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Abstract
In this paper we demonstrate the spatial and spectral dynamics of pulse propagation in a step-index few-mode optical fiber, through an experimental and numerical analysis. The Kerr induced spatial self-cleaning is demonstrated by coupling a sub-nanosecond pulsed laser at 532nm into the fiber supporting above 10 modes. A bell-shaped and approximately single mode beam can be obtained for peak powers above 6kW and it remained relatively unchanged up to 25kW. But at significantly higher input peak powers, the spatial contents of spectral sidebands change dramatically, because of intermodal four wave mixing effect.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the last several years the research on pulse propagation in multi-mode fibers (MMFs) are extensively interested, regarding their enhanced transmission capacity and also using MMFs to perform lens-less endoscopic imaging [1–2] or quantum processing (multiphoton quantum interferences) [3], etc. In applications where MMFs are used for beam delivery, an important problem to be solved is the effective coupling and stable transport of a diffraction limited optical beam from a single mode (SM) laser source into the fundamental mode of the MMF. The study of complex spatiotemporal dynamics of nonlinear light propagation in MMFs reveals the specific modal properties that mediate a number of spatiotemporal nonlinear effect including the observation of multimode solitons [4], cascaded intermodal four-wave mixing (FWM) and modulation instability [5], geometric parametric instability [6], spin orbit interaction [7], multimode fiber lasers [8,9], and supercontinuum generation [10,11]. Experiments on nonlinear propagation in MMFs have recently revealed an unexpected effect that was called Kerr beam self-cleaning. It consists in the reshaping, above a certain threshold, of the output transverse beam profile into a bell-shaped beam in a graded-index (GRIN) MMF [12–14]. The power level in this nonlinear process is below the threshold for frequency conversion or self-focusing of sub nanosecond to femtosecond pulses propagating in the normal dispersion regime [15,16]. So far, most papers reported nonlinear beam reshaping and other complex nonlinear effects in MMFs with a parabolic refractive index profile or GRIN MMFs. Guenard et al. have been observed self-cleaning effect in a nearly step-index dissipative fiber with and without amplification [17]. So, it has been shown experimentally that Kerr self-cleaning is not restricted to GRIN fibers, where periodic self-imaging was guessed to facilitate mode coupling process and resulting beam cleaning. However, the observation of self-cleaning effect in step-index fiber is still challenging. To our knowledge, Kerr-beam spatial reshaping have not ever been reported in a step-index fiber. In this work, we extended the study of nonlinear spatial reshaping effects to a step-index few-mode optical fiber in normal dispersion regime. One of the initial phenomena allowing for spatial mode mixing and cleaning is spatial self-imaging process, which can periodically modulate the core refractive index of the fiber when associated with the Kerr nonlinearity. It has been shown that the self-imaging effect can be observed in step-index fibers but with a self-imaging quality factor less than that of equivalent GRIN fiber [18]. So, in this case, a high amount of peak power with respect to the power threshold for self-cleaning in GRIN fibers is needed for mode coupling processes, consequently, by controlling the excited mode numbers in a certain dispersion regime, it is possible to reach from a speckled output profile to a bell-shaped transverse beam in a few-mode step-index fiber. With a few-mode fiber it is easier to have quasi phase matching condition between modes than highly multi-mode fibers and consequently receiving to spatial beam self-cleaning will be more difficult by increasing mode numbers propagating in the fiber. Here it is demonstrated experimentally and numerically that for a step-index fiber with 20µm core diameter and numerical aperture 0.065 supporting more than 10 spatial modes at 532nm wavelength, we can observe a bell-shaped profile at the output for above 6kW input peak powers. Bell-shaped spatial pattern remains relatively unchanged until 25kW peak powers. However, at high power levels, where new frequencies generate, irregular spacing between wave constants of different modes, has a strong impact on the spatial and spectral properties of parametric side-band generation, and so there is intermodal four wave mixing (FWM) with side-bands at higher order transverse modes. Consequently, it is difficult to have a bell-shaped beam all over the spectral range. It has been demonstrated in several papers that in ideal GRIN MMFs, all side-bands are generated with the same bell-shaped transverse beam profile and modal contents as the self-cleaned pump [19–21]. These observations will be useful for nonlinear inter-modal interaction analysis in step-index fibers. Our results are well supported by numerical simulations based on the multidimensional nonlinear Schrodinger equation.
