Abstract

We present a unified theoretical framework for paraxial and wide-angle beam propagation methods in inhomogeneous birefringent media based on a minimal set of physical assumptions. The advantage of our schemes is that they are based on differential operators with a clear physical interpretation and easy numerical implementation based on sparse matrices. We demonstrate the validity of our schemes on three simple two-dimensional birefringent systems and introduce an example of application on complex three-dimensional systems by showing that topological solitons in frustrated cholesteric liquid-crystals can be used as light waveguides.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
  2. K. Fan, W. Cai, and X. Ji, “A full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for nonsmooth electromagnetic fields in waveguides,” J. Comput. Phys. 227(15), 7178–7191 (2008).
    [Crossref]
  3. J. Lopez-Dona, J. Wanguemert-Perez, and I. Molina-Fernandez, “Fast-Fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett. 17(11), 2319–2321 (2005).
    [Crossref]
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    [Crossref]
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    [Crossref]
  6. E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707 (2000).
    [Crossref]
  7. G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40(10), 733–748 (2008).
    [Crossref]
  8. P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17(13), 10895 (2009).
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    [Crossref]
  11. J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive Indices of Liquid Crystals for Display Applications,” J. Disp. Technol. 1(1), 51–61 (2005).
    [Crossref]
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    [Crossref]
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    [Crossref]
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  19. I. Nys, V. Nersesyan, J. Beeckman, and K. Neyts, “Complex liquid crystal superstructures induced by periodic photo-alignment at top and bottom substrates,” Soft Matter 14(33), 6892–6902 (2018).
    [Crossref]
  20. F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101(2), 760–766 (1997).
    [Crossref]
  21. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17(24), 1743 (1992).
    [Crossref]
  22. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
    [Crossref]
  23. M. A. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66(3), 036603 (2002).
    [Crossref]
  24. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19(3), 405–413 (2001).
    [Crossref]
  25. P. Oswald and P. Pieranski, Nematic and cholesteric liquid crystals: concepts and physical properties illustrated by experiments (CRC, 2006).
  26. G. Poy, F. Bunel, and P. Oswald, “Role of anchoring energy on the texture of cholesteric droplets: Finite-element simulations and experiments,” Phys. Rev. E 96(1), 012705 (2017).
    [Crossref]
  27. H. L. Ong, M. Schadt, and I. F. Chang, “Material Parameters and Intrinsic Optical Bistability in Room Temperature Nematics RO-TN-200, -201, -403, E7, m1, and m3,” Mol. Cryst. Liq. Cryst. 132(1-2), 45–52 (1986).
    [Crossref]
  28. P. D. Brimicombe, C. Kischka, S. J. Elston, and E. P. Raynes, “Measurement of the twist elastic constant of nematic liquid crystals using pi-cell devices,” J. Appl. Phys. 101(4), 043108 (2007).
    [Crossref]
  29. H. N. Ritland, “Relation Between Refractive Index and Density of a Glass at Constant Temperature,” J. Am. Ceram. Soc. 38(2), 86–88 (1955).
    [Crossref]
  30. J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1(1), 58 (2009).
    [Crossref]
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    [Crossref]
  32. P. J. Ackerman, Z. Qi, Y. Lin, C. W. Twombly, M. J. Laviada, Y. Lansac, and I. I. Smalyukh, “Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities,” Sci. Rep. 2(1), 414 (2012).
    [Crossref]
  33. G. Poy, A. J. Hess, I. I. Smalyukh, and S. Žumer, “Chirality-enhanced periodic self-focusing of light in soft birefringent media,” Phys. Rev. Lett. (to be published).

