Abstract

We extend the theory of Kassam et al. [J. Opt. Soc. Am. A 12, 2009 (1995) [CrossRef]  ] for scattering by oblique columnar structure thin films to include the induced form birefringence and the propagation of radiation in those films. We generalize the 4×4 matrix theory of Berreman [J. Opt. Soc. Am. 62, 502 (1972) [CrossRef]  ] to include arbitrary sources in the layer, which are necessary to determine the Green function for the inhomogeneous wave equation. We further extend first-order vector perturbation theory for scattering by roughness in the smooth surface limit, when the layer is anisotropic. Scattering by an inhomogeneous medium is approximated by a distorted Born approximation, where effective medium theory is used to determine the effective properties of the medium, and strong fluctuation theory is used to determine the inhomogeneous sources. In this manner, we develop a model for scattering by inhomogeneous films, with anisotropic correlation functions. The results are compared with Mueller matrix bidirectional scattering distribution function measurements for a glancing-angle deposition (GLAD) film. While the results are applied to the GLAD film example, the development of the theory is general enough that it can guide simulations for scattering in other anisotropic thin films.

© 2017 Optical Society of America

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References

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  2. N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
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    [Crossref]
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    [Crossref]
  27. L. Tsang and J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
    [Crossref]
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    [Crossref]
  29. E. Compain, S. Poirier, and 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]
  30. H. J. Patrick, L. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
    [Crossref]
  31. I. J. Hodgkinson, P. I. Bowmar, and Q. Wu, “Scatter from tilted-columnar birefringent thin films: observation and measurement of anisotropic scatter distributions,” Appl. Opt. 34, 163–168 (1995).
    [Crossref]
  32. J. B. Oliver, C. Smith, J. Spaulding, A. L. Rigatti, B. Charles, S. Papernov, B. Taylor, J. Foster, C. W. Carr, R. Luthi, B. Hollingsworth, and D. Cross, “Glancing-angle-deposited magnesium oxide films for high-fluence applications,” Opt. Mater. Express 6, 2291–2303 (2016).
    [Crossref]
  33. M. O. Jensen and M. J. Brett, “Porosity engineering in glancing angle deposition thin films,” Appl. Phys. A 80, 763–768 (2005).
    [Crossref]
  34. D. Saulnier, B. Taylor, K. L. Marshall, T. J. Kessler, and S. D. Jacobs, “Liquid crystal chiroptical polarization rotators for the near-UV region: theory, materials, and device applications,” Proc. SPIE 8828, 882807 (2013).
    [Crossref]

2016 (1)

2015 (1)

L. Pazynin, S. Sautbekov, Y. Sirenko, A. Vertiy, and N. Yashina, “Green’s function for an infinite anisotropic medium. Review,” Telecommun. Radio Eng. 74, 1039–1050 (2015).
[Crossref]

2014 (1)

2013 (1)

D. Saulnier, B. Taylor, K. L. Marshall, T. J. Kessler, and S. D. Jacobs, “Liquid crystal chiroptical polarization rotators for the near-UV region: theory, materials, and device applications,” Proc. SPIE 8828, 882807 (2013).
[Crossref]

2012 (1)

H. J. Patrick, L. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[Crossref]

2006 (1)

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

2005 (1)

M. O. Jensen and M. J. Brett, “Porosity engineering in glancing angle deposition thin films,” Appl. Phys. A 80, 763–768 (2005).
[Crossref]

2001 (1)

2000 (2)

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

J. E. Rothenberg, “Polarization beam smoothing for inertial confinement fusion,” J. Appl. Phys. 87, 3654–3662 (2000).
[Crossref]

1999 (3)

S. Skupsky and R. S. Craxton, “Irradiation uniformity for high-compression laser-fusion experiments,” Phys. Plasmas 6, 2157–2163 (1999).
[Crossref]

E. Compain, S. Poirier, and 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]

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

1997 (2)

1995 (3)

1988 (1)

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

1981 (1)

L. Tsang and J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
[Crossref]

1978 (1)

C. G. Granqvist and O. Hunderi, “Conductivity of inhomogeneous materials: effective-medium theory with dipole-dipole interaction,” Phys. Rev. B 18, 1554–1561 (1978).
[Crossref]

1972 (1)

1964 (1)

J. R. Wait, “Theory of radiation from sources immersed in anisotropic media,” J. Res. Natl. Bur. Stand. 68B, 119–136 (1964).
[Crossref]

1951 (1)

N. Marcuvitz and J. Schwinger, “On the representation of the electric and magnetic fields produced by currents and discontinuities in wave guides. I,” J. Appl. Phys. 22, 806–819 (1951).
[Crossref]

1935 (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416, 636–664 (1935).
[Crossref]

Asmail, C. C.

