Emission of multiple resonant radiations by spatiotemporal oscillation of multimode dark pulses

Zhixiang Deng; Zhixiang Deng; Yu Chen; Jun Liu; Jun Liu; Chujun Zhao; Dianyuan Fan

doi:10.1364/OE.27.036022

1. Introduction

The landscape of nonlinear fiber optics field has been mainly dominated by single-mode fibers because of their high bandwidth and structural simplicity over the past few years. Recently, nonlinear multimode and multicore optical fibers (MMFs) have attracted tremendous attention since they are reemerging as a promising avenue for addressing many significant long-standing issues, such as the capacity promotion regime of telecommunications systems through the spatial division multiplexing technique [1], and the average output power scaling scheme of ultra-short fiber lasers with higher order transverse modes locked coherently [2]. Triggered by these developments [1–3], a critical rethinking regarding how complex nonlinear interaction processes can be explored in multimode environments has been prompted in recent years. All the spatial modes contributing to the propagating pulses in MMFs dynamically interact with each other through the third-order nonlinear processes such as self-phase and cross-phase modulation, four-wave mixing, and other intermodal linear and nonlinear processes [4]. Therefore, MMFs can offer a versatile platform to explore numerous new nonlinear propagation phenomena such as spatiotemporal pulse dynamics [5–7], spatial beam self-cleaning [8,9], and spatiotemporal mode-locking [2,10]. Particularly, the graded-index (GRIN) MMFs play a prominent role because it features very low inter-modal group dispersion and controllable spatiotemporal nonlinear effects [3,5–8]. During the propagation, the spatial breathing state of the optical beam can be formed with regular compression and stretching process, which is also referred to the spatial self-imaging effect [11] and resulting from the equally spaced propagation constant of the Gauss-Laguerre modes of the GRIN MMF. Under appropriate nonlinear conditions, this typical periodically evolving behavior of the pulse beam can cause a particular class of optical spectrum with many peaks [6,12] that are otherwise impossible in the regime of single-mode waveguide. Compared with traditional dispersive wave generation caused by higher-order effects [13–15], this special class of spectrum can be produced by quasi-phase matching (QPM) via periodic modulations of the nonlinearity. Indeed, as shown in recent studies [16–18], the QPM process can play an important role in the mechanism of supercontinuum generation in nonlinear mediums, especially when operating in the normal dispersion region.

In general, supercontinuum generation in Kerr nonlinear mediums is a complex problem to analyze because several nonlinear processes are involved, such as Raman-induced soliton self-frequency shift, modulation instability, dispersive wave generation, and so on [19,20]. Several numerical and experimental studies have been carried out to describe the spatial-temporal dynamics in GRIN MMFs based on perturbative schemes [5,6,12]. Although ultra-broadband dispersive radiations have been predicted to be existed in GRIN MMFs, we would like to emphasize that all the studied reported so far have focused on multimode solitons in the anomalous dispersion region [6,12]. Similarly, it is well known that dark solitons in the normal dispersion region can also emit radiations as they are perturbed by the higher-order effects in single-mode fibers, such as the third- [21–23] or higher-order [24–26] dispersion, but the excitation of multimode dark solitons and its dynamics perturbed by the periodic spatial beam oscillation within GRIN MMFs has not been reported so far, which are the subject of this paper. In these configurations, there are multiple values of the phase mismatching for a single nonlinear process due to the periodic nature of the spatial beam evolution.

Here, we show that, the train of multimode dark pulses can be excited with the aid of spatiotemporal nonlinear interactions in normally dispersive GRIN MMFs, and the periodic modulations of the effective mode area induced by the spatial self-imaging effect can enable the generation of QPM resonant radiations from the excited dark pulses, thus enhancing the intensity of the supercontinuum in specific spectral regions. This paper is organized as follows: In Sec. 2, the theoretical model including the governing equations has been introduced, and the mechanism of multiple resonant radiations generation in GRIN MMFs is discussed. Section 3 is devoted for the numerical results and discussions. Finally, our conclusions are summarized in Sec. 4.

