**2. Condensed-gain dissipative soliton model and simulation for 2 μm fiber lasers**

spectroscopy, optical sensing, medical treatment, material processing, and nonlinear microscopy [1–8]. In application, ultrashort pulses are usually required to have high pulse energy, which is very important for both scientific and industrial aims. In addition, achieving high

Compared to traditional solid-state lasers (SSLs), fiber lasers (FLs) are better candidates for generation of ultrashort laser pulses due to their advantages of compactness, robustness, and good laser beam quality. Conventionally, generation of short pulses from fiber systems is achieved by the soliton mode-locking mechanism. Various passive mode-locking techniques can be employed, including the nonlinear polarization rotation (NPR) [9, 10], the nonlinear loop mirror [11, 12], and the saturable absorber method [13, 14]. However, pulse energy of traditional solitons (with anomalous net cavity dispersion), which is based on the balance of dispersion and nonlinearity, is usually limited by the soliton area theorem [15, 16] or the pulse peak power clamping effect [17, 18] to less than 1 nJ. Therefore, fiber lasers still produce much

To improve the pulse energy of fiber lasers, many techniques have been proposed and explored [20–36], among which four kinds of mechanisms have played important roles: dispersionmanaged soliton [20–22, 37, 38], all normal dispersion mode-locking [39], self-similar soliton [27–30], and dissipative soliton (DS) [31–36]. By taking advantage of the balance between not only nonlinearity and dispersion but also gain and loss, DS mode-locked fiber lasers have realized pulse energy 1–2 orders of magnitude larger than those from conventional soliton mode-locking [31, 32]. However, although the DS pulse energy from 1 to 1.5 μm fiber lasers has exceeded 10 nJ [40–42] and even over 20 nJ [33–35], pulse energy of 2 μm DS fiber lasers still remains at a low level. This is because the currently available gain fibers (GFs) in the 2 μm region show relatively large anomalous dispersion, resulting in conventional soliton modelocking operation of 2 μm fiber lasers [43–48]. Therefore, the pulse energy is still governed by

DS mode-locking has been widely adopted as an efficient method to improve the pulse energy of 2 μm fiber lasers. To implement DS mode-locking, the whole cavity's dispersion has to be pushed into the normal dispersion region. To that end, various methods have been proposed, e.g., inserting a chirped fiber Bragg grating into the cavity to provide normal dispersion [49] or incorporating specially designed dispersion-compensating fibers (DCFs) into the cavity [50–52]. However, these methods only improve pulse energy to around 1 nJ, and the great

Here, we will first present a new model to investigate the intracavity pulsing dynamics of a 2 μm DS mode-locked fiber laser and show that (different from the 1 to 1.5 μm counterparts) the pulse energy of 2 μm DS fiber lasers is mainly limited by the nonlinear phase shift caused by the gain fiber, and thereafter we propose that the anomalous dispersive GF should be condensed as short as possible to efficiently decouple gain from dispersion and nonlinearity. We name it the condensed-gain fiber mode-locking (CGFML). By avoiding too much phase accumulation, numerical simulations show that over 10 nJ DSs at 2 μm are readily feasible. After that, we carry out experimental operation of such CGFML of a 2 μm fiber laser, and a 4.9 nJ DS with 579 fs dechirped pulse duration is achieved. By further optimizing the cavity, the pulse

energy short pulses at various wavelengths is the persistent pursuit of laser scientists.

lower pulse energy than their solid-state counterparts [19].

112 High Power Laser Systems

the soliton area theorem and clamped by peak power [15, 17].

potential of DS mode-locking mechanism has not been fully explored.

Based on detailed simulation of the dynamics of short pulse propagating in various fiber circumstances, we found that the main factor that limited the pulse energy in 2 μm DS fiber lasers was related with nonlinear phase shift, which was primarily accumulated in the gain fiber. If we can efficiently control the nonlinear phase shift generated in the gain fiber, then the pulse energy of 2 μm DS fiber lasers probably can be significantly scaled. Therefore, we propose a condensed-gain fiber model where the gain fiber should be shortened as much as possible, and in the following, we give a detailed description about the model and carry out simulation about the pulse dynamics happened in a 2 μm DS fiber laser.

