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Abstract
Numerical simulations of a simple and direct method to generate soliton spectral tunneling (SST) based on two input pulses are reported in the paper. An intense pump pulse and a weak probe pulse with a time delay are transmitted in a photonic crystal fiber with three zero-dispersion wavelengths. Our results demonstrate that the distance and the state of soliton tunneling are obviously influenced by the probe-pump delay. Therefore, the velocity and efficiency of SST can be effectively regulated by varying the relative time delay, thus affecting the SST formation. This scenario appears promising for designing a “soliton ejector”, in which real-time control of the soliton ejection process can be achieved through phase modulation between pulses.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The shape, amplitude, and speed of optical solitons can remain unaltered before and after the collision with other solitons and barriers. Thus, the optical soliton is a kind of stable solitary wave and its robust nonlinear translation originates from the perfect balance between nonlinear interaction forces and linear dispersion [1]. Tunneling, on the other hand, is one of the most typical characteristics of wave mechanics, expressed in fascinating ways including the matter waves tunneling [2], the charge carriers tunneling [3], and even superfluid states [4]. Although the above two phenomena seem to be unrelated to each other, the paper [5] points out that the matter-wave soliton may tunnel through the potential dispersion barrier, thereby gradually establishing the correlation between them. The theoretical basis stems mainly from the fact that the soliton in the optical fiber can pass through the longitudinal junction in the temporal domain as well as the forbidden normal dispersion barrier in the frequency domain [6,7], which is referred to as soliton spectral tunneling (SST).
In recent years, the research on the mechanisms and influencing factors of SST has made a breakthrough [8,9], and the effective regulation appears to be a desirable goal of SST study. It is shown that frequency chirp [10], the self-steepening effect [11], the potential barrier [12], and the dispersion slope [13] all affect the tunneling process of optical solitons. Since the nonlinear interaction between the co-propagating pulses introduces cross-phase modulation (XPM), which transfers the intensity modulation of one pulse to the phase modulation of the other [14]. Based on this fact, double-pulse injection by changing the relative parameters between pulses has been applied broadly in the study of spectral generation characteristics [15–20]. Therefore, an SST effect based on double input pulses with the same central wavelength will be investigated in detail in this paper. When an intense pump pulse and a weak probe pulse with a time delay interact in a photonic crystal fiber, the energy change of the spectrum is observed in the time domain and frequency domain. More importantly, the use of statistical analysis enables effective modulation of the SST effect within a certain range and obtains the optimal condition for optical soliton tunneling. This regulation scheme has more general conditions of application and is easy to operate.
2. Numerical model and phase-matching topology analysis
2.1 Propagation model
We consider using the generalized nonlinear Schrödinger equation (GNLSE) to capture the evolution dynamics of ultrashort pulses in an optical fiber [21]:
Two hyperbolic-secant pulses with a time delay ${t_{del}}$ excite the optical fiber at the same pump wavelength ${\lambda _0} = 804$ nm, named the pump pulse and the probe pulse respectively. The initial electric field envelope $A({0,t} )$ takes the following form:
2.2 Fiber properties
The advantage of photonic crystal fiber (PCF) is that its controllable dispersion and nonlinear features can be achieved by properly setting the fiber geometry (like hole and pitch sizes), according to the method presented in Ref. [22,23]. We use a 1.2 m silica PCF with three zero dispersion wavelengths (ZDWs) as the transmission medium, and its ZDWs are located at ${\lambda _1} = 770$ nm, ${\lambda _2} = 924$ nm, and ${\lambda _3} = 974$ nm, respectively. The related dispersion coefficients are up to the tenth order at the pump wavelength of 804 nm, as follows: ${\beta _2} ={-} 15.8126\; p{s^2}/km$, ${\beta _3} = 0.10025\; p{s^3}/km$, ${\beta _4} = 1.0582 \times {10^{ - 3}}\; p{s^4}/km$, ${\beta _5} ={-} 1.5686 \times {10^{ - 6}}\; p{s^5}/km$, ${\beta _6} = 2.4280 \times {10^{ - 9}}\; p{s^6}/km$, ${\beta _7} = 4.0260 \times {10^{ - 10}}\; p{s^7}/km$, ${\beta _8} ={-} 1.7693 \times {10^{ - 12}}\; p{s^8}/km$, ${\beta _9} = 6.4322 \times {10^{ - 15}}\; p{s^9}/km$, ${\beta _{10}} = 3.1990 \times {10^{ - 20}}\; p{s^{10}}/km$.
