2 Mode and scheme

The schematic diagram of the scheme is shown in Figure 1. The projections at the bottom represent the profile of the plasma. In the scheme, the generation of a mid-IR pulse by the NCLP, that is, the high-frequency component first and low-frequency second, can be divided into three stages: (I) the wakefield excitation stage; (II) the electron layer compression stage; and (III) the pulse converter stage. In stage (I), the NCLP excites a bubble to move forward at the group velocity of the drive laser pulse. At the same time, each frequency component of the NCLP produces a special curving profile of the refractive index, as shown by the green curve, and the photons in the range of $\partial \eta /\partial \xi <0$ move backward and others in the range of $\partial \eta /\partial \xi >0$ move forward, as directed by the blue arrows, which together with the plasma etching lead to the rapid compression of the NCLP in the longitudinal direction. Finally, the pulse width becomes shorter and the laser peak intensity increases dramatically after propagation of a few Rayleigh lengths at stage (II), so that the compressed pulse enters the mid-IR converter stage. At this point, stage (III), the photons are rapidly decelerated due to the greater negative refractive index gradient, and then the generated mid-IR pulse slips backward into the bubble and propagates forward together with the bubble.

Figure 1 Schematic of laser chirp controlled few-cycle mid-IR pulse generation. Due to the special curving profile of the refractive index of the NCLP and the plasma etching, the pulse is rapidly compressed longitudinally. As a result, a large number of photons approach the photon deceleration phase, and produce the mid-infrared frequency component, which then slips backwards into the bubble and moves forward together with the bubble. The red curves represent the distribution of the laser electric field on-axis, and the green curves represent the corresponding distribution of the refractive index of the NCLP. The blue arrows denote the photon emission directions relative to the bubble.

Photon deceleration is caused essentially by plasma optical modulation. The basic physics of mid-IR generation in plasma by the NCLP can be interpreted by 1D nonlinear theory^[ Reference Nie, Pai, Hua, Zhang, Wu, Wan, Li, Zhang, Cheng, Su, Liu, Ma, Ning, He, Lu, Chu, Wang, Mori and Joshi ^{29 ]}. The electric field of a CLP with the polarization along the y-direction that propagates along the x-direction can be expressed as follows:

(1) $$ \begin{align}{E}_{\mathrm{L}}(t)={E}_{\mathrm{L}0}\exp \left(-\frac{r^2}{w_0^2}\right){\sin}^2\left(\frac{\pi t}{\tau_{\mathrm{d}}}\right)\sin \left[\omega (t)t\right],\end{align}$$

where ${E}_{\mathrm{L}0}$ is the initial peak electric field, ${w}_0$ is the spot size and ${\tau}_{\mathrm{d}}$ is the full length of the laser pulse. Here, ${\omega (t)={\omega}_0\left[1-b\left(t/{T}_0-{t}_{\mathrm{c}}/{T}_0\right)\right]}$ is the local longitudinal frequency of a linearly CLP, where ${\omega}_0=2\pi c/{\lambda}_0$ is the center angular frequency, ${\lambda}_0$ is the center wavelength of the CLP, $c$ is the speed of light in a vacuum, ${T}_0$ is the laser period, ${t}_{\mathrm{c}}={\tau}_{\mathrm{d}}/2$ is the initial center of the laser pulse and $b$ is the linear chirp parameter. Here, we assume $b>0$ for a PCLP and $b<0$ for an NCLP.

