Abstract

We investigate Landau-Zener all-optical tunneling in a voltage-controlled waveguide array realized in undoped nematic liquid crystals. From the material governing equations we derive the original Zener model and demonstrate a novel approach to Floquet-band tunneling.

© 2006 Optical Society of America

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References

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  1. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
    [CrossRef] [PubMed]
  2. S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1972).
    [CrossRef]
  3. A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31-50 (2003).
    [CrossRef]
  4. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from fibers to photonic crystals (Academic Press, San Diego, 2003).
  5. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4760 (1999).
    [CrossRef]
  6. B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).
    [CrossRef]
  7. R. Khomeriki and S. Ruffo, "Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index," Phys. Rev. Lett. 94, 113904 (2005).
    [CrossRef]
  8. V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).
    [CrossRef]
  9. S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, Berlin, 2001).
  10. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).
  11. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure ofWaveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053902 (2003).
    [CrossRef]
  12. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Optical Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," in Proc. Top. Meet. On Nonlinear Guided Waves and their Applications, ThD1 (Opt. Soc. Am., Dresden, Germany, 2005).
  13. C. Zener, "Non-adiabatic crossing of energy levels," Proc. R. Soc. London Ser. A 137, 696-702 (1932).
    [CrossRef]
  14. I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, New York, 1995).
  15. F. Simoni, Nonlinear Optical Properties of Liquid Crystals (World Scientific, Singapore, 1997).
  16. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Discrete light propagation and self-trapping in liquid crystals," Opt. Express 13, 1808-1815 (2005), http://www.opticsexpress.org/abstract.cfm?id=82980.
    [CrossRef] [PubMed]
  17. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530-1532 (2004).
    [CrossRef] [PubMed]
  18. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Optical multiband vector breathers in tunable waveguide arrays," Opt. Lett. 30, 174-176 (2005).
    [CrossRef] [PubMed]
  19. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "All-optical switching and beam steering in tunable waveguide arrays," Appl. Phys. Lett. 86, 051112 (2005).
    [CrossRef]
  20. A. Fratalocchi and G. Assanto, "Discrete light localization in one dimensional nonlinear lattices with arbitrary non locality," Phys. Rev. E 72, 066608 (2005).
    [CrossRef]
  21. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Optically-induced Zener tunneling in one dimensional lattices," Opt. Lett., to be published.
    [PubMed]

2005

R. Khomeriki and S. Ruffo, "Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index," Phys. Rev. Lett. 94, 113904 (2005).
[CrossRef]

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).
[CrossRef]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Discrete light propagation and self-trapping in liquid crystals," Opt. Express 13, 1808-1815 (2005), http://www.opticsexpress.org/abstract.cfm?id=82980.
[CrossRef] [PubMed]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Optical multiband vector breathers in tunable waveguide arrays," Opt. Lett. 30, 174-176 (2005).
[CrossRef] [PubMed]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "All-optical switching and beam steering in tunable waveguide arrays," Appl. Phys. Lett. 86, 051112 (2005).
[CrossRef]

A. Fratalocchi and G. Assanto, "Discrete light localization in one dimensional nonlinear lattices with arbitrary non locality," Phys. Rev. E 72, 066608 (2005).
[CrossRef]

2004

2003

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure ofWaveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31-50 (2003).
[CrossRef]

2000

B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).
[CrossRef]

1999

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4760 (1999).
[CrossRef]

1988

1972

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1972).
[CrossRef]

1932

C. Zener, "Non-adiabatic crossing of energy levels," Proc. R. Soc. London Ser. A 137, 696-702 (1932).
[CrossRef]

Aitchison, J. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure ofWaveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4760 (1999).
[CrossRef]

Assanto, G.

Brzdakiewicz, K. A.

Christodoulides, D. N.

Eisenberg, H. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure ofWaveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4760 (1999).
[CrossRef]

Fratalocchi, A.

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1972).
[CrossRef]

Garvin, H.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1972).
[CrossRef]

Hunsperger, R.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1972).
[CrossRef]

Joseph, R. I.

Karpierz, M. A.

Kevrekidis, P. G.

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).
[CrossRef]

Khomeriki, R.

R. Khomeriki and S. Ruffo, "Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index," Phys. Rev. Lett. 94, 113904 (2005).
[CrossRef]

Kivshar, Y. S.

A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31-50 (2003).
[CrossRef]

Konotop, V. V.

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).
[CrossRef]

Mandelik, D.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure ofWaveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

Morandotti, R.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure ofWaveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4760 (1999).
[CrossRef]

Niu, Q.