The dispersion and group delay curves corresponding to the pump wavelength are depicted in Fig. 1. It follows from the figure that the three ZDWs split the entire spectrum into four regimes: short-wavelength normal group velocity dispersion (GVD) regime (N1: λ ≤ λ1), short-wavelength anomalous GVD regime (A1: λ1 ≤ λ ≤ λ2), long-wavelength normal GVD regime (N2: λ2 ≤ λ ≤ λ3), and long-wavelength anomalous GVD regime (A2: λ > λ3). This is an ideal dispersion distribution for studying SST, that is, a normal dispersion regime serving as a dispersion barrier is sandwiched between two anomalous dispersion regimes. It is worth noting that PCF is not the only transport medium for soliton tunneling, other fibers with suitable tunneling barriers are also available for the study. In addition, the nonlinear coefficient γ is 0.35 W−1m−1. At the same time, since the material loss α in the silica fiber only increases sharply when λ > 2.4 μm [24] and the length of the fiber we use is very short, the loss α can be neglected here.
Fig. 1. The dispersion (a) and relative group delay (b) profiles as a function of the wavelength in the PCF. The bright and shaded areas respectively represent the normal and anomalous GVD regimes. The orange dotted line in Figure (a) shows the position of the pump wavelength of two pulses. The green-filled area in Figure (b) indicates the phase-matching regime for solitons and dispersive waves (DWs).
2.3 Topology of phase matching
High-order solitons are split into a series of fundamental solitons under the combined perturbation of dispersion and stimulated Raman scattering, and the energy is continuously transferred from solitons to DWs [25,26]. The phase matching condition for the DW with center frequency ${\omega _d}$ is determined by the non-dispersive soliton phase with spectrum centered at ${\omega _s}$, which can be written as [9]:
where ${q_s}$ denotes the soliton wave number, which can be negligible because its contribution to the fundamental solitons is tiny. Using the constant transformation, add $({\omega - {\omega_d}} )v_{g.d}^{ - 1}$ to both sides of the above equation. When ${v_{g.d}} = {v_{g.s}}$ is satisfied, Eq. (3) can be expanded as:Therefore, if the group velocities of the solitons (${v_{g.s}}$) and DWs (${v_{g.d}}$) match, it means that the solitons at ${\omega _s}$ and the DWs at ${\omega _d}$ achieve phase matching over the entire frequency domain, i.e., ${\beta _{{\omega _s}}}(\omega )= {\beta _{{\omega _d}}}(\omega )$. As seen in the green-filled area of Fig. 1(b), the distribution of DWs generated by the soliton S1 located near λ2 (central wavelength range of approximately 896 ∼ 924 nm) can be obtained from the corresponding group delay curve. The soliton S1 radiates the DW soliton S2 in the A2 region (central wavelength range of approximately 974 ∼ 998 nm) and DWs in the N1 region (central wavelength range of approximately 712 ∼ 714 nm).
3. Results and discussions
3.1 Temporal and spectral evolution of SST with double pulses injection
Let us first illustrate the nonlinear dynamics of two input pulses injected into the 1.2 m long PCF with three zero-dispersion wavelengths. The output temporal profiles and corresponding temporal evolution of double pulses with different time delays are shown in Fig. 2. In the initial stage of propagation, the input pulse remains a constant waveform by the combined effect of GVD and self-phase modulation (SPM). Shortly thereafter, high-order soliton splits into three distinct fundamental solitons, namely FS1, FS2 and FS3. The first Raman soliton FS3 emitted from the pulse during the splitting process undergoes the strongest self-frequency downshifts.
Fig. 2. (a-c) Output temporal profiles (top) and temporal evolution (middle) of double pulses with different time delays in the fiber. FS1, FS2, and FS3 stand for fundamental solitons. Enlarged figures (bottom) show details of the pulse inputs. The red arrows indicate the time center of the probe pulse.
The comparison shows that the delay variation of the fundamental solitons and the corresponding dispersive wave radiation are clearly influenced by the relative probe-pump delay. Figure 3 depicts the output temporal profiles of different transmission distances with positive and negative time delay (tdel) in detail. For the case of positive tdel, the probe at the input is 0.13 ps behind the pump and the intensity is normalized [Fig. 3(b)]. The pump remains stationary in the absence of XPM and the probe appears to be moving slower with the lag of the time center (the arrow). With the transmission of pulses, the total pulse compresses owing to perturbations of various nonlinear effects, as seen at z = 0.06 m. With further transmission, the probe is modulated only by the back of the pump. Due to the XPM interaction, the trailing edge of the total pulse has a negative frequency sweep and the leading edge has a positive one [14]. As a result, the pulse front edge is broadened and the pulse trailing edge is compressed, which causes the spectrum to shift toward higher frequencies. The case for negative tdel is the exact opposite [Fig. 3(a)]. The probe is modulated only by the front of the pump, so the leading edge of the total pulse has a negative frequency sweep and is compressed, while the trailing edge has a positive one and is broadened. The spectrum is therefore shifted to lower frequencies.
