Abstract

A single nonlinear differential equation describing a cw synchronously pumped, singly resonant optical parametric oscillator is derived. This equation accounts for the important effects associated with this kind of ultrashort tunable-pulse generation, including the effects of phase mismatch, pulse walk-off, group-velocity dispersion, cavity-length detuning, and pump depletion. The various effects are investigated in detail through numerical solutions. The formalism is general enough that the results obtained are applicable to a wide variety of systems.

© 1990 Optical Society of America

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  1. T. Kushida, Y. Tanaka, and M. Ojima, “Tunable picosecond pulse generation by optical parametric oscillator,” Jpn. J. Appl. Phys. 16, 2227–2235 (1977).
    [CrossRef]
  2. Y. Tanaka, T. Kushida, and S. Shionoya, “Broadly tunable, repetitive, picosecond parametric oscillator,” Opt. Commun. 25, 273–276 (1978).
    [CrossRef]
  3. G. I. Onishchukov, A. A. Fomichev, and A. I. Kholodnykh, “Picosecond optical parametric oscillator pumped by radiation from a continuously excited YAG:Nd3+laser,” Sov. J. Quantum Electron. 16, 1001–1002 (1983).
    [CrossRef]
  4. A. Piskarskas, V. Smil’gyavichyus, A. Umbrasas, and I. Yuodishyus, “Parametric generation of picosecond light pulses in an LiNbO3crystal at repetition frequencies up to 10 kHz,” Sov. J. Quantum Electron. 16, 841–843 (1986).
    [CrossRef]
  5. A. Piskarskas, V. Smil’gyavichyus, and A. Umbrasas, “Continuous parametric generation of picosecond light pulses,” Sov. J. Quantum Electron. 18, 155–156 (1988).
    [CrossRef]
  6. D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
    [CrossRef]
  7. E. S. Wachman, D. C. Edelstein, and C. L. Tang, “Continuous-wave mode-locked and dispersion-compensated femtosecond optical parametric oscillator,” Opt. Lett. 15, 136–138 (1990).
    [CrossRef] [PubMed]
  8. M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expressions for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys. 45, 3996–4005 (1974).
    [CrossRef]
  9. H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
    [CrossRef]
  10. H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–747 (1975).
    [CrossRef]
  11. H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. QE-11, 323–330 (1975).
    [CrossRef]
  12. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  13. H. J. Bakker, P. C. M. Planken, and H. G. Müller, “Numerical calculation of optical frequency-conversion processes: a new approach,” J. Opt. Soc. Am. B 6, 1665–1672 (1989).
    [CrossRef]
  14. J. P. Gordon and R. L. Fork, “Optical resonator with negative dispersion,” Opt. Lett. 9, 153–155 (1984).
    [CrossRef] [PubMed]
  15. The intracavity signal electric field that appears in Eq. (1) is the electric field outside the nonlinear crystal, while that used in the derivation of the gain expression G(t) is the field inside the crystal. Since the final expression for G(t) as given by Eq. (27) depends on the signal intensity rather than on the electric field, the proportionality constant between the electric fields inside and outside the crystal ultimately cancels and does not appear in the final equations describing the system.
  16. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  17. M. J. Weber, CRC Handbook of Laser Science and Technology (CRC, Boca Raton, Fla., 1987), Vol. III, Part 1.
  18. A. Piskarskas, A. Stabinis, A. Umbrasas, and A. Yankauskas, “Parametric chirp and 20-fold compression of pulses from a quasi-cw picosecond optical parametric oscillator,” Sov. J. Quantum Electron. 15, 1539–1541 (1985).
    [CrossRef]

1990 (1)

1989 (2)

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

H. J. Bakker, P. C. M. Planken, and H. G. Müller, “Numerical calculation of optical frequency-conversion processes: a new approach,” J. Opt. Soc. Am. B 6, 1665–1672 (1989).
[CrossRef]

1988 (1)

A. Piskarskas, V. Smil’gyavichyus, and A. Umbrasas, “Continuous parametric generation of picosecond light pulses,” Sov. J. Quantum Electron. 18, 155–156 (1988).
[CrossRef]

