Chapter 3: Experimental setup and results
3.3 Comparison and discussion
Before performing the discussion and comparison, we need to introduce some definitions in order to make the following discussion more clear. In figure 3.27, it was commonly believed that the deviation frequency is simply the difference between the modulation frequency and the cavity harmonic frequency .However the theory of the frequency pulling effect predicts that will not be equal to and the value of will be a function of the modulation depth.
Fig. 3.27 Originally defined modulation frequency and harmonic frequencies
Fig. 3.28 Defined modulation frequency and pulse repetition frequency
In Figure 3.28, we have the modulation frequency , the cavity harmonic frequency , and the pulse repetition frequency . The difference between and is the detuning frequency , and the difference between and is the asynchronous mode-locking deviation frequency . is the pulling frequency by repetition frequency pulling effect [3.1].
From figure 3.21, 3.22, 3.23, 3.24, 3.25, and 3.26, one can observe the synchronous to asynchronous transition. We can also calculate the pulling frequency and plot it versus the detuning frequency .
About the repetition frequency pulling effect, in order to understand the physical meaning of such a linear drift, it is important to note that the master equation model [3.2][3.3] has pre-assumed a fixed laser round trip time. Therefore a constant pulse timing position drift per round trip will correspond to a change of the pulse repetition frequency. Mathematically, the mechanism for producing such a linear drift is as follows. First, all the pulse parameters oscillate sinusoidally due to the sinusoidal excitation of the asynchronous phase modulation, that is, the first term in the right-hand side of Eq. (3.1).
Then, due to the nonlinear characteristics of Eqs. (3.1) and (3.2), small dc components will also appear in the right-hand side of both equations through the nonlinear terms. Finally, the dc components cause the timing to drift linearly since there is no restoring force for the timing position in Eq. (3.2).
-40
Detuning frequency (kHz)
Pulling frequency (kHz)
Fig. 3.29 Detuning frequency and the pulling frequency ,at 10GHz,-5dBm
-40
Detuning frequency (kHz)
Pulling frequency (kHz)
Fig. 3.30 Detuning frequency and the pulling frequency ,at 20GHz,-5dBm
-40
Fig. 3.31 Detuning frequency and the pulling frequency ,at 10GHz,-6.5dBm
-40
Fig. 3.32 Detuning frequency and the pulling frequency ,at 20GHz,-6.5dBm
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Fig. 3.33 Detuning frequency and the pulling frequency ,at 10GHz,-8dBm
-40 -30 -20 -10
00 20 40 60 80 100
Detuning frequency (kHz)
Pulling frequency (kHz)
Fig. 3.34 Detuning frequency and the pulling frequency ,at 20GHz,-8dBm Figure 3.29, 3.30, 3.31, 3.32, 3.33, and 3.34 are the diagrams of detuning frequency versus the pulling frequency for different
strengths and frequencies. However, it should be noted that the cavity length drift cannot be ignored if the measurement time is too long.
Therefore in the above figures the plotted pulling frequency values may also be offset by the cavity harmonic frequency change due to the cavity length drift.
We have also used the pulse parameter evolution equations in Chapter 2.4 to perform numerical simulation for studying the synchronous-to-asynchronous transition and the calculated results are shown in Fig. 3.35.
Detuning frequency fd(kHz)
Pulling frequency fdelta(kHz)
Fig. 3.35 Simulation results for synchronous/asynchronous transition
Here the simulation parameters are estimated based on our 10 GHz asynchronous mode-locked Er-fiber soliton laser and can be summarized as follows: = 0.25, = 0.05, = 0.4, = 0.1, = 4, = 0.8, = 0.47. The normalization unit for time is 0.5 ps, the cavity fundamental frequency is 8.3 MHz, and the mode-locked frequency is 10 GHz.
The transition from asynchronous to synchronous mode-locking when the modulation is large can also be seen from Eqs. (3.1) and (3.2).
To the leading order, one can treat and as constants for simplicity.
If synchronous mode-locking is to be reached with , then one has .Substituting it into Eq. (3.2), the steady state value is . Then for Eq. (3.1) to have a steady state solution, the following criterion needs to be satisfied:
(3.3)
Here is the modulation depth, is the modulation frequency, and is equal to the detuning frequency .So when | | is larger than a threshold, Eq (3.3) cannot be satisfied and thus synchronous operation is no longer possible.
In figure 3.29, 3.30, 3.31, 3.32, 3.33, 3.34, and 3.35, the experimentally observed trends agree with the simulation results. But in the experimental data, the pulling frequency is offset by the cavity drift and cannot be determined exactly. In each set of measurements the total measurement time is approximately 30 minutes. This is why the cavity drift cannot be ignored. But since we now know the synchronous to asynchronous transition trends, we can design new experiments to determine the transition point more directly for avoiding the cavity drift effect. Basically we fix the modulation frequency around the transition point and change the modulation intensity to observe the transition. We find that when the modulation strength becomes large, the original asynchronous state is converted to synchronous. On the other hand, when the strength becomes small it is maintained as the asynchronous state.In this way, within a short time period we can change to different modulation strengths to find the corresponding transition points. The results are recorded in Table 3.2 and plotted Fig. 3.36.
