1. Introduction

Snell’s law and the Fresnel formulae, which describe the reflection and refraction of a spatial beam at a dielectric interface, are essential topics in physics textbooks [1,2]. Owing to the space-time duality, it is possible to find the time analogy of the spatial beam behavior and vice versa. Recent research has indicated that an optical pulse encounters a temporal boundary where the refractive index changes suddenly in time [3–9]. A temporal boundary separates two regions with different refractive indices in time and can be regarded as an analog of a spatial interface [7–9]. Momentum conservation at temporal boundaries provides a method for calculating reflected and transmitted frequencies [7,8]. Approximate analytical expressions for the electric fields of reflected and transmitted pulses have been provided [9]. The pulse evolution of different input pulses [10,11] or different temporal boundary shapes [12] has been analyzed, as well as the interaction between the pulse and temporal boundary [13]. Currently, many analogies of concepts and phenomena are known, including waveguides [8,14,15], antireflective coatings [16], Fabry–Perot resonators [17], Lloyd’s mirrors [18], and the Goos–Hänchen shift [19]. In practice, employing the nonlinear phenomenon of XPM induced by a pump-probe configuration [20] and intermodal nonlinear interaction in multimode fibers [21,22] are used to create temporal boundary.

This study investigated a temporal boundary and a dispersive wave. The generation of dispersive wave, also known as resonant radiation or Cherenkov radiation, is a general process in nonlinear fiber optics [23–25]. The frequency of dispersive wave is determined by the phase-matching condition, which depends on the dispersion relationship and nonlinearity of the fiber [26–28]. Dispersive radiation is widely used for wavelength conversion [29–32], supercontinuum generation [33,34], and optical frequency combing [35,36]. It also plays a role in temperature measurement [37] and imaging [38,39]. In addition, the solitonization of dispersive waves [40] and the interaction of dispersive waves with solitons [41–43] have been analyzed. In optical resonators, dispersive waves can stabilize the bound states formed by cavity solitons [44]. Solitons, dark solitons [45,46], self-accelerating pulses [47], and Pearcey Gaussian pulses [48] are light sources that can produce dispersive radiation. The power variation of dispersive waves has been conducted [49] and approaches to enhancing the energy conversion efficiency of dispersive waves have been explored, such as phase modulation [50–53] and modified genetic algorithm [54].

The energy distribution at the temporal boundary under second-order dispersion and the evolution of Gaussian pulses at the temporal boundary have been elucidated. However, the evolution of soliton pulses at the temporal boundary under third-order dispersion remains unknown. In this study, we investigate the soliton dynamics at the temporal boundary under third-order dispersion. We demonstrate a phenomenon in which second-order soliton form after fundamental soliton cross a temporal boundary. According to the momentum conservation condition, the frequency of second-order soliton varies with the incident frequency. The resonant frequency corresponding to the second-order soliton can be accurately predicted using the phase-matching condition provided.

2. Propagation model

We consider propagation of quasi-monochromatic optical pulses inside a dispersive medium (such as optical fibers) with the propagation constant $\bar \beta \left (\omega \right )$. For a pulse containing multiple optical cycles, we can expand $\bar \beta \left (\omega \right )$ around a reference frequency in a Taylor series as

In general, the temporal boundary moves at speed $\nu _B$; therefore, we consider a moving reference frame with the same velocity as the boundary. In this reference frame, the boundary is stationary. By using the coordinate transformation $t=T-z/\nu _B$, where $T$ is the time in the laboratory frame, the dispersion relation in the moving frame becomes

By adjusting the reference frequency, the group velocity at that frequency matches the speed of the temporal boundary. In this case $\Delta \beta _1=0$, and the dispersion relation becomes

Using Maxwell’s equations and the dispersion relation in Eq. (3), and making a slowly varying envelope approximation, we obtain the following time-domain equation [7,8]:

