1. Introduction

The phenomena of reflection and refraction of spatial beam at a dielectric interface is described by the Snell’s law and Fresnel formulae, which have been known for centuries and are important topics in physics textbooks [1,2]. Owing to space-time duality, it is possible to find the time analogy of the spatial beam behavior and vice versa. In recent years, it is demonstrated that an optical pulse encounters a temporal boundary where the refractive index changes suddenly in time [3–14]. A temporal boundary is treated as an interface in time and leads to two regions with different refractive indices [5]. This configuration can be regarded as a temporal analog of the reflection and refraction of a spatial beam. The temporal boundary can be generated by cross-phase modulation of co/counter-propagating pulse [15–18]. The input pulse exhibits an effective wavelength conversion after reflecting from the boundary [19,20] and may represent a strong light–light interaction [21,22]. The light behavior at the temporal boundary, and in time-varying media in general, can be exploited to study many fascinating fundamental phenomena in the time domain, such as optical nonreciprocity [23,24], negative refraction [25], photonic topological insulators [26], photonic time crystals [27], temporal boundary solitons [28] and time reversal [29]; it can also be applied in different exciting areas such as temporal waveguiding [30,31], frequency conversion [32], all-optical signal processing [33], and reconfigurable photonics [34,35].

More recently, the temporal analogs of the laws of reflection and refraction have been derived as optical pulse colliding with a moving step refractive-index boundary inside a dispersion medium [5,36]. The changes of pulse frequency correspond to the angles of reflection and refraction in space domain. Therefore, the expressions of the relation among the reflected frequency, transmitted frequency, and input frequency are derived as the law of temporal reflection and refraction. The conditions for temporal total internal reflection (TIR) are also determined. A comb-like reflected spectrum and an Airy-like transmitted spectrum are formed as an Airy pulse crosses a temporal boundary, and the energy of each lobe of the comb-like reflected spectra can be controlled by the decay factor of Airy pulse [37]. Double moving temporal boundaries can be used to create a temporal waveguide to confine an optical pulse to a narrow temporal region [30,31]. In a practically relevant case, cross-phase modulation in pump-probe pulse configuration can be used to create such a moving temporal boundary. When a dispersive medium exhibits Kerr nonlinearity, the refractive index is higher in the temporal region inside the pulse, and we can view the edge of the strong pump pulse as a moving refractive-index boundary. As a probe pulse crosses the temporal boundary created by either the leading or trailing edge of the pump (the soliton), it undergoes reflection and refraction [38].

Pulses emitted from laser sources are often chirped, and the chirp can also be externally imposed. Chirped pulses are particularly useful in pulse compression or amplification [39,40], supercontinuum generation [41], and filamentation [42]. However, to the best of our knowledge, only the propagation phenomena of unchirped pulses at temporal boundary have been examined to date. In this paper, we study propagation dynamics of initial chirped pulses at temporal boundary to reveal the different phenomena resulting from the effect of chirp parameter.

2. Propagation model

In our study, we considered a chirped Gaussian pulse crossing a moving step refractive-index boundary moving inside a medium with a dispersion relation given by $\beta \left ( \omega \right ) = n\left ( \omega \right )\omega /c$, where $n\left ( \omega \right )$ is the refractive index at frequency $\omega$ and $c$ is the speed of light. For a temporal boundary moving at speed ${\upsilon _B}$, we considered a reference frame moving at the same speed as the boundary. Thus, the temporal boundary was stationary in this moving reference frame. We use the coordinate transform $t = T - z/{\upsilon _B}$, where $T$ is the time in the laboratory frame. Then, the dispersion relation in the moving frame is given as follows [43]

Using Maxwell’s equations together with the dispersion relation in Eq. (1) and making the slowly varying envelope approximation, we obtain the following time-domain equation [5]

In the moving frame, the momentum of a photon is conserved while crossing the temporal boundary, thus

In the case of a linearly chirped Gaussian pulse, the incident field can be expressed as

Solving Eq. (3), we obtain the analytic expressions for the reflected and transmitted frequencies, as follows [44]

Notably, we modeled pulse propagation using the split-step Fourier method, which is based on numerical solutions of Eq. (3).

