1. INTRODUCTION

Structured (i.e., inhomogeneous) wave fields are ubiquitous in nature and modern wave-based devices. Such waves are attracting ever growing attention in optics [1] and other fields of wave physics [2]. For a proper understanding of structured waves and their potential applications, it is important to characterize their local dynamical and propagational properties. In this context, the momentum density, or energy flux density, or wave current becomes the chief quantity of interest. Ideally, this quantity should (i) be well defined theoretically, (ii) be measurable in experiment, (iii) determine the wave-induced force (momentum transfer) in local wave-matter interactions, and (iv) determine the propagational evolution of the wave field. However, as it often happens, instead of a single well-defined quantity there are several competing alternatives with different properties and specific advantages in different types of problems.

There is a vast literature on various aspects of electromagnetic momentum density, including numerous controversies and paradoxes [3–8]. Here we consider only one class of problems, namely, propagation of structured monochromatic light in free space. Most textbooks on optics and electromagnetism introduce the Poynting vector as the main and only quantity characterizing the momentum or energy-flux density. Typical pictures of the structure of an inhomogeneous optical field (e.g., focused [9], diffracted [10], or scattered [11]) show the electric-field intensity (energy-density) distribution and streamlines of the Poynting vector as if these were the fluid-like currents determining the propagational evolution of light.

More recently, an alternative canonical (also called “orbital”) momentum density was introduced for monochromatic structured optical fields [6–8]. Compared to the Poynting vector this quantity has some advantages and disadvantages. The main features of the Poynting momentum density are as follows:

In turn, the main features of the canonical momentum density are as follows:

Note that the last item is closely related to the fluid-like picture of propagation of light (or quantum-mechanical waves) within the Madelung-Bohm treatment of wave equations [20,21]. The streamlines of the canonical momentum current correspond to the experimentally measurable Bohmian trajectories of photons [19,22].

Importantly, the differences between the Poynting and canonical momenta are only local; their integral values for localized fields coincide. Yet, they coincide locally for linearly polarized (i.e., spinless) fields. Still, these local differences can be crucial and cause a number of controversies, such as the circular-polarization-dependent components of the Poynting vector orthogonal to the wavevectors in the problem [8,23–26].

The dilemma between the Poynting-like (sometimes called “kinetic”) and canonical momentum densities is not unique for electromagnetic waves; it equally appears in quantum-mechanical [27,28], acoustic [29,30], elastic [31–33], and water-wave [34] fields. (Incidentally, the acoustic analogue of the Poynting vector for waves in solids and fluids was derived by Umov [35] 10 years before the seminal Poynting work [36].) For acoustic and water waves the canonical momentum is also known as “pseudomomentum” [37–39], and it determines the Stokes drift of the medium particles [34,40,41]. Thus, in this case the canonical momentum is directly connected to the observable mechanical motion and momentum density of the medium. Therefore, thorough understanding of the Poynting versus canonical momentum properties of optical fields is also important for other fields of wave physics.

Despite the great progress in the description of the Poynting versus canonical momentum dilemma in the past decade, it still causes active debates in the structured-waves community. Furthermore, many scientists outside of this community are still unfamiliar with the problems and alternatives to the Poynting-vector description of structured light. Motivated by this, here we revisit the momentum density and propagational evolution of structured light using one of the most familiar examples: vortex beams [7,12,13,42]. It is now well established that direct optomechanical methods using small probe particles or atoms measure local optical forces proportional to the canonical momentum density (while the contribution proportional to the Poynting vector is much weaker) [8,25,26,43–45]. Moreover, “quantum weak measurements” of “Bohmian trajectories” of photons also correspond to the canonical momentum density [19,22]. In contrast to optomechanical and standard weak-measurement methods, here we analyze more traditional optical techniques based on (A) the Shack-Hartmann wavefront sensor and (B) propagational diffraction phenomena.

