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Abstract
Dissipative soliton resonance (DSR) phenomenon is a recent concept of pulse formation proposed to achieve a high-energy passively mode-locked fiber laser. Such pulses have general features, e.g., flat-top temporal profile, energy scalability, and temporal coherence. To identify the nature of various rectangular pulses which share most of DSR features, it is important to verify their coherence. When a pulse is in the nanosecond range, the experimental study of the coherence is a complex task, which is overlooked in the literature. In this work, a novel approach is proposed for the first time to study the coherence of nanosecond rectangular pulses. It is based on two different methods: Mach-Zehnder interferometer (MZI) and dispersive Fourier transformation (DFT).
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The demand for high-energy lasers is continuously growing in various fields such as medicine [1–4], industry [5], and defense and security [6,7]. Until few years ago, chirped pulse amplification (CPA) [8] was the most common and effective technique to generate ultrashort high-energy optical pulses in mode-locked oscillators. However, the implementation of this technique is limited by the need of several amplifiers at the oscillator output. In 2004, Fernandez et al. introduced the concept of chirped-pulse oscillators as a new route to high-energy femtosecond pulses in solid-state lasers without external amplification [9]. In fiber lasers, Chong et al. identified the nonlinear phase shift, spectral filter bandwidth, and group-velocity dispersion as key factors for pulse shaping and energy optimization [10]. Since then, several studies have emerged, offering a multitude of solutions for increasingly compact, efficient and highly energetic lasers. In this regard, Chang et al. studied numerically a new regime called dissipative soliton resonance (DSR) in which the energy of pulses increases indefinitely for particular values of the system parameters of the cubic-quintic complex Ginzburg-Landau equation (CGLE) [11]. Then, they demonstrated that this DSR regime can exist in normal and anomalous dispersion region by modifying few parameters of the equation [12]. Building on the theoretical reports, the generation of high-energy square-pulses has been extensively investigated in various configurations of passively mode-locked fiber lasers. Numerical results were reported in [13–18], whereas the experimental generation of such pulses has been reported in [19–29]. In the experimental literature, rectangular pulses are assumed to be a manifestation of DSR when they exhibit some behavior with the variation of the pump power. It is characterized by a linear increase of the pulse width and energy with a clamped peak power without experiencing any wave-breaking, and with invariant optical spectrum. To ensure that these pulses are DSR, they must be temporally coherent without any fine structure within the rectangular envelope. Despite the necessity of the latter feature, only few researchers have investigated the coherence of such rectangular pulses by providing an autocorrelation trace [13,15,30]. It is worth mentioning that their work was concerning relatively short pulses (in few hundreds of picoseconds) with relatively high repetition rates. However, most of the obtained DSR square pulses are in the nanosecond regime with low repetition rates, in kHz. In general, the autocorrelation trace displays dips due to the vanishing of the signal between pulses when the pulses repetition rate is lower than a threshold set by the manufacturer. In our case, the autocorrelation trace is not accurate for repetition rates lower than 1 MHz. In addition, let us mention that even when the repetition rate of the laser is compatible with the optical autocorrelator, the resulting trace is not sufficient to state about the coherence of the pulse. Indeed, in theory, if the pulse is not coherent, a peak appears in the center of the autocorrelation trace whereas it does not occur for coherent square-pulse. However, in order to optimize the signal, a polarization controller is inserted at the entrance of the optical autocorrelator. The existence of a central peak in the autocorrelation trace depends on the adjustment of the polarization controller thus leading to the impossibility to conclude on the coherence of the rectangular pulses since the absence of a central peak can have several origins. To overcome these issues, we present here, a reliable experimental setup to investigate the coherence of large rectangular-shaped pulses in passively mode-locked Er:Yb co-doped double-clad fiber laser in anomalous dispersion regime. The experimental setup consists of two different measurement methods: Mach-Zehnder interferometer (MZI) and Dispersive Fourier Transformation (DFT). The manuscript is organized as follows: in section 2., we present a detailed experimental setup including the fiber oscillator and the characterization methods. Experimental results on the generation of DSR rectangular pulses and their characteristics are discussed in section 3.. In section 4., the coherence analysis of the MZI method is presented followed by the DFT investigations in section 5.. A general conclusion is made in the last section.
