# Molecular alignment echo for controlling the readout time of vortex beams

A Voisine, P Béjot, F Billard, O Faucher, E Hertz

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A Voisine, P Béjot, F Billard, O Faucher, E Hertz. Molecular alignment echo for controlling the readout time of vortex beams. Physical Review Letters, In press. hal-05384988

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Molecular alignment echo for controlling the readout time of vortex beams

A. Voisine,¹ P. Béjot,¹ F. Billard,¹ O. Faucher,¹ and E. Hertz¹, *

¹Université Bourgogne Europe, CNRS, Laboratoire Interdisciplinaire Carnot de Bourgogne ICB UMR 6303, 21000 Dijon, France

(Dated: October 17, 2025)

We present an experimental study of molecular alignment echoes driven by ultrashort structured light fields carrying orbital angular momentum (OAM). By combining two time-delayed pump pulses with spatially varying polarization, we produce a molecular “q-plate” in a gaseous CO₂ sample that can be rephased in a controllable manner. This approach enables the storage of OAM information within the rotational coherences of molecules and its retrieval at adjustable times. The concept is validated using vortex beams of various topological charges $ℓ = \pm 1, 2$. These results extend the mechanism of molecular alignment echoes to vector light fields and establish a new route for optical processing of ultrashort OAM pulses, advancing molecular-based quantum interfaces for structured light.

The ability to precisely control the spatial structure of light beams has opened a wide range of applications in optical processing and light-matter interactions. Among the most prominent examples of structured light are optical vortex beams, which carry orbital angular momentum (OAM) [1–4]. Unlike spin angular momentum (SAM), which is associated with the polarization state, OAM arises from a helical phase distribution of the wavefront. The phase of vortex beams varies azimuthally as $e^{i\ell\varphi}$, where $\ell$ is an integer called the topological charge and $\varphi$ is the azimuthal angle in the cross section of the beam. In such a beam, each photon carries an OAM of $\ell\hbar$, independent of its polarization [1]. The unique properties of OAM fields have attracted significant interest, driven both by their fundamental implications and their broad range of applications [3–5]. The OAM degree of freedom offers for instance an unbounded state space, enabling high-dimensional encoding and boosting channel capacity in classical [6, 7] and quantum communications [8, 9]. Beyond communications, OAM beams have been exploited for optical manipulation of micro- and nano-particles [10, 11], rotational Doppler effect [12], nonlinear interaction [13], high harmonic generation [14–16], and chirality and optical activity [17–19], underscoring their versatility across various fields. Since the seminal discovery by Allen *et al.* that helically phased beams carry OAM [1], numerous approaches have been developed for their production [3, 4]. Conventional methods include mode converters based on cylindrical lenses [20], spiral phase plates [21] that imprint a helical phase profile via a thickness gradient, and fork grating holograms [22] that induce topological charge transformation through beam diffraction. The latter two approaches can be readily implemented using computer-controlled spatial light modulators, thereby enabling dynamic and programmable beam shaping. Another compact technique relies on q-plates [23], birefringent components whose neutral axis varies continuously with the azimuthal angle. Such an optical element, also known as vortex plate, generates OAM beams through spin-orbit coupling occurring in both anisotropic and inhomogeneous media.

Together these generation techniques provide a powerful toolkit for harnessing the OAM degree of freedom of light in fundamental studies and emerging photonic applications. However, their versatility, speed of reconfiguration, damage threshold, and multimode capability remain limited, particularly when applied to broadband ultrashort pulses.