T. A. Germer and 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, and B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
[Crossref]

Berreman, D. W.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), p. 145.

Bowmar, P. I.

Brett, M. J.

M. O. Jensen and M. J. Brett, “Porosity engineering in glancing angle deposition thin films,” Appl. Phys. A 80, 763–768 (2005).
[Crossref]

M. M. Hawkeye, M. T. Taschuk, and M. J. Brett, Glancing Angle Deposition of Thin Films: Engineering the Nanoscale (Wiley, 2014).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light, A Statistical Optics Approach (Wiley, 1998).

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416, 636–664 (1935).
[Crossref]

Carr, C. W.

Charles, B.

J. B. Oliver, C. Smith, J. Spaulding, A. L. Rigatti, B. Charles, S. Papernov, B. Taylor, J. Foster, C. W. Carr, R. Luthi, B. Hollingsworth, and D. Cross, “Glancing-angle-deposited magnesium oxide films for high-fluence applications,” Opt. Mater. Express 6, 2291–2303 (2016).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, and B. Charles, “Electron-beam-deposited distributed polarization rotator for high-power laser application,” Opt. Express 22, 23883–23896 (2014).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Cloughley, S. C.

Collings, P. J.

P. J. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics (Taylor & Francis, 2009).

Compain, E.

Craxton, R. S.

S. Skupsky and R. S. Craxton, “Irradiation uniformity for high-compression laser-fusion experiments,” Phys. Plasmas 6, 2157–2163 (1999).
[Crossref]

Cross, D.

Damjanovic, D.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Drevillon, B.

Elson, J. M.

Eng, L.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Engheta, N.

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

Foster, J.

J. B. Oliver, C. Smith, J. Spaulding, A. L. Rigatti, B. Charles, S. Papernov, B. Taylor, J. Foster, C. W. Carr, R. Luthi, B. Hollingsworth, and D. Cross, “Glancing-angle-deposited magnesium oxide films for high-fluence applications,” Opt. Mater. Express 6, 2291–2303 (2016).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Fox, G.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Germer, T. A.

H. J. Patrick, L. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[Crossref]

T. A. Germer, “Polarized light scattering by microroughness and small defects in dielectric layers,” J. Opt. Soc. Am. A 18, 1279–1288 (2001).
[Crossref]

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

T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (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, and B. W. Scheer, “Polarization of out-of-plane scattering from microrough silicon,” Opt. Lett. 22, 1284–1286 (1997).
[Crossref]

Gevorgian, S.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Granqvist, C. G.

C. G. Granqvist and O. Hunderi, “Conductivity of inhomogeneous materials: effective-medium theory with dipole-dipole interaction,” Phys. Rev. B 18, 1554–1561 (1978).
[Crossref]

Gruschow, V.

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, and B. Charles, “Electron-beam-deposited distributed polarization rotator for high-power laser application,” Opt. Express 22, 23883–23896 (2014).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Hanssen, L.

H. J. Patrick, L. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[Crossref]

Hawkeye, M. M.

M. M. Hawkeye, M. T. Taschuk, and M. J. Brett, Glancing Angle Deposition of Thin Films: Engineering the Nanoscale (Wiley, 2014).

Heeger, A. J.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

Hettrick, J.

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, and B. Charles, “Electron-beam-deposited distributed polarization rotator for high-power laser application,” Opt. Express 22, 23883–23896 (2014).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Hird, M.

P. J. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics (Taylor & Francis, 2009).

Hodgkinson, I. J.

Hollingsworth, B.

Hong, S.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), p. 145.

Hunderi, O.

C. G. Granqvist and O. Hunderi, “Conductivity of inhomogeneous materials: effective-medium theory with dipole-dipole interaction,” Phys. Rev. B 18, 1554–1561 (1978).
[Crossref]

Jacobs, S. D.

D. Saulnier, B. Taylor, K. L. Marshall, T. J. Kessler, and S. D. Jacobs, “Liquid crystal chiroptical polarization rotators for the near-UV region: theory, materials, and device applications,” Proc. SPIE 8828, 882807 (2013).
[Crossref]

Jensen, M. O.

M. O. Jensen and M. J. Brett, “Porosity engineering in glancing angle deposition thin films,” Appl. Phys. A 80, 763–768 (2005).
[Crossref]

Kassam, S.

Kessler, T. J.

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, and B. Charles, “Electron-beam-deposited distributed polarization rotator for high-power laser application,” Opt. Express 22, 23883–23896 (2014).
[Crossref]

D. Saulnier, B. Taylor, K. L. Marshall, T. J. Kessler, and S. D. Jacobs, “Liquid crystal chiroptical polarization rotators for the near-UV region: theory, materials, and device applications,” Proc. SPIE 8828, 882807 (2013).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Kingon, A.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Kivelson, S.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

Kohlstedt, H.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Kong, J. A.