The interaction of relativistic CLPs with underdense plasma can be described by the coupled equations under the Coulomb gauge^[ Reference Zhang, Shen, Ji, Wang, Xu, Yu, Yi, Wang, Hafz and Kulagin ^{40 ,} Reference Afhami and Eslami ^{41 ,} Reference Sprangle, Esarey and Ting ^{48 –} Reference Khachatryan, van Goor and Boller ^{51 ]}:

(2) $$\begin{align}\left(2c\frac{\partial^2a}{\partial \xi \partial \tau }-\frac{\partial^2a}{\partial {\tau}^2}\right)={c}^2{k}_{\mathrm{p}}^2\frac{a}{1+\phi },\end{align}$$

(3) $$\begin{align}2\frac{\partial^2\phi }{\partial {\xi}^2}={k}_{\mathrm{p}}^2\left[\frac{1+{p}_y^2}{{\left(1+\phi \right)}^2}-1\right],\end{align}$$

where ${a}\left(\xi \right)=\left[e{A}\left(\xi \right)\right]/\left({m}_{\mathrm{e}}{c}^2\right)$ and $\phi \left(\xi \right)=\left[e\Phi \left(\xi \right)\right]/\left({m}_{\mathrm{e}}{c}^2\right)$ are the normalized vector and scalar potentials associated with the laser pulse and the wakefield, respectively. Here, $\xi =x- ct$ and $\tau =t$ are in the laser-rest frame, ${k}_{\mathrm{p}}={\omega}_{\mathrm{p}}/c$ is the plasma wave number, ${\omega}_{\mathrm{p}}=\sqrt{\left(4\pi {n}_{\mathrm{e}}{e}^2\right)/{m}_{\mathrm{e}}}$ is the plasma frequency, $c$ is the speed of light in a vacuum, $e$ is the unit charge, ${n}_{\mathrm{e}}$ is the plasma density and ${m}_{\mathrm{e}}$ is the electron mass. Equation (3) can be solved numerically by the fourth-order Runge–Kutta method. In Equation (3), ${p}_y={p}_{y0}+A$ is the transverse momentum component of electrons and $A$ is defined as $A\sim {\int}_{\xi_0}^{\xi }{E}_{\mathrm{L}}\left({\xi}^{\prime}\right)\kern0.1em \mathrm{d}{\xi}^{\prime }$ , where ${\xi_0=\xi \left(\tau =0\right)}$ and ${p}_{y0}$ are the initial position and momentum of the plasma electron, respectively. Considering a cold plasma, ${p}_{y0}=0$ , the transverse momentum completely depends on the laser pulse. Compared with the un-chirped laser pulse, electrons driven by a CLP can still have a nonzero transverse momentum, which is constant behind the pulse.

From Equation (2), we can get the refractive index of a CLP as follows:

(4) $$\begin{align}\eta \left(\tau \right)\simeq 1-\frac{\omega_{\mathrm{p}}^2}{2{\omega}^2\left(\tau \right)}\frac{1}{1+\phi},\quad 0<\tau <{\tau}_{\mathrm{d}}.\end{align}$$

Therefore, the phase velocity and the group velocity can be expressed as ${v}_{\mathrm{p}}\left(\tau \right)\simeq c\left\{1+\left[{\omega}_{\mathrm{p}}^2/2{\omega}^2\left(\tau \right)\right]\left[1/\left(1+\phi \right)\right]\right\}$ and ${v}_{\mathrm{g}}\left(\tau \right)\simeq c\left\{1-\left[{\omega}_{\mathrm{p}}^2/2{\omega}^2\left(\tau \right)\right]\left[1/\left(1+\phi \right)\right]\right\}$ , respectively. Here, $\phi$ is the solution of Equation (3).

The wavelength changes of the drive laser in a short time duration $\Delta \tau$ can be estimated by $\lambda ={\lambda}_0+{\lambda}_0\Delta \tau \partial {v}_{\mathrm{p}}/\partial \xi$ ^[ Reference Mori ^{37 ]}, and ${\lambda}_0\partial {v}_{\mathrm{p}}/\partial \xi$ is the difference of the phase velocity between the two adjacent crests. Then, using ${v}_{\mathrm{p}}=c{\eta}^{-1}$ , the change of $\lambda$ gives the following:

(5) $$\begin{align}\lambda ={\lambda}_0\left(1-c{\int}_0^{t_{\mathrm{i}}}{\eta}^{-2}\frac{\partial \eta }{\partial \xi } \mathrm{d}\tau \right),\end{align}$$

where ${t}_{\mathrm{i}}$ is the interaction time, $\partial \eta /\partial \xi >0$ implies photon acceleration and $\partial \eta /\partial \xi <0$ means photon deceleration.

