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Abstract
We report the dispersive wave (DW) emission from the Gaussian pulse with temporal sinusoidal phase (TSP) modulation. The TSP-induced chirp can enhance or cancel the chirp generated by self-phase modulation by properly selecting the modulation parameters of TSP, which can influence the nonlinear propagation of the TSP-modulated pulse. It is shown that the TSP can effectively control the resonant frequency and energy conversion efficiency of the DW emission. We give a modified phase-matching condition to predict the resonant frequencies, which agree with the simulation results obtained by numerically solving the nonlinear Schrödinger equation. The enhanced conversion efficiency of the DWs can be increased up to 28% with only TSP modulation. Our results can extend the application of temporal phase modulation technology for wavelength conversion, and broadband supercontinuum generation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The dispersive wave (DW), first identified several decades ago [1,2] as a nonlinear phenomenon that emerges when the optical pulse propagates in nonlinear optical fibers. The DW, also known as resonant radiation and a fiber counterpart of Cherenkov radiation, has the unique ability to transfer energy from incident frequency to resonant frequency from ultraviolet to mid-infrared regime [3,4]. The ability of DW to control energy conversion raises widespread applications ranging from optical frequency combs [5] to supercontinuum generation [6]. Including higher-order solitons [7,8], several novel light sources have been employed to stimulate DWs such as dark soliton [9], Peregrine soliton [10], as well as self-accelerating wave packets [11] and the wave breaking process [12].
The resonant frequency of DWs is strongly governed by the phase matching (PM) requirement. That is, the pump wave only transfers its energy to the resonant frequency when the linear wavenumber equals the nonlinear wavenumber [2], whereas the linear and nonlinear wavenumbers are determined by dispersion and nonlinearity, respectively. Various sorts of nonlinear media, such as dispersion varying [13] or oscillating [14] fiber and photonic crystal fiber [7], multimode [15] fiber as well as optical waveguide [16] are also used to manipulate DW generation for the parametric dependence of PM. Not only the resonant frequency of the DW, but also its energy conversion efficiency, is tunable in the emission process [17], and how to improve the conversion efficiency is one of the study topics.
The potential of self-phase modulation (SPM), a crucial nonlinear effect in propagation, to influence pulse frequency shift, compression [18,19], and supercontinuum generation [20], as well as the interplay of dispersion [21], has been researched. At the same time, phase modulation has become a critical theme of pulse dynamics and DW manipulation because the chirp of temporal, spatial, and spectral phase may operate along with the chirp of dispersion and SPM [22]. The temporal phase modulation maintains the temporal structure of pulses but changes their spectra, while the spectral phase modulation causes the opposite result. Among them, sinusoidal phase is employed in numerous studies for its unique properties.
Spatial sinusoidal phase modulation has long been used to control pulse dynamics [23,24]. As its analog in time domain, temporal sinusoidal phase (TSP) has also been widely applied, including pulse train generated by continuous wave [25], pulse spectrum compression [26] and broadening cancellation [27], as well as multi-wave mixing [28] and self-mixing interferometer [29] system.
In this paper, the investigation of the DW emission manipulated by temporal sinusoidal phase (TSP) modulation is reported. Because the tendency of TSP-induced chirp around the central region of the pulse is quite similar to the SPM-induced chirp [26], the TSP can be utilized to enhance or cancel the SPM effects, allowing the resonant frequency and energy of DW to be controlled. Despite the maximum resonant frequency being limited by input power, the energy of DW can still be substantially enhanced. The resonant frequency predicted by the modified PM condition is in a great agreement with the numerical results obtained from the nonlinear Schrödinger equation (NLSE).
2. Numerical model
Optical pulse propagation in single-mode fiber can be described by the generalized NLSE [22]
The incident Gaussian pulse with TSP is
where TSP is $\varphi (T )= A\cos ({{\omega_m}T + {\phi_0}} )$ with modulation depth A, modulation frequency ${\omega _m}$ and initial phase ${\phi _0}$. The incident pulse with TSP maintains its shape but its spectrum is reshaped to the form
Fig. 1. (a) and (b) the temporal chirp induced by the SPM and the TSP as well as (b) their sum. Spectrum evolution with only SPM for modulation interval $B = 0.3$ for (c) $\delta \omega _m^ + $, (d) $\delta \omega _m^ - $, (e) and (f) $\delta {\omega _m} = 0$. The nonlinearity is $N = 1$ in (c), (d), (f) and $N = \sqrt 2 $ in (e).
