Semiclassical trajectory perspective of glory rescattering in strong-field photoelectron holography

We investigate theoretically the photoelectron momentum distribution (PMD) of ionized atoms irradiated by a linearly polarized intense laser, focusing on the holography interference patterns in PMD that carry important information of the initial wavefunction of a tunneled electron and its experienced atomic potential in rescattering. With the help of Dyson series and a semiclassical propagator, we calculate the scattering amplitudes in the cylindrical coordinate representation. In contrast to conventional recognitions that photoelectron holography is the interference of two branches of electron trajectories, however, we find strikingly that infinitely many semiclassical trajectories can be deflected by the combined Coulomb potential and laser field into the same final momentum. The initial momenta are found to be distributed on a ring-shaped curve in the transverse momentum plane and the initial positions of these trajectories are perpendicular to their initial momentum vectors. For the zero final transverse momentum, the above ring-source trajectories degenerate into the point-source axial caustic trajectories (or glory trajectories) and the quantum interference of these trajectories will dramatically alter the scattering amplitudes, which is termed the glory rescattering effect. By following Berry's spirit of uniform approximation for glory scattering in optics, we can finally derive a uniform formulation of the rescattering amplitude in the Bessel functions for the strong-field photoelectron holography patterns. Our results are in good agreement with solutions of the time-dependent Schrödinger equation and can account for recent photoelectron holography experiments. Important applications of our theory are also discussed.

Figure 1

Photoelectron semiclassical trajectories in (a) coordinate space and (b) momentum space for the final state of $p_{zf}=0.66$ a.u., $p_{xf}=0.12$ a.u., $p_{yf}=0$. (c) The initial positions on the projected $x-y$ plane and (d) the initial transverse momenta on the projected $p_x-p_y$ plane. The laser parameters are $\lambda =1200$ nm and $I=8.7\times {10}^{13}$ W/${\mathrm{cm}}^2$.

Figure 2

For $p_{zf}=0.6$ a.u., $p_{yf}=0$, and $p_{xf}$ varied from 0 to 0.2 a.u., we present (a) the calculated initial transverse momentum of $p_{\rho 0}$ depending on azimuthal angle of ${\varphi }_0$, and (b) the initial displacement of ${\rho }_0$ to the $z$ axis depending on azimuthal angle of ${\varphi }_0$. (c) For $p_{xf}=0$ corresponding to the glory caustic singularity, the initial positions are exactly at the origin of the projected $x-y$ plane; the initial transverse momenta are schematically shown by arrows. (d) The same as (c) but $p_{xf}=0.03$ a.u. (e) The initial transverse momenta of photoelectron trajectories when $p_{xf}=0.2$ a.u., and (f) the corresponding initial positions on the projected $x-y$ plane. The black areas in (a) and (b) represent the classical trajectory forbidden regions. The dashed lines in (e) and (f) are the analytical extensions of the solutions to the forbidden regions. The laser parameters are $\lambda =1200$ nm and $I=8.7\times {10}^{13}$ W/${\mathrm{cm}}^2$.

Figure 3

[(a) and (b)] The PMDs for the hydrogen ionized by laser pulses with $8.7\times {10}^{13}$ W/${\mathrm{cm}}^2$ intensity calculated by TDSE, and the interference fringes predicted by our UGRT (black lines). [(c) and (d)] The momentum distributions corresponding to fixed final momenta $p=0.6$ a.u. from calculations of TDSE (solid black lines), UGRT (solid red lines), and GRT (dashed blue lines), respectively. The wavelength of laser pulses are [(a) and (c)] $\lambda$ = 1600 nm and [(b) and (d)] $\lambda$ = 1200 nm . In (c) and (d), scattering angle $\theta =arctan\left(\frac{p_{\rho f}}{p_{zf}}\right)$.

Figure 4

Experimental holographic pattern and positions of the dark fringes calculated by UGRT [red lines in (a)], GRT (dashed black lines), and CCSFA (purple triangles). Positions of the bright fringes calculated by UGRT [black squares in (b)]. The experimental data are extracted from Ref. [8], where the metastable ($6s$) Xe atoms with ionization potential of 0.14 a.u. are ionized with linearly polarized mid-infrared laser pulses with duration of 5 to 20 cycles. The laser parameters are $\lambda =7\mu \mathrm{m}$ and $I=7.1\times {10}^{11}$ W/${\mathrm{cm}}^2$.