Abstract

Linear and nonlinear light propagation in an array of waveguides with a periodically bent axis is theoretically investigated. In the linear propagation regime, it is shown that a self-imaging effect at periodic planes may occur, a phenomenon analogous to that of dynamic localization observed when an electron in a periodic potential is subjected to an ac field. In the nonlinear propagation regime, it is shown that periodic waveguide bending under the self-imaging condition inhibits the phenomenon of discrete modulational instability.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef] [PubMed]
  2. A. Sukhorukov, Y. S. Kivshar, S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
    [CrossRef]
  3. G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999).
    [CrossRef]
  4. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
    [CrossRef] [PubMed]
  5. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
    [CrossRef]
  6. M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
    [CrossRef]
  7. M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
    [CrossRef]
  8. R. Ulrich, Opt. Commun. 13, 259 (1975).
    [CrossRef]
  9. D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 (1986).
    [CrossRef]
  10. R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, Opt. Quantum Electron. 30, 751 (1998).
    [CrossRef]
  11. A. A. Sukhorukov and Y. S. Kivshar, Opt. Lett. 28, 2345 (2003).
    [CrossRef] [PubMed]
  12. For a constant curvature[3] (?0=constant), Eq. (3) describes optical Bloch oscillations [see, e.g., U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998)].
    [CrossRef]
  13. The condition for self-imaging beyond the NNA was studied in M. M. Dignam and C. M. de Sterke, Phys. Rev. Lett. 88, 046806 (2002). According to that analysis, beyond the NNA, self-imaging is met only approximately in a sinusoidally bent array.
    [CrossRef]
  14. Y. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992).
    [CrossRef] [PubMed]
  15. For straight waveguides (?=0), the Floquet expon- ents read explicitly as ?±(q)=?2i?sinpsinq±4i??sin(q?2)?[cos2psin2(q?2)??I0cosp?(2?)]1?2. For ?>0 (self-focusing), an instability arises at any intensity level when cosp>0, i.e., for ?p?<??2.

2003 (4)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

A. Sukhorukov, Y. S. Kivshar, S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, Opt. Lett. 28, 2345 (2003).
[CrossRef] [PubMed]

2002 (2)

The condition for self-imaging beyond the NNA was studied in M. M. Dignam and C. M. de Sterke, Phys. Rev. Lett. 88, 046806 (2002). According to that analysis, beyond the NNA, self-imaging is met only approximately in a sinusoidally bent array.
[CrossRef]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

2001 (1)

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

1999 (1)

G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999).
[CrossRef]

1998 (2)

1992 (1)

Y. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992).
[CrossRef] [PubMed]

1986 (1)

D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 (1986).
[CrossRef]

1975 (1)

R. Ulrich, Opt. Commun. 13, 259 (1975).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Binder, R.

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, Opt. Quantum Electron. 30, 751 (1998).
[CrossRef]

Bräuer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

Burak, D.

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, Opt. Quantum Electron. 30, 751 (1998).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

de Sterke, C. M.

The condition for self-imaging beyond the NNA was studied in M. M. Dignam and C. M. de Sterke, Phys. Rev. Lett. 88, 046806 (2002). According to that analysis, beyond the NNA, self-imaging is met only approximately in a sinusoidally bent array.
[CrossRef]

G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999).
[CrossRef]

Dignam, M. M.

The condition for self-imaging beyond the NNA was studied in M. M. Dignam and C. M. de Sterke, Phys. Rev. Lett. 88, 046806 (2002). According to that analysis, beyond the NNA, self-imaging is met only approximately in a sinusoidally bent array.
[CrossRef]

Dunlap, D. H.

D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 (1986).
[CrossRef]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Eisenberg, S.

A. Sukhorukov, Y. S. Kivshar, S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

Kenkre, V. M.

D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 (1986).
[CrossRef]

Kivshar, Y. S.

A. A. Sukhorukov and Y. S. Kivshar, Opt. Lett. 28, 2345 (2003).
[CrossRef] [PubMed]

A. Sukhorukov, Y. S. Kivshar, S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, Opt. Quantum Electron. 30, 751 (1998).
[CrossRef]

Y. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992).
[CrossRef] [PubMed]

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

For a constant curvature[3] (?0=constant), Eq. (3) describes optical Bloch oscillations [see, e.g., U. Peschel, T. Pertsch, and F. Lederer, Opt. Lett. 23, 1701 (1998)].
[CrossRef]

Lenz, G.

G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999).
[CrossRef]

Love, J. D.