2018 (2)

P. Del Moral and A. Niclas, “A Taylor expansion of the square root matrix function,” J. Math. Anal. Appl. 465(1), 259–266 (2018).
[Crossref]

I. Nys, V. Nersesyan, J. Beeckman, and K. Neyts, “Complex liquid crystal superstructures induced by periodic photo-alignment at top and bottom substrates,” Soft Matter 14(33), 6892–6902 (2018).
[Crossref]

2017 (1)

G. Poy, F. Bunel, and P. Oswald, “Role of anchoring energy on the texture of cholesteric droplets: Finite-element simulations and experiments,” Phys. Rev. E 96(1), 012705 (2017).
[Crossref]

2012 (1)

P. J. Ackerman, Z. Qi, Y. Lin, C. W. Twombly, M. J. Laviada, Y. Lansac, and I. I. Smalyukh, “Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities,” Sci. Rep. 2(1), 414 (2012).
[Crossref]

2010 (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[Crossref]

2009 (2)

2008 (2)

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40(10), 733–748 (2008).
[Crossref]

K. Fan, W. Cai, and X. Ji, “A full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for nonsmooth electromagnetic fields in waveguides,” J. Comput. Phys. 227(15), 7178–7191 (2008).
[Crossref]

2007 (1)

P. D. Brimicombe, C. Kischka, S. J. Elston, and E. P. Raynes, “Measurement of the twist elastic constant of nematic liquid crystals using pi-cell devices,” J. Appl. Phys. 101(4), 043108 (2007).
[Crossref]

2005 (3)

J. Lopez-Dona, J. Wanguemert-Perez, and I. Molina-Fernandez, “Fast-Fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett. 17(11), 2319–2321 (2005).
[Crossref]

I. Farago and A. Havasi, “On the convergence and local splitting error of different splitting schemes,” Prog. Comput. Fluid Dyn. 5(8), 495 (2005).
[Crossref]

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive Indices of Liquid Crystals for Display Applications,” J. Disp. Technol. 1(1), 51–61 (2005).
[Crossref]

2003 (1)

S. Gauza, H. Wang, C.-H. Wen, S.-T. Wu, A. J. Seed, and R. Dabrowski, “High Birefringence Isothiocyanato Tolane Liquid Crystals,” Jpn. J. Appl. Phys. 42(Part 1, No. 6A), 3463–3466 (2003).
[Crossref]

2002 (1)

M. A. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66(3), 036603 (2002).
[Crossref]

2001 (1)

2000 (1)

1997 (1)

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101(2), 760–766 (1997).
[Crossref]

1996 (1)

1995 (1)

J. Yamauchi, J. Shibayama, and H. Nakano, “Modified finite-difference beam propagation method based on the generalized Douglas scheme for variable coefficients,” IEEE Photonics Technol. Lett. 7(6), 661–663 (1995).
[Crossref]

1994 (1)

C. Xu, W. Huang, J. Chrostowski, and S. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

1992 (2)

1991 (1)

1986 (1)

H. L. Ong, M. Schadt, and I. F. Chang, “Material Parameters and Intrinsic Optical Bistability in Room Temperature Nematics RO-TN-200, -201, -403, E7, m1, and m3,” Mol. Cryst. Liq. Cryst. 132(1-2), 45–52 (1986).
[Crossref]

1985 (1)

M. Rubin, “Optical properties of soda lime silica glasses,” Sol. Energy Mater. 12(4), 275–288 (1985).
[Crossref]

1974 (1)

1955 (1)

H. N. Ritland, “Relation Between Refractive Index and Density of a Glass at Constant Temperature,” J. Am. Ceram. Soc. 38(2), 86–88 (1955).
[Crossref]

Ackerman, P. J.

P. J. Ackerman, Z. Qi, Y. Lin, C. W. Twombly, M. J. Laviada, Y. Lansac, and I. I. Smalyukh, “Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities,” Sci. Rep. 2(1), 414 (2012).
[Crossref]

Beeckman, J.

