Abstract

Passively synchronized Er-doped and Yb-doped mode-locked fiber lasers with a master–slave configuration are theoretically investigated based on the pulse propagation model for simulating pulse collision in the common fiber section and on the master equation model for simulating laser dynamics. Computational results indicate that the central optical frequency of the slave Er laser will be shifted by the significant cross phase modulation (XPM) effect for the laser to become synchronized with the injected master Yb laser pulse train. Pulse duration change caused by fiber dispersion in the common fiber section will distort the ideal anti-symmetric characteristics of the XPM-induced frequency shift versus the relative timing position of the two color pulses. The relative timing jitter noises of the two synchronized lasers can be minimized by adjusting the relative pulse timing position, and the predicted dependence agrees well with the experimental observation.

© 2014 Optical Society of America

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References

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  1. J. Kim and F. X. Kärtner, “Attosecond-precision ultrafast photonics,” Laser Photon. Rev. 4, 432–456 (2010).
    [CrossRef]
  2. J. A. Cox, W. P. Putnam, A. Sell, A. Leitenstorfer, and F. X. Kärtner, “Pulse synthesis in the single-cycle regime from independent mode-locked lasers using attosecond-precision feedback,” Opt. Lett. 37, 3579–3581 (2012).
    [CrossRef]
  3. D. Yoshitomi, Y. Kobayashi, M. Kakehata, H. Takada, K. Torizuka, T. Onuma, H. Yokoi, T. Sekiguchi, and S. Nakamura, “Ultralow-jitter passive timing stabilization of a mode-locked Er-doped fiber laser by injection of an optical pulse train,” Opt. Lett. 31, 3243–3245 (2006).
    [CrossRef]
  4. W.-W. Hsiang, W.-C. Chiao, C.-Y. Wu, and Y. Lai, “Direct observation of two-color pulse dynamics in passively synchronized Er and Yb mode-locked fiber lasers,” Opt. Express 19, 24507–24515 (2011).
    [CrossRef]
  5. J. Biegert, P. K. Bates, and O. Chalus, “New mid-infrared light sources,” IEEE J. Sel. Topics Quantum Electron. 18, 531–540 (2012).
    [CrossRef]
  6. C. Fürst, A. Leitenstorfer, and A. Laubereau, “Mechanism for self-synchronization of femtosecond pulses in a two-color Ti:sapphire laser,” IEEE J. Sel. Topics Quantum Electron. 2, 473–479 (1996).
    [CrossRef]
  7. A. M. Weiner, Ultrafast Optics (Wiley, 2009).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
    [CrossRef]
  13. M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458–1470 (2004).
    [CrossRef]
  14. R. Paschotta, “Noise of mode-locked lasers (Part I): numerical model,” Appl. Phys. B 79, 153–162 (2004).
    [CrossRef]
  15. R. Paschotta, “Timing jitter and phase noise of mode-locked fiber lasers,” Opt. Express 18, 5041–5054 (2010).
    [CrossRef]
  16. J. Chen, J. W. Sickler, E. P. Ippen, and F. X. Kärtner, “High repetition rate, low jitter, low intensity noise, fundamentally mode-locked 167  fs soliton Er-fiber laser,” Opt. Lett. 32, 1566–1568 (2007).
    [CrossRef]
  17. J. A. Cox, A. H. Nejadmalayeri, J. Kim, and F. X. Kärtner, “Complete characterization of quantum-limited timing jitter in passively mode-locked fiber lasers,” Opt. Lett. 35, 3522–3524 (2010).
    [CrossRef]
  18. Y. Song, K. Jung, and J. Kim, “Impact of pulse dynamics on timing jitter in mode-locked fiber lasers,” Opt. Lett. 36, 1761–1763 (2011).
    [CrossRef]
  19. Y. Song, C. Kim, K. Jung, H. Kim, and J. Kim, “Timing jitter optimization of mode-locked Yb-fiber lasers toward the attosecond regime,” Opt. Express 19, 14518–14525 (2011).
    [CrossRef]
  20. W.-W. Hsiang, H.-C. Chang, and Y. Lai, “Laser dynamics of a 10  GHz 0.55  ps asynchronously harmonic modelocked Er-doped fiber soliton laser,” IEEE J. Quantum Electron. 46, 292–299 (2010).
    [CrossRef]
  21. W.-W. Hsiang, B.-W. Tsai, C. Hu, S.-Y. Wu, and Y. Lai, “Subfemtosecond synchronization between Yb-fiber and Er-fiber lasers by controlling the relative injection timing,” in Conferences on Lasers and Electro-Optics (Optical Society of America, 2013), paper JTh2A.27.
  22. B.-W. Tsai, S.-Y. Wu, C. Hu, W.-W. Hsiang, and Y. Lai, “Subfemtosecond hybrid synchronization between ultrafast Yb- and Er-fiber laser systems by controlling the relative injection timing,” Opt. Lett. 38, 3456–3459 (2013).
    [CrossRef]
  23. S.-Y. Wu, W.-W. Hsiang, and Y. Lai, “Relative timing jitter of passively synchronized Er-doped and Yb-doped mode-locked fiber lasers,” in IEEE Photonics Conference (IEEE Photonics Society, 2013), paper ThB.1.2.
  24. P. L. Baldeck, R. R. Alfano, and G. P. Agrawal, “Induced-frequency shift of copropagating ultrafast optical pulses,” Appl. Phys. Lett. 52, 1939–1941 (1988).
    [CrossRef]
  25. N. G. Usechak and G. P. Agrawal, “Rate-equation approach for frequency-modulation mode locking using the moment method,” J. Opt. Soc. Am. B 22, 2570–2580 (2005).
    [CrossRef]

