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Abstract
The figure-9 fiber laser exhibits excellent performance, but improvement of its output pulse energy is restricted by the laser structure design that ensures self-starting mode-locking. In this paper, we propose and verify a novel method to increase the pulse energy of the self-starting figure-9 fiber laser. By reducing the linear phase shift step-by-step in a self-starting figure-9 laser and synchronously increasing the pump power, the output pulse energy can be increased while the laser can always operate in the single-pulse mode-locking region. Using a 112-MHz dispersion-managed soliton figure-9 fiber laser, the effectiveness of our proposed method is verified, and the laser output pulse energy has been successfully increased to 1.4 nJ, which is 5.6 times the pulse energy before the boost. The entire self-starting mode-locking of the laser including the program-controlled joint adjustment is less than 1s with 100% success rate of more than 100 tests. This method can in principle solve the limitation on the output pulse energy caused by the self-start of the figure-9 laser.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Femtosecond (fs) laser pulses are essential for many applications in fields such as precision micromachining, precision metrology, and nonlinear optics [1,2]. Through dispersion and nonlinearity management, different pulse-formation mechanisms, such as solitons, dispersion-managed solitons, similaritons, and dissipative solitons, can be generated with mode-locked fiber lasers [3]. Existing mode-locking mechanisms include real saturable absorbers, cascaded Mamyshev regeneration, nonlinear polarization evolution (NPE), and nonlinear amplifying loop mirror (NALM) [4,5]. Among them, the NALM mode-locking mechanism is the same as NPE, with fast response time and large modulation depth, which is conducive to the generation of high-performance fs pulses. Moreover, all polarization-maintaining mode-locked fiber laser can be constructed based on NALM [6,7], which makes the laser has a strong ability to resist environmental disturbance. Especially, the figure-9 fiber laser based on NALM can not only easily self-start mode-locking by the introduction of non-reciprocal phase shifter and maintain long-term stable operation, but also generate fs pulses with high repetition rate, narrow pulse width, and low noise [8,9], which meets the requirements of various applications in unstable and even extreme environments [10] and is one of the most promising fs laser oscillations for practical applications.
Despite its advantages, the figure-9 fiber laser still suffers from problems in low output pulse energy which is difficult to improve. So far, the reported maximum output pulse energies for the 1.55-µm dispersion managed soliton figure-9 single-mode fiber laser and the 1-µm dissipative soliton figure-9 large-mode fiber laser can only reach 0.4nJ and 28nJ [11,12], both of which are far lower than the output pulse energies of 2.7 nJ and ∼µJ of the corresponding NPE mode-locked fiber lasers [13,14]. In fact, to achieve self-starting mode-locked figure-9 fiber laser, a non-reciprocal phase shifter may have to be inserted into the laser cavity that is an asymmetric NALM [7], which would cause an accumulated nonlinear phase shift difference ($\Delta {\phi _{\textrm{NL}}}$) between the travelling optical fields in clockwise (cw) and counter-clockwise (ccw) directions along NALM from the power fluctuations of continuous wave (CW) lasing. In other words, the self-starting of laser requires that the slope of low-power SA transmittance curve should be as large as possible [7]. Then, the $\Delta {\phi _{\textrm{NL}}}$ must be adjusted to locate in the certain range of the SA transmittance curve, where SA transmission is higher, for achieving the single pulse operation [15,16]. However, the asymmetric cavity structure may cause the nonlinear phase shift (NPS) of the cw and ccw optical fields in the cavity to experience different trends or slopes with the increase of pump power. Increasing pump power would then easily exceed the value of $\Delta {\phi _{\textrm{NL}}}$ allowed for the single pulse operation defined by the SA transmittance curve, which would restrict the increase of output pulse energy. In addition, the asymmetric NALM structure required for the self-starting mode-locking equally restricts using the intra-cavity dispersion and nonlinearity manipulation to increase the pulse energy [3]. In a weakly asymmetric cavity, theoretically, under extremely strong pumping one can still ensure the value of $\Delta {\phi _{\textrm{NL}}}$ do not exceed the single pulse operating range, while making the cw and ccw optical fields in the cavity accumulate NPS high enough, so that the pulse energy can be increased. Yet, the self-starting mode-locked pulse originates from the power fluctuations of the CW lasing. Under strong pumping, the CW lasing power is extremely high, and the intra-cavity gain fiber operates in deep saturation, giving rise to the “self-healing” effect for the weak power fluctuations of CW lasing, which is harmful for amplifying weak power fluctuations [17], and makes self-starting mode-locking in weakly asymmetric cavity based figure-9 laser difficult to achieve. Although the pulse energy of self-starting mode-locked figure-9 fiber lasers may be increased by using large mode field fibers to reduce the NPS in the cavity [12] and introducing the real saturable absorber to build a hybrid mode-locked laser to help self-start [18], the limitation of output pulse energy due to linear phase shifter and the asymmetric NALM for the purpose of self-starting is still in principle unresolved.
