Abstract

We present a simulation method for light emitted in uniaxially anisotropic light-emitting thin film devices. The simulation is based on the radiation of dipole antennas inside a one-dimensional microcavity. Any layer in the microcaviy can be uniaxially anisotropic with an arbitrary orientation of the optical axis. A plane wave expansion for the field of an elementary dipole inside an anisotropic medium is derived from Maxwell’s equations. We employ the scattering matrix method to calculate the emission by dipoles inside an anisotropic microcavity. The simulation method is applied to calculate the emission of dipole antennas in a number of cases: a dipole antenna in an infinite medium, emission into anisotropic slab waveguides and waveguides in liquid crystals. The dependency of the intensity and the polarization on the direction of emission is illustrated for a number of anisotropic microcavities.

© 2011 OSA

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  5. K. Driesen, D. Moors, J. Beeckman, K. Neyts, C. Gorller-Walrand, and K. Binnemans, “Near-infrared luminescence emitted by an electrically switched liquid crystal cell,” J. Lumin. 127(2), 611–615 (2007).
    [CrossRef]
  6. M. O'Neill and S. M. Kelly, “Liquid crystals for charge transport, luminescence, and photonics,” Adv. Mater. (Deerfield Beach Fla.) 15(14), 1135–1146 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. T. Setälä, M. Kaivola, and A. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(1), 1200–1206 (1999).
    [CrossRef]
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    [CrossRef]

2010

2009

W. De Cort, J. Beeckman, R. James, F. A. Fernández, R. Baets, and K. Neyts, “Tuning of silicon-on-insulator ring resonators with liquid crystal cladding using the longitudinal field component,” Opt. Lett. 34(13), 2054–2056 (2009).
[CrossRef] [PubMed]

P. De Visschere, “Electromagnetic source transformations and scalarization in stratified gyrotropic media,” Prog. Electromag. Res. B 18, 165–183 (2009).
[CrossRef]

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[CrossRef] [PubMed]

D. Yokoyama, A. Sakaguchi, M. Suzuki, and C. Adachi, “Horizontal orientation of linear-shaped organic molecules having bulky substituents in neat and doped vacuum-deposited amorphous films,” Org. Electron. 10(1), 127–137 (2009).
[CrossRef]

2007

K. Driesen, D. Moors, J. Beeckman, K. Neyts, C. Gorller-Walrand, and K. Binnemans, “Near-infrared luminescence emitted by an electrically switched liquid crystal cell,” J. Lumin. 127(2), 611–615 (2007).
[CrossRef]

P. Bienstman, P. Vandersteegen, and R. Baets, “Modelling gratings on either side of the substrate for light extraction in light-emitting diodes,” Opt. Quantum Electron. 39(10-11), 797–804 (2007).
[CrossRef]

2006

T. Suzuki, “Flat panel displays for ubiquitous product applications and related impurity doping technologies,” J. Appl. Phys. 99(11), 111101 (2006).
[CrossRef]

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73(16), 165125 (2006).
[CrossRef]

J. Beeckman, K. Neyts, and M. Haelterman, “Patterned electrode steering of nematicons,” J. Opt. A, Pure Appl. Opt. 8(2), 214–220 (2006).
[CrossRef]

2005

M. P. Aldred, A. E. A. Contoret, S. R. Farrar, S. M. Kelly, D. Mathieson, M. O'Neill, W. C. Tsoi, and P. Vlachos, “A full-color electroluminescent device and patterned photoalignment using light-emitting liquid crystals,” Adv. Mater. (Deerfield Beach Fla.) 17(11), 1368–1372 (2005).
[CrossRef]

2004

P. Bermel, J. D. Joannopoulos, Y. Fink, P. A. Lane, and C. Tapalian, “Properties of radiating pointlike sources in cylindrical omnidirectionally reflecting waveguides,” Phys. Rev. B 69(3), 035316 (2004).
[CrossRef]

2003

M. O'Neill and S. M. Kelly, “Liquid crystals for charge transport, luminescence, and photonics,” Adv. Mater. (Deerfield Beach Fla.) 15(14), 1135–1146 (2003).
[CrossRef]