3. Numerical results

We demonstrate the evolution in which the incident pulse crosses the temporal boundary inside a medium. At the reference frequency $\omega _0$, the group velocity dispersion and the third-order dispersion are $\beta _2=0.05{ps}^2/m$ and $\beta _3=0.0045{ps}^3/m$, respectively [8]. The nonlinear parameter $\gamma =1W^{-1}{km}^{-1}$ [55]. The temporal boundary is located at $T_B=-10ps$, and the change in propagation constant is $\beta _B=0.75m^{-1}$. We choose the fundamental soliton as the input pulse, and its power, pulse width, and frequency shift relative to the reference frequency are $P_0=\left |\beta _2+\beta _3\Delta \omega _i\right |/\gamma T_0^2$, $T_0=1ps$, and $\mathrm {\Delta }\omega _i=-17.22rad\cdot {ps}^{-1}$, respectively. The selected incident frequency has a negative dispersion slope, so the pulse moves towards negative time. Because we are in a reference frame moving with the boundary, this only means that the pulse is moving faster than the boundary [8]. In order to achieve pulse crossing the temporal boundary, the temporal boundary must be moved to $T_ B=-10ps$.

As shown in Fig. 1(a), the reflection and refraction of optical pulses at the temporal boundary are similar to those of light beams at the spatial interface. The incident pulse splits into three pulses at the temporal boundary: the reflected pulses labeled R1 and R2, and the transmitted pulse labeled T. The dispersive wave produced by the transmitted pulse is labeled DW. Notably, in the case of second-order soliton formed by the transmitted pulse, the transmitted pulse radiates dispersive wave. Figure 1(b) shows the evolution of the spectrum. The reflected spectrum labeled R2 shifts toward the red side, whereas the transmitted spectrum, dispersive wave spectrum, and reflected spectrum labeled R1 shift toward the blue side. Figure 1(c) depicts the functional relationship between the propagation constant and the frequency.

 

Fig. 1. (a) Temporal evolution of fundamental soliton at a temporal boundary. This figure illustrates the process of an input pulse generating two reflected pulses and one transmitted pulse. The symbols R1, R2, T, and DW in the figure represent the first reflected pulse, second reflected pulse, transmitted pulse, and dispersive wave, respectively. Black dashed lines indicate temporal boundaries. $z_s$ represents the splitting distance of the incident pulse. (b) Spectrum evolution of fundamental soliton at a temporal boundary. (c) Dispersion curves for $t>T_B$ (solid blue) and $t<T_B$ (dashed red). The symbol I in the figure represents the input pulse. The solid green line corresponds to zero-dispersion frequency. The parameters corresponding to the graph are as follows, $\beta _B=0.75m^{-1}$, $\mathrm {\Delta }\omega _i=-17.22rad\cdot {ps}^{-1}$, $\Delta v_i=\Delta \omega _i/2\pi =-2.74THz$, and $T_B=-10ps$.

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The spectral shift in Fig. 1(b) originates from a temporal boundary that breaks the translational symmetry in time. Therefore, the photon momentum remains constant in the reference frame and the photon energy may change [7,8]. Mathematically, the momentum conservation is expressed as follows:

The above equation implies that the propagation constants of $\Delta \omega$ and $\Delta \omega _i$ are equal. To conserve the propagation constant when transitioning from the $t>T_B$ region to the $t<T_B$ region, each frequency component must shift from point I on the solid blue curve in Fig. 1(c) to points A, B, and T on the dashed red curve. Of these three points, only T is physically meaningful because the slopes of the IT points related to pulse velocity have the same sign. The reflected pulse is caused by two points on the solid blue curve with the same propagation constant, marked R1 and R2 in Fig. 1(c). The slopes of these two points must be opposite the slope of I to ensure that the pulse is reflected back into the $t>T_B$ region. It is important to emphasize that the reflected pulse does not propagate backward in time or space, but rather changes its velocity to remain in the $t>T_B$ region.

