Abstract

A simple qualitative description of nonlinear couplers is presented whereby the maximum and minimum values of core power are immediately apparent from a graphical representation, here called the power flow portrait. Each coupler has a characteristic portrait, revealing its unique physics. We believe that this approach is novel to all branches of nonlinear theory. Full details are given for nonlinear couplers with cores composed of different materials, Kerr law being a special case only. The study is motivated from the familiar perspective of linear couplers and is augmented by a comprehensive quantitative description leading to new analytical results.

© 1991 Optical Society of America

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References

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  1. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
    [CrossRef]
  2. A. A. Maier, “Optical transistors and bistable devices utilizing nonlinear transmission of light in systems with unidirectional coupled waves,” Sov. J. Quantum Electron. 12, 1490–1494 (1982).
    [CrossRef]
  3. B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
    [CrossRef]
  4. S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
    [CrossRef]
  5. S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, “Instabilities and all-optical phase-controlled switching in a nonlinear directional coupler,” Appl. Phys. Lett. 49, 834–840 (1986).
    [CrossRef]
  6. Y. Silberberg and G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
    [CrossRef]
  7. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
    [CrossRef]
  8. S. Trillo and S. Wabnitz, “Coupling instability and power induced switching with two-core dual polarizations fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1988).
    [CrossRef]
  9. S. R. Friberg, A. M. Weiner, Y. Silberberg, F. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 10, 904–906 (1988).
    [CrossRef]
  10. G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
    [CrossRef]
  11. A. Ankiewicz, “Design features for an optical fiber transistor,” Proc. Inst. Electr. Eng. Part. J 136, 111–117 (1989).
  12. A. W. Snyder and Y. Chen, “Nonlinear fiber couplers: switches and polarization beam splitters,” Opt. Lett. 14, 517–519 (1989); A. W. Snyder and D. R. Rowland, “Low power fiber coupler devices: few vs many period operation,” submitted to Opt. Quantum Electron.
    [CrossRef] [PubMed]
  13. Y. Chen and A. W. Snyder, “Chaos in a conventional nonlinear coupler,” Opt. Lett. 14, 1237–1239 (1989).
    [CrossRef] [PubMed]
  14. R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
    [CrossRef] [PubMed]
  15. H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
    [CrossRef]
  16. A. S. Davidson and I. H. White, “Initial demonstration of a novel broadband optical amplifier using the Kerr effect in an optical fiber,” Opt. Lett. 14, 802–804 (1989).
    [CrossRef]
  17. A. W. Snyder, Y. Chen, D. R. Rowland, and D. J. Mitchell, “Unification of nonlinear optical fiber devices,” Opt. Lett. 15, 171–173 (1990).
    [CrossRef] [PubMed]
  18. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [CrossRef]
  19. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 387–392, 397–399.
  20. A. W. Snyder and D. J. Mitchell, “Description of nonlinear couplers by power conservation,” Opt. Lett. 14, 1146–1148 (1989).
    [CrossRef] [PubMed]
  21. A. W. Snyder, Y. Chen, D. R. Rowland, and D. J. Mitchell, “Mismatched directional couplers,” Opt. Lett. 15, 357–359 (1990).
    [CrossRef] [PubMed]
  22. D. J. Mitchell and A. W. Snyder, “Modes of nonlinear couplers—building blocks for physical insight,” Opt. Lett. 14, 1143–1145 (1989); D. J. Mitchell, A. W. Snyder, and Y. Chen, “Nonlinear triple core couplers,” Electron. Lett. 26, 1164–1166 (1990).
    [CrossRef] [PubMed]
  23. D. J. Mitchell, Y. Chen, and A. W. Snyder, “Directional couplers composed of non-Kerr law material,” Opt. Lett. 15, 535–537 (1990).
    [CrossRef] [PubMed]
  24. A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler used as an amplifier with twin biasing beams,” Int. J. Optoelectron. 5, 381–386 (1990).
  25. K. Kitayama, Y. Kimura, and S. Seikei, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
    [CrossRef]
  26. D. R. Rowland, “All-optical devices using nonlinear fiber couplers,” IEEE J. Lightwave Technol. (to be published).

