1. Introduction

Passively mode-locked lasers have become an increasingly reliable tool empowering an expanding field of research and applications. As the ultrashort pulse lasers find their way out of laboratories, they are expected to operate under more extreme conditions. All-PM-fiber lasers offer an inherent protection from such environmental influences. Avoiding free space optics can potentially further increase this stability, however, it also increases the demand for improved initial layout robustness as flexibility through later adjustment is discarded. In terms of scalability of the technology, all of the required components should be available in a preferably constant quality and without long-term degradation effects.

During the past decades, a plethora of different mode-locking mechanisms for ultrashort pulse fiber lasers have been demonstrated, which comprise nonlinear polarization rotation (NPR) [1], various material saturable absorbers, such as semiconductor saturable absorber mirrors (SESAMs) [2,3], graphene [4–6], carbon-nanotubes [7,8], black phosphorus [9,10], topological insulators [11,12], or transition metal dichalcogenides [13,14], as well as hybrid versions thereof [15–17]. However, all of these mode-locking techniques currently exhibit dissatisfactory properties, concerning either environmental stability (e.g. NPR is typically implemented using non-PM fibers), large-scale reproducibility, or degradation issues [18] which impede production upscaling of turn-key systems. In search of alternatives, NALMs [19] have often been overlooked as they were known for an unreliable starting behaviour. Nonreciprocal phase-biasing or dispersion engineering of the NALM, however, proved to be elegant ways to circumvent this problem [20–22].

In 2001, Cairns et al. suggested a different approach using a 3×3-coupler (cf. Figure 1) instead of a 2×2-coupler for the construction of the NALM [23]. Lately, this proposal has found renewed interest as an Er-based soliton fiber laser [24] and an Yb-based all-normal dispersion fiber laser [25] have been demonstrated. The significant difference compared to the conventional NALM lies in the phase change experienced by the coupled electromagnetic fields, which is $\pi /2$ for the ideal 2×2-coupler and $2\pi /3$ for an ideal, even 3×3-coupler [26,27]. As a result, mode-locked operation can be initiated more easily and thus reliably [24] without the need for additional measures such as nonreciprocal phase-biasing of the NALM. In this article, we take on this approach. By performing a theoretical examination we come up with novel realizations, comprising a Tm-oscillator mode-locked by a 3×3-coupler-NALM and first demonstrations of dispersion adjusted oscillators mode-locked by a 3×3-coupler-NALM, which exhibit excellent performance.

 

Fig. 1. Scheme of a NALM based on a 3×3-coupler. An input pulse $A_0$ enters the coupler on port 1 (P1) and is evenly split into three partial pulses $A_1$, $A_2$, and $A_3$ at the three ports 4, 5, and 6, respectively. $A_2$ and $A_3$ counter-propagate in the fiber loop and acquire different nonlinear phases, due to the asymmetrically placed active fiber (red fiber). Back at the coupler, $A_2$ and $A_3$ interfere and generate the pulses $A_\textrm{Port1}$, $A_\textrm{Port2}$, and $A_\textrm{Port3}$ at the ports 1, 2, and 3, respectively.

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2. Model of a NALM based on a 3×3-coupler

The general situation of an input pulse interacting with the NALM generates a vast parameter space. Fiber dispersion, nonlinear phase shift, pulse shape, initial pulse chirp, and interdependencies thereof can influence the resulting transmission behaviour. The emerging time and frequency response of the NALM as artificial saturable absorber can thus only be determined by a rigorous numerical treatment, where the partial pulses are propagated through the fibers of the NALM and where the NALM is part of the complete cavity.

Here, a simpler model as compared to a more elaborate split-step Fourier method (SSFM) approach is applied, which allows for the identification of important characteristics. In the model, we want to mimic the situation in Fig. 1, where port 1 is the input and port 3 is the output of the saturable absorber. Therefore, a Gaussian pulse envelope $A_0(t)$ with 200 fs FWHM pulse duration which enters the NALM on port 1 is chosen as starting point. The pulse is split into three partial pulses of equal amplitude. The first pulse ($A_1$) exits the NALM on port 4, whereas the two coupled partial pulses on port 5 ($A_2$) and port 6 ($A_3$) both experience a phase shift of $2\pi /3$ due to the coupler, i.e. there is no resulting phase difference between these two pulses at this point. Afterwards, pulse $A_3$ experiences a nonlinear phase shift proportional to the normalized pulse envelope squared [28]