A simple schematic diagram for the CGFML model is shown in **Figure 1(a)** [53]. The fiber laser cavity mainly includes five elements: output coupler (OC), single-mode fiber (SMF), gain fiber (GF), dispersion-compensating fiber (DCF), and saturable absorber (SA). Here, we use a single-mode highly doped 2 μm thulium fiber as the GF. Light evolution (pulse shape, pulse intensity, and spectrum) in the laser cavity is traced through solving the well-known nonlinear Schrodinger equation (NLSE) [27], which needs the original equation:

**Figure 1.** (a) Schematic diagram of the condensed-gain fiber laser model shows the light flow in the cavity. (b) Experimental setup of the SESAM mode-locked fiber laser system with a linear cavity. OC, output coupler; SMF, single-mode fiber; GF, gain fiber; DCF, dispersion-compensating fiber; SA, saturable absorber [53].

$$\frac{\partial \mathcal{L}(\mathbf{z}, \tau)}{\partial \mathbf{z}} + \dot{\mathbf{i}} \frac{\beta\_2}{2} \frac{\partial^2 \mathcal{L}(\mathbf{z}, \tau)}{\partial \tau^2} = \dot{\mathbf{i}} \chi \left| \mathcal{L}(\mathbf{z}, \tau) \right|^2 \mathcal{L}(\mathbf{z}, \tau) + \mathbf{g} \mathcal{L}(\mathbf{z}, \tau) \tag{1}$$

where U(z, τ) is the envelope of the light field, z is the propagation coordinate, and τ is the time-delay parameter. The SMF (8.2/125 μm, 0.14 NA) has a length of 1.4 m, with β2 = −67 ps2 / km and γ = 0.001 (Wm)−1, while the DCF (2.2/125 μm, 0.35 NA) is 1.5 m long with β2 = 93 ps2 /km and γ = 0.007 (Wm)−1, respectively. The 0.2 m GF (5/125 μm, 0.24 NA), with β2 = − 12 ps2 /km and γ = 0.003 (Wm)−1, has the gain (including saturation) as.

$$\mathbf{g} = \mathbf{g}\_0 / \left[ \mathbf{1} + \mathbf{E}\_{\text{pulse}} / \mathbf{E}\_{\text{sat}} + (\boldsymbol{\omega} - \boldsymbol{\omega}\_0)^2 / \Delta \boldsymbol{\omega}^2 \right] \tag{2}$$

where g0 is the small-signal gain (here it is taken to be 30 dB), Epulse is the pulse energy, Esat is the gain saturation energy, ω0 is the gain-center angular frequency, and Δω is the gain bandwidth (assume 90 nm).

The saturable absorption effect of SA is included by using the transfer function:

$$\mathbf{T} = \mathbf{1} - \mathbf{1}\_{\mathbf{v}} / \left[ \mathbf{1} + \mathbf{P} \text{(\tau)} / \mathbf{P}\_{\text{sat}} \right] \tag{3}$$

**Figure 2.** Evolution dynamics of pulse duration (black triangles) and spectral bandwidth (red circles) through different elements inside the laser cavity (a) and temporal phase of the pulse after DCF (black solid), GF (red dashed), and SMF

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**Figure 3.** Schematic diagram for the amplitude and phase balances in a 2 μm fiber laser (insets show the counterparts of

1 μm and 1.5 μm systems) [53].

(blue dotted) (b) and temporal (c) and spectral profiles (d) of the pulse after successive elements [53].

where l<sup>0</sup> is the unsaturated loss (take 0.7), P(τ) is the instantaneous power, and Psat is the saturation power. A critical factor for achieving DS is that spectral filtering is required to balance gain and loss. To that end, a 150-nm-bandwidth spectral filter (SF) is exerted on the SA.

Here, the split-step Fourier method is used to solve the NLSE. Simulation is started as the following procedure. With an initial white noise, the light is calculated in both temporal and spatial regions until a steady state is reached. The pump level and saturable effect are controlled through changing the values of Esat and Psat. When we take Psat = 3.5 kW and Esat = 3.4 nJ, the pulse's temporal and spectral evolution characteristics are shown in **Figure 2** [53]. In this case, the stable solution of pulse energy is 5 nJ. Detailed variations of pulse shape/width and spectral shape are clearly shown. In the DCF, owing to the combined effects of normal group velocity dispersion (GVD) and nonlinearity (NL), the pulse propagates with its duration increasing monotonically. The broadened pulse is then compressed by the SMF and GF with anomalous GVD. The pulse's spectrum has steep edges, and the bandwidth negligibly changes during the pulse circulating inside the cavity. However, the spectrum shape shows characteristic changes during the pulse evolution. The GF tends to amplify the spectrum's center more, and thus, the spectrum top becomes more arched. At the same time, the amplified pulse gives rise to increased self-phase modulation, thus leading to sharp edge peaks of the spectrum. Then, after being successively shaped by SA, DCF, GF, and SMF, the spectrum recovers its nearly flat-top shape. We can also see the advantages of the condensed-gain fiber model from the phase shift during the pulse evolution. As shown in **Figure 2(b)** [53], after passing through the three different kinds of fibers (DCF, GF, and SMF), very little phase shift is accumulated by the GF, which is finally compensated by both the DCF and the SMF.