2000

W. M. V. Wan, R. H. Friend, and N. C. Greenham, “Modelling of interference effects in anisotropic conjugated polymer devices,” Thin Solid Films 363(1-2), 310–313 (2000).
[CrossRef]

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, “Effects of dipole orientation and birefringence on the optical emission from thin films,” Opt. Commun. 183(1-4), 109–121 (2000).
[CrossRef]

1999

T. Setälä, M. Kaivola, and A. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(1), 1200–1206 (1999).
[CrossRef]

1998

1988

1981

1980

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin-layers,” Phys. Rev. B 22(6), 3030–3038 (1980).
[CrossRef]

1972

1963

P. C. Clemmow, “The theory of electromagnetic waves in a simple anisotropic medium,” Electr. Engin. Proc. Inst. 110, 101–106 (1963).
[CrossRef]

Adachi, C.

D. Yokoyama, A. Sakaguchi, M. Suzuki, and C. Adachi, “Horizontal orientation of linear-shaped organic molecules having bulky substituents in neat and doped vacuum-deposited amorphous films,” Org. Electron. 10(1), 127–137 (2009).
[CrossRef]

Aldred, M. P.

M. P. Aldred, A. E. A. Contoret, S. R. Farrar, S. M. Kelly, D. Mathieson, M. O'Neill, W. C. Tsoi, and P. Vlachos, “A full-color electroluminescent device and patterned photoalignment using light-emitting liquid crystals,” Adv. Mater. (Deerfield Beach Fla.) 17(11), 1368–1372 (2005).
[CrossRef]

Baets, R.

W. De Cort, J. Beeckman, R. James, F. A. Fernández, R. Baets, and K. Neyts, “Tuning of silicon-on-insulator ring resonators with liquid crystal cladding using the longitudinal field component,” Opt. Lett. 34(13), 2054–2056 (2009).
[CrossRef] [PubMed]

P. Bienstman, P. Vandersteegen, and R. Baets, “Modelling gratings on either side of the substrate for light extraction in light-emitting diodes,” Opt. Quantum Electron. 39(10-11), 797–804 (2007).
[CrossRef]

Baldo, M. A.

Barnes, W. L.

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, “Effects of dipole orientation and birefringence on the optical emission from thin films,” Opt. Commun. 183(1-4), 109–121 (2000).
[CrossRef]

Beeckman, J.

W. De Cort, J. Beeckman, R. James, F. A. Fernández, R. Baets, and K. Neyts, “Tuning of silicon-on-insulator ring resonators with liquid crystal cladding using the longitudinal field component,” Opt. Lett. 34(13), 2054–2056 (2009).
[CrossRef] [PubMed]

K. Driesen, D. Moors, J. Beeckman, K. Neyts, C. Gorller-Walrand, and K. Binnemans, “Near-infrared luminescence emitted by an electrically switched liquid crystal cell,” J. Lumin. 127(2), 611–615 (2007).
[CrossRef]

J. Beeckman, K. Neyts, and M. Haelterman, “Patterned electrode steering of nematicons,” J. Opt. A, Pure Appl. Opt. 8(2), 214–220 (2006).
[CrossRef]

Bermel, P.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73(16), 165125 (2006).
[CrossRef]

P. Bermel, J. D. Joannopoulos, Y. Fink, P. A. Lane, and C. Tapalian, “Properties of radiating pointlike sources in cylindrical omnidirectionally reflecting waveguides,” Phys. Rev. B 69(3), 035316 (2004).
[CrossRef]

Berreman, D. W.

Bienstman, P.

P. Bienstman, P. Vandersteegen, and R. Baets, “Modelling gratings on either side of the substrate for light extraction in light-emitting diodes,” Opt. Quantum Electron. 39(10-11), 797–804 (2007).
[CrossRef]

Binnemans, K.

K. Driesen, D. Moors, J. Beeckman, K. Neyts, C. Gorller-Walrand, and K. Binnemans, “Near-infrared luminescence emitted by an electrically switched liquid crystal cell,” J. Lumin. 127(2), 611–615 (2007).
[CrossRef]

Clemmow, P. C.

P. C. Clemmow, “The theory of electromagnetic waves in a simple anisotropic medium,” Electr. Engin. Proc. Inst. 110, 101–106 (1963).
[CrossRef]

Coles, H.

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4(10), 676–685 (2010).
[CrossRef]

Contoret, A. E. A.

