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Abstract
It is well known that optics and classical mechanics are intimately related. One of the most important concepts in classical mechanics is that of a particle in a central potential that leads to the Newtonian description of the planetary dynamics. Within this, a relevant result is Kepler’s second law that is related to the conservation of orbital angular momentum, one of the fundamental laws in physics. In this paper, we demonstrate that it is possible to find the conditions that allow us to state Kepler’s second law for optical beams with orbital angular momentum by analyzing the streamlines of their energy flow. We find that the optical Kepler’s law is satisfied only for cylindrical symmetric beams in contrast to the classical mechanics situation that is satisfied for the other conic geometries, namely, parabolic, elliptical and hyperbolic. We propose a novel approach to confirm our analytic results: we observe the propagation of the Arago’s spot created by a beam with orbital angular momentum as a local “light-tracer” instead of looking at the propagation of the whole beam. The observed patterns fully agree with the prediction of our formalism.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
During the early 17th century, Johannes Kepler published three laws describing the planetary dynamics. One of them, particularly the second law, states that the line that joins a planet and the sun sweeps out equal areas in equal time intervals. In other words, the areal velocity of a planet is a constant of motion. Kepler’s second law is intimately related to the conservation of the total angular momentum of the system. Moreover, the latter is guaranteed for spherically symmetric systems. Therefore, Kepler’s second law is not restricted to planetary motion; any particle subject to a central force must satisfy this law [1].
In the early nineties, it was demonstrated that optical Laguerre-Gauss beams (LGB) can carry orbital angular momentum (OAM), since they have a rotating periodic phase dependence [2]. High order Bessel beams have also this property, making them subject to carry a definite amount of OAM. These beams possess an advantage over LGB, within a defined volume they do not present changes in their transverse intensity distribution, neither as they propagate, simplifying the investigations of the properties of optical OAM [3–5]. Particularly, since the phase of Bessel beams does not depend on the radial coordinate during propagation, the Poynting vector follows a helix trajectory of constant radius [6–8]. Therefore, the energy flow shows rotational symmetry, suggesting a conserved quantity. However, there is a drawback with any of these beams, it is not straightforward to observe their rotating features as their intensity distributions are also rotational symmetric.
Structured beams, like Laguerre-Gauss and Bessel beams, have an apparent intriguing property; when they are partially obstructed, they may recover their initial intensity distribution after some propagation distance. This property is known as self-healing, and has been used particularly to follow the transverse displacement of the energy circulation of a beam when it is obstructed off-axis. This has allowed to observe that the intensity of a beam carrying OAM can follow curved trajectories in the transverse plane [9–14].
An alternative way to visualize the rotation of the energy in a beam carrying OAM, is by interference with a reference wave, and observe the longitudinal propagation of the interference pattern [8]. For instance, it has been shown that the superposition of a Laguerre-Gauss and a Gaussian beam produces a transverse intensity pattern that rotates as it propagates. Considering the rotating intensity pattern, it can be tempting to consider that the Laguerre-Gauss beams are analogous to a rigid rotating body in Classical Mechanics [15].
However, it has been shown that such consideration is not quite correct as the beam, instead of behaving as a rigid body, needs to be considered as formed by radial layers rotating at different angular velocities [15]. The problem with this consideration lies in disregarding the local dynamics of the beam’s energy but integrates the beam’s intensity over the whole space.
A suitable way to study the dynamics involving the rotation of an optical beam is through its local circulation of energy. In the general case, the total energy flow of a light field comprehends both the contribution of extrinsic and intrinsic energy flows, the former associated to the transverse displacement of the centroid and to the total angular momentum of the light beam [16]. As mentioned above, the local rather than the global properties of the circulation of energy provides an insightful picture on the rotation properties of optical beams. On the other hand, the intrinsic energy flow is constituted by both the contributions due to the polarization of the field, known as spin contribution, and to the variations on the amplitude and phase distributions of the field, known as orbital contribution [16]. The former does not depend on the spatial distribution of the beam; it vanishes for linearly polarized beams. For this reason, the internal energy flow is characterized only by the orbital contribution, which is described in terms of the trajectories everywhere tangent to the Poynting vector, better known as optical current streamlines [17].