1990 (4)

1989 (7)

1988 (3)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

S. Trillo and S. Wabnitz, “Coupling instability and power induced switching with two-core dual polarizations fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1988).
[CrossRef]

S. R. Friberg, A. M. Weiner, Y. Silberberg, F. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 10, 904–906 (1988).
[CrossRef]

1987 (1)

Y. Silberberg and G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

1986 (2)

S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
[CrossRef]

S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, “Instabilities and all-optical phase-controlled switching in a nonlinear directional coupler,” Appl. Phys. Lett. 49, 834–840 (1986).
[CrossRef]

1985 (3)

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

K. Kitayama, Y. Kimura, and S. Seikei, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

1982 (3)

R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
[CrossRef] [PubMed]

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

A. A. Maier, “Optical transistors and bistable devices utilizing nonlinear transmission of light in systems with unidirectional coupled waves,” Sov. J. Quantum Electron. 12, 1490–1494 (1982).
[CrossRef]

1972 (1)

Ankiewicz, A.

A. Ankiewicz, “Design features for an optical fiber transistor,” Proc. Inst. Electr. Eng. Part. J 136, 111–117 (1989).

Ashkin, A.

Binh, L. N.

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler used as an amplifier with twin biasing beams,” Int. J. Optoelectron. 5, 381–386 (1990).

Botineau, J.

Chen, Y.

Daino, B.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

Davidson, A. S.

Finlayson, N.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Friberg, S. R.

S. R. Friberg, A. M. Weiner, Y. Silberberg, F. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 10, 904–906 (1988).
[CrossRef]

Gregori, G.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Kimura, Y.

K. Kitayama, Y. Kimura, and S. Seikei, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Kitayama, K.

K. Kitayama, Y. Kimura, and S. Seikei, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Love, J. D.

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

Maier, A. A.

A. A. Maier, “Optical transistors and bistable devices utilizing nonlinear transmission of light in systems with unidirectional coupled waves,” Sov. J. Quantum Electron. 12, 1490–1494 (1982).
[CrossRef]

Mitchell, D. J.

Pham, A. T.

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler used as an amplifier with twin biasing beams,” Int. J. Optoelectron. 5, 381–386 (1990).

Rowland, D. R.

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, “Instabilities and all-optical phase-controlled switching in a nonlinear directional coupler,” Appl. Phys. Lett. 49, 834–840 (1986).
[CrossRef]

Seikei, S.

K. Kitayama, Y. Kimura, and S. Seikei, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

Sfez, F. G.

S. R. Friberg, A. M. Weiner, Y. Silberberg, F. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 10, 904–906 (1988).
[CrossRef]

Silberberg, Y.

S. R. Friberg, A. M. Weiner, Y. Silberberg, F. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 10, 904–906 (1988).
[CrossRef]

Y. Silberberg and G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

Smith, P. S.

S. R. Friberg, A. M. Weiner, Y. Silberberg, F. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 10, 904–906 (1988).
[CrossRef]

Snyder, A. W.

A. W. Snyder, Y. Chen, D. R. Rowland, and D. J. Mitchell, “Mismatched directional couplers,” Opt. Lett. 15, 357–359 (1990).
[CrossRef] [PubMed]

D. J. Mitchell, Y. Chen, and A. W. Snyder, “Directional couplers composed of non-Kerr law material,” Opt. Lett. 15, 535–537 (1990).
[CrossRef] [PubMed]

A. W. Snyder, Y. Chen, D. R. Rowland, and D. J. Mitchell, “Unification of nonlinear optical fiber devices,” Opt. Lett. 15, 171–173 (1990).
[CrossRef] [PubMed]

