Circularly polarized high harmonics generated by a bicircular field from inert atomic gases in the $p$ state: A tool for exploring chirality-sensitive processes

D. B. Milošević

Phys. Rev. A 92, 043827 – Published 19 October 2015

Abstract

$S$ -matrix theory of high-order harmonic generation (HHG) is generalized to multielectron atoms. In the multielectron case the harmonic power is expressed via a coherent sum of the time-dependent dipoles, while for the one-electron models a corresponding incoherent sum appears. This difference is important for the inert atomic gases having a $p$ ground state as used in a recent HHG experiment with a bicircular field [Nat. Photonics 9, 99 (2015)]. We investigate HHG by such a bicircular field, which consists of two coplanar counter-rotating circularly polarized fields of frequency $r ω$ and $s ω$ . Selection rules for HHG by a bicircular field are analyzed from the aspects of dynamical symmetry of the system, conservation of the projection of the angular momentum on a fixed quantization axis, and the quantum number of the initial and final atomic ground states. A distinction is made between the selection rules for atoms with closed [J. Phys. B 48, 171001 (2015)] and nonclosed shells. An asymmetry in emission of the left- and right-circularly polarized harmonics is found and explained by using a semiclassical model and the electron probability currents which are related to a nonzero magnetic quantum number. This asymmetry can be important for the application of such harmonics to the exploration of chirality-sensitive processes and for generation of elliptic or even circular attosecond pulse trains. Such attosecond pulse trains are analyzed for longer wavelengths than in Opt. Lett. 40, 2381 (2015), and for various field-component intensities.

Received 30 March 2015
Revised 20 August 2015

DOI:https://doi.org/10.1103/PhysRevA.92.043827

Authors & Affiliations

D. B. Milošević

Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Herzegovina;
Academy of Sciences and Arts of Bosnia and Herzegovina, Bistrik 7, Sarajevo, Bosnia and Herzegovina;
and Max-Born-Institut, Max-Born-Strasse 2a, 12489 Berlin, Germany

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Vol. 92, Iss. 4 — October 2015

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Images

Figure 1
Harmonic power as a function of the harmonic order for HHG by bicircular field for Ne atoms and the field-component wavelengths $λ_{1} = 1300$ nm and $λ_{2} = 650$ nm and intensities expressed in multiples of $10^{14} W / {cm}^{2}$ and denoted by $j_{1} - j_{2}$ , where $j_{1}, j_{2} = 3, 4, 5$ . The $n th$ harmonic ellipticity is $ɛ_{n} = \pm 1$ for $n = 3 q \pm 1$ , $q$ integer. (a) Comparison of the results obtained by using the realistic $2 p$ ground state and by using the hydrogen-like-atom model with the $s$ ground state. The laser intensities of the components equal $4 \times 10^{14} W / {cm}^{2}$ ( $j_{1} = j_{2} = 4$ ). The cutoff at $n = 212$ is also denoted. (b), (c) The results obtained for $j_{1}, j_{2} = 3, 4, 5$ are presented for the values of the initial- and final-state magnetic quantum numbers $m_{i}$ and $m_{f}$ as denoted.
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Figure 2
The $ω - 2 ω$ bicircular electric field vector (upper-left panel), vector potential (lower left), and electron trajectory (upper right) and velocity (lower right) between the ionization (I) and recombination (R) times which correspond to the cutoff harmonic $n = 212$ . The relevant electron probability current $j_{-}$ is shown in the right-hand panels.
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Figure 3
Ratio of the coherent to incoherent harmonic intensities as a function of the time expressed in optical cycles $T$ . The results for the groups of harmonics having the ellipticity $ɛ_{n} = + 1$ ( $n = 151, 154, ..., 196, 199$ ) and $ɛ_{n} = - 1$ ( $n = 152, 155, ..., 197, 200$ ) are presented. For the curves denoted by $s$ (black curve; $s$ ground state) and $p$ (red curve; $2 p$ ground state) all harmonics are taken into account, while for the green (blue) curve denoted by $p +$ ( $p -$ ) only the harmonics having the ellipticity $ɛ_{n} = + 1$ ( $ɛ_{n} = - 1$ ) are included. The laser and atomic parameters are as in Fig. 1. The laser component intensities are $I_{i} = j_{i} \times 10^{14} W / {cm}^{2}$ ( $i = 1, 2$ ), where $j_{1} = j_{2} = 4$ for all curves except for the magenta dot-dashed curve for which $j_{1} = 5$ and $j_{2} = 3$ and the cyan dashed curve for which $j_{1} = 3$ and $j_{2} = 5$ (for these two curves the ground state is $2 p$ and all harmonics are taken into account).
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Figure 4
Electric-field vector of a group of harmonics ( $n = 151, 152, ..., 199, 200$ ) during one-third of the cycle of the driving field. In each panel three traces are shown: all harmonics (black bold lines), all harmonics having the ellipticity $ɛ_{n} = + 1$ (red curves), and all harmonics having the ellipticity $ɛ_{n} = - 1$ (green curves). In the upper-left panel, the results obtained by using the $s$ ground state are shown, while for the remaining three panels, the $p$ ground state is used for the following intensities of the bicircular-field components: $I_{1} = I_{2} = 4 \times 10^{14} W / {cm}^{2}$ (upper panels), $I_{1} = 5 \times 10^{14} W / {cm}^{2}$ and $I_{2} = 3 \times 10^{14} W / {cm}^{2}$ (lower-left panel), and $I_{1} = 3 \times 10^{14} W / {cm}^{2}$ and $I_{2} = 5 \times 10^{14} W / {cm}^{2}$ (lower-right panel). The other laser and atomic parameters are as in Fig. 1.
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Figure 5
Harmonic power (upper panel) and ellipticity (lower panel) as functions of the harmonic order for HHG by the same field as in Fig. 1 and for Ne atoms with the $2 p$ ground state but for the $x$ axis as axis of quantization and for $m_{i} = m_{f} = 0$ .
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