The earlier studies of vortex beams using the Shack-Hartmann wavefront sensor [46,47] and diffraction of truncated beams [48–51] dealt with linearly polarized paraxial beams and hence could not sense the difference between the Poynting and canonical momentum densities. That is why all these works mentioned the Poynting vector as the measured quantity. Here we revisit these approaches for the case of circularly polarized vortex beams and show that in fact both the Shack-Hartmann method and diffraction evolution reveal the canonical momentum density. Furthermore, we analyze the streamlines of the canonical momentum density (i.e., Bohmian trajectories) and their relation to the diffraction phenomena: despite the claim that “the rotation angle of the trajectory is unrelated to the Gouy phase” [52], we show that the mean rotation angle over all the trajectories exactly corresponds to the Gouy phase in the case of the Laguerre-Gaussian (LG) beams and always produces the same $\pi$ rotation in the $z \in (- \infty ,\infty)$ range for any vortex beams. Finally, we briefly consider the relation of the Poynting-canonical dilemma to “supermomentum” (i.e., anomalously high local wavevector) near the vortex core [6,53] and “backflow” (i.e., local retrograde propagation) [54,55]. These phenomena are currently attracting great attention [56–61] and represent examples of superoscillations [19,62]. We argue that these phenomena are determined by the canonical momentum, while the Poynting vector cannot produce supermomentum, although can exhibit “false backflow.”

2. CANONICAL AND POYNTING MOMENTUM DENSITIES IN VORTEX BEAMS

The time-averaged energy density, Poynting momentum density, and canonical momentum density for monochromatic paraxial optical fields in free space can be written as [6–8,13,63]

We now consider the well-known example of the LG vortex beams with circular polarization. Assuming propagation along the $z$ axis and the zero radial index, the transverse components of their electric field can be written as [7,12,42]

Substituting Eq. (5) into Eqs. (1) and (3), and neglecting quadratic contribution from the longitudinal field ${E_z}$, we find the canonical momentum density and energy density distributions:

Figure 1(a) shows the distributions of the transverse components of the canonical and Poynting momentum densities, as well as of the energy density, in the focal $z = 0$ plane of the LG beams with $\ell = 1$ and $\sigma = \pm 1$. The $\sigma$-dependent difference between the azimuthal momentum components, Eqs. (8) and (9), depends on the radius $r$ and generally is not small: $|{\Pi _\varphi} - {P_\varphi}| \sim |{P_\varphi}|$. Moreover, for $\ell \sigma \lt 0$ and radii $r \gt {w_0}\sqrt {|\ell |}$, the directions of the canonical and Poynting azimuthal momentum densities are opposite: ${\rm sgn}({P_\varphi}) = \def\LDeqbreak{}- {\rm sgn}({\Pi _\varphi})$. Thus, the choice of the canonical or Poynting momentum densities crucially determines the picture of the local propagational evolution of structured light.

 

Fig. 1. (a) Distributions of the transverse components of the canonical and Poynting momentum densities, Eqs. (8) and (9), as well as of the amplitude $|{\Psi _\ell}|$, in the focal $z = 0$ plane of the circularly-polarized LG beams, Eq. (5) with $\ell = 1$ and $\sigma = \pm 1$. The simultaneous sign flip $(\ell ,\sigma) \to (- \ell , - \sigma)$ reverses the azimuthal momentum components: ${P_\varphi} \to - {P_\varphi}$, ${\Pi _\varphi} \to - {\Pi _\varphi}$. (b) The streamlines $(r(z),\varphi (z))$ of the canonical and Poynting momentum densities, Eqs. (8) and (9), in the same LG beams propagating in the $z \in (0,3{z_R})$ range and (c) their projections onto the transverse $(x,y)$ plane. The streamlines with the same initial azimuthal positions ${\varphi _0} = \pi /2$ and different radial positions ${r_0}/{w_0}$ (indicated by colored numbers in the lower left panel) are shown.

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The streamlines of the momentum density, Eq. (8) or Eq. (9), are the curves whose tangent vectors constitute the corresponding momentum density field. These curves can be regarded as “Bohmian trajectories” of the light propagation [19,22,52,67], by analogy with the streamlines of the velocity field in fluid mechanics. Figures 1(b) and 1(c) show such streamlines $(r(z),\varphi (z))$ in the LG beams with $\ell = 1$ and $\sigma = \pm 1$ (see Appendix A for their analytical calculations). One can see that the canonical and Poynting streamlines differ dramatically from each other, except for the straight-line trajectory at the radial intensity maximum: $\partial W/\partial r = 0$, ${r_0} = {w_0}\sqrt {|\ell |/2}$. The difference between the canonical and Poynting momentum densities vanishes there, because, according to Eq. (4), it requires gradients of the spin density (proportional to the intensity for paraxial uniformly polarized beams). The rectilinear propagation of the intensity-maximum points in diffracting LG beams was also noticed in other studies [68,69]. Note also that in the $\ell \sigma \gt 0$ case, the interplay between the azimuthal and radial components of the Poynting vector results in globally rectilinear streamlines; see the middle panels in Figs. 1(b) and 1(c). The role of the momentum streamlines in the propagation and diffraction of vortex beams will be discussed in Section 4.

