Abstract

We numerically study supercontinuum generation in photonic crystal fibers pumped with low-power 30-ps pulses close to the zero dispersion wavelength. We show how the efficiency is significantly improved by designing the dispersion to allow widely separated spectral lines generated by degenerate four-wave mixing directly from the pump to broaden and merge, resulting in a 800-nm-wide supercontinuum. Full-vectorial plane-wave calculations show that a cobweb photonic-crystal-fiber structure can realize the dispersion profiles under consideration. The predicted efficient supercontinuum generation is more robust and survives fiber imperfections modeled as random fluctuations of the dispersion coefficients along the fiber.

© 2003 Optical Society of America

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    [CrossRef]
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2003

2002

2001

2000

T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000).
[CrossRef]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

K. Tamura, H. Kubota, and M. Nakazawa, “Fundamentals of stable continuum generation at high repetition rates,” IEEE J. Quantum Electron. 36, 773–779 (2000).
[CrossRef]

J. Garnier and F. Kh. Abdullaev, “Modulational instability by random varying coefficients for the nonlinear Schrödinger equation,” Physica D 145, 65–83 (2000).
[CrossRef]

J. K. Ranka, R. S. Windler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
[CrossRef]

A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000).
[CrossRef]

1998

1997

K. Mori, H. Takara, S. Kawanishi, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fiber with convex dispersion profile,” Electron. Lett. 33, 1806–1808 (1997).
[CrossRef]

1996

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett. 8, 776–778 (1996).
[CrossRef]

1995

R. Knapp, “Transmission of solitons through random media,” Physica D 85, 496–508 (1995).
[CrossRef]

1991

1990

N. Kuwaki and M. Ohashi, “Evaluation of longitudinal chromatic dispersion,” J. Lightwave Technol. 8, 1476–1481 (1990).
[CrossRef]

1989

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” Proc. IEEE 25, 2665–2673 (1989).

1987

P. L. Baldeck and R. R. Alfano, “Intensity effects on the stimulated four photon spectra generated by picosecond pulses in optical fibers,” J. Lightwave Technol. 5, 1712–1715 (1987).
[CrossRef]

1976

C. Lin and R. H. Stolen, “New nanosecond continuum for excited-state spectroscopy,” Appl. Phys. Lett. 28, 216–218 (1976).
[CrossRef]

1970

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970).
[CrossRef]

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

Abdullaev, F. Kh.

J. Garnier and F. Kh. Abdullaev, “Modulational instability by random varying coefficients for the nonlinear Schrödinger equation,” Physica D 145, 65–83 (2000).
[CrossRef]

Alfano, R. R.

P. L. Baldeck and R. R. Alfano, “Intensity effects on the stimulated four photon spectra generated by picosecond pulses in optical fibers,” J. Lightwave Technol. 5, 1712–1715 (1987).
[CrossRef]

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970).
[CrossRef]

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

Andrekson, P. A.

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett. 8, 776–778 (1996).
[CrossRef]

Andres, M. V.

Andres, P.

Arriaga, J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

Baldeck, P. L.

P. L. Baldeck and R. R. Alfano, “Intensity effects on the stimulated four photon spectra generated by picosecond pulses in optical fibers,” J. Lightwave Technol. 5, 1712–1715 (1987).
[CrossRef]

Birks, T. A.

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” Proc. IEEE 25, 2665–2673 (1989).

Chau, A.

Chau, A. H. L.

Chen, H. H.

Coen, S.

Dinda, P. T.

Dudley, J. M.

Eggeleton, B. J.

Ferrando, A.

Garnier, J.

J. Garnier and F. Kh. Abdullaev, “Modulational instability by random varying coefficients for the nonlinear Schrödinger equation,” Physica D 145, 65–83 (2000).
[CrossRef]

Griebner, U.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

Grossard, N.

Hansen, K. P.

Harvey, J.

Harvey, J. D.

Hedekvist, P. O.

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett. 8, 776–778 (1996).
[CrossRef]

Herrmann, J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

A. V. Husakou and J. Herrmann, “Supercontinuum generation, four-wave mixing, and fission of higher-order solitons in photonic-crystal fibers,” J. Opt. Soc. Am. B 19, 2171–2182 (2002).
[CrossRef]

Husakou, A.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

Husakou, A. V.

