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Abstract
This paper presents an approach that combines the generalized multimode nonlinear Schrodinger equation with a transmission model to analyze spatiotemporal characteristics of multimode interference in single mode/large mode area fiber–graded-index multimode fiber–single mode fiber (SMF/LMA-GIMF-SMF) structures for the first time. Approximated self-imaging (ASIM) behavior in GIMF and the study of the latter structure used in spatiotemporal mode-locked fiber lasers are first demonstrated. Simulations show that these structures can work as saturable absorbers enabling high-energy pulse output due to nonlinear intermodal interactions and intensity-dependent multimode interference. Otherwise, underlying ASIM is proven that it can perturb the transmission of SMF/LMA-GIMF-SMF, causing instability of their saturable-absorption characteristics. This paper provides a theoretical guide for many applications, such as beam shaping, mode conversion, and high-energy ultrafast fiber laser.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Multimode interference (MMI) refers to the coherent superposition of different guided modes to form periodic transverse electromagnetic field distribution in space. It was observed first in free-space propagation in 1836 [1], then in planar waveguides [2] and fiber waveguides [3]. The phenomenon of periodic distribution of electromagnetic fields is well-known as self-imaging (SIM). The SIM has been recently measured in multimode fiber (MMF) by using photoluminescence stimulated by high-intensity ultrashort pulses [4,5]. MMI in MMF has attracted great attention. Numerous fiber devices based on MMI have been proposed, among which single mode fiber (SMF)-MMF-SMF structure is a critical component. It is capable of achieving multiple functions. For instance, SMF-MMF-SMF has been developed as a sensor by exploiting the correlation between its MMI phenomenon and environmental parameters [6,7]. It can also function as filters [8,9] and saturable absorbers (SA) [10] due to the wavelength-dependent and intensity-dependent MMI. Because of the simple form, large modulation depth, and high damage threshold, SMF-MMF-SMF-based SA has significant practical implications for ultrafast fiber lasers.
Since SMF-GIMF-SMF was found as SA, many studies have applied it as SA in single-mode mode-locked fiber lasers [11–16]. Recently, SMF-GIMF-SMF was also used in spatiotemporal mode-locked fiber lasers (STML) [17–19]. For tolerating higher soliton energy, the core sizes of two-side fibers connected with GIMF were expanded. The core-expanded fiber could support a few modes, which we usually call large-mode-area fiber (LMA). Hence, the LMA-GIMF-LMA was proposed and extensively used in high-power mode-locked fiber lasers as a SA, filter, or mode exciter and filter for STML [20–22]. But LMA-GIMF-LMA works only when the LMA has a limited core size with very few modes because of the large modal walk-off, usually no more than three linearly polarized modes according to the previous reports. To achieve mode-locking while using step-index LMA with larger-core, strong spatial filtering is required to guarantee spatial self-consistency. For instance, an STML with large modal dispersion was proposed lately [23], in which a large-core LMA was spliced to a GIMF, and the in-cavity spatial light back to the LMA is spatially filtered by the LMA itself. But the spatial light structure is complex and hard to maintain. This process can be achieved by an all-fiber structure, namely LMA-GIMF-SMF. More recently, an all-fiber STML with large modal dispersion was first reported [24]. The cavity was partially composed of long LMA containing six modes to form a large-modal-dispersion cavity. The large modal walk-off was mainly compensated by strong virtual spatial filtering provided by LMA-GIMF-SMF.
However, there are still many disputes about the spatiotemporal dynamic of MMI inside SMF-GIMF-SMF-based SA according to previous reports. The knowledge of its saturable-absorption characteristics and the impact of spatiotemporal dynamics on MMI inside the structure are still very limited. For another aspect, the theoretical study of LMA-GIMF-SMF used in STML has not been performed. Its potential theory and functionality are waiting to be investigated. Because of the lack of theoretical validation for designing specific SMF/LMA-GIMF-SMF structures, their performances are mainly determined by experimental skills and design experience. So their properties are hard to be optimized. Until now, numerous ideas have been put up to explain the MMI in multimode waveguides, such as guided-mode propagation analysis (MPA) [25], a type of numerical beam propagation technique [7], a lindrically symmetric method of lineas (MoLs) [26], coupled nonlinear equations [10]. However, these approaches are either only applicable to the MMI that results from a small number of ideal modes or not considering the impact of full linear and nonlinear effects on MMI. Moreover, these methods are developed only under the condition of continuous waves, so the spatiotemporal dynamics of MMI cannot be explored.
