The thesis consists of four chapters. Chapter 1 is an overview and describes the motivation for doing this research. Chapter 2 describes the methods of different mode-locking techniques and the solution of the mode-locked laser master equation by the variational method. Chapter 3 describes the experimental setup and results. Finally Chapter 4 presents the summary and possible future work.
Reference
[1.1] J. Lee, J. Koo, Y. M. Chang, P. Debnath, Y.W. Song, and J. H.
Lee,” Experimental investigation on a Q-switched, mode-locked
fiber laser based on the combination of active mode locking and passive Q switching” J. Opt. Soc. Am. B. 6 ,1479 (2012).
[1.2] I. Pavlov, E. Ilbey, E. Dülgergil, A. Bayri, and F. Ö . Ilday,
“High-power high-repetition-rate single-mode Er- Yb-doped fiber laser system” Optics Express, 9, 9471 (2012).
[1.3] 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. 3, 292 (2010).
[1.4] H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. 6, 1173 (2000).
Chapter 2
Principles of mode-locked fiber lasers
2.1 Synchronously active mode-locking
The active mode-locking is achieved by modulating the loss or phase of the laser cavity. When the modulation frequency is synchronized with the cavity’s round trip time, the pulse train in the resonator will continue
to grow and then achieve stable mode-locking. Generally, active modulation can be achieved with an acousto-optic or electro-optic modulator.
As explained above, active mode-locked fiber lasers can use two different methods for optical modulation. The two different methods will be given in the following two sub-sections (2.1.1 and 2.1.2). One is the amplitude modulation mode-locking (AM) and the other is the phase modulation mode-locking.
Fig. 2.1 Schematic of an active mode-locked laser
2.1.1 Phase modulation mode-locking
A phase modulator is an optical modulator which can be used to control the optical phase of a laser beam. By modulating the phase of the optical field in the laser cavity periodically, it can achieve mode-locking without changing the amplitude of the optical field and then produce short pulse trains.This mechanism can be analyzed both in the frequency domain and the time domain [2.1].
In the time domain, the phase modulator provides a phase change for the optical pulse. Assume the pulsewidth is much smaller than the modulation period, then the optical phase changed by the phase modulator can be expressed as,
d d
t t t
dt dt
( )0 22 2 (2.1)
This term0 is only a constant phase that has no effect on optical pulses.
When the optical pulse passes through the modulator, the central
shifted by modulation.
Actually, d 0 dt
have two solution states, but in most situations
only one state can be stable and the other can’t. Which solution state can
exist in the cavity depends on the sign of the cavity dispersion. This is
In the frequency domain, let us assume ω0 is the angular frequency of the longitudinal mode. When this mode passes through the phase modulator, the optical signal can be written below:
(2.2)
where M is the modulation index, is the modulation frequency.
It can be expanded as:
(2.3),
is the n-th order Bessel function. From the formula it can be observed that the optical signal consist of unlimited sidebands. These sidebands can phase-lock adjacent modes through injection locking and then the periodic pulse train will be formed.
2.1.2 Amplitude modulation mode-locking
Amplitude modulation mode-locking is a method to produce a short pulse train by modulating the optical amplitude of the light. It also can be analyzed both in the time domain and frequency domain.
In the time domain, the modulation provides a time dependent loss.
When the gain is greater than the loss, the optical pulse trains will continue to grow and get shorter. However, short pulses will experience larger dispersion and thus in the end, the two forces balance each other to form the steady state pulse shape. The modulation time period must be equal to a multiple of the roundtrip time for stable pulses to be generated.
Fiqure2.2 shows the active mode-locking process in the time domain.
Fig. 2.2 Actively mode-locked pulses in the time domain and the time
In the frequency domain, as the gain level is above threshold, the net gain of the system is greater than zero, then the longitudinal modes will begin to lase. The longitudinal modes are separated equally in frequency domain and the frequency interval is , where is the round trip time. Again these sidebands will injection-lock the neighboring modes sequentially and the mode-locking is formed eventually. Figure 2.3 shows the active mode-locking process in the frequency domain.
Fig. 2.3 Actively mode-locked modes in the frequency domain.[2.1]
If the central mode without amplitude modulation is expressed as E0
cos(ω0t), then the amplitude modulated optical signal can be expressed as:
0
1+ cos
Mcos
0We denoted the modulation depth is M and the modulation frequency is .