In this paper, by studying the optical current streamlines of an optical beam carrying OAM, we analyze the local energy flow to investigate and establish, for the first time to our knowledge, Kepler’s second law for optical beams. We find the conditions under which Kepler’s second law is satisfied for optical beams. Particularly, we show that the current streamlines of Bessel beams, as well as their associated Hankel waves, satisfy Kepler’s law, whereas the Transverse-parabolic beams do not satisfy it. This is in contrast to what occurs for conic trajectories of the gravitational potential case in Classical Mechanics. In order to observe the optical current streamlines, we came with the idea of partially obstructing the propagation of the corresponding optical beams, to create and use the Arago’s spot as a “light-particle” tracer that allows a numerical verification of our theoretical predictions. Our results demonstrate that Kepler’s second law is satisfied for optical beams with a definite amount of OAM.
2. Kepler’s second law in classical mechanics
Consider a single particle of mass $\mu$ subject to a conservative central force due to a potential $V(r)$, that only depends on the radial distance $r$. The force is considered to be emanating from the origin of the coordinate system. Since the potential function only depends on the radial distance, the system is spherically symmetric, that is, it is invariant under any rotation transformation. This has strong implications; the Lagrangian describing this system does not depend on any angle variable and the total angular momentum of the particle at the position $\mathbf {r}$ with momentum $\mathbf {p}$, expressed as
is a conserved quantity [1]. Since $d\mathbf {L}/dt=0$, $\mathbf {r}$ is always orthogonal to the constant vector $\mathbf {L}$, that is, the particle is confined to move on a plane whose normal is parallel to $\mathbf {L}$. If it was the case that $\mathbf {L}=0$, the cross product of the vector position and its derivative must be zero, that is, $\mathbf {r}$ must be parallel to $\mathbf {p}$, and the particle follows a straight line in the same plane orthogonal to $\mathbf {L}$. In any case, a particle subject to a central potential is confined to move on a plane. Then, in spherical polar coordinates, the polar/azimuthal axis can be chosen to be in the direction of $\mathbf {L}$, and the Lagrangian $\mathcal {L}$ of the system can be written in polar coordinates $(r, \varphi )$ as Notice that the angle variable $\varphi$ is not present in $\mathcal {L}$, only its derivative, implying that the corresponding canonical momentum, is a conserved quantity. Besides, it is precisely the magnitude of the angular momentum of Eq. (1), $L_{CM}=|\mathbf {L}|$. Indeed, the Lagrange equation for the angle variable $\varphi$ can be written as3. Kepler’s second law in optics
In the optical regime, instead of studying the trajectories of particles subject to a central potential, we will analyze the streamlines of energy flow of optical beams. In what follows, we will establish a connection between the Classical Mechanics and Optics systems.
3.1 Optical currents
The Poynting vector describes the direction and magnitude of the electromagnetic energy flux. Locally, this flux follows trajectories, or streamlines, that are tangent to the Poynting vector at every point of space, and are known as optical current streamlines [18]. Since the Poynting vector is normal to the wavefronts of a light beam, the optical current streamlines are also normal to them. In most cases, as it is shown in Fig. 1, the wavefront changes its form as the beam propagates in free space, causing the optical current streamlines to bend, unlike the case of geometric rays associated to light propagating in free space, which are always straight lines. Since on one side, in Classical Mechanics the Angular Momentum is associated to Kepler’s second law [1], and on the other the OAM of a beam is closely related to the optical currents, we will follow the approach developed by M. V. Berry in Refs. [17,18] to get the Optical Kepler’s law.
Fig. 1. Schematic representation of the wavefronts, Poynting vector and current streamlines. For visualization purposes, we show an helicoidal wavefront and a generic streamline.
Consider a beam described by a scalar wave $\psi (\mathbf {r})$. The corresponding Poynting vector is everywhere tangent to the optical current $\mathbf {P}(\mathbf {r})=|\psi (\mathbf {r})|^2\nabla \arg (\psi (\mathbf {r}))$. From this definition, we notice that the local wavevector is
In order to establish a connection with the central force problem in Classical Mechanics, in which a particle subject to a central force is confined to move on a plane, we propose, instead of studying the propagation of the streamlines in the three-dimensional space, to analyze the evolution of the projection of the streamlines onto the transverse plane of the beam. That is, we will study, in that plane, the two-dimensional streamlines given by $(r(z)\cos {\varphi (z)}, r(z)\sin {\varphi (z)})$, generated by the local transverse wavevector
3.2 Symmetry of orbital angular momentum in optics and in classical mechanics
In this section, we state the Optical analogous to Kepler’s second law in Classical Mechanics, Eqs. (5)–(6), and find the conditions under which it is satisfied. Also, we are going to show that there is a symmetry between the propagation of optical Bessel beams, as well as Hankel waves, and the dynamics of a particle moving in a central potential in Classical Mechanics described in Section 2.
