Bright high-order harmonic generation with controllable polarization from a relativistic plasma mirror

Abstract

Ultrafast extreme ultraviolet (XUV) sources with a controllable polarization state are powerful tools for investigating the structural and electronic as well as the magnetic properties of materials. However, such light sources are still limited to only a few free-electron laser facilities and, very recently, to high-order harmonic generation from noble gases. Here we propose and numerically demonstrate a laser-plasma scheme to generate bright XUV pulses with fully controlled polarization. In this scheme, an elliptically polarized laser pulse is obliquely incident on a plasma surface, and the reflected radiation contains pulse trains and isolated circularly or highly elliptically polarized attosecond XUV pulses. The harmonic polarization state is fully controlled by the laser-plasma parameters. The mechanism can be explained within the relativistically oscillating mirror model. This scheme opens a practical and promising route to generate bright attosecond XUV pulses with desirable ellipticities in a straightforward and efficient way for a number of applications.

ARTICLE

Received 19 Jan 2016 | Accepted 11 Jul 2016 | Published 17 Aug 2016

Ultrafast extreme ultraviolet (XUV) sources with a controllable polarization state are powerful tools for investigating the structural and electronic as well as the magnetic properties of materials. However, such light sources are still limited to only a few free-electron laser facilities and, very recently, to high-order harmonic generation from noble gases. Here we propose and numerically demonstrate a laserplasma scheme to generate bright XUV pulses with fully controlled polarization. In this scheme, an elliptically polarized laser pulse is obliquely incident on a plasma surface, and the reected radiation contains pulse trains and isolated circularly or highly elliptically polarized attosecond XUV pulses. The harmonic polarization state is fully controlled by the laserplasma parameters. The mechanism can be explained within the relativistically oscillating mirror model. This scheme opens a practical and promising route to generate bright attosecond XUV pulses with desirable ellipticities in a straightforward and efcient way for a number of applications.

DOI: 10.1038/ncomms12515 OPEN

Bright high-order harmonic generation with controllable polarization from a relativistic plasma mirror

Zi-Yu Chen1,2 & Alexander Pukhov1

1 Institut fr Theoretische Physik I, Heinrich-Heine-Universitat Dsseldorf, Dsseldorf D-40225, Germany. 2 National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621999, China. Correspondence and requests for materials should be addressed to Z.-Y.C. (email: mailto:[email protected]

Web End [email protected] ) or to A.P. (email: mailto:[email protected]

Web End [email protected] ).

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Ultrafast radiation sources in the extreme ultraviolet (XUV) range have become a major tool to study electronic structures and the dynamics of atoms, molecules and

condensed matter. Because polarization is a fundamental property of light and controls its interaction with matter, it is particularly important that these light sources have a tunable polarization. Furthermore, polarization control opens a wider range of applications. For instance, the magnetic as well as the electronic and phononic properties of materials can be studied using circularly polarized (CP) or elliptically polarized (EP) XUV pulses using techniques such as magnetic circular dichroism spectroscopy1. The magnetic circular dichroism technique has proven to be very useful to probe the spin-resolved features in magnetic materials in an element-specic manner2, and thus is of great interest for understanding correlated systems in condensed matter physics. In addition, CP/EP XUV pulses also enable a unique probe of chiral molecules3,4, for example, measuring the photoionization process via photo-electron circular dichroism5. As such, CP/EP XUV pulses also nd a wide range of applications in studying chemical and biological systems.

To date, signicant efforts have been devoted to generating the ultrafast XUV with a variable polarization state. The rst free-electron laser facility that was specially designed to produce such a light source, named Free Electron laser Radiation for Multidisciplinary Investigations (FERMI, Trieste, Italy), has become accessible very recently6. Polarization control at FERMI is achieved by adjusting the conguration of the undulators. Although powerful, these large-scale facilities are expensive and complex, thus limiting their wide accessibility. Therefore, there remains a strong need for sources of coherent CP/EP XUV radiation at the table-top scale.

