Abstract

We investigate Landau–Zener tunneling in one-dimensional liquid crystalline waveguide arrays by all-optical impression of acceleration with an additional beam. We derive the Zener model from the governing equations and demonstrate a novel approach to Floquet–Bloch band tunneling.

© 2006 Optical Society of America

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  1. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988).
    [CrossRef] [PubMed]
  2. S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1972).
    [CrossRef]
  3. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).
  4. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
    [CrossRef] [PubMed]
  5. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
    [CrossRef]
  6. R. Khomeriki and S. Ruffo, Phys. Rev. Lett. 94, 113904 (2005).
    [CrossRef] [PubMed]
  7. V. V. Konotop, P. G. Kevrekidis, and M. Salerno, Phys. Rev. A 72, 023611 (2005).
    [CrossRef]
  8. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.
  9. A. Fratalocchi, K. A. Brzdakiewicz, M. A. Karpierz, and G. Assanto, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper WD21.
  10. M. Ghulinyan, C. J. Oton, Z. Gaburro, L. P. and C. Toninelli, and D. S. Wiersma, Phys. Rev. Lett. 94, 127401 (2005).
    [CrossRef] [PubMed]
  11. B. Wu and Q. Niu, New J. Phys. 5, 104 (2003).
    [CrossRef]
  12. M. Cristiani, O. Morsch, J. Mueller, D. Ciampini, and E. Arimondo, Phys. Rev. A 65, 063612 (2002).
    [CrossRef]
  13. C. Zener, Proc. R. Soc. London Ser. A 137, 696 (1932).
    [CrossRef]
  14. F. Simoni, Nonlinear Optical Properties of Liquid Crystals (World Scientific, 1997).
  15. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Opt. Express 13, 1808 (2005).
    [CrossRef] [PubMed]
  16. B. Wu and Q. Niu, Phys. Rev. A 61, 023402 (2000).
    [CrossRef]
  17. A. Fratalocchi and G. Assanto, Phys. Rev. E 72, 066608 (2005).
    [CrossRef]

2005 (5)

R. Khomeriki and S. Ruffo, Phys. Rev. Lett. 94, 113904 (2005).
[CrossRef] [PubMed]

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, Phys. Rev. A 72, 023611 (2005).
[CrossRef]

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. P. and C. Toninelli, and D. S. Wiersma, Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef] [PubMed]

A. Fratalocchi and G. Assanto, Phys. Rev. E 72, 066608 (2005).
[CrossRef]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Opt. Express 13, 1808 (2005).
[CrossRef] [PubMed]

2003 (2)

B. Wu and Q. Niu, New J. Phys. 5, 104 (2003).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

2002 (1)

M. Cristiani, O. Morsch, J. Mueller, D. Ciampini, and E. Arimondo, Phys. Rev. A 65, 063612 (2002).
[CrossRef]

2000 (1)

B. Wu and Q. Niu, Phys. Rev. A 61, 023402 (2000).
[CrossRef]

1999 (1)

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

1988 (1)

1972 (1)

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1972).
[CrossRef]

1932 (1)

C. Zener, Proc. R. Soc. London Ser. A 137, 696 (1932).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Aitchison, J. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Arimondo, E.

M. Cristiani, O. Morsch, J. Mueller, D. Ciampini, and E. Arimondo, Phys. Rev. A 65, 063612 (2002).
[CrossRef]

Assanto, G.

A. Fratalocchi and G. Assanto, Phys. Rev. E 72, 066608 (2005).
[CrossRef]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Opt. Express 13, 1808 (2005).
[CrossRef] [PubMed]

A. Fratalocchi, K. A. Brzdakiewicz, M. A. Karpierz, and G. Assanto, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper WD21.

Brzdakiewicz, K. A.

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Opt. Express 13, 1808 (2005).
[CrossRef] [PubMed]

A. Fratalocchi, K. A. Brzdakiewicz, M. A. Karpierz, and G. Assanto, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper WD21.

