1. INTRODUCTION

Light beams with azimuthal (helical) phase dependence ${e^{{il\phi}}}$ were identified to be carrying orbital angular momentum (OAM) by Allen et al. in 1992 [1]. They were experimentally generated for the first time in 1993, by employing cylindrical lenses [2], and since then beams carrying OAM have found applications in numerous fields such as optical tweezers [3], optical microscopy [4], interactions with chiral molecules [5], etc. States of light with azimuthal phase dependence are also analogous to the eigenstates of the angular momentum operator in quantum mechanics–${L_z}$. In the current work, we demonstrate that a peculiar phenomenon called backflow, known from quantum mechanics, is present in the superposition of beams carrying OAM.

The phenomenon of backflow was first encountered in the context of arrival times in quantum mechanics [6] and is a manifestation of interference. In this counter-intuitive phenomenon, a quantum particle prepared in specific superposition states of only positive momenta, having a wavefunction, for example, centered in $x \lt 0$, may have an increased probability, with time, of remaining in $x \lt 0$ [7]. While, for particles moving on a line, only about 4% of the total probability can flow in the "wrong" direction [8], this probability increases to around 12% for charged particles moving on a ring [9]. To overcome such bounds that make the experimental observation of the phenomenon difficult, researchers studied backflow in two dimensions for a charged particle moving either in a uniform magnetic field in the infinite $(x,y)$ plane [10,11] or on a finite disk such that a magnetic flux line passes through the center of the disk [12]. In such two-dimensional systems, the probability of backflow can be unbounded.

Although backflow in quantum systems has not yet been experimentally realized, it has been demonstrated with optical beams [13,14] in one dimension, by exploiting its connection to the concept of superoscillations in waves, as established by Berry et al. [15,16]. In a superoscillatory function, local Fourier components are not contained in the global Fourier spectrum [17,18]. For example, in classical electromagnetism, this manifests as follows: the local Poynting vector of a superposition state can point in directions not contained between those of the constituent plane waves, leading to counter-flow or backflow of the energy density [19,20].

In the recent experimental observations, one-dimensional transverse local momentum of a superposition of beams was measured by scanning a slit [13] or by using the Shack–Hartmann wavefront sensor technique [14], respectively. The Shack–Hartmann wavefront sensor technique also allows for one-shot measurement of the two-dimensional transverse local momentum, as reported for the case of azimuthally phased beams in [21]. Another method, employing digital-hologram-based modal decompositions, for measuring local OAM is given in [22,23]. In the present work, we use the Shack–Hartman sensor to measure the local OAM of the superposition of two beams with helical phases, thereby moving from linear optical backflow to azimuthal backflow. In practice, we examine the superposition of two beams carrying only negative orbital angular momentum and observe, in the dark fringes of such an interference pattern, positive local OAM. This is what we term azimuthal backflow. We clarify that, by “local OAM” of a scalar field at each point, we refer to the product of the azimuthal component of the local momentum at that point and its corresponding radius.

Zacharias and Bahabad [24] have previously utilized the superposition of Bessel beams with OAM to realize transverse super-oscillatory intensity patterns. Our current demonstration of azimuthal backflow in beams carrying OAM can be interpreted as superoscillations in phase. The backflow presented here is thus a manifestation of rapid changes in phase, which could be of importance in applications that involve light–matter interactions such as in optical trapping or in enhancing chiral response of molecules [5,25]. Apart from these, our demonstration is a step in the direction of observing quantum backflow in two dimensions, which has been theoretically found to be more robust than one-dimensional backflow.

 