High-order harmonic generation (HHG) from noble gases has been explored extensively as a route to generate an ultrafast XUV source7. This mechanism, however, encounters intrinsic difculties in generating a CP XUV pulse. This is because the HHG is based on the tunnel ionization, acceleration and recombination of electrons ripped from an atom in the presence of a laser eld, which is explained by the so-called three-step model8. As a consequence, the emission of HHG decreases exponentially with increasing the laser ellipticity because the lateral motion of the detached electron induced by the ellipticity makes the electron less likely to recollide with its parent ion. To be exact, the electron never returns to the parent ion with a CP driving laser. To overcome this drawback, several techniques have been proposed and demonstrated recently to generate quasi-CP or highly EP HHG, such as using pre-aligned molecule targets9, resonant HHG in EP laser elds5, a co-propagating bi-chromatic EP or CP driving laser with opposite helicity1,1012 and a co-propagating bi-chromatic linearly polarized (LP) driving laser with orthogonal polarization13. However, owing to their low ionization thresholds and conversion efciencies, these sources typically suffer from low photon yields.

To ll the gap between large-scale facilities and HHG from gas, XUV via HHG14 and other mechanisms1518 from laser-irradiated plasma surfaces offers a promising alternative to generate an XUV source with high brightness. In principle, with plasma targets there is no limitation on the applicable laser intensity and thus the XUV intensity14. Several radiation mechanisms have been theoretically and experimentally identied as responsible for the HHG process, including coherent wake emission (dominant in the weakly relativistic regime)1922, relativistically oscillating mirror (ROM; dominant in the strongly relativistic regime)2328 and coherent synchrotron emission2931. It has been demonstrated that LP HHG can be generated relatively efciently using an LP driving laser at oblique incidence.

It has been commonly assumed up to now that the ROM mechanism fails for a CP driving laser. Moreover, a polarization gating (aka relativistic coherent control) has been proposed to select a single LP attosecond pulse from a pulse train3234. According to the ROM theory25, these attosecond pulses are emitted when electrons at the plasma surface are moving towards the observer and their tangential momenta vanish. This is never the case for an EP laser pulse normally incident on the plasma surface, which causes HHG to be strongly suppressed. For an EP laser pulse at oblique incidence, two experimental groups have observed HHG at relatively small angles of incidence35,36. However, only the harmonic intensities were measured, with no information about the harmonic polarization states.

In this paper, we propose and numerically demonstrate the generation of intense HHG with fully controlled polarization from laser plasmas. We show that this can be achieved using a CP laser obliquely incident onto a plasma surface. Both pulse trains and isolated circular or highly elliptic attosecond XUV pulses can be obtained. By changing the incidence angle, the harmonic polarization state can be tuned from quasi-circular through elliptical and linear to an elliptical polarization of opposite helicity. Switching the helicity of the incident laser, the handedness of the harmonics can be easily reversed. The scheme works for a wide range of laser and plasma parameters, and the efciency is comparable to that using an LP laser. This very promising new procedure thus provides a straightforward and efcient way to obtain a bright attosecond XUV source with desirable ellipticities and holds the potential of making a very large avenue of research more accessible for a number of laser laboratories worldwide.

ResultsScheme. Figure 1a shows the scheme of the proposed conguration for the HHG with a desired polarization state. The basic idea is to use a CP relativistic laser pulse obliquely incident on a solidplasma surface using a radiation mechanism known as the ROM model. In the ROM model, under the combined action of the ponderomotive force of the laser and the electrostatic restoring force resulting from charge separation, the surface electrons oscillate with relativistic speeds and reect the laser pulse like mirrors. During this nonlinear process, harmonics of the fundamental laser frequency are generated as a result of Doppler up-shifting. Except for some special cases (that is, few-cycle laser pulse interactions with near-critical density plasmas37), normal incidence CP laser pulses cannot generate harmonics for two reasons. Firstly, CP laser pulses lack the fast oscillating component in the ponderomotive force. Secondly, driven by CP pulses, electrons always have a relativistically large tangential momentum: when one tangential component vanishes, the other reaches its maximum. As a result, electrons never move towards the observer and do not emit high harmonics efciently. This difference between CP and LP pulses forms the basis of the polarization gating3234, which is the method proposed herein to obtain an isolated single attosecond pulse from a train of attosecond pulses of ROM harmonics. This is true for a CP laser pulse at normal incidence.

