SCIENCE ADVANCES

14 Apr 2023

Vol 9, Issue 15

DOI: 10.1126/sciadv.ade0953

Symmetries and their associated selection rules are extremely useful in many fields of science. For systems of electromagnetic (EM) fields interacting with matter, the symmetries of matter and the EM fields’ time-dependent polarization determine the properties of the nonlinear responses, and they can be facilitated for controlling light emission and enabling ultrafast symmetry breaking spectroscopy of various properties. Here, we formulate a general theory that describes the macroscopic and microscopic dynamical symmetries (including quasicrystal-like symmetries) of EM vector fields, revealing many previously unidentified symmetries and selection rules in light-matter interactions. We demonstrate an example of multiscale selection rules experimentally in the framework of high harmonic generation. This work paves the way for novel spectroscopic techniques in multiscale systems and for imprinting complex structures in extreme ultraviolet–x-ray beams, attosecond pulses, or the interacting medium itself.

Symmetry is regularly used to derive conservation laws and selection rules in interacting systems (1). In the field of nonlinear optics, symmetries are standardly used to determine whether a particular nonlinear process is allowed or forbidden according to the medium’s point group (2, 3). Recently, a more general group theory was developed to describe the symmetries of the electromagnetic (EM) field’s time-dependent polarization, named dynamical symmetries (DSs) (4). Such DSs and their associated selection rules have been applied to shaping the waveforms of extreme ultraviolet (XUV) and x-ray radiation emitted from high harmonic generation (HHG) (5–7) and have enabled ultrafast symmetry breaking spectroscopy of molecular (8, 9) and solid orientation (10), molecular symmetries (9), and chirality (11, 12). However, this theory of DSs is local (operating solely on a microscopic scale) (13) and, thus, fully neglects light’s macroscopic structure. Moreover, it does not account for composite microscopic-macroscopic (multiscale) DSs.

Here, we formulate a general theory for EM fields and their interactions with matter, where the multiscale symmetries of the full light-matter Hamiltonian are analyzed. We describe spatiotemporal DSs as generalized unitary transformations and study systematically all possible symmetry operations that close under group multiplication. Various combinations of EM fields are cataloged into different groups that are composed of one or more DSs. We assign each DS an associated selection rule that indicates the allowed frequencies, polarizations, momenta, and angular momenta of the harmonic emission. Our theory generalizes many previous results, such as complex structured XUV emission generated by spatiotemporal structure beams in the beams’ longitudinal axis (14–17) or profile (5–7, 18, 19). We also found previously unidentified types of symmetries, including simultaneous spin-orbit angular momentum conservation, and periodic (20) and aperiodic (21) space-time crystals of a vector field. We explore several new multiscale DSs numerically and experimentally in the framework of HHG to demonstrate the richness of this approach for light-matter interactions.

We begin by describing the multiscale DSs of an EM vector field, which are combinations of temporal, microscale, and macroscale spatial building blocks. Then, we derive a general equation that determines the selection rules of the polarization and frequencies (temporal and spatial) of generated harmonics.

We presented a theory for symmetries and selection rules in (extreme) nonlinear optics for multiscale systems. We introduced symmetries that couple time, macroscopic, and microscopic DOFs. We showed how these symmetries are transferred to the induced polarization and lead to constraints, i.e., selection rules, on physical observables. Multiscale DSs and selection rules investigations in three different systems are presented: with spin-orbit nonlinear interaction, with quasi-periodic structures, and experimental example with multiscale DS in time, polarization, and propagation axis.

