Abstract

We consider the problem of continuous-wave parametric amplification by a strong pump of a signal wave tuned to a Bragg grating. We find spectral regions of enhanced and reduced gain compared with gain in a uniform medium. The changes are explained by local bending of the dispersion relation induced by the grating. Calculations of pump powers suggest that the effects may be observed in optical fiber Bragg gratings.

© 1995 Optical Society of America

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References

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  1. C. M. de Sterke, "Optical push broom," Opt. Lett. 17, 914–916 (1992).
    [CrossRef] [PubMed]
  2. M. J. Steel and C. M. de Sterke, "Schrödinger equation description for cross-phase modulation in highly dispersive media," Phys. Rev. A 49, 5048–5055 (1994).
    [CrossRef] [PubMed]
  3. M. J. Steel, D. G. A. Jackson, and C. M. de Sterke, "Approximate model for optical pulse compression by cross-phase modulation in Bragg gratings," Phys. Rev. A 50, 3447–3452 (1994).
    [CrossRef] [PubMed]
  4. S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, "All-optical switching of grating transmission using crossphase modulation in optical fibres," Electron. Lett. 26, 1459–1460 (1990).
    [CrossRef]
  5. J. Lauzon, S. LaRochelle, and F. Ouellette, "Numerical analysis of all-optical switching of a fiber Bragg grating induced by a short-detuned pump pulse," Opt. Commun. 92, 233–239 (1992).
    [CrossRef]
  6. N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483–485 (1970).
    [CrossRef]
  7. C. L. Tang and P. P. Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron. QE-9, 9–17 (1973).
    [CrossRef]
  8. A. Yariv and P. Yeh, "Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching and x-ray lasers," J. Opt. Soc. Am. 67, 438–448 (1977).
    [CrossRef]
  9. A. A. Maïer and A. P. Sukhorukov, "Synchronous nonlinear wave interaction in Bragg diffraction in media with periodic structure," Sov. Phys. JETP 50, 645–652 (1979).
  10. V. Belyakov and N. V. Shipov, "On the enhancement of the nonlinear frequency transformation in periodic media," Phys. Lett. 86A, 94–97 (1981).
  11. V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, New York, 1992), Chap. 6, pp. 188–205.
    [CrossRef]
  12. J. Martorell and R. Corbalán, "Enhancement of second harmonic generation in a periodic structure with a defect," Opt. Commun. 108, 319–323 (1994).
    [CrossRef]
  13. J. P. van der Ziel and M. Ilegems, "Optical second harmonic generation in periodic multilayer GaAs–Al0.3Ga0.7As structures," Appl. Phys. Lett. 28, 437–439 (1976).
    [CrossRef]
  14. P. St. J. Russell and J.-L. Archambault, "Field microstructure and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media," J. Phys. III France 4, 2471–2491 (1994).
    [CrossRef]
  15. C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Vol. XXXIII, Chap. III, pp. 203–260.
    [CrossRef]
  16. J. E. Sipe, Department of Physics, University of Toronto, Toronto, Canada M5S 1A7 (personal communication, 1994).
  17. H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  18. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989), Chap. 10.
  19. J. E. Sipe, L. Poladian, and C. M. de Sterke, "Propagation through nonuniform grating structures," J. Opt. Soc. Am. A 11, 1307 (1994).
    [CrossRef]
  20. B. E. Eggleton, School of Physics, University of Sydney, New South Wales 2006, Australia (personal communication, 1995).
  21. N. G. R. Broderick, C. M. de Sterke, and J. E. Sipe, "Power estimates for the launching of gap solitons in nonuniform gratings," Opt. Commun. 113, 118–124 (1994).
    [CrossRef]
  22. C. M. de Sterke and J. E. Sipe, "Coupled modes and the nonlinear Schrödinger equation," Phys. Rev. A 42, 550–555 (1990).
    [CrossRef]

1994 (6)

M. J. Steel and C. M. de Sterke, "Schrödinger equation description for cross-phase modulation in highly dispersive media," Phys. Rev. A 49, 5048–5055 (1994).
[CrossRef] [PubMed]

M. J. Steel, D. G. A. Jackson, and C. M. de Sterke, "Approximate model for optical pulse compression by cross-phase modulation in Bragg gratings," Phys. Rev. A 50, 3447–3452 (1994).
[CrossRef] [PubMed]

