Intermodal dispersive wave generation and soliton linear-wave interaction in optical fibers

Yi Qiu; Mohammad Raihan Subhan; M. D. Shamim Ahsan; Shuxin Du; Yiqing Xu; Yiqing Xu

doi:10.1364/OPTCON.474725

1. Introduction

The emission of dispersive waves (DWs) by optical pulses is a well-known process in nonlinear fiber optics [1,2]. It is one of the key processes toward many other nonlinear phenomena such as optical rogue waves, supercontinuum generation and broadband frequency comb generation [3–6]. Particularly by engineering microstructured optical fibers, the generation of dispersive waves offers a flexible way of generating broadband sources in different spectral regions [7,8]. Traditionally, the emission of dispersive waves has been described in the context of soliton propagation in the presence of higher order dispersive effects [2,9–13]. The precise frequency shifts of DWs can be determined from the dispersion profiles of optical fibers which effectively govern the phase matching condition [4,12,14]. On the other hand, a dispersive wave emission has also been interpreted as a result of a cascaded four-wave mixing process [15,16]. In the picture of cascaded four-wave mixing, it elucidates the underlying mechanism of dispersive wave emission by mimicking the effect of a higher order nonlinear susceptibility. This frequency domain interpretation has been used to further describe the nonlinear soliton linear-wave (SLW) interaction [17,18]. Alongside the cascaded four-wave mixing, previous studies have shown that the origin of the driving process of the dispersive wave and soliton linear-wave interaction can be as simple as a cross-phase modulation (XPM) process, while the spectral profile is still governed by the phase matching condition [10,19,20].

Notably, in a soliton linear-wave interaction, the linear/probe waves used should have an arbitrary phase with respect to the soliton [5,6,18,21]. This implies that the linear-wave only experiences the intensity envelope of the soliton/driving field. In fact, the phase-insensitive process has not only been observed in the intermodal dispersive wave generation [22,23], but also demonstrated in polarization modal dispersive wave generation [24,25]. In this work, we will further investigate the relationship between the driving field and phase matching condition by separating the driving field and linear-wave into two spatial modes. We will show that the role of the driving field is to provide optical XPM as suggested by the earlier works, while the phase matching condition of the idler-wave is governed by the dispersion profile of the mode in which the linear-wave propagates. By investigating the linear-wave individually, we further show dispersive wave generation is a special case of soliton linear-wave interaction in which the linear-wave is degenerate from the driving field. The role of the dispersion that the linear-wave experiences is to detrimentally ’shape’ the spectrum of the purely cross-phase modulated linear-wave into an idler-wave.

We will organize this work as the following. We will first revisit the dispersive wave emission and soliton linear-wave interaction dynamic using a frequency comb driving field through the intermodal cross-phase modulation process. By comparing the results obtained between the intermodal and intramodal propagation, we would like to point out that dispersive wave generation is akin to soliton linear-wave interaction. Later, we will present a simple theoretical model which enables us to calculate semi-analytically the spectral profiles for the dispersive wave generation as well as for soliton linear-wave interaction. Then, we will more specifically investigate the spectral features of which the driving field has a solitary solution, i.e., soliton propagation. We will further verify our model using numerical simulations.

2. Theory

Before proceeding our study on dispersive waves (DW) generation as well as soliton linear-wave interaction (SLW), we introduce the nonlinear Schrödinger equation (NLSE) that is used to describe an optical field along an optical fiber [26]

(1)$$\frac{\partial A(z,T)}{\partial z} = i\gamma|A|^2A + i\sum_{k\geq2}i^k\frac{\beta_k}{k!}\frac{\partial^k A}{\partial T^k}.$$

where $A(z,T)$ which is the slowly varying amplitude of the pulse envelope propagating at the group velocity frame $T=t-\beta _1z$, the coefficient $\beta _k$ are the $k$-th order dispersion terms of the optical fiber, and $\gamma$ is the nonlinear parameter of the fibre. We at this stage omit the fiber loss and Raman effect for the simplicity of this study. Meanwhile, the well-known phase matching conditions for DW and SLW states that a dispersive wave is generated at the frequency of $\omega _{\rm DW}$, or an idler-wave (IW) at the frequency of $\omega _{\rm IW}$ when the dispersive wave propagates in phase with the soliton pump at frequency $\omega _S$, or the idler-wave propagates in phase with the linear-wave at frequency $\omega _{\rm LW}$. These two phase matching conditions are commonly expressed as [9,12,18]

