Abstract

A theory is presented to treat depolarized light scattering in a planar waveguide consisting of an ordered block copolymer thin film on a fused-silica substrate. In guided-wave depolarized light scattering (GWDLS), light is coupled into a transverse-magnetic mode of the film, which acts as a planar waveguide. Scattering of the incident light from randomly oriented, optically anisotropic grains results in the coupling of light into propagating transverse-electric (TE) modes in the sample. A dyadic Green’s function approach is employed to derive a quantitative relationship between the total optical power of the scattered TE wave and the average size of the grains. The grain size thus calculated from GWDLS experiments is consistent with the position space results of atomic force microscopy.

© 2007 Optical Society of America

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  1. F. S. Bates and G. H. Fredrickson, "Block copolymer thermodynamics: theory and experiment," Annu. Rev. Phys. Chem. 41, 525-557 (1990).
    [CrossRef] [PubMed]
  2. M. Goldstein and E. R. Michalik, "Theory of scattering by an inhomogeneous solid possessing fluctuations in density and anisotropy," J. Appl. Phys. 26, 1450-1457 (1955).
    [CrossRef]
  3. R. S. Stein and P. R. Wilson, "Scattering of light by polymer films possessing correlated orientation fluctuations," J. Appl. Phys. 33, 1914-1922 (1962).
    [CrossRef]
  4. N. Hadjichristidis, S. Pispas, and G. Floudas, Block Copolymer (Wiley-Interscience, 2003), Chap. 17.
  5. N. P. Balsara, B. A. Garetz, and H. J. Dai, "Relationship between birefringence and the structure of ordered block copolymer materials," Macromolecules 25, 6072-6074 (1992).
    [CrossRef]
  6. G. Floudas, G. Fytas, N. Hadjichristidis, and M. Pitsikalis, "Metastable states below the order-disorder transition in asymmetric diblock copolymer. A time-resolved depolarized light-scattering study," Macromolecules 28, 2359-2362 (1995).
    [CrossRef]
  7. M. C. Newstein, B. A. Garetz, N. P. Balsara, M. Y. Chang, and H. J. Dai, "Growth of grains and correlated grain clusters in a block copolymer melt," Macromolecules 31, 64-76 (1998).
    [CrossRef]
  8. H. Wang, M. C. Newstein, M. Y. Chang, N. P. Balsara, and B. A. Garetz, "Birefringence and depolarized light scattering of and ordered block copolymer melt under shear flow," Macromolecules 33, 3719-3730 (2000).
    [CrossRef]
  9. B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
    [CrossRef]
  10. C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1994).
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    [CrossRef]
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    [CrossRef]
  13. M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Accurate and efficient computation of the Green's tensor for stratified media," Phys. Rev. E 62, 5797-5807 (2000).
    [CrossRef]
  14. M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Green's tensor technique for scattering in two-dimensional stratified media," Phys. Rev. E 63, 066615 (2001).
    [CrossRef]
  15. M. Paulus and O. J. F. Martin, "Light propagation and scattering in stratified media: a Green's tensor approach," J. Opt. Soc. Am. A 18, 854-861 (2001).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. XIII.
  17. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984), Chap. 11.
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    [CrossRef]

2005 (1)

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

2002 (1)

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

2001 (2)

M. Paulus and O. J. F. Martin, "Light propagation and scattering in stratified media: a Green's tensor approach," J. Opt. Soc. Am. A 18, 854-861 (2001).
[CrossRef]

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Green's tensor technique for scattering in two-dimensional stratified media," Phys. Rev. E 63, 066615 (2001).
[CrossRef]

2000 (2)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Accurate and efficient computation of the Green's tensor for stratified media," Phys. Rev. E 62, 5797-5807 (2000).
[CrossRef]

H. Wang, M. C. Newstein, M. Y. Chang, N. P. Balsara, and B. A. Garetz, "Birefringence and depolarized light scattering of and ordered block copolymer melt under shear flow," Macromolecules 33, 3719-3730 (2000).
[CrossRef]

1998 (1)

M. C. Newstein, B. A. Garetz, N. P. Balsara, M. Y. Chang, and H. J. Dai, "Growth of grains and correlated grain clusters in a block copolymer melt," Macromolecules 31, 64-76 (1998).
[CrossRef]

1995 (1)