I. Nys, V. Nersesyan, J. Beeckman, and K. Neyts, “Complex liquid crystal superstructures induced by periodic photo-alignment at top and bottom substrates,” Soft Matter 14(33), 6892–6902 (2018).
[Crossref]

P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17(13), 10895 (2009).
[Crossref]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[Crossref]

Brimicombe, P. D.

P. D. Brimicombe, C. Kischka, S. J. Elston, and E. P. Raynes, “Measurement of the twist elastic constant of nematic liquid crystals using pi-cell devices,” J. Appl. Phys. 101(4), 043108 (2007).
[Crossref]

Brooke, G. H.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101(2), 760–766 (1997).
[Crossref]

Bunel, F.

G. Poy, F. Bunel, and P. Oswald, “Role of anchoring energy on the texture of cholesteric droplets: Finite-element simulations and experiments,” Phys. Rev. E 96(1), 012705 (2017).
[Crossref]

Cai, W.

K. Fan, W. Cai, and X. Ji, “A full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for nonsmooth electromagnetic fields in waveguides,” J. Comput. Phys. 227(15), 7178–7191 (2008).
[Crossref]

Chang, I. F.

H. L. Ong, M. Schadt, and I. F. Chang, “Material Parameters and Intrinsic Optical Bistability in Room Temperature Nematics RO-TN-200, -201, -403, E7, m1, and m3,” Mol. Cryst. Liq. Cryst. 132(1-2), 45–52 (1986).
[Crossref]

Chaudhuri, S.

C. Xu, W. Huang, J. Chrostowski, and S. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

Chrostowski, J.

C. Xu, W. Huang, J. Chrostowski, and S. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

Cramer, C.

Dabrowski, R.

S. Gauza, H. Wang, C.-H. Wen, S.-T. Wu, A. J. Seed, and R. Dabrowski, “High Birefringence Isothiocyanato Tolane Liquid Crystals,” Jpn. J. Appl. Phys. 42(Part 1, No. 6A), 3463–3466 (2003).
[Crossref]

Del Moral, P.

P. Del Moral and A. Niclas, “A Taylor expansion of the square root matrix function,” J. Math. Anal. Appl. 465(1), 259–266 (2018).
[Crossref]

Elston, S. J.

P. D. Brimicombe, C. Kischka, S. J. Elston, and E. P. Raynes, “Measurement of the twist elastic constant of nematic liquid crystals using pi-cell devices,” J. Appl. Phys. 101(4), 043108 (2007).
[Crossref]

E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707 (2000).
[Crossref]

Fan, K.

K. Fan, W. Cai, and X. Ji, “A full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for nonsmooth electromagnetic fields in waveguides,” J. Comput. Phys. 227(15), 7178–7191 (2008).
[Crossref]

Farago, I.

I. Farago and A. Havasi, “On the convergence and local splitting error of different splitting schemes,” Prog. Comput. Fluid Dyn. 5(8), 495 (2005).
[Crossref]

Fernandez, F. A.

Gajic, Z.

Z. Gajic and M. T. J. Qureshi, Lyapunov matrix equation in system stability and control (Courier Corporation, 2008).

Gauza, S.

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive Indices of Liquid Crystals for Display Applications,” J. Disp. Technol. 1(1), 51–61 (2005).
[Crossref]

S. Gauza, H. Wang, C.-H. Wen, S.-T. Wu, A. J. Seed, and R. Dabrowski, “High Birefringence Isothiocyanato Tolane Liquid Crystals,” Jpn. J. Appl. Phys. 42(Part 1, No. 6A), 3463–3466 (2003).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier optics (Roberts and Company Publishers, 2005).

Hadley, G. R.

Havasi, A.

I. Farago and A. Havasi, “On the convergence and local splitting error of different splitting schemes,” Prog. Comput. Fluid Dyn. 5(8), 495 (2005).
[Crossref]

Herzinger, C. M.

Hess, A. J.

G. Poy, A. J. Hess, I. I. Smalyukh, and S. Žumer, “Chirality-enhanced periodic self-focusing of light in soft birefringent media,” Phys. Rev. Lett. (to be published).

