## Abstract

Tracking the kinematics of fast-moving objects is an important diagnostic tool for science and engineering. Here, we demonstrate an approach to positional and directional sensing based on the concept of classical entanglement in vector beams of light [Found. Phys. **28**, 361–374 (1998) [CrossRef] ]. The measurement principle relies on the intrinsic correlations existing in such beams between transverse spatial modes and polarization. The latter can be determined from intensity measurements with only a few fast photodiodes, greatly outperforming the bandwidth of current CCD/CMOS devices. In this way, our setup enables two-dimensional real-time sensing with temporal resolution in the GHz range. We expect the concept to open up new directions in metrology and sensing.

© 2015 Optical Society of America

## 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.

## 2. THEORY BACKGROUND

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

*nonseparable*; namely, it is not possible to rewrite it as the simple product of only one polarization vector and a single scalar function. In this sense, Eq. (1) has the same mathematical structure as a two-qubit entangled quantum state [20]. It is a well-established result of mathematical physics that any two-dimensional field in the form of Eq. (1) can be recast in the so-called Schmidt form, $\mathit{E}(\mathit{\rho},z)=\sqrt{{\lambda}_{1}}{\widehat{\mathit{u}}}_{1}{v}_{1}(\mathit{\rho},z)+\sqrt{{\lambda}_{2}}{\widehat{\mathit{u}}}_{2}{v}_{2}(\mathit{\rho},z)$, where $\{{\widehat{\mathit{u}}}_{1},{\widehat{\mathit{u}}}_{2}\}$ and $\{{v}_{1},{v}_{2}\}$ form complete orthonormal bases in the polarization and spatial mode vector spaces, respectively, with ${\lambda}_{1}\ge {\lambda}_{2}\ge 0$. If either ${\lambda}_{1}=0$ or ${\lambda}_{2}=0$, the expression of $\mathit{E}(\mathit{\rho},z)$ is factorable and the beam is uniformly polarized. Vice versa, if ${\lambda}_{1}{\lambda}_{2}\ne 0$, the beam displays a nonuniform polarization pattern and is said to be “classically entangled.” Thus, analogously to a bona fide quantum state, in a nonuniformly polarized beam, polarization and spatial degrees of freedom are so strongly correlated that if, by any means, one alters the beam’s spatial profile, then the polarization changes accordingly. Our sensing technique relies precisely upon this peculiar phenomenon. Owing to the classical entanglement exhibited by the beam, we are able to retrieve information about the position of a moving object partially obstructing the beam only by measuring the polarization of the latter: no spatially resolving measurements are needed. Since polarization measurements can be performed at GHz rates, we are able to track very fast objects with our system.

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

where ${a}_{x}={\widehat{\mathit{u}}}_{1}\xb7\widehat{\mathit{x}}$ and ${a}_{y}={\widehat{\mathit{u}}}_{1}\xb7\widehat{\mathit{y}}$. In a radially polarized beam, one has ${\lambda}_{1}={\lambda}_{2}$ and thus ${s}_{1}={s}_{2}={s}_{3}=0$, reflecting the fact that such a beam appears completely unpolarized in the absence of an obstruction.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 $\lambda =1550\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ 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.

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.

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 $(27\pm 2)\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ms}}^{-1}$ 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.

## 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].

## Acknowledgment

The authors would like to thank Tobias Röthlingshöfer for help with the experiment, Irina Harder for phase plate fabrication, and Thomas Bauer for useful discussions. The authors recently became aware of the work by Fade and Alouini [33], which applies polarization-frequency nonseparability to measurements of a material’s depolarization strength.

See Supplement 1 for supporting content.