Fig. 1. Visual representation of azimuthal backflow in the superposition of two beams carrying helical phases. (a) Two Gaussian beams—beam 1 and beam 2—with intensities ${I_1}$ and ${I_2}$, respectively, and amplitude ratio $b = 0.6$ between them, each of waist ${w_0} = 1\;{\rm{mm}}$, carrying negative helical phases ${l_1} = - 1$ and ${l_2} = - 3$, respectively, are superposed. Then, the field of their superposition is examined in the plane without propagation ($z = 0$) and in a plane with propagation ($z \gt 0$). Inset A is a conceptual setup in which such a superposition can be realized. The fields of our interest are at the image plane (${{z}} = {{0}}$) of the phase plates at the output of the beam splitter (BS) and further away from it (${{z}} \gt {{0}}$). The normalized azimuthal components of local wave-vectors–${k_{\phi ,1,2}}/|{k_{\phi ,1,2}}|$ are indicated with gray arrows on top of the intensity pattern of each beam. (b) Two-dimensional cross-section of the intensity distribution on the plane of the superposition without propagation (grayscale map) and normalized azimuthal components of local wave-vectors–${k_{\phi ,s}}/|{k_{\phi ,s}}|$ (scale bar indicated at the bottom right corner). While the gray arrows, in the bright fringes, point in clockwise direction (defined by the signs of ${l_1}$ and ${l_2}$), the orange arrows, in the dark fringes, point in the counter-clockwise direction, thus illustrating backflow. One such region of backflow, in a given dark fringe, is marked by the white triangle labelled A. (c) The local OAM $r{k_\phi}$ for each constituent (red, green constant lines) and the superposition (blue) and the intensity (orange) at a constant radius as functions of the azimuthal angle $\phi$. The values of the blue curve, indicating positive local OAM (above the gray line), i.e., backflow, coincide with the minima of the orange curve, i.e., the dark fringes. (d) Two-dimensional cross-section of the intensity distribution (grayscale map) on a plane after propagation (${{z}} = {{20}}\;{\rm{mm}}$ in this example) and normalized ${k_{\phi ^\prime ,s}}$. As in (b), the azimuthal component of the local wave-vector exhibits backflow outside the central vortex. (e) Quantitative plots of local OAM $r^\prime {k_{\phi ^\prime}}$ considering local amplitude ratio $B(r^\prime)$ and local phase $C(r^\prime)$. The red and green lines represent the constants $r^\prime {k_{\phi ^\prime ,1}}$ and $r^\prime {k_{\phi ^\prime ,2}}$, respectively. Three different values of $r^\prime $: ${r_1} = 0.2\;{\rm{mm}}$ (black solid line), ${r_2} = 1.5\;{\rm{mm}}$ (brown dashed line), ${r_3} = 2.5\;{\rm{mm}}$ (purple dotted line) are used to plot their respective $r^\prime {k_{\phi ^\prime ,s}}$. The black, brown, and purple curves peak at the minima of the respective black, brown, and purple curves of the intensity cross-section in the upper panel. Again, the positive values of $r^\prime {k_{\phi ^\prime ,s}}$ (above the gray line) are indicative of backflow. Note that in (b) and (d), there’s a non-zero amount of energy in the dark regions because the beams have unequal amplitudes. The intensity in these regions is relatively low but nonetheless detectable.

Download Full Size | PDF

2. THEORETICAL MODEL

Measuring azimuthal backflow in the superposition of conventional beams with helical phases such as Laguerre–Gauss (LG) and higher order Bessel–Gauss (BG) beams can be challenging due to the sparsity of local regions in which such backflow can be observed (c.f. Supplement 1 Section 2 for detailed theoretical derivations and illustrations of azimuthal backflow in superpositions of LG beams). Therefore, here, for the sake of simplicity, we searched for superpositions of other beams with helical phases that would show a more frequent azimuthal backflow. Figure 1(a) is a schematic of the superposition of two Gaussian beams with unequal amplitudes, carrying helical phases of orders ${l_1}$ and ${l_2}$ (both negative or positive) respectively. In inset (A), we provide a schematic of an interferometer setup in which such a demonstration can be realized. Here we provide a mathematical description of the propagation of this superposition along the $z$-axis. We restrict ourselves to quasi-monochromatic scalar fields under the paraxial approximation, instead of the more rigourous approach using Maxwell’s equations. For $z = 0$, i.e., no propagation from the image plane of the helical phase plates at the output of the beam splitter in inset A, the scalar field is given by

We stress that azimuthal backflow can already be observed using the field in Eq. (1). However, we wish to provide a complete description of the field’s propagation and to theoretically study the azimuthal backflow at other planes. The field at any $(z \gt 0)$, (i.e., after it propagates) is given by solving the Fresnel diffraction integral [26,27], considering free propagation of the field in Eq. (1):

To observe azimuthal backflow, let us first consider the specific case of $z = 0$, where

