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Abstract
We introduce the space-time (ST) vector light sheet. This unique one-dimensional ST wave packet is characterized by classical entanglement (CE), a correlation between at least two non-separable intrinsic degrees-of-freedom (DoFs), which in this case are the spatiotemporal DoFs in parallel with the spatial-polarization DoFs. We experimentally confirm that the ST vector light sheet maintains the intrinsic features of the uniformly polarized ST light sheet, such as near-diffraction-free propagation and self-healing, while also maintaining the intrinsic polarization structure of common vector beams, such as those that are radially polarized and azimuthally polarized. We also show that the vector beam structure of the ST vector light sheet is maintained in the subluminal and superluminal regimes.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
While entanglement is traditionally understood as a phenomenon of quantum mechanics, a classical analogy was made by Spreeuw [1]. A classically entangled optical field is one that exhibits a correlation between at least two non-separable intrinsic degrees-of-freedom (DoFs) [1,2] but does not share quantum entanglement’s nonlocality [3,4]. The original analogy of classical entanglement (CE) has since offered new perspectives on structured light [5–10] and led to classical analogs of important quantum information processes [3]. In quantum mechanics, the term entanglement is reserved to describe the state of a system $|\psi \rangle $, comprised of two or more subsystems, that cannot be expressed as a tensor product, in any basis, of its subsystems. In the usual bipartite example, the composite system of two separated particles A and B, represented respectively by state vectors $|{\psi _A}\rangle $ and $|{\psi _B}\rangle $, cannot be described by $|{{\psi_A}\rangle \; \otimes \; } |{\psi _B}\rangle $, but rather as a superposition state proportional to $|{{\psi_A}\rangle \pm \; } |{\psi _B}\rangle .$ In this case, the entanglement is in a single DoF, e.g., polarization, between two particles. This may lead to nonlocal effects, and thus would be considered a uniquely quantum phenomenon. CE borrows this mathematical structure only to describe the relationship between two or more DoFs of a field. In this way, the entanglement is considered intra- rather than inter-, and is a local phenomenon. Therefore, for example, the spatial and polarization DoFs of a classical optical field can be considered classically entangled if their corresponding state vectors cannot be decomposed into a product of the state vectors for space and polarization. This is the case for vector beams [11]. Furthermore, CE has enabled the generation and characterization of more complex structured vector beams including vector fields with four DoFs [12], extended dimensionality of DoFs [13], and propagation-dependent nonseparability of DoFs [4]. In contrast to a traditional scalar-polarized beam, a radially polarized vector beam has been shown to exhibit a circularly symmetric point-spread function and an enhanced (electric) longitudinal (z-) polarization at the focus of a high-numerical aperture (NA) lens [14,15]. Conversely, an azimuthally polarized vector beam produces an on-axis optical intensity null upon high-NA focusing or equivalently an enhanced longitudinal magnetic polarization. Generally, vector beams have been the subject of approaches to enhance propagation of the z-polarization component [15], utilization in metrology [16,17] and optical tweezers [18], and investigation of their fundamental properties and methods of generation [19–21].
Another type of classically entangled field is that of a space-time (ST) wave packet which exploits correlations between spatial frequencies and temporal frequencies, kx and ω, respectively, to achieve diffraction-free and dispersion-free properties [22–28]. Advances in the generation and characterization of fields with space-time nonseparability include flying doughnut pulses [29], and a novel measurement of nonseparability [30]. In the case of recently developed ST light sheets, the diffraction-free behavior is exhibited along one spatial dimension, where the maximum achievable propagation distance in a medium is inversely proportional to the spectral uncertainty (δω) of spatio-temporal correlation [31]. Moreover, self-healing has been experimentally demonstrated for ST light sheets, as well as a controllable group velocity in non-absorbing media [32–35]. The latter feature is especially interesting because it can result in ST light sheets with arbitrary group velocities in free space, including subluminal, superluminal, and negative group velocities [36].
In this work, we present an experimental demonstration of a classically double-entangled field in the form of a ST vector light sheet, comprising a combined ST light sheet and vector beam. A question arises about how to describe a classical system composed of two subsystems, each of which is classically entangled. Some consideration can be given to the quantum case of hyperentanglement, whereby at least two particles are entangled simultaneously in multiple DoFs, e.g., $|{{\psi_P}\rangle \otimes \; } |{\psi _k}\rangle \otimes \; |{\psi _{e - t}}\rangle $, corresponding to the state vectors for multipartite polarization (p) entanglement, momentum (k) entanglement, and energy-time (e-t) entanglement, respectively [37]. However, a ST vector light sheet is not quite the classical analog of the hyperentangled case, but it is also clearly classically entangled in more than a single pair of DoFs. Thus, we refer to the ST vector light sheet as simply classically double-entangled because the entanglement exists between two pairs of DoFs in parallel.