However, for oblique incidence interactions, HHG can be efciently generated even by CP laser pulses. The force acting on the plasma surface does contain a fast oscillating component owing to the normal (p-polarization) component of the laser electric eld. Further, the oblique incidence can be reduced to a normal incidence case using Lorentz transformation to a moving frame of reference where plasma is streaming along the surface (see Methods section). In this frame of reference, electrons have an initial tangential momentum. When the angle of incidence is

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a c d

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Figure 1 | Scheme and 2D simulation results. (a) The proposed experimental conguration for generation of the polarization-controlled harmonics by a CP laser pulse obliquely incident on a plasma surface. The red and purple arrows represent the incident laser pulse and the reected XUV harmonics, respectively. (b) A snapshot of the electric eld component Ez of the reected pulse from the 2D simulation results at time t 36T0. The green dashed line

marks the specular reection direction. (c) Temporal waveform and (d) the corresponding Fourier spectra of the reected pulse along the specular reection. The green dashed line in panel (d) corresponds to the predicted scaling law IROM(n)pn 8/3 by the ROM theory. (e) The reconstructed 3D

image of the electric eld vector of the attosecond pulses (purple), obtained after spectral ltering by selecting the 10th20th harmonic orders (indicated by the harmonics within the dashed grey box in panel d). Waveform of the two orthogonal electric eld components Ey (green) and Ez (blue), as well as the projection of Ey Ez (grey), are also shown.

adjusted correctly, this momentum can exactly compensate for the momentum induced by the laser eld so that there exist moments when the plasma surface electrons move exactly towards the observer and reect the CP laser. However, the different laser polarizations may have different phase lags at the nonlinear reection from the plasma, which makes it necessary to perform simulations to clarify whether the properties of the incident laser, such as polarization and coherence, are preserved.

Two-dimensional simulation results. We rst carried out two-dimensional (2D) particle-in-cell simulations to show a general picture of the ellipticity of HHG from a CP laser obliquely irradiated onto plasma surfaces. The laser and plasma parameters are chosen to match realistic experiments, wherein the laser with a normalized amplitude of a0 30 is obliquely incident at an

angle of y 40 onto a plasma of density ne 100nc, where nc is

the critical density (see Methods section). Figure 1b presents a snapshot of the electric eld component Ez of the reected pulse in the x y plane at time t 36T0, where T0 is the laser period.

The green dashed line marks the direction of specular reection of the incident laser. A temporal waveform of the radiation in the specular reection is shown in Fig. 1c, where both the Ey and Ez components are depicted. It is apparent that both Ey and Ez have an amplitude level the same as that of the incident laser.

Figure 1d shows the Fourier spectra corresponding to Fig. 1c. The green dashed line corresponds to the scaling law for the spectral intensity IROM as a function of the harmonic order n:

IROM(n)pn 8/3, which is given by the BaevaGordienko Pukhov (BGP) theory25 of ROM. The excellent agreement of

the spectra with the theoretically predicted power law suggests that the HHG mechanism here is within the ROM regime. Harmonic structures up to the 20th order can be clearly observed for both the Ey and Ez components. Beyond that, however, the spectral line structure is not periodic, indicating that the periodicity of the attosecond pulses changes with time. It is worth noting that the 2D simulations for HHG from solidplasma surfaces are computationally expensive and the resolution is limited. Here, we only resolve the HHG up to the 20th order for demonstration purposes, but harmonic spectra with well-dened periodic structures up to much higher orders can be generated and have been observed experimentally. For example, well-dened harmonic structures up to at least the 46th order have been observed with almost the same laser (excepting that the polarization is linear) and plasma parameters38. Another experiment with a much lower intensity of a0 3.5 has also

demonstrated that harmonic comb structures up to about the 40th order can be observed39. Therefore, periodic harmonic orders higher than the 20th can be expected. In addition, different harmonic orders are appropriate for different applications. For example, the harmonics of the 7th20th orders (photon energies around 1030 eV) are of particular interest for studies such as molecular photoionization, because this frequency range is close to the ionization thresholds of most molecular systems5. Also, harmonics of the 35th42nd orders (photon energies around 5565 eV) are required for investigating the magnetic properties of solids, because this frequency range covers the M absorption edges of the magnetic elements Fe, Co and Ni (ref. 40). We leave the HHG with higher orders to be investigated with one-dimensional (1D) simulations later.

