Extracting an electron's angle of return from shifted interference patterns in macroscopic high-order-harmonic spectra of diatomic molecules

T. Das, B. B. Augstein, C. Figueira de Morisson Faria, L. E. Chipperfield, D. J. Hoffmann, and J. P. Marangos

Phys. Rev. A 92, 023406 – Published 10 August 2015

Abstract

We investigate high-order-harmonic spectra from aligned diatomic molecules in intense driving fields whose components have orthogonal polarizations. We focus on how the driving-field ellipticity influences structural interference patterns in a macroscopic medium. In a previous publication [Phys. Rev. A 88, 023404 (2013)] we have shown that the nonvanishing ellipticity introduces an effective dynamic shift in the angle for which the two-center interference maxima and minima occur, with regard to the existing condition for linearly polarized fields. In this work we show through simulation that it is still possible to observe this shift in harmonic spectra that have undergone macroscopic propagation, and discuss the parameter range for doing so. These features are investigated for $H_{2}$ in a bichromatic field composed of two orthogonally polarized waves. The shift is visible both in the near- and in the far-field regime, so that, in principle, it can be observed in experiments.

Received 15 June 2015

DOI:https://doi.org/10.1103/PhysRevA.92.023406

Authors & Affiliations

T. Das, B. B. Augstein, and C. Figueira de Morisson Faria

Department of Physics and Astronomy, University College London, Gower Street, London WC1E, 6BT, United Kingdom

L. E. Chipperfield, D. J. Hoffmann, and J. P. Marangos

Quantum Optics and Laser Science Group, Department of Physics, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom

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Vol. 92, Iss. 2 — August 2015