We assume an average refractive index gradient of $-\partial \eta /\partial \xi \sim 3\times {10}^{-3}$ μm ${}^{-1}$ for the case presented here when ${a}_0>1$ , and thus the wavelength modulation can be estimated by Equation (5), which is as high as $10{\lambda}_0$ over an interaction distance of about 3 mm. Due to the longitudinal compression of the NCLP, the drive pulse can excite a larger negative refractive index gradient.

3 Relativistic mid-infrared pulse generation

For the quantitative description for the generation of mid-IR pulses by NCLP interaction with underdense plasma, we first perform a series of 2D particle-in-cell (PIC) simulations by using the open source PIC code EPOCH^[ Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers ^{52 ]}. The normalized vector potential of the laser pulse is ${a}_0=2$ , the peak intensity is ${I}_0=8.6\times {10}^{18}\;\mathrm{W}\;{\mathrm{cm}}^{-2}$ and the center wavelength of the CLPs is ${\lambda}_0=0.8\;\unicode{x3bc} \mathrm{m}$ , while ${w}_0=15\;\unicode{x3bc} \mathrm{m}$ and ${{\tau}_{\mathrm{d}}=20{T}_0}$ are the spot size and pulse duration, respectively, where ${T}_0\approx 2.64$ fs is the laser period. The effective chirp parameter range of the CLP, that is, $-0.1<b<0.1$ , can be obtained by considering $1-b\left(t/{T}_0-{t}_{\mathrm{c}}/{T}_0\right)>0$ when $0<t<20{T}_0$ and ${t}_{\mathrm{c}}=10{T}_0$ . The incident laser pulse has a spatial-temporal profile as defined by Equation (1) with the chirp parameter $b=-0.07$ , which corresponds to a pulse energy of about 803 mJ, a full width at half maximum (FWHM) of the spectrum of about 500 nm and a chirp rate of $R=W/T\approx 18.9\;\mathrm{nm}/\mathrm{fs}$ ^[ Reference Brodzik ^{53 ]}. Here, $W$ is the FWHM of the spectrum and $T$ is the FWHM of the pulse duration. Such a bandwidth pulse may be achieved by induced-phase modulation (IPM)^[ Reference Su, Fang, Gao, Zhao, Chang and Wei ^{54 ]} with fused silica plates. A plasma channel is used to guide the propagation of focused laser pulses over many Rayleigh lengths^[ Reference Esarey, Schroeder and Leemans ^{55 ]} and to excite a stable bubble structure. The simulation box has a total longitudinal length of $3500\;\unicode{x3bc} \mathrm{m}$ , and the moving window technology is used. The size of the moving window is $80\;\unicode{x3bc} \mathrm{m}$ (x) $\times$ $70\;\unicode{x3bc} \mathrm{m}$ (y) with grid cells of 2400 $\times$ 1400 and 16 macroparticles per cell. The parabolic plasma channel has the transverse density distribution of ${n}_{\mathrm{e}}={n}_0+\Delta {nr}^2/{w}_0^2$ with a $100\;\unicode{x3bc} \mathrm{m}$ up ramp, a $3200\;\unicode{x3bc} \mathrm{m}$ plateau and a $50\;\unicode{x3bc} \mathrm{m}$ down ramp, where ${n}_0=3\times {10}^{18}\;{\mathrm{cm}}^{-3}$ is the background electron density on-axis, $\Delta n=n\left({w}_0\right)-n(0)=1.13\times {10}^{20}/{w}_0^2\left(\unicode{x3bc} \mathrm{m}\right)$ is the channel depth with the matched laser spot size^[ Reference Esarey, Schroeder and Leemans ^{55 ]} and $r$ is the radial distance from the channel axis. This plasma channel can be generated in several ways, which have been widely used in laser–plasma experiments^[ Reference Esarey, Schroeder and Leemans ^{55 –} Reference Mizuta, Hosokai, Masuda, Zhidkov, Makito, Nakanii, Kajino, Nishida, Kando, Mori, Kotaki, Hayashi, Bulanov and Kodama ^{57 ]}.