B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4760 (1999).
[CrossRef]

Ruffo, S.

R. Khomeriki and S. Ruffo, "Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index," Phys. Rev. Lett. 94, 113904 (2005).
[CrossRef]

Salerno, M.

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).
[CrossRef]

Silberberg, Y.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure ofWaveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4760 (1999).
[CrossRef]

Somekh, S.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1972).
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31-50 (2003).
[CrossRef]

Wu, B.

B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).
[CrossRef]

Yariv, A.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1972).
[CrossRef]

Zener, C.

C. Zener, "Non-adiabatic crossing of energy levels," Proc. R. Soc. London Ser. A 137, 696-702 (1932).
[CrossRef]

Appl. Phys. Lett.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1972).
[CrossRef]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "All-optical switching and beam steering in tunable waveguide arrays," Appl. Phys. Lett. 86, 051112 (2005).
[CrossRef]

IEEE J. Quantum Electron.

A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31-50 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000).
[CrossRef]

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005).
[CrossRef]

Phys. Rev. E

A. Fratalocchi and G. Assanto, "Discrete light localization in one dimensional nonlinear lattices with arbitrary non locality," Phys. Rev. E 72, 066608 (2005).
[CrossRef]

Phys. Rev. Lett.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure ofWaveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef]

R. Khomeriki and S. Ruffo, "Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index," Phys. Rev. Lett. 94, 113904 (2005).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4760 (1999).
[CrossRef]

Proc. R. Soc. London Ser. A

C. Zener, "Non-adiabatic crossing of energy levels," Proc. R. Soc. London Ser. A 137, 696-702 (1932).
[CrossRef]

Other

I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, New York, 1995).

F. Simoni, Nonlinear Optical Properties of Liquid Crystals (World Scientific, Singapore, 1997).

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Optical Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," in Proc. Top. Meet. On Nonlinear Guided Waves and their Applications, ThD1 (Opt. Soc. Am., Dresden, Germany, 2005).

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Optically-induced Zener tunneling in one dimensional lattices," Opt. Lett., to be published.
[PubMed]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from fibers to photonic crystals (Academic Press, San Diego, 2003).

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

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).

Supplementary Material (1)

» Media 1: MOV (1012 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Sketch of optically induced Landau-Zener tunneling: an intense pump beam, launched straight into the lattice (left), impresses a non-adiabatic acceleration thru an index decrease in the transition regions (right). A probe, initially coupled to an upper FB band, is accelerated in one of the transition regions and transfers energy to a lower band through LZ tunneling.

Fig. 2.
Fig. 2.

Sketch of the NLC periodic lattice: (a) front view; (b) side view.

Fig. 3.
Fig. 3.

(a) Band-gap diagram corresponding to Eq. (7) with Vp = 0,V 0 = 1 and δ = 0.5; (b) propagation of FB modes in band 1 at the maximum transverse velocity; (c) Zener tunneling between band 1 and band 2: in the transition region (dotted line) light is accelerated and transfers energy to band 2.

Fig. 4.
Fig. 4.

Signal beam propagating in the cell for V = 1.19V: (a) discrete diffraction for Pprobe = 1mW and (b) self-defocusing for Pprobe = 6mW.

Fig. 5.
Fig. 5.

(1.012 KB). Movie displaying the optically driven LZ-tunneling between FB bands 1 and 2. Light is initially coupled to band 1 with Pprobe = 1mW,Ppump = 0mW and V = 1.19V at the maximum transverse velocity. As the pump (dotted line) is turned on (green line) and reaches Ppump = 25mW (red line), the probe is accelerated and LZ-tunnels to band 2.

Equations (9)

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

xyz 2 A probe + ω 2 μ 0 ε x y A probe = 0
K xy 2 θ 0 + Δ ε RF E x 2 2 sin ( 2 θ 0 ) = 0
κ xy 2 Δ T + α I pump = 0
2 i β x A z + 2 A y 2 + ω 2 μ 0 Δ ε B 2 dx B 2 dx = 0
γ x y = sin [ π d ( x ± d 2 ) ] [ 1 + σ γ 1 ( y ) + O ( σ 2 ) ]
Δ T x y = α κ I ( x ) [ m ( iy ) m m ! I ˜ ν m 2 ]
i ψ Z + [ 1 2 ( Y + iZ V p Y ) 2 + V ( Y ) ] ψ = 0
b 1 Z = i v 1 2 e [ 2 i Θ ( Z ) dZ ] b 2
b 2 Z = i v 1 2 e [ 2 i Θ ( Z ) dZ ] b 1

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