The energy conservation condition determines the energy and width of the reflected and transmitted pulses. A fundamental soliton is selected as the input pulse. When the input pulse crosses the boundary, the transmitted pulse forms a second-order soliton. The second-order soliton radiates dispersive wave under third-order dispersion. According to Fig. 1(c), the frequency of the second-order soliton is close to the zero-dispersion frequency; therefore, the influence of self-phase modulation (SPM) induced spectal broadening should be considered.

Next, we elaborate on the evolution of the fundamental soliton before and after the temporal boundary. Sections 3.1 and 3.2 focus on the reflection and transmission coefficients and energy conservation condition, respectively. In Section 3.3, we determine whether the reflected and transmitted pulses are solitons. In Section 3.4, we explore resonance frequency. We analyze the predictive ability of the modified phase-matching condition in Section 3.5.

3.1 Reflection and transmission coefficients under third-order dispersion

In this section, we refer to the method in [9] to derive expressions for the reflection and transmission coefficients in the presence of third-order dispersion.

Figure 1(a) shows the reflection and transmission of the input pulse at the temporal boundary, and Fig. 1(b) shows the frequency shift of the reflected and transmitted pulses. We anticipate that plane waves with a frequency $\omega _0+\Delta \omega _i$ will also exhibit the same phenomenon at the temporal boundary. Therefore, the following formula is used to describe the amplitude of the plane waves:

The dispersion relation in the reference frame is obtained by substituting Eq. (7) into Eq. (4), neglecting the nonlinear terms.

From Eq. (4), $A\left (z,t\right )$ should be continuous for all values of $z$. When the input pulse reaches the temporal boundary, all four propagation constants will become equal:

These conditions result from momentum conservation in the reference frame [7]. Combining Eq. (8) and Eq. (9) yields three cubic equations. The solutions of ${\mathrm {\Delta \omega }}_{R1}$, ${\mathrm {\Delta \omega }}_{R2}$, and ${\mathrm {\Delta \omega }}_T$ are represented as follows:

To determine the reflection coefficients ($R1$ and $R2$) and transmission coefficient ($T$), we must utilize the temporal boundary condition at $t=T_B$. It is required that for any $z$, $A$, $\partial A/\partial t$, and $\partial ^2A/\partial t^2$ remain continuous across the temporal boundary. This condition produces the following three equations.

By using Eq. (9) to solve the above equations, we obtain the expressions for $R1$, $R2$, and $T$:

The above expression contains a linear phase shift that depends on the boundary position, $T_B$. Setting $T_B=0$ eliminates the linear phase shift. Figure 2 shows how the moduli and phases of Eq. (13) change with frequency shift $\Delta \omega _i$. As the frequency shift gradually decreases, the moduli of $R1$ and $T$ gradually increase, and their phases become zero. In contrast, the modulus of $R2$ gradually decreases, and its phase is $\pi$. For $\Delta v_i=-2.74THz$, it is evident from Fig. 2(a) that $\left |T\right |$ has the largest in value, and $\left |R1\right |$ has the smallest value. If $\mathrm {\Delta }\omega _i$ is in the monotone-decreasing interval of $\beta _{ge}\left (\omega \right )$ and the transmitted pulse exists, Eq. (10) and Eq. (13) hold, and $R1$, $R2$, and $T$ are real numbers.

 

Fig. 2. (a) Relationship between the modulus of reflection and transmission coefficients and the incident frequency. The solid blue line, dashed red line, and dotted yellow line represents $\left |R1\right |$, $\left |R2\right |$, and $\left |T\right |$, respectively. (b) The relationship between the phase of reflection and transmission coefficients and the incident frequency. The phase of $R1$ (blue box) and $T$ (yellow solid line) is 0, while the phase of $R2$ (red dashed line) is $\pi$.

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3.2 Energy conservation

Next, we use an energy conservation condition to study the energy distribution of the reflected and transmitted pulses. Here, we consider whether the sum of the energies of the reflected and transmitted pulses equals the incident pulse energy. However, $\left |R1\right |^2+\left |R2\right |^2+\left |T\right |^2=1$ is invalid; therefore, we need to search for other conservation relationships.

