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Spatiotemporal Evolutions of Similariton Pulses in Multimode Fibers with Raman Amplification

Spatiotemporal Evolutions of Similariton Pulses in Multimode Fibers with Raman Amplification hv photonics Communication Spatiotemporal Evolutions of Similariton Pulses in Multimode Fibers with Raman Amplification 1 , 2 2 , Leila Graini and Bülend Ortaç * Telecommunications Laboratory, 8 Mai 1945—Guelma University, Guelma 24000, Algeria; graini.leila@univ-guelma.dz National Nanotechnology Research Center and Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey * Correspondence: ortac@bilkent.unam.edu.tr Abstract: This paper aims to pave the way towards the demonstration of spatiotemporal similariton pulses’ evolution in passive multimode fibers with Raman amplification. We numerically present this issue in graded-index and step-index multimode fibers and provide a first look at the complex spatiotemporal dynamics of similariton pulses. The results showed that the similariton pulses could be generated in both multimode fibers. The temporal and spectral evolution of the pulses can be characterized as parabolic shapes with linear chirp and kW peak power. By compressing these, high-energy femtoseconds pulses can be obtained, starting initial picosecond pulses. A spatial beam profile could be preserved in both multimode fibers with the most energy coupled to the fundamental mode. Specifically, the similariton pulses’ generation with Raman amplification in a graded-index multimode fiber improves the spatial beam self-cleaning process under the different initial modes’ excitation. The observation of a new beam self-cleaning process is another attractor feature of propagation in graded-index multimode fibers. Keywords: spatiotemporal similariton; Raman amplification; multimode fibers Citation: Graini, L.; Ortaç, B. Spatiotemporal Evolutions of Similariton Pulses in Multimode 1. Introduction Fibers with Raman Amplification. In recent years, there has been increased interest in nonlinear wave propagation in Photonics 2021, 8, 354. https:// multimode optical fibers (MM-Fibers). The traditional view of MM-Fibers as undesirable doi.org/10.3390/photonics8090354 for nonlinear applications is gradually changing since MM-Fibers support a variety of new spatiotemporal phenomena, such as multimode solitons, supercontinuum generation, the Received: 2 July 2021 multiple filamentation process, and wave condensation [1]. The formation of multimode Accepted: 23 August 2021 solitons is limited by different group delays associated with different propagating modes Published: 27 August 2021 and nonlinear coupling terms, as was reported by early workers [2]. On the other hand, graded-index multimode fibers (GRIN-Fs) reduce the complexity due to the self-imaging Publisher’s Note: MDPI stays neutral effect resulting from the equal spacing of the modal wave with low intermodal disper- with regard to jurisdictional claims in sion. The theory of multimode light propagation in GRIN-Fs was established in the early published maps and institutional affil- 1970s [3]; very recently, the studies of the complex nonlinear spatiotemporal effects in these iations. MM-Fibers have been investigated [4]. Spatiotemporal optical solitons [5], supercontinuum generation [6], spatiotemporal instability [7], and self-similar fiber lasers [8] are already reported. The above-mentioned numerical studies and experimental observations are remarkable examples showing the complex nonlinear pulse dynamics in MM-Fibers. Moti- Copyright: © 2021 by the authors. vated by these works and the others, one may thus naturally wonder whether similariton Licensee MDPI, Basel, Switzerland. pulses may also occur in MM-Fibers. This article is an open access article Similariton pulses are shape-preserving waves that appear due to the interaction of distributed under the terms and nonlinearity, normal dispersion, and gain in optical fibers. This causes the shape of an conditions of the Creative Commons arbitrary input pulse to converge to a parabolic pulse shape and evolves in a self-similar Attribution (CC BY) license (https:// manner. Temporal similaritons have been studied extensively in the context of single-mode creativecommons.org/licenses/by/ optical fibers (SMFs) with different amplification processes (Raman, Erbium, and Ytterbium 4.0/). Photonics 2021, 8, 354. https://doi.org/10.3390/photonics8090354 https://www.mdpi.com/journal/photonics Photonics 2021, 8, x FOR PEER REVIEW 2 of 11 mode optical fibers (SMFs) with different amplification processes (Raman, Erbium, and Ytterbium amplification) [9–12]. These studies have already demonstrated the interesting properties of amplifiers’ similaritons pulses in conventional SMFs. The platform based on MM-Fibers could be a good candidate for the amplification process as well and the ques- tion of whether spatiotemporal similariton pulses could be formed inside the MM-Fibers. In this paper, we propose a new configuration to demonstrate the existence of the spatiotemporal similariton pulses in two different passive MM-Fiber types (GRIN-F and Step-Index Fiber (STEP-F)) with Raman amplification. For our numerical studies, we used Photonics 2021, 8, 354 2 of 11 the generalized multimode nonlinear Schrödinger equation (GMMNLSE) considering the interaction of nonlinearity, normal dispersions, gain, and modal interactions represented by mode coupling. The gain is generated by Raman amplification owing to their ad- amplification) [9–12]. These studies have already demonstrated the interesting properties vantages. It may occur at any passive fiber type and exists at any wavelength as long as of amplifiers’ similaritons pulses in conventional SMFs. The platform based on MM-Fibers an appropriate pump laser is available. The results showed that the characteristics of the could be a good candidate for the amplification process as well and the question of whether similariton pulses generated in both MM-Fibers present similar behavior, as obtained in spatiotemporal similariton pulses could be formed inside the MM-Fibers. SMFs with the most energy coupled to the fundamental mode. High energy similariton In this paper, we propose a new configuration to demonstrate the existence of the pulses have temporal and spectral parabolic shapes with a linear chirp. The pulses gener- spatiotemporal similariton pulses in two different passive MM-Fiber types (GRIN-F and ated in both cases can also be compressible down to a femtosecond pulse duration with Step-Index Fiber (STEP-F)) with Raman amplification. For our numerical studies, we used ~100 kW of peak power. However, the spatiotemporal dynamics, as well as the spatial the generalized multimode nonlinear Schrödinger equation (GMMNLSE) considering the beam profiles under different initial modes’ excitation in both MM-Fibers, are very differ- interaction of nonlinearity, normal dispersions, gain, and modal interactions represented by ent. A new type of beam self-cleaning process based on similariton pulse generation with mode coupling. The gain is generated by Raman amplification owing to their advantages. It Raman amplification in GRIN-F could be observed. may occur at any passive fiber type and exists at any wavelength as long as an appropriate pump laser is available. The results showed that the characteristics of the similariton pulses 2. Numerical Setup and Results generated in both MM-Fibers present similar behavior, as obtained in SMFs with the most To investigate the spatiotemporal evolutions of similariton pulses in MM-Fibers with energy coupled to the fundamental mode. High energy similariton pulses have temporal Raman amplification, we used the platform presented in Figure 1. The MM-Fibers used in and spectral parabolic shapes with a linear chirp. The pulses generated in both cases our numerical studies have parabolic and step refractive index profiles in the GRIN-F and can also be compressible down to a femtosecond pulse duration with ~100 kW of peak the STEP-F with an identical core radius of 25 µm and an index contrast of Δ = 0.0068, power. However, the spatiotemporal dynamics, as well as the spatial beam profiles under where NA = 0.17, respectively. To reduce the computational time, we considered the first different initial modes’ excitation in both MM-Fibers, are very different. A new type of six linearly polarized modes (LPmn), as shown in Figure 1. beam self-cleaning process based on similariton pulse generation with Raman amplification We calculated the dispersions, the nonlinear, and the coupling coefficients using in GRIN-F could be observed. codes introduced by Wright et al. [13]. The MM-Fibers characterized by the dispersions 2 2 coefficients β2 for the six modes are as follows: 16.76 ps /km (LP01), 16.75 ps /km (LP11a and 2. Numerical Setup and Results 2 2 LP11b), and 16.74 ps /km (LP21a, LP21b, and LP02) for the GRIN-F and 16.49 ps /km (LP01), 2 2 2 16.0T 6 p o investigate s /km (LP11athe and LP spatiotemporal 11b), 15.50 psevolutions /km (LP21a an of similariton d LP21b), and pulses 15.31 p insMM-Fibers /km (LP02) for with Raman the STEP amplification, -F at 1060 nm. Th we used e effective the platform areas, A pr effesented , and the nonline in Figurear 1 coeffic . The MM-Fibers ients, γ, of the used 2 −1 −1 2 −1 GRIN-F and the STEP-F are 155.09 µm and 1.22 W km , and 966.43 µm and 0.2 W in our numerical studies have parabolic and step refractive index profiles in the GRIN-F −1 and kmthe , rSTEP-F espectively. A with an s the identical effective coreare radius a of the fundamenta of 25 m and an l mode i index contrast n the MM-Fi of D = bers 0.0068, is scaled as the core radius, the effective nonlinearity of the MM-Fibers is comparable with where NA = 0.17, respectively. To reduce the computational time, we considered the first SMFs [4]. These features make both MM-Fibers suitable for Raman amplification. six linearly polarized modes (LP ), as shown in Figure 1. mn Figure 1. Simulations set-up for spatiotemporal evolutions of similariton pulses in MM-Fibers with Figure 1. Simulations set-up for spatiotemporal evolutions of similariton pulses in MM-Fibers with Raman amplification. Raman amplification. We calculated the dispersions, the nonlinear, and the coupling coefficients using codes introduced by Wright et al. [13]. The MM-Fibers characterized by the dispersions 2 2 coefficients b for the six modes are as follows: 16.76 ps /km (LP ), 16.75 ps /km (LP 2 01 11a 2 2 and LP ), and 16.74 ps /km (LP , LP and LP ) for the GRIN-F and 16.49 ps /km 11b 21a 21b, 02 2 2 2 (LP ), 16.06 ps /km (LP and LP ), 15.50 ps /km (LP and LP ), and 15.31 ps /km 01 11a 11b 21a 21b (LP ) for the STEP-F at 1060 nm. The effective areas, A , and the nonlinear coefficients, , 02 eff 2 1 1 2 of the GRIN-F and the STEP-F are 155.09 m and 1.22 W km , and 966.43 m and 1 1 0.2 W km , respectively. As the effective area of the fundamental mode in the MM- Fibers is scaled as the core radius, the effective nonlinearity of the MM-Fibers is comparable with SMFs [4]. These features make both MM-Fibers suitable for Raman amplification. Numerical simulations for the spatiotemporal nonlinear dynamics of similariton pulses are conducted using the GMMNLSE [14] and solved using the split-step Fourier Photonics 2021, 8, x FOR PEER REVIEW 3 of 11 Numerical simulations for the spatiotemporal nonlinear dynamics of similariton pulses are conducted using the GMMNLSE [14] and solved using the split-step Fourier Photonics 2021, 8, 354 3 of 11 method [15]. The GMM-NLSE (1) is including the effects of stimulated Raman scattering, higher-order dispersion, self-steepening, and intermodal effects. ( ) ( ) ( ) ( ) ( ) method [15]. The GMM-NLSE (1) is including the effects of stimulated Raman scattering, 𝜕 𝐴 𝑖𝛽 ℜ𝛽 𝐴 𝛽 ℜ𝛽 𝑖 ∑ 𝑖 𝐴 𝑖 (1 𝜕𝑡) ∑ (1 ,, higher-order dispersion, self-steepening, and intermodal effects. (1) h i  ∗ h i  ∗ (p) n ) ( ) 𝑓 𝑆 𝐴 𝐴 𝐴 𝑓 𝐴 𝑆 𝑑𝜏 𝐴 𝑧,𝑡𝜏 𝐴 (𝑧,𝑡𝜏)ℎ n (𝜏) (p) (p) ( 0) (0) ¶A b 2 w n ¶ i P 0 ¶ A = i b < b A b < b + i i A + i 1 + ¶t z p p å p å n2 l,m,n 0 0 1 1 n! c w ¶t ¶t n o (1) k  R (1 f )S A A A + f A S dtA (z, t t)A (z, t t)h (t) m m R l n R l n R plmn plmn where Ap(z,t) is the electric field temporal envelope for spatial mode p; and the first, the second, and the third term on the right-hand side indicate the propagation constant mis- where Ap(z,t) is the electric field temporal envelope for spatial mode p; and the first, the match responsible for multimode interference or mode beating, the modal dispersion or second, and the third term on the right-hand side indicate the propagation constant mis- modal walk-off and the higher-order dispersion coefficients, respectively. In the fourth match responsible for multimode interference or mode beating, the modal dispersion or −20 2 −1 term, n2 represents the nonlinear refractive index (n2 = 3.2 × 10 m W ), fR is the fractional modal walk-off and the higher-order dispersion coefficients, respectively. In the fourth contribution of the Raman effect (fR = 0.18), 𝑆 and 𝑆 represent the nonlinear cou- 20 2 1 term, n represents the nonlinear refractive index (n = 3.2  10 m W ), f is the frac- 2 2 R pling coefficients for the Raman and Kerr effect, krespectiveRly, hR is the delayed Raman tional contribution of the Raman effect (f = 0.18), S and S represent the nonlinear plmn plmn response function, and shock terms are neglected. coupling coefficients for the Raman and Kerr effect, respectively, h is the delayed Raman The gain can be added into the GMMNLSE by an additional term, gpAp, where gp is response function, and shock terms are neglected. the small-signal gain for the electric field in mode p. For short pulses, gp is defined as gp = The gain can be added into the GMMNLSE by an additional term, g A , where g p p p g0/(1 + Ep/Esat), where g0 is the initial gain, Ep is the energy of mode p, and Esat is the satura- is the small-signal gain for the electric field in mode p. For short pulses, g is defined as tion energy. Esat, for both MM-Fibers, is estimated to be