3. MEASUREMENTS OF THE TRANSVERSE MOMENTUM DENSITY BY SHACK-HARTMANN SENSOR

We performed experimental measurements of the radial distributions of the azimuthal component of the transverse momentum density in circularly polarized vortex beams using a Shack-Hartmann wavefront sensor. This is a direct purely optical method to measure the transverse components of the momentum density, without involving optomechanical effects or quantum weak measurements. It was previously applied to linearly polarized vortex beams [46], where there is no difference between the canonical and Poynting momentum densities, so that the results were associated with the Poynting vector. Here we show that such a technique in fact measures the canonical momentum density. We used LG beams with $\ell = \pm 1$ and $\sigma = \pm 1$ in the focal plane $z = 0$ where the radial component of the momentum density vanishes.

The experimental setup is shown in Fig. 2. The LG beams, Eq. (5), with $\ell = 0, \pm 1$ were produced using phase masks on a phase-only spatial light modulator (Holoeye Pluto 2.0 SLM), as shown in the inset A of Fig. 2. A continuous wave laser with the wavelength $\lambda = 780\;{\rm nm} $ (Thorlabs CLD1015) was expanded and reflected off the SLM. In order to simultaneously modulate phase and amplitude using a phase-only SLM, we adopted the technique described in [61], such that the desired field was obtained after filtering the first diffraction order [70]. Polarizer P1 was used to set the linear polarization of the incident beam. Then, the light reflected from the SLM was circularly polarized using a quarter wave plate (QWP) oriented at 45° with respect to P1. Polarizer P2 was used to determine the orientations of the fast and slow axes of the QWP, and it was removed after that. Also, to determine the circular-polarization helicity $\sigma$, a $Q$-plate (QP) of the order $q = 1/2$ (Thorlabs-WPV10L-780) was employed [71] and removed afterwards. This process allowed us to create LG beams with desired circular polarization.

 

Fig. 2. Schematic of the experimental setup consisting of polarizers P1 and P2; spatial light modulator (SLM); iris (I); quarter wave plate (QWP); Q-plate (QP); micro-lens array (MLA); lenses L1 ($f = 50\;{\rm mm} $), L2 ($f = 500\;{\rm mm} $), L3 ($f = 250\;{\rm mm} $), and L4 ($f = 150\;{\rm mm} $); and complementary metal-oxide semiconductor sensor (CMOS). The Shack-Hartmann wavefront sensor (SHWFS) consists of the MLA and the CMOS. Inset A shows a sample hologram to produce the desired LG beam, Eq. (5) with $|\ell | = 1$. Inset B shows the corresponding spotfield observed on the CMOS sensor.

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Fig. 3. Experimental (symbols) versus theoretical (curves) results for the radial distributions of the intensity and azimuthal component of the momentum density in LG beams with $\ell = 0, \pm 1$ and circular polarization $\sigma = \pm 1$ in the focal $z = 0$ plane (plotted in arbitrary units). The experimental results are obtained using the Shack-Hartmann wavefront sensor, as explained in the text and Fig. 2. These are consistent with the $\sigma$-independent canonical momentum density, Eqs. (5) and (8), rather than with the $\sigma$-dependent Poynting vector, Eq. (9) (see also Fig. 1).

