Vectorial optical field reconstruction by attosecond spectral interferometry

TL;DR

This paper presents a novel interferometric method using attosecond pulses to fully reconstruct the complex electric field, including amplitude and polarization, of few-cycle optical pulses with high sensitivity and potential applications in spectroscopy.

Contribution

The work introduces a new technique combining extreme ultraviolet interferometry and attosecond electron wave packet control for complete optical field reconstruction.

Findings

Successful reconstruction of complex electric fields of few-cycle pulses.

Ability to characterize pulses with energies as low as hundreds of nanojoules.

Potential for full electric field characterization in visible spectroscopy applications.

Abstract

An electrical pulse E(t) is completely defined by its time-dependent amplitude and polarisation direction. For optical pulses the manipulation and characterisation of the light polarisation state is fundamental due to its relevance in several scientific and technological fields. In this work we demonstrate the complete temporal reconstruction of the electric field of few-cycle pulses with a complex time-dependent polarisation. Our experimental approach is based on extreme ultraviolet interferometry with isolated attosecond pulses and on the demonstration that the motion of an attosecond electron wave packet is sensitive to perturbing fields only along the direction of its motion. By exploiting the sensitivity of interferometric techniques and by controlling the emission and acceleration direction of the wave packet, pulses with energies as low as few hundreds of nanojoules can be reconstructed. Our approach opens the possibility to completely characterise the electric field of the pulses typically used in visible pump-probe spectroscopy.

Equations

$$S\propto {\int }_{t_{ion}}^{t_{rec}}[{\mathbf{p}}_{\mathbf{s}}+e{\mathbf{A}}_{dr}(t)+e{\mathbf{A}}_{unk}(t){]}^{2}dt\simeq {\int }_{t_{ion}}^{t_{rec}}[{\mathbf{p}}_{\mathbf{s}}+e{\mathbf{A}}_{dr}(t){]}^2+2[{\mathbf{p}}_{\mathbf{s}}+e{\mathbf{A}}_{dr}(t)]\cdot e{\mathbf{A}}_{unk}(t)dt$$

Full text

The complete characterisation of optical pulses requires the possibility to sample in time oscillating electric fields on the sub-femtosecond timescale. In the last decade, the development of attosecond science and technology has made available the technological tools towards this goal${}^{\cite[cite]{\textsuperscript{\@@bibref{Number}{RMP-Krausz-2009, PQE-Nisoli-2009, SCIENCE-Kapteyn-2007}{}{}}}}$. In general, the electric field $\mathbf{E}(t)$ is a vector quantity and the measurement of the (time-dependent) polarisation is required for its complete reconstruction. The polarisation state of the radiation is a key parameter in the light-matter interaction, as it determines selection rules in photoexcitation and photoionisation of atoms and molecules, and spin properties of photoelectrons emitted from surfaces. Moreover, polarisation shaping and synthesisation of complex time-dependent polarisations are widely adopted schemes to control laser-matter interaction at high-intensities 4; 5, at the nanoscale level 6, and for the optimal control of quantum systems 7.

An electric field with a time-dependent polarisation can be retrieved by measuring its components along two (perpendicular) directions. For ultrashort pulses approaching the single-cycle regime 8, the rotation of the polarisation axis through wave plates can introduce distortions due to the large spectral bandwidth and higher-order phase dispersion terms. Therefore, the characterisation along these two directions should avoid the use of such additional dispersive elements.

A technique capable to fully characterise weak probe pulses with energy of a few tens or hundreds of nanojoules would be also desirable. In this way the full reconstruction of electric fields would not be limited to intense pulses 9, but it could be extended to the probe pulses typically used in visible and infrared pump-probe spectroscopy.

Two complementary approaches have been demonstrated so far for the complete characterisation of optical waveforms: the attosecond streak camera 10; 11 and the petahertz optical oscilloscope 12. Electric fields with linear 13 and time-dependent polarisation 14 were retrieved using the first approach, which, however, requires the use of intense fields. On the other hand, the second technique introduces a systematic distortion of the reconstructed waveform due to the non-collinear geometry 12.