2. Theoretical analysis

2.1 Propagation model in GRIN MMFs

In GRIN MMFs, a suitable model for the refractive index is [11,27]

(1)$$n(\omega ) = {n_0}(\omega ) + {n_2}|A{|^2} - \frac{{g\,{r^2}}}{2},$$

where A is the electric field envelope, $r = \sqrt {{x^2} + {y^2}}$ is the transverse radial coordinate, ${n_0}(\omega )$ is the linear fiber refractive index, ${n_2}$ is the nonlinear refractive index coefficient, $g = 2\Delta /{\rho ^2},\;\Delta = (n_{co}^2 - n_{cl}^2)/2n_{co}^2, n_{co}$ and ${n_{cl}}$ are the core and cladding refractive indexes, $\rho$ is the fiber core radius. As can be seen from Eq. (1), the refractive index profile in the spatial domain can be approximately described by ${n^2}(x, y) = n_0^2(1 - 2\Delta {r^2}/{\rho ^2})$, which indicates that the fiber can be modeled as a pure harmonic potential. The complex envelope $A(x, y, z, T)$ of the electric field referred to a time frame moving at the group velocity of the pulse obeys the following (3 + 1)-D generalized nonlinear Schrödinger equation (GNLSE) [3,5,8]:

(2)$$\begin{array}{l} \frac{{\partial A}}{{\partial z}} - i\frac{1}{{2{k_0}}}\nabla _ \bot ^2A - i\sum\limits_{n \ge 2} {\frac{{{\beta _n}}}{{n!}}{{(i{\partial _t})}^n}} A + i\frac{{{k_0}\Delta }}{{{\rho ^2}}}{r^2}A = i{k_0}\frac{{{n_2}}}{{{n_{co}}}} \times \\ \quad \quad \quad \quad \left[ {(1 - {f_R})|A{|^2}A + {f_R}\int {h(\tau )|A(t - \tau ){|^2}\textrm{d}\tau } } \right] \end{array}$$

where ${k_0} = {\omega _0}{n_{co}}/c,\;\;{\omega _0}$ is the carrier angular frequency of the optical beam, ${\beta _n}(n \ge 2)$ represents the different orders of dispersion coefficient at carrier frequency ${\omega _0}$, $h(t)$ is the Raman temporal response of the fiber and ${f_R} = 0.18$. The two terms on the right-hand side of the equation represent the instantaneous and delayed nonlinear Raman contribution, respectively. In the theoretical analysis and simulations reported in the paper we set the Raman term to zero to study the dynamics in the presence of only instantaneous Kerr effect.

In weakly or moderate nonlinear regimes, the (3 + 1) D pulse-propagation problem can be reduced to an effective quasi-(1 + 1) D NLSE involving only the z and t variables [27–29], as follows (see Appendix A.1):

(3)$$\frac{{\partial F}}{{\partial z}} - i\sum\limits_{n \ge 2} {\frac{{{\beta _n}}}{{n!}}{{(i{\partial _t})}^n}F} = i\gamma (z)|F{|^2}F,$$

where the nonlinear parameter is $\gamma (z) = \frac{{{\omega _0}{n_2}}}{{2\pi c{w^2}(z)}}$, and its average value can be calculated by

(4)$${\gamma _0} = \frac{1}{{\sqrt g }}\int_0^{\sqrt g } {\gamma (z)\textrm{d}z} = \frac{{{\omega _0}{n_2}{\beta _0}\sqrt g }}{{2\pi c}},$$

The evolution of the optical beam width $w(z)$ appearing in the nonlinear parameter $\gamma$ can be expressed as [27]:

(5)$$w(z) = {w_0}\sqrt {{{\cos }^2}(\sqrt g z) + {\mathbb C}{{\sin }^2}(\sqrt g z)} ,\quad {\mathbb C} \equiv \left( {\frac{{{n_2}w_0^2A_0^2}}{{2{n_0}}} - \frac{1}{{k_0^2}}} \right),$$

The Eq. (3) incorporates the spatial beam self-imaging effect through a periodically varying effective-mode area. Figure 1 presents the optical beam evolution process of the typical Gaussian solution of Eq. (2) in terms of peak intensity and beam size. The upper images in Fig. 1 give the snapshots of the optical beam in transverse plane at different positions over one spatial oscillation period. The lower one in Fig. 1 shows the relative peak intensity and beam width as a function of normalized distance $z/\xi$, $\xi = \pi \rho /\sqrt {2\Delta }$. In contrast to the case of heavily step-index multimode fibers [30,31], the pulsed beam in the typical GRIN MMFs periodically squeezes in space. Its compression in space is equivalent to a population of many higher-order spatial modes. Assuming that all modal components contributing to the formation of a multimode soliton are lying closely to each other, the periodic nonlinear interaction between the soliton modes can cause a grouping of dispersive wave lines and form a discrete dispersive wave radiation pattern.

