Abstract

The polarization of light scattered by the surface of a material contains information that can be used to identify the sources of that scatter. Theories for light scattering from interfacial roughness of a dielectric layer and from defects in that dielectric layer are reviewed. Methods for calculating the Mueller matrix or the Stokes vector for scatter from multiple sources and for decomposing a Stokes vector into contributions from two nondepolarizing scattering sources are derived. The theories are evaluated for a specific sample and geometry. Results show that some incident polarizations are more effective than others at discriminating among scattering sources, with s-polarized light being least effective. The polarization of light scattered from interfacial roughness depends upon the relative roughness of the two interfaces and the degree of correlation between the two interfaces. The scattering from defects in the film depends on the depth of the defect and differs from that from any one of the cases of interfacial roughness. The scattering from defects randomly distributed in the film and for small dielectric permittivity variations in the film is also calculated. Experimental results are presented for a 52-nm SiO2 film thermally grown on microrough silicon.

© 2001 Optical Society of America

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References

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  1. 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]
  2. 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]
  3. T. A. Germer, C. C. Asmail, “Polarization of light scattered by microrough surfaces and subsurface defects,” J. Opt. Soc. Am. A 16, 1326–1332 (1999).
    [CrossRef]
  4. L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light-scattering measurements of dielectric spheres upon a silicon surface,” Opt. Lett. 24, 866–868 (1999).
    [CrossRef]
  5. 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]
  6. J. C. Stover, Optical Scattering: Measurement and Analysis (SPIE Optical Engineering Press, Bellingham, Wash., 1995).
  7. T. A. Germer, C. C. Asmail, “Microroughness-blind optical scattering instrument,” U.S. patent6,034,776 (March7, 2000).
  8. T. A. Germer, “Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry,” Phys. Rev. Lett. 85, 349–352 (2000).
    [CrossRef] [PubMed]
  9. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical consderations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).
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    [CrossRef]
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  21. D. Rönnow, “Interface roughness statistics of thin films from angle-resolved light scattering at three wavelengths,” Opt. Eng. 37, 696–704 (1998).
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2000 (1)

T. A. Germer, “Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry,” Phys. Rev. Lett. 85, 349–352 (2000).
[CrossRef] [PubMed]

1999 (3)

1998 (1)

D. Rönnow, “Interface roughness statistics of thin films from angle-resolved light scattering at three wavelengths,” Opt. Eng. 37, 696–704 (1998).
[CrossRef]

1997 (2)

1996 (1)

1995 (2)

J. M. Elson, “Multilayer-coated optics: guided-wave coupling and scattering by means of interface random roughness,” J. Opt. Soc. Am. A 12, 729–742 (1995).
[CrossRef]

D. S. Flynn, C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function matrix,” Opt. Eng. 34, 1646–1650 (1995).
[CrossRef]

1993 (2)

1992 (1)

1980 (1)

1979 (1)

1977 (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]

Alexander, C.

D. S. Flynn, C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function matrix,” Opt. Eng. 34, 1646–1650 (1995).
[CrossRef]

Amra, C.

Asmail, C. C.

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

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

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, C. C. Asmail, “Microroughness-blind optical scattering instrument,” U.S. patent6,034,776 (March7, 2000).

Bennett, J. M.

Bohren, C. F.

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

Bruel, L.

Bussemer, P.

Deumié, C.

Duparré, A.

Elson, J. M.

Flynn, D. S.

D. S. Flynn, C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function matrix,” Opt. Eng. 34, 1646–1650 (1995).
[CrossRef]

Germer, T. A.

T. A. Germer, “Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry,” Phys. Rev. Lett. 85, 349–352 (2000).
[CrossRef] [PubMed]

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

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

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light-scattering measurements of dielectric spheres upon a silicon surface,” Opt. Lett. 24, 866–868 (1999).
[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, B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
[CrossRef]

L. Sung, G. W. Mulholland, T. A. Germer, “Polarization of light scattered by spheres on a dielectric film,” in Rough Surface Scattering and Contamination, P.-T. Chen, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3784, 296–303 (1999).
[CrossRef]

T. A. Germer, C. C. Asmail, “Microroughness-blind optical scattering instrument,” U.S. patent6,034,776 (March7, 2000).

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical consderations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

Giovannini, H.

Grèzes-Besset, C.

Hehl, K.

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical consderations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

Huffman, D. R.

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

Kassam, S.

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]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical consderations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

Mulholland, G. W.

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light-scattering measurements of dielectric spheres upon a silicon surface,” Opt. Lett. 24, 866–868 (1999).
[CrossRef]

L. Sung, G. W. Mulholland, T. A. Germer, “Polarization of light scattered by spheres on a dielectric film,” in Rough Surface Scattering and Contamination, P.-T. Chen, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3784, 296–303 (1999).
[CrossRef]

Neubert, J.

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical consderations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

Rahn, J. P.

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical consderations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

Rönnow, D.

