1. Introduction

Ultrashort circularly polarized (CP) or elliptically polarized (EP) radiation in the extreme ultraviolet and soft X-ray region has proven to be an unique and powerful tool for investigating ultrafast dynamics of chiral systems and magnetic materials [1–6]. It enables studies of fundamental process such as photoemission and magnetization in a completely new regime. Not surprisingly, there has been a great interest among the community to realize polarization control of such radiation sources. Over the past few years, significant progress has been made, from the free-electron-laser-based large-scale facilities [7–12], to the breakthroughs of gas-harmonic-based table-top sources [13–20].

Recently, an alternative approach to generate photon-rich EP harmonics at laboratory scale has been proposed [21, 22], promising to fill the gap between gas harmonics and large-scale facilities. It is based on high harmonic generation (HHG) from relativistic laser driven plasma mirrors [23–34], which can be interpreted as a result of Doppler up-shifting of the laser light when being reflected. Using an EP laser pulse obliquely incident onto a plasma surface, EP harmonics in the form of a train of attosecond bursts in the time domain can be obtained when the laser-plasma parameters are suitable. To better serve ultrafast measurements, the next step will be to select an isolated single attosecond pulse from the pulse train to achieve unprecedented time resolution. This can be done by angularly separating the attosecond pulses into different propagation directions via rotating laser wavefronts, known as the attosecond lighthouse technique [35, 36]. Limited by the rotation speed, the achieved applicable laser pulse duration is typically less than 10 fs. More naturally, isolated attosecond pulse can be achieved by shortening the driving pulse to a few cycles [37–39], e.g., reducing pulse duration to less than 5 fs [21]. However, the required few-cycle laser with relativistically ultrahigh intensity is still technically challenging to produce. Therefore, using existing high-peak-power multi-cycle lasers to achieve isolation of a single attosecond pulse is of great importance.

For normal incidence interactions, the relativistic oscillation of plasma mirrors is driven under the action of the laser ponderomotive force F_p and the charge-separation electrostatic force. Thanks to the strong laser-ellipticity dependence of F_p [40], the concept of polarization gating, initially proposed for laser-driven-gas HHG [41], can also be applied to HHG from relativistic plasmas [42, 43]. Indeed, for a linearly polarized (LP) laser pulse, there exists a fast oscillation component (twice the laser frequency) in F_p that drives the plasma oscillation and emits harmonics; while F_p of CP laser contains only a slowly-varying component that follows the laser envelope, and thus the HHG is absent. Therefore, a laser pulse with polarization state varying from CP to LP and back to CP enables gating one single attosecond pulse only during the LP period [43].

Unfortunately, the above scheme is effective only for normal or near-normal incidence. As the increase of incidence angle θ, the laser electric field component normal to the plasma surface grows proportionally to sin θ, which can also drive plasma oscillation. As a consequence, even CP lasers can generate HHG as efficiently as LP lasers do at large-angle incidence, as demonstrated in [21]. Therefore, up to now, polarization gating of harmonics from relativistic plasmas has been considered effective only under normal or near-normal incidence interactions [43–46]. However, the laser-plasma EP HHG scheme requires to work under oblique incidence interactions. Moreover, the HHG efficiency is much higher at oblique incidence than at normal incidence.

In this paper, we show for the first time that one can achieve polarization gating of harmonics from relativistic plasmas under large-angle oblique incidence interactions. In contrast to using CP-LP-CP polarization-varying laser pulse, we use CP laser to generate harmonics and LP laser to suppress them. This surprising approach turns out an elegant way to solve the problem: it is suitable for CP HHG and polarization gating simultaneously. Through particle-in-cell (PIC) simulations, we demonstrate the generation of an isolated single attosecond soft X-ray pulse with high ellipticity, which is also ultraintense (∼ 10¹⁹ W/cm²). Compared to the conventional case of polarization gating at normal incidence, the present scheme offers one order of magnitude higher attosecond pulse generation efficiency and thus enhanced photon flux.