In addition to these conventional methods, we have recently demonstrated the relevance of gas-phase molecules as light-matter interface for the storage and optical processing of ultrashort OAM beams [24]. Using a coupling scheme analogous to holographic imaging, the helical phase of light beams was encoded into the rotational coherences of linear molecules. The storage mechanism exploits the spatial structuring of the molecular ensemble. To this end, a writing field of inhomogeneous polarization induces a transverse distribution of molecular axis orientations along a well-defined spatial pattern. This distribution replicates the azimuthal structure of a q-plate’s optical axis, thereby forming a “molecular q-plate” that memorizes the spatial information of the field to be stored. Owing to the quantum beatings of the rotational wavepacket, this molecular q-plate revives periodically after the interaction, under field-free conditions [25, 26]. This post-pulse molecular alignment is then exploited to restore the encoded spatial phase information onto a probe beam, as a standard q-plate. The strategy has been successfully applied to store pure OAM states [24] as well as coherent superpositions of different OAM modes [27], namely OAM-based photonic qubits, carried by ultrashort laser pulses. Notably, the storage of OAM superpositions, which embeds both spatial phase and amplitude modulation, required more sophisticated molecular structuring with the creation of a three-dimensional pattern of aligned molecules. The concept of molecular waveplate opens promising perspectives. It can be used for instance to encode time-varying spatial structures by leveraging the intrinsic temporal evolution of rotational wavepackets, enables access to nonlinear interactions produced within such ensembles of aligned molecules [28, 29], or expands the scope of OAM manip-
ulation through advanced control of molecular dynamics via pulse shaping [30, 31] or polarization shaping [32, 33], to name a few. Very recently, Xu et al. [34] further highlighted this potential by achieving near-unity conversion efficiency and generating ultrashort vortex pulses spanning a broad wavelength range.

A key limitation of such molecular platform is that, for a given molecular species, the readout time is constrained by the periodicity of the quantum beatings, which depends on the rotational constant. To overcome this constraint, we propose exploiting the recently discovered concept of molecular alignment echoes [35]. The echo phenomenon, first observed by Hahn [36] in precessing nuclear spins, is ubiquitous in many areas of physics. It is a classical effect in which a system produces a collective response (an echo) after excitation by two time-delayed stimuli. In the context of field-free molecular alignment, echoes were first demonstrated in CO₂, excited by two successive nonresonant and linearly polarized ultrashort laser pulses. The first pulse P₁ generates a rotational wavepacket, which rephases into well-defined alignment revivals after excitation. The periodicity of these revivals is fixed by the rotational constant of the molecule, preventing significant alignment between them. However, if the sample interacts with a second pulse P₂, applied after a delay τ₁₂ from P₁, rephasing of the molecular wavepacket and alignment echo occurs at t = 2τ₁₂ after P₁ (or t = τ₁₂ after P₂), an effect that is immune to strong centrifugal distorsion [37]. Extensive studies on this effect have demonstrated the existence of higher-order echoes (observed at t = nτ₁₂) [35], fractional echoes [38], imaginary echoes, angular echoes [39], as well as molecular orientation echoes [40]. As a flagship application, rotational echoes have emerged as a powerful tool for investigating collisional molecular dynamics in high-pressure gases, granting access to the early stages of collisional relaxation [41, 42], even beyond the secular-approximation limit [43]. To date, all studies of rotational echoes have employed pump fields with uniform polarization states. Here, we extend this phenomenon to vector beams with spatially varying polarization, enabling tunable retrieval of spatial information. Using a two-pump-pulse sequence, we demonstrate the retrieval of OAM modes, stored in a room-temperature sample of CO₂ molecules, at controllable delay times. Within this framework, rotational echoes can be viewed as the rotational-coherence analogue of the CRIB (Controlled-Reversible Inhomogeneous Broadening) protocol [44], originally developed to realize quantum memories with multimode capacity in inhomogeneously broadened optical transitions.