2. Experimental setup
Figure 1 shows the experimental setup of the proposed coherence measurement of rectangular pulses. It consists of three distinct parts: (a) the laser cavity, (b) the MZI, and (c) the DFT. The laser cavity is a unidirectional ring (UR) enabling the generation of mode-locked rectangular pulses using the nonlinear polarization evolution mechanism [31]. The cavity includes a C-band double-clad co-doped Er:Yb 30 dBm fiber amplifier (EYDFA) from Lumibird company (KPS-BT2-C-30-BO-FA), a set of polarizer and polarization controllers (PC1 and PC2), a polarization-insensitive isolator (PI-ISO) ensuring the unidirectional propagation of light and an additional coil of single-mode fiber (SMF 28). The total cavity length is about 290 m (corresponding to a round-trip time of $1.45~\mu$s), including 5 m-long double-clad fiber with second-order dispersion $\beta _2=-0.021~ps^2/m$ and red285 m of SMF 28 with $\beta _2=-0.022~ps^2/m$. The total net cavity dispersion is estimated at $-6.375~ps^2$. An optical coupler (OC) is used to extract 20 % of the signal to study the characteristics of the rectangular pulses at the reference output. The laser is adjusted to deliver rectangular pulses in the nanosecond range, typically few tens of ns. Since the spatial extension of the pulse is few meters, it is not possible to scan it entirely by standard interferometric redsystems. Thus, the coherence is characterized with two different methods. The first method is based on a Mach-Zehnder interferometer (MZI). redIt is widely used in a large variety of applications, including medicine, fiber optic sensing, light detection and ranging, and fundamental research in lasers and optics [32–37]. To the best of our knowledge, no MZI-based study has ever been undertaken to measure the coherence of nanosecond DSR-like pulses. As illustrated in part (b) of Fig. 1, a 50 % coupler splits the beam into two equivalent signals. One part propagates through one arm, which includes two collimating lenses fixed on a translation stage (TS). The second part of the signal goes through a piezoelectric (PZT) fiber stretcher which superposes a modulated delay $\delta (t)$ around the fixed path difference $\Delta$. The total path difference is then $\Delta +\delta (t)$. The oval form PZT-based fiber stretcher consists of two U-shaped caps made of polyethylene resin designed for winding the optical fiber around it. One of the caps is fixed while the other is mounted on a PZT translation stage driven by a function generator with a triangular function at 200 Hz. Finally, both signals are recombined with a 50 % OC and local temporal coherence of the rectangular pulses is studied. The second method is based on the use of the DFT technique as displayed in part (c) of Fig. 1. It consists in propagating the pulse through a simple delay line of 30 km of SMF 28. DFT is a real-time measurement technique. In fiber lasers, it aims either to measure the dispersion D (GVD per unit length) of optical fiber or the spectrum waveform of a pulse [38], by using the mapping relation between time and wavelength, given by $\Delta \tau = L~\cdot ~D~\cdot ~\Delta ~\lambda$, where L is the length of the fiber, $\Delta \lambda$ is the bandwidth of the laser, and $\Delta \tau$ is the time duration into which the laser spectrum is mapped [39]. For ultrashort pulses of few tens of picoseconds or less, the effect of DFT is to map the optical spectrum into a temporal waveform, i.e., the temporal intensity trace of the pulses mimics the spectrum. redIn fact, the DFT is widely used, inter alia, to study pulse buildup dynamics [40–44], soliton breathing phenomena [45–47], laser instabilities [48,49], and multipulse regimes [50–52]. In our work, we use the DFT technique in a non-standard way to study the coherence of large nanosecond rectangular pulses, to confirm the results obtained using the MZI. The laser output characteristics and the coherence measurements are monitored by a 13-GHz oscilloscope (Agilent infiniium DSO8130B) combined with two 12-GHz photo-detectors (TTI, Model TIA1200 O/E Converter), an optical spectrum analyzer (Anritsu MS9740A, $0.6~\mu$m - $1.7~\mu$m, 30 pm resolution bandwidth), and an electronic spectrum analyzer (Rohde & Schwarz FSP Spectrum Analyzer 9 kHz to 13.6 GHz) to investigate the temporal trace, the optical spectrum, and the radio-frequency signal, respectively. Finally, a high-power integrating sphere (Thorlabs S146C) is used to measure the average power of the laser.
