Abstract

Quantum backflow is a counterintuitive phenomenon in which a forward-propagating quantum particle propagates locally backwards. The actual counter-propagation property associated with this delicate interference phenomenon has not been observed to date in any field of physics, to the best of our knowledge. Here, we report the observation of an analog optical effect, namely, transverse optical backflow where a beam of light propagating to a specific transverse direction is measured locally to propagate in the opposite direction. This observation is relevant to any physical system supporting coherent waves.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Quantum backflow (also known as retro-propagation) is a surprising phenomenon, first pointed out in 1969 [13] by Allcock in the context of the time-of-arrival problem in quantum theory. Allcock found that a local quantum probability current may become negative even for positive momenta quantum states, and thus cannot be a valid measure for the time of arrival. Further advances with regards to the time-of-arrival problem were made by Muga et al. [46]. The phenomenon was studied in detail in 1994 by Bracken and Melloy [7] who found a limit on the total amount of backflow. This led them to introduce a new dimensionless quantum number whose value has been reproduced more accurately in subsequent years [8,9]. Recently, there has been a renewed interest in backflow with various studies reintroducing and exploring various aspects of the phenomenon [10,11].

 figure: Fig. 1.

Fig. 1. Finite backflow function $ {f_{{\rm FBF}}}(\xi ) $. (Left) Backflow function spatial spectrum. $ {k_0} $ is the fundamental spatial frequency. The dashed black line represents the center of the $ k $ axis, related to zero transverse momentum. (Center) Backflow function in real space. (Right) Local spatial frequency of $ {f_{{\rm FBF}}}(\xi ) $. The rows correspond to (a) $ a = 1.0 $, (b) $ a = 0.7 $, and (c) $ a = 0.4 $.

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A recent development in the field explored the relation between backflow and superoscillation [12]. Superoscillation is the phenomenon in which a band-limited signal locally oscillates faster than its fastest Fourier component. The phenomenon appeared first in the microwave community [1315] and later was revived by Aharonov et al. in 1988 [16] while establishing the theory of quantum weak measurements. Berry and Popescu further developed the mathematical theory of the phenomenon and proposed to use it to realize far-field sub-wavelength optical focusing [17,18]. This suggestion was verified experimentally shortly afterward [19]. Various experimental works involving superoscillations, which appeared in the last two decades, include super-resolution microscopy [2022], optical beam shaping [2325], nano-focusing of light [26], particle trapping [27], electron beam shaping [28], nonlinear optical frequency conversion [29], as well as optical temporal super-resolution and sub-Fourier focusing [30,31]. A complementary phenomenon to superoscillation, termed suboscillation, where a lower-bound-limited signal oscillates locally slower than its lowest Fourier component, was discovered recently [32].

Berry [12] analyzed the evolution of backflow regions in the interference of quantum wavepackets. He demonstrated that these regions are dependent on the overall momenta distribution of the wavepackets and showed that the superposition of many waves creates wider and stronger backflow regions when the Fourier components are highly correlated. Moreover, he also found this phenomenon to be extremely vulnerable, since the evolution in space–time causes the destruction of the delicate phase relations critical for backflow. In 2013, Palmero et al. [33] proposed an experiment to detect quantum backflow by applying a Bragg pulse to a Bose–Einstein condensate. A recent work investigated the effect of reflection and transmission processes on backflow [34]. Importantly, the first experimental observation of the local momentum associated with backflow near optical superoscillatory foci was reported [35]. Still, to the best of our knowledge, no experimental observation of any actual backward movement associated with backflow in any wave system has been reported to date.

Here, we report on the observation of counter-propagation due to optical backflow. We construct a light beam, based on a spectrally shifted suboscillatory function [32], made of the superposition of modes having negative transverse momentum relative to a chosen axis of propagation. While the expectation value of the transverse momentum of the beam is negative (i.e., the beam travels at a negative angle relative to the chosen axis of propagation), in certain locations, the local value of the transverse momentum is positive. Isolating these regions with a slit causes the local transverse momentum to be “projected” onto the expectation value of the beam’s transverse momentum, realizing a beam propagating at an overall positive angle relative to the propagation axis. The equivalence of the paraxial optical wave equation to the Schrodinger equation (where the propagation coordinate plays the role of time) [36] ensures that our experiment reported below constitutes a reliable simulator for the quantum phenomenon.