This mimics the effect of self phase modulation (SPM) for $A_3$ during propagation in the loop, whereas pulse $A_2$ is simplistically left unchanged $A_2(t) = A_0(t)/\sqrt {3}$. Thereby, the effect of a differential nonlinear phase, caused by the asymmetrically placed gain fiber is modeled. The parameter $\kappa$ represents a direct measure for the nonlinear phase difference $\Delta \phi _\textrm{NL}$ between the two pulses and is varied to study the NALM characteristics. Both pulses are now distributed back to the ports 1, 2, and 3. Thereby only the coupled fields receive a phase shift of $2\pi /3$. The resulting time signals are obtained by a coherent summation of the pulses $A_2$ and $A_3$:

Figure 2 shows results acquired by the model. In contrast to the 2×2-coupler-NALM, a non-vanishing relative transmission to the output port (port 3) is obtained for $\Delta \phi _\textrm{NL}=0$, accompanied by a non-vanishing slope [see Fig. 2(a)]. This benefits the pulse build-up from noise and ultimately leads to a reduced pump power required for starting the mode-locked operation compared to the 2×2-coupler-NALM configuration [24]. Another notable result is that the maximum relative transmission to port 3 (39 % of the input pulse energy) is not obtained at $\Delta \phi _\textrm{NL}=2\pi /3\approx {2.09}$ rad (where constructive interference for the pulse peak prevails), but at 2.53 rad. This reveals that overall maximum constructive interference at port 3 is obtained when the saturable absorber is moderately over-driven, i.e. constructive interference for the pulse peak has already declined and advanced towards the pulse shoulders. This causes a slight flattening of the pulse peak compared to the original pulse [see Fig. 2(c)]. Nevertheless, the pulse duration is reduced by 6 % due to the NALM, as the wings, which experience less nonlinear phase shift, do not interfere constructively at port 3. Overall, port 3 provides a pulse width reduction for $\Delta \phi _\textrm{NL}$ between 0 and 2.78 rad [see Fig. 2(b)]. The modulation depth is $\Delta q = 1- \min {\left (P_\textrm{Port3}\right )}/\max {\left (P_\textrm{Port3}\right )}\approx$ 71 %. Even though ports 1 and 2 provide larger pulse duration reductions and modulation depths, they are not suitable as saturable absorber exits since they also exhibit a pulse duration increase and negative transmission slope for lower $\Delta \phi _\textrm{NL}$ which will inhibit the pulse build up [29].

 

Fig. 2. Investigation of the pulse transmission characteristics of the 3×3-coupler in the model. (a) Relative transmission to port 1, 2, and 3 as a function of the applied nonlinear phase difference $\Delta \phi _\textrm{NL}$ between the split pulses. The vertical dashed line indicates the position of maximum relative transmission to port 3. (b) Pulse width change of the pulses at port 1, 2, and 3 as a function of $\Delta \phi _\textrm{NL}$. The vertical dashed line indicates the position of maximum relative transmission to port 3. (c) Time profiles of the original pulse and the recombined pulses at port 1, 2, and 3 for the case of maximum transmission to port 3, i.e. $\Delta \phi _\textrm{NL} = {2.53}$ rad. The original pulse amplitude is scaled down to the amplitude at port 3 for better comparison. (d)+(e) Corresponding power spectral densities (PSDs) for different initial chirps: (d) GDD $=$ 0 ps², (un-)chirped FWHM pulse duration 200 fs (e) GDD $=$ 0.1 ps ², chirped FWHM pulse duration 1.4 ps. The original spectrum is scaled down to the spectrum at port 3 for better comparison. (f) Relative change of the FWHM bandwidth of the spectra at port 3 at maximum transmission in dependence of the initial GDD.

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The behaviour discussed so far remains unchanged when adding a linear initial chirp to the pulse, as this does not alter the Gaussian pulse shape. However, the spectra of the coupled pulses significantly change when the input pulse is initially chirped. The spectra in Fig. 2(d), with long tails at port 3, a dipped spectrum at port 2, and a narrow spectrum with shoulders at port 1 result from an unchirped input pulse. Adding a positive group delay dispersion (GDD) of 0.1 ps² (up-chirped pulse duration 1.4 ps), the spectra change to a strongly dipped spectrum now observed at port 1 and a narrow spectrum with shoulders at port 2 [see Fig. 2(e)]. Overall, these spectra strongly resemble the temporal evolution of the corresponding pulses [compare to Fig. 2(c)]. This time-domain to frequency-domain correspondence is caused by the linear chirp of the pulse which approximately linearly maps every point in time of the pulse to a corresponding instantaneous frequency [28]. For moderately positive values of the initial GDD, this results in a broadening of the spectrum at port 3 compared to the initial spectrum, with a maximum broadening of 60 % at 0.017 ps² (up-chirped pulse duration 310 fs) [see Fig. 2(f)]. On the other hand, negative values of the initial GDD below −0.01 ps² will lead to a reduction of the spectral width.

