## 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 [5–8].

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 $B\ge 50\text{\hspace{0.17em}}\mathrm{T}$. 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.

## 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]

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

In the case of an ultrashort laser pulse, however, the frequency spread must be taken into account. For instance, $\mathrm{\Delta}\omega /{\omega}_{0}\ge 0.441\tau /{t}_{p}$ holds for a Gaussian pulse [12], where $\tau =2\pi /{\omega}_{0}$ is the laser wave period and ${\omega}_{0}$, $\mathrm{\Delta}\omega $, and ${t}_{p}$ 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

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 [17–19]. 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 $x\ge 0$. The initial LP pulses are polarized along the $z$ axis with $\lambda =1\text{\hspace{0.17em}}\mathrm{\mu m}$. 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 $\mathrm{35,000}\lambda $, the spatial and temporal resolutions are $\mathrm{\Delta}x=\lambda /16$ and $\mathrm{\Delta}t\simeq \mathrm{\Delta}x/c$, each cell has 16 macroparticles, and the electron density ${n}_{e}=0.5{n}_{c}$. In a 3D simulation, the simulation box has a size of $210\lambda \times \mathrm{24,000}\lambda \times \mathrm{24,000}\lambda $. The spatial resolutions are $\mathrm{\Delta}x=\lambda /16$ and $\mathrm{\Delta}y=\mathrm{\Delta}z=100\lambda $, the temporal resolution is $\mathrm{\Delta}t\simeq \mathrm{\Delta}x/c$, each cell has four macroparticles, and the electron density ${n}_{e}=0.1{n}_{c}$.

Figure 3 compares 1D simulation results with varying laser pulse duration ${t}_{p}$ and magnetic field $B$. In Fig. 3(a), the magnetic splitting condition (${\omega}_{c}>\mathrm{\Delta}\omega $) holds well with ${t}_{p}=500\text{\hspace{0.17em}}\mathrm{fs}$ and $B=50\text{\hspace{0.17em}}\mathrm{T}$. Consequently, the initial pulse splits into two discrete subpulses at $t=300$ picosecond (ps). The first subpulse peaking at $x\simeq 63145\lambda $ is LCP, since its Stokes parameter $V<0$, while the second subpulse peaking at $x\simeq 62735\lambda $ 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 $t=30\text{\hspace{0.17em}}\mathrm{ps}$. 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 $\mathrm{\Delta}\omega \simeq 0.029\omega >{\omega}_{c}\simeq 0.005\omega $ in this relatively weak magnetic field. As illustrated in Fig. 3(d), at $t=30\text{\hspace{0.17em}}\mathrm{ps}$ 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 $\mathrm{\Delta}\omega >{\omega}_{c}$.

Figure 4 displays the simulation result with an extremely strong magnetic field of $B=6000\text{\hspace{0.17em}}\mathrm{T}$ (${\omega}_{c}/{\omega}_{0}\simeq 0.6$). In this case, it becomes impossible for the RCP wave to propagate into the magnetized plasma, since ${\omega}_{c}/\omega >(1-{\omega}_{p}^{2}/{\omega}^{2})$. Figure 4 shows that the incident LP pulse has been separated into two subpulse as well. However, here only the LCP subpulse (peaking at $x\simeq 88\lambda $ with $V<0$) can propagate into the magnetized plasma. The RCP subpulse (peaking at $x\simeq -100\lambda $ with $V>0$) is completely reflected and propagates backward.

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 $I={I}_{0}/4$ appears as two separate ellipsoids at $t=5\text{\hspace{0.17em}}\mathrm{ps}$ (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 ${E}_{y}$ and ${E}_{z}$ components of the electric field. Here a relatively lower plasma density ${n}_{e}=0.1{n}_{c}$ is used to alleviate nonlinear effects [22], and a stronger magnetic field $B=1000\text{\hspace{0.17em}}\mathrm{T}$ is employed in order to save the computation time. The laser intensity ${I}_{0}=1.37\times {10}^{16}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$ (${a}_{0}=0.1$), and the peak power is 10 PW, with a waist ${r}_{0}\simeq 6800\lambda $. The intensity distribution on the $x$ axis in Fig. 5(b) suggests that two subpulses have FWHM durations $\approx 47\text{\hspace{0.17em}}\mathrm{fs}$ and peak intensities of ${I}_{\mathrm{max}}\approx {I}_{0}/2$, 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 $x\simeq 1452\lambda $ has a group velocity ${v}_{g,L}\simeq 0.955c$, while the second one at $x\simeq 1422\lambda $ has ${v}_{g,R}\simeq 0.935c$. 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).

