Abstract

We experimentally investigate characteristics of saturable absorbers (SAs) based on nonlinear Kerr beam cleanup effect (NL-KBC). The SAs are formed by a long graded-index multimode fiber (GRIN MMF) with a short single-mode fiber served as a diaphragm. We studied the evolution of output spectrum and beam profiles from the GRIN MMF in order to investigate the mechanism of these SAs. We further performed saturable absorption measurements to evaluate their modulation depths and saturation intensities. We experimentally observed and first theoretically analyzed the “relaxation oscillation” behavior of the optical transmittance with increasing input intensity. We also studied their nonlinear polarization dynamics and observed the repolarized effect in NL-KBC regime. Our results confirm the optical properties of the SAs based on NL-KBC, and these SAs can find applications in Q-switched and mode-locked lasers.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Multimode optical fibers (MMFs) play an important role in various practical applications including high-resolution imaging [13], space-division multiplexing [4,5] and high-power fiber lasers [6]. In addition, due to the sophisticated interplay of spatiotemporal processes in beam propagation, MMF can provide a convenient platform for studies of complex nonlinear phenomena such as spatiotemporal mode-locking [7] and parametric instability [8]. Therefore, with the explosive improvement of laser power, nonlinear propagation in MMFs has received recent research interests [919].

Krupa et al. studied the nonlinear Kerr beam cleanup effect (NL-KBC) when laser pulses with increasing power propagated in graded-index (GRIN) MMFs [11]. The nature of this effect is that energy of high-order modes (HOMs) is irreversibly coupled into the fundamental mode where Kerr effect in GRIN MMFs is the driving mechanism. NL-KBC has also been observed in many optic systems with high-peak-power laser propagation in MMFs [1115,1820]. It can be used for high-energy and high-beam-quality laser generation [1214] and spatiotemporal beam compression [21]. Moreover, analogous to Kerr lens mode-locking in solid-state lasers [22], it is easy to imagine that GRIN MMF with a diaphragm can be served as a saturable absorber (SA) in the NL-KBC regime. These SAs based on NL-KBC effect have potential to realize all-fiber configuration when using optical fiber with smaller mode area as the diaphragm in them. Although SAs based on nonlinear multimodal interference (NL-MMI) in a short GRIN MMF segment have been theoretically proposed [23] and experimentally demonstrated [24], their performances are exceptionally sensitive to the changes of GRIN MMF length, which can be a limitation in practical applications. Instead, saturable absorption behavior of the longer GRIN MMF based on NL-KBC has not been reported, and hence a further investigation is required.

In this paper, we experimentally investigated characteristics of the SA based on NL-KBC. For the purpose of studying the mechanism of the NL-KBC and the SA, we firstly studied the evolution of output spectrum and beam profiles from the GRIN MMF with the increasing launched laser power. Then we built up the SA which consisted of a long GRIN MMF segment and a short single-mode fiber (SMF) segment as a diaphragm. We performed saturable absorption measurements on these SAs in order to evaluate their optical properties. We first experimentally and theoretically studied the “relaxation oscillation” behavior of the optical transmittance with increasing input intensity in these SAs. We also studied the nonlinear polarization dynamics of these SAs in order to ensure their environmental stability. Our results show that, based on NL-KBC, a long GRIN MMF segment with a diaphragm can be served as a SA which can be applied to Q-switched or mode-locked lasers.

2. Experimental setup

A schematic of our experimental setup is depicted in Fig. 1. The laser source used in this system was a passively Q-switched Nd:YAG microchip laser which produced intense 1064 nm pulses with its duration of about 700 ps at 10.4 kHz. The linearly polarized laser beam was launched into a GRIN MMF by using a lens with its focal length of 100 mm; an isolator was used to protect the laser source from the reflected beam. The focused beam was in the center of the input facet, and it had a Gaussian shape with a diameter (full width at half maximum, FWHM) of about 38 µm. The launched power could be adjusted by a combination of a half wave plate (HWP) and a polarizing beam splitter (PBS), and the maximum average power was measured to be 93 mW. Since the insertion losses of the isolator, HWP, PBS and lens were small, a power meter was used to measure the reflective laser power from the PBS in order to estimate the launched power. The commercial GRIN MMF (Nufern GR-50/125-23HTA) has a core diameter of 50 µm and numerical aperture of 0.230. The length of this GRIN MMF segment was adjusted to be 30 m, 11.2 m, and 2 m in order to investigate its influence on NL-KBC and characteristics of the SAs based on NL-KBC. Output beam from the GRIN MMF would be propagated into different optical systems in different series of experiments; details will be shown in the following section. It should be noted that the output beam profiles from GRIN MMFs and the transmittance of the SAs would be changed by the bending condition of GRIN MMFs [11], and hence the GRIN MMFs used in our experiments were loosely coiled into circle of about 25 cm in diameter.

 figure: Fig. 1.

Fig. 1. Experimental setup for the mechanism investigation and characteristics measurements of the saturable absorber which is formed by a graded-index multimode fiber with a single-mode fiber as a diaphragm: the measurements for (a) output spectrum and (b) output near-field intensity spatial profiles from the graded-index multimode fiber; the measurements for (c) optical transmittance and (d) polarization states of the saturable absorber.