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Next, the SLM was imaged using lenses L3 and L4 onto the microlens array (ThorLabs-MLA-150-5C-M; each lens has a pitch of 150 µm and a focal length of 5.6 mm) that focused the beam onto the CMOS camera (mvBlueFOX-200wG, pixel size 6 µm). The inset B in Fig. 2 shows the spotfield generated on the CMOS when the mask shown in the inset A was displayed on the SLM. (The spotfield is a 2D-grid-like arrangement of focal spots generated by a Shack-Hartmann wavefront sensor. This pattern is formed as each lens of the microlens array focuses the corresponding small area of the incoming wavefront onto the corresponding region on the CMOS sensor.) In accordance with the principle of the Shack-Hartmann sensor [72], a reference spotfield is generated by creating the hologram corresponding to a wide Gaussian beam on the SLM and subsequently imaging the light reflected from the SLM plane onto the microlens array. While generating the spotfield of the reference (a wide Gaussian beam), the polarizations of the light incident on and reflected from the SLM were once again set according to the aforementioned procedure.

The data analysis was performed as follows. The displacements of the centroids of each spot in the spotfield generated by the LG beam were measured with respect to the corresponding spots in the spotfield generated by the reference. The Cartesian coordinates of these displacements were used to obtain the azimuthal component of the displacement [61]. Then, the azimuthal displacement of the $i$th spot centroid, $\Delta {\varphi _i}$, is divided by the focal length ${f_m}$ of each microlens and multiplied by the corresponding intensity ${I_i}$ of the spot. The resulting quantity $P_{\varphi i}^{{\exp}} = {I_i}\Delta {\varphi _i}/{f_m}$ provides the experimentally measured azimuthal component of the momentum density, which is in agreement with the canonical momentum, Eq. (8), ${P_\varphi} \propto I\ell /kr$ (see Fig. 3). We collected several frames of the same spotfield image. For each such frame, the measured intensity and canonical momentum density are grouped into bins corresponding to segments of radii ${r_j}$. The average over all the frames of intensity ${\langle I\rangle _j}$ and canonical momentum density ${\langle P_\varphi ^{{\exp}}\rangle _j}$ are obtained for each ${r_j}$.

The results of the measurements are shown in Fig. 3. The grey symbols correspond to the measured intensity ${\langle I\rangle _j}$ as a function of the radius $r$, while the solid gray curves correspond to the theoretical dependences $|{\Psi _\ell}(r{)|^2}$, Eq. (5), optimally fitted to the experimental measurements. The red and blue curves correspond to the theoretical distributions of the azimuthal components of the canonical momentum density ${P_\varphi}(r)$ and Poynting momentum ${\Pi _\varphi}(r)$, Eqs. (8) and (9), respectively. Evidently, the measured azimuthal momentum densities (red symbols) are consistent with the canonical rather than Poynting momentum. While the azimuthal canonical momentum is independent of the polarization helicity $\sigma$ and does not change its sign across the beam radius, the azimuthal component of the Poynting vector depends on $\sigma$ and alters its sign with $r$. Our experimental results also confirm recent theoretical arguments [73] that the Shack-Hartmann wavefront sensor measures the weak value of the canonical momentum operator, i.e., the canonical momentum density.

 

Fig. 4. (a) Experimental pictures from [48] of the transverse intensity distributions of an LG beam, Eq. (5) with $\ell = 2$, diffracted at the knife-edge aperture (located in the focal $z = 0$ plane) at different distances $z$. Reprinted by permission of Taylor & Francis Ltd from J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” Journal of Modern Optics (2003) [48]. (b) Numerical calculations of this knife-edge diffraction for the LG beam with $\ell = 2$ and $k{w_0} = 20$. (c) The integral angle of rotation from the $z = 0$ plane, given by the Gouy phase ${\Phi _G} = \arctan (z/{z_R})$, Eq. (12).

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We finally note that for the experimental data shown in Fig. 3, the statistical error in the measured azimuthal momentum density (red points) ranges from $5 \times {10^{- 3}}$ to $6 \times {10^{- 2}}$ across radius $r$. The maximum statistical error is typically observed near the vortex core as the intensities of the detected spots are low in these regions, resulting in imprecise detection of the corresponding spot centroids. It can also be observed that these data points for vortex beams ($\ell \ne 0$) exhibit some systematic errors around $r/{w_0} = 0.5$ and deviate from the theoretical red curves. These radii correspond to high intensity gradients (as seen from the gray curves), where systematic errors may appear due to cross-talks between neighboring microlenses (see supplementary section of [61] for more details).