about 1.2 nJ. In our numerical g = g /(1 + E E ), where g is the initial gain, E is the energy of mode p, and E is p sat p sat 0 p/ 0 studies, we selected the CW-pump source operating at 1010 nm for Raman gain. A Gauss- the saturation energy. E , for both MM-Fibers, is estimated to be about 1.2 nJ. In our sat ian-shaped input pulse operating at 1060 nm with a pulse duration of 2 ps, pulse energy numerical studies, we selected the CW-pump source operating at 1010 nm for Raman gain. of 0.4 nJ, peak power of 200 W, and spectral bandwidth of 0.8 nm is used. The shift of 13.2 A Gaussian-shaped input pulse operating at 1060 nm with a pulse duration of 2 ps, pulse THz (corresponding to 50 nm) between the pump and the signal is selected for optimum energy of 0.4 nJ, peak power of 200 W, and spectral bandwidth of 0.8 nm is used. The Raman gain in the silica fibers. The fiber length of 130 m is chosen to ensure sufficient shift of 13.2 THz (corresponding to 50 nm) between the pump and the signal is selected Raman gain. for optimum Raman gain in the silica fibers. The fiber length of 130 m is chosen to ensure In MM-Fibers with gain, the different modes receive gain and usually compete since sufficient Raman gain. they overlap with the same gain medium. In our case, the competition between the pump In MM-Fibers with gain, the different modes receive gain and usually compete since and signal modes could not be considered due to the CW-pump used for Raman gain. The they overlap with the same gain medium. In our case, the competition between the pump Raman gain for each mode is defined as Gp (dB) = exp (gR P0 Leff/Aeff) [16], where Leff is the and signal modes could not be considered due to the CW-pump used for Raman gain. The effective fiber length, P0 is the CW-pump power, and gR is the Raman gain coefficient at Raman gain for each mode is defined as G = exp (g P L /A ) [16], where L is (dB) R 0 eff eff eff the CW-pump wavelength. Note that the modal gain can be controlled by the effective the effective fiber length, P is the CW-pump power, and g is the Raman gain coefficient 0 R area of modes, at the CW-pump wavelength. Note that the modal gain can be controlled by the effective Aeff, and the Raman gain coefficient (considering the CW-pump power is fixed). Con- area of modes, A , and the Raman gain coefficient (considering the CW-pump power is eff fixed). sequentl Consequently y, the variat ,ithe on of variation the Ram of an gai the Raman n for ea gain ch sp for atia each l mospatial de as a f mode unctias on o a function f the pro- of posed the pr MM oposed -Fiber’s length MM-Fiberis ev ’s length aluated is evaluated in Figure 2a,b in Figur for the e 2a,b GRI for N-F the and GRIN-F the STand EP-Fthe , re- STEP-F, respectively, considering the overfilling nature of the CW-pump and the power is spectively, considering the overfilling nature of the CW-pump and the power is 100 W. 100 We W also c . We a also lculated calculated on the t onw the o plots two plots in Figur in Fi e gur 2c, the e 2c,effect the ef ive fective area of mo area ofdes modes in the pro- in the proposed MM-Fibers confirm that the effective area of modes in the STEP-F is larger than posed MM-Fibers confirm that the effective area of modes in the STEP-F is larger than in in the GRIN-F of the same radius [14]. the GRIN-F of the same radius [14]. (b) (c) (a) Figure 2. (a,b) Variation of modal gain for all modes in the GRIN-F and the STEP-F, respectively, (c) and the effective area of modes. From Figure 2a,b, we note that the distribution of gain inside both MM-Fibers is different due to the difference in the structures and the mechanism of the propagations between them. In the GRIN-F, the fundamental mode, LP , received the largest gain Photonics 2021, 8, 354 4 of 11 compared to the higher-order modes (HOMs). Primarily, because the LP mode is more confined in the higher gain region, since the relatively high GeO dopants in the center of the GRIN-F core result in a higher peak Raman gain coefficient. The peak value of the Raman gain coefficient is estimated as g = 7  10 m/W (at a 1060-nanometer wavelength and considering the GeO2 doping). Second, since the modal gain is inversely proportional to the effective modal area, the latter being higher for the LP mode with the smaller effective area, as shown in Figure 2c (upper figure). The overlap between other HOMs and the pump results in a smaller gain. The modal gain at the fiber length of 130 m is equal to 20, 2, 1.5, and 0.5 dB for the LP , LP , LP and LP mode, respectively. 01 02 11ab 21aba, In the case of STEP-F, the different modes could receive the same gain since the Raman gain is flat and uniformly distributed along with the STEP-F core. Note, however, that the LP and LP modes have the larger and the smaller effective areas (Figure 2c), respectively, confirming that the effective area of the LP modes decreases with the mode order in the 0n STEP-F [13]. The effective areas of the other modes, LP , LP , LP , and LP , are 11a 11b 21a 21b nearly equal. As a result, the overlap between the CW-pump and all the modes under consideration leads to modal gain variations, as shown in Figure 2b. The gain of the LP mode is equal to 20 dB and that of the LP mode is 25 dB, where the gain of LP and 02 11a-b LP is 21.3 and 21.5 dB, respectively. 21a-b To study the dynamics of similariton pulses in both MM-Fibers, and to present their characteristics and behaviors compared to those generated in SMFs, we first started by exciting only the fundamental mode, LP . We assume the Gaussian-shaped input pulse with a pulse duration of 2 ps and energy of 0.4 nJ propagates through 130 m of the proposed MM-Fibers and under the Raman amplification effect. We set the pulse energy, mostly coupled in the LP mode, as 99%. We consider the weak excitation of the other modes, which could be the case in an experiment. The temporal and the spectral forms of all the modes at the MM-Fibers’ output can be obtained, as shown in Figure 3a–e, which is presented in uniform and normalized scales. These results show that only the LP mode evolves towards a parabolic form and, subsequently, develops a wide temporal and spectral expansion. As expected for all the other modes, only the LP mode is strongly amplified and its energy increased exponentially with the fiber length to reach a value of 37.8 nJ, as shown in Figure 3c,f, which validates the amplifier similariton pulse formation of the LP mode, while the other HOMs did not converge to the similariton pulses. As the similaritons depend only on the pulse energy and on the gain of the amplifier, the most initial pulse energy coupling into the LP mode results in a higher nonlinearity compared with the other HOMs. Therefore, even with the same amount of gain distributed among all the modes, as in the case of the STEP-F, only the LP mode transforms into the parabolic shape and gains energy. The other HOMs did not transform into the parabolic shape and then lost their energy, which is also the result of the inadequate nonlinear phase accumulation. It can be understood as follows: The interplay of nonlinearity, normal dispersion, and gain is important for reshaping and maintaining the parabolic pulse shape during propagation in the fiber. It should be noticed that the energy exchange is observed between the HOMs in the GRIN-F (Figure 3a,b) because the intermodal group delays are smaller inside this type of fiber. However, the energy level of HOMs (0.072 nJ) is extremely low compared with that of the LP mode and, thus, any perturbation on the similariton pulse evolution did not occur during its propagation in the GRIN-F. On the other hand, the energy coupling to the HOMs is not observed in the STEP-F (Figure 3d,e) and each of them evolve as in a single-mode case. Photonics 2021, 8, 354 5 of 11 Photonics 2021, 8, x FOR PEER REVIEW 5 of 11 (a) (b) (c) (d) (e) (f) Figure 3. (a–c) Spectral, temporal forms and pulse energy evolutions for 130 m of pulse propagation Figure 3. (a–c) Spectral, temporal forms and pulse energy evolutions for 130 m of pulse propagation for all the modes in the GRIN-F and (d–f) in the STEP-F, respectively, where the most initial energy for all the modes in the GRIN-F and (d–f) in the STEP-F, respectively, where the most initial energy is is coupled in the LP01 mode. Inset: output spatial beams. coupled in the LP mode. Inset: output spatial beams. Figure 4 shows the formation of the similariton pulses of the LP01 mode in temporal Figure 4 shows the formation of the similariton pulses of the LP mode in temporal and spectral domains inside the proposed MM-Fibers. Figure 4a,c are highlighting the and spectral domains inside the proposed MM-Fibers. Figure 4a,c are highlighting the reshaping of the pulse during its propagation inside the GRIN-F, followed by its self-sim- reshaping of the pulse during its propagation inside the GRIN-F, followed by its self- ilar evolution that appeared according to two parts of separate regimes. The first part is similar evolution that appeared according to two parts of separate regimes. The first part is corresponding to the restructuring and the reshaping of the LP01 mode and the appearance corresponding to the restructuring and the reshaping of the LP mode and the appearance of characteristics of the parabolic pulse profile. A typical parabolic shape is obtained at 30 of characteristics of the parabolic pulse profile. A typical parabolic shape is obtained at m of pulse propagation inside the GRIN-F, which is defined as the characteristic length Zc 30 m of pulse propagation inside the GRIN-F, which is defined as the characteristic length = (3/2g0) ln (Ng0/6γPc) [10], calculated for N~100. In the second part, beyond ZC, the LP01 Z = (3/2g ) ln (Ng /6 P ) [10], calculated for N~100. In the second part, beyond Z , the c c 0 0 C mode has the characteristics of similariton and evolves in a self-similar manner with an LP mode has the characteristics of similariton and evolves in a self-similar manner with increase in its wide temporal and spectral expansion. We note the absence of any pulse an increase in its wide temporal and spectral expansion. We note the absence of any pulse deformation or oscillation caused by wave breaking, and modulation instability side- deformation or oscillation caused by wave breaking, and modulation instability sidebands did bands d not occur id not occur in the in the generated gener spectr ated sp um. ectrum At the . At the output of t output of the GRIN-F he GRIN-F, t , the input Gaussian he input Gaussian pulse is now transformed into a parabolic shape with a linear chirp (Figure 4d). pulse is now transformed into a parabolic shape with a linear chirp (Figure 4d). The pulse duration The pulse d is u 32 ration ps and is 32 p the spulse and the pulse ener energy is about gy is 37.8 about 37.8 n nJ. The spectr J. The spectrum um generat generated ed by the self-similar by the self-simi amplificat lar amion plifica regime tion regi alsome has ala so parabolic has a parabol form i(Figu c form re(4 Fb) igure with 4b a) w bandwidth ith a band- of 16 width of 16 n nm. The small m. The small ripples seen ripples in the seen in the middle of middle the spectr of the spectrum are the result of the um are the result of the limited bandwidth limited banby dwidt theh by finite th spectral e finite spect bandwidth ral bandw of Raman idth ogai f Ram n (about an gain 23 (nm about accor 23 n ding m accor to [17 d]). - At ing t the o [ same 17]). time, At the s Figur ame e t 4ie–h me, show Figure the 4eself-similar –h show the regime self-sim and ilar similariton regime and formation similarito for n 130 format m of ion pulse for 1 pr 30 opagat m of pu ionls inside e propa the gat STEP-F ion ins , alt ide hough the ST the EP-F mechanism , although t ofhthe e mech propagations anism of is the propa different. gaSpecifically tions is different. , the time Specif duration ically, th and e time the d ener uration gy of and t the output he energ pulse y of tar he output e 15.5 ps and 37.8 nJ, respectively (Figure 4h). Here, the input pulse has evolved as a parabolic shape pulse are 15.5 ps and 37.8 nJ, respectively (Figure 4h). Here, the input pulse has evolved beyond as a par55 abm oliof c sh prap opagation, e beyond 5 which 5 m of is just prop a transient agation, state which ofis j theupulse st a tr evolution ansient stin ate o theffiber the and under amplification (Figure 4e,g). The ratio between the nonlinearity coefficient ( ) pulse evolution in the fiber and under amplification (Figure 4e,g). The ratio between the and the dispersion ( ) is high in the GRIN-F compared to that in the STEP-F. Therefore, we nonlinearity coefficient (γ) and the dispersion (β2) is high in the GRIN-F compared to that observed that the input pulse converges to a parabolic shape rapidly in the GRIN-F (30 m) in the STEP-F. Therefore, we observed that the input pulse converges to a parabolic shape than in the STEP-F (55 m). Likewise, the decrease in the nonlinearity coefficient caused rapidly in the GRIN-F (30 m) than in the STEP-F (55 m). Likewise, the decrease in the less spectral broadening (about 9 nm) for the STEP-F (Figure 4f). The above results imply nonlinearity coefficient caused less spectral broadening (about 9 nm) for the STEP-F (Fig- that the output pulses have a parabolic pulse shape with a linear chirp and a high-energy ure 4f). The above results imply that the output pulses have a parabolic pulse shape with amplification. These pulses generated in the GRIN-F and the STEP-F can be compressed a linear chirp and a high-energy amplification. These pulses generated in the GRIN-F and down to 255 and 340 fs, respectively (inset Figure 4d,h). The compressed pulses’ peak Photonics 2021, 8, x FOR PEER REVIEW 6 of 11 Photonics 2021, 8, 354 6 of 11 the STEP-F can be compressed down to 255 and 340 fs, respectively (inset Figure 4d,h). power is ~100 KW and that, when compared to the original input pulse (pulse duration of The compressed pulses’ peak power is ~100 KW and that, when compared to the