1. Introduction

All-optical control of light group velocity is not only an interesting topic in optical physics but also a key enabling technology for high-speed photonics applications such as optical telecommunication and signal processing [1–4]. The construction of data buffers, logic switches, and tunable signal delays would all benefit from such group-velocity control with a broad response bandwidth, a large tuning range, and an ability to both delay and advance optical pulses [5–8]. However, there are few methods that can satisfy all these requirements simultaneously [9–18]. The well-studied methods, such as electromagnetically induced transparency or coherent population oscillation, are only suitable for controlling the group delay of nanosecond optical pulses because of their very narrow bandwidth of dispersion resonance [13,14]. In contrast, nonlinear optics usually possess a broader bandwidth and thus may be used to control the group delay of femtosecond pulses [15–18]. One example is a nonlinear optical fiber where the refraction index resonance, resulting from the gain of stimulated Raman scattering, leads to a fast change in the group velocity, demonstrating a group delay close to the pulse duration for 430-fs lasers [15]. Quadratic nonlinear processes can also be applied for ultrafast group-velocity control. In the femtosecond second-harmonic generation, the fundamental and harmonic waves in the soliton regime are mutually dragged to overcome the effect of group-velocity mismatch (GVM) and thus can be delayed and advanced, respectively, demonstrating a delay of 50 fs for 40-fs lasers [16]. Recently, we demonstrated ultrafast group-velocity control based on resonant cascaded nonlinearity (RCN) with optical parametric amplification (OPA) in a chirped quasi-phase-matching (QPM) chip [17]. This method based on RCN can not only delay the femtosecond signal by about one pulse duration but also present high fidelity with unchanged pulse shape and duration [ Fig. 1(a)]. Nevertheless, these known nonlinear methods cannot tune group delay in a range much larger than one pulse duration or flexibly switch between delay and advance.

 

Fig. 1. Tandem RCN configuration for ultrafast group-velocity control. (a) Single-stage RCN configuration, where the signal is delayed by τ₀ with respect to its linear propagation time (without pumping). (b) N-stages tandem RCN configuration for extending signal group delay to Nτ₀. (c) N-stages tandem RCN configuration for switching signal delay to advance. Different to the configuration (b), there are two additional frequency-conversion (FC) stages before and after the OPA control stages, which do not produce delay themselves. FC₁ converts the signal to an idler for seeding the OPA control stages, while FC₂ converts the advanced idler back to a signal. As a result, the signal can achieve a group advance of Nτ₀.

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Herein we propose a tandem RCN configuration with multiple OPA stages that can produce large-magnitude and sign-controllable group delays. It consists of several identical chirped-QPM-based OPA stages, in which each OPA stage works in the small-signal amplification regime to ensure a linear superposition of group delay [Fig. 1(b)]. The tuning range of delay in a single-stage OPA can be increased to three-fold pulse duration by optimizing the pump intensity and the QPM chirp rate. With a tandem configuration of two such OPA stages, we numerically demonstrate a group delay of six-fold pulse duration for 100-fs pulses at 1550 nm. We also simulate switching the signal pulse from delay to advance using two OPA stages in a tandem RCN configuration seeded by an idler generated beforehand. As shown in Fig. 1(c), two additional frequency-conversion (FC) stages are inserted to generate the idler before the OPA control stages and for converting back to the signal after the OPA control stages. The two FC stages have the same chip design as the OPA control stages, but with much lower pump intensities. The magnitude of delay or advance for the signal can be further enhanced by applying more OPA control stages. Within the large tuning range, the signal delay can be finely controlled by the pump intensity.

2. Single-stage optimization for tandem RCN configuration

The basic component of the proposed tandem RCN configuration is a chirped-QPM-based OPA that has a large GVM between the signal and idler pulses. The chirped QPM with a large GVM produces a resonant broadband gain [Fig. 2(a)] and a linearly varied spectral phase [Fig. 2(b)], which underlie the ultrafast group-velocity control. To obtain a maximum delay in the tandem RCN configuration, the optimization of the single-stage OPA is needed because it may reduce the number of OPA stages required for a long delay. In this section, we optimize the single-stage RCN design and link its tuning range to the pump intensity and the chip parameters.

