Abstract

Optical anisotropy plays a fundamental role on light propagation in nematic liquid crystals. With specific reference to nematicons, we investigate the transverse dynamics due to the interplay of nonlinear self-confinement, birefringent walk-off and a bias-dependent transverse index profile.

© 2004 Optical Society of America

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References

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  1. N. V. Tabiryan, A.V. Sukhov, and B. Ya. Zel’dovich, “Orientational Optical Nonlinearity of Liquid Crystals,” Mol. Cryst. Liq. Cryst. 136, 1–131 (1986)
    [CrossRef]
  2. I. C. Khoo, Liquid Crystals: Physical Properties and Optical Phenomena (Whiley & Sons, New York, 1995)
  3. F. Simoni, Nonlinear Optical Properties of Liquid Crystals, (World Scientific, London, 1997)
    [CrossRef]
  4. R. Asquini and A. d’Alessandro, “BPM Analysis of an integrated optical switch using polymeric optical waveguides and SSFLC at 1.55 µm,” Mol. Cryst. Liq. Cryst. 375, 243–247 (2002)
    [CrossRef]
  5. G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003)
    [CrossRef]
  6. G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical Spatial Solitons in Nematic Liquid Crystals,” Optics Photon. News 14, 44–48 (2003)
    [CrossRef]
  7. X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Comm. 233, 211–217 (2004)
    [CrossRef]
  8. C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91, 73901 (2003)
    [CrossRef]
  9. C. Conti, M. Peccianti, and G. Assanto, “Observation of Optical Spatial Solitons in a Highly Nonlocal Medium,” Phys. Rev. Lett. 92, 113902 (2004)
    [CrossRef] [PubMed]
  10. M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003)
    [CrossRef]
  11. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518 (1999)
    [CrossRef] [PubMed]
  12. S. Trillo and W. E Torruellas, Spatial Solitons (Springer, Berlin, 2001)
  13. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, London, 2003)
  14. E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993)
    [CrossRef] [PubMed]
  15. J. F. Henninot, M. Debailleul, F. Derrien, G. Abbate, and M. Warenghem, “(2D+1) Spatial optical solitons in dye doped liquid crystals,” Synth. Met. 8915, 1–5 (2001)
  16. J. F. Henninot, M. Debailleul, and M. Warenghem, “Tunable non-locality of thermal non-linearity in dye doped nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 375, 1538–1547 (2002)
    [CrossRef]
  17. M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000)
    [CrossRef]
  18. CRC Handbook of Laser Science and Technology: Optical Materials, Suppl. 2, (ed. M. J. Weber, CRC Press, New York, 1995).
  19. M. Peccianti and G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. 26, 1690–1692 (2001)
    [CrossRef]
  20. M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002)
    [CrossRef]

2004 (2)

C. Conti, M. Peccianti, and G. Assanto, “Observation of Optical Spatial Solitons in a Highly Nonlocal Medium,” Phys. Rev. Lett. 92, 113902 (2004)
[CrossRef] [PubMed]

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Comm. 233, 211–217 (2004)
[CrossRef]

2003 (4)

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91, 73901 (2003)
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003)
[CrossRef]

G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003)
[CrossRef]

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical Spatial Solitons in Nematic Liquid Crystals,” Optics Photon. News 14, 44–48 (2003)
[CrossRef]

2002 (3)

R. Asquini and A. d’Alessandro, “BPM Analysis of an integrated optical switch using polymeric optical waveguides and SSFLC at 1.55 µm,” Mol. Cryst. Liq. Cryst. 375, 243–247 (2002)
[CrossRef]

J. F. Henninot, M. Debailleul, and M. Warenghem, “Tunable non-locality of thermal non-linearity in dye doped nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 375, 1538–1547 (2002)
[CrossRef]

M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002)
[CrossRef]

2001 (2)

J. F. Henninot, M. Debailleul, F. Derrien, G. Abbate, and M. Warenghem, “(2D+1) Spatial optical solitons in dye doped liquid crystals,” Synth. Met. 8915, 1–5 (2001)

