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Abstract
Interference of a fundamental vortex-free Gaussian beam with a co-propagating plane wave leads to nucleation of a series of vortex rings in the planes transverse to the optical axis; the number of rings grows with vanishing amplitude of the plane wave. In contrast, such interference with a beam carrying on-axis vortex with winding number l results in the formation of |l| rings elongated and gently twisted in propagation direction. The twist handedness of the vortex lines is determined by the interplay between dynamic and geometric phases of the Gaussian beam and the twist angle grows with vanishing amplitude of the plane wave. In the counter-propagating geometry the vortex rings nucleate and twist with half-wavelength period dominated by the interference grating in propagation direction.
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1. Introduction
The term “vortex rings” is associated, historically, with the study of smoke rings by Helmholtz [1], which inspired Kelvin to propose that each type of atoms was a knotted vortex ring in an all-pervading aether fluid [2]. It also inspired Maxwell’s vortex analogy for the electromagnetic field [3] and Tait’s inquiry into knot theory and first classification of knots [4]. Vortex rings remain the basic subject in theory of fluid and gas motion and as elementary excitations or coherent structures in classical turbulence [5]. The importance of vortex rings has been recognised in different areas of science and technology, from meteorology, erupting volcanoes and cardiology to acoustics and fighting oil well fires [5,6]. However, controlled creation of knotted vortex rings in hydrodynamics requires elaborate experimental techniques and such vortex knots suffer from instabilities [7]. Nevertheless, the state of the art vortex rings now find their use in controlling confined states of turbulence in water [8].
The difficulties in laboratory studies of vortex rings in viscous fluids can be contrasted with a rapid progress in theory and experimental generation of “thin-core” vortex rings in quantum fluids, such as superfluid helium and atomic Bose–Einstein condensates (BEC) [9]. The vorticity in superfluids and matter waves is quantized and the turbulent motion of quantised vortices distinguishes quantum from classical turbulence [10]. The spontaneous ring nucleation was demonstrated experimentally with the decay of dark solitons into vortex rings in a two-component condensate [11] and it was modelled as a result of collapse of ultrasound bubbles in a single-component BEC [12]. Various methods were suggested for controlled generation of quantised vortex rings, such as creation of torus voids in a single-component BEC [13] and phase-imprinting technique [14,15], achieved by driving electromagnetic fields inducing transitions between internal atomic level components.
Quantised vortices exemplify the fundamental phenomenon of wave dislocations, discovered by Nye and Berry [16], and they manifest geometric Berry phase [17]. The interest to wave singularities was greatly induced when an assessable technique was demonstrated for experimental generation of the screw type dislocations in laser beams [18,19]. Screw dislocations are associated with points of vanishing field amplitude, where phase is undetermined or singular, and the optical current circulating the dislocation was termed an “optical vortex”. Corresponding field of research, Singular Optics [20], was pioneered by M. Soskin [21]. Despite its maturity, the field continues to attract significant attention with growing number of published review papers [22–25], special issues [26–30], and books [31–35].
Rotating phase fronts and current of an optical vortex is a manifestation of a fundamental property of light, the optical orbital angular momentum [33,36,37], which finds numerous applications in modern photonics [38–40]. Further generalisation lead to the concept of structured light [34], now actively developing [41,42] well beyond the paraxial laser vortex beams to the nanoscale and higher dimensions [43–45]. In the non-paraxial systems the electric and magnetic components of light are structured with polarisation textures, where geometric phases determine novel topologies in light [46], such as knot bundles of polarization singularities in laser beams [47,48] and ultra-short pulses [49]. In particular, the particle-like optical polarization knots [50] provide a link to knotted solitons in the nonlinear field theory [51]. Indeed, the ideas of optical structured waves now extend beyond optics to waves of different nature, such as quantum matter, electronic, plasmonic, and acoustic waves, to name a few [52].
