1. INTRODUCTION

Mode-locked lasers based on single-mode fibers (SMFs) have revolutionized many applications spanning from microscopy, micromachining, and spectroscopy to time keeping [1–4]. Although single-mode femtosecond fiber lasers enjoy the advantages of a compact form factor, alignment free operation, and easy maintenance, pulse energy scaling is limited by the tight mode confinement and strong nonlinearity. Therefore, spatiotemporal mode-locking (STML) in multi-mode fiber (MMF) lasers has been proposed and demonstrated to leverage the large modal content and mode area for further energy scaling [5–11]. Besides boosting pulse energy, the introduction of rich spatial mode content brings a new dimension for the study of dissipative soliton physics. Due to both practical and fundamental interests, STML has attracted considerable attention [12–24]. For example, an attractor dissection theory has been proposed to understand the mode-locking mechanism for STML recently [22]. Moreover, a combination of spatial sampling and a dispersive Fourier transform (DFT) technique has been developed to discern the buildup dynamics of spatiotemporal dissipative solitons [23,24]. Stable mode-locking of spatiotemporal dissipative solitons needs the intermode dispersion (on group velocity) to be balanced by nonlinearity or dissipation [6]. Due to the relatively weak intermode dispersion, graded-index (GRIN) MMFs were first used for STML [6,13,14,18]. We recently showed that larger intermode dispersion from a step-index (STIN) MMF can also be counteracted for STML [17]. The intermode dispersion between the LP01 mode and LP11 mode of the Yb-doped STIN MMF is 1 ps/m at 1030 nm in that work, about an order of magnitude higher than GRIN MMFs [6,14,24–27]. Both STIN and GRIN MMFs were used to form a hybrid laser cavity, and free space optics were employed to use nonlinear polarization rotation (NPR) for mode-locking in our previous work [17]. More recently, STML in an all-fiber laser consisting of STIN, GRIN MMFs, and a short piece of SMF has been reported [21]. Although STIN MMFs were used, the measured autocorrelation (AC) traces have no additional peaks, suggesting that mode-resolved pulses remain bound in the hybrid cavity as in GRIN MMF lasers [17,21].

These reported lasers used either GRIN MMF solely or together with STIN MMF for mode-locking. Whether an all-STIN MMF laser can be used for STML remains an open question. Associated with this question, how large intermode dispersion can be tolerated for STML also needs to be investigated. Moreover, GRIN MMFs have nearly evenly spaced wave vectors (propagation constants) for transverse modes that can enable efficient intermode four-wave mixing (FWM) [28]. Nevertheless, the propagation constants for different modes in STIN MMFs differ largely and are not evenly spaced. How will this propagation constant difference (intermode phase velocity dispersion) impact STML dynamics in MMF lasers has not been studied yet, to our knowledge. Furthermore, all-STIN MMF lasers can have larger mode areas than GRIN MMF-based lasers, thus having the potential to reach higher pulse energy [7].

 

Fig. 1. Mode-locking in an all-step-index multi-mode fiber laser. (a) Schematic of the all-step-index (STIN) multi-mode fiber (MMF) spatiotemporally mode-locked laser. Lens1 and Lens2, lenses; HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarization beam splitter; ISO, isolator; Filter, spectral filter; SC, spatial coupling; M1, M2, M3, mirrors. (b) Comparison of spatiotemporal pulse propagation in GRIN and STIN MMF-based fiber lasers. For GRIN MMFs, the walk-off for mode-resolved pulses is relatively small, and the Kerr effect can be important to bind them. The large intermode dispersion of STIN MMFs cannot be balanced by the Kerr effect, and mode-resolved pulses walk off significantly in the fiber. SC and mother–child coupling [see also Section 4.B and Fig. 4(d)], responsible for synchronizing the spatiotemporal pulses again for self-consistent propagation in the laser with very large intermode dispersion. The inset shows a comparison of the fundamental modes among single-mode fiber (Nufern XP-1060), GRIN MMF (YOFC OM4), and STIN MMF (Nufern LMA-GDF-25/250-M).

