Ultrashort high-power light sources in the mid-infrared (mid-IR) to terahertz (THz) range not only provide powerful tools for fundamental research, but also are of great importance to numerous applications.1–3 First, their relatively long carrier wavelengths offer a unique advantage in studying strong-field physics in the quasistatic regime,4 generation of high harmonics in gases and solids,5–7 and electron acceleration in conventional waveguides.8 Second, they are of broad interest for ultrafast imaging of molecular structures1,9 and IR spectroscopy10 of biological systems, since the frequencies of the intramolecular and intermolecular vibrations are situated in their spectral range. Motivated by these applications, a great deal of progress has been made in the generation of ultrashort high-power mid-IR or THz pulses in recent decades.1,3,11–19 Nowadays, state-of-the-art lasers can deliver mid-IR pulses with a high power of tens of terawatts over a relatively long duration of a few picoseconds11 or with a relatively low power of the order of watts over a few-cycle duration.1 However, conventional optical components using nonlinear crystals are then normally subjected to optical breakdown damage, and conventional optical amplifiers suffer from limited gain bandwidth. Therefore, it is challenging for current laser technology to generate light pulses in the long-wavelength region beyond 10 μm with simultaneous ultrahigh power and few-cycle duration.

In contrast to nonlinear crystals, plasmas can sustain ultra-intense light and hence have become a unique nonlinear optical medium for the amplification and manipulation of high-power light pulses. In particular, recent numerical and experimental studies have shown that plasmas can also be utilized in the generation of ultrashort high-power mid-IR or THz pulses via so-called photon deceleration in laser-driven plasma wakes.20–23 In this photon deceleration, a plasma wake will be first excited by the ponderomotive force of a short intense laser pulse propagating in an underdense plasma. When the laser pulse is relativistically intense, the plasma wake will become highly nonlinear as a train of bubbles. Further, if the laser pulse is initially short enough or pre-compressed sufficiently,21–23 it will sit at the front of the first wake bubble with a positive plasma density gradient (i.e., a negative refractive-index gradient). Then, the frequency of the laser pulse will be gradually downshifted via self-phase modulation (SPM) by the plasma wake in this scenario.24–26 To enhance the energy conversion efficiency in the photon deceleration, a scheme using two laser pulses has also been proposed in which a signal laser pulse trails an intense driver laser pulse. The signal laser pulse can be frequency-downshifted efficiently when it is located at the front of the second wake bubble that is driven by the driver laser pulse in the plasma.27 To contain the generated light pulses with an increasing wavelength, the plasma density profile can be tailored with a moderately low density at the later stage.28 However, there are still some challenges to be faced in extending the photon deceleration for the generation of ultrashort high-power pulses in the long-wavelength IR or THz region. The main reasons are as follows. First, the photon deceleration to the long-wavelength IR or THz region needs to be rapidly completed in a short distance, since the Rayleigh length, which is inversely proportional to the square of the wavelength, is very limited in this case. Second, the wake bubble should be large enough to contain the generated long-wavelength IR or THz pulse. To some extent, these two requirements are contradictory, i.e., the low plasma density favors enlargement of the wake bubble but reduces the rate and efficiency of the photon deceleration.

In this paper, we propose a fast and efficient photon deceleration scheme for the generation of ultrashort high-power mid-IR or THz pulses by using two laser pulses at different frequencies. Specifically, our scheme utilizes an intense driver laser pulse with a relatively short wavelength to produce a nonlinear plasma wake. A signal laser pulse with a much longer wavelength trails the driver pulse. The time interval between the two pulses can be optimized to enable the signal pulse to be located at the front of the second wake bubble that is driven by the driver pulse, i.e., the signal pulse sits in a region with a positive density (negative refractive-index) gradient. Since the critical plasma density of a light pulse is inversely proportional to the square of its wavelength, the gradient of the refractive index in the plasma wake will become very steep for a signal pulse with a long wavelength, which results in fast photon deceleration of the signal pulse in a short distance of the order of the Rayleigh length. As a result, the initial signal pulse with a relatively long wavelength (∼10 μm) can be efficiently frequency-downshifted into the THz range via such a fast photon deceleration.

