Real-time multispeckle spectral-temporal measurement unveils the complexity of spatiotemporal solitons

The dynamics of three-dimensional (3D) dissipative solitons originated from spatiotemporal interactions share many common characteristics with other multi-dimensional phenomena. Unveiling the dynamics of 3D solitons thus permits new routes for tackling multidisciplinary nonlinear problems and exploiting their instabilities. However, this remains an open challenge, as they are multi-dimensional, stochastic and non-repeatable. Here, we report the real-time speckle-resolved spectral-temporal dynamics of a 3D soliton laser using a single-shot multispeckle spectral-temporal technology that leverages optical time division multiplexing and photonic time stretch. This technology enables the simultaneous observation on multiple speckle grains to provide long-lasting evolutionary dynamics on the planes of cavity time (t) – roundtrip and spectrum (λ) – roundtrip. Various non-repeatable speckly-diverse spectral-temporal dynamics are discovered in both the early and established stages of the 3D soliton formation.

The beam size of the extracted laser is enlarged by a 5× magnification telescope composed of lenses L1 and L2 (30 mm and 150 mm focal lengths, respectively). The magnified laser beam is then launched to the MUST measurement system. In the MUST system, the laser signal is split by two BSs with ratios of 30:70 and 50:50, respectively.
The signals of three speckle grains (SGs) in the multimode laser beam, as indicated in the left inset of Supplementary Figure 1  The photonic time stretch technology, also known as dispersive Fourier transform (DFT), is a powerful tool for real-time spectroscopy of soliton dynamics. So far, extensive efforts have been devoted to studying the spectral dynamics of bound solitons using the photonic time stretch technology. Here, we consider a general situation with two pulses, where, and Δ are the temporal separation and central frequency difference, respectively. After performing photonic time stretch, the field envelope becomes where, 2 is the chromatic dispersion used in the photonic time stretch. By applying Fourier transform to 0 ( ), Eq. ( 2) can be rewritten as By further exploiting the stationary-phase approximation, ( ) becomes Eq. ( 4) shows that, the time-stretched waveform exhibits additional features that oscillate at a frequency of 2 ⁄ . It is also worth noting that the presence of Δ results in a relative phase shift of the oscillation.

Spectral overlapping of multipulse time stretch
In the multipulse scenario, spectral overlapping can occur after the photonic time stretch, particularly when the pulses are closely located. Regarding this potential issue, we here investigate a pulse pair with varying temporal separation. As shown in Supplementary Specifically, the case of ii) can be written as where, Δ is the temporal resolution of the detection system, is the central wavelength, is the speed of light, and Ω is the spectral width of the optical pulses.
The FAC technology is applied to explore more details about the interference fringe   Figure 9d).
The relative phase evolution of the soliton-molecule can be retrieved from the spectral interferograms using the method described in Section 7, and the results are shown in Supplementary Figures 9b,e. A phenomenological function 4 can be applied to fit the results,  The asymmetry of the soliton-molecule, mainly the relative intensity and temporal separation, can largely influence the visibility of the interference fringe (Supplementary Figure 10). Thus, the shallow intensity modulation of the interference fringe pattern of Fig. 4 and Supplementary Figure 9 can be mainly attributed to the asymmetric intensities of the soliton pair.

Supplementary Note 7: Calculation of relative phase between solitons in solitonmolecules
In addition to the temporal separation, the relative phase is another key characteristic of the soliton-molecule, which can be calculated from two maximum peaks of the spectral interferogram and the carrier frequency. However, it is challenging to precisely obtain the carrier frequency when many solitons are involved. Here, we explore a universal method 5 to retrieve the relative phase from the spectral interferogram of soliton-molecules.
First, the spectral intensity of a soliton-molecule can be expressed as where, ℑ denotes the FT operation, and ~ represents the result of the FT operation.

( 11)
The function 1 ( + ) is readily obtained by isolating the sidelobes of ( ) located at = − , and it is then Fourier transformed to extract the value of Δ , where, ( + ) is the time gating function at = − . For the prerequisite of = 0, it is not necessary to temporally shift 1 ( + ) by before performing FT. To validate the proposed method, numerical studies are performed and the results are shown in Supplementary Figure 11, which exhibits a good agreement with the preset.
It is worth noting that this method is valid for any carrier frequenciesoffering a general means to evaluate the relative phase of the soliton-molecule.

Supplementary Note 9: 3D numerical studies
In our numerical studies, we consider a ring laser cavity that consists of a GRIN gain fiber (50 cm length, 62.5 μm core size), an artificial saturable absorber, a beam splitter, a bandpass filter, and a space filter. Note that, to reduce the calculation time, the simulation condition of the laser cavity has been simplified from the experimental setup, mainly combining all different fibers of the experimental laser cavity into a single GRIN gain fiber and using a saturable absorption function for STML.

MHz
A numerical model of the STML laser has been proposed in recent theoretical work 6 .
We adopt this numerical model in our numerical investigation. In brief, the propagation of the multimode light field in the GRIN gain fiber is described by the generalized multimode nonlinear Schrödinger equations (GMMNLSEs) 7 , i.e., where, ( , ) is the electric field of the spatial mode , where, is the saturation fluence (0.3 nJ/μm 2 in this work), 0 is the small signal gain coefficient, i.e., 35 m -1 here. The saturable absorption effect is established using an ideal transfer function after the gain fiber propagation, i.e., where, is the saturation intensity, which is set to be 50 GW/cm 2 .
The oscillation signal is extracted with a constant ratio, i.e., As shown in Supplementary Figure 18, two asynchronously pulsating pulses are generated, i.e., P1 and P2 (Supplementary Figure 18b). Please note that, to mimic the modal dispersion, a time delay is applied to the pulses of the degenerate transverse modes, i.e., the groups of {LP11a, LP11b} and {LP21a, LP21b}. The simulation results clearly show that the co-propagation of time-delayed pulses in intrinsic degenerate modes can generate the spectral-temporal dynamics analogous to the experimental observation (Fig. 5). Supplementary Figure 19, internal spatiotemporal transient dynamics of the 3D solitons are recognized, where the transverse mode rotating and time delay between the degenerate modes play important roles. To understand the physical origin, we refer to the minimum loss principle (or the maximum gain extraction) 6 , and a qualitative understanding can be intuitively obtained from the gain function Eq. ( 14).

As illustrated in
The pulse energy tends to be equally distributed over the spatial modes, otherwise, a strong gain saturation can be expected. If there is no time delay between the degenerate LP12a LP12b a b c modes, higher peak power is generated, which in turn leads to the destructive wave collapse 8 . Therefore, to obtain the maximum gain extraction without wave destruction, a hybrid form of 3D solitons with both internal modal variation and time offset is favorable. In addition to the internal spatiotemporal dynamics within 3D solitons, pulse-to-pulse dynamics have also been observed when the multimode fiber laser operates in the partial-STML regime, as shown in Supplementary Figures 20,21.
These numerical results suggest that single-shot characterization technologies are especially important for unveiling the dynamics of STML multimode fiber lasers and understanding their physical origins.   Here, the temporal resolution in the temporal domain is defined by the sampling rate of the real-time oscilloscope (80 GS/s), while the temporal resolution in the spectral domain is