original 2 ps and peak power of 200 W), represents a factor of ~8 and ~6 temporal compressions, input pulse (pulse duration of 2 ps and peak power of 200 W), represents a factor of ~8 and an amplification factor of 500 could be achieved in both cases. It should be noted here and ~6 temporal compressions, and an amplification factor of 500 could be achieved in that the pedestals on the compressed pulses are being observed. However, their energy is both cases. It should be noted here that the pedestals on the compressed pulses are being extremely low compared to that of the main lobes. We have estimated 70 and 90% of the observed. However, their energy is extremely low compared to that of the main lobes. We total energy is contained in the main lobes of the compressed pulses, while 30 and 10% is in have estimated 70 and 90% of the total energy is contained in the main lobes of the com- the pedestals by comparing the energies of the similariton pulses (37.8 nJ) with those of the pressed pulses, while 30 and 10% is in the pedestals by comparing the energies of the compressed pulses (25.5 and 34 nJ), in the case of the GRIN-F and the STEP-F, respectively. similariton pulses (37.8 nJ) with those of the compressed pulses (25.5 and 34 nJ), in the We can mention that the pedestals could be observed as the features of the other HOMs. case of the GRIN-F and the STEP-F, respectively. We can mention that the pedestals could Thus, the pedestals on the compressed pulses are more relevant in the case of the GRIN-F, be observed as the features of the other HOMs. Thus, the pedestals on the compressed because the walk-off of the modes is smaller inside this type of fiber (Figure 3a,b), and pulses are more relevant in the case of the GRIN-F, because the walk-off of the modes is thereby the generated temporal profile of the compressed pulse is more affected by the smaller inside this type of fiber (Figure 3a,b), and thereby the generated temporal profile presence of the other HOMs. of the compressed pulse is more affected by the presence of the other HOMs. (b) (f) (a) (e) (c) (g) (d) (h) Figure 4. (a–d) The spectral and the temporal similariton evolutions and output forms of the LP mode in the GRIN-F, and Figure 4. (a–d) The spectral and the temporal similariton evolutions and output forms of the LP01 mode in the GRIN-F, (e–h) in the STEP-F. Inset: the compressed pulses. and (e–h) in the STEP-F. Inset: the compressed pulses. The output spatial profiles of the total field containing all the modes during the pulse The output spatial profiles of the total field containing all the modes during the pulse propagation inside the GRIN-F and the STEP-F are presented in the inset of Figure 3c,f. propagation inside the GRIN-F and the STEP-F are presented in the inset of Figure 3c,f. The excitation of the LP mode with the most energy results in the Gaussian intensity The excitation of the LP01 mode with the most energy results in the Gaussian intensity distributions being preserved along the both MM-Fibers. Due to the different mode field distributions being preserved along the both MM-Fibers. Due to the different mode field diameter behavior of the LP mode for both MM-Fibers, we also observed the different diameter behavior of the LP01 mode for both MM-Fibers, we also observed the different beam sizes. beam sizes. Later we consider the effect of different launch conditions on the spatiotemporal pulse Later we consider the effect of different launch conditions on the spatiotemporal evolution. In this part, the simulations are performed to investigate the effect of the initial pulse evolution. In this part, the simulations are performed to investigate the effect of the input energy distribution between the modes. To create a significant change, we consider initial input energy distribution between the modes. To create a significant change, we the first six modes with equal initial energy distributions (total peak power of 200 W among consider the first six modes with equal initial energy distributions (total peak power of all the modes). Indeed, such a configuration can be achieved using a spatial light modulator 200 W among all the modes). Indeed, such a configuration can be achieved using a spatial (SLM) [18]. We considered the same GRIN–F and STEP–F parameters. For a propagation light modulator (SLM) [18]. We considered the same GRIN–F and STEP–F parameters. length of 130 m, we noticed that the initial pulse is converged to a similariton pulse in the For a propagation length of 130 m, we noticed that the initial pulse is converged to a sim- GRIN-F for the LP mode only, and it is significantly amplified compared to the HOMs, ilariton pulse in the GRIN-F for the LP01 mode only, and it is significantly amplified com- as shown in Figure 5a–c. While the HOMs did not gain energy, even they have the same pared to the HOMs, as shown in Figure 5a–c. While the HOMs did not gain energy, even amount of initial energy. It can be explained that the similariton pulse formation in the they have the same amount of initial energy. It can be explained that the similariton pulse GRIN-F significantly depends on the Raman gain of modes, which favors the LP mode formation in the GRIN-F significantly depends on the Raman gain of modes, which favors since it is received the largest gain compared to HOMs (Figure 2a). On the other hand, in the LP01 mode since it is received the largest gain compared to HOMs (Figure 2a). On the the STEP-F, the pulse is broken up into sub-pulses, presented by different modes, whose other hand, in the STEP-F, the pulse is broken up into sub-pulses, presented by different separation grows linearly with the fiber length owing to the large walk-off between modes, Photonics 2021, 8, x FOR PEER REVIEW 7 of 11 Photonics 2021, 8, 354 7 of 11 modes, whose separation grows linearly with the fiber length owing to the large walk-off between modes, because the modes have rather different group velocities, and each of because the modes have rather different group velocities, and each of them travels with them travels with their group velocity as in a single-mode case. Therefore, all the modes their group velocity as in a single-mode case. Therefore, all the modes evolved towards the evolved towards the respective similariton forms, and they are amplified with a different respective similariton forms, and they are amplified with a different modal gain coefficient modal gain coefficient (Figure 5d–f). The greater walk-off of the modes in the STEP-F (Figure 5d–f). The greater walk-off of the modes in the STEP-F means that the coupling means that the coupling between HOMs may be less efficient. However, when the energy between HOMs may be less efficient. However, when the energy of the modes increased of the modes increased during the similaritons’ formation process, the coupling between during the similaritons’ formation process, the coupling between the degenerate modes, the degenerate modes, which travel at the same group velocity, seems very relevant in the which travel at the same group velocity, seems very relevant in the case of LP and 21a case of LP21a and LP21b, as shown in Figure 5f, in which the energy oscillations between LP , as shown in Figure 5f, in which the energy oscillations between this pair of modes 21b this pair of modes is observed. is observed. (b) (a) (c) (d) (e) (f) Figure 5. (a–c) Spectral, temporal forms and pulse energy evolutions for 130 m of pulse propagation Figure 5. (a–c) Spectral, temporal forms and pulse energy evolutions for 130 m of pulse propagation for all the modes in the GRIN-F, and (d–f) in the STEP-F, respectively, where the initial energy is for all the modes in the GRIN-F, and (d–f) in the STEP-F, respectively, where the initial energy is equally distributed among all the modes. equally distributed among all the modes. We also simulated the spatial profiles’ evolution of the total field containing all the We also simulated the spatial profiles’ evolution of the total field containing all the modes equally exciting inside the GRIN-F and the STEP-F. Due to the similariton pulse modes equally exciting inside the GRIN-F and the STEP-F. Due to the similariton pulse generation that occurred only on the LP01 mode, an initial spatial beam profile, even com- generation that occurred only on the LP mode, an initial spatial beam profile, even posed by the LP01 and HOMs, is progressively cleaned to a centered beam profile beyond composed by the LP and HOMs, is progressively cleaned to a centered beam profile 50 m of propagation, as shown in Figure 6a. The centered beam has Gaussian-like intensity beyond 50 m of propagation, as shown in Figure 6a. The centered beam has Gaussian-like distributions of the fundamental mode called a bell-shape beam [19]. On the other hand, intensity distributions of the fundamental mode called a bell-shape beam [19]. On the the spatial beam evolution in the STEP-F presented in Figure 6b shows that the spatial other hand, the spatial beam evolution in the STEP-F presented in Figure 6b shows that profile containing all the modes did not converge to a Gaussian-like intensity distribution the spatial profile containing all the modes did not converge to a Gaussian-like intensity in this case, and the spatial energy transfer between pairs of modes LP21a and LP21b are distribution in this case, and the spatial energy transfer between pairs of modes LP and 21a visible. As a result, the spatiotemporal nonlinear dynamics and the output spatial beam LP are visible. As a result, the spatiotemporal nonlinear dynamics and the output spatial 21b profiles are very different. The similariton generation with Raman amplification in the beam profiles are very different. The similariton generation with Raman amplification in GRIN-F occurred only on the LP01, improving the spatial beam self-cleaning and using the GRIN-F occurred only on the LP , improving the spatial beam self-cleaning and using very low energetic input pulses, unlike the cases using the stimulated Raman scattering very low energetic input pulses, unlike the cases using the stimulated Raman scattering (SRS) effect [20]. It is important to underline that the beam self-cleaning reported in this (SRS) effect [20]. It is important to underline that the beam self-cleaning reported in this work is fundamentally different from that observed in [20], which is induced via the SRS work is fundamentally different from that observed in [20], which is induced via the SRS process in MM-Fibers. The signal used to be strong enough to induce a Raman conversion, process in MM-Fibers. The signal used to be strong enough to induce a Raman conversion, and the cleaned beam appears at the Stokes’ wavelength. The modal gain competition and the cleaned beam appears at the Stokes’ wavelength. The modal gain competition between the Stokes’ modes is considered, where only red-shift wavelengths have been between the Stokes’ modes is considered, where only red-shift wavelengths have been carried by the LP mode. In contrast, here, the beam self-cleaning appears without any wavelength conversion. The modal gain competition and the walk-off between pump Photonics 2021, 8, x FOR PEER REVIEW 8 of 11 Photonics 2021, 8, 354 8 of 11 carried by the LP01 mode. In contrast, here, the beam self-cleaning appears without any wavelength conversion. The modal gain competition and the walk-off between pump and and signal could not be considered due to the strong CW-pump used for Raman gain. In signal could not be considered due to the strong CW-pump used for Raman gain. In ad- addition, the initial pulse energy used in this work (0.4 nJ) is not strong enough to generate dition, the initial pulse energy used in this work (0.4 nJ) is not strong enough to generate mode stokes for the SRS process. mode stokes for the SRS process. (a) (b) Figure 6. (a,b) Spatial beam evolution in the GRIN-F and the STEP-F, respectively, where the initial Figure 6. (a,b) Spatial beam evolution in the GRIN-F and the STEP-F, respectively, where the initial energy is equally distributed among all the modes. energy is equally distributed among all the modes. 