Hu, C.

Hu, J.

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1(1), 58 (2009).
[Crossref]

Huang, W.

C. Xu, W. Huang, J. Chrostowski, and S. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12(11), 1926–1931 (1994).
[Crossref]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[Crossref]

James, R.

Ji, X.

K. Fan, W. Cai, and X. Ji, “A full vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for nonsmooth electromagnetic fields in waveguides,” J. Comput. Phys. 227(15), 7178–7191 (2008).
[Crossref]

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[Crossref]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[Crossref]

Johs, B.

Karpierz, M. A.

M. A. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66(3), 036603 (2002).
[Crossref]

Kischka, C.

P. D. Brimicombe, C. Kischka, S. J. Elston, and E. P. Raynes, “Measurement of the twist elastic constant of nematic liquid crystals using pi-cell devices,” J. Appl. Phys. 101(4), 043108 (2007).
[Crossref]

Koshiba, M.

Kriezis, E. E.

G. D. Ziogos and E. E. Kriezis, “Modeling light propagation in liquid crystal devices with a 3-D full-vector finite-element beam propagation method,” Opt. Quantum Electron. 40(10), 733–748 (2008).
[Crossref]

E. E. Kriezis and S. J. Elston, “Wide-angle beam propagation method for liquid-crystal device calculations,” Appl. Opt. 39(31), 5707 (2000).
[Crossref]

Lansac, Y.

P. J. Ackerman, Z. Qi, Y. Lin, C. W. Twombly, M. J. Laviada, Y. Lansac, and I. I. Smalyukh, “Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities,” Sci. Rep. 2(1), 414 (2012).
[Crossref]

Laviada, M. J.

P. J. Ackerman, Z. Qi, Y. Lin, C. W. Twombly, M. J. Laviada, Y. Lansac, and I. I. Smalyukh, “Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities,” Sci. Rep. 2(1), 414 (2012).
[Crossref]

Li, J.

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive Indices of Liquid Crystals for Display Applications,” J. Disp. Technol. 1(1), 51–61 (2005).
[Crossref]

Lifante Pedrola, G.

G. Lifante Pedrola, Beam Propagation Method for Design of Optical Waveguide Devices (John Wiley & Sons, 2015).

Lin, Y.

P. J. Ackerman, Z. Qi, Y. Lin, C. W. Twombly, M. J. Laviada, Y. Lansac, and I. I. Smalyukh, “Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities,” Sci. Rep. 2(1), 414 (2012).
[Crossref]

Lopez-Dona, J.

J. Lopez-Dona, J. Wanguemert-Perez, and I. Molina-Fernandez, “Fast-Fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett. 17(11), 2319–2321 (2005).
[Crossref]

Lu, R.

J. Li, C.-H. Wen, S. Gauza, R. Lu, and S.-T. Wu, “Refractive Indices of Liquid Crystals for Display Applications,” J. Disp. Technol. 1(1), 51–61 (2005).
[Crossref]

Menyuk, C. R.

J. Hu and C. R. Menyuk, “Understanding leaky modes: slab waveguide revisited,” Adv. Opt. Photonics 1(1), 58 (2009).
[Crossref]

Milinazzo, F. A.

F. A. Milinazzo, C. A. Zala, and G. H. Brooke, “Rational square-root approximations for parabolic equation algorithms,” J. Acoust. Soc. Am. 101(2), 760–766 (1997).
[Crossref]

Molina-Fernandez, I.

J. Lopez-Dona, J. Wanguemert-Perez, and I. Molina-Fernandez, “Fast-Fourier-based three-dimensional full-vectorial beam propagation method,” IEEE Photonics Technol. Lett. 17(11), 2319–2321 (2005).
[Crossref]

Nakano, H.

