Abstract

The Faraday effect, caused by a magnetic-field-induced change in the optical properties, takes place in a vast variety of systems from a single atomic layer of graphenes to huge galaxies. Currently it plays a pivotal role in many applications such as the manipulation of light and the probing of magnetic fields and materials’ properties. Basically, this effect causes a polarization rotation of light during its propagation along the magnetic field in a medium. Here, we report an extreme case of the Faraday effect where a linearly polarized ultrashort laser pulse splits in time into two circularly polarized pulses of opposite handedness during its propagation in a highly magnetized plasma. This offers a new degree of freedom for manipulating ultrashort and ultrahigh-power laser pulses. Together with the technologies of ultra-strong magnetic fields, it may pave the way for novel optical devices such as magnetized plasma polarizers. In addition, it may offer a powerful means to measure strong magnetic fields in laser-produced plasmas.

© 2017 Optical Society of America

1. INTRODUCTION

As the hallmark of magneto-optics, the Faraday effect or Faraday rotation observed in 1846 was the first experimental evidence of the electromagnetic wave nature of light [1]. Importantly, it provides an ingenious method for manipulating light, and becomes the basic principle underlying the operation of a number of magneto-optical devices [2,3]. In principle, the Faraday rotation is caused by magneto-chiral dichroism of left-circularly polarized (LCP) and right-circularly polarized (RCP) electromagnetic waves propagating at differential speeds in magnetized materials. Since the magneto-chiral dichroism in most materials is very weak, considerable Faraday rotation generally happens only after a long propagation distance. This severely limits the miniaturization and integration of magneto-optical devices. Therefore, there has been a growing interest in the search for enhanced Faraday rotation. As a collection of charged particles, a dense plasma responds strongly to electromagnetic waves and thus often gives rise to a strong Faraday rotation under the influence of a magnetic field [4]. Furthermore, the plasma optical devices are particularly suitable for the fast manipulation of ultrashort high-power laser pulses due to their ultrahigh damage threshold [58].

In this work, we report an extreme case of the Faraday effect in which not only the polarization direction but also the polarization state of ultrashort laser pulses can be completely changed in strongly magnetized plasmas with magnetic fields B50T. The underlying physics is that a linearly polarized (LP) laser pulse can be considered as the superposition of a RCP subpulse and a LCP subpulse, while the eigen electromagnetic waves propagating along the magnetic field in plasmas are RCP and LCP waves, which have differential group velocities as well as differential phase velocities. Therefore, under appropriate conditions a LP laser pulse will split into a RCP subpulse and a LCP subpulse, as shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Sketch of the magnetic splitting of an ultrashort LP laser pulse, which is incident along the magnetic field B into plasma. The incident LP pulse will split into RCP and LCP subpulses due to their differential group velocities. The RCP subpulse follows the LCP subpulse in time. Here the electric field vector of the RCP pulse at a fixed position rotates clockwise in time as viewed along the wave vector of the laser pulse, and vice versa for the LCP pulse.

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2. THEORY

We first provide a set of formulas to describe the propagation of electromagnetic waves in magnetized plasmas. The electromagnetic wave propagation along the magnetic field in a plasma is mainly governed by the dispersion relation [4]

c2k2ω2=1ωp2ω2(1±ωc/ω),
where ± are respectively for the LCP (+) and RCP (−) waves, ω and k are the waves’ angular frequency and wavenumber, the plasma frequency ωp(nee2/ϵ0me)1/2 is defined by the plasma density, and the electron cyclotron frequency ωceB/me is proportional to the magnetic field strength B. From the dispersion relation, one can easily get the phase velocities vp=[1ωp2/(ω2±ωωc)]1/2c for the LCP (+) and RCP (−) waves [4]. The differential phase velocities will induce a rotation of the polarization plane of a LP wave, since it can be considered as the sum of a RCP wave and a LCP wave. Under the limitations of low plasma density (ωpω) and small magnetic field (ωcω), the Faraday rotation angle can be estimated as ΔϕRMλ2, where RM=e3ne(x)B(x)dx/8π2ϵ0me2c3 is the so-called rotation measure in astronomy [9,10] and λ is the wavelength. This is the scenario of the familiar Faraday rotation, in which the rotation angle is proportional to the magnitude of the magnetic field. However, this linear Faraday effect can only be applied for a relatively small magnetic field with a low plasma density. As long as ωc/ω approaches (1ωp2/ω2), the phase velocity for the RCP wave will quickly become infinite. Therefore, the RCP wave cannot propagate in a strong magnetized plasma if (1ωp2/ω2)ωc/ω1. But the propagation of the RCP wave becomes possible again in the whistler-mode region (ωc/ω>1). In the latter case, the RCP wave can even penetrate into an overcritical density plasma, but is accompanied by a strong heating of the plasma [11]. For the sake of simplicity, we will not discuss the wave propagation in the whistler-mode region here. Nevertheless, one can conclude from the above analysis that the Faraday rotation angle is no longer linearly proportional to the magnitude of the magnetic field if the latter is strong enough, i.e., if one enters a nonlinear regime of the Faraday effect.

In addition to the differential phase velocities, more importantly, we notice that the group velocities are also different for the LCP and RCP waves in a magnetized plasma. From the dispersion relation, one can deduce the group velocities

vg,Lc=(1ωp2/ω21+ωc/ω)1/2[1ωcωp2/ω32(1+ωc/ω)2]1,
vg,Rc=(1ωp2/ω21ωc/ω)1/2[1+ωcωp2/ω32(1ωc/ω)2]1,
for the LCP and RCP waves, respectively. As shown in Fig. 2, the former increases with the magnetic field, while the latter behaves in the opposite way. So for a LP short laser pulse, its LCP and RCP components will gradually split apart. Assuming the pulse initially has a duration tp, the time delay between the peaks of LCP and RCP subpulses (Δvgt/vg,R) will be larger than tp after
ts=vg,RΔvgtp,
where Δvg=vg,Lvg,R is the difference in the group velocities. Figure 2(c) indicates that the stronger the magnetic field and the higher the plasma density, the larger the difference in the group velocities will be. If the magnetic field is small enough (ωcω) and the plasma density is low enough (ωpω), we can get
Δvgc2nencωcω,
where ncϵ0meω2/e2 is the critical plasma density.

 figure: Fig. 2.

Fig. 2. (a, b) Group velocities from Eqs. (2) and (3) for the LCP and RCP waves, respectively; (c) the difference in the group velocities, (d) the minimum field (ωc,min) required for an obvious magnetic splitting as a function of the frequency spread (Δω) of the pulse.