3. Rectangular Pulses in Anomalous Dispersion Regime
In this section, the characteristics of rectangular-shape pulses are studied at the reference laser cavity output as in Fig. 1. The pulse temporal trace is given in Fig. 2(a), obtained at pump power of 2 W. There is one pulse per cavity round-trip, with a pulse width of 35.4 ns, as shown in the inset of Fig. 2(a). The corresponding optical spectrum is centered at a wavelength of 1612 nm as shown in Fig. 2(b). The spectral 3 dB bandwidth is equal to 9.2 nm. Figure 2(c) represents the RF spectrum trace with a signal-to-noise ratio of 68 dB, indicating high signal stability. The inset depicts a sinc function modulated signal corresponding to the Fourier transform of a rectangular pulse. By increasing the pump power from 0.8 W to 4.02 W without any PC readjustment, the pulse remains perfectly rectangular without suffering from any wave-breaking as illustrated in Fig. 3(a). The same observation is made concerning the optical spectrum, which also keeps its initial shape, as shown in Fig. 3(b), by holding the spectral bandwidth at -3 dB around the same value as depicted in Fig. 3(c).
Fig. 1. Experimental setup. EYDFA: Erbium-Ytterbium co-doped fiber Amplifier, OC: optical coupler, PC1,2: polarization controller, SMF 28: single-mode fiber, PI-ISO: polarization-insensitive isolator, DFT: dispersive Fourier transformation, PZT: Piezoelectric translator, MZI: Mach-Zehnder interferometer, L1,2: lens, TS: translation stage.
Fig. 2. Temporal trace of rectangular pulses (a), the corresponding optical spectrum (b), and the radio-frequency trace (c).
Fig. 3. (a) Temporal trace of rectangular pulses for pump power ranging from 0.8 W to 4.02 W. (b) The corresponding optical spectrum for the same range of pump power. (c) 3-dB bandwidth of the spectrum as function of pump power.
Other characteristics are required to confirm the eventual DSR nature of these pulses. The most common characteristics of a DSR rectangular pulse are the linear evolution of the pulse width and the pulse energy versus the pump power, and the peak power clamping effect. In Fig. 4(a), the pulse width (red stars fitted with a linear red curve) increases linearly with the pump power, whereas the peak power (blue dots fitted with a linear blue curve) remains nearly constant. Pulse energy also follows the linear evolution with the pump power, as shown in Fig. 4(b).
Fig. 4. Characteristics of rectangular pulses. Evolution of the (a) pulse width (red stars linearly fitted by red curve), the peak power (blue dots fitted by linear blue curve) and (b) the pulse energy (black dots with linear fit curve) with pump power ranging from 0.8 W to 4.02 W.
These three characteristics are as expected for a DSR pulse, and in many publications, these criteria were enough to conclude that the obtained rectangular pulses are in the DSR regime regardless of their coherence. Since temporal coherence for nanosecond pulses has never been studied and taken into consideration, we complete our study by verifying it, so that we could conclude whether these rectangular pulses are DSR or not.