Arguably, the first appearance of vortex rings in optics can be dated back to the studies of Airy rings [53] – dark rings near the focal plane of a diffraction pattern on a circular aperture. The interest in detailed analysis of focal area was stimulated, in part, by Gouy “phase anomaly” [54]. The initial scalar treatment for small apertures [55] followed by vectorial non-paraxial theory for wide apertures [56] and the analysis of the energy flow revealed the appearance of vortex rings [57,58]. Theoretical and experimental studies demonstrated nucleation and topological reactions of vortex rings in non-paraxial beams [59–63] and the gradual transition to paraxial beams free of any dislocations [60].
Further distinction between paraxial and non-paraxial light was suggested by Berry and Dennis, namely that higher-order dislocation loops, allowed as solutions to the non-paraxial Helmholtz equation, do not exist in paraxial Schroedinger-type equation [64]. In this study, the non-paraxial Bessel beams were used to untie higher-order vortex rings into links and knots [64]. Nevertheless, elaborate superpositions of paraxial Laguerre-Gaussian beams [64–68] and quantum eigenfunctions [69,70] also allow to construct linked and knotted vortex rings. Optical vortex knots were generated in experiment with a precise control of the field by spatial light modulator [71,72], embedding knot function in a propagating light beam [73,74]; this approach was verified in single photon quantum regime [75].
Generation of vortex and polarization knots in paraxial fields requires state of the art experimental techniques to structure the light. Yet, the spontaneous, or generic, vortex rings appear naturally in non-paraxial light or in the nonlinear systems, such as viscous fluid and BEC superfluids, discussed above. In nonlinear optical media the transversely-structured beams form spatial solitons, including solitons carrying quantized vortices [76–78], and the nonlinearity supports twisted toroidal vortex solitons in inhomogeneous media [79] and knotted solitons in dissipative laser system [80]. In particular, the self-focusing of paraxial Gaussian beam leads to the spontaneous generation of a vortex ring [81], as a result of destructive interference due to nonlinear phase difference, accumulated by beam’s peak intensity, reaching and exceeding value of $\pi$ with respect to phase of low intensity beam’s tails, propagating in linear regime. Similar mechanism was demonstrated experimentally for spontaneous generation of spatio-temporal vortex rings [82] and it was suggested as the means to generate temporal dynamics of vortex rings in BEC [83]. Another nonlinear phenomenon, the symmetry-breaking instability, leads to the spontaneous splitting of initially cylindrically symmetric edge dislocations into a series of vortex rings with alternating handedness [84].
The question arises if generic vortex rings can be also found in accessible paraxial linear systems. Ultimately, of course, phase dislocations and vortex rings, as any field nodes, appear as the result of destructive interference of several waves. For example, the perturbation analysis shows that the nonlinear self-knotting of vortex rings in self-trapped paraxial laser beams can be explained by the destructive interference of a stable soliton with its perturbation modes [81], similar to the vortex knots in superpositions of purely linear modes [65,68–70,73].
Perhaps the simplest way to create a single vortex ring is to interfere two focused co-propagating Gaussian laser beams of slightly different diameters or amplitudes [15]. We can further simplify and reduce the parameter domain by making one of the beams much broader than the first and replace it with a plane wave. In these settings, the dynamic and geometric Gouy phases of the Gaussian beam dictate the interference pattern and the two key parameter are the relative amplitude and relative phase of two waves. As shown below, despite its apparent simplicity, the system reveals somewhat unexpected complexity. Intuitively, when the amplitude of the plane wave is made to decrease to small values, it can be regarded as a perturbation to the Gaussian beam, and any additional features such as vortex rings should move to infinity and disappear. In contrast, we observe the number of vortex rings increase with vanishing plane wave amplitude, although their locations move to large radii or far from focal plane. By varying relative phase the vortex rings nucleate on optical axis or “shrink and disappear” [16,66]. Furthermore, varying control parameters we observe nucleation of pairs of counter-rotating rings, similar to Airy rings reactions in the non-paraxial beams [59–61].
