1. Introduction

Interest in stable, all-fiber optical ultra-short laser pulse sources has fueled rapid development of rare-earth doped fiber lasers and amplifiers over the last a few decades. Ultra-short pulse fiber lasers are already a well-known alternative to the solid-state lasers due to its relatively lower cost, simplicity, and robustness. The primary challenge to generate high-quality laser pulses from fiber systems lays in the optimization of pulse dynamics, which has always been known driven by sophisticated manipulation of intracavity dispersion, nonlinearity, and dissipation. Empirically, different operation regimes are categorized to represent substantially different output characteristics in terms of pulse duration, pulse energy, and noise features [1, 2]. These regimes representing substantially different intracavity pulse dynamics include soliton mode-locking, stretched pulse soliton mode-locking, all-normal dispersion mode-locking, dissipative soliton mode-locking, self-similar mode-locking, etc [3–11]. Generally speaking, in fiber lasers the intracavity pulse dynamic can be manipulated by cutting/adding fiber segments with various dispersion, inserting a tunable dispersion compensation component (grating pairs or prism pairs), or using a spectral pulse shaping device, such as spatial light modulator (SLM) [12] and the digital micro-mirror device (DMD) [13, 14]. Spectral pulse shapers attract the most interests in recent years because they introduce the least cavity reconfiguration and provide a precise control of the pulse shaping. However, the side effects are also obvious like its high cost, the introduction of high complexity in construction and incompatibility with high pulse energy or high average power operation.

In this paper, we propose a new method to finely manipulate the intracavity pulse dynamics with a simple phase bias in a nonlinear amplifying loop mirror (NALM). Phase biases in NALM have been proposed to control the transmission of the fiber loops in Ref. [17]. Here, we found that the phase bias affects both the equivalent saturable absorbing and the spectral filtering effects of the cavity. Taking advantage of the broad gain spectrum of thulium-doped fiber [15, 16], central wavelength can be tuned over 50 nm range by manipulating the intracavity phase bias. Combining with the zero-crossing dispersion curve, the thulium-doped fiber laser operation can freely transit from soliton-like regime to stretch-pulse soliton and thence to dissipative soliton regimes without using bulky grating pairs or SLMs. The mechanism of the phase bias in the NALM on pulse dynamics manipulation has been discussed in detail.

2. Theoretical analysis

NALM-based fiber laser, typically configured as in Fig. 1, was chosen to realize fine pulse dynamics manipulation with a phase bias. The reason is as follows. Compared with the nonlinear polarization rotation (NPR) method [18], NALM-based laser is less sensitive to the environmental perturbation as it does not rely on the polarization for mode-locking. NALM is an artificial saturable absorber which shows much faster response to most real saturable absorber composed of different materials [19–22]. Additionally, the NALM-based cavity is sensitive to the phase offset in the loop [17], allowing finely manipulation of intracavity pulse dynamics through phase tuning.

2.1. Theoretical model

Theoretical analysis was performed before the experimental investigation. As the evolution of pulse in the NALM cavity may be so intriguing to explore, the detailed cavity properties are included in the model to achieve mode-locking instead of directly using an equivalent saturable absorber, with a cavity simplified as in Fig. 1. Initially, from port A_in with a unified field, we follow the circulation of the optical pulses inside the cavity and consider the possible action of the cavity components on the pulses. Concretely, the light propagating in the weakly birefringent optical fibers is described by the coupled cubic complex Ginzburg-Landau equation (CGLE, Eq. (1)), where the two orthogonal polarization components on slow and fast axes u and v are coupled. β₂ and β₃ are the group velocity dispersion (GVD) and the third order dispersion (TOD) parameter, and the nonlinear parameter γ is responsible for nonlinearity. Here the self-steepening and the Raman effects are omitted for simplicity. The gain effects are incorporated with the light amplification and gain bandwidth limitation in the gain medium, governed by g₀. T₂ is the dipole relaxation time and contributes to the parabolic gain profile. α represents fiber loss. $\mathrm{\Delta }\beta ={\beta }_{0x}-{\beta }_{0y}$ is related to the linear birefringence of the fiber. The problem becomes even more complicated when the bidirectional propagating fields in the NALM are included, which means the full vector equation consists of u_cw, u_ccw, v_cw and v_ccw, as well as the interaction in between, where the subscripts cw and ccw stand for the clockwise and counter-clockwise direction. In the strict sense, the clockwise field and the counter-clockwise field will interact in the form of cross-phase modulation (XPM) as well as gain sharing, making the problem so tough that iteration should be applied. Indeed, for pulses shorter than 5 ps, the interaction length between the two waves is shorter than 1 mm, negligible compared to the loop length. The asymmetrical position of the gain fiber in the loop also makes the gain sharing trivial. As a result, the XPM effect is taken into account only between the co-propagation fields.

