Fig. 2: The scheme and spectrum of generated optical quasi-linearly polarized SNHPs.
a The laser setup for the characterization of the generated optical SNHPs. A quarter-wave plate decomposes the toroidal pulse into its circularly polarized components, while a linear polarizer is used to isolate the quasi-linearly polarized SNHP. The spatial spectrum of the SNHP is obtained by imaging the wavefront at the focal region using a series of narrowband spectral filters. The vortex characteristics of the generated waves were examined through edge diffraction analysis. The intensity distributions of light detected by a charge coupled device (CCD) camera at different wavelengths in the absence and presence of an opaque edge are shown in (b, c), respectively. The color indicates the wavelength, and the brightness represents the intensity. The ring-shaped intensity and fork-shaped pattern indicate the presence of optical vortices, consistent with those of SNHPs. The red and blue regions correspond to electric fields Ex with opposite phases in (d1–d3): d1 SNHP decomposed from the canonical toroidal pulse, exhibiting a single-cycle structure; d2 analytical SNHP decomposed from the toroidal pulse with a partial spectrum, exhibiting a few-cycle structure; d3 SNHP reconstructed from experimental results, showing a few-cycle structure consistent with (d2). The concurrence (con) and entanglement of formation (EoF) evolution of the measured transverse electric field components in (e) indicate that the generated SNHPs exhibit strong space-time nonseparability. The inserted figure in (e) represents the measured tracking curves of the maximum field positions for different wavelengths. The trajectories of different wavelengths do not cross when the incident wave is a toroidal pulse, demonstrating isodiffraction characteristics. The state-tomography matrix of generated pulses when the incident wave is a toroidal pulse is shown in (f). In (f), \(|{\lambda }_{1}\rangle\)–\(|{\lambda }_{5}\rangle\) and \(|{r}_{1}\rangle\)–\(|{r}_{5}\rangle\) represent the spectral states and spatial states, respectively. The color indicates the normalized intensity of the state-tomography matrix. When the incident wave is a toroidal pulse, the diagonalized matrix confirms isodiffraction characteristics. The measured fidelity (F) is 0.82, indicating a good match with canonical SNHPs.