3. Discussions 3. Discussions The present study reveals that the spatiotemporal similariton pulse evolutions in The present study reveals that the spatiotemporal similariton pulse evolutions in MM-Fibers with the Raman amplification process and its new fundamental spatiotemporal MM-Fibers with the Raman amplification process and its new fundamental spatiotem- nonlinear dynamics is an open question for investigation. Several aspects of spatiotemporal poral nonlinear dynamics is an open question for investigation. Several aspects of spatio- similariton pulse evolution were studied, such as whether spatiotemporal similariton in temporal similariton pulse evolution were studied, such as whether spatiotemporal simi- different kinds of MM-Fibers (GRIN-F and STEP-F) can exist, how they propagate under lariton in different kinds of MM-Fibers (GRIN-F and STEP-F) can exist, how they propa- various initial conditions, and the impact of the multimode propagation regime on its gate under various initial conditions, and the impact of the multimode propagation re- spatial behavior. We first started with the case where most of the initial energy (0.4 nJ) is gime on its spatial behavior. We first started with the case where most of the initial energy coupled on the fundamental LP mode. In that case, the energy transfer from the LP 01 01 (0.4 nJ) is coupled on the fundamental LP01 mode. In that case, the energy transfer from mode into the other HOMs is limited, and the other HOMs being decoupled since the the LP01 mode into the other HOMs is limited, and the other HOMs being decoupled since coupling effect is mainly limited to energy-dependent effects [14], and thereby, the MM- the coupling effect is mainly limited to energy-dependent effects [14], and thereby, the Fibers behave as single-mode fibers and observe an attraction towards similariton pulses MM-Fibers behave as single-mode fibers and observe an attraction towards similariton presented by the LP mode only. The other HOMs did not converge to the similariton pulses presented by the LP01 mode only. The other HOMs did not converge to the similar- pulses with a very low initial energy; even they have nearly the same amount of gain in iton pulses with a very low initial energy; even they have nearly the same amount of gain the STEP-F. The LP mode evolved into a linearly chirped pulse with a parabolic intensity in the STEP-F. The LP01 mode evolved into a linearly chirped pulse with a parabolic inten- shape, and its energy is significantly increased to 37.8 nJ after 130 m of propagation in sity shape, and its energy is significantly increased to 37.8 nJ after 130 m of propagation both MM-Fibers with Raman amplification. The similariton pulse duration (FWHM) and in both MM-Fibers with Raman amplification. The similariton pulse duration (FWHM) spectral bandwidth are 32 ps and 16 nm, respectively in the case of the GRIN-F, and 15.5 ps and spectral bandwidth are 32 ps and 16 nm, respectively in the case of the GRIN-F, and and 8 nm in the case of the STEP-F, respectively. These results allow for efficient pulse 15.5 ps and 8 nm in the case of the STEP-F, respectively. These results allow for efficient compression since the similariton experienced spectral broadening and high pulse energies. pulse compression since the similariton experienced spectral broadening and high pulse The resulting linear chirp makes it relatively easy to obtain high peak power pulses after energies. The resulting linear chirp makes it relatively easy to obtain high peak power compression in a subsequent dispersive optical element. The original input pulses after pulses after compression in a subsequent dispersive optical element. The original input shaping into similaritons are then compressed using an anomalous dispersion fiber. After a pulses after shaping into similaritons are then compressed using an anomalous dispersion certain propagation distance, a strongly compressed pulse can be obtained. The similariton fiber. After a certain propagation distance, a strongly compressed pulse can be obtained. pulses are compressed to 255 and 340 fs; a factor of ~8 and ~6 temporal compressions are The similariton pulses are compressed to 255 and 340 fs; a factor of ~8 and ~6 temporal achieved in the case of the GRIN-F and the STEP-F, respectively. The compressed pulses compressions are achieved in the case of the GRIN-F and the STEP-F, respectively. The peak power is ~100 KW, and an amplification factor of 500 could be achieved in both cases. compressed pulses peak power is ~100 KW, and an amplification factor of 500 could be Our second attention is notably in the spatiotemporal behavior of similariton pulses achieved in both cases. under the condition of the equal energy distributed among all the modes. In that case, Our second attention is notably in the spatiotemporal behavior of similariton pulses the results were different between the GRIN-F and the STEP-F. The main reason is that under the condition of the equal energy distributed among all the modes. In that case, the the distribution of the gain inside each MM-Fiber is different. In the case of the GRIN-F, results were different between the GRIN-F and the STEP-F. The main reason is that the the Raman gain is higher in the center of the GRIN-F core due to the relatively high GeO distribution of the gain inside each MM-Fiber is different. In the case of the GRIN-F, the content and high nonlinearity. Therefore, the LP mode has a larger gain and is strongly Raman gain is higher in the center of the GRIN-F core due to the relatively high GeO2 amplified compared to HOMs due to its greater overlap with the higher gain. As a result, only the LP mode transformed to a similariton pulse and it became dominant. The energy Photonics 2021, 8, 354 9 of 11 exchange is observed between the HOMs because the walk-off of the modes, which is (p) calculated by Db of modes 2–6, is smaller (0.013 ps/m) inside this type of fiber. However, the energy of HOMs is too weak and does not affect the similariton formation process. Moreover, an efficient spatial conversion of a multimode (speckled pattern) beam into a single-mode beam of near-Gaussian shape (bell-shape beam), resembling that of the LP mode, which results in beam quality enhancement (called beam clean-up), occurred. The beam clean-up is accompanied by similariton pulses at an output energy of 5.5 nJ that results in a brightness enhancement for the Raman amplification based on GRIN-F, while in the STEP-F, because the gain is nearly flat and distributed uniformly in the core of the fiber, all the modes transformed to the similariton forms, and the spatial beam (p) clean-up does not occur. The modal walk-off (Db of modes 2–6 are 0.38, 0.38, 0.89, 0.89, and 1.07 ps/m, respectively) leads the modes to separate temporally, and spatial energy oscillations between the degenerate modes. We can note the significance of these results by the fact that the similariton generation with Raman amplification in the GRIN-F occurred only on the LP , improving the spatial beam self-cleaning and using very low energetic input pulses, unlike cases that used rare Yb-doped MM-Fibers, where spatial beam self-cleaning has also been reported, even in the case of a non-parabolic refractive-index profile [19]. In addition, the usage of tapered Yb-doped GRIN-F [21] also enhances the nonlinear effect and improves the spatial Kerr self-cleaning and the decrease in the core-diameter accelerating self-image, and therefore, the nonlinear interaction between the modes is accelerated when injecting a beam into the large core side. However, the power threshold for self-induced spatial cleaning via the Kerr beam self-cleaning is increased (500 W and more), and the amount of power transferred to the cleaning mode is reduced. Note also that Kerr beam self-cleaning may be accompanied by a complex temporal pulse break-up and significant temporal reshaping, as well as a shortening of the input pulses [22]. In summary, we have introduced new results for the control of nonlinear beam reshaping in the spatial, temporal, and spectral domains. Combining similariton pulse and beam self-cleaning in GRIN-F permits obtaining not only high brightness but also high beam quality pulses. We anticipate that our results will be of general fundamental interest because they provide the first example of spatiotemporal similariton generation in MM-Fibers with Raman amplification. We can furthermore expect a particular interest in the development of higher pulse energy and short-pulse sources for high-power spatiotemporal multimode fiber lasers [23]. MM-Fibers allow for a considerably higher pulse energy than single-mode fibers and spatiotemporal similariton may have higher energy than those generated in single-mode fibers as well. 4. Conclusions In conclusion, we numerically provided a first look at the spatiotemporal similariton pulses’ evolution in passive MM-Fibers with Raman amplification. We presented these issues in a GRIN-F and STEP-F using the same configuration. Under the condition of fundamental mode excitation with the most initial energy, the similariton pulses have a temporal and spectral parabolic shape with a linear chirp and high energy in both MM-Fibers, although the mechanisms of the propagations between them are different. Compressions of the output amplified pulses lead to the generation of 255 and 340 fs pulses in the case of the GRIN-F and the STEP-F, respectively. This approach has the potential in a generation of ultra-short pulses with ~100 kW of peak power corresponding to an amplification factor of ~500. The transverse distribution intensity of the beams retains a Gaussian-like profile along both the MM-Fibers. However, the spatiotemporal evolutions, as well as the spatial beam profiles, are very different when the initial pulse energy is equally distributed among all the modes. We have found that in both MM-Fibers, similariton-like solutions exist, but they differ in their nature. In the case of the GRIN-F, the propagation is essentially single-mode since the amplification of the fundamental Photonics 2021, 8, 354 10 of 11 mode is much higher than the amplification of higher-order modes. The initial spatial beam profile is progressively cleaned into a bell-shaped beam, resembling that of the fundamental mode. These features can be useful for the new beam self-cleaning process in the GRIN-F using a spatiotemporal similariton evolution. In the case of the STEP-F, all the modes considered in the simulations have nearly similar amplification coefficients and the interplay between them along the propagation distance determines the spatial and temporal structure of the generated pulses. Moreover, we can also use this platform for the similariton generation for all the modes. From these results, we expect that similariton pulses generated in MM-Fibers will open new directions in studies of nonlinear wave propagation and can offer a new route to mode-area scaling for different applications with controllable spatiotemporal properties. Author Contributions: B.O. and L.G. conceived of the presented idea, L.G. performed the analytic calculations and the numerical simulations. Both L.G. and B.O. contributed to the final version of the manuscript. B.O. supervised the project. B.O. and L.G. have read and agreed to the published version of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: We gratefully acknowledge the Research Fellowship support by Türkiye Bursları, and the Directorate General for Scientific Research and Technological Development (DG-RSDT) of Algeria. Conflicts of Interest: The authors declare no conflict of interest. References 1. Picozzi, A.; Millot, G.; Wabnitz, S. Nonlinear optics: Nonlinear virtues of multimode fibre. Nat. Photonics 2015, 9, 289–291. [CrossRef] 2. Crosignani, B.; di Porto, P. Soliton propagation in multimode optical fibers. Opt. Lett. 1981, 6, 329–330. [CrossRef] 3. Gloge, D.; Marcatili, E.A.J. Multimode Theory of Graded-Core Fibers. Bell Syst. Tech. J. 1973, 52, 1563–1578. [CrossRef] 4. Mafi, A. Pulse Propagation in a Short Nonlinear Graded-Index Multimode Optical Fiber. J. Light Technol. 2012, 30, 2803–2811. [CrossRef] 5. Ahsan, A.S.; Agrawal, G.P. Graded-index solitons in multimode fibers. Opt. Lett. 2018, 43, 3345–3348. [CrossRef] 6. Tegin, U.; Ortaç, B. Spatiotemporal Instability of Femtosecond Pulses in Graded-Index Multimode Fibers. IEEE Photonics Technol. Lett. 2017, 29, 2195–2198. [CrossRef] 7. Tegin, U.; Ortaç, B. Cascaded Raman scattering based high power octave-spanning supercontinuum generation in graded-index multimode fibers. Sci. Rep. 2018, 8, 12470. [CrossRef] 8. Tegin, U.; Kakkava, E.; Rahmani, B.; Psaltis, D.; Moser, C. Spatiotemporal self-similar fiber laser. Optica 2019, 6, 1412–1415. [CrossRef] 9. Fermann, M.E.; Kruglov, V.I.; Thomsen, B.C.; Dudley, J.M.; Harvey, J.D. Self-similar propagation and amplification of parabolic pulses in optical fibers. Phys. Rev. Lett. 2000, 84, 6010–6013. [CrossRef] 10. Kruglov, V.I.; Peacock, A.C.; Harvey, J.D.; Dudley, J.M. Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers. J. Opt. Soc. Am. B 2002, 19, 461–491. [CrossRef] 11. Finot, C.; Millot, G.; Dudley, J.M. Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers. Opt. Lett. 2004, 29, 2533–2535. [CrossRef] 12. Ozeki, Y.; Takushima, Y.; Aiso, K.; Taira, K.; Kikuchi, K. Generation of 10 GHz similariton pulse trains from 1.2 Km-long erbium-doped fiber amplifier for application to multiwavelength pulse sources. Electron. Lett. 2004, 40, 1103–1104. [CrossRef] 13. Wright, L.G.; Ziegler, Z.M.; Lushnikov, P.M.; Zhu, Z.; Eftekhar, M.A.; Christodoulides, D.N.; Wise, F.W. Multimode nonlinear fiber optics: Massively parallel numerical solver, tutorial and outlook. IEEE J. Sel. Top. Quantum Electron. 2018, 24, 1–16. [CrossRef] 14. Poletti, F.; Horak, P. Description of ultrashort pulse propagation in multimode optical fibers. J. Opt. Soc. Am. B 2008, 25, 1645–1654. [CrossRef] 15. Hult, J. A fourth-order Runge–Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers. J. Light. Technol. 2007, 25, 3770–3775. [CrossRef] 16. Agrawal, G.P. Nonlinear Fiber Optics, 5th ed.; Academic Press: Cambridge, MA, USA, 2012. Photonics 2021, 8, 354 11 of 11 17. Dussauze, M.; Cardinal, T. Nonlinear Optical Properties of Glass. In Springer Handbook of Glass; Musgraves, D.J., Hu, J., Calvez, L., Eds.; Springer Handbooks: Cham, Switzerland, 2019. 18. Tegin, U.; Rahmani, B.; Kakkava, E.; Borhani, N.; Moser, C.; Psaltis, D. Controlling spatiotemporal nonlinearities in multimode fibers with deep neural networks. APL Photonics 2020, 5, 030804. [CrossRef] 19. Guenard, R.; Krupa, K.; Dupiol, R.; Fabert, M.; Bendahmane, A.; Kermene, V.; Desfarges-berthelemot, A.; Auguste, J.L.; Tonello, A.; Barthélémy, A.; et al. Kerr self-cleaning of pulsed beam in an Ytterbium doped multimode fiber. Opt. Exp. 2017, 25, 4783–4792. [CrossRef] 20. Terry, N.B.; Alley, T.G.; Russell, T.H. An explanation of SRS beam cleanup in graded-index fibers and the absence of SRS beam cleanup in step-index fibers. Opt. Exp. 2007, 15, 17509–17519. [CrossRef] 21. Niang, A.; Mansuryan, T.; Krupa, K.; Tonello, A.; Fabert, M.; Leproux, P.; Modotto, D.; Egorova, O.N.; Levchenko, A.E.; Lipatov, D.S.; et al. Spatial beam self-cleaning and supercontinuum generation with Yb-doped multimode graded-index fiber taper based on accelerating self-imaging and dissipative landscape. Opt. Exp. 2019, 27, 24018–24028. [CrossRef] 22. Krupa, K.; Tonello, A.; Couderc, V.; Barthélémy, A.; Millot, G.; Modotto, D.; Wabnitz, S. Spatiotemporal light-beam compression from nonlinear mode coupling. Phys. Rev. A 2018, 97, 043836. [CrossRef] 23. Wright, L.G.; Christodoulides, D.N.; Wise, F.W. Spatiotemporal mode-locking in multimode fiber lasers. Science 2017, 358, 94–97. [CrossRef] [PubMed] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Photonics Multidisciplinary Digital Publishing Institute

Spatiotemporal Evolutions of Similariton Pulses in Multimode Fibers with Raman Amplification