The storage mechanism is realized by superimposing the input field carrying the OAM with an auxiliary plane-wave beam, in analogy with holographic imaging [24, 27]. The input and auxiliary fields are circularly polarized with opposite helicities, thereby generating a pump field

with a spatially varying polarization vector distribution. For instance, to store an input beam carrying an OAM state of charge ℓ, the pump field can be expressed as:

$$ \vec{E}_P(t) = \varepsilon_0(t)e^{-i\omega_0 t} (\hat{\sigma}_+ + e^{i\ell\varphi}\hat{\sigma}_-), \quad (1) $$

where $\omega_0$ corresponds to the angular frequency, and $\hat{\sigma}_+$ ($\hat{\sigma}_-$) to the left (right)-handed circular polarization in the $(x-y)$ transverse plane of the beam, i.e. $\hat{\sigma}_\pm = \frac{1}{\sqrt{2}}(1, \pm i)$. It is readily to show that, in the Cartesian basis, this pump field can be written as:

$$ \vec{E}_P(t) = \sqrt{2}\varepsilon_0(t)e^{-i(\omega_0 t - \frac{\ell\varphi}{2})} \begin{bmatrix} \cos(\frac{\ell\varphi}{2}) \\ \sin(\frac{\ell\varphi}{2}) \end{bmatrix} \quad (2) $$

The global field exhibits a polarization vector distribution that varies continuously with the azimuthal angle $\varphi$, following an angle $\alpha = \ell\varphi/2$. This reproduces the spatial pattern of the neutral axis of a q-plate [23] designed to generate an OAM state of topological charge $\ell$. In such a plate, a left-circularly polarized beam is converted into a right-circularly polarized beam while simultaneously acquiring OAM of order $\ell$. When interacting with a molecular sample, the pump field of Eq. 2 induces molecular alignment [25, 26] with a spatial distribution that follows the azimuthal pattern of the polarization directions. It therefore creates a molecular q-plate that stores the OAM information of the input beam through molecular anisotropy, as illustrated in Fig. 1(a). If the pump duration is short compared to the molecular rotational period, alignment occurs in the non-adiabatic regime, giving rise to periodic revivals of molecular alignment after the excitation. Consequently, the molecular q-plate also revives periodically, enabling the retrieval of the stored OAM onto a probe pulse. As detailed in our earlier work [24], the underlying excitation relies on a non-resonant Λ-type scheme. The spatial information is stored into rotational coherences via a stimulated non-resonant Raman process, involving the absorption (resp. emission) of a photon from the input (resp. auxiliary) beam. Retrieval occurs through the same mechanism. A probe pulse, polarized as the auxiliary field, is injected into the molecular sample at a rephasing time of alignment. The interaction with the aligned molecules generates a field of opposite helicity, which carries the spatial phase of the input field. Overall, the excitation scheme exploits the interplay between SAM and OAM. Conservation of SAM dictates the helicity sequence of the pulses in the parametric cascaded process, while OAM conservation determines the topological charge $\ell$ of the retrieved signal field. The use of alignment echoes via a two-pulse sequence broadens the versatility of the approach, offering a tunable readout of stored spatial information as depicted in Fig. 1(b).

The experimental setup is illustrated in Fig. 2. A Ti:Sapphire laser system delivers pulses of 35 fs duration,
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FIG. 1. Schematic representation of spatial information storage in rotational coherences. (a) A beam carrying an OAM ($\ell = 1$ in this example) is coupled to an auxiliary beam ($\ell = 0$). The resulting spatially varying polarization of the beam produces an alignment pattern equivalent to the one of a molecular q-plate (of charge $\ell = 1$), which periodically rephases. At each revival of the q-plate, the encoded information can be restored using a probe field. (b) Molecular echo control: a sequence of two pump pulses controls the rephasing of the molecular q-plate, enabling tunable readout time of the stored information.