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3. Results and discussions

3.1 Output spectral evolution from the GRIN MMF

Firstly, we investigated the output spectral evolution from the GRIN MMFs with different length by using a commercial optical spectrum analyzer (Anritsu, MS9740A); the results are shown in Fig. 2. It should be noted that this optical spectrum analyzer has a SMF fibered input. The output spectrum from the GRIN MMF shown in Fig. 2 can just qualitatively illustrate the relative intensity variation of each wavelength component because the spectra of the HOMs could hardly be included, and the minimum resolution of the spectra was only 0.03 nm which may probably be larger than the bandwidth of the microchip laser. As shown in Fig. 2, the evolution can be divided into three distinct stages as the launched peak power increased. In the linear regime where the launched power was low, the output spectrum was almost identical to the input one which had two longitudinal modes: a strong mode centered in about 1064.29 nm, and a much weaker one centered in about 1064.44 nm. Then, as the launched peak power further increased, the bandwidth of the strong mode was broadened due to the self-phase modulation. Finally, the stimulated Raman scattering (SRS) occurred. Moreover, as shown in Fig. 2(a), the evolution of relative intensity between the strong mode and the weak mode indicates that the strong mode was noticeably depleted with the increasing input power. Comparing Figs. 2(b)–2(d) shows that the SRS threshold was lower in longer GRIN MMFs; the SRS could not be observed from the 2-m GRIN MMF with the maximum input power because the product of “Input Peak Power × GRIN MMF Length” is small in this situation; output spectrum with the second-order SRS peak (centered in 1174.3 nm), the third-order SRS peak (centered in 1238.2 nm), and even super-continuum spectrum could be observed from the 30-m GRIN MMF when the launched peak power reached a high level. These phenomena are consistent with SRS dynamics, which is similar to the case of Raman effect in SMF. We will find that the spectrum evolution has significant influences on optical properties of the SA based on NL-KBC in the following sections.

 figure: Fig. 2.

Fig. 2. The output spectra from the GRIN MMFs with the increasing input peak power, when the fiber length was (a)-(b) 30 m, (c) 11.2 m, and (d) 2 m. The spectral resolution was set to be 0.03 nm in (a) and 0.3 nm in (b)-(d). The numerical value in the panel indicates the input peak power.

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3.2 Output beam profile evolution from the GRIN MMF

In order to study the mechanism of the SAs based on NL-KBC, we then investigated the evolution of output near-field intensity spatial profiles from the GRIN MMF. As illustrated by Fig. 1(b), these beam profiles were imaged on a complementary metal-oxide-semiconductor camera (CINOGY, CinCam CMOS-1202) through an aspherical lens with a magnification of about 279. The length of GRIN MMF was chosen to be 11.2 m in this case. As shown in Fig. 3, before the launched peak power increased to the SRS threshold (4632 W), the output near-field intensity spatial profile gradually evolved from irregular pattern to a centered, bell-shape structure with weak speckled background. This beam profile evolution indicated the appearance of NL-KBC with which the laser power was gradually coupled into the fundamental mode from the HOMs. Therefore, the laser intensity randomly distributed in the output facet of GRIN MMF when the input peak power was low, but more intensity would gather near the center of the GRIN MMF when the input power was higher. It could be inferred that, based on NL-KBC, a GRIN MMF with an appropriate diaphragm can be served as a SA.

 figure: Fig. 3.

Fig. 3. The output near-field intensity spatial profile evolution from the GRIN MMF with the increasing launched peak power; the length of the GRIN MMF is 11.2 m. The beam profiles were measured without any spectral filters. The numerical value in the panel indicates the input peak power.

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After the SRS occurred (4632 W in this case), the beam profile kept a well-defined bell-shape structure with an even smaller diameter at first, and then it gradually evolved to asymmetry pattern with a larger FWHM diameter again. This phenomenon can be explained by the nonlinear dynamics of SRS and the nonlinear non-reciprocity of NL-KBC which leads to irreversibility of the energy flow from HOMs into the fundamental mode in the GRIN MMF. The diameter of the fundamental mode in GRIN MMFs can be calculated by

$${D_0} = 2{(\frac{{a\lambda }}{{n\pi \sqrt {2\Delta } }})^{\frac{1}{2}}},$$
where a is core radius of the fiber, λ is the propagation wavelength, n is the core refractive index, and Δ is the relative index difference. When the first-order SRS (1116.5 nm) occurred but its power was low, the power of propagating fundamental wave (FW, 1064.29 nm) was increased with the increasing input power. Therefore, the FW could be further influenced by the NL-KBC, and hence their beam power maintained in their fundamental mode with a diameter of only 7.39 µm. Moreover, the Stokes wave might be influenced by the Raman-induced beam cleanup effect (RBC) [25,26], and hence its beam profile may also maintain a well-defined bell-shape. When the input power further increased, the power of Stokes wave became larger, and the strong mode of the FW was noticeably depleted, as shown in Fig. 2. Although the Stokes wave could be induced by NL-KBC and RBC, it had larger fundamental-mode diameter (7.48 µm) than the FW according to Eq. (1) since its wavelength is much longer. Therefore, the output beam profile became larger in this case. After the launched peak power increased to a high level, for example 12380 W, a considerable amount of propagation wave components might not reach the threshold of NL-KBC due to the serious dissipation of fundamental wave as shown in Fig. 2. As a result, linear energy exchanges regained the upper hand between the fundamental mode and the HOMs of these components, and hence the output beam profile evolved to asymmetry pattern again.

In conclusion of the results from Fig. 2 and Fig. 3, we think that constructing SAs based on NL-KBC is possible and feasible. When these SAs are used for the mode-locked or Q-switched operation of the input laser, one can define that the threshold of SRS as the “damage threshold” of the SAs since no exceptional wavelength components are expected in this case. However, when these SAs are used for the generation of Raman laser with high beam quality or the nonlinear switch in the optical systems, one should concern the launched power at which the output beam profile has the smallest FWHM diameter; this launched power level can be defined as the instability attractor of the SA. However, in this paper we mainly focus on the former function of these SAs, and thus there will be little detailed discussion on the instability attractor in the following sections.