The spectrum broadening enhancement and cancellation shown in Figs. 1(c) and 1(e) demonstrate that TSP can effectively enhance or cancel SPM. It is worth noting that the evolution in Fig. 1(c) is very comparable to that for $N = \sqrt 2 $ (double peak power) but without TSP in Fig. 1(e), implying that TSP can boost SPM proportionally. For various pulse durations, similar results can be obtained for a different pulse duration ${T_2}$ by scaling the modulation frequency as ${\omega _{m2}} = {\omega _{m1}}{{{T_1}} / {{T_2}}}$ where ${\omega _{m1}}$ is the modulation frequency for the original pulse with duration ${T_1}$. However, in the presence of higher-order dispersion, it is impossible to replace Kerr nonlinearity with TSP for reconstructing the SPM.
3. Boosting DW emission
Next the dispersion comes into our consideration. For ${\beta _2} ={-} 0.41p{s^2}k{m^{ - 1}}$, ${\beta _3} = 0.0687p{s^3}k{m^{ - 1}}$ at $1060nm$ input wavelength, $\gamma = 15{W^{ - 1}}k{m^{ - 1}}$ and ${T_0} = 279fs$, ${P_0} = 3.5W$ we have ${\delta _3} = 0.1$, and $z \cdot {L_D} = B$ for $N = 1$ [31]. The Raman effect can be barely observed for such low power and large third-order dispersion [32]. And the TSP modulation can neutralize the Raman frequency shift, which will be discussed in Sec. 4. Hence, TSP modulation can enhance DW energy while minimizing Raman frequency shift that is possibly detrimental in DW emission.
We use the same modulation as Figs. 1(c), 1(d) with periodic modulation interval ${z_m}$, which can be realized by a fiber-loop system with phase-modulator and Raman pump, to exert a steady chirp to utilize the enhanced and cancelled SPM via the TSP. After several numerical experiments, the results indicate that the cumulative modulation instability for modulation interval ${z_m} \ge 0.4{L_D}$ is out of control, corresponding to the requirement ${B_m} \le - 4$ of the PM equation, in which the resonant frequency is nonexistent. Here the temporal and spectral evolution for the periodic TSP that is equal to or opposite to the SPM-induced chirp ($\delta \omega _m^ +$ or $\delta \omega _m^ -$) and modulation interval ${z_m} = 0.3{L_D}$ are plotted in Fig. 2, accompanied with energy conversion efficiency $\mu = {{{E_{DW}}} / {{E_{total}}}}$ as a function of distance are plotted in Fig. 2(c). The energy is numerically integrated based on the corresponding spectrum components. The output spectra of the two scenarios and fundamental soliton input are plotted in Fig. 2(d), in which the intuitive comparison reveals a drastic enhancement of DW intensity by TSP.
Fig. 2. Temporal and spectral evolution for $\delta \omega _m^ - $ (top row), $\delta \omega _m^ + $ (middle row). (c) Energy conversion efficiency $\mu $. (d) Output spectrum for the modulated pulses (a2), (b2) and Soliton input.
The modulated spectrum with the TSP, as expressed in Eq. (4), is made up of numerous spectral peaks with the wavelength interval of ${\omega _{_m}} = {\pi / {\sqrt 2 }}$, i.e. ${\approx} 9nm$. In other words, every TSP modulation transfers energy from incident frequency to these peaks. Even if the energy transmitted each time is negligible for a small modulation depth, some peaks after several modulations may still be seen during the propagation, which is especially noticeable in periodic modulation. Around $1069nm$, the first-order peak is mixed into the interference fringe with the pump wave. While the second-order peak at ${\approx} 1078nm$, which is continually shed from the pump wave, presents as a noticeable spectral peak marked by the arrow in the right column of Fig. 2. For this reason, the spectrum peak vanishes for smaller modulation interval, despite the remainder of the propagation nearly unchanged. However, for more experimental complexity associated with a smaller modulation interval, ${z_m} = 0.3{L_D}$ remains the optimum.
