Abstract

Ellipsometric light scattering (ELS) is shown to selectively extract the coherent scattering contribution representing the averaged properties of a particle ensemble. This property is essential for the previously reported [Erbe et al., Phys. Rev. E 73, 031406 (2006)] high sensitivity of ELS to the refractive index profile at particle interfaces. Two mechanisms for coherence loss in ELS measurements are discussed: sample polydispersity and illumination by a Gaussian beam. Suitable experimental quantities for a distinction of coherent and incoherent scattering contributions are introduced. Furthermore, the application of the concepts to reflection ellipsometry at rough surfaces is discussed.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  27. J. Lekner, Theory of Reflection (Martinus Nijhoff, 1987).
  28. R. Sigel and G. Strobl, “Light scattering by fluctuations within a nematic wetting layer in an isotropic phase of a liquid crystal,” J. Chem. Phys. 112, 1029-1039 (2000).
    [CrossRef]
  29. G. E. Yakubov, B. Loppinet, H. Zhang, J. Rühe, R. Sigel, and G. Fytas, “Collective dynamics of an end-grafted polymer brush in solvents of varying quality,” Phys. Rev. Lett. 92, 115501(2004).
    [CrossRef] [PubMed]
  30. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge, 2002); http://www.giss.nasa.gov/~crmim/publications/(2004).

2007

J. Oberdisse, “Adsorption and grafting on colloidal interfaces studied by scattering techniques,” Curr. Opin. Colloid Interface Sci. 12, 3-8 (2007).
[CrossRef]

N. Kučerka, M.-P. Nieh, J. Pencer, T. Harroun, and J. Katsaras, “The study of liposomes, lamellae and membranes using neutrons and X-rays,” Curr. Opin. Colloid Interface Sci. 12, 17-22 (2007).
[CrossRef]

A. Erbe, K. Tauer, and R. Sigel, “Ion distribution around electrostatically stabilized polystyrene latex particles studied by ellipsometric light scattering,” Langmuir 23, 452-459(2007).
[CrossRef] [PubMed]

A. Erbe and R. Sigel, “Tilt angle of lipid acyl chains in unilamellar vesicles determined by ellipsometric light scattering,” Eur. Phys. J. E 22, 303-309 (2007).
[CrossRef] [PubMed]

M. I. Mishchenko, L. Liu, and G. Videen, “Conditions of applicability of the single-scattering approximation,” Opt. Express 157522-7527 (2007).
[CrossRef] [PubMed]

2006

A. Erbe, K. Tauer, and R. Sigel, “Ellipsometric light scattering for the characterization of thin layers on dispersed colloidal particles,” Phys. Rev. E 73, 031406 (2006).
[CrossRef]

2005

M. Zackrisson, A. Stradner, P. Schurtenberger, and J. Bergenholtz, “Small-angle neutron scattering on a core-shell colloidal system: a contrast variation study,” Langmuir 21, 10835-10845 (2005).
[CrossRef] [PubMed]

2004

M. I. Mishchenko, J. W. Hovenier, and D. W. Mackowski, “Singe scattering by a small volume element,” J. Opt. Soc. Am. A 21, 71-87 (2004).
[CrossRef]

M. Stieger, W. Richtering, J. S. Pedersen, and R. Lindner, “Small-angle neutron scattering study of structural changes in temperature sensitive microgel colloids,” J. Chem. Phys. 120, 6197-6206 (2004).
[CrossRef] [PubMed]

G. E. Yakubov, B. Loppinet, H. Zhang, J. Rühe, R. Sigel, and G. Fytas, “Collective dynamics of an end-grafted polymer brush in solvents of varying quality,” Phys. Rev. Lett. 92, 115501(2004).
[CrossRef] [PubMed]

2000

R. Sigel and G. Strobl, “Light scattering by fluctuations within a nematic wetting layer in an isotropic phase of a liquid crystal,” J. Chem. Phys. 112, 1029-1039 (2000).
[CrossRef]

1995

1991

1988

G. Gouesbet, G. Grehan, and B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35-48 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “Light-scattering from a sphere arbitrarily located in a Gaussian-beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427-1443(1988).
[CrossRef]

B. Maheu, G. Gouesbet, and G. Grehan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Op. (Paris) 19, 59-67 (1988).
[CrossRef]

1985

G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a Gaussian-beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83-93 (1985).
[CrossRef]

1982

G. Gouesbet and G. Grehan, “On the scattering of light by a Mie scatter center located on the axis of an axisymmetric light profile,” J. Opt. (Paris) 13, 97-103 (1982).
[CrossRef]

1979

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292-3302(1979).
[CrossRef]

1978

1977

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618-5626 (1977).
[CrossRef]

Appl. Opt.

