Abstract

We present the operational principles and analytical characteristics of several all-optical modulators and switches that incorporate nonlinear directional couplers with either single or double biasing beams. Graphic representation and potential-well models are used effectively to describe the device’s operation and enhance the ability to anticipate its performance. Phase-to-amplitude modulation is considered, which leads to the special case of phase-controlled switching. Amplitude-to-amplitude modulation is studied, which leads to optical amplification phenomena. An intensity-sensitive switch is also examined.

© 1991 Optical Society of America

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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, “Self-switching of light in a directional coupler,” Sov. J. Quantum Electron. 14, 101–104 (1984).
    [CrossRef]
  3. B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
    [CrossRef]
  4. A. A. Maier, “Switching of radiation in tunnel-coupled optical waveguides by weak radiation frequency,” Sov. J. Quantum Electron. 16, 892–897 (1986).
    [CrossRef]
  5. M. D. Feit and J. A. Fleck, “Three-dimensional analysis of a directional coupler exhibiting a Kerr nonlinearity,” IEEE J. Quantum Electron. 24, 2081–2086 (1988).
    [CrossRef]
  6. K. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, 17–19 (1983).
    [CrossRef]
  7. R. Hoffe and J. Chrostowski, “Optical pulse compression and breaking in nonlinear fibre couplers,” Opt. Commun. 57, 34–38 (1986).
    [CrossRef]
  8. P. Li, Kam Wa, J. E. Stich, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Electron. Lett. 21, 26–27 (1985).
    [CrossRef]
  9. D. D. GusovskiiE, M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
    [CrossRef]
  10. P. R. Berger, Y. Chen, P. Bhattachaya, J. Pamulapati, and G. C. Vezzoli, “Demonstration of all-optical modulation in a vertical guided-wave nonlinear coupler,” Appl. Phys. Lett. 52, 1125–1127 (1988).
    [CrossRef]
  11. S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, “Femotosecond switching in a dual-core-fibre nonlinear coupler,” Opt. Lett. 13, 904–905 (1988).
    [CrossRef] [PubMed]
  12. S. R. Friberg and P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
    [CrossRef]
  13. A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler—two-input operations,” Int. J. Optoelectron. 5, 367–380 (1990).
  14. A. T. Pham and L. N. Binh, “Graphical and stability analysis of nonlinear optical directional couplers using two-input biasing beams,” submitted to Int. J. Opt. Comput.
  15. A. Ankiewicz, “Design features for an optical fibre transistor,” Proc. Inst. Electr. Eng. 136, 111–117 (1989).
  16. 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, 838–840 (1986).
    [CrossRef]
  17. S. Trillo and S. Wabnitz, “Coupling instability and power-induced switching with two-core dual-polarization fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1988).
    [CrossRef]

1990 (1)

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler—two-input operations,” Int. J. Optoelectron. 5, 367–380 (1990).

1989 (1)

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

1988 (4)

P. R. Berger, Y. Chen, P. Bhattachaya, J. Pamulapati, and G. C. Vezzoli, “Demonstration of all-optical modulation in a vertical guided-wave nonlinear coupler,” Appl. Phys. Lett. 52, 1125–1127 (1988).
[CrossRef]

S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, “Femotosecond switching in a dual-core-fibre nonlinear coupler,” Opt. Lett. 13, 904–905 (1988).
[CrossRef] [PubMed]

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

M. D. Feit and J. A. Fleck, “Three-dimensional analysis of a directional coupler exhibiting a Kerr nonlinearity,” IEEE J. Quantum Electron. 24, 2081–2086 (1988).
[CrossRef]

1987 (1)

S. R. Friberg and P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
[CrossRef]

1986 (3)

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, 838–840 (1986).
[CrossRef]

R. Hoffe and J. Chrostowski, “Optical pulse compression and breaking in nonlinear fibre couplers,” Opt. Commun. 57, 34–38 (1986).
[CrossRef]

A. A. Maier, “Switching of radiation in tunnel-coupled optical waveguides by weak radiation frequency,” Sov. J. Quantum Electron. 16, 892–897 (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]

P. Li, Kam Wa, J. E. Stich, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Electron. Lett. 21, 26–27 (1985).
[CrossRef]

D. D. GusovskiiE, M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

1984 (1)

A. A. Maier, “Self-switching of light in a directional coupler,” Sov. J. Quantum Electron. 14, 101–104 (1984).
[CrossRef]

1983 (1)

K. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, 17–19 (1983).
[CrossRef]

1982 (1)

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

Ankiewicz, A.

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

Berger, P. R.

P. R. Berger, Y. Chen, P. Bhattachaya, J. Pamulapati, and G. C. Vezzoli, “Demonstration of all-optical modulation in a vertical guided-wave nonlinear coupler,” Appl. Phys. Lett. 52, 1125–1127 (1988).
[CrossRef]

Bhattachaya, P.

P. R. Berger, Y. Chen, P. Bhattachaya, J. Pamulapati, and G. C. Vezzoli, “Demonstration of all-optical modulation in a vertical guided-wave nonlinear coupler,” Appl. Phys. Lett. 52, 1125–1127 (1988).
[CrossRef]

Binh, L. N.

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler—two-input operations,” Int. J. Optoelectron. 5, 367–380 (1990).

A. T. Pham and L. N. Binh, “Graphical and stability analysis of nonlinear optical directional couplers using two-input biasing beams,” submitted to Int. J. Opt. Comput.

Chen, Y.

P. R. Berger, Y. Chen, P. Bhattachaya, J. Pamulapati, and G. C. Vezzoli, “Demonstration of all-optical modulation in a vertical guided-wave nonlinear coupler,” Appl. Phys. Lett. 52, 1125–1127 (1988).
[CrossRef]

Chrostowski, J.