The central frequency ωo produces two sidebands (ω0±ΩM) which
have the same phase after the modulation. When these modes pass through the amplitude modulator again, the other two inphase sidebands (ω0±2ΩM) will be produced. This process will repeat until all the
longitudinal modes in the gain bandwidth are locked. Due to mode-locking, these frequency components that are separated ΩM will have fixed phase relation and will produce a periodical pulse train in the time domain.
2.2 Asynchronously mode-locked fiber lasers
In normal active harmonic mode-locked lasers, the optical modulator is driven synchronously with respect to the cavity harmonic frequency.
The optical pulses will be shifted by a frequency if not synchronized.
With the finite bandwidth filter and gain, the pulse trains cannot achieve stable mode-locking. However, in the asynchronous soliton mode-locked laser [2.2], the modulation frequency and the cavity harmonic frequency have a small deviation frequency from several kHz to tens kHz and stable mode-locking still can be achieved [2.3] .
The fiber laser is consisted of the gain, the optical bandpass filter, group velocity (GVD), self phase modulation (SPM) and the phase modulator driven asynchronously. Because of asynchronous modulation, the optical pulses reach the optical modulator not always at the peaks of the modulation signal. When the pulse passes through the phase modulator, the central wavelength of the pulse can get a small frequency shift. When the nonlinear effects are not considered, the asynchronous modulation will shift the central frequency of the pulses periodically with accumulation. The gain medium and the filter are both fixed with finite
bandwidth, the overdeveloped amount of central frequency shift can make the pulse leave the center of the gain or the filter. Then the pulses can’t exist in the cavity since they will experience quite large loss.
Fig. 2.5 Laser cavity with the gain, filter, (group velocity dispersion) GVD, (self phase modulation) SPM and the phase modulation driven asynchronously.
Fig. 2.6 The noise-cleanup effect in the asynchronous soliton mode-locked laser.
The noise clean up mechanism is similar to the effects of the sliding-frequency guiding filter in the soliton communication systems [2.4]. The central frequency of soliton and the central frequency of the filters will vary together in the fiber cavity during propagation, but the central frequency of the linear noise keeps fixed and will be filtered out by the optical filter. So a higher SMSR (Side Mode suppression ratio) can be obtained [2.5], even when there is no explicit intracavity optical device to suppress the supermode noises.
2.3 Master equation for asynchronously mode-locked fiber Lasers
The master equation model for asynchronous mode-locked lasers is given as [2.1][2.3][2.6][2.7]:
The equation is derived under the assumption of small round-trip change.
Here is the complex field envelope of the pulse, is the number of the cavity round trip time, is time, is linear gain, is saturation energy, is linear loss, is the effect of the optical filtering,
is group velocity dispersion, is the effect of equivalent fast saturable absorber, is the self-phase modulation coefficient, is the phase modulation depth, is the modulation frequency, and is the timing walk-off per round trip.
is the deviation frequency between the N-th cavity harmonic frequency and the modulation frequency . In analyses, the sinusoidal modulation curve of the phase modulator can be expanded by Taylor’s series as:
where
.
2.4 Solution of the master equation by the variational method The master equation can be solved approximately by using the variational method developed in [2.7]. By assuming the following solution ansatz :
where is the optical center frequency, is timing, is chirp, is the amplitude, and is the pulserwidth, one can derive the evolution equations for all the pulse parameters:
These ordinary differential equations can then be solved numerically to study the laser dynamics of asynchronous mode-locking.
Reference
[2.1] H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. 6, 1173 (2000).
[2.2] C. R. Doerr, H. A. Haus, and E. P. Ippen, “Asychronous soliton mode locking”, Opt. Lett. 19, 1958 (1994).
[2.3] H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, “Theory of soliton stability in asynchronous modelocking,” IEEE J. Lightwave
Technol. 14, 622 (1996).
[2.4] L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, ‘‘The sliding frequency guiding filter: an improved form of soliton jitter control,’’ Opt. Lett. 17, 1575 (1992).
[2.5] G. T. Harvey and L. F. Mollenauer, “Harmonically mode-locked fiber ring laser with an internal Fabry-Perot stabilizer for soliton transmission,” Opt. Lett. 2, 107 (1993)
[2.6] H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Amer. B, vol. 8, pp.
2068–2076, 1991.
[2.7] 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).
[2.8] 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).