1986 (1)

A. Piskarskas, V. Smil’gyavichyus, A. Umbrasas, and I. Yuodishyus, “Parametric generation of picosecond light pulses in an LiNbO3crystal at repetition frequencies up to 10 kHz,” Sov. J. Quantum Electron. 16, 841–843 (1986).
[CrossRef]

1985 (1)

A. Piskarskas, A. Stabinis, A. Umbrasas, and A. Yankauskas, “Parametric chirp and 20-fold compression of pulses from a quasi-cw picosecond optical parametric oscillator,” Sov. J. Quantum Electron. 15, 1539–1541 (1985).
[CrossRef]

1984 (1)

1983 (1)

G. I. Onishchukov, A. A. Fomichev, and A. I. Kholodnykh, “Picosecond optical parametric oscillator pumped by radiation from a continuously excited YAG:Nd3+laser,” Sov. J. Quantum Electron. 16, 1001–1002 (1983).
[CrossRef]

1978 (1)

Y. Tanaka, T. Kushida, and S. Shionoya, “Broadly tunable, repetitive, picosecond parametric oscillator,” Opt. Commun. 25, 273–276 (1978).
[CrossRef]

1977 (1)

T. Kushida, Y. Tanaka, and M. Ojima, “Tunable picosecond pulse generation by optical parametric oscillator,” Jpn. J. Appl. Phys. 16, 2227–2235 (1977).
[CrossRef]

1975 (3)

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–747 (1975).
[CrossRef]

H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. QE-11, 323–330 (1975).
[CrossRef]

1974 (1)

M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expressions for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys. 45, 3996–4005 (1974).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bakker, H. J.

Becker, M. F.

M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expressions for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys. 45, 3996–4005 (1974).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Edelstein, D. C.

E. S. Wachman, D. C. Edelstein, and C. L. Tang, “Continuous-wave mode-locked and dispersion-compensated femtosecond optical parametric oscillator,” Opt. Lett. 15, 136–138 (1990).
[CrossRef] [PubMed]

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

Fomichev, A. A.

G. I. Onishchukov, A. A. Fomichev, and A. I. Kholodnykh, “Picosecond optical parametric oscillator pumped by radiation from a continuously excited YAG:Nd3+laser,” Sov. J. Quantum Electron. 16, 1001–1002 (1983).
[CrossRef]

Fork, R. L.

Gordon, J. P.

Haus, H. A.

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–747 (1975).
[CrossRef]

H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. QE-11, 323–330 (1975).
[CrossRef]

Kholodnykh, A. I.

G. I. Onishchukov, A. A. Fomichev, and A. I. Kholodnykh, “Picosecond optical parametric oscillator pumped by radiation from a continuously excited YAG:Nd3+laser,” Sov. J. Quantum Electron. 16, 1001–1002 (1983).
[CrossRef]

Kuizenga, D. J.

M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expressions for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys. 45, 3996–4005 (1974).
[CrossRef]

Kushida, T.

Y. Tanaka, T. Kushida, and S. Shionoya, “Broadly tunable, repetitive, picosecond parametric oscillator,” Opt. Commun. 25, 273–276 (1978).
[CrossRef]

T. Kushida, Y. Tanaka, and M. Ojima, “Tunable picosecond pulse generation by optical parametric oscillator,” Jpn. J. Appl. Phys. 16, 2227–2235 (1977).
[CrossRef]

Müller, H. G.

Ojima, M.

T. Kushida, Y. Tanaka, and M. Ojima, “Tunable picosecond pulse generation by optical parametric oscillator,” Jpn. J. Appl. Phys. 16, 2227–2235 (1977).
[CrossRef]

Onishchukov, G. I.

G. I. Onishchukov, A. A. Fomichev, and A. I. Kholodnykh, “Picosecond optical parametric oscillator pumped by radiation from a continuously excited YAG:Nd3+laser,” Sov. J. Quantum Electron. 16, 1001–1002 (1983).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phillion, D. W.

M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expressions for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys. 45, 3996–4005 (1974).
[CrossRef]

Piskarskas, A.