L. Tsang and J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
[Crossref]

Luthi, R.

Marcuvitz, N.

N. Marcuvitz and J. Schwinger, “On the representation of the electric and magnetic fields produced by currents and discontinuities in wave guides. I,” J. Appl. Phys. 22, 806–819 (1951).
[Crossref]

Marshall, K. L.

D. Saulnier, B. Taylor, K. L. Marshall, T. J. Kessler, and S. D. Jacobs, “Liquid crystal chiroptical polarization rotators for the near-UV region: theory, materials, and device applications,” Proc. SPIE 8828, 882807 (2013).
[Crossref]

McCall, M. W.

M. W. McCall, Q. Wu, and I. J. Hodgkinson, Birefringent Thin Films and Polarizing Elements, 2nd ed. (Imperial College, 2014).

Mitchell, G.

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Noll, T.

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Oliver, J. B.

J. B. Oliver, C. Smith, J. Spaulding, A. L. Rigatti, B. Charles, S. Papernov, B. Taylor, J. Foster, C. W. Carr, R. Luthi, B. Hollingsworth, and D. Cross, “Glancing-angle-deposited magnesium oxide films for high-fluence applications,” Opt. Mater. Express 6, 2291–2303 (2016).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, and B. Charles, “Electron-beam-deposited distributed polarization rotator for high-power laser application,” Opt. Express 22, 23883–23896 (2014).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Papernov, S.

J. B. Oliver, C. Smith, J. Spaulding, A. L. Rigatti, B. Charles, S. Papernov, B. Taylor, J. Foster, C. W. Carr, R. Luthi, B. Hollingsworth, and D. Cross, “Glancing-angle-deposited magnesium oxide films for high-fluence applications,” Opt. Mater. Express 6, 2291–2303 (2016).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Park, N. Y.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Patrick, H. J.

H. J. Patrick, L. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[Crossref]

Pazynin, L.

L. Pazynin, S. Sautbekov, Y. Sirenko, A. Vertiy, and N. Yashina, “Green’s function for an infinite anisotropic medium. Review,” Telecommun. Radio Eng. 74, 1039–1050 (2015).
[Crossref]

Poirier, S.

Rigatti, A. L.

J. B. Oliver, C. Smith, J. Spaulding, A. L. Rigatti, B. Charles, S. Papernov, B. Taylor, J. Foster, C. W. Carr, R. Luthi, B. Hollingsworth, and D. Cross, “Glancing-angle-deposited magnesium oxide films for high-fluence applications,” Opt. Mater. Express 6, 2291–2303 (2016).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Rothenberg, J. E.

J. E. Rothenberg, “Polarization beam smoothing for inertial confinement fusion,” J. Appl. Phys. 87, 3654–3662 (2000).
[Crossref]

Saulnier, D.

D. Saulnier, B. Taylor, K. L. Marshall, T. J. Kessler, and S. D. Jacobs, “Liquid crystal chiroptical polarization rotators for the near-UV region: theory, materials, and device applications,” Proc. SPIE 8828, 882807 (2013).
[Crossref]

Sautbekov, S.

L. Pazynin, S. Sautbekov, Y. Sirenko, A. Vertiy, and N. Yashina, “Green’s function for an infinite anisotropic medium. Review,” Telecommun. Radio Eng. 74, 1039–1050 (2015).
[Crossref]

Scheer, B. W.

Schrieffer, J. R.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

Schwinger, J.

N. Marcuvitz and J. Schwinger, “On the representation of the electric and magnetic fields produced by currents and discontinuities in wave guides. I,” J. Appl. Phys. 22, 806–819 (1951).
[Crossref]

Setter, N.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Sharma, K. A.

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

Sirenko, Y.

L. Pazynin, S. Sautbekov, Y. Sirenko, A. Vertiy, and N. Yashina, “Green’s function for an infinite anisotropic medium. Review,” Telecommun. Radio Eng. 74, 1039–1050 (2015).
[Crossref]

Skupsky, S.

S. Skupsky and R. S. Craxton, “Irradiation uniformity for high-compression laser-fusion experiments,” Phys. Plasmas 6, 2157–2163 (1999).
[Crossref]

Smith, C.

J. B. Oliver, C. Smith, J. Spaulding, A. L. Rigatti, B. Charles, S. Papernov, B. Taylor, J. Foster, C. W. Carr, R. Luthi, B. Hollingsworth, and D. Cross, “Glancing-angle-deposited magnesium oxide films for high-fluence applications,” Opt. Mater. Express 6, 2291–2303 (2016).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, and B. Charles, “Electron-beam-deposited distributed polarization rotator for high-power laser application,” Opt. Express 22, 23883–23896 (2014).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Spaulding, J.