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Images

Figure 1
Real parts of the effective shifts $ζ (t, t^{'})$ as functions of the driving-field intensity $I_{ω}$ of the fundamental for a harmonic of frequency $Ω = 31 ω$ , using $H_{2}$ $(I_{p}$ = 0.59 a.u.) in two-color laser fields of increasing ellipticity. The fundamental and the second harmonic have been approximated by monochromatic waves where $I_{2 ω} = ξ I_{ω}$ . The ellipticities have been increased from $ξ = 0$ to $ξ = 0.3$ in increments of $Δ ξ = 0.05$ . A lighter color indicates a higher ellipticity. The dashed and solid lines refer to the short and long orbits, respectively. Panels (a) and (b) refer to relative phases of $ϕ = 0$ and $ϕ = 0.4 π$ , respectively.
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Figure 2
HHG macroscopic response of $H_{2}$ in a linearly polarized field of wavelength $λ = 800 nm$ , plotted in a logarithmic scale. The beam waist is $w = 30 μ m$ , and the gas jet is placed at $z_{g} = 4 mm$ after the focus. The FWHM of the gas jet is 0.5 mm and the FWHM of the intensity envelope is 30 fs. Panels (a) and (b) show the individual contributions of the long and short orbits, for a driving-field intensity $I_{ω} = 2 \times 10^{14} W / {cm}^{2}$ , and alignment angle $θ_{L} = 0$ ; note that the oscillations in (b) are caused by an artifact due the inaccuracy of the standard saddle-point approximation in the cutoff region. Panels (c) and (d) show the spectra from the coherent superpositions of the long and short orbits, for a driving-field intensity $I_{ω} = 2.5 \times 10^{14} W / {cm}^{2}$ , and alignment angles $θ_{L} = 0$ (c) and $θ_{L} = 3 π / 16$ (d). Panels (e) and (f) show the far-field harmonic spectra, for the same intensity and alignment angles as in panels (c) and (d). The white dashed line indicates the position of the structural two-center minimum. The labels $s, l$ , and $l + s$ indicate contributions from the short orbit, long orbit, or from a coherent superposition of both, respectively.
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Figure 3
Propagated HHG spectra for $H_{2}$ in a Gaussian pulse, where the peak intensities of the $ω$ and $2 ω$ waves are $I_{ω} = 2.5 \times 10^{14} W / {cm}^{2}$ and $I_{2 ω} = 7.5 \times 10^{13} W / {cm}^{2}$ defined at the gas jet (their intensity ratio is around 0.3) and relative phase $ϕ = - 0.2 π$ . The wavelength of the fundamental is $λ = 800$ nm. The beam waist is $w = 30 μ m$ and the center of the gas jet is located at $z_{g} = 4$ mm after the focus. The FWHM of the intensity envelope is 30 fs. Panels (a) and (b) show the far-field result for a coherent superposition of orbits denoted by the labels $l + s$ , where (b) is a close-up of the region around the shifted minimum seen in (a). Panels (c) and (d) show the individual contributions to far-field spectra for the long and short orbit, respectively. The dashed lines and the solid curves indicate the position of the unshifted and shifted two-center minima, respectively. The red and the green solid curves give the positions of the shifted minima for the short and long orbits, respectively. The curves have been computed for a driving-field amplitude using $z$ spread equally across the interaction region such that $z_{g} - Δ z \leq z \leq z_{g} + Δ z$ , with $Δ z = 0.5$ mm. All spectra have been plotted in arbitrary units and in a logarithmic scale. The alignment angle between the molecule and the field is $θ_{L} = π / 16$ in all panels.
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Figure 4
Individual contributions of the long and short orbits (left and right panels, respectively) for the HHG macroscopic response of $H_{2}$ in a Gaussian pulse composed of parallel and perpendicular waves given by Eqs. (21) and (22), respectively. The peak intensities of the $ω$ and $2 ω$ waves are $I_{ω} = 2.5 \times 10^{14} W / {cm}^{2}$ and $I_{2 ω} = 7.5 \times 10^{13} W / {cm}^{2}$ defined at the gas jet (their intensity ratio is around 0.3) and relative phase $ϕ = 0$ . The center of the gas jet is located at $z_{g} = 2$ mm after the focus. This phase has been found to give a very large residual shift $ζ (t, t^{'})$ at the cutoff. The wavelength of the fundamental is $λ = 800$ nm. The beam waist is $w = 30 μ m$ , and the FWHM of the intensity envelope is 30 fs. Panels (a) and (b) show the HHG yield from the individual orbits; panels (c) and (d) show a zoom-in of the upper panels close to the interference minimum. The white dashed lines and the solid curves indicate the position of the unshifted and shifted two-center minima, respectively. The red and the green curves give the positions of the shifted minima for the short and long orbits, respectively. The curves have been computed for a driving-field amplitude using $z$ spread equally across the interaction region such that $z_{g} - Δ z \leq z \leq z_{g} + Δ z$ , with $Δ z = 1$ mm. All panels have been plotted in arbitrary units and in a logarithmic scale. The labels $l$ and $s$ are associated with the long and the short orbit, respectively. The alignment angle between the molecule and the field is $θ_{L} = π / 16$ in all panels.
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Figure 5
Propagated spectra considering a coherent superposition of the long and short orbits, for the same field parameters as in Fig. 4. Most propagation conditions have also been kept as in Fig. 4 except the center of the gas jet, which has been chosen to be at $z_{g} = 2 mm$ and $z_{g} = 4 mm$ after the focus (left and right panels, respectively). Panels (a) and (b) show the spectra in the interaction region, while (c) and (d) show the far-field results. The white dashed lines and the solid curves indicate the position of the unshifted and shifted two-center minima, respectively. The red and the green curves give the positions of the shifted minima for the short and long orbits, respectively. The curves have been computed for a driving-field amplitude using $z$ spread equally across the interaction region such that $z_{g} - Δ z \leq z \leq z_{g} + Δ z$ , with $Δ z = 1 mm$ . All spectra have been plotted in arbitrary units, and in a logarithmic scale. The alignment angle between the molecule and the field is $θ_{L} = π / 16$ in all panels.
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Figure 6
Propagated, far-field HHG spectra considering two half cycles, for the same driving-field and molecular parameters in Fig. 3, but different FWHM of the intensity envelope. In panels (a) and (c), this width is 30 fs, while in panels (b) and (d) it is 5.5 fs. The upper and lower panels show the whole spectra, and a close-up near the interference minimum, respectively. In the lower panels we show only the shift related to the short orbit, which dominates throughout. These shifts are displayed as the thick solid curves in the figure, while the static interference condition is given by the white dashed line. The shifts associated with the first and second half cycle are shown as the red curves in panels (c) and (d), and the pink curve in panel (c), respectively. All spectra have been plotted in arbitrary units and in a logarithmic scale.
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