Abstract

Müller matrix ellipsometry measurements are performed on accurately sized polystyrene latex and silicon oxide spherical particles deposited on a crystalline silicon surface. The mean particle diameter ranges from 0.2 to 1.5 µm. An argon laser beam (wavelength, 515 nm) impinges on the sample at a fixed near-grazing-incidence angle. The Müller matrix of the diffuse light scattered by the particles is measured in the plane of incidence as a function of the scattering angle. Results are presented and compared with Bobbert and Vlieger’s theory. In particular the validity range of Videen’s model is estimated.

© 2002 Optical Society of America

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References

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  1. 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]
  2. L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light scattering measurements of dielectric spheres on a silicon surface,” Opt. Lett. 24, 866–868 (1999).
    [CrossRef]
  3. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Phys. A 137, 209–242 (1986). See also the excellent Ph.D. dissertion by P. A. Bobbert, “On the optical properties of spheres and small spheroids on a substrate” (Rijksuniversiteit te Leiden, Leiden, The Netherlands, 1988).
    [CrossRef]
  4. E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
    [CrossRef]
  5. E. Compain, B. Drévillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
    [CrossRef]
  6. E. Compain, B. Drévillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
    [CrossRef]
  7. E. Compain, S. Poirier, B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller matrix ellipsometers,” Appl. Opt. 38, 3490–3502 (1999).
    [CrossRef]
  8. L. Mouche, F. Tardif, J. Derrien, “Particle deposition on silicon wafers during wet cleaning processes,” J. Electrochem. Soc. 141, 1684–1691 (1994).
    [CrossRef]
  9. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  10. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).
  12. T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
    [CrossRef]
  13. J. L. de la Peña, J. M. Saiz, G. Videen, F. Gonzàlez, P. J. Valle, F. Moreno, “Scattering from particles on surfaces: visibility factor and polydispersity,” Opt. Lett. 24, 1451–1453 (1999).
    [CrossRef]
  14. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991); errata 9, 844–845 (1992).
    [CrossRef]
  15. W. H. Press, Numerical Recipes (Cambridge U. Press, New York, 1988).

1999 (3)

1998 (3)

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

E. Compain, B. Drévillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drévillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
[CrossRef]

1997 (1)

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

1994 (1)

L. Mouche, F. Tardif, J. Derrien, “Particle deposition on silicon wafers during wet cleaning processes,” J. Electrochem. Soc. 141, 1684–1691 (1994).
[CrossRef]

1991 (1)

1986 (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Phys. A 137, 209–242 (1986). See also the excellent Ph.D. dissertion by P. A. Bobbert, “On the optical properties of spheres and small spheroids on a substrate” (Rijksuniversiteit te Leiden, Leiden, The Netherlands, 1988).
[CrossRef]

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Phys. A 137, 209–242 (1986). See also the excellent Ph.D. dissertion by P. A. Bobbert, “On the optical properties of spheres and small spheroids on a substrate” (Rijksuniversiteit te Leiden, Leiden, The Netherlands, 1988).
[CrossRef]

Bohren, C. F.

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

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Compain, E.

E. Compain, S. Poirier, B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller matrix ellipsometers,” Appl. Opt. 38, 3490–3502 (1999).
[CrossRef]

E. Compain, B. Drévillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
[CrossRef]

E. Compain, B. Drévillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

de la Peña, J. L.

Derrien, J.

L. Mouche, F. Tardif, J. Derrien, “Particle deposition on silicon wafers during wet cleaning processes,” J. Electrochem. Soc. 141, 1684–1691 (1994).
[CrossRef]

Doicu, A.

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Drevillon, B.

E. Compain, S. Poirier, B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller matrix ellipsometers,” Appl. Opt. 38, 3490–3502 (1999).
[CrossRef]

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

Drévillon, B.

E. Compain, B. Drévillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

E. Compain, B. Drévillon, “Broadband division of amplitude polarimeter based on uncoated prisms,” Appl. Opt. 37, 5938–5944 (1998).
[CrossRef]

Germer, T. A.

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light scattering measurements of dielectric spheres on 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]

Gonzàlez, F.

Huffman, D. R.

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

Moreno, F.

Mouche, L.

L. Mouche, F. Tardif, J. Derrien, “Particle deposition on silicon wafers during wet cleaning processes,” J. Electrochem. Soc. 141, 1684–1691 (1994).
[CrossRef]

Mulholland, G. W.

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light scattering measurements of dielectric spheres on 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]

Poirier, S.

Press, W. H.

W. H. Press, Numerical Recipes (Cambridge U. Press, New York, 1988).

Saiz, J. M.

Sung, L.

