1. INTRODUCTION

Vector beams of light with cylindrical, nonuniform polarization patterns [1] have found application in diverse areas of optics such as improved focusing [2,3], laser machining [4], plasmon excitation [5], metrology [6], optical trapping [7] and nano-optics [8–10]. Recently, they have attracted attention [11–16] due to a simple but striking property: when viewed as a superposition of transverse electromagnetic modes with orthogonal linear polarizations, the nonseparable mode function of a radially polarized vector beam is mathematically equivalent to a maximally entangled Bell state of two qubits known from quantum mechanics [17–20]. In contrast with the canonical Bell states in quantum optics, where two photons are entangled in polarization and exhibit nonlocal correlations when spatially separated, this “classical entanglement” in vector beams is necessarily local as it exists only between different degrees of freedom of one and the same physical system.

However, these correlations have recently been shown to represent a valuable resource. Vector beams have been used to violate an analog of Bell’s inequality for spin-orbit modes [12,13] and have led to continuous variable entanglement between different degrees of freedom [21]. In addition, vector beams have been used to implement classical counterparts of quantum protocols [22,23]. Promising proposals include an application to the study of quantum random walks [24] and real-time single-shot Mueller matrix measurements [25]. In the present work, we demonstrate for the first time, to our knowledge, a fully operational application of classical entanglement to high-speed kinematic sensing.

Several techniques are available nowadays for sensing the kinematics of fast-moving objects [26–30]. Each comes with its own strengths and drawbacks. For example, high-speed imaging is typically limited to capturing only a small number of frames, while pump-probe techniques require the recorded event to be repeated identically many times. Ideally, one seeks a solution that is capable of performing fast sensing continuously, in real-time, and from a simple setup, employing only standard equipment and offering flexibility in the choice of wavelength. By using the nonseparable mode structure of cylindrically polarized beams (see Fig. 1), one only needs to detect changes in polarization, thus fulfilling all the just-mentioned requirements at the same time. In the following work, we first discuss the physics of vector beams and then introduce the technique of sensing and show the results of our experimental investigations.

 

Fig. 1. Classical entanglement. (a) (Top) Instantaneous transverse electric field distribution of a radially polarized beam of light. The orange density plot shows the beam’s doughnut-shaped intensity distribution, while black arrows indicate the position-dependent instantaneous local direction of the electric field vector. (Bottom) The beam’s global polarization state is shown in a Poincaré sphere representation. Initially, the light field is globally unpolarized, with the Poincaré vector located at the origin. (b) If an opaque obstacle is brought into the beam, the global polarization takes on nonzero values according to the obstacle’s position within the beam. This method allows the object’s kinematics to be inferred from a polarization measurement alone.

Download Full Size | PDF

2. THEORY BACKGROUND

The electric field of a general nonuniformly polarized paraxial beam can be written as

For a field $\mathit{E}(\mathit{\rho },z)$ in the Schmidt form, the measurable Stokes parameters can be written as

When an opaque object cuts across a nonuniformly polarized beam, the spatial and polarization patterns of the latter vary with time according to the obstructing object’s instantaneous position, as described by its central coordinates $x_0(t),y_0(t)$. It is not difficult to show that, for such a modified beam, Eqs. (2) are still valid provided that ${\lambda }_1,{\lambda }_2,a_x,a_y$ are regarded now as functions of $x_0(t),y_0(t)$. When the values of the Stokes parameters $s_0,s_1,s_2,s_3$ are replaced by the measured ones on the left side of Eqs. (2), these can be regarded as a nonlinear algebraic system of four equations for the two variables $x_0(t),y_0(t)$, which can be solved by means of suitable algorithms. In this way, the instantaneous trajectory of the object is recovered.

