Abstract

We present a theory and experiments on active mode locking in the presence of negative group-velocity dispersion (GVD) and self-phase modulation (SPM). It is shown that beyond a critical value of GVD a solitonlike pulse can be stabilized by the mode locker. The width of the soliton can be shorter than the width of the Gaussian pulse produced by the mode locker in the absence of soliton shaping. We establish analytically that the pulse shortening possible by addition of SPM and GVD is limited only by the requirement that the phase shift of the soliton per round trip be limited. Parameter ranges allowing for stable solitary-pulse formation and shortening are derived and discussed for different gain media and compared with numerical simulations and experimental results.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
    [Crossref]
  2. J. D. Kafka and T. Baer, “Mode-locked erbium-doped fiber laser with soliton pulse shaping,” Opt. Lett. 14, 1269–1271 (1989).
    [Crossref] [PubMed]
  3. K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (Institution of Electrical Engineers, London, 1992), p. 1/1-4.
  4. F. Fontana, N. Ridi, M. Romagnoli, and P. Franco, “Fully integrated 30 ps modelocked fiber laser electronically tunable over 1530–1560 nm,” Opt. Commun. 107, 240–244 (1994).
    [Crossref]
  5. J. D. Kafka, M. L. Watts, and J.W. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti:sapphire laser,” IEEE J. Quantum Electron. QE-28, 2151–2162 (1992).
    [Crossref]
  6. D. Kopf, F. X. Kärtner, and U. Keller, “Pulse shortening in a Nd:glass laser by gain reshaping and soliton formation,” Opt. Lett. 19, 2146–2148 (1994).
    [Crossref] [PubMed]
  7. A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode-locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174–176 (1994).
    [Crossref]
  8. L. Yan, P.-T. Ho, C. H. Lee, and G. L. Burdge, “Generation of ultrashort pulses form a neodymium glass laser system,” IEEE J. Quantum Electron. 25, 2431–2440 (1989).
    [Crossref]
  9. D. W. Hughes, M. W. Phillips, J. R. M. Barr, and D. C. Hanna, “A laser-diode-pumped Nd:glass laser: mode-locked, high power, and single frequency performance,” IEEE J. Quantum Electron. 28, 1010–1017 (1989).
    [Crossref]
  10. H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. QE-11, 323–330 (1975).
    [Crossref]
  11. H. A. Haus and A. Mecozzi, “Long-term storage of a bit stream of solitons,” Opt. Lett. 17, 1500–1502 (1992).
    [Crossref] [PubMed]
  12. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 34, 61–68 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].
  13. D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
    [Crossref]
  14. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
    [Crossref] [PubMed]
  15. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [Crossref]
  16. J. N. Elgin and S. M. J. Kelly, “Spectral modulation and the growth of resonant modes associated with periodically amplified solitons,” Opt. Lett. 21, 787–789 (1993).
    [Crossref]
  17. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
    [Crossref] [PubMed]
  18. S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average solitons,” Electron. Lett. 28, 806–807 (1992).
    [Crossref]
  19. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, “Sideband instability induced by periodic power variation in long-distance fiber links,” Opt. Lett. 18, 1499–1501 (1993).
    [Crossref] [PubMed]
  20. U. Keller, T. H. Chiu, and J. F. Ferguson, “Self-starting femtosecond mode-locked Nd:glass laser using intracavity saturable absorber,” Opt. Lett. 18, 1077 (1993).
    [Crossref]
  21. F. X. Kärtner and U. Keller, “Stabilization of solitonlike pulses with a slow saturable absorber,” Opt. Lett. 20, 16–18 (1995).
    [Crossref] [PubMed]

1995 (1)

1994 (3)

F. Fontana, N. Ridi, M. Romagnoli, and P. Franco, “Fully integrated 30 ps modelocked fiber laser electronically tunable over 1530–1560 nm,” Opt. Commun. 107, 240–244 (1994).
[Crossref]