M. Peccianti and G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. 26, 1690–1692 (2001)
[CrossRef]

2000 (1)

M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000)
[CrossRef]

1999 (1)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518 (1999)
[CrossRef] [PubMed]

1993 (1)

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993)
[CrossRef] [PubMed]

1986 (1)

N. V. Tabiryan, A.V. Sukhov, and B. Ya. Zel’dovich, “Orientational Optical Nonlinearity of Liquid Crystals,” Mol. Cryst. Liq. Cryst. 136, 1–131 (1986)
[CrossRef]

Abbate, G.

J. F. Henninot, M. Debailleul, F. Derrien, G. Abbate, and M. Warenghem, “(2D+1) Spatial optical solitons in dye doped liquid crystals,” Synth. Met. 8915, 1–5 (2001)

Adamski, A.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Comm. 233, 211–217 (2004)
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, London, 2003)

Asquini, R.

R. Asquini and A. d’Alessandro, “BPM Analysis of an integrated optical switch using polymeric optical waveguides and SSFLC at 1.55 µm,” Mol. Cryst. Liq. Cryst. 375, 243–247 (2002)
[CrossRef]

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, “Observation of Optical Spatial Solitons in a Highly Nonlocal Medium,” Phys. Rev. Lett. 92, 113902 (2004)
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91, 73901 (2003)
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003)
[CrossRef]

G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003)
[CrossRef]

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical Spatial Solitons in Nematic Liquid Crystals,” Optics Photon. News 14, 44–48 (2003)
[CrossRef]

M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002)
[CrossRef]

M. Peccianti and G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. 26, 1690–1692 (2001)
[CrossRef]

M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000)
[CrossRef]

Braun, E.

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993)
[CrossRef] [PubMed]

Brzdkiewicz, K. A.

Cambournac, C.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Comm. 233, 211–217 (2004)
[CrossRef]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, “Observation of Optical Spatial Solitons in a Highly Nonlocal Medium,” Phys. Rev. Lett. 92, 113902 (2004)
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91, 73901 (2003)
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003)
[CrossRef]

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical Spatial Solitons in Nematic Liquid Crystals,” Optics Photon. News 14, 44–48 (2003)
[CrossRef]

d’Alessandro, A.

R. Asquini and A. d’Alessandro, “BPM Analysis of an integrated optical switch using polymeric optical waveguides and SSFLC at 1.55 µm,” Mol. Cryst. Liq. Cryst. 375, 243–247 (2002)
[CrossRef]

De Luca, A.

M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000)
[CrossRef]

Debailleul, M.

J. F. Henninot, M. Debailleul, and M. Warenghem, “Tunable non-locality of thermal non-linearity in dye doped nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 375, 1538–1547 (2002)
[CrossRef]

J. F. Henninot, M. Debailleul, F. Derrien, G. Abbate, and M. Warenghem, “(2D+1) Spatial optical solitons in dye doped liquid crystals,” Synth. Met. 8915, 1–5 (2001)

Derrien, F.

J. F. Henninot, M. Debailleul, F. Derrien, G. Abbate, and M. Warenghem, “(2D+1) Spatial optical solitons in dye doped liquid crystals,” Synth. Met. 8915, 1–5 (2001)

Faucheux, L. P.

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993)
[CrossRef] [PubMed]

Haelterman, M.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Comm. 233, 211–217 (2004)
[CrossRef]

Henninot, J. F.

J. F. Henninot, M. Debailleul, and M. Warenghem, “Tunable non-locality of thermal non-linearity in dye doped nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 375, 1538–1547 (2002)
[CrossRef]

J. F. Henninot, M. Debailleul, F. Derrien, G. Abbate, and M. Warenghem, “(2D+1) Spatial optical solitons in dye doped liquid crystals,” Synth. Met. 8915, 1–5 (2001)

Hutsebaut, X.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Comm. 233, 211–217 (2004)
[CrossRef]

Khoo, I. C.