D. J. Mitchell and A. W. Snyder, “Modes of nonlinear couplers—building blocks for physical insight,” Opt. Lett. 14, 1143–1145 (1989); D. J. Mitchell, A. W. Snyder, and Y. Chen, “Nonlinear triple core couplers,” Electron. Lett. 26, 1164–1166 (1990).
[CrossRef] [PubMed]

A. W. Snyder and D. J. Mitchell, “Description of nonlinear couplers by power conservation,” Opt. Lett. 14, 1146–1148 (1989).
[CrossRef] [PubMed]

Y. Chen and A. W. Snyder, “Chaos in a conventional nonlinear coupler,” Opt. Lett. 14, 1237–1239 (1989).
[CrossRef] [PubMed]

A. W. Snyder and Y. Chen, “Nonlinear fiber couplers: switches and polarization beam splitters,” Opt. Lett. 14, 517–519 (1989); A. W. Snyder and D. R. Rowland, “Low power fiber coupler devices: few vs many period operation,” submitted to Opt. Quantum Electron.
[CrossRef] [PubMed]

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
[CrossRef]

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

Stegeman, G. I.

G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Y. Silberberg and G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, “Instabilities and all-optical phase-controlled switching in a nonlinear directional coupler,” Appl. Phys. Lett. 49, 834–840 (1986).
[CrossRef]

Stolen, R. H.

Trillo, S.

S. Trillo and S. Wabnitz, “Coupling instability and power induced switching with two-core dual polarizations fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1988).
[CrossRef]

S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
[CrossRef]

Wabnitz, S.

S. Trillo and S. Wabnitz, “Coupling instability and power induced switching with two-core dual polarizations fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1988).
[CrossRef]

S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
[CrossRef]

S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, “Instabilities and all-optical phase-controlled switching in a nonlinear directional coupler,” Appl. Phys. Lett. 49, 834–840 (1986).
[CrossRef]

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

Weiner, A. M.

S. R. Friberg, A. M. Weiner, Y. Silberberg, F. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 10, 904–906 (1988).
[CrossRef]

White, I. H.

Winful, H. G.

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

Wright, E. M.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, “Instabilities and all-optical phase-controlled switching in a nonlinear directional coupler,” Appl. Phys. Lett. 49, 834–840 (1986).
[CrossRef]

Zanoni, R.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Appl. Phys. Lett. (5)

S. Trillo and S. Wabnitz, “Nonlinear nonreciprocity in a coherent mismatched directional coupler,” Appl. Phys. Lett. 49, 752–754 (1986).
[CrossRef]

S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, “Instabilities and all-optical phase-controlled switching in a nonlinear directional coupler,” Appl. Phys. Lett. 49, 834–840 (1986).
[CrossRef]

Y. Silberberg and G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

K. Kitayama, Y. Kimura, and S. Seikei, “Fiber-optic logic gate,” Appl. Phys. Lett. 46, 317–319 (1985).
[CrossRef]

IEEE J. Lightwave Technol. (1)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” IEEE J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Int. J. Optoelectron. (1)

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler used as an amplifier with twin biasing beams,” Int. J. Optoelectron. 5, 381–386 (1990).

J. Appl. Phys. (1)

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Lett. (10)

S. R. Friberg, A. M. Weiner, Y. Silberberg, F. G. Sfez, and P. S. Smith, “Femtosecond switching in a dual-core-fiber nonlinear coupler,” Opt. Lett. 10, 904–906 (1988).
[CrossRef]

A. S. Davidson and I. H. White, “Initial demonstration of a novel broadband optical amplifier using the Kerr effect in an optical fiber,” Opt. Lett. 14, 802–804 (1989).
[CrossRef]

A. W. Snyder, Y. Chen, D. R. Rowland, and D. J. Mitchell, “Unification of nonlinear optical fiber devices,” Opt. Lett. 15, 171–173 (1990).
[CrossRef] [PubMed]

A. W. Snyder and Y. Chen, “Nonlinear fiber couplers: switches and polarization beam splitters,” Opt. Lett. 14, 517–519 (1989); A. W. Snyder and D. R. Rowland, “Low power fiber coupler devices: few vs many period operation,” submitted to Opt. Quantum Electron.
[CrossRef] [PubMed]