.1 XUV spectral interferometry with isolated attosecond pulses

In our experimental approach we take advantage of extreme ultraviolet (XUV) spectral interferometry driven by isolated attosecond pulses to demonstrate the full reconstruction of unknown fields characterised by a complex polarisation pattern. XUV spectral interferometry of high-order harmonic sources was already demonstrated with trains of attosecond pulses for the investigation of plasma 15 and molecular 16 dynamics. In these approaches one XUV pulse acts as a reference, while the characteristics (phase or amplitude) of the second one are modified by the ongoing dynamics.

The application of XUV interferometry for the reconstruction of optical pulses is presented in Fig. 1a: two coherent intense driving fields are focused in two closely-spaced focal spots, where two attosecond electronic wave packets are released by tunnel ionisation in the continuum and accelerated (a,b). When the electric field drives back the wave packets, two closely-spaced coherent isolated attosecond pulses are generated upon recombination of the electrons with the ground state of the parent ions. The polarisation direction of the isolated attosecond pulses is linked to the direction of motion of the electronic wave packets ($x$ direction in Fig. 1a), which is determined by the direction of polarisation of the driving field. The two XUV pulses interfere in the far-field along the vertical direction, while the spectra are dispersed by a suitable optical setup in the perpendicular direction (Fig. 1b). The unknown, perturbing optical pulse $\mathbf{E}_{unk}$ is overlapped in one of the two focal spots. The weak field perturbs the generation of XUV radiation leading to a modulation in the amplitude ($\Delta E$) and phase ($\Delta\varphi$) of one of the two isolated attosecond pulses. This modulation is imprinted in the strength (amplitude) and position (phase) of the interference fringes (see Movie 1 in the Supplementary Information), and it is determined by the instantaneous unknown field during the motion of the electronic wave packet 12. Indeed $\Delta E$ and $\Delta\varphi$ are related to the imaginary and real part of the phase accumulated by the electron wave packet in the continuum ($S$) 17, which is determined by the ionisation ($t_{ion}$) and recombination ($t_{rec}$) times, and by the total field experienced by the electron wave packet in the continuum. We have verified that these three quantities depend linearly on the amplitude of the unknown field (in a suitable intensity range, see discussion below).

By changing the arrival time $\tau$ of the pulse, the fringe position and the total signal of the XUV spectrum oscillate as shown in Figs. 1c and 1d, respectively. The electric field of the perturbing field can be reconstructed from either the amplitude or the phase of the oscillation of the interference pattern, as shown in Fig. 1e, which reports the input field (solid lines) and the electric field reconstructed from the amplitude and phase of the oscillations. The input field was shifted in time and normalised to match the reconstructed fields. The amplitude modulation of the interference pattern can be attributed to the variation of the ionisation probability, due to the presence of the unknown field. Therefore, the field reconstructed from this modulation corresponds to the unknown field at the ionisation instant. On the other hand, the phase modulation of the interference pattern is due to the variation of the phase accumulated by the electron wave packet between the ionisation and recombination instants. The reconstructed field corresponds to an average value of the unknown field between these two instants. For this reason, the reconstructions from the amplitude and phase modulations present a temporal offset, as shown in Fig. 1e.

We have verified that the effect on the phase of the interference pattern scales linearly with the electric field for peak intensities between $I=5\times 10^{8}\leavevmode\nobreak\ \mathrm{W/cm^{2}}$ and $I=9\times 10^{12}\leavevmode\nobreak\ \mathrm{W/cm^{2}}$. The effect on the amplitude of the modulation is linear in a smaller intensity range (up to about $I=5\times 10^{11}\leavevmode\nobreak\ \mathrm{W/cm^{2}}$). This difference can be attributed to the exponential dependence of the tunnelling ionisation rate on the electric field, which limits the intensity range for the linear dependence on the unknown field. For this reason, in the following we will focus on the electric fields reconstructed from the phase of the interference pattern.

We have also verified that pulses with a carrier-wavelength as short as $200\leavevmode\nobreak\ \mathrm{nm}$ can be reproduced with an error between 5% and 15% (evaluated on the single reconstructed electric field value) depending on the duration of the unknown pulse.