Download Full Size | PDF

In this paper, we report experimentally and numerically on STML in an all-STIN MMF laser. Stable STML with multi-transverse modes sharing a single repetition rate ${f_r}$ was experimentally observed. Distinct from previous reports on STML, our laser operates in a new regime with mode-resolved pulses walking off significantly. The AC measurements show that mode-resolved pulses are no longer bound by the Kerr effect in the fibers, and their walk-off generally follows net intermode dispersion. Simulations confirm this large walk-off and reveal that it is the spatial coupling (SC) [i.e., coupling from free space into the MMF; see blue arrow in Fig. 1(a)] that compensates for the ps-scale walk-off for self-consistent propagation. A mother–child mode coupling picture is proposed to elucidate this mechanism [see Fig. 1(b)]. In principle, this mechanism can enable stable STML with infinitely large intermode dispersion for STML states with a single ${f_r}$ if there is only a single degenerate mother mode. The influence of intermode phase velocity dispersion is also analyzed in simulations. Our work is an important step towards understanding spatiotemporal dissipative soliton dynamics in the very large intermode dispersion limit and contributes to tailoring ultrafast spatiotemporal waves in multi-mode photonic systems [10].

2. DESIGN OF ALL-STIN MMF LASER

The layout for our all-STIN MMF laser is shown in Fig. 1(a), and mode-locking was initiated by NPR. The laser consisted of a 1.5 m long STIN Yb-doped fiber (Nufern LMA-YDF-25/250-VIII) and a 2.4 m STIN passive fiber (Nufern LMA-GDF-25/250-M), i.e., total fiber length $L = {3.9}\;{\rm{m}}$. Both the gain fiber and passive fiber have a 25 µm core diameter and ${\rm{NA}} = {0.06}$, supporting six modes. These modes belong to the LP01 mode (mode 1), LP11 mode (mode 2 and mode 3), LP21 mode (mode 4 and mode 5), and LP02 mode (mode 6); see Fig. 1(b). The two MMFs were spliced without offset [gray arrow in Fig. 1(a)]; thus, there is no strong intermode coupling in the splicing. A relatively small mode number was chosen as the first step to investigate mode-locking dynamics in all-STIN MMF lasers.

Intermode dispersion (meaning intermode dispersion on group velocity in the following, if not specially noted) is represented by a vector $\delta \beta _1^{(m)} (m = 1, 2, \ldots ,6)$, which is the difference of the inverse of group velocity with respect to mode 1. $\delta \beta _1^{(m)}$ of the used STIN MMFs is (0, 0.6, 0.6, 0.9, 0.9, 0) ps/m. The net intermode dispersion of the current laser (evaluated between LP01 and LP11 modes at 1060 nm) is about three times that of our previous work [17]. An illustration of the mode-resolved pulse propagation in the all-STIN MMF laser is shown in Fig. 1(b). Nonlinear interaction between different modes via cross-phase modulation (XPM) among modes may counteract the intermode dispersion for low intermode dispersion GRIN MMFs and sustain the mode-resolved pulses as an entity [22,29,30]. Previously reported STML in MMF lasers operated in this negligible walk-off regime. In the all-STIN MMF laser, the Kerr effect cannot bind them together, and they walk off strongly. Since walk-off is not balanced by nonlinearity, dissipation including a saturable absorber (SA), spectral filter (SF), and SC should be responsible for compensating for the large walk-off. In particular, a spatial filter (or SC in our case) has been noted to be critical to counteracting weak walk-off in GRIN MMF lasers [6,22]. Here, we show that SC can work to counteract very large pulse walk-off [ps-scale walk-off; see Fig. 1(b) and Section 4.B].