3. Numerical results

For our numerical simulations, we consider a pulse propagating inside a dispersive waveguide (such as an optical fiber). The pulse was in the normal dispersion region where ${\beta _2 = 0.005ps^2/m}$ and the pulse width was set to ${T_0} = 1ps$ [5,45]. As shown in Fig. 1, the temporal boundary was located at ${t_B} = 5ps$ with the change in the propagation constant ${\beta _B} = 0.5{m^{ - 1}}$. And we choose $\Delta {\beta _1} = 0.1ps/m$, $z = 200m$. We normalize the parameters so ${\delta _2} = 1$, $\Delta {\delta _1} = 20$, ${\tau _B} = 5$, ${\delta _B} = 100$ and $Z = 1$.

 

Fig. 1. (a) Temporal and (b) spectral shape of the chirped Gaussian pulse after collision with a moving step refractive-index boundary. Panels (c, e, g) and (d, f, h) are the simulated temporal and spectral evolutions of the chirped Gaussian pulse with a temporal boundary located a ${\tau _B} = 5$, and ${\delta _B} = 100$.

Download Full Size | PDF

Figures 1(a) and (b) display the shape of the output pulse and spectra at propagation distance $Z = 1$ when the chirp values are 0, −5, and 5. As shown in Fig. 1(a), the incident chirped pulse is split into two parts: a reflected pulse and a transmitted pulse. The transmitted pulse width is wider than that of the initial unchirped pulse. The transmitted pulse width with a positive initial chirp is wider than that with a negative initial chirp. The change in the reflected pulse is similar to that in the transmitted pulse. In Fig. 1(b), the spectra of the incident pulse are also split into two parts: reflected spectra and transmitted spectra. The initial unchirped reflected spectra and the initial unchirped transmitted spectra retained a Gaussian shape. For $\left | C \right | = 5$, the transmitted spectra retain a Gaussian shape but the reflected spectra are distorted. The spectra of the transmitted pulse with a positive initial chirp are the same as those of the transmitted pulse with a negative initial chirp. As shown in Fig. 1(b), the curves overlap. Whether $C$ is positive or negative, the reflection spectra with $\left | C \right |=5$ are the same. Compared with the initial case of $C=0$, the reflection spectra of the initial chirped pulse had more energy. The effect of the initial chirp parameters on the energy distribution and spectra shape in Section 3.1.

Figures 1(c, e, g) show the temporal evolution and are analogous to the reflection and refraction of the optical beam at the spatial boundary. Figures 1(d, f, h) show that the spectrum shifts to the red side and splits into two different spectral bands that travel at different speeds. These spectral shifts occur because a temporal boundary breaks translational symmetry in time [5,36]. As a result, the photon momentum (or $\beta '$) conserved whereas photon energy (or $\Delta \omega$) may change. These features are analogous to the occurrence at a spatial interface, where a beam crosses a boundary between two optical media and its propagation direction changes, as described by the usual Snell’s law. Hence, a change in the angle at a spatial interface translates into a change in the frequency of incident light when reflection and refraction occur at a temporal interface. In Figs. 1(e, f), the initial chirp parameter is 0, which is the same as in Ref. [5]. The same temporal analogy exists for the reflection and refraction of light. A chirped pulse may broaden or compress, depending on whether ${\delta _2}$ and $C$ have the same or opposite signs. For ${\delta _2}C > 0$, the chirped pulse broadens monotonically at a rate faster than that for the unchirped pulse, as shown in Fig. 1(g). For ${\delta _2}C < 0$, the pulse width initially decreases, then becomes a minimum depending on the values of $C$, and then broadens, as shown in Fig. 1(c). When the leading edge of the pulse reaches the temporal boundary, it begins to split until the trailing edge of the pulse leaves the boundary. Therefore, we defined the splitting distance ${Z_s}$ as the distance from the point at which the leading edge of the pulse reaches the temporal boundary to the point at which the trailing edge of the pulse reaches the temporal boundary. ${Z_s}$ is marked in Fig. 1(e). The effect of the initial chirp parameters on the pulse splitting distance is discussed in Section 3.3.