R. Hoffe and J. Chrostowski, “Optical pulse compression and breaking in nonlinear fibre couplers,” Opt. Commun. 57, 34–38 (1986).
[CrossRef]

Daino, B.

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

Dianov, M.

D. D. GusovskiiE, M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Feit, M. D.

M. D. Feit and J. A. Fleck, “Three-dimensional analysis of a directional coupler exhibiting a Kerr nonlinearity,” IEEE J. Quantum Electron. 24, 2081–2086 (1988).
[CrossRef]

Fleck, J. A.

M. D. Feit and J. A. Fleck, “Three-dimensional analysis of a directional coupler exhibiting a Kerr nonlinearity,” IEEE J. Quantum Electron. 24, 2081–2086 (1988).
[CrossRef]

Friberg, S. R.

S. R. Friberg, A. M. Weiner, Y. Silberberg, B. G. Sfez, and P. S. Smith, “Femotosecond switching in a dual-core-fibre nonlinear coupler,” Opt. Lett. 13, 904–905 (1988).
[CrossRef] [PubMed]

S. R. Friberg and P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
[CrossRef]

Gregori, G.

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

GusovskiiE, D. D.

D. D. GusovskiiE, M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Hoffe, R.

R. Hoffe and J. Chrostowski, “Optical pulse compression and breaking in nonlinear fibre couplers,” Opt. Commun. 57, 34–38 (1986).
[CrossRef]

Jensen, S. M.

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

Kitayama, K.

K. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, 17–19 (1983).
[CrossRef]

Li, P.

P. Li, Kam Wa, J. E. Stich, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Electron. Lett. 21, 26–27 (1985).
[CrossRef]

Maier, A. A.

A. A. Maier, “Switching of radiation in tunnel-coupled optical waveguides by weak radiation frequency,” Sov. J. Quantum Electron. 16, 892–897 (1986).
[CrossRef]

D. D. GusovskiiE, M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

A. A. Maier, “Self-switching of light in a directional coupler,” Sov. J. Quantum Electron. 14, 101–104 (1984).
[CrossRef]

Mason, N. J.

P. Li, Kam Wa, J. E. Stich, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Electron. Lett. 21, 26–27 (1985).
[CrossRef]

Neustruev, V. B.

D. D. GusovskiiE, M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Pamulapati, J.

P. R. Berger, Y. Chen, P. Bhattachaya, J. Pamulapati, and G. C. Vezzoli, “Demonstration of all-optical modulation in a vertical guided-wave nonlinear coupler,” Appl. Phys. Lett. 52, 1125–1127 (1988).
[CrossRef]

Pham, A. T.

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler—two-input operations,” Int. J. Optoelectron. 5, 367–380 (1990).

A. T. Pham and L. N. Binh, “Graphical and stability analysis of nonlinear optical directional couplers using two-input biasing beams,” submitted to Int. J. Opt. Comput.

Roberts, J. S.

P. Li, Kam Wa, J. E. Stich, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Electron. Lett. 21, 26–27 (1985).
[CrossRef]

Robson, P. N.

P. Li, Kam Wa, J. E. Stich, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Electron. Lett. 21, 26–27 (1985).
[CrossRef]

Seaton, C. T.

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, 838–840 (1986).
[CrossRef]

Sfez, B. G.

Shcherbakov, I. A.

D. D. GusovskiiE, M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Shklovskii, E. I.

D. D. GusovskiiE, M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Silberberg, Y.

Smith, P. S.

Smith, P. W.

S. R. Friberg and P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
[CrossRef]

Stegeman, G. I.

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, 838–840 (1986).
[CrossRef]

Stich, J. E.

P. Li, Kam Wa, J. E. Stich, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Electron. Lett. 21, 26–27 (1985).
[CrossRef]

Trillo, S.

Vezzoli, G. C.

P. R. Berger, Y. Chen, P. Bhattachaya, J. Pamulapati, and G. C. Vezzoli, “Demonstration of all-optical modulation in a vertical guided-wave nonlinear coupler,” Appl. Phys. Lett. 52, 1125–1127 (1988).
[CrossRef]

Wa, Kam

P. Li, Kam Wa, J. E. Stich, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Electron. Lett. 21, 26–27 (1985).
[CrossRef]

Wabnitz, S.

S. Trillo and S. Wabnitz, “Coupling instability and power-induced switching with two-core dual-polarization fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (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, 838–840 (1986).
[CrossRef]

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

Wang, S.

K. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, 17–19 (1983).
[CrossRef]

Weiner, A. M.

Wright, E. M.

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, 838–840 (1986).
[CrossRef]

Appl. Phys. Lett. (3)

K. Kitayama and S. Wang, “Optical pulse compression by nonlinear coupling,” Appl. Phys. Lett. 43, 17–19 (1983).
[CrossRef]

P. R. Berger, Y. Chen, P. Bhattachaya, J. Pamulapati, and G. C. Vezzoli, “Demonstration of all-optical modulation in a vertical guided-wave nonlinear coupler,” Appl. Phys. Lett. 52, 1125–1127 (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, 838–840 (1986).
[CrossRef]

Electron. Lett. (1)

P. Li, Kam Wa, J. E. Stich, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Electron. Lett. 21, 26–27 (1985).
[CrossRef]

IEEE J. Quantum Electron. (3)

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

M. D. Feit and J. A. Fleck, “Three-dimensional analysis of a directional coupler exhibiting a Kerr nonlinearity,” IEEE J. Quantum Electron. 24, 2081–2086 (1988).
[CrossRef]

S. R. Friberg and P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. QE-23, 2089–2094 (1987).
[CrossRef]

Int. J. Optoelectron. (1)

A. T. Pham and L. N. Binh, “Nonlinear optical directional coupler—two-input operations,” Int. J. Optoelectron. 5, 367–380 (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. B (1)

Opt. Commun. (1)

R. Hoffe and J. Chrostowski, “Optical pulse compression and breaking in nonlinear fibre couplers,” Opt. Commun. 57, 34–38 (1986).
[CrossRef]

Opt. Lett. (1)

Proc. Inst. Electr. Eng. (1)

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

Sov. J. Quantum Electron. (3)

D. D. GusovskiiE, M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

A. A. Maier, “Switching of radiation in tunnel-coupled optical waveguides by weak radiation frequency,” Sov. J. Quantum Electron. 16, 892–897 (1986).
[CrossRef]

A. A. Maier, “Self-switching of light in a directional coupler,” Sov. J. Quantum Electron. 14, 101–104 (1984).
[CrossRef]

Other (1)

A. T. Pham and L. N. Binh, “Graphical and stability analysis of nonlinear optical directional couplers using two-input biasing beams,” submitted to Int. J. Opt. Comput.