4. Coherence measurement of rectangular pulses with MZI
Before studying the coherence of rectangular pulses, the MZI is tested and validated using a Distributed Feed-Back (DFB) laser as a continuous wave laser source emitting at a wavelength of 1556.5 nm. Figure 5(a) represents the optical spectrum of the DFB laser and Fig. 5(b) exhibits the temporal trace of the signal at the output of the MZI.
The temporal trace shows the interferometric fringes with high contrast. The period $T= 5~ms$ corresponds to the modulated delay performed by the PZT fiber stretcher driven by a $200~Hz$ triangular function generator. The temporal curve is not a perfect sine as expected for a continuous phase variation. Phase jumps are localized at every minimum and maximum of the triangular function (around $1~ms$, $3.5~ms$, and $6~ms$.), which causes an inhomogeneous response of the fiber to the PZT fiber stretcher. Apart from these phase jumps, the MZI is operational and could be used to measure the coherence of more complex laser regimes. From Fig. 5(b) we deduce that there are about 33 fringes for one period of modulation of the PZT so that a $2\pi$ -phase delay between the two arms of the interferometer corresponds to about $150~\mu s$. The period of the interferometric fringes is about 100 times greater than the round-trip time of the cavity ($1.45~\mu s$). Therefore, the interferometric fringes will not be visible along a rectangular pulse but will modulate a large number of pulses. Figure 6 is a schematic representation of the expected temporal intensity evolution of the rectangular pulses in the fiber laser at the MZI output.
In the case of coherent pulses, it is expected that the intensity of interfering beams will experience a sinusoidal evolution corresponding to consecutive destructive and constructive interference (as observed for the DFB laser). So, when we measure the intensity of the top center of each two interfering pulses (see red crosses in the upper curve of Fig. 6(a)), the resulting curve would be sinusoidal, as shown in the same figure (lower red curve of Fig. 6(a)), representing the envelope versus time). Alternatively, in the case of incoherent pulses, the intensity reached by interfering part of each pulse remains constant, i.e., as the pulses are not coherent, the resulting intensity is merely the sum of intensities of the random interfering pulses (see Fig. 6(b)). Figure 7(a) represents the interferometric fringes of the square pulses with $26.5~ns$ pulse width, obtained at a pump power of $1.67~W$, for different fixed optical delays $\Delta (\sim 0, \sim 100, ~\sim 200,$ and $\sim 300~\mu m$). For a fixed delay of $\Delta \sim 0~\mu m$, i.e., when the path difference is nearly zero, the interferometric fringes are visible with relatively high contrast. When $\Delta$ increases to $100~\mu m$ and $200~\mu m$ the contrast of the fringes decreases and disappears for $\Delta \sim 300~\mu m$. Fig. 7(b) represents the corresponding peak intensity (envelope) traces for each value of path difference, in which the evolution process of fringe visibility is clearer. Considering the discontinuities introduced by the fiber response to the stretching, we can see that for $\Delta \sim 0~\mu m$, the contrast of the fringes is high and mimics the sine function as it is expected in the schematic of Fig. 6(a). When $\Delta$ increases, the contrast of interferometric fringes decreases. For $\Delta \sim 300~ \mu m$, the fringes completely disappear as in the case of incoherent pulses shown in Fig. 6(b), where the peak intensity is constant versus time.
The same measurements are done for different pulse widths. For example, we plot the results obtained at a pump power of $4.02~W$, for a pulse width of $69~ns$, in Fig. 8. We notice a relatively high contrast for $\Delta \sim 0~\mu m$, but the coherence pattern disappears when $\Delta$ is around $300~\mu m$. From Fig. 7 and 8, we conclude that when the path difference, $\Delta \geq 300~\mu$m, corresponding to a time delay of $\tau = 1~ ps$, the coherence completely disappears. This means that the square pulses are not coherent and might consist of sub-picosecond pulses under the square envelope. The duration of these pulses is estimated to be about 1 ps.
Fig. 5. (a) Optical spectrum trace of the DFB laser source. (b) Temporal trace of interferometric fringes of the DFB laser at the output of the MZI.
