Recently, the pulse fiber lasers have become a focus of extensive attention and research. Among them, the passively mode-locked fiber lasers have the distinctive properties of high stability, low loss, and so on and are becoming more important and prominent^[1,2]. The saturable absorbers (SAs) are key devices used to obtain a passively mode-locked fiber laser, which consists of two types of real SAs based on physical material properties and synthetic saturable absorbers. The former is mainly a semiconductor saturable absorber mirror (SESAM) and various nanomaterials^[3,4], while the latter includes nonlinear polarization rotation (NPR)^[5], nonlinear optical loop mirror (NOLN)^[6], nonlinear amplifying loop mirror (NALM)^[7], and nonlinear multimode interference (NL-MMI)^[8]. In 2022, Li et al. proposed and demonstrated experimentally harmonic mode-locked dissipative soliton resonance in a net-normal-dispersion Er/Yb co-doped fiber laser. In this work, the SESAM and an equivalent narrow-band spectral filter served as the mode-locker and were used to trigger multi-pulsing. The dispersion compensating fiber (DCF) with different lengths is incorporated into the laser cavity to investigate the pulse characteristics^[9]. However, there is a lack of global analysis and description of the pulse characteristic changes in the intracavity net dispersion variations. In 2022, Mkrtchyan et al. presented a scheme for studying a dispersion-managed mode-locked ultrashort-pulse in a Nd-doped all-fiber laser. In this work, the linear laser cavity is adopted with a chirped fiber Bragg grating (CFBG), which is utilized as the dispersion compensation device, and a semitransparent reflection mirror. The SESAM serves as the mode-locker and the second non-transparent mirror. In addition, the mode-locked regimes in the net-cavity-dispersion in the −0.05−0.24ps2 range were investigated in detail^[10].

Furthermore, the higher-order modes with unique spatial and polarization properties, which are obtained by mode conversion elements in the fiber laser, are currently of great interest. Among them, the orbital angular momentum (OAM) modes characterized by a helical phase front of exp(±ilϕ), which can be generated by vector modes (VMs) or linear polarization (LP) modes on the basis of mode superposition, are particularly important. The OAM modes have been studied in a variety of promising applications, such as optical tweezers, optical manipulation and trapping, and motion sensing^[11–13].

The fiber laser that can realize the combination of transverse modes and longitudinal modes and output is called a spatiotemporal fiber laser^[14–16], which can gather the advantages of two modes, which not only broadens the diversified application scenarios of the original fiber laser but also opens up a new way of thinking for its correlated research. For example, in the field of spectroscopic detection, applying the transverse mode in the spatial domain to Raman spectra can obtain the unique chemical fingerprints of specific molecules or materials, which can be used to quickly confirm different substances or materials. While applying the longitudinal mode to spectral detection, the ultra-fast characteristic of its pulse can make up for the defects of the transient spectra under the steady-state spectral distribution, which complements the time-domain characteristic of spectral detection and then perfects the temporal and spatial completeness of the spectral detection application field. Therefore, it is of great research value and significance to explore the output characteristics of pulsed erbium-doped fiber lasers with higher-order mode output by combining the horizontal and vertical dimensions of mode analysis in optical fibers and focusing on two key points, namely, the mode conversion as well as the passive pulse realization.

However, limited by the mode conversion element, pulse fiber lasers cannot output high-order modes, and there is a lack of research on mode-locked fiber lasers with high-order modes output. Therefore, the dispersion-managed mode-locked pulse fiber laser with OAM modes emission based on mode superposition principle is still challenging.

In 2017, Jiang et al. presented and demonstrated experimentally the generation of OAM mode on the basis of mode superposition^[17], which provides theoretical guidance for experimental feasibility. In this work, we propose an all-fiber dispersion-managed mode-locked Er-doped fiber laser with an OAM output based on mode superposition. The pulse characteristics during the dispersion-managed process are investigated in detail. At the same time, this mode-locked fiber laser can produce switchable outputs LP01, LP11, LP21, and LP31 and the corresponding topological charge of the OAM modes.