J. Yamauchi, J. Shibayama, and H. Nakano, “Modified finite-difference beam propagation method based on the generalized Douglas scheme for variable coefficients,” IEEE Photonics Technol. Lett. 7(6), 661–663 (1995).
[Crossref]

Nersesyan, V.

I. Nys, V. Nersesyan, J. Beeckman, and K. Neyts, “Complex liquid crystal superstructures induced by periodic photo-alignment at top and bottom substrates,” Soft Matter 14(33), 6892–6902 (2018).
[Crossref]

Neyts, K.

I. Nys, V. Nersesyan, J. Beeckman, and K. Neyts, “Complex liquid crystal superstructures induced by periodic photo-alignment at top and bottom substrates,” Soft Matter 14(33), 6892–6902 (2018).
[Crossref]

P. J. M. Vanbrabant, J. Beeckman, K. Neyts, R. James, and F. A. Fernandez, “A finite element beam propagation method for simulation of liquid crystal devices,” Opt. Express 17(13), 10895 (2009).
[Crossref]

Niclas, A.

P. Del Moral and A. Niclas, “A Taylor expansion of the square root matrix function,” J. Math. Anal. Appl. 465(1), 259–266 (2018).
[Crossref]

Nys, I.

I. Nys, V. Nersesyan, J. Beeckman, and K. Neyts, “Complex liquid crystal superstructures induced by periodic photo-alignment at top and bottom substrates,” Soft Matter 14(33), 6892–6902 (2018).
[Crossref]

Ong, H. L.

H. L. Ong, M. Schadt, and I. F. Chang, “Material Parameters and Intrinsic Optical Bistability in Room Temperature Nematics RO-TN-200, -201, -403, E7, m1, and m3,” Mol. Cryst. Liq. Cryst. 132(1-2), 45–52 (1986).
[Crossref]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[Crossref]

Oswald, P.

G. Poy, F. Bunel, and P. Oswald, “Role of anchoring energy on the texture of cholesteric droplets: Finite-element simulations and experiments,” Phys. Rev. E 96(1), 012705 (2017).
[Crossref]

P. Oswald and P. Pieranski, Nematic and cholesteric liquid crystals: concepts and physical properties illustrated by experiments (CRC, 2006).

Pieranski, P.

P. Oswald and P. Pieranski, Nematic and cholesteric liquid crystals: concepts and physical properties illustrated by experiments (CRC, 2006).

Poy, G.

G. Poy, F. Bunel, and P. Oswald, “Role of anchoring energy on the texture of cholesteric droplets: Finite-element simulations and experiments,” Phys. Rev. E 96(1), 012705 (2017).
[Crossref]

G. Poy, A. J. Hess, I. I. Smalyukh, and S. Žumer, “Chirality-enhanced periodic self-focusing of light in soft birefringent media,” Phys. Rev. Lett. (to be published).

Qi, Z.

P. J. Ackerman, Z. Qi, Y. Lin, C. W. Twombly, M. J. Laviada, Y. Lansac, and I. I. Smalyukh, “Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities,” Sci. Rep. 2(1), 414 (2012).
[Crossref]

Qureshi, M. T. J.

Z. Gajic and M. T. J. Qureshi, Lyapunov matrix equation in system stability and control (Courier Corporation, 2008).

Raynes, E. P.

P. D. Brimicombe, C. Kischka, S. J. Elston, and E. P. Raynes, “Measurement of the twist elastic constant of nematic liquid crystals using pi-cell devices,” J. Appl. Phys. 101(4), 043108 (2007).
[Crossref]

Rheinländer, B.

Ritland, H. N.

H. N. Ritland, “Relation Between Refractive Index and Density of a Glass at Constant Temperature,” J. Am. Ceram. Soc. 38(2), 86–88 (1955).
[Crossref]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the fdtd method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[Crossref]

Rubin, M.

