Abstract

The accurate analysis of optical waveguides is an important issue when designing devices for optical communication. Waveguides combined with liquid crystals have great potential because they allow waveguide tuning over a wide range using low voltages. In this paper, we present calculations that combine an advanced algorithm for calculating liquid crystal behavior and a finite-element mode solver that is able to incorporate the full anisotropy of the materials. Calculation examples demonstrate the validity of our program.

© 2009 IEEE

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  1. N. Clark, M. Handshy, "Surface-stabilized ferroelectric liquid-crystal electro-optic waveguide switch," Appl. Phys. Lett. 57, 1852-1854 (1990).
  2. A. d'Alessandro, R. Asquini, "Liquid crystal devices for photonic switching applications: State of the art and future developments," Mol. Cryst. Liq. Cryst. 398, 207-221 (2003).
  3. H. Desmet, W. Bogaerts, A. Adamski, J. Beeckman, K. Neyts, R. Baets, "Silicon-on-insulator optical waveguides with liquid crystal cladding for switching and tuning," Proc. ECOC (2003) pp. 430-431.
  4. L. Sirleto, G. Coppola, G. Breglio, "Optical multimode interference router based on a liquid crystal waveguide," J. Opt. A: Pure Appl. Opt. 5, S298-S304 (2003).
  5. C. Gizzi, R. Asquini, A. d'Alessandro, "A polarization independent liquid crystal assisted vertical coupler switch," Mol. Cryst. Liq. Cryst. 421, 95-105 (2004).
  6. A. d'Alessandro, B. Bellini, D. Donisi, R. Beccherelli, R. Asquini, "Nematic liquid crystal optical channel waveguides on silicon," IEEE J. Quantum Electron. 42, 1084-1090 (2006).
  7. A. Fratalocchi, R. Asquini, G. Assanto, "Integrated electro-optic switch in liquid crystals," Opt. Exp. 13, 32-36 (2005).
  8. S. V. Serak, N. V. Tabiryan, M. Peccianti, G. Assanto, "Spatial soliton all-optical logic gates," IEEE Photon. Technol. Lett. 18, 1287-1289 (2006).
  9. J. Beeckman, K. Neyts, M. Haelterman, "Patterned electrode steering of nematicons," J. Opt. A: Pure Appl. Opt. 8, 214-220 (2006).
  10. J. Henninot, M. Debailleul, R. Asquini, A. d'Alessandro, M. Warenghem, "Self-waveguiding in an isotropic channel induced in dye doped nematic liquid crystal and a bent self-waveguide," J. Opt. A: Pure Appl. Opt. 6, 315-323 (2004).
  11. R. James, E. Willman, F. A. Fernández, S. E. Day, "Finite-element modeling of liquid-crystal hydrodynamics with a variable degree of order," IEEE Trans. Electron Dev. 53, 1575-1582 (2006).
  12. P. G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford Univ. Press, 1995).
  13. R. Asquini, A. d'Alessandro, "Bpm analysis of an integrated optical switch using polymeric optical waveguides and ssflc at 1.55 $\mu{\hbox {m}}$," Mol. Cryst. Liq. Cryst 375, 243-247 (2002).
  14. G. Ntogari, D. Tsipouridou, E. Kriezis, "A numerical study of optical switches and modulators based on ferroelectric liquid crystals," J. Opt. A: Pure Appl. Opt. 7, 82-87 (2005).
  15. E. Gros, L. Dupont, "Beam deflector using double-refraction in ferroelectric liquid crystal waveguides," Ferroelectrics 246, 219-226 (2000).
  16. J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, M. Haelterman, "Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect," Opt. Quant. Electron. 35, 95-106 (2005).
  17. J. Beeckman, K. Neyts, X. Hutsebaut, M. Haelterman, "Observation of out-coupling of a nematicon," Opto-Electron. Rev. 14, 263-267 (2006) 12.
  18. S. Selleri, L. Vincetti, A. Cucinotta, M. Zoboli, "Complex FEM modal solver of optical waveguides with PML boundary conditions," Opt. Quant. Electron. 33, 359-371 (2001).
  19. A. Fallahkhair, S. Li, T. Murphy, "Vector finite difference modesolver for anisotropic dielectric waveguides," J. Lightw. Technol. 26, 1423-1431 (2008).
  20. L. Valor, J. Zapata, "An efficient finite element formulation to analyze waveguides with lossy inhomogeneous bi-anisotropic materials," IEEE Trans. Microw. Theory Tech. 44, 291-296 (1996).
  21. S. Hsu, M. Chen, H. Chang, "Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm," Opt. Exp. 15, 5416-5430 (2007).
  22. V. Schulz, "Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides," IEEE Trans. Microw. Theory Tech. 51, 1086-1095 (2003).
  23. F. Bertazzi, O. Peverini, M. Goano, G. Ghione, R. Orta, R. Tascone, "A fast reduced-order model for the full-wave fem analysis of lossy inhomogeneous anisotropic waveguides," IEEE Trans. Microw. Theory Tech. 50, 2108-2114 (2002).
  24. P. Savi, I. Gheorma, R. Graglia, "Full-wave high-order FEM model for lossy anisotropic waveguides," IEEE Trans. Microw. Theory Tech. 50, 495-500 (2002).
  25. Y. Lu, F. A. Fernandez, "An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides," IEEE Trans. Microw. Theory Tech. 41, 1215-1223 (1993).
  26. J. A. M. Svedin, "Propagation analysis of chirowaveguides using the finite element method," IEEE Trans. Microw. Theory Tech. 38, 1488-1496 (1990).
  27. J. A. M. Svedin, "A numerically efficient finite-element formulation for the general waveguide problem without spurious modes," IEEE Trans. Microw. Theory Tech. 37, 1708-1715 (1989).
  28. T. Angkaew, M. Matsuhara, N. Kumagai, "Finite-element analysis of waveguide modes: A novel approach that eliminates spurious modes," IEEE Trans. Microw. Theory Tech. MTT-35, 117-123 (1987).
  29. L. Valor, J. Zapata, "Efficient finite element analysis of waveguides with lossy inhomogeneous anisotropic materials characterized by arbitrary permittivity and permeability tensors," IEEE Trans. Microw. Theory Tech. 43, 2452-2459 (1995).
  30. G. Tartarini, H. Renner, "Efficient finite-element analysis of tilted open anisotropic optical channel waveguides," IEEE Trans. Microw. Theory Tech. 9, 389-391 (1999).
  31. L. Nuno, J. Balbastre, H. Castane, "Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite-element method (FEM) using edge elements," IEEE Trans. Microw. Theory Tech. 45, 446-449 (1997).
  32. S. Hsu, H. Chang, "Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3d anisotropy," Opt. Exp. 15, 15 797-15 811 (2007).
  33. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).
  34. M. Koshiba, Y. Tsuji, "Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems," J. Lightw. Technol. 18, 737-743 (2000).
  35. F. Tisseur, K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev. 43, 235-286 (2001).
  36. A. F. Peterson, "Vector finite element formulation for scattering from two-dimensional heterogeneous bodies," IEEE Trans. Antenna Propag. 43, 357-365 (1994).
  37. V. Chigrinov, V. Kozenkov, H. Kwok, Photoalignment of Liquid Crystalline Materials (Wiley, 2008).
  38. H. Desmet, K. Neyts, R. Baets, "Modeling nematic liquid crystals in the neighborhood of edges," J. Appl. Phys. 98, 123517 (2005).
  39. P. Yeh, C. Gu, Optics of Liquid Crystal Displays (Wiley-Interscience, 1999).
  40. Y. Satomura, M. Matsuhara, N. Kumagai, "Analysis of electromagnetic-wave modes in anisotropic slab waveguide," IEEE Trans. Microw. Theory Tech. MTT-22, 86-92 (1974).