To gain deeper insight into the intracavity pulsing dynamics, a qualitative illustration for 2 μm DSs is summarized in **Figure 3** [53], along with their 1 and 1.5 μm counterparts (insets) [54].

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\_\_\_\_\_\_

114 High Power Laser Systems

the gain saturation energy, ω0

width (assume 90 nm).

where g0

where l<sup>0</sup>

∂*U*(*z*, *τ*) <sup>∂</sup>*<sup>z</sup>* <sup>+</sup> *<sup>i</sup>*

γ = 0.003 (Wm)−1, has the gain (including saturation) as.

*g* = g0 /[1 + Epulse /Esat + (ω − ω0)

*β* \_\_<sup>2</sup> 2 ∂ \_\_\_\_\_\_\_ <sup>2</sup> *U*(*z*, *τ*)

<sup>∂</sup> τ2 <sup>=</sup> *<sup>i</sup>*|*U*(*z*, *<sup>τ</sup>*)|

km and γ = 0.001 (Wm)−1, while the DCF (2.2/125 μm, 0.35 NA) is 1.5 m long with β2 = 93 ps2

and γ = 0.007 (Wm)−1, respectively. The 0.2 m GF (5/125 μm, 0.24 NA), with β2 = − 12 ps2

The saturable absorption effect of SA is included by using the transfer function:

T = 1 − l<sup>0</sup> /[1 + P(τ)/Psat] (3)

Here, the split-step Fourier method is used to solve the NLSE. Simulation is started as the following procedure. With an initial white noise, the light is calculated in both temporal and spatial regions until a steady state is reached. The pump level and saturable effect are controlled through changing the values of Esat and Psat. When we take Psat = 3.5 kW and Esat = 3.4 nJ, the pulse's temporal and spectral evolution characteristics are shown in **Figure 2** [53]. In this case, the stable solution of pulse energy is 5 nJ. Detailed variations of pulse shape/width and spectral shape are clearly shown. In the DCF, owing to the combined effects of normal group velocity dispersion (GVD) and nonlinearity (NL), the pulse propagates with its duration increasing monotonically. The broadened pulse is then compressed by the SMF and GF with anomalous GVD. The pulse's spectrum has steep edges, and the bandwidth negligibly changes during the pulse circulating inside the cavity. However, the spectrum shape shows characteristic changes during the pulse evolution. The GF tends to amplify the spectrum's center more, and thus, the spectrum top becomes more arched. At the same time, the amplified pulse gives rise to increased self-phase modulation, thus leading to sharp edge peaks of the spectrum. Then, after being successively shaped by SA, DCF, GF, and SMF, the spectrum recovers its nearly flat-top shape. We can also see the advantages of the condensed-gain fiber model from the phase shift during the pulse evolution. As shown in **Figure 2(b)** [53], after passing through the three different kinds of fibers (DCF, GF, and SMF), very little phase shift is accumulated by the GF, which is finally compensated by both the DCF and the SMF.

To gain deeper insight into the intracavity pulsing dynamics, a qualitative illustration for 2 μm DSs is summarized in **Figure 3** [53], along with their 1 and 1.5 μm counterparts (insets) [54].

 is the unsaturated loss (take 0.7), P(τ) is the instantaneous power, and Psat is the saturation power. A critical factor for achieving DS is that spectral filtering is required to balance gain and loss. To that end, a 150-nm-bandwidth spectral filter (SF) is exerted on the SA.

where U(z, τ) is the envelope of the light field, z is the propagation coordinate, and τ is the time-delay parameter. The SMF (8.2/125 μm, 0.14 NA) has a length of 1.4 m, with β2 = −67 ps2

is the small-signal gain (here it is taken to be 30 dB), Epulse is the pulse energy, Esat is

2

<sup>2</sup> /Δω2

is the gain-center angular frequency, and Δω is the gain band-

*U*(*z*, *τ*) + g*U*(*z*, *τ*) (1)

] (2)

/

/km

/km and

**Figure 2.** Evolution dynamics of pulse duration (black triangles) and spectral bandwidth (red circles) through different elements inside the laser cavity (a) and temporal phase of the pulse after DCF (black solid), GF (red dashed), and SMF (blue dotted) (b) and temporal (c) and spectral profiles (d) of the pulse after successive elements [53].

**Figure 3.** Schematic diagram for the amplitude and phase balances in a 2 μm fiber laser (insets show the counterparts of 1 μm and 1.5 μm systems) [53].

