Torus-knot angular momentum in twisted attosecond pulses from high-order harmonic generation

Björn Minneker, Birger Böning, Anne Weber, and Stephan Fritzsche

Phys. Rev. A 104, 053116 – Published 29 November 2021

Abstract

Bicircular twisted Laguerre-Gaussian beams possess a definite torus knot angular momentum (TKAM) $\hat{J_{γ}} = \hat{L} + γ \hat{S}$ as an alternative a form of angular momentum. TKAM is conserved in nonlinear atomic processes such as high harmonic generation and can be classified by a time delay parameter $τ$ and a coordination parameter $γ$ . These parameters are defined by the respective projected orbital angular momentum $ℓ_{i}$ and the energy $ℏ p_{i} ω_{0}$ of the two superimposed Laguerre-Gaussian beams. We derive a consistent geometric method to determine $τ$ and $γ$ from the driving beam as well as from the high harmonic radiation. The method relates both invariance parameters ( $τ$ and $γ$ ) to a torus knot which can be constructed from the emitted high harmonic radiation. These knots are constructed from the spatiotemporal evolution of the electric field of the respective high harmonic radiation or the driving beam. We demonstrate the classification of the invariance parameters for a planar atomic gas target irradiated by bicircular Laguerre-Gaussian beams explicitly. In addition, we demonstrate that the respective torus knots determined by $τ$ and $γ$ can be mapped onto each other within minor modifications. The numerical calculations are done within the strong-field approximation and the associated quantum orbit approach. Therefore, we also briefly review high harmonic generation by bicircular twisted light beams. This introduced geometric method is a different approach to interpret the invariance parameters $τ$ and $γ$ , as well as their underlying relations, compared to a purely formal derivation. The investigations presented in this work are in good agreement with previous findings and provide insight into the dynamical symmetry of TKAM in the context of high harmonic generation induced by bicircular twisted Laguerre-Gaussian beams.

Received 9 July 2021
Accepted 12 November 2021
Corrected 7 January 2022
Corrected 20 January 2022

DOI:https://doi.org/10.1103/PhysRevA.104.053116

Physics Subject Headings (PhySH)

Angular momentum of light High-order harmonic generation Laser dynamics Spatial profiles of optical beams

Atomic, Molecular & OpticalNonlinear Dynamics

Corrections

7 January 2022

Correction: In the front matter, affiliations were not ascribed properly to the second and third authors; the affiliation attributions have now been corrected.

20 January 2022

Second Correction: Minor errors in Eqs. (7b) and (7c) have been fixed.

Authors & Affiliations

Björn Minneker ^1,2,3,*, Birger Böning ^2,3, Anne Weber ¹, and Stephan Fritzsche ^1,2,3

¹Theoretisch Physikalisches Institut, Friedrich-Schiller-Universität, Jena, Germany
²GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany
³Helmholtz Institut, Jena, Germany

^*bjoern.minneker@uni-jena.de

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 104, Iss. 5 — November 2021