2. THEORY

Consider the following backflow function built using a spectrally shifted suboscillatory function $ {f_{{\rm Sub}}}(\xi ) $ [32] in the coordinate $ \xi $:

$${f_{{\rm BF}}}(\xi ) = {f_{{\rm Sub}}}(\xi )\exp (iL\xi ) = \frac{{\exp (iL\xi )}}{{{{\left[ {\cos ({k_0}\xi ) + ia\sin ({k_0}\xi )} \right]}^N}}},$$
where $ a \in \{ 0 \lt {\mathbb R} \le 1\} $, $ N \in \{ {\mathbb N} \gt 0\} $, $ {k_0} $ is the fundamental spatial frequency of the suboscillatory function, and $ \exp (iL\xi ) $ acts as a spectral shifting function. The Fourier transform of $ {f_{{\rm BF}}}(\xi ) $ is given with
$${F_{{\rm BF}}}(k) = 2\pi \sum\limits_{m = - \infty }^{ + \infty } {C_m}(a) \cdot \delta (k - L - m{k_0}),$$
where $ k $ denotes the spatial frequency, $ {C_m}(a) $ are Fourier coefficients, and $ L $ is the spectral shift parameter. By applying the local frequency operator on Eq. (1), we get
$$\begin{split}{k_{{\rm local}}}(\xi ) &= {\rm Im}\frac{{\partial \ln [{f_{{\rm BF}}}(\xi )]}}{{\partial \xi }}\\& = L - \frac{{Na{k_0}}}{{\mathop {\cos }\nolimits^2 ({k_0}\xi ) + {a^2}\mathop {\sin }\nolimits^2 ({k_0}\xi )}}.\end{split}$$

It is obvious that $ {k_{{\rm local}}}(\xi ) \in [L - \frac{{N{k_0}}}{a},L - N{k_0}a] $. The global spectrum of $ {f_{{\rm Sub}}}(\xi ) $ is a negatively valued single-sided spectrum with the highest harmonic at $ - N{k_0} $ [32] (which sets it as the slowest frequency component). For the function $ {f_{{\rm BF}}}(\xi ) $, the $ \exp (iL\xi ) $ term shifts the highest available spectral frequency to $ M = - N{k_0} + L $. This implies that for a proper selection of the parameters $ L $, $ N $, $ {k_0} $, and $ a $, it is possible to obtain a single-sided spectrum composed of only negative components $ M \lt 0 $ while having a local positive frequency $ (L - N{k_0}a) \gt 0 $, which is the hallmark of the backflow phenomenon. Consider, for example, the Fourier coefficients of Eq. (1) for $ N = 3 $, which are calculated by complex integration to be

$${C_m}(a) = \left\{ {\begin{array}{lc}{\left( {{m^2} - 1} \right)\frac{{{{\left( {a + 1} \right)}^{\frac{m}{2} - \frac{3}{2}}}}}{{{{\left( {a - 1} \right)}^{\frac{m}{2} + \frac{3}{2}}}}},}&{m \in \left\{ {{\rm odd} \lt 0} \right\},}\\{0,}&{{\rm otherwise}{\rm .}}\end{array}} \right.$$

Together with $ {k_0} = 1 $ and $ L = 2 $, the spectrum in Eq. (2) is completely single sided, having only negative components, with the highest harmonic at $ M = - 1 $. The local frequency, however, as defined by Eq. (3), is positive in the regions close to $ \xi = 2\pi n $ ($ n \in {\mathbb N} $) for $ a \lt 2/3 $.

The backflow function in Eq. (1) is periodic and extends from $ - \infty $ to $ \infty $ and so cannot be used in an experiment. We therefore derive a bounded signal by convolving the Fourier transform in Eq. (2) with a narrow Gaussian spectral distribution. The resulting spectrum of this finite backflow function is given with

$${F_{{\rm FBF}}}\left( k \right) = \sum\limits_{m = - P}^{ - N} {C_m}(a)\exp \left( { - \frac{{{{\left[ {k - L - m{k_0}} \right]}^2}}}{{2{\sigma _0}^2}}} \right),$$
where $ {C_m}(a) $ are the coefficients defined in Eq. (4), $ L $ is the spectral shift parameter, $ {k_0} $ is the fundamental spatial frequency, and $ {\sigma _0}^2 $ is the spectral variance of each harmonic. Since $ {C_m}(a) $ decay rapidly, we synthesize our function using a finite number of negative harmonics, where $ ( - P) $ and $ ( - N) $ represent the indices corresponding to the lowest and highest harmonics, respectively. A sufficiently narrow spectral variance is chosen such that theoretically only a negligible fraction ($ 3.82 \times {10^{ - 12}} $) of the energy of the highest mode $ M $ is positive (practically ensuring a one-sided spectrum). Experimentally (see below), we realize our beam using discrete sampling of the theoretical smooth function for which there is no energy at all associated with the Gaussian envelopes at positive transverse frequency values. The resulting spectrum is shown in Fig. 1 (left) for several values of the parameter $ a $. The inverse Fourier transform of this spectrum, which corresponds to the finite backflow function $ {f_{{\rm FBF}}}(\xi ) $, is shown in Fig. 1 (center). Figure 1 (right) shows the local momentum calculated by Eq. (3) for each case. It can be seen that for the case of $ a = 0.4 $, while the entire spectrum is negative [Fig. 1(c) (left)], positive local momentum regions are clearly evident [Fig. 1(c) (right)].