In this paper, we propose and demonstrate a new approach to increase the pulse energy of the mode-locked figure-9 fiber laser without affecting its self-starting function and single pulse mode-locking operation. By reducing the linear phase shift (LPS) of non-reciprocal phase shifter step-by-step in a self-starting figure-9 laser to right shift the SA transmittance curve, and synchronously increasing the pump power, the output pulse energy can be gradually increased under single pulse operation. With a typical 112-MHz dispersion-managed soliton figure-9 fiber laser, we verify the effectiveness of this method, and demonstrate the improvements on the laser output pulse energy, as well as the success rate of mode-locking start-up and the total time consumption.
2. Methods
For a figure-9 fiber laser with given cavity asymmetry, the value of LPS ${\phi _\textrm{L}}$ introduced in the cavity determines the self-starting mode locking ability of the laser [7]. The optimal ${\phi _\textrm{L}}$ to obtain self-starting is 1.5π because the slope at low-power region of SA transmittance curve is the largest. As ${\phi _\textrm{L}}$ decreases, the difficulty for the laser to self-start mode-locking increases because the slope at low-power region of SA transmittance curve becomes smaller, until the laser cannot self-start mode-locking ultimately. However, as shown in Fig. 1, the relationships between $\Delta {\phi _{\textrm{NL}}}$ and the equivalent SA transmittance assuming the splitting ratio and gain in the equivalent NALM of 0.5 and 1 for different ${\phi _\textrm{L}}$ indicate that, for any value of ${\phi _\textrm{L}}$, there is a value range corresponding to single-pulse mode-locking operation which can be theoretically defined as the regions that the transmittance of pulse is greater than that of CW component [15] (see the thickened parts in Fig. 1). In addition, the smaller value of ${\phi _\textrm{L}}$ in figure-9 laser, the larger $\Delta {\phi _{\textrm{NL}}}$ required for single pulse mode locking. If the cavity asymmetry remains unchanged, the laser output single pulse energy can be larger. But the problem is that as the value of ${\phi _\textrm{L}}$ becomes smaller, the laser will not be able to self-start mode locking at all, let alone increase the pulse energy output.
Fig. 1. SA transmittance curves of figure-9 fiber laser with different LPSs assuming that the splitting ratio and gain in NALM are 0.5 and 1, respectively. The ${\phi _\textrm{L}}$ for the red, green, and black curves are respectively ${\phi _{\textrm{L,1}}} = 1.5\pi $, ${\phi _{\textrm{L,2}}} = \pi $ and ${\phi _{\textrm{L,3}}} = 0.5\pi $, and the thickened parts represent to the single pulse mode-locking operation regions, with corresponding $\Delta {\phi _{\textrm{NL}}}$ ranges of $0 < \Delta {\phi _{\textrm{NL}}}\textrm{ < 2}({2\pi - {\phi_{\textrm{L,1}}}} )$, $0 < \Delta {\phi _{\textrm{NL}}}\textrm{ < 2}({2\pi - {\phi_{\textrm{L,2}}}} )$, and $2({\pi - {\phi_{\textrm{L,3}}}} )< \Delta {\phi _{\textrm{NL}}}\textrm{ < }2\pi$, respectively. The LPS corresponding to the purple curve is $1.5\pi - \Delta {\phi _\textrm{L}}$.