A. Piskarskas, V. Smil’gyavichyus, and A. Umbrasas, “Continuous parametric generation of picosecond light pulses,” Sov. J. Quantum Electron. 18, 155–156 (1988).
[CrossRef]

A. Piskarskas, V. Smil’gyavichyus, A. Umbrasas, and I. Yuodishyus, “Parametric generation of picosecond light pulses in an LiNbO3crystal at repetition frequencies up to 10 kHz,” Sov. J. Quantum Electron. 16, 841–843 (1986).
[CrossRef]

A. Piskarskas, A. Stabinis, A. Umbrasas, and A. Yankauskas, “Parametric chirp and 20-fold compression of pulses from a quasi-cw picosecond optical parametric oscillator,” Sov. J. Quantum Electron. 15, 1539–1541 (1985).
[CrossRef]

Planken, P. C. M.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Shionoya, S.

Y. Tanaka, T. Kushida, and S. Shionoya, “Broadly tunable, repetitive, picosecond parametric oscillator,” Opt. Commun. 25, 273–276 (1978).
[CrossRef]

Siegman, A. E.

M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expressions for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys. 45, 3996–4005 (1974).
[CrossRef]

Smil’gyavichyus, V.

A. Piskarskas, V. Smil’gyavichyus, and A. Umbrasas, “Continuous parametric generation of picosecond light pulses,” Sov. J. Quantum Electron. 18, 155–156 (1988).
[CrossRef]

A. Piskarskas, V. Smil’gyavichyus, A. Umbrasas, and I. Yuodishyus, “Parametric generation of picosecond light pulses in an LiNbO3crystal at repetition frequencies up to 10 kHz,” Sov. J. Quantum Electron. 16, 841–843 (1986).
[CrossRef]

Stabinis, A.

A. Piskarskas, A. Stabinis, A. Umbrasas, and A. Yankauskas, “Parametric chirp and 20-fold compression of pulses from a quasi-cw picosecond optical parametric oscillator,” Sov. J. Quantum Electron. 15, 1539–1541 (1985).
[CrossRef]

Tanaka, Y.

Y. Tanaka, T. Kushida, and S. Shionoya, “Broadly tunable, repetitive, picosecond parametric oscillator,” Opt. Commun. 25, 273–276 (1978).
[CrossRef]

T. Kushida, Y. Tanaka, and M. Ojima, “Tunable picosecond pulse generation by optical parametric oscillator,” Jpn. J. Appl. Phys. 16, 2227–2235 (1977).
[CrossRef]

Tang, C. L.

E. S. Wachman, D. C. Edelstein, and C. L. Tang, “Continuous-wave mode-locked and dispersion-compensated femtosecond optical parametric oscillator,” Opt. Lett. 15, 136–138 (1990).
[CrossRef] [PubMed]

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

Umbrasas, A.

A. Piskarskas, V. Smil’gyavichyus, and A. Umbrasas, “Continuous parametric generation of picosecond light pulses,” Sov. J. Quantum Electron. 18, 155–156 (1988).
[CrossRef]

A. Piskarskas, V. Smil’gyavichyus, A. Umbrasas, and I. Yuodishyus, “Parametric generation of picosecond light pulses in an LiNbO3crystal at repetition frequencies up to 10 kHz,” Sov. J. Quantum Electron. 16, 841–843 (1986).
[CrossRef]

A. Piskarskas, A. Stabinis, A. Umbrasas, and A. Yankauskas, “Parametric chirp and 20-fold compression of pulses from a quasi-cw picosecond optical parametric oscillator,” Sov. J. Quantum Electron. 15, 1539–1541 (1985).
[CrossRef]

Wachman, E. S.

E. S. Wachman, D. C. Edelstein, and C. L. Tang, “Continuous-wave mode-locked and dispersion-compensated femtosecond optical parametric oscillator,” Opt. Lett. 15, 136–138 (1990).
[CrossRef] [PubMed]

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

Weber, M. J.

M. J. Weber, CRC Handbook of Laser Science and Technology (CRC, Boca Raton, Fla., 1987), Vol. III, Part 1.

Yankauskas, A.

