The polarization state of tightly focused radially polarized ultrashort pulses (RPUPs) has not been sufficiently investigated. Previous studies have mainly focused on the intensity distribution and polarization state distribution near the focus of radially polarized pulses after focusing, or on the polarization state in specific planes of tightly focused linearly polarized ultrashort pulses. However, there have been few reports on the polarization state distribution at the focal plane of tightly focused RPUPs. The polarization state of tightly focused RPUPs at the focal plane can be studied using the complex sink source method (CSSM) and Richards-Wolf vector diffraction integral theory (RWVDIT). The CSSM can be used to analyze the far-field diffraction effects of tightly focused RPUPs, thereby deriving analytical expressions for these far-field effects. These studies are of great importance for understanding the physical phenomena in the interaction between RPUPs and matter.

CSSM was employed to obtain the expressions of RPUPs. Based on the expressions, the intensity distributions, spectral peak frequency distribution, and time-domain waveform variations of RPUPs in the far field were analyzed. The spectral peak frequency of RPUPs exhibited a blueshift near the optical axis in the far field, and a redshift away from the optical axis, and the time-domain waveform was broadened compared to that in the focal region. These far-field physical characteristics of the pulses should be considered when calculating the focusing properties of ultrashort pulses using RWVDIT. For few-cycle pulses, the calculation accuracy of frequency-domain RWVDIT was an order of magnitude higher than that of time-domain RWVDIT. For sub-cycle pulses, time-domain RWVDIT produced significant errors and was therefore no longer applicable, whereas frequency-domain RWVDIT could still maintain high calculation accuracy. For sub-cycle, single-cycle, and few-cycle pulses, the light intensity varies steeply in both time and space, so that the first-order approximation of the optical path difference in the RWVDIT was no longer applicable. Instead, the optical path difference was directly derived from the spherical wave focusing model, thereby improving the accuracy of the calculation results.

The case of a tightly focused beam with a beam waist of w₀=0.4 μm was analyzed. For few-cycle pulses, the time differences between the peaks of the radial electric field and the longitudinal electric field were 0.65\;\mathrmfs and 0.67\;\mathrmfs at radial positions of \rho =0.160\;\textμm and \rho =0.864\;\textμm (Fig.6), and the corresponding phase differences between the radial electric field and the longitudinal electric field were about \textπ /2 . This finding indicated that the total electric field vector rotated in the plane composed of the longitudinal electric field vector and the radial electric field vector, which could be referred to as a rotating elliptic spiral polarization state. At different radial positions, the electric field vector exhibited forward or backward rotation (Fig.7). For single-cycle pulses, the time differences between the peaks of the radial electric field and the longitudinal electric field were 0.58\;\mathrmfs , 0.46\;\mathrmfs , and 0.68\;\mathrmfs for \rho =0.160\;\textμm , \rho =0.500\;\textμm , and \rho =0.864\;\textμm (Fig.8a, Fig.8b, Fig.8c), with the corresponding phase differences of 0.44\textπ , 0.34\textπ , and 0.51\textπ . The polarization states appeared to be the forward rotating elliptic spiral polarization state and the backward rotating elliptic spiral polarization state, respectively. These polarization states showed cylindrical symmetry around the optical axis. The spin angular momentum (SAM) vector S_\rho could be used to measure the rotation of the polarization vector. For example, if S_\rho =0 , the polarization state was the radial polarization state. For both tightly focused and non-tightly focused sub-cycle pulses, the radial coordinates corresponding to S_\rho =0 significantly deviated from the beam waist position of w_0 .

After considering the blueshift and redshift of spectral peak frequencies, and the waveform changes of RPUPs in the far field, the spatiotemporal configurations of the pulses in the focal region are calculated theoretically, which agree well with the results of CSSM. Regarding the sub-cycle and single-cycle pulses, the calculation results of frequency-domain RWVDIT are consistent with those of CSSM. With variation of radial coordinates at the focal plane, the electric field vector shows elliptic spiral polarization, circular spiral polarization, and radial polarization, all of which are cylindrically symmetric. The position of the radial polarization state at the focal plane significantly deviates from the beam waist position under the condition of tight focusing. For sub-cycle pulses, the position of the radial polarization state significantly deviates from the beam waist position under both tightly focused and non-tightly focused conditions, which is attributed to the ultra-wide spectrum and the blueshift of the spectral center frequency. The research results are of great significance for generating the required ultrashort target pulses in the focal region, and for investigating chiral molecular recognition and ultrafast dynamic control of chiral molecules.