High-harmonic generation (HHG) is a unique source of ultrashort pulses for time-resolved experiments with femtosecond to attosecond resolution [1]. HHG occurs when an infrared (IR) laser field is focused in an atomic or molecular gas with an intensity in the ${10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ range. This results in the emission of a comb of odd harmonics of the IR, extending into the extreme ultraviolet (XUV). HHG can be used as a structural or dynamic probe of an external medium, e.g., by photoionization [2] or transient absorption [3]. Alternatively, the generation process itself can serve as a probe of the emitting medium by encoding structural and dynamical information onto the emitted light [4,5]. In both cases, accurately characterizing the XUV is crucial. HHG is generally considered a scalar quantity: when harmonics are produced in an isotropic medium by a linearly polarized laser, they are linearly polarized, parallel to the driving polarization. The extension of HHG to a vectorial quantity opens new perspectives for applications. The vectorial properties of HHG have been used to probe atomic [6] and molecular [7] orbitals, or to detect anisotropies of the generating medium [8]. The vectorial properties of XUV light are of particular interest to probe anisotropic media by transient absorption. Last, the recent production of quasi-circularly polarized HHG [5,9–12] will enable time-resolved experiments on chiral molecules or magnetic materials in the XUV range [13].

The characterization of fully polarized, transverse vectorial light requires the measurement of the amplitude and phase of its two components perpendicular to the propagation axis. For HHG, this can be achieved by sequentially measuring two projections of the electric field using an XUV polarizer. Here we propose an alternative solution that involves no polarizing optics and can be operated in single-shot: Combined High-harmonic Interferometries for VEctorial Spectroscopy (CHIVES). CHIVES relies on the combination of two synchronized interferometric setups for amplitude and phase measurements, namely, two-source interferometry [14–18] and a transient grating modulating HHG in only one source [19,20]. Harmonic source $S_1$ [up in Fig. 1(a)] is spatially modulated by a sinusoidal perturbation induced by the optical interference of two crossed pump beams. These pumps can either be perturbative fields synchronized with the generation field [19], modulating its polarization state (the perturbative and driving fields being orthogonally polarized), or intense pulses inducing a spatial modulation of the target medium itself (molecular excitation grating [20]). In both cases, the pumps create a spatial modulation of the HHG polarization state along the grooves of the grating, resulting in the emission of two orthogonal components. Harmonic source $S_2$, spatially separated from and temporally synchronized with $S_1$, serves as a reference for the measurement (isotropic unexcited medium, linear polarization parallel to the driving field).

 

Fig. 1. (a) CHIVES principle (see text). The bluish fields are polarized along $y$ while the reddish ones are polarized along the $x$ axis. (b) Experimental setup: a $0-\pi$ phase plate creates two HHG sources in a $N_2$ gas jet. Two pumps create a grating of molecular excitation in $S_1$ only. (c) Spatially resolved spectra. Pump off: two-source interference fringe pattern. Pump on: fringe pattern superposed with incoherent sum of Bragg peaks. Regions of interest are highlighted. They correspond to the harmonic emission through the shortest quantum paths. We discarded the outer rings resulting from long trajectories. The latter shows different behavior.

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For each polarization component, the sinusoidal modulation of $S_1$ leads to diffraction peaks up and down the main HHG beam in the far field. The two polarization components are summed incoherently on the detector. In the following, we refer to them, respectively, as parallel and perpendicular, relative to the direction of their polarization with respect to the polarization of the generation. Because $S_2$ is not modulated, it produces only a central beam that interferes with the parallel component of the undiffracted light from $S_1$. The resulting fringe pattern provides the phase difference between $S_2$ and the parallel component of $S_1$. Implementing the two interferometric techniques yields the parallel component from the fringes but also the sum of both components from the diffracted intensities. A fitting algorithm is used to extract both components from the data. In this Letter, we present successively the experimental implementation of the technique using a grating of molecular alignment, its analysis, and the interpretation of the results.