M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000)
[CrossRef]

I. C. Khoo, Liquid Crystals: Physical Properties and Optical Phenomena (Whiley & Sons, New York, 1995)

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, London, 2003)

Libchaber, A.

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993)
[CrossRef] [PubMed]

Neyts, K.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Comm. 233, 211–217 (2004)
[CrossRef]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, “Observation of Optical Spatial Solitons in a Highly Nonlocal Medium,” Phys. Rev. Lett. 92, 113902 (2004)
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91, 73901 (2003)
[CrossRef]

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003)
[CrossRef]

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical Spatial Solitons in Nematic Liquid Crystals,” Optics Photon. News 14, 44–48 (2003)
[CrossRef]

G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003)
[CrossRef]

M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002)
[CrossRef]

M. Peccianti and G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. 26, 1690–1692 (2001)
[CrossRef]

M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000)
[CrossRef]

Segev, M.

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518 (1999)
[CrossRef] [PubMed]

Simoni, F.

F. Simoni, Nonlinear Optical Properties of Liquid Crystals, (World Scientific, London, 1997)
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518 (1999)
[CrossRef] [PubMed]

Sukhov, A.V.

N. V. Tabiryan, A.V. Sukhov, and B. Ya. Zel’dovich, “Orientational Optical Nonlinearity of Liquid Crystals,” Mol. Cryst. Liq. Cryst. 136, 1–131 (1986)
[CrossRef]

Tabiryan, N. V.

N. V. Tabiryan, A.V. Sukhov, and B. Ya. Zel’dovich, “Orientational Optical Nonlinearity of Liquid Crystals,” Mol. Cryst. Liq. Cryst. 136, 1–131 (1986)
[CrossRef]

Torruellas, W. E

S. Trillo and W. E Torruellas, Spatial Solitons (Springer, Berlin, 2001)

Trillo, S.

S. Trillo and W. E Torruellas, Spatial Solitons (Springer, Berlin, 2001)

Umeton, C.

M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000)
[CrossRef]

Warenghem, M.

J. F. Henninot, M. Debailleul, and M. Warenghem, “Tunable non-locality of thermal non-linearity in dye doped nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 375, 1538–1547 (2002)
[CrossRef]

J. F. Henninot, M. Debailleul, F. Derrien, G. Abbate, and M. Warenghem, “(2D+1) Spatial optical solitons in dye doped liquid crystals,” Synth. Met. 8915, 1–5 (2001)

Zel’dovich, B. Ya.

N. V. Tabiryan, A.V. Sukhov, and B. Ya. Zel’dovich, “Orientational Optical Nonlinearity of Liquid Crystals,” Mol. Cryst. Liq. Cryst. 136, 1–131 (1986)
[CrossRef]

Appl. Phys. Lett. (1)

M. Peccianti, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000)
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Assanto and M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003)
[CrossRef]

Mol. Cryst. Liq. Cryst. (3)

N. V. Tabiryan, A.V. Sukhov, and B. Ya. Zel’dovich, “Orientational Optical Nonlinearity of Liquid Crystals,” Mol. Cryst. Liq. Cryst. 136, 1–131 (1986)
[CrossRef]

R. Asquini and A. d’Alessandro, “BPM Analysis of an integrated optical switch using polymeric optical waveguides and SSFLC at 1.55 µm,” Mol. Cryst. Liq. Cryst. 375, 243–247 (2002)
[CrossRef]

J. F. Henninot, M. Debailleul, and M. Warenghem, “Tunable non-locality of thermal non-linearity in dye doped nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 375, 1538–1547 (2002)
[CrossRef]

Opt. Comm. (1)

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, “Single-component higher-order mode solitons in liquid crystals,” Opt. Comm. 233, 211–217 (2004)
[CrossRef]

Opt. Lett. (2)

Optics Photon. News (1)

G. Assanto, M. Peccianti, and C. Conti, “Nematicons: Optical Spatial Solitons in Nematic Liquid Crystals,” Optics Photon. News 14, 44–48 (2003)
[CrossRef]