(2)$$\hat{D}(\omega_{\rm DW} - \omega_S) = \hat{D}(\omega_S),$$

(3)$$\hat{D}(\omega_{\rm IW} - \omega_S) = \hat{D}(\omega_{\rm LW} - \omega_S).$$

Here $\hat {D}(\omega - \omega _S)$ is the wavenumbers of a spectral component at $\omega$ in a reference frame moving with the soliton at frequency $\omega _S$, which is related to the fibre propagation constant $\beta (\omega )$ as

(4)$$\hat{D}(\omega-\omega_S) = \beta(\omega) - \beta_0(\omega_S) - \beta_1(\omega-\omega_S).$$

Thus, $\hat {D}$ represents terms above the second order in the Taylor series expended at frequency $\omega _S$ and can be further reduced to

(5)$$\hat{D}(\omega - \omega_S) = \sum_{k\geq2}\frac{\beta_k}{k!}(\omega-\omega_S)^k.$$

It corresponds to the dispersion operator on the right-hand side of Eq. (1). The evolution of the optical field along the optical fiber can be obtained by numerically solving the NLSE Eq. (1) using the split-step Fourier method [26].

2.1 Numerical simulations

To facilitate the later study of the intermodal DW and SLW, we perform our numerical simulations using a frequency comb soliton as the pump field which has a periodic intensity profile in the time domain. The reasons for using frequency comb has three folds: (1) reduce the potential walk-off effect between spectral components, especially for the XPM process; (2) improve the conversion efficiency in a SLW process when using a CW linear-wave; (3) enable us to differentiate the spectral comb lines between pump and linear-wave spectral components. We first consider a simulation of a soliton comb pump without a weak linear-wave propagating in a 20 m long optical fibre. In our simulation, the dispersion parameters of the optical fiber are accounted for up to the third order with $\beta _2$=$-4.6$ ps$^2$/km and $\beta _3$=$0.18$ ps$^3$/km at the soliton pump frequency, and the nonlinear parameter is $\gamma$=2.5 W$^{-1}$/km. With these parameters, a common fundamental soliton with a sech temporal profile $A_S(T)=\sqrt {P_0}$ sech $(T/T_0)$ can be approximately calculated when only the second order dispersion parameters is considered, and the corresponding pulse width is $T_0 = \sqrt {|\beta _2|/\gamma P_0}$ where $P_0$ is the peak power of the soliton. As we are considering propagating a frequency comb soliton, the temporal spacing between the periodic soliton train as shown in Fig. 1(a) is determined by the spectral spacing of the comb lines as shown in Fig. 1(b).

Fig. 1. (a) Input temporal intensity profile of the sech soliton pulse train (blue) for DW and SLW simulations. Red curve is the temporal window for a single sech soliton monitoring. (b) Input spectral intensity profile of the sech soliton comb (blue) and the corresponding spectrum (red envelope) of a single sech soliton.

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We will focus on the spectral evolution of one of the soliton pulses indicated by the temporal window in the red curve in Fig. 1(a). The peak power of the soliton pulse train is set to $P_0$=160 W and the corresponding pulse width is $T_0$=100 fs. We plot in Fig. 2(b) the output spectral of the soliton, and in Fig. 2(a) the dispersion profile $\hat {D}(\omega -\omega _S)$ of the fiber calculated using Eq. (5). Clearly, the output spectral profile of the single soliton matches well with that of soliton comb, which verify that the temporal window is sufficiently broad for monitoring the spectral evolution over the given fiber distance. As expected, a DW spectral component with a frequency shift of 13 THz (indicated by the blue dashed line) is generated at the phase-matching point as described by Eq. (2). This frequency of the dispersive wave is readily to calculate using $\omega _{\rm DW}=\omega _S-3\beta _2(\omega _S)/\beta _3$ [14,15]. Figure 2(c) shows the typical spectral evolution of the soliton as well as the growth of dispersive wave spectral component along the propagation distance, and the dispersive wave is commonly visualized as the energy shedding from the soliton. We will further address this phenomenon in the later section.