3. EXPERIMENT

The experimental setup is shown in Fig. 2. We prepare a continuous-wave laser beam with wavelength $\lambda =1550\mathrm{nm}$ in a radially polarized mode. The beam impinges on a moving sample. Subsequently, half-waveplates and polarizing beam splitters are used to project the beam onto its horizontal, vertical, diagonal, and antidiagonal polarization components. Finally, a network of four InGaAs photodetectors with a 3 dB bandwidth of 4 GHz measures the individual projections, from which the Stokes parameters $s_0$, $s_1$, and $s_2$ can be straightforwardly obtained. For the particular case of a radially polarized mode, the $s_3$ parameter is always zero. An auxiliary camera is used for additional visual verification and beam characterization. Further details can be found in Supplement 1.

 

Fig. 2. Experimental setup. (a) A continuous-wave laser beam is prepared in a radially polarized mode by a liquid crystal mode converter. (b) The beam impinges on an opaque object whose motion in space modulates the beam’s polarization Stokes parameters. (c) A polarization-independent beamsplitter (BS1) taps off 10% of the beam for inspection by a conventional camera. This allows for mode pattern characterization and independent verification measurements. (d) A polarization-independent 50/50 beamsplitter (BS2) divides the beam up for projection onto its linear polarization components via a pair of polarizing beamsplitters (PBS) and a half-wave plate ($\lambda /2$). The projections are simultaneously measured by four InGaAs detectors with 4 GHz bandwidth. By linear combination of the projection signals, the beam’s Stokes parameters are obtained. Knowledge of the Stokes parameters allows the object’s instantaneous trajectory to be reconstructed (see Fig. 3).

Download Full Size | PDF

In order to demonstrate the system’s broad applicability, three types of measurement are carried out. First, a metal rotor is made to turn about the beam axis [Fig. 3(a)]. By sampling the Stokes parameters during the motion of the rotor, the instantaneous value of its angle of rotation ${\theta }_0$ is successfully inferred. An accuracy of 4.1° (mean error) is achieved without correcting for beam imperfections and detector coupling.

 

Fig. 3. Rotation sensing and position tracking. (a) A metal rotor [width $m=(0.79\pm 0.01)\mathrm{mm}$] turns about the center of a radially polarized beam [width $w_1=(1.95\pm 0.10)\mathrm{mm}$]. Due to the beam’s classically entangled structure, the rotation in space induces a sinusoidal oscillation of the beam’s Stokes parameters. Measurements of the $s_0$, $s_1$, and $s_2$ Stokes parameters allow the instantaneous angle of rotation to be inferred. Each data point was obtained by integrating over 200 ns, so that electronic noise is averaged out to within the data point width. Dotted curves show the theoretically expected values under the assumption of an ideal mode function. (b) (Left) A metal sphere [diameter $d=(1.00\pm 0.01)\mathrm{mm}$] traverses a radially polarized light beam [width $w_2=(2.84\pm 0.10)\mathrm{mm}$]. (Center) The Stokes parameters $s_0$, $s_1$, and $s_2$ vary as a function of the sphere’s position. Solid lines show the expected Stokes parameters as obtained from simulation. (Right) The sphere’s trajectory is inferred from the measured Stokes parameters. The sphere is moved gradually in discrete steps of 50 μm, providing a calibrated, reproducible reference motion. To allow for a realistic comparison with a fast object, the acquisition time at each point is only 250 ps. The blue contours show the combined Bayesian 68% credible region $R$, while the gray shadow shows the sphere’s dimensions to scale. In both plots, $s_0$ is normalized to its initial value, while $s_1$ and $s_2$ are normalized to the instantaneous value of $s_0$. The theoretical model used to obtain the theory and simulation curves for (a) and (b), respectively, and the position-tracking algorithm are detailed in Supplement 1.

Download Full Size | PDF

Second, a metal sphere is moved across the beam [Fig. 3(b)]. We measure the Stokes parameters with an acquisition time of 250 ps at each position. A Bayesian algorithm is used to estimate the sphere’s position from these data (see Supplement 1). The inferred trajectory shown in Fig. 3(b) is seen to be in good agreement with the actual trajectory. As one expects, the inference is particularly successful in areas where the beam has a high intensity, i.e., where the Stokes parameter modulation introduced by the sphere results in a higher signal-to-noise ratio.

