Complete spatiotemporal and polarization characterization of ultrafast vector beams

The use of structured ultrashort pulses with coupled spatiotemporal properties is emerging as a key tool for ultrafast manipulation. In particular, the ultrafast vector beams are opening exciting opportunities in different fields such as microscopy, time-resolved imaging, nonlinear optics, particle acceleration or attosecond science. We propose and demonstrate a technique for the full characterization of structured time-dependent polarization light waveforms with spatiotemporal resolution. We have developed and implemented a compact twofold spectral interferometer, based on in-line bulk interferometry and fibre-optic coupler assisted interferometry. We have experimentally measured structured infrared femtosecond vector beams, including radially polarized beams and complex-shaped beams exhibiting both temporal and spatial evolving polarization. Our measurements confirm that light waveforms with polarization evolving at the micrometer and femtosecond scales can be achieved through the use of s-waveplates and polarization gates. This new scale of measurement achieved will open the way to predict, check and optimize applications of structured vector beams at the ultrafast -femtosecond- and micrometer scales.


Introduction
During the last decades, the development of laser technology has boosted our ability to control the properties of ultrafast light pulses. Nowadays it is possible to routinely generate coherent radiation from the near-infrared to the soft x-rays 1 , that can be emitted in the form of few-cycle fs laser pulses [2][3][4] , or even attosecond pulses 5 . Furthermore, it is possible not only to tailor their spatiotemporal properties [6][7][8] , but also to structure ultrafast light fields in their angular momentum 2/18 properties, including both polarization and orbital angular momentum [9][10][11] . The ultimate control of the angular momentum properties of ultrafast laser pulses have opened new routes for the study of chiral structures [12][13][14] , topological systems [15][16][17] , or magnetic materials 18,19 at the ultrafast timescales.
One example of structured ultrafast light fields with tailored spatiotemporal and angular momentum properties are the so-called vector beams, linearly polarized fields whose polarization direction changes along the transverse spatial profile of the light beam 20 . The paradigm of vector beams is constituted by radially (RP) an azimuthally (AP) polarized beams. Interestingly, RP beams allow to focus light below the diffraction limit 21 , which enables applications in different fields such as laser machining 22,23 or particle acceleration 24,25 , among others. AP beams can induce longitudinal magnetic fields at the singularity of the electric field 26 which offers potential applications in spectroscopy and microscopy 27 .
Nowadays vector beams can be routinely produced as continuum waves in the infrared (IR) and visible regimes through the use of uniaxial and biaxial crystals [28][29][30] , spatial light modulators 31 , optical fibres 32 , electrically-tuned q-plates 33 or azimuthally dependent half-waveplates fabricated by ultrafast laser nanostructuring of silica glass, also known as s-waveplates 34 , among others.
Recently, the generation of short pulsed, femtosecond vector beams 35,36 has gained interest due to their application in high harmonic generation and attosecond science [37][38][39] or particle acceleration 40 .
However, the advances of new laser sources and their applications is tied up to the development of characterization techniques. Since the 1990s, different techniques have been used for the temporal characterization of scalar -i.e., linearly polarized (LP)-ultrashort laser pulses 41 . In the last two decades, the problem of measuring spatiotemporal couplings in scalar beams has been tackled with new strategies [42][43][44][45][46] . In parallel, the reconstruction of time-evolving polarization pulses has also been addressed [47][48][49][50][51] . However, the necessity for the full spatiotemporal characterization of structured ultrafast laser pulses, which includes both spatiotemporal and polarization properties, remained a challenge up to now.
In this work we propose and demonstrate a technique to characterize the arbitrarily space-time (and the space-frequency) polarization dependence of structured ultrafast light pulses. In particular we perform the full characterization of infrared femtosecond vector beams generated through an swaveplate, which allow us: (i) to measure the spatiotemporal quality of RP pulses; (ii) to shape and characterize time-dependent vector beams, structured through the use of polarization gates; and (iii) to monitorize the focusing properties of structured vector beams. Our technique is based on 3/18 twofold spectral interferometry, both for the spatiotemporal reconstruction through spatiallyresolved spectral interferometry assisted by a fibre-optic coupler 43 and for the polarization analysis through in-line bulk interferometry 51 . The complete knowledge of the vector beam amplitude and phase allows to reconstruct the polarization state of the beam profile (including intensity, azimuth, and relative phase between the polarization components) both in the space-frequency and spacetime domains. We demonstrate that spectral interferometry is a powerful technique for the characterization of ultrafast vector beams, opening the route for a new set of characterization techniques of structured light waveforms, to be used in a diverse range of applications 49,52,53 that make use of spatial and temporal polarization shaping.