2008

A. Fallahkhair, S. Li, T. Murphy, "Vector finite difference modesolver for anisotropic dielectric waveguides," J. Lightw. Technol. 26, 1423-1431 (2008).

2007

S. Hsu, M. Chen, H. Chang, "Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm," Opt. Exp. 15, 5416-5430 (2007).

S. Hsu, H. Chang, "Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3d anisotropy," Opt. Exp. 15, 15 797-15 811 (2007).

2006

J. Beeckman, K. Neyts, X. Hutsebaut, M. Haelterman, "Observation of out-coupling of a nematicon," Opto-Electron. Rev. 14, 263-267 (2006) 12.

R. James, E. Willman, F. A. Fernández, S. E. Day, "Finite-element modeling of liquid-crystal hydrodynamics with a variable degree of order," IEEE Trans. Electron Dev. 53, 1575-1582 (2006).

A. d'Alessandro, B. Bellini, D. Donisi, R. Beccherelli, R. Asquini, "Nematic liquid crystal optical channel waveguides on silicon," IEEE J. Quantum Electron. 42, 1084-1090 (2006).

S. V. Serak, N. V. Tabiryan, M. Peccianti, G. Assanto, "Spatial soliton all-optical logic gates," IEEE Photon. Technol. Lett. 18, 1287-1289 (2006).