Y. Chen and A. W. Snyder, “Chaos in a conventional nonlinear coupler,” Opt. Lett. 14, 1237–1239 (1989).
[CrossRef] [PubMed]

R. H. Stolen, J. Botineau, and A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512–514 (1982).
[CrossRef] [PubMed]

A. W. Snyder and D. J. Mitchell, “Description of nonlinear couplers by power conservation,” Opt. Lett. 14, 1146–1148 (1989).
[CrossRef] [PubMed]

A. W. Snyder, Y. Chen, D. R. Rowland, and D. J. Mitchell, “Mismatched directional couplers,” Opt. Lett. 15, 357–359 (1990).
[CrossRef] [PubMed]

D. J. Mitchell and A. W. Snyder, “Modes of nonlinear couplers—building blocks for physical insight,” Opt. Lett. 14, 1143–1145 (1989); D. J. Mitchell, A. W. Snyder, and Y. Chen, “Nonlinear triple core couplers,” Electron. Lett. 26, 1164–1166 (1990).
[CrossRef] [PubMed]

D. J. Mitchell, Y. Chen, and A. W. Snyder, “Directional couplers composed of non-Kerr law material,” Opt. Lett. 15, 535–537 (1990).
[CrossRef] [PubMed]

Proc. Inst. Electr. Eng. Part. J (1)

A. Ankiewicz, “Design features for an optical fiber transistor,” Proc. Inst. Electr. Eng. Part. J 136, 111–117 (1989).

Sov. J. Quantum Electron. (1)

A. A. Maier, “Optical transistors and bistable devices utilizing nonlinear transmission of light in systems with unidirectional coupled waves,” Sov. J. Quantum Electron. 12, 1490–1494 (1982).
[CrossRef]

Other (2)

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

D. R. Rowland, “All-optical devices using nonlinear fiber couplers,” IEEE J. Lightwave Technol. (to be published).

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Figures (15)

Fig. 1
Fig. 1

(a) Schematic of coupler geometry showing the input conditions; P1 and P2 are the powers, and ϕ is the relative phase difference between the cores. (b) Normal modes of matched and mismatched linear couplers, where β1 and β2 are defined in relation to Eq. (1).

Fig. 2
Fig. 2

Power flow portrait for a linear coupler with mismatched (β1 > β2) cores. The arrows show the directions of initial power flow for excitations with phase difference ϕ = 0 or π. The input conditions that excite modes are shown as solid curves, with P1/P given by Q±(M) from Eq. (2b).

Fig. 3
Fig. 3

Power flow portrait showing the modes of a Kerr law coupler with identical cores. The solid curve, as in Fig. 2, is for stable modes. The dotted curve indicates unstable modes. The bottom two diagrams label the various regions of maxima and minima referred to in the text.

Fig. 4
Fig. 4

Power flow portrait for the Kerr law coupler of Fig. 3 but now showing the separatrix (dashed curves). The corresponding maxima and minima must lie in a similarly shaded region. The bottom two diagrams label the various regions of maxima and minima referred to in the text.

Fig. 5
Fig. 5

(a) Mechanism for complete power exchange on a nonlinear coupler when the initial power induces a mismatch (β1ϕ2) between the cores that permits at least 50% power transfer over some length. (b) Mechanism for power transfer when the initial power induces a mismatch that is too large for 50% power exchange. The shading in the cores represents the magnitude of the intensity-dependent refractive index.

Fig. 6
Fig. 6

Kerr law coupler with identical cores. (a) Variation of power in core 1 as a function of distance z along the coupler for six different input powers when all the power is initially in core 1 and when LC is the length over which complete power transfer occurs for a linear coupler. (b) Period zp of power transfer between the cores, where z p ¯ is the period for a linear (P → 0) coupler. Numerical results are found by the solution of Eq. (A3).