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Applying a band-pass spectral lter that selects harmonics between the 10th and 20th orders, we obtain a train of attosecond XUV pulses, as shown in Fig. 1e. From the helical structures of the electric eld contour EHEHy EHz plotted in this three-dimen

sional (3D) image, we can see directly that each attosecond HHG pulse is elliptically polarized. The HHG pulses reach a peak electric eld amplitude of EH 5 (normalized to meo0c/eE4 1012

V m 1), which corresponds to a dimensional value of EH 2 1013 V m 1. This clearly demonstrates the potential of

the ROM mechanism to obtain a bright helical XUV source. The averaged amplitude ratio between the two electric components in this frequency range is e min EHy; EHz

n o=max EHy; EHz

n o0:96,

indicating that a high ellipticity can be reached. The phase shift DfH between the two electric eld components fHz and fHy is

DfH fHz fHy 0.36p for the HHG pulse around x 30l0 in

Fig. 1e. The sign of DfH also demonstrates that the HHG pulse generated here has the same helicity as the incident laser pulses

fLz fLyp=2

.

Parametric study. In the following, we use a series of 1D simulations with a higher resolution to study the parametric dependence of the HHG ellipticity. From Fig. 2a we can see that, for a broad range of laser amplitudes from a0 5 to a0 30, both

the amplitude ratio e and the phase shift DfH between the two electric components of HHG change insignicantly. Similarly, e and DfH exhibit only a weak dependence on the initial plasma density in the range of 50400nc (see Fig. 2b). Compared with the laser amplitude and plasma density, the plasma density scale length Ls plays a more important role in changing the HHG amplitude ratio e, as shown in Fig. 2c. For laser plasma interactions in the ultrarelativistic regime (a0c1), the dynamics are largely determined by the dimensionless similarity parameter

S ne/a0nc. Because the reection mainly occurs at the critical

surface with a relativistic density of nRelc nca0, that is, at a xed

S 1, the system is expected to be self-similar41. Therefore the

dynamics of harmonic generation do not depend separately upon a0 and ne, but instead upon the scale length through the electron density prole nenRelcexp S

p x=Ls (ref. 38). A previous study has suggested that there exists an optical scale length whose value is about c/o0 (ref. 38), where c is the light speed in vacuum and o0 is the laser angular frequency. The parametric study here shows that the helical HHG exists for a wide range of laser and plasma parameters given that the scale length is well controlled. Furthermore, the feasibility of scale length control is conrmed in experiments38.

Polarization control. Based on the parametric studies above, we choose an optimal scale length of Ls 0.1, a moderate laser

amplitude of a0 5 and a plasma density of ne 200nc to study

polarization state controllability in the HHG pulse. Figure 2d shows the amplitude ratio and phase shift of the HHG eld components as a function of the laser incidence angle y. The

HHG in the frequency range of the 20th30th orders are selected, except that the 5th10th orders for y 22.5 and the 15th20th

orders for y 40 are used owing to a lower cutoff of well-dened

harmonic structures at these relatively small incidence angles. Nevertheless, we found that, even when using the higher orders of 35th42th in the case of y 40, elliptical HHG pulses can be

generated, although with a smaller ellipticity value e. Notably, an amplitude ratio of e close to unity, together with a phase shift of fHEp/2, is obtained at the angle of y 22.5. This indicates that

intense quasi-CP HHG pulses are generated with this simple geometry. Moreover, as the angle of incidence y increases, the phase shift DfH changes continuously from p/2 to 0, and

eventually to o p/2. This signies that the polarization state of

the HHG pulses varies as y increases, from a polarization state

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Figure 2 | Parametric study and polarization control. (ad) Amplitude ratio e and phase shift DfH between the two orthogonal components of the harmonic electric elds as a function of (a) laser amplitude a0, (b) plasma density ne, (c) plasma density scale length Ls and (d) laser incidence angle y. The other parameters are: (a) ne 100nc, y 40 and Ls 0.2; (b) a0 5, y 40 and Ls 0.2; (c) a0 5, y 40 and ne 100nc; and (d) a0 5, ne 200nc

and Ls 0.1. The Ls value is normalized by l in the moving frame for convenience in the 1D simulations. The different shaded areas in panel (d) represent

harmonics possessing opposite helicities. LH, left-handed; RH, right-handed.