.2 Attosecond electron dynamics as time-gated directional field detector

The strength of the effect and, therefore, the reconstruction of the electric field strongly depends on the relative orientation between the direction of the electronic motion (determined by the direction of polarisation of the driving field) and the instantaneous polarisation direction of the unknown field. This is demonstrated in Fig. 2a, which shows a linearly polarised field (continuous black line) and the reconstructed fields in the case of polarisation parallel (blue open square) and perpendicular (green open circle) to the motion of the electronic wave packet. The reconstruction of a (almost) zero field in the perpendicular case reveals that the electronic motion is (almost) unaffected by the unknown field in this configuration. This can be explained by observing that the phase $S$ accumulated by the electronic wave packet depends on the integral of the kinetic energy between the ionisation and recombination instants 17:

[TABLE]

where $e$, $\mathbf{p_{s}}$, $\mathbf{A}_{dr}(t)$, and $\mathbf{A}_{unk}(t)$ indicate the electron charge, the stationary momentum, and the vector potential of the driving and unknown fields, respectively. The driving field steers the motion of the electronic wave packet along its polarisation direction ($\mathbf{p_{s}}$ is parallel to $\mathbf{A}_{dr}(t)$ with excellent approximation), thus making negligible the contribution of the second term of Eq. 1 and the influence of the unknown, perturbing field, when the two pulses are perpendicularly polarised. This conclusion holds true up to intensities for which the unknown field cannot be considered a perturbation of the driving one, any longer.

As the electronic motion is sensitive to the parallel, but not to the perpendicular perturbation, the phase and amplitude of the isolated attosecond pulse encode information only on the projection of the instantaneous electric field along the direction of motion. For the reconstruction of the unknown field, the electronic dynamics acts as a time-gated directional field detector, which is sensitive only to the field component along its main axis (which coincides with the direction of motion of the electron) within a time-gating window. Due to the dynamics of the electronic wave packet in the continuum, the duration of the gate amounts to a few hundreds of attoseconds.

This conclusion is confirmed in Fig. 2b, which shows the maximum of the projection of the linearly polarised field of Fig. 2a, along a reference direction ($x$), as a function of the angle $\eta$ between this direction and the polarisation axis of the field. As expected, the curve evolves as $|\cos(\eta)|$ (orange line), with a small difference between $\eta=0^{\circ}$ and $\eta=180^{\circ}$ due to the asymmetry of the few-cycle field.

The maximum field of the pulse, reconstructed by the analysis of the evolution of the interference pattern (magenta points), as a function of the angle $\theta$ between the direction of the electronic wave packet and the perturbing field, matches perfectly the evolution of the maximum of the projected field. We also verified that, not only the maximum amplitudes, but also the evolution in time of the complete field is well reproduced for all relative angles.

The component of the electric field perpendicular to the electronic motion does neither affect the reconstruction for a generic polarisation state of the unknown pulse. This is demonstrated by the reconstruction of the field for a pulse with circular polarisation (red full circle), shown in Fig. 2a. Also in this case, the reconstructed field is given by the projection of the vectorial field along the axis of the time-gated directional field detector. A similar conclusion holds true also for elliptically polarised pulses.

These observations open the possibility for the characterisation of pulses with a complex time-dependent polarisation, by applying the reconstruction along two (perpendicular) polarisation directions of the isolated attosecond pulses. As an example (see Fig. 2c), we considered two perpendicularly polarised few-cycle pulses with different carrier frequencies. Due to the different periods of oscillation, the polarisation state of the total field varies from circular on the edges of the pulses (with opposite helicities) to linear in the centre. The reconstruction of the pulse along perpendicular directions ($x$ and $y$; blue squares) follows perfectly the input fields, thus giving access to the complete characterisation of the vectorial field.

.3 Experiment and full reconstruction of linearly polarised electric fields

In the experimental setup (see Fig. 3a), we used the polarisation gating (PG) technique to implement XUV spectral interferometry with isolated attosecond pulses 18; 19.