Our approach to increase pulse energy of the laser while ensuring fast self-starting mode locking function is as follows. For a figure-9 fiber laser with excellent self-starting function, e.g., ${\phi _\textrm{L}}$ is near 1.5$\pi$, if ${\phi _\textrm{L}}$ is adjusted to be gradually decreased from its initial value while the pump of the laser is simultaneously increased to make sure the value of $\Delta {\phi _{\textrm{NL}}}$ always located in the region for the single pulse mode-locking operation defined by the continuously right-shifted SA transmittance curve, the output pulse energy can be increased without affecting the original self-starting function of the laser. In principle, the method may be achieved with the following operations. For a self-starting mode locked figure-9 fiber laser, the SA transmittance curve of the laser may be described by the red curve in Fig. 1. By adjusting the pump, the laser can stably operate near point A on the SA transmittance curve which corresponds to be close to but not exceed the maximum allowable $\Delta {\phi _{\textrm{NL}}}$ defined by the SA transmittance curve for achieving the single pulse mode-locking operation. If ${\phi _\textrm{L}}$ is superimposed with a small increment $-\Delta {\phi _\textrm{L}}$, the SA transmittance curve moves slightly to the right purple curve in Fig. 1. If reducing ${\phi _\textrm{L}}$ does not introduce excessive additional loss, $\Delta {\phi _{\textrm{NL}}}$ remains basically constant at the moment when ${\phi _\textrm{L}}$ decreases $\Delta {\phi _\textrm{L}}$. However, the saturable absorption loss experienced by the intra-cavity pulse decreases by the slightly right-shifting of the SA transmittance curve, causing the increasing of peak power of the pulse. This may also increase the value of $\Delta {\phi _{\textrm{NL}}}$, switching the laser to operate near point B on the slightly right-shifted SA transmittance curve. As long as $\Delta {\phi _{\textrm{NL}}}$ is small enough, point B must be within the single-pulse mode-locking region defined by the right-shifted SA transmittance curve. Moreover, the pump power can still be allowed to be increased by $\Delta {P_\textrm{p}}$ to make the laser stably operate near point C on the right-shifted SA transmittance curve which corresponds to be close to but not exceed the maximum allowable $\Delta {\phi _{\textrm{NL}}}$ defined by the SA transmittance curve for achieving the single pulse mode-locking operation. Note, since the pulse is always adjusted to operate in the single pulse operation range of the right-shifted SA transmittance curve, the loss experienced by the pulse is less than that experienced by the CW (or ASE) components, which can suppress the CW development in the cavity to avoid the occurrence of CW components accompanying the pulses or multi-pulse phenomenon. By repeating the above process of reducing ${\phi _\textrm{L}}$ and increasing the pump power, the operation point of the laser can be moved towards the single pulse mode-locking region defined by the multiple right-shifted SA transmittance curve, for instance, near point D on the green curve shown in Fig. 1(a), still without any influence on the self-starting mode-locking function of the laser. When the LPS continues to decrease from $\pi $ to right-shift the SA transmittance curve, as shown in Fig. 1, the single pulse mode-locking region gradually shrinks due to the decrease of the cavity loss experienced by the CW lasing. However, as long as the set LPS and pump power increment is appropriate, using the above synchronous joint adjustment steps, the laser can still be switched to the single pulse mode locking region defined by a very far right-shifted SA transmittance curve, for example, point E on the black curve in Fig. 1, while still maintains the single pulse operation for the laser. Of course, for different mode-locking regimes, the nonlinear phase shift of the laser allowed for single pulse mode-locking may also depend on the dispersion and nonlinear management of the cavity, which results that point E shown in Fig. 1 may not be reached. In this way, the pulse energy of the figure-9 fiber laser can be greatly increased, and if the above-mentioned fine adjustment process of ${\phi _\textrm{L}}$ and pump power is automatically realized by the program-control, the method for enhancing the pulse energy can remain the fast self-starting mode-locking function for the laser.