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
High-harmonic generation driven by bicircular LG beams. (a) Physical setup: A driving beam (orange), consisting of two superimposed LG beams, irradiates the physical target atoms (grey dots) from a gas jet (green) that is emitted perpendicularly to the optical axis (dash-dot line). The target is approximated by a two-dimensional distribution of atoms in the $x − y$ plane (black dots). After the interaction of the target atoms with the driving beam, high-harmonic radiation (blue) is emitted from the interaction region. (b) Intensity distribution of the driving beam ( $ℓ_{1} = ℓ_{2} = 1, ω_{1} = ω_{2} / 2 = ω$ ) in the target plane. Lissajous figures indicate the polarization of the superimposed LG beams for fixed azimuthal angles $φ$ . The black dots indicate the electric field vector for a fixed time $t$ and azimuthal angle $φ$ . The colored dots display the electric field maxima for fixed $φ$ and $t$ . (c) The phase of the electric field of the driving beam, together with the Lissajous figures also shown in (b). (d) The blue Lissajous figures are identical to the one in panels (b) and (c), where $φ = 0$ denotes the position at which the electric field vector is maximized (blue dot). The green, red, and orange lines denote the positions of the field maxima. (e) If the open ends in (d) are connected, the green, red, and orange lines form a torus knot. (f) Projection of the torus knot constructed in (e) onto a torus.
Reuse & Permissions
Figure 2
(a) Bicircular field as a superposition of a circularly polarized harmonic (red) and its counter-rotating second harmonic (blue), which gives rise to the Lissajous figure (trefoil shape) if the position of the electric field vector is followed in time. (b) Absolute values for the electric field at azimuthal angles $φ = 0$ and $φ = π / 2$ as a function of time. (c)–(f) Orientation of the Lissajous figures for various values of the azimuthal angle $φ$ of a bicircular driving beam ( $ℓ_{1} = ℓ_{2} = 1, ω_{1} = ω_{2} / 2 = ω$ ). The background shows the cycle averaged intensity distribution of the driving beam with the characteristic singularity on the beam axis. Each panel (c)–(f) shows the electric field at the target for increasing times $t = 0, T / 12, T / 6, T / 3$ . The black dots indicate the electric field at the respective angle in the target plane. The colored dots denote the maxima of the electric field concerning the azimuthal angle similar to Figs. 1 and 1. (g) Temporal evolution of the HHs with respect to the black axis (time axis). The separate helices indicate the HH intensity maximum with respect to the azimuthal angle at the target (light gray plane at the bottom). The dots are the field maxima that we also show in (c)–(f). (h) The time axis in (g) is bent and connected for the times $t = 0$ and $t = T$ . (i) The resulting closed line of intensity maxima has the topology of a torus knot.
Reuse & Permissions
Figure 3
(a), (b) Temporal evolution of the light springs formed by the HHs. The figures display the contour obtained at 80% of the maximum electric field strength. Therefore, we included the 16th up to the 25th harmonics. The parameters of the driving beam are (a) $ℓ_{1} = ℓ_{2} = 1, ω_{1} = ω_{2} / 2 = ω$ and (b) $ℓ_{1} = 2, ℓ_{2} = 1, ω_{1} = ω_{2} / 2 = ω$ . (c)–(k) Intensity distribution of (a) for the times $t = j \frac{T}{9}$ with the integers $j \in [1, 9]$ , so that $t$ increases from (c) $t = \frac{T}{9}$ to (k) $t = T$ .
Reuse & Permissions
Figure 4
(a) Absolute value of the electric field of the driving beam (blue) and the superimposed electric fields of the HHs at a fixed position (black). (b) Temporal evolution of the HHs in the near field and their associated two-dimensional projections where the changing color indicates the temporal evolution. (c) Spatial evolution of the emitted HHs with resolved angle-dependent polarization. The polarization figure is also shown in (b) as projection onto the $E_{x} − E_{y}$ plane.
Reuse & Permissions
Figure 5
Schematic two-dimensional representation of the electric field maxima of the superimposed HHs generated from electric field maxima of the driving beams with (a), (b) $ℓ_{1} = ℓ_{2} = 1, ω_{1} = ω_{2} / 2 = ω$ and (c), (d) $ℓ_{1} = - 2, ℓ_{2} = 1, ω_{1} = ω_{2} / 2 = ω$ . (a) and (c) are associated with the temporal torus knot (light spring) and correspond to the parameters $τ = 2 / 3 ω^{- 1}, γ = - 1 / 3$ , and ${\tilde{α}}_{\max} = π$ while (b) and (d) are associated with the spatial torus knot and the corresponding parameters $τ = - 1 / 3 ω^{- 1}, γ = 5 / 3$ , and ${\tilde{α}}_{\max} = 10 π$ . Panels (b) and (c) are obtained by keeping only the respective left sides of the dashed line in panels (a) and (d). In order to guide the eye, equal colors represent equal toroidal coordinates of the resulting torus knot. Therefore, the dimension with evolving color is associated with the toroidal axis and the remaining dimension with the poloidal axis.
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information