Fig. 2. (a) Dispersion profile $\hat {D}(\omega -\omega _S)$ expanded around soliton frequency $\omega _S$. (b) Simulated output spectrum of one sech soliton pulse (red curve) and pulse train (blue comb lines). (c) Simulated spectra evolution along the optical fiber distance.

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Similarly, for the SLW observation simulated again using Eq. (1), a linear-wave continuous-wave (CW) with a power of 100 mW and frequency at $\omega _{\rm LW}-\omega _S$=6.1 THz is combined with the soliton train and propagates along the fiber. The simulated output spectrum is as shown in Fig. 3(b), and an induced idler-wave component is generated at 11 THz at which $\hat {D}(\omega _{\rm IW}-\omega _S)$ of the idler-wave is identical as of the linear-wave $\hat {D}(\omega _{\rm LW}-\omega _S)$ as indicated in Fig. 3(a). Note that the dispersive wave should have been emitted during the SLW interaction [12,18], but the amplitude of the generated dispersive wave is overwhelmed by the linear-wave and idler-wave. In order to have better visibility and thus further distinguish the roles between the soliton pump and fiber dispersion, we will separately propagate the soliton comb pump and linear-wave in two different spatial modes in the next section.

Fig. 3. (a) Dispersion profile $\hat {D}(\omega -\omega _S)$ with indications of the phase matching linear-wave (LW) at $\hat {D}(\omega _{\rm LW}-\omega _S)$ and idler-wave (IW) at $\hat {D}(\omega _{\rm IW}-\omega _S)$. (b) Simulated output spectrum of soliton linear-wave (SLW) interaction.

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3. Intermodal DW and SLW

Unlike co-propagating the sech soliton comb and linear-wave, we now consider the two fields are propagating incoherently in two spatial modes. Thus, the linear-wave can only experience the intensity envelope of the soliton comb, and the evolution of the two fields can be described by the coupled-NLSEs as [26]

(6)$$\frac{\partial A_1}{\partial z} = \left[i\gamma(1-f_R)(|A_1|^2+2|A_2|^2) + i\gamma f_R R_1 + i\sum_{k\geq2}i^k\frac{\beta_{{\rm M1},k}}{k!}\frac{\partial^k}{\partial T^k} \right]A_1,$$

(7)$$\frac{\partial A_2}{\partial z} = \left[i\gamma(1-f_R)(|A_2|^2+2|A_1|^2) + i\gamma f_R R_2 + i\sum_{k\geq2}i^k\frac{\beta_{{\rm M2},k}}{k!}\frac{\partial^k}{\partial T^k} \right]A_2,$$

where $A_1$ and $A_2$ are the pulse envelopes of the field of Mode 1 and Mode 2, and the corresponding $\beta _{{\rm M1},k}$ and $\beta _{{\rm M2},k}$ are the $k$-th order dispersion parameter experienced in Mode 1 and Mode 2, respectively. The field in Mode 1 and Mode 2 is propagating within the same group velocity frame, and the temporal walk-off is only contributed from the higher-order dispersion $k\geq 2$. The terms $R_1$ and $R_2$ are the terms of Raman effect and expressed as

(8)$$R_j = \int_{0}^{\infty}h_R(\tau)|A_j(t-\tau)|^2 d\tau + \int_{0}^{\infty}h_R(\tau)|A_{3-j}(t-\tau)|^2 d\tau$$

where $j=$ 1 or 2 is the subscript corresponding to $A_1$ and $A_2$, and $h_R$ is the Raman response function [26]. $f_R$ is the fractional Raman contribution, and typical $f_R$=0.18 for silica fibers. The first term of $R_j$ is responsible for the intramodal-induced Raman scattering, and the second term is responsible for the intermodal-induced Raman scattering. Again, we omit the fiber loss as the loss of the optical field is negligible over a short distance of propagation.