3.3 Transmittance of the SA based on NL-KBC

In this study, we built up the SAs with a short SMF segment (Nufern, PM980-XP) spliced to the GRIN MMF. The mode field diameter of this SMF is calculated to be 7.14 µm which is close to the fundamental-mode diameter of the GRIN MMF, and hence the SMF could be served as a diaphragm in the SA. Its length was chosen to be 0.35 m for almost completely stripping out HOM components of the propagation wave.

In order to evaluate the optical properties of the SAs based on NL-KBC, we performed saturable absorption measurements on the SAs with GRIN MMF length of 30 m, 11.2 m, and 2 m. The experimental setup is shown in Fig. 1(c), and the calculated optical transmittance of these SAs under different input intensity are presented in Fig. 4. The transmittance of the SA presented here is defined as the ratio of the output average power of the SMF to the launched average power of the GRIN MMF. Comparing Figs. 4(a)–4(c) shows that the optical transmittance of the SAs with different GRIN MMF length had a similar change trend which could be classified into three distinct regions. In Region 1, the optical transmittance grew monotonically as the input intensity increased due to the appearance of NL-KBC and hence the gradual centralization of the output beam profile from the GRIN MMF. When the input power further increased in Region 2, the optical transmittance curve performed a “relaxation oscillation” behavior, which can be explained as follow. In our experiment, the Gaussian launched beam had a FWHM diameter of about 38 µm, and thus it excited a large number of transverse modes of the GRIN MMF. According to the Ref. [23], comparing with launched beam diameter (about 38 µm) and fundamental mode diameter (7.39 µm) of the GRIN MMF, we can find that the power of exciting fundamental mode did not dominate in the initial mode power distribution in our case. Caused by this initial mode power distribution and Bragg-type quasi-phase-matching conditions in the GRIN MMF, for a sufficiently large product of “Input Peak Power × GRIN MMF Length”, the periodic power exchange between the fundamental mode and HOMs (the “oscillation” behavior) could be observed as increasing launched intensity. This periodic phenomenon has been independently predicted by the theoretical and numerical studies in Nazemosadat’s [23] and Krupa’s work [11]. In addition, according to the numerical models presented in the Refs. [11,23], the “relaxation” behavior indicated the perturbation of Bragg-type quasi-phase-matching conditions which may probably be caused by the cross-phase modulation (XPM) on the fundamental mode originating from HOMs, XPM among the HOMs, the four wave mixing among the HOMs, and other high-order nonlinear effects among the propagation modes. As a typical phenomenon in NL-KBC regime, this “relaxation oscillation” behavior has significant impacts on the output laser performance from the GRIN MMF and hence the optical properties of SAs based on NL-KBC. However, little attention has been paid to its influences in previous experimental studies. Here, a qualitative explanation for its mechanism is presented above, and its influences on the modulation stability and output polarization dynamics of the SAs based on NL-KBC will be discussed in the following section. In Region 3, after the SRS occurred, the optical transmittance first increased and then decreased, which was in good agreement with the beam profile evolution presented in Fig. 3.

 figure: Fig. 4.

Fig. 4. The optical transmittance of the SAs with the GRIN MMF length of (a) 30 m, (b) 11.2 m, and (c) 2 m as a function of the increasing input intensity. The dash lines in the figures are used to indicate different regions, and the solid lines are the fitting curves of the experimental data in Region 1 and Region 2.

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In order to evaluate the modulation depths and saturation intensities of these SAs, the transmittance curve in Region 1 and Region 2 can be fitted by

$$T(I) = {T_0} + q \cdot [{1 - \exp ( - {I \mathord{\left/ {\vphantom {I {{I_{\textrm{sat}}}}}} \right.} {{I_{\textrm{sat}}}}})} ],$$
where T is the optical transmittance, T0 is the initial transmittance, q can be defined as the saturable absorption or the modulation depth, I and Isat are the input and saturation intensity, respectively. As shown in Figs. 4(a)–4(c), the modulation depths of the SAs with GRIN MMF length of 30 m, 11.2 m, and 2 m are fitted to be 28.94%, 38.51%, and 24.54%; their saturation intensities are fitted to be 3.43, 1.66, and 16.55 MW/cm2; their initial transmittance are fitted to be −6.88%, −17.88%, and 3.85%, respectively. It should be noted that the fitted parameters in the SAs with 30-m and 11.2-m GRIN MMF are not accurate because their recorded data in Region 1 was scarce, which was limited by the minimum rotation degree of the input HWP. However, we find that these fitted parameters can provide guidance for the design of optical properties of the SAs. According to the performance of NL-KBC, the parameter T0 + q which indicates the transmittance when the input intensity is sufficiently high should have similar value for these SAs, since their input laser power distributions were almost the same. The calculated T0 + q of 22.06%, 20.63%, and 28.39% for the SAs with GRIN MMF length of 30 m, 11.2 m, and 2 m were in good agreement with this theoretical analysis. Surprisingly, due to the inherent and environmental disturbances in the long GRIN MMF, their initial transmittance in linear regime can be independent of the input power distribution and the length of GRIN MMF. Therefore, the modulation depth of the SA based on NL-KBC can be adjusted by changing the input power distribution. Comparing fitting curves in Figs. 4(b) and 4(c) shows that, the SA with longer GRIN MMF length had lower saturation intensity in Region 1. This behavior is constant with the performance of NL-KBC because longer GRIN MMF means lower threshold for NL-KBC. Therefore, the saturation intensity of the SA based on NL-KBC can be adjusted by changing the length of GRIN MMF. Table 1 illustrates the modulation depth and saturation intensity of commercial or reported SAs. As shown in the table, the SA based on NL-KBC with 2-m GRIN MMF has larger modulation depth which is comparable to the commercial SESAM when compared with other two-dimensional materials. Moreover, according to the analysis mentioned above, its saturation intensity can be lower with longer GRIN MMF, and its modulation depth can be even larger with smaller input beam diameter. This indicates that SAs based on NL-KBC, like other conventional SAs, can be applied to mode-locked lasers.

