1 Introduction

Frequency resolved optical gating (FROG) has been well developed as one of the most widely used and reliable methods for ultrashort laser pulse characterization [1–4]. Various FROG geometries such as second and third harmonic [5, 6] as well as polarization and transient optical grating FROG [7, 8] have been successfully applied to meet different requirements or experimental situations. Most of FROG geometries are designed for linearly polarized pulse characterization but not available for elliptically polarized pulse directly. While dynamic control of polarization states has been demonstrated to play a vital role in various ongoing experiments like high-order harmonic generation [9], above-threshold multi-photon ionization [10], molecular dissociation [11], and single attosecond pulse generation [12], thus, the complete characterization of polarization states of ultra-short laser pulse is highly desired. In Ref. [13], the technique of polarized light interference versus wavelength of only a glint (POLLIWOG) was employed to characterize the target pulse by measuring the spectral interference between the reference pulse and two orthogonal polarization components of the target pulse. In this case, the given bandwidth of the reference pulse is a restriction of the method for the reason that the bandwidth of target pulse must be comparable with (or less than) that of the reference pulse. Alternatively, the self-referenced method of tomographic ultrafast retrieval of transverse light E fields (TURTLE) [14, 15] could be used to characterize the polarization state of various laser pulses by the use of an appropriate FROG configuration. In TURTLE, two measurements are implemented to characterize the orthogonal polarization components of the target pulse while one additional measurement is used for a polarization projection at an arbitrary angle, from which all the phase information of the target pulse could be attained completely.

In this paper, we employed the technique of molecular-alignment-based cross-correlation frequency resolved optical gating (M-XFROG) [16] to measure the elliptically polarized pulse and investigated its polarization state with the Stokes parameters. Impulsive rotational Raman excitation with a femtosecond exciting pulse enforced air molecules to be aligned and produces a transient birefringence, which functioned as a linear optical gating for the target pulse without phase-matching restriction within the transparent spectral range of air molecular gas. Meanwhile, the M-XFROG technique was realized with gaseous air molecules, so that the temporal and spectral distortions of the target pulse in the measurement could be neglected owing to the negligible absorption and dispersion of the gaseous molecules. Here, we tried to employ the M-XFROG technique to fully characterize the elliptically polarized target pulses.

Due to molecular polarizability anisotropy with unequal polarizability components parallel and perpendicular to the molecular axis, air molecules were forced to align along the laser field polarization under impulsive excitation by the broadband femtosecond laser pulse. The non-resonant impulsive interaction excites the initially randomly aligned air molecules into a series of rotational states and the quantum beatings among these rotational wave-packets result in a periodic field-free molecular alignment phenomena [17–20]. The degree of molecular alignment could be depicted by the average value of ≪cos2 θ≫, where θ denotes the angle between the exciting field polarization and molecular axis. The pre-aligned molecules show an orientation-dependent refractive index change as δn = 2π(ρ 0Δα/n 0)(≪cos2 θ≫ − 1/3) [21], where ρ 0 and n 0 denote the number density of the gaseous molecules and linear refractive index of randomly aligned molecules, respectively. Because of such molecular-alignment-induced transient birefringence, a linearly polarized target pulse experiences a polarization rotation as its field components parallel or perpendicular to the molecular alignment feels different refractive indexes. Therefore, the pre-aligned air molecules equivalently function as a transient optical gating and such ultrafast optical gating could be properly applied to measure linearly polarized supercontinuum femtosecond laser pulses covering the range from ultraviolet to near-infrared, without phase-matching restriction and measurable spectral bandwidth limitation [16]. Quite similarly, in polarization gating FROG (PG-FROG), the refractive index change induced by optical Kerr effect is δn = n 2 I, where n 2 and I denote the nonlinear refractive index of the solid-state nonlinear media and intensity of the exciting field. However, the conventional PG-FROG is more susceptible to the nonlinear characteristic of the used solid nonlinear media, which may typically cause unpredictable temporal and spectral distortions or absorption to the target pulse and thus decrease the accuracy. This problem becomes even serious in the ultraviolet range. Compared with the solid state nonlinear media, such as fused silica [22], the low absorption and dispersion of gaseous air molecule [23] in M-XFROG show a significant advantage in the cases of extremely weak pulses and deep blue region, which enable us to perform the characterization of weak fourth harmonic pulses centered around 200 nm [24]. As molecular-alignment-based optical gating works efficiently for linearly polarized pulses, and therefore, we tried to carry out a full characterization of elliptically polarized pulses by M-XFROG.

