Experimental Observations of 10 GHz Bound Soliton Pairs

Figure 5-3 shows the SHG autocorrelation trace of the soliton bound state, which clearly indicates the existence of two pulses. From the plot one can infer that the two pulses are almost identical and the FWHM of the pulse width is 1.3 ps. The time separation between the two pulses is only about three times the individual pulse width, i.e., 4 ps. Such small separation suggests that there should be an attractive or repulsive force exerted on the respective soliton due to direct soliton interaction. The sign of the force (attractive or repulsive) will depend on the phase difference between the two solitons [11].

The solid curve in Fig. 5-4 shows the optical spectrum of the bound solitons. The optical spectrum is periodically modulated, which results from the interference between the two solitons. The dots in Fig. 5-4 show the fitting of the interference in the optical spectrum with the assumption of two identical but time-shifted pulses. The parameters used in the fitting include the interference period of 1.95 nm and the soliton’s optical FWHM bandwidth of 1.94 nm. The observed interference period in the optical spectrum is consistent with the pulse separation measured from the SHG autocorrelation. The time-bandwidth product of the individual pulse is 0.32, indicating that the pulse is with a sech pulse-shape and with little chirp. Since in

Fig.5-3 the central interference dip is in the center of the optical power spectrum, it indicates that the phase difference of the two solitons is equal to π . The superior stability is also one of the noticeable characteristics when the laser is working in the bound state regime. As shown in Fig. 5-5, when the cavity length is kept locked, the supermode suppression ratio (SMSR) is more than 70 dB. Figure 5-6 shows the 10 GHz bound soliton pairs measured from the fast sampling scope. The rising/fall time of the photodiode and electronic is not fast enough to observe the individual pulses.

However, the equal heights of the measured pulse shapes indicate the 10 GHz bound soliton pairs are very stable and have a high SMSR, as same as the measurement result from the RF spectrum annlyzer.

5.4 Theoretical Model of 10 GHz Bound Soliton Pairs

Conceptually the formation of stable bound soliton states can be described by the following two steps. First of all, similar to multiple pulse operation in passive mode-locking, pulse splitting can occur through the effect of soliton energy quantization, provided that the gain medium can support enough pulse energy [12]. In our laser, pulse splitting can also be triggered by adjusting the polarization controllers to modify the threshold of APM or APL. Secondly, to achieve stable bound soliton pairs, each split soliton should experience the balanced attractive and repulsive forces simultaneously when propagating along the laser cavity. In our case, because of the

π phase difference and the relatively close separation, the two solitons should repel each other due to direct soliton interaction. However, as to be explained below, we believe an effective attractive force will be introduced by the phase modulator and the anomalous group velocity dispersion (GVD) of the cavity. In an actively FM mode-locked fiber laser with net anomalous GVD, the pulses should be formed in the

center of the up-chirped modulation cycle [13]. This should be also true for the bound soliton case, as shown in Fig. 5-7. Under such situation, the front soliton will experience a negative frequency shift through the modulator, which leads to the decreasing of its group velocity through the anomalous GVD of the cavity. On the contrary, the back soliton will experience the increasing of the group velocity.

Therefore the phase modulator and the anomalous GVD can provide the necessary equivalent attractive force to balance the repulsive force from direct soliton interaction.

The above description can be further examined quantitatively by a simple soliton perturbation theory [14],in which the evolution equations of the soliton parameters, including amplitudes a_i(z), positions s_i(z), frequencies ω_i(z)and phases θ_i(z), can be obtained during direct soliton interaction (without phase modulation). The bound soliton state is described by the sum of u_i(z,t) , where

amplitude and the π phase difference, the sum and the difference of the soliton parameters of the two solitons remain unchanged, except for the position difference and frequency difference. They are described by

ω

2 s∆ = 4 ps and soliton’s FWHM pulse width = 1.3 ps , the accumulated 0

frequency difference during one cavity round trip is estimated to be 10.32 rad⋅GHz. On the other hand, when the soliton pair passes through the up-chirped modulation cycle once, the two solitons obtain a relative frequency difference of 2M(2πf_M)²∆s₀,

2 π ∆ should cancel each other at the steady state, we can infer that the modulation depth M is about 0.654 rad. This inferred value is in reasonable agreement with the estimated value from the driving RF power. The accumulated change of the position difference during one round trip will be zero when the accumulated frequency difference is equal to zero. Although the above simple theory does yield reasonable predictions agreed with experimental observations, we want to note that it should still be worthwhile to carry out theoretical studies based on a more complete laser model in the future.