Fig. 3. Output temporal profiles at various propagation length for double pulses with (a) tdel = −0.13 ps and (b) tdel = 0.13 ps propagating in the fiber. The black arrows indicate the time center of the probe pulse.
Our results show that FS1 and FS2 in Fig. 3(a) have greater time delay and redshift compared with Fig. 3(b), while FS3, with the maximum energy, decelerates so fast that the inter-pulse XPM occurs very early, and thus the subsequent spectral evolution causes a different degree of time delay. Here the XPM is most efficient because the relative group velocity between the pump and the edge of probe is close to zero [27]. This results from the group velocity mismatch and broadening velocity of the pulse edges cancelling each other out, resulting in probe and pump edge synchronization. It can be concluded that the negative tdel promotes the soliton self-frequency shift (SSFS) while the positive tdel has an inhibitory effect on it.
Another noteworthy phenomenon in Fig. 2 is that the output intensity with positive tdel is much greater than that of FS2 with negative tdel and no time delay. Such a phenomenon comes from the fact that FS2 with positive tdel is undergoing spectral transformation between two anomalous dispersion regimes (A1 and A2) due to inhibition of SSFS and therefore compressing in the time domain. In this case, FS2 radiates blue-shifted dispersive waves (B-DWs) with higher intensity in the N1 regime, as shown in Fig. 2(c).
Figure 4 depicts the output spectral profiles and spectral evolution of double pulses with different time delays. SPM brings about an initial broadening of the spectrum by generating new frequency components. However, in addition to the SPM there are perturbations of other nonlinear effects, so that the spectrum of ultrashort pulses is greatly extended, which is referred as supercontinuum (SC). During this dramatic broadening of spectrum, the fundamental solitons are emitted in the A1 region due to the Raman effect and radiate the corresponding DWs due to the higher-order dispersion effect. Each fundamental soliton obtained by splitting undergoes SSFS to longer wavelengths. When it reaches λ2 and satisfies the condition of group velocity matching (GVM) and phase matching (PM), as described in Eq. (4), the fundamental soliton continues to couple energy towards the potential barrier (the regime N2) until a new DW soliton is formed in A2. This soliton transport process is called the SST effect.
Fig. 4. Output spectral profiles (top) and spectral evolution (bottom) of double pulses with different time delays in the fiber. The vertical dashed lines imply the three ZDWs. The white boxes mark residual dispersive waves.
The results demonstrate two things. First, the negative tdel promotes the SST effect, which can be confirmed by the fact that the residual DW energy near the potential barrier after soliton tunneling (marked in white boxes) is less in Fig. 4(a) than in Fig. 4(b). That is, the negative tdel here make the energy conversion in soliton tunneling more adequate. Second, the positive tdel makes the soliton transmit faster and thus its SSFS is suppressed, so a longer distance is required to complete tunneling in Fig. 4(c). Therefore, FS2 in the positive tdel case has a smaller redshift and weaker intensity compared to the negative tdel case.
We also plot the spectrogram traces in Fig. 5 for three different transmission distances to better demonstrate the influence of time delay between two pulses on spectral changes. The spectrogram trajectory shows an irregular M-shape, implying that the spectrum has abundant frequency components. The first emitted fundamental soliton, FS3, has a stronger peak power and is therefore the first to complete soliton tunneling. During transmission, the frequency of the fundamental solitons is continuously red-shifted due to the Raman effect, and thus their group velocity continuously decreases. Spectrum broadening causes the redshifted solitons in the anomalous dispersion regime and the DWs in the normal dispersion regime to separate, but they might still overlap in the temporal domain, whereby the B-DWs can be captured by the fundamental solitons due to the action of four-wave mixing (FWM). The higher the energy of the fundamental solitons, the higher the corresponding captured wave energy. As can be seen from Figs. 5(a3), 5(b3) and 5(c3), the negative tdel case results in more DW energy being captured and also the formation of a fourth fundamental soliton (FS), while the positive tdel case has the weakest captured wave energy. Thus, the negative tdel case promotes the energy transfer.
Fig. 5. The corresponding spectrograms at various propagation length (1) z = 0.4 m, (2) z = 0.7 m, and (3) z = 1.2 m for double pulses with different time delay propagating in the fiber.
On the other hand, how the time delay affects the soliton tunneling process is also well illustrated in Fig. 5. Taking the tunneling process of FS2 as an example, negative tdel always causes the soliton to tunnel first at the dispersion barrier because of its contribution to SSFS, while positive tdel causes the SST effect to appear at a slower rate in comparison.