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Figures (28)

Fig. 1
Fig. 1

(a) (Pt, Γ) diagram with C/Q = 1, representing all possible states of operation of a NLDC. (b) Product of optical power P1P2 as a function of P1, for two different values of total power Pt and represented by operating points M and N, respectively.

Fig. 2
Fig. 2

Input configuration for the NLDC used as a phase-to-amplitude modulator. P1(0)P2(0) = ≪ 1 and −1 ≤ cos ψ(0) ≤ 1 [sin ψ(0) > 0].

Fig. 3
Fig. 3

(Pt, Γ) diagram for the NLDC used as a phase-to-amplitude modulator. At R P1(0)P2(0) = , cos ψ(0)= 0; at S P1(0)P2(0) = , cos ψ(0) = −1; at T P1(0)P2(0) = , cos ψ(0) = 1. The arrows on segment ST indicate the directions of movement of the operating point when cos ψ(0) ≠ 0.

Fig. 4
Fig. 4

Potential function for the operation at point shown in Fig. 3. (a) At R cos ψ(0) = 0; (b) at S cos ψ(0) = −1; (c) at T cos ψ(0) = 1.

Fig. 5
Fig. 5

Power evolution along the propagation direction z of the NLDC used as a phase-to-amplitude modulator with five different values of cos ψ(0). Curve 1, cos ψ(0) = 1, type-2 operation; curve 2, cos ψ(0) = −1, type-1 operation; curve 3, cos ψ(0) = 0, critical operation; curve 4, cos ψ(0) = x, x > 0, type-2 operation; curve 5, cos ψ(0) = −x, x > 0, type-1 operation.

Fig. 6
Fig. 6

Modulated output power (P1, L) as a function of the initial input phase difference x = cos ψ(0).

Fig. 7
Fig. 7

Power evolution along the first guide with (curve 1) cos ψ(0) = −1 and (curve 2) cos ψ(0) = 1 and P1(0) + P2(0) = Pc = 4, P2(0) = 4 × 10−4, C/Q = 1.

Fig. 8
Fig. 8

Second input configuration for the NLDC used as an all-optical phase-controlled switch.

Fig. 9
Fig. 9

(Pt, Γ) diagram for the input configuration in Fig. 8. Note that point Q is identical to point D of Fig. 1. At point N, P1(0) = P2(0) = Pb/2, cos ψ(0)= 1; at point 0, P1(0) = P2(0) = Pb/2, cos ψ(0) = cos ( may be positive or negative); at point P, P1(0) = P2(0) = Pb/2, cos ψ(0)= Pb/Pc.

Fig. 10
Fig. 10

Potential function for operating point O in Fig. 9, with (a) ψ(0) = > 0 and (b) ψ(0)= < 0.

Fig. 11
Fig. 11

Power evolution along guide 1 for the operating point O of Fig. 9 with initial conditions P1(0) = P2(0) = Pb/2, Pb > Pc/2, and (curve 1) ψ(0)= , (curve 2) ψ(0) = −, (curve 3) ψ(0) = 0.

Fig. 12
Fig. 12

Power evolution along guide 1 for the operating point C of Fig. 1 with initial input conditions P(0) = P2(0) = Pb/2 and Pb < Pc and (curve 1) ψ(0) = cos−1(Pb/Pc) and (curve 2) ψ(0) = −cos−1(Pb/Pc).

Fig. 13
Fig. 13

Input configuration for application of the NLDC as an amplitude-to-amplitude modulator in the single-input excitation mode.

Fig. 14
Fig. 14

Graphic representation of the NLDC used as an amplitude-to-amplitude modulator with the first guide being excited. Note that G1 is identical to point D of Fig. 1, the critical point.

Fig. 15
Fig. 15

Potential function for the system operating at (a) G1, (b) K1, (c) H1.

Fig. 16
Fig. 16

Power evolution of guide 1 of the NLDC used as an amplitude-to-amplitude modulator with the initial input conditions P2(0) = 0 and P1(0)= Pc ± |ΔPin|; the upper and the lower signs correspond to the upper and the lower curves. Curves 1 and 2 have equal magnitudes |ΔPin|; Curves 3 and 4 have equal magnitudes |ΔPin| that are smaller than those of curves 1 and 2.

Fig. 17
Fig. 17

Initial input configuration for the NLDC used as an amplitude-to-amplitude modulator with both inputs being excited and the small signal ΔPin superimposed upon (a) both inputs with reverse polarity and (b) the input of the second guide.

Fig. 18
Fig. 18

(Pt, Γ) diagram of the NLDC used as an amplitude-to-amplitude modulator with both guides being excited and the small-signal input ΔPin superimposed upon both pump beams. Note that G2 is identical to D of Fig. 1. At H2 P1(0) = P2(0) = Pb/2, Cos ψ(0)= 1; at K2 P1(0) = (Pb/2) + |ΔPin|, P2(0) = (Pb/2) − |ΔPin|, cos ψ(0)= 1 or P1(0) = (Pb/2) − |ΔPin|, P2(0) = (Pb/2) + |ΔPin|, cos ψ(0) = 1.