Fig. 1. Solution of Eq. (2) assuming a Gaussian initial beam profile with w₀ = 20 µm. (a-e) Snapshots of the optical beam in transverse plane at different propagation distances over one period of spatial self-imaging effect. (f) On-axis normalized intensity |A₂(0,0,z)|/ |A₂(0,0,0)| and beam width versus propagation distance z. The parameters results from typical commercially available GRIN fibers: $A_0^2 = 20\,\textrm{GW}/\textrm{c}{\textrm{m}^2}, \,\Delta = 8.8 \times {10^{ - 3}},{n_0} = 1.47,{n_2} = 3.2 \times {10^{ - 8}}$ µ${\textrm{m}^\textrm{2}}/\textrm{W}, \, \rho = 26$ µ$\textrm{m},{\lambda_0} = 1.064$ µm.

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2.2 QPM relation for multiple resonant radiations generation from dark pulses

The above discussion shows that the multimode beams in the GRIN MMFs undergo a periodic beating (or self-imaging) along the longitudinal direction, and the relative refractive index changes periodically due to the Kerr effect in a manner that is similar to the equivalent long-period fiber Bragg gratings [32–34]. This dynamic grating permits the generation of a series of resonant radiation sidebands shed by temporal multimode solitons through the QPM conditions.

If we only consider the normal group-velocity dispersion ($n = 2$ and ${\beta _2} > 0$) condition and a constant nonlinear parameter ${\gamma _0}$, Eq. (3) can admit a dark soliton solution ${F_{ds}}(z, t)$ of the form

(6)$${F_{ds}}(z, t) = {F_0} \cdot (\cos \phi \tanh \Theta - i\sin \phi ),$$

(7)$$\Theta \equiv {F_0} \cdot (t - z \cdot \sqrt {{\gamma _0}{\beta _2}} {F_0}\sin \phi )/{T_0} \cdot \cos \phi ,$$

where ${F_0}$ is the amplitude of the soliton background, ${T_0}$ is the duration of the dark soliton and $\phi$ is the internal phase angle related to the grayness parameter, which represents an internal degree of freedom with its particular physics and satisfies the requirement of $\cos \phi \ge 0$ [35–37], i.e., $\phi \in [ - \pi /2,\pi /2]$. $\phi = 0$ represents the black soliton having zero amplitude in the middle and propagating at the group velocity, while $\phi \in ( - \pi /2,0)$ and $\phi \in (0,\pi /2)$ correspond to the gray solitons propagating faster and slower than the black solitons, respectively.

When higher-order dispersion effects ($n > 2$) are introduced, dark solitons can be perturbed and emit radiations in a similar way to the case of bright solitons. Recently, the dispersive radiation emission from dark soliton was explicitly observed. Here, we focus on the emission of multiple dispersive radiations by spatiotemporal oscillation of dark pulses in GRIN MMFs. These radiation frequencies can be determined by the following QPM condition (see Appendix A.2):

(8)$$\sum\limits_{n \ge 2} {\frac{{{\beta _n}}}{{n!}}} {\Omega ^n} - \sin \phi \sqrt {{\gamma _0}{\beta _2}} {F_0}\Omega + {\gamma _0}F_0^2/2 = 2\sqrt g \cdot m,\quad m = 0,\; \pm 1,\; \pm 2, \cdots$$

where all of the left-hand terms of Eq. (8) account for the effective wavenumber of the localized dark wave-packet and the right-hand term represents the virtual momentum carried by the periodicity of the beam effective area oscillation. The case of $m = 0$ corresponds to the traditional resonant radiation which can also be emitted in single-mode fibers [13,14]. Whereas, the multiple resonant radiations can also be yielded when $m \ne 0$, even without including the effect of the third (or higher) order dispersion. This QPM method via the effective area oscillation of the optical beam in GRIN MMFs can be understood by considering that the dark solitons and the resonant radiations are phase-mismatched, and that the beam effective area oscillation determines the degree of mismatch. As a result, the resonant radiation light can experience multimode interference effect when propagating along the GRIN MMFs. Due to the efficient nonlinear parameter modulation induced by the spatial beam oscillation, the regions of constructive interference can be made slightly longer than the destructive interference regions. The consequence is the ultimate constructive build-up of light intensity despite the existence of intensity oscillation. Therefore, the spatial self-imaging effect within GRIN MMFs can provide the QPM relation for the generation of multiple resonant radiations.