D. Kopf, F. X. Kärtner, and U. Keller, “Pulse shortening in a Nd:glass laser by gain reshaping and soliton formation,” Opt. Lett. 19, 2146–2148 (1994).
[Crossref] [PubMed]

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode-locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174–176 (1994).
[Crossref]

1993 (3)

1992 (4)

H. A. Haus and A. Mecozzi, “Long-term storage of a bit stream of solitons,” Opt. Lett. 17, 1500–1502 (1992).
[Crossref] [PubMed]

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
[Crossref]

J. D. Kafka, M. L. Watts, and J.W. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti:sapphire laser,” IEEE J. Quantum Electron. QE-28, 2151–2162 (1992).
[Crossref]

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average solitons,” Electron. Lett. 28, 806–807 (1992).
[Crossref]

1991 (1)

1990 (1)

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[Crossref] [PubMed]

1989 (3)

J. D. Kafka and T. Baer, “Mode-locked erbium-doped fiber laser with soliton pulse shaping,” Opt. Lett. 14, 1269–1271 (1989).
[Crossref] [PubMed]

L. Yan, P.-T. Ho, C. H. Lee, and G. L. Burdge, “Generation of ultrashort pulses form a neodymium glass laser system,” IEEE J. Quantum Electron. 25, 2431–2440 (1989).
[Crossref]

D. W. Hughes, M. W. Phillips, J. R. M. Barr, and D. C. Hanna, “A laser-diode-pumped Nd:glass laser: mode-locked, high power, and single frequency performance,” IEEE J. Quantum Electron. 28, 1010–1017 (1989).
[Crossref]

1986 (1)

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
[Crossref]

1975 (1)

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

1971 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 34, 61–68 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

1970 (1)

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
[Crossref]

Baer, T.

Barr, J. R. M.

D. W. Hughes, M. W. Phillips, J. R. M. Barr, and D. C. Hanna, “A laser-diode-pumped Nd:glass laser: mode-locked, high power, and single frequency performance,” IEEE J. Quantum Electron. 28, 1010–1017 (1989).
[Crossref]

Burdge, G. L.

L. Yan, P.-T. Ho, C. H. Lee, and G. L. Burdge, “Generation of ultrashort pulses form a neodymium glass laser system,” IEEE J. Quantum Electron. 25, 2431–2440 (1989).
[Crossref]

Chiu, T. H.

Davey, R. P.

K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (Institution of Electrical Engineers, London, 1992), p. 1/1-4.

DeMaria, A. J.

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode-locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174–176 (1994).
[Crossref]

Elgin, J. N.

J. N. Elgin and S. M. J. Kelly, “Spectral modulation and the growth of resonant modes associated with periodically amplified solitons,” Opt. Lett. 21, 787–789 (1993).
[Crossref]

Ferguson, J. F.

Fontana, F.

F. Fontana, N. Ridi, M. Romagnoli, and P. Franco, “Fully integrated 30 ps modelocked fiber laser electronically tunable over 1530–1560 nm,” Opt. Commun. 107, 240–244 (1994).
[Crossref]

Franco, P.

F. Fontana, N. Ridi, M. Romagnoli, and P. Franco, “Fully integrated 30 ps modelocked fiber laser electronically tunable over 1530–1560 nm,” Opt. Commun. 107, 240–244 (1994).
[Crossref]

Gordon, J. P.

Greer, E. J.

K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (Institution of Electrical Engineers, London, 1992), p. 1/1-4.

Hanna, D. C.

D. W. Hughes, M. W. Phillips, J. R. M. Barr, and D. C. Hanna, “A laser-diode-pumped Nd:glass laser: mode-locked, high power, and single frequency performance,” IEEE J. Quantum Electron. 28, 1010–1017 (1989).
[Crossref]

Haus, H. A.