2. Methods

We study the HHG and polarization gating using the one-dimensional (1D) PIC code VLPL (Virtual Laser Plasma Lab) [47]. We exploit the HHG in the coherent synchrotron emission (CSE) regime [49–51]. The laser and plasma parameters are taken referred to [49]. The temporal laser pulse profile has a Gaussian envelope $a(t)=a_0\exp (-t^2/{\tau }_0^2)$, where a₀ = eE₀/m_eω₀c is the normalized laser amplitude, τ₀ is the temporal width, E₀ is the laser amplitude, ω₀ is the laser angular frequency, e is the elementary charge, and m_e is the electron mass. In the simulations, either one or two pulses (for the gating scheme) with a₀ = 30 are used. The laser wavelength is λ₀ = 800 nm. The full-width at half-maximum (FWHM) pulse duration is given by ${\tau }_{\text{FWHM}}=2\sqrt{\ln 2}{\tau }_0=30$ fs. The incidence angle is θ = 63°. To model oblique incidence in 1D geometry, Bourdier’s method is adopted to take Lorentz transformations from the laboratory frame to a moving frame [48], which streams along the plasma surface in the plane of incidence (x − y plane) with a velocity of ${\mathbf{v}}_{\mathbf{M}}=c\sin \theta {\widehat{\mathbf{e}}}_y$, where c is the light speed in vacuum and ${\widehat{\mathbf{e}}}_y$ is the unit vector along the y-axis. As such, the laser pulse is transformed to be at normal incidence (with wave vector along the x-axis). The plasma has an exponential density ramp with a scale length of L_s = 0.15λ₀ at the front of the target (with a thickness of 200 nm), leading to a peak density of n_e = 95n_c, where $n_c=m_e{\omega }_0^2/4\pi e^2$ is the critical density. The ions are taken to be immobile. The cell size is λ₀/1000 and each cell is filled with 100 macroparticles. Absorption boundary condition is used for both fields and particles.

Here, the normally incident laser light E_L(x, t) in the moving frame is assumed to have two orthogonal transverse electric field components ${\mathbf{E}}_y(x,t)=E_{0y}\cos (\zeta +{\delta }_y){\widehat{\mathbf{e}}}_y$ and ${\mathbf{E}}_z(x,t)=E_{0z}\cos (\zeta +{\delta }_z){\widehat{\mathbf{e}}}_z$, where ζ = ω₀(t – x/c) is the propagator, E_0y and E_0z are the maximum amplitudes, δ_y and δ_z are the arbitrary phases, respectively. Here, we consider the special case of the two orthogonal field components with equal amplitudes, i.e., E_0y = E_0z = a₀. As such, the polarization ellipse of the resultant pulse is always rotated 45° in the polarization plane. In this case, the ellipticity of the laser pulse can be expressed as [52]: ε = tan(δ_ph/2), where δ_ph = δ_z − δ_y is the phase shift between the orthogonal field components. This means via control of the relative phase δ_ph of the two field components, the laser polarization state can vary from LP (i.e., δ_ph = 0 and ε = 0) to CP (i.e., δ_ph = π/2 and ε = 1). We emphasize that the 45°-rotated LP laser we used here is only partially p-polarized, unlike the purely p-polarized LP lasers used in Fig. 5(a) in [21].

3. Results and discussion

We first examine the HHG yield for the two extreme cases of ε = 0 and ε = 1. The complete dependence of HHG efficiency on ε is discussed later. The Fourier spectra of the reflected fields using the LP driving laser (ε = 0) and the CP driving laser (ε = 1) are compared in Figs. 1(a) and 1(b) respectively. Clear harmonic structure of the fundamental laser frequency can be seen from both figures. For the case of ε = 0, the initial power-law-decay only extends to about the 10th harmonic order before it merges into a more rapid decay. In contrast, for the case of ε = 1, the HHG spectral intensity scaling is much closer to the synchrotron-like spectrum estimated by the CSE model calculation [49], indicating a significantly higher HHG efficiency (more than two orders of magnitude higher at the 100th harmonic order). We have checked for a broad range of laser strength, e.g., from a₀=5 to a₀=60, this conclusion holds that the HHG efficiency is much higher driven by the CP lasers than by the 45°-rotated LP lasers.

 

Fig. 1 (a)–(b) Fourier spectra of the reflected pulses. The dashed lines correspond to the theoretically predicted scaling laws I(n) ∝ n^−8/3 by the ROM γ-spike model [26] (green) and I(n) ∝ n^−4/3 by the CSE model [49] (purple), respectively. (c)–(d) Electron density n_e and intensity of the generated attosecond pulses after spectral filtering to select the 100th–150th harmonic orders (black-white color scale) plotted against time t and position x. t_e1 = 19.30T₀ and t_e2 = 19.37T₀ denote the time instant when the electron transverse momenta reach minimum (significant attosecond pulse generation) for the cases of (a) and (b), respectively. Here T₀ is the laser period. Distribution of the (e)–(f) momenta and (g)–(h) longitudinal velocity of the electron macroparticles. (e) and (g) correspond to t_e1; (f) and (h) correspond to t_e2. The insets in (a) and (b) show the polarization ellipses of the driving laser in the y − z plane with ε = 0 and ε = 1, respectively. The dashed circles in (g) and (h) show the majority of radiating electrons of the reflective plasma mirrors that give rise to the harmonic emission. (a), (c), (e) and (g) are for the case of ε = 0; analogously, (b), (d), (f) and (h) are for the case of ε = 1.