D. Rönnow, “Interface roughness statistics of thin films from angle-resolved light scattering at three wavelengths,” Opt. Eng. 37, 696–704 (1998).
[CrossRef]

Scheer, B. W.

Stover, J. C.

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

Sung, L.

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light-scattering measurements of dielectric spheres upon a silicon surface,” Opt. Lett. 24, 866–868 (1999).
[CrossRef]

L. Sung, G. W. Mulholland, T. A. Germer, “Polarization of light scattered by spheres on a dielectric film,” in Rough Surface Scattering and Contamination, P.-T. Chen, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3784, 296–303 (1999).
[CrossRef]

van de Hulst, H. C.

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

Appl. Opt. (6)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Eng. (2)

D. S. Flynn, C. Alexander, “Polarized surface scattering expressed in terms of a bidirectional reflectance distribution function matrix,” Opt. Eng. 34, 1646–1650 (1995).
[CrossRef]

D. Rönnow, “Interface roughness statistics of thin films from angle-resolved light scattering at three wavelengths,” Opt. Eng. 37, 696–704 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

T. A. Germer, “Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry,” Phys. Rev. Lett. 85, 349–352 (2000).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

T. A. Germer, C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

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 (6)

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

T. A. Germer, C. C. Asmail, “Microroughness-blind optical scattering instrument,” U.S. patent6,034,776 (March7, 2000).

L. Sung, G. W. Mulholland, T. A. Germer, “Polarization of light scattered by spheres on a dielectric film,” in Rough Surface Scattering and Contamination, P.-T. Chen, Z.-H. Gu, A. A. Maradudin, eds., Proc. SPIE3784, 296–303 (1999).
[CrossRef]

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical consderations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr. 160 (National Bureau of Standards, Gaithersburg, Md., 1977).

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).

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Figures (7)

Fig. 1
Fig. 1

(a) Scattering and sample coordinate system and (b) diagram showing the thin film with an embedded defect.

Fig. 2
Fig. 2

Polarization parameters PC, P, and η for scattering out of the plane of incidence from the limiting cases of correlated and equal roughness (solid curves), uncorrelated and equal roughness (long-dashed curves), roughness of the exposed interface (short-dashed curves), and roughness of the buried interface (dotted–dashed curves) and from experimental results from a SiO2 layer grown on microrough silicon (squares). Each column refers to a different incident polarization scheme. Other parameters in the model are described in the text.

Fig. 3
Fig. 3

Roughness parameters extracted from polarized light-scattering measurements from the 52-nm SiO2 layer thermally grown on silicon. The results are obtained from measurements out of the plane of incidence (solid symbols) and in the plane of incidence (open symbols).

Fig. 4
Fig. 4

Polarization parameters PC, P, and η for scattering in the plane of incidence from the limiting cases of correlated and equal roughness (solid curves), uncorrelated and equal roughness (long-dashed curves), roughness of the exposed interface (short-dashed curves), and roughness of the buried interface (dotted-dashed curves) and from experimental results from a SiO2 layer grown on microrough silicon (squares). The incident polarizations were left circularly polarized (left column) and linearly polarized (right column) along the direction sˆi-pˆi.

Fig. 5
Fig. 5

Polarization parameters PC, P, and η for scattering from different locations in the film by using the varying-incident-polarization scheme described in the text. The defects are (a) a Rayleigh defect just above the film, (b) roughness of the air/film interface, (c) a Rayleigh defect just below the air/film interface, (d) a Rayleigh defect just above the film/substrate interface, (e) roughness of the film/substrate interface, and (f) a Rayleigh defect in the substrate.

Fig. 6
Fig. 6

Intensity, PC, and η for scattering from different locations near the dielectric film. The incident and scattering angles are θi=θr=60°, and the out-of-plane angle is ϕr=90°. The curves represent the positional dependence of a Rayleigh scatterer, while the symbols represent roughness at one of each of the two interfaces. The relative scales for the intensity are arbitrary between each layer.

Fig. 7
Fig. 7

Polarization parameters PC, P, and η for scattering from a random distribution of Rayleigh defects in the film (solid curves), from small variations in the dielectric permittivity of the film for which 2(x, y, d)=2(x, y, 0) (long-dashed curves), and from the roughness of the exposed interface (short-dashed curves).

Equations (71)