J. B. Oliver, C. Smith, J. Spaulding, A. L. Rigatti, B. Charles, S. Papernov, B. Taylor, J. Foster, C. W. Carr, R. Luthi, B. Hollingsworth, and D. Cross, “Glancing-angle-deposited magnesium oxide films for high-fluence applications,” Opt. Mater. Express 6, 2291–2303 (2016).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Stephenson, G. B.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Stolitchnov, I.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Strang, G.

G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, 1980).

Streiffer, S.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Su, W.-P.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

Taganstev, A. K.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Taschuk, M. T.

M. M. Hawkeye, M. T. Taschuk, and M. J. Brett, Glancing Angle Deposition of Thin Films: Engineering the Nanoscale (Wiley, 2014).

Taylor, B.

J. B. Oliver, C. Smith, J. Spaulding, A. L. Rigatti, B. Charles, S. Papernov, B. Taylor, J. Foster, C. W. Carr, R. Luthi, B. Hollingsworth, and D. Cross, “Glancing-angle-deposited magnesium oxide films for high-fluence applications,” Opt. Mater. Express 6, 2291–2303 (2016).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, and B. Charles, “Electron-beam-deposited distributed polarization rotator for high-power laser application,” Opt. Express 22, 23883–23896 (2014).
[Crossref]

D. Saulnier, B. Taylor, K. L. Marshall, T. J. Kessler, and S. D. Jacobs, “Liquid crystal chiroptical polarization rotators for the near-UV region: theory, materials, and device applications,” Proc. SPIE 8828, 882807 (2013).
[Crossref]

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

Taylor, D. V.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Tsang, L.

L. Tsang and J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
[Crossref]

Vertiy, A.

L. Pazynin, S. Sautbekov, Y. Sirenko, A. Vertiy, and N. Yashina, “Green’s function for an infinite anisotropic medium. Review,” Telecommun. Radio Eng. 74, 1039–1050 (2015).
[Crossref]

Wait, J. R.

J. R. Wait, “Theory of radiation from sources immersed in anisotropic media,” J. Res. Natl. Bur. Stand. 68B, 119–136 (1964).
[Crossref]

Wu, Q.

Wu, Q. H.

Yamada, T.

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

Yashina, N.

L. Pazynin, S. Sautbekov, Y. Sirenko, A. Vertiy, and N. Yashina, “Green’s function for an infinite anisotropic medium. Review,” Telecommun. Radio Eng. 74, 1039–1050 (2015).
[Crossref]

Zeng, J.

H. J. Patrick, L. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[Crossref]

Ziolkowski, R. W.

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

Ann. Phys. (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416, 636–664 (1935).
[Crossref]

Appl. Opt. (3)

Appl. Phys. A (1)

M. O. Jensen and M. J. Brett, “Porosity engineering in glancing angle deposition thin films,” Appl. Phys. A 80, 763–768 (2005).
[Crossref]

J. Appl. Phys. (3)

N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and S. Streiffer, “Ferroelectric thin films: review of materials, properties, and applications,” J. Appl. Phys. 100, 051606 (2006).
[Crossref]

J. E. Rothenberg, “Polarization beam smoothing for inertial confinement fusion,” J. Appl. Phys. 87, 3654–3662 (2000).
[Crossref]

N. Marcuvitz and J. Schwinger, “On the representation of the electric and magnetic fields produced by currents and discontinuities in wave guides. I,” J. Appl. Phys. 22, 806–819 (1951).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Res. Natl. Bur. Stand. (1)

J. R. Wait, “Theory of radiation from sources immersed in anisotropic media,” J. Res. Natl. Bur. Stand. 68B, 119–136 (1964).
[Crossref]

Metrologia (1)

H. J. Patrick, L. Hanssen, J. Zeng, and T. A. Germer, “BRDF measurements of graphite used in high-temperature fixed point blackbody radiators: a multi-angle study at 405 nm and 658 nm,” Metrologia 49, S81–S92 (2012).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Opt. Mater. Express (1)

Phys. Plasmas (1)

S. Skupsky and R. S. Craxton, “Irradiation uniformity for high-compression laser-fusion experiments,” Phys. Plasmas 6, 2157–2163 (1999).
[Crossref]

Phys. Rev. B (1)

C. G. Granqvist and O. Hunderi, “Conductivity of inhomogeneous materials: effective-medium theory with dipole-dipole interaction,” Phys. Rev. B 18, 1554–1561 (1978).
[Crossref]

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]

Proc. SPIE (1)