3.1 Effect of the initial chirp parameter on the output spectra and energy

In this section, we discuss the effect of the initial chirp on the output spectral shape and energy distribution at the temporal boundary. According to Ref. [44], we can obtain the moduli of reflection coefficient $\left | R \right |$ and transmission coefficient $\left | T \right |$, and the conservation formula of energy integral after the incident pulse crosses the temporal boundary as follows:

Figure 2(a) shows the frequency dependence of the reflected and transmitted energy coefficients according to Eqs. (7), (8), and (9). The parameters in Eqs. (7), (8), and (9) are identical to those in Fig. 1. The most striking feature in Fig. 2(a) occurs near ${\Delta v_c} = \left ( {{{\left ( {2{\delta _2}{\delta _B}} \right )}^{1/2}} - \Delta {\delta _1}} \right )/\left ( {2\pi {\delta _2}} \right ) = - 0.932$, where the coefficient of the transmitted energy is 0 according to Eq. (6). When $\Delta v < \Delta v_c$, the energy of incident pulse will not be transferred to the transmitted pulse. Therefore, the pulse in this part will experience TIR at the temporal boundary. When $\Delta v > {\Delta v_c}$, with the increasing $\Delta v$, the transmitted energy coefficient increases and the reflected energy coefficient decreases. Figure 2(a) also shows the spectrum of the incident pulse based on the Fourier transform of Eq. (4). Noticeably, the width of the spectrum increases with increasing chirp.

 

Fig. 2. (a) Frequency dependence of the reflection and transmission energy coefficients for $\Delta {\delta _1} = 20$, ${\delta _2} = 1$, and ${\delta _B} = 100$. And the spectrum of the input chirped Gaussian pulse under different initial chirp parameters. (b) The influence of the initial chirp parameters on the energy distribution of reflected and transmitted pulse.

Download Full Size | PDF

In Fig. 2(b), the energy value is normalized by the total energy of the input spectra. As shown in Fig. 2(b), when $C > 0$, the energy of the reflected pulse increases with the increasing the chirp and the energy of the transmitted pulse decreases with the increasing the chirp. When $C < 0$ (not shown), the energy of the reflected pulse increases with an increasing $\left | C \right |$ and the energy of the transmitted pulse decreases with an increasing $\left | C \right |$. When $\left | C \right |$ of two incident pulses are equal, the energy values of the reflected pulses generated by them are equal, and the energy values of the transmitted pulses generated by them are equal. As shown in Fig. 2(a), ${\left | T \right |^2}/S$ and ${\left | R \right |^2}$ are nearly constants when $\Delta v > -0.5$. However, in the range of $\Delta v < -0.5$, the reflected and transmitted energy coefficients change sharply. With an increase in the chirp, the bandwidth of the pulse widens. The reflected energy coefficient under the newly contained bandwidth will be larger than that under the original bandwidth. Therefore, the incident pulse allocates more energy to the reflected pulse in this new bandwidth. However, when the value of $C$ increases from 0 to 2, the incident pulse has little energy in the part of $\Delta v < - 0.5$. Therefore, the reflected energy will increase slightly. When the value of $C$ increases to 5, the pulse contains more energy in the part of $\Delta v < - 0.5$. The most energy at $\Delta v < - 0.5$ is allocated to the reflected pulse. As a result, the reflected energy will increase sharply when $C>2$. However, it is impossible to transfer all incident energy to the reflected pulse by adjusting the values of the chirp because the pulse always split to produce a transmitted pulse when $\Delta v > 0$. The TIR of the entire chirped pulse is discussed in Section 3.2.

Figure 3 shows the simulation of the output spectra at $Z=1.2$. All of the parameters are set to the same as Fig. 1, but we increased the number of chirp values. Figures 3(a) shows the reflected spectra and Figs. 3(b) shows the transmitted spectra. We only show the output spectral shape under the positive chirp parameters. The final output spectra are the same when the values of $\left | C \right |$ are equal. As shown Figs. 3(a), when $C \le 2$, the reflected spectra retain the Gaussian shape of the input pulse. However, when $C>2$, there is a raised packet near ${\Delta v_0} = - 5.4$. As the chirp increases, this packet will gradually become larger. The maximum power of the reflected spectra will eventually appear near ${\Delta v_0}$ when $C \ge 4$. However, there is no packet in the transmitted spectra, as shown in Figs. 3(b). This phenomenon is attributed to the energy coefficients and energy values being different for different frequency components. For ${\Delta v_0}$, we can obtain the frequency of the input pulse ${\Delta v_1} = - 0.917$ according to Eq. (5). The reflection energy coefficient of $\Delta v_1$ tends to be $1$. This implies that the incident pulse around this frequency component allocates most of the energy to the reflected pulse. As shown in Fig. 2(a), with an increase in chirp, the energy of the incident pulse increases at $\Delta v_n < {\Delta v_1}$. Therefore, the energy of the reflected spectra will increase near $\Delta v_0$. However, the energy of the incident pulse is very small at the part of $\Delta v < \Delta v_1$ when $C < 2$. Therefore, the reflected spectra will not appear as a raised packet near $\Delta v_0$ when $C<2$. As the energy of the incident spectra at the part of $\Delta v < {\Delta v_1}$ increases further, the distortion of the reflected spectra becomes more serious. Therefore, the maximum power of the reflected spectra appears near $\Delta v_0$ when $C \ge 4$, as shown in Fig. 3(a). However, the incident pulse contains much more energy at $\Delta v > - 0.5$ than it does at $\Delta v < - 0.5$. As the energy coefficient in Fig. 2(a) shows, most of the energy of the incident pulse at $\Delta v > - 0.5$ will be allocated to the transmitted spectra and the lower energy of the incident pulse at the part of $\Delta v < - 0.5$ will be allocated to the transmitted spectra. According to Eq. (6), the energy of the incident pulse is allocated to the left edge of the transmitted spectra when $\Delta v < -0.5$, and the energy of the incident pulse is allocated to the other part of the transmitted spectra when $\Delta v > -0.5$. This results in significantly less energy being allocated to the left edge of the transmitted spectra than to other parts of the transmitted spectra. Therefore, the distortion on the left edge of the transmitted spectra is not marked, even if the transmitted energy coefficient changes significantly. The transmitted energy coefficient closes to a constant when $\Delta v > - 0.5$. Therefore, the transmitted spectra will remain as a Gaussian shape as shown in Figs. 3(b).

