Abstract

We employed the integral equation method (IEM) to simulate optical scattering by a randomly rough surface for spectroscopic ellipsometry. An explicit Mueller-matrix expression of the IEM for single scattering by moderately small surface roughness makes it possible to calculate the depolarization effect. The IEM allows a relatively rigorous assessment of the surface-scattering effect in a wide spectral range.

© 2003 Optical Society of America

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References

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  1. L. Tsang, J. A. Kong, in Scattering of Electromagnetic Waves, J. A. Kong, ed. (Wiley, New York, 2001), Vol. III, Chaps. 1 and 2.
  2. A. K. Fung, Microwave Scattering and Emission Model and Their Applications (Artech House, Norwood, Mass., 1994).
  3. A. K. Fung, Z. Li, K. S. Chen, “Back scattering from a randomly rough dielectric surface,” IEEE Trans. Geosci. Remote Sens. 30, 356–369 (1992).
    [CrossRef]
  4. D. E. Aspnes, “Optical response of microscopically rough surfaces,” Phys. Rev. B 41, 10334–10343 (1990).
    [CrossRef]
  5. S. Krishnan, P. C. Nordine, “Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects,” Appl. Opt. 33, 4184–4192 (1994).
    [CrossRef] [PubMed]
  6. R. M. A. Azzam, N. M. Bashara, “Polarization characteristics of scattered radiation from a diffraction grating by ellipsometry with application to surface roughness,” Phys. Rev. B 5, 4721–4729 (1972).
    [CrossRef]
  7. I. Ohlı́dal, F. Lukeš, “Ellipsometric parameters of randomly rough surfaces,” Opt. Commun. 5, 323–326 (1972).
    [CrossRef]
  8. M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986).
    [CrossRef] [PubMed]
  9. G. Videen, M. G. Turner, V. J. Iafelice, W. S. Bickel, W. L. Wolfe, “Scattering from a small sphere near a surface,” J. Opt. Soc. Am. A 10, 118–126 (1993).
    [CrossRef]
  10. P. I. Rovira, R. W. Collins, “Analysis of specular and textured SnO2:F films by high speed four-parameter Stokes vector spectroscopy,” J. Appl. Phys. 85, 2015–2025 (1999).
    [CrossRef]
  11. C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
    [CrossRef]
  12. D. L. Jordan, G. D. Lewis, E. Jakeman, “Emission polarization of roughened glass and aluminum surfaces,” Appl. Opt. 35, 3583–3590 (1996).
    [CrossRef] [PubMed]

1999

P. I. Rovira, R. W. Collins, “Analysis of specular and textured SnO2:F films by high speed four-parameter Stokes vector spectroscopy,” J. Appl. Phys. 85, 2015–2025 (1999).
[CrossRef]

1996

1994

C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
[CrossRef]

S. Krishnan, P. C. Nordine, “Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects,” Appl. Opt. 33, 4184–4192 (1994).
[CrossRef] [PubMed]

1993

1992

A. K. Fung, Z. Li, K. S. Chen, “Back scattering from a randomly rough dielectric surface,” IEEE Trans. Geosci. Remote Sens. 30, 356–369 (1992).
[CrossRef]

1990

D. E. Aspnes, “Optical response of microscopically rough surfaces,” Phys. Rev. B 41, 10334–10343 (1990).
[CrossRef]

1986

1972

R. M. A. Azzam, N. M. Bashara, “Polarization characteristics of scattered radiation from a diffraction grating by ellipsometry with application to surface roughness,” Phys. Rev. B 5, 4721–4729 (1972).
[CrossRef]

I. Ohlı́dal, F. Lukeš, “Ellipsometric parameters of randomly rough surfaces,” Opt. Commun. 5, 323–326 (1972).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, “Optical response of microscopically rough surfaces,” Phys. Rev. B 41, 10334–10343 (1990).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, “Polarization characteristics of scattered radiation from a diffraction grating by ellipsometry with application to surface roughness,” Phys. Rev. B 5, 4721–4729 (1972).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, “Polarization characteristics of scattered radiation from a diffraction grating by ellipsometry with application to surface roughness,” Phys. Rev. B 5, 4721–4729 (1972).
[CrossRef]

Bickel, W. S.

Chen, K. S.

A. K. Fung, Z. Li, K. S. Chen, “Back scattering from a randomly rough dielectric surface,” IEEE Trans. Geosci. Remote Sens. 30, 356–369 (1992).
[CrossRef]

Collins, R. W.

P. I. Rovira, R. W. Collins, “Analysis of specular and textured SnO2:F films by high speed four-parameter Stokes vector spectroscopy,” J. Appl. Phys. 85, 2015–2025 (1999).
[CrossRef]

Fung, A. K.

A. K. Fung, Z. Li, K. S. Chen, “Back scattering from a randomly rough dielectric surface,” IEEE Trans. Geosci. Remote Sens. 30, 356–369 (1992).
[CrossRef]

A. K. Fung, Microwave Scattering and Emission Model and Their Applications (Artech House, Norwood, Mass., 1994).

Iafelice, V. J.