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In the case of an ultrashort laser pulse, however, the frequency spread must be taken into account. For instance, Δω/ω00.441τ/tp holds for a Gaussian pulse [12], where τ=2π/ω0 is the laser wave period and ω0, Δω, and tp are the center frequency, FWHM frequency spread, and FWHM duration of the pulse, respectively. Thus the group velocities are not constant, and the pulse temporal broadening due to dispersion must be considered. Consequently, the magnetic splitting of the pulse is observable only under the condition

vg,R|ω=ω0+Δω/2<vg,L|ω=ω0Δω/2.
Otherwise, the dispersive broadening will dominate over the magnetic splitting. The above inequality prescribes a lower limit of the magnetic field (BBmin=meωc,min/e). Under the same assumptions as for Eq. (5), we can get ωcωc,minΔω/2, which is in good agreement with the numerical solution at ne=0.1nc in Fig. 2(d). With a relatively higher plasma density such as ne=0.5nc, however, the required ωc,min increases very fast with an increasing Δω. From the numerical solutions, we find that
ωcωc,min=Δω
is a sufficient condition for inequality (6) if ne/nc0.5, as shown in Fig. 2(d). That is to say, the pulse splitting will be quicker than the dispersive broadening if the electron cyclotron frequency (ωc) in the magnetic field is larger than the frequency spread (Δω) of the laser pulse. The latter is inversely proportional to the pulse duration. This implies that the shorter the pulse duration is, the stronger the magnetic field is required to be to split the pulse, while Eq. (4) implies that the longer the pulse duration is, the thicker the required magnetized plasma has to be. These two aspects grimly prescribe that the magnetic splitting of a laser pulse can be clearly observed only if the pulse duration is modest and the magnetic field is strong enough.

With the invention of novel laser techniques such as chirped-pulse amplification [13], it becomes possible to generate laser pulses as short as a femtosecond (fs). At the same time, as 20 T magnets become commercially available, magnetic fields above 100 T are recorded in some laboratories [14]. In particular, the interaction of high-power laser pulses with matter can generate kilotesla-level magnetic fields [15,16]. Not only are such kilotesla-level magnetic fields of fundamental interest, but they also demonstrate potential for various applications [1719]. The breathtaking advances in the pulsed laser and high magnetic field sciences combine to provide a good opportunity to achieve the magnetic splitting of an ultrashort laser pulse.

3. SIMULATION

To verify the magnetic splitting of short laser pulses, we perform a series of particle-in-cell (PIC) simulations using the code OSIRIS [20]. In simulations, laser pulses are incident along the magnetic field into semi-infinite plasmas at x0. The initial LP pulses are polarized along the z axis with λ=1μm. For reference, the pulse peaks are all assumed to arrive at the vacuum–plasma interface (x=0) at t=0. The moving-window technique is employed with a simulation box moving along the x axis at the speed of light in vacuum. The simulation box is set large enough to contain the laser pulse for the whole process of each simulation. In 1D simulations, the sizes of the simulation boxes range from 500 to 35,000λ, the spatial and temporal resolutions are Δx=λ/16 and ΔtΔx/c, each cell has 16 macroparticles, and the electron density ne=0.5nc. In a 3D simulation, the simulation box has a size of 210λ×24,000λ×24,000λ. The spatial resolutions are Δx=λ/16 and Δy=Δz=100λ, the temporal resolution is ΔtΔx/c, each cell has four macroparticles, and the electron density ne=0.1nc.

Figure 3 compares 1D simulation results with varying laser pulse duration tp and magnetic field B. In Fig. 3(a), the magnetic splitting condition (ωc>Δω) holds well with tp=500fs and B=50T. Consequently, the initial pulse splits into two discrete subpulses at t=300 picosecond (ps). The first subpulse peaking at x63145λ is LCP, since its Stokes parameter V<0, while the second subpulse peaking at x62735λ is RCP with V>0. The degrees of circular polarization exceed 94% for both the LCP and RCP subpulses. The simulation shows that the difference in the group velocities for these two subpulses is about 0.0046c, which is in rough agreement with the prediction of 0.0050c by Eq. (5). In Fig. 3(b), the difference in the group velocities is increased roughly by an order of magnitude with a 500 T magnetic field. Consequently, the LCP and RCP subpulses are clearly separated at a much earlier time of t=30ps. By such a 500 T magnetic field, we find that the laser pulses with much shorter durations such as 50 fs can also be separated, although each sub-pulse is a little longer than the initial pulse due to dispersion as shown in Fig. 3(c). However, a 50 fs laser pulse cannot be separated by a 50 T magnetic field, since the pulse frequency spread is Δω0.029ω>ωc0.005ω in this relatively weak magnetic field. As illustrated in Fig. 3(d), at t=30ps the pulse duration has been stretched to about 600 fs, which is an order of magnitude longer than the estimated time delay between the RCP and LCP subpulses. This confirms that the dispersive broadening will dominate over the magnetic splitting of the pulse if Δω>ωc.

 figure: Fig. 3.

Fig. 3. Stokes parameters from 1D PIC simulations with varying laser pulse duration tp and magnetic field B (a) at t=300ps with tp=500fs (Δω/ω00.0029) and B=50T (ωc/ω00.005), (b) at t=30ps with tp=500fs and B=500T, (c) at t=3ps with tp=50fs and B=500T, and (d) at t=30ps with tp=50fs and B=50T. The Stokes parameter I denotes the intensity regardless of polarization, Q and U describe the state of linear polarization, and V represents the circular polarization [21]. All parameters are normalized to the instantaneous peak intensity Imax. Here the laser intensity is low enough (the dimensionless amplitude a|eE/ωmec|=0.01) that the nonlinear effects [22,23] due to the laser field itself can be ignored.

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Figure 4 displays the simulation result with an extremely strong magnetic field of B=6000T (ωc/ω00.6). In this case, it becomes impossible for the RCP wave to propagate into the magnetized plasma, since ωc/ω>(1ωp2/ω2). Figure 4 shows that the incident LP pulse has been separated into two subpulse as well. However, here only the LCP subpulse (peaking at x88λ with V<0) can propagate into the magnetized plasma. The RCP subpulse (peaking at x100λ with V>0) is completely reflected and propagates backward.

 figure: Fig. 4.

Fig. 4. Stokes parameters at t=333fs with an extremely strong magnetic field B=6000T (ωc/ω00.6). Other parameters are the same as those in Fig. 3(c).