2. Experimental results
In our experiments, we used from the second harmonic of a home-maid longitudinally single-mode Nd:YAG microchip laser with Gaussian spatial beam shape [22], generating 450ps pulses at the repetition rate of 1kHz, with 35µJ maximum pulse energy at 532nm wavelength. The fiber we have used in all of the experiments reported bellow has a core diameter D=20µm and a numerical aperture 0.065. Figure 1(a) shows the refractive index profile of the fiber measured at 532nm wavelength. Our calculations confirm that the considered fiber can support more than 10 modes at 532nm wavelength at both polarization components. Simulation results for propagation constant and the invers of group velocity for first six modes propagating in considered few-mode step-index fiber has been illustrated in Figs. 1(c)–1(d). Self-imaging period in considered fiber is between 2 to 3.5mm which is much shorter than that of highly multi-mode fibers (10–43 mm in a step-index MMF with 50–105 µm core diameter [23]). On the other hand, it is larger than self-imaging period in GRIN fibers (less than 1mm). The relative group delay among modes in this fiber has been obtained between 3 to 6 (ps/m) at 532nm which is greater than the case of GRIN fibers (<1ps/m). It should be considered that the temporal pulse width of the input beam is 450ps and even after 5m of propagation, different propagating modes will not experience a significant walk-off and it cannot disrupt the modal interactions. A schematic view of experimental setup used for analysis of pulse propagation in considered fiber is illustrated in Fig. 1(b). The laser beam was injected into the fiber using a focusing lens f=50mm controlled by using a 3-axis translation stage, imaged the laser beam on the input face of the fiber with a 15µm beam diameter and coupling efficiency close to %60. The input pump power was controlled by using a variable attenuator including a half-wave plate and a polarizing beam splitter. A two lenses optical imaging system by magnification 3 has been used at the output end of the fiber, and we obtained the beam spatial image from a CCD camera and the spectrum was measured by an optical spectrometer with a 0.2nm resolution. In our coupling conditions more than %99 of guided input power coupled into the first six modes.
Fig. 1. (a) Refractive index profile measured at 532 nm wavelength. (b) Schematic view of experimental setup (c) and (d) propagation constant and inverse of group velocity for six first modes.
We recorded the output intensity pattern both in the linear and nonlinear regimes. Figures 2(a)–2(g) illustrate the nonlinear dynamics of beam pattern after propagation in 5m long few-mode step-index fiber. By adjusting the input condition while viewing a given (output) measurement in real time, we observed, tuned and enhanced the same features observed in our simulations. The initial peak power of the incident signal was set to 0.3kW (coupled power), and it gradually increased up to 12kW. At low input powers, the speckled output intensity pattern resulting from the coherent superposition of modes can be observed. By increasing the input power, the near field pattern from the fiber output shows a bell-shaped smooth central beam. The self-cleaned beam started to form approximately at a 6kW input peak power and it remained preserved up to 25kW. The observed behavior in spatial distribution evolution with increasing the peak-power is referred to Kerr-induced spatial self-cleaning, where the main part of the launched power was transferred toward the fundamental mode. Kerr self-cleaning leads to a narrowing of the output beam diameter to 9µm, equal to the diameter of the fundamental mode, measured at full width at half maximum in intensity (FWHMI) for 532nm pump wavelength. Besides the evolution of the output spatial patterns, we checked that there is no significant change in the output spectral width and so there is not any SRS induced frequency conversion. Near field and their corresponding far field images in linear and nonlinear regime of propagation is illustrated in Figs. 3(a)–3(d). It can be seen that there is same behavior in spatial reshaping toward bell-shaped beam in far-field and near-field images.
Fig. 2. Experimental output two-dimensional near-field pattern as a function of input guided power measured at the pump wavelength 532 nm in 5 m long few-mode step-index fiber, scale bars 10 µm. The injected power is a)0.3 kW, b)1.5 kW, c)3 kW, d)4.8 kW, e)6 kW, f)7.8 kW, g)10 kW. Fiber length, 5 m.
Fig. 3. Near field (a,b) and far field (c,d) intensity patterns at the few-mode step-index fiber output recorded in linear propagation regime (a,c) and in the nonlinear regime (b,d), scale bars 10 µm.