Joannopoulos, J. D.

Johnson, S. G.

Karlsson, M.

M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B 15, 2269–2275 (1998).
[CrossRef]

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett. 8, 776–778 (1996).
[CrossRef]

Kawanishi, S.

K. Mori, H. Takara, and S. Kawanishi, “Analysis and design of supercontinuum pulse generation in a single-mode optical fiber,” J. Opt. Soc. Am. B 18, 1780–1790 (2001).
[CrossRef]

K. Mori, H. Takara, S. Kawanishi, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fiber with convex dispersion profile,” Electron. Lett. 33, 1806–1808 (1997).
[CrossRef]

Knapp, R.

R. Knapp, “Transmission of solitons through random media,” Physica D 85, 496–508 (1995).
[CrossRef]

Knight, J. C.

Korn, G.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

Kubota, H.

K. Tamura, H. Kubota, and M. Nakazawa, “Fundamentals of stable continuum generation at high repetition rates,” IEEE J. Quantum Electron. 36, 773–779 (2000).
[CrossRef]

Kuwaki, N.

N. Kuwaki and M. Ohashi, “Evaluation of longitudinal chromatic dispersion,” J. Lightwave Technol. 8, 1476–1481 (1990).
[CrossRef]

Leonardt, R.

Lin, C.

C. Lin and R. H. Stolen, “New nanosecond continuum for excited-state spectroscopy,” Appl. Phys. Lett. 28, 216–218 (1976).
[CrossRef]

Maillotte, H.

Man, T. P. M.

Menyuk, C. R.

Millot, G.

Miret, J. J.

Mori, K.

K. Mori, H. Takara, and S. Kawanishi, “Analysis and design of supercontinuum pulse generation in a single-mode optical fiber,” J. Opt. Soc. Am. B 18, 1780–1790 (2001).
[CrossRef]

K. Mori, H. Takara, S. Kawanishi, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fiber with convex dispersion profile,” Electron. Lett. 33, 1806–1808 (1997).
[CrossRef]

Morioka, T.

K. Mori, H. Takara, S. Kawanishi, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fiber with convex dispersion profile,” Electron. Lett. 33, 1806–1808 (1997).
[CrossRef]

Nagel, J. A.

Nakazawa, M.

K. Tamura, H. Kubota, and M. Nakazawa, “Fundamentals of stable continuum generation at high repetition rates,” IEEE J. Quantum Electron. 36, 773–779 (2000).
[CrossRef]

Nickel, D.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

Ohashi, M.

N. Kuwaki and M. Ohashi, “Evaluation of longitudinal chromatic dispersion,” J. Lightwave Technol. 8, 1476–1481 (1990).
[CrossRef]

Ortigosa-Blanch, A.

W. J. Wadsworth, A. Ortigosa-Blanch, J. C. Knight, T. A. Birks, T. P. M. Man, and P. St. J. Russell, “Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source,” J. Opt. Soc. Am. B 19, 2148–2155 (2002).
[CrossRef]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

Pole, C. D.

Provino, L.

Ranka, J. K.

Reeves, W. H.

Russell, P. S. J.

Russell, P. St. J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

W. J. Wadsworth, A. Ortigosa-Blanch, J. C. Knight, T. A. Birks, T. P. M. Man, and P. St. J. Russell, “Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source,” J. Opt. Soc. Am. B 19, 2148–2155 (2002).
[CrossRef]

W. H. Reeves, J. C. Knight, and P. St. J. Russell, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express 10, 609–613 (2002).
[CrossRef] [PubMed]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000).
[CrossRef]

Shapiro, S. L.

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970).
[CrossRef]

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

Silvestre, E.

Stentz, A. J.

Stolen, R. H.

C. Lin and R. H. Stolen, “New nanosecond continuum for excited-state spectroscopy,” Appl. Phys. Lett. 28, 216–218 (1976).
[CrossRef]

Takara, H.