G. Floudas, G. Fytas, N. Hadjichristidis, and M. Pitsikalis, "Metastable states below the order-disorder transition in asymmetric diblock copolymer. A time-resolved depolarized light-scattering study," Macromolecules 28, 2359-2362 (1995).
[CrossRef]

1992 (1)

N. P. Balsara, B. A. Garetz, and H. J. Dai, "Relationship between birefringence and the structure of ordered block copolymer materials," Macromolecules 25, 6072-6074 (1992).
[CrossRef]

1990 (1)

F. S. Bates and G. H. Fredrickson, "Block copolymer thermodynamics: theory and experiment," Annu. Rev. Phys. Chem. 41, 525-557 (1990).
[CrossRef] [PubMed]

1976 (1)

T. G. Giallorenzi, J. A. Weiss, and J. P. Sheridan, "Light scattering from smectic liquid-crystal waveguides," J. Appl. Phys. 47, 1820-1826 (1976).
[CrossRef]

1975 (1)

T. G. Giallorenzi and J. P. Sheridan, "Light scattering from nematic liquid crystal waveguides," J. Appl. Phys. 46, 1271-1282 (1975).
[CrossRef]

1971 (1)

1962 (1)

R. S. Stein and P. R. Wilson, "Scattering of light by polymer films possessing correlated orientation fluctuations," J. Appl. Phys. 33, 1914-1922 (1962).
[CrossRef]

1955 (1)

M. Goldstein and E. R. Michalik, "Theory of scattering by an inhomogeneous solid possessing fluctuations in density and anisotropy," J. Appl. Phys. 26, 1450-1457 (1955).
[CrossRef]

Abuzaina, F. M.

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

Balsara, N. P.

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

H. Wang, M. C. Newstein, M. Y. Chang, N. P. Balsara, and B. A. Garetz, "Birefringence and depolarized light scattering of and ordered block copolymer melt under shear flow," Macromolecules 33, 3719-3730 (2000).
[CrossRef]

M. C. Newstein, B. A. Garetz, N. P. Balsara, M. Y. Chang, and H. J. Dai, "Growth of grains and correlated grain clusters in a block copolymer melt," Macromolecules 31, 64-76 (1998).
[CrossRef]

N. P. Balsara, B. A. Garetz, and H. J. Dai, "Relationship between birefringence and the structure of ordered block copolymer materials," Macromolecules 25, 6072-6074 (1992).
[CrossRef]

Bates, F. S.

F. S. Bates and G. H. Fredrickson, "Block copolymer thermodynamics: theory and experiment," Annu. Rev. Phys. Chem. 41, 525-557 (1990).
[CrossRef] [PubMed]

Bellare, A.

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

Boontongkong, Y.

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. XIII.

Chang, M. Y.

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

H. Wang, M. C. Newstein, M. Y. Chang, N. P. Balsara, and B. A. Garetz, "Birefringence and depolarized light scattering of and ordered block copolymer melt under shear flow," Macromolecules 33, 3719-3730 (2000).
[CrossRef]

M. C. Newstein, B. A. Garetz, N. P. Balsara, M. Y. Chang, and H. J. Dai, "Growth of grains and correlated grain clusters in a block copolymer melt," Macromolecules 31, 64-76 (1998).
[CrossRef]

Cohen, R. E.

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

Dai, H. J.

M. C. Newstein, B. A. Garetz, N. P. Balsara, M. Y. Chang, and H. J. Dai, "Growth of grains and correlated grain clusters in a block copolymer melt," Macromolecules 31, 64-76 (1998).
[CrossRef]

N. P. Balsara, B. A. Garetz, and H. J. Dai, "Relationship between birefringence and the structure of ordered block copolymer materials," Macromolecules 25, 6072-6074 (1992).
[CrossRef]

Durkee, D. A.

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

Floudas, G.

G. Floudas, G. Fytas, N. Hadjichristidis, and M. Pitsikalis, "Metastable states below the order-disorder transition in asymmetric diblock copolymer. A time-resolved depolarized light-scattering study," Macromolecules 28, 2359-2362 (1995).
[CrossRef]

N. Hadjichristidis, S. Pispas, and G. Floudas, Block Copolymer (Wiley-Interscience, 2003), Chap. 17.

Fredrickson, G. H.