H. A. Haus and A. Mecozzi, “Long-term storage of a bit stream of solitons,” Opt. Lett. 17, 1500–1502 (1992).
[Crossref] [PubMed]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
[Crossref] [PubMed]

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
[Crossref]

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

Heynau, H.

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode-locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174–176 (1994).
[Crossref]

Ho, P.-T.

L. Yan, P.-T. Ho, C. H. Lee, and G. L. Burdge, “Generation of ultrashort pulses form a neodymium glass laser system,” IEEE J. Quantum Electron. 25, 2431–2440 (1989).
[Crossref]

Hughes, D. W.

D. W. Hughes, M. W. Phillips, J. R. M. Barr, and D. C. Hanna, “A laser-diode-pumped Nd:glass laser: mode-locked, high power, and single frequency performance,” IEEE J. Quantum Electron. 28, 1010–1017 (1989).
[Crossref]

Kafka, J. D.

J. D. Kafka, M. L. Watts, and J.W. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti:sapphire laser,” IEEE J. Quantum Electron. QE-28, 2151–2162 (1992).
[Crossref]

J. D. Kafka and T. Baer, “Mode-locked erbium-doped fiber laser with soliton pulse shaping,” Opt. Lett. 14, 1269–1271 (1989).
[Crossref] [PubMed]

Kärtner, F. X.

Kaup, D. J.

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[Crossref] [PubMed]

Keller, U.

Kelly, S. M. J.

J. N. Elgin and S. M. J. Kelly, “Spectral modulation and the growth of resonant modes associated with periodically amplified solitons,” Opt. Lett. 21, 787–789 (1993).
[Crossref]

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average solitons,” Electron. Lett. 28, 806–807 (1992).
[Crossref]

Kopf, D.

Kuizenga, D. J.

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
[Crossref]

Lai, Y.

Lee, C. H.

L. Yan, P.-T. Ho, C. H. Lee, and G. L. Burdge, “Generation of ultrashort pulses form a neodymium glass laser system,” IEEE J. Quantum Electron. 25, 2431–2440 (1989).
[Crossref]

Matera, F.

Mecozzi, A.

Moores, J. D.

Nelson, B. P.

K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (Institution of Electrical Engineers, London, 1992), p. 1/1-4.

Phillips, M. W.

D. W. Hughes, M. W. Phillips, J. R. M. Barr, and D. C. Hanna, “A laser-diode-pumped Nd:glass laser: mode-locked, high power, and single frequency performance,” IEEE J. Quantum Electron. 28, 1010–1017 (1989).
[Crossref]

Pieterse, J.W. J.

J. D. Kafka, M. L. Watts, and J.W. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti:sapphire laser,” IEEE J. Quantum Electron. QE-28, 2151–2162 (1992).
[Crossref]

Ridi, N.

F. Fontana, N. Ridi, M. Romagnoli, and P. Franco, “Fully integrated 30 ps modelocked fiber laser electronically tunable over 1530–1560 nm,” Opt. Commun. 107, 240–244 (1994).
[Crossref]

Romagnoli, M.

F. Fontana, N. Ridi, M. Romagnoli, and P. Franco, “Fully integrated 30 ps modelocked fiber laser electronically tunable over 1530–1560 nm,” Opt. Commun. 107, 240–244 (1994).
[Crossref]

F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, “Sideband instability induced by periodic power variation in long-distance fiber links,” Opt. Lett. 18, 1499–1501 (1993).
[Crossref] [PubMed]

Settembre, M.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 34, 61–68 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Siegman, A. E.

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
[Crossref]

Silberberg, Y.

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
[Crossref]

Smith, K.

K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (Institution of Electrical Engineers, London, 1992), p. 1/1-4.

Stetser, D. A.

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode-locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174–176 (1994).
[Crossref]

Watts, M. L.

J. D. Kafka, M. L. Watts, and J.W. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti:sapphire laser,” IEEE J. Quantum Electron. QE-28, 2151–2162 (1992).
[Crossref]

Yan, L.