Without loss of generality, in the frequency domain, an arbitrary elliptically polarized pulse could be describe as

$$ \widetilde{E}(\Upomega ) = \widetilde{E}_{H} (\Upomega )\widehat{x} + r\widetilde{E}_{V} (\Upomega )e^{ - i(\Upomega \tau + \theta )} \widehat{y},\quad \Upomega = \omega - \omega_{0} $$
(1)

where \( \widetilde{E}_{H} (\Upomega ) \) and \( \widetilde{E}_{V} (\Upomega ) \) represent the complex amplitude of horizontal and vertical polarization components while r, τ and θ denote the relative amplitude, delay and phase, respectively. Both horizontal and vertical polarization components of elliptically polarized target pulse could be steadily measured by M-XFROG, respectively. However, in order to determine the values of τ and θ, a third measurement should be implemented for the polarization projection at an arbitrary degree γ [14, 15], whose field could be written as

$$ \widetilde{E}_{\gamma } (\Upomega ) = \cos \gamma \widetilde{E}_{H} (\Upomega ) + r\sin \gamma \widetilde{E}_{V} (\Upomega )e^{ - i(\Upomega \tau + \theta )} $$
(2)

Obviously, the information of τ and θ was contained in the measured M-XFROG trace of \( \widetilde{E}_{\gamma } (\Upomega ) \) and their values could be retrieved by searching the minimum root-mean-square (RMS) error between the measured M-XFROG trace and the reconstructed M-XFROG trace of \( \widetilde{E}_{\gamma } (\Upomega ) \) [15]. In general, the RMS error is defined as

$$ e = \frac{{\sqrt {\sum\nolimits_{i,j}^{N} {[I_{\gamma }^{\text{meas}} (i,j) - I_{\gamma }^{\text{recon}} (i,j)]^{2} } } }}{{\sqrt {\sum\nolimits_{i,j}^{N} {[I_{\gamma }^{\text{meas}} (i,j)]^{2} } } }}, $$
(3)

where \( I_{\gamma }^{\text{recon}} \) denotes the reconstructed M-XFROG trace and it could be numerically expressed by [1]

$$ I(\omega ,\tau ) = \left| {\int\limits_{ - \infty }^{ + \infty } {E(t)G(t - \tau )} {\text{e}}^{ - i\omega t} {\text{d}}t} \right|^{2} , $$
(4)

where G(t) is the gating function, which is provided by pre-aligned molecules in our case. The complex amplitude and phase of E(t) H(V) can be retrieved by employing the widely used principle component generalized projections algorithm [1], with a acceptable trace error after hundreds of iterations.

2 Experimental setup

As schematically shown in Fig. 1a, our experiments were carried out with the output of an amplified Ti: sapphire laser system (800 nm, 1 kHz), which was split into two parts: one was used as exciting pulse to align the air molecules and the other was used to produce an elliptically polarized target pulse by propagating through an 800-nm quarter-wave plate with its optical axis randomly orientated. After the quarter-wave plate, the elliptically polarized target pulse was sent to an 800-nm Glan–Taylor polarizer with an extinction ratio of ~1 × 10−6. As shown in Fig. 1b, the first polarizer was properly rotated to set its transmission direction parallel to the horizontal (E H) or vertical (E V) polarization components of the elliptically polarized target pulse. In this case, the linearly polarized components E H and E V could be independently characterized. The exciting and target pulses were focused by two separate convex lenses of f = 60 and 40 cm, respectively, and crossed with each other by a small angle of ~10°. The positions of the convex lenses were adjusted to make sure the foci were spatially overlapped in the interaction region. The energy of the exciting pulse was measured to be ~810 μJ while those of the horizontal and vertical polarization components of the target pulse were ~91 and ~198 μJ, respectively. The transmission direction of the second polarizer was set to be perpendicular to the linearly polarized E H (E V) so that the time-dependent molecular-alignment-induced polarization rotated spectrum of E H (E V) could get thought the polarizer and be carefully focused into a fiber-coupled spectrometer (HR4000, Ocean Optics) with a lens (f = 10 cm) for the measurement of E H (E V) M-XFROG trace. The polarization of the exciting pulse was rotated to be 45° with respect to E H(V), as shown in the inset of Fig. 1b, so that the aligned molecules exerted an identical optical gating on E H and E V, which helped to improve the accuracy. Finally, the M-XFROG trace of E γ was also measured in the identical manner, with the polarizers in Fig. 1 rotating to the corresponding angles. For convenience, we chose γ to be 45 degree in our experiment. The time delay between the exciting and target pulses was controlled by a delay line stage with a step of 5 fs. Moreover, the measured background (without exciting pulse) was subtracted from the measured M-XFROG traces (with exciting pulse) before retrieval, allowing us to further increase the signal-to-noise ratio of the measured signals.