The VMs are the eigenstates in the optical fiber. Depending on the azimuthal index, the high-order VMs can be classified into the four cases: TE, TM, HEe,o, and EHe,o, where “e” and “o” describe the even and odd modes, respectively. In a Cartesian coordinate system, the transverse electric field distributions of these VMs can be represented by the following equations^[18]: $$\begin{gathered} {HEl+1,meHEl+1,mo}=Fl,m(r){x^cos(lϕ)−y^sin(lϕ)x^sin(lϕ)+y^cos(lϕ)}(l≥1), \\ {EHl−1,meEHl−1,mo}=Fl,m(r){x^cos(lϕ)+y^sin(lϕ)x^sin(lϕ)−y^cos(lϕ)}(l≥1), \\ {TM0,mTE0,m}=Fl,m(r){x^cos(ϕ)+y^sin(ϕ)x^sin(ϕ)−y^cos(ϕ)}(l=1), \end{gathered}$$(1)where the subscripts l and m denote the azimuthal and radial index, respectively, Fl,m describes the solution of the Bessel equation, x^ and y^ are the x- and y-polarization directions, respectively, and ϕ represents the azimuth coordinates.

Next, the transverse electric field distributions of the LP modes can be generated by VMs by considering the weakly guiding approximation, {LPl,mc,xLPl,ms,xLPl,mc,yLPl,ms,y}=Fl,m(r){x^cos(lϕ)x^sin(lϕ)y^cos(lϕ)y^sin(lϕ)}(l≥0),(2)where the superscript c(s) presents the cosine (sine) function, and x(y) denotes the x(y)-polarization direction. Considering the electric field distributions and polarization direction, LP modes can be classified as even and odd modes in both the x- and y-directions.

Finally, according to our previous work^[19], the linear x(y)-polarization OAM (LP-OAM) modes with x^ or x^ polarization direction can be generated by overlapping the same polarization direction even and odd LP modes with a +π/2 phase difference, $$\begin{gathered} {x^V±l,my^V±l,m}=Fl,m(r){x^cos(lϕ)±ix^sin(lϕ)y^cos(lϕ)±iy^sin(lϕ)} \\ ={LPl,mc,x+iLPl,ms,xLPl,mc,y+iLPl,ms,y}(l≥1). \end{gathered}$$(3)

To prove its experimental feasibility, a linear laser cavity scheme is designed and proposed, shown in Fig. 1(a). Here, a 980 nm laser diode (LD, 850 mW maximum pump power) pumps the 1.9 m-long Er-doped active fiber (CETC, Er-5/125-33) serving as the gain medium with 38.8 dB/m at 1530 nm via a 980 nm/1550 nm wavelength division multiplexer (WDM). The length of the Er-doped fiber is already optimized to 1.9 m for maximum efficiency. The SESAM (BATOP, SAM-1550-20-3 ps) is utilized as the mode-locker and the non-transparent mirror. The optical coupler (OC1) with a 9:1 coupling ratio is employed to connect the optical paths and to monitor the pulse characteristics and mode pattern at output 2 port, and another same OC (OC2) serves as a total reflection mirror. The net-cavity-dispersion is changed in the −0.495−+0.197ps2 range by incorporating different lengths of DCF (YOFC, NDCF-G652C/250, +178×10−3ps2/m).

The mode selective coupler (MSC) is used to realize the mode conversion, which provides prerequisites for later OAM mode generation based on mode superposition. Meanwhile, the schematic of the MSC is shown in Fig. 1(b). This MSC is made using a single-mode fiber (SMF, YOFC, G652D) and a ring-core fiber (RCF, YOFC, SI2117-F) by the fusion tapering method. The ring-core fiber with a high refractive index ring supports the transmission of multiple LP modes (LP01, LP11, LP21, and LP31) in the C-band, and the relative refractive index difference versus the radius is shown in Fig. 2. According to the coupled mode theory, the fundamental mode in the SMF and the high-order LP modes in the ring-core fiber undergo alternating energy transitions along the longitudinal direction of the optical fiber when the phase-matching conditions are satisfied, thus achieving mode conversion.

The linear cavity possesses a line length of ∼11.82m and a loop length of ∼2m, leading to a 25.64 m optical path of light traveling a cycle in the laser cavity. Here, a two-branch interference scheme is used to verify the OAM modes. The high-order LP modes generated by the MSC are transmitted through the coupling arm and output at the output 1 port, which are collimated and amplified by the objective lens (obj) and then combined with the fundamental mode output at the output 2 port by the beam splitter (BS) and finally detected by the charge-coupled device (CCD, Xenics Xeva). It should be emphasized that, unlike the tri-loop polarization controller (PC1), the squeezed polarization controller (PC2) is used to adjust the phase difference between the high-order LP modes of the same azimuthal index, ultimately resulting in the corresponding topological charges of the OAM modes on the basis of mode superposition.