Here, we provide a method for analyzing spatiotemporal characteristics of MMI in SMF/LMA-GIMF-SMF using the generalized multimode nonlinear Schrodinger equation (GMMNLSE) and a transmission model for the first time. Linear and nonlinear influences are taken into account inside the GMMNLSE. This method is applicable for the analysis of MMI between multiple combinations of complicated modes and can also describe the impact of comple x spatiotemporal dynamics on MMI. Simulations unveil several results. First, Strong nonlinear intermodal interactions restrain modal walk-off and mitigate the nonlinear phase mismatch of modes, which makes the phase shift of transmission of SMF-GIMF-SM-based SA within $\pi $, enabling it to support high-energy pulse output. Second, underlying approximated self-imaging (ASIM) behavior in GIMF is first revealed. ASIM can perturb the transmission of SMF/LMA-GIMF-SMF, causing instability of their saturable-absorption characteristics. Third, LMA-GIMF-SMF can be used not only as a fiber-based mode exciter and filter, but as well as an additional SA for STML, which can improve soliton energy and stability. Last, the transmittance of SMF-GIMF-SMF increases with the output beam profile from GIMF enlarging, while that of LMA-GIMF-SMF shows the opposite.
2. Methods
2.1 Generalized multimode nonlinear Schrodinger equation
To simulate and analyze the MMI in GIMF, we adopt the generalized multimode nonlinear Schrodinger equation (GMMNLSE) [27,28], which is widely used to describe the dynamics of multimode solitons in GIMF [29]. The GMMNLSE addressing the variation of modal field envelopes in GIMF can be written as follows:
To increase the calculation speed, the GMMNLSE is numerically solved by using a parallel computing approach [30] that is based on the graphics processing units of computers. Additionally, we only consider the ${0^{th}}$ to ${4^{th}}$ order dispersion in the equation, the default number of modes considered is set as 20 in GIMF, and the iterative minimum step is set as 50 $\mu m$ to consider intermodal beating length.
2.2 model overlap integral for modal excitation and filtering
To determine which modes are excited or filtered at spliced interfaces, here we use modal overlap integral to calculate modal coupling coefficients from source modes into target modes. When the laser travels from SMF or LMA into GIMF, mode excitation will happen at the spliced interface to produce more modes in the GIMF. Instead, as the laser propagates from the GIMF into an SMF or LMA, spatial filtering will occur to eliminate a portion of modes. The expression of modal overlap integral is given by [27]:
2.3. Transmission model for SMF/LMA-GIMF-SMF
The transmission of SMF/LMA-GIMF-SMF is a very important parameter when it is seen as a coupler, SA, and bandpass filter. To understand the process of energy transfer, the corresponding Sankey diagram is illustrated in Fig. 1. The links determined by modal overlap integrals in the Sankey diagram represent the transfers of energy between modes. Specifically, the transmission of SMF/LMA-GIMF-SMF can be expressed as follows:
Fig. 1. Sankey diagram of the transmission model. ${M_i}$ represents mode i; ${\eta _m}$ represent energy ratio of mode m in LMA; ${a_{mn}}\,and\,{b_n}$ represent overlap integrals between connected modes.
3. Results and discussion
3.1 SMF-GIMF-SMF
Although SMF-GIMF-SMF has been widely used in applications, such as mode-locked fiber lasers and wavelength-tunable lasers, the internal dynamics of MMI in GIMF are still needed to be explored to meet various application requirements. SMF-GIMF-SMF comprises a segment of SMF, GIMF, and another SMF. Two SMF are spliced to the end of GIMF, as shown in Fig. 2. SMF (8/125 $\mu m$ NA = 0.14) is on-axis coupled to one end of GIMF (50/125 $\mu m$ NA = 0.2), and another identical SMF is on-axis spliced to another end of GIMF. The length of GIMF is set as 1 m. SMF has one mode at the central wavelength of 1550 nm, while GIMF can support hundreds of modes. But the first 20 eigenmodes in GIMF are chosen for simulation, which is sufficient to reveal internal modal dynamics in GIMF. And the initial pulse in SMF was set as 0.5 ps long with the energy of 5 nJ.