D. Saulnier, B. Taylor, K. L. Marshall, T. J. Kessler, and S. D. Jacobs, “Liquid crystal chiroptical polarization rotators for the near-UV region: theory, materials, and device applications,” Proc. SPIE 8828, 882807 (2013).
[Crossref]

Radio Sci. (1)

L. Tsang and J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
[Crossref]

Rev. Mod. Phys. (1)

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

Rev. Sci. Instrum. (1)

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

Telecommun. Radio Eng. (1)

L. Pazynin, S. Sautbekov, Y. Sirenko, A. Vertiy, and N. Yashina, “Green’s function for an infinite anisotropic medium. Review,” Telecommun. Radio Eng. 74, 1039–1050 (2015).
[Crossref]

Other (9)

G. Strang, Linear Algebra and Its Applications, 2nd ed. (Academic, 1980).

C. Brosseau, Fundamentals of Polarized Light, A Statistical Optics Approach (Wiley, 1998).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), p. 145.

M. W. McCall, Q. Wu, and I. J. Hodgkinson, Birefringent Thin Films and Polarizing Elements, 2nd ed. (Imperial College, 2014).

J. B. Oliver, T. J. Kessler, C. Smith, B. Taylor, V. Gruschow, J. Hettrick, B. Charles, J. Spaulding, T. Noll, A. L. Rigatti, S. Papernov, K. A. Sharma, G. Mitchell, and J. Foster, “Development of a glancing-angle-deposited distributed polarization rotator,” in Advanced Photonics, OSA Technical Digest (online) (Optical Society of America, 2015), paper NS4B.1.

P. J. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics (Taylor & Francis, 2009).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

M. M. Hawkeye, M. T. Taschuk, and M. J. Brett, Glancing Angle Deposition of Thin Films: Engineering the Nanoscale (Wiley, 2014).

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Figures (5)

Fig. 1.
Fig. 1.

Diagram showing an anisotropic film on a substrate.

Fig. 2.
Fig. 2.

Diagram showing the waves for a boundary condition imposed at z = z .

Fig. 3.
Fig. 3.

Mueller matrix measured from the GLAD coating described in the text. The light is incident from the substrate, and the scatter is evaluated above the coating in transmission. The data are shown on projected-cosine space, so that a horizontal slice through the data represents the plane containing the columns and the surface normal. The 11-element is the regular BTDF on a scale in inverse steradians, while the other elements are shown normalized to the 11-element.

Fig. 4.
Fig. 4.

Calculated Mueller matrix for the scatter by a GLAD coating. The light is incident from the substrate, and the scatter is evaluated above the coating in transmission. The data are shown on projected-cosine space, so that a horizontal slice through the data represents the plane containing the columns. The 11-element is the regular BTDF on a scale in inverse steradians, while the other elements are shown normalized to the 11-element.

Fig. 5.
Fig. 5.

Calculated Mueller matrix for the scatter by roughness at the exposed interface of a GLAD coating. The light is incident from the substrate, and the scatter is evaluated above the coating. The transverse correlation lengths were the same as for the volume homogeneity and the rms roughness was 10 nm.

Equations (82)

Equations on this page are rendered with MathJax. Learn more.