L. Yan, P.-T. Ho, C. H. Lee, and G. L. Burdge, “Generation of ultrashort pulses form a neodymium glass laser system,” IEEE J. Quantum Electron. 25, 2431–2440 (1989).
[Crossref]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 34, 61–68 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Appl. Phys. Lett. (1)

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode-locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174–176 (1994).
[Crossref]

Electron. Lett. (1)

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average solitons,” Electron. Lett. 28, 806–807 (1992).
[Crossref]

IEEE J. Quantum Electron. (6)

D. J. Kuizenga and A. E. Siegman, “FM and AM mode locking of the homogeneous laser—Part I: Theory,” IEEE J. Quantum Electron. QE-6, 694–708 (1970).
[Crossref]

L. Yan, P.-T. Ho, C. H. Lee, and G. L. Burdge, “Generation of ultrashort pulses form a neodymium glass laser system,” IEEE J. Quantum Electron. 25, 2431–2440 (1989).
[Crossref]

D. W. Hughes, M. W. Phillips, J. R. M. Barr, and D. C. Hanna, “A laser-diode-pumped Nd:glass laser: mode-locked, high power, and single frequency performance,” IEEE J. Quantum Electron. 28, 1010–1017 (1989).
[Crossref]

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

H. A. Haus and Y. Silberberg, “Laser mode locking with addition of nonlinear index,” IEEE J. Quantum Electron. QE-22, 325–331 (1986).
[Crossref]

J. D. Kafka, M. L. Watts, and J.W. J. Pieterse, “Picosecond and femtosecond pulse generation in a regeneratively mode-locked Ti:sapphire laser,” IEEE J. Quantum Electron. QE-28, 2151–2162 (1992).
[Crossref]

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

Opt. Commun. (1)

F. Fontana, N. Ridi, M. Romagnoli, and P. Franco, “Fully integrated 30 ps modelocked fiber laser electronically tunable over 1530–1560 nm,” Opt. Commun. 107, 240–244 (1994).
[Crossref]

Opt. Lett. (8)

Phys. Rev. A (1)

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[Crossref] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 34, 61–68 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Other (1)

K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (Institution of Electrical Engineers, London, 1992), p. 1/1-4.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Pulse-width reduction as a function of normalized dispersion. Below Dn,crit = 0.652 no stable soliton can be formed.

Fig. 2
Fig. 2

Time evolution of the pulse intensity in a Nd:YAG laser for the parameters in Table 2, D = −17 ps2, for the first 1,000 round trips in the laser cavity, starting with a 68-ps-long Gaussian pulse.

Fig. 3
Fig. 3

Time evolution of (a) the intensity and (b) the spectrum for the same parameters as in Fig. 2 over 10,000 round trips. The laser reaches steady state after ∼4000 round trips.

Fig. 4
Fig. 4

Time evolution of the intensity in a Nd:YAG laser for the parameters in Table 2. The amount of negative dispersion is reduced to D = −10 ps2, starting again from a 68-ps-long pulse. The continuum in this case does not decay as in Figs. 2 and 3 because of insufficient dispersion.

Fig. 5
Fig. 5

Parameters chosen as for Fig. 4 but for 50,000 round trips.

Fig. 6
Fig. 6

Solid curves, the normalized fluorescence spectrum of the Nd:glass LG-760 used in our experiment, measured with an optical spectrum analyzer, when pumped below threshold. LG-760 is a quasi-homogeneous laser material; thus when it is lasing the saturated gain is proportional to the fluorescence spectrum. From the refractive index of the quartz prism and the beam waist at the knife position we can calculate the low-path filter function attributable to the knife-edge relative to the knife position, which is an error function. The several dashed curves show the difference between the saturated gain determined from the measured fluorescence spectrum and the computed low-path filter losses for different knife positions, which we call the net gain. For lasing at a certain power level, moving the knife edge into the cavity reduces the overall gain somewhat but also flattens the net gain profile over a range of ∼10 nm.