J. Martorell and R. Corbalán, "Enhancement of second harmonic generation in a periodic structure with a defect," Opt. Commun. 108, 319–323 (1994).
[CrossRef]

P. St. J. Russell and J.-L. Archambault, "Field microstructure and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media," J. Phys. III France 4, 2471–2491 (1994).
[CrossRef]

J. E. Sipe, L. Poladian, and C. M. de Sterke, "Propagation through nonuniform grating structures," J. Opt. Soc. Am. A 11, 1307 (1994).
[CrossRef]

N. G. R. Broderick, C. M. de Sterke, and J. E. Sipe, "Power estimates for the launching of gap solitons in nonuniform gratings," Opt. Commun. 113, 118–124 (1994).
[CrossRef]

1992 (2)

J. Lauzon, S. LaRochelle, and F. Ouellette, "Numerical analysis of all-optical switching of a fiber Bragg grating induced by a short-detuned pump pulse," Opt. Commun. 92, 233–239 (1992).
[CrossRef]

C. M. de Sterke, "Optical push broom," Opt. Lett. 17, 914–916 (1992).
[CrossRef] [PubMed]

1990 (2)

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, "All-optical switching of grating transmission using crossphase modulation in optical fibres," Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

C. M. de Sterke and J. E. Sipe, "Coupled modes and the nonlinear Schrödinger equation," Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

1981 (1)

V. Belyakov and N. V. Shipov, "On the enhancement of the nonlinear frequency transformation in periodic media," Phys. Lett. 86A, 94–97 (1981).

1979 (1)

A. A. Maïer and A. P. Sukhorukov, "Synchronous nonlinear wave interaction in Bragg diffraction in media with periodic structure," Sov. Phys. JETP 50, 645–652 (1979).

1977 (1)

1976 (1)

J. P. van der Ziel and M. Ilegems, "Optical second harmonic generation in periodic multilayer GaAs–Al0.3Ga0.7As structures," Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

1973 (1)

C. L. Tang and P. P. Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron. QE-9, 9–17 (1973).
[CrossRef]

1972 (1)

H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

1970 (1)

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989), Chap. 10.

Archambault, J.-L.

P. St. J. Russell and J.-L. Archambault, "Field microstructure and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media," J. Phys. III France 4, 2471–2491 (1994).
[CrossRef]

Belyakov, V.

V. Belyakov and N. V. Shipov, "On the enhancement of the nonlinear frequency transformation in periodic media," Phys. Lett. 86A, 94–97 (1981).

Belyakov, V. A.

V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, New York, 1992), Chap. 6, pp. 188–205.
[CrossRef]

Bey, P. P.

C. L. Tang and P. P. Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron. QE-9, 9–17 (1973).
[CrossRef]

Bloembergen, N.

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Broderick, N. G. R.

N. G. R. Broderick, C. M. de Sterke, and J. E. Sipe, "Power estimates for the launching of gap solitons in nonuniform gratings," Opt. Commun. 113, 118–124 (1994).
[CrossRef]

Corbalán, R.

J. Martorell and R. Corbalán, "Enhancement of second harmonic generation in a periodic structure with a defect," Opt. Commun. 108, 319–323 (1994).
[CrossRef]

Eggleton, B. E.

B. E. Eggleton, School of Physics, University of Sydney, New South Wales 2006, Australia (personal communication, 1995).

Hibino, Y.

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, "All-optical switching of grating transmission using crossphase modulation in optical fibres," Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

Ilegems, M.

J. P. van der Ziel and M. Ilegems, "Optical second harmonic generation in periodic multilayer GaAs–Al0.3Ga0.7As structures," Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

Jackson, D. G. A.

M. J. Steel, D. G. A. Jackson, and C. M. de Sterke, "Approximate model for optical pulse compression by cross-phase modulation in Bragg gratings," Phys. Rev. A 50, 3447–3452 (1994).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

LaRochelle, S.

J. Lauzon, S. LaRochelle, and F. Ouellette, "Numerical analysis of all-optical switching of a fiber Bragg grating induced by a short-detuned pump pulse," Opt. Commun. 92, 233–239 (1992).
[CrossRef]

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, "All-optical switching of grating transmission using crossphase modulation in optical fibres," Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

Lauzon, J.

J. Lauzon, S. LaRochelle, and F. Ouellette, "Numerical analysis of all-optical switching of a fiber Bragg grating induced by a short-detuned pump pulse," Opt. Commun. 92, 233–239 (1992).
[CrossRef]

Maïer, A. A.