F. S. Bates and G. H. Fredrickson, "Block copolymer thermodynamics: theory and experiment," Annu. Rev. Phys. Chem. 41, 525-557 (1990).
[CrossRef] [PubMed]

Fytas, G.

G. Floudas, G. Fytas, N. Hadjichristidis, and M. Pitsikalis, "Metastable states below the order-disorder transition in asymmetric diblock copolymer. A time-resolved depolarized light-scattering study," Macromolecules 28, 2359-2362 (1995).
[CrossRef]

Garetz, B. A.

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

H. Wang, M. C. Newstein, M. Y. Chang, N. P. Balsara, and B. A. Garetz, "Birefringence and depolarized light scattering of and ordered block copolymer melt under shear flow," Macromolecules 33, 3719-3730 (2000).
[CrossRef]

M. C. Newstein, B. A. Garetz, N. P. Balsara, M. Y. Chang, and H. J. Dai, "Growth of grains and correlated grain clusters in a block copolymer melt," Macromolecules 31, 64-76 (1998).
[CrossRef]

N. P. Balsara, B. A. Garetz, and H. J. Dai, "Relationship between birefringence and the structure of ordered block copolymer materials," Macromolecules 25, 6072-6074 (1992).
[CrossRef]

Gay-Balmaz, P.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Green's tensor technique for scattering in two-dimensional stratified media," Phys. Rev. E 63, 066615 (2001).
[CrossRef]

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Accurate and efficient computation of the Green's tensor for stratified media," Phys. Rev. E 62, 5797-5807 (2000).
[CrossRef]

Giallorenzi, T. G.

T. G. Giallorenzi, J. A. Weiss, and J. P. Sheridan, "Light scattering from smectic liquid-crystal waveguides," J. Appl. Phys. 47, 1820-1826 (1976).
[CrossRef]

T. G. Giallorenzi and J. P. Sheridan, "Light scattering from nematic liquid crystal waveguides," J. Appl. Phys. 46, 1271-1282 (1975).
[CrossRef]

Gido, S. P.

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

Goldstein, M.

M. Goldstein and E. R. Michalik, "Theory of scattering by an inhomogeneous solid possessing fluctuations in density and anisotropy," J. Appl. Phys. 26, 1450-1457 (1955).
[CrossRef]

Gupton, J. P.

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

Hadjichristidis, N.

G. Floudas, G. Fytas, N. Hadjichristidis, and M. Pitsikalis, "Metastable states below the order-disorder transition in asymmetric diblock copolymer. A time-resolved depolarized light-scattering study," Macromolecules 28, 2359-2362 (1995).
[CrossRef]

N. Hadjichristidis, S. Pispas, and G. Floudas, Block Copolymer (Wiley-Interscience, 2003), Chap. 17.

Kim, W. G.

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

Liddle, J. A.

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

Martin, O. J. F.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Green's tensor technique for scattering in two-dimensional stratified media," Phys. Rev. E 63, 066615 (2001).
[CrossRef]

M. Paulus and O. J. F. Martin, "Light propagation and scattering in stratified media: a Green's tensor approach," J. Opt. Soc. Am. A 18, 854-861 (2001).
[CrossRef]

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Accurate and efficient computation of the Green's tensor for stratified media," Phys. Rev. E 62, 5797-5807 (2000).
[CrossRef]

Michalik, E. R.

M. Goldstein and E. R. Michalik, "Theory of scattering by an inhomogeneous solid possessing fluctuations in density and anisotropy," J. Appl. Phys. 26, 1450-1457 (1955).
[CrossRef]

Newstein, M. C.

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

H. Wang, M. C. Newstein, M. Y. Chang, N. P. Balsara, and B. A. Garetz, "Birefringence and depolarized light scattering of and ordered block copolymer melt under shear flow," Macromolecules 33, 3719-3730 (2000).
[CrossRef]

M. C. Newstein, B. A. Garetz, N. P. Balsara, M. Y. Chang, and H. J. Dai, "Growth of grains and correlated grain clusters in a block copolymer melt," Macromolecules 31, 64-76 (1998).
[CrossRef]

Patel, A. J.

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

Paulus, M.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Green's tensor technique for scattering in two-dimensional stratified media," Phys. Rev. E 63, 066615 (2001).
[CrossRef]

M. Paulus and O. J. F. Martin, "Light propagation and scattering in stratified media: a Green's tensor approach," J. Opt. Soc. Am. A 18, 854-861 (2001).
[CrossRef]

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Accurate and efficient computation of the Green's tensor for stratified media," Phys. Rev. E 62, 5797-5807 (2000).
[CrossRef]

Pispas, S.