3.1 Non-degenerate intermodal SLW

We initially omit the Raman effect by setting the fractional Raman contribution $f_R$=0 for our simulation. We first simulate the soliton comb pump at the center frequency propagates in Mode 1 and only consider the second order dispersion, such that the soliton comb will propagate without any distortion. A CW linear-wave with a power of 100 mW at frequency shift of 6.1 THz propagates in Mode 2 with the dispersion parameters taken up the third order. Thus, $\hat {D}_{\rm M1}$ and $\hat {D}_{\rm M2}$ have an identical second order dispersion parameters as the previous section, and $\hat {D}_{\rm M2}$ includes the third order parameter. Since the linear-wave only experiences the cross-phase modulation from the soliton comb pump, the dispersion parameter of Mode 2 can be rewritten into the group velocity frame for a linear-wave with an arbitrary frequency shift from the soliton center frequency as

(9)$$\hat{D'}_{\rm M2}(\omega' - \omega_{\rm LW}) = \hat{D}_{\rm M2}(\omega - \omega_S) + \beta_{\rm M1,\ 1}[\omega - \omega_S].$$

where $\beta _{\rm M1,\ 1}=\sum _{k\geq 2}\beta _{{\rm M1}, k}(\omega _{\rm LW}-\omega _S)^{k-1}/(k-1)!$ is relative to the group velocity of in Mode 1 at frequency of $\omega _{\rm LW}-\omega _S$, and the second term on the right hand side is to calculate the extra phase shift experiences by the field in Mode 2 after transforming the frequency frame centered at the linear-wave. Under this transformation, we can now set the linear-wave as the center frequency component in Mode 2, and visualize the soliton comb pump in Mode 1 as a travelling XPM across the linear-wave. We first verify the transformation of Mode 2 by numerically solving the coupled NLSEs Eq. (6) and (7) at the new spectral frame of $\omega - \omega _{\rm LW}$. The dispersion profile experienced by the linear-wave after the transformation is plotted in Fig. 4(a). Figure 4(b) and (c) show the output spectrum and the corresponding spectral evolution of the linear-wave along the fiber distance, respectively.

Fig. 4. (a) Mode 2 dispersion profile $\hat {D}'(\omega '-\omega _{\rm LW})$ (blue curve) after transforming from $\hat {D}(\omega -\omega _S)$ (red dash curve). (b) Simulated output spectrum of intermodal linear-wave in Mode 2 by cross-phase modulating soliton comb pump in Mode 1. (c) Simulated spectra evolution of linear-wave along the optical fiber distance.

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As compared with the output spectrum around the LW and IW region in the SLW interaction in Fig. 3(a), the output spectrum in Fig. 4(b) displays almost identical spectral features. The frequency shift of the idler-wave with respect to the linear-wave is 4.9 THz after transforming the frequency coordinate of Mode 2, which is exactly equal to that in the SLW interaction. Note that the soliton comb pump propagates without any significant distortion (The spectrum is not shown). We would like to stress that the role of the soliton comb pump in Mode 1 is to only provide a travelling cross-phase modulation on the linear-wave in Mode 2, resulting the spectral broadening of the linear-wave. Meanwhile, the frequency shift of the generated idler-wave is invariant and determined by the resonant phase-matching condition. This suggests that the spectral broadening process of the linear-wave by XPM should be prior before the dispersion of the fiber can influence a non-monochromatic linear-wave.

3.2 Degenerate intermodal SLW

We now consider the linear-wave is degenerate ($\omega _{\rm LW}=\omega _S$) from the soliton comb pump, which has identical pulse profile and propagate in Mode 2. Again, the dispersion parameters are identical to those used in the previous sections, which accounts for only the second order in Mode 1, but up to the third order in Mode 2. In this case, $\hat {D'}_{\rm M2}(\omega ' - \omega _{\rm LW})$ is identical to $\hat {D}(\omega - \omega _S)$ used in the intramodal section. The peak power of the linear-wave is set to 100 mW as of the CW case, as at this point we don’t have the exact amount of degeneracy of the linear-wave from the soliton comb pump. For comparison, we also simulate using a CW linear-wave with a power of 100 mW.