A. A. Maïer and A. P. Sukhorukov, "Synchronous nonlinear wave interaction in Bragg diffraction in media with periodic structure," Sov. Phys. JETP 50, 645–652 (1979).

Martorell, J.

J. Martorell and R. Corbalán, "Enhancement of second harmonic generation in a periodic structure with a defect," Opt. Commun. 108, 319–323 (1994).
[CrossRef]

Mizrahi, V.

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, "All-optical switching of grating transmission using crossphase modulation in optical fibres," Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

Ouellette, F.

J. Lauzon, S. LaRochelle, and F. Ouellette, "Numerical analysis of all-optical switching of a fiber Bragg grating induced by a short-detuned pump pulse," Opt. Commun. 92, 233–239 (1992).
[CrossRef]

Poladian, L.

Russell, P. St. J.

P. St. J. Russell and J.-L. Archambault, "Field microstructure and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media," J. Phys. III France 4, 2471–2491 (1994).
[CrossRef]

Shank, C. V.

H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Shipov, N. V.

V. Belyakov and N. V. Shipov, "On the enhancement of the nonlinear frequency transformation in periodic media," Phys. Lett. 86A, 94–97 (1981).

Sievers, A. J.

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

Sipe, J. E.

J. E. Sipe, L. Poladian, and C. M. de Sterke, "Propagation through nonuniform grating structures," J. Opt. Soc. Am. A 11, 1307 (1994).
[CrossRef]

N. G. R. Broderick, C. M. de Sterke, and J. E. Sipe, "Power estimates for the launching of gap solitons in nonuniform gratings," Opt. Commun. 113, 118–124 (1994).
[CrossRef]

C. M. de Sterke and J. E. Sipe, "Coupled modes and the nonlinear Schrödinger equation," Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Vol. XXXIII, Chap. III, pp. 203–260.
[CrossRef]

J. E. Sipe, Department of Physics, University of Toronto, Toronto, Canada M5S 1A7 (personal communication, 1994).

Steel, M. J.

M. J. Steel and C. M. de Sterke, "Schrödinger equation description for cross-phase modulation in highly dispersive media," Phys. Rev. A 49, 5048–5055 (1994).
[CrossRef] [PubMed]

M. J. Steel, D. G. A. Jackson, and C. M. de Sterke, "Approximate model for optical pulse compression by cross-phase modulation in Bragg gratings," Phys. Rev. A 50, 3447–3452 (1994).
[CrossRef] [PubMed]

Stegeman, G. I.

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, "All-optical switching of grating transmission using crossphase modulation in optical fibres," Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

Sterke, C. M. de

M. J. Steel, D. G. A. Jackson, and C. M. de Sterke, "Approximate model for optical pulse compression by cross-phase modulation in Bragg gratings," Phys. Rev. A 50, 3447–3452 (1994).
[CrossRef] [PubMed]

M. J. Steel and C. M. de Sterke, "Schrödinger equation description for cross-phase modulation in highly dispersive media," Phys. Rev. A 49, 5048–5055 (1994).
[CrossRef] [PubMed]

N. G. R. Broderick, C. M. de Sterke, and J. E. Sipe, "Power estimates for the launching of gap solitons in nonuniform gratings," Opt. Commun. 113, 118–124 (1994).
[CrossRef]

J. E. Sipe, L. Poladian, and C. M. de Sterke, "Propagation through nonuniform grating structures," J. Opt. Soc. Am. A 11, 1307 (1994).
[CrossRef]

C. M. de Sterke, "Optical push broom," Opt. Lett. 17, 914–916 (1992).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, "Coupled modes and the nonlinear Schrödinger equation," Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Vol. XXXIII, Chap. III, pp. 203–260.
[CrossRef]

Sukhorukov, A. P.

A. A. Maïer and A. P. Sukhorukov, "Synchronous nonlinear wave interaction in Bragg diffraction in media with periodic structure," Sov. Phys. JETP 50, 645–652 (1979).

Tang, C. L.

C. L. Tang and P. P. Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron. QE-9, 9–17 (1973).
[CrossRef]

Yariv, A.

Yeh, P.

