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 [810]. Recently, they have attracted attention [1116] 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 [1720]. 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 [2630]. 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.

 figure: Fig. 1.

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.

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2. THEORY BACKGROUND

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

E(ρ,z)=e1f1(ρ,z)+e2f2(ρ,z),
where the vectors e1,e2 determine the beam’s polarization, the scalar functions f1(ρ,z),f2(ρ,z) set the wavefront, and ρ=x^x+y^y is the transverse position vector (see Supplement 1). The expression in Eq. (1) is 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, E(ρ,z)=λ1u^1v1(ρ,z)+λ2u^2v2(ρ,z), where {u^1,u^2} and {v1,v2} form complete orthonormal bases in the polarization and spatial mode vector spaces, respectively, with λ1λ20. If either λ1=0 or λ2=0, the expression of E(ρ,z) is factorable and the beam is uniformly polarized. Vice versa, if λ1λ20, 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 E(ρ,z) in the Schmidt form, the measurable Stokes parameters can be written as

s0=λ1+λ2,
s1=(λ1λ2)(|ax|2|ay|2),
s2=(λ1λ2)(axay*+ax*ay),
s3=i(λ1λ2)(axay*ax*ay),
where ax=u^1·x^ and ay=u^1·y^. In a radially polarized beam, one has λ1=λ2 and thus s1=s2=s3=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 x0(t),y0(t). It is not difficult to show that, for such a modified beam, Eqs. (2) are still valid provided that λ1,λ2,ax,ay are regarded now as functions of x0(t),y0(t). When the values of the Stokes parameters s0,s1,s2,s3 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 x0(t),y0(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 λ=1550nm 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 s0, s1, and s2 can be straightforwardly obtained. For the particular case of a radially polarized mode, the s3 parameter is always zero. An auxiliary camera is used for additional visual verification and beam characterization. Further details can be found in Supplement 1.

 figure: Fig. 2.

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 (λ/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).

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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 θ0 is successfully inferred. An accuracy of 4.1° (mean error) is achieved without correcting for beam imperfections and detector coupling.

 figure: Fig. 3.

Fig. 3. Rotation sensing and position tracking. (a) A metal rotor [width m=(0.79±0.01)mm] turns about the center of a radially polarized beam [width w1=(1.95±0.10)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 s0, s1, and s2 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±0.01)mm] traverses a radially polarized light beam [width w2=(2.84±0.10)mm]. (Center) The Stokes parameters s0, s1, and s2 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, s0 is normalized to its initial value, while s1 and s2 are normalized to the instantaneous value of s0. 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.

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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±2)ms1 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 s0, 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.

 figure: Fig. 4.

Fig. 4. Real-time sensing. A metal knife edge of thickness (3±2)μm cuts across a focused radially polarized mode [theoretically estimated width w3=(2.0±0.5)μm] at (27±2)ms1. The plot shows a sequence of single-shot measurements of the beam’s s0, s1, and s2 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.

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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 s3 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]  

References

  • View by:

  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]

2015 (4)

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]

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]

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]

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]

2014 (4)

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

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]

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]

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

2013 (1)

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]

2012 (2)

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

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]

2011 (3)

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

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

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]

2010 (4)

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

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

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]

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]

2009 (4)

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

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]

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

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]

2007 (2)

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]

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]

2005 (1)

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]

2003 (1)

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

2001 (1)

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

2000 (1)

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

1998 (1)

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

Abouraddy, A. F.

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]

Agio, M.

Aiello, A.

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]

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

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]

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).

Alonso, M. A.

Alouini, M.

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

Andersen, U. L.

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]

Banzer, P.

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]

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

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]

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]

Bauer, T.

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

Bawendi, M.

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.

Beckley, A. M.

Borges, C. V. S.

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]

Boyd, R. W.

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]

Brown, T. G.

Chen, L.

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]

de Oliveira, A. N.

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]

Di Giuseppe, G.

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]

Dorn, R.

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

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

Eberler, M.

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

Eberly, J. H.

Elser, D.

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]

Euser, T. G.