A schematic of photon deceleration using two laser pulses at different frequencies is displayed in Fig. 1, where the driver laser pulse (blue curve) has a relatively short initial wavelength λ_d0 and the signal pulse (red curve) with a relatively long initial wavelength λ_s0 is trailing the driver pulse. Owing to its strong ponderomotive force, the driver pulse will excite a plasma wake when it is propagating in an underdense plasma. If the driver pulse is ultrashort and relativistically intense, the plasma wake will enter into the bubble regime. In this scenario, the plasma electron density drops nearly to zero at the center of every bubble. Meanwhile, the electron density has a steep positive gradient at the front of each bubble and a steep negative gradient at the rear.

From the dispersion relation ${\omega }^2(\xi ,\tau )={{\omega }_p}^2(\xi ,\tau )+c^2{k}^2(\xi ,\tau )$, the local phase velocity of the laser pulse can be derived as $v_p/c=1/\sqrt{1-{\omega }_p^2/{\omega }^2}\simeq 1+{\omega }_p^2/2{\omega }^2$, where ω(ξ, τ) is the instantaneous laser frequency defined in the light frame (ξ = x − ct, τ = t), the instantaneous plasma frequency is determined by the density n_e(ξ, τ) as ${\omega }_p(\xi ,\tau )=\sqrt{e^2{n}_e(\xi ,\tau ){ϵ}_0/m_e}$, c is the speed of light in vacuum, and m_e and e are the electron mass and charge, respectively. This analysis reveals that a plasma density gradient will lead to a spatial variation in the local phase velocity of the laser pulse. Consequently, the wavelength (or frequency) of a signal laser pulse can be modulated as the crests (or troughs) of the laser wave move at the different local phase velocities. The rate of change in the laser wavelength can be estimated as dλ/dτ = Δv_p, where Δv_p ≃ λ∂v_p/∂ξ is the difference in phase velocity between two adjacent wave crests. Under the assumption n_e(ξ, τ) ≪ n_sc, the spatial partial derivative of the phase velocity can be approximately written as$$\frac{\partial v_p}{\partial \xi }\simeq \frac{c}{2n_{sc}}\frac{{\lambda }^2}{{\lambda }_{s0}^2}\frac{\partial n_e(\xi ,\tau )}{\partial \xi },$$(1)where λ_s0 = 2πc/ω_s0 is the wavelength of the initial signal pulse, and $n_{sc}=m_e{\omega }_{s0}^2{ϵ}_0/e^2$ is the critical plasma density corresponding to λ_s0 (or ω_s0). Using Eq. (1), the instantaneous laser wavelength due to the photon deceleration in a plasma wake can be estimated as28$$\begin{array}{ll}\hfill \frac{\lambda }{{\lambda }_{s0}}& \simeq {\left(1-\frac{c}{n_{sc}}{\int }_0^T\frac{\partial n_e}{\partial \xi }d\tau \right)}^{-1/2}\hfill \\ \hfill & ={\left(1+2c\eta {\int }_0^T\frac{\partial \eta }{\partial \xi }d\tau \right)}^{-1/2},\hfill \end{array}$$(2)where $\eta =c/v_p=\sqrt{1-n_e/n_{sc}}$ is the refractive index of the signal pulse in the plasma. Equation (2) indicates that the wavelength of a signal laser pulse will be elongated if it sits at the front of the plasma bubble with a positive plasma density gradient as shown in Fig. 2, where $\partial \eta /\partial \xi \simeq -{(2n_{sc})}^{-1}\partial n_e/\partial \xi$. More importantly, the variation in the signal laser wavelength increases with increasing gradient of the refractive index. It is worth pointing out that for a given plasma density profile, a laser pulse with a longer carrier wavelength will feel a steeper gradient of the refractive index. This is because the critical plasma density is inversely proportional to the square of the laser wavelength. In Fig. 2, gradients of refractive indices are compared for signal laser pulses with different carrier wavelengths, with the plasma density profile being taken around the density peak between the first and second wake bubbles excited by the driver pulse. It is clear that a laser pulse with a longer initial carrier wavelength will experience a faster and more efficient photon deceleration, since it is subjected to from a steeper gradient of the refractive index. This is the most important merit of the photon deceleration scheme using two laser pulses at different frequencies, in which the signal laser pulse has a longer initial wavelength.