Ishimaru, A.

C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
[CrossRef]

Jakeman, E.

Jordan, D. L.

Kong, J. A.

L. Tsang, J. A. Kong, in Scattering of Electromagnetic Waves, J. A. Kong, ed. (Wiley, New York, 2001), Vol. III, Chaps. 1 and 2.

Krishnan, S.

Lam, C. M.

C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
[CrossRef]

Lewis, G. D.

Li, Z.

A. K. Fung, Z. Li, K. S. Chen, “Back scattering from a randomly rough dielectric surface,” IEEE Trans. Geosci. Remote Sens. 30, 356–369 (1992).
[CrossRef]

Lukeš, F.

I. Ohlı́dal, F. Lukeš, “Ellipsometric parameters of randomly rough surfaces,” Opt. Commun. 5, 323–326 (1972).
[CrossRef]

Nordine, P. C.

Ohli´dal, I.

I. Ohlı́dal, F. Lukeš, “Ellipsometric parameters of randomly rough surfaces,” Opt. Commun. 5, 323–326 (1972).
[CrossRef]

Rovira, P. I.

P. I. Rovira, R. W. Collins, “Analysis of specular and textured SnO2:F films by high speed four-parameter Stokes vector spectroscopy,” J. Appl. Phys. 85, 2015–2025 (1999).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, in Scattering of Electromagnetic Waves, J. A. Kong, ed. (Wiley, New York, 2001), Vol. III, Chaps. 1 and 2.

Turner, M. G.

Videen, G.

Williams, M. W.

Wolfe, W. L.

Appl. Opt.

IEEE Trans. Antennas Propag.

C. M. Lam, A. Ishimaru, “Mueller matrix calculation for a slab of random medium with both random rough surfaces and discrete particles,” IEEE Trans. Antennas Propag. 42, 145–156 (1994).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

A. K. Fung, Z. Li, K. S. Chen, “Back scattering from a randomly rough dielectric surface,” IEEE Trans. Geosci. Remote Sens. 30, 356–369 (1992).
[CrossRef]

J. Appl. Phys.

P. I. Rovira, R. W. Collins, “Analysis of specular and textured SnO2:F films by high speed four-parameter Stokes vector spectroscopy,” J. Appl. Phys. 85, 2015–2025 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

I. Ohlı́dal, F. Lukeš, “Ellipsometric parameters of randomly rough surfaces,” Opt. Commun. 5, 323–326 (1972).
[CrossRef]

Phys. Rev. B

R. M. A. Azzam, N. M. Bashara, “Polarization characteristics of scattered radiation from a diffraction grating by ellipsometry with application to surface roughness,” Phys. Rev. B 5, 4721–4729 (1972).
[CrossRef]

D. E. Aspnes, “Optical response of microscopically rough surfaces,” Phys. Rev. B 41, 10334–10343 (1990).
[CrossRef]

Other

L. Tsang, J. A. Kong, in Scattering of Electromagnetic Waves, J. A. Kong, ed. (Wiley, New York, 2001), Vol. III, Chaps. 1 and 2.

A. K. Fung, Microwave Scattering and Emission Model and Their Applications (Artech House, Norwood, Mass., 1994).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the surface-scattering problem.

Fig. 2
Fig. 2

(a) Spectra of the range factor (k0h)(k0L)/|1|0.5, (b) degree of polarization, and (c) reflectance of bulk ZnO with different surface roughnesses simulated at different angles of incidence in the energy range of 1.5–3.5 eV. As in conventional ellipsometry, the output vector was collected at the specular reflected direction. Note that the calculation is valid when the range factor is approximately below 1.2 as a rule of thumb. For the EMA layer approach the Bruggeman model was used, and the volume fraction of the void is 50%.

Fig. 3
Fig. 3

Degree of polarization of ZnO with different surface roughnesses as a function of incident angle. The wavelength of the incident beam is 561 nm (2.2 eV). As in conventional ellipsometry, the output vector was collected at the specular reflected direction.

Fig. 4
Fig. 4

(a) Spectra of the range factor (k0h)(k0L)/|1|0.5, (b) degree of polarization, and (c) reflectance of bulk c-Si with different surface roughnesses simulated at different angles of incidence in the energy range of 1.5–3.5 eV. As in conventional ellipsometry, the output vector was collected at the specular reflected direction. Note that the calculation is valid when the range factor is approximately below 1.2 as a rule of thumb. For the EMA layer approach the Bruggeman model was used, and the volume fraction of the void is 50%.

Fig. 5
Fig. 5

(a) Degree of polarization and (b) reflectance of c-Si with different surface roughnesses as a function of incident angle. The wavelength of the incident beam is 561 nm (2.2 eV). As in conventional ellipsometry, the output vector was collected at the specular reflected direction.