M. P. Aldred, A. E. A. Contoret, S. R. Farrar, S. M. Kelly, D. Mathieson, M. O'Neill, W. C. Tsoi, and P. Vlachos, “A full-color electroluminescent device and patterned photoalignment using light-emitting liquid crystals,” Adv. Mater. (Deerfield Beach Fla.) 17(11), 1368–1372 (2005).
[CrossRef]

De Cort, W.

De Visschere, P.

P. De Visschere, “Electromagnetic source transformations and scalarization in stratified gyrotropic media,” Prog. Electromag. Res. B 18, 165–183 (2009).
[CrossRef]

Driesen, K.

K. Driesen, D. Moors, J. Beeckman, K. Neyts, C. Gorller-Walrand, and K. Binnemans, “Near-infrared luminescence emitted by an electrically switched liquid crystal cell,” J. Lumin. 127(2), 611–615 (2007).
[CrossRef]

Farrar, S. R.

M. P. Aldred, A. E. A. Contoret, S. R. Farrar, S. M. Kelly, D. Mathieson, M. O'Neill, W. C. Tsoi, and P. Vlachos, “A full-color electroluminescent device and patterned photoalignment using light-emitting liquid crystals,” Adv. Mater. (Deerfield Beach Fla.) 17(11), 1368–1372 (2005).
[CrossRef]

Fernández, F. A.

Fink, Y.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73(16), 165125 (2006).
[CrossRef]

P. Bermel, J. D. Joannopoulos, Y. Fink, P. A. Lane, and C. Tapalian, “Properties of radiating pointlike sources in cylindrical omnidirectionally reflecting waveguides,” Phys. Rev. B 69(3), 035316 (2004).
[CrossRef]

Friberg, A.

T. Setälä, M. Kaivola, and A. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(1), 1200–1206 (1999).
[CrossRef]

Friend, R. H.

W. M. V. Wan, R. H. Friend, and N. C. Greenham, “Modelling of interference effects in anisotropic conjugated polymer devices,” Thin Solid Films 363(1-2), 310–313 (2000).
[CrossRef]

Gorller-Walrand, C.

K. Driesen, D. Moors, J. Beeckman, K. Neyts, C. Gorller-Walrand, and K. Binnemans, “Near-infrared luminescence emitted by an electrically switched liquid crystal cell,” J. Lumin. 127(2), 611–615 (2007).
[CrossRef]

Greenham, N. C.

W. M. V. Wan, R. H. Friend, and N. C. Greenham, “Modelling of interference effects in anisotropic conjugated polymer devices,” Thin Solid Films 363(1-2), 310–313 (2000).
[CrossRef]

Haelterman, M.

J. Beeckman, K. Neyts, and M. Haelterman, “Patterned electrode steering of nematicons,” J. Opt. A, Pure Appl. Opt. 8(2), 214–220 (2006).
[CrossRef]

James, R.

Joannopoulos, J. D.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73(16), 165125 (2006).
[CrossRef]

P. Bermel, J. D. Joannopoulos, Y. Fink, P. A. Lane, and C. Tapalian, “Properties of radiating pointlike sources in cylindrical omnidirectionally reflecting waveguides,” Phys. Rev. B 69(3), 035316 (2004).
[CrossRef]

Kaivola, M.

T. Setälä, M. Kaivola, and A. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(1), 1200–1206 (1999).
[CrossRef]

Kelly, S. M.

M. P. Aldred, A. E. A. Contoret, S. R. Farrar, S. M. Kelly, D. Mathieson, M. O'Neill, W. C. Tsoi, and P. Vlachos, “A full-color electroluminescent device and patterned photoalignment using light-emitting liquid crystals,” Adv. Mater. (Deerfield Beach Fla.) 17(11), 1368–1372 (2005).
[CrossRef]

M. O'Neill and S. M. Kelly, “Liquid crystals for charge transport, luminescence, and photonics,” Adv. Mater. (Deerfield Beach Fla.) 15(14), 1135–1146 (2003).
[CrossRef]

Kim, H.

Ko, D. Y. K.

Lane, P. A.

