Since the generation of high power femtosecond laser pulses, numerous investigations have been devoted to the study of the interaction of high-intensity lasers with plasma. One application of the laser–plasma interaction is the generation of the high brightness and the high-energy electron beams in which electrons with energy up to 4.2 GeV, 6-pC charge, and 0.3-mrad divergence have been produced from a 9-cm-long capillary discharge waveguide.^[1–12] The propagation of an intense laser pulse in an under-dense plasma induces a plasma wake that is suitable for the acceleration of the electrons to relativistic energies.^[13–16] The electrons with sufficient initial kinetic energy can ride the plasma wake, staying in phase with the longitudinal electric field inside the wake and gaining energy. Because of the small structure of the wake field, i.e., its micron-scale longitudinal and transverse size, the bunches of a duration less than one plasma period have to be injected into the laser wake field in order to get high-quality electron bunches. However, the effective acceleration mechanism occurs when the ambient plasma electrons contribute to the injection process (self-injection). The self-injection process is explained by the two physical mechanisms: wave-breaking (the longitudinal self-injection),^[17,18] and the bubble regime (transverse self-injection).^[19–21] The bubble regime is a multidimensional effect, where the symmetric transverse ponderomotive force of the laser expels the electrons radially from the propagation axis and creates a spherical region (bubble) devoid of the plasma electrons behind the laser. This bubble moves with a velocity equal to the group velocity of the laser pulse in the plasma. An intense charge separation electric field inside the moving bubble can capture the electrons at the base of the bubble and accelerate them with a narrow energy spread.The quality of the accelerated electrons is mainly determined by the evolution of the bubble structure. Recently, there have been several theoretical^[22,23] and experimental^[24] investigations to study the influence of the bubble evolution on the wake field structure and hence on the quality of the electron beam.It has been shown that in the fully relativistic regime in which the normalized vector potential of laser a ₀ = eE ₀/m _e cω ₀ approaches 1 (where E ₀ and ω₀ are the peak electric field and the frequency of the laser), the interaction of the laser with the under-dense plasma can lead to several nonlinear processes such as laser self-steepening.^[25] In laser self-steepening, the peak region of the pulse with high-intensity moves faster than those with low intensity at the pulse head. This leads to optical shock formation and then a nonlinear wake is excited. As a result,the plasma wavelength elongates and uninterrupted self-injection follows.^[26–28] The laser pulse is depleted due to the excitation of a plasma wave. For a situation where the laser is ultra-intense (a ₀ ≫ 1), the pump deletion occurs at a narrow region in front of the pulse corresponding to a few cycles of the laser front.^[29] A consequence of the localized pulse depletion is induction of the transverse oscillations in the bubble sheath leading to multi-injection, a large energy spread and collective transverse oscillation of the electron beams.^[30] Therefore, it is important to control these instabilities to achieve high-quality electron beams. In this work the interaction of a chirped laser pulse with the under-dense plasma is studied via two-dimensional (2D) particle-in-cell (PIC) simulations. The aim of this work is to propose a new approach to suppress the induced transverse oscillations in the bubble structure at ultra-relativistic regimes (a ₀ ≈ 10) using the frequency-chirped laser pulse. Based on this technique, a considerable difference in pulse depletion process is observed for positively and negatively chirped laser pulses. This difference can be used to tune the bubble structure and hence, the quality of the injected bunch in the nonlinear bubble regime.