 

Fig. 1 Typical cavity configuration of the NALM-based fiber laser. WDM: wavelength division multiplexer; Δϕ: phase bias device.

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Most of the time, the phase bias device consists of polarization changing components. We can present this stack of polarization changing elements in terms of Jones Matrix [23] when encountered. Suppose the angle between slow axis and horizontal line is α, then a quarter-wave-plate can be expressed using Eq. (2), where the superscript r represents the reverse direction. The equation indicates that the matrix for counter-propagating field are different. On the other hand, for a Faraday rotator, the matrix M_FR itself has a companion matrix, $M_{\text{FR}}^r$, shown in Eq. (3), that describes the effect of the same cell on a counter-propagating wave [23]. And Faraday rotator stands for the most commonly used non-reciprocal element.

For a 2 × 2 coupler with the coupling ratio of ρ, the transfer matrix is represented by Eq. (4). Coupler is also a reciprocal element.

2.2. Qualitative discussion

From the model above, we can cover all the fibers and the other components inside the cavity. However, it’s still impossible to deduce the analytical solution of the transfer function with gain effect fully considered. Here the gain effect is firstly ignored, which is the situation of a nonlinear optical loop mirror (NOLM). Multiplying the matrix of the output from the NOLM by that of the coupler, the transmission part of the result is the output of the cavity, while the cavity transmission modulation function can be derived from the reflection part, given as below:

 

Fig. 2 The transmission function of a NALM with respect to the phase difference, when different coupling ratios are compared.

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Typically, total phase difference δ between the counter-propagating light fields originates from three parts, the contribution from the birefringent effect ${\mathrm{\Delta }}_{\text{birefringent}}$, the nonlinear effect ${\mathrm{\Delta }}_{\text{nl}}$, and the phase bias ${\mathrm{\Delta }}_{\text{bias}}$, presented in Eq. (6). However, the birefringent phase offset has little to do with the propagation direction, determined mainly by the fiber itself, meaning that ${\mathrm{\Delta }}_{\text{birefringent}}$ is fixed and trivial. In consequence, the combination of ${\mathrm{\Delta }}_{\text{nl}}$ and ${\mathrm{\Delta }}_{\text{bias}}$ can be mapped two-dimensionally for fix-point examination, as shown in Fig. 3(a), in which the maximum nonlinear phase can be estimated through ${\mathrm{\Delta }}_{\text{nl}}=2\pi n_2{P}_{\text{peak}}L/\left(\lambda \cdot A_{\text{eff}}\right)$. The blue curves represent different phase offsets introduced by the phase bias device, while the red ones correspond to phase offsets provided by nonlinearity. The summation determines the transmission characteristic of the NALM.

 

Fig. 3 Illustration of (a) the fix point and (b) the wavelength-related transmission curve shift after the change of phase bias. The phase delay introduced by the polarization element stacks, with linear polarization incident light and elliptical polarized one considered, respectively. α and β represent the orientation of the QWPs.