Figures 2(a)–2(c) show the evolution of the transverse electric field ( ${E}_y$ ) of the drive laser and the electron density distribution ( ${n}_{\mathrm{e}}$ ), together with longitudinal electron density on-axis. In stage (I), one can see from Figure 2(a) that a stable bubble can be excited, and the laser pulse is located in front of the bubble. The length of the bubble corresponds to the plasma wavelength ${\lambda}_{\mathrm{p}}=\left(2\pi c\right)/\sqrt{\left(4\pi {n}_{\mathrm{e}}{e}^2\right)/{m}_{\mathrm{e}}}=19.3\;\unicode{x3bc} \mathrm{m}$ , and the maximum density perturbation is only $6\times {10}^{18}\;{\mathrm{cm}}^{-3}$ . Here, the density up ramp of stage (I) can guide the NCLP propagation into the plasma without causing boundary electron injection, and it has almost no effect on the frequency profile of the laser pulses. After that, in stage (II) of Figure 1, the FWHM of the NCLP rapidly compresses to 9 fs when the propagation distance is 1.8 mm, which is well below the initial pulse width, as shown in Figure 2(b). Meanwhile, a small frequency downshift (about $2\;\unicode{x3bc} \mathrm{m}$ ) can be seen, represented by the black curve in Figure 2(e). With the compression of the drive laser in stage (II), the peak electric field of the NCLP can approach $1.7\times {10}^{13}\;\mathrm{V}\;{\mathrm{m}}^{-1}$ , and the plasma density in front of the bubble is increased in excess of $3\times {10}^{19}\;{\mathrm{cm}}^{-3}$ at $t=2250{T}_0$ . The enhanced density peak provides a large negative refractive index gradient $\partial \eta /\partial \xi$ , as shown in Figure 1, leading to more efficient photon deceleration. For example, at $t=2250{T}_0$ , one can estimate $\partial \eta /\partial \xi \approx -8\times {10}^{-3}\;{\unicode{x3bc} \mathrm{m}}^{-1}$ . Therefore, compared with the un-chirped pulse, the NCLP can rapidly obtain a large refractive index gradient in a shorter plasma by severe compression. After $t=2250{T}_0$ , the pulse enters into stage (III), where the large refractive index gradient leads to a rapid frequency downshift of photons. Assuming the number of photons to be constant, that is, regardless of the energy loss, the frequency downshift of photons will lead to a reduction of the laser pulse energy, which manifests as deceleration. Therefore, the low-frequency mid-IR pulse slips backward, and fills the entire bubble, as shown in Figure 2(c). The on-axis spectral evolution is shown in Figure 2(d), where one can see that the laser experiences a red-shift steadily before the pulse is compressed. Then the photons constantly slow down. As a result, the spectrum shows multiple mid-IR components. Figure 2(e) shows the spectral distributions at different times. A mid-IR pulse is generated with the maximum center wavelength of ${\lambda}_{\mathrm{c}}=7.9\;\unicode{x3bc} \mathrm{m}$ at $t=3600{T}_0$ . Figure 2(f) shows the temporal profile of the mid-IR pulse at $t=3600{T}_0$ . The temporal FWHM of the mid-IR pulse is approximately 56.8 fs and the number of optical cycles is roughly 2.2. Considering the peak electric field of mid-IR pulses with ${E}_{\mathrm{MIR}}\approx 1.1\times {10}^{12}\;\mathrm{V}\;{\mathrm{m}}^{-1}$ , the normalized pulse intensity can be estimated as ${{a}_{\mathrm{MIR}}=\left(e{\lambda}_{\mathrm{c}}{E}_{\mathrm{MIR}}\right)/\left(2\pi {m}_{\mathrm{e}}{c}^2\right)\approx 2.9}$ . Therefore, it is a relativistic, few-cycle and long-wavelength mid-IR pulse.