 

Fig. 3. The output spectral profile at the distance of 1.2 with different chirp parameters. (a) is reflected spectral profile and (b) is transmitted spectral profile.

Download Full Size | PDF

3.2 Total inter reflection of the chirped Gaussian pulse

In the previous part, we discussed that the chirped Gaussian pulse will allocate more energy to the reflected pulse with the increasing the chirp. However, the TIR of the entire incident pulse does not occur. According to Eq. (6), we obtain the condition of the temproal TIR as

Figure 4 shows the propagation of a chirped Gaussian pulse under the different ${\delta _B}$ values when $C = 3$. In Figs. 4(a, b), for a small value of ${\delta _B}$, most of the energy from the incident pulse allocates to the transmitted pulse. With an increasing value of ${\delta _B}$, the energy allocated to the transmitted pulse decreases gradually. As shows in Figs. 4(g, h), when ${\delta _B} = 400$, the transmitted pulse disappeares, and all the energy is allocated to the reflected pulse. Therefore, the chirped Gaussian pulse undergoes a TIR at the temporal boundary. This condition is similar to a Gaussian pulse causing TIR at temporal boundary. According to Eq. (10), when ${\delta _B} = 200$, the center frequency of the pulse causes a TIR at the temporal boundary. However, as shows in Figs. 4(c, d), the entire pulse does not undergo TIR. However, the width of the transmitted spectra decreases compared that in Fig. 4(b). This is because the condition of TIR depends on the the incident pulse frequency. Therefore, only frequency components smaller than the center frequency will undergo TIR. The incident spectra which frequency larger than the center frequency also splits into reflected and transmitted spectra bands. In Ref. [5], the Gaussian pulse undergoes TIR at the temporal boundary when ${\delta _B} = 280$. As shows in Figs. 4(e, f), the transmitted pulse still exists when ${\delta _B} = 300$, even though its energy is very small. As shown in Fig. 2(a), when the chirp value is $3$, the spectrum of the pulse ranges of $-1.132\sim 1.132$. According to Eq. (6), we obtain the critical point ${\Delta v_c} = 0.716$ when the ${\delta _B} = 300$. Therefore, the incident pulse at the frequency component ($\Delta v > \Delta v_c$) crosses the temporal boundary, and the incident pulse at the frequency component ($\Delta v < \Delta v_c$) undergoes TIR. This results in a narrower transmitted spectrum than that shown in Fig. 4(d), as shown in Figs. 4(e, f). When ${\delta _B} = 400$, we obtain the critical point ${\Delta v_c} = 1.319$. Thus, TIR will occur at each frequency component of the incident pulse, as shown in Fig. 4(g, h).

 

Fig. 4. (a, c, e, g) Temporal and (b, d, f, h) sepectral evolution of the chirped Gaussian pulse for different ${\delta _B}$ vaules.