Fig. 7
Fig. 7

Autocorrelation of the mode-locked pulse (solid curve) and the corresponding sech2 fit (dashed curve). The output power was 70 mW at 930-mW absorbed pump power.

Fig. 8
Fig. 8

Sampling signal of the fast detector when the mode-locked laser operates at the transition to instability. The short, femtosecond pulse cannot be resolved by the detector and therefore results in a sharp spike corresponding to the detector response time. In advance of the femtosecond pulse there is a roughly 100-ps-long pulse.

Tables (2)

Tables Icon

Table 1 Maximum Pulse-Width Reduction and Necessary Normalized GVD for Different Laser Systemsa

Tables Icon

Table 2 Parameters Used for Numerical Simulations

Equations (103)

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

T R A ( T , t ) T = i D 2 A T 2 + i δ | A | 2 A + { g ( 1 + 1 Ω g 2 2 T 2 ) l M [ 1 cos ( ω M t ) ] } A
g = g 0 1 + ( W / E L ) ,
W = T R / 2 T R / 2 | A ( T , t ) | 2 d t ,
A s ( T , t ) = A 0 sech ( x ) exp ( i θ ) ,
x = ( 1 / τ ) ( t 2 | D | p 0 T t 0 )
θ = p 0 ( t t 0 ) + | D | ( 1 τ 2 p 0 2 ) T + θ 0 ,
Φ 0 = ½ δ A 0 2 = | D | τ 2 ,
W = | A s ( T , t ) | 2 d t = 2 A 0 2 τ .
T R A T = { g ( 1 + 1 Ω g 2 2 t 2 ) l M [ 1 cos ( ω M t ) ] } A .
A n ( T , t ) = A n ( t ) exp ( λ n T / T R ) ,
A n ( t ) = [ W n / ( 2 n π n ! τ a ) ] 1 / 2 × H n ( t / τ a ) exp ( t 2 2 τ a 2 ) ,
τ a = D g / M s 4 ,
D g = g Ω g 2 ,
M s = M ω M 2 2 .
λ n = g n l 2 M s τ a 2 ( n + ½ ) ,
g n = 1 1 + ( W n / E L ) .
τ a , FWHM = 1.66 τ a .
A ( T , t ) = [ a ( x ) + a c ( T , t ) ] exp ( i θ + Δ p t ) ,
a ( x ) = A sech ( x ) , x = ( 1 / τ ) [ t Δ t ( T ) ] ,
θ = 1 T R 0 T Φ ( T ) d T + Δ θ ( T ) ,
Φ = | D | / τ 2 = δ A 2 / 2 .
T R [ a c T + W T f w + Δ θ T f θ Δ p T f p + Δ t T f t ] = Φ L ( a c + Δ p f p ) + R ( a + Δ p f p + a c ) M ω M sin ( ω M τ x ) Δ t a ( x ) .
L = i σ 3 [ ( 2 x 2 1 ) + 2 sech 2 ( x ) ( 2 + σ 1 ) ] ,
R = g ( 1 + 1 Ω g 2 τ 2 2 x 2 ) l M [ 1 cos ( ω M τ x ) ] ,
f w ( x ) = 1 W [ 1 x tanh ( x ) ] a ( x ) ( 1 1 ) ,
f θ ( x ) = i a ( x ) ( 1 1 ) ,