3. EXPERIMENT

In our experiment (see Fig. 2), a reflective phase-only spatial light modulator (SLM, Holoeye Pluto) was used to realize finite backflow functions, set according to Eq. (5) for different values of the tuning parameter $ a = \{ 0.4,0.5,0.6,0.7,0.8,0.9,1\} $, with $ P = 21 $, $ N = 3 $, and $ L = 0.733\;{{\rm mm}^{ - 1}} $. The fundamental frequency and each harmonic spectral variance were set to $ {k_0} = 0.391\;{{\rm mm}^{ - 1}} $ and $ {\sigma _0} = 0.0611\;{{\rm mm}^{ - 1}} $, respectively. The resulting complex-valued spectral functions were encoded into a set of phase-only masks using a known phase-encoding technique [37]. The masks were equipped with a phase-only blazed grating (with period $ \Lambda = 64\; \unicode{x00B5}{\rm m} $) whose first-diffraction-order optical axis is the propagation direction associated with the transverse frequency $ k = 0 $. The intensity of the encoded masks corresponding to the values of $ a = 0.4 $, $ a = 0.7 $, and $ a = 1 $ appear in Fig. 3 (left).

 figure: Fig. 2.

Fig. 2. Experimental setup. BE, beam expander; SLM, spatial light modulator; MS, moving stage; SL, slit; M, mirror; $ {L_1},{L_2},{L_3},{L_4} $, lenses. $ {d_1} + {d_2} $ equals lens $ {L_3} $ focal length. $ {Z_1} $ and $ {Z_2} $ mark the locations of the first and second focal planes, respectively. (Inset) (I) Realization of one of the phase-only masks used in the experiment. (II) Corresponding intensity of the beam at the first diffraction order at the first focal plane.

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 figure: Fig. 3.

Fig. 3. Experimental measurements. (Left) Generated SLM phase-only masks. Each line creates a propagating mode with a well-defined negative transverse momentum. The dotted line represents the center of the $ x $ axis, related to zero transverse momentum. (Center) Measured intensity distribution (in counts) in the first focal plane (backflow beam). The two dashed white lines represent the width of the slit. (Right) Measured beam image at the second focal plane, averaged over the $ y $ coordinate for each slit position. The dashed-dotted black line denotes the center of the propagation axis. Continuous red line: measured expectation value of the beam position (equal to momentum in the first focal plane). Dashed blue line: analytically calculated expectation value for a theoretical infinite periodic backflow beam after it is slit-filtered. Dotted green line: expectation value derived from Fourier transforming the SLM image and then Fourier-transforming again the slit-filtered image. (a) $ a = 1 $, (b) $ a = 0.7 $, and (c) $ a = 0.4 $.

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In the experiment, the laser beam (Quantum Ventus 532 Solo) was expanded and collimated before the SLM, reflected off it (embedded with the phase masks), and Fourier transformed using a 50 cm focal lens into the first focal plane where the finite backflow functions are realized. The intensity patterns of the beams in this plane were measured using a CMOS camera (Ophir Spiricon SP620U) and are presented in Fig. 3 (center). Next, the camera was removed from the first focal plane and replaced by a movable 100 µm wide slit mounted on top of a 25 mm long stepper motor linear stage (Newport MFA-PPD) that was set to move in steps of 5 µm along $ \xi $. The spatially filtered beam in the first focal plane was Fourier transformed again by a 50 cm focal lens into the second focal plane. Multiple intensity images were taken at the second focal plane for different positions of the slit. The right column in Fig. 3 shows the beam intensity in the second focal plane averaged over the $ y $ axis for different positions of the slit. Note that the application of an optical Fourier transform twice using two consecutive $ 2f $ systems results in a coordinate inversion. The curves in Fig. 3 (right) represent the expectation value of the beam’s position and hence describe the measured deflection of the beam in the second focal plane, relative to the propagation axis. These expectation values were calculated directly from the captured beam images (red continuous line), compared to an analytical expression based on the infinite backflow beam after it was slit-filtered (dashed blue line), and to a numerical calculation based on two consecutive optical Fourier transforms of the SLM patterns (dotted green line). All results show a very high level of agreement. Notice that we start in the SLM plane with a momentum representation of the backflow function and end back at a plane representing momentum. Thus, the beam deflection represents expectation values of the transverse momentum. Note that the exact location of the propagation (optical) axis at the second focal plane, associated with $ k = 0 $, was calibrated beforehand by applying a single slit mask to the center of the SLM and measuring the expectation value of the image at that plane.

A comparison of the expectation curve in the cases of $ a = 1 $ (a), $ a = 0.7 $ (b), and $ a = 0.4 $ (c) indicates that, as expected, the degree of deflection due to backflow increases as the value of $ a $ decreases (and the beam does not contain backflow for $ a = 1 $). Note that the $ a = 0.4 $ case achieves the maximal backflow value, and the beam’s deflection completely crosses the propagation axis for certain positions of the slit, thus materializing local positive momentum. It is also clear that a larger backflow entails lower intensity in the beam, which is a defining characteristic of super- and suboscillating functions [3842]. We further define a quantitative backflow probability, $ {\mu _{{\rm BF}}} $, as the maximum ratio (over slit positions) of the power detected over negative values of the camera axis (Fig. 3, right) to the total detected power. The calculated values of this backflow measure are $ {\mu _{{\rm BF}}} = 0.34,0.48,0.54 $ for $ a = 1,0.7,0.4 $, respectively. When $ {\mu _{{\rm BF}}} \gt 0.5 $, the expectation value of the beam position crosses the propagation axis, and there is backflow.