It is worth noting that Eq. (8) indicates the frequencies of the emitted resonant radiations which are largely dependent on the grayness parameter of dark solitons. Therefore, one should know the escape velocity of the dark solitons to predict the exact frequencies of the dispersive wave comb-like lines using this QPM technique. Figure 2 shows the roots of the QPM relation within the grayness phase of the dark solitons, which can provide some tailorability for the resonant radiation generation. This is the first demonstration of QPM to produce resonant radiations from the dark solitons in the normal dispersion regime, but it is interesting to note that the resonant radiations induced by the QPM have been observed in a variety of situations. Indeed, the “Kelly sideband” [38] seen in laser cavities and sidebands seen during supercontinuum generation (SCG) in integrated photonic waveguides with width modulations [16] are both examples of QPM resonant radiations. QPM has also been demonstrated for both geometric parametric instability [5] as well as bright soliton-radiations phase matching [6,12] in GRIN MMFs. The GRIN multimode waveguides provide a powerful new platform for QPM resonant radiation generation, allowing for efficient engineering of the nonlinear parameter within a range of modulation periods.

Fig. 2. The frequencies of QPM resonant radiations change as a function of the grayness of dark solitons. Parameters: ${\beta _2} = 7.508 \times {10^{ - 27}}{\textrm{s}^\textrm{2}}\textrm{/m},{\beta _3} = 5.912 \times {10^{ - 41}}{\textrm{s}^\textrm{3}}/\textrm{m},{\beta _4} ={-} 7.273 \times {10^{ - 54}}{\textrm{s}^\textrm{4}}/\textrm{m}$; all other parameters are similar to those used in Fig. 1. It can be easily found that the spectral locations of the multiple resonant radiations are weakly dependent on the grayness parameter as the orders of QPM are higher.

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3. Numerical results and discussions

3.1 Multimode dark soliton formation and its resonant radiation

A multimode soliton is a composite structure in which several transverse modes are actually superimposed. (Here, the fiber is considered to be weakly guiding so it allows for neglecting longitudinal component.) Although the multimode soliton might not have unique distribution for the spatial mode, through cross phase modulation between different modes within GRIN MMFs [3,8], the spectrum of each mode shifts slightly to help equalize the group-velocity across the modes (or rather mode fusion), $v_g^{ - 1} = \textrm{d}\beta /\textrm{d}\omega$, where $\beta$ represents the modal propagation constant. This is an essential condition for the formation of multimode localized states. In the anomalous dispersion regime, the formation of multimode bright solitons and its multimode dispersive wave radiations over an ultrabroadband spectral range have been reported recently [6,12]. As a parallel counterpart, the dark solitons, possessing a localized intensity dip compared to the continuous wave background, can also exist in GRIN MMFs in the regime of normal dispersion.

Even though a true continuous wave background cannot be achieved in practice, the dark pulses on a background pulse of finite width can still propagate adiabatically as solitons [20,37,39]. Therefore, to verify the excitation of multimode dark solitons within a GRIN MMF, we begin to study the evolution of an initial temporal condition given as an in-phase sum of two equal Gaussian intensity pulses:

(9)$$F(t) = \sqrt {{P_{in}}} \left[ {\exp \left( { - \frac{{{{(t - \Delta t)}^2}}}{{2\tau_0^2}}} \right) + \exp \left( { - \frac{{{{(t + \Delta t)}^2}}}{{2\tau_0^2}}} \right)} \right],$$