2013 (1)

2012 (2)

2011 (3)

2010 (5)

2009 (1)

2007 (2)

2006 (1)

2005 (1)

2004 (2)

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458–1470 (2004).
[CrossRef]

R. Paschotta, “Noise of mode-locked lasers (Part I): numerical model,” Appl. Phys. B 79, 153–162 (2004).
[CrossRef]

2003 (1)

1996 (1)

C. Fürst, A. Leitenstorfer, and A. Laubereau, “Mechanism for self-synchronization of femtosecond pulses in a two-color Ti:sapphire laser,” IEEE J. Sel. Topics Quantum Electron. 2, 473–479 (1996).
[CrossRef]

1993 (1)

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

1988 (1)

P. L. Baldeck, R. R. Alfano, and G. P. Agrawal, “Induced-frequency shift of copropagating ultrafast optical pulses,” Appl. Phys. Lett. 52, 1939–1941 (1988).
[CrossRef]

Agrawal, G. P.

N. G. Usechak and G. P. Agrawal, “Rate-equation approach for frequency-modulation mode locking using the moment method,” J. Opt. Soc. Am. B 22, 2570–2580 (2005).
[CrossRef]

P. L. Baldeck, R. R. Alfano, and G. P. Agrawal, “Induced-frequency shift of copropagating ultrafast optical pulses,” Appl. Phys. Lett. 52, 1939–1941 (1988).
[CrossRef]

Alfano, R. R.

P. L. Baldeck, R. R. Alfano, and G. P. Agrawal, “Induced-frequency shift of copropagating ultrafast optical pulses,” Appl. Phys. Lett. 52, 1939–1941 (1988).
[CrossRef]

Baldeck, P. L.

P. L. Baldeck, R. R. Alfano, and G. P. Agrawal, “Induced-frequency shift of copropagating ultrafast optical pulses,” Appl. Phys. Lett. 52, 1939–1941 (1988).
[CrossRef]

Bao, X.

Bates, P. K.

J. Biegert, P. K. Bates, and O. Chalus, “New mid-infrared light sources,” IEEE J. Sel. Topics Quantum Electron. 18, 531–540 (2012).
[CrossRef]

Biegert, J.

J. Biegert, P. K. Bates, and O. Chalus, “New mid-infrared light sources,” IEEE J. Sel. Topics Quantum Electron. 18, 531–540 (2012).
[CrossRef]

Chalus, O.

J. Biegert, P. K. Bates, and O. Chalus, “New mid-infrared light sources,” IEEE J. Sel. Topics Quantum Electron. 18, 531–540 (2012).
[CrossRef]

Chang, H.-C.