To demonstrate that the beam self-cleaning effect in step-index fiber occurs prior to frequency conversion, in Fig. 4 we show a series of spectra for increasing peak powers. These spectra correspond directly to the near field spatial patterns of the output beam reported in Fig. 2. We can see at a power level of 6kW, which bell-shaped transverse beam profile appears, there is a very weak spectral broadening, the discrete frequency peak appearing above 16kW peak power corresponds to the first Raman stokes sideband. We also did not observe any noticeable change in the input/output power-transmission ratio in all of the intensity interval. So, we confirm that Kerr self-cleaning also accurse here as a result of nonreciprocal nonlinear mode coupling. The self-cleaned beam also remained robust against external disturbances (e.g., intentional bending of the fiber), similar to the case of GRIN MMFs [11].
Fig. 4. Spectra of the step-index fiber output beam as a function of input peak power. Fiber length, 5 m.
In Fig. 5 we illustrated the dependence of the output near-field beam FWHMI diameters on the output power. It clearly demonstrates that a significant reduction of the beam diameter is achieved. Figure 5 also demonstrate the brightness enhancement of the output beam by increasing the peak power up to 12kW. In this experiment we have used from a diaphragm with less than 10µm diameter to measure the fraction of power carried by central part of the beam. The concentration of energy in the center of the fiber indicates that the self-induced spatial-cleaning observed at the output pattern enhances significantly the beam spatial quality. However, we consider in new frequency generation projects, it is possible that sideband generation to be no longer in LP01 mode in few-mode step-index fiber. Figure 6 summarizes the spectrum and associated transverse beam shapes at some selected wavelengths from step-index fiber at 532nm pump for significant increased input peak power (Pp-p=35kW). Raman peaks can be seen in the experimental spectrum. There are also some peaks that are generated as a consequence of IMFWM process (a signal at 599nm in LP02 mode that its predicted idler is at 456nm in LP01 mode). In the next section it is discussed analytically how nonlinear intermodal FWM process leads to the energy transfer from LP01 mode at 532nm pump wavelength into higher order modes at sideband wavelengths in step-index few-mode fiber. In these different series of experiments, it is visible how the pump beam has a bell-shaped profile, whereas the generated new frequencies are carried by different beam shapes. In ideal GRIN fibers, geometric parametric instability, owing to quasi-phase matching from the dynamic grating generated via the Kerr effect by pump self-imaging, leads to frequency multicasting of beam self-cleaning across a wideband array of sidebands. However, any deviation from perfectly parabolic refractive index profile, causes the spatial beam to be no longer bell shaped at the whole of the spectrum [20,24].
Fig. 5. Output power at 532 nm measured in the central part of the fiber as a function of the total output power (blue squares), and corresponding FWHMI diameter of the near-field beam at 532 nm (red circles). Fiber length, 5 m.
Fig. 6. The spectrum of sideband generation in step-index few-mode fiber for 35kW input peak power. Inset: Near field output profile with different bandpass filters, scale bars 10µm.
It should be noted that the output pattern quality at pump wavelength decreases slightly for high input peak power. This is due to the energy transfer at the central part of the beam towards Raman stokes band [15].
3. Numerical results
Simulations to guide the experiments were calculated using generalized multimode nonlinear Schrödinger equation [25] (GMM-NLSE):
Fig. 7. Numerical results of beam propagation in step-index few-mode fiber, scale bars 10 µm. (a-h) spatial reshaping as a function of input peak power for a) 0.3 kW, b) 1.5 kW, c) 3.6 kW, d) 4.2 kW, e) 6 kW, f) 10 kW. (g-k) spatial reshaping as a function of propagation distance in fiber, g) 0.02 m, h) 0.5 m, i) 3 m, j) 4 m, k) 5m
As it is evident from figure, the input Gaussian beam transfers into the several guided modes at the beginning of the propagation in step-index few-mode fiber. Our analysis reveals that by further propagating along fiber the intensity distribution modifies to a high brightness bell-shaped beam. The power of the fundamental mode increases by a factor of 2.3 after 5m of propagation distance along the fiber. We did not observe any significant change in spectral width in our simulations. It shows that beam self-cleaning is not result from SRS effect. To investigate the re-localization of light at the center of fiber core due to nonlinear interactions, we analyze the contribution of guided modes at the input and output of the fiber. In Fig. 8 we illustrate the energy distribution among LPlm modes after only 0.2mm of propagation, for the case of low intensity beam pulse [Fig. 8(a)] and high intensity beam pulse [Fig. 8(b)]. Figures 8(c), 8(d) show the corresponding spatial distribution of the beam, for the case of a low intensity and high intensity pulse, respectively, but after 5m of propagation. As it can be seen, at high intensities the presence of nonlinear propagation substantially modifies the modal distribution of the beam, by strongly increasing the contribution of the fundamental mode, whereas the relative contributions of higher order modes reduce.