M. Rubin, “Optical properties of soda lime silica glasses,” Sol. Energy Mater. 12(4), 275–288 (1985).
[Crossref]

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P. J. Ackerman, Z. Qi, Y. Lin, C. W. Twombly, M. J. Laviada, Y. Lansac, and I. I. Smalyukh, “Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities,” Sci. Rep. 2(1), 414 (2012).
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[Crossref]

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H. L. Ong, M. Schadt, and I. F. Chang, “Material Parameters and Intrinsic Optical Bistability in Room Temperature Nematics RO-TN-200, -201, -403, E7, m1, and m3,” Mol. Cryst. Liq. Cryst. 132(1-2), 45–52 (1986).
[Crossref]

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Supplementary Material (1)

NameDescription
» Supplement 1       Additional formulas for wide-angle and index operators.

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Figures (6)

Fig. 1.
Fig. 1. Schematic representation of the geometry of our computational scheme: the arbitrary birefringent system is split into a set of slabs normal to $z$ and we assume that in each slab the permittivity is $z$-independent; inside a slab, forward and backward propagating modes have their evolution controlled by two independent Schrödinger equations; continuity equations allow the calculation of the transfer of these modes through interfaces between different slabs.
Fig. 2.
Fig. 2. (a) Color-graded representation of the real part of $E_y$ when a plane wave propagates through a system of birefringent slabs with homogeneous permittivity tensor. In this particular system, our formalism in the main text is fully equivalent to the simple Jones method and solely relies on the phase operator $\tilde {\boldsymbol \epsilon }$. (b) Same as (a) with a Gaussian beam propagating through a uniaxial media with tilted optical axis $\boldsymbol {n}$; the Poynting vector $\boldsymbol {S}_p$ is not aligned with the wavevector $\boldsymbol {k}$ due to the action of the walk-off operator $\mathbf {W}$ in the propagation Eq. (14). (c) Natural spread of the intensity of a Gaussian beam in a homogeneous birefringent media, which is physically modeled by the diffraction operator $\mathbf {L}$ in the propagation Eq. (14). In (a,b,c), the white bar represents $0.5$ µm, and in (a,c) the optical axis is normal to the plane of the figure.
Fig. 3.
Fig. 3. (a,b,c) Color-graded plot of the real part of $E_x$ in systems A, B and C respectively. The optical axis variations are represented schematically with cylinders below the plots, and the axes orientations on the right apply to all plots. The white bars represent $0.5$ µm. (d) Computational error $\mu$ as a function of the mesh spacing $\Delta _y$ for system A. The reference solution for the calculation of the error is analytical. (e) Same as (d) for systems B and C. The reference solution is calculated from a FDTD simulation on a very fine mesh. In (d,e), dashed (dotted) curves correspond to wide-angle (paraxial) beam propagation simulations and the solid curve correspond to a FDTD simulation.
Fig. 4.
Fig. 4. (a) Representation of the transverse profile of the director field of a $z$-invariant CF2 with cylinders. The thickness of the liquid crystal layer is $h$ and homeotropic boundary conditions are imposed on the top and bottom confining surface along $x$. Light eigenmodes are calculated inside the dashed region. Color-graded representation of $\textrm {Re}~E_x$ (b), $\textrm {Im}~E_x$ (c), $\textrm {Re}~E_y$ (d) and $\textrm {Im}~E_y$ (e) are shown below for the fundamental eigenmode in a $30$ µm-thick sample.
Fig. 5.
Fig. 5. (a) Fraction of lost intensity of the fundamental eigenmode as a function of propagation distance $z$ inside a straight CF2. From right to left, $h=10$$30$ µm by increments of $2$ µm. The dashed line symbolizes the transition to the linear loss regime. (b) Scattering loss $\Gamma$ as a function of sample thickness $h$. The error bars are obtained from the standard deviation of the slope of $f$ above the dashed line of (a). The solid line correspond to a fit with an exponential law.
Fig. 6.
Fig. 6. (a–c) Simulated crossed-polarizers optical micrographs of sine-like CF2s with a typical radius of curvature of $100$ µm, $70$ µm and $40$ µm respectively. (d–f) Associated $x$-averaged beam intensity when the fundamental eigenmode of Figs. 4(c)–(e) is sent inside the CF2s of (a–c). The white bars represent $30$ µm.