N. Hadjichristidis, S. Pispas, and G. Floudas, Block Copolymer (Wiley-Interscience, 2003), Chap. 17.

Pitsikalis, M.

G. Floudas, G. Fytas, N. Hadjichristidis, and M. Pitsikalis, "Metastable states below the order-disorder transition in asymmetric diblock copolymer. A time-resolved depolarized light-scattering study," Macromolecules 28, 2359-2362 (1995).
[CrossRef]

Segalman, R. A.

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

Sheridan, J. P.

T. G. Giallorenzi, J. A. Weiss, and J. P. Sheridan, "Light scattering from smectic liquid-crystal waveguides," J. Appl. Phys. 47, 1820-1826 (1976).
[CrossRef]

T. G. Giallorenzi and J. P. Sheridan, "Light scattering from nematic liquid crystal waveguides," J. Appl. Phys. 46, 1271-1282 (1975).
[CrossRef]

Stein, R. S.

R. S. Stein and P. R. Wilson, "Scattering of light by polymer films possessing correlated orientation fluctuations," J. Appl. Phys. 33, 1914-1922 (1962).
[CrossRef]

Tai, C. T.

C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1994).

Tien, P. K.

Wang, H.

H. Wang, M. C. Newstein, M. Y. Chang, N. P. Balsara, and B. A. Garetz, "Birefringence and depolarized light scattering of and ordered block copolymer melt under shear flow," Macromolecules 33, 3719-3730 (2000).
[CrossRef]

Weiss, J. A.

T. G. Giallorenzi, J. A. Weiss, and J. P. Sheridan, "Light scattering from smectic liquid-crystal waveguides," J. Appl. Phys. 47, 1820-1826 (1976).
[CrossRef]

Wilbur, J. D.

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

Wilson, P. R.

R. S. Stein and P. R. Wilson, "Scattering of light by polymer films possessing correlated orientation fluctuations," J. Appl. Phys. 33, 1914-1922 (1962).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. XIII.

Yang, L.

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984), Chap. 11.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984), Chap. 11.

Annu. Rev. Phys. Chem. (1)

F. S. Bates and G. H. Fredrickson, "Block copolymer thermodynamics: theory and experiment," Annu. Rev. Phys. Chem. 41, 525-557 (1990).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Appl. Phys. (4)

T. G. Giallorenzi and J. P. Sheridan, "Light scattering from nematic liquid crystal waveguides," J. Appl. Phys. 46, 1271-1282 (1975).
[CrossRef]

T. G. Giallorenzi, J. A. Weiss, and J. P. Sheridan, "Light scattering from smectic liquid-crystal waveguides," J. Appl. Phys. 47, 1820-1826 (1976).
[CrossRef]

M. Goldstein and E. R. Michalik, "Theory of scattering by an inhomogeneous solid possessing fluctuations in density and anisotropy," J. Appl. Phys. 26, 1450-1457 (1955).
[CrossRef]

R. S. Stein and P. R. Wilson, "Scattering of light by polymer films possessing correlated orientation fluctuations," J. Appl. Phys. 33, 1914-1922 (1962).
[CrossRef]

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

Macromolecules (6)

M. Y. Chang, F. M. Abuzaina, W. G. Kim, J. P. Gupton, B. A. Garetz, M. C. Newstein, N. P. Balsara, L. Yang, S. P. Gido, R. E. Cohen, Y. Boontongkong, and A. Bellare, "Analysis of grain structure in partially ordered block copolymers by depolarized light scattering and transmission electron microscopy," Macromolecules 35, 4437-4447 (2002).
[CrossRef]

N. P. Balsara, B. A. Garetz, and H. J. Dai, "Relationship between birefringence and the structure of ordered block copolymer materials," Macromolecules 25, 6072-6074 (1992).
[CrossRef]

G. Floudas, G. Fytas, N. Hadjichristidis, and M. Pitsikalis, "Metastable states below the order-disorder transition in asymmetric diblock copolymer. A time-resolved depolarized light-scattering study," Macromolecules 28, 2359-2362 (1995).
[CrossRef]