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The magnetic splitting of a 50 fs laser pulse is also verified by a 3D simulation, as displayed in Fig. 5(a), where the isosurface of intensity I=I0/4 appears as two separate ellipsoids at t=5ps (see Visualization 1 for the entire splitting process). At an early stage in Visualization 1, a conventional Faraday rotation as large as many cycles is also evidenced by the quick variations in the Ey and Ez components of the electric field. Here a relatively lower plasma density ne=0.1nc is used to alleviate nonlinear effects [22], and a stronger magnetic field B=1000T is employed in order to save the computation time. The laser intensity I0=1.37×1016W/cm2 (a0=0.1), and the peak power is 10 PW, with a waist r06800λ. The intensity distribution on the x axis in Fig. 5(b) suggests that two subpulses have FWHM durations 47fs and peak intensities of ImaxI0/2, as expected according to energy conservation. Since the laser intensity now is already weakly relativistic, each subpulse is a little shorter than the initial pulse due to the self-compression of intense laser pulses in plasmas [24]. The first subpulse centered at x1452λ has a group velocity vg,L0.955c, while the second one at x1422λ has vg,R0.935c. They are in good quantitative agreement with the predictions by Eqs. (2) and (3), respectively. The difference between these two group velocities is close to the estimation by Eq. (5).

 figure: Fig. 5.

Fig. 5. (a) Isosurface of intensity I=I0/4 at t=5ps (yellow ellipsoids). Ey and Ez cross sections at z=0 are given on the rear and the bottom of the box, respectively, while the right side displays the transversal distribution of the intensity. (b) The distributions of the Stokes parameters I, Q, U, and V on the x axis, where all parameters are normalized to the initial peak intensity I0. (c, d) The time evolutions of the endpoint of the electric-field vector E in the yz plane in the time intervals (c) 54λ<(ctx)<66λ and (d) 85λ<(ctx)<97λ; the arrows indicate that the electric-field vectors rotate anticlockwise and clockwise, respectively, as viewed along B. Here Ey and Ez are normalized to meωc/e.

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Regardless of the temporal splitting of the pulse, the transversal distribution of the laser intensity is kept as a Gaussian function, as shown in Fig. 5(a). Furthermore, Fig. 5(a) illustrates that the y component of the electric field Ey at t=5ps becomes as strong as the z component Ez, although the pulse is initially polarized along the z axis only. Figure 5(c) shows that the endpoint of the electric-field vector rotates counterclockwise, as viewed along B in the time interval 54λ<(ctx)<66λ. This time interval corresponds to the rising stage of the first subpulse, and the electric-field vector at its falling stage also rotates counterclockwise and thus is omitted here. Therefore, we are convinced that the first subpulse is a LCP pulse. and hence it propagates faster. Conversely, Fig. 5(d) confirms that the endpoint of the electric-field vector rotates clockwise during 85λ<(ctx)<97λ and the second subpulse is a RCP pulse.

4. DISCUSSION

In comparison with a typical Faraday rotation, the extreme case of the Faraday effect reported above offers a new degree of freedom for manipulating ultrashort high-power laser pulses. Therefore, it may form the basis of new types of optical devices, such as magnetized plasma polarizers. Since the laser gain of amplifiers and the loss of resonators such as the Brewster plate are usually polarization dependent, the laser emissions are often LP [12]. To get a circularly polarized pulse, a quarter-wave plate is usually employed [2]. For a high-power laser pulse, however, the quarter-wave plate suffers from the problem of optically induced damage [12]. The state-of-the-art laser facilities under construction will provide a peak power as high as 10 PW [25], where the diameter of the quarter-wave plate should be larger than a few decimeters to avoid the laser-induced damage. To the best of our knowledge, it is extremely challenging to manufacture such a large-diameter quarter-wave plate. Fortunately, one may realize a different type of magnetized plasma polarizer for such high-power lasers based on the above extremely strong Faraday effect. Thanks to the ultrahigh damage threshold of plasmas, this magnetized plasma polarizer is nearly free from laser-induced damage. It is worth noting that in the above 3D simulation the laser pulse already has a peak power of 10 PW, and this pulse has been converted into circularly polarized subpulses by a magnetized plasma on the centimeter scale (a waist of 0.68 cm). The resultant high-power circularly polarized pulses are particularly attractive for laser-driven ion acceleration [26,27], optical control of mesoscopic objects [28], and the ultrahigh acceleration of plasma blocks for fusion ignition [1719,29].

Although the magnetized plasma polarizer is nearly free from laser-induced damage, it also has its own limitations due to nonlinear effects in intense laser–plasma interactions [22,23]. Above all, the laser pulse may collapse at a distance zR(P/Pc)1/2 if its power exceeds the critical power for relativistic self-focusing (Pc17.5nc/ne GW) [22], where zR=πr02/λ is the Rayleigh length. Therefore, the distance for the magnetic splitting (cts) must be shorter than zR(P/Pc)1/2. Using Eqs. (4) and (5), we get

r02ωcω(nenc)1/2>λtpvg,R2π[P17.5GW]1/2.
This prescribes a lower limit for the pulse waist r0. Assuming ωc=0.01ω (B100T) and ne/nc=0.1, we find that a waist r0>1700λ is required for the magnetic splitting of a 500 fs 10 PW laser pulse. Setting a0=0.1 for a 10 PW laser pulse, we will have a pulse waist r06800λ that satisfies the above requirement well. With r06800λ, the Rayleigh length zR1.44×108λ. On the other hand, such a large waist and a long Rayleigh length are also crucial in postponing the self-modulational instability, which is due to the laser-driven plasma wakefield and becomes significant at the time scale of laser self-focusing [22,30,31]. Second, besides the relativistic self-focusing, the laser pulse could also be focused by a transversely inhomogeneous plasma with dn(r)/dr>0 or defocused with dn(r)/dr<0. Analogous to the geometric optics picture of self-focusing in Ref. [22], we get Δvp/cΔne/2nc, where Δvp (Δne) is the difference between the phase velocities (plasma densities) at the center and at the edge of the pulse. Then the focusing (or defocusing) angle of the laser pulse is given by αΔvp/c=Δne/2nc. Further, the condition α<r0/cts should be satisfied in order to split the laser pulse before it is focused (or defocused). Combining this condition with Eqs. (4) and (5), we get
Δnene<8ne(r0ωc)2nc(ctpω)2.
Under the conditions ωc=0.01ω and ne/nc=0.1, it is required that Δne/ne<16.2% for the magnetic splitting of a 500 fs 10 PW laser pulse with a waist r06800λ. Similarly, we get the difference in phase velocity due to the transverse inhomogeneity of the magnetic field as Δvp/cneΔωc/2ncω, where Δωc is the difference between the magnetic fields at the center and at the edge of the pulse. In the case of ne=0.1nc and ωc/ω=0.01, we find that Δvp/c0.0005Δωc/ωc will be very small. Consequently, the focusing or defocusing effect due to the transverse inhomogeneity of the magnetic field could be negligible in this case. However, a magnetic field inhomogeneity that is less than a few tens of percentage points would be of great benefit to the quality of the resultant LCP and RCP subpulses. The magnetic splitting of laser pulses should not be sensitive to the longitudinal inhomogeneity of plasma density or magnetic field. For a longitudinally inhomogeneous plasma or/and magnetic field, we find that the distance between the peaks of LCP and RCP subpulses is close to ΔvgdtΔvgdx/cne(x)B(x)dx, and the magnetic splitting emerges if this distance is larger than ctp. Third, if gaseous targets are used, one should also take into account the nonlinear effects due to ionization and Kerr nonlinearity. The former could induce a defocusing effect, since usually more electrons are produced via ionization on the laser axis, while the latter could induce a self-focusing effect, since the higher intensity at the pulse center leads to a larger refractive index. It is worth noting that these nonlinear effects sometimes may counteract each other. For instance, a plasma channel as long as a few kilometers in the atmosphere could be created if the Kerr effect balances the diffraction and the ionization-induced defocusing [22,32].