Table 3.2 Synchronous/Asynchronous transition points under different modulation deviation frequency (kHz)
2.19926 -2 34
Asynchronous mode-locking deviation frequency (kHz)
Modulation depth
Fig. 3.36 Synchronous/Asynchronous transition points under different modulation
strengths
Table 3.3 Synchronous/Asynchronous transition points under theoretical results
Modulation depth
Modulation strength(dBm)
Asynchronous mode-locking deviation frequency (kHz)
2.23493 -2 34
Fig. 3.37 Synchronous/Asynchronous transition points under theoretical results
Next we plot the asynchronous mode-locking deviation frequency with the detuning frequency in figure 3.37, 3.38, 3.39, 3.40, 3.41 and 3.42. They basically exhibit a linear relationship.
30 40 50 60 70 80 90 100 110
Deviation frequency fASM (kHz)
Detuning frequency fd(kHz)
Fig. 3.38 Detuning frequency versus asynchronous mode-locking deviation frequency ,at 10GHz,-5dBm
30 40 50 60 70 80
Deviation frequency fASM (kHz)
Detuning frequency fd(kHz)
Fig. 3.39 Detuning frequency versus asynchronous mode-locking deviation
frequency ,at 20GHz,-5dBm
Deviation frequency fASM (kHz)
Detuning frequency fd(kHz)
Fig. 3.40 Detuning frequency versus asynchronous mode-locking deviation
frequency ,at 10GHz,-6.5dBm
30 40 50 60 70 80 90
Deviation frequency f ASM (kHz)
Detuning frequency fd(kHz)
frequency ,at 20GHz,-6.5dBm
Deviation frequency fASM (kHz)
Detuning frequency fd(kHz)
Fig. 3.42 Detuning frequency versus asynchronous mode-locking deviation
frequency ,at 10GHz,-8dBm
30 40 50 60 70 80 90
Deviation frequency fASM (kHz)
Detuning frequency fd(kHz)
Fig. 3.43 Detuning frequency versus asynchronous mode-locking deviation
frequency ,at 20GHz,-8dBm
We also want to know the transition point when the asynchronous mode-locking disappears. As in the synchronous/asynchronous transition case, the impact of the cavity drift is not small. So we again use the same experiment method described previously. The observed transition deviation frequency value also has a positive correlation with the modulation strength. The recorded data are listed in the Table 3.4 and plotted in Fig. 3.44.
Table 3.4 Transition points when the asynchronous mode-locking disappear.
Modulation depth
Modulation strength(dBm)
Asynchronous mode-locking deviation frequency (kHz)
2.19926 -2 60
2.10754 -4 58
1.96687 -6 52
1.76107 -8 48
1.5093 -10 42
1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3
Asynchronous mode-locking deviation frequency (kHz)
Modulation depth
Fig. 3.44 Transition points when the asynchronous mode-locking disappear.
Under the synchronous mode-locking operation states we have also found cases with a relatively large deviation frequency (about 170~200kHz), as shown in Fig. 3.44. This large deviation frequency should be due to the relaxation oscillation effects. The laser still operates synchronously but the relaxation oscillation effects produce side peaks similar to asynchronous mode-locking with a relatively large deviation frequency. We have also observed cases under which the laser is mode-locked with bound pulses, as shown in Fig. 3.45. These results indicate that the laser also has other interesting laser dynamics to be explored more.
10.0048 10.0050 10.0052 10.0054 10.0056 10.0058 10.0060 -120
-100 -80 -60 -40 -20 0 20
X = 10.0055153, Y = -18.586 X = 10.005352, Y = 3.624
Strength(dBm)
Frequency(GHz)
Fig. 3.45 RF spectrum of the laser output with large deviation frequency.
Fig. 3.46 Optical spectrum of the laser output with bound pulses.
Reference
[3.1] S. S. Jyu, and Y. Lai, “Repetition frequency pulling effectsin asynchronous mode-locking”, Opt. Lett. 3, 347(2013).
[3.2] 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. 3, 292 (2010).
[3.3] H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. 6, 1173 (2000).
[3.4] B. Razavi, “A study of injection locking and pulling in oscillators,” IEEE J. Solid-State Circuits 39, 1415 (2004).
Chapter 4 Conclusions
4.1 Summary
In the thesis study we have accomplished the following main achievements:
First, we experimentally investigate the relationship among the synchronous mode-locking, asynchronous mode-locking, the modulation intensity and the modulation frequency with the experimental laser structure of hybrid active/passive mode-locking mechanism.
Second, by using the variational and numerical solution of the master equation, we theoretically study the transition between the synchronous and asynchronous operation states.
Third, the experimental observations have been carefully compared with the theoretical predictions. Good agreement has been found, which should be greatly helpful for understanding more deeply the laser dynamics of asynchronous mode-locking.
4.2 Future work
The thesis has demonstrated the relationship among the synchronous mode-locking, asynchronous mode-locking, the modulation intensity and the modulation frequency up to 20 GHz. We are very interested in going for even higher modulation frequencies (40 GHz or higher). We now know the synchronous/asynchronous transition boundary will depend on the modulation frequency. It is interesting to see whether asynchronous mode-locking can still work well under higher modulation frequencies and what are the criteria for laser design in order to achieve asynchronous mode-locking under higher modulation frequencies.
Recently, the asynchronous mode-locking mechanism has been applied to a Yb-doped fiber laser with an all normal dispersion fiber cavity and to a Er-doped fiber laser mode-locked with bound pulses. It is also interesting to see whether the synchronous/asynchronous transition properties can exhibit different behaviors under these different laser systems.