J. Beeckman, K. Neyts, M. Haelterman, "Patterned electrode steering of nematicons," J. Opt. A: Pure Appl. Opt. 8, 214-220 (2006).

2005

A. Fratalocchi, R. Asquini, G. Assanto, "Integrated electro-optic switch in liquid crystals," Opt. Exp. 13, 32-36 (2005).

G. Ntogari, D. Tsipouridou, E. Kriezis, "A numerical study of optical switches and modulators based on ferroelectric liquid crystals," J. Opt. A: Pure Appl. Opt. 7, 82-87 (2005).

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, M. Haelterman, "Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect," Opt. Quant. Electron. 35, 95-106 (2005).

H. Desmet, K. Neyts, R. Baets, "Modeling nematic liquid crystals in the neighborhood of edges," J. Appl. Phys. 98, 123517 (2005).

2004

J. Henninot, M. Debailleul, R. Asquini, A. d'Alessandro, M. Warenghem, "Self-waveguiding in an isotropic channel induced in dye doped nematic liquid crystal and a bent self-waveguide," J. Opt. A: Pure Appl. Opt. 6, 315-323 (2004).

C. Gizzi, R. Asquini, A. d'Alessandro, "A polarization independent liquid crystal assisted vertical coupler switch," Mol. Cryst. Liq. Cryst. 421, 95-105 (2004).

2003

A. d'Alessandro, R. Asquini, "Liquid crystal devices for photonic switching applications: State of the art and future developments," Mol. Cryst. Liq. Cryst. 398, 207-221 (2003).

L. Sirleto, G. Coppola, G. Breglio, "Optical multimode interference router based on a liquid crystal waveguide," J. Opt. A: Pure Appl. Opt. 5, S298-S304 (2003).

V. Schulz, "Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides," IEEE Trans. Microw. Theory Tech. 51, 1086-1095 (2003).

2002

F. Bertazzi, O. Peverini, M. Goano, G. Ghione, R. Orta, R. Tascone, "A fast reduced-order model for the full-wave fem analysis of lossy inhomogeneous anisotropic waveguides," IEEE Trans. Microw. Theory Tech. 50, 2108-2114 (2002).

P. Savi, I. Gheorma, R. Graglia, "Full-wave high-order FEM model for lossy anisotropic waveguides," IEEE Trans. Microw. Theory Tech. 50, 495-500 (2002).

R. Asquini, A. d'Alessandro, "Bpm analysis of an integrated optical switch using polymeric optical waveguides and ssflc at 1.55 $\mu{\hbox {m}}$," Mol. Cryst. Liq. Cryst 375, 243-247 (2002).

2001

S. Selleri, L. Vincetti, A. Cucinotta, M. Zoboli, "Complex FEM modal solver of optical waveguides with PML boundary conditions," Opt. Quant. Electron. 33, 359-371 (2001).

F. Tisseur, K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev. 43, 235-286 (2001).

2000

M. Koshiba, Y. Tsuji, "Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems," J. Lightw. Technol. 18, 737-743 (2000).

E. Gros, L. Dupont, "Beam deflector using double-refraction in ferroelectric liquid crystal waveguides," Ferroelectrics 246, 219-226 (2000).

1999

G. Tartarini, H. Renner, "Efficient finite-element analysis of tilted open anisotropic optical channel waveguides," IEEE Trans. Microw. Theory Tech. 9, 389-391 (1999).

1997

L. Nuno, J. Balbastre, H. Castane, "Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite-element method (FEM) using edge elements," IEEE Trans. Microw. Theory Tech. 45, 446-449 (1997).

1996

L. Valor, J. Zapata, "An efficient finite element formulation to analyze waveguides with lossy inhomogeneous bi-anisotropic materials," IEEE Trans. Microw. Theory Tech. 44, 291-296 (1996).

1995

L. Valor, J. Zapata, "Efficient finite element analysis of waveguides with lossy inhomogeneous anisotropic materials characterized by arbitrary permittivity and permeability tensors," IEEE Trans. Microw. Theory Tech. 43, 2452-2459 (1995).

1994

A. F. Peterson, "Vector finite element formulation for scattering from two-dimensional heterogeneous bodies," IEEE Trans. Antenna Propag. 43, 357-365 (1994).

1993

Y. Lu, F. A. Fernandez, "An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides," IEEE Trans. Microw. Theory Tech. 41, 1215-1223 (1993).

1990

J. A. M. Svedin, "Propagation analysis of chirowaveguides using the finite element method," IEEE Trans. Microw. Theory Tech. 38, 1488-1496 (1990).

N. Clark, M. Handshy, "Surface-stabilized ferroelectric liquid-crystal electro-optic waveguide switch," Appl. Phys. Lett. 57, 1852-1854 (1990).