Extending analysis to the vortex Gaussian beams we can also treat the small amplitude auxiliary plane wave as perturbation. As a result of this perturbation, in the focal plane, the phase dislocation of topological charge $l$ will split into $|l|$ single charge optical vortices. However, the full three-dimensional structure of these newly created vortex lines is complex: there are $|l|$ vortex rings twisted along the optical axis, the two parts of the ring can twist in opposite directions, and the twist angle diverges with vanishing amplitude of the plane wave. We demonstrate how the competition of the dynamic and Gouy phases defines these nontrivial features.
Exploring the effects of different phases of Gaussian beams on the generation of vortex rings, we also consider the counter-propagating geometry [85,86]. In this case the interference grating in axial direction dominates, the transverse vortex rings in fundamental Gaussian beams are spaced half-wavelength apart, and the $|l|$ vortex rings in vortex Gaussian beams are tightly wound about the optical axis.
2. Paraxial wave equation
Paraxial approximation assumes that the electric field of a monochromatic laser beam $\mathcal {E}$ has dominating carrier wavenumber $k=2\pi /\lambda$ in the direction of the optical axis $\tilde {z}$, $\mathcal {E}(\tilde {x},\tilde {y},\tilde {z})= E(\tilde {x},\tilde {y},\tilde {z}) \exp (ik\tilde {z})$, here tilde indicates the dimensional variables. We will use dimensionless transverse variables $x=\tilde {x}/w_0$ and $y=\tilde {y}/w_0$ with characteristic beam size $w_0$. The beam waist $w_0$ determines the diffraction length $z_d=kw_0^2/2$ and the dimensionless propagation coordinate $t=\tilde {z}/z_d$. The equation governing slowly varying envelope $E(x,y,t)$ is the Schrödinger-type paraxial wave equation
In the following we will distinguish amplitude, $|E(x,y,t)|$, and phase of paraxial beams, ${\rm Arg}\,E=\psi (x,y,t) + \alpha$, here $\alpha$ is an arbitrary phase constant. Similarly, the plane wave solution to Eq. (1) will be denoted with real positive amplitude $P$ and phase constant $\beta$, i.e. we will consider the superposition
here the dimensionless wavenumber $\kappa =kz_d$ and co- and counter-propagating geometries correspond, respectively, to $\kappa _1=\kappa$ and $\kappa _1=-\kappa$.The nodal lines $\mathcal {E}=0$ lie on the intersections of two surfaces in three-dimensional space, ${\rm Re}\,\mathcal {E}=0$ and ${\rm Im}\,\mathcal {E}=0$, which translate into a system of equations
here the relative phase difference $\phi =\pi +\beta -\alpha$ and $n$ is an arbitrary integer. We note that the first of Eq. (3) describes level surfaces for paraxial beam envelope [87] and it does not depend on the choice of geometry, co-propagating or counter-propagating, thus the surface is universal for both geometries.3. Fundamental Gaussian beam
Cylindrically symmetric fundamental mode $E = G_0(r,t)$ is given by
here polar radius $r^2=x^2+y^2$ and phase includes the diffraction and geometric Gouy terms. We have set the amplitude of the fundamental mode Eq. (4) to unity so that $P$ is a relative amplitude of the two interfering beams.The first of Eq. (3) defines the Gaussian field iso-surfaces [87], $|G_0(r,t)|= P$, with the solution in the form of a closed prolate surface of revolution, $r=\rho (t)$,
plotted in Fig. 1(a). For simplicity, we will refer to the family of such surfaces as “spheroids”, to distinguish them from “toroids” in the next section. The extend of spheroids in propagation direction can be calculated by noting that $\rho \to 0$ for $|t|\to t_{\rm max}=\sqrt {P^{-2}-1}$. Furthermore, we distinguish two types of shapes: a simple spheroid with one maximum at $t=0$ for $P>1/\sqrt {e}\simeq 0.61$ and a “dumbbell” shape with two maxima, ${\rm max}\,(\rho ) = 1/P\sqrt {2e}$, located at $t=\pm \sqrt {P^{-2}/e-1}$ for $P<1/\sqrt {e}$. We note that these maxima, although diverge as $P\to 0$, remain well “inside” the beam, ${\rm max}\,(\rho ) <\sqrt {1+t^2}$. At the same time, the only regions “outside” of the beam, $\rho >\sqrt {1+t^2}$, appear for $P<1/e\simeq 0.37$ near the focal area, where spheroids in Fig. 1(a) (solid lines) are wider than the hyperboloid of the beam radius (dashed line). The two surfaces cross at the rings, $\rho (\pm \tau )=\sqrt {1+\tau ^2} = 1/Pe$, located at $t=\pm \tau$ with $\tau = \sqrt {(Pe)^{-2}-1}$; these crossings play an important role discussed below.