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EsscatEpscat=exp(ikR)RSssSpsSspSppEsincEpinc,
dΦr=FrΦi cos θr dΩ,
Fr=M(S)/(A cos θi cos θr),
η=arctan(Φ1, Φ2)/2,
P=(Φ12+Φ22+Φ32)1/2/Φ0,
PL=(Φ12+Φ22)1/2/Φ0,
PC=Φ3/Φ0.
Φ=Φ0×(1, PL, 0, PC)T.
Suv(1)=(4/π)(1-2)exp[i(qi2+qr2-qr3-qi3)τ]×qr3qi3A1/2ΔZ1(q)suv(1)
spp(1)=-2qi2qr2(1kikr-2qi1qr1 cos ϕr)/(ΓpiΓpr),
sps(1)=(2kqi2qr2qi1 sin ϕr)/(ΓpiΓsr),
ssp(1)=(2kqi2qr2qr1 sin ϕr)/(ΓsiΓpr),
sss(1)=-(k2qi2qr2 cos ϕr)/(ΓsiΓsr),
Γpβ=2Fpβ(+)qβ3-Fpβ(-)qβ2,
Γsβ=Fsβ(+)qβ3-Fsβ(-)qβ2,
Fpβ(±)=2Kβ(±)qβ1-1Kβ(±)qβ2,
Fsβ(±)=Kβ(±)qβ1-Kβ(±)qβ2,
Kβ(±)=exp(2iqβ2τ)±1,
ΔZm(q)=A-1/2Ad2r Δzm(r)exp(iq·r),
qx=kr cos ϕr-ki,
qy=kr sin ϕr.
Suv(2)=(1/π)(2-1)qr3qi3 exp[i(-qi3-qr3)τ]A1/2ΔZ2(q)suv(2),
spp(2)=-[2kikrFpi(+)Fpr(+)-qi2qr2Fpi(-)Fpr(-)×cos ϕr]/(ΓpiΓpr),
sps(2)=-[kqi2Fpi(-)Fsr(+) sin ϕr]/(ΓpiΓsr),
ssp(2)=-[kqr2Fsi(+)Fpr(-) sin ϕr]/(ΓsiΓpr),
sss(2)=-[k2Fsi(+)Fsr(+) cos ϕr]/(ΓsiΓsr).
α=a3(sph-2)/(sph+22).
Suvdef=4α exp[i(qi2+qr2)(τ-d)-i(qi3+qr3)τ]23/4qi3qr3suvdef(d),
sppdef(d)=[kikrGpi(+)Gpr(+)-qi2qr2Gpi(-)Gpr(-)×cos ϕr]/(ΓpiΓpr),
spsdef(d)=[kqi2Gpi(-)Gsr(+) sin ϕr]/(ΓpiΓsr),
sspdef(d)=[kqr2Gsi(+)Gpr(-) sin ϕr]/(ΓsiΓpr),
sssdef(d)=[k2Gsi(+)Gsr(+) cos ϕr]/(ΓsiΓsr),
Gpβ(±)=2Lβ(±)qβ1-1Lβ(±)qβ2,
Gsβ(±)=Lβ(±)qβ1-Lβ(±)qβ2,
Lβ(±)=exp(2iqβ2d)±1.
dα=Δ2/(4π2)d3r,
SuvRG=exp[i(qi2+qr2-qi3-qr3)τ]π-12-1/4qi3qr3×Vd3rΔ2(r)suvdef(rz)exp(-iq·r),
qx=kr cos ϕr-ki,
qy=kr sin ϕr,
qz=qi2+qr2,
I=j=1nEm exp(iαm)2,
I=m=1nIm+m=2nm=1m-1{[Imm(1)-Im-Im]Re cmm+[Imm(2)-Im-Im]Im cmm},
I=I1+I2+[I12(1)-I1-I2]Re c12+[I12(2)-I1-I2]Im c12.
|κ1|=(Ψˆ1TΦ)1/2,
|κ2|=(Ψˆ2TΦ)1/2.
b1=a11 Re c12+a12 Im c12,
b2=a21 Re c12+a22 Im c12,
ajj=Ψ¯12(j)T[Ψ12(j)-|κ1|2Ψ1-|κ2|2Ψ2],
bj=Ψ¯12(j)T[Φ-|κ1|2Ψ1-|κ2|2Ψ2].
Re c12=(a12b2-a22b1)/(a21a12-a11a22),
Im c12=(a21b1-a11b2)/(a21a12-a11a22).
0=(0/μ0)1/2(|Js|2+|Jp|2),
1=(0/μ0)1/2(|Js|2-|Jp|2),
2=2(0/μ0)1/2 Re(Js*Jp),
3=2(0/μ0)1/2 Im(Js*Jp).
m00=(|Spp|2+|Sss|2+|Ssp|2+|Sps|2)/2,
m01=(|Sss|2-|Spp|2+|Ssp|2-|Sps|2)/2,
m02=Re(SssSps*+SppSsp*),
m03=Im(SssSps*-SppSsp*),
m10=(|Sss|2-|Spp|2-|Ssp|2+|Sps|2)/2,
m11=(|Sss|2+|Spp|2-|Ssp|2-|Sps|2)/2,
m12=Re(SssSps*-SppSsp*),
m13=Im(SssSps*+SppSsp*),
m20=Re(SssSsp*+SppSps*),
m21=Re(SssSsp*-SppSps*),
m22=Re(SppSss*+SpsSsp*),
m23=Im(SssSpp*+SspSps*),
m30=Im(SspSss*+SppSps*),
m31=Im(SspSss*-SppSps*),
m32=Im(SppSss*-SpsSsp*),
m33=Re(SppSss*-SpsSsp*).

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