Tables Icon

Table 1. Optical properties of different saturable absorbers.

In conclusion of these transmittance measurements, we confirm the saturable absorption effect of a long GRIN MMF with a diaphragm. The saturation intensity and modulation depth of the SAs based on NL-KBC can be designed by the input beam diameter and the length of GRIN MMF, respectively. The “relaxation oscillation” behavior of optical transmittance has been observed with increasing the input intensity before the SRS occurred. However, this behavior in Region 2 may not be a desirable feature for an SA since it may introduce instabilities in the mode-locked and Q-switched operation. Therefore, the ideal operation region of these SAs can be Region 1, but the operating pulse energy may be limited to a low level when the length of GRIN MMF is long. Comparing with other conventional SAs shows that the SAs based on NL-KBC can be applied to mode-locked lasers. It should be noted that the SA based on NL-KBC is similar to the SA based on NL-MMI because Kerr effect in GRIN MMF is the driving mechanism for nonlinear energy exchanges between fundamental mode and HOMs in both situations. However, the most significant difference is the length of GRIN MMF which leads to distinct initial states in linear regime. For SA based on NL-MMI in which the GRIN MMF length is short (10−3∼10−1 m), the power transmission induced by the linear MMI is a periodic function of the GRIN MMF length when the input power is low [23]. Therefore, the transmittance evolution with the increasing input power is significantly influenced by the GRIN MMF length and hence the initial output state of the SA. For SA based on NL-KBC in which the GRIN MMF length is sufficiently long (>100 m), the propagating laser is subjected to more inherent and environmental disturbances in the long GRIN MMF. As a result, the beam profile presents an irregular pattern at the output facet of GRIN MMF when the input power is low (as shown in Fig. 3). In a specific case, for example changing the bending condition of the GRIN MMF, the initial transmittance of the SA based on NL-KBC can be relatively low, and it will be independent of the input power distribution and the length of GRIN MMF. With this initial condition, the optical transmittance evolution induced by NL-KBC can be smoothened as shown in Region 1 of Fig. 4. Therefore, unlike the SA based on NL-MMI, a seriously specific length is not necessary in the SA based on NL-KBC. Moreover, multimode interference-based filtering effect [31] may have important influences on the properties of the SAs based on NL-KBC. However, we do not have a high-power white light source, and hence these experiments are not included here.

3.4 Nonlinear polarization dynamics of the SA based on NL-KBC

Finally, we have also studied polarization dynamics of the SAs based on NL-KBC in order to ensure if these SAs are environmental stable. A similar approach with Krupa’s work [15] was taken, and our experimental setup is shown in Fig. 1(d). A 0.35-m polarization-maintaining (PM) SMF (Nufern, PM980-XP) was spliced to the GRIN MMF and served as a diaphragm in the SA as mentioned above. By using a quarter wave plate (QWP), a linear polarizer (a combination of a HWP and a PBS in our case), and a power meter, we measured and calculated the Stokes parameters (S1, S2, S3, and S4) of the output beam from the SAs with their GRIN MMF length of 30 m, 11.2 m, and 2 m. Based on these Stokes parameters, we further calculated a set of polarization degrees to characterize output time-average polarization states of the SAs. The calculated polarization degrees consisted of the total degree of polarization (DOP), the degree of linear polarization (DOLP), and the degree of circular polarization (DOCP); their evolution with the increasing input peak power is shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. The output polarization degrees of the SAs with the GRIN MMF length of (a) 30 m, (b) 11.2 m, and (c) 2 m as a function of the increasing input peak power. The dash lines in the figures are used to indicate the SRS threshold, and the dotted curves are used to link the adjacent data points and guide the eye.

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Here, we firstly focus on the evolution of DOP. Before the SRS occurred, the DOP first increased and then fluctuated around a certain level with the increasing input peak power. This behavior was caused by the performance of NL-KBC. When the input peak power was low, the output beam had a low DOP due to random polarization mode coupling. After the input peak power reached the threshold of NL-KBC, the energy was gradually coupled into the fundamental mode from HOMs, which led to the repolarization of output beam; this result is in good agreement with Krupa’s work [15]. When further increasing the input power, the fluctuation of output DOP is consistent with “relaxation oscillation” behavior of optical transmittance as mentioned above. The slight differences between both evolutions might be caused by measuring errors, random polarization mode coupling, nonlinear polarization rotation and hence temporal averaging depolarization. Comparing Figs. 5(a)–5(c) shows that the SA with longer GRIN fiber length had a lower DOP at the same input peak power. This phenomenon can be explained as follow: the SA with longer GRIN MMF length has lower transmittance at the same input power level (as shown in Fig. 4), and hence the unavoidable disturbance caused by random polarization mode coupling in the MMF had a more significant impact on the polarization state of output beam. It should be noted that the DOP was slightly larger than 1 when the GRIN MMF length was 2 m, which might be caused by the measuring errors. Moreover, the output DOP should probably be larger in the case with shorter MMF [15,32]; the initial DOPs with low input peak power may not be accurate because the recorded data was scarce in our cases.