Fig. 8. Numerical results of beam propagation in step-index few-mode fiber for modal contribution at a)z=2 mm, b)z=5 m for 0.5 kW input peak power and c)z=2 mm, d)z=5 m for 10 kW input peak power.
As it was mentioned in the experimental results, at input peak powers between 6kW-25kW the intensity of higher order modes is negligible, but at high energy levels by generating FWM induced side-band frequencies, there is not a bell-shaped beam at whole spectral range. Parametric side-band generation in step-index fibers has been reported and analyzed in several papers [26,27]. To better understand the physical origin behind energy transfer of the beam towards higher order modes at high peak powers in our considered fiber, we review the theory of intermodal FWM and the phase matching equations. For two pump photons generating symmetrically detuned signal and idler photons, the energy conservation yields: $2{\omega _p} = {\omega _s} + {\omega _i}$ and with ${\omega _p}$, ${\omega _s}$ and ${\omega _i}$ being the pump, signal and idler angular frequencies, and phase-matching conditions for two pump photons in the same LP modes is $\Delta \beta = \beta _0^{{j_{(i )}}} + \beta _0^{{k_{(s )}}} - 2\beta _0^{{l_{(p )}}} = 0$, where for example $\beta _0^{{j_{(p )}}}$ is propagation constant of pump wave in mode j. So, it is evident that in an intermodal FWM process, pump and sideband waves can be carried by different fiber modes, even if both pump photons be in the same mode. In order to numerically analyze the ultimate stability of Kerr self-cleaning into the LP01 mode upon large variation of input power, we repeated the simulation with the same input beam condition, but with significantly higher input pump power 35kW (Fig. 9).
Fig. 9. Numerical spectra for all of the modes propagating in step-index fiber obtained at 35 kW input peak power, Raman effect is not included. Fiber length, 5 m.
In obtaining the spectrum shown in Fig. 9, Raman scattering was not included (${f_R}$ = 0) as it does not affect the FWM spectral positions. So only peaks generated due to IMFWM process are shown. It can be seen that in this power level the pump wavelength remains on LP01 mode but new frequencies generated due to intermodal FWM process will not be Gaussian any more. Note that the intermodal FWM signal side band is obtained at 598nm on the LP02 mode whereas the pump wave at 532nm still emerges in LP01 mode. Therefor Kerr-induced beam reshaping is only observed for a certain range of input powers and it can lead to the generation of a multimode supercontinuum with some induced sidebands in higher order modes.
4. Conclusion
In conclusion, we demonstrated experimentally and numerically the nonlinear reshaping and frequency conversion in a step-index few-mode fiber. We observed Kerr-beam self-cleaning of the pump wave in a certain range of input powers. At high input peak powers, intermodal FWM sidebands generation leads to the energy transfer from LP01 mode in pump wavelength into the induced sidebands carried by higher order modes. The analysis has been done for sub-nanosecond 532nm pump wavelength. Power dependent beam self-cleaning results from the Kerr nonlinearity and intermodal coupling, since we did not observe any spectral broadening at the powers fine above self-cleaning threshold and also, modal contents calculated numerically in different power levels, indicates that the contribution of the fundamental mode increases strongly, whereas the relative contributions of higher order modes reduce. Our observations will be useful in developing novel photonic devices for applications based on combining high-power beam translation and a good spatial beam quality. On the other hand, the analysis performed in this work is helpful in understanding spatiotemporal dynamics in multi-mode step-index fibers which has been weakly reported.
Disclosures
The authors declare no conflicts of interest.
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