M. C. Newstein, B. A. Garetz, N. P. Balsara, M. Y. Chang, and H. J. Dai, "Growth of grains and correlated grain clusters in a block copolymer melt," Macromolecules 31, 64-76 (1998).
[CrossRef]

H. Wang, M. C. Newstein, M. Y. Chang, N. P. Balsara, and B. A. Garetz, "Birefringence and depolarized light scattering of and ordered block copolymer melt under shear flow," Macromolecules 33, 3719-3730 (2000).
[CrossRef]

B. A. Garetz, M. C. Newstein, J. D. Wilbur, A. J. Patel, D. A. Durkee, R. A. Segalman, J. A. Liddle, and N. P. Balsara, "Grain structure in block copolymer thin films studied by guided wave depolarized light scattering," Macromolecules 38, 4282-4288 (2005).
[CrossRef]

Phys. Rev. E (2)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Accurate and efficient computation of the Green's tensor for stratified media," Phys. Rev. E 62, 5797-5807 (2000).
[CrossRef]

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, "Green's tensor technique for scattering in two-dimensional stratified media," Phys. Rev. E 63, 066615 (2001).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. XIII.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1984), Chap. 11.

C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, 2nd ed. (IEEE, 1994).

N. Hadjichristidis, S. Pispas, and G. Floudas, Block Copolymer (Wiley-Interscience, 2003), Chap. 17.

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Figures (5)

Fig. 1
Fig. 1

AFM phase image from an annealed MSI film.

Fig. 2
Fig. 2

Schematic of the configuration of the waveguide and the three components of the scattered field as given in Eq. (3); solid dots and dashes represent the polarization of the electric field.

Fig. 3
Fig. 3

Dependence of the depolarized power on correlation length w. Points: theoretically calculated values; bold curve: linear interpolation for 0.3 < w d < 2 .

Fig. 4
Fig. 4

Dependence of the depolarized power on the path length L. Circles: the depolarized power at the exit spot; squares: the total depolarized power at a rectangular exit area with infinite length in the y direction.

Fig. 5
Fig. 5

Propagation of the output coupling ray.

Tables (1)

Tables Icon

Table 1 Interpolation Coefficients in Eq. (19) for SI or MSI Samples with n = 0 and m = 0

Equations (29)