In a sense, the filtering by the slit applied on the backflow beam realizes a nonlinear “projection” operation of a local property (local transverse momentum, which is not an eigenvalue of the momentum operator $ \hat k \to - i{\partial _x} $) to a global property (eigenvalue of the momentum operator), thus allowing to observe the backflow as an actual deflection of the beam. More formally, our beams comprise a superposition of plane waves (eigenvectors of the momentum operator), and without the slit, measurement of the momentum amounts to a selection of one of these plane waves. With the slit, we first select the beam at a specific position and then measure the momentum. This no longer yields one of the momentum eigenvalues of the original beam, as it was changed by going through the slit, and the result can be a value outside the spectrum (set of all eigenvalues) of the original beam. This is related to the formalism of weak measurements [12,43]: the local momentum is considered as the result of a weak measurement, giving rise to a “weak value” observable $ {A_{{\rm weak}}} $ through the operation of the momentum operator $ \hat k $ on a preselected state, which is, in our case, the backflow beam $ \psi $ (whose representation in momentum space is $ \langle k|\psi \rangle = {F_{{\rm FBF}}}(k) $ and in coordinate space $ \langle \xi |\psi \rangle = {f_{{\rm FBF}}}(\xi ) $), and after post selecting with a coordinate state $ |\xi \rangle $ (describing the position of the slit) [12],

$${k_{{\rm local}}}(\xi ) = {\rm Im}\frac{{\partial \ln [\psi (\xi )]}}{{\partial \xi }} = {\rm Re}\frac{{\langle \xi |\hat k|\psi \rangle }}{{\langle \xi |\psi \rangle }} = {A_{{\rm weak}}}.$$

To examine the deflection’s sensitivity to the slit’s width, $ W $, we have used a beam propagation simulation to calculate the beam’s deflection as a function of the slit’s position and width. Figure 4 shows the simulated beam image at the second focal plane, averaged over the $ y $ coordinate, for different values of the slit’s width: $ W = 50\;{\unicode{x00B5}{\rm m}} $ (a), $ W = 100\;{\unicode{x00B5}{\rm m}} $ (b), and $ W = 200\;{\unicode{x00B5}{\rm m}} $ (c), all cases with the backflow parameter $ a = 0.4 $. The backflow measure $ {\mu _{{\rm BF}}} $ is found to be 0.59, 0.54, 0.50 for $ W = 50,100,200\;{\unicode{x00B5}{\rm m}} $, respectively. The green dotted curves represent the expectation value of the beam’s position along $ x $. It is evident from the curves that the degree of backflow decreases as the slit width increases. Of course, reducing the slit width increases the variance in the position of the beam (its width), but the change in position of the center of mass of the beam (its expectation value) is due to the local momentum at the slit position.

 figure: Fig. 4.

Fig. 4. Intensity distribution for different slit widths. Simulated beam image at the second focal plane, averaged over the $ y $ coordinate for each slit position, for the case of $ a = 0.4 $ and for different values of the slit’s width: (left) $ W = 50\;{\unicode{x00B5}{\rm m}} $, (center) 100 µm, and (right) 200 µm. The dotted green curve represents the expectation value of the beam's position.

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4. CONCLUSION

We experimentally constructed, measured, and observed beam deflections associated with backflowing beams. We first designed a backflow beam based on the mathematical form of suboscillatory functions that were discovered recently. This allowed controlling the degree of backflow in the beam. Slit-filtering the generated beams allowed, counterintuitively, their deflection towards a direction opposite to that associated with the momentum states comprising the original beam. This effect is the result of a delicate interference phenomenon that, until now, hindered the observation of movement or deflection associated with backflow in any wave system (from quantum particles to optical waves to acoustic waves, etc.). Our results are also relevant to single photons, where each photon is in a backflow state comprising a superposition of different transverse momentum states. The backflow we demonstrated in this work is transverse optical backflow. It would be more challenging to demonstrate longitudinal optical backflow [12] along the axis of propagation of a light beam, which is more in line with the original concept of retro-propagation. In the future, backflow and possible generalizations of the phenomenon might be used for unique applications. Controlling the far-field radiation patterns of light going through small apertures is an obvious direction. It can also be very interesting to use backflow for stand-off spectroscopy [44,45], where optical sensing of remote areas is based on information propagating backwards towards the source. It might also be relevant to nonlinear and ultrafast optics, especially if this phenomenon is manifested in the time domain, generating local negative frequencies.

Acknowledgment

We would like to thank Liran Hareli for fruitful discussions.