Graini, Leila; Ortaç, Bülend
Photonics , Volume 8 (9) – Aug 27, 2021
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hv photonics Communication Spatiotemporal Evolutions of Similariton Pulses in Multimode Fibers with Raman Amplification 1 , 2 2 , Leila Graini and Bülend Ortaç * Telecommunications Laboratory, 8 Mai 1945—Guelma University, Guelma 24000, Algeria; graini.leila@univ-guelma.dz National Nanotechnology Research Center and Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey * Correspondence: ortac@bilkent.unam.edu.tr Abstract: This paper aims to pave the way towards the demonstration of spatiotemporal similariton pulses’ evolution in passive multimode fibers with Raman amplification. We numerically present this issue in graded-index and step-index multimode fibers and provide a first look at the complex spatiotemporal dynamics of similariton pulses. The results showed that the similariton pulses could be generated in both multimode fibers. The temporal and spectral evolution of the pulses can be characterized as parabolic shapes with linear chirp and kW peak power. By compressing these, high-energy femtoseconds pulses can be obtained, starting initial picosecond pulses. A spatial beam profile could be preserved in both multimode fibers with the most energy coupled to the fundamental mode. Specifically, the similariton pulses’ generation with Raman amplification in a graded-index multimode fiber improves the spatial beam self-cleaning process under the different initial modes’ excitation. The observation of a new beam self-cleaning process is another attractor feature of propagation in graded-index multimode fibers. Keywords: spatiotemporal similariton; Raman amplification; multimode fibers Citation: Graini, L.; Ortaç, B. Spatiotemporal Evolutions of Similariton Pulses in Multimode 1. Introduction Fibers with Raman Amplification. In recent years, there has been increased interest in nonlinear wave propagation in Photonics 2021, 8, 354. https:// multimode optical fibers (MM-Fibers). The traditional view of MM-Fibers as undesirable doi.org/10.3390/photonics8090354 for nonlinear applications is gradually changing since MM-Fibers support a variety of new spatiotemporal phenomena, such as multimode solitons, supercontinuum generation, the Received: 2 July 2021 multiple filamentation process, and wave condensation [1]. The formation of multimode Accepted: 23 August 2021 solitons is limited by different group delays associated with different propagating modes Published: 27 August 2021 and nonlinear coupling terms, as was reported by early workers [2]. On the other hand, graded-index multimode fibers (GRIN-Fs) reduce the complexity due to the self-imaging Publisher’s Note: MDPI stays neutral effect resulting from the equal spacing of the modal wave with low intermodal disper- with regard to jurisdictional claims in sion. The theory of multimode light propagation in GRIN-Fs was established in the early published maps and institutional affil- 1970s [3]; very recently, the studies of the complex nonlinear spatiotemporal effects in these iations. MM-Fibers have been investigated [4]. Spatiotemporal optical solitons [5], supercontinuum generation [6], spatiotemporal instability [7], and self-similar fiber lasers [8] are already reported. The above-mentioned numerical studies and experimental observations are remarkable examples showing the complex nonlinear pulse dynamics in MM-Fibers. Moti- Copyright: © 2021 by the authors. vated by these works and the others, one may thus naturally wonder whether similariton Licensee MDPI, Basel, Switzerland. pulses may also occur in MM-Fibers. This article is an open access article Similariton pulses are shape-preserving waves that appear due to the interaction of distributed under the terms and nonlinearity, normal dispersion, and gain in optical fibers. This causes the shape of an conditions of the Creative Commons arbitrary input pulse to converge to a parabolic pulse shape and evolves in a self-similar Attribution (CC BY) license (https:// manner. Temporal similaritons have been studied extensively in the context of single-mode creativecommons.org/licenses/by/ optical fibers (SMFs) with different amplification processes (Raman, Erbium, and Ytterbium 4.0/). Photonics 2021, 8, 354. https://doi.org/10.3390/photonics8090354 https://www.mdpi.com/journal/photonics Photonics 2021, 8, x FOR PEER REVIEW 2 of 11 mode optical fibers (SMFs) with different amplification processes (Raman, Erbium, and Ytterbium amplification) [9–12]. These studies have already demonstrated the interesting properties of amplifiers’ similaritons pulses in conventional SMFs. The platform based on MM-Fibers could be a good candidate for the amplification process as well and the ques- tion of whether spatiotemporal similariton pulses could be formed inside the MM-Fibers. In this paper, we propose a new configuration to demonstrate the existence of the spatiotemporal similariton pulses in two different passive MM-Fiber types (GRIN-F and Step-Index Fiber (STEP-F)) with Raman amplification. For our numerical studies, we used Photonics 2021, 8, 354 2 of 11 the generalized multimode nonlinear Schrödinger equation (GMMNLSE) considering the interaction of nonlinearity, normal dispersions, gain, and modal interactions represented by mode coupling. The gain is generated by Raman amplification owing to their ad- amplification) [9–12]. These studies have already demonstrated the interesting properties vantages. It may occur at any passive fiber type and exists at any wavelength as long as of amplifiers’ similaritons pulses in conventional SMFs. The platform based on MM-Fibers an appropriate pump laser is available. The results showed that the characteristics of the could be a good candidate for the amplification process as well and the question of whether similariton pulses generated in both MM-Fibers present similar behavior, as obtained in spatiotemporal similariton pulses could be formed inside the MM-Fibers. SMFs with the most energy coupled to the fundamental mode. High energy similariton In this paper, we propose a new configuration to demonstrate the existence of the pulses have temporal and spectral parabolic shapes with a linear chirp. The pulses gener- spatiotemporal similariton pulses in two different passive MM-Fiber types (GRIN-F and ated in both cases can also be compressible down to a femtosecond pulse duration with Step-Index Fiber (STEP-F)) with Raman amplification. For our numerical studies, we used ~100 kW of peak power. However, the spatiotemporal dynamics, as well as the spatial the generalized multimode nonlinear Schrödinger equation (GMMNLSE) considering the beam profiles under different initial modes’ excitation in both MM-Fibers, are very differ- interaction of nonlinearity, normal dispersions, gain, and modal interactions represented by ent. A new type of beam self-cleaning process based on similariton pulse generation with mode coupling. The gain is generated by Raman amplification owing to their advantages. It Raman amplification in GRIN-F could be observed. may occur at any passive fiber type and exists at any wavelength as long as an appropriate pump laser is available. The results showed that the characteristics of the similariton pulses 2. Numerical Setup and Results generated in both MM-Fibers present similar behavior, as obtained in SMFs with the most To investigate the spatiotemporal evolutions of similariton pulses in MM-Fibers with energy coupled to the fundamental mode. High energy similariton pulses have temporal Raman amplification, we used the platform presented in Figure 1. The MM-Fibers used in and spectral parabolic shapes with a linear chirp. The pulses generated in both cases our numerical studies have parabolic and step refractive index profiles in the GRIN-F and can also be compressible down to a femtosecond pulse duration with ~100 kW of peak the STEP-F with an identical core radius of 25 µm and an index contrast of Δ = 0.0068, power. However, the spatiotemporal dynamics, as well as the spatial beam profiles under where NA = 0.17, respectively. To reduce the computational time, we considered the first different initial modes’ excitation in both MM-Fibers, are very different. A new type of six linearly polarized modes (LPmn), as shown in Figure 1. beam self-cleaning process based on similariton pulse generation with Raman amplification We calculated the dispersions, the nonlinear, and the coupling coefficients using in GRIN-F could be observed. codes introduced by Wright et al. [13]. The MM-Fibers characterized by the dispersions 2 2 coefficients β2 for the six modes are as follows: 16.76 ps /km (LP01), 16.75 ps /km (LP11a and 2. Numerical Setup and Results 2 2 LP11b), and 16.74 ps /km (LP21a, LP21b, and LP02) for the GRIN-F and 16.49 ps /km (LP01), 2 2 2 16.0T 6 p o investigate s /km (LP11athe and LP spatiotemporal 11b), 15.50 psevolutions /km (LP21a an of similariton d LP21b), and pulses 15.31 p insMM-Fibers /km (LP02) for with Raman the STEP amplification, -F at 1060 nm. Th we used e effective the platform areas, A pr effesented , and the nonline in Figurear 1 coeffic . The MM-Fibers ients, γ, of the used 2 −1 −1 2 −1 GRIN-F and the STEP-F are 155.09 µm and 1.22 W km , and 966.43 µm and 0.2 W in our numerical studies have parabolic and step refractive index profiles in the GRIN-F −1 and kmthe , rSTEP-F espectively. A with an s the identical effective coreare radius a of the fundamenta of 25 m and an l mode i index contrast n the MM-Fi of D = bers 0.0068, is scaled as the core radius, the effective nonlinearity of the MM-Fibers is comparable with where NA = 0.17, respectively. To reduce the computational time, we considered the first SMFs [4]. These features make both MM-Fibers suitable for Raman amplification. six linearly polarized modes (LP ), as shown in Figure 1. mn Figure 1. Simulations set-up for spatiotemporal evolutions of similariton pulses in MM-Fibers with Figure 1. Simulations set-up for spatiotemporal evolutions of similariton pulses in MM-Fibers with Raman amplification. Raman amplification. We calculated the dispersions, the nonlinear, and the coupling coefficients using codes introduced by Wright et al. [13]. The MM-Fibers characterized by the dispersions 2 2 coefficients b for the six modes are as follows: 16.76 ps /km (LP ), 16.75 ps /km (LP 2 01 11a 2 2 and LP ), and 16.74 ps /km (LP , LP and LP ) for the GRIN-F and 16.49 ps /km 11b 21a 21b, 02 2 2 2 (LP ), 16.06 ps /km (LP and LP ), 15.50 ps /km (LP and LP ), and 15.31 ps /km 01 11a 11b 21a 21b (LP ) for the STEP-F at 1060 nm. The effective areas, A , and the nonlinear coefficients, , 02 eff 2 1 1 2 of the GRIN-F and the STEP-F are 155.09 m and 1.22 W km , and 966.43 m and 1 1 0.2 W km , respectively. As the effective area of the fundamental mode in the MM- Fibers is scaled as the core radius, the effective nonlinearity of the MM-Fibers is comparable with SMFs [4]. These features make both MM-Fibers suitable for Raman amplification. Numerical simulations for the spatiotemporal nonlinear dynamics of similariton pulses are conducted using the GMMNLSE [14] and solved using the split-step Fourier Photonics 2021, 8, x FOR PEER REVIEW 3 of 11 Numerical simulations for the spatiotemporal nonlinear dynamics of similariton pulses are conducted using the GMMNLSE [14] and solved using the split-step Fourier Photonics 2021, 8, 354 3 of 11 method [15]. The GMM-NLSE (1) is including the effects of stimulated Raman scattering, higher-order dispersion, self-steepening, and intermodal effects. ( ) ( ) ( ) ( ) ( ) method [15]. The GMM-NLSE (1) is including the effects of stimulated Raman scattering, 𝜕 𝐴 𝑖𝛽 ℜ𝛽 𝐴 𝛽 ℜ𝛽 𝑖 ∑ 𝑖 𝐴 𝑖 (1 𝜕𝑡) ∑ (1 ,, higher-order dispersion, self-steepening, and intermodal effects. (1) h i  ∗ h i  ∗ (p) n ) ( ) 𝑓 𝑆 𝐴 𝐴 𝐴 𝑓 𝐴 𝑆 𝑑𝜏 𝐴 𝑧,𝑡𝜏 𝐴 (𝑧,𝑡𝜏)ℎ n (𝜏) (p) (p) ( 0) (0) ¶A b 2 w n ¶ i P 0 ¶ A = i b < b A b < b + i i A + i 1 + ¶t z p p å p å n2 l,m,n 0 0 1 1 n! c w ¶t ¶t n o (1) k  R (1 f )S A A A + f A S dtA (z, t t)A (z, t t)h (t) m m R l n R l n R plmn plmn where Ap(z,t) is the electric field temporal envelope for spatial mode p; and the first, the second, and the third term on the right-hand side indicate the propagation constant mis- where Ap(z,t) is the electric field temporal envelope for spatial mode p; and the first, the match responsible for multimode interference or mode beating, the modal dispersion or second, and the third term on the right-hand side indicate the propagation constant mis- modal walk-off and the higher-order dispersion coefficients, respectively. In the fourth match responsible for multimode interference or mode beating, the modal dispersion or −20 2 −1 term, n2 represents the nonlinear refractive index (n2 = 3.2 × 10 m W ), fR is the fractional modal walk-off and the higher-order dispersion coefficients, respectively. In the fourth contribution of the Raman effect (fR = 0.18), 𝑆 and 𝑆 represent the nonlinear cou- 20 2 1 term, n represents the nonlinear refractive index (n = 3.2  10 m W ), f is the frac- 2 2 R pling coefficients for the Raman and Kerr effect, krespectiveRly, hR is the delayed Raman tional contribution of the Raman effect (f = 0.18), S and S represent the nonlinear plmn plmn response function, and shock terms are neglected. coupling coefficients for the Raman and Kerr effect, respectively, h is the delayed Raman The gain can be added into the GMMNLSE by an additional term, gpAp, where gp is response function, and shock terms are neglected. the small-signal gain for the electric field in mode p. For short pulses, gp is defined as gp = The gain can be added into the GMMNLSE by an additional term, g A , where g p p p g0/(1 + Ep/Esat), where g0 is the initial gain, Ep is the energy of mode p, and Esat is the satura- is the small-signal gain for the electric field in mode p. For short pulses, g is defined as tion energy. Esat, for both MM-Fibers, is estimated to be about 1.2 nJ. In our numerical g = g /(1 + E E ), where g is the initial