Abstract

The propagation of an optical pulse in a coupled-resonator optical waveguide may be calculated nonperturbatively to all orders of dispersion, in the conventional tight-binding approximation, even though the dispersion relationship is nonlinear. Working in this framework, we discuss limits of the physical parameters and approximations to the exact formulation that highlight the conditions under which pulse distortion can be minimized. The results are fundamental to the design of coupled-resonator optical waveguides and are also relevant to other applications of the tight-binding method.

© 2002 Optical Society of America

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References

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  1. S. Mookherjea and A. Yariv, “Coupled-resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. (to be published).
  2. A. Yariv, Y. Xu, R. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999).
    [CrossRef]
  3. Y. Xu, R. Lee, and A. Yariv, Opt. Lett. 25, 755 (2000).
    [CrossRef]
  4. T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, Phys. Rev. Lett. 82, 4623 (1999).
    [CrossRef]
  5. S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterlé, Opt. Lett. 26, 1019 (2001).
    [CrossRef]
  6. M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
    [CrossRef]
  7. See, e.g., K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2001), Chap. 6.
    [CrossRef]
  8. S. Mookherjea and A. Yariv, Opt. Express 9, 91 (2001), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  9. S. Mookherjea and A. Yariv, Phys. Rev. E 64, 066602 (2001).
    [CrossRef]
  10. S. Mookherjea and A. Yariv, “Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion,” Phys. Rev. E (to be published).
  11. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  12. G. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, England, 1944).

2001

2000

1999

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, Phys. Rev. Lett. 82, 4623 (1999).
[CrossRef]

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

A. Yariv, Y. Xu, R. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

Bayer, M.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Benisty, H.

Forchel, A.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Gutbrod, T.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Houdré, R.

Jimba, Y.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, Phys. Rev. Lett. 82, 4623 (1999).
[CrossRef]

Knipp, P. A.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Krauss, T.

Kuwata-Gonokami, M.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, Phys. Rev. Lett. 82, 4623 (1999).
[CrossRef]

Lee, R.

Miyazaki, H.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, Phys. Rev. Lett. 82, 4623 (1999).
[CrossRef]

Mookherjea, S.

S. Mookherjea and A. Yariv, Opt. Express 9, 91 (2001), http://www.opticsexpress.org .
[CrossRef] [PubMed]

S. Mookherjea and A. Yariv, Phys. Rev. E 64, 066602 (2001).
[CrossRef]

S. Mookherjea and A. Yariv, “Coupled-resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. (to be published).

S. Mookherjea and A. Yariv, “Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion,” Phys. Rev. E (to be published).

Mukaiyama, T.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, Phys. Rev. Lett. 82, 4623 (1999).
[CrossRef]

Oesterlé, U.

Olivier, S.

Rattier, M.

Reinecke, T. L.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Rethmaier, J. P.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Sakoda, K.

See, e.g., K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2001), Chap. 6.
[CrossRef]

Scherer, A.

Smith, C.

Takeda, K.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, Phys. Rev. Lett. 82, 4623 (1999).
[CrossRef]

Watson, G.

G. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, England, 1944).

Weisbuch, C.

Wener, R.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

Xu, Y.

Yariv, A.

S. Mookherjea and A. Yariv, Phys. Rev. E 64, 066602 (2001).
[CrossRef]

S. Mookherjea and A. Yariv, Opt. Express 9, 91 (2001), http://www.opticsexpress.org .
[CrossRef] [PubMed]

Y. Xu, R. Lee, and A. Yariv, Opt. Lett. 25, 755 (2000).
[CrossRef]

A. Yariv, Y. Xu, R. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999).
[CrossRef]

S. Mookherjea and A. Yariv, “Coupled-resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. (to be published).

S. Mookherjea and A. Yariv, “Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion,” Phys. Rev. E (to be published).

Opt. Express

Opt. Lett.

Phys. Rev. E

S. Mookherjea and A. Yariv, Phys. Rev. E 64, 066602 (2001).
[CrossRef]

Phys. Rev. Lett.

M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Wener, and J. P. Rethmaier, Phys. Rev. Lett. 83, 5374 (1999).
[CrossRef]

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, and M. Kuwata-Gonokami, Phys. Rev. Lett. 82, 4623 (1999).
[CrossRef]

Other

See, e.g., K. Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2001), Chap. 6.
[CrossRef]

S. Mookherjea and A. Yariv, “Pulse propagation in a coupled-resonator optical waveguide to all orders of dispersion,” Phys. Rev. E (to be published).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

G. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, England, 1944).

S. Mookherjea and A. Yariv, “Coupled-resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. (to be published).

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Figures (2)

Fig. 1
Fig. 1

Schematic of an infinitely long CROW with periodicity R along the z direction, consisting of defect cavities embedded in a two-dimensional photonic crystal.

Fig. 2
Fig. 2

Temporal evolution of a Gaussian envelope at specific distances inside a CROW, showing the effects of dispersive propagation. At greater depths, the peak of the envelope arrives at a later time, and ripples in the trailing edge indicate higher-order distortion.

Equations (13)

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

ϕkr=nexp-inkRlψlr-nRez,
ωk0+K=Ω1-Δα/2+Ωκ cosKRω0+Δω cosKR,
Δω=Ωd3rϵresr-Rez-ϵwgr-Rezψlr·ψlr-Rez,
Er,t=expiω0t-π/Rπ/RdK2π×expiΔωt cosKRck0+Kϕk0+Kr.
Er,z=0,t=expiω0tEz=0,tGr,
Ez=0,tGr=-π/Rπ/RdK2π×expiΔωt cosKRck0+Kϕk0+Kr,z=0.
βn=12Rck0n=0nin0dttE0,t-E0,0JnΔωtn1,
Er,t=14expiω0tm=0bmJmΔωtn=0βnGr×exp-i±m±n+k0Rlψlr-±m±n+Rez,
bm=1m=02imm1.
Ez,t=14n=-z/Rb˜nJmΔωtβz/R-nlψlz=0×14n=z/RbnJnΔωtβn-z/Rlψlz=0,
b˜n=2b0n=0bnn0.
n0dxxJnbx=1
βn=i-nEz=0,n/Δω,  n1;

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