Download Full Size | PDF

3.3 Effect of the initial chirp parameters on pulse splitting distance

In this section, we will study the effect of the initial chirp parameters on the splitting distance of the incident pulse at the temporal boundary. Pulse splitting occurs from the distance at which the leading edge of the pulse reaches the temporal boundary to the distance at which the trailing edge of the pulse reaches the temporal boundary. To obtain the relation between the splitting distance and the chirp parameter, we ignore the range $\tau > {\tau _B}$. Then the Eq. (2) becomes

According to the Eq. (14), we can theoretically predict the relation of the width $\tau _1$ and initial width $\tau _0$ as the pulse travels through the distance $Z$ [43]:

According to the Eq. (16), the relationship betwen splitting distance and the chirp values was obtianed. And it is shown in Fig. 5. The parameters are the same as those in Fig. 1; however, we increase the number of chirp values. When $C>0$, the splitting distance increases as the value of $C$ increases. Because according to Eq. (17), when ${\delta _2}C > 0$, the value of $C$ is larger, the speed of pulse broadening increases. Therefore, for larger values of the chirp, the leading edge of the pulse reaches the boundary at a smaller $Z$, and the trailing edge of the pulse reaches the boundary at a larger $Z$. Hence, the splitting distance will continually grow. However, this situation changes when $C<0$. According to Eq. (17), the pulse was compressed and then expanded when ${\delta _2}C < 0$. The pulse compression distance increases with an increasing $C$ when $C<-1$. Therefore, as the value of $\left | C \right |$ increases, the splitting distance decreases and then increases. The minimum value is approximately 0.051, which appears when $C=-4$.

 

Fig. 5. The relationship between the values of the chirp parameter C and pulse splitting distance $Z_s$.

Download Full Size | PDF

3.4 Effect of anomalous dispersion

Figure 6 shows the temporal and spectral evolutions of the chirped Gaussian pulse in the normal dispersion regime (a, b, c, d, e, and f) and anomalous dispersion regime (g, h, i, j, k, and l). The temporal evolution, as shown in Figs. 6(g, i, k) also hit the temporal boundary. However, compared with the normal dispersion regime, as shown in Figs. 6(a, c, e), the transmitted pulse does not "bend" toward the incident pulse and its speed decreases. This phenomenon is strikingly analogous to that of an optical beam transmitted from an optically thicker medium to an optically denser medium in spatial. The reason for the difference in speed is apparent from the spectral evolution in Figs. 6(h, j, l), which shows that the reflected spectra shift toward the blue side and the transmitted spectra shift toward the red side because of the GVD. This means that the speed of reflected spectra increases, and the speed of transmitted spectra decreases. In the normal dispersion regime, the spectrum shifts toward the red side and the speed increases, as shown in Figs. 6(b, d, f). The frequency shift can be determined by Eq. (3). This reason for the above phenomenon is independence of the initial chirp of the pulse. This difference is related to the region of dispersion. However, the splitting distance changes because the symbol of the ${\delta _2}C$ will change. Therefore, the splitting distance will decrease when $C<0$ and the splitting distance will first decrease and then increase when $C>0$.

 

Fig. 6. (a, c, e, g, i, k) Temporal and (b, d, f, h, j, l) spectral evolution of the chirped Gaussian pulse for different initial chirp parameters. (a-f) in the normal dispersion regime and (g-l) in the anomalous dispersion regime.

Download Full Size | PDF

4. Conclusion

In this study, we investigated the reflection and refraction of a chirped Gaussian pulse crossing the temporal boundary inside a dispersion medium. It was discovered that the dynamics of a chirped Gaussian pulse could be controlled during reflection and refraction. We have investigated the impact of the chirp on the pulse width of reflected pulse and transmitted pulse. The transmitted pulse width becomes wider than that of the initial unchirped pulse. The transmitted pulse width with a positive initial chirp becomes wider than that with a negative initial chirp. The change in the reflected pulse is similar to that in the transmitted pulse. We found that the initial chirp parameter can affect the splitting distance $Z_s$ at the temporal boundary. When $C>0$, $Z_s$ becomes increasingly larger with increasing $C$. But when $C<0$, as the value of $\left | C \right |$ increases, the splitting distance decreases and then increases. Thus, we can control the magnitude of $Z_s$ by adjusting the chirp value. We also show that, for a highly chirped pulse, the reflected spectra are distorted. Compared with the unchirped pulse, the chirped reflected spectra have more energy and the chirped transmitted spectra have less energy. By adjusting the magnitude of $C$, the energy of the reflected and refracted pulses can be controlled. In addition, we discussed the influence of initial chirp on the conditions of TIR. For an initial chirped pulse, a larger ${\delta _B}$ is needed to make the whole pulse experience TIR at the temporal boundary.