P. Bermel, J. D. Joannopoulos, Y. Fink, P. A. Lane, and C. Tapalian, “Properties of radiating pointlike sources in cylindrical omnidirectionally reflecting waveguides,” Phys. Rev. B 69(3), 035316 (2004).
[CrossRef]

Leo, K.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[CrossRef] [PubMed]

Lidorikis, E.

P. Bermel, E. Lidorikis, Y. Fink, and J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73(16), 165125 (2006).
[CrossRef]

Lindner, F.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[CrossRef] [PubMed]

Lukosz, W.

W. Lukosz, “Light-emission by multipole sources in thin-layers. 1. Radiation-patterns of electric and magnetic dipoles,” J. Opt. Soc. Am. 71(6), 744–754 (1981).
[CrossRef]

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin-layers,” Phys. Rev. B 22(6), 3030–3038 (1980).
[CrossRef]

Lüssem, B.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[CrossRef] [PubMed]

Mathieson, D.

M. P. Aldred, A. E. A. Contoret, S. R. Farrar, S. M. Kelly, D. Mathieson, M. O'Neill, W. C. Tsoi, and P. Vlachos, “A full-color electroluminescent device and patterned photoalignment using light-emitting liquid crystals,” Adv. Mater. (Deerfield Beach Fla.) 17(11), 1368–1372 (2005).
[CrossRef]

Moors, D.

K. Driesen, D. Moors, J. Beeckman, K. Neyts, C. Gorller-Walrand, and K. Binnemans, “Near-infrared luminescence emitted by an electrically switched liquid crystal cell,” J. Lumin. 127(2), 611–615 (2007).
[CrossRef]

Morris, S.

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photonics 4(10), 676–685 (2010).
[CrossRef]

Mulder, C. L.

Neyts, K.

W. De Cort, J. Beeckman, R. James, F. A. Fernández, R. Baets, and K. Neyts, “Tuning of silicon-on-insulator ring resonators with liquid crystal cladding using the longitudinal field component,” Opt. Lett. 34(13), 2054–2056 (2009).
[CrossRef] [PubMed]

K. Driesen, D. Moors, J. Beeckman, K. Neyts, C. Gorller-Walrand, and K. Binnemans, “Near-infrared luminescence emitted by an electrically switched liquid crystal cell,” J. Lumin. 127(2), 611–615 (2007).
[CrossRef]

J. Beeckman, K. Neyts, and M. Haelterman, “Patterned electrode steering of nematicons,” J. Opt. A, Pure Appl. Opt. 8(2), 214–220 (2006).
[CrossRef]

K. Neyts, “Simulation of light emission from thin-film microcavities,” J. Opt. Soc. Am. A 15(4), 962–971 (1998).
[CrossRef]

O'Neill, M.

M. P. Aldred, A. E. A. Contoret, S. R. Farrar, S. M. Kelly, D. Mathieson, M. O'Neill, W. C. Tsoi, and P. Vlachos, “A full-color electroluminescent device and patterned photoalignment using light-emitting liquid crystals,” Adv. Mater. (Deerfield Beach Fla.) 17(11), 1368–1372 (2005).
[CrossRef]

M. O'Neill and S. M. Kelly, “Liquid crystals for charge transport, luminescence, and photonics,” Adv. Mater. (Deerfield Beach Fla.) 15(14), 1135–1146 (2003).
[CrossRef]

Reineke, S.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[CrossRef] [PubMed]

Reusswig, P. D.

Rotschild, C.

Safonov, A.

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, “Effects of dipole orientation and birefringence on the optical emission from thin films,” Opt. Commun. 183(1-4), 109–121 (2000).
[CrossRef]

Sakaguchi, A.

D. Yokoyama, A. Sakaguchi, M. Suzuki, and C. Adachi, “Horizontal orientation of linear-shaped organic molecules having bulky substituents in neat and doped vacuum-deposited amorphous films,” Org. Electron. 10(1), 127–137 (2009).
[CrossRef]

Sambles, J. R.

Samuel, I. D. W.

J. A. E. Wasey, A. Safonov, I. D. W. Samuel, and W. L. Barnes, “Effects of dipole orientation and birefringence on the optical emission from thin films,” Opt. Commun. 183(1-4), 109–121 (2000).
[CrossRef]

Schwartz, G.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[CrossRef] [PubMed]

Seidler, N.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[CrossRef] [PubMed]

Setälä, T.