f p ( x ) = i x τ a ( x ) ( 1 1 ) ,
f t ( x ) = 1 τ tanh ( x ) a ( x ) ( 1 1 ) .
L f w = 1 W f θ ,
L f θ = 0 ,
L f p = 2 τ 2 f t ,
L f t = 0 .
L f k = λ k f k ,
λ k = i ( k 2 + 1 ) ,
f k ( x ) = exp ( ikx ) ( ( k + i tanh x ) 2 sech 2 x ) ,
L f ¯ k = λ ¯ k f ¯ k ,
λ ¯ k = + i ( k 2 + 1 ) ,
f ¯ k = σ 1 f k .
u v = 1 2 + u + ( x ) v ( x ) d x .
L + = σ 3 L σ 3 ,
L + f k ( + ) = λ k ( + ) f k ( + ) ,
λ k ( + ) = i ( k 2 + 1 ) ,
f k ( + ) = 1 π ( k 2 + 1 ) 2 σ 3 f ¯ k ;
L + f ¯ k ( + ) = λ ¯ k ( + ) f ¯ k ( + ) ,
λ ¯ k ( + ) = i ( k 2 + 1 ) ,
f ¯ k ( + ) = 1 π ( k 2 + 1 ) 2 σ 3 f k .
f k ( + ) f k = δ ( k k ) , f ¯ k ( + ) f ¯ k = δ ( k k ) , f ¯ k ( + ) f k = f k ( + ) f ¯ k = 0 .
f w ( + ) ( x ) = i 2 τ σ 3 f θ ( x ) = 2 τ a ( x ) ( 1 1 ) ,
f θ ( + ) ( x ) = 2 τ σ 3 f w ( x ) = 2 i τ W [ 1 x tanh ( x ) ] a ( x ) ( 1 1 ) ,
f P ( + ) ( x ) = 2 i τ W σ 3 f t ( x ) = 2 i W tanh ( x ) a ( x ) ( 1 1 ) ,
f t ( + ) ( x ) = 2 i τ W σ 3 f P ( x ) = 2 τ 2 W x a ( x ) ( 1 1 ) .
δ ( x x ) = d k [ | f k f k ( + ) | + | f ¯ k f ¯ k ( + ) | ] + | f w f w ( + ) | + | f θ f θ ( + ) | + | f P f P ( + ) | + | f t f t ( + ) | .
a c = d k [ g ( k ) | f k + ( k ) | f ¯ k ] .
σ 1 a c = a c * ,
( k ) = g ( k ) * .
a c = 2 G ( x ) x 2 + 2 tanh ( x ) G ( x ) x tanh 2 ( x ) G ( x ) + G * ( x ) sech 2 ( x ) ,
G ( x ) = g ( k ) exp ( ikx ) d k .
T R W T = 2 ( g l g 3 Ω g 2 τ 2 π 2 24 M ω M 2 τ 2 ) W + f w ( + ) R a c .
g l = π 2 24 M ω M 2 τ 2 + g 3 Ω g 2 τ 2 ,
g = g 0 1 + W 0 / E L .
T R Δ W T = 2 [ g ( 1 + W 0 / E L ) ( W 0 E L + 1 3 Ω g 2 τ 2 ) + π 2 12 M ω M 2 τ 2 ] Δ W + f w ( + ) R a c ,
T R Δ θ T = f θ ( + ) R a c ,
T R Δ p T = 4 g 3 Ω g 2 τ 2 Δ p + f p ( + ) R a c ,
T R Δ t T = π 2 4 M ω M 2 τ 2 Δ t 2 | D | Δ p + f t ( + ) R a c ,
T R g ( k ) T = i Φ 0 ( k 2 + 1 ) g ( k ) + f k ( + ) R a c + f k ( + ) R [ a 0 ( x ) + Δ w f w + Δ p f p ] + f k ( + ) M ω M sin ( ω M τ x ) a 0 ( x ) Δ t .
ω M τ 1 Ω g τ
T R G T = { g l i Φ 0 + g Ω g 2 ( 1 + i D n ) 2 t 2 M [ 1 cos ( ω M t ) ] } G + 1 [ f k ( + ) R a 0 ( x ) f k ( + ) M ω M sin ( ω M τ x ) a 0 ( x ) Δ t ] ,