In terms of DOLP and DOCP, their evolutions were obviously different in these three SAs. These results might be caused by nonlinear polarization rotation and the misalignment between the slow axis of PM SMF and the linear polarization direction of input laser. Therefore, although it is exciting to find that the DOP and DOLP of the SA with short GRIN MMF length (2 m in our case) were closed to 1, a polarization controller may be needed when this SA is used in the laser cavity, which means the mode-locked or Q-switched operation may not be environmentally stable.

After the SRS occurred, the data of polarization degree was somehow distorted since the HWP and QWP used in the experiments are only applicable to the input laser wavelength (1064 nm). Although the polarization state of the output beam with Stokes wave is in a more complicated and interesting situation, a detailed discussion will not be included here since it is not within the scope of this study as mentioned above.

In conclusion of the study on nonlinear polarization dynamics of the SAs based on NL-KBC, we found that the output polarization state of the SA could be partially repolarized in the NL-KBC regime. Especially when the GRIN MMF length was short, the output DOP and DOLP could be close to 1, which indicated the SA was mainly PM. However, a polarization controller will be needed when the SA is used in the laser cavity, unless the slow axes of PM SMF in the SA and the input PM fiber can be matched with an appropriate angel to neutralize nonlinear rotation effect in the GRIN MMF.

4. Conclusion

In conclusion, we experimentally investigated the mechanism and the optical properties of the SAs based on NL-KBC. The evolution of output spectrum and beam profiles from the GRIN MMF with the increasing input laser power indicates the potential of constructing SAs based on a long GRIN MMF with a diaphragm. The optical transmittance measurements have further confirmed the saturable absorption properties of these SAs. In addition, we find that the modulation depth and saturation intensity of these SAs can be designed by the input power distribution and the length of GRIN MMF, respectively. Their saturation absorption performances are remarkable when compared with other conventional SAs. We also study nonlinear polarization dynamics of these SAs, and we found that they can be mainly PM with a short GRIN MMF length. Moreover, we first experimentally observed and theoretically studied the “relaxation oscillator” behavior of the optical transmittance in these SAs and its influences on the output polarization states. However, it should be noted that these SAs have several undesired features which may limit their performance for mode-locked or Q-switched operation in lasers. First, the “relaxation oscillation” behavior of optical transmittance has been observed in NL-KBC regime, which may introduce instabilities and limit the operating pulse energy to a low level. Second, due to the nonlinear polarization rotation effect in the GRIN MMF, a polarization controller would be required in the laser cavity, which means the mode-locked or Q-switched operation based on these SAs may not be environmentally stable. Despite these weaknesses, our results prove that the SAs based on NL-KBC can be applied to the modulation of lasers, which may provide new methods for mode-locking and Q-switching. Our future work will focus on the numerical model to design the SA based on NL-KBC and the performance of lasers with these SAs. Moreover, after the SRS occurred, some interesting phenomena from these SAs have been observed, and they require further investigations. Therefore, this study should be extended to the situation with a sufficiently high input power level, and the investigation on these complicated multimode spatiotemporal dynamics may provide new approaches to understand and utilize the complex multimode nonlinearities.

Funding

Young Teacher Foundation of Sun Yat-sen University (20174500031610017); Natural Science Foundation of Guangdong Province (2017A030310305, 2018A030310092).

Acknowledgments

The authors would like to acknowledge Qiang Zhong, Fujuan Wang, and Jiaoyang Li for the technical support in this work.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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16. W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013). [CrossRef]  

17. L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015). [CrossRef]  

18. L. G. Wright, Z. Liu, D. A. Nolan, M. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016). [CrossRef]  

19. Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016). [CrossRef]  

20. U. Teğin, E. Kakkava, B. Rahmani, D. Psaltis, and C. Moser, “Spatiotemporal self-similar fiber laser,” Optica 6(11), 1412–1415 (2019). [CrossRef]  

21. K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A 97(4), 043836 (2018). [CrossRef]  

22. T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17(18), 1292–1294 (1992). [CrossRef]  

23. E. Nazemosadat and A. Mafi, “Nonlinear multimodal interference and saturable absorption using a short graded-index multimode optical fiber,” J. Opt. Soc. Am. B 30(5), 1357–1367 (2013). [CrossRef]  

24. Z. Wang, D. N. Wang, F. Yang, L. Li, C. Zhao, B. Xu, S. Jin, S. Cao, and Z. Fang, “Er-doped mode-locked fiber laser with a hybrid structure of a step-index–graded-index multimode fiber as the saturable absorber,” J. Lightwave Technol. 35(24), 5280–5285 (2017). [CrossRef]  

25. W. B. Baek and Roh, “Single-mode Raman fiber laser based on a multimode fiber,” Opt. Lett. 29(2), 153–155 (2004). [CrossRef]  

26. B. M. Flusche, T. G. Alley, T. H. Russell, and W. B. Roh, “Multi-port beam combination and cleanup in large multimode fiber using stimulated Raman scattering,” Opt. Express 14(24), 11748–11755 (2006). [CrossRef]  

27. https://www.batop.de/products/saturable-absorber/saturable-absorber-mirror/data-sheet/saturable-absorber-mirror-1064nm/saturable-absorber-mirror-SAM-1064-70-3ps.pdf