Due to the nonlinear effects discussed above, the laser pulse will lose energy as it propagates in plasma, even if the collisional damping is ignored. From the simulations, we find that it is crucial to set a01 and nenc in order to reduce the collisionless losses. Therefore, we use a0=0.1 and ne=0.1nc in the 3D simulation, shown in Fig. 5. Then about 95.092% of the laser energy can be preserved in the LCP (48.069%) and RCP (47.023%) pulses. In particular, only about 0.062% of the laser energy is lost after t=100fs when the pulse propagates inside the plasma; another 4.846% of the laser energy is lost near the vacuum–plasma interface before t=100fs. Therefore, one can expect that the collisionless losses can be controlled at a level of a few percentage points with a much longer propagation distance when a relatively weaker magnetic field (100T) and a longer laser pulse (500fs) are used. The collisional losses, which are not treated in our PIC simulations, can be estimated as Kib=1exp(κibL) [33], where Lcts is the distance required for the magnetic splitting and κibνei(ne/nc)2(1ne/nc)1/2/c is the spatial damping rate by inverse bremsstrahlung. At high laser intensities, e.g., I>1015W/cm2, the electron–ion collision frequency should be modified as νeiZie4nelnΛ/(4πϵ02me2veff3) [22,33,34], where Zi is the ionization state, lnΛ is the Coulomb logarithm, and the effective electron thermal velocity veff=(vte2+vos2)1/2a0c is defined by the electron thermal velocity vte and the electron oscillatory velocity vosa0c in the laser field. Assuming ωc=0.01ω, ne=0.1nc, and ZilnΛ10, we get Kib7.2% for a 500 fs laser pulse with a0=0.1. With a decreasing plasma density, we find that both the collisionless losses and the collisional losses can be reduced. With a decreasing laser intensity, however, the collisional losses will increase, although the collisionless losses can be reduced. A moderate laser intensity of 1016W/cm2 (a00.1) may be appropriate to keep both the collisonal and collisionless losses at a tolerable level.

Besides its applications in optical devices, this extremely strong Faraday effect may be applied to measure ultra-strong magnetic fields. Although the Faraday rotation is widely used in the measurement of magnetic fields, it essentially has three limitations. First, the magnetic field should be small enough (ωcω) to guarantee its linear relation with the Faraday rotation angle. Second, there may be an n×180° ambiguity of the Faraday rotation angle. Third, the exact information for the initial polarization direction is required. In laser-produced plasmas with strong magnetic fields (B1000T) [15,16], sometimes it may be difficult to meet all the above requirements simultaneously. In these scenarios, however, the probe pulse may split into two circularly polarized pulses due to the extremely strong Faraday effect if the plasma thickness >100λnc/ne (the corresponding areal density ρR>104g/cm2). Then the magnetic field could be estimated from the time delay between two resultant circularly polarized pulses. Therefore, this extremely strong Faraday effect could be a powerful alternative to the conventional Faraday rotation in the measurement of ultra-strong magnetic fields in plasmas. Such strongly magnetized plasmas may be encountered in magnetically assisted fast ignition [35], which is advantageous in depositing the laser energy into the core of the fuel target in inertial confinement fusion.

It is worth pointing out that the higher the plasma density is, the more obvious this extremely strong Faraday effect is. This is because the light is slowed down more obviously and the difference in the group velocities is larger at a higher plasma density. We notice that the temporal splitting of laser pulses can also be achieved in other slow-light media such as atomic vapors [36], although there the pulse duration is usually longer than nanosecond. In contrast to a bandwidth of gigahertz for a tunable pulse with atomic vapors [37], femtosecond laser pulses with terahertz (THz) bandwidths can be manipulated by the magnetized plasmas. In principle, this extremely strong Faraday effect can be applied to manipulate electromagnetic radiation from radio waves to gamma rays for numerious potential applications [3840]. However, this effect is observable only when ωc/ω is not too small, which presents a practical limit for experiments at a high wave frequency, while for THz radiation, magnetic fields on the order of tesla are already high enough to achieve this effect.

5. CONCLUSION

In summary, an extreme case of the Faraday effect has been found in magnetized plasma due to its remarkable chiral dichroism. With this, the magnetic splitting of a LP short laser pulse into a LCP pulse and a RCP pulse can be realized. This opens the way for advanced applications, such as a magnetized plasma polarizer. The latter could allow the generation of circularly polarized laser pulses as high power as 10 PW in up-to-date laser facilities. Moreover, this eliminates some limitations of the Faraday rotation for the measurement of magnetic fields, thus offering a way to measure ultrahigh magnetic fields broadly existing in objects in the universe and in laser–matter interactions in laboratories.

Funding

National Basic Research Program of China (2013CBA01504); National Natural Science Foundation of China (NSFC) (11129503, 11374210, 11405108, 11421064, 11675108); National 1000 Youth Talent Project of China; Leverhulme Trust.

Acknowledgment

The authors thank L. J. Qian, J. Q. Zhu, J. Fuchs, S. Chen, Y. T. Li, Z. Zhang, T. Sano, H. C. Wu, X. H. Yuan, Y. P. Chen, G. Q. Xie, and L. L. Zhao for fruitful discussions. Simulations have been carried out at the Pi cluster of Shanghai Jiao Tong University.