E ( 0 ) ( r ) = E ( 0 ) exp ( i k 0 , z z ) exp ( i κ 0 · ρ ) .
E ( r ) = 1 4 π 2 d 2 κ E ( κ ) exp ( i k z z ) exp ( i κ · ρ ) .
s ^ ( k ) = z ^ × k / | z ^ × k | , p ^ ( k ) = k × s ^ ( k ) / | k × s ^ ( k ) | .
E ( 0 ) = s ^ ( k 0 ) E s ( 0 ) + p ^ ( k 0 ) E p ( 0 ) ,
E ( κ ) = s ^ ( k ) E s ( κ ) + p ^ ( k ) E p ( κ ) .
[ E p ( κ ) E s ( κ ) ] = S [ E p ( 0 ) E s ( 0 ) ] ,
M = A S S * A 1 ,
A = 1 2 [ 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ] .
E i = cos θ i 2 n i Z 0 | E ( 0 ) | 2 ,
L r = ( ω c ) 2 n r 3 8 π 2 Z 0 A cos θ r | E ( k ) | 2 ,
f r = ( ω c ) 2 n r 3 n i 1 4 π 2 cos θ r A cos θ i | E ( k ) | 2 | E ( 0 ) | 2 .
f r = ( ω c ) 2 n r 3 n i 1 4 π 2 cos θ r A cos θ i A S S * A 1 .
( S 1 + S 2 ) ( S 1 + S 2 ) * = S 1 S 1 * + S 1 S 2 * + S 2 S 1 * + S 2 S 2 * .
( a S ) S * = S ( a S * ) = a ( S S * ) .
S = S ( r ) d 3 r ,
f r = ( ω c ) 2 n r 3 n i 1 4 π 2 cos θ r A cos θ i × A [ S ( r ) S * ( r ) d 3 r d 3 r ] A 1 .
ξ ^ ( k ) = p ^ ( k ) cos ϕ ± s ^ ( k ) sin ϕ , η ^ ( k ) = p ^ ( k ) sin ϕ + s ^ ( k ) cos ϕ ,
[ 0 × × 0 ] [ E H ] = i ω c [ D B ] + [ J E J M ] ,
[ D B ] = M [ E H ] ,
M = [ ε σ σ μ ] ,
[ 0 × × 0 ] [ E H ] = i ω c M [ E H ] + [ J E J M ] .
E ( r ) = E ( z ) exp ( i κ · ρ ) ,
z [ E x H y E y H x ] = i ω c Δ κ [ E x H y E y H x ] + Γ κ [ J E J M ] ,
[ E z H z ] = F κ [ E x H y E y H x ] ,
[ J E ( r ) J M ( r ) ] = [ j E j M ] δ ( r r ) .
[ J E ( r ) J M ( r ) ] = 1 ( 2 π ) 2 d 2 κ [ J E ( κ , z ) J M ( κ , z ) ] exp ( i κ · ρ ) ,
[ J E ( κ , z ) J M ( κ , z ) ] = d 2 ρ [ J E ( r ) J M ( r ) ] exp ( i κ · ρ ) .
[ J E ( κ , z ) J M ( κ , z ) ] = [ j E j M ] exp ( i κ · ρ ) δ ( z z ) .
[ Δ E x ( z ) Δ H y ( z ) Δ E y ( z ) Δ H x ( z ) ] = Γ κ [ j E j M ] exp ( i κ · ρ ) .
n ^ = z ^ Z x x ^ Z y y ^ ,
E = E ( 0 ) + Z ( E ( 0 ) / z ) + E ( 1 ) , H = H ( 0 ) + Z ( H ( 0 ) / z ) + H ( 1 ) ,
n ^ × Δ E = 0 n ^ × Δ H = 0 .
Δ E x ( 1 ) = ( Z / x ) Δ E z ( 0 ) Z Δ ( E x ( 0 ) / z ) , Δ E y ( 1 ) = ( Z / y ) Δ E z ( 0 ) Z Δ ( E y ( 0 ) / z ) , Δ H x ( 1 ) = ( Z / x ) Δ H z ( 0 ) Z Δ ( H x ( 0 ) / z ) , Δ H y ( 1 ) = ( Z / y ) Δ H z ( 0 ) Z Δ ( H y ( 0 ) / z ) .
E ( 0 ) = E ( 0 ) ( z ) exp ( i κ 0 · ρ ) , H ( 0 ) = H ( 0 ) ( z ) exp ( i κ 0 · ρ ) .
E x ( 0 ) / z = i ( ω / c ) B y ( 0 ) + i κ 0 , x E z ( 0 ) , E y ( 0 ) / z = i ( ω / c ) B x ( 0 ) + i κ 0 , y E z ( 0 ) , H x ( 0 ) / z = i ( ω / c ) D y ( 0 ) + i κ 0 , x H z ( 0 ) , H y ( 0 ) / z = i ( ω / c ) D x ( 0 ) + i κ 0 , y H z ( 0 ) .
E ( 1 ) = 1 4 π 2 d 2 κ E ( 1 ) ( z ) exp ( i κ · ρ ) .