We assume that the incident pulse $A$ has a narrowband spectrum with a central frequency at $\omega _0+{\mathrm {\Delta \omega }}_i$, where $\omega _0$ is the reference frequency and $\Delta \omega _i$ is the incident frequency shift. From Eq. (10), there is no linear relationship. Their linear relationship can be obtained by expanding them in a Taylor series [9].

According to the method in Ref. [9], the frequency-domain expressions of the reflected and transmitted pulses are

Using the inverse Fourier transform, we obtain the time-domain expressions of the reflected and transmitted pulses.

To obtain the analytical expressions for $U$, $V$, and $W$, we consider the momentum conservation of the reflected and transmitted pulses. Combining Eq. (8) with Eq. (9), we obtain

Taking the derivative of $\Delta \omega$ on both sides, we get

Substituting $\Delta \omega _i$ into Eq. (19), $U$, $V$, and $W$ may be negative. Equation (17) indicates that the temporal width of the two reflected pulses is reduced by a factor of $\left |U\right |$ and $\left |V\right |$, respectively, and the transmitted pulse is reduced by a factor of $\left |W\right |$. This can also be derived from Eq. (16), which shows that the bandwidth of the two reflected pulse varies by a factor of $\left |U\right |$ and $\left |V\right |$, respectively, and that of the transmitted pulse varies by a factor of $\left |W\right |$. As the energy entering the reflected pulse is $\left |R1\left ({\mathrm {\Delta \omega }}_i\right )\right |^2/\left |U\left ({\mathrm {\Delta \omega }}_i\right )\right |$ and $\left |R2\left ({\mathrm {\Delta \omega }}_i\right )\right |^2/\left |V\left ({\mathrm {\Delta \omega }}_i\right )\right |$ respectively, and the energy going into the transmitted pulse is $\left |T\left ({\mathrm {\Delta \omega }}_i\right )\right |^2/\left |W\left ({\mathrm {\Delta \omega }}_i\right )\right |$. The conservation of the energy relation is

Equation (20) holds when ${\mathrm {\Delta \omega }}_i$ is in the monotone-decreasing interval of $\beta _{ge}\left (\omega \right )$ and the transmitted pulse is present.

3.3 Soliton formation

We obtain the power expression of the reflected and transmitted pulses from the time-domain expressions.

The change in power with $\Delta \omega _i$ is shown in Fig. 3(a). It can be seen that the power variation processes of the three pulses differ. In the given frequency shift range, $P_{R1}$ is quite small and varies slightly, $P_{R2}$ decreases rapidly and then slowly, and $P_T$ increases and then decreases. $P_T$ decreases because the power $P_0$ required to maintain the fundamental soliton decreases. Figure 3(b) shows the percentage of power, which considers the ratio of the reflected and transmitted powers to the incident power. As the frequency shift decreases, the proportion of $P_{R1}$ increases slowly, the proportion of $P_{R2}$ decreases rapidly, and the proportion of $P_T$ increases steadily.

 

Fig. 3. (a) Relationship between power and the incident frequency shift. The solid blue, red, and yellow lines correspond to $P_T$, $P_{R1}$, and $P_{R2}$ respectively, while the purple dashed line corresponds to the incident power $P_0$. (b) The relationship between the percentage of power and incident frequency shift. The solid blue, red, and yellow lines are $P_T/P_0$, $P_{R1}/P_0$, and $P_{R2}/P_0$, respectively. It should be noted that the sum of the three percentages does not equal one hundred percent.