 

Fig. 2. Optimization of single-stage RCN for femtosecond signal with a bandwidth of 4.4 THz in a chirped QPM PPLN chip. (a) Signal gain evolution within the chip. The upper panel shows three frequency components at Δν=4.5 THz (blue), 0 (black), and −4.5 THz (red). The lower panel displays the two-dimensional map of the gain as a function of the signal frequency and propagation length, and the pink curve gives the gain spectrum at the output. (b) Evolution of nonlinear spectral phase within the chip. Simulation parameters: λ_p=1064 nm, λ_s=1550 nm, τ₀=100 fs, GVM_si=−96 fs/mm, I_p=0.27 GW/cm², L = 100 mm, and κ₀=16 cm⁻². (c) The group delay Δτ_s/τ₀ (red) and pulse fidelity τ_s/τ₀ (blue) versus the chirp rate κ at I_p=0.27 GW/cm². τ₀, τ_s, and Δτ_s represent the seeding signal duration, output signal duration, and the group delay of signal, respectively. (d) The Δτ_s/τ₀ (red) and τ_s/τ₀ (blue) versus pump intensity I_p at κ₀=16 cm⁻².

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Consider a chirped QPM chip with a linearly varied domain period Λ_g(z) along the longitudinal dimension z, which contributes a z-dependent wave-vector K_g(z)=K_g(z₀)+κ(z‒z₀), where κ is the chirp rate [19]. A narrowband pump pulse is assumed. By neglecting the second- and higher-order dispersions, the phase-mismatch among the interacting waves can be written as,

To verify the above analytical results, we ran numerical simulations using nonlinear coupled-wave equations [17,23] by considering the dispersion terms up to the third order. The femtosecond signal at 1550 nm was seeded to the OPA pumped by a 1064-nm narrowband laser with an intensity of 0.27 GW/cm². The chirped QPM chip used was a 100-mm-long chirped periodically-poled LiNbO₃ (CPPLN) chip with a chirp rate of κ₀=16 cm⁻². All the pump, signal and idler are extraordinary in polarization as in the traditional QPM design, which maximizes their nonlinear interactions. This QPM chip allows the broadband amplification of the Gaussian signal pulse with a bandwidth of ∼4.4 THz and a pulse duration of τ₀=100 fs. It has a negative GVM_si of −96 fs/mm [24], so the signal pulse is delayed. As concluded in our previous study [17], the signal pulse can achieve sufficient delay only in the case of the small-signal amplification regime, so the signal intensities in the following simulations were always set to satisfy this condition. Figure 2 summarizes the simulation results of the signal group-velocity control. In Figs. 2(c) and 2(d), the signal delay was measured by the folds of seeding duration, that is, Δτ_s/τ₀, which is also known as the fractional group delay. In addition, we also defined a pulse fidelity parameter τ_s/τ₀ for characterizing the output pulse duration deviated from the input value, which ideally should be equal to 1 and practically can be acceptable up to 2. The degradation of pulse fidelity was mainly caused by the high-order dispersion and the gain narrowing effect. To improve the pulse fidelity, the second-order nonlinear spectral phase induced in the OPA should be compensated by optimizing the compressor, and the gain narrowing effect should be minimized by compensating the phase mismatch of higher-order dispersions with a CPPLN chip of a nonlinearly varied domain period Λ_g(z).

The simulation proved the analytical results of Eq. (5). The red curve in Fig. 2(c) shows the rapid increase in fractional delay with a decrease in the chirp rate κ in an exponential manner, which is consistent with the result of Δτ_s∝1/κ² given in Eq. (5). This shows that reducing the chirp rate κ is effective in increasing the group delay. However, in the case of a fixed chip length, the decrease in κ will also reduce the phase-matching bandwidth, so it may distort the signal pulse and hence degrade the pulse fidelity, as shown in the blue curve in Fig. 2(c). To ensure both a large delay and high fidelity, we set κ = 16 cm⁻² in the following simulations. Under this condition, we calculated the fractional delay and pulse fidelity as a function of pump intensity, I_p. The red curve in Fig. 2(d) shows a nearly linear dependence of the fractional delay on the I_p: a higher I_p leads to a larger delay. As I_p∝Γ², the result in Fig. 2(d) is also in good agreement with Eq. (5). However, the increase in I_p may induce a gain-narrowing effect, which will degrade fidelity. Therefore, we adopted an I_p=0.81 GW/cm² to balance the delay and fidelity. With these parameters, the fraction delay can reach 3 while the attained pulse fidelity of 1.3 is acceptable. The three-fold fraction delay obtained by optimization exceeds our previous result [17] by about three times.