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16. S. Fujioka, Z. Zhang, K. Ishihara, K. Shigemori, Y. Hironaka, T. Johzaki, A. Sunahara, N. Yamamoto, H. Nakashima, T. Watanabe, H. Shiraga, H. Nishimura, and H. Azechi, “Kilotesla magnetic field due to a capacitor-coil target driven by high power laser,” Sci. Rep. 3, 1170 (2013). [CrossRef]  

17. H. Hora, G. Korn, L. Giuffrida, D. Margarone, A. Picciotto, J. Krasa, K. Jungwirth, J. Ullschmied, P. Lalousis, S. Eliezer, G. H. Miley, S. Moustaizis, and G. Mourou, “Fusion energy using avalanche increased boron reactions for block ignition by ultrahigh power picosecond laser pulses,” Laser Part. Beams 33, 607–619 (2015). [CrossRef]  

18. S. Eliezer, H. Hora, G. Korn, N. Nissim, and J. M. M. Val, “Avalanche proton-boron fusion based on elastic nuclear collisions,” Phys. Plasmas 23, 050704 (2016). [CrossRef]  

19. G. H. Miley, H. Hora, and G. Kirchhoff, “Reactor for boron fusion with picosecond ultrahigh power laser pulses and ultrahigh magnetic field trapping,” J. Phys. 717, 012095 (2016). [CrossRef]  

20. R. A. Fonseca, L. O. Silva, F. S. Tsung, V. K. Decyk, W. Lu, C. Ren, W. B. Mori, S. Deng, S. Lee, T. Katsouleas, and J. C. Adam, “OSIRIS, a three-dimensional fully relativistic particle in cell code for modeling plasma based accelerators,” Lect. Notes Comput. Sci. 2331, 342–351 (2002). [CrossRef]  

21. J. Tinbergen, Astronomical Polarimetry (Cambridge University, 1996).

22. P. Gibbon, Short Pulse Laser Interactions with Matter (Imperial College, 2000).

23. H. Hora, Laser Plasma Physics (SPIE, 2016).

24. O. Shorokhov, A. Pukhov, and I. Kostyukov, “Self-compression of laser pulses in plasma,” Phys. Rev. Lett. 91, 265002 (2003). [CrossRef]  

25. D. N. Papadopoulos, J. P. Zou, C. Le Blanc, G. Chériaux, P. Georges, F. Druon, G. Mennerat, P. Ramirez, L. Martin, A. Fréneaux, A. Beluze, N. Lebas, P. Monot, F. Mathieu, and P. Audebert, “The Apollon 10 PW laser: experimental and theoretical investigation of the temporal characteristics,” High Power Laser Sci. Eng. 4, e34 (2016). [CrossRef]  

26. H. Daido, M. Nishiuchi, and A. S. Pirozhkov, “Review of laser-driven ion sources and their applications,” Rep. Prog. Phys. 75, 056401 (2012). [CrossRef]  

27. A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751–793 (2013). [CrossRef]  

28. Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013). [CrossRef]  

29. S. M. Weng, M. Liu, Z. M. Sheng, M. Murakami, M. Chen, L. L. Yu, and J. Zhang, “Dense blocks of energetic ions driven by multi-petawatt lasers,” Sci. Rep. 6, 22150 (2016). [CrossRef]  

30. T. M. Antonsen Jr. and P. Mora, “Self-focusing and Raman scattering of laser pulses in tenuous plasmas,” Phys. Rev. Lett. 69, 2204–2207 (1992). [CrossRef]  

31. E. Esarey, J. Krall, and P. Sprangle, “Envelope analysis of intense laser pulse self-modulation in plasmas,” Phys. Rev. Lett. 72, 2887–2890 (1994). [CrossRef]  

32. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003). [CrossRef]  

33. S. Eliezer, The Interaction of High-Power Lasers with Plasmas (Institute of Physics, 2002).

34. S. M. Weng, Z. M. Sheng, and J. Zhang, “Inverse bremsstrahlung absorption with nonlinear effects of high laser intensity and non-Maxwellian distribution,” Phys. Rev. E 80, 056406 (2009). [CrossRef]  

35. W. M. Wang, P. Gibbon, Z. M. Sheng, and Y. T. Li, “Magnetically assisted fast ignition,” Phys. Rev. Lett. 114, 015001 (2015). [CrossRef]  

36. D. Grischkowsky, “Adiabatic following and slow optical pulse propagation in rubidium vapor,” Phys. Rev. A 7, 2096–2102 (1973). [CrossRef]  

37. R. M. Camacho, M. V. Pack, J. C. Howell, A. Schweinsberg, and R. W. Boyd, “Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow light in cesium vapor,” Phys. Rev. Lett. 98, 153601 (2007). [CrossRef]  

38. P. Gibbon, “Harmonic generation by femtosecond laser-solid interaction: a coherent “water-window” light source?” Phys. Rev. Lett. 76, 50–53 (1996). [CrossRef]  

39. U. Teubner and P. Gibbon, “High-order harmonics from laser-irradiated plasma surfaces,” Rev. Mod. Phys. 81, 445–479 (2009). [CrossRef]  

40. W. M. Wang, P. Gibbon, Z. M. Sheng, and Y. T. Li, “Tunable circularly polarized terahertz radiation from magnetized gas plasma,” Phys. Rev. Lett. 114, 253901 (2015). [CrossRef]  