where ${P_{in}}$ is the input peak power, ${\tau _0}$ is the input pulse width and $2\Delta t$ represents the temporal spacing between both initial input pulses. Figure 3 summarizes the simulation results in this configuration for input pulses with a similar peak power of 100kW and a delay of 0.32 ps between them. As can be seen from Fig. 3(a), for small propagation distances ($\le$ 0.2 m), both temporal pulses broaden and acquire a linear frequency chirp because of the normal dispersion and Kerr nonlinearity. Once the temporal overlapping for both pulses occurs during the propagation, a new pulse can be produced with a sinusoidally modulated intensity due to the interference between the largely shifted spectral components of the leading and trailing edges of the colliding pulses [39]. As the propagation distance further increases, one accelerated dark pulse can be observed at the leading edge of the newly formed pulse, asymptotically approaching the theoretical shape of fundamental dark solitons. The corresponding spectral evolution behavior is shown in Fig. 3(b). After the formation of the dark pulse the spectrum asymmetrically broadened owing to the generation of the traditional dispersive wave via the perfect phase-matching relation, and a grouping of discrete narrowband dispersive wave lines are present on both sides of the spectrum as indicated by the red dashed arrow. These discrete comb-like lines may be generated by the spatial-temporal oscillation of multimode dark pulses.

Fig. 3. Dark soliton formation and its multiple resonant radiations emission in GRIN MMFs. (a) Simulated temporal evolution as a function of the fiber length. The red dashed line corresponds to the tangent of the dark soliton trajectory at the distance where the multimode resonant radiations are emitted. (b) The corresponding simulated spectral dynamics. DW: dispersive wave. The black and red arrows correspond, respectively, to the traditional DW by perfect phase matching and several discrete radiation lines via QPM.

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To predict the spectral positions of the radiation DW lines using the QPM relation, one should know the grayness parameter (or equivalently their escape velocity) and the amplitude of the dark pulses at the propagation distance where the resonant radiation is emitted. Using the numerical simulation of the temporal evolution versus fiber length [Fig. 3(a)], a very good approximation can be made based on the red dashed line, which corresponds to the tangent of the dark pulse moving path. The slope of the dashed line is the velocity ${v_{ds}}$. The amplitude of the dark pulse can be estimated directly in the temporal profile which is not given here. Then, by applying the approximated values of the velocity and the amplitude in Eq. (8), we can calculate the frequencies of the radiation DW lines. Figure 4 presents both the simulated spectra (blue curve) at the output of a 3-m long GRIN MMF and the theoretical predictions are represented by the red vertical dashed lines. The slight discrepancy between the simulation result and solution of Eq. (8) can be attributed to the uncertainty in determining the amplitude and grayness parameter of the dark soliton, as discussed above. These results highlight the generation of discrete radiation DW lines originating from the dark pulses which can be identified through the QPM relation (8) associated with the grayness parameter.

Fig. 4. The predicted spectral locations of the radiation DW lines in contrast to the simulated spectrum obtained at the output of 3-m long GRIN MMF.

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3.2 Formation of a train of multimode dark pulses and its continuum radiation

We now turn to study the formation of a train of dark pulses and their continuum radiation emission process within the GRIN MMFs. By increasing the temporal delay $\Delta t$ between the two initial Gaussian pulses, a train of dark pulses with different amplitudes, widths and phase angles can be generated [39]. The numerical results obtained for a temporal delay of 0.4 ps between two pulses are summarized in Fig. 5. All other parameters are identical to those in the situation of Fig. 3, except for the peak power that is increased to 150 kW. The temporal evolution plotted in Fig. 5(a) exhibits that five dark pulses with acceleration or deceleration on a variable background are formed within 1m propagation distance. It should be noted that the distortion at the leading edge of the pulse is possibly attributed to the formation of a shock wave front [15]. The corresponding spectral characterization is shown in Fig. 5(b), where the spectral components (plotted using the logarithmic scale) cover a remarkably wide range starting from −140 THz to 160 THz. It is not difficult to see in Fig. 5(b) that several groups of the resonant radiation spectral sidebands are present at both sides of pump spectrum, and each group consists of a series of non-uniformly spaced spectral comb-like lines, which is attributed to the periodic spatial compression and expansion of propagating multimode solitons in GRIN MMFs. Using the QPM condition related with the grayness and the amplitude of dark pulses, the emission of each one in a group of spectra comb-like lines can be identified to one of the formed dark pulse train. As can be seen in Fig. 6, there is an excellent agreement between our predicted positions of the resonant radiation spectral sidebands and the numerical results at the output of 3-m long GRIN MMF.