The interaction of the short-pulse and the ultra-high intense lasers (I > 10¹⁹ W/cm²) with the undersense plasma presents different features compared to moderately high intensities (I < 10¹⁸ W/cm²). At moderately high intensity, the nonlinear instabilities like relativistic self-focusing,[31] relativistic self-phase modulation,[32] Raman backward and forward scattering,[33] and envelope self-modulation[34] cause the laser pulse to be absorbed and scattered in a distance of the order of a Rayleigh length. At ultra-high intensities, due to relativistic effects,[35–38] the growth rate for the above-mentioned instabilities decreases. This leads to the depletion of only the leading edge of the pulse and the main body of the pulse undergoes little modification. Therefore, the laser pulse eventually evolves to a state in which the leading edge becomes a step function. The phase and the group velocity of the laser in a dispersive medium such as plasma are not equal. Therefore, propagation of a laser pulse through plasma induces a phase shift between the carrier wave and the envelope known as the carrier–envelope phase (CEP). The time variation of the electric field for extremely short laser pulses of few-oscillations depends on the carrier–envelope phase. The localized depletion of a laser pulse at ultra-high intensities causes the varying electric field of the laser at the front of the depleted pulse to act like a few-cycle pulse[39,40] in which the asymmetric transverse ponderomotive force leads to the occurrence of the transverse oscillations in the bubble structure.[41] To stabilize the bubble shape, the depletion of the pulse leading edge must be mitigated in order to avoid the effect of the carrier phase shift on the transverse ponderomotive force. Since the depletion velocity of the pulse, i.e., ,^[29] depends on the laser frequency, i.e., ω(ξ_depl), it is possible to postpone (speed up) the pulse depletion rate by locating the higher frequency (lower frequency) at the front of the pulse. Here, ξ_depl is the position of the localized depletion within the laser in dimensionless delayed time ξ, coordinates and ω _p is the plasma frequency. By this technique, we would be able to significantly suppress/enhance the transverse cavity oscillations when the pulse propagates through the plasma. In order to demonstrate the proposed scheme, the 2D PIC simulations have been carried out by EPOCH^[42] code to show the effect of pulse chirping on the evolution of the bubble structure. In these simulations, the total simulation box is 200 µm (x) × 70 µm (y) which corresponds to a window with 2000 × 900 cells and 6 × 10⁶ particles. The under-dense hydrogen plasma (T _e = 0) is initially located in the region of 30 µm < x < 175 µm, –30 µm < y < 30 µm, with the density of n_e = 0.02n_c, where n_c = 1.1 × 10²¹ cm^–3 is the critical density for the laser pulse with a wave length of λ_Laser = 1 µm. The electric field of the chirped laser pulse that propagates along the x axis can be expressed as[43]

Here the electric field amplitude of the laser pulse is defined as E ₀ which is a constant value. The local frequency of the linearly chirped pulse, the longitudinal laser length, and the laser beam waist are denoted by ω (ξ) = (1 – bξ), σ _x and σ _y respectively. Here, b > 0 (b < 0) denotes the dimensionless chirp parameter for the positively (negatively) chirped pulse and ξ = [(x – ct) ω0/c] is the dimensionless delayed time in which c, m _e, ω0 and e are the speed of light in vacuum, the plasma electron mass, the local frequency in the center of the pulse, and the absolute charge of the plasma electron, respectively. Also, t and x represent the time and spatial coordinates, respectively. In this work the laser beam waist, the longitudinal laser length, and the normalized incident laser amplitude are set equal to σ_y = 8 µm, σ_x = 10 µm, and a₀ = eE ₀/m _e cω₀ = 10 respectively.

The bubble structure at the three different instances during the propagation of the laser pulse through the under-dense plasma are shown in Fig. 1. Here the unchirped laser pulse with a normalized amplitude of a ₀ = 10, a longitudinal size of σ _x = 10 µm, and a spot size of σ _y = 8 enters the simulation box from the left boundary.

Fig. 1.

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	Fig. 1. (color online) The evolution of THE bubble structure when the un-chirped laser pulse propagates through the under-dense plasma (n e = 0.02n c) at the times of: (a) t = 300 fs, (b) t = 500 fs, and (c) t = 600 fs. The laser pulse with the normalized amplitude of a 0 = 10, the longitudinal size of σ x = 10 µm, and the spot size of σ y = 8 µm enters the simulation box from the left boundary.

As is clear from Fig. 1(a), at the initial stage of the bubble regime, i.e., at t = 300 fs, the symmetric bubble structure is established whereas, at the later times, the instabilities appear in the surrounding sheath of the bubble as seen in Figs. 1(b) and 1(c). When the ultra-intense laser pulse interacts with the plasma, a density spike is produced by the ponderomotive force of the front of the laser pulse and subsequently a large plasma wake is excited behind the laser. The longitudinal electric field, the plasma density at the front of the pulse, and the transverse electric field of laser are plotted in Figs. 2(a), 2(b), and Fig. 3, respectively, for t = 300 fs, t = 500 fs, and t = 600 fs.