Δ E x ( 1 ) = i [ k x Δ E z ( 0 ) + ( ω / c ) Δ B y ( 0 ) ] Z ( κ κ 0 ) / ( 2 π ) 2 , Δ E y ( 1 ) = i [ k y Δ E z ( 0 ) ( ω / c ) Δ B x ( 0 ) ] Z ( κ κ 0 ) / ( 2 π ) 2 , Δ H x ( 1 ) = i [ k x Δ H z ( 0 ) ( ω / c ) Δ D y ( 0 ) ] Z ( κ κ 0 ) / ( 2 π ) 2 , Δ H y ( 1 ) = i [ k y Δ H z ( 0 ) + ( ω / c ) Δ D x ( 0 ) ] Z ( κ κ 0 ) / ( 2 π ) 2 ,
Z ( κ κ 0 ) = d 2 ρ Z ( ρ ) exp [ i ( κ κ 0 ) · ρ ] .
Δ κ = W κ Σ κ W κ 1 ,
[ E x ( z ) H y ( z ) E y ( z ) H x ( z ) ] = W κ exp ( i ω Σ κ z / c ) a ,
[ E x ( z ) H y ( z ) E y ( z ) H x ( z ) ] = W κ u exp ( i ω z cos θ / c ) E u ,
W κ u = [ cos θ cos ϕ sin ϕ cos ϕ cos θ sin ϕ cos θ sin ϕ cos ϕ sin ϕ cos θ cos ϕ ] ,
[ E x ( z ) H y ( z ) E y ( z ) H x ( z ) ] = W κ d exp ( i ω z cos θ / c ) E d ,
W κ d = [ cos θ cos ϕ sin ϕ n s cos ϕ n s cos θ sin ϕ cos θ sin ϕ cos ϕ n s sin ϕ n s cos θ cos ϕ ] ,
[ W κ W κ u 0 42 W κ σ κ 0 42 W κ d ] [ a E u E d ] = [ Δ E x ( 0 ) Δ H y ( 0 ) Δ E y ( 0 ) Δ H x ( 0 ) Δ E x ( τ ) Δ H y ( τ ) Δ E y ( τ ) Δ H x ( τ ) ] ,
σ κ = exp ( i ω Σ κ τ / c ) ,
[ Δ E x ( 0 ) Δ H y ( 0 ) Δ E y ( 0 ) Δ H x ( 0 ) ] = [ cos θ cos ϕ sin ϕ cos ϕ cos θ sin ϕ cos θ sin ϕ cos ϕ sin ϕ cos θ cos ϕ ] [ E p d ( 0 ) E s d ( 0 ) ] ,
[ Δ E x ( τ ) Δ H y ( τ ) Δ E y ( τ ) Δ H x ( τ ) ] = [ cos θ cos ϕ sin ϕ n s cos ϕ n s cos θ sin ϕ cos θ sin ϕ cos ϕ n s sin ϕ n s cos θ cos ϕ ] [ E p u ( 0 ) E s u ( 0 ) ] ,
[ W κ 0 44 W κ u 0 42 W κ σ κ W κ σ κ 0 42 0 42 0 44 W κ σ κ 0 42 W κ d ] [ a a E u E d ] = [ 0 4 Δ E x ( z ) Δ H y ( z ) Δ E y ( z ) Δ H x ( z ) 0 4 ] ,
σ = exp ( i ω Σ κ z / c ) ,
α i = 4 π r 1 r 2 r 3 ε c ε h 3 ε h + 3 L i ( ε c ε h ) ,
L i = r 1 r 2 r 3 2 0 d q ( r i 2 + q ) f ( q )
f ( q ) = [ ( q + r 1 2 ) ( q + r 2 2 ) ( q + r 3 2 ) ] 1 / 2 .
L 1 + L 2 + L 3 = 1 .
L 1 = r 1 r 2 2 0 1 ( r 1 2 + q ) 3 / 2 ( r 2 2 + q ) 1 / 2 d q
L 1 = 1 1 + ( r 1 / r 2 ) , L 2 = 1 1 + ( r 2 / r 1 ) .
f ε c ε eff , i 3 ε eff , i + 3 L i ( ε c ε eff , i ) + ( 1 f ) ε h ε eff , i 3 ε eff , i + 3 L i ( ε h ε eff , i ) = 0 .
ε = U y ( α ) [ ε eff , 1 0 0 0 ε eff , 2 0 0 0 ε eff , 3 ] U y ( α ) ,
U y ( α ) = [ cos α 0 sin α 0 1 0 sin α 0 cos α ] .
[ 0 × × 0 ] [ E H ] = i ω c M eff [ E H ] i ω c Δ M [ E H ] ,
[ J E J M ] = i ω c Δ M [ E H ] .
[ J E J M ] = i ω c Δ M [ E ( 0 ) H ( 0 ) ] ,
[ E H ] = 1 4 π 2 d 2 κ [ E ( κ ) H ( κ ) ] exp ( i κ · ρ ) ,
[ E ( κ ) H ( κ ) ] = i ω c d 3 r × G ˜ ( κ , r , r ) Δ M ( r ) [ E ( 0 ) ( r ) H ( 0 ) ( r ) ]
[ E ( 0 ) ( r ) H ( 0 ) ( r ) ] = K ( κ 0 , r ) [ E p ( 0 ) E s ( 0 ) ] .
S ( r ) = i ω c G ( κ , z ) e i κ · ρ Δ M ( r ) K ( κ 0 , z ) e i κ 0 · ρ ,