K. Mori, H. Takara, and S. Kawanishi, “Analysis and design of supercontinuum pulse generation in a single-mode optical fiber,” J. Opt. Soc. Am. B 18, 1780–1790 (2001).
[CrossRef]

K. Mori, H. Takara, S. Kawanishi, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fiber with convex dispersion profile,” Electron. Lett. 33, 1806–1808 (1997).
[CrossRef]

Tamura, K.

K. Tamura, H. Kubota, and M. Nakazawa, “Fundamentals of stable continuum generation at high repetition rates,” IEEE J. Quantum Electron. 36, 773–779 (2000).
[CrossRef]

Wabnitz, S.

Wadsworth, W. J.

Wadworth, W. J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

Wai, P. K. A.

Windler, R. S.

Winters, J. H.

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” Proc. IEEE 25, 2665–2673 (1989).

Zhavoronkov, N.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

Appl. Phys. Lett.

C. Lin and R. H. Stolen, “New nanosecond continuum for excited-state spectroscopy,” Appl. Phys. Lett. 28, 216–218 (1976).
[CrossRef]

Electron. Lett.

K. Mori, H. Takara, S. Kawanishi, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fiber with convex dispersion profile,” Electron. Lett. 33, 1806–1808 (1997).
[CrossRef]

IEEE J. Quantum Electron.

K. Tamura, H. Kubota, and M. Nakazawa, “Fundamentals of stable continuum generation at high repetition rates,” IEEE J. Quantum Electron. 36, 773–779 (2000).
[CrossRef]

IEEE Photon. Technol. Lett.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett. 8, 776–778 (1996).
[CrossRef]

J. Lightwave Technol.

N. Kuwaki and M. Ohashi, “Evaluation of longitudinal chromatic dispersion,” J. Lightwave Technol. 8, 1476–1481 (1990).
[CrossRef]

P. L. Baldeck and R. R. Alfano, “Intensity effects on the stimulated four photon spectra generated by picosecond pulses in optical fibers,” J. Lightwave Technol. 5, 1712–1715 (1987).
[CrossRef]

J. Opt. Soc. Am. B

P. T. Dinda, G. Millot, and S. Wabnitz, “Polarization switching and suppression of stimulated Raman scattering in birefringent optical fibers,” J. Opt. Soc. Am. B 15, 1433–1441 (1998).
[CrossRef]

M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B 15, 2269–2275 (1998).
[CrossRef]

K. Mori, H. Takara, and S. Kawanishi, “Analysis and design of supercontinuum pulse generation in a single-mode optical fiber,” J. Opt. Soc. Am. B 18, 1780–1790 (2001).
[CrossRef]

S. Coen, A. Chau, R. Leonardt, J. Harvey, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Supercontinuum generation via stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers,” J. Opt. Soc. Am. B 19, 753–764 (2002).
[CrossRef]

J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windler, B. J. Eggeleton, and S. Coen, “Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B 19, 765–771 (2002).
[CrossRef]

W. J. Wadsworth, A. Ortigosa-Blanch, J. C. Knight, T. A. Birks, T. P. M. Man, and P. St. J. Russell, “Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source,” J. Opt. Soc. Am. B 19, 2148–2155 (2002).
[CrossRef]

A. V. Husakou and J. Herrmann, “Supercontinuum generation, four-wave mixing, and fission of higher-order solitons in photonic-crystal fibers,” J. Opt. Soc. Am. B 19, 2171–2182 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970).
[CrossRef]

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002).
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Figures (12)

Fig. 1
Fig. 1

Dispersion profiles d1 (left) and d2 (right). Top row: phase mismatch Δβ for direct degenerate FWM (dashed curve) and spectrum at L=10 cm (solid curve). Bottom row: spectrum at L=20 cm, 30 cm, 1 m, 2 m, and 3.7 m (down to up).

Fig. 2
Fig. 2

Dispersion profile d1 (thick solid curve), d2 (dashed curve), d3 (dash-dotted curve), d4 (dotted curve), d5 (circles), and d6 (thin solid curve).

Fig. 3
Fig. 3

Phase mismatch Δβ(λ) for dispersion profile d1 (thick solid curve), d2 (dashed curve), d3 (dash-dotted curve), d4 (dotted curve), d5 (circles), and d6 (thin solid curve).