In the propagation without modulation, the Gaussian pulse will be compressed in the beginning and then broadened because the interplay of SPM and dispersion. What is surprising is that the continuous chirp $\delta \omega _m^ - $ induced by TSP the constantly cancels the SPM-induced chirp, and the pulse is dispersed into two distinct peaks. The two peaks then collide and produce a compression point because the SPM chirp accumulates over the propagation distance and the TSP chirp can only cancel the SPM chirp well within a narrow range in the temporal center. As a result, two DW emissions can be clearly observed in Fig. 2(a2). The two spectral peaks of DW are caused by the interference of coherent fields. In the case $\delta \omega _m^ + $, around the central region of the pulse, the TSP-induced chirp is almost identical to the SPM-induced chirp. They collaborate to keep the velocity and the short duration of the pulse. The energy is steadily transferred from the pump wave to the resonant frequency, which brings an incredible conversion efficiency reaching $\mu = 29.12\%$ at $z = 3km$. Compared with $\mu = 0.5\%$ for Gaussian pulse, the efficiency for periodic TSP is increased up to near 60 times. We can anticipate a higher $\mu $ for longer propagation distance. And for shorter modulation intervals, the results above remain unchanged.
4. PM requirement
Although periodic modulation can provide significant benefits, particularly the incredible dispersive wave strength, its experimental use is more complicated and costly. And we have found that the resonant frequency is not much impacted by periodic modulation. Therefore, we argue that performing a single modulation at the input is preferable. The results demonstrated that a single modulation can considerably control the intensity especially resonant frequency of DW as well. With respect to the PM requirement, the wavenumber of the linear wave is ${k_{lin}} = {\delta _3}{\omega ^3} - 0.5{\omega ^2}$. Considering the cancellation of SPM is proportional to ${B_m}$, the wavenumber of the Gaussian pulse is $\frac{1}{2}{B_m}{\omega / \pi }$ [2,32]. By equaling the two wavenumbers we obtain
where $\omega = 2\pi ({\nu - {\nu_0}} ){T_0}$. Unfortunately, to keep $N = 1$, the maximal frequency shift of DWs is limited at the resonant frequency for Gaussian pulse (${B_m} = 0$) since the pulse cannot be compressed to a higher intensity. However, $\mu $ is drastically increased for ${B_m} > 0$. To reveal the interconnections, we modulate the pulse by the TSP with different modulation depths. For $- 4 < {B_m} < 0$, the PM condition Eq. (7) can be satisfied by three distinct frequencies (only one solution $\omega = 0$ for ${B_m} \le - 4$). As shown in Fig. 3(a), the resonant frequency of DW ${\omega _D}$ is marked by the dashed curve, and the curve calculated by Eq. (7) is in great agreement with results. With an increasing of ${B_m}$, more energy is redistributed to the wings of spectrum (broadened to $- A{\omega _m} \le \omega \le A{\omega _m}$) rather than central frequency. In this situation, however, limited by the PM requirement, the frequency of the pump wave was moved back around the central frequency, which results in a frequency shift of the pump wave. That is, the soliton formation in DW emission process is always accompanied by pulse acceleration or deceleration. It can also be understood as a weaker spectral recoil caused by less frequency shift of the DW.Fig. 3. (a) Output spectra as a function of ${B_m}$ and resonant frequency predicted by the PM requirement (dashed line). Results showing 500 individual spectra and the mean spectra (solid black curves) for (b) ${B_m} = 1$, (c) ${B_m} ={-} 1$.
Considering quantum noise induced by quantum fluctuation, the output spectra corresponding to ${B_m} ={\pm} 1$ are plotted in Figs. 3(b) and 3(c), where also the intensity of DW can be clearly observed. Input noise is spectrally induced on each spectral discretization bin with the intensity obeying normal distribution [30]. The mean spectra are obtained by taking the average first and then the logarithm. In these individual examples, the strongest DW noise is accompanied with the weakest DW intensity, that is, for ${B_m} ={-} 1$. Due to the influence of noise, the intensity difference between the greatest and weakest DWs is $4dB$. In sharp contrast, for ${B_m} = 1$ the strongest DW, this difference is only $2dB$. Similarly, the noise is not that obvious near the pulse center. On the one hand, it is related to the logarithmic scale we take; on the other hand, it can be understood that the stronger the signal, the lager the signal-to-noise ratio.