Next, we examine the behavior of azimuthal backflow at $z \gt 0$. We use Eqs. (2) and (3) to plot the intensity distribution $|\Psi (r^\prime ,\phi ^\prime ,z{)|^2}$ and the normalized local wave-vectors ${k_{\phi ^\prime ,s}}/|{k_{\phi ^\prime ,s}}|$, respectively. The two-dimensional plot is given in Fig. 1(d). Comparing Fig. 1(d) to Fig. 1(b), we see on the grayscale map of the intensity distribution, that for $z \gt 0$, a vortex around $r^\prime = 0$ is formed and no azimuthal backflow exists within this region. The value of $z$ determines the radius of this vortex. Apart from this observation, the arrow-fields in both the figures are similar. However, from the quantitative point of view, for $z \gt 0$, we see from Eq. (3) that the local OAM depends on the radius $r^\prime $. In contrast to a single plot in the case of $z = 0$ [c.f. Fig. 1(c)], here, for each radius there is a correponding plot of local OAM and intensity cross-section as functions of $\phi ^\prime $ [c.f. Fig. 1(e)].

It is thus understood that for $z \gt 0$, suitable radii ought to be chosen in order to observe azimuthal backflow utilizing the field in Eq. (2). As the purpose of our experiment is to demonstrate azimuthal backflow, we limit our experimental demonstration to the field in Eq. (1), i.e., at $z = 0$, wherein the local wave-vector has only an azimuthal component and this component in turn has no radial dependence.

 

Fig. 2. Schematic of the experimental setup. (a) POL, polarizer; SLM, spatial light modulator; MLA, micro-lens array; I, iris to spatially filter the first order of diffraction; ${L_1}$, ${L_2}$, ${L_3}$, and ${L_4}$ are lenses. The laser beam is polarized and expanded by a factor of eight by lenses ${L_1}$ ($f = {{50}}\;{\rm{mm}}$) and ${L_2}$ ($f = {{400}}\;{\rm{mm}}$) to cover the spatial extent of the SLM. Inset A shows a sample hologram to produce the desired superposition field in Eq. (1) with ${l_1} = - 1,{l_2} = - 3$, $b = 0.6$. This hologram is encoded on the plane of the SLM using the method described in [31]. The first diffraction order of the beam reflected from the SLM is spatially filtered by an iris (I) in the Fourier plane of the lens ${L_3}$ ($f = {{250}}\;{\rm{mm}}$). The filtered beam is Fourier transformed once again by the lens ${L_4}$ ($f = {{125}}\;{\rm{mm}}$) onto the microlens array (ThorLabs-MLA-150-5C-M), which is placed at the image plane of the SLM ($z = 0$). The micro lens array (each lens has a pitch of 150 µm and a focal length of 5.6 mm) focuses the light onto the CMOS camera (mvBlueFOX-200wG; pixel size 6 µm). Inset B shows the corresponding spotfield observed on the CMOS sensor. (b) On every spot in inset B, an arrow corresponding to the normalized direction of the total local wave-vector $\vec k/|\vec k|$ is displayed. The arrows are generated by combining the x and y displacements of the centroids of the spotfield in inset B relative to the reference. Due to imperfections in imaging and the finite size of the microlenses, the arrows contain both radial and azimuthal components. While the gray arrows point in the clock-wise direction in accordance with the negative values of ${l_1}$ and ${l_2}$, the yellow arrows, predominantly pointing in the counter-clockwise direction, indicate local regions in which backflow occurs.

Download Full Size | PDF

3. EXPERIMENT

The detailed setup of the experiment is presented in Fig. 2(a). The field in Eq. (1) is realized by using phase masks on a phase-only spatial light modulator (Holoeye Pluto 2.0 SLM), as shown in inset A of Fig. 2(a). A 780 nm continuous wave laser (Thorlabs CLD1015) is reflected off the SLM. Since we use a phase-only SLM to simultaneously modulate phase and amplitude, we adopt the technique discussed in [31], so that the desired field is generated after filtering the first diffraction order. The SLM is imaged using lenses ${L_3}$ and ${L_4}$ onto the microlens array (ThorLabs-MLA-150-5C-M) that focuses the beam onto the CMOS camera (mvBlueFOX-200wG). By definition, the image plane of the SLM refers to plane I ($z = 0$), as mentioned in the previous section. Inset B shows the spotfield generated on the CMOS when the mask in inset A is displayed on the SLM. Following the Shack–Hartmann sensor principle [32,33], a reference spotfield is generated by illuminating the microlens array with a wide Gaussian beam.