3. Experimental setup
The output pulse energy of a mode-locked fiber laser is mainly limited by the two factors of cavity nonlinearity and saturable absorption effect [3]. Excessive nonlinear phase shift accumulated in the cavity can cause effects such as wave breaking or soliton splitting for the intracavity pulses, which restricts the increase of the output pulse energy [3,19]. By managing the dispersion and nonlinearity to shape the intracavity pulse, the limitation of the output pulse energy due to cavity nonlinearity may be alleviated. Because of this, the maximum achievable pulse energies for fiber lasers based on different pulse-formation mechanisms are also different. However, the limitation of the saturable absorption effect on the achievable output pulse energy is independent on the specific pulse-formation mechanism of fiber lasers. For simplicity and generality, therefore, here we use the dispersion-managed soliton figure-9 fiber laser to verify our proposed method of increasing output pulse energy.
Figure 2 shows the schematic diagram of the dispersion-managed soliton figure-9 fiber laser we used for experimental verifications. The gain fiber is a 0.85-m normal dispersion single-mode PM erbium-doped fiber (Liekki-Er80-4/125-HD-PM). The dispersion coefficient of the gain fiber is 0.02 ps^2/m, and its absorallption at 1530 nm is ∼80 dB/m. The erbium-doped fiber (EDF) is pumped by two 975-nm laser diodes, which are combined with a polarization beam combiner to give a maximum pump power of 1.3 W, through a PM wavelength division multiplexer (WDM). The insertion loss of the WDM for the pump is ∼1 dB. In order to ensure the asymmetry of the fiber cavity to facilitate the self-starting of the laser mode-locking, and to avoid difficulties in fusion of short fibers, EDF is directly used as the pigtailed fiber of the common port of the WDM. The pigtailed fiber of the WDM signal port is a 0.2-m PM1550-XP (Nufern) fiber with a dispersion coefficient of $\textrm{ - }0.022\textrm{p}{\textrm{s}^\textrm{2}}\textrm{/m}$. To compensate the dispersion in the cavity, a segment of 0.55-m PM1550-XP fiber is used after the other end of the EDF. Therefore, the net dispersion in the fiber ring is about +0.0005 ps^{2}. Note that the design of near-zero net normal dispersion in the cavity enables the laser to tolerate a larger nonlinear phase shift for the single pulse mode-locking [3], so that the verification experiments for the method we proposed will not be affected by the pulse-breaking. The fiber in the ring is twisted by 90$^{\circ}$ [7]. The linear arm of the figure-9 fiber laser consists of a linear phase shifter inserted between two polarization beam splitters (PBS-1 and PBS-2) and M [7]. The physical length of the linear arm is ∼10 cm. The total cavity length of the laser is ∼1.8 m, giving pulse repetition rate of ∼112 MHz. The total cavity loss, including device insertion loss, fiber splicing loss, and polarization-related loss, is estimated to be ∼8 dB.
Fig. 2. Schematic diagram of the program controlled all-PM figure-9 mode-locked fiber laser. WDM: wavelength-division-multiplexing coupler; PBS: polarization beam splitter; HWP: half-waveplate; QWP: quarter-waveplate; FR: ${45^ \circ }$ faraday rotator; EDF: erbium-doped fiber; M: reflecting mirror; cw: clockwise; ccw: counter-clockwise.