T. Setälä, M. Kaivola, and A. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(1), 1200–1206 (1999).
[CrossRef]

Suzuki, M.

D. Yokoyama, A. Sakaguchi, M. Suzuki, and C. Adachi, “Horizontal orientation of linear-shaped organic molecules having bulky substituents in neat and doped vacuum-deposited amorphous films,” Org. Electron. 10(1), 127–137 (2009).
[CrossRef]

Suzuki, T.

T. Suzuki, “Flat panel displays for ubiquitous product applications and related impurity doping technologies,” J. Appl. Phys. 99(11), 111101 (2006).
[CrossRef]

Tapalian, C.

P. Bermel, J. D. Joannopoulos, Y. Fink, P. A. Lane, and C. Tapalian, “Properties of radiating pointlike sources in cylindrical omnidirectionally reflecting waveguides,” Phys. Rev. B 69(3), 035316 (2004).
[CrossRef]

Tsoi, W. C.

M. P. Aldred, A. E. A. Contoret, S. R. Farrar, S. M. Kelly, D. Mathieson, M. O'Neill, W. C. Tsoi, and P. Vlachos, “A full-color electroluminescent device and patterned photoalignment using light-emitting liquid crystals,” Adv. Mater. (Deerfield Beach Fla.) 17(11), 1368–1372 (2005).
[CrossRef]

Vandersteegen, P.

P. Bienstman, P. Vandersteegen, and R. Baets, “Modelling gratings on either side of the substrate for light extraction in light-emitting diodes,” Opt. Quantum Electron. 39(10-11), 797–804 (2007).
[CrossRef]

Velázquez, A. M.

Vlachos, P.

M. P. Aldred, A. E. A. Contoret, S. R. Farrar, S. M. Kelly, D. Mathieson, M. O'Neill, W. C. Tsoi, and P. Vlachos, “A full-color electroluminescent device and patterned photoalignment using light-emitting liquid crystals,” Adv. Mater. (Deerfield Beach Fla.) 17(11), 1368–1372 (2005).
[CrossRef]

Walzer, K.

S. Reineke, F. Lindner, G. Schwartz, N. Seidler, K. Walzer, B. Lüssem, and K. Leo, “White organic light-emitting diodes with fluorescent tube efficiency,” Nature 459(7244), 234–238 (2009).
[CrossRef] [PubMed]

Wan, W. M. V.

W. M. V. Wan, R. H. Friend, and N. C. Greenham, “Modelling of interference effects in anisotropic conjugated polymer devices,” Thin Solid Films 363(1-2), 310–313 (2000).
[CrossRef]

Wasey, J. A. E.

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Figures (15)

Fig. 1
Fig. 1

Coordinate system and definition of inclination and azimuth angles.

Fig. 2
Fig. 2

A stack of anisotropic films with a dipole antenna p leads to interference with reflections of the cavity interfaces above and below the dipole.

Fig. 3
Fig. 3

Sketch of the input-output relation of the scattering matrices. S i N ¯ ¯ is constructed by step by step addition of layers to S i i ¯ ¯ .

Fig. 4
Fig. 4

Sketch of the input and output waves of the matrix I j ¯ ¯ . The phase propagation matrix D j ¯ ¯ and the matrices B j ¯ ¯ and B j + 1 ¯ ¯ relating the waves in medium j to the waves in medium j+1.

Fig. 5
Fig. 5

Simulated configuration: a dipole embedded in a thin isotropic layer between two semi-infinite anisotropic layers with c x .

Fig. 6
Fig. 6

Simulated total emitted power into the anisotropic medium for a dipole in the middle of an isotropic film with thickness d. Solid lines: anisotropic plane wave expansion in a homogeneous medium. Dashed lines: isotropic plane wave expansion in a layer as a function of the layer thickness.

Fig. 7
Fig. 7

Optically thick glass substrate with thin emitting layer, anisotropic film and aluminum.

Fig. 8
Fig. 8

K + vs. κ / k 0 for the isotropic reference (see Fig. 7) . a) Complete κ / k 0 region b) close-up of waveguiding region.

Fig. 9
Fig. 9

K + vs. κ / k 0 for the anisotropic structure with c z . a) Complete κ / k 0 region b) close-up of waveguiding region.

