Abstract

We study the optical oscillations of supermodes in planar optical waveguide arrays with a parabolically graded propagation constant in individual waveguide interacting through nearest neighbor couplings. In these arrays, we have identified a transition between a symmetric dipole oscillation (DO) and a symmetry-breaking Bloch oscillation (BO) under appropriate conditions. There exist obvious correspondences between gradon localization and various optical oscillations. By virtue of an analog between the oscillations of optical system and that of a plane pendulum, we propose a shift of the graded profile to cause a transition from BO to DO. We confirm the optical transition by means of Hamiltonian optics, as well as by the field evolution of the supermodes. The results offer great potential applications in optical switching, which can be applied to design suitable optical devices.

© 2010 Optical Society of America

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References

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  1. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
    [CrossRef] [PubMed]
  2. U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. 23, 1701–1703 (1998).
    [CrossRef]
  3. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
    [CrossRef] [PubMed]
  4. A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Optical Bloch oscillations in general waveguide lattices,” J. Opt. Soc. Am. B 24, 2632–2639 (2007).
    [CrossRef]
  5. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3, 243–261 (2008).
    [CrossRef]
  6. R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
    [CrossRef]
  7. S. Longhi, “Bloch dynamics of light waves in helical optical waveguide arrays,” Phys. Rev. B 76, 195119 (2007).
    [CrossRef]
  8. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
    [CrossRef]
  9. 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, 4756–4759 (1999).
    [CrossRef]
  10. G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
    [CrossRef]
  11. V. Lousse and S. Fan, “Tunable terahertz Bloch oscillations in chirped photonic crystals,” Phys. Rev. B 72, 075119 (2005).
    [CrossRef]
  12. E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
    [CrossRef]
  13. G. Wang, J. P. Huang, and K. W. Yu, “Tunable photonic Bloch oscillations in electrically modulated photonic crystals,” Opt. Lett. 33, 2200–2202 (2008).
    [CrossRef] [PubMed]
  14. G. Wang, J. P. Huang, and K. W. Yu, “Long-living photonic dipole oscillations in photonic crystals,” Opt. Lett. 34, 1777–1779 (2009).
    [CrossRef] [PubMed]
  15. A. V. Ponomarev and A. R. Kolovsky, “Dipole and Bloch oscillations of cold atoms in a parabolic lattice,” Laser Phys. 16, 367–370 (2006).
    [CrossRef]
  16. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).
  17. M. J. Zheng, J. J. Xiao, and K. W. Yu, “Tunable localization and oscillation of coupled plasmon waves in graded plasmonic chains,” J. Appl. Phys. 106, 113307 (2009).
    [CrossRef]
  18. M. J. Zheng, J. J. Xiao, and K. W. Yu, “Controllable optical Bloch oscillation in planar graded optical waveguide arrays,” Phys. Rev. A 81, 033829 (2010).
    [CrossRef]

2010 (1)

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Controllable optical Bloch oscillation in planar graded optical waveguide arrays,” Phys. Rev. A 81, 033829 (2010).
[CrossRef]

2009 (3)

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

G. Wang, J. P. Huang, and K. W. Yu, “Long-living photonic dipole oscillations in photonic crystals,” Opt. Lett. 34, 1777–1779 (2009).
[CrossRef] [PubMed]

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Tunable localization and oscillation of coupled plasmon waves in graded plasmonic chains,” J. Appl. Phys. 106, 113307 (2009).
[CrossRef]

2008 (2)

2007 (2)

2006 (3)

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[CrossRef]

A. V. Ponomarev and A. R. Kolovsky, “Dipole and Bloch oscillations of cold atoms in a parabolic lattice,” Laser Phys. 16, 367–370 (2006).
[CrossRef]

2005 (1)

V. Lousse and S. Fan, “Tunable terahertz Bloch oscillations in chirped photonic crystals,” Phys. Rev. B 72, 075119 (2005).
[CrossRef]

2003 (1)

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

1999 (3)

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[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, 4756–4759 (1999).
[CrossRef]

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[CrossRef]

1998 (1)

Aitchison, J. S.

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, 4756–4759 (1999).
[CrossRef]

Brauer, A.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Costantino, P.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

de Sterke, C. M.

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[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, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Eisenberg, H. S.