4. PROPAGATIONAL DYNAMICS OF TRUNCATED VORTEX BEAMS

Another method to reveal the local propagational evolution of vortex beams is to consider diffraction of such beams after a suitable (e.g., knife-edge) aperture. This method has been applied in optics [48–51] and later for electron vortex beams [74,75]. However, the optical case was considered for linearly polarized beams, and the observed evolution was attributed to the Poynting vector. We argue that in fact this evolution is determined by the canonical momentum density.

We first recall the known facts. A vortex beam truncated by a knife edge or a similar aperture, usually placed in the focal plane, experiences an azimuthal rotation upon the $z$-propagation, as shown in Fig. 4. The direction of this rotation is determined by ${\rm sgn}(\ell)$ and its magnitude for LG beams is given by the Gouy phase ${\Phi _G}(z)$ [48–50,74]. The latter fact hints at the connection between the azimuthal rotation and diffraction, although similar rotations have also been observed for non-diffracting Bessel beams [48]. However, despite the general association with the Poynting vector [48,49], an accurate quantitative relation between this rotational evolution and the momentum density in vortex beams remains somewhat elusive. Even for linearly polarized beams, Berry and McDonald remarked that the rotation of the streamlines (trajectories) determined by the momentum density in LG beams “is unrelated to the Gouy phase” [52].

A. Laguerre-Gaussian Beams

To characterize the rotational evolution of truncated vortex beams within the optical-current paradigm, we employ the method used in [75] for electron vortex beams. Namely, assuming that the local velocity of the propagation in a structured optical field is given by ${\textbf v} = {c^2}{\textbf P}/W$ [19,76], we introduce the local angular velocity of the beam rotation with respect to the $z$-coordinate (rather than time):

Equations (10)–(12) establish the direct relation between the azimuthal momentum density and the Gouy-phase rotation of truncated LG beams. It appears in the integral expectation values rather than in the local momentum density (i.e., individual Bohmian trajectories or weak values). Remarkably, if we use the Poynting momentum density ${\Pi _\varphi}$, Eq. (9), instead of the canonical momentum ${P_\varphi}$, Eq. (8),

Thus, the global rotation cannot discriminate between the Poynting and canonical momentum densities. To do this, one has to measure the local $r$-dependent rotations in vortex beams. In [48], following [67], this was done by involving the concept of the “radial maximum trajectory” $r = {r_{{\max}}}(z)$ based on the $z$-evolution of the radial maximum of the intensity distribution and by measuring rotations of the intensity rings with different ${r_{{\max}}}$ in beams with multiple radial maxima (e.g., Bessel beams). However, this concept of a single radial trajectory is elusive (the momentum density streamlines generate a continuum of trajectories [52], Fig. 1), and we would like to deal with the continuous momentum density field.

B. Bessel and Other Vortex Beams

Note that Expressions (8) and (10) for the $\ell$-dependent azimuthal component of the canonical momentum and the corresponding local angular velocity are universal for vortex beams with the azimuthal dependence $\propto \exp (i\ell \varphi)$. Consider, for example, paraxial non-diffracting Bessel beams [78]:

In contrast, the azimuthal component of the Poynting vector differs considerably for different types of vortex beams. For paraxial Bessel beams, we have ${E_z} \propto - i\sigma (\kappa /\sqrt 2 k){J_{\ell + \sigma}}(\kappa r)\def\LDeqbreak{}{e^{i(\ell + \sigma)\varphi}}$ [66], which results in the Poynting momentum [79]:

Figure 5 shows the transverse distributions of the canonical and Poynting momentum and energy densities, as well as the streamlines of the canonical and Poynting currents for circularly polarized Bessel beams with $\ell = 1$ and $\sigma = \pm 1$. All the streamlines are spirals in this case because of the absence of the radial momentum components (i.e., the absence of diffraction); see Appendix A. Similar to the LG case, the canonical and Poynting streamlines coincide at the radial intensity maxima, i.e., for $d{J_\ell}(\kappa r)/dr = 0$. Furthermore, akin to the LG-beam case, the azimuthal component of the Poynting current has $r$-regions with positive and negative ${\Pi _\varphi}$. Thus, if the Poynting momentum density were to determine the $z$-evolution of truncated vortex beams, we could observe local negative rotations in certain $r$-ranges. Moreover, such rotations would strongly depend on the circular polarization $\sigma$. Unfortunately, the experiments [48–50] were performed for linear polarizations, but it is well known that the Kirchhoff-Fresnel integral determining diffraction of paraxial fields [81] does not depend on the polarization. Therefore, we can conclude that the local ($r$-dependent) diffraction evolution of truncated vortex beams cannot be described by the Poynting momentum density.