Fig. 19
Fig. 19

Sketch of potential function for operating points (a) H2 and (b) K2.

Fig. 20
Fig. 20

Power evolution along the first waveguide, induced by the initial conditions P1(0) = (Pb/2) ±Pin|, P2(0) = (Pb/2) |ΔPin|, and cos ψ(0) = 1. The upper signs correspond to curves 1 and 3, and the lower signs correspond to curves 2 and 4. Curves 1 and 2 have equal values of |ΔPin| that are larger than those of the curves 3 and 4, which have equal values of |ΔPin|.

Fig. 21
Fig. 21

(Pt, Γ) diagram for the NLDC used as an amplitude-to-amplitude modulator with both guides being excited and the small-signal input ΔPin superimposed upon the input of waveguide 2. At A2 P1(0) = P2 (0) = Pb/2, cos ψ(0)= 1; at B2 P1(0) − Pb/2, P2(0) = (Pb/2) + |ΔPin| cos ψ(0) = 1; at C2 P1(0) = Pb/2, P2(0) = (Pb/2) − |ΔPin|, cos ψ(0)= 1.

Fig. 22
Fig. 22

Potential function for the motion of power P1 for the operating points (a) B2 and (b) C2 of Fig. 21.

Fig. 23
Fig. 23

Power evolutions along the first guide, induced by the initial conditions P1(0) = Pb/2, P2(0) = (Pb/2) ∓ |ΔPin| and cos ψ/(0) = 1. The upper sign corresponds to curves 1 and 3, and the lower sign corresponds to curves 2 and 4. Curves 1 and 2 have equal values of |ΔPin| that are larger than those of curves 3 and 4, which have equal values of |ΔPin|.

Fig. 24
Fig. 24

Modulation characteristic for the NLDC used as an amplitude-to-amplitude modulator with only its first guide being excited.

Fig. 25
Fig. 25

Modulation characteristics for the NLDC used as an amplitude-to-amplitude modulator with both of its inputs being excited and the signal ΔPin being superimposed upon both inputs.

Fig. 26
Fig. 26

Three different input configurations for using the NLDC as an intensity-sensitive switch. Pc = 4(C/Q), Pb > Pc/2, and ΔP is the control signal.

Fig. 27
Fig. 27

(Pt, r) diagram for the NLDC operation for the input configuration in Fig. 26(c). At point A ΔP = 0; at point B ΔP > 0. Point C is identical to D of Fig. 1.

Fig. 28
Fig. 28

Power evolution along the first waveguide for (curve 1) ΔP = 0 and (curve 2) ΔP > 0.

Equations (132)