D n = | D | Ω g 2 / g .
τ c = τ a ( 1 + i D n ) 4 ,
λ m = i Φ 0 + g l M ω M 2 τ a 2 ( 1 + i D n ) ( m + ½ ) .
λ m = i Φ 0 + 1 3 D g M s [ ( τ a τ ) 2 + π 2 4 ( τ a τ ) 2 6 ( 1 + i D n ) ( m + ½ ) ] .
( τ a τ ) 2 + π 2 4 ( τ τ a ) 2 < 3 Re { ( 1 + i D n ) } .
ξ 2 3 Re { ( 1 + i D n ) } ξ + ( π 2 / 4 ) < 0 .
D n , crit = 0.652 .
R = 1.66 1.76 ξ .
ξ < 9 D n 2 or R < 1.66 1.76 9 D n 2 4 .
τ 2 = | D | / Φ 0 .
R max = 1.66 1.76 ( 9 Φ 0 / 2 ) 2 D g M s 12
τ min = 2 D g 2 9 Φ 0 M s 6 .
D n = 2 9 ( 9 Φ 0 / 2 ) 2 D g M s 3 .
τ trans T R = 1 Re { λ 0 } 3 D g M s R 2 .
G ( x ) = i Φ 0 1 { f k ( + ) R a 0 ( x ) M s τ 2 f k ( + ) x a 0 ( x ) ( Δ t / τ ) } .
| G ( x ) | A 0 Φ 0 D g τ 2 = A 0 D n ,
τ a , FWHM = 68 ps ,
Ω g 2 π ( THz )
ω M 2 π ( MHz )
D g ( ps 2 )
τ a , FWHM ( ps )
T min , FWHM ( ps )
τ trans T R
f k ( + ) | cos ( ω M τ x ) | f k , w , θ , p , t = ½ f k ( + ) | [ f k ω M τ ( + ) + f k + ω M τ ( + ) + O ( ω M τ ) ] | f k , w , θ , p , t ,
f k ( + ) | cos ( ω M τ x ) | f k = ½ [ δ ( k k + ω M τ ) + δ ( k k ω M τ ) ] + O ( ω M τ ) ,
f k ( + ) | cos ( ω M τ x ) | f ¯ k = O ( ω M τ ) ,
f ¯ k ( + ) | cos ( ω M τ x ) | f k = ½ [ δ ( k k + ω M τ ) + δ ( k k ω M τ ) ] + O ( ω M τ ) ,
f ¯ k ( + ) | cos ( ω M τ x ) | f k = O ( ω M τ ) ;
( f , f ¯ ) k ( + ) | cos ( ω M τ x ) | f w , θ , p , t = O ( ω M τ ) ,
f w , θ , p , t ( + ) | cos ( ω M τ x ) | ( f , f ¯ ) k = O ( ω M τ ) .
f k ( + ) ( x ) | F ( 1 Ω g τ x ) | f k , w , θ , p , t ( x ) ,
f k ( + ) ( ω ) | F ( i ω Ω g τ ) | f k , w , θ , p , t ( ω ) ,
f k ( ω ) = 2 π δ ( ω k ) + π ω k sinh [ π / 2 ( ω k ) ] + 2 k P . V . { 2 ω k + π sinh [ ( π / 2 ) ( ω k ) ] } ,
f k ( + ) ( ω ) | F ( i ω Ω g τ ) | f k , w , θ , p , t ( ω ) = F ( i k Ω g τ ) f k ( + ) ( ω ) f k , w , θ , p , t ( ω ) . + F ( i k Ω g τ ) f k ( + ) ( ω ) | ( ω ) | ( ω k ) f k , w , θ , p , t ( ω ) + .
f k ( + ) ( ω ) | F ( i ω Ω g τ ) | f k , w , θ , p , t ( ω ) = F ( i k Ω g τ ) δ ( k k ) + O ( 1 Ω g τ ) .
f k ( + ) | 2 x 2 | f k , w , θ , p , t ( ω ) = k 2 ( k k ) + O ( 1 Ω g τ ) .

Metrics