Fig. 4
Fig. 4

Degenerate FWM gain g(λ) for dispersion profile d1 (thick solid curve), d2 (dashed curve), d3 (dash-dotted curve), d4 (dotted curve), d5 (circles), and d6 (thin solid curve).

Fig. 5
Fig. 5

Dispersion profiles d2 (left) and d3 (right). Top row: phase mismatch Δβ for direct degenerate FWM (dashed curve) and the spectrum at L=10 cm (solid curve). Bottom row: spectrum at L=20 cm, 30 cm, 1 m, and 2 m (down to up).

Fig. 6
Fig. 6

Dispersion profiles d4 (left) and d5 (right). Top row: phase mismatch Δβ for direct degenerate FWM (dashed curve) and the spectrum at L=10 cm (solid curve). Bottom row: spectrum at L=20 cm, 30 cm, 1 m, and 2 m (down to up).

Fig. 7
Fig. 7

Dispersion profiles d3 (left) and d6 (right). Top row: phase mismatch Δβ for direct degenerate FWM (dashed curve) and the spectrum at L=10 cm (solid curve). Bottom row: spectrum at L=20 cm, 30 cm, 1 m, and 2 m (down to up).

Fig. 8
Fig. 8

Left: dispersion profiles for the cobweb PCF structures with pitch 8.5 μm, wall thickness 130 nm, and core diameter 1600 nm (solid curve), 1500 nm (dashed curve), 1450 nm (dash-dotted curve), and 1400 nm (dotted curve). Right: Corresponding phase-mismatch curves.

Fig. 9
Fig. 9

Dispersion profiles d3 (left) and cob3 (right). Top row: phase mismatch Δβ for direct degenerate FWM (dashed curve) and spectrum at L=10 cm (solid curve). Bottom row: spectrum at L=20 cm, 30 cm, 1 m, 2 m, and 3.7 m (down to up).

Fig. 10
Fig. 10

Random fluctuations of the FWM Stokes gain band g(Δβ) (gray region), given by Eq. (4), for constant pump Ip and ρ=1% (left) and ρ=10% (right). The upper row is for case d1 and the bottom row for case cob3.

Fig. 11
Fig. 11

Average FWM Stokes gain gav over L=20 cm, as given by Eq. (9) in the undepleted pump approximation. Shown is case d1 (left) and cob3 (right) for ρ=0 (solid), ρ=1% (dashed), and ρ=10% (dotted).

Fig. 12
Fig. 12

Dispersion profiles d1 (left) and cob3 (right). Spectrum under influence of fluctuations with strength ρ=0 (top row) and ρ=10% (bottom row) at L=20 cm, 30 cm, 1 m, 2 m, and 3.7 m (down to up).

Tables (1)

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Table 1 Dispersion Profiles a

Equations (11)

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Ajz=iμAj+i(j-1)δβAj+(-1)jΔ2Ajτ-i2k=27βkk!kAjτk+iγ1+iωpτ×AjfRhR(τ-s)(|Aj(s)|2+|A3-j(s)|2)ds+(1-fR)|Aj|2+23 |A3-j|2Aj+13 Aj*A3-j2.
hR(t)=τ12+τ22τ1τ22exp(-t/τ2)sin(t/τ1),
Δβ=2Ω2β22!+Ω4β44!+Ω6β66!+γ(1-fR)Ip,
g={[(1-fR)γIp]2-(Δβ/2)2}1/2.
D(λ)=-2πcλ2β2+2πcβ31λ-1λp.
β2=λz4Dz2πc1λp-1λz,β3=λz4Dz4π2c2,
βkβkmax-βkmin(βkmax+βkmin)/2,k=1,2.
δβ(z)=δβ+σ0(z),βk,x(z)=βk+σk,x(z),
Δ(z)=Δ+σ1(z), βk,y(z)=βk+σk,y(z),
σ02(z)δβ=σ12(z)Δ=σk,x2(z)βk=σk,y2(z)βk=ρ,
gav1L0Lg[Δβ(z)]dz,

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