To achieve controllable adjustment of LPS that our method required, we designed a non-reciprocal phase shifter based on a ${\lambda / 4}$ waveplate, a ${\lambda / 2}$ waveplate, and a ${45^ \circ }$ Faraday rotator. This non-reciprocal phase shifter, together with PBS1, PBS2, and the ${90^ \circ }$ twisted PM-fiber loop, can be regarded as an equivalent NALM. The vertical components cw and ccw transmission optical fields interfere at PBS1 and then exit from the cavity. According to the transmissions of cw and ccw optical fields in the equivalent NALM and Jones matrix theory [7,20], the transmittance of the equivalent NALM can be obtained:
According to Eq. (1), the transmittance curve of the equivalent NALM can be shifted to the left or right by setting different LPS, and LPS and splitting ratio are both related to ${\theta _1}$ and ${\theta _2}$. When $\rho $ is fixed, value of LPS can be adjusted continuously and linearly by varying ${\theta _2}$ and ${\theta _1}$ (see Figs. 3(a) and (b)). In addition, for a fixed ${\theta _2}$, the monotonous and continuous adjustment for both the LPS value and $\rho $ can be achieved by changing ${\theta _\textrm{1}}$ only (see Fig. 3(c)).
Fig. 3. ${\theta _2}$ (red curve) and LPS (blue curve) as functions of ${\theta _1}$ when $\rho $ is maintained at (a) 0.5 and (b) 0.43; (c) Splitting ratio (red curve) and LPS (blue curve) with respect to ${\theta _\textrm{1}}$ when ${\theta _2}$ is ${0^ \circ }$.
The two waveplates are installed and fixed on two stepping motors (ELL14 K, Thorlabs). By controlling the stepping rotation of the motors through the real-time program control of the host computer, ${\theta _\textrm{1}}$, ${\theta _\textrm{2}}$ can be tuned in real time, so that the LPS can be controlled precisely. The control sequence of the joint adjustment of the two stepper motors and the pump power is shown in Fig. 4(a). To meet the requirements of laser self-starting mode locking for LPS and $\rho $, the initial values of ${\theta _\textrm{1}}$ and ${\theta _\textrm{2}}$ and their step angles $\Delta {\theta _\textrm{1}}$ and $\Delta {\theta _\textrm{2}}$ are assigned respectively according to the Eqs. (2) and (4). Set a proper pump power ${P_{\textrm{start}}}$ to ensure the laser to self-start mode-locking. After the mode-locking is stabilized, reduce the pump power to ${P_{\textrm{single}}}$ corresponding to single pulse operation. Since the minimum response time of serial communication used here is 120 ms, the time ${T_1}$ for the single pulse self-starting process is 360 ms. The time from 0 to ${T_1}$ is the self-started mode locking region. At $t = {T_1}$, ${\theta _\textrm{1}}$ is stepped by $\Delta {\theta _\textrm{1}}$, followed by ${\theta _\textrm{1}}$ stepped by $\Delta {\theta _\textrm{1}}$ at every time interval T. After each step of ${\theta _\textrm{1}}$, ${\theta _2}$ is stepped by $\Delta {\theta _2}$ after $\tau$ (30 ms) delay. Then, following a delay of ${T / 2}$, the pump power is increased to ${P_{\textrm{Pk}}} (\textrm{k}=1,2\ldots \textrm{n}$, ${P_{\textrm{Pk}}}$ is determined by experiments, taking closing to but not exceeding the maximum pump power corresponding to single pulse operation as the criterion). Moreover, changing the rotation angle of a single waveplate can also achieve continuous tuning of LPS. In this case, set $\Delta {\theta _\textrm{2}} = 0$, similarly, two independent child threads are used to control the rotation angle ${\theta _\textrm{1}}$ and pump power respectively. The corresponding joint adjustment sequence is shown in Fig. 4(b). T can be adjusted within 240 ms to 2 s. By changing T, the total time of joint adjustment can be tuned for examining the success rate of starting the laser mode-locking.
Fig. 4. (a) Timing diagram of ${\theta _\textrm{1}}$, ${\theta _\textrm{2}}$, and pump power under constant splitting ratio. (b) Timing diagram of pump power and ${\theta _\textrm{1}}$ under constant ${\theta _\textrm{2}}$.