Technique for complete spatial, temporal and polarization resolved characterization.
The technique that we introduce to perform the complete spatiotemporal and polarization characterization is based on spectral interferometry. A scheme of the experimental setup is shown in Fig. 1. The laser output is divided into two replicas, one of them is used as a reference (known), while the other beam is shaped in its polarization components -using an s-waveplate and retarder waveplates-, conforming the unknown beam to be characterized.
In the unknown beam, a thick birefringent plate is placed to delay the horizontal and vertical polarization components of the beam. Afterwards, a linear polarizer (LP) is used to sample different projections of the beam. The reference beam is collected by a fixed position fibre port, while the unknown beam is spatially scanned (in the transverse XY plane) with a motorized fibre port. Both single-mode fibres are combined in a broadband fibre-optic coupler. The delay between the reference and the unknown beam is adjusted and fixed with the longitudinal position of the reference fibre. The known reference phase is measured with a standard temporal characterization technique 41 (e.g., in the present case, the SPIDER technique 54 ).
In order to perform the complete characterization, first we measure the spatially-resolved spectral interferometry between the 0º-projection (X-component) of the unknown beam and the reference pulse, + ( , , ), by setting the LP horizontal. Then we obtain their relative phase using Fourier analysis 55 (see details in Methods). Afterwards, we measure the spectrum profile of the same beam projection, ( , , ). As the reference phase is known, we obtain the spatiospectral (and spatiotemporal) amplitude and phase of the X-component of the unknown 4/18 beam. This strategy 43 has been shown to be very versatile by use in the measurement of diffractive focusing, nonlinear processes and few-cycle pulse characterization 8,56-58 , among others.
Second, we acquire the 90º-projection of the unknown beam, ( ; , ) by placing the LP vertically. Finally, with the LP at 45º, we measure an intermediate projection of the X and Y components, + ( ; , ), which encodes their relative phase (see details in Methods). This allows us to accurately retrieve the phase of the Y component, and thus, the frequency-dependent and time-dependent polarization 51 . The phase introduced by the birefringent plate is calibrated as described in the Methods Section. As the fibre scans the transverse profile of the unknown beam, we retrieve the full spatiotemporal (and spatiospectral) polarization dependence of the beam. beam are delayed with a birefringent plate. The unknown beam is spatially scanned with a fibre coupler. The X projection spectrally interferes with a known reference pulse, while the Y and X components spectrally interfere after a 45º linear polarizer (LP). Inset: definition of the polarization ellipse through the azimuthal angle and the ellipticity = ⁄ .

Characterization of radially polarized laser pulses
First, we have characterized a femtosecond RP beam created through an s-waveplate placed after the output of a chirped pulse amplification Ti:sapphire laser (see Methods for further details).
In order to explore the polarization distribution of the beam along the azimuthal coordinate , we scanned the XY plane through a circumference of radius R = 3 mm (corresponding to the halfmaximum of the beam intensity) around the optical axis. In figure 2 we show the ellipticity ( ), azimuth ( ) and intensity profiles of the pulse in the space-frequency (first row) and space-time  following figures are cut below the 1% of the spectral/temporal peak signal.