Fig. 6. Temporal evolution of the pulse intensity, and the envelope, in case of (a) coherent pulses and (b) incoherent pulses.
Fig. 7. (a) Temporal evolution of the interferometry fringes of the $26.5~ns$ pulses at the pump power of $1.67~W$, for different optical delays $\Delta$ (ranging from $0~\mu m$ to $300~ \mu m$), at the output of MZI. (b) The corresponding peak intensity evolution.
Fig. 8. (a) Temporal evolution of the interferometry fringes of the $69~ns$ pulses at the pump power of $4.02~W$, for different optical delays $\Delta$ (ranging from $0~\mu m$ to $300~\mu m$), at the output of MZI. (b) Intensity envelop of the interferometry fringes.
Finally, by using a MZI, we prove that although the obtained rectangular pulses show most of the DSR pulse features, they are not necessarily DSR but may be packets of shorter pulses. With this method, we scan a short part of the rectangular pulse, so the coherence measurement is local. It is possible to adjust the delay $\Delta$ with the free-space collimators and scan different parts of the pulse as shown previously for different path difference $\Delta$ ($\sim 0$, $100$, $200$, and $300~\mu m$). However, the studied pulses have spatial extension of few meters and it is not possible to create such delays with collimators. To confirm the previous results, other measurements are made by employing the DFT technique which allows more global coherence characterization. By studying the shape evolution of the whole pulse after propagating along a fiber coil, it is possible to easily discriminate between coherent and incoherent rectangular pulses.
5. Coherence characterization of rectangular pulses with DFT technique
As previously explained, the usual DFT technique allows performing real-time measurement of the spectral profile. This occurs if the fiber length L satisfies the condition $L$ $>>$ $L_D$ [53], where $L_D$ is the dispersion length. Experimentally, this condition can be easily satisfied for femtosecond or picosecond pulses, while it is unattainable in the nanosecond range. For example, if we consider a Gaussian pulses with $\tau _0=10~ns$, the required fiber length to perform an efficient DFT will exceed $10^9 ~km$. However, our goal is not to realize real-time spectroscopy but rather to obtain informations about the temporal coherence of the pulses and we will see that the DFT allows to easily discriminate between a coherent and incoherent pulse. Indeed, let us consider a rectangular pulse in the nanosecond range and a dispersive line of few tens of km of fiber. If the pulse is coherent, the amount of chromatic dispersion is not sufficient to satisfy the temporal far-field condition and the pulse propagating through the dispersive medium is not modified. The time-encoded spectral features cannot be captured in time-domain. Conversely, if the pulse undergoes significant modifications in the temporal profile and mimics the spectrum waveform, we can conclude that the pulse is composed of subsequent fine structures, hence it is incoherent. In our experiment, a rectangular pulse launched into $30~km$ of SMF (see part (c) Fig. 1). Figure 9(a) shows the temporal trace of the initial rectangular pulses at the laser output, while Fig. 9(b) represents the temporal trace of the pulses at the output of the DFT line. We can see that the wings of the pulse propagating through $30~km$ of dispersive medium are stretched, and the pulse loses its rectangular shape. For pump powers of $0.46~W$ and $0.8~W$, where the initial pulse width is $6.5~ns$ and $12.2~ns$, respectively, the temporal trace after propagation mimics the spectral waveform as in standard DFT technique (see spectrum of Fig. 2(b)). For higher pump powers and larger pulses, the central part of the pulse remains flat. However, the rise time of the initial rectangular pulses is around $0.1~ns$ while the rise time after the DFT line is around $10~ns$ as depicted in Fig. 9(c). These observations are in contrast with the theoretical redpredictions if the redrectangular pulses are assumed to be coherent. Therefore, we can conclude that the studied rectangular pulses are not temporally coherent but consist of many shorter pulses that allow the envelope to be stretched by only $30~km$ of fiber. For further investigation of the DFT-line effect on the pulses, additional numerical simulations have been performed. In our simulations, we have considered the optical fiber as purely second-order dispersive medium. In addition, we consider the initial peak power of the pulse relatively low so that the nonlinear effects can be neglected which is a fundamental assumption for DFT to be performed. In the reference frame moving at the group velocity, the electric field envelope at the abscissa z is related to the incident electric field envelope through the integral formula [53]:
Since DSR pulses are highly linearly chirped on the edges of the pulse [54], many values of the linear chirp have been tested in the simulations such as $C = 15,~100,~500$. As the result does not change significantly in the simulations, it is chosen arbitrarily as $C = 15$. The order of the super-Gaussian is $m = 100$ to better fit the rectangular shape of the experimental pulses. We take $t_0=2.5~ns$ leading to initial pulse duration, defined as the full width at half maximum, equals to $5~ns$. The group velocity dispersion value of silica single-mode fiber at the wavelength of $1.55~\mu m$ is $\beta _2=-0.022~ps^2/m$.
