Abstract

The frequency comb from a mode-locked fiber laser can be stabilized through feedback to the pump power. An understanding of the mechanisms and bandwidth governing this feedback is of practical importance for frequency comb design and of basic interest since it provides insight into the rich nonlinear laser dynamics. We compare experimental measurements of the response of a fiber-laser frequency comb to theory. The laser response to a pump-power change follows that of a simple low-pass filter with a time constant set by the gain relaxation time and the system-dependent nonlinear loss. Five different effects contribute to the magnitude of the response of the frequency comb spacing and offset frequency but the dominant effects are from the resonant contribution to the group velocity and intensity-dependent spectral shifts. The origins of the intensity-dependent spectral shifts are explained in terms of the laser parameters.

© 2005 Optical Society of America

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References

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Appl. Phys. B (2)

N. Haverkamp, H. Hundertmark, C. Fallnich, and H. R. Telle, "Frequency stabilization of mode-locked Erbium fiber lasers using pump power control," Appl. Phys. B 78, 321-324 (2004).
[CrossRef]

L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, "Ultrashort-pulse fiber ring lasers," Appl. Phys. B 65, 277-94 (1997).
[CrossRef]

Electron. Lett. (2)

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fiber ring laser," Electron. Lett. 28, 2226-7 (1992).
[CrossRef]

S. M. J. Kelly,"Characteristic sideband instability of periodically amplified average soliton," Electron. Lett. 28, 806-807 (1992).
[CrossRef]

IEEE J. of Quantum Electron. (1)

N. R. Newbury and B. R. Washburn, "Theory of the frequency comb output from a femtosecond fiber laser," IEEE J. of Quantum Electron. 41, 1388-1402 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

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

Opt. Commun. (1)

J. D. Moores, "On the Ginzburg-Landau laser mode-locking model with fifth-order saturable absorber term," Opt. Commun. 96, 65-70 (1993).
[CrossRef]

Opt. Express (4)

Opt. Lett. (8)

Phys. Rev. Lett. (1)

S. A. Diddams, D. J. Jones, S. T. C. J. Ye, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102 (2000).
[CrossRef] [PubMed]

Other (2)

B. R. Washburn, S. Diddams, N. R. Newbury, J. W. Nicholson, M. F. Yan, and C. G. Jørgensen, "A phase locked, fiber laser-based frequency comb: limit on optical linewidth," in Proceedings of Conference on lasers and Electro-optics, (Optical Society of America, 2004)

E. Desurvire, Erbium-Doped Fiber Amplifiers. (Wiley, New York, 1994).

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Figures (10)

Fig. 1.
Fig. 1.

A schematic of the fiber ring laser. The laser consisted of 1 m of Erbium doped fiber (EDF) that was pumped with a 980 nm diode, and 3 m of single-mode fiber with a net anomalous dispersion of -0.06 ps2. Self-amplitude modulation is provided by a polarizer/isolator combination that is sandwiched between two polarization controllers.

Fig. 2.
Fig. 2.

(a) Schematic representation of the lumped gain, loss, and SAM experienced by the pulse electric field as it traverses the cavity. Saturation of the SAM is added to insure stability. Here it is convenient to describe the SAM and its saturation heuristically in terms of the pulse energy, w, and the two parameters γSAM and γSAT . In Ref. [6] a more rigorous treatment is given where the SAM and its saturation are described in terms of the electric field amplitude, which is proportional to w for a soliton, and a pair of parameters. (b) Without saturation of the SAM (and without gain saturation), the system is unstable to pulse energy perturbations (black line). Inclusion of SAM saturation leads to a stable solution (red line). In terms of the variables in Fig. 2(a), η = 4 (g-l) + 4γSATw 4.

3.
3.

(a) Normalized response of the gain medium and two different lasers as a function of the modulation frequency of the pump power. The laser relaxation times are always greater than that of the Erbium gain medium. (b) Measured fCEO beat note for the figure-eight laser (red) and the ring laser (blue).

Fig. 4.
Fig. 4.

(a) The different mechanisms contributing to the change in φCEO . Note that a spectral shift in the carrier frequency effectively enters only as a shift in the envelope arrival time. The four mechanisms contributing to changes in the envelope arrival time (i.e. repetition frequency) are discussed in more detail in Section 5. (b) The expected linear relationship between the response of the offset frequency and the repetition frequency from Eq. (2) .

Fig. 5.
Fig. 5.

Counted values of fr and fCEO for a 0.5 Hz square-wave modulation of the pump power. The slight slope on the fr data is a result of thermal effects. The counter gate time was 100 ms. The average repetition rate was about 50 MHz.

Fig. 6.
Fig. 6.

The measured dfCEO /dP versus dfr /dP. The two symbols refer to data taken with different polarizer settings. For each polarizer setting, the data correspond to different pump powers. The solid line is a straight-line fit.

Fig. 7.
Fig. 7.

The measured dfr /dP versus pump power. The strengths of the various contributions are shown, with the colors corresponding to the colored terms in Eq. (4). The sum of the effects yields the solid black line, to be compared to the experimental measurements (solid black squares).

Fig. 8.
Fig. 8.

(a) Definition of frequency terms: ω 0 is the gain peak, ωc is the pulse spectral peak, ωrms is the pulse spectral width, and ω Δ=ωc - ω 0. (b)Schematic showing the spectral shift effects following Ref. [6]. The pulse can shift higher or lower in frequency if there is a frequency dependent loss. It will shift lower in frequency due to the Raman self-frequency shift. In the EDF, it will shift back toward the gain peak at ω 0 due to gain filtering.

Fig. 9.
Fig. 9.

(a) Example OSA spectra taken at pump powers (above threshold) of 38 mW (green), 43 mW (brown) and 50 mW (black). The peaks on the sides are Kelly sidebands. The bump in the center of the spectrum at high pump powers (black trace) is from cw breakthrough that occurs at higher pump powers. (b) Center of the spectrum from a fit to the OSA spectra as a function of pump power (triangles) and a polynomial fit (solid line).

Fig. 10.
Fig. 10.

Pump-induced spectral shifts: The solid line is the “measured” values taken as the derivative of the polynomial fit to the data in Fig. 9(b). These data are used in calculating the spectral-shift contribution to dfr /dP given in Fig. 7. The dashed line is a fit to the data using the theoretical expression Eq. (6) and the measured dw/dP.

Equations (7)

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T Δ w = 1 T r [ η Δ w 2 Δ g w ] ,
T Δ g = 1 T g [ Δ g + g w Δ w w g P Δ P P ]
df CEO dP = β 0 2 π ( df r dP ) + f r 2 π ( spm dP )
T r = 1 f r = β 1 + ω Δ β 2 + 1 2 ω rms 2 β 3 + g Ω g + μ A 2 δ ω 0
d f r dP = f r 2 { β 2 d ω Δ dP ͆ Spectral Shift + ω rms 2 β 3 ͆ TOD 2 P + ν 3 dB Er ͆ Gain 2 P ν 3 dB Ω g + 3 μ A 2 δ ͆ SS 2 P ω 0 } ,
ω Δ , NL = 2 A 2 δ 5 D g ( 1 + C 2 ) ( τ R + μ ω 0 1 C ) and ω Δ , L = l ω 2 D g ,
d ω Δ dP = ( ω Δ , NL + ω Δ , L ) 1 P + 3 ω Δ , NL 2 w dw dP .

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