## REFERENCES

**1. **Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. **1**, 1–57 (2009). [CrossRef]

**2. **S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. **179**, 1–7 (2000). [CrossRef]

**3. **R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. **91**, 233901 (2003). [CrossRef]

**4. **M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A **86**, 329–334 (2007). [CrossRef]

**5. **N. M. Mojarad and M. Agio, “Tailoring the excitation of localized surface plasmon-polariton resonances by focusing radially-polarized beams,” Opt. Express **17**, 117–122 (2009). [CrossRef]

**6. **F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express **19**, 25143–25150 (2011). [CrossRef]

**7. **Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express **18**, 10828–10833 (2010). [CrossRef]

**8. **J. Kindler (née Müller), P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Waveguide properties of single subwavelength holes demonstrated with radially and azimuthally polarized light,” Appl. Phys. B **89**, 517–520 (2007). [CrossRef]

**9. **M. Neugebauer, T. Bauer, P. Banzer, and G. Leuchs, “Polarization tailored light driven directional optical nanobeacon,” Nano Lett. **14**, 2546–2551 (2014). [CrossRef]

**10. **P. Woźniak, P. Banzer, and G. Leuchs, “Selective switching of individual multipole resonances in single dielectric nanoparticles,” Laser Photon. Rev. **9**, 231–240 (2015). [CrossRef]

**11. **A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Poincaré sphere representation for classical inseparable Bell-like states of the electromagnetic field,” arXiv:1007.2528 (2010).

**12. **C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A **82**, 033833 (2010). [CrossRef]

**13. **E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A **82**, 022115 (2010). [CrossRef]

**14. **X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. **36**, 4110–4112 (2011). [CrossRef]

**15. **K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics **7**, 72–78 (2012). [CrossRef]

**16. **X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica **2**, 611–615 (2015). [CrossRef]

**17. **R. J. Spreeuw, “A classical analogy of entanglement,” Found. Phys. **28**, 361–374 (1998). [CrossRef]

**18. **R. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A **63**, 062302 (2001). [CrossRef]

**19. **A. Luis, “Coherence, polarization, and entanglement for classical light fields,” Opt. Commun. **282**, 3665–3670 (2009). [CrossRef]

**20. **A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. **17**, 043024 (2015). [CrossRef]

**21. **C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, C. Marquardt, P. St.J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. **106**, 060502 (2011). [CrossRef]

**22. **A. N. de Oliveira, S. P. Walborn, and C. H. Monken, “Implementing the Deutsch algorithm with polarization and transverse spatial modes,” J. Opt. B **7**, 288–292 (2005). [CrossRef]

**23. **S. M. H. Rafsanjani, M. Mirhosseini, O. S. Magaña-Loaiza, and R. W. Boyd, “State transfer based on classical nonseparability,” Phys. Rev. A **92**, 023827 (2015). [CrossRef]

**24. **S. K. Goyal, F. S. Roux, A. Forbes, and T. Konrad, “Implementing quantum walks using orbital angular momentum of classical light,” Phys. Rev. Lett. **110**, 263602 (2013). [CrossRef]

**25. **F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. **16**, 073019 (2014). [CrossRef]

**26. **A. Whybrew, “High-speed imaging,” in *Handbook of Laser Technology and Applications*, C. Webb and J. Jones, eds. (Institute of Physics, 2004), Chap. D2.7.

**27. **A. Velten, R. Raskar, and M. Bawendi, “Picosecond camera for time-of-flight imaging,” in *Imaging Systems and Applications* (Optical Society of America, 2011), paper IMB4.

**28. **C. Weitkamp, *Lidar: Range-Resolved Optical Remote Sensing of the Atmosphere*, Springer Series in Optical Sciences (Springer, 2005).

**29. **K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature **458**, 1145–1149 (2009). [CrossRef]

**30. **K. Nakagawa, A. Iwasaki, Y. Oishi, R. Horisaki, A. Tsukamoto, A. Nakamura, K. Hirosawa, H. Liao, T. Ushida, K. Goda, F. Kannari, and I. Sakuma, “Sequentially timed all-optical mapping photography (STAMP),” Nat. Photonics **8**, 695–700 (2014). [CrossRef]

**31. **A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express **18**, 10777–10785 (2010). [CrossRef]

**32. **S. Kheifets, A. Simha, K. Melin, T. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science **343**, 1493–1496 (2014). [CrossRef]

**33. **J. Fade and M. Alouini, “Depolarization remote sensing by orthogonality breaking,” Phys. Rev. Lett. **109**, 043901 (2012). [CrossRef]