Fig. 1. (a) Spheroids $\rho (t)$ Eq. (6) (solid lines) and Gaussian radius $\sqrt {1+t^2}$ (dashed line). (b) Graphic solution to Eq. (7). In both (a) and (b) the numbers next to solid curves indicate values of the relative amplitude $P = 0.1$, 0.2, 0.3, 0.5, and 0.8. In (b) the horizontal thin line $\phi =\pi /10$ determines locations of vortices in Fig. 2: square in Fig. 2(a,c) and dots in Fig. 2(b,d); the dashed line, $-\tan ^{-1}t$, corresponds to the limits of the axial extent of spheroids in (a).
For the co-propagating geometry we set $\kappa _1=\kappa$ and the second equation of Eq. (3) translates into relation for the phase at the spheroids Eq. (6),
plotted in Fig. 1(b).We begin our analysis from a simple case of large relative amplitude $1>P>1/e$, when the spheroids Eq. (6) are “inside” the beam and phase on the spheroids has a simple shape without extrema, see Fig. 1(b) for $P=0.5$, 0.8. There is one solution to Eq. (7) for $n=0$ and $|\phi |\le \tan ^{-1}(t_{\rm max})$, describing single vortex ring, as shown in Fig. 2(a) for $P=0.5$, together with distributions of phase and current in Fig. 2(c). As we scan the values from $\phi =-\pi$ to $\pi$, we observe in Visualization 1 a vortex ring bifurcating on optical axis at $t=t_{\rm max}$, moving backwards on the spheroid in Fig. 2(a), and shrinking to a point at $t=-t_{\rm max}$. The direction of the Poynting vector in Fig. 2(c) does not take into account the carrier wave, $\exp (i\kappa z)$, and is dominated by the negative Gouy phase in Eq. (7), thus vortex ring has a clockwise circulation in the plane $(t,r)$. The actual energy flow is directed forward because of the factor $\exp (i\kappa z)$; this does not change the position or the handedness of the vortex ring.
Fig. 2. (a,b) Vortex rings on spheroids and (c,d) corresponding phase (hue colormap) and current (white arrows) for the plane-wave amplitudes $P=0.5$ in (a,c) and $P=0.2$ in (b,d). Gray surfaces in (a,b) and dashed lines in (c,d) show the half-width hyperboloid, $r=\sqrt {1+t^2}$. In both cases $\phi =\pi /10$, as indicated in Fig. 1(b). Changing the relative phase $\phi$ leads to the motion and nucleation of vortex rings as shown in Visualization 1 and Visualization 2 for (a) and (b), respectively.
Decreasing the value of $P$ below $1/e$ we observe the development of two extrema at $t=\pm \tau$ in Fig. 1(b) with the values of $\psi _0(\rho,\pm \tau )=\pm (\tau -\tan ^{-1}\tau )$. For example, at $P=0.2$ and $\phi =\pi /10$, there are three roots in Fig. 1(b) and three vortex rings in Fig. 2(b,d). Scanning the values $\phi \in [-\pi,\pi ]$ we observe in Visualization 2 the nucleation of a pair of vortex rings at $t=\pm \tau$, i.e. at the rings of crossings of the spheroid Eq. (6) and half-width hyperboloid $r=\sqrt {1+t^2}$. Figure 2(d) shows that the two rings have opposite circulations.