Phys. Rev. A (1)

E. Braun, L. P. Faucheux, and A. Libchaber, “Strong self-focusing in nematic liquid crystals,” Phys. Rev. A 48, 611–622 (1993)
[CrossRef] [PubMed]

Phys. Rev. E (1)

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602 (2003)
[CrossRef]

Phys. Rev. Lett. (2)

C. Conti, M. Peccianti, and G. Assanto, “Route to Nonlocality and Observation of Accessible Solitons,” Phys. Rev. Lett. 91, 73901 (2003)
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, “Observation of Optical Spatial Solitons in a Highly Nonlocal Medium,” Phys. Rev. Lett. 92, 113902 (2004)
[CrossRef] [PubMed]

Science (1)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518 (1999)
[CrossRef] [PubMed]

Synth. Met. (1)

J. F. Henninot, M. Debailleul, F. Derrien, G. Abbate, and M. Warenghem, “(2D+1) Spatial optical solitons in dye doped liquid crystals,” Synth. Met. 8915, 1–5 (2001)

Other (5)

CRC Handbook of Laser Science and Technology: Optical Materials, Suppl. 2, (ed. M. J. Weber, CRC Press, New York, 1995).

S. Trillo and W. E Torruellas, Spatial Solitons (Springer, Berlin, 2001)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, London, 2003)

I. C. Khoo, Liquid Crystals: Physical Properties and Optical Phenomena (Whiley & Sons, New York, 1995)

F. Simoni, Nonlinear Optical Properties of Liquid Crystals, (World Scientific, London, 1997)
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Sketch of the NLC cell and experimental geometry.

Fig. 2.
Fig. 2.

Calculated maximum (on-axis) walk-off versus cell bias

Fig. 3.
Fig. 3.

Bias induced index profile in a cell filled with E7 and an applied voltage V=1.48V, providing maximum walk-off.

Fig. 4.
Fig. 4.

Simulated propagation of a 3mW X-polarized gaussian beam launched in a biased cell (as in Fig. 3) with k-vector parallel to Z and a) no phase front tilt ; b) a 7° tilt in order to compensate walk-off on axis.

Fig. 5.
Fig. 5.

Nematicon transverse profile in the observation plane at ϕ=45° with respect to X. a) For V 0=1.0V the small walk-off (about 2°) mediates an oscillation of modest amplitude; b) at V 0=1.6V a larger walk-off (about 7°) corresponds to a shorter period with larger elongation across X. c) By launching the input beam with a phase front tilt in order to compensate the walk-off, the nematicon at V 0=1.6V can be generated with no motion across X

Fig. 6.
Fig. 6.

Soliton trajectories for P=3.2mW versus bias V 0. The scale Δx quantifies the deviation from input position X=0.

Fig. 7.
Fig. 7.

Calculated (solid line) and measured (dashed line with dots) periodicity Λ of the nematicon transverse oscillation versus applied bias V0.

Equations (5)

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

( K 1 cos 2 Θ + K 3 sin 2 Θ ) d 2 Θ d X 2 + K 3 K 1 2 sin 2 ξ ( d Θ d X ) 2 + 1 2 ε a ( d V d X ) 2 sin 2 Θ = 0
( ε sin 2 Θ + ε cos 2 Θ ) d 2 V d X 2 + ε a sin 2 Θ d Θ d X d V d X = 0
δ ( Θ ) = arctan ( Δ n 2 sin ( 2 Θ ) Δ n 2 + 2 n 2 + Δ n 2 cos ( 2 Θ ) )
j 2 k 0 n ( Θ ) E Z = 2 E + k 0 2 ( n 2 ( θ ) n 2 ( Θ ) ) E + j 2 k 0 n ( Θ ) tan δ ( θ ) E X
K θ + ε 0 ( 1 2 Δ ε a d V d X 2 + 1 4 Δ n 2 E 2 ) sin 2 θ = 0

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