As shown in Figs. 2(a) and 2(b), the amplitude of wake and also the density spike continuously enhances during the time. As a result, the laser energy absorption by the electrons in the spike or pulse depletion will increase too. Viewed from a photon description, the photons localized within the density spike lose energy as they excite the wake and therefore their wavelength increases leading to frequency red-shifting at the front of the pulse (Fig. 3).

Fig. 2.

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	Fig. 2. (color online) (a) The excited wake field behind the un-chirped pulse, (b) The excited density spike at the pulse head of the un-chirped laser pulse when it propagates through the under-dense plasma (n e = 0.02n c) at the times of t = 300 fs, t = 500 fs, and t = 600 fs.

Fig. 3.

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	Fig. 3. (color online) The evolution of the laser’s electric field of the unchirped laser pulse during its propagation through the under-dense plasma at the times of t = 300 fs, t = 500 fs, and t = 600 fs.

In this situation the front of the pulse is approximately converted to a one-cycle profile. Once the leading edge of the pulse erodes, the bubble structure becomes susceptible to transverse instabilities. In fact, considering the plasma as a dispersive medium, due to the induced phase shift between the carrier waves and the pulse envelope, the role of the time average of the laser field at the front of the depleted pulse becomes dominant on the intensity envelope of the pulse in laser–plasma interactions. Therefore, changing the field direction at the pulse front, as it is depicted by the red and blue arrows in Fig. 3, causes the electrons to be asymmetrically expelled from the laser propagation axis leading to transverse oscillations in the bubble structure. In order to control the evolution of the bubble structure we benefit from the negatively chirped laser pulse to reduce the pulse depletion rate at the pulse’s leading edge. The interaction of a negatively chirped laser pulse with an under-dense plasma (n _e = 0.02n _c) is compared with a positively chirped one in Fig. 4.

Fig. 4.

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Fig. 4. (color online) The evolution of bubble structure when the chirped laser pulse propagates through the under-dense plasma (n e = 0.02n c). The evolution of the bubble in the case of the negatively chirped pulse with the chirped parameter of b = –0.08 at the times of: (a1) t = 300 fs, (a2) t = 500 fs, and (a3) t = 600 fs. The evolution of the bubble in the case of the positively chirped pulse with the chirped parameter of b = 0.08 at the times of: (b1) t = 300 fs, (b2) t = 500 fs, and (b3) t = 600 fs. The other laser parameters are the same as the un-chirped case.

As can be seen for the negatively chirped pulse the symmetric structure of the moving bubble remains stable with time (Figs. 4(a1)–4(a3)), whereas as Figures 4(b1)– 4(b3) show the cavity oscillations in the surrounding sheath of the bubble for the positively chirped pulse occur even sooner than the bubble for the un-chirped pulse. The density spike at the pulse head and the longitudinal electric field for both the negatively and the positively chirped pulse are depicted in Fig. 5 and Fig. 6, respectively for t = 300 fs, t = 500 fs, and t = 600 fs.

Fig. 5.

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	Fig. 5. (color online) The electron density at the pulse head of: (a) the negatively chirped pulse, (b) the positively chirped pulse, when propagating through the under-dense plasma (n e = 0.02n c) at the times of t = 300 fs, t = 500 fs, and t = 600 fs.

In fact, with the higher frequencies located in front of the pulse, the plasma transparency increases at the leading part of the pulse that results in the excitation of a small density spike (Fig. 5(a)) and also a weaker wake field compared with the un-chirped pulse (Fig. 6(a)), whereas the low frequency cycles at the front of the positively chirped pulse initiate the largest density spike (Fig. 5(b)) and the strongest wake field behind the laser (Fig. 6(b)).

Fig. 6.

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	Fig. 6. (color online) The excited wake field behind the chirped laser pulse propagating through the under-dense plasma: (a) the negatively chirped laser pulse, (b) the positively chirped laser pulse at the times of t = 300 fs, t = 500 fs, and t = 600 fs.