A. Piskarskas, A. Stabinis, A. Umbrasas, and A. Yankauskas, “Parametric chirp and 20-fold compression of pulses from a quasi-cw picosecond optical parametric oscillator,” Sov. J. Quantum Electron. 15, 1539–1541 (1985).
[CrossRef]

Yuodishyus, I.

A. Piskarskas, V. Smil’gyavichyus, A. Umbrasas, and I. Yuodishyus, “Parametric generation of picosecond light pulses in an LiNbO3crystal at repetition frequencies up to 10 kHz,” Sov. J. Quantum Electron. 16, 841–843 (1986).
[CrossRef]

Appl. Phys. Lett. (1)

D. C. Edelstein, E. S. Wachman, and C. L. Tang, “Broadly tunable high repetition rate femtosecond optical parametric oscillator,” Appl. Phys. Lett. 54, 1728–1730 (1989).
[CrossRef]

IEEE J. Quantum Electron. (2)

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–747 (1975).
[CrossRef]

H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. QE-11, 323–330 (1975).
[CrossRef]

J. Appl. Phys. (2)

M. F. Becker, D. J. Kuizenga, D. W. Phillion, and A. E. Siegman, “Analytic expressions for ultrashort pulse generation in mode-locked optical parametric oscillators,” J. Appl. Phys. 45, 3996–4005 (1974).
[CrossRef]

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

T. Kushida, Y. Tanaka, and M. Ojima, “Tunable picosecond pulse generation by optical parametric oscillator,” Jpn. J. Appl. Phys. 16, 2227–2235 (1977).
[CrossRef]

Opt. Commun. (1)

Y. Tanaka, T. Kushida, and S. Shionoya, “Broadly tunable, repetitive, picosecond parametric oscillator,” Opt. Commun. 25, 273–276 (1978).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Sov. J. Quantum Electron. (4)

A. Piskarskas, A. Stabinis, A. Umbrasas, and A. Yankauskas, “Parametric chirp and 20-fold compression of pulses from a quasi-cw picosecond optical parametric oscillator,” Sov. J. Quantum Electron. 15, 1539–1541 (1985).
[CrossRef]

G. I. Onishchukov, A. A. Fomichev, and A. I. Kholodnykh, “Picosecond optical parametric oscillator pumped by radiation from a continuously excited YAG:Nd3+laser,” Sov. J. Quantum Electron. 16, 1001–1002 (1983).
[CrossRef]

A. Piskarskas, V. Smil’gyavichyus, A. Umbrasas, and I. Yuodishyus, “Parametric generation of picosecond light pulses in an LiNbO3crystal at repetition frequencies up to 10 kHz,” Sov. J. Quantum Electron. 16, 841–843 (1986).
[CrossRef]

A. Piskarskas, V. Smil’gyavichyus, and A. Umbrasas, “Continuous parametric generation of picosecond light pulses,” Sov. J. Quantum Electron. 18, 155–156 (1988).
[CrossRef]

Other (3)

The intracavity signal electric field that appears in Eq. (1) is the electric field outside the nonlinear crystal, while that used in the derivation of the gain expression G(t) is the field inside the crystal. Since the final expression for G(t) as given by Eq. (27) depends on the signal intensity rather than on the electric field, the proportionality constant between the electric fields inside and outside the crystal ultimately cancels and does not appear in the final equations describing the system.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

M. J. Weber, CRC Handbook of Laser Science and Technology (CRC, Boca Raton, Fla., 1987), Vol. III, Part 1.

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of a cw synchronously pumped, singly resonant OPO. The solid, dotted, and dashed Lines correspond to the signal, idler, and pump pulses, respectively.

Fig. 2
Fig. 2

Loci of constant σ’s on the signal-intensity–pump-intensity plane.

Fig. 3
Fig. 3

Product of nondegeneracy factor and signal pulse width fΔt1 for type I phase matching in AgGaS2.

Fig. 4
Fig. 4

(a) Intensity profiles of pump, signal, and idler at various effective detuning times, (b) output signal efficiency as a function of effective detuning time at various peak pump intensities. For (a), Î30 = 4L. The values listed in Table 1 are used for other parameters.