4. Verification results and discussion
To easily self-start mode-locking and achieve pulse energy as high as possible, we fixed the LPS and $\rho $ of the equivalent NALM to $1.5\pi $ and 0.5 by setting ${\theta _\textrm{1}}$ and ${\theta _2}$ to ${22^ \circ }$ and ${11^ \circ }$, respectively. Under this condition, the laser self-starting mode-locking with the success rate of 100% for 400 times switching on and only ∼100-ms pulse build-up time. The pump power range for the single pulse operation is [350 mW, 550 mW]. Figure 5(a) shows the measured output pulse spectrum for the pump power of 550 mW. It can be seen that the spectral coverage of the pulse is 1520-1613 nm, with 3-dB spectral width of the 1591 nm peak as ∼25 nm, and the output pulse spectrum is slightly modulated by the interference between the cw and ccw transmission optical fields. The single pulse mode-locked state is very stable, and the duration of the output pulse is ∼1 ps (∼92 fs after fiber dispersion compensation). The measured average power of the output pulse is 32 mW, giving the pulse energy of 0.28 nJ.
Fig. 5. (a) Output spectrum of the laser for ${\phi _\textrm{L}} = 1.5\pi $ and pump power of 550 mW. The inset is the pulse after fiber compression. (b) RF spectrum of output pulses with a resolution bandwidth (RBW) of 300 kHz.
Then, with $\rho $ fixed at 0.5, ${\theta _\textrm{1}}$ and ${\theta _2}$ are adjusted from their initial angles of ${22^ \circ }$ and ${11^ \circ }$ with $\Delta {\theta _\textrm{1}} ={-} {1^ \circ }$ and $\Delta {\theta _2} ={-} {0.5^ \circ }$, respectively. T is set at 2 s. The curve labeled with black solid squares in Fig. 6(a) shows the relationship between experimentally obtained $\Delta {P_\textrm{p}}$ and LPS. In the adjustment process that the LPS is reduced to $1.1\pi $ and the pump power is increased to its maximum value of 1.3 W (corresponding to point K in the Fig. 6(a)) by the program-controlled joint adjustment, the laser can always maintain stable single-pulse mode locking, with the output pulse energy increasing from 0.28 to 1.2 nJ. And, as expected, during the joint adjustment process, no CW component or multi-pulse phenomenon are observed. Figure 6(b) is the output spectrum when the laser is adjusted to point K, which is significantly wider than that at point O, corresponding to the initial state (see Fig. 5). Since the minimum LPS ensuring a high self-starting success rate is measured to be $1.3\pi $ for the laser, at point K the laser can no longer self-start mode-locking as the LPS of $1.1\pi $ is much less than $1.3\pi $. Therefore, under the condition that self-starting mode-locking function of the laser is maintained, the method we proposed to right-shift the SA transmittance curve by reducing LPS have effectively increased the output pulse energy.
Fig. 6. (a) Joint adjusting curve between pump power and LPS. Points O and P are the initial and final states, respectively. The O-K section of the black solid square curve describes the joint adjustment mentioned in the method, and the K-P section is to reduce only the LPS while keeping the pump power unchanged; the red solid circular curve is another joint adjustment path; (b) Output pulse spectrum at the joint adjustment end point P; (c) Output pulse spectrum at the end point P when $\rho $ is 0.43.
Fig. 7. When the split ratio is 0.43 and the LPS is adjusted to $0.9\pi $, (a) the RF spectrum of the 1.4 nJ output pulse (black) with a RBW of 30 Hz and a span range of 120 kHz, along with the noise floor (red). The inset is the RF spectrum measured with a RBW of 300 kHz; (b) the pulse after fiber compression. The inset is time-frequency characteristics of the compressed pulse.