Curr. Opin. Colloid Interface Sci.

J. Oberdisse, “Adsorption and grafting on colloidal interfaces studied by scattering techniques,” Curr. Opin. Colloid Interface Sci. 12, 3-8 (2007).
[CrossRef]

N. Kučerka, M.-P. Nieh, J. Pencer, T. Harroun, and J. Katsaras, “The study of liposomes, lamellae and membranes using neutrons and X-rays,” Curr. Opin. Colloid Interface Sci. 12, 17-22 (2007).
[CrossRef]

Eur. Phys. J. E

A. Erbe and R. Sigel, “Tilt angle of lipid acyl chains in unilamellar vesicles determined by ellipsometric light scattering,” Eur. Phys. J. E 22, 303-309 (2007).
[CrossRef] [PubMed]

J. Chem. Phys.

M. Stieger, W. Richtering, J. S. Pedersen, and R. Lindner, “Small-angle neutron scattering study of structural changes in temperature sensitive microgel colloids,” J. Chem. Phys. 120, 6197-6206 (2004).
[CrossRef] [PubMed]

R. Sigel and G. Strobl, “Light scattering by fluctuations within a nematic wetting layer in an isotropic phase of a liquid crystal,” J. Chem. Phys. 112, 1029-1039 (2000).
[CrossRef]

J. Op.

B. Maheu, G. Gouesbet, and G. Grehan, “A concise presentation of the generalized Lorenz-Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Op. (Paris) 19, 59-67 (1988).
[CrossRef]

J. Opt.

G. Gouesbet and G. Grehan, “On the scattering of light by a Mie scatter center located on the axis of an axisymmetric light profile,” J. Opt. (Paris) 13, 97-103 (1982).
[CrossRef]

G. Gouesbet, G. Grehan, and B. Maheu, “Scattering of a Gaussian-beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83-93 (1985).
[CrossRef]

G. Gouesbet, G. Grehan, and B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz-Mie theory using finite series,” J. Opt. (Paris) 19, 35-48 (1988).
[CrossRef]

J. Opt. Soc. Am. A

Langmuir

M. Zackrisson, A. Stradner, P. Schurtenberger, and J. Bergenholtz, “Small-angle neutron scattering on a core-shell colloidal system: a contrast variation study,” Langmuir 21, 10835-10845 (2005).
[CrossRef] [PubMed]

A. Erbe, K. Tauer, and R. Sigel, “Ion distribution around electrostatically stabilized polystyrene latex particles studied by ellipsometric light scattering,” Langmuir 23, 452-459(2007).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. B

D. E. Aspnes, J. B. Theeten, and F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292-3302(1979).
[CrossRef]

F. Toigo, A. Marvin, V. Celli, and N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618-5626 (1977).
[CrossRef]

Phys. Rev. E

A. Erbe, K. Tauer, and R. Sigel, “Ellipsometric light scattering for the characterization of thin layers on dispersed colloidal particles,” Phys. Rev. E 73, 031406 (2006).
[CrossRef]

Phys. Rev. Lett.

G. E. Yakubov, B. Loppinet, H. Zhang, J. Rühe, R. Sigel, and G. Fytas, “Collective dynamics of an end-grafted polymer brush in solvents of varying quality,” Phys. Rev. Lett. 92, 115501(2004).
[CrossRef] [PubMed]

Other

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge, 2002); http://www.giss.nasa.gov/~crmim/publications/(2004).

J. Lyklema, Fundamentals of Interface and Colloid Science. Volume II: Solid-Liquid Interfaces (Academic, 1995).

J. D. Gaskill, “Gaussian Beams,” in Optical Engineer's Desk Reference, W. L. Wolfe, ed. (Optical Society of America, 2003), pp. 161-174. Note that in his definition of the Gaussian beam differs by a factor of π from the definition used by Gouesbet and co-workers and, therefore, also in this paper.