The characteristics of dispersion-managed pulses are measured and displayed by the following devices: a 200 MHz InGaAs photodetector (PD12C-200M) connected to a digital phosphor oscilloscope (Tektronix, DPO3032 with a 300 MHz bandwidth), an optical spectrum analyzer (YOKOGAWA, AQ6375) with a resolution of 0.01 nm, an optical power meter, an electrical spectrum analyzer (R&S, FSH8) with dynamic tunable resolution frequency, and an interferometric autocorrelator (Femtochrome, FR-103XL) with a 200 ps measurement range.

When this proposed fiber laser operates in the mode-locked regime without a DCF under 150 mW pump power, the net-cavity-dispersion is around −495×10−3ps2, and the pulse performance generated is shown in Fig. 3. The proposed fiber laser operates at 1549 nm center wavelength with an around 3 dB bandwidth of 8 nm, shown in Fig. 3(a). Figure 3(b) displays the pulse train in the 1500 ns time window, which indicates the 128.2 ns pulse interval and the 7.8 MHz repetition rate, and corresponds to the 25.64 m optical path. The radio frequency (RF) spectrum with a 54.53 dB signal-to-noise ratio (SNR) at the fundamental frequency demonstrates the stability of this mode-locked fiber laser. Since the interferometric autocorrelator cannot display the actual autocorrelation curve due to the limitation of the detection range, the actual curve is presented here with the help of sech2 fitting. The autocorrelation (AC) traces shown in Fig. 3(d) exhibit a double-scale structure with a narrow coherent spike of 2.15 ps duration located on top of a 142.58 ps wide pedestal, which indicates the noise-like pulse (NLP).

Next, the different length DCFs are incorporated to compensate for net anomalous dispersion in the cavity, resulting in the −0.495−+0.197ps2 net-cavity-dispersion range. Here, we present the mapping of mode-locked generation regimes depending on the pump power, net-cavity-dispersion, and polarization state, which is shown in Fig. 4(a). When the net dispersion is −0.159ps2 under 450 mW pump power, we obtain up to a second-harmonic mode-locked pulse with a 12.76 MHz pulse repetition rate and a 78.4 ns pulse interval [see Fig. 4(b)].

The anomalous group velocity dispersion in the fiber results in a high frequency at the pulse front and a low frequency at the back edge, while the frequency chirp induced by the self-phase modulation (SPM) effect is just the opposite. The conventional solitons (CS) will be generated in the laser when the two effects are mutually balanced ^[20]. In the frequency domain, the conventional soliton exhibits approximately symmetric spectral Kelly sidebands with a 1561 nm center wavelength^[21], as shown in Fig. 5(a). Meanwhile, the CS shows a typical hyperbolic secant curve in the time domain and is close to the Fourier transform limit^[22], see Fig. 5(c). The 67.4 dB SNR at the fundamental frequency in the RF spectrum in Fig. 5(b) implies a reasonably low noise level of the CS.

According to the soliton area theorem, both the pulse generation threshold and the soliton splitting threshold are lowered at near-zero net anomalous dispersion, leading to a decrease in the maximum achievable single-pulse energy, which promotes the generation of multiple pulses^[23,24]. In the case of zero net dispersion, no pulses are generated. Typically, the saturable absorption of absorbers is limited by pulse energies that are too small to satisfy. Therefore, the dramatic increase of modulation depth SAs aiming for achieving stable mode locking would be a forward-looking solution^[25].

Further increasing the length of the DCF incorporated in the laser cavity makes the change of negative to positive net dispersion that supports the single pulse generation. Meanwhile, the occurrence of single- to double-harmonic mode-locked pulses happens through the Q-switched mode-locked (QML) regime at 0.019ps2 net-cavity dispersion. Since the positive net dispersions are larger and are induced by longer DCFs, the pump power required for the laser to start resonant operation and for pulses to appear increases.