Due to the relatively wide single pulse operation region given by the right-shifted SA transmittance curve, after the joint adjustment to point K, although the pump power is already at the maximum available pump power of 1.3 W, the LPS can still be allowed to reduce to $0.9\pi $ (corresponding to point P) while the single pulse operation still maintains for the laser. The average power and the pulse energy are almost unchanged at 134 mW and 1.2 nJ respectively, and the output spectrum is also close to that at point K. The reason may be explained as follows. Since the pump power and $\rho $ remain unchanged during the joint adjustment process from point K to P, the value of $\Delta {\phi _{\textrm{NL}}}$ is almost unchanged either. When the LPS is adjusted from $1.1\pi $ to $0.9\pi $, the only effect is to right-shift the SA transmittance curve. If the unchanged $\Delta {\phi _{\textrm{NL}}}$ is still located in the single-pulse mode-locked region specified by the continuously right-shifted SA transmittance curve, the laser can maintain the single-pulse mode-locked operation, and the output spectrum and pulse energy are basically unchanged. Therefore, if the available pump power is sufficiently high and a highly doped gain fiber is used, a larger value of $\Delta {\phi _{\textrm{NL}}}$ can be provided, and the LPS can then be allowed to be further reduced (see Fig. 1), which in turn enables the laser to obtain greater output pulse energy. Moreover, due to the larger range of possible $\Delta {\phi _{\textrm{NL}}}$ value for the single pulse mode-locking operation specified by the SA transmittance curve, there are different paths for the joint adjustment between LPS and pump power. For example, the red curve with a solid circle in Fig. 6(a) also allows the LPS to be reduced to $0.9\pi $ while maintaining the single pulse operation, and the average power and the pulse energy at the end point P are also 134 mW and 1.2 nJ respectively. The output spectrum is as same as that shown in Fig. 6(b).
It was also found that, when $\rho $ is fixed at any value within [0.43, 0.74], our method of joint adjustment for LPS and pump power can be equally used to improve the output energy while maintaining the mode-locking self-starting and single pulse operation. However, for different $\rho $, the maximum achievable output pulse energy is different. It is interesting that, when the LPS is adjusted from $1.5\pi $ to $0.9\pi $ for a fixed $\rho $ of 0.43, the output pulse energy increases from 0.25 to 1.4 nJ. As expected, the initial pulse energy (0.28 nJ) for the $\rho $ of 0.5 is higher than the initial pulse energy of 0.25 nJ at the $\rho $ of 0.43. However, the maximum pulse energy (1.2 nJ) of the $\rho $ of 0.5 is lower than the maximum pulse energy of 1.4 nJ at the $\rho $ of 0.43. The reason may be that, in the deep gain saturation region, the doped fiber offers almost the same gains for the counter-propagating pulses in the cavity, thus, for a given fixed value of $\Delta \phi _\textrm{L}^{cw - ccw}$, according to Eq. (3), the peak power of the pulse entering the equivalent NALM for the splitting ratio of 0.43 must be greater than that for the splitting ratio of 0.5, i.e., the switching power of SA for the splitting ratio of 0.43 is higher [21], corresponding to the higher output pulse energy. Figure 7(a) shows the RF spectrum of the 1.4 nJ pulse. Figure 7(b) shows the pulse after fiber compression. The inset shows the time-frequency characteristics of the compressed pulse. It can be seen that the signal-to-noise ratio of the RF spectrum of the output pulse with a resolution bandwidth of 30 Hz is as high as 80 dB, indicating that the laser is working in a stable single pulse mode-locked state, and the compressed pulse width is about 70 fs.
Moreover, the above joint adjustment can also be realized by adjusting ${\theta _\textrm{1}}$ only. When ${\theta _2}$ is fixed at ${\textrm{0}^ \circ }$, as ${\theta _\textrm{1}}$ is reduced from $\textrm{1}{\textrm{9}^ \circ }$ to $\textrm{ - }{4^ \circ }$, the LPS can also be reduced from $1.5\pi $ monotonically to $0.9\pi $, and the splitting ratio is reduced from 0.74 to 0.43. We found that while the laser is kept mode-locked self-starting and stable single-pulse operation, the average power and energy of the output pulse can be increased to 154 mW and 1.4 nJ respectively for different joint adjustment paths. In addition, since the split ratio for the end point of the joint adjustment is 0.43, the output spectrum is consistent with the spectrum shown in Fig. 6(c).