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, 4756–4759 (1999).
[CrossRef]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

Fan, S.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

V. Lousse and S. Fan, “Tunable terahertz Bloch oscillations in chirped photonic crystals,” Phys. Rev. B 72, 075119 (2005).
[CrossRef]

Ghulinyan, M.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

Huang, J. P.

Istrate, E.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[CrossRef]

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, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Kolovsky, A. R.

A. V. Ponomarev and A. R. Kolovsky, “Dipole and Bloch oscillations of cold atoms in a parabolic lattice,” Laser Phys. 16, 367–370 (2006).
[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, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Lederer, F.

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Optical Bloch oscillations in general waveguide lattices,” J. Opt. Soc. Am. B 24, 2632–2639 (2007).
[CrossRef]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. 23, 1701–1703 (1998).
[CrossRef]

Lenz, G.

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[CrossRef]

Longhi, S.

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3, 243–261 (2008).
[CrossRef]

S. Longhi, “Bloch dynamics of light waves in helical optical waveguide arrays,” Phys. Rev. B 76, 195119 (2007).
[CrossRef]

Lousse, V.

V. Lousse and S. Fan, “Tunable terahertz Bloch oscillations in chirped photonic crystals,” Phys. Rev. B 72, 075119 (2005).
[CrossRef]

Morandotti, R.

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, 4756–4759 (1999).
[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, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Nolte, S.

Oton, C. J.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Pavesi, L.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Pertsch, T.

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Optical Bloch oscillations in general waveguide lattices,” J. Opt. Soc. Am. B 24, 2632–2639 (2007).
[CrossRef]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. 23, 1701–1703 (1998).
[CrossRef]

Peschel, U.

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, “Optical Bloch oscillations in general waveguide lattices,” J. Opt. Soc. Am. B 24, 2632–2639 (2007).
[CrossRef]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

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, 4756–4759 (1999).
[CrossRef]

U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. 23, 1701–1703 (1998).
[CrossRef]

Ponomarev, A. V.

A. V. Ponomarev and A. R. Kolovsky, “Dipole and Bloch oscillations of cold atoms in a parabolic lattice,” Laser Phys. 16, 367–370 (2006).
[CrossRef]

Sapienza, R.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Sargent, E. H.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[CrossRef]

Silberberg, Y.

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, 4756–4759 (1999).
[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, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Szameit, A.

Talanina, I.

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (1999).
[CrossRef]

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, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

Tunnermann, A.

Verslegers, L.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Wang, G.

Wiersma, D.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

Xiao, J. J.

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Controllable optical Bloch oscillation in planar graded optical waveguide arrays,” Phys. Rev. A 81, 033829 (2010).
[CrossRef]

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Tunable localization and oscillation of coupled plasmon waves in graded plasmonic chains,” J. Appl. Phys. 106, 113307 (2009).
[CrossRef]

Yu, K. W.

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Controllable optical Bloch oscillation in planar graded optical waveguide arrays,” Phys. Rev. A 81, 033829 (2010).
[CrossRef]

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Tunable localization and oscillation of coupled plasmon waves in graded plasmonic chains,” J. Appl. Phys. 106, 113307 (2009).
[CrossRef]

G. Wang, J. P. Huang, and K. W. Yu, “Long-living photonic dipole oscillations in photonic crystals,” Opt. Lett. 34, 1777–1779 (2009).
[CrossRef] [PubMed]

G. Wang, J. P. Huang, and K. W. Yu, “Tunable photonic Bloch oscillations in electrically modulated photonic crystals,” Opt. Lett. 33, 2200–2202 (2008).
[CrossRef] [PubMed]

Yu, Z.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Zheng, M. J.

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Controllable optical Bloch oscillation in planar graded optical waveguide arrays,” Phys. Rev. A 81, 033829 (2010).
[CrossRef]

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Tunable localization and oscillation of coupled plasmon waves in graded plasmonic chains,” J. Appl. Phys. 106, 113307 (2009).
[CrossRef]

J. Appl. Phys. (1)

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Tunable localization and oscillation of coupled plasmon waves in graded plasmonic chains,” J. Appl. Phys. 106, 113307 (2009).
[CrossRef]

J. Opt. Soc. Am. B (1)

Laser Photonics Rev. (1)

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3, 243–261 (2008).
[CrossRef]

Laser Phys. (1)

A. V. Ponomarev and A. R. Kolovsky, “Dipole and Bloch oscillations of cold atoms in a parabolic lattice,” Laser Phys. 16, 367–370 (2006).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