The dynamic change of optical spectra under different net-cavity-dispersion is shown in Fig. 6(a). With the variations of net-cavity-dispersion from anomalous to normal dispersion, the optical spectra tend to be narrower and is accompanied by a blue shift of the center wavelength, which would be originated from the breathing process^[26]. The increase of the cavity length leads to the pulse broadening and the peak power decreasing, which in turn narrows the pulse optical spectra due to the reduction of self-phase modulation. Finally, the pulse energy as a function of the net-cavity-dispersion is investigated in Fig. 6(b). The pulse energy shows a large drop and reaches a minimum, which is caused by the power loss in the laser cavity due to different core diameters of the DCF and SMF alignment fusion. Subsequently, with the increase of the cavity length, the mode-locked pulse repetition frequency decreases, eventually promoting a gradual growth in the pulse energy.

In this work, the mode-locked pulses appeared in the NLP and CS states, and the high-order harmonic pulses reach up to only two. The most critical factor is that the gain in the laser cavity is unsatisfied for the excitation of the higher-order harmonic pulses. Future research can explore the following two aspects: The first is realizing the lower-loss alignment fusion of the DCF and the SMF by setting appropriate fusion modes; The second is using the double- or multi-stage pumping amplification scheme to realize the higher gain of the higher-order harmonic mode-locked pulses in the cavity.

Following the dispersion-managed mode-locking operation, the mode manipulation patterns are also gradually surfaced and detected by the CCD. First, three homemade MSCs achieving different high-order LP mode conversions are sequentially connected to the laser cavity. The fundamental mode pattern is obtained by masking the output 1 port in Fig. 1(a). Then, shutting off the output 2 port, the odd and even modes of different high-order LP modes are measured by adjusting the PC2 appropriately, as shown in Figs. 7(a1) and 7(b1), 7(a2) and 7(b2), and 7(a3) and 7(b3). The symmetrically distributed two-, four-, and six-valve mode intensity patterns correspond to the high-order LP modes LP11, LP21, and LP31, respectively. Finally, the PC2 is carefully turned so that the phase difference between the odd and even modes of the same azimuthal index LP modes reaches π/2, and the OAM modes carrying corresponding topological charges are formed based on the mode superposition principle. The experimental results can be seen in Figs. 7(c1)–7(c3).

In order to validate the generated OAM, a two-branch scheme is used here in Fig. 1(a), that is, interfering the generated OAM mode with the fundamental mode and then observing the interference patterns in order to further determine the topological charge and sign of the generated OAM mode. Figures 7(d1)–7(d3) give the interference patterns, which confirm that these anti-clockwise spirals are the OAM modes with +1, +2, and +3 charges, respectively. Thus, this proposed mode-locked all-fiber laser can output pulsed LP01, LP11, LP21, LP31, and OAM modes, which proves its feasibility.

Taking the OAM+1 mode as an example, the mode purity is calculated to further explore the generated OAM modes. The OAM+1 mode intensity pattern is shown in Fig. 8(a), and Fig. 8(b) shows the azimuthal intensity gray value profile of the OAM+1 mode for the data ring with radius r. Then, the amplitude and phase profile are obtained by applying Fourier transform analysis^[27], shown in Figs. 8(c) and 8(d). The final calculated mode purity of the OAM+1 mode is around 93.40%. Similarly, the mode purity of the OAM+2 and OAM+3 modes is 91.67% and 91.59%, respectively.

In summary, we propose a dispersion-managed mode-locked pulse Er-doped fiber laser with high-mode purity OAM output. In this work, the linear cavity scheme is adopted with a SESAM serving as the mode-locker and an all-fiber ring loop as the total reflection mirror. The pulse characteristics are investigated within the −0.495−+0.197ps2 net-cavity-dispersion range, which is induced by incorporating different lengths of the DCF to compensate for the cavity dispersion from negative to positive. The NLP with double-scale structure autocorrelation traces and the two-order harmonic CS mode-locked pulse with a 6 ps pulse duration, a 12.76 MHz repetition rate, and a 78.4 ns pulse interval are obtained. In addition, the adjustable high-order modes output among LP11, LP21, LP31, and the high-mode purity OAM with corresponding topological charges are generated on the basis of mode superposition. The intracavity gain sets the limits for the maximum repetition rate of high-order harmonic pulses and the occurrence of multiple mode-locked pulse regimes, which could be expanded by more precise dispersion management schemes and multi-stage pumping amplification.