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( d P 1 d z ) 2 = - Q 2 ( P 1 - a ) ( P 1 - b ) ( P 1 - c ) P 1 - d ) ,
a = P t + d 2 , b = P t + δ 2 , c = P t - δ 2 , d = P t - α 2 ,
α = [ P t 2 - 8 C 2 Q 2 ( 1 - Q C Γ ) + 8 C Q ( C 2 Q 2 - 2 Γ C Q ) 1 / 2 ] 1 / 2 , δ = [ P t 2 - 8 C 2 Q 2 ( 1 - Q C Γ ) - 8 C Q ( C 2 Q 2 - 2 Γ C Q ) 1 / 2 ] 1 / 2 .
1 2 ( d U d t ) 2 = E - U ( y )
E = - 2 C 2 Γ 2 ,
2 U ( P 1 ) = Q 2 P 1 2 ( P t - P 1 ) 2 - ( 1 - ( Q / C ) Γ ) 4 C 2 P 1 ( P t - P 1 ) .
ζ 1 / 4 P 1 P 2 ζ 2 / 4             for all z ,
ζ 1 = 8 C 2 Q 2 ( 1 - Q C Γ ) - 8 C Q ( C 2 Q 2 - 2 Γ C Q ) 1 / 2 ,
ζ 2 = 8 C 2 Q 2 ( 1 - Q C Γ ) + 8 C Q ( C 2 Q 2 - 2 Γ C Q ) 1 / 2 .
P 1 ( 0 ) P 2 ( 0 ) = 1 ,
P t 2 = [ P 1 ( 0 ) + P 2 ( 0 ) ] 2 = ζ 2 Γ = Γ R = ζ 2 R ,
ψ ( 0 ) = π / 2.
Γ - Γ A = [ P 1 ( 0 ) P 2 ( 0 ) ] 1 / 2 cos ψ ( 0 ) = 1 / 2 cos ψ ( 0 ) .
cos ψ ( 0 ) = x .
E - U P 1 = P t / 2 = - ( Γ - Γ A ) [ Q C P t 2 2 + 2 C 2 ( Γ + Γ A ) ] .
E - U P 1 = P t / 2 - 1 / 2 x ( Q C P t 2 2 + 4 C 2 Γ A ) .
P t P 1 , 2 ( z ) 0.
P t P 1 ( z ) P t / 2 ,
0 P 2 ( z ) P t / 2.
ζ 2 ζ 2 D + d ζ 2 d Γ | Γ = 0 Γ = P c 2 + d ζ 2 d Γ | Γ = 0 Γ ,
ζ 2 A P c 2 + 8 .
P t 2 = ζ 2 A P c 2 + 8 ,
P 1 ( 0 ) = P t + [ P t 2 - 4 P 1 ( 0 ) P 2 ( 0 ) ] 1 / 2 2 P c + P c ,
P 2 ( 0 ) = P 1 ( 0 ) P 2 ( 0 ) P 1 ( 0 ) P c + P c .
ζ 1 0.
P 1 ( L ) = 1 2 { P t - ( P t 2 - ζ 1 ) 1 / 2 × cn [ - Q L g 1 + cn - 1 ( cos θ 0 , k 1 ) , k 1 ] } ,
k 1 2 = P t 2 - ζ 1 ζ 2 - ζ 1 P c 2 + 8 P c 2 - 4 P c 1 / 2 x + 8 1 ;
1 - k 1 2 = - 4 P c 1 / 2 x P c 2 - 4 P c 1 / 2 x + 8 - 4 1 / 2 x P c , g 1 = 2 ( ζ 2 - ζ 1 ) 1 / 2 2 P c , cos θ 0 = P t - 2 P 1 ( 0 ) ( P t 2 - ζ 1 ) 1 / 2 - 1.
K 1 = 0 π / 2 ( 1 - k 2 sin 2 t ) - 1 / 2 d t .
K 1 ln [ 4 ( 1 + k 1 2 ) 1 / 2 ] 1 2 ln ( - 4 P c 1 / 2 x ) , cn ( u , k 1 ) sech u .
Q L g 1 = 2 K 1 | x = - 1 ln ( 4 P c 1 / 2 ) .
L 1 2 C ln ( 4 P c 1 / 2 ) ,
P 1 ( L ) ( P c / 2 ) { 1 - sech [ ln ( - x ) ] } ,
P 1 ( L ) P c 2 ( 1 + 2 x 1 + x 2 ) ,
- 1 x = cos ψ ( 0 ) < 0.
P 1 ( L ) = a + k 2 d sn 2 [ Q L g 2 + sn - 1 ( sin θ 0 , k 2 ) , k 2 ] 1 + k 2 sn 2 [ Q L g 2 + sn - 1 ( sin θ 0 , k 2 ) , k 2 ] ,
a = P t + ( P t 2 - ζ 1 ) 1 / 2 2 P t P c , d = P t - ( P t 2 - ζ 1 ) 1 / 2 2 0 , k 2 = ( P t 2 - ζ 1 ) 1 / 2 - ( P t 2 - ζ 2 ) 1 / 2 ( P t 2 - ζ 1 ) 1 / 2 + ( P t 2 - ζ 2 ) 1 / 2 ( P c 2 + 8 ) 1 / 2 - ( 4 P c 1 / 2 x ) 1 / 2 ( P c 2 + 8 ) 1 / 2 + ( 4 P c 1 / 2 x ) 1 / 2 1 , 1 - k 2 2 8 ( 1 / 2 x / P c ) 1 / 2 , K 2 ln [ 4 ( 1 - k 2 2 ) 1 / 2 ] 1 4 ln ( 4 P c 1 / 2 x ) , g 2 = 4 ( P t 2 - ζ 1 ) 1 / 2 + ( P t 2 - ζ 2 ) 1 / 2 4 P c = Q C , sin θ 0 0.
sn - 1 ( sin θ 0 , k 2 ) - 2 K 2 ,             sn ( u , k 2 ) tanh u .
Q L g 2 = 2 K 2 x = 1 1 2 ln ( 4 P c 1 / 2 )
L = 1 2 C ln ( 4 P c 1 / 2 ) ,
P 1 ( L ) = P c 1 + tanh 2 ( ½ ln x ) = P c 2 ( 1 + 2 x 1 + x 2 ) ,
0 x = cos ψ ( 0 ) 1.
P 2 ( L ) = P t - P 1 ( L ) P c 2 ( 1 - 2 x 1 + x 2 ) ,
- 1 x = cos ψ ( 0 ) 1.
P 1 ( 0 ) = P 2 ( 0 ) = P b / 2 ,             P b > P c / 2 ,             cos ψ ( 0 ) = 1.
d N P 1 , 2 ( z ) a N ,