Fig. 5
Fig. 5

Peak pump intensity threshold as a function of effective detuning time at various values of crystal bandwidths. The values listed in Table 1 are used for other parameters. The detuning range for fixed pump intensity and crystal bandwidth can be inferred from this figure.

Fig. 6
Fig. 6

(a) Intensity profiles of signal, idler, and pump at various values of crystal bandwidth, (b) output signal efficiency versus crystal bandwidth curves at various peak pump intensities. For (a), Î30 = 6L. The values listed in Table 1 are used for other parameters.

Fig. 7
Fig. 7

Minimum peak pump intensity threshold as a function of crystal bandwidth at various values of cavity bandwidths for output mirror transmission T2 = (a) 3%, (b) 8%. The values listed in Table 1 are used for other parameters.

Fig. 8
Fig. 8

(a) Intensity profiles of signal, idler, and pump at various values of cavity bandwidths, (b) output signal efficiency versus cavity bandwidth curves at various peak pump intensities. For (a), Î30 = 6L. The values listed in Table 1 are used for other parameters.

Fig. 9
Fig. 9

(a) Output signal efficiency, (b) output signal peak intensity, and (c) output signal pulse width as functions of peak pump intensity at various normalized effective detuning times. The curves from the lowest pump threshold to the highest threshold correspond to effective detuning times of 0, 0.005, 0.01, 0.015, and −0.005, respectively. In (b), constant-σ curves are plotted as dashed curves to indicate the regions of validity. The values Usted in Table 1 are used.

Fig. 10
Fig. 10

(a) Output signal efficiency and (b) output signal pulse width as functions of output mirror transmission at various peak pump intensities. The values listed in Table 1 are used.

Fig. 11
Fig. 11

(a) Output signal efficiency and (b) output signal pulse width as functions of pulse-walk-off time at various peak pump intensities. The values listed in Table 1 are used.

Fig. 12
Fig. 12

Output signal efficiency as a function of signal-frequency-to-pump-frequency ratio at various peak pump intensities. The values listed in Table 1 are used for other parameters.

Tables (1)

Tables Icon

Table 1 List of Parameters and Their Most Used Values in the Numerical Solutions

Equations (51)