Fig. 10
Fig. 10

K o + vs. κ / k 0 for 3 azimuth directions for o-polarized waves in the anisotropic structure with c x . a) Complete κ / k 0 region b) close-up of waveguiding region.

Fig. 11
Fig. 11

K e + vs. κ / k 0 for 3 azimuth directions for e-polarized waves in the anisotropic structure with c x . a) Complete κ / k 0 region b) close-up of waveguiding region.

Fig. 12
Fig. 12

A stack thin anisotropic layers enclosed by glass and air. Because of the different orientations of the c-axis coupling occurs between the ordinary and extra-ordinary waves.

Fig. 13
Fig. 13

K o + and K e + vs. κ / k 0 due to coupling of e- and o-waves there is a negative flux of o-waves b) K + vs. κ / k 0 , the total flux remains positive.

Fig. 14
Fig. 14

Orientation of the liquid crystal molecules in a one dimensional waveguide.

Fig. 15
Fig. 15

a) K + vs. κ / k 0 in one dimensional LC waveguide (Gaussian distribution of the liquid crystal inclination angle α with FWHM=1177nm). b) Effective indices of the e-polarized modes vs. FWHM of the Gaussian distribution.

Equations (45)

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E = E 0 e x p ( j ω t j k x x j k y y j k z z )
k = ( k x k y k z ) = κ 1 κ + k z 1 z = k 1 + k 1 c
2 Θ + 2 Θ c 2 μ ε 2 Θ t 2 = μ v t
2 Ψ ε ε 2 Ψ c 2 + μ ε 2 Ψ t 2 = ε ε u c + J
J = p t δ ( r )
2 u = · p t δ ( r )
2 v = c ( × p t δ ( r ) )
Θ = j μ ω 2 k 2 c · ( k × p ) k 2 k 2 + μ ε ω 2
Ψ = j ω k k 2 k · p j ε ε ω p ε ε k 2 k 2 + μ ε ω 2
Π = k μ ω Θ
Φ = 1 ε ω ( j u k Ψ )
E = k ε ω ( u + j k Ψ ) j k × Θ c j μ ω Ψ c
E ( k ) = k ε ( k · p ) k ( k · p ) μ ω 2 k 2 μ ε ε ω 2 p c ε ε k 2 k c 2 + μ ε ω 2 j k × j μ ω 2 k 2 c · ( k × p ) k 2 k 2 + μ ε ω 2 c
E ( x , y , z ) = 1 ( 2 π ) 3 + + + d k x d k y d k z E ( k ) exp [ j ( k x x + k y y + k z z ) ]
k z = μ ε ε [ ε + Δ ε c z 2 ] ω 2 κ 2 ε [ ε + Δ ε ( c z 2 + c t 2 ) ] ε + Δ ε c z 2 Δ ε c t c z κ ε + Δ ε c z 2
κ c r i t = μ ε ω 2 ε + Δ ε c z 2 ε + Δ ε ( c z 2 + c t 2 )
k z = ± μ ε ω 2 κ 2
k z , e , + = μ ε ε [ ε + Δ ε c z 2 ] ω 2 κ 2 ε [ ε + Δ ε ( c z 2 + c t 2 ) ] ε + Δ ε c z 2 Δ ε c t c z κ ε + Δ ε c z 2
k z , o , + = μ ε ω 2 κ 2
E o = j 8 π 2 + + d k x d k y [ k × μ ω 2 k 2 c · ( k × p ) k z , o , + c ] exp [ j ( k x x + k y y + k z , o , + z ) ]
E e = j 8 π 2 + + d k x d k y ( k p ε k 2 ( k · p ) k 2 ε ) k k 2 ε k 2 ε k ε + Δ ε c z 2 ε ( k z , e , + + Δ ε c t c z κ ε + Δ ε c z 2 ) × exp [ j ( k x x + k y y + k z , e , + z ) ]