As an aside, the propagation constants with respect to mode 1 for the used MMFs is represented by a vector $\delta \beta _0^{(m)} = ({0,{-}2.1,{-}2.1,{-}4.7,{-}4.7,{-}5.3})\;{{\rm{mm}}^{- 1}}$. This $\delta \beta _0^{(m)}$ will inhibit inter-LP-mode FWM (see simulations below). For the mode area, the fundamental mode area of our STIN MMFs is about 3.5 times that of typical GRIN MMF (YOFC OM4) and 5.5 times that of SMF (Nufern 1060-XP); see a comparison in Fig. 1(b). Therefore, the nonlinearity is relatively weak, which can be beneficial to boost pulse energy. More details of the laser and mode-locking diagnosis methods can be found in Supplement 1 Section 1.

 

Fig. 2. Spatiotemporal mode-locking with a single repetition rate. (a) RF spectrum of the mode-locking state shows a single RF tone. The RF spectra labeled by samples 1, 2, 3, and 4 were measured with a pinhole sampling four positions indicated in (b). The inset shows an example of the measured pulse train with a period of 30 ns. (b) The output beam profile is LP11-mode-like but is asymmetric. (c) Optical spectrum of the output pulse. The inset is the Fourier transform of the spectrum, namely, its first-order autocorrelation (AC). This first-order AC also shows additional peaks. As a feature of spatiotemporal mode-locking, the beam profile varies with spectral filtering. The center wavelengths of the three filters are 1061,1063, and 1067 nm. Scale bar: 500 µm. (d) The bottom purple spectrum was measured without spatial sampling. The spectra labeled by samples 1, 2, 3, and 4 were measured with a pinhole sampling four positions indicated in (b). (e) Second-order intensity AC traces of the whole pulse and pulses sampled at different positions. The traces exhibit additional peaks at ${\pm}{1.1}$ and ${\pm}{2.2}\;{\rm{ps}}$.

Download Full Size | PDF

3. EXPERIMENTAL OBSERVATION OF STML IN THE ALL-STIN MMF LASER

A. Spatiotemporal Characterization of the STML State

Self-starting mode-locking with a pulse energy of 4.8 nJ was obtained by increasing the pump power to 4.2 W with an appropriate orientation of the wave plates. The pulse energy is similar to typical single-mode all-normal-dispersion fiber lasers at 1 µm [31]); higher pulse energy may be possible by reducing the cavity loss. Only a single RF tone at ${f_r} = 33.1\;{\rm{MHz}}$ was observed [Fig. 2(a)]. The corresponding mode-locked pulse train with a period of 30 ns is shown as the inset. The measured beam profile is LP11-mode-like, but the right lobe is weaker than the left lobe [see Fig. 2(b)]. This suggests that the modal content is dominated by the LP11 mode but is not a pure LP11 mode.

The mode-locked optical spectrum was observed to exhibit periodic modulation [see Fig. 2(c)]. Inspired by the so-called spatially and spectrally resolved imaging (${S^2}$ imaging) method [32], we took a Fourier transform of the output optical power spectrum (but without passing through an external MMF as in typical ${S^2}$ imaging measurements) to estimate the modal content. Fourier transform of the spectrum, which is the first-order AC of the output pulse, contains secondary peaks at ${\rm{\pm 1.0}}$ and ${\rm{\pm 2.0}}\;{\rm{ps}}$ [inset of Fig. 2(c)]. This delay is close to the net intermode group delays between LP11 and LP21, and LP01/02 modes, which are 1.2 and 2.3 ps. The slight difference can be attributed to the nonlinear interaction between spatial modes and the finite resolution in the Fourier transform (the measured spectrum span sets the resolution to 0.12 ps). Based on the intensities of the peaks, we estimated the power contribution from modes LP11, LP21, and LP01/02 to be 78.6%, 14.8%, and 6.6% [32], respectively. As a caveat, this can be regarded only as a rough estimation, and more rigorous mode decomposition is needed to determine the modal content experimentally [33–36]. Moreover, the output was coupled into a SMF for the spectrum measurement, which may also change the mode ratio. Nevertheless, this estimation together with the measured LP11-mode-like beam profile strongly suggests that the LP11 mode is the dominant mode in the observed STML state.