As shown in the insets, in the case of 1 and 1.5 μm, the GFs (Yb-doped or Er-doped) have normal dispersion and introduce positive phase shift, which can be compensated (even if the shift is large) by the negative phase shift provided by the SMF. However, it is quite different in the 2 μm wavelength regime, where the GF (Tm-doped) has anomalous dispersion and, thus, negative phase shift. To achieve soliton mode-locking, normally dispersive fibers (DCF and SMF) are thus required to be integrated into the cavity. However, too large normal dispersion value (long fibers) will induce large phase shift and consequently pulse splitting. Therefore, small net normal dispersion (caused both by DCF and SMF) places a tolerant phase shift region for the GF (purple area). A longer GF will induce much more significant phase shift (red-dashed arrow) than that incurred by the DCF or SMF (yellow- or green-dashed arrow). In the phase limitation range, shorter GF (red arrow) has a larger slope and hence can achieve higher pulse energy. On the contrary, longer GF (orange arrow), due to its smaller slope, has to sacrifice a large part of amplitude to reduce its phase shift under the tolerable level. Therefore, short GF should be adopted to achieve high energy pulses from a cavity in the 2 μm spectral region.

Short lengths of single-cladding Tm-doped fiber (tens of centimeters) are chosen as the GF. Under stable mode-locking operation, maximum pulse energies with different GF lengths are shown in **Figure 4(b)** [53]. The experimental results clearly follow the trend of the simulation prediction; that is, the pulse energy increases quickly as the length of GF decreases. When the GF length is shortened to ~15 cm, pulse energy of ~5 nJ is achieved, and detailed laser

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With the 15 cm GF, stable CW mode-locking is self-started when pump power is increased to ~650 mW. Owing to the large output coupling ratio suppressing the intermediate transitions between the CW laser operation and the CW mode-locking regime [55], no Q-switching or Q-switched mode-locking is observed. The stable CW mode-locked operation maintains when pump power is increased up to the maximum 1 W available pump power. The maximum average output power of this 2 μm DS fiber laser is 158 mW. **Figure 5(a)** [53] shows the 2 μm DS pulse train at the maximum output. The repetition rate is ~32 MHz, giving a pulse

The laser spectrum, detected with a spectrometer (0.1 nm resolution), is shown in **Figure 5(b)**. The center wavelength is 1918 nm and the 3 dB bandwidth is 15 nm. Steep spectral edges indicate the typical characteristics of DSs [31, 32]. The radio-frequency (RF) spectrum (**Figure 5(c)**) has a signal-to-noise ratio of ~52 dB, showing that the mode-locking state is very stable. We also use an autocorrelator to measure the pulse characteristics at the maximum output, and the pulse shape (autocorrelation (AC) trace) directly outputted from the laser cavity is indicated in **Figure 5(d)**. The autocorrelation trace is fitted well by a Gaussian curve, giving a pulse duration of 16 ps. Therefore, the time-bandwidth product of the 2 μm DS pulse is calculated to be 18, which is highly chirped. For compressing this chirped pulse, we couple the output pulse directly into a ~25 m length of SMF-28 fiber. After dispersion compensation, the pulse is compressed to 579 fs (**Figure 5(e)**), and the time-bandwidth product reduces to 0.7.

This CGFML model can be readily extended to beyond 2 μm, e.g., mid-infrared fiber lasers to scale DS energy. According to this model, to achieve high-energy DSs, the GF length should

**Figure 4.** Simulated (circle dots) (a) and measured (asterisk dots) (b) maximum pulse energy under different lengths of

GF under the pump power of 1 W. The curves are exponential fittings [53].

characteristics are shown in the following.

energy of ~4.9 nJ.

Based on the above analysis, we propose the condensed GF (shortened to a small length while providing adequate gain at the same time) to scale the pulse energy of DSs in the 2 μm and mid-infrared spectral regions. Within the phase limitation range, a condensed GF has a large slope (**Figure 3** [53]) and thereby can provide high pulse energy.

To verify the advantages of CGFML in the 2 μm regime, we carry out simulations in a semiconductor saturable absorber mirror (SESAM) mode-locked fiber laser (**Figure 1** [53]). The simulated maximum output pulse energies with various GF lengths are indicated in **Figure 4(a)** [53]. It is clear that decreasing the GF length will dramatically increase the pulse energy. Shortening the GF to 0.2 m, as high as 11 nJ pulses, is achieved, which is much higher than the pulse energy of tradition solitons (usually less than 1 nJ). This thus confirms that CGFML is an effective route for generating high-energy soliton pulses in laser systems with anomalous dispersion GFs (shortening the anomalous dispersion GF as much as possible).