W.-W. Hsiang, H.-C. Chang, and Y. Lai, “Laser dynamics of a 10  GHz 0.55  ps asynchronously harmonic modelocked Er-doped fiber soliton laser,” IEEE J. Quantum Electron. 46, 292–299 (2010).
[CrossRef]

Chen, J.

Chen, L.

Chen, Y.

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458–1470 (2004).
[CrossRef]

Chiao, W.-C.

Cox, J. A.

Fürst, C.

C. Fürst, A. Leitenstorfer, and A. Laubereau, “Mechanism for self-synchronization of femtosecond pulses in a two-color Ti:sapphire laser,” IEEE J. Sel. Topics Quantum Electron. 2, 473–479 (1996).
[CrossRef]

Grein, M. E.

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458–1470 (2004).
[CrossRef]

Haus, H. A.

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458–1470 (2004).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

Heidt, A. M.

Holzlöhner, R.

Hsiang, W.-W.

B.-W. Tsai, S.-Y. Wu, C. Hu, W.-W. Hsiang, and Y. Lai, “Subfemtosecond hybrid synchronization between ultrafast Yb- and Er-fiber laser systems by controlling the relative injection timing,” Opt. Lett. 38, 3456–3459 (2013).
[CrossRef]

W.-W. Hsiang, W.-C. Chiao, C.-Y. Wu, and Y. Lai, “Direct observation of two-color pulse dynamics in passively synchronized Er and Yb mode-locked fiber lasers,” Opt. Express 19, 24507–24515 (2011).
[CrossRef]

W.-W. Hsiang, H.-C. Chang, and Y. Lai, “Laser dynamics of a 10  GHz 0.55  ps asynchronously harmonic modelocked Er-doped fiber soliton laser,” IEEE J. Quantum Electron. 46, 292–299 (2010).
[CrossRef]

W.-W. Hsiang, B.-W. Tsai, C. Hu, S.-Y. Wu, and Y. Lai, “Subfemtosecond synchronization between Yb-fiber and Er-fiber lasers by controlling the relative injection timing,” in Conferences on Lasers and Electro-Optics (Optical Society of America, 2013), paper JTh2A.27.

S.-Y. Wu, W.-W. Hsiang, and Y. Lai, “Relative timing jitter of passively synchronized Er-doped and Yb-doped mode-locked fiber lasers,” in IEEE Photonics Conference (IEEE Photonics Society, 2013), paper ThB.1.2.

Hu, C.

B.-W. Tsai, S.-Y. Wu, C. Hu, W.-W. Hsiang, and Y. Lai, “Subfemtosecond hybrid synchronization between ultrafast Yb- and Er-fiber laser systems by controlling the relative injection timing,” Opt. Lett. 38, 3456–3459 (2013).
[CrossRef]

W.-W. Hsiang, B.-W. Tsai, C. Hu, S.-Y. Wu, and Y. Lai, “Subfemtosecond synchronization between Yb-fiber and Er-fiber lasers by controlling the relative injection timing,” in Conferences on Lasers and Electro-Optics (Optical Society of America, 2013), paper JTh2A.27.

Hult, J.

Ippen, E. P.

J. Chen, J. W. Sickler, E. P. Ippen, and F. X. Kärtner, “High repetition rate, low jitter, low intensity noise, fundamentally mode-locked 167  fs soliton Er-fiber laser,” Opt. Lett. 32, 1566–1568 (2007).
[CrossRef]

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458–1470 (2004).
[CrossRef]

Jung, K.

Kakehata, M.

Kärtner, F. X.

Kim, C.

Kim, H.

Kim, J.

Kobayashi, Y.

Lai, Y.

B.-W. Tsai, S.-Y. Wu, C. Hu, W.-W. Hsiang, and Y. Lai, “Subfemtosecond hybrid synchronization between ultrafast Yb- and Er-fiber laser systems by controlling the relative injection timing,” Opt. Lett. 38, 3456–3459 (2013).
[CrossRef]

W.-W. Hsiang, W.-C. Chiao, C.-Y. Wu, and Y. Lai, “Direct observation of two-color pulse dynamics in passively synchronized Er and Yb mode-locked fiber lasers,” Opt. Express 19, 24507–24515 (2011).
[CrossRef]

W.-W. Hsiang, H.-C. Chang, and Y. Lai, “Laser dynamics of a 10  GHz 0.55  ps asynchronously harmonic modelocked Er-doped fiber soliton laser,” IEEE J. Quantum Electron. 46, 292–299 (2010).
[CrossRef]

S.-Y. Wu, W.-W. Hsiang, and Y. Lai, “Relative timing jitter of passively synchronized Er-doped and Yb-doped mode-locked fiber lasers,” in IEEE Photonics Conference (IEEE Photonics Society, 2013), paper ThB.1.2.