Fig. 7.

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	Fig. 7. (color online) The evolution of the laser’s electric field: (a) The negatively chirped pulse, (b) The positively chirped pulse, during its propagation through the under-dense plasma at the times of t = 300 fs, t = 500 fs, and t = 600 fs.

As a result, the pulse depletion rate at the leading edge of the negatively chirped (positively chirped) pulse decreases (increases) and thus, the carrier phase shift becomes negligible (significant) in the leading edge of the pulse as depicted in Fig. 7(a) (Fig. 7(b)). Therefore, the ponderomotive force of laser keeps (breaks down) its symmetric properties in transverse direction during the pulse propagation.

The longitudinal electric field together with the plasma density are shown in Figs. 8(a)–8(c) for negatively chirped pulse at times t = 410, 510, and 570 fs. The electrostatic fields are denoted by the blue curves and the density distributions of injected electrons are specified by the black curves. Also, the energy spectrum of the injected electrons at the corresponding time is plotted in Fig. 8(d). Figures 9(a)–9(d) are the same as Figs. 8(a)–8(d), but for positively chirped pulse. In these figures the electron-accelerating phase buckets (where E _x < 0) and the electron-deceleration buckets (where E _x > 0) of the wake wave are crosshatched by the green lines and the red lines, respectively. It is seen from Fig. 8(a) that the electrons are trapped in plasma wake field, as pointed out by the blue arrow. These trapped electrons stay in the accelerating phase of the field as can be seen in Figs. 8(b) and 8(c). This leads to continuous acceleration of electrons (Fig. 8(d)) to higher energies.

Fig. 8.

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Fig. 8. (color online) The superposition of one-dimensional (1D) diagrams of the wake field structure and plasma density during the propagation of the negatively chirped pulse in plasma at the times of: (a) t = 410 fs, (b) t = 510 fs, and (c) t = 570 fs. The blue curves show the electrostatic field of wake field and the black curves specify the density distribution of injected electrons. Also the green region determines the acceleration phase of the wake field. (d) The energy spectrum of injected electrons at the corresponding times of: t = 410 fs, t = 510 fs, and t = 570 fs.

Fig. 9.

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Fig. 9. (color online) The superposition of 1D diagrams of the wake field structure and the plasma density during the propagation of the positively chirped pulse in plasma at the times of: (a) t = 400 fs, (b) t = 510 fs, and (c) t = 570 fs. The blue curves show the electrostatic field of the wake field and the black curves specify the density distribution of the injected electrons. Also the green and red regions determine the acceleration phase and deceleration phase of the wake field, respectively. (d) The energy spectrum of the injected electrons at the corresponding times of: t = 410 fs, t = 510 fs, and t = 570 fs.

However, as can be clearly seen in Figs. 9(a)–9(d), for the positively chirped pulse, due to the evolution of the bubble base and the loading of a large number of electrons in the moving bubble, the accelerating region of the wake field turns into a deceleration region frequently and the injected electrons sometimes enter the deceleration region and lose energy.

By comparing the energy spectrum of the electrons for the negatively and the positively chirped pulse (Figs. 8(d) and 9(d)), we can see that higher-energy electrons are obtained from the former and a higher number of electrons is obtained from the latter due to the multiple self-injection.

In the nonlinear bubble regime, the transverse oscillations of the cavity structure as a result of the depletion of the pulse leading edge can be controlled by applying the chirped laser pulses. The simulation results show that by locating the higher frequencies in front of the pulse, the pulse depletion process is postponed, thus the cavity structure stays stable for a longer time and a single electron bunch with a higher energy can be generated. In the case in which the lower frequencies are located at the front of the pulse, the pulse depletion speeds up, thus the transverse instability of the bubble appears in an earlier stage of laser plasma interaction. The unstable bubble results in the accelerating phase of the wake field being turned into the decelerating phase, frequently. As a consequence the energy loss of the injected electrons causes the generation of the electron bunches with lower mean energy. Therefore, the utilizations of the chirped laser pulse provides a method to control the quality of the generated bunch in the nonlinear bubble regime.