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As a result, enough transmission coefficient is guaranteed even with relatively minor phase offset provided by birefringent and nonlinearity, and further optimization of the mode-locking state is available by adjusting the phase bias. What’s more, the nonlinearity-related phase can now be shifted to a position with a positive slope, indicating that the cavity is preferred to transmit high peak power field. With a positive slope, the transmission curve enables the cavity to pick a major pulse from noise fluctuation and to consequently preferentially amplify the pulse, which is equivalent to a fast saturable absorber with a recovery time comparable to the pulse duration. With part of the required phase offset provided by the phase bias device, the initialization of mode-locking will be much easier to trigger, even using a coupler with a coupling ratio $\rho =0.5$.

Apart from the fast saturable absorb effect, the phase bias is also responsible for the wavelength tuning and pulse dynamic manipulating, illustrated in Fig. 3. According to the analysis above, the transfer function of the NALM is dependent on the summation of phase offset mainly provided by the nonlinearity and the phase bias device, with birefringent contribution omitted. It’s reasonable to assume that the fiber laser achieves stable mode-locking at an original position, i.e. point A in Fig. 3(a). Adjusting the phase bias will deviate the phase shift provided by the phase bias from A on the contour. In favor of maximizing the transmission, the nonlinear phase shift changes accordingly, to a position where the total phase difference δ remains nearly unchanged, assuming point A^′. According to the equation for the nonlinear phase shift above, the peak intensity P_peak as well as the central wavelength λ may be tuned accordingly. Here two driving forces are responsible for approaching the new fix point A^′. First, the equivalent filter function shifts some amount on the wavelength axis right after the change of the phase bias, shown in Fig. 3(b). The shifted transmission curve tries to drive the central wavelength from λ₁

to λ₂. Besides, as the net cavity dispersion is usually sloped, central wavelength shifting will slightly change the net cavity dispersion. Consequently, the pulse peak power changes as it’s directly related to the pulse dynamics change. Afterward, a new fix point A^′ is approached, where the central wavelength is shifted, the pulse peak power is changed, the net cavity dispersion is tuned, and the pulse dynamic control is accomplished. We could imagine that cavity with larger TOD accumulated can exaggerate the influence of net cavity dispersion within the gain bandwidth, enabling much distinguishable dynamics manipulation without breaking the cavity configuration or breaking the mode-locking state.

Further, we would like to investigate the rules during phase bias adjusting. The phase bias device can be composed in several different configurations, and the core component is the non-reciprocal element, usually a Faraday rotator (FR). Here, a phase bias composed of a Faraday rotator sandwiched by two quarter-wave-plates is taken as an example. Two situations are considered. The incident lights into the polarization changing stacks are (1) horizontally polarized or (2) elliptically polarized. The resulted phase difference accumulation is calculated using the Jones matrix, shown in Figs. 3(c) and 3(d), respectively. The results are presented in $\left[0,2\pi \right)$ with colors, where α and β stand for the orientation of the two quarter-wave-plates, in the range of $\left[0,\pi \right]$. From the figures, when one wave-plate is fixed, the adjustment of the other one cannot always provide the necessary phase difference fully covering $\left[0,2\pi \right)$. In other words, the composition of Faraday rotator and two quarter-wave-plate can fulfill the completeness for specific phase difference contribution.

However, the direction to approach the extremum of transmission determines whether the feedback for saturable absorb effect is positive or negative. We always expect that the transmission increases when the phase shift caused by the nonlinearity increases. Specific to the situation in Fig. 3(d), without loss of generality, when the input light is elliptically polarized, two different locations are considered in Fig. 3(d). Forpoint A, the phase offset provided by the phase bias is a little bit larger than 0, or a little bit larger than $2\pi$ (these two are identical), the straight-forward way to maximize the transmission curve is to lower the nonlinear phase shift, which is equivalent to lower the peak power to get larger transmission, resulting in the cavity working in a CW state. So point A corresponds to the negative feedback. As a contrary, point B corresponds to the situation where the phase offset provided by the phase bias is a little less than $2\pi$. For a pulse, the higher the intensity is, the larger nonlinear phase shift it will accumulate. As a consequence, the operation point will be pushed closer to the maximum, just as the arrow indicates in Figs. 3(c) and 3(d), leading to positive feedback (the artificial saturable absorber type). Correspondingly, in Figs. 3(c) and 3(d), the blue regions stand for the negative feedback areas, while the red regions are on behalf of the positive feedback.