Finally, the influence of the coupler’s parameters on the transmission to port 3 is studied. In general, the 3×3-coupler does not need to be fully symmetric in terms of the transmitted and coupled fields. However, we restrict the analysis to a geometrically symmetric cross section of the coupler, i.e. the three fibers inside the coupler are arranged as in Fig. 3(a). For this symmetric arrangement, the coupling coefficient $C$, for example, from port 1 to port 5 is equal to the one from port 1 to port 6 [30]. An input pulse $A_0$ on port 1 is thus divided to ports 4, 5, and 6 according to

If furthermore energy conservation is assumed, the transmission/coupling coefficients ($T$ and $C$) and the coupler phase shift $\phi _\textrm{C}$ are connected by [30]

Figure 3(b) shows $T$ and $C$ in dependence of $\phi _\textrm{C}$. For the case of a fully symmetric coupler $C = T = 1/\sqrt {3}$ and $\phi _\textrm{C}=2\pi /3$ are restored. The phase shift $\phi _\textrm{C}$ is bounded by $\pi /2$ and $\pi$, for weak ($C \rightarrow 0$) and maximum ($C=2T=2/3$) coupling between the fibers, respectively. In contrast to the 2×2-coupler, 100%-coupling is not possible with this type of coupler and a minimum of $1/3^2 \approx$ 11% of the launched power stays in the input fiber even for maximum coupling.

 

Fig. 3. Investigation of the pulse transmission of the 3×3-coupler for varying coupler parameters in the model. (a) Scheme of the considered 3×3-coupler. (b) Relationship between the coupler phase shift $\phi _\textrm{C}$ and the transmission / coupling coefficients $T$ and $C$ of the coupler. (c) Transmission to port 3 as a function of the applied nonlinear phase difference $\Delta \phi _\textrm{NL}$ between the split pulses. Different lines correspond to different coupler phase shifts $\phi _\textrm{C}$, ranging from $\pi /2$ (brightest) to $\pi$ (darkest) in steps of $0.02\pi$. The transmission curve of a fully symmetric coupler is plotted in red for comparison.

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Intuitively, one could assume that a lower transmission coefficient $T$ might increase the efficiency of the 3×3-coupler-NALM, as less power is lost to port 4. However, on the return path also less power is transmitted from port 6 to port 3. Figure 3(c) shows the transmission curves of the 3×3-coupler-NALM to port 3 for varying coupler parameters in the model. For maximum coupling (black curve, $\phi _\textrm{C}=\pi$) the NALM has a vanishing transmission slope to port 3 at $\Delta \phi _\textrm{NL} =$ 0 rad and a maximum transmission of 36% at $\Delta \phi _\textrm{NL} \approx$ 3.7 rad, which refutes the previous conjecture. A weaker coupling (brighter curves) results in larger transmission and increasing slope for $\Delta \phi _\textrm{NL} =$ 0 rad and a shift of the transmission maximum towards lower $\Delta \phi _\textrm{NL}$. The maximum transmission of 40% is obtained for $\phi _\textrm{C}\approx$ 2.26 rad at $\Delta \phi _\textrm{NL}\approx$ 2.71 rad. Further decreasing the phase bias then again results in a shift of the transmission maximum’s position towards even lower $\Delta \phi _\textrm{NL}$ and a lower maximum transmission. For $C=0$ the transmission to port 3 vanishes independent of the nonlinear phase shift, as the power is completely transmitted to port 4.

In summary, a shift of the coupling parameters away from the symmetric parameters ($C=T$) is not expected to increase the performance of the 3×3-coupler-NALM noteworthily. Overall, the maximum transmission can be increased from 39 to 40%, however at the cost of an increased nonlinear phase shift of 2.71 rad instead of 2.53 rad within the NALM.