 

Fig. 5. The same as in Fig. 1 but for paraxial Bessel beams, Eqs. (15)–(17). The colored numbers in the lower left panel indicate the values of $\kappa {r_0}$.

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Although the local $r$-dependent angular velocity, Eq. (10), determined by the canonical momentum is independent of the form of the vortex beam ${\Psi _\ell}$, the resulting $z$-evolution can depend on it. Hence, we can write the local angle of the truncated vortex rotation as

 

Fig. 6. (a) Experimental pictures from [48] of the transverse intensity distributions of a Bessel beam, Eq. (15) with $\ell = 2$, diffracted at the knife-edge aperture (located in the $z = 0$ plane) at different distances $z$. Theoretical fits of the knife-edge profile distorted by the $r$-dependent azimuthal rotation, Eq. (18), are shown in red. Reprinted by permission of Taylor & Francis Ltd from J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” Journal of Modern Optics (2003) [48]. (b) Numerical calculations of the similar knife-edge diffraction of the Bessel-Gauss beam, Eq. (19) with $\ell = 1$, $\kappa /k = 0.2$, and $k{w_0} = 150$. The characteristic $z$-scale of the rotation of the inner maximum is ${z_0} = 2k/{\kappa ^2}$. (c) Numerically calculated integral angle of rotation from the $z = 0$ plane, Eq. (20), in the Bessel-Gauss beam, Eq. (19) with $\ell = 1$, $\kappa /k = 0.2$, and $k{w_0} = 100$. The characteristic $z$-scale of this integral rotation is ${z_1} = k{w_0}/2\kappa \gg {z_0}$. Although the $z$-dependence of this rotation differs from the Gouy-phase $\arctan$ function [for reference, an example of such function, $\arctan (2z/{z_1})$, is plotted by a dashed line], it asymptotically approaches the same $\pi /2$ limit (dotted line), in agreement with the universal Eq. (21).

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The theoretical model of [48], based on the “radial maximum trajectory”, used Expression (18) with $F(z) = {w^2}(z){\Phi _G}(z)/2$ involving the width and Gouy phase for the LG beam, Eq. (5). Apparently, for Bessel or other beams, this function should have a different form. Since Bessel beams are non-normalizable (the integral $\int_0^\infty J_\ell ^2(\kappa r) r {\rm d}r$ diverges), let us now consider normalizable and diffracting Bessel-Gauss beams [82]:

To understand this result we calculate the integral (over all rings) rotation angle for the Bessel-Gauss beam, Eq. (19), similar to Eqs. (10)–(12):

We have found analytically that the total integral rotation for the whole range $z \in (- \infty ,\infty)$ equals

5. ON THE “SUPERMOMENTUM” AND “BACKFLOW”

Here we briefly discuss the canonical and Poynting currents in the context of the “supermomentum” [6,53] and “backflow” [54,55]. Both phenomena are currently attracting significant attention [56–61] and represent examples of superoscillations [62]. Namely, if the wavefield consists of plane waves (Fourier spectrum) with some component of the wavevectors limited by the range ${k_i} \in (0,k)$, then the spatial regions where the local phase gradient (i.e., the local wavevector or normalized momentum density $\omega {P_i}/W$) is larger than $k$ or less than zero correspond to the supermomentum and backflow, respectively [19]. In other terms, these conditions correspond to the superluminal or negative local velocity ${v_i} = {c^2}{P_i}/W$: ${v_i} \gt c$ and ${v_i} \lt 0$. In the context of vortex beams, we consider these phenomena for the azimuthal momentum component, or angular momentum.