28. M. B. Hisyam, M. F. M. Rusdi, A. A. Latiff, and S. W. Harun, “Generation of Mode-locked Ytterbium doped fiber ring laser using few-layer black phosphorus as a saturable absorber,” IEEE J. Sel. Top. Quantum Electron. 23(1), 39–43 (2017). [CrossRef]  

29. Z. Wang, H. Mu, J. Yuan, C. Zhao, Q. Bao, and H. Zhang, “Graphene-Bi2Te3 heterostructure as broadband saturable absorber for ultra-short pulse generation in Er-doped and Yb-doped fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 23(1), 195–199 (2017). [CrossRef]  

30. X. Li, Y. Wang, Y. Wang, Y. Zhang, K. Wu, P. Shum, X. Yu, Y. Zhang, and Q. Wang, “All-normal-dispersion passively mode-locked Yb-doped fiber ring laser based on a graphene oxide saturable absorber,” Laser Phys. Lett. 10(7), 075108 (2013). [CrossRef]  

31. A. Mafi, P. Hofmann, C. J. Salvin, and A. Schülzgen, “Low-loss coupling between two single-mode optical fibers with different mode-field diameters using a graded-index multimode optical fiber,” Opt. Lett. 36(18), 3596–3598 (2011). [CrossRef]  

32. Y. Zhan, Q. Yang, H. Wu, J. Lei, and P. Liang, “Degradation of beam quality and depolarization of the laser beam in a step-index multimode optical fiber,” Optik 120(12), 585–590 (2009). [CrossRef]  

References

  • View by:

  1. Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
    [Crossref]
  2. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “High-resolution, lensless endoscope based on digital scanning through a multimode optical fiber,” Biomed. Opt. Express 4(2), 260–270 (2013).
    [Crossref]
  3. M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
    [Crossref]
  4. H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
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  5. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
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  6. J. Nilsson and D. N. Payne, “High-power fiber lasers,” Science 332(6032), 921–922 (2011).
    [Crossref]
  7. L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
    [Crossref]
  8. K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
    [Crossref]
  9. S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Technol. 31(3), 398–406 (2013).
    [Crossref]
  10. A. Mafi, “Pulse propagation in a short nonlinear graded-index multimode optical fiber,” J. Lightwave Technol. 30(17), 2803–2811 (2012).
    [Crossref]
  11. K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
    [Crossref]
  12. R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25(5), 4783–4792 (2017).
    [Crossref]
  13. R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber,” Opt. Express 25(19), 22219–22227 (2017).
    [Crossref]
  14. D. Ceoldo, K. Krupa, A. Tonello, V. Couderc, D. Modotto, U. Minoni, G. Millot, and S. Wabnitz, “Second harmonic generation in multimode graded-index fibers: spatial beam cleaning and multiple harmonic sideband generation,” Opt. Lett. 42(5), 971–974 (2017).
    [Crossref]
  15. K. Krupa, G. G. Castañeda, A. Tonello, A. Niang, D. S. Kharenko, M. Fabert, V. Couderc, G. Millot, U. Minoni, D. Modotto, and S. Wabnitz, “Nonlinear polarization dynamics of Kerr beam self-cleaning in a graded-index multimode optical fiber,” Opt. Lett. 44(1), 171–174 (2019).
    [Crossref]
  16. W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
    [Crossref]
  17. L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
    [Crossref]
  18. L. G. Wright, Z. Liu, D. A. Nolan, M. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
    [Crossref]
  19. Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
    [Crossref]
  20. U. Teğin, E. Kakkava, B. Rahmani, D. Psaltis, and C. Moser, “Spatiotemporal self-similar fiber laser,” Optica 6(11), 1412–1415 (2019).
    [Crossref]
  21. K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A 97(4), 043836 (2018).
    [Crossref]
  22. T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17(18), 1292–1294 (1992).
    [Crossref]
  23. E. Nazemosadat and A. Mafi, “Nonlinear multimodal interference and saturable absorption using a short graded-index multimode optical fiber,” J. Opt. Soc. Am. B 30(5), 1357–1367 (2013).
    [Crossref]
  24. Z. Wang, D. N. Wang, F. Yang, L. Li, C. Zhao, B. Xu, S. Jin, S. Cao, and Z. Fang, “Er-doped mode-locked fiber laser with a hybrid structure of a step-index–graded-index multimode fiber as the saturable absorber,” J. Lightwave Technol. 35(24), 5280–5285 (2017).
    [Crossref]
  25. W. B. Baek and Roh, “Single-mode Raman fiber laser based on a multimode fiber,” Opt. Lett. 29(2), 153–155 (2004).
    [Crossref]
  26. B. M. Flusche, T. G. Alley, T. H. Russell, and W. B. Roh, “Multi-port beam combination and cleanup in large multimode fiber using stimulated Raman scattering,” Opt. Express 14(24), 11748–11755 (2006).
    [Crossref]
  27. https://www.batop.de/products/saturable-absorber/saturable-absorber-mirror/data-sheet/saturable-absorber-mirror-1064nm/saturable-absorber-mirror-SAM-1064-70-3ps.pdf
  28. M. B. Hisyam, M. F. M. Rusdi, A. A. Latiff, and S. W. Harun, “Generation of Mode-locked Ytterbium doped fiber ring laser using few-layer black phosphorus as a saturable absorber,” IEEE J. Sel. Top. Quantum Electron. 23(1), 39–43 (2017).
    [Crossref]
  29. Z. Wang, H. Mu, J. Yuan, C. Zhao, Q. Bao, and H. Zhang, “Graphene-Bi2Te3 heterostructure as broadband saturable absorber for ultra-short pulse generation in Er-doped and Yb-doped fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 23(1), 195–199 (2017).
    [Crossref]
  30. X. Li, Y. Wang, Y. Wang, Y. Zhang, K. Wu, P. Shum, X. Yu, Y. Zhang, and Q. Wang, “All-normal-dispersion passively mode-locked Yb-doped fiber ring laser based on a graphene oxide saturable absorber,” Laser Phys. Lett. 10(7), 075108 (2013).
    [Crossref]
  31. A. Mafi, P. Hofmann, C. J. Salvin, and A. Schülzgen, “Low-loss coupling between two single-mode optical fibers with different mode-field diameters using a graded-index multimode optical fiber,” Opt. Lett. 36(18), 3596–3598 (2011).
    [Crossref]
  32. Y. Zhan, Q. Yang, H. Wu, J. Lei, and P. Liang, “Degradation of beam quality and depolarization of the laser beam in a step-index multimode optical fiber,” Optik 120(12), 585–590 (2009).
    [Crossref]