1989

J. A. M. Svedin, "A numerically efficient finite-element formulation for the general waveguide problem without spurious modes," IEEE Trans. Microw. Theory Tech. 37, 1708-1715 (1989).

1987

T. Angkaew, M. Matsuhara, N. Kumagai, "Finite-element analysis of waveguide modes: A novel approach that eliminates spurious modes," IEEE Trans. Microw. Theory Tech. MTT-35, 117-123 (1987).

1974

Y. Satomura, M. Matsuhara, N. Kumagai, "Analysis of electromagnetic-wave modes in anisotropic slab waveguide," IEEE Trans. Microw. Theory Tech. MTT-22, 86-92 (1974).

Appl. Phys. Lett.

N. Clark, M. Handshy, "Surface-stabilized ferroelectric liquid-crystal electro-optic waveguide switch," Appl. Phys. Lett. 57, 1852-1854 (1990).

Ferroelectrics

E. Gros, L. Dupont, "Beam deflector using double-refraction in ferroelectric liquid crystal waveguides," Ferroelectrics 246, 219-226 (2000).

IEEE Trans. Microw. Theory Tech.

F. Bertazzi, O. Peverini, M. Goano, G. Ghione, R. Orta, R. Tascone, "A fast reduced-order model for the full-wave fem analysis of lossy inhomogeneous anisotropic waveguides," IEEE Trans. Microw. Theory Tech. 50, 2108-2114 (2002).

L. Valor, J. Zapata, "Efficient finite element analysis of waveguides with lossy inhomogeneous anisotropic materials characterized by arbitrary permittivity and permeability tensors," IEEE Trans. Microw. Theory Tech. 43, 2452-2459 (1995).

G. Tartarini, H. Renner, "Efficient finite-element analysis of tilted open anisotropic optical channel waveguides," IEEE Trans. Microw. Theory Tech. 9, 389-391 (1999).

IEEE J. Quantum Electron.

A. d'Alessandro, B. Bellini, D. Donisi, R. Beccherelli, R. Asquini, "Nematic liquid crystal optical channel waveguides on silicon," IEEE J. Quantum Electron. 42, 1084-1090 (2006).

IEEE Photon. Technol. Lett.

S. V. Serak, N. V. Tabiryan, M. Peccianti, G. Assanto, "Spatial soliton all-optical logic gates," IEEE Photon. Technol. Lett. 18, 1287-1289 (2006).

IEEE Trans. Microw. Theory Tech.

P. Savi, I. Gheorma, R. Graglia, "Full-wave high-order FEM model for lossy anisotropic waveguides," IEEE Trans. Microw. Theory Tech. 50, 495-500 (2002).

J. A. M. Svedin, "Propagation analysis of chirowaveguides using the finite element method," IEEE Trans. Microw. Theory Tech. 38, 1488-1496 (1990).

Y. Satomura, M. Matsuhara, N. Kumagai, "Analysis of electromagnetic-wave modes in anisotropic slab waveguide," IEEE Trans. Microw. Theory Tech. MTT-22, 86-92 (1974).

IEEE Trans. Antenna Propag.

A. F. Peterson, "Vector finite element formulation for scattering from two-dimensional heterogeneous bodies," IEEE Trans. Antenna Propag. 43, 357-365 (1994).

IEEE Trans. Electron Dev.

R. James, E. Willman, F. A. Fernández, S. E. Day, "Finite-element modeling of liquid-crystal hydrodynamics with a variable degree of order," IEEE Trans. Electron Dev. 53, 1575-1582 (2006).

IEEE Trans. Microw. Theory Tech.

J. A. M. Svedin, "A numerically efficient finite-element formulation for the general waveguide problem without spurious modes," IEEE Trans. Microw. Theory Tech. 37, 1708-1715 (1989).

T. Angkaew, M. Matsuhara, N. Kumagai, "Finite-element analysis of waveguide modes: A novel approach that eliminates spurious modes," IEEE Trans. Microw. Theory Tech. MTT-35, 117-123 (1987).

Y. Lu, F. A. Fernandez, "An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides," IEEE Trans. Microw. Theory Tech. 41, 1215-1223 (1993).

L. Valor, J. Zapata, "An efficient finite element formulation to analyze waveguides with lossy inhomogeneous bi-anisotropic materials," IEEE Trans. Microw. Theory Tech. 44, 291-296 (1996).

L. Nuno, J. Balbastre, H. Castane, "Analysis of general lossy inhomogeneous and anisotropic waveguides by the finite-element method (FEM) using edge elements," IEEE Trans. Microw. Theory Tech. 45, 446-449 (1997).

