Abstract

We use Hamiltonian optics to investigate chaotic ray dynamics in a photonic crystal whose lattice parameters vary slowly with position. The ray dynamics are chaotic even in regimes where only stable motion has been found in previous studies of energy band transport. Stable ray paths provide dynamical barriers that localize chaotic motion to certain regions of the photonic crystal.

© 2003 Optical Society of America

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  1. T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
    [CrossRef]
  2. R. G. Scott, S. Bujkiewicz, T. M. Fromhold, P. B. Wilkinson, and F. W. Sheard, Phys. Rev. A 66, 023407 (2002).
    [CrossRef]
  3. A. Patenè, School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK (private communication, December12, 2002).
  4. P. St. J. Russell and T. A. Birks, J. Lightwave Technol. 17, 1982 (1999).
    [CrossRef]
  5. P. B. Wilkinson, Phys. Rev. E 65, 056616 (2002).
    [CrossRef]
  6. T. J. Shepherd, P. J. Roberts, and R. Loudon, Phys. Rev. E 55, 6024 (1997).
    [CrossRef]
  7. P. B. Wilkinson and T. M. Fromhold are preparing a manuscript called “The transition to chaos in chirped two-dimensional photonic crystals.”
  8. S. Sridhar and E. J. Heller, Phys. Rev. A 46, R1728 (1992).
    [CrossRef]
  9. H.-J. Stöckmann, J. Mod. Optics 49, 2045 (2002).
    [CrossRef]
  10. P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. Lett. 86, 5466 (2001).
    [CrossRef] [PubMed]
  11. P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. E 64, 026203 (2001).
    [CrossRef]

2002 (3)

H.-J. Stöckmann, J. Mod. Optics 49, 2045 (2002).
[CrossRef]

P. B. Wilkinson, Phys. Rev. E 65, 056616 (2002).
[CrossRef]

R. G. Scott, S. Bujkiewicz, T. M. Fromhold, P. B. Wilkinson, and F. W. Sheard, Phys. Rev. A 66, 023407 (2002).
[CrossRef]

2001 (3)

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. Lett. 86, 5466 (2001).
[CrossRef] [PubMed]

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. E 64, 026203 (2001).
[CrossRef]

T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
[CrossRef]

1999 (1)

1997 (1)

T. J. Shepherd, P. J. Roberts, and R. Loudon, Phys. Rev. E 55, 6024 (1997).
[CrossRef]

1992 (1)

S. Sridhar and E. J. Heller, Phys. Rev. A 46, R1728 (1992).
[CrossRef]

Birks, T. A.

Bujkiewicz, S.

R. G. Scott, S. Bujkiewicz, T. M. Fromhold, P. B. Wilkinson, and F. W. Sheard, Phys. Rev. A 66, 023407 (2002).
[CrossRef]

T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
[CrossRef]

Eaves, L.

T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
[CrossRef]

Fromhold, T. M.

R. G. Scott, S. Bujkiewicz, T. M. Fromhold, P. B. Wilkinson, and F. W. Sheard, Phys. Rev. A 66, 023407 (2002).
[CrossRef]

T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
[CrossRef]

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. Lett. 86, 5466 (2001).
[CrossRef] [PubMed]

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. E 64, 026203 (2001).
[CrossRef]

P. B. Wilkinson and T. M. Fromhold are preparing a manuscript called “The transition to chaos in chirped two-dimensional photonic crystals.”

Heller, E. J.

S. Sridhar and E. J. Heller, Phys. Rev. A 46, R1728 (1992).
[CrossRef]

Krokhin, A. A.

T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
[CrossRef]

Loudon, R.

T. J. Shepherd, P. J. Roberts, and R. Loudon, Phys. Rev. E 55, 6024 (1997).
[CrossRef]

Micolich, A. P.

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. E 64, 026203 (2001).
[CrossRef]

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. Lett. 86, 5466 (2001).
[CrossRef] [PubMed]

Patenè, A.

A. Patenè, School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK (private communication, December12, 2002).

Roberts, P. J.

T. J. Shepherd, P. J. Roberts, and R. Loudon, Phys. Rev. E 55, 6024 (1997).
[CrossRef]

Russell, P. St. J.

Scott, R. G.

R. G. Scott, S. Bujkiewicz, T. M. Fromhold, P. B. Wilkinson, and F. W. Sheard, Phys. Rev. A 66, 023407 (2002).
[CrossRef]

Sheard, F. W.