gain, E is the energy of mode p, and E is p sat p sat 0 p/ 0 studies, we selected the CW-pump source operating at 1010 nm for Raman gain. A Gauss- the saturation energy. E , for both MM-Fibers, is estimated to be about 1.2 nJ. In our sat ian-shaped input pulse operating at 1060 nm with a pulse duration of 2 ps, pulse energy numerical studies, we selected the CW-pump source operating at 1010 nm for Raman gain. of 0.4 nJ, peak power of 200 W, and spectral bandwidth of 0.8 nm is used. The shift of 13.2 A Gaussian-shaped input pulse operating at 1060 nm with a pulse duration of 2 ps, pulse THz (corresponding to 50 nm) between the pump and the signal is selected for optimum energy of 0.4 nJ, peak power of 200 W, and spectral bandwidth of 0.8 nm is used. The Raman gain in the silica fibers. The fiber length of 130 m is chosen to ensure sufficient shift of 13.2 THz (corresponding to 50 nm) between the pump and the signal is selected Raman gain. for optimum Raman gain in the silica fibers. The fiber length of 130 m is chosen to ensure In MM-Fibers with gain, the different modes receive gain and usually compete since sufficient Raman gain. they overlap with the same gain medium. In our case, the competition between the pump In MM-Fibers with gain, the different modes receive gain and usually compete since and signal modes could not be considered due to the CW-pump used for Raman gain. The they overlap with the same gain medium. In our case, the competition between the pump Raman gain for each mode is defined as Gp (dB) = exp (gR P0 Leff/Aeff) [16], where Leff is the and signal modes could not be considered due to the CW-pump used for Raman gain. The effective fiber length, P0 is the CW-pump power, and gR is the Raman gain coefficient at Raman gain for each mode is defined as G = exp (g P L /A ) [16], where L is (dB) R 0 eff eff eff the CW-pump wavelength. Note that the modal gain can be controlled by the effective the effective fiber length, P is the CW-pump power, and g is the Raman gain coefficient 0 R area of modes, at the CW-pump wavelength. Note that the modal gain can be controlled by the effective Aeff, and the Raman gain coefficient (considering the CW-pump power is fixed). Con- area of modes, A , and the Raman gain coefficient (considering the CW-pump power is eff fixed). sequentl Consequently y, the variat ,ithe on of variation the Ram of an gai the Raman n for ea gain ch sp for atia each l mospatial de as a f mode unctias on o a function f the pro- of posed the pr MM oposed -Fiber’s length MM-Fiberis ev ’s length aluated is evaluated in Figure 2a,b in Figur for the e 2a,b GRI for N-F the and GRIN-F the STand EP-Fthe , re- STEP-F, respectively, considering the overfilling nature of the CW-pump and the power is spectively, considering the overfilling nature of the CW-pump and the power is 100 W. 100 We W also c . We a also lculated calculated on the t onw the o plots two plots in Figur in Fi e gur 2c, the e 2c,effect the ef ive fective area of mo area ofdes modes in the pro- in the proposed MM-Fibers confirm that the effective area of modes in the STEP-F is larger than posed MM-Fibers confirm that the effective area of modes in the STEP-F is larger than in in the GRIN-F of the same radius [14]. the GRIN-F of the same radius [14]. (b) (c) (a) Figure 2. (a,b) Variation of modal gain for all modes in the GRIN-F and the STEP-F, respectively, (c) and the effective area of modes. From Figure 2a,b, we note that the distribution of gain inside both MM-Fibers is different due to the difference in the structures and the mechanism of the propagations between them. In the GRIN-F, the fundamental mode, LP , received the largest gain Photonics 2021, 8, 354 4 of 11 compared to the higher-order modes (HOMs). Primarily, because the LP mode is more confined in the higher gain region, since the relatively high GeO dopants in the center of the GRIN-F core result in a higher peak Raman gain coefficient. The peak value of the Raman gain coefficient is estimated as g = 7  10 m/W (at a 1060-nanometer wavelength and considering the GeO2 doping). Second, since the modal gain is inversely proportional to the effective modal area, the latter being higher for the LP mode with the smaller effective area, as shown in Figure 2c (upper figure). The overlap between other HOMs and the pump results in a smaller gain. The modal gain at the fiber length of 130 m is equal to 20, 2, 1.5, and 0.5 dB for the LP , LP , LP and LP mode, respectively. 01 02 11ab 21aba, In the case of STEP-F, the different modes could receive the same gain since the Raman gain is flat and uniformly distributed along with the STEP-F core. Note, however, that the LP and LP modes have the larger and the smaller effective areas (Figure 2c), respectively, confirming that the effective area of the LP modes decreases with the mode order in the 0n STEP-F [13]. The effective areas of the other modes, LP , LP , LP , and LP , are 11a 11b 21a 21b nearly equal. As a result, the overlap between the CW-pump and all the modes under consideration leads to modal gain variations, as shown in Figure 2b. The gain of the LP mode is equal to 20 dB and that of the LP mode is 25 dB, where the gain of LP and 02 11a-b LP is 21.3 and 21.5 dB, respectively. 21a-b To study the dynamics of similariton pulses in both MM-Fibers, and to present their characteristics and behaviors compared to those generated in SMFs, we first started by exciting only the fundamental mode, LP . We assume the Gaussian-shaped input pulse with a pulse duration of 2 ps and energy of 0.4 nJ propagates through 130 m of the proposed MM-Fibers and under the Raman amplification effect. We set the pulse energy, mostly coupled in the LP mode, as 99%. We consider the weak excitation of the other modes, which could be the case in an experiment. The temporal and the spectral forms of all the modes at the MM-Fibers’ output can be obtained, as shown in Figure 3a–e, which is presented in uniform and normalized scales. These results show that only the LP mode evolves towards a parabolic form and, subsequently, develops a wide temporal and spectral expansion. As expected for all the other modes, only the LP mode is strongly amplified and its energy increased exponentially with the fiber length to reach a value of 37.8 nJ, as shown in Figure 3c,f, which validates the amplifier similariton pulse formation of the LP mode, while the other HOMs did not converge to the similariton pulses. As the similaritons depend only on the pulse energy and on the gain of the amplifier, the most initial pulse energy coupling into the LP mode results in a higher nonlinearity compared with the other HOMs. Therefore, even with the same amount of gain distributed among all the modes, as in the case of the STEP-F, only the LP mode transforms into the parabolic shape and gains energy. The other HOMs did not transform into the parabolic shape and then lost their energy, which is also the result of the inadequate nonlinear phase accumulation. It can be understood as follows: The interplay of nonlinearity, normal dispersion, and gain is important for reshaping and maintaining the parabolic pulse shape during propagation in the fiber. It should be noticed that the energy exchange is observed between the HOMs in the GRIN-F (Figure 3a,b) because the intermodal group delays are smaller inside this type of fiber. However, the energy level of HOMs (0.072 nJ) is extremely low compared with that of the LP mode and, thus, any perturbation on the similariton pulse evolution did not occur during its propagation in the GRIN-F. On the other hand, the energy coupling to the HOMs is not observed in the STEP-F (Figure 3d,e) and each of them evolve as in a single-mode case. Photonics 2021, 8, 354 5 of 11 Photonics 2021, 8, x FOR PEER REVIEW 5 of 11 (a) (b) (c) (d) (e) (f) Figure 3. (a–c) Spectral, temporal forms and pulse energy evolutions for 130 m of pulse propagation Figure 3. (a–c) Spectral, temporal forms and pulse energy evolutions for 130 m of pulse propagation for all the modes in the GRIN-F and (d–f) in the STEP-F, respectively, where the most initial energy for all the modes in the GRIN-F and (d–f) in the STEP-F, respectively, where the most initial energy is is coupled in the LP01 mode. Inset: output spatial beams. coupled in the LP mode. Inset: output spatial beams. Figure 4 shows the formation of the similariton pulses of the LP01 mode in temporal Figure 4 shows the formation of the similariton pulses of the LP mode in temporal and spectral domains inside the proposed MM-Fibers. Figure 4a,c are highlighting the and spectral domains inside the proposed MM-Fibers. Figure 4a,c are highlighting the reshaping of the pulse during its propagation inside the GRIN-F, followed by its self-sim- reshaping of the pulse during its propagation inside the GRIN-F, followed by its self- ilar evolution that appeared according to two parts of separate regimes. The first part is similar evolution that appeared according to two parts of separate regimes. The first part is corresponding to the restructuring and the reshaping of the LP01 mode and the appearance corresponding to the restructuring and the reshaping of the LP mode and the appearance of characteristics of the parabolic pulse profile. A typical parabolic shape is obtained at 30 of characteristics of the parabolic pulse profile. A typical parabolic shape is obtained at m of pulse propagation inside the GRIN-F, which is defined as the characteristic length Zc 30 m of pulse propagation inside the GRIN-F, which is defined as the characteristic length = (3/2g0) ln (Ng0/6γPc) [10], calculated for N~100. In the second part, beyond ZC, the LP01 Z = (3/2g ) ln (Ng /6 P ) [10], calculated for N~100. In the second part, beyond Z , the c c 0 0 C mode has the characteristics of similariton and evolves in a self-similar manner with an LP mode has the characteristics of similariton and evolves in a self-similar manner with increase in its wide temporal and spectral expansion. We note the absence of any pulse an increase in its wide temporal and spectral expansion. We note the absence of any pulse deformation or oscillation caused by wave breaking, and modulation instability side- deformation or oscillation caused by wave breaking, and modulation instability sidebands did bands d not occur id not occur in the in the generated gener spectr ated sp um. ectrum At the . At the output of t output of the GRIN-F he GRIN-F, t , the input Gaussian he input Gaussian pulse is now transformed into a parabolic shape with a linear chirp (Figure 4d). pulse is now transformed into a parabolic shape with a linear chirp (Figure 4d). The pulse duration The pulse d is u 32 ration ps and is 32 p the spulse and the pulse ener energy is about gy is 37.8 about 37.8 n nJ. The spectr J. The spectrum um generat generated ed by the self-similar by the self-simi amplificat lar amion plifica regime tion regi alsome has ala so parabolic has a parabol form i(Figu c form re(4 Fb) igure with 4b a) w bandwidth ith a band- of 16 width of 16 n nm. The small m. The small ripples seen ripples in the seen in the middle of middle the spectr of the spectrum are the result of the um are the result of the limited bandwidth limited banby dwidt theh by finite th spectral e finite spect bandwidth ral bandw of Raman idth ogai f Ram n (about an gain 23 (nm about accor 23 n ding m accor to [17 d]). - At ing t the o [ same 17]). time, At the s Figur ame e t 4ie–h me, show Figure the 4eself-similar –h show the regime self-sim and ilar similariton regime and formation similarito for n 130 format m of ion pulse for 1 pr 30 opagat m of pu ionls inside e propa the gat STEP-F ion ins , alt ide hough the ST the EP-F mechanism , although t ofhthe e mech propagations anism of is the propa different. gaSpecifically tions is different. , the time Specif duration ically, th and e time the d ener uration gy of and t the output he energ pulse y of tar he output e 15.5 ps and 37.8 nJ, respectively (Figure 4h). Here, the input pulse has evolved as a parabolic shape pulse are 15.5 ps and 37.8 nJ, respectively (Figure 4h). Here, the input pulse has evolved beyond as a par55 abm oliof c sh prap opagation, e beyond 5 which 5 m of is just prop a transient agation, state which ofis j theupulse st a tr evolution ansient stin ate o theffiber the and under amplification (Figure 4e,g). The ratio between the nonlinearity coefficient ( ) pulse evolution in the fiber and under amplification (Figure 4e,g). The ratio between the and the dispersion ( ) is high in the GRIN-F compared to that in the STEP-F. Therefore, we nonlinearity coefficient (γ) and the dispersion (β2) is high in the GRIN-F compared to that observed that the input pulse converges to a parabolic shape rapidly in the GRIN-F (30 m) in the STEP-F. Therefore, we observed that the input pulse converges to a parabolic shape than in the STEP-F (55 m). Likewise, the decrease in the nonlinearity coefficient caused rapidly in the GRIN-F (30 m) than in the STEP-F (55 m). Likewise, the decrease in the less spectral broadening (about 9 nm) for the STEP-F (Figure 4f). The above results imply nonlinearity coefficient caused less spectral broadening (about 9 nm) for the STEP-F (Fig- that the output pulses have a parabolic pulse shape with a linear chirp and a high-energy ure 4f). The above results imply that the output pulses have a parabolic pulse shape with amplification. These pulses generated in the GRIN-F and the STEP-F can be compressed a linear chirp and a high-energy amplification. These pulses generated in the GRIN-F and down to 255 and 340 fs, respectively (inset Figure 4d,h). The compressed pulses’ peak Photonics 2021, 8, x FOR PEER REVIEW 6 of 11 Photonics 2021, 8, 354 6 of 11 the STEP-F can be compressed down to 255 and 340 fs, respectively (inset Figure 4d,h). power is ~100 KW and that, when compared to the original input pulse (pulse duration of The compressed pulses’ peak power is ~100 KW and that, when compared to the original 2 ps and peak power of 200 W), represents a factor of ~8 and ~6 temporal