Shaping and characterization of time-dependent ultrafast vector beams
In order to show the ability of our technique to characterize vector beams whose polarization distribution varies temporally in the femtosecond timescale, we structure a laser pulse through the use of polarization gates. When combining an s-waveplate with different types and sets of waveplates, the resulting beam presents a strong spatiospectral and spatiotemporal coupling in its intensity and polarization parameters. One set of waveplates that is particularly interesting is the so-called narrow polarization gate, which is used to effectively generate shorter pulses in certain applications as e.g. isolated attosecond pulse generation [59][60][61] . In the Methods Section we detail the operation of the narrow gate using a multiple-order waveplate (QWM) and a zero-order waveplate (QW0).
In this experiment, instead of using a spatially-uniform LP beam, we illuminate the narrow gate setup (i.e. the two waveplates, first QWM with fast axes at 0º and next QW0 at −45º) using the RP

Focusing monitorization of structured time-dependent vector beam pulses
Many applications of vector beams, e.g. particle trapping, microprocessing, particle acceleration or nonlinear optics, are carried out at their focus position. Our characterization technique allows us to perform the full characterization both at the far-field and at the near-field. Here we analyse and compare the spatiotemporal polarization dynamics at the focus of the two previously presented vector beams, i.e., the RP vector beam and the RP beam followed by the narrow polarization gate.
In both cases, the beam was focused using an achromatic lens with a focal length of 50 cm. In these experiments the ultrafast vector beam presents a rich evolution both in the radial and azimuthal coordinates. Therefore, we did a two-dimensional spatial scan along the transverse XY plane. 8/18 First, when using a purely RP beam, the ellipticity ( , , ) is ideally zero for every position x,y and time t. As shown before, although there is a spatiotemporal intensity modulation, there is no substantial temporal dependence of the polarization parameters of the beam. In Fig. 4 c1, we show the spatial profile at the focus position for a temporal instant that corresponds to the peak of the pulse. We notice that, instead of an ideally homogeneous ring shape of a RP, the measured beam presents spatial intensity modulations due to the above-mentioned inhomogeneities of the input spatial profile. On the other hand, the spatial distributions of the X and Y polarization projections (Figs. 4 a1, b1, respectively) correspond to that of a RP with the singularity at the centre (see the azimuth profile in Fig. 4 e1). In the Supplementary Movie 1 we show that the structure of these magnitudes of the focused RP beam is preserved in time. In the spectral domain, the dependence of the homologous magnitudes is analogous, also being wavelength independent, as shown in the Supplementary Section S.IV and in Supplementary Movie 2.
Contrarily to this case, when focusing the RP beam after illuminating the narrow gate, the beam exhibits a temporal/spectral polarization evolution together with the spatial dependence. In this second case the intensity ring is split into two lobes with time-dependent orientations, except for the centre of the pulse (the mean propagating time of the beam components passing through the fast and slow axes of the QWM), where the ring is recovered (Fig. 4 c3), as in the case of a focused RP beam (Fig. 4 c1). The X and Y projections of this temporal snapshot of the intensity corresponds to two lobes oriented at ±45° (Fig. 4 a3, b3). In the temporal leading edge of the beamcorresponding to the fast axis component at QWM-the two lobes are oriented in the x-axis for the total intensity as well as for the X and Y projections (Fig. 4 a2-c2). Contrarily, in the temporal trailing edge, the two lobes are oriented in the y-axis (Fig. 4 a4-c4), as the slow axis of the QWM is oriented vertically. The complete temporal evolution is shown in Supplementary Movie 3, where the total intensity evolves from two vertical lobes until they completely fill the ring and then they split into two horizontal lobes. Thus, the X and Y intensity projections consist in two spatial lobes evolving from vertical to horizontal but rotating in opposite direction. Regarding the ellipticity ( Fig. 4 d3), it describes symmetrical CP in the x and y axes directions (where there is spatial superposition of the X and Y projections), while presenting LP out of those axes. The polarization azimuth (Fig. 4 e3) of the LP contributions mentioned before is oriented at 0º and 90º corresponding to the ±45° quadrants of the X (+45°) and Y (−45°) intensity projections (Fig. 4 a3-b3). We also 9/18 found that the ellipticity of the two lobes is = 1 (CP) both for the leading and trailing edges of the beam (Fig. 4 d2, d4), and there is a gradual temporal evolution (Supplementary Movie 3).
In the spatiospectral domain, the frequency dependence presents some similarities (as shown in the Supplementary Section S.IV and in Supplementary Movie 4). The X and Y projections of the spatially resolved spectrum correspond to two lobes with different orientations in the XY-plane rotating as a function of the wavelength and forming a ring for certain intermediate wavelengths, while the total spectrum forms a ring with the cross shape in the ellipticity described for Fig. 4 d3 (CP in the x and y axes, LP elsewhere), being frequency-independent. comparison with the focus for the radially polarized beam at the pulse peak (at 20 fs). Rows 2-4: selections for three different times, respectively, -100, 20 and 120 fs, for the radially polarized beam followed by a narrow gate. The spatial scan was performed in a square grid with 21 x 21 points using a step of 18 .

