Optical Modulation of Electron Beams in Free Space

We exploit free-space interactions between electron beams and tailored light fields to imprint on-demand phase profiles on the electron wave functions. Through rigorous semiclassical theory involving a quantum description of the electrons, we show that monochromatic optical fields focused in vacuum can be used to correct electron beam aberrations and produce selected focal shapes. Stimulated elastic Compton scattering is exploited to imprint the required electron phase, which is proportional to the integral of the optical field intensity along the electron path and depends on the transverse beam position. The required light intensities are attainable in currently available ultrafast electron microscope setups, thus opening the field of free-space optical manipulation of electron beams.

Figure 1

Optical free-space electron modulator (OFEM). (a) The proposed element is placed in the electron microscope column right before the objective lens. (b) The OFEM incorporates a parabolic mirror that focuses light with a high numerical aperture on a vacuum region that intersects the electron beam. The electric field distribution at the optical focal spot is patterned by using a far-field spatial light modulator (SLM). (c) A phase is imprinted on the electron wave function, whose dependence on transverse coordinates $\mathbf{R}$ is proportional to the field intensity integrated along $z$.

Figure 2

1D electron focus shaping. We plot the OFEM-imprinted electron phase [(a),(c),(e)] and the corresponding wave function at the focal plane [(b),(d),(f)]. Dashed curves in (a),(b) and red curves in (c)–(f) correspond to ideal target profiles, while solid curves in (a),(b) and blue curves in (c)–(f) stand for the result obtained by introducing optical diffraction in the OFEM illumination. We consider a linear phase variation (a) leading to single-peak wave functions (b), as well as more complex phase patterns (c),(e) producing symmetric (d) and antisymmetric (f) double-peak wave functions. We take a ratio of the objective-lens semiaperture to the light wavelength $R_{\max }/{\lambda }_0=12.5$. The in-plane OFEM and focal coordinates $x$ and $x_f$ are normalized to $R_{\max }$ and the projected electron Abbe wavelength ${\lambda }_{e\perp }={\lambda }_e/\mathrm{NA}$, respectively, where ${\lambda }_e=2\pi /q_0$ is the electron wavelength and $\mathrm{NA}=R_{\max }/(z_f-z_L)$ is the microscope numerical aperture. The electron probability density $|\psi {|}^2$ is shown as color-matching thick-light curves in (b),(d),(f). Upper horizontal scales correspond to 60 keV electrons, $R_{\max }=10\mu \mathrm{m}$, and $\mathrm{NA}=0.01$.

Figure 3

2D electron focus shaping. (a) Designated focal shape. (b) Phase of the Fourier transform of (a) (target phase). (c) Diffraction-limited phase to be imprinted by OFEM assuming $R_{\max }/{\lambda }_0=30$. (d) Finally obtained focal profile. The bar shows the scale of (a),(d) in units of ${\lambda }_{e\perp }$. The phase plots have a radius $R_{\max }$.