Recalling that $z$ is the evolution variable, and that the element of area in polar coordinates has the form $dA=\frac {1}{2}r^2d\varphi$, the areal velocity of the optical current of paraxial light beams, in cylindrical coordinates, can be written as
where Eq. (10) has been used. If the areal velocity is to be constant, namely, $C_{0}$, it can be obtained that This equation can be rewritten, using the right equation in Eq. (11), as Thus, Kepler’s second law has been proved for optical beams; the azimuthal variation of the phase must be constant. For azimuthally periodic beams, the constant can be the integer $m$. In other words, the phase has a linear dependence of the type $m\varphi$. Laguerre-Gauss and Bessel-Gauss beams have this property and then must satisfy the Optical Second Kepler’s law. Interestingly, although second Kepler’s law in Classical Mechanics applies to particles, and to rigid bodies as well, its optical version applies for light beams that cannot be considered as a rigid body. The key in this analogy is to observe that a point following a current streamline, describes a trajectory with constant areal velocity, just as the center of mass of a rigid body subject to a central force, or as a planet orbiting the Sun.Now, we illustrate our results through the study of the areal velocity of the optical current of Bessel beams and their associated Hankel waves.
Bessel beams arise as solutions, in cylindrical coordinates, to the Helmholtz wave equation $(\rho ,\varphi ,z)$. Solving it using separation of variables, leads to the Bessel equation for the radial coordinate, yielding the solutions to Helmholtz equation in the following way [3,4]
where $k_r$ and $k_z$ the transverse and longitudinal components of the wave vector. It has been demonstrated that the above representation of Bessel beams is rather incomplete and that, in fact, those beams are the result of the linear superposition of two conical waves, one propagating as an outgoing conical wave and one that does it as an incoming conical wave [5,12] In the above equations, $\psi _m^{+}(\bullet )$ represents the outgoing conical wave, and $\psi _m^{-}(\bullet )$ the incoming conical wave. In these waves, the sums in square brakets are the Hankel functions $H^{(1)}_m(\bullet )$ and $H^{(2)}_m(\bullet )$, respectively, and for this reason Eqs. (16) and (17) are referred to as conical Hankel waves. The singularity present in both Hankel functions at $r= 0$, can be interpreted in the following way: since the incoming and outgoing cylindrical waves, described by the Hankel functions, converge at and diverge from the propagation axis simultaneously, the axis acts as a sink and as a source of both waves [5]. This is also the case for the physical description of formation of Bessel modes in cylindrical waveguides and optical fibers [19]. Hence, Bessel beams can only exist as nondiffracting beams, in the region of space in which both conical waves interfere [5,19]. Furthermore, when a Bessel beam is partially obstructed during propagation, the dynamics of the Hankel waves is disclosed; the outgoing and incoming conical waves propagate producing two shadows, one that moves outwards and another one moving towards the center of the beam, after which the incoming conical wave becomes into an outgoing conical wave [11].Now we will study the current streamlines of paraxial Bessel beams. Since the radial part of the Bessel beam is the superposition of counter-propagating waves with the same frequency, the transverse amplitude is a real stationary wave. Then, the phase of $\psi _m(\bullet )$ does not depend on the radial coordinate $r$,
Then, using Eqs. (10) and (11), and noticing that for paraxial beams $k_z\simeq k$, we get the equations of the streamlines, namely, From these last two equations, we find that the angular velocity, is constant. Hence, the streamlines follow helices in the three dimensional space, while in the two-dimensional projection space they describe circular trajectories of radius $r_0$ rotating at a constant angular velocity, with a constant OAM.Now, let us analyze the case of the streamlines of Hankel conical waves. We notice that, due to the conical nature of their phase, we can anticipate that the projected streamlines will not follow two-dimensional circular trajectories, but spiral. The phase of the Hankel waves can be written as (see Eqs. (16) and (17))
4. Observation of the optical Kepler’s law
In the previous sections, we have established rigorous conditions to establish which type of optical beams satisfy Kepler’s second law. Now we will verify numerically these predictions. For this purpose we resource to Arago’s spot, also known as Poisson’s spot, and use it as “light-particle” tracer. For plane wave illumination on a circular opaque obstacle, the Arago’s transverse and longitudinal intensity distribution is fully characterized, so it can be a reliable tracer [20]. Our hypothesis is that the Arago’s spot follows the optical current streamlines, so it can be used as a tracer to test Kepler’s second law [21].