Our ST vector light sheet should not be confused with other reported spatiotemporal optical vortices [38–40] that show circularly symmetric intensity profiles and do not exhibit self-healing. Like its scalar counterpart, the ST vector light sheet demonstrates near-diffraction-free propagation and self-healing. Radial and azimuthal polarization structures are also exhibited, similarly to common vector beams. We also show that the spatially inhomogeneous polarization distribution of the ST vector light sheet is maintained in both the subluminal and superluminal regimes.
This work is organized as follows. Section 2 presents a theoretical framework for describing cylindrical vector beams, ST light sheets, and ST vector light sheets. Section 3 is devoted to the experimental synthesis of the ST vector light sheet. Section 4 provides the results and discussion. Finally, the conclusion of this paper is given in Section 5.
2. Theory
Cylindrical vector beams are axially symmetric amplitude and phase vector beam solutions of Maxwell's equations. The superposition of orthogonally polarized first-order Hermite–Gauss modes, HG01 and HG10, may be used to describe vector beams [41,42]:
where ${\vec{E}_\rho }$ and ${\vec{E}_\phi }$ indicate vector beams of radial and azimuthal polarization, respectively. Radially and azimuthally polarized beams of axial symmetry, or cylindrical vector beams of radial and azimuthal polarization, can be expressed as Jones matrices given byFor simplicity, we will assume one-dimensional (1D) light sheets with an electric field E(x, z, t), where x denotes the transverse coordinate, z the axial coordinate, and t the time. A linear relationship between spatial frequencies and temporal frequencies, kx and ω, respectively, of the form $\omega /c = {k_0} + ({{k_z} - {k_0}} )\tan \theta $ can be established by appropriately designing the function $ \omega ({|{{k_x}} |} ),$ where ${k_0}$ is a fixed wave number and $\theta $ is the spectral tilt angle and ${k_{Z }}$ is also established only by $|{{k_x}} |$, and the spatial bandwidth $\Delta {k_x}$ is now correlated to the temporal bandwidth $\Delta \omega $ (see Supplement 1, Fig. S1(c) for more details). In this scenario, the ST beam has the form [24]:
A theory of ST vector light sheets is produced by applying polarization unit vectors of cosine and sine dependence to Eq. (5), for the example of radially polarized ST vector light sheets. The theory is as follows:
3. Experiment
The experimental setup used in this study is depicted in Fig. 1(a). We utilize an ultrafast laser (InSight X3, Spectra Physics; bandwidth of ∼8.5 nm, operating at 800 nm) and a linear polarizer P1 to create horizontally polarized pulses, then we expand the beam’s size to a diameter of 25 mm. The pulsed spectrum is spatially dispersed along the horizontal axis by G1. The first diffraction order is directed to a cylindrical lens L1 followed by a polarization-dependent SLM (Hamamatsu, X15213-07) that displays a 2D phase distribution. The spectral uncertainty of the spatially resolved spectrum at the SLM plane depends on the spectral resolving power of the grating and the size of the beam encountering the grating. The reflected beam from the SLM is then passed through L1 and directed to G2 (identical to G1), after which the ST light sheets are formed. We then determine the Stokes parameters after to confirm the horizontal polarization state of the ST light sheet, see Fig. 1(b). Following the grating G2, the reflected beam is imaged to an output plane with a 4× demagnification along the z direction by a 4f two-lens system (lenses L3 and L4 with focal lengths of 40 cm and 10 cm, respectively). After L3 and L4, the ST light sheet is formed once the temporal-spatial frequencies are superposed, see Fig. 1(c). The width of the ST-beam is Δx = 0.096 mm. Note that we use a flip mirror between BS and G2 to direct the beam to spherical lens L2, which upon a spatial Fourier transform, results in the spatio-temporal spectrum of the ST light sheet in (${k_x},\lambda )$-domain at the CMOS (Thorlabs, DCC1545M), as shown in Fig. 1(d). Upon rotating the polarization analyzer by 90°, a zero-order vortex half-wave plate (VWP) (Thorlabs, WPV10L-780) is added to entangle the spatial and polarization DoFs, which results in the ST vector light sheet. We use a scientific CMOS (sCMOS) camera (Hamamatsu, ORCA-Fusion BT Digital CMOS camera: C15440-20UP) to record the time-averaged intensity of the ST light sheet and the ST vector light sheet along the propagation axis z. The spatio-temporal spectrum curve appears as a parabola. Here, we measure the laser spectral bandwidth Δλ = 2 nm, the spatial bandwidth Δkx = 18 rad/mm and the spectral uncertainty δλ = 50 pm with the selected spectral tilt angle θ = 44.97°, see Fig. 1(d). These parameters are the basis for preserving the beam and mitigating its spreading. The concept of ST wave-packets is described in detail in [24], and briefly summarized in the supplementary material (Supplement 1, Fig. S1). A broader spectrum is needed to protect a beam with larger spatial bandwidth [34]. In Fig. 1(e) we insert an opaque obstruction of 0.2 mm diameter along the center of the ST vector light sheet to demonstrate self-healing characteristics.