A potential application of our theory is ultrafast spectroscopy for the detection of the symmetry of the medium. This is done using a driving field that exhibits a known DS. When the medium lacks that DS, the symmetry of the total system is reduced so that some restrictions in the selection rules are removed (38). One important medium that breaks symmetries is a chiral medium, i.e., a medium that is asymmetric under any reflection or inversion. Probing the chirality of molecules can be a challenging task, and using multiscale DS consideration can help in choosing the right microscopic (11, 12, 39) and macroscopic field parameters to enhance the far-field chiral signal for molecular chirality detection and discrimination as shown in section S6. Our work also paves the way for several interesting directions beyond harmonic generation. Extending the theory to nonlocal interactions, which are especially important for HHG with long-wavelength high-power lasers (40) and condensed matter (41), may lead to insights regarding multiscale matter, light, and their interaction. Extensions to complexed structured laser ablation (42, 43) or controlled optoelectronic (44) to easily shaping complex structures should also be possible and exciting, leading to inducing symmetries in the media. Overall, we expect that the use of multiscale symmetries will lead to extended understanding of, and novel findings in, various multiscale systems.

The experimental setup is illustrated in Fig. 3E. The output beam of a 1-kHz, 35-fs full width at half maximum, 800-nm carrier wavelength, Ti:sapphire amplifier (Coherent Legend USX) is split into two paths. The first beam, with 2.25 mJ per pulse, retains its spatial Gaussian profile and is focused with a lens (fLG = 300 mm, Rayleigh range of the Gaussian beams = 5 mm) through the MAZEL TOV apparatus (37). This apparatus consists of (i) an SHG crystal (0.5-mm-thick BBO crystal), which transfers ~15% of the energy to the second harmonic beam; (ii) calcite plates, which precompensate for group delays induced by normally dispersive optics down the beam path; and (iii) a single achromatic QWP (for the two spectral components) that converts the linear s-polarized fundamental and perpendicular p-polarized SH incoming beams to counter-rotating circularly polarized beams. This intense beam also drills a hole in the aluminum foil, terminating a semi-infinite gas cell (SIGC) filled with argon (45 torr). The second beam, with 0.5 mJ per pulse, undergoes an amplitude modulation using two perpendicularly oriented polarizers on either side of a phase-only SLM (HOLOEYE PLUTO). This beam acquires the spatial distribution of a ring, which is then imaged and focused (fLB3 = 150 mm) to form a Bessel beam with β/k = 0.0015. Here, k = 2π/800 nm is the wave vector, and β is the on-axis difference between the wave vectors of the Gauss and Bessel beams. This experimental condition corresponds to a periodicity of L ≈ 0.5 mm, which is one order of magnitude smaller than the Rayleigh range. A QWP is used to scan the polarization of the Bessel beam. Both beams are combined using a holed mirror and are focused close to the output of the SIGC, where the high harmonic is generated. An aluminum filter downstream of the SIGS removes the pump beams before the spectrum of the HHG beam is measured using the XUV spectrometer. We first inserted only the bicircular Gaussian beam fields (i.e., without the Bessel beam) and phase-matched the HHG process by tuning the gas pressure, adjusting the location of the focus, and changing the opening of the aperture before the lens to maximize the 3n ± 1 harmonics. Then, we added the Bessel beam and measured the harmonic intensity generated by the total driver field as a function of the QWP angle (i.e., ϕ).

**Funding: **O.C. acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (819440-TIMP). A.B. acknowledges support by the Israeli Science Foundation, grant no. 537/19. A.F. acknowledges support by the Israeli Science Foundation, grant no. 524/19. D.P. acknowledges support by the Israeli Science Foundation, grant no. 1803/18. O.N. acknowledges support of the Adams Fellowship Program of the Israel Academy of Sciences and Humanities, support from the Alexander von Humboldt foundation, and support from a Schmidt Science Fellowship.

**Author contributions: **G.L., O.N., and O.C. initiated the research. G.L., O.N., A.F., and D.P. developed the theory. G.L. applied the theory to produce all the presented theoretical results. G.L., L.H., and G.S. conducted the experiment, supervised by A.B. G.L. and E.B. analyzed the experimental data. G.L., O.N., and O.C. wrote the first draft. All authors contributed to the writing the paper. O.C. supervised the project.