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References

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  1. G. R. Allcock, “The time of arrival in quantum mechanics I. Formal considerations,” Ann. Phys. (N. Y.) 53, 253–285 (1969).
    [Crossref]
  2. G. R. Allcock, “The time of arrival in quantum mechanics II. The individual measurement,” Ann. Phys. (N. Y.) 53, 286–310 (1969).
    [Crossref]
  3. G. Allcock, “The time of arrival in quantum mechanics III. The measurement ensemble,” Ann. Phys. (N. Y.) 53, 311–348 (1969).
    [Crossref]
  4. J. Muga, J. Palao, and C. Leavens, “Arrival time distributions and perfect absorption in classical and quantum mechanics,” Phys. Lett. A 253, 21–27 (1999).
    [Crossref]
  5. J. G. Muga and C. R. Leavens, “Arrival time in quantum mechanics,” Phys. Rep. 338, 353–438 (2000).
    [Crossref]
  6. J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
    [Crossref]
  7. A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197 (1994).
    [Crossref]
  8. S. P. Eveson, C. J. Fewster, and R. Verch, “Quantum inequalities in quantum mechanics,” in Annales Henri Poincaré (Springer, 2005), Vol. 6, pp. 1–30.
  9. M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
    [Crossref]
  10. J. Yearsley and J. Halliwell, “An introduction to the quantum backflow effect,” in Journal of Physics: Conference Series (IOP Publishing, 2013), Vol. 442, p. 012055.
  11. F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
    [Crossref]
  12. M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
    [Crossref]
  13. S. A. Schelkunoff, “A mathematical theory of linear arrays,” Bell Syst. Tech. J. 22, 80–107 (1943).
    [Crossref]
  14. D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
    [Crossref]
  15. G. T. Di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
    [Crossref]
  16. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
    [Crossref]
  17. M. Berry, Quantum Coherence and Reality: In Celebration of the 60th Birthday of Yakir Aharonov (World Scientific, 1994).
  18. M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965 (2006).
    [Crossref]
  19. F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
    [Crossref]
  20. E. T. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
    [Crossref]
  21. F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
    [Crossref]
  22. A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
    [Crossref]
  23. E. Greenfield, R. Schley, I. Hurwitz, J. Nemirovsky, K. G. Makris, and M. Segev, “Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams,” Opt. Express 21, 13425–13435 (2013).
    [Crossref]
  24. Y. Eliezer and A. Bahabad, “Super-oscillating airy pattern,” ACS Photon. 3, 1053–1059 (2016).
    [Crossref]
  25. T. Zacharias, B. Hadad, A. Bahabad, and Y. Eliezer, “Axial sub-Fourier focusing of an optical beam,” Opt. Lett. 42, 3205–3208 (2017).
    [Crossref]
  26. A. David, B. Gjonaj, Y. Blau, S. Dolev, and G. Bartal, “Nanoscale shaping and focusing of visible light in planar metal-oxide-silicon waveguides,” Optica 2, 1045–1048 (2015).
    [Crossref]
  27. B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
    [Crossref]
  28. R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
    [Crossref]
  29. R. Remez and A. Arie, “Super-narrow frequency conversion,” Optica 2, 472–475 (2015).
    [Crossref]
  30. Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
    [Crossref]
  31. Y. Eliezer, B. K. Singh, L. Hareli, A. Arie, and A. Bahabad, “Experimental realization of structured super-oscillatory pulses,” Opt. Express 26, 4933–4941 (2018).
    [Crossref]
  32. Y. Eliezer and A. Bahabad, “Super defocusing of light by optical sub-oscillations,” Optica 4, 440–446 (2017).
    [Crossref]
  33. M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
    [Crossref]
  34. H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
    [Crossref]
  35. G. Yuan, E. T. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8, 2 (2019).
    [Crossref]
  36. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
    [Crossref]
  37. E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
    [Crossref]
  38. Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.
  39. B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7–17 (2013).
    [Crossref]
  40. J. Tollaksen, “Novel relationships between superoscillations, weak values, and modular variables,” J. Phys. Conf. Ser. 70, 012016 (2007).
    [Crossref]
  41. M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2011).
    [Crossref]
  42. Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
    [Crossref]
  43. M. Berry, “Five momenta,” Eur. J. Phys. 34, 1337 (2013).
    [Crossref]
  44. V. C. Coffey, “Advances in standoff detection make the world safer,” Photon. Spectra 47, 44–47 (2013).
  45. J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
    [Crossref]

2019 (1)

G. Yuan, E. T. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8, 2 (2019).
[Crossref]

2018 (1)

2017 (6)

Y. Eliezer and A. Bahabad, “Super defocusing of light by optical sub-oscillations,” Optica 4, 440–446 (2017).
[Crossref]

T. Zacharias, B. Hadad, A. Bahabad, and Y. Eliezer, “Axial sub-Fourier focusing of an optical beam,” Opt. Lett. 42, 3205–3208 (2017).
[Crossref]

B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
[Crossref]

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
[Crossref]

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
[Crossref]

H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
[Crossref]

2016 (2)

Y. Eliezer and A. Bahabad, “Super-oscillating airy pattern,” ACS Photon. 3, 1053–1059 (2016).
[Crossref]

F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
[Crossref]

2015 (2)

2013 (8)

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
[Crossref]

E. Greenfield, R. Schley, I. Hurwitz, J. Nemirovsky, K. G. Makris, and M. Segev, “Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams,” Opt. Express 21, 13425–13435 (2013).
[Crossref]