Figures 3(a) and (b) are simulated mode-solved pulses intensity in SMF and GIMF at their interface, respectively, and corresponding near-field beam profiles are shown in Figs. 3(c) and (d). The excited beam profile is circularly symmetric rings because the coaxial connection between SMF and GIMF only excites radial modes [31] whose propagation constants are marked with red circles in Fig. 3(e). Figure 3(e) presents propagation constants of the first 20 eigenmodes relative to mode 1, where the propagation constants distribution manifests several discrete mode groups. Modes in the same group have nearly identical propagation constant, which differs from SIMF and makes SIM length shorter [10,32]. The shorter SIM decreases the tuning length of GIMF to reach the regime of SA.
Fig. 3. Simulated results of MMI in SMF-GIMF-SMF. (a) and (b) are simulated mode-solved pulses intensity in SMF and GIMF at their interface, respectively. (c) and (d) are corresponding beam profiles of (a) and (b), respectively. (e) Propagation constants of the first 20 modes in GIMF. Excited modes are mode 1, 6, and, 15 marked with red cycles. (f) Evolution of x-axis cross sections of beam profiles along the propagation direction of GIMF at the last 5 mm, when nonlinear terms are considered. (g) Six chosen beam profiles at different positions. See more in Visualization 1.
Using GMMNLSE, the evolution of the x-axis cross sections of beam profiles along the propagation direction of GIMF at the last 5 mm is derived in Fig. 3(f), and six chosen beam profiles are displayed in Fig. 3(g). The beam profile evolution exhibits a periodic pattern with a period of 600 $\mu m$, which is the SIM length. Interestingly, the SIM length is approximately equal to the beating length between mode 1 and mode 6. This observation can be explained by equispaced propagation constants of modes in GIMF. As we know that SIM occurs when each mode satisfies ${\beta _1}{L_s} + q2\pi = {\beta _n}{L_s}$ ($q$ is an integer, ${L_s}$ is SIM length). Because of the equispaced propagation constants of modes in GIMF, the SIM length is determined by the longest beating length, namely the beating length between mode 1 and mode 6 (${L_s} = 2\pi /\Delta {\beta _6} = 585\; \mu m$, $\Delta {\beta _6}$ is the propagation constant of mode 6 relative to mode 1). Hence, the SIM occurs at an integer multiple beating length $Z = 2\pi k/\Delta {\beta _6}$ (k = 1,2,3…).
Nonlinear effects play a significant role in GIMF. Here we simulate MMI in GIMF with and without nonlinear terms to determine the impact of nonlinearity on MMI. If only linear terms are considered, the mode-solved pulses have a noticeable walk-off due to modal dispersion after traveling through 1m GIMF, as shown in Fig. 4(a). Instead, when nonlinear terms are included, the pulses are compressed and the walk-off between pulses is very small, as shown in Fig. 4(b). On the one hand, the chirps caused by dispersion and nonlinearity are compensated at the negative dispersion region, which restrains pulse broadening. On the other hand, the walk-off is compensated by chromatic dispersion because the XPM results in an asymmetric spectral broadening of pulses, enabling pulses to have different frequency shifts [29,30]. Figure 4(c) shows the calculated nonlinear phase for each mode. Notably, Although the energy of the fundamental mode (mode 1) is nearly 4 times higher than mode 6, their nonlinear phase still keeps synchronously. This observation can be simply explained by XPM (Despite the SPM of mode 6 being small, nonlinear phase (${|{{A_1}} |^2}{A_6}$) induced by XPM is large). Besides, the small nonlinear phase of mode 15 is contributed to its low intensity and small overlap with other modes in the time window. To access the area variation of the beam profile in GIMF, as shown in Figs. 4(d)-(f), we calculate the area of beam profiles (the beam profile is normalized) in the first 5 mm, the middle 5 mm, and the final 5 mm GIMF with and without nonlinearity. Because the nonlinear phase mismatch between modes increases gradually with modes propagating through GIMF, the two curves gradually deviate from each other. The area evolves periodically with the position of GIMF because of SIM. However, the SIM process is not perfect but shows jitter as modes propagate along 1 m GIMF. This phenomenon is shown as ASIM behavior, which stems from the not strictly-equispaced propagation constants among modes.
