Abstract

Starting from Lagrangian principles we develop a formalism suitable for describing coupled optical parity-time symmetric systems.

© 2007 Optical Society of America

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References

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  1. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
    [CrossRef]
  2. C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
    [CrossRef]
  3. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
    [CrossRef]
  4. A. Mostafazadeh, J. Phys. A 36, 7081 (2003).
    [CrossRef]
  5. B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).
    [CrossRef]
  6. A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
    [CrossRef]
  7. K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).
  8. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 14, 817 (2003).
    [CrossRef]
  9. T. Tamir, Guided-Wave Optoelectronics (Springer-Verlag, 1990).
    [CrossRef]

2003 (2)

A. Mostafazadeh, J. Phys. A 36, 7081 (2003).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 14, 817 (2003).
[CrossRef]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

2001 (1)

B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).
[CrossRef]

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

1998 (1)

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

1973 (1)

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[CrossRef]

J. Math. Phys. (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).
[CrossRef]

J. Phys. A (1)

A. Mostafazadeh, J. Phys. A 36, 7081 (2003).
[CrossRef]

Mod. Phys. Lett. A (1)

B. Bagchi, C. Quesne, and M. Znojil, Mod. Phys. Lett. A 16, 2047 (2001).
[CrossRef]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 14, 817 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Other (2)

T. Tamir, Guided-Wave Optoelectronics (Springer-Verlag, 1990).
[CrossRef]

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000).

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Figures (3)

Fig. 1
Fig. 1

PT-coupled waveguide system: (a) waveguide configuration (green represents gain region while yellow stands for loss region) and (b) refractive index (blue) and gain/loss profile (red). M stands for the geometric symmetry axis.

Fig. 2
Fig. 2

Normalized coupling length calculated from supermode analysis (solid curve) compared with that obtained from the PT-CMT (dots) as a function of waveguide separation D. Inset shows a simulation of beam propagation when the separation between the two waveguides is D = 4 .

Fig. 3
Fig. 3

Discrete diffraction in a PT-waveguide array resulting from a single channel excitation.

Equations (7)

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i ϕ ( η , ξ ) ξ + 2 ϕ ( η , ξ ) η 2 + V ( η ) ϕ ( η , ξ ) = 0 ,
i ϕ * ( η , ξ ) ξ + 2 ϕ * ( η , ξ ) η 2 + V ( η ) ϕ * ( η , ξ ) = 0 .
L = i 2 [ ϕ ( η ) ϕ ξ * ( η ) ϕ ξ ( η ) ϕ * ( η ) ] + ϕ η ( η ) ϕ η * ( η ) V ( η ) ϕ ( η ) ϕ * ( η ) .
ϕ ( η , ξ ) = [ a ( ξ ) u 1 ( η ) + b ( ξ ) u 2 ( η ) ] exp ( i μ ξ ) ,
i d a d ξ + Δ a a + κ b = 0
i d b d ξ + Δ b b + κ a = 0 ,
Δ μ = u * ( η ) Δ V ( η ) u ( η ) d η u * ( η ) u ( η ) d η ,

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