 

Fig. 7. Radial dependences of the normalized azimuthal components of the canonical and Poynting momentum densities in circularly polarized Bessel beams with $\kappa /k = 0.2$, $\ell = 1,2,3$, and $\sigma = \pm 1$. Exact expressions are given by Eqs. (25)–(27), while paraxial approximation, shown by dashed curves, corresponds to $a \simeq 1$ and $b \simeq 0$. The areas of the canonical supermomentum $c|{P_\varphi}|/W \gt 1$ are highlighted by pink. The gray profiles indicate the energy-density distributions $W(r)$. The simultaneous sign flip $(\ell ,\sigma) \to (- \ell , - \sigma)$ reverses the azimuthal momentum components: ${P_\varphi} \to - {P_\varphi}$, ${\Pi _\varphi} \to - {\Pi _\varphi}$.

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A. Supermomentum Near the Vortex Core

For the azimuthal component of the canonical momentum density, Eq. (8), the supermomentum occurs in the subwavelength vicinity of the vortex center:

However, the azimuthal component of the Poynting vector in Eqs. (9) and (17) for ${\rm sgn}(\sigma \ell) = - 1$ exhibits the same $1/r$ divergence (when divided by the intensity) as the canonical momentum and hence contradicts the general condition, Eq. (23). This means that our paraxial-approximation equations become inapplicable near the vortex center, and the second-order terms become important in this region. Indeed, near the vortex core (e.g., for $r \ll {w_0}$ in LG beams), $\partial {\Psi _\ell}/\partial r \simeq (|\ell |/r){\Psi _\ell}$, and the longitudinal field component is estimated as $|{E_z}| \sim (|\ell |/kr)|{\Psi _\ell}|$. Then, its contribution becomes of the order of the main field, $|{E_z}| \sim |{\Psi _\ell}|$, for $r \lesssim |\ell |{k^{- 1}}$, i.e., exactly in the supermomentum area, Eq. (22).

Importantly, this longitudinal-field contribution includes spin-orbit interaction effects [65,66], and all the beam properties become strongly $\sigma$-dependent in the supermomentum area. To take these effects into account, we calculate the azimuthal components of the canonical and Poynting momentum, and also the energy density, for circularly polarized Bessel beams without using paraxial approximation. In doing so, we use the exact Bessel-beam solutions described in [66]. These are helicity eigenmodes, i.e., represent a superposition of plane waves with different wave vectors and the same circular polarization (helicity) $\sigma = \pm 1$. Owing to this, the magnetic field exactly satisfies Eq. (7): ${\textbf H} = - i\sigma {\textbf E}$, whereas the Cartesian components of the electric field can be written as

Figure 7 shows the normalized values $c{P_\varphi}/W$ and $c{\Pi _\varphi}/W$ as functions of $r$, both exact and in the paraxial approximation, for Bessel beams with $\ell = 1,2,3$ and $\sigma = \pm 1$. One can see a drastic difference between the paraxial and exact curves for ${\rm sgn}(\sigma \ell) = - 1$. In agreement with the general Relation (23), the exact Poynting vector never corresponds to supermomentum. Furthermore, the exact canonical momentum does not produce supermomentum in the case of ${\rm sgn}(\sigma \ell) = - 1$ and $|\ell | \le 2$. Under these conditions $W(0) \ne 0$, and, hence, the azimuthal supermomentum can be achieved only in the case when the energy density (including all the field components) exactly vanishes in the vortex center. One can say that it is the vanishing energy density, rather than the anomalous momentum density, that provides for the supermomentum.

B. Azimuthal Backflow

Although, the backflow is usually considered for Cartesian components of the wave current, recently it was shown that it can be equally considered for the local azimuthal phase gradient [61]. In this case, one has to consider the expansion of the wavefield in the orbital-angular-momentum (vortex) eigenmodes instead of plane waves. In particular, if the field represents a superposition of the modes with $\ell \gt 0$, then the negative local azimuthal phase gradient corresponds to the azimuthal backflow.

In this context, we note that the azimuthal component of the Poynting vector in a circularly polarized vortex beam with $\ell \gt 0$ and $\sigma = - 1$, Eqs. (9) and (17), becomes negative in certain $r$-regions; see Figs. 1 and 5. However, one should not interpret it as an azimuthal backflow [60]. Such vortex beams are orbital-angular-momentum eigenmodes with fixed eigenvalues $\ell$ (in the paraxial regime one can consider the orbital and spin parts of the angular momentum independently [12,13,65]), while superoscillations are interference phenomena that appear only in superpositions of several momentum or angular-momentum eigenmodes [62]. Moreover, as we have shown, the Poynting momentum density does not correspond to the local phase gradient, or wavevector, or propagation direction of light, and the actual optical current corresponds to the canonical momentum density. Its azimuthal component is always positive in a vortex beam with fixed $\ell \gt 0$, and the interference of at least two vortex beams is needed to achieve the true azimuthal backflow [61].