[2.9] W. W. Hsiang, C. Lin, N. Sooi, and Y. Lai, “Long-term stabilization of a 10 GHz 0.8 ps asynchronously mode-locked Er-fiber soliton laser by deviation-frequency locking,” Opt. Exp. 5, 1822 (2006).
[2.10] M. C. Chan, “Hybrid Mode-locking Er-doped Fiber Laser,”
Institute of Electro-Optical engineering in National Chiao-Tung University, master thesis (2002).
[2.11] M. Nakazawa and M. Yoshida, “Scheme for independently stabilizing the repetition rate and optical frequency of a laser using a regenerative mode-locking technique”, Opt. Lett. 33, 1059 (2008).
[2.12] H. G. Weber and M. Nakazawa, “Ultrahigh-Speed Optical Transmission Technology” ,(2007).
[2.13] B. E. A. Saleh and M. C. Teich , Fundamentals of Photonics , Wiley-Interscience, Canada, (2007).
[2.14] D. J. KUIZENGA, and A .E. SIEGMAN “FM and AM mode locking of the homogeneous laser,” IEEE J. Quantum Electron. 6, 694
(1970).
[2.15] M. Nakazawa, K. Kasai, and M. Yoshida,” C2h2 absolutely optical frequency-stabilized and 40 GHz repetition rate stabilized regeneratively mode-locked picosecond erbium fiber laser at 1.53 μm” Opt. Lett. 33, 2641(2008).
Chapter 3
Experimental setup and results 3.1 Experimental setup
The system setup of our asynchronous mode-locked Er-doped fiber lasers is shown in Fig 3.1.
Fig. 3.1 Experimental setup
The Er-doped fiber is pumped by two 980 nm laser diodes. The isolator in the cavity is for single direction optical wave propagation.
There are two couplers used here to distribute optical signal. The coupler
1 divides 80% optical signal into the cavity and 20% into the coupler 2, which divides 10% into the PD. The optical power incident on PD is converted to electrical signals and then into the RF spectrum analyzer.
The rest 90% of the optical power is for laser output. The two polarization controllers and the in-line polarizer are used to achieve polarization additive-pulse mode-locking (P-APM). The phase modulator is the core element of the experiment for achieving asynchronous mode-locking. In the experiment we change the modulation frequency and modulation strength to explore the laser dynamics. The devices that have been used in the fiber ring cavity are listed in the Table 3.1
Table 3.1 Devices in the fiber ring cavity
Device specification
980nm laser diode JDSU 3000 Series 980 nm pump * 2 Phase modulator Vπ=4.7 volt @1GHz
Erbium-doped fiber LIEKKI™ Er80-8/125, ~5.5m Single mode fiber SMF28, ~19.5m
Polarization Thorlabs manual fiber polarization controllers (FPC030) *2
Tunable optical bandpass filter
Bandwidth : 13.5 nm
Central wavelength : 1530~1570 nm
Coupler 80/20*1 , 90/10*1
3.2 Parameter tuning
3.2.1 10 GHz operation
From Chapter 2, we know how to generate synchronously and asynchronously mode-locking. The changes of the RF spectra and the transition from synchronization to non-synchronization are the main focuses of observation. We change the modulation frequency and modulation strength to explore the laser dynamics. In the experiment, we first set both the pump lasers at 500 mA. Then we set the phase modulation frequency at 10GHz. With the change of the modulation frequency, we record all the spectral data from the RF spectrum analyzer.
The experiment is explained by the schematic diagram in Fig. 3.2 and the measured data are listed afterwards.