3. Tandem RCN configuration

3.1 Extending group delay via tandem configuration

Optimization of the single-stage RCN configuration produces a three-fold fractional delay for the femtosecond signal at 1550 nm. Further enhancement of the group delay is required, but it is difficult to achieve with only a single-stage RCN. To overcome this problem, we proposed a tandem RCN configuration and expected to achieve 3N-fold fractional delay by using a tandem of N identical OPA stages. In this section we describe how we determined whether and how the group delays from multiple OPA stages can be superposed linearly, and evaluated the pulse fidelity.

A tandem configuration with two OPA stages was first studied and evaluated, as shown in Fig. 3(a). Both stages use the same CPPLN chip of L = 100 mm and κ₀=16 cm⁻², and pump of λ_p=1064 nm and I_p=0.81 GW/cm². To ensure a good temporal overlap between the pump and signal pulses across the length of the chip, we adopted a super-Gaussian pump pulse of a duration of 100 ps and stretched the 1550-nm, 100-fs Gaussian signal to a chirped pulse of a duration of 30 ps. After passing through the OPA stages, the chirped signal pulse was compressed by compensating both the imposed chirp and linear dispersions in multiple OPA chips. This enabled the signal pulse to recover its initial duration perfectly after passing through the multiple OPA stages without the pump. Once the OPAs were activated, the first-order nonlinear spectral phase induced by the RCN imprinted on the signal [Eq. (4)] and led to a group delay [Eq. (5)] compared to the situation without a pump. The signal delay produced could be dynamically controlled by the pump intensity.

 

Fig. 3. The tandem configuration with two OPA stages. (a) The signal under control propagates through the two OPA stages successively with an attenuator after each OPA stage for restoring the signal intensity. The pump, signal and idler pulses are all polarized along the optical axis of CPPLN. The signal spectrum (red) and nonlinear spectral phase (blue) produced (b) in the first OPA, (c) at the output of the second OPA without attenuation, (d) at the output of the second OPA with attenuation, the output signal pulses (red) and the signal pulse without OPA pump (black dashed line) (e) for case (b), (f) for case (c), and (g) for case (d). The blue dashed curve in (g) represents the output signal pulse after compensating the second-order nonlinear spectral phase. Simulation parameters: I_p0=0.81 GW/cm², I_s0/I_p0=10⁻³⁰, L = 100 mm and κ₀=16 cm⁻².

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Figures 3(b)–3(g) summarize the simulation results for the two-stage tandem configuration. After the first OPA working under small-signal amplification, a first-order nonlinear spectral phase was superposed on the signal because of the GVM-induced RCN [blue line in Fig. 3(b)], and it caused a time delay of ∼300 fs, corresponding to a fraction delay of three for the 100-fs signal [Fig. 3(e)]. The pulse fidelity was kept at a tolerable level of 1.3, as shown in Fig. 3(e). The amplified signal from the first stage was then seeded to the second stage directly without attenuation. Owing to the strong seeding, the second OPA reached amplification saturation, which generated a negligible nonlinear spectral phase [blue curve in Fig. 3(c)] and nearly no delay [Fig. 3(f)]. Furthermore, both the spectral and the temporal profiles of the signal from the saturated second OPA deteriorated. These results agreed with our previous studies [17] and show the need for small-signal amplification to manipulate group velocity efficiently. To retain the small-signal amplification for the second OPA, the signal from the first stage was attenuated by a factor equals to the first OPA gain, such that both OPA stages operated under the same conditions. In this way, the second OPA added an identical nonlinear spectral phase to the signal as that in the first OPA [blue curve in Fig. 3(d)], which doubled the signal delay to approximately 600 fs [Fig. 3(g)]. The attained pulse fidelity of 1.6 after the second OPA was still considered acceptable. The pulse fidelity was further improved to 1.38 [blue dashed curve in Fig. 3(g)] once we compensated the second-order nonlinear spectral phase induced in two OPA stages by optimizing the compressor. Some spectrum distortions of the signal from both OPA stages [red curves in Figs. 3(b) and 3(d)] are an expected result of the adiabatic amplification in the CPPLN chips [25]. These results verify that the delays produced by a tandem nonlinear RCN can be superposed linearly if each OPA works in the small-signal amplification regime.