S z = 1 2 Re [ E × H * ] z
+ + S z d x d y = + + 2 π 2 R e [ E ( k x , k y ) × H ( k x , k y ) * ] z d k x d k y = + + K ( k x , k y ) d k x d k y
F = + + [ K o + + K e + + K o + K e ] d k x d k y
1 o = k × c | k × c |
1 e = k k 2 ε k ε c | k k 2 ε k ε c |
E c a v , T E / T M + = E , T E / T M + + A T E / T M E , T E / T M 1 A T E / T M + A T E / T M
E c a v , T E / T M = E , T E / T M + A T E / T M + E , T E / T M + 1 A T E / T M A T E / T M +
[ E c a v , o + E c a v , e + ] = ( 1 A ¯ ¯ A + ¯ ¯ ) 1 ( [ E , o + E , e + ] + A ¯ ¯ [ E , o E , e ] )
[ E c a v , o E c a v , e ] = ( 1 A + ¯ ¯ A ¯ ¯ ) 1 ( [ E , o E , e ] + A + ¯ ¯ [ E , o + E , e + ] )
A ± ¯ ¯ = [ A o o ± A e o ± A o e ± A e e ± ]
( E o , N , + E e , N , + E o , i , E e , i , ) = [ S i N , 11 ¯ ¯ S i N , 12 ¯ ¯ S i N , 21 ¯ ¯ S i N , 22 ¯ ¯ ] ( E o , i , + E e , i , + E o , N , E e , N , )
T + ¯ ¯ = S i N , 11 ¯ ¯
A + ¯ ¯ = S i N , 21 ¯ ¯
[ E o u t , o ± E o u t , e ± ] = T ± ¯ ¯ [ E c a v , o ± E c a v , e ± ]
( E o , j + 1 , + E e , j + 1 , + E o , j + 1 , E e , j + 1 , ) = I j ¯ ¯ ( E o , j , + E e , j , + E o , j , E e , j , )
S i j + 1 , 11 ¯ ¯ = ( I j ¯ ¯ 1 11 S i j , 12 ¯ ¯ I j ¯ ¯ 1 21 ) 1 S i , j , 11 ¯ ¯
S i j + 1 , 12 ¯ ¯ = ( I j ¯ ¯ 1 11 S i j , 12 ¯ ¯ I j ¯ ¯ 1 21 ) 1 ( S i , j , 12 ¯ ¯ I j , 22 ¯ ¯ 1 I j ¯ ¯ 1 12 )
S i j + 1 , 21 ¯ ¯ = S i j , 22 ¯ ¯ I j ¯ ¯ 1 21 S i j + 1 , 11 ¯ ¯ + S i j , 21 ¯ ¯
S i j + 1 , 22 ¯ ¯ = S i j , 22 ¯ ¯ I j ¯ ¯ 1 21 S i j + 1 , 12 ¯ ¯ + S i j , 22 ¯ ¯ I j ¯ ¯ 1 22
D j ¯ ¯ = exp ( j d j [ k z , o , j , + 0 0 0 0 k z , e , j , + 0 0 0 0 k z , o , j , 0 0 0 0 k z , e , j , ] )
( E x E y H x H y ) = [ 1 o , + , j + 1 , x 1 e , + , j + 1 , x 1 o , , j + 1 , x 1 e , , j + 1 , x 1 o , + , j + 1 , y 1 e , + , j + 1 , y 1 o , , j + 1 , y 1 e , , j + 1 , y Y o , + , j + 1 , x Y e , + , j + 1 , x Y o , , j + 1 , x Y e , , j + 1 , x Y o , + , j + 1 , y Y e , + , j + 1 , y Y o , , j + 1 , y Y e , , j + 1 , y ] ( E o , j + 1 , + E e , j + 1 , + E o , j + 1 , E e , j + 1 , ) = B j + 1 ¯ ¯ ( E o , j + 1 , + E e , j + 1 , + E o , j + 1 , E e , j + 1 , )
B j + 1 ¯ ¯ I j ¯ ¯ ( E o , j , + E e , j , + E o , j , E e , j , ) = ( E x , j + 1 E y , j + 1 H x , j + 1 H y , j + 1 ) = ( E x , j E y , j H x , j H y , j ) = B j ¯ ¯ D j ¯ ¯ ( E o , j , + E e , j , + E o , j , E e , j , )
I j ¯ ¯ = B j + 1 ¯ ¯ 1 B j ¯ ¯ D j ¯ ¯
F = n o ( 1 + n e 2 n o 2 4 n o 2 sin 2 ρ )

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