5. Raman scattering
In nonlinear fibers, the Raman scattering is the second-most common nonlinear phenomenon only after the SPM. Previous research has revealed that the presence of Raman scattering typically results in a considerable attenuation of DW emission with negligible disruption on the resonant frequency [33]. Even though this weakening deteriorates as TOD increases, the Raman scattering nevertheless blocks the generation of high-intensity DWs. Therefore, it is necessary to consider avoiding Raman scattering, which can be realized in hollow-core photonic crystal fibers filled with Raman-free gas [3]. However, we find that the attenuation of DW emission by Raman scattering can be significantly reduced by the continuous chirp given by the TSP modulation. Figure 4 shows the spectral peak intensity of DW as a function of ${\delta _3}$ for chirp $\delta \omega _m^ + $, $\delta \omega _m^ - $ without (solid lines) and with (dashed lines) the Raman scattering included, normalized by input spectral intensity. We ignore the Raman scattering by setting ${f_R} = 0$ in Eq. (3), and the results are mean intensity after adding noise, like Figs. 3(b) and 3(c). For the case $N \ne 1$, ${N^2}$ times the modulation depth in Eq. (6) is applied. It makes sense that the DW intensity for $N = \sqrt 2 $ is substantially higher than that for $N = 1$. It is worth noting that despite the DW intensity being so tremendous, this is merely due to its incredibly short bandwidth. Hence, that the drop of the DW intensity for $N = \sqrt 2 $ in Fig. 4(a) is because of broadened bandwidth is easy to be understood. The DW energy still grows even while the peak intensity drops. In Fig. 4(b), the second DW emission interferes with the first DW emission in the frequency domain, resulting in a reduction and fluctuation in the DW intensity, which can also be seen in Fig. 2(a2).
Fig. 4. DW peak intensity at 3 km propagation distance as a function of TOD parameter for (a) $\delta \omega _m^ + $, and (b) $\delta \omega _m^ - $.
In Fig. 4, it can be observed that whether Raman scattering is considered or not, the DW intensity first increases rapidly as TOD increases but subsequently reaches saturation with larger TOD. And it can be observed that larger N encourages the saturation state to arrive earlier. The comparison shows that, the DW intensity is weaker when Raman scattering is present, while the DW intensity barely differs with or without Raman (even the largest difference is $\lt 2dB$) no matter $N = 1$ or $\sqrt 2 $. It is first partially owing to the conspicuous enhancement of DW by TSP modulation. The crucial factor is that the continuous TSP-induced chirp periodically balances the spectral energy of shorter and longer wavelength components, corresponding to the central wavelength, through which the Raman frequency shift can be effectively neutralized. Thus, the pulse is confined to a restricted temporal regime, as shown in Figs. 2(a1) and 2(b1). The bound state of the pump pulse powerfully counters the Raman self-frequency shift and enhances the DW intensity. As a result, when the pulse is continuously chirped by the TSP modulation, the attenuation of DW intensity by Raman scattering can be significantly reduced, but not completely eliminated.
6. Conclusion
In conclusion, we investigate the DW emission from Gaussian pulse with TSP modulation. The TSP-induced chirp can be similar or opposite to the SPM-induced chirp decided by the modulation depth. The resonant frequency can be altered by TSP for modulation depth $A < 0$ (${B_m} < 0$), and the energy conversion efficiency can be drastically enhanced by TSP for $A > 0$ (${B_m} > 0$). We give the modified PM requirement, which is proved by the results obtained by NLSE. The implementation of periodic TSP is more complicated but brings a significantly optimized result. The DW spectrum is further compressed and concentrated to resonant frequency, and the conversion efficiency can reach $\mu = 29.12\%$ at distance $z = 3km$. Our results are helpful to development of wavelength conversion technology in fiber optics, which is useful to the generation of supercontinuum and frequency combs.
Funding
National Natural Science Foundation of China (61975130); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515010084); Natural Science Foundation of Hunan Province (2021JJ30202);
Acknowledgment
We thank the referees for their enlightening comments to improve this work.
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 reasonable request.
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