The few-cycle pulses passed through a binary plate, which imposed a $\pi$ phase difference between the lower and upper halves of the beam 16. In the focus, this phase difference caused two closely-spaced coherent focal spots, which were the source points of the two isolated attosecond pulses. The far-field interference pattern of the two XUV pulses is presented in Fig. 3b. Clear interference fringes were observed in the vertical direction, while the spectrum was dispersed in the horizontal direction by a grazing incidence XUV spectrometer. The deviation of the interference fringes from the horizontal direction can be attributed to small misalignments in the grazing incidence spectrometer.

The unknown, perturbing field $\mathbf{E}_{unk}$ (initially linearly polarised along the $x$ direction) was overlapped with one of the two foci. In the experiment, we typically used unknown pulses with energies on the order of a few hundreds of nanojoule corresponding to intensities of about $10^{10}-10^{11}\leavevmode\nobreak\ \mathrm{W/cm^{2}}$. A peak intensity as low as a few $\sim 10^{9}\leavevmode\nobreak\ \mathrm{W/cm^{2}}$ still produced a significant shift of the fringes ($\sim 0.1\leavevmode\nobreak\ \mathrm{rad}$), which could be experimentally measured with our setup.

Figures 3c and 3d show the divergence and the XUV signal as a function of the relative delay between the driving and perturbing fields, respectively. The delay-dependence are in good agreement with the simulations presented in Figs. 1c,d (see Movie 2 in Supplementary Information). The field retrieved from the amplitude and phase modulations is shown in Fig. 3e. These measurements demonstrate the reconstruction of few-cycle linearly polarised pulses using spectral interferometry based on isolated attosecond pulses. The robustness of the reconstruction was verified using different XUV wavelengths and different algorithms (see Supplementary Information).

To demonstrate that the dynamics of the electronic wave packet is sensitive only to the weak field parallel to its motion, we measured the unknown field for different rotation angle $\psi$ of a half-wave plate inserted in the beam path (see Fig. 3a). The maxima of the reconstructed field (red circles in Fig. 3f) well reproduced, the expected evolution ($|\cos(2\psi+\psi_{0})|$, where $\psi_{0}$ is an offset angle), confirming that the isolated attosecond pulse encodes information only on the perturbing field parallel to its polarisation. The lowest intensity of the component of the perturbing field along the electronic direction used in Fig. 3f was estimated in $I_{unk}=\leavevmode\nobreak\ 1.5\times 10^{9}\leavevmode\nobreak\ \mathrm{W/cm^{2}}$.

.4 Full reconstruction of electric fields with time-dependent polarisation

The reconstruction of complex vectorial fields requires the characterisation of two (usually perpendicular) components. The PG technique offers a straightforward way to implement the reconstruction of optical fields along different directions, because the polarisation directions $\gamma_{1,2}$ of the isolated attosecond pulses can be modified by simply rotating the angle of the first plate of the PG unit from $\alpha_{1}=+45^{\circ}$ to $\alpha_{2}=-45^{\circ}$ (see Fig. 3a and Supplementary Information). This approach presents the advantage that the axis of our directional field sensitive detector can be rotated without any additional dispersive optical element neither on the driving nor on the perturbing field. By measuring the projections $E_{1}(t)$ and $E_{2}(t)$ of the vector field $\mathbf{E}_{unk}(t)$ along the two directions identified by $\gamma_{1}$ and $\gamma_{2}$, the two perpendicular components $E_{x}(t)$ and $E_{y}(t)$ can be reconstructed (see Supplementary Information).

To demonstrate the full capability of the technique, we synthesised fields with a complex time-dependent polarisation state, by replacing the half-wave plate with a replica of the PG unit on the unknown pulse (see Fig. 3a). The axis of the plates of this second PG unit were initially aligned along the polarisation of the incoming linearly polarised field. By rotating the angle $\beta$ of the first plate to $\beta^{\pm}=\pm 45^{\circ}$, a sharp transition from circular (elliptical) to linear and back to circular(elliptical) is expected. For opposite values of $\beta$, the helicity of the transition is expected to reverse 21; 22. For each $\beta^{\pm}$ angle, two measurements for $\alpha_{1}$ and $\alpha_{2}$ were acquired.