Fig. 7
Fig. 7

Maximum and minimum values of the power in core 1 a half-period along the coupler corresponding to initial excitations that are small perturbations from P1/P = 0.5. The curves shown for the subsequent maximum or minimum are given directly by the location of the separatrix in Fig. 4. The actual numerical values are found by the solution of the algebraic equation (A13) for the Kerr law coupler with identical cores.

Fig. 8
Fig. 8

(a) Power flow portrait illustrating a particular fixed input ratio P10/P (open boxes) and the associated maximum or minimum values a half-period along the coupler (filled boxes). (b) These maximum or minimum values of P1/P shown explicitly versus input power P.

Fig. 9
Fig. 9

(a) Power flow portrait illustrating a fixed total input power P = P0 and a changing fraction of initial core power P10/P, given by the labeled squares. (b) Resultant values of P1/P after a half-period versus initial P10/P.

Fig. 10
Fig. 10

(a) Power flow portrait for the saturating nonlinearity given in the text with 2 P B < P S < ( 2 + 1 ) P B. (b) Power flow portrait for the saturating nonlinearity with P S > ( 2 + 1 ) P B. (c) Power flow portrait for the power-law nonlinearity of Subsection 3.A.1 with exponent n 4.

Fig. 11
Fig. 11

Power flow portrait for a degenerate Kerr law coupler with β1 > β2 and K1, K2 satisfying Eq. (5). This portrait, like that in Fig. 4, is called degenerate. PB = (β+β+)/K2.

Fig. 12
Fig. 12

Power flow portraits for nearly degenerate asymmetric Kerr law couplers. The degeneracy of the bifurcation can be broken in two ways: the portrait in (a) is obtained if the linear mismatch M ¯ of Table 3 is greater than that required to satisfy Eq. (5), and the portrait in (b) is obtained if M ¯ is less than that required to satisfy Eq. (5). PB = (β+β+)/K2.

Fig. 13
Fig. 13

Different generic power flow portraits obtained for general Kerr law couplers. Only the modes on the ϕ = 0 graph are shown. PB = (β+β+)/K2. Each portrait is associated with a range of parameters indicated in Fig. 14.

Fig. 14
Fig. 14

Complete classification of qualitatively different types of portrait possible for Kerr law couplers with nonidentical cores in terms of two parameters: the linear mismatch M ¯ of Table 3 and the ratio K1/K2 of Kerr coefficients in cores 1 and 2. The solid curves correspond to those parameters giving nongeneric portraits. The regions between the curves correspond to the generic portraits. Representative portraits are given for the various regions and boundaries for ϕ = 0 excitation.

Fig. 15
Fig. 15

(a) Power flow portrait for a nonidentical core Kerr law coupler of the type in Fig. 13(c). (b) Power in core 1 as a function of distance along the coupler for inputs corresponding to those labeled in (a).

Tables (3)

Tables Icon

Table 1 Constructing the Portrait

Tables Icon

Table 2 Reading the Portrait

Equations (48)