2.3. Intracavity dispersion management

The designed oscillator is configured as Fig. 4(c). It consists of a NALM connected with a fiber loop mirror retroreflector by a 2 × 2 broadband fused coupler with $\rho =0.5$. Without isolator in the loop, the light travels along both clockwise (cw) and counter-clockwise (ccw) directions, and the accumulated phase shift differs due to the asymmetric amplification effect. The phase bias is inserted to further adjust the phase difference, to control the intracavity dynamics, and to provide ameliorated self-starting mode-locking property in the experiment [17, 25].

 

Fig. 4 (a) Schematic of the NALM mode-locked Tm-fiber laser, and (b) the corresponding group dispersion delay in cavity as well as the normalized ASE spectrum (weak pump condition). FR: Faraday rotator; λ/4: quarter-wave plate; WDM: wavelength division multiplexer; NDF: normal dispersion fiber; C: collimator.

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As the potential laser sources into mid-infrared [26], a thulium-doped fiber laser is investigated theoretically and experimentally here, while the method can also be spread to some other wavelengths. To match between the gain and dispersion profile in the cavity, the amplified stimulated emission (ASE) spectrum of 0.15 m gain fiber is first measured as the orange line in Fig. 4(b), indicating the broadband gain spectrum. The sharp dips accompanied represent the strong absorption peaks of water [27]. To make the pulse dynamic change distinguishable, the dispersion curve is designed across zero in this region, shown as the blue trace in Fig. 4(b) (the detailed parameters of the fiber laser are given in Table 1). Normal dispersion fiber is added to shift the zero-dispersion point to around 1935 nm. And the accumulation of the third order dispersion (TOD) tilts the GDD curve to exhibit odd-symmetry in the vicinity of zero-crossing point [28]. We expect that combining with wavelength tuning, the net cavity dispersion (NCD) is shifted. The oscillator is forced to work in different mode-locking regimes subsequently, and thus the intracavity pulse dynamics manipulation is achieved.

Table 1. Cavity configuration

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3. Results

3.1. Numerical simulation

With the model considered, we are able to cover all the experimental devices intracavity. As the model itself is very complicated, we have to numerically solve it and find the eigenstate of the laser under certain operating conditions. In the numerical simulations, we start the calculation from port 1 of the coupler. After one round-trip circulation in the cavity, the calculated result is applied as the input of the next round of calculation until a steady state is achieved. It should be emphasized that we use the saturable absorb effect caused by NALM in the cavity, instead of an equivalent real saturable absorber. A unified pulse with picosecond width was used as input so that we are able to achieve a mode-locking state with several hundreds of round-trips in the cavity. The typical evolution of temporal as well as spectral pulse propagating in the cavity is shown below. As the fields on both polarization axes experience nearly the same evolution, here we present with the sum of two polarizations. As the same, in the NALM, the light intensities in both directions were added up.

To explore the pulse dynamics in different NCD regime, we scan the cavity central wavelength from 1960 nm to 1910 nm, where the NCD is tuned from −0.006 to 0.005 ps${ }^2$. Three typical evolution dynamics are shown in Fig. 5, where the (a, b and c) depict the temporal evolution, while (d, e and f) illustrate the evolution in the spectral domain. We can deduce by combining the evolution of pulse and spectrum that the pulse dynamic is manipulated with different NCD. The undistorted spectrum evolution in Fig. 5(d) is clear-cut evidence that the cavity works in the soliton-like regime for NCD −0.006 ps${ }^2$. The slight breathing process in the temporal domain is understandable with NDF considered. With central wavelength shifting shorter, the NCD approaches zero, the pulse breathing action in both temporal and spectral domain aggravates (Figs. 5(b) and 5(e)), and a wider spectrum can be acquired here. The NCD becomes positive when the central wavelength shifted even shorter. The much longer pulse duration in Fig. 5(c), the sharp edge of the spectrum, and the undistorted propagation of spectrum intracavity in Fig. 5(f) all indicate that the oscillator works in dissipative soliton regime. After all, the simulation shows that it’s reasonable to switch the mode-locking regime combining the zero-crossing dispersion with central wavelength tuning, without changing the cavity configuration.