To evaluate the handedness of the vortex rings we will use the following result [61]: in the right-handed coordinate system $(\xi,\eta )$ the winding number $m\equiv \frac {1}{2\pi } \oint \nabla {\rm Arg}\,\psi \,d{\bf l}$ of the zero of complex field $\psi =\alpha \xi +\beta \eta$ is given by $m={\rm sign\,Im}(\alpha ^* \beta )$. Linearizing total field Eq. (2) around node at the spheroid $(t,\rho (t))$ we obtain $m={\rm sign} (\rho ^2-1-t^2)$. As seen in Fig. 2(d) the middle vortex ring outside of the Gaussian beam half-width hyperboloid, $\rho >\sqrt {1+t^2}$, carries topological charge opposite to the two vortices “inside” the beam.
Thus far, we identified two different topological transformations leading to the appearance of vortex rings. The first mechanism is the bifurcation (annihilation) of a single vortex ring at the optical axis, when the phase $\phi$ passes through the end points $\psi _0(\rho,\pm t_{\rm max})=\mp \tan ^{-1} t_{\rm max}$ in Fig. 1(b), namely
valid for any value of $P$ and $n$ in Eq. (7). This relation is shown by a dashed curve in Fig. 3(a); crossing this curve in parameter domain $(P,\phi )$ changes the number of vortex rings by one.Fig. 3. (a) Number of vortex rings $N(P,\phi )$ in the parameter domain $0.02\le P\le 1$ and $\phi \in [-\pi,\pi ]$, zoned by dashed line Eq. (8) and solid line Eq. (9). (b) Vortex rings (circles) on spheroids for $P=0.02$, 0.01 and $\phi =\pi /10$. (c) Solid lines: maximal and minimal numbers of vortex rings at the interval $\phi \in [-\pi,\pi ]$ and dotted trend line $N\simeq 2 (1/P-2)/e\pi$. (d) Rings near focal area for $P=0.02$, 0.01, 0.005, and 0.001. Dashed line in (c,d) shows the Gaussian beam radius, $r=\sqrt {1+t^2}$.
The second mechanism is the nucleation (mutual annihilation) of a pair of counter-rotating vortex rings at the half-width hyperboloid $\rho (\pm \tau )=\sqrt {1+\tau ^2}$ when the phase $\phi$ passes through the extrema of $\psi _0$ in Eq. (7)
This relation is shown with solid lines in Fig. 3(a); crossing these curves in parameter domain $(P,\phi )$ changes the number of vortex rings by two.
Crucial difference between the two scenarios above is that, while the first is structurally stable and thus generic, the second is not. The proof can be visualised in terms of crossing of two abstract surfaces, ${\rm Re}\,\mathcal {E}=0$ and ${\rm Im}\,\mathcal {E}=0$. The two convex surfaces, e.g. two spheres, first touching and then crossing as they move closer together, generate single ring, however distorted by the deformations of the surfaces. In contrast, for the pairwise nucleation of rings [59–61] the outlines of two surfaces have to be exactly similar, e.g. a sphere centered on axis of a cylinder crosses it at two parallel rings as in Fig. 2(b). Yet, the deformations of the surfaces prevent such an ideal nucleation and can lead only to the local point-to-ring bifurcations, as in the first robust mechanism. For example, if the cylinder has a slightly elliptic profile, it crosses the sphere at a pair of disconnected rings. Nevertheless, the rings reconnect and form parallel rings within a very small change of control parameter, thus distinguishing the two scenarios is limited by spatial resolution. For the completeness of the present study here we further explore the radially symmetric Gaussian beam Eq. (4); an investigation of the perturbed beams will be presented elsewhere.