Discussion
Tailoring light beams in full dimensionality, i.e., both spatial and temporal shaping of the individual light waveforms on a femtosecond time-scale, is nowadays possible. Our results demonstrate spectral interferometry as a suitable technique for performing a complete spatiotemporal and spatiospectral characterization of such ultrafast beams, whose polarization changes in time and space. The use of spectral interferometry is advantageous as the detection is fully linear (except for the reference measurement) and the data processing is fast, direct and univocal, as well as the acquired data being minimal for this level of measurements. The use of a birefringent plate and a fibre-optic coupler to implement a twofold interferometer avoids using multiple standard interferometers and alignment of beam recombination.
Regarding the vector beam shaping, we show that the combination of spatially varying polarization with temporal polarization shaping can produce singular spatiotemporal polarization dependences. By using a radially polarized beam to illuminate a narrow polarization gate, we create a complex vector beam with different orientations of the polarization gate or constant circular polarization, depending on the azimuthal angle in the transverse plane. We show experimentally that the temporal evolution of the focus of such a beam presents rich dynamics in contrast to focused radially polarized beams. The typical ring mode of focusing is effectively shortened in time because of the narrow polarization gate, which could be used for example to manipulate or trap nanoparticles during shorter times. This can be advantageous when using few-cycle pulses, due to the dispersion in media for ultrabroadband pulses.
In conclusion, the technique presented here constitutes the first accomplishment, to the best of The fibre-coupler is made of broadband single-mode fibres centred at 800 nm, being both input arms almost equal-length so that their dispersion is compensated. The relative dispersion due to small difference (~1 mm) is calibrated with spectral interferometry.
The thick birefringent plate used was a 3-mm calcite plate (Altechna) with the fast axis oriented vertically. The multiple-order waveplate QWM is a 3-mm quartz plate operating as quarter-wave plate for 806 nm. The zero-order waveplate QW0 is a 1.3-mm quartz plate designed for quarterwave operation at 800 nm.
The spectra were acquired with a fibre-coupled spectrometer (Avantes). The spatial scan in the XY plane was done with a two-axes motorized stage (Thorlabs). The reference pulse was characterized with a SPIDER measurement 54 .
The s-waveplate for 800 nm wavelength was fabricated by ultrafast laser nanostructuring of silica glass.