Γ N = [ P 1 ( 0 ) P 2 ( 0 ) ] 1 / 2 cos ψ ( 0 ) - 2 P 1 ( 0 ) P 2 ( 0 ) P c = P b 2 ( 1 - P b P c ) , ζ 1 N = 8 C 2 Q 2 ( 1 - Q C Γ N ) - 8 C Q ( C 2 Q 2 - 2 Γ N C Q ) 1 / 2 = ( P b - P c ) 2 .
a N = P t + ( P t 2 - ζ 1 N ) 1 / 2 2 = P b + ( 2 P c P b - P c 2 ) 1 / 2 2 ,
d N = P t - ( P t 2 - ζ 1 N ) 1 / 2 2 = P b - ( 2 P c P b - P c 2 ) 1 / 2 2 .
Γ O = P b 2 ( cos - P b P c ) , ζ 1 O ζ 1 N + d ζ 1 d Γ Γ = Γ N ( Γ O - Γ N ) = ( P b - P c ) 2 + ( 2 P c - P c 2 P b - P c 2 ) P b 2 ( 1 + cos ) , ζ 2 O ζ 2 N + d ζ 2 d Γ Γ = Γ N ( Γ O - Γ N ) = P b 2 + ( 2 P c + P c 2 P b - P c 2 ) P b 2 ( 1 - cos ) .
L = ¼ L p = ( g / Q ) K ,
g = 2 ( ζ 2 O - ζ 1 O ) 1 / 2 2 P c ( 2 P b P c - 1 ) 1 / 2 , 1 - k 2 = 1 - P t 2 - ζ 1 B ζ 2 B - ζ 1 B = ζ 2 B - P t 2 ζ 2 B - ζ 1 B 2 P b 2 ( 1 - cos ) ( 2 P b - P c ) 2 0.
K = ln [ 4 ( 1 - k 2 ) 1 / 2 ] ln [ 4 ( 2 P c - P b ) P b ] .
L 1 2 C [ 2 ( P b / P c ] - 1 ) 1 / 2 ln [ 4 ( 2 P b - P c ) P b ] .
L 1 2 C ln ( 4 ) ,
a N = P c ,             d N = 0.
P 1 ( 0 ) = P 2 ( 0 ) = P b / 2 ,             P b < P c ,
cos ψ ( 0 ) = P b / P c .
0 P 1 , 2 ( z ) P b .
L = ¼ L p = ( g / Q ) K ,
g = 2 ( ζ 2 c - ζ 1 c ) 1 / 2 = 2 P c , k = ( P t 2 - ζ 1 c ζ 2 c - ζ 1 c ) 1 / 2 = P b P c = cos ψ ( 0 ) ,
K π / 2.
L π / ( 4 C ) ,
ψ ( 0 ) ± π / 2.
k = ln [ 4 ( 1 - k 2 ) 1 / 2 ] = ln { 4 [ 1 - cos 2 ψ ( 0 ) ] 1 / 2 } = ln [ 4 ψ ( 0 ) ] .
L 1 2 C ln [ 4 ψ ( 0 ) ] ,
d = 0 P 1 , 2 ( z ) a = P c - Δ P in P c ,
b P c / 2 P 1 ( z ) a = P c + Δ P in P c , d = 0 P 2 ( z ) c P c / 2.
P 1 ( 0 ) = P 2 ( 0 ) = P b / 2 ,             P b > P c / 2 , cos ψ ( 0 ) = 1.
b = P b / 2 + Δ P in P b / 2 P 1 ( z ) a a H 2 , d d H 2 P 2 ( z ) c = P b / 2 - Δ P in P b / 2.
d d H 2 P 1 ( z ) c = P b / 2 - Δ P in P b / 2 , b = P b / 2 + Δ P in P b / 2 P 2 ( z ) a a H 2 .
d B 2 d A 2 P 1 ( z ) c B 2 = P b / 2 , b B 2 c B 2 = P b / 2 P 2 ( z ) a B 2 a A 2 .
b C 2 = P b / 2 P 1 ( z ) a C 2 a A 2 , d C 2 d A 2 P 2 ( z ) c C 2 b C 2 = P b / 2.
P 1 ( L ) = ½ { P t - ( P t 2 - ξ 1 ) 1 / 2 cn [ ( Q / g 1 ) L - cn - 1 ( cn θ 0 , k 1 ) , k 1 ] } .
P t = P c + Δ P in P c , ζ 1 = 0 ,             ζ 2 = P c 2 = 16 ( C 2 / Q 2 ) , k 1 2 = P t 2 - ζ 1 ζ 2 - ζ 1 P c + 2 Δ P in P c 1 ,             1 - k 1 2 - 2 Δ P in P c , g 1 = 2 ( ζ 2 - ζ 1 ) 1 / 2 = 2 P c , cos θ 0 = P t - 2 P 1 ( 0 ) ( P t 2 - ζ 2 ) 1 / 2 = - 1 ,             cn - 1 ( cos θ 0 , k 1 ) = 2 K 1 ,
K 1 ln [ 4 ( 1 - k 1 2 ) 1 / 2 ] ln ( - 8 P c Δ P in ) 1 / 2 , cn ( u , k 1 ) sech u .
P 1 ( L ) P c 2 { 1 - sech [ 2 C L + ln ( - Δ P in 8 P c ) , k 1 ] } .
Q L g 1 = 2 K 1 min = ln ( 8 P c Δ P in max ) ,
Δ P in max = 8 P c / exp ( 2 C L ) .
P 1 ( L ) P c 2 ( 1 + 2 Δ P in ¯ 1 + Δ P in ¯ 2 ) ,
P 1 ( L ) = a + k 2 d sn 2 [ - Q g 2 L + sn - 1 ( sin θ 0 , k 2 ) , k 2 ] 1 + k 2 sn 2 [ - Q g 2 L + sn - 1 ( sin θ 0 , k 2 ) , k 2 ] .
P t = P c + Δ P in P c , a = P c + Δ P in P c = 4 ( C / Q ) ,             d = 0 , ζ 1 = 0 ,             ζ 2 = P c 2 = 16 ( C 2 / Q 2 ) , k 2 = ( P t 2 - ζ 1 ) 1 / 2 - ( P t 2 - ζ 2 ) 1 / 2 ( P t 2 - ζ 1 ) 1 / 2 + ( P t 2 - ζ 2 ) 1 / 2 P c - ( 2 Δ P in P c ) 1 / 2 P c + ( 2 Δ P in P c ) 1 / 2 1 , 1 - k 2 2 ( 32 Δ P in P c ) 1 / 2 , g 2 = 4 ( P t 2 - ζ 1 ) 1 / 2 + ( P t 2 - ζ 2 ) 1 / 2 4 P c , sin 2 θ 0 = ( + δ ) [ a - P 1 ( 0 ) ] ( - δ ) [ P 1 ( 0 ) - d ] = 0 , sn - 1 ( sin θ 0 , k 2 ) = sn - 1 ( 0 , k 2 ) = 2 K 2 , K 2 ln [ 4 ( 1 - k 2 2 ) 1 / 2 ] ln 4 ( P c 32 Δ P in ) , sn ( u , k 2 ) tanh u .
