1/σ2=1/σ2+1/so2τ2,
χ2(vz, -vz)=(1+vz2σ2)-2=(1+g)-2,
δχ2(vz, τ)=χ2(vz, -vz)-χ2(vz)=(g-g)(2+g+g)(1+g)-2(1+g)-2,
g=gso2τ2/(σ2+so2τ2).
δχ2(vz, τ)2g(1+so2τ2/σ2)-1forg12(1+gso2τ2/σ2)-2forg1g(g+2)(1+g)-2[1+(1+g)so2τ2/σ2]-3/2forg2.
D{ρ}=(2πF2/A)0J0(vxyτ)δχ2(vz, τ)τdτ=2πF2g(g+2)(g+1)-2×exp{-vxy/[vzso(1+1/g)1/2]}/[Avz2so2(1+1/g)].
T1=TΔΩf(s)(zˆ  k^o)-1(zˆ  nˆ)-4,
f(s)=g(g+2)(g+1)-2(2πso2)-1exp(-s/so),
so2=so2(1+1/g),
f(s)=(2πso2)-1exp(-s/so).

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