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Spectral interferometry and data analysis. In spectral interferometry, two delayed collinear pulses interfere in the spectral domain and their spectral fringes encode their relative phase as given The relative phase can be extracted by Fourier-transform Spectral Interferometry (FTSI) analysis of the fringes 55 . The pulse delay must be high enough to separate the signals in the time domain after Fourier-transform, as well as small enough to resolve the interferences in the spectrometer.
The delay introduced by the birefringent plate between the X and Y components of the unknown beam is 1.8 ps, determined by the plate thickness. In the spectral interferometry between the reference and the X component of the beam, we manually introduced a 2 ps delay. The spectral bandwidth of the unknown pulse must be less or equal than the reference spectrum that interfere, and their relative amplitude must be of the same order to obtain better contrast fringes. The spatial resolution of the technique is related to the mode-field diameter of the single-mode optical fibre, being in the present case 4 .
As the reference pulse is characterized -the phase ( ) is known-, the phase of the horizontal projection of the unknown vector beam, ( ; , ), is obtained from the spatiallyresolved spectral interferences with the reference pulse, given by where , is the relative phase of the birefringent plate eigenaxes. The calibration of this phase is described below.
The spatial scans of the individual spectra, and , are optional as they can be obtained with the FTSI algorithm from the measurement of and the spectral interferometry scans given above, i.e., + and + . Nevertheless, as they can be directly measured, we acquired them, since the 13/18 performance of the FTSI algorithm is improved when subtracting the individual spectra from the interferences before data processing.
Calibrations. The global dispersion of the birefringent plate can be calculated from the thickness and the refractive index using Sellmeier equations. However, the accurate knowledge of the relative phase between the fast and slow axes is critical for the correct retrieval of the beam polarization.
This calibration depends on the thickness and alignment of the birefringent plate. In order to calibrate the system we used a linearly polarized pulse, with no time evolving polarization, at 45º before the birefringent plate 51 . From its own interferences, with the projection + , we retrieved accurately the relative dispersion of the birefringent plate. In our measurements, we repeated this calibration after any realignment. We also found that the calibration of the birefringent plate did not depend on the transverse spatial position. Furthermore, with the same calibrating pulse (linearly polarized at 45º), we measured and at the same sampling position, to calibrate the amplitude response of the system, which we used to correct the measurements of the individual spectra, and , of the unknown beam.

Models for the simulations.
To simulate the shaped vector beam shown in this work, we started from a homogeneous beam, plane wave, using the experimental spectral amplitude of the laser output. To model the zero-order and multiple-order quartz waveplates, we firstly calibrated their thickness and retardation (if previously unknown) using spectral interferometry in combination with our detection (birefringent plate, LP and spectrometer). In the simulations, we calculated the dispersion of their eigenaxes from Sellmeier equations and then we imposed the known retardation for the corresponding operation wavelength. Naturally, we applied every element considering the described orientations.
For the s-waveplate, we modelled it as a half-wave plate with the fast axis orientation depending on the azimuthal angle . As a reference, when the s-waveplate is oriented to create RP from input horizontal linear polarization, the fast axis orientation is 2 ⁄ . We operated in the space-frequency domain, and at the end we obtained the space-time dependence by Fourier transformation.
Generation of the narrow polarization gate. The experimental implementation for the narrow polarization gate consists of using two consecutive quarter-waveplates, the first multiple-order 14/18 QWM and the second zero-order QW0, with relative eigenaxes at 45º. In the scheme of Fig. 5

Summary
In this Supplementary material we show enlarged and detailed information regarding the results presented in the main manuscript.

S.I. Characterization of the input laser beam
In this Section we show the measurement of the laser beam used for the experiments. As in the first part of the manuscript, we perform a circular scan with radius R=3 mm.

S.II. Characterization and simulation of radially polarized laser pulses
In this Section we show the experimental measurements compared to the theoretical simulations of a radially polarized beam, corresponding to the manuscript section: Results || Characterization of radially polarized laser pulses.

S.III. Shaping and characterization of time-dependent ultrafast vector beams
In this Section we show the experimental measurements compared to the theoretical simulations of a radially polarized beam followed by a narrow polarization gate, corresponding to the manuscript section: Results || Shaping and characterization of time-dependent ultrafast vector beams.

S.IV. Focusing monitorization of structured time-dependent vector beam pulses
In this Section we show the spectral results for the experimental measurements of the focusing of a radially polarized beam followed by a narrow polarization gate (Fig. S5, rows 2-4), corresponding to the manuscript section: Results || Focusing monitorization of structured timedependent vector beam pulses. The results are compared to the focusing of a radially polarized beam (Fig. S5, row 1). The full spatio-spectral results are expanded to see the wavelengthdependent evolution in Supplementary Movies 2 and 4, already described in the manuscript.