As a signature of STML, the inset of Fig. 2(c) shows that the beam profiles change considerably when using filters to select different portions of the spectra [arrows in Fig. 2(c) illustrate the passband center of the filters]. To further characterize the spatiotemporal features of the STML state, we spatially sampled four positions of the output beam [see Fig. 2(b)] using a pinhole to measure the corresponding output. All the sampled outputs have the same repetition rate as the unsampled output, showing that different modes are synchronized [see Fig. 2(a)]. However, different samples show distinct spectra as plotted in Fig. 2(d), which is evidence of a STML state occupying a multi-transverse mode [6,13,14,18]. Due to environmental perturbations, a cw component may arise randomly during measurements. These measurements validate that STML with a multi-transverse mode excited and a single repetition rate is feasible in an all-STIN MMF laser, despite the large net intermode dispersion.

 

Fig. 3. Evidence of large intermode pulse walk-off. (a) Optical spectra of the output pulse with different fiber lengths, all showing some modulation. The inset shows the mode profiles. Scale bar: 500 µm. (b) Due to modulation, Fourier transform of the spectra shows additional peaks. The locations of the peaks generally follow the net intermode dispersion indicated by the solid lines: $(\delta \beta _1^{(1)} - \delta \beta _1^{(2)})L$ for the blue line and $(\delta \beta _1^{(4)} - \delta \beta _1^{(2)})L$ for the green line. (c) For the 3.9 m long fiber laser, mode-locking without strong spectral modulation can be observed by adjusting the SC condition. (d) Output beam profile for the state in (c), and white curves are horizontal and vertical slices, which are Gaussian-like. Scale bar: 500 µm. (e) Second-order autocorrelation traces with the input being the whole pulse or pulses sampled at four positions indicated in (d). The traces are smooth since the mode-locking contains negligible high-order modes and negligible intermode walk-off.

Download Full Size | PDF

All the measured spectra show periodic modulation. To verify that the modulation arises from additional mode-resolved temporal pulse peaks, we also measured the second harmonic intensity AC (i.e., second-order AC), as shown in Fig. 2(e). Note that the output pulses are chirped and the AC traces were measured without dispersion compensation so as to reflect the intracavity pulse width. Consistent with the first-order AC, all second-order AC traces (with or without spatial sampling) show observable additional peaks at ${\pm}{1.1}$ and ${\pm}{2.2}\;{\rm{ps}}$. Such additional peaks were not observed in GRIN MMF lasers for non-multi-pulse states [6,16,18,19] or hybrid GRIN–STIN MMF lasers [17,21]. As a result of multi-transverse-mode excitation, the contrast of these additional peaks differs from sample to sample. The strongest contrast was measured for sample 3, which can have a relatively strong contribution from the LP21 mode [Figs. 2(b) and 2(e)]. We also fitted the center second-order AC peak by Gaussian curves. The fitted width of the chirped pulse varied between 1.1 and 1.5 ps for different sampling positions, which is relatively short compared to previous reports [6,13–18,21]. The observed spectra and second-order AC variations due to spatial sampling are consistent with simulations (see Fig. 4 and Supplement 1 Section 4.).

B. Confirmation of Intermode Dispersion Induced Walk-Off

To confirm that the observed additional AC peaks result from large mode-resolved pulse walk-off, we adjusted the total fiber length of the laser by adding (cutting) 1 m passive fiber to (from) the laser. Figure 3(a) shows the mode-locked optical spectra under different total fiber lengths. Due to the different laser configuration as well as different SC condition, the new measured spectra are relatively narrow and the beam profiles also change. However, both spectra exhibit modulation, and the first-order AC shows additional peaks at nonzero delays [Fig. 3(b)]. The peak locations deviate farther from the zero delay with increasing total fiber length, and the locations reasonably follow the theoretical net intermode dispersion $\delta \beta _1^{(m)}L$. Note that the ${2.9}\;{\rm{m}}$ cavity was mode-locked with a very weak LP21 mode, and the corresponding first-order AC peak is absent. This fiber length control measurement strongly suggests intermode nonlinear interaction is weak in the fibers, and mode-resolved pulses walk off significantly in our all-STIN MMF lasers.