Scale length Ls/[afii9838] Incidence angle [afii9835] (deg)

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that is circular (DfH p/2) through one that is elliptical

(0oDfHop/2) and linear (DfH 0) and nally to one that is

elliptical of opposite helicity ( p/2oDfHo0). In this way,

therefore, a practical and straightforward method to control the ellipticity of the HHG pulses is produced by simply adjusting the incidence angle of the laser pulse, which is important to a number of applications.

Circular attosecond pulses using elliptic laser. In addition to obtaining quasi-circular or highly elliptic HHG using CP laser pulses at a small-angle incidence, here we show CP harmonics and/or attosecond XUV pulses can also be generated using EP laser pulses at an oblique incidence. Considering the above case of y 40 in Fig. 2d for example, the HHG amplitude ratio is

e 0.97 and the phase shift is DfH 0.42p. For a perfect CP

attosecond XUV pulse; however, the phase shift must be DfH p/2; we therefore compensate for the necessary additional

phase shift of df 0.08p by using EP laser pulses that have a

phase shift of fLz fLy p=2 df1:822. The other parameters

are the same as those used in the case of y 40 in Fig. 2d. The

waveform of the generated attosecond XUV pulse train is given in Fig. 3a, showing a pulse train whose amplitude ratio is eD1.0 and the phase shift is DfHDp/2. As a result, an attosecond XUV pulse train with nearly perfect circular polarization has been generated using EP laser pulses. This approach is also very promising and is easy to implement experimentally.

Isolated attosecond helical XUV pulse. Attosecond HHG pulse trains have proven to be useful in studying ultrafast XUV nonlinear processes, such as the photodissociation of molecules42 and chiral experiments1. However, an isolated single attosecond helical XUV pulse holds the potential for time-resolved dichroism measurements with unprecedented temporal resolutions43. For instance, important questions regarding the timescale of magnetization dynamics in correlated materials may be addressed with such a tool40. Here, we demonstrate how an isolated single attosecond elliptical/circular HHG pulse can be generated with the present scheme using a few-cycle laser pulse. Figure 3b shows a resulting waveform of the attosecond HHG pulse after spectral ltering whereby the 35th42nd orders are selected. Here, the incident EP laser pulse has a duration of 5 fs full-width at half-maximum, an amplitude of a0 5, an initial

phase of DfL 1.822 and an incidence angle of y 40, while the

plasma density is ne 200nc. It can be seen that an isolated single

attosecond XUV pulse with quasi-circular polarization has been

generated. This shows the possibility of applying this technique to ultrafast dichroism measurements.

Switching HHG handedness. For dichroism study applications, the difference in the absorption of left-handed (LH) and right-handed (RH) light is measured; that is, DA(lL)

ALH(lL) ARH(lL), where lL is the light wavelength. Thus it is

important to generate helical light with opposite handedness. This can be easily achieved by changing the handedness of the incident laser pulse in our scheme because the VlasovMaxwell equations are symmetric about the handedness of electromagnetic elds. To demonstrate this, we compared the HHG produced by CP laser pulses possessing opposite handedness, where both laser pulses possess an amplitude of a0 5 and are obliquely incident

at the angle of y 40 onto a plasma of density ne 200nc.

Figures 4a,b show the HHG waveform generated by an LH laser and an RH laser, respectively, where a band-pass lter was used to select the 15th20th harmonic orders. It is seen that the two HHG pulses are nearly the same except for the opposite phase shift DfH. This shows the feasibility of reversing the rotation direction of the HHG pulse by simply switching the handedness of the incident CP laser pulses.