P 1 ( L ) P c 1 + tanh 2 [ - C L + ln 16 ( P c 32 Δ P in ) 1 / 2 ] ,
P 1 ( L ) P c 2 { 1 + sech [ 2 C L + ln ( Δ P in 8 P c ) ] } .
Q L g 2 = 2 K 2 min = ln [ 16 ( P c 32 Δ P in max ) 1 / 2 ] .
Δ P in max = 8 P c / exp ( 2 C L ) ,
P 1 ( L ) P c 2 ( 1 + 2 Δ P in ¯ 1 + Δ P in ¯ 2 )
0 Δ P in ¯ = Δ P in Δ P in max 1.
P 1 ( 0 ) = P 2 ( 0 ) = P b / 2 ,             cos ψ ( 0 ) = 1.
Γ A = [ P 1 ( 0 ) P 2 ( 0 ) ] 1 / 2 cos ψ ( 0 ) - 2 P 1 ( 0 ) P 2 ( 0 ) P c = P b 2 ( 1 - P b P c ) , ζ 1 A = 8 C 2 Q 2 ( 1 - Q C Γ A ) - 8 C Q ( C 2 Q 2 - 2 Γ C Q ) 1 / 2 = ( P b - P c ) 2 , a H 2 = P b + ( P b 2 - ζ 1 A ) 1 / 2 2 = P b + ( 2 P c P b - P c 2 ) 1 / 2 2 b H 2 = c A = P b 2 , d H 2 = P b - ( P b 2 - ζ 1 A ) 1 / 2 2 = P b - ( 2 P c P b - P c 2 ) 1 / 2 2 .
P 1 ( 0 ) = P b / 2 + Δ P in ,             P 2 ( 0 ) = P b / 2 - Δ P in , cos ψ ( 0 ) = 1.
P 1 ( L ) = a K + k 2 d K 2 sn 2 [ - Q L g 2 + sn - 1 ( sin θ 0 , k 2 ) , k 2 ] 1 + k 2 sn 2 [ - Q L g 2 + sn - 1 ( sin θ 0 , k 2 ) , k 2 ] ,
b K 2 P 1 ( z ) a K 2 .
P t = P 1 ( 0 ) + P 2 ( 0 ) = P b , a K 2 a H 2 = P B + ( 2 P c P b - P c 2 ) 1 / 2 2 , b K 2 = P b 2 + Δ P in = P 1 ( 0 ) , c K 2 = P b 2 - Δ P in = P 2 ( 0 ) , d K 2 d H 2 = P B - ( 2 P c P b - P c 2 ) 1 / 2 2 , sin 2 θ 0 = ( b K 2 - d K 2 ) [ a K 2 - P 1 ( 0 ) ] ( a K 2 - b K 2 ) ( P 1 ( 0 ) - d K 2 ) = ( b K 2 - d K 2 ) ( a K 2 - b K 2 ) ( a K 2 - b K 2 ) ( b K 2 - d K 2 ) = 1 , sn - 1 ( sin θ 0 , k 2 ) = K 2 , k 2 2 = ( a K 2 - b K 2 ) ( c K 2 - d K 2 ) ( a K 2 - c K 2 ) ( b K 2 - d K 2 ) = { [ ( 2 P c P b - P c 2 ) 1 / 2 / 2 ] - Δ P in } { [ ( 2 P c P b - P c 2 ) 1 / 2 / 2 ] + Δ P in } 1 , 1 - k 2 2 = 8 Δ P in ( 2 P c P b - P c 2 ) 1 / 2 , g 2 = 2 [ ( a K 2 - c K 2 ) ( b K 2 - d K 2 ) ] 1 / 2 4 ( 2 P c P b - P c 2 ) 1 / 2 , K 2 ln [ 4 ( 1 - k 2 2 ) 1 / 2 ] = 1 2 ln [ 2 ( 2 P c P b - P c 2 ) 1 / 2 Δ P in ] , sn ( u , k 2 ) tanh u .
P 1 ( L ) a H 2 + d H 2 tanh 2 ( - Q L g 2 + K 2 ) 1 + tanh 2 ( - Q L g 2 + K 2 ) ,
P 1 ( L ) P b 2 + ( 2 P c P b - P c 2 ) 1 / 2 2 sech ( - 2 Q L g 2 + 2 K 2 ) .
Q L g 2 = K 2 min = 1 2 ln [ 2 ( 2 P c P b - P c 2 ) 1 / 2 Δ P in max ] .
Δ P in max = 2 ( 2 P c P b - P c 2 ) 1 / 2 exp { 2 C L [ 2 ( P b / P c ) - 1 ] 1 / 2 } .
P 1 ( L ) P b 2 + ( 2 P c P b - P c 2 ) 1 / 2 Δ P in ¯ 1 + Δ P in ¯ 2 ,
0 Δ P in ¯ = Δ P in Δ P in max 1 ,
P 2 ( L ) = P b - P 1 ( L ) P b 2 - ( 2 P c P b - P c 2 ) 1 / 2 Δ P in ¯ 1 + Δ P in ¯ 2 .
P 1 ( 0 ) = P b 2 ,             P 2 ( 0 ) = P b 2 + Δ P in , cos ψ ( 0 ) = 1.
P 1 ( L ) = a C 2 + k 2 d C 2 sn 2 [ - Q L g 2 + sn - 1 ( sin θ 0 , k 2 ) , k 2 ] 1 + k 2 sn 2 [ - Q L g 2 + sn - 1 ( sin θ 0 , k 2 ) , k 2 ] ,