Fig. 1
figure 1

a The experimental setup. HWP 800-nm half-wave plate, QWP 800-nm quarter-wave plate. The inset shows that the gating function G(t) is acted by the first alignment signal (shortly after the zero time delay) of air molecules. b The measurement strategy: the linearly polarized horizontal polarization component (E H) and vertical components (E V) of the elliptically polarized target pulse were extracted and measured by M-XFROG technique, respectively. The inset shows that the polarization of the exciting pulse was set to be 45° with respect to the horizontal (vertical) polarization component. c The orientation angle θ and the ellipticity angle χ of the elliptically polarized pulse (Color online)

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3 Results and discussions

Figure 2a, b shows the measured and retrieved M-XFROG traces of E H with a trace error of ~9.13 × 10−3 after 400 iterations. Figure 2c, d shows the retrieved temporal profile, phase, and the measured (blue dash–dot curve) and retrieved (pink-solid curve) spectra of E H. The corresponding results of E V are displayed in Fig. 2e–h. The retrieval algorithm and settings for E H and E V were identical. As expected, the M-XFROG trace, temporal and spectra profiles of E H were almost identical to those of E V. The temporal duration of E H (~54 fs, full width at half maximum, FWHM) was also nearly the same as that of E V (~56 fs). Because of the non-collinear geometry in our experiments, a time smearing δt [25] about 1.4 fs for the ~10° crossing angle should be considered.

Fig. 2
figure 2

a, e The measured and b, f the retrieved M-XFROG traces of the horizontal (vertical) polarization component. c, g The retrieved pulse temporal profile (black-solid curve) and phase (red dash-dotted curve), and d, h the measured (blue dash-dotted curve) and retrieved (pink-solid curve) spectra of the horizontal (vertical) polarization component

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It should be emphasized the molecular-alignment-induced optical gating (M-gating) functions as G(t) in the M-XFROG technique. In some traditional cases, such as second harmonic generation based FROG or PG-FROG, the time duration of the gating functions are as wide as that of the target pulse. It has been proved to be reliable to employ such a pulse-gating function to measure the target pulse. However, the FWHM duration of the M-gating function in M-XFROG, which was mainly determined by the intrinsic properties of the gaseous molecules rather than the exciting pulse, was ~200 fs and it was much longer than that of the target pulse in our case. The gating function is very important for pulse retrieval so that its reliability should be carefully investigated. Therefore, we made a comparison between the measured molecular alignment signal and the retrieved M-gating functions. Figure 3 shows the retrieved M-gating functions for E H (red-solid curve) and E V (blue-solid curve) by using the twin retrieval of the excitation electric fields frequency-resolved optical gating (TREEFROG) algorithm [26]. It indicates that the two retrieved M-gating functions agree well with the measured molecular alignment signal (black dash–dotted curve) in air. Moreover, due to the characteristic of the delayed Raman response of air molecules, there existed a ~100 fs delay between the zero time delay and the peak of the first molecular alignment signal in air.