References

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  7. C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, “Plasma mirrors for ultrahigh-intensity optics,” Nat. Phys. 3, 424–429 (2007).
    [Crossref]
  8. R. M. G. M. Trines, F. Fiúza, R. Bingham, R. A. Fonseca, L. O. Silva, R. A. Cairns, and P. A. Norreys, “Simulations of efficient Raman amplification into the multipetawatt regime,” Nat. Phys. 7, 87–92 (2011).
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    [Crossref]
  16. S. Fujioka, Z. Zhang, K. Ishihara, K. Shigemori, Y. Hironaka, T. Johzaki, A. Sunahara, N. Yamamoto, H. Nakashima, T. Watanabe, H. Shiraga, H. Nishimura, and H. Azechi, “Kilotesla magnetic field due to a capacitor-coil target driven by high power laser,” Sci. Rep. 3, 1170 (2013).
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  17. H. Hora, G. Korn, L. Giuffrida, D. Margarone, A. Picciotto, J. Krasa, K. Jungwirth, J. Ullschmied, P. Lalousis, S. Eliezer, G. H. Miley, S. Moustaizis, and G. Mourou, “Fusion energy using avalanche increased boron reactions for block ignition by ultrahigh power picosecond laser pulses,” Laser Part. Beams 33, 607–619 (2015).
    [Crossref]
  18. S. Eliezer, H. Hora, G. Korn, N. Nissim, and J. M. M. Val, “Avalanche proton-boron fusion based on elastic nuclear collisions,” Phys. Plasmas 23, 050704 (2016).
    [Crossref]
  19. G. H. Miley, H. Hora, and G. Kirchhoff, “Reactor for boron fusion with picosecond ultrahigh power laser pulses and ultrahigh magnetic field trapping,” J. Phys. 717, 012095 (2016).
    [Crossref]
  20. R. A. Fonseca, L. O. Silva, F. S. Tsung, V. K. Decyk, W. Lu, C. Ren, W. B. Mori, S. Deng, S. Lee, T. Katsouleas, and J. C. Adam, “OSIRIS, a three-dimensional fully relativistic particle in cell code for modeling plasma based accelerators,” Lect. Notes Comput. Sci. 2331, 342–351 (2002).
    [Crossref]
  21. J. Tinbergen, Astronomical Polarimetry (Cambridge University, 1996).
  22. P. Gibbon, Short Pulse Laser Interactions with Matter (Imperial College, 2000).
  23. H. Hora, Laser Plasma Physics (SPIE, 2016).
  24. O. Shorokhov, A. Pukhov, and I. Kostyukov, “Self-compression of laser pulses in plasma,” Phys. Rev. Lett. 91, 265002 (2003).
    [Crossref]
  25. D. N. Papadopoulos, J. P. Zou, C. Le Blanc, G. Chériaux, P. Georges, F. Druon, G. Mennerat, P. Ramirez, L. Martin, A. Fréneaux, A. Beluze, N. Lebas, P. Monot, F. Mathieu, and P. Audebert, “The Apollon 10 PW laser: experimental and theoretical investigation of the temporal characteristics,” High Power Laser Sci. Eng. 4, e34 (2016).
    [Crossref]
  26. H. Daido, M. Nishiuchi, and A. S. Pirozhkov, “Review of laser-driven ion sources and their applications,” Rep. Prog. Phys. 75, 056401 (2012).
    [Crossref]
  27. A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751–793 (2013).
    [Crossref]
  28. Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
    [Crossref]
  29. S. M. Weng, M. Liu, Z. M. Sheng, M. Murakami, M. Chen, L. L. Yu, and J. Zhang, “Dense blocks of energetic ions driven by multi-petawatt lasers,” Sci. Rep. 6, 22150 (2016).
    [Crossref]
  30. T. M. Antonsen and P. Mora, “Self-focusing and Raman scattering of laser pulses in tenuous plasmas,” Phys. Rev. Lett. 69, 2204–2207 (1992).
    [Crossref]
  31. E. Esarey, J. Krall, and P. Sprangle, “Envelope analysis of intense laser pulse self-modulation in plasmas,” Phys. Rev. Lett. 72, 2887–2890 (1994).
    [Crossref]
  32. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
    [Crossref]
  33. S. Eliezer, The Interaction of High-Power Lasers with Plasmas (Institute of Physics, 2002).
  34. S. M. Weng, Z. M. Sheng, and J. Zhang, “Inverse bremsstrahlung absorption with nonlinear effects of high laser intensity and non-Maxwellian distribution,” Phys. Rev. E 80, 056406 (2009).
    [Crossref]
  35. W. M. Wang, P. Gibbon, Z. M. Sheng, and Y. T. Li, “Magnetically assisted fast ignition,” Phys. Rev. Lett. 114, 015001 (2015).
    [Crossref]
  36. D. Grischkowsky, “Adiabatic following and slow optical pulse propagation in rubidium vapor,” Phys. Rev. A 7, 2096–2102 (1973).
    [Crossref]
  37. R. M. Camacho, M. V. Pack, J. C. Howell, A. Schweinsberg, and R. W. Boyd, “Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow light in cesium vapor,” Phys. Rev. Lett. 98, 153601 (2007).
    [Crossref]
  38. P. Gibbon, “Harmonic generation by femtosecond laser-solid interaction: a coherent “water-window” light source?” Phys. Rev. Lett. 76, 50–53 (1996).
    [Crossref]
  39. U. Teubner and P. Gibbon, “High-order harmonics from laser-irradiated plasma surfaces,” Rev. Mod. Phys. 81, 445–479 (2009).
    [Crossref]
  40. W. M. Wang, P. Gibbon, Z. M. Sheng, and Y. T. Li, “Tunable circularly polarized terahertz radiation from magnetized gas plasma,” Phys. Rev. Lett. 114, 253901 (2015).
    [Crossref]

2016 (5)

L. L. Yu, Y. Zhao, L. J. Qian, M. Chen, S. M. Weng, Z. M. Sheng, D. A. Jaroszynski, W. B. Mori, and J. Zhang, “Plasma optical modulators for intense lasers,” Nat. Commun. 7, 11893 (2016).
[Crossref]

S. Eliezer, H. Hora, G. Korn, N. Nissim, and J. M. M. Val, “Avalanche proton-boron fusion based on elastic nuclear collisions,” Phys. Plasmas 23, 050704 (2016).
[Crossref]

G. H. Miley, H. Hora, and G. Kirchhoff, “Reactor for boron fusion with picosecond ultrahigh power laser pulses and ultrahigh magnetic field trapping,” J. Phys. 717, 012095 (2016).
[Crossref]

D. N. Papadopoulos, J. P. Zou, C. Le Blanc, G. Chériaux, P. Georges, F. Druon, G. Mennerat, P. Ramirez, L. Martin, A. Fréneaux, A. Beluze, N. Lebas, P. Monot, F. Mathieu, and P. Audebert, “The Apollon 10 PW laser: experimental and theoretical investigation of the temporal characteristics,” High Power Laser Sci. Eng. 4, e34 (2016).
[Crossref]

S. M. Weng, M. Liu, Z. M. Sheng, M. Murakami, M. Chen, L. L. Yu, and J. Zhang, “Dense blocks of energetic ions driven by multi-petawatt lasers,” Sci. Rep. 6, 22150 (2016).
[Crossref]

2015 (4)

W. M. Wang, P. Gibbon, Z. M. Sheng, and Y. T. Li, “Magnetically assisted fast ignition,” Phys. Rev. Lett. 114, 015001 (2015).
[Crossref]

W. M. Wang, P. Gibbon, Z. M. Sheng, and Y. T. Li, “Tunable circularly polarized terahertz radiation from magnetized gas plasma,” Phys. Rev. Lett. 114, 253901 (2015).
[Crossref]