HHG efciency. As is known, the HHG efciency using a CP laser is low at normal or small-angle incidence3234. This result changes drastically, however, as the angle of incidence increases. Experimental results by Yeung et al. have shown that at 22.5 incidence the harmonic (13th28th) efciency using a CP laser is at least two orders of magnitude lower than that using an LP laser36. On the other hand, experimental results by Easter et al. have shown that at 35 incidence the harmonic (13th19th) efciency using a CP laser is just a factor of 3 lower than that using an LP laser with the same harmonic orders35. To verify the dependence of the efciency upon the incidence angle, we carry out a series of simulations, as shown in Fig. 5a, in which the laser amplitude is a0 5, the plasma density is ne 200nc and the

harmonic orders are 13th30th. At incidence angles of 22.5 and 35, the simulation results are in excellent agreement with the above-mentioned experimental results35,36. In addition, the simulation results predict that the harmonic efciency with the CP laser increases with the angle of incidence, and reaches the same value as when using an LP laser at 45 incidence. As the incidence angle is further increased, the efciencies with the CP and the LP lasers tend to stay at the same level. Figure 5b shows the Ey components of the HHG spectra when using CP and LP lasers at y 55 incidence, where it is also seen that the HHG efciency

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Figure 3 | Attosecond helical XUV pulses. (a) Waveform of a CP XUV attosecond pulse train after spectral ltering where the 15th20th harmonic orders are selected. Here an EP laser pulse of 30 fs duration is used. (b) Waveform of a CP XUV isolated single attosecond pulse after spectral ltering where the 35th42nd harmonic orders are selected. Here a few-cycle EP laser pulse of 5 fs duration is used. The other parameters are: laser amplitude a0 5; laser

initial phase fLz fLy1:822; laser incidence angle y 40; plasma density ne 200nc; and plasma scale length Ls 0.1. In both panels, waveform of the 3D

electric eld vector (purple), the two orthogonal electric eld components Ey (green) and Ez (blue) and the projection of Ey Ez (grey) are displayed.

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a b

Using left-handed laser Using right-handed laser

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Figure 4 | Switching the harmonic handedness. Electric eld waveform of the harmonics using a laser with helicity of (a) left-handedness and(b) right-handedness. Spectral ltering is applied where the 15th20th harmonic orders are selected. The laser amplitude is a0 5 and the incidence angle

is y 40. The plasma density is ne 200nc with a scale length of Ls 0.1.

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Figure 5 | Harmonic efciency and plasma dynamics. (a) Inuence of the incidence angle upon the efciency of the harmonics (13th30th) with CP and LP laser pulses. The laser amplitude is a0 5 for the CP laser and a0 7.07 for the LP laser to keep the intensity and pulse energy the same. (b) HHG

spectra of Ey component, compared between the cases using CP and LP laser pulses at 55 incidence. The green dashed line corresponds to the predicted scaling law IROM(n)pn 8/3 by the ROM theory. (c,d) Spatial distribution of electron longitudinal (px) and transverse momenta (py and pz) for the case of a

CP laser at 55 incidence at two different times of (c) t 28.64T0 and (d) t 30.40T0. In these simulations, the other parameters are: laser amplitude

a0 5, plasma density ne 200nc and scale length Ls 0.1. The momenta are normalized by mec. The dashed black lines in panels (c,d) mark the zero

momenta.

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10 70.6 70.8 71.0 X / [afii9838]0

with the CP laser is comparable to that with the LP laser. These results show the potential of achieving efcient helical HHG with the present scheme. Moreover, as mentioned, intense HHG can be generated even with a low efciency, because the applied laser intensity is high.

Regime of validity. Figure 5b again shows that the harmonic spectra agree well with the IROM(n)pn 8/3 scaling law, which is

predicted by the BGP theory (also termed the g-spikes model) of ROM25. This theory implies the existence of zeros in the transverse momenta of plasma surface electrons44. To further demonstrate that the HHG mechanism here is within the g-spikes

ROM regime, we plot the spatial distribution of the electron longitudinal (px) and transverse momenta (py and pz) at two different times from the simulation results with the CP laser at 55 incidence, as shown in Fig. 5c,d. It can be clearly seen that moments do exist when both of the transverse momenta of the

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plasma surface electrons become zero simultaneously, and it is at these moments that the harmonics are efciently emitted. In addition, the parametric studies above also show that our scheme works well for a broad range of laser amplitudes (a0 from 5 to 30)

at oblique incidence. These results suggest that the mechanism here is in accordance with the g-spikes model of ROM, and thus is relevant to the strongly relativistic regime with a20 1 and

relativistically overdense plasmas with S 1.