2019 (2)

2018 (1)

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A 97(4), 043836 (2018).
[Crossref]

2017 (8)

Z. Wang, D. N. Wang, F. Yang, L. Li, C. Zhao, B. Xu, S. Jin, S. Cao, and Z. Fang, “Er-doped mode-locked fiber laser with a hybrid structure of a step-index–graded-index multimode fiber as the saturable absorber,” J. Lightwave Technol. 35(24), 5280–5285 (2017).
[Crossref]

M. B. Hisyam, M. F. M. Rusdi, A. A. Latiff, and S. W. Harun, “Generation of Mode-locked Ytterbium doped fiber ring laser using few-layer black phosphorus as a saturable absorber,” IEEE J. Sel. Top. Quantum Electron. 23(1), 39–43 (2017).
[Crossref]

Z. Wang, H. Mu, J. Yuan, C. Zhao, Q. Bao, and H. Zhang, “Graphene-Bi2Te3 heterostructure as broadband saturable absorber for ultra-short pulse generation in Er-doped and Yb-doped fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 23(1), 195–199 (2017).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25(5), 4783–4792 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber,” Opt. Express 25(19), 22219–22227 (2017).
[Crossref]

D. Ceoldo, K. Krupa, A. Tonello, V. Couderc, D. Modotto, U. Minoni, G. Millot, and S. Wabnitz, “Second harmonic generation in multimode graded-index fibers: spatial beam cleaning and multiple harmonic sideband generation,” Opt. Lett. 42(5), 971–974 (2017).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref]

2016 (3)

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

L. G. Wright, Z. Liu, D. A. Nolan, M. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref]

2015 (2)

L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
[Crossref]

M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
[Crossref]

2013 (6)

I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “High-resolution, lensless endoscope based on digital scanning through a multimode optical fiber,” Biomed. Opt. Express 4(2), 260–270 (2013).
[Crossref]

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” J. Lightwave Technol. 31(3), 398–406 (2013).
[Crossref]

W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

E. Nazemosadat and A. Mafi, “Nonlinear multimodal interference and saturable absorption using a short graded-index multimode optical fiber,” J. Opt. Soc. Am. B 30(5), 1357–1367 (2013).
[Crossref]

X. Li, Y. Wang, Y. Wang, Y. Zhang, K. Wu, P. Shum, X. Yu, Y. Zhang, and Q. Wang, “All-normal-dispersion passively mode-locked Yb-doped fiber ring laser based on a graphene oxide saturable absorber,” Laser Phys. Lett. 10(7), 075108 (2013).
[Crossref]

2012 (2)

A. Mafi, “Pulse propagation in a short nonlinear graded-index multimode optical fiber,” J. Lightwave Technol. 30(17), 2803–2811 (2012).
[Crossref]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref]

2011 (2)

2009 (1)

Y. Zhan, Q. Yang, H. Wu, J. Lei, and P. Liang, “Degradation of beam quality and depolarization of the laser beam in a step-index multimode optical fiber,” Optik 120(12), 585–590 (2009).
[Crossref]

2006 (1)

2004 (1)

2000 (1)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[Crossref]

1992 (1)

Agrawal, G. P.

Alley, T. G.

Auguste, J. L.

Baek, W. B.

Bao, Q.

Z. Wang, H. Mu, J. Yuan, C. Zhao, Q. Bao, and H. Zhang, “Graphene-Bi2Te3 heterostructure as broadband saturable absorber for ultra-short pulse generation in Er-doped and Yb-doped fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 23(1), 195–199 (2017).
[Crossref]

Barthélémy, A.

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A 97(4), 043836 (2018).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber,” Opt. Express 25(19), 22219–22227 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25(5), 4783–4792 (2017).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Bendahmane, A.

Brabec, T.

Cao, S.

Castañeda, G. G.

Ceoldo, D.

Choi, W.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref]

Choi, Y.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref]

Christodoulides, D. N.

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358(6359), 94–97 (2017).
[Crossref]

L. G. Wright, Z. Liu, D. A. Nolan, M. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
[Crossref]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016).
[Crossref]

L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23(3), 3492–3506 (2015).
[Crossref]

Cižmár, T.

M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
[Crossref]

Couderc, V.