The experiment was carried out using the 1 kHz Ti:sapphire Aurore laser system at CELIA, which delivers 7 mJ, 25 fs, 800 nm pulses. The laser is split in three parallel beams focused by a 50 cm lens into a continuous gas jet (70 μm nozzle) backed with 4 bars of $N_2$. The central beam (probe beam) is linearly polarized and its profile is shaped with a phase plate, which introduces a $\pi$ phase difference between its upper and lower halves. As a result, the IR focus splits into two spots separated by $a\sim 100\mathrm{\mu m}$, creating two spatially separated harmonic sources [21]. The two other beams (pump beams) cross in the molecular jet over $S_1$. Their interference pattern creates a sinusoidal grating of molecular excitation (rotational wavepacket) with a spatial period $\mathrm{\Lambda }\approx 10\mathrm{\mu m}$ (set by the angle between the two pumps, see [19]). The pumps are advanced by 4.1 ps with respect to the probe in order to maximize the degree of molecular alignment when HHG occurs. The alignment angle (between the driving polarization and the main axis of the molecular ensemble) is controlled by rotating a half-wave plate in each pump beam. The harmonic emission is analyzed by an XUV spectrometer consisting of a $1200{\mathrm{mm}}^{-1}$ grating, a set of dual microchannel plates, a phosphor screen, and a charge coupled device camera. The cylindrical grating lets the light diverge in the vertical dimension and spectrally resolves it in the horizontal dimension. This provides a spatiospectral image of the harmonic emission. Figure 1(c) shows typical spectra. When the pumps are off (bottom half), we observe horizontal fringes resulting from the two source interference. When the pumps are on (top half), the fringe position shifts (see e.g., the amplitude at $x=0$ of $H_9$, the ninth harmonic order, is zero for pump on and almost max for pump off) and extra spots appear above and below each harmonic, corresponding to the first order diffraction of the transient grating.

To retrieve the phase and amplitude of the two orthogonal components of the harmonic emission, we extend the model of Ref. [22]. We first consider the two sources separated by a distance $a$ along the $x$ axis. We assume that the electric field at the exit of the medium factorizes as

In $S_2$, the sample is isotropic and ${\mathrm{E⃗}}_2(\mathrm{\Omega },x)$ is parallel to the driving polarization (unit vector ${\mathrm{e⃗}}_{\parallel }$): ${\mathrm{E⃗}}_2(\mathrm{\Omega },x)=E_2^{\parallel }(\mathrm{\Omega },x){\mathrm{e⃗}}_{\parallel }=D_{\mathrm{\Omega }}^{\mathrm{iso}}{e}^{i{\phi }_{\mathrm{\Omega }}^{\mathrm{iso}}}{\mathrm{e⃗}}_{\parallel }$, where $D_{\mathrm{\Omega }}^{\mathrm{iso}}$ and ${\phi }_{\mathrm{\Omega }}^{\mathrm{iso}}$ are the amplitude and phase of the harmonic dipole moment, respectively.

By contrast, HHG from $S_1$ has two vectorial components, $E_1^{\parallel }(\mathrm{\Omega },x)$ and $E_1^{\perp }(\mathrm{\Omega },x)$, due to the anisotropy introduced by the molecular alignment. We model ${\mathrm{E⃗}}_1(\mathrm{\Omega })$ assuming that the degree of molecular alignment follows linearly the local intensity of the excitation. This is correct for pump intensities up to $8\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ [23]. We also assume that the harmonic amplitude and phase follow linearly the degree of molecular alignment [23]. The parallel component of the electric field is given by

The sinusoidal modulation of the HHG amplitude and phase in the near field produces Bragg peaks in the far field. Here we consider only the first order diffraction peaks. This holds only in the low pump energy regime and for HHG phase modulations smaller than 2 rad [23]. The intensity on the detector is the incoherent sum of $I^{\parallel }$ and $I^{\perp }$. For pumps off, it is typical of two-source interference $I_0(\xi ,\mathrm{\Omega })=2{|\tilde{A}(\xi ,\mathrm{\Omega })D_{\mathrm{\Omega }}^{\mathrm{iso}}|}^2(1+\cos (a\xi ))$. Averaging over the slowly varying envelope of the fringes yields $〈I_0(\mathrm{\Omega })〉=2〈{|\tilde{A}(\xi ,\mathrm{\Omega })D_{\mathrm{\Omega }}^{\mathrm{iso}}|}^2〉$. $I_0$ is normalized by $〈I_0(\mathrm{\Omega })〉$. For pumps on, the zeroth and first order intensities are given by

