Abstract

The physics of nonlinear couplers is dictated by its normal modes, modes that are found from linear (axially uniform) couplers. Elementary power-flow arguments establish whether the mode is stable or unstable. These facts provide the bifurcation diagram that fully characterizes nonlinear coupling.

© 1989 Optical Society of America

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References

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  1. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 387–392, 397–399.
  2. S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 (1982).
    [CrossRef]
  3. B. Daino, G. Gregori, S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
    [CrossRef]
  4. Y. Silverberg, G. I. Stegeman, Appl. Phys. Lett. 61, 1135 (1987).
  5. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  6. A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
    [CrossRef]
  7. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zamont, C. T. Seaton, IEEE J. Lightwave Technol. 6, 953 (1988).
    [CrossRef]
  8. A. W. Snyder, D. J. Mitchell, in Digest of Conference on Nonlinear Guided-Wave Phenomena (Optical Society of America, Washington, D.C., 1989), paper PD2.
  9. R. Ulrich, Opt. Lett. 1, 109 (1977).
    [CrossRef] [PubMed]
  10. N. Minorski, Nonlinear Oscillations (Van Nostrand, Princeton, N.J., 1962).
  11. A. W. Snyder, Y. Chen, Opt. Lett. 14, 517 (1989).
    [CrossRef] [PubMed]
  12. A. W. Snyder, D. J. Mitchell, Opt. Lett. 14, 1146 (1989).
    [CrossRef] [PubMed]

1989 (2)

1988 (1)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zamont, C. T. Seaton, IEEE J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

1987 (1)

Y. Silverberg, G. I. Stegeman, Appl. Phys. Lett. 61, 1135 (1987).

1985 (1)

B. Daino, G. Gregori, S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

1982 (1)

S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

1977 (1)

1972 (1)

Chen, Y.

Daino, B.

B. Daino, G. Gregori, S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

Finlayson, N.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zamont, C. T. Seaton, IEEE J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Gregori, G.

B. Daino, G. Gregori, S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

Jensen, S. M.

S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 387–392, 397–399.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Minorski, N.

N. Minorski, Nonlinear Oscillations (Van Nostrand, Princeton, N.J., 1962).

Mitchell, D. J.

A. W. Snyder, D. J. Mitchell, Opt. Lett. 14, 1146 (1989).
[CrossRef] [PubMed]

A. W. Snyder, D. J. Mitchell, in Digest of Conference on Nonlinear Guided-Wave Phenomena (Optical Society of America, Washington, D.C., 1989), paper PD2.

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zamont, C. T. Seaton, IEEE J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Silverberg, Y.

Y. Silverberg, G. I. Stegeman, Appl. Phys. Lett. 61, 1135 (1987).

Snyder, A. W.

A. W. Snyder, D. J. Mitchell, Opt. Lett. 14, 1146 (1989).
[CrossRef] [PubMed]

A. W. Snyder, Y. Chen, Opt. Lett. 14, 517 (1989).
[CrossRef] [PubMed]

A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 387–392, 397–399.

A. W. Snyder, D. J. Mitchell, in Digest of Conference on Nonlinear Guided-Wave Phenomena (Optical Society of America, Washington, D.C., 1989), paper PD2.

Stegeman, G. I.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zamont, C. T. Seaton, IEEE J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Y. Silverberg, G. I. Stegeman, Appl. Phys. Lett. 61, 1135 (1987).

Ulrich, R.

Wabnitz, S.

B. Daino, G. Gregori, S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

Wright, E. M.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zamont, C. T. Seaton, IEEE J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Zamont, R.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zamont, C. T. Seaton, IEEE J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Silverberg, G. I. Stegeman, Appl. Phys. Lett. 61, 1135 (1987).

IEEE J. Lightwave Technol. (1)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zamont, C. T. Seaton, IEEE J. Lightwave Technol. 6, 953 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 (1982).
[CrossRef]

J. Appl. Phys. (1)

B. Daino, G. Gregori, S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (3)

Other (4)

A. W. Snyder, D. J. Mitchell, in Digest of Conference on Nonlinear Guided-Wave Phenomena (Optical Society of America, Washington, D.C., 1989), paper PD2.

N. Minorski, Nonlinear Oscillations (Van Nostrand, Princeton, N.J., 1962).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 387–392, 397–399.

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Figures (4)

Fig. 1
Fig. 1

P 1 ° and P 2 ° are the initial power of modes of cores 1 and 2, respectively. The initial phase between the modal amplitudes is ϕ°. We denote the linear refractive index as n ¯ and nonlinear fluctuations as δn.

Fig. 2
Fig. 2

Normal modes of linear couplers for identical cores (even and odd symmetric modes) and nonidentical cores. The nonidentical cores are characterized by incomplete power transfer.

Fig. 3
Fig. 3

The input conditions necessary for the ϕ° = 0 normal modes of nonlinear couplers, where P = P 1 ° + P 2 °. The arrows represent the direction of power flow resulting from any perturbation from the initial conditions required to excite a normal mode, (a) For identical cores; (b) for the ϕ° = 0 modes of cores with n ¯1 > n ¯2.

Fig. 4
Fig. 4

Two identical parallel cores excited by a fixed ratio P1/P2 of core power at zero phase difference ϕ. Since the core refractive index depends on intensity, at low input power (P → 0) the refractive index n1 = n2, and power initially flows from the core with the greater input to that with the smaller input. For large power (P → ∞), then n1 is significantly greater than n2, and power initially flows from core 2 to core 1. At some intermediate input power (P = P0), n1 is sufficiently greater than n2 which results in zero power flow and thus creates a normal mode.

Equations (4)

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β i ( P i ° ) = B ¯ i + δ β i ( P i ° ) ,
P 1 ° P 2 ° ( P 1 ° P 2 ° ) 1 / 2 = β 1 ( P 1 ° ) β 2 ( P 2 ° ) C ,
P 1 ° P 2 ° = P c 2 / 4 ,
P c = 2 C / κ .

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