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]

Fade, J.

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

Fatemi, F. K.

Feurer, T.

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]

Forbes, A.

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]

Förtsch, M.

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]

Franke-Arnold, S.

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]

Gabriel, C.

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]

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).

Giacobino, E.

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]

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

Glöckl, O.

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

Goda, K.

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]

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]

Goyal, S. K.

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]

Hirosawa, K.

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]

Holleczek, A.

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).

Horisaki, R.

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]

Hor-Meyll, M.

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]

Howell, J. C.

Huguenin, J. A. O.

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]

Iwasaki, A.

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]

Jalali, B.

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]

Joly, N. Y.

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]

Kagalwala, K. H.

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]

Kannari, F.

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]

Karimi, E.

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]

Kheifets, S.

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]

Khoury, A. Z.

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]

Kindler (née Müller), J.

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]

Konrad, T.

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]

Kozawa, Y.

Leach, J.

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]

Leuchs, G.

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]

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]

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

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

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]

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]

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

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

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).

Li, T.

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]

Liao, H.

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]

Little, B.

Luis, A.

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

Magaña-Loaiza, O. S.

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]

Marquardt, C.

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]

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

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]

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).

Marrucci, L.

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]

Meier, M.

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]

Melin, K.

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]

Mirhosseini, M.

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]

Mojarad, N. M.

Monken, C. H.

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]

Nakagawa, K.

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]

Nakamura, A.

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]

Neugebauer, M.

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

Oishi, Y.

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]

Padgett, M. J.

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]

Peschel, U.

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]

Piccirillo, B.

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]

Qian, X.-F.

Quabis, S.

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]

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

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

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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).
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Raizen, M. G.

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).
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Raskar, R.

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.

Romano, V.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys. A 86, 329–334 (2007).
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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).
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Russell, P. St.J.

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).
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Sakuma, I.

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]

Saleh, B. E. A.

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).
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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]

Sato, S.

She, W.

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).
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Simha, A.

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]

Slussarenko, S.

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

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

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]

Velten, A.

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.

Walborn, S. P.

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

Whybrew, A.

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.

Wozniak, P.

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

Zhong, W.

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]

Adv. Opt. Photon. (1)

Appl. Phys. A (1)

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]

Appl. Phys. B (1)

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]

Found. Phys. (1)

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

J. Opt. B (1)

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]

Laser Photon. Rev. (1)

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]

Nano Lett. (1)

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

Nat. Photonics (2)

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]

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]

Nature (1)

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]

New J. Phys. (2)

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]

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

Opt. Commun. (2)

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

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

Opt. Express (4)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (4)

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

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]

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]

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]

Phys. Rev. Lett. (4)

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]

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

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]

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

Science (1)

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]

Other (4)

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.

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.

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

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).

Supplementary Material (1)

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Supplement 1: PDF (1463 KB)      Supplemental document

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Figures (4)

Fig. 1.
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.
Fig. 2.
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 (λ/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).
Fig. 3.
Fig. 3. Rotation sensing and position tracking. (a) A metal rotor [width m=(0.79±0.01)mm] turns about the center of a radially polarized beam [width w1=(1.95±0.10)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 s0, s1, and s2 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±0.01)mm] traverses a radially polarized light beam [width w2=(2.84±0.10)mm]. (Center) The Stokes parameters s0, s1, and s2 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, s0 is normalized to its initial value, while s1 and s2 are normalized to the instantaneous value of s0. 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.
Fig. 4.
Fig. 4. Real-time sensing. A metal knife edge of thickness (3±2)μm cuts across a focused radially polarized mode [theoretically estimated width w3=(2.0±0.5)μm] at (27±2)ms1. The plot shows a sequence of single-shot measurements of the beam’s s0, s1, and s2 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.

Equations (5)

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E(ρ,z)=e1f1(ρ,z)+e2f2(ρ,z),
s0=λ1+λ2,
s1=(λ1λ2)(|ax|2|ay|2),
s2=(λ1λ2)(axay*+ax*ay),
s3=i(λ1λ2)(axay*ax*ay),

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