3.2 Statistical characteristics of the time delay effects
For deeper analysis, we performed a series of numerical simulations by varying only the time delay tdel, and the output pulse shape and spectrum at z = 1.2 m are plotted in Fig. 6. From Figs. 6(a1) and 6(b1), one can observe that the output profiles with identical |tdel| are not perfectly symmetrical. It is worth noting that positive tdel (0 < tdel < 0.18) suppresses SSFS making the soliton energy conversion inadequate, leaving higher intensity red-shifted dispersive waves (R-DWs) in N2, while negative tdel (−0.18 < tdel < 0) facilitates SSFS making more energy from the fundamental soliton transfer to A2, leaving lower intensity R-DWs. These R-DWs are marked with boxed lines as shown in Fig. 6(b1). If the time delay |tdel| is too large, the edges of the probe pulse will not receive much phase modulation. Hence the facilitation of negative tdel has a saturation state, which is expressed by the appearance of the sharp corners (marked with white arrows) in Figs. 6(a1) and 6(b1). It can be seen that as tdel decreases, the time delay of the fundamental solitons successively appears to gradually increase, which indicates the slowing down of transmission speed and the enhancement of SSFS. This simulation shows that the maximum SSFS (i.e., the optimal tunneling conditions) of FS1 and FS2 appear at tdel = −0.18 ps and tdel = −0.11 ps, respectively. However, since FS3 completes tunneling very early at z ≈ 0.2 m, the influence of tdel at the output z = 1.2 m does not show up. Thus, positive and negative time delays will cause different delay and energy distribution in the spectrum, the details of which are shown in Figs. 6(a2) and 6(b2).
Fig. 6. Output (a1) pulse shapes and (b1) spectrum versus time delay tdel varying from −0.3 ps to 0.3 ps in the 1.2 m long PCF. The related output (a2) temporal and (b2) spectral profiles under 5 different tdel values. The white boxes and white arrows mark R-DWs and “sharp corners” respectively.
Finally, the data statistics of different delay effects are plotted in Fig. 7, which show how the normalized intensity of the fundamental solitons and the energy conversion efficiency of both tunneled solitons and B-DW at z = 1.2 m vary in time delay. As described above the results show that positive tdel inhibits SSFS, while as tdel decreases more of the probe receives a negative sweep, resulting in a larger proportion of the pulse trailing edge compression and thus inducing a fading blue frequency shift. Similarly, as the negative tdel increases, more of the pulse leading edge compresses, thus inducing a fading red frequency shift. Therefore, on the whole a decreasing tdel will lead to an enhancement of SSFS, and thus a gradual increase in the normalised intensity of FS1 and FS2, as shown in Figs. 7(a1) and 7(a2). On the other hand, Fig. 7(a3) shows that if |tdel| is small, the energy superposition of probe and pulse makes FS3 stronger, which indicates that the pulse power also affects the SST effect.
Fig. 7. (a) The normalized intensity of the fundamental solitons (FS1 for the red curve, FS2 for the blue curve, FS3 for the green curve) and (b) the energy conversion efficiency of tunneled solitons (square marked, orange line) and B-DW (triangles marked, blue line) as a function of the time delay tdel.
Figure 7(b) confirms the findings of this study, namely that the power ratio of tunneled solitons (FS2 and FS3) increases for smaller negative tdel, while their power ratio decreases for larger positive tdel. Due to energy conservation, the power ratio of B-DW has the opposite trend. Our results indicate that the threshold condition for XPM generation between the probe and the pump is |tdel| ≈ 0.18. If |tdel| exceeds this threshold, the phase modulation between pulses will disappear.
4. Conclusion
Throughout this work, we investigate analytically a mechanism to acquire the tunable SST effect via injecting an intense pump pulse and a weak probe pulse with an appropriate time delay in a PCF with three ZDWs. As the time delay decreases within a certain degree, the transmission speed of the fundamental soliton slows down and thus its redshift accelerates, which facilitates the appearance of SST effect. In this case, a positive time delay causes the pulse to undergo an XPM-induced relative blue frequency shift, thereby suppressing the SSFS, while a negative time delay causes the pulse to undergo a larger red frequency shift, thereby enhancing the SSFS. It has been shown that the energy conversion efficiency of each part of the SC can be regulated effectively by arranging the relative probe-pump delay. Therefore, this regulation scheme has excellent adaptability and flexibility, which allows the speed and efficiency of the SST effect to be managed very easily and enables the optimum tunnelling conditions for different solitons to be determined precisely. This controllable generation of SST effect has potential application value in the design of “soliton ejector”, which allows real-time control of the soliton ejection process.
Funding
Scientific Research Foundation of Hunan Provincial Education Department (21C0702); Subject of Hunan Provincial Social Science Achievement Review Committee (XSP22YBC233); National Natural Science Foundation of China (62172159); Natural Science Foundation of Hunan Province (2021JJ30294).
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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