We note that, for a given $P$, both types of topological transformations happen exactly two times on the interval $\phi \in [-\pi,\pi ]$, i.e. one bifurcation and one collapse of a vortex ring on the optical axis as well as one nucleation and one mutual annihilation of a pair of vortex rings on the half-width hyperboloid, see Visualization 2. However, the “average” number of vortex rings grows with the decrease of $P$, as summarised in Fig. 3. Indeed, as we decrease $P$ in Fig. 1(b), the phase on the spheroid Eq. (7) grows, and the new solutions became available for $n\ne 0$ in Eq. (9). Asymptotically, for $P\to 0$, we obtain $\tau \to 1/Pe$, $\tan ^{-1}\tau \to \pi /2$, and we derive from Eq. (9) the values of $P$ at which the number of vortex rings, $N(P,\phi )$, changes by four, $P_n\simeq 2/e\pi (4n+s)$, here $n$ is an arbitrary integer, while $s=1$ for $\phi =0$ and $s=3$ for $\phi =\pm \pi$. Figure 3(a) confirms this asymptotic relation to a good accuracy: the crossings of solid lines correspond to the simultaneous nucleation of two pairs of vortex rings at $t=\pm \tau$ and lead to the changes of $N$ by four.
Figure 3(b) illustrates two examples with large numbers of vortex rings, $N(0.02,\pi /10)=11$ and $N(0.01,\pi /10)=23$. In both cases the total number of rings is odd with one ring close to the focal plane and the number of vortex and antivortex rings differs by one, cf. empty and filled circles in Fig. 3(b). Similar structure is expected for small relative phase $|\phi | <\pi /2$, see Fig. 3(a). In contrast, for $\pi /2<|\phi |\le \pi$, the total number of rings is even and the numbers of vortex and anti-vortex rings are equal.
Figure 3(c) shows the trend in growing total number of vortex rings with $P\to 0$, namely $N\simeq 2/e\pi P$. Since the propagation extend of spheroids Eq. (6) also grows as $t_{\rm max}\simeq 1/P$, we can estimate the density of vortex rings per unit propagation length as $N/2t_{\rm max}=1/e\pi \simeq 0.12$, or about one vortex ring per ten diffraction lengths. However, this density is not uniform on the spheroids, as seen in Fig. 3(b), it is higher closer to the focal plane. Indeed, while the maximal radius of vortex rings (spheroids Eq. (6)) also grows as $1/P$, the ring radius at the focal plane grows much slower, $\rho (0)=\sqrt {\ln (1/P)}$, developing the pronounced dumbbell shape. Figure 3(d) illustrates the structure of vortex rings near the focal area.
4. Vortex Gaussian beams
The vortex Gaussian beams $G_l(\rho,\varphi,t)$ represent a subset of Laguerre-Gaussian beams excluding radial modes, i.e. we consider
Fig. 4. (a) Graphic solution to $|G_l|=P$ with two roots for each profile with $l=1,3$ and $P=0.6$, dotted line. The two roots form toroids with the crossections shown in (b) for $P=0.3$, 0.6, and 0.9. Dashed lines show the hyperboloids of vortex envelopes maxima, $r=\sqrt {l (1+t^2)/2}$.
An example of the graphic solution to the first of Eq. (3), $|G_l|=P$, is presented in Fig. 4(a). There are two roots, $\rho _1<\rho _2$, for every $P<1$, forming a toroid with inner ($r=\rho _1(t)$) and outer ($r=\rho _2(t)$) surfaces joined at the hyperboloids, see Fig. 4(b) and Ref [87]. The merger of two roots, $\rho _1=\rho _2=\sqrt {l/2}/P$, defines the axial extend of these toroids, $|t|\le t_{\rm max}$. This length does not depend on topological charge $l$ and coincides with the axial length of spheroids Eq. (6), namely $t_{\rm max}=\sqrt {P^{-2}-1}$.