To display the efficiency of the photon deceleration scheme using two laser pulses at different frequencies, a series of two-dimensional (2D) particle-in-cell (PIC) simulations are conducted using the Osiris code.29 To reduce computational cost, the simulation box is set as a window that contains two laser pulses and moves at the speed of light. The moving window has a size of 200 μm (x) × 100 μm (y), and it is divided into 16 000 × 1600 cells. The plasma is located in the region x ≥ 0 with a uniform electron number density n₀ = 0.0005n_dc, where $n_{dc}=m_e{\omega }_{d0}^2{ϵ}_0/e^2$ is the critical plasma density corresponding to the wavelength λ_d0 of the driver pulse. For the plasma region, 16 macroparticles per cell are allocated. The plasma is initially cold, and the plasma ions are protons with m_p = 1836m_e. The driver and signal laser pulses are successively incident into the plasma from the left side. The driver laser initially has a wavelength λ_d0 = 1 μm, a temporal FWHM duration of 16.5 fs, and a spot size (FWHM in intensity) of 30 μm. To stimulate the plasma wake in the bubble regime, the driver laser is assumed to have a relativistic intensity of normalized amplitude a_d0 = 4. The normalized amplitude a_d0 of the linearly polarized driver laser is related to its intensity I as $I({\mathrm{W}/\mathrm{c}\mathrm{m}}^2)=1.37\times 10^{18}{a}_{d0}^2/{\lambda }_{d0}^2{(\mu \mathrm{m})}^2$. The signal laser initially has a relatively long wavelength λ_s0 ≥ 1 μm, a temporal FWHM duration of 33 fs, and a spot size of 25 μm. For comparison, in the simulations, signal lasers with different initial wavelengths of λ_s0 = 1, 5, and 10 μm and different initial normalized amplitudes of a_s0 = 1, 2, 3, 5, and 8 are employed, where a_s0 is related to the intensity I as $I({\mathrm{W}/\mathrm{c}\mathrm{m}}^2)=1.37\times 10^{18}{a}_{s0}^2/{\lambda }_{s0}^2{(\mu \mathrm{m})}^2$. The signal laser is launched after the driver laser with a time delay of 182 fs, so that the signal laser pulse is mainly located at the front of the second wake bubble driven by the driver pulse.

The photon decelerations of signal laser pulses with the same initial intensity a_s0 = 1 but three different initial carrier wavelengths are compared in Fig. 3. As can be seen in Figs. 3(a)–3(c), the bubble regime of plasma wake generation is achieved with similar plasma density structures, since the same driver laser with a relativistic intensity a_d0 = 4 is used in these three cases. The signal laser pulses are all located at the front of the second wake bubble, and their influences on the plasma wake structures are not obvious, because their intensity a_s0 = 1 is much lower than that of the driver pulse. However, Figs. 3(d)–3(f) illustrate that the effects of the plasma wake upon the signal lasers are clearly different owing to their different initial wavelengths. The longer the initial wavelength of the signal pulse, the steeper will be the gradient of the refractive index to which it is subjected. As a result, the Wigner spectrograms in Figs. 3(g)–3(i) demonstrate that the signal pulse with the longest initial carrier wavelength λ_s0 = 10 μm will be clearly red-shifted, whereas the modulation of the signal pulse with the shortest initial carrier wavelength λ_s0 = 1 μm is almost negligible. It is also indicated by Figs. 3(h) and 3(i) that the photon deceleration is most efficient at the front of the wake bubble, where the gradient of the plasma density is steepest, and the decelerated photons with the longest wavelengths will slip backward to the center of the wake bubble, owing to their lower group velocities. More importantly, we notice that the photon deceleration of the signal pulse with λ_s0 = 10 μm is achieved efficiently within an extremely short propagation distance (∼150 μm). This propagation distance is shorter than the corresponding Rayleigh length Z_R = πw²/λ ≃ 196 μm in this case. Therefore, no plasma channel is required in the photon deceleration scheme using two laser pulses at different frequencies to guide the propagation of laser pulses. By contrast, the photon deceleration of a signal pulse whose wavelength is equal to that of the driver pulse takes a much longer time and is not accomplished by the end of the simulation.

The spectra of the modulated signal pulses with different initial carrier wavelengths λ_s0 = 1, 5 and 10 μm are displayed in Figs. 4(a)–4(c), respectively, in which the initial spectra are also shown for comparison. For the signal pulse with λ_s0 = λ_d0 = 1 μm, the central wavelength is only slightly modulated to ∼1.04 μm at t = 3000T_d0, owing to the slow photon deceleration in this case. The spectrum of the signal pulse with λ_s0 = 5 μm is obviously modulated, and its center wavelength is about 14.5 μm at t = 1500T_d0. For the signal pulse with the longest initial wavelength λ_s0 = 10 μm, the center wavelength can be rapidly modulated to be as long as 45 μm at t = 230T_d0, which is about 4.5 times longer than its initial center wavelength. Furthermore, the duration of the modulated signal pulse is as short as two cycles, as shown by the inset in Fig. 3(f).