Download Full Size | PDF

Figures 1(c) and 1(d) show the electron density profiles as a function of position x and time t. For the case of ε = 1, we can observe highly-compressed dense electron bunches of nanometer spatial scale formed at the front of the plasma surface, which is a feature of the CSE regime [49]. While the surface electrons rapidly diverge into low density bunches and move incoherently for the case of ε = 0. The black-white color scales show the spatial-temporal distribution of electric field E_y of the generated attosecond pulses after spectral filtering whereby the 100th–150th orders (corresponding to photon energy of 155 eV – 248 eV) are selected. It is clearly seen the attosecond pulses are generated when the dense electron bunches formed at the front target surface are accelerated away from the plasma and moving towards the reflected laser direction.

Efficient HHG from a relativistic plasma mirror requires (i) the longitudinal (along the reflected direction) electron velocity reaching maximum in absolute value (a “gamma spike”) [26], which can lead to higher frequencies due to Doppler up-shifting, and (ii) at the same time, the transverse electron momentum reaching minimum [26], or equivalently, the transverse acceleration being around maximum [51], which can result in stronger radiation. The difference in HHG efficiency between the CP and LP driving laser cases can be attributed to the phase difference between the two orthogonal driving field components, which leads to different plasma dynamics. Specifically, at the moment of harmonic emission, the CP laser pulses can result in smaller transverse momenta (see Figs. 1(e) and 1(f))) and at the same time larger longitudinal electron velocities (see Figs. 1(g) and 1(h))) than the LP laser pulses do. And thus the HHG can be more efficient driven by the CP laser pulses than by the LP laser pulses. In addition, a higher electron-bunch density (see Figs. 1(c) and 1(d))) further enhances the harmonic intensity driven by the CP laser pulses.

The above results suggest the concept of polarization gating may also be applicable to HHG in the highly efficient oblique-incidence regime. To determine the threshold ellipticity ε_th for the polarization gating, we next investigate the generation efficiency for different harmonic range as a function of the laser ellipticity. Here the laser ellipticity at which the harmonic efficiency drops to 1/e² of its maximum value is adopted as the threshold ellipticity ε_th. The efficiency η is calculated as $\eta ={\displaystyle \int \left[E_y^2({\zeta }_{\mathrm{H}})+E_z^2({\zeta }_{\mathrm{H}})\right]\mathrm{d}{\zeta }_H/{\displaystyle \int \left[E_y^2({\zeta }_{\mathrm{L}})+E_z^2({\zeta }_{\mathrm{L}})\right]\mathrm{d}{\zeta }_L}}$, where ζ_H denotes the propagator of the harmonics within the spectral range between the n₁th and n₂th order and ζ_L denotes the propagator of the laser. The results are shown in Fig. 2. For different harmonic ranges, the general trends are similar. For small ellipticity values, the generation efficiencies stay at low levels. After a certain turning point, however, η grows dramatically by several orders of magnitude with the increase of laser ellipticity until η reaches the highest level at ε = 1. In addition, the threshold ellipticity ε_th also increases with increasing the harmonic orders. In other words, the dependence of the HHG efficiency on the ellipticity of the driving field is stronger in the higher-spectral range, which makes the higher-spectral range more suitable for polarization gating in the present scheme. For harmonics with > 100th orders, we have ε_th ≳ 0.65 approximately. We note the efficiency decreases slightly with ellipticity for moderate ε between 0 and 0.3, which is not well understood yet and needs further clarification.

 

Fig. 2 Harmonic generation efficiency η as a function of the driving laser ellipticity ε for different harmonic ranges. For each harmonic range, the efficiencies are normalized to the highest value reached at ε = 1. The dash dotted gray line corresponds to 1/e² of the highest normalized efficiency, ε at which is adopted as the threshold ellipticity ε_th. The inset shows the polarization ellipse of the driving fields in the y − z plane. Colors from purple to blue represent ε from 0 to 1.