The two-stage tandem can be extended to an N-stage tandem owing to the linear superposition nature of RCN. The N-stage tandem configuration can expand the maximum fractional delay to 3N. The stage number N will be limited by the pulse fidelity. Because each OPA stage has an identical design, it is straightforward to add or remove OPA stages according to the practical requirements. Such an N-stage tandem configuration removes the delay limitation of a single stage, and opens up the opportunity for ultrafast group-velocity control with a long delay.

3.2 Group advance via tandem configuration

In the previous section, we extended the tuning range of the group delay via a tandem RCN configuration. It is of interest to determine whether such a tandem configuration can support a group advance. Generally, group delays and advances require different physical conditions, so they are difficult to achieve simultaneously without modifying the control parameters (e.g., chip structure and/or pump conditions). Some key physical conditions must be altered to switch the delay to advance. For example, in resonant media, signal delay and advance can be observed under gain and absorption resonances, respectively [4]. In resonant nonlinearity conditions, such as stimulated Brillouin scattering, the signal delay and advance occur in the Stokes and Anti-Stokes bands, respectively [8]. In our method, the sign of GVM_si must be changed to produce a signal delay and advance according to Eq. (5). However, it is impossible to change the sign of GVM_si in a single-stage RCN with the same chip and seeding conditions. In contrast, the tandem RCN configuration with multiple OPA stages provides more design freedom. The signal pulse at 1550 nm can be advanced if the OPAs are seeded by the idler at 3.4 µm. These idler pulses at 3.4 µm, as byproducts of OPA, are blocked directly from producing a group delay in the previous case, as shown in Fig. 3. Here, we show that these idler pulses in tandem configuration can be adopted as a control degree of freedom to switch the delay to advance for the 1550-nm signal pulses under control.

As indicated by Eq. (5), the signal pulse is delayed under the condition of signal seeding in a CPPLN OPA with GVM_si<0, but the idler pulse is advanced under the condition of idler seeding. To achieve signal advance, we produced an advanced idler first in a normal tandem configuration and then converted the advanced idler into an advanced signal with an additional device. Two FC stages were added before and after the OPA stages, which converted the 1550-nm signal under control to a 3.4-µm idler, and then converted it back to the 1550-nm signal. Both FC stages adopted the same chip design as in the OPA stages to ensure FC bandwidth, but pump intensities were kept low so that the nonlinear phase and group advance were negligible in the FC stages. The large size of the 3.4-µm idler group advance produced in the tandem RCN configuration, was completely transferred to the 1550-nm signal under control in the final FC stage.

We verified this proposal by simulating a tandem configuration, as shown in Fig. 4(a). It included two FC stages with I_p=0.11 MW/cm² and two OPA stages with I_p=0.81 GW/cm². As expected, the first FC stage converted the 1550-nm signal under control to the 3.4-µm idler and imposed a negligible nonlinear spectral phase on the idler [Fig. 4(b)]. Both the following two OPA stages worked in the small-signal amplification regime and imprinted equal first-order nonlinear spectral phases on the seeded idler pulses successively [Figs. 4(c) and 4(d)]. A group advance of 300 fs was imposed on the 3.4-µm idler in the first OPA, which was then doubled to 600 fs at the output of the second OPA [Fig. 4(f)]. After the second FC stage, the nonlinear spectral phase on the idler was transferred to the signal under control [Fig. 4(e)]. This spectral phase had an inverse slope to the delay case in Fig. 3(d), so it rendered an advance for the signal under control. Although four nonlinear stages were adopted in the tandem configuration, only the middle two OPA stages contributed to the group advance. Therefore, the attainable maximum group advance was approximately 600 fs, corresponding to a six-fold fractional advance for the signal under control [Fig. 4(g)]. The pulse fidelity was as high as 1.38 after compensating the second-order nonlinear spectral phase [blue dashed curve in Fig. 4(g)]. With the help of the two FC stages, the tandem configuration of the N-stage OPA can produce a 3N fractional advance. The only difference between the tandem configurations used for group delay and advance is the seeding to the N-stage OPA, which suggests that we can flexibly switch the delay to advance for the pulse under control by altering the seeding at the OPA stages.