Experimentally, we access $I_0$ (resp. $I_1$) by integrating the undiffracted (resp. diffracted) light over the spatial and spectral dimensions for each harmonic order [plain lines in Figs. 2(b) and 2(c)]. ${\phi }_{\mathrm{obs}}$ is measured by Fourier-transform of the zeroth order fringe pattern [plain lines in Fig. 2(a)]. Equations (6)–(11) are then used to fit the experimental values of ${\phi }_{\mathrm{obs}}$, $I_0$, and $I_1$ and retrieve $d_{\mathrm{\Omega },\theta }^{\parallel }$, ${\varphi }_{\mathrm{\Omega },\theta }^{\parallel }$, $d_{\mathrm{\Omega },\theta }^{\perp }$, and ${\varphi }_{\mathrm{\Omega },\theta }^{\perp }$. In this robust procedure, we minimized the error function $\epsilon =w_0{\epsilon }_0+w_1{\epsilon }_1+w_2{\epsilon }_2$ where the $w_i$ are weights chosen as the mean values of the measured intensities and absolute phase, and ${\epsilon }_i$ are the square values of the difference between the theoretical value for a given set of parameters and the actual experimental data. We performed a theoretical study of the robustness of the retrieval by generating noisy numerical data, and found that the fitting procedure converged properly in all cases. The symmetry of the generating medium imposes $d_{\mathrm{\Omega },0°}^{\perp }=d_{\mathrm{\Omega },90°}^{\perp }=0$. We enforce this condition by imposing the value of the perpendicular intensity at these angles to be zero. The retrieved $I_0$, $I_1$, and ${\phi }_{\mathrm{obs}}$ are displayed with symbols in Figs. 2(a)–2(c). When the orthogonal component of the grating fades away, our model is less relevant. This explains the worse agreement observed for $I_1$ at 0° and 90°. However, the remaining error is extremely low, in the ${10}^{-16}$ on average. To illustrate the influence of the grating’s perpendicular component, note that when it is ignored, as in Ref. [22], the error remains high, in the ${10}^{-3}$ on average, and the simultaneous fit clearly does not converge.

 

Fig. 2. Plain lines: far-field measurement of (a) ${\phi }_{\mathrm{obs}}$, (b) $I_0$, and (c) $I_1$ as a function of alignment angle. Symbols (squares, triangles, disks, diamonds, and crosses): values retrieved through the fit.

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The values of $d_{\mathrm{\Omega },\theta }^{\parallel }$, ${\varphi }_{\mathrm{\Omega },\theta }^{\parallel }$, $d_{\mathrm{\Omega },\theta }^{\perp }$, and ${\varphi }_{\mathrm{\Omega },\theta }^{\perp }$ that fit the best the experimental values of ${\phi }_{\mathrm{obs}}$, $I_0$, and $I_1$ according to Eqs. (6)–(11) are plotted in Fig. 3. For the parallel component, the amplitude is maximized when molecules are aligned parallel to the laser field, except for the $H_{11}$, which shows a weak contrast and an inverted behavior. This inversion was already reported in Ref. [24] and attributed to the vicinity of the ionization threshold. Our measurements show that the evolution of the phase and intensity of the orthogonal component of $H_{11}$ are close to that of the higher orders. By contrast, $H_9$ shows a maximum in the orthogonal emission for higher alignment angles and an inverted phase behavior for the parallel component. These elements should help understanding the process of below-threshold HHG [25]. $H_{13}$ to $H_{17}$ show orthogonal dipole moments comparable to the parallel ones when the molecules are aligned around 45°. This is consistent with the strong ellipticity that was measured for these harmonics by conventional polarimetry [15,20]. Note that the variations of ${\varphi }_{\mathrm{\Omega },\theta }^{\parallel }$ are smoother than the ones of ${\varphi }_{\mathrm{\Omega },\theta }^{\perp }$ because the former are strongly constrained by ${\phi }_{\mathrm{obs}}$.

 

Fig. 3. Amplitude and phase of the parallel component (a) and (b) and of the perpendicular component (c) and (d) as retrieved by the CHIVES procedure. The grey curve shows the upper bound limit imposed on the fitting function to enforce a null value of the dipole at 0° and 90°.

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CHIVES could be extended to resolve the polarization state of HHG. The two missing elements are the phase relation between the parallel and perpendicular components, ${\phi }_{\mathrm{\Omega }}^{\mathrm{iso}}-{\phi }_{\mathrm{\Omega }}^{0\perp }$ and the degree of polarization of the harmonic light. ${\phi }_{\mathrm{\Omega }}^{\mathrm{iso}}-{\phi }_{\mathrm{\Omega }}^{0\perp }$ could be calibrated by one polarimetry scan (Malus law) at a given alignment angle and used for all other measurements. With this extension, CHIVES would considerably speed up polarization measurements by replacing long polarimetry scans by single shot interferometry measurements.

In conclusion, we proposed a combined arrangement for single-shot, phase-sensitive investigation of the vectorial properties of high-order harmonic emission. We derived the equations describing how it works and successfully implemented it in aligned $N_2$. We confirmed previous results and extended phase measurements toward harmonic orders close to the ionization barrier of $N_2$. We foresee that CHIVES will be useful for time-resolved, phase-sensitive vectorial studies in HHG spectroscopy, as well as for low repetition rate experiments and refined polarization studies. This technique can be slightly modified to reconstruct the excitation grating by holography, as well as to study the phase relation between HHG short and long trajectories. Also, CHIVES can be extended to study other polarization-sensitive nonlinear phenomena like cross-polarized wave generation [26] and polarization resolved SHG microscopy [27].