We investigated Bessel beams, their respective Hankel waves as well as Transverse-parabolic beams that we will refer to as T-parabolic beams (we avoid calling them Weber beams as this name can be misleading in the Mathematical Physics literature [22,23]). Bessel and T-parabolic beams have their spectrum on a ring-delta. The spectrum of the former is modulated by $\exp (i m \varphi )$, while the spectrum of the latter is characterized by $\exp (i a \ln |\tan \frac {\varphi }{2}|)$, where $a$ determines the latus rectum of the inner parabola of the intensity pattern [24]. Bessel beams possess well defined amount of OAM, unlike the T-parabolic beams. As discussed above, the Arago’s spot of a partially obstructed Bessel beam is expected to satisfy Kepler’s second law, while that of an obstructed T-parabolic beam does not.
A circular opaque obstruction, with a diameter slightly smaller than the width of a bright ring, was placed at the middle of a bright ring for the Bessel intensity pattern and in an intensity parabola of the T-parabolic beam. Behind the obstruction, after a short distance of propagation, an Arago’s spot is formed, and moves according to the features of the rotating phase of each beam, see insets in Fig. 2. This spot presents a narrow intensity maximum within the shadow zone that allows following its displacement to observe the energy flow of the beam. We define the position of the spot as the position of the maximum of its intensity.
Fig. 2. Propagation of the Arago’s spot of the following obstructed beams: (a) Bessel beam with $m=10$, (b) Hankel wave with $m=10$ and (c) $m=7$, and (d) Transverse-parabolic beam with $a=3$.
In order to obtain the areal velocity of the spot on the projection plane (which is parallel to the transverse plane of the optical beam), the displacement of the spot is measured in terms of the angle change $\Delta \varphi$ of the spot’s vector position in polar coordinates between two close propagation distances $z_0$ and $z_1$, since the propagation distance along the $z-$axis is analogous to the temporal evolution of a particle in Classical Mechanics. In this sense, the angular velocity can be approximated by $\Delta \varphi /\Delta z$, where $\Delta z=z_1-z_0$. Hence, the areal velocity is approximated by $dA/dz\approx r^2(z_1)\Delta \varphi /(2\Delta z)$ (see Eq. (5)).
In Fig. 3, it is shown the areal velocity of the spots produced by several obstructions of a Bessel beam and their associated Hankel waves, with a topological charge of $m=10$. Such obstructions are shown in Fig. 4. The behavior of the spots present constant areal velocities, within the measurement error range, and agrees with the predicted one by Eq. (24), namely, $dA/dz=m/(2k_z)=5.0357\times 10^{-3} \hbox {mm}$ (in the Classical Mechanics context, the units of the areal velocity is squared length over time, whereas in the optical regime is squared length over length, i.e., length). Furthermore, as discussed previously, Fig. 3 shows that the areal velocity of the Arago’s spot does not depend on the position of the obstruction, since every optical current streamline describes the same areal velocity. In the case of Bessel beams, the null radial velocity and constant angular velocity leads to the trivial satisfaction of Kepler’s second law, just as discussed in the previous section. This behavior is maintained even changing the size of the obstruction and the propagation distance, as long as the Arago’s spot is well defined. Since the position of the spot is measured as the position of the maximum of the spot’s intensity, the areal velocity measurement error is induced by the length of the pixel describing the maximum of the spot in the propagation simulation. The smaller the size spot, the smaller the number of pixels defining it, the larger the measurement error. That is why farthest spots possess more measurement error; as they lie farthest from the origin, the slower they are formed, the smaller they are.
Fig. 3. Numerical measurements of the Arago’s spot areal velocity, of partially obstructed (a) Bessel beams, (b) outgoing and (c) incoming Hankel waves, all of them with $m=10$. The propagation distance is $z=6.941$ mm. The obstruction positions are given, in mm, by $A=(0.352, 0)$, $B=(0.487, 0)$, $C=(0.597, 0)$, $D=(0.703, 0)$, $E=(0.803, 0)$, $F=(0.902, 0)$, and shown in Fig. 4. The diameters of the obstructions are given by $l_1=12.2\times 10^{-2}$ mm, $l_2=9.3\times 10^{-2}$ mm, and $l_3=l_4=l_5=l_6=6.7\times 10^{-2}$ mm. The theoretical prediction about the areal velocity is $dA/dz=5.0357\times 10^{-3}$ mm, which it is shown as the red line.