P t = P 1 ( 0 ) + P 2 ( 0 ) = P b + Δ P in P b , a C 2 a A 2 = P b + ( 2 P c P b - P c 2 ) 1 / 2 2 ,             b C 2 = P b 2 , c C 2 = P b 2 + Δ P in ,             d C 2 d A 2 = P b - ( 2 P c P b - P c 2 ) 1 / 2 2 sin 2 θ 0 = ( b C 2 - d C 2 ) ( a C 2 - P 1 ( 0 ) ) ( a C 2 - b C 2 ) ( P 1 ( 0 ) - d C 2 ) = ( b C 2 - d C 2 ) ( a C 2 - b C 2 ) ( a C 2 - b C 2 ) ( b C 2 - d C 2 ) = 1 , sn - 1 ( sin θ 0 , k 2 ) = K 2 , k 2 2 = ( a C 2 - b C 2 ) ( c C 2 - d C 2 ) ( a C 2 - c C 2 ) ( b C 2 - d C 2 ) ( 2 P c P b - P c 2 ) 1 / 2 - 2 Δ P in ( 2 P c P b - P c 2 ) 1 / 2 - 2 Δ P in 1 , 1 - k 2 2 - 4 Δ P in ( 2 P c P b - P c 2 ) 1 / 2 , g 2 = 2 [ ( a C 2 - c C 2 ) ( b C 2 - d C 2 ) ] 1 / 2 4 ( 2 P c P b - P c 2 ) 1 / 2 , K 2 ln ( 4 ( 1 - k 2 2 ) 1 / 2 ) = 1 2 ln [ 4 ( 2 P c P b - P c 2 ) 1 / 2 - Δ P in ] , sn ( u , k 2 ) tanh u .
P 1 ( L ) a A 2 + d A 2 tanh 2 [ - ( Q L / g 2 ) + K 2 ] 1 + tanh 2 [ - ( Q L / g 2 ) + K 2 ] ,
P 1 ( L ) P b 2 + ( 2 P c P b - P c 2 ) 1 / 2 2 sech ( - 2 Q L g 2 + 2 K 2 ) .
Δ P in max = 4 ( 2 P c P b - P c 2 ) 1 / 2 exp [ 2 C L ( 2 ( P b / P c ) - 1 ) 1 / 2 ] ,
P 1 ( L ) P b 2 - ( 2 P c P b - P c 2 ) 1 / 2 Δ P in ¯ 1 + Δ P in ¯ 2 ,
- 1 Δ P in ¯ = Δ P in / Δ P in max 0.
P 2 ( L ) = P t - P 1 ( L ) P b 2 + ( 2 P c P b - P c 2 ) 1 / 2 Δ P in ¯ 1 + Δ P in ¯ 2 .
P 1 ( 0 ) = ( P b / 2 ) + Δ P in ,             P 2 ( 0 ) = P b / 2 , cos ψ ( 0 ) = 1 ,
P 1 ( L ) P b 2 + ( 2 P c P b - P c 2 ) 1 / 2 Δ P in ¯ 1 + Δ P in ¯ 2
- 1 Δ P in ¯ = Δ P in Δ P in max 1 ,
Δ P 1 ( L ) = P 1 ( L ) - ( P c / 2 ) P c Δ P in ¯ = exp ( 2 C L ) Δ P in ,
G = Δ P 1 ( L ) / Δ P in exp ( 2 C L ) .
Δ P 1 ( L ) = P 1 ( L ) - P b 2 ( 2 P c P b - P c 2 ) 1 / 2 Δ P in ¯ = 1 2 exp [ 2 CL ( 2 P b P c - 1 ) 1 / 2 ] Δ P in ,
G = Δ P 1 ( L ) Δ P in 1 2 exp [ 2 C L ( 2 P b P c - 1 ) 1 / 2 ] .
Δ P 1 ( L ) = P 1 ( L ) - P b 2 ( 2 P c P b - P c 2 ) 1 / 2 Δ P in ¯ = 1 4 exp [ 2 C L ( 2 P b P c - 1 ) 1 / 2 ] Δ P in ,
G = Δ P 1 ( L ) Δ P in 1 4 exp [ 2 C L ( 2 P b P c - 1 ) 1 / 2 ] .
P 1 ( L ) = 0 ,
P 2 ( L ) P c ,
P 1 ( L ) ½ [ P b - ( 2 P c P c - P c 2 ) 1 / 2 ] ,
P 2 ( L ) ½ [ P b + ( 2 P c P b - P c 2 ) 1 / 2 ] .
Δ P = 2 = 2 Δ P in max = 16 P c / exp ( 2 C L ) .
Δ P = 2 = 2 Δ P in max = 8 ( 2 P c P b - P c 2 ) 1 / 2 exp [ 2 C L ( 2 P b P c - 1 ) 1 / 2 ] .
P 1 ( L ) 0 ,
P 2 ( L ) P c .
P t = P 1 ( 0 ) + P 2 ( 0 ) = P c - + Δ P P c , P 1 ( 0 ) P 2 ( 0 ) = ( P c - ) Δ P P c Δ P , Γ = [ P 1 ( 0 ) P 2 ( 0 ) ] 1 / 2 cos ψ ( 0 ) - Q 2 C P 1 ( 0 ) P 2 ( 0 ) ( P c Δ P ) 1 / 2 , ζ 1 = P 1 ( 0 ) P 2 ( 0 ) = P c Δ P , ζ 2 ζ 2 | Γ = 0 + d ζ 2 d Γ | Γ = 0 Γ = P c 2 - 4 P c ( P c Δ P ) 1 / 2 .
a = P t + ( P t 2 - ζ 1 ) 1 / 2 2 P c + ( P c 2 - P c Δ P ) 1 / 2 2 P c , b = P t + ( P t 2 - ζ 2 ) 1 / 2 2 P c + [ 4 P c ( P c Δ P ) 1 / 2 ] 1 / 2 2 P c 2 , c = P t - ( P t 2 - ζ 2 ) 1 / 2 2 P c - [ 4 P c ( P c Δ P ) 1 / 2 2 P c 2 , d = P t - ( P t 2 - ζ 1 ) 1 / 2 2 P c - ( P c 2 - P c Δ P ) 1 / 2 2 0.
Q L / g 2 = 2 K 2 ,
g 2 = 2 [ ( a - c ) ( b - d ) ] 1 / 2 4 P c , k 2 = [ ( a - b ) ( c - d ) ( a - c ) ( b - d ) ] 1 / 2 ( P c 2 - P c Δ P ) 1 / 2 - [ 4 P c ( P c Δ P ) 1 / 2 ] 1 / 2 ( P c 2 - P c Δ P ) 1 / 2 + [ 4 P c ( P c Δ P ) 1 / 2 ] 1 / 2 1 , 1 - k 2 2 4 [ 4 P c 3 ( P c Δ P ) 1 / 2 ] 1 / 2 P c 2 , K 2 ln [ 4 ( 1 - k 2 2 ) 1 / 2 ] 1 8 ln ( 16 P c Δ P ) .
Δ P 16 P c exp ( 4 C L ) = 2 exp ( 2 C L ) .

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