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One condition for soliton formation is that the pulse spectrum should be located within the anomalous GVD region. To determine whether the pulse spectrum lies in the anomalous GVD region, the second-order derivative of the dispersion curve is less than zero. The second-order derivative of the dispersion curve is expressed as follows:

Another condition for soliton formation is that the soliton order $N$ should be greater than 0.5 [55]. The soliton order is expressed as

A pulse can form a soliton when both conditions are satisfied. Substituting the incident frequency ${\mathrm {\Delta \omega }}_i$ in Fig. 1 into Eq. (10) yields the reflected and transmitted frequencies ${\mathrm {\Delta \omega }}_{R1}\left ({\mathrm {\Delta \omega }}_i\right )$, ${\mathrm {\Delta \omega }}_{R2}\left ({\mathrm {\Delta \omega }}_i\right )$, and ${\mathrm {\Delta \omega }}_T\left ({\mathrm {\Delta \omega }}_i\right )$. Substituting these three frequencies successively into Eq. (22) shows that ${\beta }^{\prime \prime }\left ({\mathrm {\Delta \omega }}_{R1}\left ({\mathrm {\Delta \omega }}_i\right )\right )$ is greater than zero and that both ${\beta }^{\prime \prime }\left ({\mathrm {\Delta \omega }}_{R2}\left ({\mathrm {\Delta \omega }}_i\right )\right )$ and ${\beta }^{\prime \prime }\left ({\mathrm {\Delta \omega }}_T\left ({\mathrm {\Delta \omega }}_i\right )\right )$ are less than zero. Furthermore, as shown in Fig. 1, the reflected pulse with angular frequency ${\mathrm {\Delta \omega }}_{R1}$ is in the normal GVD region, and the reflected pulse with angular frequency ${\mathrm {\Delta \omega }}_{R2}$ and the transmitted pulse with angular frequency ${\mathrm {\Delta \omega }}_T$ are in the abnormal GVD region. Substituting the required values into Eq. (23), we notice that the soliton order for the reflected pulse with angular frequency ${\mathrm {\Delta \omega }}_{R2}$ is below 0.5, whereas the soliton order for the transmitted pulse with angular frequency ${\mathrm {\Delta \omega }}_T$ ranges from 1.5 to 2. In summary, only the transmitted pulse shown in Fig. 1 can form second-order soliton [55].

3.4 Frequency analysis of dispersive wave

This section analyzes the evolution of soliton formed by transmitted pulse. The distance from the center of the incident pulse to the boundary is $z_B=T_B/{\beta }^\prime \left (\mathrm {\Delta }\omega _i\right )$, where ${\beta }^\prime \left (\Delta \omega _i\right )$ represents the first derivative of the dispersion curve:

Taking $1/e^{3.5}$ as the intensity of the pulse at the leading and trailing edges, $t_1$ and $t_2$ represent the times of the leading and trailing edges, respectively. We can then obtain

The splitting distance $z_s$ in Fig. 1(a) can be expressed as

 

Fig. 4. (a) Comparison between simulation and analytical results. The solid line represents the simulation result, and the dashed line represents the analytic result. (b) Comparison of two simulation results ($z_2$ and $z_d$). The spectrum of $z_2$ is represented by a solid line, and the spectrum of $z_d$ is represented by a dashed line.

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The frequency of the dominant outermost spectral peak is used to calculate the resonance frequency. The frequency change between the transmitted pulse and peak can be represented by SPM-induced spectral broadening [55].

After the distance $L=T_T^2/\left |{\beta }^{\prime \prime }\right |$, the second-order dispersion corresponding to the dominant peak frequency is ${\beta }^{\prime \prime }\left (\mathrm {\Delta }\omega _t\right )+\mathrm {\Delta }\beta _2$, and the frequency is $\mathrm {\Delta }\omega _{new}$.

In the presence of higher-order effects, the frequency of the radiation generated by soliton can be determined from the phase-matching condition. The condition used in this numerical simulation is as follows [28]:

The phase-matching condition is shown in Fig. 5. The blue dashed line represents $\eta \left (\Delta \omega \right )$, the red curve represents $\beta _{le}$, and the green straight line represents zero-dispersion frequency. The left side of the green line shows the anomalous GVD region, and the right side shows the normal GVD region. At this moment, the intersection of $\eta \left (\Delta \omega \right )$ and $\beta _{le}$ is equivalent to the solution of Eq. (29), which is the phase matching frequency $\mathrm {\Delta }\omega _r$.