M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
[Crossref]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
[Crossref]

B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7–17 (2013).
[Crossref]

E. T. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15, 094008 (2013).
[Crossref]

M. Berry, “Five momenta,” Eur. J. Phys. 34, 1337 (2013).
[Crossref]

V. C. Coffey, “Advances in standoff detection make the world safer,” Photon. Spectra 47, 44–47 (2013).

2011 (1)

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2011).
[Crossref]

2010 (2)

Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
[Crossref]

M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
[Crossref]

2009 (2)

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
[Crossref]

2007 (3)

J. Tollaksen, “Novel relationships between superoscillations, weak values, and modular variables,” J. Phys. Conf. Ser. 70, 012016 (2007).
[Crossref]

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
[Crossref]

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

2006 (1)

M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965 (2006).
[Crossref]

2005 (1)

M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
[Crossref]

2002 (1)

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

2000 (1)

J. G. Muga and C. R. Leavens, “Arrival time in quantum mechanics,” Phys. Rep. 338, 353–438 (2000).
[Crossref]

1999 (1)

J. Muga, J. Palao, and C. Leavens, “Arrival time distributions and perfect absorption in classical and quantum mechanics,” Phys. Lett. A 253, 21–27 (1999).
[Crossref]

1994 (1)

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197 (1994).
[Crossref]

1988 (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
[Crossref]

1969 (3)

G. R. Allcock, “The time of arrival in quantum mechanics I. Formal considerations,” Ann. Phys. (N. Y.) 53, 253–285 (1969).
[Crossref]

G. R. Allcock, “The time of arrival in quantum mechanics II. The individual measurement,” Ann. Phys. (N. Y.) 53, 286–310 (1969).
[Crossref]

G. Allcock, “The time of arrival in quantum mechanics III. The measurement ensemble,” Ann. Phys. (N. Y.) 53, 311–348 (1969).
[Crossref]

1961 (1)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

1952 (1)

G. T. Di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[Crossref]

1943 (1)

S. A. Schelkunoff, “A mathematical theory of linear arrays,” Bell Syst. Tech. J. 22, 80–107 (1943).
[Crossref]

Aharonov, Y.

Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
[Crossref]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
[Crossref]

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.

Albarelli, F.

F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
[Crossref]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
[Crossref]

Allcock, G.

G. Allcock, “The time of arrival in quantum mechanics III. The measurement ensemble,” Ann. Phys. (N. Y.) 53, 311–348 (1969).
[Crossref]

Allcock, G. R.

G. R. Allcock, “The time of arrival in quantum mechanics I. Formal considerations,” Ann. Phys. (N. Y.) 53, 253–285 (1969).
[Crossref]

G. R. Allcock, “The time of arrival in quantum mechanics II. The individual measurement,” Ann. Phys. (N. Y.) 53, 286–310 (1969).
[Crossref]

Arie, A.

Y. Eliezer, B. K. Singh, L. Hareli, A. Arie, and A. Bahabad, “Experimental realization of structured super-oscillatory pulses,” Opt. Express 26, 4933–4941 (2018).
[Crossref]

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
[Crossref]

B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
[Crossref]

R. Remez and A. Arie, “Super-narrow frequency conversion,” Optica 2, 472–475 (2015).
[Crossref]

Bahabad, A.

Bartal, G.

Bent, N.

Berry, M.

M. Berry, “Five momenta,” Eur. J. Phys. 34, 1337 (2013).
[Crossref]

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2011).
[Crossref]

M. Berry, “Quantum backflow, negative kinetic energy, and optical retro-propagation,” J. Phys. A 43, 415302 (2010).
[Crossref]

M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965 (2006).
[Crossref]

M. Berry, Quantum Coherence and Reality: In Celebration of the 60th Birthday of Yakir Aharonov (World Scientific, 1994).

Blau, Y.

Bolduc, E.

Bostelmann, H.

H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
[Crossref]

Boyd, R. W.

Bracken, A.

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197 (1994).
[Crossref]

Cadamuro, D.

H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
[Crossref]

Chen, Y.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
[Crossref]

Coffey, V. C.

V. C. Coffey, “Advances in standoff detection make the world safer,” Photon. Spectra 47, 44–47 (2013).

Cohen, E.

B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7–17 (2013).
[Crossref]

Colombo, F.

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.

Damborenea, J.

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

David, A.

de Abajo, F. J. G.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
[Crossref]

De Lucia, F. C.

J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
[Crossref]

Di Francia, G. T.

G. T. Di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento 9, 426–438 (1952).
[Crossref]

Dolev, S.

Dunin-Borkowski, R. E.

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
[Crossref]

Egusquiza, I.

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

Eleftheriades, G. V.

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
[Crossref]

Eliezer, Y.

Eveson, S. P.

S. P. Eveson, C. J. Fewster, and R. Verch, “Quantum inequalities in quantum mechanics,” in Annales Henri Poincaré (Springer, 2005), Vol. 6, pp. 1–30.

Fewster, C. J.