Figure 2 2D simulation of mid-IR generation with the NCLP. (a)–(c) Distributions of the plasma density ( ${n}_{\mathrm{e}}$ ) and transverse electric field ( ${E}_y$ ) at different times. The orange curve is the electron density on-axis. (d) Spectral evolution as a function of the propagation time. (e) Spectral distribution of the on-axis laser electric field at $t=150{T}_0$ (blue), $2250{T}_0$ (black) and $3600{T}_0$ (red). (f) Temporal profile of the mid-IR electric field at $t=3600{T}_0$ .

The compression of the NCLP enhances the generation efficiency of the mid-IR pulse and shortens the length of the plasma. Therefore, in order to investigate the pulse compression process in detail, we present the refractive index distribution of the drive laser. Figure 3 shows the refractive index and the evolution of pulse length of the NCLP. Meanwhile, the case without chirp is also given for comparison. Since both the ${v}_{\mathrm{p}}$ and ${v}_{\mathrm{g}}$ are a function of laser frequency, the NCLP has different distribution of refractive index $\eta$ , SPM and GVD for different frequency component. Figures 3(a) and 3(b) show the laser electric field and refractive index at $t=150{T}_0$ for the cases of the chirp parameter $b=-0.07$ and un-chirped laser pulses, respectively, where the results marked by the red curve ( ${\eta}_{\mathrm{PIC}}$ ) come from the PIC simulation, and those marked by the black curve ( ${\eta}_{\mathrm{t}}$ ) are the analytical results from Equation (4). For the NCLP, due to the change of frequency component, we have $\partial \eta /\partial \xi >0$ caused by the fall of the refractive index at the low-frequency part of the NCLP, where the fall edge is experiencing a photon acceleration. However, for the un-chirped pulse, the refractive index is close to that at the tail of pulse.

Figure 3 Comparison of refractive index and evolution of the NCLP and un-chirped pulse. Longitudinal distribution of the laser electric field and refractive index in the cases of (a) $b=-0.07$ and (b) an un-chirped laser. (c), (d) The corresponding locations of the rise edge and the fall edge, corresponding to the case with and without chirp, respectively. The insets of (c) and (d) show the evolution of the laser peak electric field ${E}_{\mathrm{p}}$ .

To record the distance between the reference points and the pulse center, the position change of the rise edge and the fall edge in the propagation process is shown in Figures 3(c) and 3(d). Due to the plasma etching in the wakefield, the rise edges of the NCLP and un-chirped pulse both compress toward the center of the pulse, as shown by the red dots in Figures 3(c) and 3(d). However, the fall edge of the NCLP also compresses toward the center due to the positive refractive index gradient (photon acceleration). Therefore, at the earlier stage of interaction, the longitudinal bunching effect of the NCLP results from the photon acceleration and plasma etching. This compression on both edges results in a dramatic increase in pulse intensity and the efficient generation of a mid-IR pulse. The insets of Figures 3(c) and 3(d) show the evolution of the peak laser electric field. One can see that the intensity of the NCLP is significantly enhanced, approaching its maximum value of about $1.8\times {10}^{13}\;\mathrm{V}\;{\mathrm{m}}^{-1}$ ( $9.2\times {10}^{12}\;\mathrm{V}\;{\mathrm{m}}^{-1}$ for the case of the un-chirped laser) at $t=2000{T}_0$ , which verifies the feasibility of the chirped pulse compression scheme in Figure 1. After that, the electric field intensity gradually decreases due to the generation of mid-IR frequency components and the defocusing effect of the drive laser.