compressions, input pulse (pulse duration of 2 ps and peak power of 200 W), represents a factor of ~8 and an amplification factor of 500 could be achieved in both cases. It should be noted here and ~6 temporal compressions, and an amplification factor of 500 could be achieved in that the pedestals on the compressed pulses are being observed. However, their energy is both cases. It should be noted here that the pedestals on the compressed pulses are being extremely low compared to that of the main lobes. We have estimated 70 and 90% of the observed. However, their energy is extremely low compared to that of the main lobes. We total energy is contained in the main lobes of the compressed pulses, while 30 and 10% is in have estimated 70 and 90% of the total energy is contained in the main lobes of the com- the pedestals by comparing the energies of the similariton pulses (37.8 nJ) with those of the pressed pulses, while 30 and 10% is in the pedestals by comparing the energies of the compressed pulses (25.5 and 34 nJ), in the case of the GRIN-F and the STEP-F, respectively. similariton pulses (37.8 nJ) with those of the compressed pulses (25.5 and 34 nJ), in the We can mention that the pedestals could be observed as the features of the other HOMs. case of the GRIN-F and the STEP-F, respectively. We can mention that the pedestals could Thus, the pedestals on the compressed pulses are more relevant in the case of the GRIN-F, be observed as the features of the other HOMs. Thus, the pedestals on the compressed because the walk-off of the modes is smaller inside this type of fiber (Figure 3a,b), and pulses are more relevant in the case of the GRIN-F, because the walk-off of the modes is thereby the generated temporal profile of the compressed pulse is more affected by the smaller inside this type of fiber (Figure 3a,b), and thereby the generated temporal profile presence of the other HOMs. of the compressed pulse is more affected by the presence of the other HOMs. (b) (f) (a) (e) (c) (g) (d) (h) Figure 4. (a–d) The spectral and the temporal similariton evolutions and output forms of the LP mode in the GRIN-F, and Figure 4. (a–d) The spectral and the temporal similariton evolutions and output forms of the LP01 mode in the GRIN-F, (e–h) in the STEP-F. Inset: the compressed pulses. and (e–h) in the STEP-F. Inset: the compressed pulses. The output spatial profiles of the total field containing all the modes during the pulse The output spatial profiles of the total field containing all the modes during the pulse propagation inside the GRIN-F and the STEP-F are presented in the inset of Figure 3c,f. propagation inside the GRIN-F and the STEP-F are presented in the inset of Figure 3c,f. The excitation of the LP mode with the most energy results in the Gaussian intensity The excitation of the LP01 mode with the most energy results in the Gaussian intensity distributions being preserved along the both MM-Fibers. Due to the different mode field distributions being preserved along the both MM-Fibers. Due to the different mode field diameter behavior of the LP mode for both MM-Fibers, we also observed the different diameter behavior of the LP01 mode for both MM-Fibers, we also observed the different beam sizes. beam sizes. Later we consider the effect of different launch conditions on the spatiotemporal pulse Later we consider the effect of different launch conditions on the spatiotemporal evolution. In this part, the simulations are performed to investigate the effect of the initial pulse evolution. In this part, the simulations are performed to investigate the effect of the input energy distribution between the modes. To create a significant change, we consider initial input energy distribution between the modes. To create a significant change, we the first six modes with equal initial energy distributions (total peak power of 200 W among consider the first six modes with equal initial energy distributions (total peak power of all the modes). Indeed, such a configuration can be achieved using a spatial light modulator 200 W among all the modes). Indeed, such a configuration can be achieved using a spatial (SLM) [18]. We considered the same GRIN–F and STEP–F parameters. For a propagation light modulator (SLM) [18]. We considered the same GRIN–F and STEP–F parameters. length of 130 m, we noticed that the initial pulse is converged to a similariton pulse in the For a propagation length of 130 m, we noticed that the initial pulse is converged to a sim- GRIN-F for the LP mode only, and it is significantly amplified compared to the HOMs, ilariton pulse in the GRIN-F for the LP01 mode only, and it is significantly amplified com- as shown in Figure 5a–c. While the HOMs did not gain energy, even they have the same pared to the HOMs, as shown in Figure 5a–c. While the HOMs did not gain energy, even amount of initial energy. It can be explained that the similariton pulse formation in the they have the same amount of initial energy. It can be explained that the similariton pulse GRIN-F significantly depends on the Raman gain of modes, which favors the LP mode formation in the GRIN-F significantly depends on the Raman gain of modes, which favors since it is received the largest gain compared to HOMs (Figure 2a). On the other hand, in the LP01 mode since it is received the largest gain compared to HOMs (Figure 2a). On the the STEP-F, the pulse is broken up into sub-pulses, presented by different modes, whose other hand, in the STEP-F, the pulse is broken up into sub-pulses, presented by different separation grows linearly with the fiber length owing to the large walk-off between modes, Photonics 2021, 8, x FOR PEER REVIEW 7 of 11 Photonics 2021, 8, 354 7 of 11 modes, whose separation grows linearly with the fiber length owing to the large walk-off between modes, because the modes have rather different group velocities, and each of because the modes have rather different group velocities, and each of them travels with them travels with their group velocity as in a single-mode case. Therefore, all the modes their group velocity as in a single-mode case. Therefore, all the modes evolved towards the evolved towards the respective similariton forms, and they are amplified with a different respective similariton forms, and they are amplified with a different modal gain coefficient modal gain coefficient (Figure 5d–f). The greater walk-off of the modes in the STEP-F (Figure 5d–f). The greater walk-off of the modes in the STEP-F means that the coupling means that the coupling between HOMs may be less efficient. However, when the energy between HOMs may be less efficient. However, when the energy of the modes increased of the modes increased during the similaritons’ formation process, the coupling between during the similaritons’ formation process, the coupling between the degenerate modes, the degenerate modes, which travel at the same group velocity, seems very relevant in the which travel at the same group velocity, seems very relevant in the case of LP and 21a case of LP21a and LP21b, as shown in Figure 5f, in which the energy oscillations between LP , as shown in Figure 5f, in which the energy oscillations between this pair of modes 21b this pair of modes is observed. is observed. (b) (a) (c) (d) (e) (f) Figure 5. (a–c) Spectral, temporal forms and pulse energy evolutions for 130 m of pulse propagation Figure 5. (a–c) Spectral, temporal forms and pulse energy evolutions for 130 m of pulse propagation for all the modes in the GRIN-F, and (d–f) in the STEP-F, respectively, where the initial energy is for all the modes in the GRIN-F, and (d–f) in the STEP-F, respectively, where the initial energy is equally distributed among all the modes. equally distributed among all the modes. We also simulated the spatial profiles’ evolution of the total field containing all the We also simulated the spatial profiles’ evolution of the total field containing all the modes equally exciting inside the GRIN-F and the STEP-F. Due to the similariton pulse modes equally exciting inside the GRIN-F and the STEP-F. Due to the similariton pulse generation that occurred only on the LP01 mode, an initial spatial beam profile, even com- generation that occurred only on the LP mode, an initial spatial beam profile, even posed by the LP01 and HOMs, is progressively cleaned to a centered beam profile beyond composed by the LP and HOMs, is progressively cleaned to a centered beam profile 50 m of propagation, as shown in Figure 6a. The centered beam has Gaussian-like intensity beyond 50 m of propagation, as shown in Figure 6a. The centered beam has Gaussian-like distributions of the fundamental mode called a bell-shape beam [19]. On the other hand, intensity distributions of the fundamental mode called a bell-shape beam [19]. On the the spatial beam evolution in the STEP-F presented in Figure 6b shows that the spatial other hand, the spatial beam evolution in the STEP-F presented in Figure 6b shows that profile containing all the modes did not converge to a Gaussian-like intensity distribution the spatial profile containing all the modes did not converge to a Gaussian-like intensity in this case, and the spatial energy transfer between pairs of modes LP21a and LP21b are distribution in this case, and the spatial energy transfer between pairs of modes LP and 21a visible. As a result, the spatiotemporal nonlinear dynamics and the output spatial beam LP are visible. As a result, the spatiotemporal nonlinear dynamics and the output spatial 21b profiles are very different. The similariton generation with Raman amplification in the beam profiles are very different. The similariton generation with Raman amplification in GRIN-F occurred only on the LP01, improving the spatial beam self-cleaning and using the GRIN-F occurred only on the LP , improving the spatial beam self-cleaning and using very low energetic input pulses, unlike the cases using the stimulated Raman scattering very low energetic input pulses, unlike the cases using the stimulated Raman scattering (SRS) effect [20]. It is important to underline that the beam self-cleaning reported in this (SRS) effect [20]. It is important to underline that the beam self-cleaning reported in this work is fundamentally different from that observed in [20], which is induced via the SRS work is fundamentally different from that observed in [20], which is induced via the SRS process in MM-Fibers. The signal used to be strong enough to induce a Raman conversion, process in MM-Fibers. The signal used to be strong enough to induce a Raman conversion, and the cleaned beam appears at the Stokes’ wavelength. The modal gain competition and the cleaned beam appears at the Stokes’ wavelength. The modal gain competition between the Stokes’ modes is considered, where only red-shift wavelengths have been between the Stokes’ modes is considered, where only red-shift wavelengths have been carried by the LP mode. In contrast, here, the beam self-cleaning appears without any wavelength conversion. The modal gain competition and the walk-off between pump Photonics 2021, 8, x FOR PEER REVIEW 8 of 11 Photonics 2021, 8, 354 8 of 11 carried by the LP01 mode. In contrast, here, the beam self-cleaning appears without any wavelength conversion. The modal gain competition and the walk-off between pump and and signal could not be considered due to the strong CW-pump used for Raman gain. In signal could not be considered due to the strong CW-pump used for Raman gain. In ad- addition, the initial pulse energy used in this work (0.4 nJ) is not strong enough to generate dition, the initial pulse energy used in this work (0.4 nJ) is not strong enough to generate mode stokes for the SRS process. mode stokes for the SRS process. (a) (b) Figure 6. (a,b) Spatial beam evolution in the GRIN-F and the STEP-F, respectively, where the initial Figure 6. (a,b) Spatial beam evolution in the GRIN-F and the STEP-F, respectively, where the initial energy is equally distributed among all the modes. energy is equally distributed among all the modes. 