Fig. 4. Spatiotemporal analysis of MMI in SMF-GIMF-SMF with and without nonlinear terms. (a) Output pulses intensity from 1 m GIMF, when nonlinear terms are ignored. (b) Output pulses intensity, when nonlinear terms are introduced. (c) Calculated nonlinear phase of modes at different positions in GIMF. (d)-(f) Beam profile area evolution at the first 5 mm, middle 5 mm, and last 5 mm GIMF for without (dash line) and with (solid line) nonlinear terms. (g) Light intensity evolution at last 5 mm. (h) Power transmission of SMF-GIMF-SMF for different lengths of GIMF. See more in Visualization 1.
The light intensity and transmission of SMF-GIMF-SMF are also studied, as shown in Figs. 4(g) and (h), respectively. The results indicate that when the area of beam profile increases, light intensity becomes lower, while the transmission of SMF-GIMF-SMF soars up. The latter can be explained by the following: the transmission reaches the local maximums at the places where modes are in phase; when modes are in the situation that they are in phase, like initially excited modes, the transmission arrives at the maximum. The result uncovers a misunderstanding that high transmission happens at the small beam profile, while low transmission happens at the large beam profile. Moreover, the transmission ranges from 0.3 to 0.85, presenting as high as 55% modulation depth. When the nonlinear effects are taken into account, the transmission curve has a small phase shift. The result proves that transmission of SMF-GIMF-SMF is nonlinear-phase dependent, which is indispensable for it to work as a SA. Moreover, because of the ASIM behavior in GIMF, disturbance exists in the transmission function of SMF-GIMF-SMF versus GIMF length, leading to the diverse transmission for every period. The disturbance caused by ASIM in turn affects its saturable-absorption properties. The influence of ASIM on the evolutions of beam profile, light intensity, and transmission, can be seen in Visualization 1 for details.
One thing needed to mention is that the initial pulse intensity we set in the simulation is higher than the common pulse intensity in 1550 nm single-mode mode-locked lasers, and the length of GIMF is longer than most of the reported length on SMF-GIMF-SMF-based mode-locked fiber lasers [11–16]. Even with that high intensity and long GIMF length, the phase shift of transmission because of nonlinear effects is still within $\pi $, which promotes SMF-GIMF-SMF-based SA to support high-energy pulse output. The small phase shift stems from strong nonlinear intermodal coupling that is explained in discussion for Fig. 4(c). This finding can be confirmed by previous reports, one of which experimentally demonstrated that the saturation intensity of SMF-GIMF-SMF can reach up to 665.8 MW/cm2 when tuning the GIMF length properly [12]. Another experiment demonstrated that SMF-GIMF-SMF with 5 m GIMF still realized mode-locking [17]. If we lower the intensity of the initial input pulse, the output pulses from GIMF gradually walk off, the nonlinear phase drop down, and the phase shift of transmission function is still within $\pi $ and smaller. Based on the above results, these features of SMF-GIMF-SMF make it to be an ideal SA that can tolerate high energy output for mode-locked fiber lasers.
3.2 LMA-GIMF-SMF
Apart from SMF-GIMF-SMF, the influence of nonlinear effects on LMA-GIMF-SMF is also investigated. We find that the LMA-GIMF-SMF cannot only work as a mode exciter and filter but can also work as an additional SA used in high-power STML. LMA-GIMF-SMF structure is formed by replacing left-hand SMF in Fig. 2. The LMA (20/125 $\mu m$ NA = 0.14) has a large core diameter, which can support six modes. Assuming initial pulses are 0.5 ps with total energy 5 nJ, the energy ratio of modes in LMA is set as [0.58, 0.15, 0.15, 0.05, 0.05, 0.02]. The mode-solved pulses intensity and corresponding beam profile in LMA are shown in Fig. 5(a). Because of the increasing number of input modes, the excited modes in GIMF are more complicated (Figs. 5(b) and (c)). The beam profile evolution as a function of the position is dramatic, and the nonlinear effects offer more influence on MMI compared with SMF-GIMF-SMF (Figs. 5(f) and (g)). As can be seen, the ASIM of LMA-GIMF-SMF is almost twice longer as SMF-GIMF-SMF, about 1200 $\mu m$ determined by $2\pi /\Delta {\beta _2} = 1181\; \mu m$. Figures 5(d) and (e) present chosen near-field beam profiles at different position without and with nonlinear terms, respectively. By contrast, the beam profiles changes when nonlinear effects are involved. The nonlinear phase mismatch between modes becomes larger as pulses go through GIMF (Fig. 6(c)), which results in the variation of the beam profile. See from these beam profiles, the obvious periodic patterns are presented with a period of 1200 $\mu m$. Notably, the phase conjugation takes place at about half ASIM length, in which modes are in antiphase, and hence the beam profile is centrosymmetric with the in-phase beam profiles (Figs. 5(d1)-(d3)).