Another measurement in the 3.9 m cavity also supports that additional temporal peaks result from walk-off between different modes. Here, we adjusted the SC condition via mirror 3 in Fig. 1(a); a mode-locked state with a smooth optical spectrum was observed [Fig. 3(c)], and the first-order AC was seen to have negligible additional peaks. A Gaussian-like output beam was observed in this state [see vertical and horizontal slices of the measured beam in Fig. 3(d)]. It suggests that this state is dominated by the LP01 mode. The output pulse energy of the LP01 mode dominated state is 2.7 nJ, lower than the state in Fig. 2. We also measured the second-order AC with and without spatial sampling in this state. The measured AC traces are smooth and have a single peak [Fig. 3(e)]. Fluctuations in the AC traces for samples 3 and 4 can be attributed to the relatively low signal-to-noise-ratio in the AC measurements. Moreover, the pulse width stays nearly constant for different sampling positions. This measurement also highlights the importance of SC in controlling the STML state in all-STIN MMF lasers.

The observations in Figs. 2 and 3 together support that the walk-off for a mode-resolved pulse can be comparable to pulse width in the all-STIN MMF laser. However, they still overlap to some extent and are mutually coupled via nonlinearity and gain/dissipation. The large walk-off of the output pulse shows that the Kerr nonlinearity in STIN MMFs cannot balance the intermode dispersion. Hence, our laser operates in a STML regime distinct from lasers containing GRIN MMFs.

 

Fig. 4. Intracavity spatiotemporal pulse propagation dynamics in simulations. (a) Simulated optical spectrum of the output pulse, and the inset is the Fourier transform of the optical spectrum, i.e., first-order autocorrelation (AC) of the output pulse, which shows peaks belonging to different modes. The other inset is the simulated output beam profile. Scale bar: 10 µm. (b) Second-order intensity AC of the whole pulse and the sampled output pulse [see cross in (a) for the sampling point]. (c) Intracavity change of the mode-resolved pulse energy. The horizontal dashed lines show that the energy of modes 1 and 4 stays fixed in the passive fiber due to the large intermode phase velocity dispersion. Right panel shows the energy change due to SC. (d) Intracavity change of the mode-resolved pulse center position. Mode-resolved pulses walk off at a rate determined by $\delta \beta _1^{(m)}$ in the fibers, and it is mainly SC that corrects the pulse position. The right panel is an illustration of mother–child coupling (see main text). (e) Mode-resolved pulses at representative positions; see indications in (d); the top panels plot the spatiotemporal features of simulated pulses. Mode-resolved pulses walk off significantly after SA (position iv), and pulse positions becomes aligned again after SC; note that position vi is the same as position i for the very next round trip. (f) Pulse interval between modes 2 and 4 at the output port with varying intermode dispersion. When the intermode dispersion is relatively large, the interval is determined to be $(\delta \beta _1^{(4)} - \delta \beta _1^{(2)})L$; the interval can be slightly smaller than this delay due to nonlinearity when the intermode dispersion is small. The inset shows the mode-resolved pulses with $\delta \beta _1^{(2)} = 1{\rm{ps/m}}$.