W.-W. Hsiang, B.-W. Tsai, C. Hu, S.-Y. Wu, and Y. Lai, “Subfemtosecond synchronization between Yb-fiber and Er-fiber lasers by controlling the relative injection timing,” in Conferences on Lasers and Electro-Optics (Optical Society of America, 2013), paper JTh2A.27.

Laubereau, A.

C. Fürst, A. Leitenstorfer, and A. Laubereau, “Mechanism for self-synchronization of femtosecond pulses in a two-color Ti:sapphire laser,” IEEE J. Sel. Topics Quantum Electron. 2, 473–479 (1996).
[CrossRef]

Leitenstorfer, A.

J. A. Cox, W. P. Putnam, A. Sell, A. Leitenstorfer, and F. X. Kärtner, “Pulse synthesis in the single-cycle regime from independent mode-locked lasers using attosecond-precision feedback,” Opt. Lett. 37, 3579–3581 (2012).
[CrossRef]

C. Fürst, A. Leitenstorfer, and A. Laubereau, “Mechanism for self-synchronization of femtosecond pulses in a two-color Ti:sapphire laser,” IEEE J. Sel. Topics Quantum Electron. 2, 473–479 (1996).
[CrossRef]

Mecozzi, A.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

Menyuk, C. R.

Nakamura, S.

Nejadmalayeri, A. H.

Onuma, T.

Paschotta, R.

R. Paschotta, “Timing jitter and phase noise of mode-locked fiber lasers,” Opt. Express 18, 5041–5054 (2010).
[CrossRef]

R. Paschotta, “Noise of mode-locked lasers (Part I): numerical model,” Appl. Phys. B 79, 153–162 (2004).
[CrossRef]

Putnam, W. P.

Sekiguchi, T.

Sell, A.

Sickler, J. W.

Sinkin, O. V.

Song, Y.

Takada, H.

Torizuka, K.

Tsai, B.-W.

B.-W. Tsai, S.-Y. Wu, C. Hu, W.-W. Hsiang, and Y. Lai, “Subfemtosecond hybrid synchronization between ultrafast Yb- and Er-fiber laser systems by controlling the relative injection timing,” Opt. Lett. 38, 3456–3459 (2013).
[CrossRef]

W.-W. Hsiang, B.-W. Tsai, C. Hu, S.-Y. Wu, and Y. Lai, “Subfemtosecond synchronization between Yb-fiber and Er-fiber lasers by controlling the relative injection timing,” in Conferences on Lasers and Electro-Optics (Optical Society of America, 2013), paper JTh2A.27.

Usechak, N. G.

Weiner, A. M.

A. M. Weiner, Ultrafast Optics (Wiley, 2009).

Wu, C.-Y.

Wu, S.-Y.

B.-W. Tsai, S.-Y. Wu, C. Hu, W.-W. Hsiang, and Y. Lai, “Subfemtosecond hybrid synchronization between ultrafast Yb- and Er-fiber laser systems by controlling the relative injection timing,” Opt. Lett. 38, 3456–3459 (2013).
[CrossRef]

W.-W. Hsiang, B.-W. Tsai, C. Hu, S.-Y. Wu, and Y. Lai, “Subfemtosecond synchronization between Yb-fiber and Er-fiber lasers by controlling the relative injection timing,” in Conferences on Lasers and Electro-Optics (Optical Society of America, 2013), paper JTh2A.27.

S.-Y. Wu, W.-W. Hsiang, and Y. Lai, “Relative timing jitter of passively synchronized Er-doped and Yb-doped mode-locked fiber lasers,” in IEEE Photonics Conference (IEEE Photonics Society, 2013), paper ThB.1.2.

Yokoi, H.