Download Full Size | PDF

Based on the ellipticity-dependence of the HHG, we can now design a scheme to test the polarization gating idea and demonstrate its feasibility to generate an isolated EP attosecond pulse. The intended ellipticity-varying laser pulse with CP or EP state only in its central part can be experimentally achieved by a simple set-up using one or two (for fine delay-tuning) multiple order quartz waveplate. The optical axis is set to be 45° with respect to the polarization direction of an initially LP driving field in the plane perpendicular to the laser propagation direction, as schematically shown in Fig. 3(a). Due to birefringence effect of the quartz plate, the electric fields parallel and perpendicular to the optical axis travel with different velocities. Consequently, one can obtain two delayed and partially overlapped laser pulses after the quartz plate. The delayed pulses are LP with mutually perpendicular polarization except within the overlapped temporal region being elliptically polarized. It is within this overlapped region that harmonics can be efficiently generated. The width of the gating time window can be expressed as [43]: ${\delta }_{\text{gating}}=\left|\ln {\epsilon }_{\text{th}}\right|{\tau }_{\text{FWHM}}^2/(4\ln (2)\Delta )$, where Δ is the delay between the two perpendicular field component. To obtain only one attosecond pulse, the gate width should be shorter than one laser optical period. As an example, here we choose the parameters as ε_th = 0.65, τ_FWHM = 19.0 fs and Δ = 16.8 fs (6.3 cycles) for demonstration. The corresponding gating time is δ_gating ≈ 3.3 fs. Besides the gating time, the phase shift δ_ph is also important. It is found from the simulations here the tolerance in the delay should be within 0.1 cycles. We note that the delay can be coarsely controlled by using different plate thickness, and finely tuned by adjusting the angle the laser wave vector makes with the plate surface and thus the effective plate thickness. Especially, the later method allows for a very high-precision tuning. For example, in experiments, a fine tuning of 0.015 cycles of delay per degree of angular adjustment can be achieved for a 440 µm thick plate [53]. The 3D waveform of the polarization-modulated laser field and its two orthogonal components are shown in Fig. 3(b). The polarization state is elliptical at the center and linear in the wings. The other simulation parameters are the same with those used in Fig. 1.

 

Fig. 3 (a) Optical setup to generate a gating laser pulse with time-varying polarization. In the overlapped region of the delayed e- and o-pulse, the resultant pulse is elliptically polarized. (b) 3D waveform (red) of the gating laser pulse for demonstration, its two orthogonal components E_y (blue) and E_z (green), and the projection of E_y−E_z (gray). (c)–(d) Attosecond pulses (E_y-components) generated respectively by using an LP (E_y-polarized) laser pulse and the gating pulse as shown in panel (b). The attosecond pulses are spectral filtering whereby the 100th–150th harmonics are selected. The dotted yellow lines correspond to the 1/e² of the maximum intensities. The inset of panel (d) shows the E_z-component of the corresponding attosecond pulses. (e) 3D waveform (purple) of the gated attosecond pulses, the two orthogonal components E_y (blue) and E_z (green), and the projection of E_y–E_z (gray).

Download Full Size | PDF

The generated attosecond pulses are shown in Figs. 3(c) and 3(d), after spectral filtering of the HHG whereby the 100th–150th orders are selected. One can clearly see that a LP (E_y-polarized) driving pulse leads to a train of attosecond pulses, as usual. In contrast, only one single strong attosecond pulse is produced when the gating laser pulse is applied, and the satellites are effectively suppressed with intensity below the 1/e² level of the strongest pulse. The isolated attosecond pulse corresponds to a frequency spectrum extending to the soft X-ray spectral region. The FWHM pulse duration is about 38 as. At the same time, its normalized intensity is boosted to an extremely high value of (eE/m_ecω₀)² ≈ 16.0, corresponding to 3.4 × 10¹⁹ W/cm². In comparison, we check the case of using conventional (CP-LP-CP) gating pulse with the same intensity and delay at normal incidence. The generated attosecond pulse in the same frequency range only has a peak intensity of (eE?/m_ecω₀)² ≈ 1.3, which is one order of magnitude less efficient than that of the obliquely incident case. From the 3D waveform of the gated pulse shown in Fig. 3(e), we see directly the attosecond pulse is indeed elliptically polarized. We mention that the HHG efficiency in CSE depends sensitively on preplasma conditions [49]. To ensure that the isolation scheme is robust and not due to the inherent unpredictability of this regime, we have checked this scheme can also work with other parameters, e.g., a different plasma density gradient. We also note that the handedness of the attosecond pulse in this scheme can be easily changed by reversing the handedness of the driving pulse in the present scheme, since the governing Vlasov-Maxwell equations are symmetric about the the helicity of electromagnetic fields [21].

4. Conclusions

In conclusion, we have presented a scheme to achieve the first polarization gating of high harmonics from relativistic plasma mirrors under large-angle oblique-incidence interaction. Efficient generation of a bright elliptically polarized isolated attosecond pulse in the soft X-ray regime has been numerically demonstrated. This scheme can be scaled to ultraintense HHG with shorter wavelength, e.g., extending to the “water window” spectral region (280–530 eV). The present results open up exciting prospects for the generation of the desired radiation sources using the large number of existing high-peak-power many-cycle lasers at laboratory scale.