In addition, we examined the success rate of the single-pulse mode-locked operation under two different linear phase-shift reduction schemes: adjusting ${\theta _\textrm{1}}$ only and simultaneously adjusting ${\theta _\textrm{1}}$ and ${\theta _2}$ (fixed split ratio of 0.5 and 0.43). If the adjustment delay between the LPS and the pump power is maintained at ${T / 2}$, our experimental results show that when reducing ${\theta _\textrm{1}}$ individually, the success rate of the over 100 times joint adjustments under the total time of 720 ms is 100%; when reducing ${\theta _\textrm{1}}$ and ${\theta _2}$ simultaneously under the fixed splitting ratio, the success rate of joint adjustment under the total time of 870 ms is also 100%. The reason may be that, the laser cavity cycle time is about ∼10 ns, which is sufficient to recover the intra-cavity gain dynamics caused by ${\theta _\textrm{1}}$ and (or) ${\theta _2}$ stepping and pump power increasing under the ms adjustment time, making the laser transit to a new stable single mode locking state easily. In the condition of the initial phase shift of $1.5\pi $, it takes 100% success rate for self-starting mode-locking and transitioning to a single pulse state within 360 ms, leading to the total time of 1 s from the laser mode-locking self-start to the end of the joint adjustment, and the success rate is as high as 100%.
It is worth noting that the above method of increasing pulse energy is verified by the typical ∼110-MHz dispersion managed soliton laser, where the length of the gain fiber is 0.85 m. The output pulse energy can be increased to 1.4 nJ for the LPS of $0.9\pi $, limited by the saturation effect of the gain fiber, which has already been verified by the measured relationship between the output pulse energy and the pump power in the single-pulse mode-locked state with the LPS adjusted to $0.9\pi $. It is worthwhile to emphasize that, to our best knowledge, the 1.4-nJ output pulse energy achieved here is the highest obtained by such a dispersion-managed soliton self-starting mode-locked figure-9 single-mode fiber laser so far, and also comparable to that by the NPE fiber laser. It is expected that the output pulse energy can be further improved by using our method if the gain fiber doping concentration is higher and the higher pump power is available. Moreover, the method we proposed to enhance pulse energy while maintaining mode-locked self-starting and single pulse operation is also applicable to the pulse energy enhancement of the figure-9 fiber laser using large-mode-area gain fiber and other kinds operating regimes of figure-9 fiber lasers. Therefore, this approach has in principle solved the limitation of output pulse energy caused by the introduction of LPS and cavity asymmetry in the laser.
5. Conclusion
We have proposed and verified an approach to high pulse energy emission of the figure-9 fiber laser without affecting its self-starting function and single pulse mode-locking operation. By reducing the LPS of the mode-locked self-starting laser step by step and continuously right-shifting the SA transmittance curve, and synchronously increasing the pump power, the laser can operate in the single pulse region of the SA transmittance curve, and finally the output pulse energy can be increased. With a typical 112-MHz dispersion-managed soliton figure-9 fiber laser, we have verified the effectiveness of this method and successfully increased the output pulse energy of the laser to 1.4 nJ with the output pulse maintaining excellent time-frequency characteristics. Note that 1.4 nJ is the highest pulse energy obtained by such figure-9 fiber lasers thus far. In addition, through program-controlled joint adjustment, the laser maintains a total self-starting time in the order of seconds with 100% success rate of more than 100 tests. This method is applicable to figure-9 fiber lasers using large mode area gain fibers, and is also compatible with the improvement of the pulse energy of the figure-9 fiber laser through the dispersion and nonlinearity management. Therefore, this method can fundamentally solve the limitation on the output pulse energy caused by the self-start of the figure-9 laser.
Funding
National Key Research and Development Program of China (2017YFB0405100, 2017YFB0405200); Strategic Priority Research Program of the Chinese Academy of Sciences (XDB21010300); National Natural Science Foundation of China (61377044, 61805258); Director Fund of Advanced Laser Technology Laboratory of Anhui Province (20191001).
Acknowledgment
The authors would like to thank Dr. Yingqiu Mao (USTC, Hefei National Laboratory for Physical Sciences at the Microscale) for constructive suggestions. All authors reviewed and edited the final paper.
Disclosures
The authors declare no conflicts of interest.
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