In summary, a scheme to generate CP or highly EP attosecond XUV pulses is proposed and numerically demonstrated, which is based on high harmonic generation from a relativistic plasma mirror. It is shown that such harmonics can be efciently generated when the laserplasma parameters are suitable. In addition, the harmonic polarization is fully controllable by the laserplasma parameters. The scheme allows the use of a relativistically intense laser, and thus it is a promising scheme to achieve a chiral XUV source with high brilliance. This provides an exciting tool with applications in a number of elds.

Methods

Particle-in-cell simulation. We carried out all simulations using the Virtual Laser Plasma Lab (VLPL) code45. For 2D simulations, the size of the simulation box is 45l0 70l0 in the x y plane, with a laser wavelength of l0 800 nm and a cell size

of l0/200 in each dimension. The laser and plasma parameters are chosen to match those used in the experiments38. The laser pulse has a normalized amplitude ofa0 eE0/meo0c 30 (corresponding to an intensity of 2 1021 W cm 2) and pulse

duration of 30 fs full-width at half-maximum, where E0 is the laser electric eld amplitude, e is the elementary charge and me is the electron mass. The pulse is focused into a Gaussian spot with a diameter of 2 mm, which requires a Ti:sapphire laser system that can deliver a pulse energy of about 1 J. The laser pulse is obliquely incident at an angle of y 40 onto the target, which is taken to be a fully ionized

plasma. The plasma slab has an electron density of ne 100nc and a thickness of

500 nm, where ncmeo20=4pe2. In the front of the plasma slab, preplasma exists

with an exponential density prole and a density scale length of Ls 0.2l0. To

simulate oblique laser incidence in the 1D setup, a Lorentz transformation from the laboratory frame to a moving frame of reference has been made24,46. As such, the laser is transformed to be at normal incidence onto a plasma slab streaming in the y direction parallel to the planar surface. For all 1D simulations, a relatively high spatial resolution of 1,000 cells per laser wavelength in the moving frame is used.

Control of polarization. A CP laser pulse can be represented as a superposition of two LP pulses with equal amplitude and a constant phase difference of p/2:

ECP Ey Ez with EyE0 cos o0t

^ey and EzE0 sin o0t

^ez, where ^ey and ^ez are

respectively the unit vector along the y and z directions. Thus the corresponding vector potential can be written as AyA0 sin o0t

^ey and Az A0 cos o0t

^ez. In

the 1D geometry, the canonical momentum in the transverse direction is conserved: p> eA>/c constant, where p> and A> are the transverse momentum

and vector potential, respectively. In the moving frame of reference, the initial momenta in the transverse directions are py0 mec tan y^ey and pz0 0. Then we

can obtain the expression for the transverse momentum as:

py mec tan y^ey eAy=c mec tan y eA0 sin o0t

=c

^ey 1 pz eAz=c eA0 cos o0t

=c^ez 2

According to the BGP theory25, the HHG is emitted when the transverse momentum of the surface electron p> reaches a minimum or vanishes. In the case of a laser normally incident with y 0, we observe that

p?eA0=c sin o0t

^ey cos o0t

^ez

, which never vanishes or reaches a minimum,

and consequently, no harmonics are generated. The situation changes with an oblique angle of incidence y, which makes it possible to have py and pz simultaneously reach a minimum or vanish. The incident angle y therefore provides a degree of freedom that can be used to adjust the relative amplitude and phase between the two components of the transverse momentum, and thus can be used to change the polarization state of the harmonics generated.

Data availability. The data that support the ndings of this study are available from the corresponding authors upon request.

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Acknowledgements

Z.Y.C. acknowledges nancial support from the China Scholarship Council(201404890001). This work was supported by the Deutsche Forschungsgemeinschaft SFB TR 18, EU FP7 project EUCARD-2 and the Science and Technology Fund of the

National Key Laboratory of Shock Wave and Detonation Physics (China) with project Nos. 077110 and 077160.

Author contributions

Z.Y.C. conceived and conducted the simulations, and drafted the manuscript. A.P. developed the code and theory, and supervised the work. All authors discussed the results and reviewed the manuscript.

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How to cite this article: Chen, Z.-Y. et al. Bright high-order harmonic generation with controllable polarization from a relativistic plasma mirror. Nat. Commun. 7:12515doi: 10.1038/ncomms12515 (2016).

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