3. Experimental results - Yb-fiber laser

In the following sections, we present experimental results of Ytterbium (Yb) and Thulium (Tm) based oscillators employing the 3×3-coupler-NALM. The experimental setup of the Yb-fiber laser is shown in Fig. 4(a). The oscillator employs a chirped fiber Bragg grating (cFBG) with a FWHM of 15 nm centered at 1030.5 nm. A second amplifier outside the NALM is used in double-pass configuration in order to ease the starting process and to compensate for the losses caused by the cFBG. It is core-pumped through the cFBG via an additional wavelength division multiplexer (WDM) at a wavelength of 976 nm. The cFBG partially compensates the normal dispersion of the fiber, such that the overall resonator dispersion is still positive (0.01 ps²). In order to also analyse the reverse signal coming from port 1, a 4-port optical circulator is used. The oscillator is started in the multi-pulse regime by increasing the pump power for both amplifiers to a level of 200 mW. Single pulse operation is obtained by decreasing both pump powers to $\approx \,$90 mW each. The output spectra of the NALM during single pulse operation on port 1 (via the optical circulator), port 2, and behind the cFBG (as port 3 is not directly accessible in this setup) are shown in Fig. 4(b). Qualitatively, the output spectra show good agreement with the expected spectra from the model for a chirped input pulse [see Fig. 2(e)], and also with the spectra obtained from a full-cavity SSFM simulation [28] [see Fig. 4(c)], in which the effects due to dispersion and SPM are included for every fiber section and both propagation directions in the NALM. The tilting of the spectra towards shorter wavelengths is likely caused by the negative third order dispersion (TOD) induced by the cFBG, i.e. excluding all TOD from the simulation results in symmetric output spectra, including only the positive TOD from the fiber results in spectra tilted towards longer wavelengths.

 

Fig. 4. (a) Scheme of an all-PM Yb-fiber laser incorporating the 3×3-coupler-NALM. (b) Experimentally obtained output spectra at port 1 via the circulator, port 2, and behind the cFBG which is an amplified version of the output at port 3. (c) Calculated spectra at port 1, 2, and 3 resulting from a full-cavity SSFM simulation. (d) Evolution of the spectral FWHM (black line) and the pulse duration (red line) in the simulation. The starting point $z=$ 0 m is chosen to be at the cFBG. The sections marked with blue/red boxes correspond to passive/active fibers, respectively. Black lines in the blue boxes mark the beginning and ending of the NALM. Only the values for the counter-clockwise propagating pulse are shown. (e) Measured spectrum at the cFBG together with parabolic and Gaussian fits. (f) Misfit parameter of the pulse during propagation. (g) Measured autocorrelation trace (ACT) of the compressed output pulses of the Yb-fiber laser at the minimum obtained pulse duration.

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Fig. 4(d) shows the simulated evolution of the spectral width and the pulse duration in the course of one round trip. The monotonic increase of the pulse duration, which is reset by the cFBG, implies that the oscillator does not operate in the dispersion-managed soliton regime [31], as the pulse is permanently up-chirped. The parabolic spectral shape of the output at the cFBG [see Fig. 4(e)] along with the large chirp (measured uncompressed pulse duration 9.2 ps), however, indicate a parabolic temporal profile enabled by self-similar pulse evolution [32,33]. Continuously fitting the simulated temporal intensity profile to a parabola $p(t)$ and calculating the misfit parameter [34]

For the spectra in Fig. 4, a comparison with the simulation suggests that the nonlinear phase difference $\Delta \phi _\textrm{NL}$ between the counter-propagating pulses is 2.7 rad. However, tuning both pump powers allows for a wide range of tolerable nonlinear phase differences, from approximately 1.8 to 3.4 rad. When compressing the output signals with a Treacy type grating compressor, the shortest pulse duration obtained in this configuration is 125 fs, assuming the same pulse shape as for the Fourier limited case [see Fig. 4(g)]. All of the output ports allow for measured autocorrelation function widths below 200 fs after pulse compression. The maximum pulse energy (1.2 nJ) in the single pulse regime can be extracted from the cFBG-port.