Fig. 1. (a) Experimental setup for synthesizing and characterizing the ST light sheet and ST vector light sheet. (b) Setup for Stokes parameters measurement. (c) Time-averaged intensity of the transverse spatial intensity profile I(x, y) of a ST light sheet obtained with the sCMOS camera. (d) Measured spatiotemporal spectrum ${|{\tilde{E}({{k_x},\lambda } )} |^2}$ of ST light sheets with spectral tilt angle of θ = 44.97° obtained with the CMOS camera. (e) Diagram of beam blocking to demonstrate self-healing. For the optical components, P: linear polarizer, HWP: half-wave plate, VWP: Vortex waveplate, QWP: Quarter waveplate, G1 and G2: diffraction grating with 1200 grooves/mm (the grating G1 is used in reflection mode but is shown here in transmission mode for simplicity), BS: Beam splitter, L1, L3, and L4: Cylindrical lens, L2: Spherical lens, FM: Flip mirror. The focal lengths of the lenses L1, L2, L3, and L4 are 50 cm, 12.5 cm, 40 cm, and 10 cm, respectively.
The form of the phase on the SLM is ψ(x, y) = (|x−xc| / xs) (y / αys)1/2 where the phase pattern is wrapped from 0 to 2π. The factors xs and ys depend on the SLM specification, xc is the central point along the x-direction on the SLM, and α is the curvature of the hyperbolic trajectories. The VWP changes the input beam to radial or azimuthal polarization, depending on the incident polarization orientation and the relative angle of the fast axis.
4. Results
To confirm that the ST vector light sheets propagate similarly to the ST light sheets, and that the polarization structures are preserved with propagation, we measure the intensity distributions of the ST vector light sheet along the propagation direction at different axial positions z = 0.1, 0.9, and 2 m using the sCMOS camera as shown in the inset of Fig. 2 (see also Supplement 1, Fig. S2). We observe that the ST light sheet and ST vector light sheet have similar full widths at half maximum (FWHM) of Δx ∼ 96 µm (where x is the vertical width) at z = 0.1 m, with negligible differences.
Fig. 2. Measured transverse FWHM along the propagation axis z for the ST light sheet and for the ST vector light sheet. Insets show the measured intensity distributions of the ST light sheet and the ST vector light sheet along the propagation direction at axial positions z = 0.1, 0.9, and 2 m using the sCMOS camera. The ST light sheet is in the top of the insets, the ST vector light sheet is in the bottom.
The width increases modestly, about 6 times, to ∼596 µm after propagating as far as z = 2 m. The results of the relative reduction in diffractive spreading for the ST light sheet agree well with the numerical simulation reported in [33]. The intensity distributions of the ST light sheet and ST vector light sheet at the aforementioned z distances are shown in the top and the bottom of each inset in Fig. 2. It is clear that the vector-beam nature of the ST vector light sheet is maintained with propagation. Importantly, this specialized vector field inherits its robustness to diffraction as a result of it being doubly entangled with the ST light sheet. This behavior is also observed when the dispersion relationship is tuned for the ST vector light sheet to produce either subluminal or superluminal versions of the ST vector light sheet. These results are shown in Fig. S3 in Supplement 1.
Figure 3 depicts the experimentally obtained intensity distributions for both radially and azimuthally polarized ST vector light sheets, as shown in rows (a) and (b), respectively. The field is switched from radial polarization to azimuthal polarization upon insertion of HWP2 (shown in Fig. 1). The local intensity orientation is indicated by the white arrows for each field. We also confirm that the ST vector light sheet has null intensity due to the polarization singularity. The orientations of linear polarization analyzer P2 are depicted by the black arrows at the top of the last four columns (from left to right: vertical, 45°, -45°, and horizontal), and the associated measured intensity profile is shown below for each type of vector beam. In contradistinction to cylindrical vector beams, we observe that the vertically and horizontally polarized eigenmodes are spatially displaced and non-overlapping. In addition, these polarization modes are not of equal magnitude, i.e., the contribution of the horizontally polarized field component is significantly greater than the vertically polarized component. These effects contribute to a more complex intensity distribution for the ${\pm} $ 45° polarization projections. It is worth noting that these results are a result of the 1-D nature of the underlying ST light sheet. A comparison with vector beams derived from a conventional 1-D Gaussian light sheet confirms this (refer to Fig. S4 in Supplement 1).