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 ${E}_{y}$ at $t=5\text{\hspace{0.17em}}\mathrm{ps}$ becomes as strong as the $z$ component ${E}_{z}$, 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 $\mathbf{B}$ in the time interval $54\lambda <(ct-x)<66\lambda $. 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 $85\lambda <(ct-x)<97\lambda $ 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 [17–19,29].

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 $\sim {z}_{R}{(P/{P}_{c})}^{-1/2}$ if its power exceeds the critical power for relativistic self-focusing (${P}_{c}\simeq 17.5{n}_{c}/{n}_{e}$ GW) [22], where ${z}_{R}=\pi {r}_{0}^{2}/\lambda $ is the Rayleigh length. Therefore, the distance for the magnetic splitting ($\sim c{t}_{s}$) must be shorter than ${z}_{R}{(P/{P}_{c})}^{-1/2}$. Using Eqs. (4) and (5), we get

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 ${a}_{0}\ll 1$ and ${n}_{e}\ll {n}_{c}$ in order to reduce the collisionless losses. Therefore, we use ${a}_{0}=0.1$ and ${n}_{e}=0.1{n}_{c}$ 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 $t=100\text{\hspace{0.17em}}\mathrm{fs}$ when the pulse propagates inside the plasma; another 4.846% of the laser energy is lost near the vacuum–plasma interface before $t=100\text{\hspace{0.17em}}\mathrm{fs}$. 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 ($\sim 100\text{\hspace{0.17em}}\mathrm{T}$) and a longer laser pulse ($\sim 500\text{\hspace{0.17em}}\mathrm{fs}$) are used. The collisional losses, which are not treated in our PIC simulations, can be estimated as ${K}_{ib}=1-\mathrm{exp}(-{\kappa}_{ib}L)$ [33], where $L\sim c{t}_{s}$ is the distance required for the magnetic splitting and ${\kappa}_{ib}\simeq {\nu}_{ei}{({n}_{e}/{n}_{c})}^{2}{(1-{n}_{e}/{n}_{c})}^{-1/2}/c$ is the spatial damping rate by inverse bremsstrahlung. At high laser intensities, e.g., $I>{10}^{15}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$, the electron–ion collision frequency should be modified as ${\nu}_{ei}\simeq {Z}_{i}{e}^{4}{n}_{e}\text{\hspace{0.17em}}\mathrm{ln}\text{\hspace{0.17em}}\mathrm{\Lambda}/(4\pi {\u03f5}_{0}^{2}{m}_{e}^{2}{v}_{\text{eff}}^{3})$ [22,33,34], where ${Z}_{i}$ is the ionization state, $\mathrm{ln}\text{\hspace{0.17em}}\mathrm{\Lambda}$ is the Coulomb logarithm, and the effective electron thermal velocity ${v}_{\text{eff}}={({v}_{te}^{2}+{v}_{os}^{2})}^{1/2}\simeq {a}_{0}c$ is defined by the electron thermal velocity ${v}_{te}$ and the electron oscillatory velocity ${v}_{os}\simeq {a}_{0}c$ in the laser field. Assuming ${\omega}_{c}=0.01\omega $, ${n}_{e}=0.1{n}_{c}$, and ${Z}_{i}\text{\hspace{0.17em}}\mathrm{ln}\text{\hspace{0.17em}}\mathrm{\Lambda}\simeq 10$, we get ${K}_{ib}\simeq 7.2\%$ for a 500 fs laser pulse with ${a}_{0}=0.1$. 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 $\sim {10}^{16}\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{cm}}^{2}$ (${a}_{0}\sim 0.1$) 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 (${\omega}_{c}\ll \omega $) to guarantee its linear relation with the Faraday rotation angle. Second, there may be an $n\times 180\xb0$ 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 ($B\sim 1000\text{\hspace{0.17em}}\mathrm{T}$) [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 $>100\lambda {n}_{c}/{n}_{e}$ (the corresponding areal density $\rho R>{10}^{-4}\text{\hspace{0.17em}}\mathrm{g}/{\mathrm{cm}}^{2}$). 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 [38–40]. However, this effect is observable only when ${\omega}_{c}/\omega $ 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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