Fig. 5. Formation of a series of discrete DW radiation pattern induced by multiple dark solitons in the GRIN MMF. (a) Simulated temporal dynamics. The red dashed lines correspond to the velocity of each dark soliton at the distance where the multimode resonant radiations are emitted. (b) The corresponding simulated spectral evolution as a function of the fiber length. The black and red arrows correspond, respectively, to the traditional DW and several grouping of DW comb-like lines through the QPM condition for each dark soliton.

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Fig. 6. A comparison between predicted QPM spectra sidebands with grayness phase of dark pulses ($\phi \in [ - \pi /2,\pi /2]$) and spectrum for numerically obtained DWs recorded in the output of a 3 m long GRIN MMF. Initial pulse delay: 0.4 ps. Red stripes: the sideband range predicted by Eq. (8) for all grayness values.

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The numerical results and analytical study presented above demonstrate that increasing the number of dark solitons formed in the pulse train can allow us to generate multiple DW radiations, resulting in the broadening of spectral sideband. By further increasing the initial temporal delay $\Delta t$, the temporal modulation formed by the collision of two bright pulses reshapes nonlinearly and produces a train of isolated dark pulses. A quantitative characterization of this configuration is facilitated in Fig. 7. The blue curve shows the spectral distribution (plotted using decibel scale) corresponding to the output temporal profile of the inset in Fig. 7, which highlights the generation of multiple (>10) dark pulses in the pulse train. The predicted frequencies of every order QPM spectral sidebands (red shadow area in Fig. 7) and the numerical results (blue curve in Fig. 7) are in excellent agreement and exhibit the radiation continuum generation due to multiple DW radiation lines emitted by a number of dark pulses with different grayness. Our results indicate that GRIN MMFs present a unique approach to realize supercontinuum generation when pumping in the normal dispersion region.

Fig. 7. Radiation continuum generation from a train of multimode dark pulses. Inset shows the simulated temporal dynamics about the formation of a train of dark pulses. Initial pulse delay: 0.46 ps.

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The spectral locations of these resonant radiations predicted by theoretical analysis are essentially supported by the direct numerical integration of Eq. (3), as evidenced by Fig. 7, which shows the output spectrum after the propagation distance of 3 m. One may also ask whether our results obtained from a quasi-(1 + 1)-D NLSE (3) can apply to the full (3 + 1)-D GNLSE (2). Although we have not made such a comparison in this paper, we prefer to assume that our results can be consistent with the full (3 + 1)-D GNLSE problem under the condition of $\xi \ll T_0^2/{\beta _2}$ in which situation the spatial variations will affect the temporal dynamics significantly while the reverse does not occur. On the other hand, the intrinsic spatio-temporal coupling in Eq. (2) will happen [5–12] while the quasi-(1 + 1)-D NLSE (3) presented here does not produce any qualitative difference in predicting the observed features in its validity regime [6,28].

4. Conclusions

In conclusion, we have demonstrated theoretically and numerically the emission of multiple resonant radiations through the spatial-temporal oscillation of dark pulses. The formation of dark solitons is completed by the nonlinear collision of two identical and delayed Gaussian pulses propagating in a normally dispersive GRIN MMF. We reveal that each dark soliton of the pulse train can emit multiple resonant radiation sidebands whose frequency location in the spectral domain can be predicted using QPM arguments involving the grayness, the amplitude of the dark pulse, and the spatial oscillation period of the beam. A large number of dark solitons with unpredicted grayness by increasing the delay between both input pulses result in the formation of a radiation continuum around the position of each order through the QPM technique. We demonstrate clear evidence that this built-in and controllable periodicity of the effective mode area induced by spatial beam oscillation within the GRIN MMFs is responsible for the parametric excitation of resonant radiation at multiple resonant frequencies. Our results provide valuable insights into the physics and complexity of nonlinear wave propagation in normally dispersive multimode environments.