We plot in Fig. 5(a) the output spectra of the pulsed linear-wave (blue curve) and the CW linear-wave (yellow curve), and superimpose on top the output spectrum of the single soliton generated DW (intramodal) from Fig. 2(b) (red dotted curve) for comparison. The spectra from the two SLW processes exhibit almost identical features as from the single soliton, having the generated idler-waves coincided with the dispersive wave generated in the intramodal soliton propagation. The spectral evolution of the linear-wave along the optical fiber plotted in Fig. 5(b) also shows almost the identical evolution as that in the intramodal dispersive wave generation in Fig. 2(c). These similar behaviours evidence that the soliton comb pump should have been degenerate during the intramodal dispersive wave generation process when a part of the soliton field is acting as a linear-wave and undergoing the cross-phase modulation by the soliton pump. Effectively, the degenerate soliton field is converted into the idler-wave and becomes the well-known dispersive wave. We further plot in Fig. 5(c) the temporal evolution of the pulsed linear-wave along the fiber distance. When the new spectral components being generated through XPM, they should start temporally disperse during the propagation as the linear-wave has insufficient peak power to sustain the soliton profile. The boundary of the spreading linear-wave indicated by the red dashed line corresponds to the group velocity of the idler-wave calculated from the first order dispersion parameter [27].

Fig. 5. (a) Simulated output spectra in Mode 2 of using a pulsed linear-wave (blue curve) and CW linear-wave (red curve), and the output spectrum of a single soliton from Fig. 2(b) (yellow dotted curve) for comparison. (b) Simulated spectral evolution of the linear-wave in Mode 2. (c) Simulated temporal evolution of linear-wave in Mode 2 where the red boundary curve indicates the group velocity of the generated idler-wave.

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3.3 Influence of Raman effects

To further investigate the influence of the Raman effects, we now include the Raman scattering by setting the fractional Raman contribution to $f_R$=0.18 in both Eq. (6) and Eq. (7) and perform numerical simulations. Particularly, We compare the output spectra of degenerate intermodal soliton linear-wave interaction with and without the inclusion of the Raman scattering in Mode 2, to the case of the dispersive wave generated from a single soliton in Mode 1.

We plot in Fig. 6 the output spectra in Mode 2 using a pulsed linear-wave (blue curve) and CW linear-wave (red curve), as well as output spectra of dispersive waves (yellow dotted curve) generated using only a single soliton pump in Mode 1 for two different second order dispersion parameters $\beta _2$. As the phase matching condition only depends on the Kerr effect, the frequency shifts of the idler-wave/dispersive wave are identical between Fig. 6(a) and (b) with $\beta _2$=$-4.6$ ps$^2$/km [or Fig. 6(c) and (d) with $\beta _2$=$-5.3$ ps$^2$/km]. When comparing Fig. 6(a) and (c) (without Raman scattering) to Fig. 6(b) and (d) (with Raman scattering), we can observe that both the idler-wave and dispersive wave simulated with Raman scattering have a slightly higher conversion than that without Raman scattering, indicating by the black dash lines. This difference is simply due to the extra Raman gain during the conversion process. Yet, as we expected, both the idler-wave and dispersive wave still resemble closely to each other in the cases with and without Raman scattering.

Fig. 6. Simulated output spectra in Mode 2 using a pulsed linear-wave (blue curve) and CW linear-wave (red curve), and the output spectrum of dispersive wave generated from a single soliton in Mode 1 (yellow dotted curve). (a) and (b) with $\beta _2$=$-4.6$ ps$^2$/km, and (c) and (d) with $\beta _2$=$-5.3$ ps$^2$/km. (a) and (c) without Raman scattering, and (b) and (d) with Raman scattering.