V. Schulz, "Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides," IEEE Trans. Microw. Theory Tech. 51, 1086-1095 (2003).

J. Lightw. Technol.

M. Koshiba, Y. Tsuji, "Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems," J. Lightw. Technol. 18, 737-743 (2000).

J. Opt. A: Pure Appl. Opt.

L. Sirleto, G. Coppola, G. Breglio, "Optical multimode interference router based on a liquid crystal waveguide," J. Opt. A: Pure Appl. Opt. 5, S298-S304 (2003).

J. Appl. Phys.

H. Desmet, K. Neyts, R. Baets, "Modeling nematic liquid crystals in the neighborhood of edges," J. Appl. Phys. 98, 123517 (2005).

J. Lightw. Technol.

A. Fallahkhair, S. Li, T. Murphy, "Vector finite difference modesolver for anisotropic dielectric waveguides," J. Lightw. Technol. 26, 1423-1431 (2008).

J. Opt. A: Pure Appl. Opt.

G. Ntogari, D. Tsipouridou, E. Kriezis, "A numerical study of optical switches and modulators based on ferroelectric liquid crystals," J. Opt. A: Pure Appl. Opt. 7, 82-87 (2005).

J. Beeckman, K. Neyts, M. Haelterman, "Patterned electrode steering of nematicons," J. Opt. A: Pure Appl. Opt. 8, 214-220 (2006).

J. Henninot, M. Debailleul, R. Asquini, A. d'Alessandro, M. Warenghem, "Self-waveguiding in an isotropic channel induced in dye doped nematic liquid crystal and a bent self-waveguide," J. Opt. A: Pure Appl. Opt. 6, 315-323 (2004).

Mol. Cryst. Liq. Cryst.

C. Gizzi, R. Asquini, A. d'Alessandro, "A polarization independent liquid crystal assisted vertical coupler switch," Mol. Cryst. Liq. Cryst. 421, 95-105 (2004).

Mol. Cryst. Liq. Cryst

R. Asquini, A. d'Alessandro, "Bpm analysis of an integrated optical switch using polymeric optical waveguides and ssflc at 1.55 $\mu{\hbox {m}}$," Mol. Cryst. Liq. Cryst 375, 243-247 (2002).

Mol. Cryst. Liq. Cryst.

A. d'Alessandro, R. Asquini, "Liquid crystal devices for photonic switching applications: State of the art and future developments," Mol. Cryst. Liq. Cryst. 398, 207-221 (2003).

Opt. Quant. Electron.

S. Selleri, L. Vincetti, A. Cucinotta, M. Zoboli, "Complex FEM modal solver of optical waveguides with PML boundary conditions," Opt. Quant. Electron. 33, 359-371 (2001).

Opt. Exp.

A. Fratalocchi, R. Asquini, G. Assanto, "Integrated electro-optic switch in liquid crystals," Opt. Exp. 13, 32-36 (2005).

S. Hsu, H. Chang, "Full-vectorial finite element method based eigenvalue algorithm for the analysis of 2D photonic crystals with arbitrary 3d anisotropy," Opt. Exp. 15, 15 797-15 811 (2007).

S. Hsu, M. Chen, H. Chang, "Investigation of band structures for 2D non-diagonal anisotropic photonic crystals using a finite element method based eigenvalue algorithm," Opt. Exp. 15, 5416-5430 (2007).

Opt. Quant. Electron.

J. Beeckman, K. Neyts, X. Hutsebaut, C. Cambournac, M. Haelterman, "Simulation of 2-D lateral light propagation in nematic-liquid-crystal cells with tilted molecules and nonlinear reorientational effect," Opt. Quant. Electron. 35, 95-106 (2005).

Opto-Electron. Rev.

J. Beeckman, K. Neyts, X. Hutsebaut, M. Haelterman, "Observation of out-coupling of a nematicon," Opto-Electron. Rev. 14, 263-267 (2006) 12.

SIAM Rev.

F. Tisseur, K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev. 43, 235-286 (2001).

Other

P. Yeh, C. Gu, Optics of Liquid Crystal Displays (Wiley-Interscience, 1999).

V. Chigrinov, V. Kozenkov, H. Kwok, Photoalignment of Liquid Crystalline Materials (Wiley, 2008).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

P. G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford Univ. Press, 1995).

H. Desmet, W. Bogaerts, A. Adamski, J. Beeckman, K. Neyts, R. Baets, "Silicon-on-insulator optical waveguides with liquid crystal cladding for switching and tuning," Proc. ECOC (2003) pp. 430-431.

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