3. Discussions 3. Discussions The present study reveals that the spatiotemporal similariton pulse evolutions in The present study reveals that the spatiotemporal similariton pulse evolutions in MM-Fibers with the Raman amplification process and its new fundamental spatiotemporal MM-Fibers with the Raman amplification process and its new fundamental spatiotem- nonlinear dynamics is an open question for investigation. Several aspects of spatiotemporal poral nonlinear dynamics is an open question for investigation. Several aspects of spatio- similariton pulse evolution were studied, such as whether spatiotemporal similariton in temporal similariton pulse evolution were studied, such as whether spatiotemporal simi- different kinds of MM-Fibers (GRIN-F and STEP-F) can exist, how they propagate under lariton in different kinds of MM-Fibers (GRIN-F and STEP-F) can exist, how they propa- various initial conditions, and the impact of the multimode propagation regime on its gate under various initial conditions, and the impact of the multimode propagation re- spatial behavior. We first started with the case where most of the initial energy (0.4 nJ) is gime on its spatial behavior. We first started with the case where most of the initial energy coupled on the fundamental LP mode. In that case, the energy transfer from the LP 01 01 (0.4 nJ) is coupled on the fundamental LP01 mode. In that case, the energy transfer from mode into the other HOMs is limited, and the other HOMs being decoupled since the the LP01 mode into the other HOMs is limited, and the other HOMs being decoupled since coupling effect is mainly limited to energy-dependent effects [14], and thereby, the MM- the coupling effect is mainly limited to energy-dependent effects [14], and thereby, the Fibers behave as single-mode fibers and observe an attraction towards similariton pulses MM-Fibers behave as single-mode fibers and observe an attraction towards similariton presented by the LP mode only. The other HOMs did not converge to the similariton pulses presented by the LP01 mode only. The other HOMs did not converge to the similar- pulses with a very low initial energy; even they have nearly the same amount of gain in iton pulses with a very low initial energy; even they have nearly the same amount of gain the STEP-F. The LP mode evolved into a linearly chirped pulse with a parabolic intensity in the STEP-F. The LP01 mode evolved into a linearly chirped pulse with a parabolic inten- shape, and its energy is significantly increased to 37.8 nJ after 130 m of propagation in sity shape, and its energy is significantly increased to 37.8 nJ after 130 m of propagation both MM-Fibers with Raman amplification. The similariton pulse duration (FWHM) and in both MM-Fibers with Raman amplification. The similariton pulse duration (FWHM) spectral bandwidth are 32 ps and 16 nm, respectively in the case of the GRIN-F, and 15.5 ps and spectral bandwidth are 32 ps and 16 nm, respectively in the case of the GRIN-F, and and 8 nm in the case of the STEP-F, respectively. These results allow for efficient pulse 15.5 ps and 8 nm in the case of the STEP-F, respectively. These results allow for efficient compression since the similariton experienced spectral broadening and high pulse energies. pulse compression since the similariton experienced spectral broadening and high pulse The resulting linear chirp makes it relatively easy to obtain high peak power pulses after energies. The resulting linear chirp makes it relatively easy to obtain high peak power compression in a subsequent dispersive optical element. The original input pulses after pulses after compression in a subsequent dispersive optical element. The original input shaping into similaritons are then compressed using an anomalous dispersion fiber. After a pulses after shaping into similaritons are then compressed using an anomalous dispersion certain propagation distance, a strongly compressed pulse can be obtained. The similariton fiber. After a certain propagation distance, a strongly compressed pulse can be obtained. pulses are compressed to 255 and 340 fs; a factor of ~8 and ~6 temporal compressions are The similariton pulses are compressed to 255 and 340 fs; a factor of ~8 and ~6 temporal achieved in the case of the GRIN-F and the STEP-F, respectively. The compressed pulses compressions are achieved in the case of the GRIN-F and the STEP-F, respectively. The peak power is ~100 KW, and an amplification factor of 500 could be achieved in both cases. compressed pulses peak power is ~100 KW, and an amplification factor of 500 could be Our second attention is notably in the spatiotemporal behavior of similariton pulses achieved in both cases. under the condition of the equal energy distributed among all the modes. In that case, Our second attention is notably in the spatiotemporal behavior of similariton pulses the results were different between the GRIN-F and the STEP-F. The main reason is that under the condition of the equal energy distributed among all the modes. In that case, the the distribution of the gain inside each MM-Fiber is different. In the case of the GRIN-F, results were different between the GRIN-F and the STEP-F. The main reason is that the the Raman gain is higher in the center of the GRIN-F core due to the relatively high GeO distribution of the gain inside each MM-Fiber is different. In the case of the GRIN-F, the content and high nonlinearity. Therefore, the LP mode has a larger gain and is strongly Raman gain is higher in the center of the GRIN-F core due to the relatively high GeO2 amplified compared to HOMs due to its greater overlap with the higher gain. As a result, only the LP mode transformed to a similariton pulse and it became dominant. The energy Photonics 2021, 8, 354 9 of 11 exchange is observed between the HOMs because the walk-off of the modes, which is (p) calculated by Db of modes 2–6, is smaller (0.013 ps/m) inside this type of fiber. However, the energy of HOMs is too weak and does not affect the similariton formation process. Moreover, an efficient spatial conversion of a multimode (speckled pattern) beam into a single-mode beam of near-Gaussian shape (bell-shape beam), resembling that of the LP mode, which results in beam quality enhancement (called beam clean-up), occurred. The beam clean-up is accompanied by similariton pulses at an output energy of 5.5 nJ that results in a brightness enhancement for the Raman amplification based on GRIN-F, while in the STEP-F, because the gain is nearly flat and distributed uniformly in the core of the fiber, all the modes transformed to the similariton forms, and the spatial beam (p) clean-up does not occur. The modal walk-off (Db of modes 2–6 are 0.38, 0.38, 0.89, 0.89, and 1.07 ps/m, respectively) leads the modes to separate temporally, and spatial energy oscillations between the degenerate modes. We can note the significance of these results by the fact that the similariton generation with Raman amplification in the GRIN-F occurred only on the LP , improving the spatial beam self-cleaning and using very low energetic input pulses, unlike cases that used rare Yb-doped MM-Fibers, where spatial beam self-cleaning has also been reported, even in the case of a non-parabolic refractive-index profile [19]. In addition, the usage of tapered Yb-doped GRIN-F [21] also enhances the nonlinear effect and improves the spatial Kerr self-cleaning and the decrease in the core-diameter accelerating self-image, and therefore, the nonlinear interaction between the modes is accelerated when injecting a beam into the large core side. However, the power threshold for self-induced spatial cleaning via the Kerr beam self-cleaning is increased (500 W and more), and the amount of power transferred to the cleaning mode is reduced. Note also that Kerr beam self-cleaning may be accompanied by a complex temporal pulse break-up and significant temporal reshaping, as well as a shortening of the input pulses [22]. In summary, we have introduced new results for the control of nonlinear beam reshaping in the spatial, temporal, and spectral domains. Combining similariton pulse and beam self-cleaning in GRIN-F permits obtaining not only high brightness but also high beam quality pulses. We anticipate that our results will be of general fundamental interest because they provide the first example of spatiotemporal similariton generation in MM-Fibers with Raman amplification. We can furthermore expect a particular interest in the development of higher pulse energy and short-pulse sources for high-power spatiotemporal multimode fiber lasers [23]. MM-Fibers allow for a considerably higher pulse energy than single-mode fibers and spatiotemporal similariton may have higher energy than those generated in single-mode fibers as well. 4. Conclusions In conclusion, we numerically provided a first look at the spatiotemporal similariton pulses’ evolution in passive MM-Fibers with Raman amplification. We presented these issues in a GRIN-F and STEP-F using the same configuration. Under the condition of fundamental mode excitation with the most initial energy, the similariton pulses have a temporal and spectral parabolic shape with a linear chirp and high energy in both MM-Fibers, although the mechanisms of the propagations between them are different. Compressions of the output amplified pulses lead to the generation of 255 and 340 fs pulses in the case of the GRIN-F and the STEP-F, respectively. This approach has the potential in a generation of ultra-short pulses with ~100 kW of peak power corresponding to an amplification factor of ~500. The transverse distribution intensity of the beams retains a Gaussian-like profile along both the MM-Fibers. However, the spatiotemporal evolutions, as well as the spatial beam profiles, are very different when the initial pulse energy is equally distributed among all the modes. We have found that in both MM-Fibers, similariton-like solutions exist, but they differ in their nature. In the case of the GRIN-F, the propagation is essentially single-mode since the amplification of the fundamental Photonics 2021, 8, 354 10 of 11 mode is much higher than the amplification of higher-order modes. The initial spatial beam profile is progressively cleaned into a bell-shaped beam, resembling that of the fundamental mode. These features can be useful for the new beam self-cleaning process in the GRIN-F using a spatiotemporal similariton evolution. In the case of the STEP-F, all the modes considered in the simulations have nearly similar amplification coefficients and the interplay between them along the propagation distance determines the spatial and temporal structure of the generated pulses. Moreover, we can also use this platform for the similariton generation for all the modes. From these results, we expect that similariton pulses generated in MM-Fibers will open new directions in studies of nonlinear wave propagation and can offer a new route to mode-area scaling for different applications with controllable spatiotemporal properties. Author Contributions: B.O. and L.G. conceived of the presented idea, L.G. performed the analytic calculations and the numerical simulations. Both L.G. and B.O. contributed to the final version of the manuscript. B.O. supervised the project. B.O. and L.G. have read and agreed to the published version of the manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: We gratefully acknowledge the Research Fellowship support by Türkiye Bursları, and the Directorate General for Scientific Research and Technological Development (DG-RSDT) of Algeria. Conflicts of Interest: The authors declare no conflict of interest. References 1. Picozzi, A.; Millot, G.; Wabnitz, S. Nonlinear optics: Nonlinear virtues of multimode fibre. Nat. Photonics 2015, 9, 289–291. [CrossRef] 2. Crosignani, B.; di Porto, P. Soliton propagation in multimode optical fibers. Opt. Lett. 1981, 6, 329–330. [CrossRef] 3. Gloge, D.; Marcatili, E.A.J. Multimode Theory of Graded-Core Fibers. Bell Syst. Tech. J. 1973, 52, 1563–1578. [CrossRef] 4. Mafi, A. Pulse Propagation in a Short Nonlinear Graded-Index Multimode Optical Fiber. J. Light Technol. 2012, 30, 2803–2811. [CrossRef] 5. Ahsan, A.S.; Agrawal, G.P. Graded-index solitons in multimode fibers. Opt. Lett. 2018, 43, 3345–3348. [CrossRef] 6. Tegin, U.; Ortaç, B. Spatiotemporal Instability of Femtosecond Pulses in Graded-Index Multimode Fibers. IEEE Photonics Technol. Lett. 2017, 29, 2195–2198. [CrossRef] 7. Tegin, U.; Ortaç, B. Cascaded Raman scattering based high power octave-spanning supercontinuum generation in graded-index multimode fibers. Sci. Rep. 2018, 8, 12470. [CrossRef] 8. Tegin, U.; Kakkava, E.; Rahmani, B.; Psaltis, D.; Moser, C. Spatiotemporal self-similar fiber laser. Optica 2019, 6, 1412–1415. [CrossRef] 9. Fermann, M.E.; Kruglov, V.I.; Thomsen, B.C.; Dudley, J.M.; Harvey, J.D. Self-similar propagation and amplification of parabolic pulses in optical fibers. Phys. Rev. Lett. 2000, 84, 6010–6013. [CrossRef] 10. Kruglov, V.I.; Peacock, A.C.; Harvey, J.D.; Dudley, J.M. Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers. J. Opt. Soc. Am. B 2002, 19, 461–491. [CrossRef] 11. Finot, C.; Millot, G.; Dudley, J.M. Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers. Opt. Lett. 2004, 29, 2533–2535. [CrossRef] 12. Ozeki, Y.; Takushima, Y.; Aiso, K.; Taira, K.; Kikuchi, K. Generation of 10 GHz similariton pulse trains from 1.2 Km-long erbium-doped fiber amplifier for application to multiwavelength pulse sources. Electron. Lett. 2004, 40, 1103–1104. [CrossRef] 13. Wright, L.G.; Ziegler, Z.M.; Lushnikov, P.M.; Zhu, Z.; Eftekhar, M.A.; Christodoulides, D.N.; Wise, F.W. Multimode nonlinear fiber optics: Massively parallel numerical solver, tutorial and outlook. IEEE J. Sel. Top. Quantum Electron. 2018, 24, 1–16. [CrossRef] 14. Poletti, F.; Horak, P. Description of ultrashort pulse propagation in multimode optical fibers. J. Opt. Soc. Am. B 2008, 25, 1645–1654. [CrossRef] 15. Hult, J. A fourth-order Runge–Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers. J. Light. Technol. 2007, 25, 3770–3775. [CrossRef] 16. Agrawal, G.P. Nonlinear Fiber Optics, 5th ed.; Academic Press: Cambridge, MA, USA, 2012. Photonics 2021, 8, 354 11 of 11 17. Dussauze, M.; Cardinal, T. Nonlinear Optical Properties of Glass. In Springer Handbook of Glass; Musgraves, D.J., Hu, J., Calvez, L., Eds.; Springer Handbooks: Cham, Switzerland, 2019. 18. Tegin, U.; Rahmani, B.; Kakkava, E.; Borhani, N.; Moser, C.; Psaltis, D. Controlling spatiotemporal nonlinearities in multimode fibers with deep neural networks. APL Photonics 2020, 5, 030804. [CrossRef] 19. Guenard, R.; Krupa, K.; Dupiol, R.; Fabert, M.; Bendahmane, A.; Kermene, V.; Desfarges-berthelemot, A.; Auguste, J.L.; Tonello, A.; Barthélémy, A.; et al. Kerr self-cleaning of pulsed beam in an Ytterbium doped multimode fiber. Opt. Exp. 2017, 25, 4783–4792. [CrossRef] 20. Terry, N.B.; Alley, T.G.; Russell, T.H. An explanation of SRS beam cleanup in graded-index fibers and the absence of SRS beam cleanup in step-index fibers. Opt. Exp. 2007, 15, 17509–17519. [CrossRef] 21. Niang, A.; Mansuryan, T.; Krupa, K.; Tonello, A.; Fabert, M.; Leproux, P.; Modotto, D.; Egorova, O.N.; Levchenko, A.E.; Lipatov, D.S.; et al. Spatial beam self-cleaning and supercontinuum generation with Yb-doped multimode graded-index fiber taper based on accelerating self-imaging and dissipative landscape. Opt. Exp. 2019, 27, 24018–24028. [CrossRef] 22. Krupa, K.; Tonello, A.; Couderc, V.; Barthélémy, A.; Millot, G.; Modotto, D.; Wabnitz, S. Spatiotemporal light-beam compression from nonlinear mode coupling. Phys. Rev. A 2018, 97, 043836. [CrossRef] 23. Wright, L.G.; Christodoulides, D.N.; Wise, F.W. Spatiotemporal mode-locking in multimode fiber lasers. Science 2017, 358, 94–97. [CrossRef] [PubMed]

Journal

Photonics – Multidisciplinary Digital Publishing Institute

Published: Aug 27, 2021

Keywords: spatiotemporal similariton; Raman amplification; multimode fibers

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