Fig. 6
Fig. 6

(a) Spectra of the range factor (k0h)(k0L)/|1|0.5, (b) degree of polarization, and (c) reflectance of bulk Al with different surface roughnesses simulated at different angles of incidence in the energy range of 1.5–4.0 eV. As in conventional ellipsometry, the output vector was collected at the specular reflected direction. Note that the calculation is valid when the range factor is approximately below 1.2 as a rule of thumb. For the EMA layer approach the Bruggeman model was used and the volume fraction of the void is 50%.

Fig. 7
Fig. 7

(a) Degree of polarization and (b) reflectance of Al with different surface roughnesses as a function of incident angle. The wavelength of the incident beam is 561 nm (2.2 eV). As in conventional ellipsometry, the output vector was collected at the specular reflected direction.

Equations (41)

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kix=k0sin θicos ϕi,
kiy=k0sin θisin ϕi,
kiz=k0cos θi,
ksx=k0sin θscos ϕs,
ksy=k0sin θssin ϕs,
ksz=k0cos θs,
R=c=exp-(kiz-ksz)2h2×|Rp|20000|Rs|20000Re{RpRs*}-Im{RpRs*}00Im{RpRs*}Re{RpRs*},
R=i=|Spp|2|Sps|2ReSppSps*-ImSppSps*|Ssp|2|Sss|2ReSspSss*-ImSspSss*2 ReSppSsp*2 ReSpsSss*ReSppSss*+SpsSsp*ImSpsSsp*-SppSss*2 ImSppSsp*2 ImSpsSss*ImSppSss*+SpsSsp*ReSppSss*-SpsSsp*,
SabScd*=k028πexp[-h2(kiz2+ksz2)]n=1h2n|IabnIcdn*|×W(n)(ksx-kix, ksy-kiy)n!,
W(n)(ksx-kix, ksy-kiy)
=12πρn(ξ, ς)exp[j(ksx-kix)ξ+j(ksy-kiy)ς]dξdς=L22nexp-[(ksx-kix)2+(ksy-kiy)2]L24n,
Iαβn=(ksz+kiz)nfαβexp(-h2kizksz)+(ksz)nFαβ(-kix, -kiy)+(kiz)nFαβ(-ksx, -ksy)2.
fpp=2Rpcos θi+cos θs×[sin θisin θs-(1+cos θicos θs)×cos(ϕs-ϕi)],
fss=2Rscos θi+cos θs×[sin θisin θs-(1+cos θicos θs)cos(ϕs-ϕi)],
fps=2R sin(ϕs-ϕi),
fsp=-2R sin(ϕs-ϕi),
Fpp(-kix, -kiy)=-(csTp-sqT1p/1)×(T1pcsf+Tp1c1)+(Tp2-csT1pTp/sq)c2,
Fss(-kix, -kiy)=(csTs-sqT1s)×(T1scsf+Tsc1)-(Ts2-csT1sTs/sq)c2,
Fsp(-kix, -kiy)=(csT-sqT1/1)×(T1/css+T1/sq)sf+(T2-csT1T/sq)s2sf,
Fps(-kix, -kiy)=(csT1-sqT)(T/css+T1/sq)sf+(T12-csT1T/sq)s2sf,
Fpp(-ksx, -ksy)=-(cssTp-sqsT1p/1)×(T1pcsf+Tp1c1s)+(T1p2-cssT1pTp/sqs)c2s,
Fss(-ksx, -ksy)=(cssTs-sqsT1s)×(T1scsf+Tsc1s)-(T1s2-cssT1sTs/sqs)c2s,
Fsp(-ksx, -ksy)=-(cssT1-sqsT)×(T/cs+T1/sqs)sf-(T2-cssT1T/sqs)ss2sf,
Fps(-ksx, -ksy)=-(cssT-sqsT1/1)×(T1/cs+T1/sqs)sf-(T12-cssT1T/sqs)ss2sf,
s=sin θi,ss=sin θs,cs=cos θi,
css=cos θs,sf=sin(ϕs-ϕi),
csf=cos(ϕs-ϕi),sq=(1-sin2 θi)1/2,
sqs=(1-sin2 θs)1/2,
c1=(csf-sss)/(sqcss),
c1s=(csf-sss)/(sqscs),
c2=s(ss-scsf )/css,
c2s=ss(s-sscsf )/cs,
T1p=1+Rp,Tp=1-Rp,
T1s=1+Rs,Ts=1-Rs,
T1=1+R,T=1-R,
I(θ, ϕ, z)=IpIsUV=EpEp*EsEs*2Re(EpEs*)2Im(EpEs*).
P=[(Ip-Is)2+U2+V2]1/2Ip+Is.
I(π-θ0, ϕ0)=1120δ(cos θ0-cos θi)δ(ϕ0),
Is(θs=θ0, ϕs=ϕ0)=(R=i+R=c)Ii(π-θ0, ϕ0).
(k0h)(k0L)|1|<1.2.

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