Then, the displacement of the centroids of the spotfield generated by the superposition field w.r.t. that of the reference are measured in the x and y directions. These are combined to find the directions of the local wave-vectors of the superposition, as plotted in Fig. 2(b) on top of each spot in the spotfield in inset B. In agreement with the theoretical two-dimensional illustrations in Figs. 1(b) and 1(d), the yellow arrows here in the dark fringes correspond to the regions of backflow. Due to imperfections in the imaging and the finite sizes of the microlenses used to sample the wave-vectors, the yellow arrows in the regions between the dark and bright fringes have radial components (and are not purely azimuthal). Hence, in to quantitatively analyze the azimuthal backflow, we generate one-dimensional plots of the local OAM [c.f. Fig. 1(c)] in Fig. 3.

The data points of the plots given in Fig. 3 are generated as follows. In the spotfields of the constituent beams or the superposition, the $i$th spot’s centroid on the reference spotfield is displaced by $\Delta {x_i}$ and $\Delta {y_i}$ in the x and y directions, respectively. The displacements in the cartesian coordinates are transformed to displacements in the polar coordinates $({r_i},{\phi _i})$. ${r_i}$ is found by calculating the distance between the spotfield’s global center of mass and the individual spot’s centroid. ${\phi _i}$ is given by the angle between the horizontal axis and the line joining the center of mass and the individual spot’s centroid. See the illustration in Fig. 3(a) for a schematic representation. Following this, $\Delta {x_i}$ and $\Delta {y_i}$ are combined to find the angular displacement of the spot–$\Delta {\phi _i} = - \sin {\phi _i}\Delta {x_i} + \cos {\phi _i}\Delta {y_i}$. To obtain the azimuthal component of the local wave-vector for the $i$-th displaced spot, the angular displacement is scaled using the focal length ${f_m}$ of each microlens–${k_{{\phi _i}}} = \frac{{2\pi}}{{\lambda {f_m}}}\Delta {\phi _i}$. The local OAM is then given by ${r_i}{k_{{\phi _i}}}$. The local OAM is plotted in Fig. 3 for each constituent beam (red and green scatter plots) and the superposition (blue scatter plot). The solid red, green, and blue are the corresponding theoretical predictions, and we find the experimental data to be in good agreement with the theory. Here, the constituent beams carry negative angular momenta; hence, all blue data points that correspond to positive values (above the black line) are indicative of azimuthal backflow.

The periodicity of the local OAM of the superposition depends on $|{l_1} - {l_2}| = \Delta l$. For higher $\Delta l = 3$ [Fig. 3(c)], the number of peaks become more frequent and are taller relative to the peaks in Fig. 3(b) (for which $\Delta l = 2$). Once $\Delta l$ is increased further, although the value of backflow increases substantially, its detection requires finer sampling, i.e., microlenses of smaller size [34,35].

4. DISCUSSION AND OUTLOOK

In this work, we have studied the phenomenon of azimuthal backflow both theoretically and experimentally, by utilizing the superposition of two beams carrying helical phases and having unequal amplitudes. We show explicitly that for two beams carrying negative OAM, the local OAM of their superposition is positive in certain spatial regions. As the angular spectra of the constituents beams are discrete, the backflow is directly certified from the measurement. This is advantageous compared to previous demonstrations [13,14], where the Fourier spectra of the constituent beams are infinite, and hence it is required to carefully certify backflow i.e., to ensure that the local linear momentum does not arise from the infinite tail of the Fourier spectrum [36]. This is because beams carrying well-defined OAM can be experimentally generated, while plane waves cannot.