Download Full Size | PDF

4. MECHANISM OF STML IN ALL-STIN MMF LASER

The measurements confirm that the additional AC peaks arise from large walk-off between mode-resolved pulses. The walk-off accumulated between two spatial modes $m$ and $n$ in the MMFs (denoted as $\Delta {t_{\textit{mn}}}$) is close to $(\delta \beta _1^{(m)} - \delta \beta _1^{(n)})L$ and can be comparable to pulse width. For self-consistent propagation in the laser, there should be a mechanism compensating for walk-off, i.e., inducing mode-resolved pulse shift ${-}\Delta {t_{\textit{mn}}}$. To understand this compensation, we used the coupled generalized multimode nonlinear Schrödinger equation (GMMNLSE) [6,7] to simulate the STML dynamics in an all-STIN MMF laser that mimics our laser layout. The details of the simulation can be found in Supplement 1 Section 2. Since it is challenging to model MMF laser parameters accurately, we aimed at corroborating the STML mechanism qualitatively by simulations rather than establishing a quantitative agreement with experimental measurements. SC is a key component of the all-STIN MMF laser and was modeled by a ${{6}} \times {{6}}$ matrix $M$ with the element $M(m,n)$ representing coupling from mode $n$ to mode $m.$ Thus, the diagonal elements stand for intramode coupling, while the off-diagonal elements stand for intermode coupling. Without loss of generality, we first considered a simple coupling matrix as the following (a similar conclusion can be reached with a more complicated coupling matrix; see Supplement 1 Section 3.A):

Mode 2 has relatively strong intramode coupling in this matrix. The actual matrix in experiments may be different from this one, and we used this matrix to have a simulated mode profile similar to measurements. Other mode profiles in simulations are possible by adjusting the matrix; see Supplement 1 Section 3.B.

A. Counteracting Large Intermode Dispersion by SC

A STML state with a single ${f_r}$ can be simulated as shown in Fig. 4. The used $\delta \beta _1^{(m)}$ vector in this figure is the same as the experimentally used STIN MMFs. The simulated optical spectrum is plotted in Fig. 4(a) exhibiting periodic modulation. The corresponding beam profile is LP11-mode-like but asymmetric [inset of Fig. 4(a)], which is similar to the measured output beam in Fig. 2(b). We also took a Fourier transform of the optical power spectrum to estimate the modal content. The Fourier transform exhibits additional peaks centered at ${\pm}{0.9}$ and ${\pm}{1.8}\;{\rm{ps}}$ [inset of Fig. 4(a)]. Based on the theoretical intermode dispersion, these additional peaks should belong to the LP21 and LP01/LP02 modes, respectively. The intensities of the first-order AC peaks suggest that modes LP01/02, LP11, and LP21 contribute about 8.4%, 58.8%, and 32.8% to the output pulse, respectively [32]. This retrieved modal content is in reasonable agreement with the directly calculated mode-resolved pulse energy ratios in the simulation, which are 2.6%, 68.6%, and 28.8% for modes LP01/02, LP11, and LP21, respectively. The discrepancy can arise from the residual overlap between pulses that makes the ${S^2}$ method less effective. The simulated second-order AC of the output pulse also shows modification of the AC trace at the delay of ${\pm}{0.9}$ and ${\pm}{1.8}\;{\rm{ps}}$ due to significant mode-resolvd pulse walk-off [Fig. 4(b)]. As an indication of the spatiotemporal features, we spatially sampled the output pulse [see mark in the inset of Fig. 4(a)], and the simulated AC of the sampled output also shows peaks at ${\pm}{0.9}$ and ${\pm}{1.8}\;{\rm{ps}}$ but with a different contrast. These second-order AC features are consistent with the measurement in Fig. 2(e), and more details of the spatially sampled spectra and ACs in simulations can be found in Supplement 1 Section 4.