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4. Approximate analytical spectral profile for DW and SLW

Finally, having shown that the dispersive wave generation and soliton linear-wave interaction can be visualized as the cross-phase modulation by the soliton comb pump, we would like to demonstrate an approximate analytical expression for the spectral profile for the undergoing XPM linear-wave. For a relatively short nonlinear distance, we can approximate the output spectrum of the linear-wave as

(10)$$\widetilde{A}_{\rm SLW}(z,\omega) = \frac{\mathfrak{F}\left\{A(0,T)\exp[i2\gamma P(z,T) L_{\rm eff}]\right\}}{1+i\hat{D}(\omega-\omega_{\rm LW})z},$$

where $\mathfrak {F}$ is the Fourier transformation operation, $A(0,T)$ is the initial input field of the linear-wave in Mode 2, $P(z,T)$ is the intensity profile of the pump pulse in Mode 1, and $L_{\rm eff}$ is the effective XPM distance experienced by the linear-wave. $L_{\rm eff}$ is calculated based on the walk-off distance between the newly generated spectral components through XPM and the pump pulse (See Appendix A. for detailed derivations).

The approximation analytical expression of Eq. (10) requires the propagation distance of the linear-wave $z>L_{\rm eff}$ larger than the effective XPM distance, such that the pump pulse can initially provide a constant amount of XPM to the linear-wave within the effective length, and the newly generated spectral components experience only dispersive evolution in the remaining propagation. For simplicity, we again consider the pump pulse is a soliton comb $P(T)=\sqrt {P_0}$ sech $(T/T_0)$ as used in the previous section. It is readily to plot in Fig. 7(a) the spectral output (red curve) of a SLW interaction using Eq. (10) with the same conditions used as in Fig. 4. We also superimpose the spectrum calculated with only pure XPM (blue dotted curve) by neglecting the dispersion term in the denominator as $\hat {D}(\omega -\omega _{\rm LW})$=0 in Eq. (10), and the simulated spectrum (yellow curve) for comparison. As can be seen, the pure XPM spectrum sets the upper limit of the spectral profile. With the inclusion of the dispersion $\hat {D}(\omega -\omega _{\rm LW})$, the spectral profile is resembled almost the same spectral feature of the simulated spectrum. Similarly, the analytically calculated DW spectrum is plotted in Fig. 7(b) together with the XPM spectrum and simulated spectrum with the parameters used in Fig. 2. As expected, the analytical DW spectrum follows closely the simulated spectrum, and we attribute the discrepancy around the central spectral amplitude to the fact that the degeneracy of the linear-wave is undefined.

Fig. 7. Output spectra of a linear-wave calculated from pure XPM (blue dotted curves), analytical expression using Eq. (10) (red curves), and simulation (yellow curves), for (a) SLW with a CW input and (b) DW with a weak pulse input, respectively.

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5. Discussion

By individually studying the behaviours of the pump and linear-wave in two different modes, it has provided us with some handiness to insight into the nonlinear interaction between the pump and linear-wave. Particularly, the intermodal dispersion can be simply transformed with respect to the other mode in the spectral domain which helps to investigate the phase matching condition for the DW generation and SLW interaction. Especially, the degenerate SLW interaction reveals that DW and SLW are essentially the same physical phenomenon, and the DW is the idler-wave converting from a degenerate pump through SLW interaction driven by the XPM process. We note that the precise degree of degeneracy between the pump and linear-wave has not yet been fully investigated. Nevertheless, the spectral feature in a DW generation that is far from the center frequency should not be overly influenced by the degeneracy.

Furthermore, we identify that the role of fiber dispersion is detrimental to the XPM broadening process. From the comparison between the XPM spectrum and analytically calculated spectrum, instead of the DW or IW being generated as new spectral components, we can visualize that the DW and IW become spectrally outstanding as a result of the concave spectral features from pure XPM spectral profiles. The concave spectral features are due to the continuously evolving spectral components which have phase mismatches with respect to the pump and thus cannot be constantly phase-modulated. Note that in order to isolate the Kerr effect from other nonlinear processes, we have initially omitted the Raman contribution in our study. As the previous studies have shown that the underlying mechanism of DW and SLW is cascaded four-wave mixing, the inclusion of the Raman effect will only contribute extra spectral-dependent gain in the simulation [15,18]. This is further confirmed in our later simulation in Section 3.3 in which the Raman effect is included.