Fig. 3
figure 3

The measured molecular alignment (black dash-dotted curve) signal in air and the retrieved M-gating functions of E H (red-solid curve) and E V (blue-solid curve). Due to the delayed characteristic of the Raman response, the peak of the measured alignment signal of air molecules has a shift of ~100 fs relative to the zero time delay

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Figure 4a shows the measured M-XFROG trace for \( E_{{45^{^\circ } }} , \) with its experimental environment being identical to E H and E V. With the retrieved G(t), we properly initialized the values of τ and θ in Eq. 2 and reconstructed the guessed M-XFROG trace for \( E_{{45^{^\circ } }} \) with Eq. 4. The minimum RMS error in Eq. 3 was searched with a minimization algorithm (The used routine fminsearch of MATLAB uses the Nelder–Mead simplex algorithm to find the minimum of a scalar function of several variables, starting at an initial estimate.) by setting τ and θ as iterated values. When the RMS error reached its optimal value ~2.71 × 10−2, τ and θ were retrieved to be ~−1.44 rad and ~1.17 fs. Figure 4b shows the reconstructed M-XFROG trace with the retrieved values of τ and θ, which shares a similar pattern with the measured one.

Fig. 4
figure 4

a The measured and b the reconstructed M-XFROG trace of E 45°. For the reconstructed M-XFROG trace, the retrieved optimal values of τ and θ have been used

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Finally, with the whole information of Eq. 1, we returned to the time domain and employed the Stokes parameters [27, 28]:

$$ S_{0} = \langle E_{H}^{2} (t)\rangle + \langle E_{V}^{2} (t)\rangle , $$
(5)
$$ S_{1} = \langle E_{H}^{2} (t)\rangle - \langle E_{V}^{2} (t)\rangle , $$
(6)
$$ S_{2} = 2\langle E_{H} (t)E_{V} (t)\cos (\Updelta \varphi )\rangle , $$
(7)
$$ S_{3} = 2\langle E_{H} (t)E_{V} (t)\sin (\Updelta \varphi )\rangle , $$
(8)

to describe the polarization state of elliptically polarized pulse, where E H(V)(t) is the horizontal (vertical) field amplitude while φ H(V)(t) is the corresponding temporal phase. The angle brackets here denote taking an average over the time domain and Δφ = φ V − φ H denotes the temporal phase difference. More intuitively, as shown in Fig. 1c, the elliptical polarization states could be depicted with the orientation angle \( \phi \in [ - \pi /2,\pi /2] \) and ellipticity angle \( \chi \in [ - \pi /4,\pi /4] \) as [28, 29]

$$ \tan (2\phi ) = S_{2} /S_{1} , $$
(9)
$$ \sin (2\chi ) = S_{3} /(S_{1}^{2} + S_{2}^{2} + S_{3}^{2} )^{1/2} . $$
(10)

The positive (negative) value of χ corresponds to a right (left) elliptically polarized state [26]. In most cases, (ϕ, χ) are the most important parameters to be manipulated when elliptically polarized pulses are applied [29, 30]. According to Eqs. 9 and 10, we could quantitatively determine the values of ϕ and χ with Δφ, E H and E V. The value of the temporal phase difference Δφ was mostly around ~1.68 rad, indicating that the target pulse was right elliptically polarized. Figure 5 shows the calculated orientation angle ϕ (−79.2°) and ellipticity angle χ (33.6°) of the target pulse. The positive value of χ also confirmed that the target pulse was in a right elliptically polarized state.

Fig. 5
figure 5

The calculated orientation angle ϕ (−79.2°) and ellipticity angle χ (33.6°) of the left elliptically polarized target pulse. The positive value of χ indicated that the target pulse was in a right elliptically polarized state

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4 Conclusion

In conclusion, we have demonstrated that elliptically polarized pulses could be fully characterized by the M-XFROG technique with the rotational excitation of the pre-aligned air molecules functioned as the optical gating function. The horizontal and vertical polarization components as well as the polarization projection at 45 degree of the elliptically polarized target pulse were measured, respectively, while the polarization state of the target pulse was obtained by combining the retrieved information. This technique was proved to be convenient and reliable for linearly polarized as well as elliptically polarized pulses measurement, which made it a useful tool for the characterization of ultrashort laser pulses.