 

Fig. 4. (a) A two-OPA tandem configuration for signal group advance using an idler seeding design. Two FC stages are added before and after the two OPA stages. All the interacting optical pulses are extraordinary in polarization. The attenuator after every OPA stage is used to restore the idler intensity. Spectrum (red) and nonlinear spectral phase (blue) (b) of idler produced in the first FC stage with I_p=0.11 MW/cm², (c) of idler produced in the first OPA stage with I_p=0.81 GW/cm², (d) of idler at the output of the second OPA stage with I_p=0.81 GW/cm², (e) of signal at the output of the second FC stage with I_p=0.11 MW/cm². (f) Output idler pulses after the first (blue) and second (red) OPA stages, respectively. (g) Output signal pulse (red) after the second FC stage. The blue dashed curve in (g) represents the output signal pulse after compensating the second-order nonlinear spectral phase. The black dashed curves in (f) and (g) represent the signal pulse without OPA pump. Other parameters are the same as those in Fig. 3, except for the pump intensities in FC stages.

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Before the end of this section, we pointed out that the signal pulse achieved a large gain (the gain of a single OPA) besides the large delay or advance, as shown in Figs. 3 and 4. In each OPA stage for group delay, the seed signal was attenuated such that the OPA is in the small-signal amplification regime. In the group-advance mode with additional FC stages, the seed was the idler produced in the previous OPA, and the final output signal in the final FC stage was approximately similar to or slightly larger than the output signal in the group-delay mode. Nevertheless, the output signal after the tandem configuration was much higher than the seed, with a magnification comparable to the single-stage OPA gain. Such a signal gain benefits the applications even in the presence of losses by stretching and compression.

4. Discussion and conclusion

We have demonstrated some of the major advantages of the tandem RCN configuration. First, it can render both group delay and advance for the signal pulse under control without modifying the chip dispersion structure and control parameters. Second, the tuning range of the group delay and advance can be significantly enhanced by increasing the number of OPA stages N. The stage number N is ultimately limited by the pulse fidelity degradation in the nonlinear process, and N = 5 can be well supported in our simulation conditions if the fidelity is restricted within 2. Third, the tuning range of the group delay and advance can be dynamically controlled by the pump intensity. Finally, the gain-resonance can ensure a high signal transmission even in the presence of loss factors. To the best of our knowledge, these performances provided by tandem RCN represent the state of the art in ultrafast group-velocity control.

Our numerical simulations were limited to 1 + 1 dimensions of the propagation (z) and time (t), which could be extended to higher dimensions including the transverse variables (x, y) as done in our previous work [22]. As an essential procedure for promoting this method toward practical application, we discuss the potential failure mode caused by the transverse effect of the pump beam. As shown by Eq. (5) and Fig. 2(d), the signal delay or advance was determined only by the pump intensity, but independent on the signal intensity itself because the OPA stages operated in the small-signal amplification regime. Thus, the proposed method is quite versatile and applicable to a Gaussian signal and also other kinds of signal beams. On the other hand, the pump beam profile should be uniform ideally. Pump beam nonuniformity will change the group-velocity control deviated from its expectation, and will also degrade the pulse fidelity. If a ±10% variation of the group delay is acceptable, the pump beam nonuniformity must be better than ±10%, as indicated by Fig. 2(d). Above discussions suggest that uniform pump beam is ideal but a Gaussian pump beam is still acceptable provided the pump beam size is much larger than the signal beam.

In conclusion, we have demonstrated a tandem RCN configuration for large-magnitude and sign-controllable ultrafast group-velocity control. Such a tandem configuration integrates multiple identical OPA stages, in which each OPA stage has been optimized according to a theoretical analysis of RCN, and the tuning range of the group delay increases proportionally with the number of stages. Both group delay and group advance for the signal under control can be produced by this tandem configuration. The switch from delay to advance only depends on the OPA seeding wavelength and does not need to change the chip dispersion structure or the control parameters. Our numerical simulations verified the tandem RCN configuration and demonstrated a large tuning range over six-fold pulse duration for both group delay and advance. The results presented in this paper show the potential and prospect of tandem RCN configuration in ultrafast group-velocity control, and may find potential applications in fields such as future high-capacity optical communications.