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[ G ( t ) L ( t ) + δ T d d t + 1 ω c 2 d 2 d t 2 ] E 1 ( t ) = 0.
( z + 1 υ g 1 t + i D 1 2 ω 1 c 2 t 2 ) E 1 ( z , t ) = i 2 π ω 1 2 k 1 c 2 χ eff E 2 * E 3 e i Δ k z , ( z + 1 υ g 2 t + i D 2 2 ω 2 c 2 t 2 ) E 2 ( z , t ) = i 2 π ω 2 2 k 2 c 2 χ eff E 1 * E 3 e i Δ k z , ( z + 1 υ g 3 t + i D 3 2 ω 3 c 2 t 2 ) E 3 ( z , t ) = i 2 π ω 3 2 k 3 c 2 χ eff E 1 E 2 e i Δ k z ,
( ζ + δ T 13 l c τ + i D 1 2 ω 1 c 2 τ 2 ) E 1 = i 2 π ω 1 2 k 1 c 2 χ eff E 2 * E 3 e i Δ k ζ , ( ζ + δ T 23 l c τ + i D 2 2 ω 2 c 2 τ 2 ) E 2 = i 2 π ω 2 2 k 2 c 2 χ eff E 1 * E 3 e i Δ k ζ , ( ζ + i D 3 2 ω 3 c 2 τ 2 ) E 1 = i 2 π ω 3 2 k 3 c 2 χ eff E 1 E 2 e i Δ k ζ ,
u 1 ( l c , t ) = u 1 ( 0 , t ) [ 1 + 1 2 g 0 2 l c 2 + 1 6 g 0 4 l c 4 1 6 ω 3 I 1 ( 0 , t ) ω 1 I 3 ( 0 , t ) g 0 4 l c 4 1 24 g 0 2 l c 4 Δ k 2 ] ,
g 0 2 = 8 π 3 ω 1 2 ω 2 2 ω 3 k 1 k 2 k 3 c 6 I 3 ( 0 , t ) χ eff 2 .
ϕ 1 ( l c , t ) = ϕ 1 ( 0 , t ) + 1 6 g 0 2 Δ k l c 3 × { 1 1 5 g 0 2 l c 2 [ 2 + ω 3 I 1 ( 0 , t ) ω 1 I 3 ( 0 , t ) ] } .
G 0 ( t ) = 1 2 g 0 2 l c 2 + 1 6 g 0 4 l c 4 1 6 ω 3 I 1 ( t ) ω 1 I 3 ( t ) g 0 4 l c 4 1 24 g 0 2 l c 4 Δ k 2 + i 1 6 g 0 2 l c 3 Δ k { 1 1 5 g 0 2 l c 2 [ 2 + ω 3 I 1 ( 0 , t ) ω 1 I 3 ( 0 , t ) ] } .
Δ k = 2 π l c Δ ω n δ ω .
Δ ω n = 2 π c l c ( n 2 n 1 + ω 2 d n 2 d ω | ω 2 ω 1 d ω 1 d ω | ω 1 ) 1 ,
E 1 ( 0 ) ( ζ , τ ) = E 1 ( 0 , t ) ( 1 + 1 2 g 0 2 ζ 2 + i 1 6 g 0 2 Δ k ζ 2 + ) , E 2 ( 0 ) ( ζ , τ ) = E 1 ( 0 , t ) [ ω 2 ω 1 ( k 1 k 2 ) 1 / 2 g 0 ζ + i 1 2 Δ k ζ + ] , E 3 ( 0 ) ( ζ , τ ) = E 3 ( 0 , τ ) + .
E i ( ζ , τ ) = E i ( 0 ) ( ζ , τ ) + E i 1 ( 1 ) ( ζ , τ ) δ T 13 + E i 2 ( 1 ) ( ζ , τ ) δ T 23 +
E i j ( k ) ( ζ , τ ) = E i j 1 ( k ) ( τ ) ζ + E i j 2 ( k ) ( τ ) ζ 2 + .
E 11 ( 1 ) ( ζ , τ ) ζ l c ( 1 + 1 3 g 0 2 ζ 2 ) δ δ τ E 1 ( 0 , τ ) ,
E 12 ( 1 ) ( ζ , τ ) 1 6 l c g 0 2 ζ 3 δ δ τ E 1 * ( 0 , τ ) .
G ( t ) = G 0 ( t ) 2 δ T 13 d d t 1 3 g 0 2 l c 2 δ T 13 d d t .
G ( t ) = G 0 ( t ) 2 δ T 13 d d t 1 3 g 0 2 l c 2 δ T 13 d d t i D 1 l c ω 1 c d 2 d t 2 .