Fig. 3.2 Change of the modulation frequency
Fig. 3.3 Schematic diagram for modulation frequency and cavity frequency
10.0044 10.0046 10.0048 10.0050 10.0052 10.0054 10.0056 -120
-100 -80 -60 -40 -20 0
20 X = 10.0050068, Y = 2.42
Intensity (dBm)
Frequency (GHz)
Fig. 3.4 RF spectrum of the laser output near 10GHz with 1MHz span
Fig. 3.5 Optical spectrum of the laser output near 1560nm with 20nm span
Fig. 3.6 Schematic diagram for modulation frequency and cavity frequency
10.0044 10.0046 10.0048 10.0050 10.0052 10.0054 10.0056 -120
-100 -80 -60 -40 -20
X = 10.0049701, Y = -17.854
Intensity (dBm)
Frequency (GHz)
Fig. 3.7 RF spectrum of the laser output near 10GHz with 1MHz span
Fig. 3.8 Optical spectrum of the laser output near 1560nm with 20nm span
Fig. 3.9 Schematic diagram for modulation frequency and cavity frequency
10.0044 10.0046 10.0048 10.0050 10.0052 10.0054 10.0056 -120
-100 -80 -60 -40 -20
0 X = 10.0050035, Y = -2.275
Intensity (dBm)
Frequency (GHz)
Fig. 3.10 RF spectrum of the laser output near 10GHz with 1MHz span
Fig. 3.11 Optical spectrum of the laser output near 1560nm with 20nm span
Fig. 3.12 Schematic diagram for modulation frequency and cavity frequency
10.0044 10.0046 10.0048 10.0050 10.0052 10.0054 10.0056 -120
-100 -80 -60 -40 -20
X = 10.0050168, Y = -21.672
Intensity (dBm)
Frequency (GHz)
Fig. 3.13 RF spectrum of the laser output near 10GHz with 1MHz span
Fig. 3.14 Optical spectrum of the laser output near 1560nm with 20nm span
Fig. 3.15 Schematic diagram for modulation frequency and cavity frequency
10.0044 10.0046 10.0048 10.0050 10.0052 10.0054 10.0056 -120
-100 -80 -60 -40 -20 0 20
X = 10.0050001, Y = 1.976
Intensity (dBm)
Frequency (GHz)
Fig. 3.16 RF spectrum of the laser output near 10GHz with 1MHz span
Fig. 3.17 Optical spectrum of the laser output near 1560nm with 20nm span
Fig. 3.18 Schematic diagram for modulation frequency and cavity frequency
10.0044 10.0046 10.0048 10.0050 10.0052 10.0054 10.0056 -120
-100 -80 -60 -40 -20
0 X = 10.0049968, Y = -5.6
Intensity (dBm)
Frequency (GHz)
Fig. 3.19 RF spectrum of the laser output near 10GHz with 1MHz span
Fig. 3.20 Optical spectrum of the laser output near 1560nm with 20nm span Figure 3.3, 3.6, 3.9, 3.12, 3.15, and 3.18 illustrate the different ways
for changing the modulation frequency. The data are recorded for each 1 kHz change. Figure 3.4, 3.7, 3.10, 3.13, 3.16, and 3.19 show the measured RF spectra and Figure 3.5, 3.8, 3.11, 3.14, 3.17, and 3.20 show the optical spectra. These experimental data are for the case of 10G Hz, -5dBm.
From the stored RF spectra, we can also read the repetition frequencies and plot them below.
10.00496 10.00498 10.00500 10.00502 10.00504 10.00506 10.00508 10.00510 10.00496 Fig. 3.21 Repetition frequency versus modulation frequency at 10GHz,-5dBm
10.00488 10.00490 10.00492 10.00494 10.00496 10.00498 10.00500
Fig. 3.22 Repetition frequency versus modulation frequency at 10GHz,-6.5dBm
10.00518 10.00520 10.00522 10.00524 10.00526 10.00528 10.00530 10.00532 10.00519
Fig. 3.23 Repetition frequency versus modulation frequency at 10GHz,-8dBm In figure 3.21, 3.22, and 3.23，the laser repetition frequency versus the modulation frequency for different modulation strength are plotted.
3.2.2 20 GHz operation
To have more data for comparison, we also operate the laser under 20 GHz and record the data as the 10 GHz case. In the following we only show the repetition frequency versus the modulation frequency diagrams.
19.98396 19.98398 19.98400 19.98402 19.98404 19.98406 19.98408 19.98410 19.98396
19.98397 19.98398 19.98399 19.98400 19.98401 19.98402 19.98403
Repetition frequency (GHz)
Modulation frequency(GHz)
Fig. 3.24 Repetition frequency versus modulation frequency at 20GHz,-5dBm
19.99304 19.99306 19.99308 19.99310 19.99312 19.99314 19.99316 19.99318
Fig. 3.25 Repetition frequency versus modulation frequency at 20GHz,-6.5dBm
19.98374 19.98376 19.98378 19.98380 19.98382 19.98384 19.98386 19.98388 19.98373
Fig. 3.26 Repetition frequency versus modulation frequency at 20GHz,-8dBm
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
-40
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
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