Equations (27)

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[ ( Z 2 + Δ ) δ α β α β + ϵ α β ] E β + [ ϵ α z α Z ] E z = 0 ,
[ ϵ z β Z β ] E β + [ Δ + ϵ z z ] E z = 0 ,
0 = [ I Z 2 + S Z + R ] E ,
E z = H 1 [ Z β ϵ z β ] E β ,
H = [ ϵ z z + Δ ] ,
S α β = [ δ α γ + α 1 ϵ z z γ ] [ γ H 1 ϵ z β + ϵ γ z H 1 β ] ,
R α β = [ δ α γ + α 1 ϵ z z γ ] [ δ γ β Δ γ β + ϵ γ β ϵ γ z H 1 ϵ z β ] .
S = 2 ( W + W ) + O ( η 2 ) , R = ϵ ~ + L + O ( η 2 ) ,
W = [ I + W ( 1 ) ] 1 W ( 0 ) ,
W α β ( 0 ) = 1 2 [ α ϵ z β ϵ z z + ϵ α z ϵ z z β ] , W α β ( 1 ) = δ α β Δ α β ϵ ref ,
L α β = δ α β Δ α β + α 1 ϵ z z γ ϵ ~ γ β , ϵ ~ α β = ϵ α β ϵ α z ϵ z β ϵ z z .
T 2 + i S T + R = 0 .
T ( ± ) ± ϵ ~ + L + i W + i W + O ( η 2 ) ,
Z E ( ± ) = i T ( ± ) E ( ± ) .
T ( ± ) = ± ( ϵ ~ + L ) + i W ( 0 ) + O ( η 2 , ρ 3 ) ,
E ( + ) | Z + Δ Z = P ( ϵ ) P ( d i f f ) P ( ϵ ) E ( + ) | Z + O ( η 2 , ρ 3 , Δ Z 3 ) ,
P ( ϵ ) = exp [ i Δ Z 2 ϵ ~ ] , P ( d i f f ) = exp [ i Δ Z ( L + i W ( 0 ) ) ] .
T ( ± ) ± ϵ ~ + L + i W + O ( η 2 ) .
E ( + ) | Z + Δ Z = P ( w P ( r ) P ( w ) E ( + ) | Z + O ( η 2 , Δ Z 3 ) ,
P ( w ) = exp [ Δ Z 2 W ] , P ( r ) = exp [ i Δ Z ϵ ~ + L ] .
P ( w ) = [ I + W ( 1 ) + Δ Z 4 W ( 0 ) ] 1 [ I + W ( 1 ) Δ Z 4 W ( 0 ) ] + O ( Δ Z 3 ) .
E n ( + ) + E n ( ) = E n + 1 ( + ) + E n + 1 ( ) ,
N n ( + ) E n ( + ) N n ( ) E n ( ) = N n + 1 ( + ) E n + 1 ( + ) N n + 1 ( ) E n + 1 ( ) ,
E t = [ N 2 ( + ) + N 1 ( ) ] 1 [ N 1 ( + ) + N 1 ( ) ] E i , E r = [ N 2 ( + ) + N 1 ( ) ] 1 [ N 1 ( + ) N 2 ( + ) ] E i ,
μ | | E (num) E (ref) | | 2 / | | E (ref) | | 2 ,
ϕ ( z ) 1 | E ( x , y , z ) E ( 0 ) ( x , y ) d x d y E ( 0 ) ( x , y ) E ( 0 ) ( x , y ) d x d y | 2 .
Γ d d z [ 10 log 10 ( 1 ϕ ) ] .