Toroids in Fig. 4(b) suggest to expect closed nodal lines Eq. (3), developing (as $P$ growths from zero) from the infinite axial vortex lines $\sim \exp (il\varphi )$. While the inner surface $r=\rho _1(t)$ carries the “original” axial vortex line(s), the outer surface $r=\rho _2(t)$ hosts additional vortices “coming from infinity”, thus completing vortex rings. We emphasise that for both the co- and counter-propagating beams the nodal lines will be located at the same toroids in Fig. 4(b).
For the co-propagating vortex beam and plane wave, $\kappa _1=\kappa$, we derive the second of Eq. (3) in the form specifying the azimuthal angle of the nodal line at the toroid, $\varphi = \phi /l + \varphi _{ns}(t)$, with
The relative phase $\phi$ contributes as a trivial shift of the azimuth base direction at $t=0$, thus we can set $\phi =0$ without loss of generality. The index $n=1,2,\dots,l$ enumerates different single-charge vortex lines splitting from the original vortex of the charge $l$. The index $s=1,2$ distinguishes two surfaces of the toroid: clearly, the azimuth of nodal lines will be different at $\rho _1(t)\ne \rho _2(t)$.
The two last terms of Eq. (12) define the twist of vortex lines about propagation axis, as we show in Fig. 5. Consider first a single charge vortex and relatively large plane wave amplitude $P=0.5$ in Fig. 5(a). We distinguish two parts of the vortex ring with color, to help the eye placing these lines at the toroid. The green vortex line can be thought of as the perturbed on-axis vortex now laying at the inner surface $r=\rho _1(t)$, while the yellow vortex line at the outer surface $r=\rho _2(t)$ is the one coming from infinity. The two lines join at the rims of the torus $t=\pm t_{\rm max}$ and form a single ring, gently twisted in the counter-clockwise direction, reflecting positivity of $l=+1$.
Fig. 5. Vortex rings in the co-propagating Gaussian vortex and plane waves. Plane wave amplitude $P=0.6$ in (a,b) and $P=0.1$ in (c,d) while the vortex beam charge $l=1$ in (a,c) and $l=3$ in (b,d). Gray surfaces show the hyperboloids, $r=\sqrt {l(1+t^2)/2}$, and the colored surfaces are the toroids of Fig. 4(b); these surfaces are cut by $x=0$ plane to help the eye. For the single charge vortex beam in (a,c) we distinguish by the green colour the vortex line at the inner part of toroid $r=\rho _1(t)$ and the yellow line at $r=\rho _2(t)$ completes the same vortex ring. For the triple charge vortex beam in (b,d) different colors distinguish three vortex rings. Visualization 3 demonstrates how the vortex ring in (a,c) grows and twists as $P$ decreases from 1. Visualization 4 and Visualization 5 employ relative phase span $0\le \phi \le 2\pi$ to rotate (b) and (d), respectively.
Similar picture we obtain for higher-charge vortex beams, e.g. with $l=3$ in Fig. 5(b). The three vortex rings, all spanned from $t=- t_{\rm max}$ to $t=t_{\rm max}$, differ from each other by a simple rotation $\varphi \to \varphi \pm 2\pi /3$, as defined by Eq. (12). The different colors helps the eye in Fig. 5(b) to distinguish the three rings, and the full span of the relative phase difference $\varphi \in [0,2\pi ]$ leads to the rotation of the whole structure by the angle $2\pi /3$. Such rotation is periodic, if we drop the distinction between the rings, see Visualization 4.
Structure of vortex rings is significantly more complex for smaller $P$, such as $P=0.1$ in Fig. 5(c,d). The inner surface gets closer to the optical axis, $\rho _1(0)\sim P^{1/l}$, and the “original” vortex lines are only slightly perturbed from the optical axis. They twist around axis in the counter-clockwise direction, at least near the focal area $t\sim 0$, where the Gouy phase contributes positively in Eq. (12). In contrast, the outer vortex lines at $r=\rho _2(t)$ are twisted strongly clockwise. Here the large radius $\rho _2$ contributes in Eq. (12) to large negative azimuth angles, namely one full clockwise turn for both, $l=1$ in Fig. 5(c) and $l=3$ in Fig. 5(d). Visualization 3 shows how the outer vortex line’s twist grows with the decreasing $P$ for $l=1$ from Fig. 5(a) to (c). Continuously changing the relative phase $\varphi \in [-\pi,\pi ]$ in Visualization 5 provides a better view of the twisted rings in Fig. 5(d).