Christodoulides, D. N.

Ciampini, D.

M. Cristiani, O. Morsch, J. Mueller, D. Ciampini, and E. Arimondo, Phys. Rev. A 65, 063612 (2002).
[CrossRef]

Cristiani, M.

M. Cristiani, O. Morsch, J. Mueller, D. Ciampini, and E. Arimondo, Phys. Rev. A 65, 063612 (2002).
[CrossRef]

Desyatnikov, A. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

Eisenberg, H. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Fratalocchi, A.

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Opt. Express 13, 1808 (2005).
[CrossRef] [PubMed]

A. Fratalocchi and G. Assanto, Phys. Rev. E 72, 066608 (2005).
[CrossRef]

A. Fratalocchi, K. A. Brzdakiewicz, M. A. Karpierz, and G. Assanto, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper WD21.

Gaburro, Z.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. P. and C. Toninelli, and D. S. Wiersma, Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef] [PubMed]

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1972).
[CrossRef]

Garvin, H.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1972).
[CrossRef]

Ghulinyan, M.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. P. and C. Toninelli, and D. S. Wiersma, Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef] [PubMed]

Hunsperger, R.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1972).
[CrossRef]

Joseph, R. I.

Karpierz, M. A.

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Opt. Express 13, 1808 (2005).
[CrossRef] [PubMed]

A. Fratalocchi, K. A. Brzdakiewicz, M. A. Karpierz, and G. Assanto, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper WD21.

Kevrekidis, P. G.

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, Phys. Rev. A 72, 023611 (2005).
[CrossRef]

Khomeriki, R.

R. Khomeriki and S. Ruffo, Phys. Rev. Lett. 94, 113904 (2005).
[CrossRef] [PubMed]

Kivshar, Y. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Konotop, V. V.

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, Phys. Rev. A 72, 023611 (2005).
[CrossRef]

Krolikowski, W.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

Lederer, F.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

Mandelik, D.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

Morandotti, R.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Morsch, O.

M. Cristiani, O. Morsch, J. Mueller, D. Ciampini, and E. Arimondo, Phys. Rev. A 65, 063612 (2002).
[CrossRef]

Mueller, J.

M. Cristiani, O. Morsch, J. Mueller, D. Ciampini, and E. Arimondo, Phys. Rev. A 65, 063612 (2002).
[CrossRef]

Neshev, D. N.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

Niu, Q.

B. Wu and Q. Niu, New J. Phys. 5, 104 (2003).
[CrossRef]

B. Wu and Q. Niu, Phys. Rev. A 61, 023402 (2000).
[CrossRef]

Oton, C. J.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. P. and C. Toninelli, and D. S. Wiersma, Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef] [PubMed]

P., L.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. P. and C. Toninelli, and D. S. Wiersma, Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef] [PubMed]

Pertsch, T.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

Ruffo, S.

R. Khomeriki and S. Ruffo, Phys. Rev. Lett. 94, 113904 (2005).
[CrossRef] [PubMed]

Salerno, M.

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, Phys. Rev. A 72, 023611 (2005).
[CrossRef]

Silberberg, Y.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

Simoni, F.

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

Somekh, S.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1972).
[CrossRef]

Sukhorukov, A. A.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

Toninelli, C.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. P. and C. Toninelli, and D. S. Wiersma, Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef] [PubMed]

Trompeter, H.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

Wiersma, D. S.

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. P. and C. Toninelli, and D. S. Wiersma, Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef] [PubMed]

Wu, B.

B. Wu and Q. Niu, New J. Phys. 5, 104 (2003).
[CrossRef]

B. Wu and Q. Niu, Phys. Rev. A 61, 023402 (2000).
[CrossRef]

Yariv, A.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1972).
[CrossRef]

Zener, C.