S. P. Eveson, C. J. Fewster, and R. Verch, “Quantum inequalities in quantum mechanics,” in Annales Henri Poincaré (Springer, 2005), Vol. 6, pp. 1–30.

Froim, S.

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
[Crossref]

Gjonaj, B.

Gottfried, J. L.

J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
[Crossref]

Greenfield, E.

Grübl, G.

M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
[Crossref]

Guaita, T.

F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
[Crossref]

Hadad, B.

Halliwell, J.

J. Yearsley and J. Halliwell, “An introduction to the quantum backflow effect,” in Journal of Physics: Conference Series (IOP Publishing, 2013), Vol. 442, p. 012055.

Hareli, L.

Y. Eliezer, B. K. Singh, L. Hareli, A. Arie, and A. Bahabad, “Experimental realization of structured super-oscillatory pulses,” Opt. Express 26, 4933–4941 (2018).
[Crossref]

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
[Crossref]

Hegerfeldt, G.

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

Huang, F. M.

F. M. Huang, Y. Chen, F. J. G. de Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A 9, S285 (2007).
[Crossref]

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

Hurwitz, I.

Javier Garcia de Abajo, F.

F. M. Huang, N. Zheludev, Y. Chen, and F. Javier Garcia de Abajo, “Focusing of light by a nanohole array,” Appl. Phys. Lett. 90, 091119 (2007).
[Crossref]

Karimi, E.

Kreidl, S.

M. Penz, G. Grübl, S. Kreidl, and P. Wagner, “A new approach to quantum backflow,” J. Phys. A 39, 423 (2005).
[Crossref]

Leavens, C.

J. Muga, J. Palao, and C. Leavens, “Arrival time distributions and perfect absorption in classical and quantum mechanics,” Phys. Lett. A 253, 21–27 (1999).
[Crossref]

Leavens, C. R.

J. G. Muga and C. R. Leavens, “Arrival time in quantum mechanics,” Phys. Rep. 338, 353–438 (2000).
[Crossref]

Lechner, G.

H. Bostelmann, D. Cadamuro, and G. Lechner, “Quantum backflow and scattering,” Phys. Rev. A 96, 012112 (2017).
[Crossref]

Lobachinsky, L.

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
[Crossref]

Longhi, S.

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. 3, 243–261 (2009).
[Crossref]

Lu, P.-H.

R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
[Crossref]

Makris, K. G.

Melloy, G.

A. Bracken and G. Melloy, “Probability backflow and a new dimensionless quantum number,” J. Phys. A 27, 2197 (1994).
[Crossref]

Miziolek, A. W.

J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
[Crossref]

Modugno, M.

M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
[Crossref]

Muga, J.

M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
[Crossref]

J. Damborenea, I. Egusquiza, G. Hegerfeldt, and J. Muga, “Measurement-based approach to quantum arrival times,” Phys. Rev. A 66, 052104 (2002).
[Crossref]

J. Muga, J. Palao, and C. Leavens, “Arrival time distributions and perfect absorption in classical and quantum mechanics,” Phys. Lett. A 253, 21–27 (1999).
[Crossref]

Muga, J. G.

J. G. Muga and C. R. Leavens, “Arrival time in quantum mechanics,” Phys. Rep. 338, 353–438 (2000).
[Crossref]

Munson, C. A.

J. L. Gottfried, F. C. De Lucia, C. A. Munson, and A. W. Miziolek, “Laser-induced breakdown spectroscopy for detection of explosives residues: a review of recent advances, challenges, and future prospects,” Anal. Bioanal. Chem. 395, 283–300 (2009).
[Crossref]

Nagar, H.

B. K. Singh, H. Nagar, Y. Roichman, and A. Arie, “Particle manipulation beyond the diffraction limit using structured super-oscillating light beams,” Light: Sci. Appl. 6, e17050 (2017).
[Crossref]

Nemirovsky, J.

Palao, J.

J. Muga, J. Palao, and C. Leavens, “Arrival time distributions and perfect absorption in classical and quantum mechanics,” Phys. Lett. A 253, 21–27 (1999).
[Crossref]

Palmero, M.

M. Palmero, E. Torrontegui, J. Muga, and M. Modugno, “Detecting quantum backflow by the density of a Bose-Einstein condensate,” Phys. Rev. A 87, 053618 (2013).
[Crossref]

Paris, M. G.

F. Albarelli, T. Guaita, and M. G. Paris, “Quantum backflow effect and nonclassicality,” Int. J. Quantum Inf. 14, 1650032 (2016).
[Crossref]

Penz, M.