centered at 800 nm at a repetition rate of 1 kHz, with a maximum pulse energy of 5 mJ. The beam is first divided into pump and probe arms using an 80/20 beam splitter. The pump beam is directed through a delay line, allowing control of the pump-probe delay. It then passes through a half waveplate, multiple-order waveplate and polarizer to generate the pulse doublet $P_1$ and $P_2$ required for the echo production. The relative energy between $P_1$ and $P_2$ can be tuned by means of the half waveplate. Each pulse of this sequence must contain the input beam (carrying the OAM to be stored) and the auxiliary beam, circularly polarized with opposite handedness. This is achieved in a single step using a q-plate with a phase retardation $\delta = \pi/2$ (equivalent to a quarter-wave plate), into which the two-pulse sequence is sent after being circularly polarized with a quarter waveplate. The q-plate converts an input beam $\hat{\sigma}_+(\ell = 0)$ into an equal superposition of $\hat{\sigma}_+(\ell = 0)$ and $\hat{\sigma}_-(\ell)$, corresponding respectively to the auxiliary and input beams. After the 80/20 splitter, the arm corresponding to the probe beam is frequency-doubled in a BBO crystal. The second-harmonic component is isolated using a polarizer and passband filter. It is then circularly polarized with a Berek compensator (BC₁). Pump and probe beams are collinearly recombined by an optical wedge and focused by lens $L_1$ ($f = 30$

<figure><img src="image_4.png" /></figure>

FIG. 2. Experimental set-up. (a) BS: beam splitter, HWP: half waveplate, P: Polarizer, BBO: type 1 doubling crystal, BC: Berek compensator, W: wedge, L: plano-convex lens, PMT: photomultiplier tube, BD: beam dump, F: 400 nm bandpass filter, ND: neutral density filter. (b) Pump shaping for producing the pump pulse sequence: q-plate ($\delta = \pi/2$), MOWP: multiple-order waveplate, QWP: quarter waveplate.

cm) into a static gas cell filled with 1 bar of CO₂. At the cell output, the beams are recollimated by lens $L_2$ ($f = 25$ cm). A circular analyzer, consisting of a Berek compensator (BC₂) and a polarizer, selects the signal field of interest, which corresponds to the polarization change of the probe beam. The signal, separated from the pump using a dichroic filter (F), is detected either by a CCD camera providing the spatially resolved signal or by a photomultiplier tube (PMT) for spatially integrated measurements. The spatial phase profile and OAM state are retrieved by recording the interference pattern with a reference Gaussian beam of identical polarization. Finally, we note that Berek compensators are used in place of conventional quarter-wave plates in the probe path. This choice is motivated by their ability to induce a tunable phase shift (adjustable as a function of wavelength) as well as to precompensate birefringence effects arising from optical reflections.

Figure 3 presents the experimental measurement for the storage of an OAM mode $\ell = 1$ with a readout time defined by $\tau_{12} = 3$ ps. The PMT signal in Fig. 3(a) exhibits two prominent peaks at delays $\tau = 0$ and 3 ps, corresponding to the arrival time of the pump pulses $P_1$ and $P_2$. These peaks arise from the combined contribution of the electronic Kerr response and the inertial component of the molecular response [26]. Subsequent transient signals, observed at 10.8 ps and 13.8 ps, correspond to the first rotational revivals of alignment induced by $P_1$ and $P_2$, respectively. In CO₂, the first revival occurs at 10.8 ps after the pulse interaction and is characterized by a distinctive three-peak structure readily identifiable. Between the pulses and the first revival, a clear signal appears at $\tau \approx 6$ ps. It corresponds to the molecular
<figure><img src="image_5.png" /><figcaption>FIG. 3. Storage of an OAM mode of $\ell = 1$ using a pump-pulse sequence delayed by $\tau_{12} = 3$ ps. (a) Alignment signal of CO$_2$ as a function of the pump-probe delay recorded with a PMT (see text), (b) signal intensity pattern at the occurrence of the echo ($\tau \approx 6$ ps), (c) interferometric intensity pattern, and (d) retrieved spatial phase.</figcaption></figure>