Yoshitomi, D.

Zhang, Z.

Zweck, J.

Appl. Phys. B (1)

R. Paschotta, “Noise of mode-locked lasers (Part I): numerical model,” Appl. Phys. B 79, 153–162 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

P. L. Baldeck, R. R. Alfano, and G. P. Agrawal, “Induced-frequency shift of copropagating ultrafast optical pulses,” Appl. Phys. Lett. 52, 1939–1941 (1988).
[CrossRef]

IEEE J. Quantum Electron. (3)

W.-W. Hsiang, H.-C. Chang, and Y. Lai, “Laser dynamics of a 10  GHz 0.55  ps asynchronously harmonic modelocked Er-doped fiber soliton laser,” IEEE J. Quantum Electron. 46, 292–299 (2010).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively modelocked lasers,” IEEE J. Quantum Electron. 40, 1458–1470 (2004).
[CrossRef]

IEEE J. Sel. Topics Quantum Electron. (2)

J. Biegert, P. K. Bates, and O. Chalus, “New mid-infrared light sources,” IEEE J. Sel. Topics Quantum Electron. 18, 531–540 (2012).
[CrossRef]

C. Fürst, A. Leitenstorfer, and A. Laubereau, “Mechanism for self-synchronization of femtosecond pulses in a two-color Ti:sapphire laser,” IEEE J. Sel. Topics Quantum Electron. 2, 473–479 (1996).
[CrossRef]

J. Lightwave Technol. (3)

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

Laser Photon. Rev. (1)

J. Kim and F. X. Kärtner, “Attosecond-precision ultrafast photonics,” Laser Photon. Rev. 4, 432–456 (2010).
[CrossRef]

Opt. Express (4)

Opt. Lett. (6)

Other (3)

A. M. Weiner, Ultrafast Optics (Wiley, 2009).

W.-W. Hsiang, B.-W. Tsai, C. Hu, S.-Y. Wu, and Y. Lai, “Subfemtosecond synchronization between Yb-fiber and Er-fiber lasers by controlling the relative injection timing,” in Conferences on Lasers and Electro-Optics (Optical Society of America, 2013), paper JTh2A.27.

S.-Y. Wu, W.-W. Hsiang, and Y. Lai, “Relative timing jitter of passively synchronized Er-doped and Yb-doped mode-locked fiber lasers,” in IEEE Photonics Conference (IEEE Photonics Society, 2013), paper ThB.1.2.

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Figures (10)

Fig. 1.
Fig. 1.

Schematic of the synchronized laser system. LD, laser diode; WDM, wavelength division multiplexer (WDM1 and WDM2, 1560/1030nm; WDM3, 1560/976nm); ISO, isolator; PBS, polarization beam splitter; FC, fiber collimator; QWP, quarter-wave plate; HWP, half-wave plate; PZT, piezoelectric transducer.

Fig. 2.
Fig. 2.

(a) Comparison of calculated XPM-induced center wavelength shifts of green pulses in [24] (without GVD and SPM) as a function of the input relative timing position between green pulses and infrared pulses. Red circles are analytic results of Baldeck et al. [24]. The blue line is our result evaluated by the derivative phase. The black line is our result of E[λ] according to the calculated optical spectrum data. (b), (c), (d) Calculated XPM-induced center frequency shifts of 1.56 μm pulses after the common HI 1060 fiber versus the initial relative timing position between the 1.56 and 1.03 μm pulses. (b) Er, τFWHM=0.2ps, Ep=0.2nJ, and Yb, τFWHM=0.2ps, Ep=0.74nJ. (c) Only adjusting Yb, τFWHM=0.2, 0.15, 0.1 ps. (d) Only adjusting Er, τFWHM=0.2, 0.15, 0.3 ps.

Fig. 3.
Fig. 3.

Comparisons of the calculated results for generating the Er laser of 0.2nJ pulse-energy and 0.2ps pulse-width (FWHM) by using the RK4IP algorithm from solving the master equation (the blue lines) and from the evolution equations of the pulse parameters [20] (the black lines).

Fig. 4.
Fig. 4.

Schematic of the numerical modeling procedure.

Fig. 5.
Fig. 5.