M. J. Zheng, J. J. Xiao, and K. W. Yu, “Controllable optical Bloch oscillation in planar graded optical waveguide arrays,” Phys. Rev. A 81, 033829 (2010).
[CrossRef]

Phys. Rev. B (2)

V. Lousse and S. Fan, “Tunable terahertz Bloch oscillations in chirped photonic crystals,” Phys. Rev. B 72, 075119 (2005).
[CrossRef]

S. Longhi, “Bloch dynamics of light waves in helical optical waveguide arrays,” Phys. Rev. B 76, 195119 (2007).
[CrossRef]

Phys. Rev. Lett. (6)

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999).
[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, 4756–4759 (1999).
[CrossRef]

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch oscillations in an array of curved optical waveguides,” Phys. Rev. Lett. 83, 963–966 (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, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. 96, 053903 (2006).
[CrossRef] [PubMed]

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[CrossRef]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103, 033902 (2009).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[CrossRef]

Other (1)

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram for the POWAs and the input Gaussian beam. The light propagates along the axis of waveguide, that is, the z direction. The waveguide array is labeled by n ( n = 1 , 2 , , N ) . The parabolic propagation constant is described by β ( x , k , S ) as in Eq. (6); H 0 = β ( x , k , 0 ) and H 1 = β ( x , k , 1 ) are applied in the corresponding ranges [ 0 , z 1 ] and [ z 1 , z 2 ] , respectively. The input Gaussian beam has the form as described in Eq. (10), whose cross section is denoted by the green circle. The parameters are N = 100 , n 0 = 86 , k 0 = 0 , σ = 1 , z 1 = 1.39 , and z 2 = 5.32 .

Fig. 2
Fig. 2

(a) Phase diagram for the POWAs with N = 100 waveguides. Separated by the critical curve β = β c , there are three regions representing three kinds of gradon modes, namely, the right-degenerate, left-degenerate, and middle-nondegenerate gradons. Insets show the mode patterns of the three gradon modes and a critical mode. (b) The plot of mean position x S versus eigenvalues β ( S = 0 ) . The abrupt variation of x S indicates the occurrence of BO-DO transition at β c = 4 .

Fig. 3
Fig. 3

(a) A possible BO-DO transition. The arrow marks the shift S. The lift-and-shift procedure is shown by the route A B C D . (b) The phase space orbits in POWA for the cases S = 0 (solid lines) and S = 1 (dashed lines). The solid (dashed) lines 1 ( 1 ) , 2 ( 2 ) , and 3 ( 3 ) are corresponding to DO, critical motion, and BO when S = 0 ( S = 1 ) , respectively. The shift S is shown by an arrow. Points A–D are also marked accordingly.

Fig. 4
Fig. 4

Comparison of Hamiltonian optics results with field-evolution analysis results for (a) x and (b) k in BO-DO transition. Contour plots of field-evolution analysis results for (c) | ψ ( x ) | 2 as a function of the waveguide index n and the propagation distance z and (d) | ϕ ( k ) | 2 as a function of the transverse wavevector k and the propagation distance z.

Equations (14)

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

[ i d d z + V n ] a n ( z ) + a n + 1 ( z ) + a n 1 ( z ) = 0 ,
β m u n m = [ α ( x S ) 2 + α ] u n m + u n + 1 m + u n 1 m ,
β | u = H | u ,
i d d z | u = H | u .
i d d t | ϕ = H | ϕ .
β ( x , k , S ) = 2 ( x S ) 2 + 2 ( 1 + cos   k ) .
H ( p θ , θ ) = p θ 2 2 m L 2 + m g L ( 1 cos   θ ) ,
x S = u m | x S | u m     ( m = 1 , 2 , , N ) ,
d x d z = β ( x , k , S ) k ,     d k d z = β ( x , k , S ) x .
ψ ( 0 ) = 1 ( 2 π σ 2 ) 1 / 4 e ( n n 0 ) 2 / 4 σ 2 e i k 0 ( n n 0 ) ,
| ψ ( 0 ) = m A m | u m ,
| ψ ( z ) = m A m e i β m z | u m .
| ϕ ( k , z ) = F [ | ψ ( x , z ) ] .
x = ψ | x | ψ ψ | ψ ,     k = ϕ | k | ϕ ϕ | ϕ .

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