H. Hora, G. Korn, L. Giuffrida, D. Margarone, A. Picciotto, J. Krasa, K. Jungwirth, J. Ullschmied, P. Lalousis, S. Eliezer, G. H. Miley, S. Moustaizis, and G. Mourou, “Fusion energy using avalanche increased boron reactions for block ignition by ultrahigh power picosecond laser pulses,” Laser Part. Beams 33, 607–619 (2015).
[Crossref]

X. H. Yang, W. Yu, H. Xu, M. Y. Yu, Z. Y. Ge, B. B. Xu, H. B. Zhuo, Y. Y. Ma, F. Q. Shao, and M. Borghesi, “Propagation of intense laser pulses in strongly magnetized plasmas,” Appl. Phys. Lett. 106, 224103 (2015).
[Crossref]

2013 (4)

M. Liu and X. Zhang, “Nano-optics: plasmon-boosted magneto-optics,” Nat. Photonics 7, 429–430 (2013).
[Crossref]

S. Fujioka, Z. Zhang, K. Ishihara, K. Shigemori, Y. Hironaka, T. Johzaki, A. Sunahara, N. Yamamoto, H. Nakashima, T. Watanabe, H. Shiraga, H. Nishimura, and H. Azechi, “Kilotesla magnetic field due to a capacitor-coil target driven by high power laser,” Sci. Rep. 3, 1170 (2013).
[Crossref]

A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751–793 (2013).
[Crossref]

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

2012 (1)

H. Daido, M. Nishiuchi, and A. S. Pirozhkov, “Review of laser-driven ion sources and their applications,” Rep. Prog. Phys. 75, 056401 (2012).
[Crossref]

2011 (1)

R. M. G. M. Trines, F. Fiúza, R. Bingham, R. A. Fonseca, L. O. Silva, R. A. Cairns, and P. A. Norreys, “Simulations of efficient Raman amplification into the multipetawatt regime,” Nat. Phys. 7, 87–92 (2011).
[Crossref]

2009 (2)

S. M. Weng, Z. M. Sheng, and J. Zhang, “Inverse bremsstrahlung absorption with nonlinear effects of high laser intensity and non-Maxwellian distribution,” Phys. Rev. E 80, 056406 (2009).
[Crossref]

U. Teubner and P. Gibbon, “High-order harmonics from laser-irradiated plasma surfaces,” Rev. Mod. Phys. 81, 445–479 (2009).
[Crossref]

2007 (2)

R. M. Camacho, M. V. Pack, J. C. Howell, A. Schweinsberg, and R. W. Boyd, “Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow light in cesium vapor,” Phys. Rev. Lett. 98, 153601 (2007).
[Crossref]

C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, “Plasma mirrors for ultrahigh-intensity optics,” Nat. Phys. 3, 424–429 (2007).
[Crossref]

2006 (1)

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006).
[Crossref]

2005 (1)

A. Brandenburg and K. Subramanian, “Astrophysical magnetic fields and nonlinear dynamo theory,” Phys. Rep. 417, 1–209 (2005).
[Crossref]

2004 (1)

U. Wagner, M. Tatarakis, A. Gopal, F. N. Bee, E. L. Clark, A. E. Dangor, R. G. Evans, M. G. Haines, S. P. D. Mangles, P. A. Norreys, M. S. Wei, M. Zepf, and K. Krushelnick, “Laboratory measurements of 0.7 GG magnetic fields generated during high-intensity laser interactions with dense plasmas,” Phys. Rev. E 70, 026401 (2004).
[Crossref]

2003 (2)

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref]

O. Shorokhov, A. Pukhov, and I. Kostyukov, “Self-compression of laser pulses in plasma,” Phys. Rev. Lett. 91, 265002 (2003).
[Crossref]

2002 (1)

R. A. Fonseca, L. O. Silva, F. S. Tsung, V. K. Decyk, W. Lu, C. Ren, W. B. Mori, S. Deng, S. Lee, T. Katsouleas, and J. C. Adam, “OSIRIS, a three-dimensional fully relativistic particle in cell code for modeling plasma based accelerators,” Lect. Notes Comput. Sci. 2331, 342–351 (2002).
[Crossref]

1996 (1)

P. Gibbon, “Harmonic generation by femtosecond laser-solid interaction: a coherent “water-window” light source?” Phys. Rev. Lett. 76, 50–53 (1996).
[Crossref]

1994 (1)

E. Esarey, J. Krall, and P. Sprangle, “Envelope analysis of intense laser pulse self-modulation in plasmas,” Phys. Rev. Lett. 72, 2887–2890 (1994).
[Crossref]

1992 (1)

T. M. Antonsen and P. Mora, “Self-focusing and Raman scattering of laser pulses in tenuous plasmas,” Phys. Rev. Lett. 69, 2204–2207 (1992).
[Crossref]

1985 (1)

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[Crossref]

1973 (1)

D. Grischkowsky, “Adiabatic following and slow optical pulse propagation in rubidium vapor,” Phys. Rev. A 7, 2096–2102 (1973).
[Crossref]

1846 (1)

M. Faraday, “On the magnetization of light and the illumination of magnetic lines of force,” Philos. Trans. R. Soc. London 136, 1–20 (1846).
[Crossref]

Adam, J. C.

R. A. Fonseca, L. O. Silva, F. S. Tsung, V. K. Decyk, W. Lu, C. Ren, W. B. Mori, S. Deng, S. Lee, T. Katsouleas, and J. C. Adam, “OSIRIS, a three-dimensional fully relativistic particle in cell code for modeling plasma based accelerators,” Lect. Notes Comput. Sci. 2331, 342–351 (2002).
[Crossref]

André, Y.-B.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref]

Antonsen, T. M.

T. M. Antonsen and P. Mora, “Self-focusing and Raman scattering of laser pulses in tenuous plasmas,” Phys. Rev. Lett. 69, 2204–2207 (1992).
[Crossref]

Arita, Y.

Y. Arita, M. Mazilu, and K. Dholakia, “Laser-induced rotation and cooling of a trapped microgyroscope in vacuum,” Nat. Commun. 4, 2374 (2013).
[Crossref]

Audebert, P.

D. N. Papadopoulos, J. P. Zou, C. Le Blanc, G. Chériaux, P. Georges, F. Druon, G. Mennerat, P. Ramirez, L. Martin, A. Fréneaux, A. Beluze, N. Lebas, P. Monot, F. Mathieu, and P. Audebert, “The Apollon 10 PW laser: experimental and theoretical investigation of the temporal characteristics,” High Power Laser Sci. Eng. 4, e34 (2016).
[Crossref]

C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, “Plasma mirrors for ultrahigh-intensity optics,” Nat. Phys. 3, 424–429 (2007).
[Crossref]

Azechi, H.