Ziel, J. P. van der

J. P. van der Ziel and M. Ilegems, "Optical second harmonic generation in periodic multilayer GaAs–Al0.3Ga0.7As structures," Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

Appl. Phys. Lett. (2)

N. Bloembergen and A. J. Sievers, "Nonlinear optical properties of periodic laminar structures," Appl. Phys. Lett. 17, 483–485 (1970).
[CrossRef]

J. P. van der Ziel and M. Ilegems, "Optical second harmonic generation in periodic multilayer GaAs–Al0.3Ga0.7As structures," Appl. Phys. Lett. 28, 437–439 (1976).
[CrossRef]

Electron. Lett. (1)

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, "All-optical switching of grating transmission using crossphase modulation in optical fibres," Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. L. Tang and P. P. Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron. QE-9, 9–17 (1973).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. III France (1)

P. St. J. Russell and J.-L. Archambault, "Field microstructure and temporal and spatial instability of photonic Bloch waves in nonlinear periodic media," J. Phys. III France 4, 2471–2491 (1994).
[CrossRef]

Opt. Commun. (3)

J. Lauzon, S. LaRochelle, and F. Ouellette, "Numerical analysis of all-optical switching of a fiber Bragg grating induced by a short-detuned pump pulse," Opt. Commun. 92, 233–239 (1992).
[CrossRef]

N. G. R. Broderick, C. M. de Sterke, and J. E. Sipe, "Power estimates for the launching of gap solitons in nonuniform gratings," Opt. Commun. 113, 118–124 (1994).
[CrossRef]

J. Martorell and R. Corbalán, "Enhancement of second harmonic generation in a periodic structure with a defect," Opt. Commun. 108, 319–323 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. (1)

V. Belyakov and N. V. Shipov, "On the enhancement of the nonlinear frequency transformation in periodic media," Phys. Lett. 86A, 94–97 (1981).

Phys. Rev. A (3)

M. J. Steel and C. M. de Sterke, "Schrödinger equation description for cross-phase modulation in highly dispersive media," Phys. Rev. A 49, 5048–5055 (1994).
[CrossRef] [PubMed]

M. J. Steel, D. G. A. Jackson, and C. M. de Sterke, "Approximate model for optical pulse compression by cross-phase modulation in Bragg gratings," Phys. Rev. A 50, 3447–3452 (1994).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, "Coupled modes and the nonlinear Schrödinger equation," Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

Sov. Phys. JETP (1)

A. A. Maïer and A. P. Sukhorukov, "Synchronous nonlinear wave interaction in Bragg diffraction in media with periodic structure," Sov. Phys. JETP 50, 645–652 (1979).

Other (5)

V. A. Belyakov, Diffraction Optics of Complex-Structured Periodic Media (Springer-Verlag, New York, 1992), Chap. 6, pp. 188–205.
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989), Chap. 10.

C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Vol. XXXIII, Chap. III, pp. 203–260.
[CrossRef]

J. E. Sipe, Department of Physics, University of Toronto, Toronto, Canada M5S 1A7 (personal communication, 1994).

B. E. Eggleton, School of Physics, University of Sydney, New South Wales 2006, Australia (personal communication, 1995).

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Figures (7)

Fig. 1
Fig. 1

Frequencies involved in the continuous-wave parametric amplification system. Only the signal at ωs is affected by the grating in the region around the Bragg resonance ωB.

Fig. 2
Fig. 2

Amplification factor η in the δ–Δ plane or α plane for fixed pump strength μ = 0.5. The + signs indicate regions of positive η, and the − sign indicate regions of negative η. The thick curves denote the contour η = 0 dividing these regions.

Fig. 3
Fig. 3

Amplification η as a function of δ for Δ = −0.2 and (a) μ = 0, (b) μ = 0.25, (c) μ = 0.5, (d) μ = 1. The two curve styles are used only to distinguish the curves.

Fig. 4
Fig. 4

Amplification η as a function of δ for a Gaussian grating with Δ = −0.2 and μ = 0.5. The inset shows the grating strength as a function of position.

Fig. 5
Fig. 5

Amplification η as a function of α for = 0 and (a) μ = 0, (b) μ = 0.25, (c) μ = 0.5, (d) μ = 1. The two curve styles are used only to distinguish the curves.

Fig. 6
Fig. 6

Amplification η as a function of α for μ = 0.5 and (a) = 0, (b) = 0.25, (c) = 0.5, (d) = 1. The two curve styles are used only to distinguish the curves.

Fig. 7
Fig. 7

Dispersion relation for uniform medium (dotted lines) and periodic medium (solid curves), neglecting material dispersion. The grating introduces an additional wave-vector mismatch Δk.