K. Krupa, G. G. Castañeda, A. Tonello, A. Niang, D. S. Kharenko, M. Fabert, V. Couderc, G. Millot, U. Minoni, D. Modotto, and S. Wabnitz, “Nonlinear polarization dynamics of Kerr beam self-cleaning in a graded-index multimode optical fiber,” Opt. Lett. 44(1), 171–174 (2019).
[Crossref]

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A 97(4), 043836 (2018).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber,” Opt. Express 25(19), 22219–22227 (2017).
[Crossref]

D. Ceoldo, K. Krupa, A. Tonello, V. Couderc, D. Modotto, U. Minoni, G. Millot, and S. Wabnitz, “Second harmonic generation in multimode graded-index fibers: spatial beam cleaning and multiple harmonic sideband generation,” Opt. Lett. 42(5), 971–974 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25(5), 4783–4792 (2017).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116(18), 183901 (2016).
[Crossref]

Curley, P. F.

Dasari, R. R.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref]

Desfarges-Berthelemot, A.

Dupiol, R.

Essiambre, R. J.

Fabert, M.

Fang, Z.

Fang-Yen, C.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref]

Farahi, S.

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Flusche, B. M.

Guenard, R.

Harun, S. W.

M. B. Hisyam, M. F. M. Rusdi, A. A. Latiff, and S. W. Harun, “Generation of Mode-locked Ytterbium doped fiber ring laser using few-layer black phosphorus as a saturable absorber,” IEEE J. Sel. Top. Quantum Electron. 23(1), 39–43 (2017).
[Crossref]

Hisyam, M. B.

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Z. Wang, H. Mu, J. Yuan, C. Zhao, Q. Bao, and H. Zhang, “Graphene-Bi2Te3 heterostructure as broadband saturable absorber for ultra-short pulse generation in Er-doped and Yb-doped fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 23(1), 195–199 (2017).
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Biomed. Opt. Express (1)

IEEE J. Sel. Top. Quantum Electron. (2)

M. B. Hisyam, M. F. M. Rusdi, A. A. Latiff, and S. W. Harun, “Generation of Mode-locked Ytterbium doped fiber ring laser using few-layer black phosphorus as a saturable absorber,” IEEE J. Sel. Top. Quantum Electron. 23(1), 39–43 (2017).
[Crossref]

Z. Wang, H. Mu, J. Yuan, C. Zhao, Q. Bao, and H. Zhang, “Graphene-Bi2Te3 heterostructure as broadband saturable absorber for ultra-short pulse generation in Er-doped and Yb-doped fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 23(1), 195–199 (2017).
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J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (1)

Laser Phys. Lett. (1)

X. Li, Y. Wang, Y. Wang, Y. Zhang, K. Wu, P. Shum, X. Yu, Y. Zhang, and Q. Wang, “All-normal-dispersion passively mode-locked Yb-doped fiber ring laser based on a graphene oxide saturable absorber,” Laser Phys. Lett. 10(7), 075108 (2013).
[Crossref]

Nat. Commun. (1)

W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013).
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Nat. Photonics (4)

L. G. Wright, Z. Liu, D. A. Nolan, M. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10(12), 771–776 (2016).
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K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11(4), 237–241 (2017).
[Crossref]

M. Plöschner, T. Tyc, and T. Čižmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Opt. Express (4)

Opt. Lett. (6)

Optica (1)

Optik (1)

Y. Zhan, Q. Yang, H. Wu, J. Lei, and P. Liang, “Degradation of beam quality and depolarization of the laser beam in a step-index multimode optical fiber,” Optik 120(12), 585–590 (2009).
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Phys. Rev. A (1)

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light-beam compression from nonlinear mode coupling,” Phys. Rev. A 97(4), 043836 (2018).
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Other (1)

https://www.batop.de/products/saturable-absorber/saturable-absorber-mirror/data-sheet/saturable-absorber-mirror-1064nm/saturable-absorber-mirror-SAM-1064-70-3ps.pdf

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Figures (5)

Fig. 1.
Fig. 1. Experimental setup for the mechanism investigation and characteristics measurements of the saturable absorber which is formed by a graded-index multimode fiber with a single-mode fiber as a diaphragm: the measurements for (a) output spectrum and (b) output near-field intensity spatial profiles from the graded-index multimode fiber; the measurements for (c) optical transmittance and (d) polarization states of the saturable absorber.
Fig. 2.
Fig. 2. The output spectra from the GRIN MMFs with the increasing input peak power, when the fiber length was (a)-(b) 30 m, (c) 11.2 m, and (d) 2 m. The spectral resolution was set to be 0.03 nm in (a) and 0.3 nm in (b)-(d). The numerical value in the panel indicates the input peak power.
Fig. 3.
Fig. 3. The output near-field intensity spatial profile evolution from the GRIN MMF with the increasing launched peak power; the length of the GRIN MMF is 11.2 m. The beam profiles were measured without any spectral filters. The numerical value in the panel indicates the input peak power.
Fig. 4.
Fig. 4. The optical transmittance of the SAs with the GRIN MMF length of (a) 30 m, (b) 11.2 m, and (c) 2 m as a function of the increasing input intensity. The dash lines in the figures are used to indicate different regions, and the solid lines are the fitting curves of the experimental data in Region 1 and Region 2.
Fig. 5.
Fig. 5. The output polarization degrees of the SAs with the GRIN MMF length of (a) 30 m, (b) 11.2 m, and (c) 2 m as a function of the increasing input peak power. The dash lines in the figures are used to indicate the SRS threshold, and the dotted curves are used to link the adjacent data points and guide the eye.

Tables (1)

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Table 1. Optical properties of different saturable absorbers.

Equations (2)

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D 0 = 2 ( a λ n π 2 Δ ) 1 2 ,
T ( I ) = T 0 + q [ 1 exp ( I / I I sat I sat ) ] ,

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