C. Zener, Proc. R. Soc. London Ser. A 137, 696 (1932).
[CrossRef]

Appl. Phys. Lett. (1)

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1972).
[CrossRef]

New J. Phys. (1)

B. Wu and Q. Niu, New J. Phys. 5, 104 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (3)

V. V. Konotop, P. G. Kevrekidis, and M. Salerno, Phys. Rev. A 72, 023611 (2005).
[CrossRef]

B. Wu and Q. Niu, Phys. Rev. A 61, 023402 (2000).
[CrossRef]

M. Cristiani, O. Morsch, J. Mueller, D. Ciampini, and E. Arimondo, Phys. Rev. A 65, 063612 (2002).
[CrossRef]

Phys. Rev. E (1)

A. Fratalocchi and G. Assanto, Phys. Rev. E 72, 066608 (2005).
[CrossRef]

Phys. Rev. Lett. (4)

M. Ghulinyan, C. J. Oton, Z. Gaburro, L. P. and C. Toninelli, and D. S. Wiersma, Phys. Rev. Lett. 94, 127401 (2005).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Phys. Rev. Lett. 83, 4756 (1999).
[CrossRef]

R. Khomeriki and S. Ruffo, Phys. Rev. Lett. 94, 113904 (2005).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

C. Zener, Proc. R. Soc. London Ser. A 137, 696 (1932).
[CrossRef]

Other (4)

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

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper ThD1.

A. Fratalocchi, K. A. Brzdakiewicz, M. A. Karpierz, and G. Assanto, in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2005), paper WD21.

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Figures (3)

Fig. 1
Fig. 1

(Color online). Sketch of optically assisted tunneling in a waveguide array. (a) A pump beam nonlinearly induces a refractive-index change that defines (b) two transition regions. A probe beam is accelerated in a transition region and experiences Zener tunneling to a lower-order band, modifying its transverse velocity.

Fig. 2
Fig. 2

(Color online). Numerical simulations: (a) bandgap spectrum of Eq. (4) with V p = 0 (relating propagation constant β to Bloch wavenumber k y ), (b) propagation of FB modes of band 1 at the maximum transverse velocity, and (c) Zener tunneling between band 1 and band 2. (c) In the transition region (dotted line), light transferred to a lower- (higher-) order band propagates with a larger angle with respect to the limit value in band 1.

Fig. 3
Fig. 3

(Color online). Summary of experimental results in a cell biased at V = 1.19 V : (a) discrete light propagation for P probe = 1 mW ; (b) self-defocusing of the probe for P probe = 6 mW and discrete probe propagation at the maximum transverse velocity for P probe = 1 mW ; (c) P pump = 0 mW , (d) P pump = 25 mW , and (e) P pump = 30 mW . (f) Enlarged detail of (e). Zener tunneling can be seen in (d)–(f) as an increased propagation angle with respect to the maximum allowed in band 1 (black arrows).

Equations (9)

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K x y 2 θ 0 + Δ ϵ rf E x 2 2 sin ( 2 θ 0 ) = 0 ,
κ x y 2 Δ T + α I pump = 0 ,
x y z 2 A probe + ω 2 μ 0 ϵ ( x , y ) A probe = 0 .
i ψ Z + [ 1 2 ( Y + i Z V p Y ) 2 + V ( Y ) ] ψ = 0 ,
V p = u 1 ( Y ) δ Y + u 1 ( Y V 0 δ ) ( V 0 δ Y ) ,
b 1 Z = i v 1 b 2 2 exp [ 2 i ϴ ( Z ) d Z ] ,
b 2 Z = i v 1 b 1 2 exp [ 2 i ϴ ( Z ) d Z ] ,
a 1 = b 1 exp { i [ 2 v 0 1 2 Z δ 2 6 Z 3 + ϴ ( Z ) d Z ] } ,
a 2 = b 2 exp { i [ 2 v 0 1 2 Z δ 2 6 Z 3 ϴ ( Z ) d Z ] } ,

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