Figure 4(c) depicts the intracavity pulse energy change for different modes within a single round trip (since modes 5 and 6 are relatively weak, they are omitted for clarity). Mode 2 has the highest energy at the exit of the gain fiber. In the passive fiber, pulse energy for modes 1 and 4 stays nearly constant, while there is a weak energy exchange between modes 2 and 3 [see horizontal dashed lines in Fig. 4(c)]. We believe this is because the relatively large phase velocity dispersion between mode 1 (or mode 4) and the other modes forbids energy exchange via intermode FWM, even though there is residual temporal overlap between these modes in the passive MMF. Modes 2 and 3 are degenerate in phase velocity, and intermode FWM between them is possible. Pulses lose energy after passing through SA and SF. The loss induced by SA is relatively high, since the NPR-based artificial SA in the experiment usually has high loss. Furthermore, the 3 nm SF induces considerable loss. The inset shows detailed energy changes before and after SC, and mode 1 is observed to have net energy gain in SC. The mode-resolved pulse center within a round trip is plotted in Fig. 4(d). Due to the large intermode dispersion, different modes gradually walk off in MMFs. The walk-off rate is close to the intermode dispersion [see dahsed lines in Fig. 4(d)]. The walk-off changes slightly when passing SA and SF. It is mainly the SC rather than SA as in [17] that balances $\Delta {t_{\textit{mn}}}$ and synchronizes the mode-resolved pulses.

To elucidate the SC-based walk-off compensation mechanism, we show the mode-resolved pulses at representative positions [labeled as i, iv, v, and vi in Fig. 4(c)] in Fig. 4(e). Position vi is also the starting point i for the next round trip. The top panels of Fig. 4(e) illustrate the spationtemporal profile of the pulses. Labels $A$, $B$, and $C$ stand for pulses of modes 1, 2, and 4, respectively (mode 3 is degenerate with mode 2 and not labeled). These pulses are short and smooth, and we believe it is because the coupling matrix is relatively simple, and more structured pulses were simulated when using a relatively complicated matrix (Supplement 1 Section 3.A).

At the beginning of a round trip, pulses are aligned and all pulses nearly have a single peak, except that mode 1 has a small observable secondary peak denoted as ${A_2}$ in Figs. 4(e), i. (see the blue vertically dashed line). At the exits of the MMFs, pulses are amplified and delayed from mode 1 due to intermode dispersion $\delta \beta _1^{(m)}$. Then the pulses are absorbed by the SA. There is no significant change of walk-off in this stage, but the tails of the pulses are strongly attenuated and pulse energy becomes more localized [pulse ${A_2}$ vanishes; see top panel in Fig. 4(e), iv]. The pulses are further dissipated by the SF. Again, no significant change in the walk-off induced by the SF is observed [see Fig. 4(d)]. In the SC, the coupling matrix redistributes energy among modes, which changes the pulse center position simultaneously. For instance, pulse $C$ is shifted from 1.3 to 0.2 ps [note that pulse $B$ has moved to 0.2 ps; see also Fig. 4(d)]. This is because the intramode coupling for mode 4 is blocked in the matrix ($M(4,4) = 0$), and the new pulse $C$ purely arises from coupling from modes 2 and 3 as shown in Fig. 4(e), vi [see also illustration in the right panel of Fig. 4(d)]. Hence, the new pulse $C$ is aligned with pulse $B.$ Similarly, a new pulse $A$ aligned with pulse $B$ is generated via intermode coupling from modes 2 and 3. In such a way, the intermode walk-off is compensated for (see Supplement 1 Section 3.A for discussion with a complex matrix).

B. Mother–Child Mode Coupling

Based on the above analysis, the gain and loss balance for modes 1 and 4 relies upon extracting energy from the LP11 mode in SC. Without this gain, these modes experience large net loss and will be dissipated [e.g., pulse ${A_2}$ in Fig. 4(e)]. Hence, we refer to the LP11 mode as the mother mode and the others as child modes. The part of child modes arising from SC will always be aligned with the mother mode. Note that delayed pulses (e.g., ${A_2}$) are dissipated in this picture (thus, $\Delta {t_{\textit{mn}}}$ erased), rather than pulled back to the mother mode pulse. This is distinct from the divided pulse laser [37], where pulses are retimed and combined. The pulse elimination may be a factor limiting the pulse energy scaling in the all-STIN MMF laser.