Appendix

A.1 Derivation of quasi-(1 + 1) D propagation model in GRIN MMFs

In this appendix, we present the derivation of a quasi-(1 + 1) D generalized nonlinear Schrodinger equation with a periodic nonlinear coefficient, which can be solved in an extremely fast and efficient way. Starting from Eq. (2):

(10)$$\begin{array}{l} \frac{{\partial A}}{{\partial z}} - i\frac{1}{{2{k_0}}}\nabla _ \bot ^2A - i\sum\limits_{n \ge 2} {\frac{{{\beta _n}}}{{n!}}{{(i{\partial _t})}^n}} A + i\frac{{{k_0}\Delta }}{{{\rho ^2}}}{r^2}A = i{k_0}\frac{{{n_2}}}{{{n_{co}}}} \times \\ \quad \quad \quad \quad \left[ {(1 - {f_R})|A{|^2}A + {f_R}\int {h(\tau )|A(t - \tau ){|^2}\textrm{d}\tau } } \right]. \end{array}$$

In weakly or moderate nonlinear regimes, the spatial evolution of the optical beam profile in GRIN MMFs can exhibit the nonlinear periodic properties, i.e. the spatial self-imaging effect, which is resulting from the equidistant distribution of propagation eigenvalues [12,28,29]. In this case, an approximated solution of Eq. (10) with ${\partial _t} = 0$ is a Gaussian beam, the width of which periodically varies along the propagation direction. The amplitude of this solution, which can be calculated by employing the method of variational techniques [29] or moments [11], can be described as follows:

(11)$$|A(x, y, z)|= {A_0}\frac{{{w_0}}}{{w(z)}}\exp \left( { - \frac{{{x^2} + {y^2}}}{{2{w^2}(z)}}} \right),$$

where ${w_0},\;{A_0}$ are the beam spot size and amplitude in $z = 0$, respectively; and the evolution of the optical beam width $w(z)$ satisfies the following equation:

(12)$$\frac{{{\textbf{d}^2}w}}{{\textbf{d}{z^2}}} + g \cdot w + \frac{{\mathbb C}}{{{w^3}}} = 0,\quad {\mathbb C} \equiv \left( {\frac{{{n_2}w_0^2A_0^2}}{{2{n_0}}} - \frac{1}{{k_0^2}}} \right).$$

This equation is a typical second-order differential equation whose solution in a closed form can be written as [27]:

(13)$$w(z) = {w_0}\sqrt {{{\cos }^2}(\sqrt g z) + {\mathbb C}{{\sin }^2}(\sqrt g z)} .$$

We now assume that the self-imaging pattern is only slightly affected by the nonlinearity and maintains stable during propagation. Equation (10) can be solved by using the method of separation of variables, namely, assuming a solution of the following form:

(14)$$A(x, y, z, t) = F(z, t) \cdot {A_s}(x, y, z),$$

where ${A_s}(x, y, z)$ is the approximated spatial optical field from Eq. (11), $F(z, t)$ represents the temporal envelope of the optical pulse. In the theoretical analysis and simulations reported in the paper we set the Raman term to zero to study the dynamics in the presence of only instantaneous Kerr effect. By substituting the ansatz (14) for the corresponding expression in Eq. (10), we obtain

(15)$${A_s}\frac{{\partial F}}{{\partial z}} - i{A_s}\sum\limits_{n \ge 2} {\frac{{{\beta _n}}}{{n!}}{{(i{\partial _t})}^n}F} = i{k_0}\frac{{{n_2}}}{{{n_{co}}}}|{A_s}{|^2}{A_s}|F{|^2}F.$$

Then we multiply all terms by $A_s^ \ast $ and subsequently integrate over the transverse plane. By doing so, we obtain

(16)$$\frac{{\partial F}}{{\partial z}} - i\sum\limits_{n \ge 2} {\frac{{{\beta _n}}}{{n!}}{{(i{\partial _t})}^n}F} = i\frac{{{\omega _0}{n_2}}}{{2\pi c \cdot {w^2}(z)}}|F{|^2}F.$$

For convenience, we can set $\gamma (z) = \frac{{{\omega _0}{n_2}}}{{2\pi c{w^2}(z)}}$, and its average value can be calculated as

(17)$${\gamma _0} = \frac{1}{{\sqrt g }}\int_0^{\sqrt g } {\gamma (z)\textrm{d}z} = \frac{{{\omega _0}{n_2}{\beta _0}\sqrt g }}{{2\pi c}}.$$

Therefore, we obtain the following quasi-(1 + 1) D NLSE with the periodically modulated nonlinear parameter:

(18)$$\frac{{\partial F}}{{\partial z}} - i\sum\limits_{n \ge 2} {\frac{{{\beta _n}}}{{n!}}{{(i{\partial _t})}^n}F} = i\gamma (z)|F{|^2}F.$$