Tables (1)

Tables Icon

Table 1 Gain η and Bandwidth of the Peak Response for Gratings of Three Lengthsa

Equations (37)

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

E = [ E + exp ( i k s x ) + E - exp ( - i k s x ) ] exp ( - i ω s t ) + [ P + exp ( i k p x ) + P - exp ( - i k p x ) ] exp ( - i ω p t ) + [ I + exp ( i k i x ) + I - exp ( - i k i x ) ] exp ( - i ω i t ) + c . c . ,
n ( ω ) = n ¯ ( ω ) + Δ n cos ( 2 π x d ) ,
+ i E + x + i v E + t + 1 2 ω v 2 E + x 2 + κ exp ( - 2 i δ x ) E - + Γ { [ E + 2 + 2 E - 2 + 2 P + 2 + 2 P - 2 + 2 I + 2 + 2 I - 2 ] E + + 2 exp ( - i Δ si x ) I * P - P + + 2 exp ( - 2 i Δ sp x ) E - P - * P + + 2 exp ( - 2 i Δ si x ) E - I - * I + + exp ( - i Δ x ) I + * P + 2 } = 0 ,
+ i E + x + i v E + t + 1 2 ω v 2 E + x 2 + κ exp ( - 2 i δ x ) E - + Γ [ 2 P 2 E + + exp ( - i Δ x ) P 2 I * ] = 0 ,
- i E - x + i v E - t + 1 2 ω v 2 E - x 2 + κ exp ( + 2 i δ x ) E + + 2 Γ P 2 E - = 0 ,
+ i P x + i v p P t + 1 2 ω p v p 2 P x 2 + Γ p P 2 P = 0 ,
+ i I x + i v i I t + 1 2 ω i v i 2 I x 2 + Γ i [ 2 P 2 I + exp ( - i Δ x ) P 2 E + * ] = 0.
+ i d E + d x + κ exp ( - 2 i δ x ) E - + Γ [ 2 P 2 E + + exp ( - i Δ x ) P 2 I * ] = 0 ,
- i d E - d x + κ exp ( + 2 i δ x ) E + + 2 Γ P 2 E - = 0 ,
+ i d P d x + Γ p P 2 P = 0 ,
+ i d I d x + Γ i [ 2 P 2 I + exp ( - i Δ x ) P 2 E + * ] = 0.
P ( x ) = P 0 exp ( i Γ p P 0 2 x ) ,
C ± = E ± exp [ i ( Δ / 2 ± δ ) x ] ,
K = I exp [ - i ( - Δ / 2 + δ + 2 Γ p P 0 2 ) x ] ,
d d x [ C + K * C - ] = i [ Δ / 2 + δ + 2 μ μ κ - μ δ - Δ / 2 0 - κ 0 Δ / 2 - δ - 2 μ ] × [ C + K * C - ] ,
C + ( L ) 2 = C + ( 0 ) 2 [ cosh 2 g L + ( μ + Δ / 2 g ) 2 sinh 2 g L ] ,
L σ L ,             κ κ / σ ,             μ μ / σ ,             δ δ / σ ,             Δ Δ / σ ,
η = log 10 C + ( L ) 2 C + ( 0 ) 2 .
0 L κ ( x ) d x 4 ,
δ = - 4 μ - δ ,
Δ = - 4 μ - Δ ,
A 1 ( x ) = C + * ( x ) exp ( - 2 i μ x ) ,
A 2 ( z ) = - K ( x ) exp ( - 2 i μ x ) ,
A 3 ( x ) = - C - * ( x ) exp ( - 2 i μ x ) ,
α = δ + 2 μ ,
= Δ / 2 + μ ,
d d x [ C + K * C - ] = i [ α + μ κ - μ α - 0 - κ 0 - α + ] [ C + K * C - ] .
2 = k s + k i - 2 k p + 2 μ = 0.
α - ( α 2 - κ 2 ) 1 / 2 = 2
α = 4 2 + κ 2 4 ,
K * ( x ) = - i C + ( x ) .
δ max κ exp ( μ L / 2 ) .
E ± exp ( i a x ) ,
E + x , ω v 2 E + x 2 , κ E - exp ( - 2 i δ x ) ,
κ E + , ω v κ 2 E + , κ E - ,
ω v κ 1
ω " v κ .

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