Fig. 5. Simulated results of MMI in LMA-GIMF-SMF. (a) and (b) are simulated mode-solved pulses intensity in LMA and GIMF at their interface, respectively. Insets are corresponding beam profiles of (a) and (b). (c) Propagation constants of the first 20 modes in GIMF; Excited modes are marked with red cycles. Six chosen beam profiles at different positions when nonlinearity is neglected (d) or considered (e). (h1) is in phase with h(3), but antiphase with h(2). Evolution of x-axis cross sections of beam profiles along the propagation direction of GIMF at the last 5 mm, when nonlinear terms are ignored (f) or considered (g). See more in Visualization 2.
Fig. 6. Spatiotemporal analysis of MMI in LMA-GIMF-SMF with and without nonlinear terms. (a) Output pulses intensity from 1 m GIMF when nonlinear terms are ignored. (b) Output pulses intensity when nonlinear terms are introduced. (c) Calculated nonlinear phase of modes at different positions in GIMF. (d)-(f) Beam profile area evolution at the first 5 mm, middle 5 mm, and last 5 mm GIMF for without (dash line) and with (solid line) nonlinear terms. (g) Light intensity evolution at last 5 mm. (h) Transmission of LMA-GIMF-SMF for different lengths of GIMF. See more in Visualization 2.
We find that the LMA-GIMF-SMF can be seen as a SA as well. The combination of a SA and the LMA-GIMF-SMF in an STML can work as a hybrid mode-locking device. The hybrid mode-locking can suppress intensity noise to improve the purity and energy of output solitons [33–35]. Figure 6(a) shows output pulses intensity from GIMF without nonlinear effects. Due to nonlinear interactions, the pulses are trapped together (see Fig. 6(b)). The area of beam profile as the function of propagation distance diverges increasingly from the area of beam profile that is gained when nonlinearity is ignored (see Figs. 6(d)-(f)), which can be explained by cumulative nonlinear phase mismatch (see Fig. 6(c)). Because of the smaller beam profile area when nonlinearity is neglected, the average light intensity is higher than that with nonlinearity (see Fig. 6(g)). Unlike SMF-GIMF-SMF, the highest transmission of LMA-GIMF-SMF happens not only at the position where modes are in phase but also in antiphase. Thus, the period of transmission is one-half of its ASIM length (see Fig. 6(h)). Additionally, the transmission exhibits a small oscillated amplitude, only ranging from 0.252 to 0.284. Similarly, When the nonlinear phase is induced, the transmission has a small shift, which ensures that it can be regarded as SA. The small phase shift of transmission results from strong nonlinear intermodal coupling that keeps the nonlinear phases of modes increasing almost synchronously along GIMF. It should be mentioned that although its modulation depth is low, a SA with as low as 0.5% modulation depth also can be used for mode-locking [36]. Besides, disturbance in transmission caused by ASIM still affects the saturable-absorption properties of LMA-GIMF-SMF, which can be seen in Visualization 2 but is not very apparent in Fig. 6(h).
To demonstrate the impact of initial energy ratio in LMA on LMA-GIMF-SMF, we simulated another situation in which the energy ratio of the fundamental mode is lowered among initial modes in LMA (energy ratio is [0.35, 0.22, 0.22, 0.09, 0.09, 0.03]). Simulations show that although the initial input changes, the period of ASIM still maintains constant, about 1200 $\mu m$, which can be seen in Visualization 3. The influence of ASIM on MMI can be seen in Visualization 3 for details. By contrast, the oscillated amplitude (modulation depth) of the transmission reduces when the energy of the fundamental mode drops down, as shown in Fig. 7. And as was expected, the average transmission of LMA-GIMF-SMF gets lower when the initial energy of the fundamental mode decreases. Because most of the energy occupied by high-order modes in GIMF is coupled into radiation and cladding modes of SMF ultimately.
Fig. 7. Transmission variation of LMA-GIMF-SMF with various lengths of GIMF for S1 (solid line) and S2 (dash-dotted line). S1: initial modal energy ratio [0.58, 0.15, 0.15, 0.05, 0.05, 0.02] in LMA; S2: initial modal energy ratio [0.35, 0.22, 0.22, 0.09, 0.09, 0.03] in LMA.