S. Fujioka, Z. Zhang, K. Ishihara, K. Shigemori, Y. Hironaka, T. Johzaki, A. Sunahara, N. Yamamoto, H. Nakashima, T. Watanabe, H. Shiraga, H. Nishimura, and H. Azechi, “Kilotesla magnetic field due to a capacitor-coil target driven by high power laser,” Sci. Rep. 3, 1170 (2013).
[Crossref]

Bee, F. N.

U. Wagner, M. Tatarakis, A. Gopal, F. N. Bee, E. L. Clark, A. E. Dangor, R. G. Evans, M. G. Haines, S. P. D. Mangles, P. A. Norreys, M. S. Wei, M. Zepf, and K. Krushelnick, “Laboratory measurements of 0.7 GG magnetic fields generated during high-intensity laser interactions with dense plasmas,” Phys. Rev. E 70, 026401 (2004).
[Crossref]

Beluze, A.

D. N. Papadopoulos, J. P. Zou, C. Le Blanc, G. Chériaux, P. Georges, F. Druon, G. Mennerat, P. Ramirez, L. Martin, A. Fréneaux, A. Beluze, N. Lebas, P. Monot, F. Mathieu, and P. Audebert, “The Apollon 10 PW laser: experimental and theoretical investigation of the temporal characteristics,” High Power Laser Sci. Eng. 4, e34 (2016).
[Crossref]

Bingham, R.

R. M. G. M. Trines, F. Fiúza, R. Bingham, R. A. Fonseca, L. O. Silva, R. A. Cairns, and P. A. Norreys, “Simulations of efficient Raman amplification into the multipetawatt regime,” Nat. Phys. 7, 87–92 (2011).
[Crossref]

Borghesi, M.

X. H. Yang, W. Yu, H. Xu, M. Y. Yu, Z. Y. Ge, B. B. Xu, H. B. Zhuo, Y. Y. Ma, F. Q. Shao, and M. Borghesi, “Propagation of intense laser pulses in strongly magnetized plasmas,” Appl. Phys. Lett. 106, 224103 (2015).
[Crossref]

A. Macchi, M. Borghesi, and M. Passoni, “Ion acceleration by superintense laser-plasma interaction,” Rev. Mod. Phys. 85, 751–793 (2013).
[Crossref]

Bougeard, M.

C. Thaury, F. Quéré, J.-P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Réau, P. d’Oliveira, P. Audebert, R. Marjoribanks, and P. Martin, “Plasma mirrors for ultrahigh-intensity optics,” Nat. Phys. 3, 424–429 (2007).
[Crossref]

Bourayou, R.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301, 61–64 (2003).
[Crossref]

Boyd, R. W.

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Supplementary Material (1)

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» Visualization 1       Movie version of Fig. 5(a).

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Figures (5)

Fig. 1.
Fig. 1. Sketch of the magnetic splitting of an ultrashort LP laser pulse, which is incident along the magnetic field B into plasma. The incident LP pulse will split into RCP and LCP subpulses due to their differential group velocities. The RCP subpulse follows the LCP subpulse in time. Here the electric field vector of the RCP pulse at a fixed position rotates clockwise in time as viewed along the wave vector of the laser pulse, and vice versa for the LCP pulse.
Fig. 2.
Fig. 2. (a, b) Group velocities from Eqs. (2) and (3) for the LCP and RCP waves, respectively; (c) the difference in the group velocities, (d) the minimum field ( ω c , min ) required for an obvious magnetic splitting as a function of the frequency spread ( Δ ω ) of the pulse.
Fig. 3.
Fig. 3. Stokes parameters from 1D PIC simulations with varying laser pulse duration t p and magnetic field B (a) at t = 300 ps with t p = 500 fs ( Δ ω / ω 0 0.0029 ) and B = 50 T ( ω c / ω 0 0.005 ), (b) at t = 30 ps with t p = 500 fs and B = 500 T , (c) at t = 3 ps with t p = 50 fs and B = 500 T , and (d) at t = 30 ps with t p = 50 fs and B = 50 T . The Stokes parameter I denotes the intensity regardless of polarization, Q and U describe the state of linear polarization, and V represents the circular polarization [21]. All parameters are normalized to the instantaneous peak intensity I max . Here the laser intensity is low enough (the dimensionless amplitude a | e E / ω m e c | = 0.01 ) that the nonlinear effects [22,23] due to the laser field itself can be ignored.
Fig. 4.
Fig. 4. Stokes parameters at t = 333 fs with an extremely strong magnetic field B = 6000 T ( ω c / ω 0 0.6 ). Other parameters are the same as those in Fig. 3(c).
Fig. 5.
Fig. 5. (a) Isosurface of intensity I = I 0 / 4 at t = 5 ps (yellow ellipsoids). E y and E z cross sections at z = 0 are given on the rear and the bottom of the box, respectively, while the right side displays the transversal distribution of the intensity. (b) The distributions of the Stokes parameters I , Q , U , and V on the x axis, where all parameters are normalized to the initial peak intensity I 0 . (c, d) The time evolutions of the endpoint of the electric-field vector E in the y z plane in the time intervals (c)  54 λ < ( c t x ) < 66 λ and (d)  85 λ < ( c t x ) < 97 λ ; the arrows indicate that the electric-field vectors rotate anticlockwise and clockwise, respectively, as viewed along B . Here E y and E z are normalized to m e ω c / e .

Equations (9)

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c 2 k 2 ω 2 = 1 ω p 2 ω 2 ( 1 ± ω c / ω ) ,
v g , L c = ( 1 ω p 2 / ω 2 1 + ω c / ω ) 1 / 2 [ 1 ω c ω p 2 / ω 3 2 ( 1 + ω c / ω ) 2 ] 1 ,
v g , R c = ( 1 ω p 2 / ω 2 1 ω c / ω ) 1 / 2 [ 1 + ω c ω p 2 / ω 3 2 ( 1 ω c / ω ) 2 ] 1 ,
t s = v g , R Δ v g t p ,
Δ v g c 2 n e n c ω c ω ,
v g , R | ω = ω 0 + Δ ω / 2 < v g , L | ω = ω 0 Δ ω / 2 .
ω c ω c , min = Δ ω
r 0 2 ω c ω ( n e n c ) 1 / 2 > λ t p v g , R 2 π [ P 17.5 GW ] 1 / 2 .
Δ n e n e < 8 n e ( r 0 ω c ) 2 n c ( c t p ω ) 2 .

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