echo generated by the pump sequence which rephases at $\tau_{12} \approx 3$ ps after $P_2$. The beam intensity distribution of the signal field at this delay is shown in Fig. 3(b). It exhibits the characteristic annular profile of an OAM state, dictated by a phase singularity at the beam center. To confirm this claim, the signal was interfered with a synchronized reference beam at a small crossing angle, as shown in Fig. 3(c). The resulting interferogram displays the typical fork-shaped dislocation, which, after spatial Fourier filtering, yields the spatial phase of the signal beam. The reconstructed phase map, presented in Fig. 3(d), reveals a counter-clockwise phase winding from 0 to $2\pi$, confirming the helical phase structure of topological charge $\ell = 1$ carried by the signal field. The OAM information stored in the molecular sample is thus successfully retrieved in the signal echoes. These results demonstrate alignment echo dynamics induced by structured light fields, thereby extending echo phenomena to the vector-beam regime.

To demonstrate the tunability of the readout time achieved with this approach, a similar experiment was carried out by varying the delay between the two pump pulses. The results obtained for a delay of $\tau_{12} = 2$ ps are presented in Fig. 4. Figure 4(a) shows the temporal dynamics of the alignment signal for the storage of an OAM mode $\ell = 1$. The main features observed in Fig. 3 are preserved, with the only noticeable difference being the echo shifted to $\tau \approx 4$ ps. The annular beam profile and reconstructed phase [Figs. 4(b) to (d)] retain the characteristic signature of an OAM state with $\ell = 1$, thereby demonstrating controllable readout timing of OAM beams. The present approach imposes no restriction on the OAM order that can be stored. To illustrate this capability, we performed an experiment storing an OAM mode $\ell = 2$. As shown in Fig. 4(f), the interference pattern exhibits a

<figure><img src="image_6.png" /><figcaption>FIG. 4. Storage of OAM modes with $\tau_{12} = 2$ ps. Panels (a–d) show measurements analogous to those in Fig. 3, obtained at $\tau \approx 4$ ps. Panels (e–g) and (h–j) display the corresponding measurements for $\ell = 2$ and $\ell = -1$, respectively.</figcaption></figure>

characteristic on-axis fork with a double bifurcation, as expected for an OAM state of $\ell = 2$ (the $\ell$ value being identified from the bifurcation order). The corresponding reconstructed phase [Fig. 4(g)] displays the expected counterclockwise phase increase from 0 to $4\pi$, forming two spiral arms. Another distinguishing feature, visible in Fig. 4(e), is the larger dark core of the doughnut-shaped intensity profile compared to the $\ell = 1$ case, consistent with the scaling of the radial node radius as $r^{\ell}$. Finally, Figs. 4(h–j) show the results for the storage and retrieval of an OAM mode with $\ell = -1$. These different results confirm that the present method is applicable to arbitrary $\ell$ values and demonstrate its potential for the optical processing of ultrashort OAM fields with controllable retrieval times.

To summarize, we have extended the concept of molecular alignment echoes to the domain of structured light, enabling control over the readout time of vortex beams stored in molecular ensembles. By employing two temporally delayed pump pulses, we demonstrate that the rephasing of rotational coherences can be precisely engineered to retrieve stored OAM information at adjustable delays, thereby overcoming the constraint imposed by the intrinsic revival period of the molecules. The suc-
cessful retrieval of vortex beams with various topolog-
ical charges confirms the robustness and versatility of
this approach. This echo-based control introduces a new
temporal degree of freedom in molecular light-matter in-
terfaces, opening pathways toward reconfigurable optical
memories. Future extensions could exploit molecular ang-
gular echoes [39] to dynamically reconfigure the molecu-
lar medium on the picosecond timescale, providing a pow-
erful platform for ultrafast OAM processing and paving
the way for dynamic, multidimensional manipulation of
light's angular momentum.

ACKNOWLEDGEMENTS

This work was supported by the CNRS, the
EIPHI Graduate School (contract “ANR-17-EURE-
0002”) and benefited from the facilities of the SMART-
LIGHT platform funded by the Agence Nationale de
la Recherche (EQUIPEX+ contract “ANR-21-ESRE-
0040”) and Région Bourgogne Franche-Comté.

* edouard.hertz@ube.fr

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