 

Fig. 5 Simulation results of the pulse dynamic manipulation leading to mode-locking regime switching. The temporal (a, b, c) and spectral (d, e, f) evolution along the fiber, indicating three different mode-locking regimes.

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3.2. Experimental results

Benefited from the robust architecture and easy implementation of phase shift tuning, NALM mode-locked laser was experimentally investigated under the guidance of the theoretical design, and the detailed parameters are listed in Table 1. The result agrees quite well with the simulation. The oscillator can obtain versatile mode-locking states with a wide combination of the orientation of the two QWPs with a fundamental repetition rate of 60 MHz. The cavity achieves stable single pulse mode-locking with 800 mW pump, emitting about 30 mW average power. Figure 6 shows the typical spectra and the corresponding temporal autocorrelations of the oscillator in different states by only changing the orientation of the two QWPs. What should be highlighted is that all these states are repeatable and the laser remains mode-locking in the tuning process. The spectra are measured with optical spectrum analyzer with a resolution of 0.05 nm, with the results from simulation shaded in orange for comparison. The temporal interference autocorrelation trace (ACT) of the pulse was acquired using a home-built interference auto-correlator. The intensity ACT of the Fourier transform-limited pulse is superposed in orange. One can clearly tell that residual chirp still remains in the pulses. Furthermore, as all of the fibers used in the cavity have positive TOD, the small wave packets beside the main peak testify that the TOD in the cavity is non-negligible, most evident in the first state.

 

Fig. 6 Typical spectra (a, c, e) and the corresponding interference ACT (b, d, f) of the oscillator in different mode-locking regimes. The orange regions in (a, c, e) are the spectra from simulation and those in (b, d, f) correspond to the calculated intensity ACT of the transform-limited pulse.

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The spectrum in Fig. 6(a) is about 35 nm in FWHM, centered at 1964 nm. From the spectrum shape and the symmetrical Kelly sidebands [3], the oscillator works in the soliton-like regime. The interference ACT (Fig. 6(b)) shows a pulse duration of 150 fs, slightly longer than the 120 fs transform-limited pulse duration. Shifting the central wavelength shorter by slightly rotating the orientation of the QWPs, spectrum of Fig. 6(c) is obtained, standing for stretched-pulse soliton. Stretched-pulse is a widely-used technique pursuing higher pulse energy and shorter pulse duration. Consequently, the spectrum obtained can be as wide as 77 nm in FWHM. The corresponding ACT is shown in Fig. 6(d), measured 114 fs with chirp, while the transform-limited pulse can be slightly shorter than 100 fs. The pedestal beside the central peak in ACT clearly verifies the residual GDD. Further adjusting the orientation of the wave plates and shifting the central wavelength to 1910 nm, a spectrum of Fig. 6(e) can be obtained, indicating that the oscillator works in the dissipative soliton regime, with sharp edges and much longer pulse duration (Fig. 6(f)). Although there exists no filter in the cavity, the transmission curve of NALM performs like a broadband filter here [29]. The corresponding transform-limited pulse width is 193 fs, while the measured pulse is highly chirped, nearly 500 fs, which is another evidence that the cavity works in dissipative soliton regime.

4. Conclusion

In summary, we presented intracavity dynamics controlling in a NALM Tm-doped fiber laser, with the help of a phase bias device consisting of non-reciprocal Faraday rotator and wave-plates. The engineered dispersion curve in the cavity combined with the central wavelength tunability makes it possible for the cavity to operate in soliton-like, stretched-pulse soliton, and dissipative soliton states. The functional phase bias device is investigated in detail. And the experiment results agree well with the simulation indication. Although the control range is limited compared to the grating pairs, its easy implementation and high energy compatibility make it suitable in some specific cases. Moreover, one can finely adjust the pulse intracavity dynamics and get the most appropriate operation state for further application, without changing the cavity configuration.