3.3 Offset-splice LMA-GIMF-SMF
For further study of the saturable-absorption and spatiotemporal features of LMA-GIMF-SMF, we simulate the situation of splicing the LMA and GIMF with a small offset (Fig. 8). By increasing the amount of offset, the modal dynamics and transmission of LMA-GIMF-SMF are investigated, respectively. To improve the property of MMI-based SA, many researchers proposed new structures. For instance, the offset splice between two different GIMFs in SMF-GIMF-GIMF-SMF is used to increase modulation depth and to get rid of the strict restrictions on the length of GIMF [37,38]. A segment of SIMF or no-core fiber(NCF) is inserted between SMF and GIMF, forming an SMF-SIMF/NCF-GIMF-SMF structure, to improve the saturable-absorption property [11,16]. The SIMF and NCF expand the mode field diameter at the entrance of the GIMF, which can excite more modes and avoid power leakage [11,16]. These approaches inspire us to ask the question that “whether offset splice is possible to increase the saturable-absorption property of LMA-GIMF-SMF?”. For another aspect, the offset splice is usually employed in STML, in which the offset splice can excite more modes and enable multimode soliton output [28,39,40]. The study on the offset splice between LMA and GIMF can offer an important implication on mode control and mode-dynamics analysis for STML.
Fig. 8. Schematic of offset-spliced large-mode-area fiber—graded-index multimode fiber—single mode fiber structure.
To begin with, the pulses in LMA are initiated with the same parameters as the first situation mentioned before. Figures 9(a)-(c) show the excited mode-solved pulses intensity for 2.5 $\mu m$ offset, 5 $\mu m$ offset, and 7.5 $\mu m$ offset, respectively, and their corresponding beam profiles are illustrated in Figs. 9(d)-(f). When the spliced offset increases, the more high-order modes will be excited. In the time domain, the output pulses from 1m GIMF for the three cases are shown in Figs. 9(g)-(i). The inserts in Figs. 9(g)-(i) are corresponding pulse intensity in SMF after mode coupling from GIMF into SMF. Notably, When the spliced offset is increasing, the walk-off between pulses becomes larger. The reason for the observation is that the frequency shifts of pulses caused by the XPM decrease—as the number of modes increases, the intensity of the original pulses decreases because of energy spreading to more modes, which leads to the weakening of the XPM effect; Thus, the modal dispersion cannot be compensated by chromatic dispersion. Furthermore, the beam profile evolutions for different spliced offsets are illustrated in Figs. 10(a)-(c), in which the variation of beam profile is getting intense as the offset gets larger. But the period of ASIM is the same, about 1200 $\mu m$. Figures 10(d)-(f) show a part of beam profiles for different spliced offsets to show ASIM.
Fig. 9. Comparison of Pulses intensity and beam profiles for different spliced offsets. Excited mode-solved pulses intensity at the entrance of GIMF for 2.5 $\mu m$ spliced offset (a), 5 $\mu m$ spliced offset (b), and 7.5 $\mu m$ spliced offset (c), respectively. (d)-(f) Corresponding beam profiles of (a)-(c). (g)-(i) Corresponding output pulses from 1 m GIMF. The inserts in (g)-(i) show corresponding pulse intensity in SMF after mode coupling from GIMF into SMF.
Fig. 10. Beam profile evolution for different spliced offsets. (a), (d) for 2.5 $\mu m$ spliced offset. (b), (e) for 5 $\mu m$ spliced offset. (c), (f) for 7.5 $\mu m$ spliced offset. (a), (b), and (c) are evolutions of x-axis cross sections of beam profiles along the propagation direction of GIMF at the last 5 mm (c). (d), (e), and (f) are chosen beam profiles at different positions. See more in Visualization 4, Visualization 5, and Visualization 6.