It is worth pointing out that the spectra in Fig. 4 are obtained according to the electric field distributions of the signal pulses inside the plasma, which may be perturbed by the wake and bubble formation. Therefore, we have performed an additional PIC simulation using a finite plasma length of 130 μm for the case of λ_s0 = 10 μm. We find that the resulting spectrum of the modulated signal pulse obtained from this PIC simulation is very similar to that obtained from the simulation with a semi-infinite plasma in the long-wavelength regime (λ ≥ 40 μm), which is the case in which we are most interested. This may be because the long-wavelength spectral part of the signal pulse is located mainly at the center of the second wake bubble, where the plasma density is very low $(\sim 0.0002n_{dc})$. Therefore, this part of the signal pulse can exit the plasma and enter the vacuum with no significant change in the spectrum.

The time evolutions of the central wavelengths (yellow curves) of the modulated signal lasers with different initial center wavelengths λ_s0 = 1, 5 and 10 μm are displayed in Figs. 4(d)–4(f), respectively. It can clearly be seen that the central wavelengths of the signal lasers increase at the beginning of the modulation owing to the photon deceleration in all cases. For the signal laser with λ_s0 = 1 μm, the central wavelength continues to increase at a very slow speed at the end of the simulation. The central wavelengths of the signal lasers with λ_s0 = 5 and 10 μm increase rapidly at the beginning, owing to the efficient photon deceleration. Conversely, however, for these signal lasers with λ_s0 = 5 and 10 μm, the central wavelengths will decrease in the later stage. This is because the efficiently modulated signal laser pulses in these cases will slip backward relative to the plasma wake. As long as these signal pulses arrive at the rear of the wake bubble, they will experience photon acceleration due to the negative plasma density gradient there. For the signal pulse with λ_s0 = 10 μm, the center wavelength achieves its maximum λ_s,max ≃ 45 μm at t = 230T_d0 after a fast and efficient photon deceleration. It is also indicated by Fig. 4(f) that the energy ratio of the modulated signal pulse in the long-wavelength regime will increase rapidly at the beginning owing to the efficient photon deceleration and then decrease owing to the reversed photon acceleration in the later stage. It is worth noting that the laser energy depletion is very weak in our simulations, since a relatively low background plasma density is employed.

The effect of the signal laser intensity on the photon deceleration is shown in Fig. 5. With increasing signal laser intensity, the effect of the signal laser pulse on the plasma wake becomes non-negligible. Under the effect of the ponderomotive force of the signal pulse, the density profile at the front of the wake bubble that is initially driven by the driver laser becomes steeper. Therefore, with a moderate increase in the signal laser intensity a_s0, the photon deceleration will become more efficient. As a result, as shown in Fig. 5(a), the central wavelength λ_c and normalized amplitude a_s of the modulated signal laser increase with increasing signal laser intensity for a_s0 ≤ 5. Meanwhile, the energy conversion efficiency ζ decreases with increasing central wavelength λ_c of the modulated signal laser. However, it is important to note that the structure of the wake bubble will be broken by the ponderomotive force of the signal pulse if this is too intense. As confirmed in Fig. 5(b), a signal laser pulse with a_s0 = 8 will destroy the structure of the wake bubble and make the plasma density profile clearly different from that in the case of a_s0 = 1. Consequently, as shown in Fig. 5(a), the central wavelength λ_c and normalized amplitude a_s of the modulated signal laser drop abruptly in the case of a_s0 = 8.

It is worth pointing out that the photon deceleration scheme using two laser pulses at different frequencies relies on the availability of intense mid-IR laser pulses with ultrashort durations ($<$100 fs) as the initial signal laser. However, it is still difficult to deliver ultrashort intense mid-IR laser pulses using conventional laser technology. For instance, the duration of currently available intense CO₂ laser pulses in the 10 μm spectral range is still limited to the order of picoseconds.11,12,17–19 Fortunately, self-compression of intense laser pulses in a plasma may offer an alternative approach to the generation of intense CO₂ laser pulses with a duration shorter than 100 fs.23

The signal laser pulse will inevitably fall into two bubbles if it has a duration longer than the length of the plasma wake bubble. Consequently, a portion of the signal pulse in the region with a positive gradient of the plasma density will be red-shifted, while another portion in the region with a negative gradient of the plasma density will be blue-shifted. The simulation result using a signal pulse with a relatively long duration of 200 fs is displayed in Fig. 6, which clearly shows that the second and third wake bubbles behind the driver pulse have become fully occupied by the signal pulse. Consequently, as illustrated in Fig. 6(b), the signal laser will be red-shifted (respectively blue-shifted) at the front (respectively rear) of a plasma bubble with positive (respectively negative) density gradient.

