Propagation of intense and short circularly polarized pulses in a molecular gas: From multiphoton ionization to nonlinear macroscopic effects

We present a detailed analysis of the propagation dynamics of short and intense circularly polarized pulses in an aligned diatomic gas. Compared to linearly polarized intense pulses, high harmonic generation (HHG) and the coherent generation of attosecond pulses in the intense-circular-polarization case are a new research area. More specifically, we numerically study the propagation of intense and short circularly polarized pulses in the one-electron ${\mathrm{H}}_2{}^{+}$ molecular gas, using a micro-macro Maxwell–Schrödinger model. In this model, the macroscopic polarization is computed from the solution of a large number of time-dependent Schrödinger equations, the source of dipole moments, and using a trace operator. We focus on the intensity and the phase of harmonics generated in the ${\mathrm{H}}_2{}^{+}$ gas as a function of the pulse-propagation distance. We show that short coherent circularly polarized pulses of same helicity can be generated in the molecular gas as a result of cooperative phase-matching effects.

Figure 1

Geometry of HHG from an incident circularly polarized pulse on aligned ${\mathrm{H}}_2{}^{+}$ molecules.

Figure 2

(a) $E_{x^{\prime }},E_{y^{\prime }}$ amplitudes of circularly polarized initial pulse in space for $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2,{\lambda }_0=800$ nm, $z^{\prime }(\mathrm{a}.\mathrm{u}.)=a_0=0.0529$ nm. (b) Total pulse in time; $E(\mathrm{a}.\mathrm{u}.)=5\times {10}^9\mathrm{V}/\mathrm{cm},\text{time}\left(\text{a.u.}\right)=24$ as.

Figure 3

(a) Homogeneous and (b) nonhomogeneous density profiles, $\mathcal{N}\left(z^{\prime }\right)$; ${\mathcal{N}}_0=8\times {10}^{-5}{a}_0^{-3}$.

Figure 4

(a) Reflection of $E_y$ component at the entering border “vacuum-gas,” $t=722$ a.u., homogeneous case. (b) Reflection of $E_y$ component at the entering border “vacuum-gas,” $t=722$ a.u., nonhomogeneous case with $\beta =3/4,\text{time}\left(\text{a.u.}\right)=24$ as.

Figure 5

Intensity of the low-order electric-field harmonic spectrum for different gas-region lengths $L=\mathrm{\Delta }{z}_M{N}_g$ for LP pulses $(\lambda =800$ nm, $I=5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2,\mathrm{\Delta }{z}_M=100$ a.u.).

Figure 6

Intensity of harmonics of the dipole moment ${\widehat{d}}_y\left(\omega \right)$ of molecule in the right end of gas region for different propagation lengths $L=\mathrm{\Delta }{z}_M{N}_g$ of LP pulses $(\lambda =800$ nm, $I=5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2,\mathrm{\Delta }{z}_M=100$ a.u.).

Figure 7

Intensity of harmonics of the transmitted electric field for different gas-region lengths $L=\mathrm{\Delta }{z}_M{N}_g$ for LP pulses.

Figure 8

Intensity of the low-order harmonics as a function of the propagation length $L=\mathrm{\Delta }{z}_M{N}_g$ of the LP pulse.

Figure 9

Intensity of the low-order harmonics of the electric $E_y$ component as a function of propagation length $L=\mathrm{\Delta }{z}_M{N}_g$ of the LP pulse in gas.

Figure 10

Low-order harmonic spectrum of the transmitted electric field for different gas-region lengths $L=\mathrm{\Delta }{z}_M{N}_g$ for a CP pulse.

Figure 11

Low-order harmonic of the dipole moment of aligned ${\mathrm{H}}_2{}^{+}$ molecules in the right (exit) end of the gas region for different propagation lengths $L=\mathrm{\Delta }{z}_M{N}_g$ of CP pulses.

Figure 12

Total harmonic spectrum of the transmitted electric field for different gas-region lengths $L=\mathrm{\Delta }{z}_M{N}_g$ for a CP pulse.

Figure 13

Low-order harmonic spectrum of the dipole-moment component $d_x\left(\omega \right)$ of ${\mathrm{H}}_2{}^{+}$ molecules in the right end of the gas region for different propagation lengths of CP pulses.

Figure 14

Low-order harmonic spectrum of the dipole-moment component $d_y\left(\omega \right)$ of ${\mathrm{H}}_2{}^{+}$ molecules in the right end of the gas region for different propagation lengths of CP pulses.

Figure 15

The level sets of electron density ${log}_{10}({\left|\psi (x,y)\right|}^2)$ for a ${\mathrm{H}}_2{}^{+}$ molecule in the left (entering) end of the gas region at six consecutive times: $t=0,302,349,451,757,1237$ a.u., $\text{time}\text{(a.u.)}=24$ as. Total number of TDSEs is $N_g=512$.

Figure 16

Intensity of the low-order harmonics as a function of the propagation length $L=\mathrm{\Delta }{z}_M{N}_g$ of the CP pulse in gas.

Figure 17

Intensity of the low-order harmonics of the $E_x$ component as a function of propagation length $L=\mathrm{\Delta }{z}_M{N}_g$ of the CP pulse in gas.

Figure 18

Intensity of the low-order harmonics of the $E_y$ component as a function of propagation length $L=\mathrm{\Delta }{z}_M{N}_g$ of the CP pulse in gas.

Figure 19

$cos(\varphi \left\{{\widehat{E}}_x\left[(2k+1){\omega }_0\right]\right\}-\varphi \left\{{\widehat{E}}_y\left[(2k+1){\omega }_0\right]\right\})$, for $k=0,1,2,3,4,5$ as a function of propagation length $L=\mathrm{\Delta }{z}_M{N}_g$ of the CP pulse. Circularly polarized harmonics are generated at zero corresponding to a $\pm \pi /2$ phase difference.

Figure 20

$cos(\varphi \left\{{\widehat{d}}_x\left[(2k+1){\omega }_0\right]\right\}-\varphi \left\{{\widehat{d}}_y\left[(2k+1){\omega }_0\right]\right\})$, for $k=0,1,2,3,4,5$ as a function of propagation length $L=\mathrm{\Delta }{z}_M{N}_g$ of the CP pulse, for ${\mathrm{H}}_2{}^{+}$ molecules at the (a) left end (incident) and (b) at the right-end (outgoing) of the gas region.

Figure 21

(a) $E_y$ as a function of space at final time with homogeneous and nonhomogeneous $[\beta =1$/3 in Eq. (13)], molecular density. (b) Same figure with zoom along $y$.

Figure 22

Transmitted electric field as a function of space at final time with homogeneous molecular density. (a) $N_g=256$. (b) $N_g=512$.