Fig. 4. Visualization of the obstructions used in Figs. 3 and 7 for (a) Bessel beam with $m=10$, (b) Hankel wave with $m=10$, and (c) T-parabolic beam with $a=3$.
In order to show a nontrivial case of a varying radial velocity, as well as to ascertain quantitatively the dynamics of the Hankel waves, we also performed the same partial obstructions of the Bessel beams over their associated Hankel waves (see Fig. 4). The singularity at the origin of the Hankel waves is filtered by multiplying a super Gaussian times the Neumann (imaginary) component of the Hankel wave (see Eqs. (16) and (17)). As shown in Fig. 3, the areal velocity of the spots of the Hankel waves also agrees, with the measurement error, with the predicted one, and does not depend on the position of the obstruction. On the other hand, in Figs. 5 and 6, it is shown the propagation behavior of the spots of obstructed Bessel beams and their corresponding Hankel waves for $m=7$ and $m=10$, respectively. As they propagate, the distance from the origin of coordinates to the position of the spot remains constant for Bessel beams, whereas slightly increases and decreases for the outgoing and incoming Hankel waves, respectively, while all maintain their areal velocity constant, satisfying Kepler’s second law. As mentioned above, the larger the propagation distance, the larger the size of the spot. In the same way, the larger the number of pixels defining it, the less the associated measurement error.
Fig. 5. Numerical measurements of the Arago’s spot areal velocity as function of the propagation distance, of partially obstructed (a) Bessel beams, (b) outgoing and (c) incoming Hankel waves, all of them with $m=7$. The initial propagation distance is $z=0$ mm, the obstruction diameter is $l_0=0.093$ mm, and the theoretical prediction about the areal velocity is $dA/dz=3.5250\times 10^{-3}$ mm, which it is shown as the red line. (d) The distance from the origin to the maximum of the intensity of the Arago’s spot, denoted by $r_s$, as function of the propagation distance.
Fig. 6. Numerical measurements of the Arago’s spot areal velocity as function of the propagation distance, of partially obstructed (a) Bessel beams, (b) outgoing and (c) incoming Hankel waves, all of them with $m=10$. The initial propagation distance is $z=0$ mm, the obstruction diameter is $l_0=0.093$ mm, and the theoretical prediction about the areal velocity is $dA/dz=5.0357\times 10^{-3}$ mm, which it is shown as the red line. (d) The distance from the origin to the maximum of the intensity of the Arago’s spot, denoted by $r_s$, as function of the propagation distance.
On the other hand, the areal velocity of the spots of the T-parabolic beams behave very differently depending on the position of the obstruction, as shown in Fig. 7 (see Fig. 4). In this case, the origin from which the angle, and therefore the areal velocity, is measured, is the origin of the parabolic coordinate system, since all the parabolas of the system are confocal with their focus lying at the origin, just as in the parabolic trajectories case of the gravitational potential in Classical Mechanics. However, since the phase of T-parabolic beams does not depend linearly on the polar azimuthal $\varphi$, the condition (14) is not satisfied and the areal velocity is non-constant. This can also be noticed during propagation; for instance, the areal velocity of the the spot formed by the obstruction placed in (0.013, 0.060) (see Fig. (7)) changes from $(1.819\pm 0.387)\times 10^{-3}$mm, at $z=0.008$mm, to $(3.302\pm 0.094)\times 10^{-3}$mm, at $z=0.034$mm.
Fig. 7. Numerical measurements of the Arago’s spot areal velocity, of a partially obstructed T-parabolic beam with $a=3$. The propagation distance is $z=0.0171$ mm. The obstruction positions are given, in mm, by $A_p=(-0.022, 0)$, $B_p=(-0.009, 0.032)$, $C_p=(0, 0.045)$, $D_p=(0.013, 0.060)$, $E_p=(0.050, 0.082)$, and shown in Fig. 4(c). The obstructions diameter is $8.1\times 10^{-3}$ mm.