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, Opt. Quantum Electron. 30, 751 (1998).
[CrossRef]

Micallef, R. W.

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, Opt. Quantum Electron. 30, 751 (1998).
[CrossRef]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Musslimani, Z. H.

M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef]

Pertsch, T.

Peschel, U.

Peyrard, M.

Y. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992).
[CrossRef] [PubMed]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

A. Sukhorukov, Y. S. Kivshar, S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Sukhorukov, A.

A. Sukhorukov, Y. S. Kivshar, S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

Sukhorukov, A. A.

Talanina, I.

G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999).
[CrossRef]

Ulrich, R.

R. Ulrich, Opt. Commun. 13, 259 (1975).
[CrossRef]

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Sukhorukov, Y. S. Kivshar, S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

R. Ulrich, Opt. Commun. 13, 259 (1975).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

R. W. Micallef, Y. S. Kivshar, J. D. Love, D. Burak, and R. Binder, Opt. Quantum Electron. 30, 751 (1998).
[CrossRef]

Phys. Rev. A (1)

Y. S. Kivshar and M. Peyrard, Phys. Rev. A 46, 3198 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (1)

D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 (1986).
[CrossRef]

Phys. Rev. Lett. (5)

The condition for self-imaging beyond the NNA was studied in M. M. Dignam and C. M. de Sterke, Phys. Rev. Lett. 88, 046806 (2002). According to that analysis, beyond the NNA, self-imaging is met only approximately in a sinusoidally bent array.
[CrossRef]

G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef]

Physica D (1)

M. J. Ablowitz and Z. H. Musslimani, Physica D 184, 276 (2003).
[CrossRef]

Other (1)

For straight waveguides (?=0), the Floquet expon- ents read explicitly as ?±(q)=?2i?sinpsinq±4i??sin(q?2)?[cos2psin2(q?2)??I0cosp?(2?)]1?2. For ?>0 (self-focusing), an instability arises at any intensity level when cosp>0, i.e., for ?p?<??2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

(a) Array of periodically curved optical waveguides. (b) Refractive index n ( x ) n s of the waveguide array (solid curve) used in numerical simulations and waveguide mode profile (dashed curve). n s = 3.4 , a = 6.4 μ m , Δ n = 0.01 , and λ = 1.55 μ m .

Fig. 2
Fig. 2

Evolution of beam intensity in an L = 10 mm long sinusoidally curved array under single-channel excitation, and corresponding output intensity profile (upper plots) for (a) Λ = 10 mm and A 43.3 μ m , (b) Λ = 2.5 mm and A 10.8 μ m , and (c) Λ = 2.5 mm and A 6 μ m . The dashed sinusoidal curves in the lower plots show, for comparison, the waveguide bending profile. In (a) and (b) the self-imaging condition is satisfied. The coupling coefficient is Δ 0.6 mm 1 .

Fig. 3
Fig. 3

Threshold value χ I 0 Δ for MI in a sinusoidally curved array [self-focusing, 2 π ( Λ Δ ) = 1 ] versus NLBW wave number p (a) far from self-imaging [solid curve, Γ = 1 ; dashed curve, Γ = 3 ] and (b) close to self-imaging [solid curve, Γ = 2.395 ; dashed curve, Γ = 2.420 ]. In (b) the shaded area is the MI domain at exact self-imaging. (c) MI threshold at Γ = 2.405 versus 2 π ( Λ Δ ) for p = 0 (solid curve), p = π 2 (dashed curve), and p = π (dotted curve).

Fig. 4
Fig. 4

MI in the self-imaging condition for (a) χ I 0 Δ = 0.4 and (b) χ I 0 Δ = 1 and for 2 π ( Λ Δ ) = 1 . The MI threshold is χ I 0 Δ 0.692 [see Fig. 3(b) at p = 0 ].

Equations (5)

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

i ψ z = 2 2 n s 2 ψ x 2 + V [ x x 0 ( z ) ] ψ σ ψ 2 ψ .
i ϕ z = 2 2 n s 2 ϕ x 2 + V ( x ) ϕ + n s x ̈ 0 ( z ) x ϕ σ ϕ 2 ϕ .
i c ̇ n = Δ ( c n + 1 + c n 1 ) + n s a x ̈ 0 ( z ) n c n χ c n 2 c n ,
c ¯ n ( z ) = exp [ i p n i γ ( z ) n ] exp [ i θ L ( z , p ) ] ,
0 Λ d ξ exp [ i γ ( ξ ) ] = 0 .

Metrics