Fig. 3. Intensity distributions of the ST vector light sheet. Cross-sectional image of a radially polarized ST vector light sheet and its intensity profiles after passing through different orientations of P2 (a). Similar results for an azimuthally polarized ST vector light sheet (b). The black arrows above indicate the polarization orientation of the polarization analyzer P2. The leftmost column shows the results without P2.
Figure 4 shows the intensity map of the ST vector light sheet obtained by inserting a QWP oriented at 45° immediately after the VWP (shown in Fig. 1, and without insertion of HWP2). The orientations of analyzer P2 are depicted by the black arrows at the top of the last four columns (from left to right: vertical, 45°, -45°, and horizontal). For the standard circularly symmetric, radially polarized vector beam case, this results in polarization states that are circularly polarized (albeit with different relative phases) at certain local positions on the transverse intensity profile. In the case of the radially polarized ST vector light sheet, we observe local elliptically polarized states of polarization.
Fig. 4. Intensity distribution of radially polarized ST vector light sheet followed by the QWP. The black arrows above indicate the polarization orientation of the analyzer P2. The leftmost column shows the results without P2.
Next, we study the self-healing property of the ST vector light sheet. The ability of diffraction-free optical fields to self-heal is one of their most intriguing characteristics [44–47]. When these fields are partially disrupted by an obstacle, they undergo a redistribution that manifests as a reconstruction of their intensity profile prior to the obstruction. Vector Bessel beams have been studied, in part, because they exhibit both the self-healing and vector-beam properties [17,41,48]. However, these are examples of the standard CE between a single pair of DoFs as these vector beams rely on quasi-monochromatic sources. To our knowledge, there has been no previous reporting of any 1-D vector beam, yet alone a 1-D self-healing vector beam. Thus, here, we investigate self-healing for double-entangled vector beams derived from an ST light sheet.
Figure 5(a) and (e) depicts the time-averaged intensity of the transverse spatial profile of the ST light sheet and the ST vector light-sheet respectively, at 1.5 cm without the beam block. Figure 5(b) and (f) shown the time-averaged intensity of the ST light sheet and the ST vector light sheet respectively, at 1.5 cm after an opaque obstruction (a monofilament fishing line) with a 0.2 mm diameter fishing line. The opaque obstruction is placed along the y-axis, see Fig. 1(e). This obstacle serves to block a significant portion of the light sheet. At 6.5 cm after the perturbation, the beam has not fully reconstructed. However, there is a visible return of the intensity in the area where the beam block temporarily scattered the field, see Fig. 5(c) and (g). After propagating about 90.5 cm, the light sheet appears to have reconstructed as shown in Fig. 5(d) and has a strong resemblance to the unperturbed ST light sheets shown in Fig. 5(a). The obstruction also appears to have minimal effect on the underlying polarization structure of the ST vector light sheet, as confirmed by the radially polarized projection in Fig. 5(h). The self-healing property of the ST vector light sheet appears to be similar to those of ST light sheets and the polarization structures are mostly preserved with propagation after the obstruction. The spectral tilt angle that is used for the self-healing ST vector light sheet is 44.97°. It is worth noting that a measure for the degree of nonseparability for a vector Bessel beam has been established and used as a metric for assessment of self-healing. We are exploring adapting a similar metric for our ST vector light sheet.
Fig. 5. Transverse intensity distributions for the ST light sheet and the ST vector light sheet after an opaque obstruction (green object) with a 0.2 mm diameter fishing line positioned at z = 0. ST light sheet without beam block at z = 1.5 cm (a), ST light sheet 1.5 cm (b) and 6.5 cm post-block (c). ST light sheet self-healing 90.5 cm post-block (d). ST vector light sheet without beam block at z = 1.5 cm (e), ST vector light sheet 1.5 cm (f) and 6.5 cm post-block (g). Radially polarized ST vector light sheet self-healing 90.5 cm post-block (h). Images of the ST light sheet and ST vector light sheet were taken with the sCMOS.
5. Conclusion
In conclusion, we experimentally investigated the first demonstration of the ST vector light sheet. These specialized vector beams rely on the in-parallel correlation between two DoFs, namely, the spatio-temporal spectrum and the polarization and spatial DoFs. We confirmed that ST vector light sheets propagate similarly to ST light sheets, and that the polarization structures are mostly preserved with propagation. In addition, self-healing of ST vector light sheets post-obstruction was confirmed. We also show that the spatially inhomogeneous polarization distribution of the ST vector light sheet is maintained in both the subluminal and superluminal regimes. Analogous to how multiparameter quantum entanglement holds promise for robust quantum communications, this special case of classical entanglement would be interesting for investigation of propagation in turbid media.
Funding
Office of Naval Research (N00014-20-1-2789).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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