A.2 Derivation of QPM relation

To study the multiple dispersive resonant radiations emitted from the dark solitons, it is effective to assume $F(z, t)$, consisting of the dark soliton envelope of ${F_{ds}}(z, t)$ as well as the small perturbation field $g(z, t)$ in the transverse plane [23,35,36], is written in the following form:

(19)$$F(z, t) = {F_{ds}}(z, t){e^{i\int_0^z {\gamma (z^{\prime})\textrm{d}z^{\prime}} F_0^2}} + g(z, t).$$

In this sense, we considered the weak time dependent perturbation term $g(z, t)$ in the form of $g(z, t) = a(z){e^{i\Omega t}} + {b^ \ast }(z){e^{ - i\Omega t}}$ in the reference frame defined by $\tau = t - \sin \phi \cdot \sqrt {{\gamma _0}{\beta _2}} {F_0}z$, where the dark solitons are at rest in the moving reference frame. It is also worth noting that when $\tau \to \infty$ in the extreme situation, the dark soliton tail approaches to a continuous wave, i.e., $\mathop {\lim }\limits_{\tau \to \infty } \partial _\tau ^n{F_{ds}} = 0,\;\forall n > 0$. We obtain after linearization

(20)$$\frac{\partial }{{\partial z}}\left( {\begin{array}{{c}} a\\ b \end{array}} \right) = i\left( {\begin{array}{{cc}} {W( - \Omega ) + \gamma (z)|{F_{ds}}{|^2}}&{\gamma (z)F_{ds}^2}\\ { - \gamma (z)F_{ds}^{ {\ast} 2}}&{ - W(\Omega ) - \gamma (z)|{F_{ds}}{|^2}} \end{array}} \right)\left( {\begin{array}{{c}} a\\ b \end{array}} \right),$$

where

(21)$$W(\Omega ) = \sum\limits_{n \ge 2} {\frac{{{\beta _n}}}{{n!}}{\Omega ^n}} - \sin \phi \sqrt {{\gamma _0}{\beta _2}} {F_0}\Omega .$$

In view of the periodic $\gamma (z)$ with a period of $\pi /\sqrt g$, we can expand the perturbation term in the form of Fourier series as follows:

(22)$$\gamma (z) = \sum\nolimits_m {{a_m}{e^{i2m\sqrt g z}}} ,$$

where ${a_m}$ represents the Fourier coefficients depending on the overall power conveyed in GRIN MMFs (this coefficient will become smaller as m increases), as described by the parameter C in Eq. (12). Importantly, in the case of sinusoidal modulation, the exponential term in Eq. (22) generates an infinite set of Fourier harmonics. The efficient coupling of energy into the resonant radiation (or dispersive wave) modes efficiently occurs when the determinant of the matrix in Eq. (20) equals to zero. Considering the momentum conservation, this leads to the following QPM condition [Eq. (8)]:

(23)$$\sum\limits_{n \ge 2} {\frac{{{\beta _n}}}{{n!}}} {\Omega ^n} - \sin \phi \sqrt {{\gamma _0}{\beta _2}} {F_0}\Omega + {\gamma _0}F_0^2/2 = 2\sqrt g \cdot m,\quad m = 0,\; \pm 1,\; \pm 2, \cdots$$

Funding

Natural Science Foundation of Hunan Province (2019JJ50530); National Natural Science Foundation of China (61805115, 61875132); China Postdoctoral Science Foundation (2019M653019); Shenzhen Science and Technology Innovation Commission (JCYJ20170302153731930, JCYJ20180305124927623).

Disclosures

The authors declare no conflicts of interest.

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Emission of multiple resonant radiations by spatiotemporal oscillation of multimode dark pulses

Abstract

1. Introduction

2. Theoretical analysis

2.1 Propagation model in GRIN MMFs

2.2 QPM relation for multiple resonant radiations generation from dark pulses

3. Numerical results and discussions

3.1 Multimode dark soliton formation and its resonant radiation

3.2 Formation of a train of multimode dark pulses and its continuum radiation

4. Conclusions

Appendix

A.1 Derivation of quasi-(1 + 1) D propagation model in GRIN MMFs

A.2 Derivation of QPM relation

Funding

Disclosures

References

Cited By

Figures (7)

Equations (23)

Optics Express