 

Fig. 5. Phase matching condition. The blue dashed line is $\eta \left (\Delta \omega \right )$, which represents the propagation constant of the soliton. The red curve is $\beta _{le}\left (\Delta \omega \right )$, which is the dispersion curve. The solid green line denotes the zero-dispersion frequency. Its left and right sides are the anomalous GVD region and normal GVD region, respectively. The frequency $\mathrm {\Delta }\omega _{new}$ is on the red curve. The intersection of the blue dashed line and the red curve represents the resonance frequency $\mathrm {\Delta }\omega _r$.

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3.5 Transmitted pulse solitons with different frequencies

For our numerical simulations, we can obtain transmitted pulses of different frequencies by varied the incident frequency. If no temporal boundary exists, the transmitted pulses do not exist. If the transmitted pulses can form solitons, the incident frequency shift must fulfill three conditions. First, the incident frequency shift is within the monotonically decreasing range of $\beta _{ge}$. Second, the incident frequency shift does not satisfy the condition of total internal reflection. Third, a transmitted frequency shift occurs in the anomalous GVD region. The variation in the soliton order formed by the transmitted pulses is shown in Fig. 6. It can be seen from the figure that fundamental solitons can form second-order solitons after the temporal boundary.

 

Fig. 6. Relationship between soliton order and frequency shift of the transmitted pulse

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We obtain the resonance frequencies of these solitons through simulations, as indicated by the red line in Fig. 7. To calculate the resonance frequency using analytical expressions, we reorganize the formulas used in Section 3.4. The resonance frequency $\Delta \omega _r$ is as follows:

The relationship between $\Delta \omega _{new}$ and $\Delta \omega _t$ is shown below:

The relationship between $\Delta \omega _t$ and $\Delta \omega _i$ is given in Eq. (10).

 

Fig. 7. Relationship between resonance frequency and transmitted frequency. The red line represents the frequency at which the peak power of dispersive waves is located, and the yellow solid circle (blue open circle) represents the resonance frequency with (without) SPM-induced spectral broadening. The horizontal coordinate is the frequency shift of the transmitted pulse.

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The trend of the above resonance frequency $\Delta \omega _r$ is shown in the yellow solid circle in Fig. 7. For comparison, if the influence of the SPM-induced spectral broadening is ignored in the resonance frequency calculation, the trend is shown by the blue open circle in Fig. 7. In the diagram, the analytical results marked by yellow solid circles are closer to the simulation results marked by a red line. The reason why the red line gradually moves away from the blue open circle is that the normalized third-order dispersion exceeds 0.022 when the transmitted frequency shift is less than -2.43THz [23]. Under a large normalized third-order dispersion, the second-order solitons formed by the transmitted pulses transfer more energy to the normal GVD region. In this case, it is necessary to consider the impact of the SPM-induced spectral broadening.

4. Conclusion

In this study, we investigate the reflection and transmission of fundamental soliton at the temporal boundary under third-order dispersion. We derive the reflection and transmission coefficients and an energy conservation formula that varies with frequency, and we also provide approximate analytical expressions for the reflected and transmitted pulses. Compared to the incident pulse, the reflected and transmitted pulses differ in terms of power and width. In this process, energy is conserved, whereas power is not.

If the incident frequency spectrum is blue-shifted, and the transmitted spectrum lies in the anomalous GVD region, the incident pulse forms two reflected pulses and one transmitted pulse at the temporal boundary. One of the two reflected pulses is in the normal GVD region and the other is in the anomalous GVD region. As the incident spectrum shifts toward the blue side, the power of the transmitted pulse first increases and then decrease. Based on the formation conditions of the soliton, it is found that the transmitted pulse can form second-order soliton. Under the influence of a large normalized third-order dispersion, second-order soliton radiates dispersive wave and the transmitted spectrum changes from a single-peak structure to an asymmetric two-peak structure. We establish a modified phase-matching condition that includes the effect of SPM-induced spectral broadening. We make a prediction for solitons with different transmitted frequencies. Comparing with the simulation results, it is found that the modified phase-matching condition can better predict the resonance frequency.