When there is only one degenerate mother mode in the all-STIN MMF laser, infinitely large intermode dispersion can be tolerated for STML. To verify this capability, we gradually increased intermode dispersion $\delta \beta _1^{(2)}$ from 0.01 to 100 ps/m in simulations (other elements of the $\delta \beta _1^{(m)}$ changed accordingly). With the above matrix $M$, pulses belonging to different spatial modes always have the same ${f_r}$. However, mode-resolved pulses can walk off significantly, and the simulated pulse interval between mode 2 and mode 4 at the output port closely follows the trend $(\delta \beta _1^{(4)} - \delta \beta _1^{(2)})L$ [Fig. 4(f)]. Pulses belonging to different LP modes may greatly walk off at the output port [e.g., see inset of Fig. 4(f)], but mode-resolved pulses share the same round trip time for intermode dispersion up to 100 ps/m. Hence, we believe infinitely large intermode dispersion can be tolerated for mode-locking with a single repetition rate in all STIN MMF lasers with an appropriate coupling matrix $M$.

The gain and loss balance can be used as a criterion to distinguish the mother mode from child modes. The gain and loss for mode $m$ should satisfy the following relationship in a STML state:

If there are several mother modes with different group velocities, mode-locking with multiple repetition rates will occur (see simulations in Section 5). Hence, $M$ should be engineered to have a single mother mode to realize STML with a single ${f_r}$ in all-STIN MMF lasers. Although we discuss a limited mode number here, mother–child mode coupling should enable STML with a single ${f_r}$ in much larger mode number cases by promoting a single mother mode $n$ [for instance, by making $M(n,n)$ larger than other diagonal elements)] and have sufficiently large elements in the $n$th column to enable mother–child mode coupling from mode $n$ to excite a multi-transverse mode.

5. DISCUSSION AND CONCLUSION

The observed spatiotemporal dissiaptive solitons are distinct from conservative multi-mode solitons in single-pass GRIN MMFs [30,38]. Those multi-mode solitons rely upon Kerr nonlinearity to balance intermode dispersion. In contrast, laser components including SA, SF, and SC add new dimensions to balance intermode dispersion to form spatiotemporal dissipative solitons. As a result, balancing intermode dispersion by Kerr nonliearity is not always required for STML in MMF lasers. The role of SC in our MMF laser is analogous to spectral filtering for dissipative soliton formation in all-normal-dispersion SMF lasers [31]. In that case, normal dispersion is not balanced by Kerr nonlinearity, and spectral filtering is leveraged to shorten highly chirped pulses for self-consistent propagation. In our case, intermode dispersion of STIN MMFs cannot be balanced by Kerr nonlinearity either, and mode-resolved pulses walk off significantly. SC synchronizes pulses and ensures self-consistent propagation in the all-STIN MMF laser. Thus, modal walk-off is important to shape the mode-locked pulses in the all-STIN MMF laser, which is also important to the formation of walk-off solitons [39]. However, walk-off solitons exist in GRIN MMFs with anomalous dispersion, distinct from our normal dispersion laser.

In summary, STML with a single repetition rate in an all-STIN MMF laser is demonstrated. The measured optical spectra show modulation, and AC traces exhibit multiple peaks due to mode-resolved pulse walk-off. Numerical simulations revealed that SC is responsible for counteracting the large intermode dispersion of STIN MMFs. For an appropriate SC condition allowing a single degenerate mother mode, infinitely large intermode dispersion can be tolerated for stable STML. However, if the SC condition promotes a multi-mother mode, spatial-mode-multiplexed multi-comb states can be possible. Hence, it is interesting to insert elements (such as spatial light modulators) to precisely control the SC condition for agile control of coherent spatiotemporal emissions from all-STIN MMF lasers. The current laser has a small mode number, and building all-STIN MMF lasers with richer modal content is under investigation.