The generated mid-IR pulse by photon deceleration is a few-cycle pulse, and its CEP is very important for subsequent applications, since the CEP can directly affect the process of laser–matter interaction. Due to the difference between the phase velocity and the group velocity of waves of different frequency, there must be a phase slip between the drive NCLP ${\mathrm{CEP}}_0$ and the generated mid-IR pulse ${\mathrm{CEP}}_{\mathrm{MIR}}$ . When the CEP is dephased by $2\pi$ , the propagation length of pulse is ${L}_{2\pi}\simeq \left({n}_{\mathrm{c}}/{n}_{\mathrm{e}}\right){\lambda}_0$ ^[ Reference Huijts, Andriyash, Rovige, Vernier and Faure ^{58 ]}. Therefore, the CEP shift of the NCLP can be expressed as follows:

(6) $$\begin{align} \begin{array}{l@{}l}{\Delta}_{\mathrm{CEP}}\,& ={\mathrm{CEP}}_0-{\mathrm{CEP}}_{\mathrm{MIR}}\\ {}& \approx 2\pi \times \left(\frac{n_{\mathrm{e}}l}{n_{\mathrm{c}}{\lambda}_0}-\lfloor\frac{n_{\mathrm{e}}l}{n_{\mathrm{c}}{\lambda}_0}\rfloor\right)\end{array},\end{align}$$

where $\lfloor \rfloor$ is the least integer function and $l$ is the length of the mid-IR pulse generation time. The phase of the generated mid-IR pulse should be consistent with that of the drive laser at the mid-IR pulse generation time. Then, the mid-IR pulse slips rapidly backwards into the bubble, where there are almost no electrons, that is, ${\omega}_{\mathrm{p}}\approx 0$ , ${v}_{\mathrm{p}}\approx {v}_{\mathrm{g}}\approx c$ . Therefore, the ${\Delta}_{\mathrm{CEP}}$ between the drive laser and the mid-IR pulse is fixed at the generation time. To illustrate this point, the dependence of the ${\mathrm{CEP}}_{\mathrm{MIR}}$ at $t=3600{T}_0$ on the ${\mathrm{CEP}}_0$ is shown in Figure 4. One can see that the phase difference between ${\mathrm{CEP}}_0$ and ${\mathrm{CEP}}_{\mathrm{MIR}}$ is fixed for the fixed plasma density and length, and there is a nearly linear relationship of the phase difference ${\Delta}_{\mathrm{CEP}}$ . The mid-IR pulse peak electric field changes with the variation of the initial $\mathrm{{CEP}}_0$ , and the direction of peak electric field changes periodically, that is, a linear change from $-\pi$ to $0$ with ${\mathrm{CEP}}_0$ changing from $0$ to $\pi$ , as shown in Figure 4(a). The inset of Figure 4(a) provides the temporal variation of the mid-IR electric field with different ${\mathrm{CEP}}_0$ ( $0$ , $\pi /2$ and $\pi$ ). One can be seen from Figure 4(b) that the ${\mathrm{CEP}}_{\mathrm{MIR}}$ is locked on the ${\mathrm{CEP}}_0$ , and the difference of ${\mathrm{CEP}}_{\mathrm{MIR}}$ and ${\mathrm{CEP}}_0$ is about 0.6 $\pi$ at $t=3600{T}_0$ , which is in excellent agreement with the theoretical calculation ${\Delta}_{\mathrm{CEP}}\approx 0.6\;\pi$ .

Figure 4 The evolution of the CEP at $t=3600{T}_0$ . (a) The generated mid-IR electric field as a function of the initial drive pulse phase $\mathrm{CEP}_0$ . The inset shows the electric field waveform for different $\mathrm{CEP}_0$ of the initial drive pulse ( $0$ , blue dashed; $\pi /2$ , black solid; $\pi$ , red dot dash). (b) The phase evolution of the mid-IR electric field with $\mathrm{CEP}_0$ variation.