6. CONCLUSIONS

In this work, we aimed to overview the roles of the canonical and Poynting momentum densities (currents) in the propagation and diffraction of structured monochromatic optical beams. In doing so, we considered vortex beams of different kinds. While the Poynting vector is determined by the well-known vector product of the electric and magnetic fields, the canonical momentum density is associated with the wave phase gradient (local wavevector) averaged over all the field components and multiplied by the wave intensity. In this manner, the canonical momentum provides a direct link to the quantum-mechanical de Broglie momentum and probability current. It is well established in other works that local optomechanical momentum effects (i.e., optical forces on small particles or momentum exchange with atoms) are determined by the canonical momentum density rather than the Poynting vector [8,13,14,17,18]. Furthermore, quantum weak measurements of the local momentum also result in the canonical momentum density [6,19,22]. In this work, we have considered the canonical-Poynting dilemma in the context of purely optical propagation and diffraction phenomena.

In the paraxial regime, assuming uniformly polarized light, the canonical current is independent of the polarization, while the Poynting vector depends on the degree of circular polarization, i.e., spin. Previous optical approaches to the optical currents in the propagation and diffraction of structured light [46,48,50,52,67] associated it with the Poynting vector. However, these works dealt with linearly polarized (zero-spin) beams where the canonical and Poynting currents coincide with each other. We have revisited these approaches using circularly polarized beams and found that it is the canonical momentum density rather than the Poynting vector that should be associated with the observable phenomena in the propagation of light.

First, we have performed experimental measurements of the transverse momentum density in vortex LG beams via a Shack-Hartmann wavefront sensor. This yielded polarization-independent results consistent with the canonical current rather than the Poynting vector. Second, we have analyzed diffraction of different kinds of vortex beams at a knife-edge aperture placed in the focal plane. The truncated beams experience a $z$-dependent rotation that depends on the sign of the vortex charge $\ell$ and can be associated with the Gouy phase. We have shown that this $z$-dependent rotation can be explained by the optical current properly integrated over the transverse cross-section of the beam. Notably, this integral current is the same in the canonical and Poynting versions; only the local $r$-dependent rotation can discriminate between these. However, since the paraxial diffraction is polarization-independent, we can conclude that it must be locally defined by the canonical rather than Poynting momentum density. Remarkably, we have derived a theoretical relation for the total integral rotation over the whole $z \in (- \infty ,\infty)$ range in a circularly symmetric vortex beam with an arbitrary $r$-profile. The angle of this rotation always equals $\pi {\rm sgn}(\ell)$, which provides a global relation between the optical currents, Gouy phase, and the geometrical-optics ray picture. While optical currents describe local propagation properties of a structured wavefield, geometrical optics rays connect far-field regions in the field evolution.

Finally, we have discussed the “supermomentum” [6,19,53,56,57] and “backflow” [19,54,55,58–61] effects in the canonical and Poynting currents in vortex beams. The former phenomenon means anomalously high momentum density normalized by the energy density, and it appears in a subwavelength area near the vortex core. We have shown that it is only possible for the canonical momentum density and strongly depends on the nonparaxial spin-dependent corrections. In this regime, the energy and canonical momentum densities become polarization-dependent. The latter phenomenon means anomalously negative local optical current, and it should be also associated with the canonical momentum density in interfering modes. While the Poynting vector can have negative azimuthal components in vortex beams with $\ell \gt 0$, it should not be associated with the backflow, because the actual propagation is determined by the canonical momentum density. Interference of at least two vortex modes with different $\ell$ is required for the proper canonical azimuthal backflow [61].

We hope that this work sheds light on important peculiarities of optical propagation and the relevant physical characteristics for their description. The canonical momentum density provides a useful theoretical tool for understanding the free-space evolution of monochromatic light fields. It would be important to further explore and extend this framework beyond its current limitations: e.g., for polychromatic optical fields and for structured light in anisotropic media.