Calculated results of the 0.2nJ and 0.2ps Er laser pulse evolution with the timing walk-off of 0.8ps per round-trip caused by the cavity repetition rate difference. (a) Unsynchronization without injecting the Yb laser pulse. (b) Synchronization with injecting the Yb laser pulse of 0.74nJ; 0.2ps.

Fig. 6.
Fig. 6.

Calculated evolution plots of the Er laser pulse parameters for achieving the synchronization with the injecting Yb laser pulse train (0.74nJ and 0.2ps) with the initial timing walk-off R=0.8ps. (a) Pulse energy. (b) Pulse-width. (c) Relative timing position at the starting of the common fiber. (d) Center frequency shift. Blue lines, at the inlet of the common fiber. Red lines, at the outlet of the common fiber.

Fig. 7.
Fig. 7.

Calculated steady state results of the Er laser pulse parameters when in synchronization with the injecting Yb laser pulse train. The Yb laser pulse is with 0.74nJ and 0.2ps. (a) Frequency shift at the inlet of the common fiber. (b) Relative timing position at the starting point of the common fiber. (c) Pulse-width. (d) Pulse energy.

Fig. 8.
Fig. 8.

(a) Induced center frequency shift function m1(t0) of Er laser pulse regarding the initial relative timing positions, t0=t0,Ert0,Yb via the pulse collision effect. The calculation is based on adjusting gradually the timing walk-off R for synchronizing with injecting the stable fRep, Yb laser pulse of 0.74nJ and 0.2ps. (b) First-order derivative of the induced frequency shift function, m1(t0).

Fig. 9.
Fig. 9.

(a) Calculated relative timing jitter (normalized) versus the relative timing position between the two color pulses in the beginning of the collision. (b) The corresponding center wavelength shift of the Er pulse at the inlet of the common fiber section.

Fig. 10.
Fig. 10.

Normalized relative timing jitter spectrum under the relative timing position t00.218ps, for different chirps: C0 (the black line), C1.27 (the green line), C1.7 (the purple line), and C2.5 (the blue line).

Tables (1)

Tables Icon

Table 1. Common Fiber Parameters of Numerical Simulations

Equations (19)

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

z|u+β^1t|uj2β^22t2|u+jn^|u=0.
β^i=(βi,u100βi,u2),i=1or2
n^=(γ(|u1|2+2|u2|2)00γ(|u2|2+2|u1|2)).
D^=β^1t+j2β^22t2;N^=jn^.
z|uI=N^I|uI,N^I=I^(z,z0)N^I^(z,z0),
I^(z,z0)=exp((zz0)D^).
|uI=exp(hD^2)|u(z,t)
|k1=exp(hD^2)[hN^|u(z,t)]
|k2=hN^(|uI+|k12)
|k3=hN^(|uI+|k22)
|k4=hN^exp(hD^2)(|uI+|k3)
|u(z+h,t)=exp(hD^2)(|uI+|k1+2|k2+2|k36)+|k46.
TRu(T,t)T=(g01+|u|2dt/Esl0)u+(dr+jdi)2ut2+(krjki)|u|2u.
u(T,t)=a(T)sech[tt0(T)τ(T)]1+jC(T)ej{ω(T)[tt0(T)]+ϑ(T)}.
TRdωdT=m1(t0)4dr(1+C2)3τ2ω
TRdt0dT=2diω+2drCω+R.
TRdΔωdT=m1(t¯0)(Δt0,ErΔt0,Yb)4dr(1+C2)3τ2Δω+TRSω(T)
TRdΔt0,ErdT=2diΔω+2drCΔω+TRSt0(T).
|Δt0,Er(Ω)Δt0,Yb(Ω)|2=4(di+drC)2TR2[(Ω2+2(di+drC)m1(t¯0)TR2)2+(Ωr(1+C2)TR)2]|Sω(Ω)|2+Ω2TR2+r2(1+C2)2TR2[(Ω2+2(di+drC)m1(t¯0)TR2)2+(Ωr(1+C2)TR)2]|St0(Ω)|2+Ω4+(Ωr(1+C2)TR)2[(Ω2+2(di+drC)m1(t¯0)TR2)2+(Ωr(1+C2)TR)2]|Δt0,Yb(Ω)|2.

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