R. G. Scott, S. Bujkiewicz, T. M. Fromhold, P. B. Wilkinson, and F. W. Sheard, Phys. Rev. A 66, 023407 (2002).
[CrossRef]

T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
[CrossRef]

Shepherd, T. J.

T. J. Shepherd, P. J. Roberts, and R. Loudon, Phys. Rev. E 55, 6024 (1997).
[CrossRef]

Sridhar, S.

S. Sridhar and E. J. Heller, Phys. Rev. A 46, R1728 (1992).
[CrossRef]

Stöckmann, H.-J.

H.-J. Stöckmann, J. Mod. Optics 49, 2045 (2002).
[CrossRef]

Taylor, R. P.

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. E 64, 026203 (2001).
[CrossRef]

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. Lett. 86, 5466 (2001).
[CrossRef] [PubMed]

Tench, C. R.

T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
[CrossRef]

Wilkinson, P. B.

P. B. Wilkinson, Phys. Rev. E 65, 056616 (2002).
[CrossRef]

R. G. Scott, S. Bujkiewicz, T. M. Fromhold, P. B. Wilkinson, and F. W. Sheard, Phys. Rev. A 66, 023407 (2002).
[CrossRef]

T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
[CrossRef]

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. Lett. 86, 5466 (2001).
[CrossRef] [PubMed]

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. E 64, 026203 (2001).
[CrossRef]

P. B. Wilkinson and T. M. Fromhold are preparing a manuscript called “The transition to chaos in chirped two-dimensional photonic crystals.”

J. Lightwave Technol. (1)

J. Mod. Optics (1)

H.-J. Stöckmann, J. Mod. Optics 49, 2045 (2002).
[CrossRef]

Phys. Rev. A (2)

S. Sridhar and E. J. Heller, Phys. Rev. A 46, R1728 (1992).
[CrossRef]

R. G. Scott, S. Bujkiewicz, T. M. Fromhold, P. B. Wilkinson, and F. W. Sheard, Phys. Rev. A 66, 023407 (2002).
[CrossRef]

Phys. Rev. E (3)

P. B. Wilkinson, Phys. Rev. E 65, 056616 (2002).
[CrossRef]

T. J. Shepherd, P. J. Roberts, and R. Loudon, Phys. Rev. E 55, 6024 (1997).
[CrossRef]

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. E 64, 026203 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

T. M. Fromhold, A. A. Krokhin, C. R. Tench, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard, and L. Eaves, Phys. Rev. Lett. 87, 046803 (2001).
[CrossRef]

P. B. Wilkinson, T. M. Fromhold, R. P. Taylor, and A. P. Micolich, Phys. Rev. Lett. 86, 5466 (2001).
[CrossRef] [PubMed]

Other (2)

P. B. Wilkinson and T. M. Fromhold are preparing a manuscript called “The transition to chaos in chirped two-dimensional photonic crystals.”

A. Patenè, School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK (private communication, December12, 2002).

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Figures (4)

Fig. 1
Fig. 1

Dispersion relations calculated for ly=0.5 µm and (a) lx=0.5 µm, m=1.0 µm, (b) lx=0.5 µm, m=0.9 µm, (c) lx=0.45 µm, m=1.0 µm, and (d) lx=0.55 µm, m=1.0 µm.

Fig. 2
Fig. 2

Schematic plot of the allowed (white) and forbidden (gray) regions for rays corresponding to fixed ω in a slowly varying photonic crystal with θ0.

Fig. 3
Fig. 3

Ray dynamics at θ=0°. (a) Poincaré section showing points corresponding to stable and chaotic ray paths. The symbols +, °, and × indicate initial conditions of ray paths shown in (b), (c), and (d), respectively. Gray regions are forbidden. (b) Stable ray path from inside the dotted–dashed curve in (a). (c) Stable ray paths from outside the dashed curves. Arrows indicate turning points. (d) Chaotic ray path from the region between the dashed and dotted–dashed curves.

Fig. 4
Fig. 4

Ray dynamics at θ=21°. (a) Poincaré section showing small dots (filled circles) corresponding to the chaotic (stable) ray paths shown in (b) [(c)]. (b) Two chaotic ray paths separated by a dynamical barrier. (c) The stable ray path that forms the barrier. Orientation of the x and y axes is shown in the insets in (b) and (c). Gray shading indicates the forbidden regions.

Equations (5)

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Hr,μ,ω=ωlocr,μ-ω,
drdt=Hμ,
dμdt=-Hr.
cosμxlxr=coskxlxr-mrωloc2c2kxsinkxlxr,
cosμylyr=coskylyr-mrωloc2c2kysinkylyr,

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