[ G ( t ) L S ( t ) + π 2 3 Δ ω n 2 a I 3 ( t ) d 2 d t 2 + 1 ω c 2 d 2 d t 2 i D 1 l c ω 1 c d 2 d t 2 δ T n ( t ) d d t 2 δ T 13 d d t + δ T d d t ] E 1 ( t ) = 0 ,
a = 4 π 3 ω 1 2 ω 2 2 ω 3 k 1 k 2 k 3 c 6 χ eff 2 l c 2
δ T n ( t ) = 2 π 3 Δ ω n a I 3 ( t ) { 1 2 5 [ 2 a I 3 ( t ) + ω 3 ω 1 a I 1 ( t ) ] } + 2 3 a I 3 ( t ) δ T 13 .
G ( t ) = a I 3 ( t ) + 2 3 a 2 I 3 2 ( t ) .
S ( t ) = 2 a 2 ω 3 3 ω 1 I 3 ( t ) I 1 ( t ) .
I s ( t ) = ( 1 R 2 ) I 1 ( t ) .
I i ( t ) = 2 ω 2 ω 1 a I 3 ( t ) I 1 ( t ) { 1 + 2 3 [ a I 3 ( t ) ω 3 ω 1 a I 1 ( t ) ] } .
I p ( t ) = I 3 ( t ) ω 3 ω 1 I i ( t ) .
| a I 3 ( 1 14 ω 3 I 1 ω 1 T 3 + ω 3 2 I 1 2 ω 1 2 I 3 2 ) 10 ( 1 ω 3 I 1 ω 1 T 3 ) | | σ | 1.
| ( π δ ω Δ ω n ) 2 4 a I 3 ( 1 ω 3 I 1 ω 1 T 3 ) | | 4 π 2 a I 3 ( 1 ω 3 I 1 ω 1 T 3 ) ( Δ ω n Δ t 3 ) 2 | 1 | κ | 1 ,
f = ( 3 ln 2 4 ) 1 / 2 ( λ 1 3 d 2 n 1 d λ | λ 1 + λ 2 3 d 2 n 2 d λ | λ 2 n 2 n 1 λ 2 d n 2 d λ | λ 2 + λ 1 d n 1 d λ | λ 1 ) × 1 π c Δ t 1 1.
I 3 ( t ) = I 30 exp ( 4 ln 2 t 2 Δ t 3 2 ) .
[ I ˆ 3 ( t ˆ ) + 2 3 I ˆ 3 2 ( t ˆ ) L 2 ω ˆ 3 3 ω ˆ 1 I ˆ 3 ( t ˆ ) I ˆ 1 ( t ˆ ) + π 2 3 Δ ω ˆ n 2 I ˆ 3 ( t ˆ ) d d t ˆ 2 + 1 ω ˆ c 2 d 2 d t ˆ 2 δ T ˆ n ( t ˆ ) d d t ˆ + δ T ˆ eff d d t ˆ ] E ˆ 1 ( t ) = 0.
δ T ˆ n ( t ˆ ) = 2 π 3 Δ ω ˆ n I ˆ 3 ( t ˆ ) { 1 2 5 [ 2 I ˆ 3 ( t ˆ ) + ω ˆ 3 ω ˆ 1 I ˆ 1 ( t ˆ ) ] } + 2 3 I ˆ 3 ( t ˆ ) δ T ˆ 13 .
E 1 ζ = i 2 π ω 1 2 k 1 c 2 χ eff E 2 * E 3 e i Δ k ζ , E 2 ζ = i 2 π ω 2 2 k 2 c 2 χ eff E 1 * E 3 e i Δ k ζ , E 3 ζ = i 2 π ω 3 2 k 3 c 2 χ eff E 1 E 2 e i Δ k ζ .
I ( τ ) = c 2 2 π [ ( k 1 ω 1 | E 1 | 2 + k 2 ω 2 | E 2 | 2 + k 3 ω 3 | E 3 | 2 ) ] = I 1 ( ζ , τ ) + I 2 ( ζ , τ ) + I 3 ( ζ , τ ) ;
u 1 ( ζ , τ ) exp [ i ϕ 1 ( ζ , τ ) ] = ( c 2 k 1 2 π ω 1 2 I ) 1 / 2 E 1 ( ζ , τ ) , u 2 ( ζ , τ ) exp [ i ϕ 2 ( ζ , τ ) ] = ( c 2 k 2 2 π ω 2 2 I ) 1 / 2 E 2 ( ζ , τ ) , u 3 ( ζ , τ ) exp [ i ϕ 3 ( ζ , τ ) ] = ( c 2 k 3 2 π ω 3 2 I ) 1 / 2 E 3 ( ζ , τ ) ;
θ ( ζ , τ ) = Δ k ζ + ϕ 3 ( ζ , τ ) ϕ 1 ( ζ , τ ) ϕ 2 ( ζ , τ ) ;