Third and finally, the setup’s real-time capability is demonstrated by focusing the beam and measuring the Stokes parameters during the transit of a knife edge moving across the beam at $(27\pm 2){\mathrm{ms}}^{-1}$ close to the focal plane. (The beam is sufficiently gently focused that it is not dominated by longitudinal field components at the waist; see Supplement 1). As seen from the captured polarization data in Fig. 4, the transit takes only 92 ns, after which the beam is fully covered. From the shape of the recorded traces, the knife edge’s direction of motion, horizontal in this case, can be inferred (up to 180° rotation; see Supplement 1). The event is captured as a sequence of single-shot measurements, requiring only a single occurrence. Furthermore, since the measurement is triggered on a change in $s_0$, the particular instant of occurrence does not have to be known in advance. As the Stokes parameters are captured continuously, there is no dead time in this measurement. This result clearly demonstrates the measurement technique’s potential for high-speed kinematic sensing.

 

Fig. 4. Real-time sensing. A metal knife edge of thickness $(3\pm 2)\mathrm{\mu m}$ cuts across a focused radially polarized mode [theoretically estimated width $w_3=(2.0\pm 0.5)\mathrm{\mu m}$] at $(27\pm 2){\mathrm{ms}}^{-1}$. The plot shows a sequence of single-shot measurements of the beam’s $s_0$, $s_1$, and $s_2$ Stokes parameters during the knife edge’s passage until the beam is fully covered (normalized to the initial power), with a total duration of 92 ns. The sampling resolution reaches up to 100 ps. From the measured traces, the knife edge’s direction of motion can be easily inferred up to 180° rotation.

Download Full Size | PDF

4. DISCUSSION

All three measurements confirm the setup’s ability to perform quantitatively meaningful kinematic sensing at very high temporal resolutions. The measurement technique allows for the use of bucket detectors rather than spatially resolving detectors, and the measurement can be as fast as the detectors. With the analog bandwidth of 4 GHz available in our setup, we should thus be capable of resolving even subnanosecond motions. We note that the measurement precision is subject to random error from electronic detector noise at high bandwidths. This becomes dominant in the regime where the measured sample has only a small overlap with the beam [as seen in Fig. 3(b)], or when the sample covers the beam completely. Some applications, such as precision sensing of objects moving within a confined region, may benefit from using a beam with a nonzero $s_3$ Stokes parameter. Such beams have been suggested for the investigation of small particle scattering [31]. Although they require an additional photodiode pair, such beams avoid the zero of intensity at the origin. We note, however, that the classical entanglement of such a beam is not maximal, and that the correlations between polarization and position are therefore necessarily weaker.

5. SUMMARY

We have demonstrated that the classical entanglement manifested by vector beams of light may be used to detect the kinematics of very fast objects with GHz temporal bandwidth. Although no explicit measurement of the spatial degree of freedom takes place, the measured object disturbs the beam in a spatially dependent but polarization-insensitive manner. The resulting spatial modulation becomes correlated with the global polarization state through the classically entangled mode structure of cylindrically polarized beams of light. The presented application thus emphasizes the utility of classical entanglement in identifying useful correlations in physical systems. The method presented requires only standard optical components which are commercially available at a wide range of optical wavelengths and can easily be extended to the microwave regime. It allows for continuous, real-time measurement of two-dimensional spatial information with unprecedented temporal resolution. We suggest that, due to its simplicity, the method may even be employed in noisy environments such as free-space channels. For example, existing lidar technologies based on time-of-flight measurements may be enhanced by the new method. On the microscale, focused classically entangled modes may provide a new approach to precision measurements, for example of Brownian motion in the ballistic regime in two dimensions [32].