4. Experimental results - Tm-fiber laser

In a further setup, an all-PM Tm-fiber laser mode-locked by a 3×3-coupler-NALM is realized [see Fig. 5(a)]. The pump source is a home-build Er/Yb-doped double-clad amplifier, seeded by a 1565 nm single-mode diode, which delivers up to 2 W of pump power. Due to the anomalous fiber dispersion at $\approx \,$2 µm wavelength and the lack of dispersion compensation, the oscillator operates in the conventional soliton regime (overall resonator dispersion: −0.58 ps² at 2000 nm) and has a repetition rate of 30.3 MHz. The oscillator is started in the multi-pulse regime by increasing the pump power to a level of 1.5 W. Single pulse operation is then obtained by decreasing the pump power to 790 mW and is preserved down to a pump power of 570 mW. The central wavelength during single pulse operation varies from 2010 to 1994 nm depending on the applied pump power. The corresponding measured output spectra of the open ports 2 and 4 for the lower pump power limits of single pulse operation are shown in Fig. 5(b). The output at port 2 shows a slight dip at the central wavelength, as predicted by the model for the case of an unchirped input pulse [see Fig. 2(d)]. Interestingly, the Kelly-sidebands [36] at the NALM output port 2 show alternating peaks and dips for different sideband orders. This behaviour can be understood when employing a full-cavity SSFM simulation again. The nonlinear phase shift generated by the counter-clockwise propagating pulse is greater than $\pi \,$rad, which leads to destructive interference between the soliton and the dispersive waves [see dips in the blue spectrum in Fig. 5(c)]. The resulting spectrum at port 3 is nearly free of spectral sidebands [see Fig. 5(d)], as the dispersive waves experience negligible SPM and are consequently inferfered out of the resonator via port 1 and 2. This showcases impressively the pulse cleaning effect due to the NALM. As the combined pulse energy at port 3 (580 pJ) is now larger than the 430 pJ which are allowed by the area theorem for the corresponding pulse duration of 540 fs, parts of the soliton energy are again shed into dispersive waves during propagation in the unidirectional part of the cavity, generating the spectrum observed at port 4. The corresponding pulse duration is reduced to 480 fs.

 

Fig. 5. (a) Scheme of an all-PM Tm-soliton fiber laser incorporating the 3×3-coupler-NALM. (b) Measured output spectra of the soliton Tm-fiber laser mode-locked by a 3×3-coupler-NALM at lower pump threshold of single pulse operation. (c) Simulated spectra of the counter-propagating pulses immediately before recombination at the 3×3-coupler. (d) Simulated spectra immediately after the combination at the 3×3-coupler. (e) Output spectra of a dispersion-managed soliton (DMS) Tm-fiber laser mode-locked by a 3×3-coupler-NALM. The scheme of the setup is similar to Fig. 4(a) except for the Yb-SC-fibers replaced by Tm-SC-fibers, the 4-port circulator replaced by a 3-port circulator, and an additional normally dispersive fiber. (f) Autocorrelation trace (ACT) of the compressed output pulses of the DMS oscillator with the corresponding spectrum in the inset.

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As a shorter pulse duration and a stabilized central wavelength are desired, a cFBG (FWHM of 35 nm centered at 1950 nm) and a normally dispersive fiber for distributed dispersion compensation are added to the resonator, resulting in a setup similar to Fig. 4(a), also including a second amplifier before the cFBG. The increased fiber length reduces the repetition rate to 23.4 MHz. The oscillator is again started in a multi-pulse regime at a pump power level of 700 mW for both amplifiers. Single pulse operation is obtained by decreasing the pump power to 300 / 450 mW for the NALM / double-pass Tm-fiber, respectively. The Tm-oscillator now operates in the dispersion-managed soliton regime (overall resonator dispersion 0.05 ps²), resulting in output spectra that offer bandwidths larger than 30 nm at port 4 [see Fig. 5(e)]. The pulses can be dechirped simply by splicing an appropriate length of passive fiber to the output. The resulting pulse duration is 215 fs, assuming the same pulse shape as for the Fourier limited case [see Fig. 5(f) for ACT].

5. Conclusions

In this article, the versatility of the 3×3-coupler-NALM as a mode-locking device is put forward. A model has been presented, which allows for the identification of numerous relevant features of the saturable absorber despite the model’s simplicity. Subsequently, exemplary fiber oscillators which employ the NALM have been demonstrated in the passive self-similar, the soliton, and the dispersion-managed soliton regime. Additional examinations using split-step Fourier method simulations revealed intricacies of the mode-locker which will help in further improving the concept. The spectral shapes offered by the oscillators are attractive for a range of applications including seed sources for nonlinear chirped pulse amplification.