The mechanism explained above suggests that the time separation should be dependent on the modulation depth M. We have experimentally verified this point and the results are shown in Fig. 5-8 and Fig. 5-9. Figure 5-8 shows measurement results of the SHG autocorrelations when the modulation depth is changed. As expected, the time separation decreases when the modulation depth is increased, which is also shown in Fig. 5-9. We believe this is a direct proof that the observed bound soliton state is a new soliton phenomenon which is fundamentally different from the previous reports in passively modelocked fiber lasers. Furthermore, the superior stability of the soliton bound state laser can also be explained within the same framework. Since the linear noises accompanied with the solitons will continuously experience the frequency shift by the phase modulator and will be eventually filtered out.

5.5 Summary

To summarize, for the first time, stable 10 GHz bound soliton pairs have been experimentally observed in a hybrid FM mode-locked Er-fiber laser. Their time separation is dependent on the modulation depth, indicating a new type of bound soliton phenomenon. We believe these closely adjacent soliton pairs are stably formed through the balance between the soliton direct interaction and the effective attracting force introduced by the phase modulation and the negative GVD of the cavity.

References for Chapter 5:

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[2] D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, “Observation of bound states of solitons in a passively mode-locked fiber laser,” Phys. Rev. A 64, 033814 (2001).

[3] F. Gutty, P. Grelu, N. Huot, G. Vienne, and G. Millot, “Stabilisation of modelocking in fibre ring laser through pulse bunching,” Electron. Lett. 37, 745 (2001).

[4] D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E 72, 016616 (2005).

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[6] N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton Solutions of the Complex Ginzburg-Landau Equation,” Phys. Rev. Lett. 79, 4047 (1997).

[7] W.-W. Hsiang, C.-Y. Lin, and Y. Lai, “Stable new bound soliton in a 10 GHz hybrid frequency modulation mode-locked Er-fiber laser,” Opt. Lett. 31, 1627 (2006).

[8] W.-W. Hsiang, C.-Y. Lin, M.-F. Tien, and Y. Lai, “Direct generation of a 10 GHz 816 fs pulse train from an erbium-fiber soliton laser with asynchronous phase modulation,” Opt. Lett. 30, 2493 (2005).

[9] C. X. Yu, H. A. Haus, E. P. Ippen, W. S. Wong, and A. Sysoliatin,

“Gigahertz-repetition-rate mode-locked fiber laser for continuum generation,” Opt.

Lett. 25, 1418 (2000).

[10] X. Shan, D. Cleland, and A. Ellis, “Stabilising Er fibre soliton laser with pulse phase locking,” Electron. Lett. 28, 182 (1992).

[11] J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596 (1983).

[12] A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28, 67 (1992).

[13] K. Tamura and M. Nakazawa, “Pulse energy equalization in harmonically FM mode-locked lasers with slow gain,” Opt. Lett. 21, 1930 (1996).

[14] T. Georges and F. Favre, “Modulation, filtering, and initial phase control of interacting solitons,” J. Opt. Soc. Am. B 10, 1880 (1993).

(a)

(b)

Fig. 5-1 Bound soliton pairs in mode-locked fiber laser. (a) Only one bound soliton pair circulates in the fiber laser cavity (the typical case in passively mode-locked fiber laser). (b) More than one (N>1) bound soliton pairs circulate simultaneously in the fiber laser cavity (the case corresponding to the 10 GHz bound soliton pairs in our hybrid FM mode-locked Er-fiber laser).

Fig. 5-2 Schematic of the mode-locked Er-fiber laser and the feedback control loop. PI, proportional and integral control module; PD, photodiode.

Fig. 5-3. SHG autocorrelation trace of the bound soliton pair.

Fig. 5-4. Optical spectrum of the bound soliton pair (solid curve, measurement; dots, fitting).

Fig. 5-5. RF spectrum of the 10 GHz bound soliton pair, SMSR > 70 dB.

Fig. 5-6. Stable 10 GHz bound soliton pairs measured form the fast sampling scope.

Fig. 5-7. Timing diagram between the phase modulation signal and the bound soliton pair. M:

modulation depth.

Fig. 5-8 Measurements of SHG autocorrelation traces when the driving RF power of the phase modulator varies.

Fig. 5-9. Time separation versus the RF power of the phase modulator (inset, timing diagram between the phase modulation signal and the bound soliton pair. M, modulation depth).

Chapter 6