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M = ( β 1 - β 2 ) / ( β + - β - ) .
± P 1 - P 2 2 P 1 P 2 - M = 0 ,
P 1 P = Q ± ( M ) = 1 2 ± M 2 1 + M 2 .
K ( P 1 - P 2 ) β + - β - = ± P 1 - P 2 2 P 1 P 2 .
β 1 - β 2 = β ¯ 1 - β ¯ 2 + K 1 P 1 - K 2 P 2 ,
K 1 - K 2 K 1 + K 2 + M ¯ 1 + M ¯ 2 = 0 ,
β 1 ( P 1 ) - β 2 ( P 2 ) β + - β - = ± P 1 - P 2 2 P 1 P 2
ψ ( x , y , z ) = a 1 ( z ) ψ 1 ( x , y ) + a 2 ( z ) ψ 2 ( x , y ) ,
β + - β - = 2 k A [ n ( x , y ) - n 1 ( x , y ) ] ψ 1 ψ 2 d A ,
S ˙ 1 = ( β + - β - ) S 3 ,
S ˙ 2 = - ( β 1 - β 2 ) S 3 ,
S ˙ 3 = ( β 1 - β 2 ) S 2 - ( β + - β - ) S 1 ,
S 1 = P 1 - P 2 ,
S 2 = 2 P 1 P 2 cos ϕ ,
S 3 = 2 P 1 P 2 sin ϕ
β i = β ¯ + k A δ n i ψ i 2 d A ,
β 1 ( P 1 ) - β 2 ( P 2 ) β + - β - = S 1 S 2 = ± P 1 - P 2 2 P 1 P 2 ,
Γ = γ ( S 1 , P ) + S 2 ,
γ ( S 1 , P ) = 0 S 1 M ( S 1 , P ) d S 1 ,
M = M ¯ + ( K 1 - K 2 ) P 2 ( β + - β - ) + ( K 1 + K 2 ) S 1 2 ( β + - β - ) ,
Γ = M ¯ S 1 + ( K 1 - K 2 ) P S 1 2 ( β + - β - ) + ( K 1 + K 2 ) S 1 2 4 ( β + - β - ) + S 2 .
γ ( 2 P 1 * - P ) ± 2 P 1 * ( P - P 1 * ) = γ ( 2 P 1 u - P ) + 2 P 1 u ( P - P 1 u ) ,
P = 2 P B 2 x - 1 [ 1 2 x ( 1 - x ) ] ,
γ ( P 1 in - P 2 in ) + 2 P 1 in P 2 in cos ϕ in = γ ( P 1 out - P 2 out ) + 2 P 1 out P 2 out cos ϕ out ,
K ( P 1 in - P 2 in ) 2 2 ( β + - β - ) ± 2 P 1 in P 2 in = K ( P 1 out - P 2 out ) 2 2 ( β + - β - ) ± 2 P 1 out P 2 out ,
S ¨ 1 = - ( β + - β - ) [ ( β + - β - ) - P d β ( P / 2 ) d P ] S 1 .
P d β ( P / 2 ) d P < β + - β - .
P B d β ( P B / 2 ) d P = β + - β - ,
β ( P 1 ) - β ( P 2 ) β + - β - < 1 2 log ( P 1 P 2 ) < P 1 - P 2 2 P 1 P 2
β ( P 1 ) - β ( P 2 ) = 0 log ( P 1 / P 2 ) P d β d P d [ log ( P P 2 ) ]
P = [ ± 2 x - 1 2 x ( 1 - x ) - M ¯ ] [ β + - β - ( K 1 + K 2 ) x - K 2 ] ,
K 1 - K 2 K 1 + K 2 = - M ¯ 1 + M ¯ 2 ,
K 1 = 0 ,
K 2 = 0 ,
K 1 - K 2 K 1 + K 2 = - M ¯ .
z p = d S 1 S ˙ 1 = 2 β + - β - min S 1 max S 1 d S 1 S 3 ,
S 3 = P 2 - S 1 2 - [ Γ - γ ( S 1 ) ] 2 .
z = 1 β + - β - S 1 ( 0 ) S 1 ( z ) d S 1 S 3 .
S 3 2 2 S 2 M ( Γ M - Γ ) + α ¯ 2 ( S 1 - S 1 M ) 2 + ,
α ¯ 2 = S 2 M d M d S 1 - 1 - M 2
z p ~ - ( 1 / α ) log ( Γ - Γ M ) + const . ,
z p ~ - ( 2 / α ) log ( Γ - Γ M ) + const .
z p ~ - 2 ( β + - β - ) P / P B - 1 log ( | Δ P B | ) .
z p ~ - 1 ( β + - β - ) P / P B - 1 log ( | Δ P B | ) .
z p ~ - 2 ( β + - β - ) P / P B - 1 log ( | Δ P B | ) .
S 1 ( z ) = S 1 ( 0 ) cosh ( α z ) ~ 1 2 S 1 ( 0 ) exp ( α z ) ,
A = ½ exp ( α L ) .
α = ( β + - β - ) P / P B - 1 .

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