ζ = 2 π c 2 χ eff ( 2 π I ω 1 2 ω 2 2 ω 3 2 c 2 k 1 k 2 k 3 ) ζ ;
Δ S = ( ζ / ζ ) Δ k .
d u 1 d ζ = u 2 u 3 sin θ , d u 2 d ζ = u 1 u 3 sin θ , d u 3 d ζ = u 1 u 2 sin θ ,
d ϕ 1 d ζ = u 2 u 3 u 1 cos θ , d ϕ 2 d ζ = u 1 u 3 u 2 cos θ , d ϕ 3 d ζ = u 1 u 2 u 3 cos θ ,
d θ d ζ = Δ S + cot θ d d ζ ln ( u 1 u 2 u 3 ) .
( ζ + δ T 13 l c τ + i D 1 2 ω 1 c 2 τ 2 ) E 1 = i 4 π ω 1 2 k 1 c 2 χ eff E 1 * E 3 e i Δ k ζ , ( ζ + i D 3 2 ω 3 c 2 τ 2 ) E 3 = i 2 π ω 3 2 k 3 c 2 χ eff E 1 2 e i Δ k ζ .
G ( t ) = 2 [ 2 a I 3 ( t ) ] 1 / 2 + 4 a I 3 ( t ) + 4 3 [ 2 a I 3 ( t ) ] 3 / 2 ω 3 ω 1 { 1 + 16 3 [ 2 a I 3 ( t ) ] 1 / 2 } a I 1 ( t ) 1 3 [ 2 a I 3 ( t ) ] 1 / 2 Δ k 2 l c 2 + i { [ 2 a I 3 ( t ) ] 1 / 2 2 3 a [ ω 3 ω 1 I 1 ( t ) + 4 I 3 ( t ) ] } Δ k l c δ T 13 d d t i D 1 l c ω 1 c d 2 d t 2 .
Δ k = 2 π l c Δ ω d 2 δ ω 2 ,
Δ ω d 2 = 2 π c l c ( 2 d n 1 d ω | ω 1 + ω 1 d 2 n 1 d ω 2 | ω 1 ) 1 .
{ G ( t ) L S ( t ) 4 π 2 3 Δ ω d 4 [ 2 a I 3 ( t ) ] 1 / 2 d 4 d t 4 + 1 ω c 2 d 2 d t 2 i D 1 l c ω 1 c d 2 d t 2 + δ T d 1 2 ( t ) d 2 d t 2 + δ T d d t δ T 13 d d t } E 1 ( t ) = 0 ,
G ( t ) = 2 [ 2 a I 3 ( t ) ] 1 / 2 + 4 a I 3 ( t ) + 4 3 [ 2 a I 3 ( t ) ] 3 / 2 ,
S ( t ) = 2 ω 3 ω 1 { 1 + 16 3 [ 2 a I 3 ( t ) ] 1 / 2 } a I 1 ( t ) ,
δ T d 1 2 ( t ) = 2 π Δ ω d 2 { [ 2 a I 3 ( t ) ] 1 / 2 2 3 a [ ω 3 ω 1 I 1 ( t ) + 4 I 3 ( t ) ] } .
{ G ( t ) L 1 S ( t ) + 2 π 2 3 Δ ω n 2 [ 2 a I 2 ( t ) I 3 ( t ) I 1 ( t ) ] 1 / 2 d 2 d t 2 + 1 ω c 2 d 2 d t 2 i D 1 l c ω 1 c d 2 d t 2 δ T d 2 2 ( t ) d d t + δ T ( t ) d d t Δ T 13 d d t } E 1 ( t ) = 0 ,
G ( t ) = [ 2 a I 2 3 ( t ) I 3 ( t ) I 1 ( t ) ] 1 / 2 + a I 3 ( t ) + 1 3 [ 2 a 3 I 2 ( t ) I 3 2 ( t ) I 1 ( t ) ] 1 / 2 ,
S ( t ) = ω 3 6 ω 1 ( 2 a I 2 ( t ) { 3 + [ 2 a I 2 ( t ) I 3 ( t ) I 1 ( t ) ] 1 / 2 + 8 [ 2 a 3 I 1 ( t ) I 2 ( t ) I 3 ( t ) ] 1 / 2 } ) ,
δ T d 2 ( t ) = π Δ ω n { [ 2 a I 2 ( t ) I 3 ( t ) I 1 ( t ) ] 1 / 2 + 1 3 [ 2 a I 3 ( t ) 2 ω 3 ω 1 a I 2 6 a I 2 ( t ) I 3 ( t ) I 1 ( t ) ] } ,

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