The above simulation result indicates that the length of the plasma wake bubble sets an upper limit on the duration of the initial signal laser pulse to guarantee efficient photon deceleration. Furthermore, the length of the plasma wake bubble also sets an upper limit on the wavelength of the final modulated signal laser pulse. To contain the modulated signal laser pulse, the wake bubble must be longer than the wavelength of the modulated signal laser. As is well known, the radius of the plasma wake bubble will increase with decreasing background plasma density n₀ for a given driver laser intensity a_d0.30,31 On the other hand, a relatively high-density background plasma is required to sustain a large gradient of plasma density for efficient photon deceleration. Therefore, the rapid and efficient photon deceleration scheme using two-color lasers could be further optimized by using a background plasma with a decreasing density profile. In this case, the photon deceleration can take place more efficiently in a relatively high-density background plasma in the early stage. Also, a relatively low-density background plasma in the later stage will enlarge the plasma wake bubble, which provides the possibility of modulating the signal pulse into the longer-wavelength spectral regime.27,28 A photon deceleration scheme optimized by using a combination of two laser pulses at different frequencies and a density-tailored plasma will be studied in detail in the future.

It is worth pointing out that a slower group velocity of the signal pulse will also set an upper limit on the deceleration length. For an initial signal pulse with a carrier wavelength λ_s0 = 10 μm, the group velocity in a background plasma with n_e = 0.0005n_dc [i.e., n_e = 0.05n_sc(λ_s0)] will be v_g ≃ 0.975c. As the length of the wake bubble exited by the driver pulse is about 50 μm, it will take about 3.3 ps for the signal pulse to slip backward from the bubble front to the bubble center. Correspondingly, the upper limit of the deceleration length is about 1000 μm. However, this estimate only holds for an initial signal pulse in an unperturbed background plasma. On the one hand, a modulated signal pulse with an increasing wavelength will greatly shorten the deceleration length. On the other hand, a low plasma density in the wake bubble will lengthen the deceleration length. Under the assumption that the modulated signal pulse has an instantaneous center wavelength λ_c = 40 μm and the plasma density in the bubble is about n_e ≃ 0.0002n_dc [i.e., n_e = 0.32n_sc(λ_c)], the instantaneous group velocity of the modulated signal pulse in the bubble can be estimated as v_g ≃ 0.825c. Correspondingly, the upper limit of the deceleration length is about 143 μm. If the propagation distance of the signal pulse is longer than this upper limit, the signal pulse will slip backward into the rear part of the wake bubble and then be frequency-upshifted owing to the negative density gradient there. This theoretical analysis is in qualitative agreement with the time evolution of the central wavelength of the modulated signal laser pulse obtained from the PIC simulation as shown in Fig. 4(f).

In conclusion, we have proposed a fast and efficient photon deceleration scheme in a laser-driven plasma wake by using a signal laser pulse with a wavelength longer than that of the driver laser pulse. The longer the wavelength of the signal pulse, the greater is the refractive index gradient to which it will be subjected in the plasma wake. Therefore, a signal laser pulse with a longer initial wavelength will experience a faster and more efficient photon deceleration. PIC simulations demonstrate that a relativistic signal laser pulse initially in the 10 μm spectral range can be efficiently modulated to become a relativistic THz pulse in the 40 μm spectral range. More importantly, this modulation is completed rapidly within a distance ($\sim 150\mu$m) that is shorter than the corresponding Rayleigh length (∼200 μm) of the generated THz pulse, which frees the signal and driver pulses from laser energy depletion and plasma channel guiding. In other words, the scheme using two laser pulses at different frequencies provides a new degree of freedom for photon deceleration in a laser-driven plasma wake, which may greatly extend the spectral range of modulated light pulses.

Acknowledgment. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11975154, 12375236, 12135009, and 12275249) and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA25050100). Simulations were carried out on the Pi supercomputer at Shanghai Jiao Tong University.