R. M.A. Azzam and N. M. Bazhara, Ellipsometry and Polarized Light (Elsevier, 1977).

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

D. Schrader, “Physical properties of poly(styrene),” in Polymer Handbook, 4th ed., J.Brandrup, E.H.Immergut, and E.A.Grulke, eds. (Wiley, 1999), pp. V/91-V/96.

J. Lekner, Theory of Reflection (Martinus Nijhoff, 1987).

J. S. Higgins and H. C. Benoît, Polymers and Neutron Scattering (Clarendon, 1996).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).

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Figures (2)

Fig. 1
Fig. 1

Simulation of tan ( Ψ I ) (dashed curve), tan ( Ψ Q ) (solid curve), and Δ (dotted curve) for particle mixtures of varying polydispersity in a plane wave illumination. From top to bottom the tan ( Ψ I ) data represent uniform size distributions of the particle radius in the ranges [ 28   – 68 nm ] , [ 52.7   – 64.7 nm ] , and [ 57.9   – 61.9 nm ] . The ranges were chosen so that the results for tan ( Ψ Q ) and Δ match the results for a monodisperse particle with 60 nm radius.

Fig. 2
Fig. 2

Simulation results for tan ( Ψ I ) (dashed curve), tan ( Ψ Q ) (solid curve), and Δ (dotted curve) from ellipsometric light scattering of a Gaussian beam by monodisperse spheres. See text for details.

Tables (1)

Tables Icon

Table 1 Nonidealities in an Ellipsometric Experiment a

Equations (45)