6. Conclusions

In conclusion, we have presented a theoretical study on the intermodal dispersive wave and soliton linear-wave interaction in an optical fiber. The generated idler-wave in a pump degenerate soliton linear-wave interaction evidences that the dispersive wave emission is a special case of soliton linear-wave interaction. We have also introduced an approximate analytical expression which can directly calculate the output spectra of the dispersive wave and soliton linear-wave interaction. We envisage our work can find potential applications in designing and optimizing broadband spectral light source generation.

A. Appendix: Derivation of approximate analytical spectral profile for DW and SLW

For a sufficiently low power linear-wave such $|A_2|^2\ll |A_1|^2$, the propagation equation without the Raman scattering (by setting $f_R$=0) for the linear-wave in Mode 2 can be reduced from Eq. (7) into

(11)$$\frac{\partial A_2}{\partial z} = i\left(2\gamma|A_1|^2 + \sum_{k\geq2}i^k\frac{\beta_{{\rm M2},k}}{k!}\frac{\partial^k}{\partial T^k}\right)A_2,$$

where the first term on the left hand side is responsible for the XPM, and the second dispersion term is effectively the wavenumber $\hat {D}(\omega -\omega _{\rm LW})$ experienced by the newly generated idler-waves. In the absence of dispersion, the linear-wave is directly cross-phase modulated by the soliton comb pump, and we can simply write an analytical expression for the linear-wave as

(12)$$A_2(z,T) = A_2(0)\exp[i2\gamma P(z,T) z],$$

where $P(z,T)$=$|A_1(z,T)|^2$ is the co-propagated pump temporal intensity profile in Mode 1, and $A_2(0)$ is the initial input field of the linear-wave at $z$=0 in Mode 2. This spectral envelope of $A_2(z,T)$ sets the ultimate spectral amplitude limit after the XPM on the linear-wave. However, with the inclusion of the dispersion in Eq. (11), the cross-phase modulated spectral components from linear-wave cannot be continuously sustained within the pump pulse envelope and spreading out. The effective XPM distance can be estimated from the walk-off distance between pump pulse and the newly generated spectral components as

(13)$$L_{\rm eff} = T_0/\beta_{\rm M2,1}(\Omega-\omega_{\rm LW}),$$

where $\Omega$ is defined as the spectral region in which the new spectral components can experience a significant amount of nonlinear phase shift as

(14)$$\hat{D}(\Omega-\omega_{\rm LW})<\gamma P_0.$$

By substituting Eq. (13) in Eq. (12) and taking the Fourier transformation, we obtain the spectral profile by including the dispersion on each of the spectral components as

(15)$$\widetilde{A}_{\rm SLW}(z,\omega) = \exp[{-}i\hat{D}(\omega-\omega_{\rm LW})z]\mathfrak{F}[A_2(L_{\rm eff},T)].$$

By approximating the dispersion term $\exp [-i\hat {D}(\omega -\omega _{\rm LW})z]$ with a Taylor series as

(16)$$\exp[i\hat{D}(\omega-\omega_{\rm LW})z] \approx 1 + i\hat{D}(\omega-\omega_{\rm LW})z,$$

we obtain an approximate spectral profile for the generated idler-wave spectrum as

(17)$$\widetilde{A}_{\rm SLW}(z,\omega) = \frac{\mathfrak{F}\left\{A(0,T)\exp[i2\gamma P(z,T) L_{\rm eff}]\right\}}{1+i\hat{D}(\omega-\omega_{\rm LW})z}.$$

Funding

National Outstanding Youth Science Fund Project of National Natural Science Foundation of China (62003137).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.

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Intermodal dispersive wave generation and soliton linear-wave interaction in optical fibers

Abstract

1. Introduction

2. Theory

2.1 Numerical simulations

3. Intermodal DW and SLW

3.1 Non-degenerate intermodal SLW

3.2 Degenerate intermodal SLW

3.3 Influence of Raman effects

4. Approximate analytical spectral profile for DW and SLW

5. Discussion

6. Conclusions

A. Appendix: Derivation of approximate analytical spectral profile for DW and SLW

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (17)

Optics Continuum