L. Sung, G. W. Mulholland, T. A. Germer, “Polarized light scattering measurements of dielectric spheres on 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]

Tardif, F.

L. Mouche, F. Tardif, J. Derrien, “Particle deposition on silicon wafers during wet cleaning processes,” J. Electrochem. Soc. 141, 1684–1691 (1994).
[CrossRef]

Valle, P. J.

Van de Hulst, H. C.

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

Videen, G.

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Phys. A 137, 209–242 (1986). See also the excellent Ph.D. dissertion by P. A. Bobbert, “On the optical properties of spheres and small spheroids on a substrate” (Rijksuniversiteit te Leiden, Leiden, The Netherlands, 1988).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

Wriedt, T.

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Appl. Opt. (2)

J. Electrochem. Soc. (1)

L. Mouche, F. Tardif, J. Derrien, “Particle deposition on silicon wafers during wet cleaning processes,” J. Electrochem. Soc. 141, 1684–1691 (1994).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. A (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Phys. A 137, 209–242 (1986). See also the excellent Ph.D. dissertion by P. A. Bobbert, “On the optical properties of spheres and small spheroids on a substrate” (Rijksuniversiteit te Leiden, Leiden, The Netherlands, 1988).
[CrossRef]

Rev. Sci. Instrum. (2)

E. Compain, B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. 68, 2671–2680 (1997).
[CrossRef]

E. Compain, B. Drévillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

Other (5)

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

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

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1975).

W. H. Press, Numerical Recipes (Cambridge U. Press, New York, 1988).

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]

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Figures (12)

Fig. 1
Fig. 1

PMME general setup.

Fig. 2
Fig. 2

Formal solution.

Fig. 3
Fig. 3

Comparison between Videen’s and Bobbert’s models for a 100-nm PSL sphere on a c-Si substrate at 515 nm and an incidence angle of 80°. M 11 is the differential scattering cross section, and its vertical scale is expressed in µm2 sr-1.

Fig. 4
Fig. 4

Comparison between Videen’s and Bobbert’s models for a 500-nm PSL sphere on a c-Si substrate at 515 nm and an incidence angle of 80°. M 11 is the differential scattering cross section, and its vertical scale is expressed in µm2 sr-1.

Fig. 5
Fig. 5

Comparison between Videen’s and Bobbert’s models for a 1-µm PSL sphere on a c-Si substrate at 515 nm and an incidence angle of 80°. M 11 is the differential scattering cross section, and its vertical scale is expressed in µm2 sr-1.

Fig. 6
Fig. 6

Measurement configuration.

Fig. 7
Fig. 7

Typical measured Müller matrix.

Fig. 8
Fig. 8

Measured (open circles) and fitted (curves) Müller matrices of 200-nm PSL spheres on a c-Si wafer (λ = 515 nm, 80° of incidence).

Fig. 9
Fig. 9

Measured (open circles) and fitted (curves) Müller matrices of 1.24-µm PSL spheres on a c-Si wafer (λ = 515 nm, 80° of incidence).

Fig. 10
Fig. 10

Measured (open circles) and fitted (curves) Müller matrices of 500-nm SiO2 spheres on a c-Si wafer (λ = 515 nm, 80° of incidence).

Fig. 11
Fig. 11

Measured (open circles) and fitted (curves) Müller matrices of 1.5-µm SiO2 spheres on a c-Si wafer (λ = 515 nm, 80° of incidence).

Fig. 12
Fig. 12

Measured (open circles) and fitted (curves) Müller matrices of 1.24-µm PSL spheres on a c-Si wafer (λ = 515 nm, 80° of incidence). This measurement differs from Fig. 9 because the laser is focused on the surface of the wafer so that only one particle at a time is measured.

Tables (1)

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Table 1 Fitted Parameters of the Gaussian and Monodisperse Size Distributions for Four Samples

Equations (13)

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hl1krYlmθ, ϕ=i-l2π02πdβ 0π/2-isin αdα×expikr cos γYlmα, β,
VSR=A · WS.
WS=BVI+VIR+VSR.
WS=1-BA-1 BVI+VIR.
1-BA-1=1+i=1BAi,
M¯ij=MijM11 for i, j1, 1.
M11=M22i.e., M¯22=1, M12=M21, M33=M44, M34=-M43, the two off-diagonal 2×2 blocks are null.
M=r=0MrNrdr.
Nrλ=density×radius×wavelength  1.
λDa=515×10-615024×10-2mm.
χ2=1Nn=1Ni,j11,12,33,34Mijmesθn-Mijtheθn2,
χ2=1Nn=1Nlog M11mesθn-log M11theθnlogM11max-log M11min2+i,j1,1M¯ijmesθn-M¯ijtheθn2,
Nr=A exp-r-r02σ2,

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