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ρ r = r p r s = tan ( Ψ r ) exp [ i Δ r ]
( E H E V ) ( f ) = exp [ i k ( d z ) ] i k d S ̲ ( 2 ) ( E H E V ) ( i ) .
S ̲ ( 2 ) = ( S 2 S 4 S 3 S 1 ) .
ρ = S 2 S 1 = tan ( Ψ ) exp [ i Δ ] .
P ( 2 ) = ( cos ( P ) sin ( P ) ) ,
A ( 2 ) = ( cos ( A ) sin ( A ) ) ,
C ̲ ( 2 ) = 1 2 ( 1 + i 1 i 1 i 1 + i ) .
E = exp [ i k ( d z ) ] / ( i k d ) A ( 2 ) T S ̲ ( 2 ) C ̲ ( 2 ) P ( 2 ) ,
2 i k d E / S 1 exp [ i k ( d z ) ] = ρ cos ( A ) ( 1 + i ) [ cos ( P ) i sin ( P ) ] + ( 1 i ) sin ( A ) [ cos ( P ) + i sin ( P ) ] .
ρ = i tan ( A 0 ) exp [ 2 i P 0 ] .
Ψ = | A 0 | ,
Δ = 2 P 0 + sign ( tan ( A 0 ) ) · 90 ° .
Ψ = 1 2 ( A 0 A 0 ) ,
Δ = P 0 + P 0 .
4 ( k r ) 2 I = | S 1 | 2 + | S 2 | 2 + cos ( 2 A ) [ | S 2 | 2 | S 1 | 2 ] + 2 sin ( 2 A ) [ sin ( 2 P ) Re ( S 1 * S 2 ) cos ( 2 P ) Im ( S 1 * S 2 ) ] .
I ¯ = E H E H * + E V E V * t ,
Q ¯ = E H E H * E V E V * t ,
U ¯ = E H E V * + E V E H * t ,
V ¯ = i E H E V * E V E H * t .
I ¯ 2 = Q ¯ 2 + U ¯ 2 + V ¯ 2 .
I ¯ 2 Q ¯ 2 + U ¯ 2 + V ¯ 2 .
S ̲ ( 4 ) = 1 2 ( | S 2 | 2 + | S 1 | 2 | S 2 | 2 | S 1 | 2 | S 2 | 2 | S 1 | 2 | S 2 | 2 + | S 1 | 2 0 0 0 0 0 0 0 0 S 2 * S 1 + S 2 S 1 * i ( S 2 S 1 * S 2 * S 1 ) i ( S 2 * S 1 S 2 S 1 * ) S 2 * S 1 + S 2 S 1 * ) .
tan ( Ψ I ) = | S 2 | 2 | S 1 | 2 .
tan ( Ψ A ) exp [ i Δ A ] = S 33 ( 4 ) + i S 43 ( 4 ) S 11 ( 4 ) S 21 ( 4 ) = S 1 * S 2 | S 1 | 2 ,
tan ( Ψ B ) exp [ i Δ B ] = S 11 ( 4 ) + S 22 ( 4 ) S 33 ( 4 ) i S 43 ( 4 ) = | S 2 | 2 S 1 S 2 * .
Δ = arg ( S 1 * S 2 ) .
s ¯ 0 = S 2 S 2 * + S 1 S 1 * ,
s ¯ 1 = S 2 S 2 * S 1 S 1 * ,
s ¯ 2 = S 2 S 1 * + S 1 S 2 * ,
s ¯ 3 = i S 2 S 1 * S 1 S 2 * .
s ¯ 0 2 s ¯ 1 2 + s ¯ 2 2 + s ¯ 3 2 .
I f 2 Q f 2 U f 2 V f 2 = ( s ¯ 0 2 s ¯ 1 2 ) ( I i 2 Q i 2 U i 2 V i 2 ) + ( s ¯ 0 2 s ¯ 1 2 s ¯ 2 2 s ¯ 3 2 ) ( U i 2 + V i 2 ) .
S ̲ ( 4 ) = 1 2 ( s ¯ 0 , c s ¯ 1 0 0 s ¯ 1 s ¯ 0 , c 0 0 0 0 s ¯ 2 s ¯ 3 0 0 s ¯ 3 s ¯ 2 ) + 1 2 ( s ¯ 0 s ¯ 0 , c 0 0 0 0 s ¯ 0 s ¯ 0 , c 0 0 0 0 0 0 0 0 0 0 ) .
tan ( Ψ I ) = tan ( Ψ A ) tan ( Ψ B )
tan ( Ψ A ) tan ( Ψ I ) tan ( Ψ B ) .
4 ( k r ) 2 I = | S 1 | 2 + | S 2 | 2 + cos ( 2 A ) [ | S 2 | 2 | S 1 | 2 ] + 2 sin ( 2 A ) | S 1 * S 2 | sin ( 2 P Δ ) .
tan ( 2 Ψ Q ) = tan ( 2 | A 0 | ) = 2 | S 1 * S 2 | | S 1 | 2 | S 2 | 2 .
4 cos ( 2 A 0 ) [ | S 1 | 2 | S 2 | 2 + 4 | S 1 * S 2 | ] > 0 .
4 ( k r ) 2 I | S 1 | 2 = 1 + tan 2 ( Ψ I ) + cos ( 2 A ) [ tan 2 ( Ψ I ) 1 ] + 2 sin ( 2 A ) tan ( Ψ A ) sin ( 2 P Δ ) .
tan ( 2 Ψ Q ) = 2 tan ( Ψ A ) 1 tan 2 ( Ψ I ) 2 tan ( Ψ A ) 1 tan 2 ( Ψ A ) = tan ( 2 Ψ A ) ,
tan ( 2 Ψ Q ) = 2 tan ( Ψ I ) 1 tan 2 ( Ψ I ) tan ( Ψ A ) tan ( Ψ B ) 2 tan ( Ψ I ) 1 tan 2 ( Ψ I ) = tan ( 2 Ψ I ) .
tan ( Ψ A ) tan ( Ψ Q ) tan ( Ψ I ) tan ( Ψ B ) .
S 2 = n = 1 m = n + n 2 n + 1 n ( n + 1 ) { a n g n , T M m τ n | m | ( cos Θ ) + i m b n g n , T E m π n | m | ( cos Θ ) } ,
S 1 = n = 1 m = n + n 2 n + 1 n ( n + 1 ) i m { m a n g n , T M m π n | m | ( cos Θ ) + i b n g n , T E m τ n | m | ( cos Θ ) } .
tan ( 2 Ψ r , Q ) = 2 | r s * r p | | r s | 2 | r p | 2 .

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