 

Fig. 3. Experimental result demonstrating azimuthal backflow. (a) illustrates the method used to extract the local OAM. The center of mass (CM) of the reference spotfield is marked in yellow. Then polar coordinates of the $i$th spot $({r_i},{\phi _i})$ are found. For the $i$th spot in the reference (Ref), there is a corresponding displaced (Dis) spot (in spotfield of the constituent beams or the superposition) marked in red. $\Delta {\phi _i}$ is found by converting the displacements in cartesian coordinates $\Delta {x_i}$ and $\Delta {y_i}$ to displacements in polar coordinates. The local OAM is then given by ${r_i}\frac{{2\pi}}{{\lambda {f_m}}}\Delta {\phi _i} = {r_i}{k_{{\phi _i}}}$. ${f_m}$ is the focal length of each microlens. In (b) and (c), the scatter plots are data points and the solid curves are theoretical predictions. The red, green, and blue scatter plots of $r{k_{\phi ,1}}$ (beam 1), $r{k_{\phi ,2}}$ (beam 2), and $r{k_{\phi ,s}}$ (superposition), respectively, are in good agreement with their corresponding theoretical predictions. In these examples, the constituent beams carry negative angular momenta; hence, all blue data points that are positive correspond to azimuthal backflow. In (b) and (c), the ratio $|b| = 0.6$ is the same, while $\Delta l = 2$ (${l_1} = - 1$, ${l_2} = - 3$) and $\Delta l = 3$ in (${l_1} = - 1$, ${l_2} = - 4$), respectively. Note that in both (a) and (b), the minima of the blue scatter plot show slight deviations from the theoretical prediction. This is a systematic error owing to cross-talks between microlenses. See Supplement 1 Section 3 for a detailed analysis of statistical errors and insights on systematic errors. The statistical error in $r{k_{\phi ,s}}$ ranges from ${\pm}0.02$ to ${\pm}1.3$. Yet, the observation of azimuthal backflow is unaffected by these errors.

Download Full Size | PDF

It is worth noting that the azimuthal backflow in superpositions of LG/BG beams is hard to observe due to complex radial dependence (c.f. Supplement 1 Section 2). For the beams that we propose, even if the azimuthal component of the local wave-vector has a radial dependence (i.e., for $z \gt 0$), the azimuthal backflow can be observed and is relatively robust.

Recently, there has been a growing interest in the study of superoscillatory behavior in intensity of structured light [24,37]. A typical feature is the existence of sub-diffraction hotspots which can be used in super-resolution imaging [38,39]. In parallel we experimentally study the superoscillatory behavior of the phase. Azimuthal backflow is a consequence of strong phase gradients of fields over small spatial extents. Such gradients can affect the interactions of fields with atoms and molecules. For example, high phase gradients present in the superpositions we generate can be used to excite higher order multipole transitions in atoms. Such higher order transitions are possible, specifically, in cold atoms with wavefunctions that sample a large region in space [40]. These transitions are relevant in designing ultra-precise atomic clocks [41,42] or in generating atom probes for photons in low-light intensity regions [43]. A study of the interplay between the size of the wavefunction of an atom placed in a superposition of negative angular momentum states of light and the region of backflow (positive local OAM) in such states would be interesting. The electric field pattern investigated by us can also be employed in manipulating small particles in optical tweezers [44] or even to enhance chiral light–matter interactions [5,25].

It is important to note that in our study we use a scalar field description where the polarization of light is not of consequence. However, the research can be extended to vector fields [45–47] as well. Particularly, in various recent theoretical works [48–52] energy backflow has been predicted in vector fields.

From the fundamental point of view, superoscillations (or backflow) and weak measurements are both a manifestation of measurement values lying outside of the system’s spectrum. Our measurement of local OAM—regarded as the normalized OAM density [21,22]—can be interpreted, therefore, as a weak measurement of the angular momentum operator in quantum mechanics with postselection in the coordinate state [53]. Following the weak measurement approach, one can reconstruct photon trajectories similar to what has been examined experimentally [54] and theoretically [29,55]. Backflow results from wave interference [16,19], which means that the mathematics also applies to single particle wavefunctions [8], thus highlighting wave-particle duality intrinsic to quantum mechanics. Related to this, there is an ongoing debate on the interpretation of backflow [56–58]. An interesting open question is to which extent a study of the transverse two-dimensional spatial degree of freedom of a single photon can emulate the more robust two-dimensional quantum backflow analysed in [12]. The current work is a step towards observing quantum optical backflow [59].