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M. Berry and S. Popescu, “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” J. Phys. A 39, 6965 (2006).
[Crossref]

M. Berry and P. Shukla, “Pointer supershifts and superoscillations in weak measurements,” J. Phys. A 45, 015301 (2011).
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G. Yuan, E. T. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8, 2 (2019).
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R. Remez, Y. Tsur, P.-H. Lu, A. H. Tavabi, R. E. Dunin-Borkowski, and A. Arie, “Superoscillating electron wave functions with subdiffraction spots,” Phys. Rev. A 95, 031802 (2017).
[Crossref]

Phys. Rev. Lett (1)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett 60, 1351 (1988).
[Crossref]

Phys. Rev. Lett. (1)

Y. Eliezer, L. Hareli, L. Lobachinsky, S. Froim, and A. Bahabad, “Breaking the temporal resolution limit by superoscillating optical beats,” Phys. Rev. Lett. 119, 043903 (2017).
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Phys. Today (1)

Y. Aharonov, S. Popescu, and J. Tollaksen, “A time symmetric formulation of quantum mechanics,” Phys. Today 63 (11), 27 (2010).
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B. Tamir and E. Cohen, “Introduction to weak measurements and weak values,” Quanta 2, 7–17 (2013).
[Crossref]

Sci. Rep. (1)

A. M. Wong and G. V. Eleftheriades, “An optical super-microscope for far-field, real-time imaging beyond the diffraction limit,” Sci. Rep. 3, 1715 (2013).
[Crossref]

Other (4)

M. Berry, Quantum Coherence and Reality: In Celebration of the 60th Birthday of Yakir Aharonov (World Scientific, 1994).

J. Yearsley and J. Halliwell, “An introduction to the quantum backflow effect,” in Journal of Physics: Conference Series (IOP Publishing, 2013), Vol. 442, p. 012055.

S. P. Eveson, C. J. Fewster, and R. Verch, “Quantum inequalities in quantum mechanics,” in Annales Henri Poincaré (Springer, 2005), Vol. 6, pp. 1–30.

Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, and J. Tollaksen, The Mathematics of Superoscillations (American Mathematical Society, 2017), Vol. 247.

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Figures (4)

Fig. 1.
Fig. 1. Finite backflow function $ {f_{{\rm FBF}}}(\xi ) $. (Left) Backflow function spatial spectrum. $ {k_0} $ is the fundamental spatial frequency. The dashed black line represents the center of the $ k $ axis, related to zero transverse momentum. (Center) Backflow function in real space. (Right) Local spatial frequency of $ {f_{{\rm FBF}}}(\xi ) $. The rows correspond to (a) $ a = 1.0 $, (b) $ a = 0.7 $, and (c) $ a = 0.4 $.
Fig. 2.
Fig. 2. Experimental setup. BE, beam expander; SLM, spatial light modulator; MS, moving stage; SL, slit; M, mirror; $ {L_1},{L_2},{L_3},{L_4} $, lenses. $ {d_1} + {d_2} $ equals lens $ {L_3} $ focal length. $ {Z_1} $ and $ {Z_2} $ mark the locations of the first and second focal planes, respectively. (Inset) (I) Realization of one of the phase-only masks used in the experiment. (II) Corresponding intensity of the beam at the first diffraction order at the first focal plane.
Fig. 3.
Fig. 3. Experimental measurements. (Left) Generated SLM phase-only masks. Each line creates a propagating mode with a well-defined negative transverse momentum. The dotted line represents the center of the $ x $ axis, related to zero transverse momentum. (Center) Measured intensity distribution (in counts) in the first focal plane (backflow beam). The two dashed white lines represent the width of the slit. (Right) Measured beam image at the second focal plane, averaged over the $ y $ coordinate for each slit position. The dashed-dotted black line denotes the center of the propagation axis. Continuous red line: measured expectation value of the beam position (equal to momentum in the first focal plane). Dashed blue line: analytically calculated expectation value for a theoretical infinite periodic backflow beam after it is slit-filtered. Dotted green line: expectation value derived from Fourier transforming the SLM image and then Fourier-transforming again the slit-filtered image. (a) $ a = 1 $, (b) $ a = 0.7 $, and (c) $ a = 0.4 $.
Fig. 4.
Fig. 4. Intensity distribution for different slit widths. Simulated beam image at the second focal plane, averaged over the $ y $ coordinate for each slit position, for the case of $ a = 0.4 $ and for different values of the slit’s width: (left) $ W = 50\;{\unicode{x00B5}{\rm m}} $, (center) 100 µm, and (right) 200 µm. The dotted green curve represents the expectation value of the beam's position.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

f B F ( ξ ) = f S u b ( ξ ) exp ( i L ξ ) = exp ( i L ξ ) [ cos ( k 0 ξ ) + i a sin ( k 0 ξ ) ] N ,
F B F ( k ) = 2 π m = + C m ( a ) δ ( k L m k 0 ) ,
k l o c a l ( ξ ) = I m ln [ f B F ( ξ ) ] ξ = L N a k 0 cos 2 ( k 0 ξ ) + a 2 sin 2 ( k 0 ξ ) .
C m ( a ) = { ( m 2 1 ) ( a + 1 ) m 2 3 2 ( a 1 ) m 2 + 3 2 , m { o d d < 0 } , 0 , o t h e r w i s e .
F F B F ( k ) = m = P N C m ( a ) exp ( [ k L m k 0 ] 2 2 σ 0 2 ) ,
k l o c a l ( ξ ) = I m ln [ ψ ( ξ ) ] ξ = R e ξ | k ^ | ψ ξ | ψ = A w e a k .

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