Figures 4a,b show the $x$ and $y$ components of vectorial electric field retrieved for $\beta^{+}$ and $\beta^{-}$, respectively, and the projection of the field in the $x-y$ plane. The pulse was characterised by a complex time-dependent ellipticity (see Figs. 4a,b), which evolves from elliptical (positive helicity) to linear and back to elliptical (negative helicity) for $\beta^{+}=+45^{\circ}$. The transition between different polarisation states can be clearly observed in the three-dimensional representation of the fields, which evidences a rotation of the polarisation vector clockwise (counter clockwise) on the leading (falling) edge of the pulse for $\beta^{+}$ (Fig. 4c) 23. For $\beta^{-}$, the rotation directions are reversed (Fig. 4d), as expected.

Our approach also allows for the characterisation of the carrier envelope phase (CEP) of the field, by analysing the variations of the electric field components in the time window of linear polarisation. The CEP was controlled by changing the amount of fused silica inserted in the beam path 24. The reconstruction for three CEPs spaced by $\pi/2$ is shown in Figs. 5a,b,c. We can observe that the ellipticity is only slightly affected by the CEP, as well as the component of the electric field along the $x$ direction (apart from a shift in time). The minor component of the field, on the other hand, is strongly modified and it is even reversed for a $\pi$ phase shift (Figs. 5a,c). The simulations are shown in Figs. 5d,e,f, which report the field for the three CEPs $\varphi=0$ (d), $\varphi=\pi/2$ (e), and $\varphi=\pi$ (f). The good matching between the simulations and the experiments allows one for a reliable determination of the CEP of the unknown field. The measured projection in the $xy$ plane is also reported in Figs. 5g,h,i. The field corresponding to the time window of Figs. 5a,b,c is shown in green. The transition through a state of linear polarisation is clearly visible, and it evidences that the projections are characterised by narrow (and opposite) region of linear polarisation for $\varphi=0,\pi$. This transition region becomes broader for intermediate CEPs ($\varphi=\pi/2$). This periodic variation is in excellent agreement with simulations (see Movie 3 in Supplementary Information).

.5 Conclusions

In conclusion, we have demonstrated the complete temporal characterisation of the time-dependent amplitudes of the two components (including their carrier-envelope phases) of a few-cycle pulse with a complex polarisation state. The experimental approach is based on the implementation of spectral XUV interferometry and on the demonstration that the dynamics of a single electron wave packet acts as an attosecond directional sensitive field detector. These results opens the possibility for the complete characterisation of an optical field with a generic polarisation state using an all-optical approach free from distortions. Combined with phase sensitive detection techniques based on lock-in amplifier, the sensitivity of the techniques can be further improved opening new perspectives for the full characterisation (amplitude, phase, and carrier-envelope phase) of probe fields used in pump-probe spectroscopy. This approach would offer the possibility to reconstruct the complete response (complex dielectric wave function) of metallic and nanostructured system in the visible and infrared spectral range, in perfect analogy to time-domain spectroscopy for semiconductors in the THz domain.

.6 Methods

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

.7 Acknowledgements

This project has received also funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 641789 Molecular Electron Dynamics investigated by Intense Fields and Attosecond Pulses (MEDEA), the European Research Council Starting Grant agreement no. 637756 Steering attosecond electron dynamics in biomolecules with UV-XUV LIGHT pulses (STARLIGHT). G. G. P. acknowledges support from the German Science Foundation (PA 730/7 within priority programme 1840).

.8 Contributions

P. C., M. R., and G. S. conceived and planned the experiment. P. C., M. R., and H. A. conducted the experiment. M. R. and A. C. analysed the data. S. K., F. C, and M. N., contributed to the development of the experimental setup. L. P. and F. F. designed and installed the XUV beamline. D. H and G. G. P. designed and installed the STEREO ATI. J. U., C. D. S and R. M. designed and installed the Reaction Microscope. G. S. performed the simulations and theoretical analysis. The manuscript was drafted by G. S. and completed in consultation with all authors.

.9 Competing financial interests

The authors declare no competing financial interests.

Correspondence and requests for materials should be addressed to G. Sansone

(giuseppe.sansone@physik.uni-freiburg.de

giuseppe.sansone@polimi.it).