Polarizing Free Electrons in Optical Near Fields

Polarizing electron beams using light is highly desirable but exceedingly challenging, as the approaches proposed in previous studies using free-space light usually require enormous laser intensities. Here, we propose the use of a transverse electric optical near field, extended on nanostructures, to efficiently polarize an adjacent electron beam by exploiting the strong inelastic electron scattering in phase-matched optical near fields. Intriguingly, the two spin components of an unpolarized incident electron beam—parallel and antiparallel to the electric field—are spin-flipped and inelastically scattered to different energy states, providing an analog of the Stern-Gerlach experiment in the energy dimension. Our calculations show that when a dramatically reduced laser intensity of $\sim {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and a short interaction length of $16\mathrm{\mu }\mathrm{m}$ are used, an unpolarized incident electron beam interacting with the excited optical near field can produce two spin-polarized electron beams, both exhibiting near unity spin purity and a 6% brightness relative to the input beam. Our findings are important for optical control of free-electron spins, preparation of spin-polarized electron beams, and applications in material science and high-energy physics.

Figure 1

Illustration of the process of polarizing electrons in an optical near field. A wide TE laser beam (blue arrow) is normally incident on a nanowire array, with the electric field oriented parallel to the nanowires. This produces a near field that is periodically modulated (bottom panel). Unpolarized electrons passing through the optical near field are scattered by either absorbing or emitting a photon of energy $E_p$, with the spins of the scattered electrons aligned along $+\widehat{y}$ and $-\widehat{y}$ (red circular arrows), respectively.

Figure 2

(a)–(f) Distributions of the (a),(c),(e) spin-preserving ($|{\psi }_n^{+}{|}^2$) and (b),(d),(f) spin-flipped electrons ($|{\psi }_n^{-}{|}^2$) among discrete energy levels (plotted in the color scale as ribbons) for different propagation distances, calculated within the diffractionless approximation ($x=\text{const}$). The input electron beams are spin-up polarized (see black arrows, i.e., ${\psi }_0^{+}$) along (a) $\pm \widehat{z}$, (c) $+\widehat{y}$, and (e) $-\widehat{y}$, respectively. (g) The spin-flip probabilities for incident electron spins along $\widehat{z}$ and $\widehat{y}$ show the same period of $L_p$. (h) The period $L_p$ for different $\mathrm{\Delta }{\mathcal{E}}_y$ (see definition in Fig. 1). We assume an electric field component $\mathrm{\Delta }{\mathcal{E}}_y=1\times {10}^7\mathrm{V}/\mathrm{cm}$ in (a)–(g), and use parameters $\lambda =1\mathrm{\mu }\mathrm{m}$ and $a=100\mathrm{nm}$ for all calculations.

Figure 3

(a) Electric (right) and magnetic (left) near field surrounding a nanowire array excited by a plane wave of wavelength $\lambda =1\mathrm{\mu }\mathrm{m}$ ($E_p=1.24\mathrm{eV}$). We assume a platinum nanowire array, with an experimentally measured optical permittivity $-105.76+16.49i$ [86] for the chosen photon energy, radius of $R=20\mathrm{nm}$, and a lattice constant of $a=100\mathrm{nm}$. (b)–(d) The spatial distributions of the (b) spin-preserving and (c) spin-flipped electron density, and the (d) spin-flip probability, calculated for an electron beam interacting with the optical near field shown in (a). An electron beam, spin-polarized along $\pm \widehat{y}$ and with a phase-matched velocity, i.e., $\beta =1/10$, is initially focused at $z=0$ and $x=0.3a$, with a waist of 3 nm. In (c), the spin-flipped electrons are scattered to the state of energy $E_p\mp E_0$. The unperturbed beam width is shown by the white dashed curves in (b). The electron beam is slightly deflected due to a reshaping effect in the optical field, and the nanowire array is adaptively rearranged to avoid crossing (the top of the nanowires are shown by the white solid curve). The incident electric field is assumed to be ${\mathcal{E}}_y^{\mathrm{in}}=4\times {10}^7\mathrm{V}/\mathrm{cm}$.