S S * = ( ω c ) 2 d 3 r d 3 r e i ( κ κ 0 ) · ( ρ ρ ) × [ G ( κ , z ) Δ M ( r ) K ( κ 0 , z ) ] [ G ( κ , z ) Δ M ( r ) K ( κ 0 , z ) ] * .
Δ M ( r ) = Δ M 0 ϕ ( r ) ,
S S * = d 3 r d 3 r e i ( κ κ 0 ) · ( ρ ρ ) ] × f ( r , r ) S 0 ( z ) [ S 0 ( z ) ] * ,
S 0 ( z ) = i ω c G ( κ , z ) Δ M 0 K ( κ 0 , z )
f ( r , r ) = ϕ ( r ) ϕ * ( r ) .
f ( r , r ) = exp ( Δ r · T · Δ r )
T = U y ( α ) [ τ 1 2 0 0 0 τ 2 2 0 0 0 τ 3 2 ] U y ( α )
S S * = A d z d z F ( κ κ 0 , z , z ) S 0 ( z ) [ S 0 ( z ) ] * ,
F ( κ , z , z ) = π τ 1 τ 2 τ 3 τ z exp [ ( z z ) 2 / τ z 2 ] × exp [ i ( τ 1 2 τ 3 2 ) κ x ( z z ) sin α cos α τ z 2 ] × exp [ ( τ 1 τ 3 κ x / 2 / τ z ) 2 ] × exp [ ( τ 2 κ y / 2 ) 2 ] ,
Δ M 0 = M f Δ f ,
( Δ M 0 ) 2 = f ( 1 f ) ( ε c ε h ) 2 .
( Δ M 0 ) 2 = f [ ε eff , i ( ε c ε eff , i ) ε eff , i + L i ( ε c ε eff , i ) ] 2 + ( 1 f ) [ ε eff , i ( ε h ε eff , i ) ε eff , i + L i ( ε h ε eff , i ) ] 2 .
Δ 11 = M 51 + ( M 53 + ξ ) F 11 + M 56 F 21 , Δ 12 = M 55 + ( M 53 + ξ ) F 12 + M 56 F 22 , Δ 13 = M 52 + ( M 53 + ξ ) F 13 + M 56 F 23 , Δ 14 = M 54 ( M 53 + ξ ) F 14 M 56 F 24 , Δ 21 = M 11 + M 13 F 11 + ( η + M 16 ) F 21 , Δ 22 = M 15 + M 13 F 12 + ( η + M 16 ) F 22 , Δ 23 = M 12 + M 13 F 13 + ( η + M 16 ) F 23 , Δ 24 = M 14 + M 13 F 14 + ( η + M 16 ) F 24 , Δ 31 = M 41 ( M 43 η ) F 11 M 46 F 21 , Δ 32 = M 45 ( M 43 η ) F 12 M 46 F 22 , Δ 33 = M 42 ( M 43 η ) F 13 M 46 F 23 , Δ 34 = M 44 + ( M 43 η ) F 14 + M 46 F 24 , Δ 41 = M 21 + M 23 F 11 + ( M 26 ξ ) F 21 , Δ 42 = M 25 + M 23 F 12 + ( M 26 ξ ) F 22 , Δ 43 = M 22 + M 23 F 13 + ( M 26 ξ ) F 23 , Δ 44 = M 24 M 23 F 14 ( M 26 ξ ) F 24 ,
Γ 11 = Γ 31 = Γ 41 = Γ 12 = Γ 22 = Γ 32 = Γ 14 = Γ 24 = Γ 44 = Γ 25 = Γ 35 = Γ 45 = Γ 16 = 0 , Γ 21 = Γ 42 = Γ 15 = 1 , Γ 34 = 1 , Γ 13 = [ ( ξ + M 53 ) M 66 M 56 M 63 ] / d , Γ 16 = [ M 33 M 56 M 36 ( ξ + M 53 ) ] / d , Γ 23 = [ M 13 M 66 ( η + M 16 ) M 63 ] / d , Γ 26 = [ ( η + M 16 ) M 33 M 13 M 36 ] / d , Γ 33 = [ M 46 M 63 ( η M 43 ) M 66 ] / d , Γ 36 = [ M 36 ( M 43 η ) M 33 M 46 ] / d , Γ 43 = [ ( ξ M 26 ) M 63 + M 23 M 66 ] / d , Γ 46 = [ ( M 26 ξ ) M 33 M 23 M 36 ] / d ,
d = M 33 M 66 M 36 M 63 .
F 11 = [ M 36 ( η + M 61 ) M 31 M 66 ] / d , F 12 = [ M 36 M 65 ( ξ M 35 ) M 66 ] / d , F 13 = [ ( M 62 ξ ) M 36 M 32 M 66 ] / d , F 14 = [ ( M 34 η ) M 66 M 36 M 64 ] / d , F 21 = [ M 31 M 63 M 33 ( η + M 61 ) ] / d , F 22 = [ ( ξ + M 35 ) M 63 M 33 M 65 ] / d , F 23 = [ ( ξ M 62 ) M 33 + M 32 M 63 ] / d , F 24 = [ M 33 M 64 + ( η M 34 ) M 63 ] / d .

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