5. Conclusion
In this work, we have established Kepler’s second law for optical wavefields. We presented a formal approach based on the local energy flow of optical beams. It was proved that the optical current streamlines of Bessel beams and Hankel waves, describe streamlines in a fixed plane transverse to the optical axis of the beam whose areal velocity is constant, as well as that every streamline describes the same areal velocity. We established the conditions under which Kepler’s second law is satisfied for optical beams; the azimuthal variation of the beam’s phase must be constant. Furthermore, the modulus of the gradient of the beam’s phase must not depend on the azimuthal angle variable, in analogy to the problem of a central potential in Classical Mechanics that also does not depend on the angle variable, fact that guarantees the conservation of the total Angular Momentum, and hence Kepler’s second law. Although it is well known that the total Orbital Angular Momentum of light is a conserved quantity, we demonstrated that only optical beams with a constant azimuthal variation of their phase satisfy Kepler’s second law. Furthermore, we used the Arago’s spot created by the diffraction of obstructed beams as a “light-particle” tracer to verify our predictions. With this, we showed that, contrary to what might be thought, T-parabolic beams, which indeed conserve their total orbital Angular Momentum, do not satisfy Kepler’s law. We have demonstrated that the areal velocity for optical beams has an homologous form as that of a particle in Classical Mechanics proportional to the amount of Angular Momentum, which may provide further insights to extend the analogies between Beam Optics and Classical Mechanics.
Funding
National Natural Science Foundation of China (11674111, 61575070).
Acknowledgments
SCHC acknowledges support from Huaqiao University, China.
Disclosures
The authors declare no conflicts of interest.
References
1. H. Goldstein, Classical Mechanics, 2nd. Ed. (Addison-Wesley, 1980).
2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of ligth the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
3. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef]
4. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]
5. S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).
6. S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123(1-3), 225–233 (1996). [CrossRef]
7. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclassical Opt. 4(2), S82–S89 (2002). [CrossRef]
8. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995). [CrossRef]
9. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998). [CrossRef]
10. J. Mendoza-Hernández, M. Arroyo-Carrasco, M. Iturbe-Castillo, and S. Chávez-Cerda, “Laguerre-Gauss beams versus Bessel beams showdown: peer comparison,” Opt. Lett. 40(16), 3739–3742 (2015). [CrossRef]
11. M. Anguiano-Morales, M. M. Méndez-Otero, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Conical dynamics of Bessel beams,” Opt. Eng. 46(7), 078001 (2007). [CrossRef]
12. I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]
13. R. Rop, I. A. Litvin, and A. Forbes, “Generation and propagation dynamics of obstructured and unobstructured rotating orbital angular momentum carrying helicon beams,” J. Opt. 14(3), 035702 (2012). [CrossRef]
14. R. A. Terborg and K. Volke-Sepúlveda, “Quantitave characterization of the energy circulation in helical beams by means of near-field diffraction,” Opt. Express 21(3), 3379–3387 (2013). [CrossRef]
15. A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “An optical vortex as a rotating body: mechanical features of a singular light beam,” J. Opt. A: Pure Appl. Opt. 6(5), S170–S174 (2004). [CrossRef]
16. A. Bekshaev, K. Y. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011). [CrossRef]
17. M. V. Berry and K. T. McDonald, “Exact and geometrical optics energy trajectories in twisted beams,” J. Opt. A: Pure Appl. Opt. 10(3), 035005 (2008). [CrossRef]
18. M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11(9), 094001 (2009). [CrossRef]
19. J. E. Gómez-Correa, S. E. Balderas-Mata, V. Coello, N. P. Puente, J. Rogel-Salazar, and S. Chávez-Cerda, “On the physics of propagating Bessel modes in cylindrical waveguides,” Am. J. Phys. 85(5), 341–345 (2017). [CrossRef]
20. R. L. Lucke, “Rayleigh-Sommerfeld diffraction and Poisson’s spot,” Eur. J. Phys. 27(2), 193–204 (2006). [CrossRef]
21. M. Gondran and A. Gondran, “Energy flow lines and the spot of Poisson-Arago,” Am. J. Phys. 78(6), 598–602 (2010). [CrossRef]
22. Some authors refer to Transverse-parabolic beams introduced in [23] as Weber beams. We refrain from doing so since it can be misleading. In the mathematical physics literature the parabolic cylinder Weber functions $D_\nu (\xi )$ are the solutions of the Helmholtz equation when the wavenumber is pure imaginary or k2 < 0 that correspond to evanescent fields and not to propagating fields. Also, Weber functions are referred to the Bessel function of the second type or Neumann function [23].
23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standars, 1972).
24. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef]