The nonlinear phases for the three cases are also calculated and presented in Figs. 11(a)-(c). Because of the decrease in pulse intensity as the offset increases, the nonlinear phase decreases and tends to be linear with propagation length. Figures 11(d)-(f) show the area variation of beam profiles with propagation distance for different spliced offsets. Because of the growing number of high-order modes as the offset increases, the average area of the beam profile gradually expands. By contrast, the average light intensity weakens when the offset increases, as shown in Fig. 11(g). Otherwise, the variations of transmission with different GIMF lengths for these cases are shown in Fig. 11(h). As is apparent in Fig. 11(h), the average transmission of offset-spliced LMA-GIMF-SMF decreases when the offset enlarges due to more energy leakage into radiation and cladding modes. Nonetheless, the modulation depth of the offset-spliced LMA-GIMF-SMF increases as the offset enlarges. The improving modulation depth can promote high-energy pulse output in STML. Furthermore, as the offset increases, the degree of disturbance induced by ASIM in transmission changes. The transmission for the intermediate offset (5 $\mu m$) exhibits the smallest disturbance and maintains a nearly standard cosine (see Visualization 4, Visualization 5, and Visualization 6).
Fig. 11. Characterization of MMI in offset-spliced LMA-GIMF-SMF for different offsets. Accumulative nonlinear phase along GIMF for 2.5 $\mu m$ offset (a), 5 $\mu m$ offset (b), and 7.5 $\mu m$ offset (c). (d)-(f) Beam profile area evolution at the first 5 mm, middle 5 mm, and last 5 mm of GIMF for different offsets. (g) Light intensity evolution. (h) Transmission of LMA-GIMF-SMF for different offsets. See more in Visualization 4, Visualization 5, and Visualization 6.
4. Conclusion
Using GMMNLSE and the transmission model, we conduct the theoretical study of spatiotemporal characteristics of MMI in SMF/LMA-GIMF-SMF. Simulations unveil several results. (I) The longest beating length in GIMF determines the SIM length. The SIM is imperfect due to the non-strictly equispaced propagation constants of modes in GIMF, which is shown as ASIM behavior. The ASIM can perturb the transmission of SMF/LMA-GIMF-SMF, causing instability of their saturable-absorption characteristics. (II) Nonlinear effects play an important role in affecting MMI. Significant nonlinear effects and mode coupling caused by the high pulse intensity prevent modal walk-off and decrease the nonlinear phase mismatch of the modes. This nonlinear phase makes the phase shift of transmission function within $\pi $, facilitating SMF-GIMF-SMF-based SA to support high-energy pulse output. (III) In the case of SMF-GIMF-SMF, the transmittance rises as the output beam profile from GIMF enlarges. When the local maximum area of beam profile in GIMF reaches, all modes are in phase, so the transmittance of SMF-GIMF-SMF is highest. Moreover, its modulation depth can reach as high as 55%. These characteristics of SMF-GIMF-SMF make it to be an ideal SA that can tolerate high energy output for mode-locked fiber lasers. (IV) For LMA-GIMF-SMF, it has been proved that it can be used as a fiber-based mode exciter and filter, as well as an additional SA for STML to improve the energy and purity of multimode solitons. Because the excited modes are intricate, the beam profile evolution in GIMF is more dramatic, and its ASIM length is twice longer compared with SMF-GIMF-SMF. Different from SMF-GIMF-SMF, when modes are in phase, the beam profile is the smallest. Its transmission and modulation depth can be further improved by raising the energy proportion of the fundamental mode in LMA. (V) For further study of the saturable-absorption and spatiotemporal characters of LMA-GIMF-SMF, we splice the LMA and GIMF with a small offset. Apart from facilitating multimode output for STML as spliced offset increases, the modulation depth of LMA-GIMF-SMF-based SA increases as well. The improving modulation depth will further promote high-energy pulse output in STML. Additionally, Simulation shows that the degree of disturbance induced by ASIM in its transmission changes with offset increases. The transmission for the intermediate offset (5 $\mu m$) exhibits the smallest disturbance and maintains a variation of nearly standard cosine.
We believe the reported numerical approach and results provide new perspectives for the MMI in SMF/LMA-GIMF-SMF. Although MMI has been extensively used in plenty of applications, the understanding of its internal spatiotemporal dynamics was very limited before. This work expands the knowledge of both linear and nonlinear spatiotemporal dynamics in SMF/LMA-GIMF-SMF, as well as intuitively explains their principle and functionality. Besides, the proposed method can be an effective tool for simulating MMI in other multimode waveguides, Other than fibers. In terms of application, the presented results provide theoretical guidance for MMI-based devices, such as MMI-based SA and filters used in mode-locked fiber lasers. Moreover, the study of LMA-GIMF-SMF with large cores makes several contributions to STML or LMA mode-locked laser to improve its pulse energy and stability.
Funding
National Natural Science Foundation of China (61875103, 62075113, 62122040).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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