Fig. 9. Temporal traces of the rectangular pulses for different pump powers (a) at the reference output and (b) at the output of the DFT-line. (c) Rise time of the initial pulses (red squares fitted with linear red curve) and after the DFT-line (blue dots fitted with linear blue curve).
The simulations results are shown in Fig. 10. The temporal profile and the spectrum of the initial pulse are represented in blue curve. The red and black curves in Fig. 10(a) represent the pulse profile after propagating via $30~km$ and $2.5 \times 10^8~km$ of fiber, respectively. One can note that the pulse shape after the $30~km$ of DFT line remains almost rectangular. The rise time difference is very low, increasing from $0.42~ns$ for the initial pulse to $0.57~ns$ for the DFT pulse. In the case of propagation in $2.5 \times 10^8~km$ of fiber, the temporal trace of the pulse is modified and mimics the optical spectrum of the pulse. These observations demonstrate that, to perform an efficient DFT on a coherent pulse with initial duration as large as $5~ns$, at least $2.5 \times 10^8~km$ fiber length is required. The use of $30~km$ of SMF is not sufficient and does not allow time-mapping of the optical spectrum. Based on these observations, as the $30~km$ of fiber was experimentally enough to strongly modify the shape of the pulses and mimic the optical spectrum, we conclude that the studied pulses are not coherent. This is in agreement with the conclusions of the former study using the Mach-Zehnder interferometer.
Fig. 10. (a) Temporal trace of the rectangular pulses representing the initial pulse in blue, the pulse propagation after $30~km$ in red, and the pulse propagated in $2.5 \times 10^8~km$ in black. (b) The corresponding optical waveform for the initial rectangular pulse.
6. Conclusion
In conclusion, we provide an experimental study of the temporal coherence of large rectangular pulses in a passively mode-locked fiber laser using both MZI and DFT techniques. In the case of MZI, we were able to circumvent the limitations due to the repetition rate and the pulse duration, and to visualize in real-time the interferometric fringes on the oscilloscope. We observed that when the time delay of the interferometer reaches $1~ps$, the interferometric fringes disappear indicating that the obtained pulses contain shorter pulses under the rectangular envelope. Through the DFT technique, we confirm that these pulses cannot be coherent since their temporal shape is altered after only $30~km$ of propagation. The results of the DFT are more significant for the shortest pulses with duration of $6.5~ns$ and $12.2~ns$ as they tend to mimic the optical spectrum. Based on the numerical simulation, this would be impossible to occur for large coherent pulses because the length of SMF needed for such effect is five orders of magnitude larger than what is implemented in our experimental setup. Hence, we have demonstrated that rectangular pulses sharing most of DSR regime common features, are not necessarily of the same nature if they are not temporally coherent.
Acknowledgments
The authors are grateful to Dr. P. Le Boudec from IDIL (Lannion, France) for kindly providing the DFB semiconductor laser.
Disclosures
The authors declare no conflicts of interest.
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