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× × E 1 ( r ) + k 1 2 E 1 ( r ) = 0 , z < d ,
× × E 2 ( r ) + k 2 2 E 2 ( r ) = k 0 2 ε 0 j f j ( r ) δ ε j E 2 ( r ) , d z 0 ,
× × E 3 ( r ) + k 3 2 E 3 ( r ) = 0 , z > 0 ,
δ ε j = ε 0 ( n e 2 n o 2 ) ( g ̂ j g ̂ j 1 3 I ) ,
E scattered ( r ) = k 0 2 ε 0 V d r G ( r , r ) j f j ( r ) δ ε j E m ( r ) ,
G ( r , r ) = a ̂ z a ̂ z k 2 2 δ ( r r ) + i 4 π 0 d k ρ [ g T E ( k ρ ; r , r ) + g T M ( k ρ ; r , r ) ] ,
G ( r , r ) = G D ( r , r ) + n G n ( r , r ) + m G m ( r , r ) ,
g α β T E ( k ρ ; r , r ) = c α β T E ( k ρ ; ρ , ϕ ) l T E ( k ρ ; z , z ) ,
c y x T E ( k ρ ; r , r ) = 1 k 2 z [ k ρ J 0 ( k ρ ρ ) + 2 ρ J 1 ( k ρ ρ ) ] sin ϕ cos ϕ ,
c y y T E ( k ρ ; r , r ) = 1 k 2 z [ k ρ J 0 ( k ρ ρ ) cos 2 ϕ 1 ρ J 1 ( k ρ ρ ) cos ( 2 ϕ ) ] ,
c y z T E ( k ρ ; r , r ) = 0 .
l T E ( k ρ ; z , z ) = A 2 T E exp ( i k 2 z z ) + B 2 T E exp ( i k 2 z z ) = 1 1 F 21 T E F 23 T E exp ( 2 i k 2 z d ) { 2 F 21 T E F 23 T E exp ( 2 i k 2 z d ) cos [ k 2 z ( z z ) ] + F 21 T E exp [ i k 2 z ( 2 d + z + z ) ] + F 23 T E exp [ i k 2 z ( z + z ) ] } ,
1 F 21 T E F 23 T E exp ( 2 i k 2 z d ) = 0 ,
k 2 z n d = arctan k 2 z n ( k 1 z n + k 3 z n ) k 2 z n 2 k 1 z n k 3 z n + n π , n = 0 , 1 , 2 , ,
g α β n ( k ρ n ; r , r ) = 1 2 c α β n ( k ρ n ; ρ , ϕ ) L n ( k ρ n ; z , z ) ,
L n ( k ρ n ; z , z ) = l T E ( k ρ n ; z , z ) [ 1 F 21 n F 23 n exp ( 2 i k 2 z n d ) ] [ 1 F 21 n F 23 n exp ( 2 i k 2 z d ) ] k ρ k ρ n .
E y n ( r ) = k 0 2 ε 0 V d r j f j ( r ) α = x , y β = x , z g y α n ( r , r ) δ ε α β j E β m ( r ) .
E x m ( r ) = i k 2 z m E 0 [ n 3 2 k 2 z m n 2 2 k 3 z m sin ( k 2 z m z ) + cos ( k 2 z m z ) ] exp ( i k ρ m x ) ,
E z m ( r ) = k ρ m E 0 [ n 3 2 k 2 z m n 2 2 k 3 z m cos ( k 2 z m z ) + sin ( k 2 z m z ) ] exp ( i k ρ m x ) .
I n ( r ) = k 0 4 ( n e 2 n o 2 ) 2 K V d r V d r α = x , y β = x , z α = x , y β = x , z g y α n ( r , r ) E β m ( r ) g y α n * ( r , r ) E β m * ( r ) j j f j ( r ) f j ( r ) δ ε α β j δ ε α β j ε 0 2 = k 0 4 ( n e 2 n o 2 ) 2 K [ V d r V d r α = x , y β = x , z α = x , y β = x , z g y α n ( r , r ) E β m ( r ) g y α n * ( r , r ) E β m * ( r ) j δ ε α β j δ ε α β j ε 0 2 f j ( r ) f j ( r ) + V d r V d r α = x , y β = x , z α = x , y β = x , z g y α n ( r , r ) E β m ( r ) g y α n * ( r , r ) E β m * ( r ) j j j [ f j ( r ) f j ( r ) δ ε α β j δ ε α β j ε 0 2 ] ,
I n ( r ) = k 0 4 ( n e 2 n o 2 ) 2 K V d r V d r α = x , y β = x , z α = x , y β = x , z g y α n ( r , r ) E β m ( r ) g y α n * ( r , r ) E β m * ( r ) j δ ε α β j δ ε α β j ε 0 2 f j ( r ) f j ( r ) .
I n ( r ) = Φ grain k 0 4 ( n e 2 n o 2 ) 2 K α = x , y β = x , z 0 2 π d ϕ g 0 π d θ g sin θ g ( a ̂ α g ̂ g ̂ a ̂ β 1 3 δ α β ) 2 × V d r V d r g y α n ( r , r ) E β m ( r ) g y α n * ( r , r ) E β m * ( r ) exp ( r r 2 2 w 2 ) ,
( P n ) Exit = S d S I n ( r ) ,
( P m ) Exit = S d S I m ( 0 ) = S K [ E x m E x m * ( 0 ) + E z m E z m * ( 0 ) ] .
( P n P m ) Exit = Φ grain k 0 4 ( n e 2 n o 2 ) 2 S [ E x m E x m * ( 0 ) + E z m E z m * ( 0 ) ] S d S α = x , y β = x , z 0 2 π d ϕ g 0 π d θ g sin θ g ( a ̂ α g ̂ g ̂ a ̂ β 1 3 δ α β ) 2 × V d r V d r g y α n ( r , r ) E β m ( r ) g y α n * ( r , r ) E β m * ( r ) exp ( r r 2 2 w 2 ) .
( P n P m ) Exit = ( n e n o ) 2 ( a w d + b ) ,
T m = 4 ( n p n 3 ) cos θ i m cos θ t m [ cos θ i m + ( n p n 3 ) cos θ t m ] 2 ,
T n = 4 ( n p n 3 ) cos θ i n cos θ t n [ ( n p n 3 ) cos θ i n + cos θ t n ] 2 ,
( P n P m ) Measured = T n T m ( P n P m ) Exit .

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