5. Counter-propagating beams
Fundamental Gaussian beam. Setting $\kappa _1=-\kappa$ in Eq. (3) we consider the destructive interference of counter-propagating beams [85,86]. The first condition $|E(x,y,t)|=P$ is solved by Eq. (6), thus the vortex rings can be located at the spheroids discussed above. The second equation of Eq. (3) is dominated by the very large term $2\kappa t$; e.g. for the beam radius of $w_0=5\,\mu$m and wavelength $\lambda =0.5\,\mu$m, we obtain $\kappa \simeq 2000$. Therefore, we neglect $\psi _0(\rho,t)\ll 2\kappa t$, which means that we assume $P\gg \exp (-2\kappa )$, reasonable for any practical purpose. We obtain the set of vortex ring positions numbered with $n$,
here the propagation increment $\Delta t=\pi /\kappa$ corresponds to $\Delta \tilde {z}=\lambda /2$. Therefore, the change of phase $\phi$ from $-\pi$ to $\pi$ shifts the interference grating forward by $\lambda /2$, equivalently $n\to n+1$.Figure 6 shows an example of the total intensity and vortex rings in the small propagation window around the focal plane. Any region within a spheroid would essentially show the same picture at this propagation scale, except the regions near the end points where the ring diameter vanishes, $\rho (t_{\rm max})=0$. Therefore, as in the case of co-propagating beams above, scanning of phase $\phi$ leads to one bifurcation and one collapse of a vortex ring on the optical axis.
Fig. 6. Total intensity isosurfaces $|\mathcal {E}|^2=1$ (gray) and transverse vortex rings for the superposition of the counter-propagating fundamental Gaussian beam and a plane wave with the amplitude $P=0.5$; propagation length of one wavelength.
In contrast to the co-propagating beams, the nucleation of vortex-antivortex ring pairs does not occur in counter-propagating scheme. To show it, we calculated the winding number $m$ of a ring, introduced above, by linearising the total field around arbitrary nodal point $(\rho,t)$,
All the rings have the same circulation thus pairwise nucleation is impossible.
We can estimate the total number of rings on the spheroid as $N=2t_{\rm max}/\Delta t$ and, for vanishing $P\to 0$ and $t_{\rm max}\simeq 1/P$, we obtain $N\simeq 2\kappa /\pi P$. Although it grows as $\sim 1/P$ with $P\to 0$, similar to the co-propagating beams above, the actual number is four orders of magnitude larger; the density of two rings per wavelength corresponds to $1/\Delta t \simeq 600$ for $w_0/\lambda =10$, or $\sim 10^3$ rings per diffraction length.
Vortex Gaussian beams. Setting $\kappa _1=-\kappa$ in Eq. (3) and $\phi =0$, as in Eq. (12), we can also neglect the variation of phase $\psi _l$ in $r$ and $t$ with respect to dominating term $2\kappa t$, similar to Eq. (13). It follows that the difference in azimuth is lost for two parts of each ring, i.e. vortex lines at $\rho _1\ne \rho _2$, in contrast to Eq. (12). We arrive at the following angular coordinates for both parts of a vortex ring with index $n=1,2\dots l$:
Direct visualisation of exact solution in Fig. 7 confirms the approximate relation Eq. (14). The negative angular velocity $-2\kappa /l$ leads to clockwise winding of vortex rings with the same azimuth for both parts at $\rho _1\ne \rho _2$ in Fig. 7(a,c). The two parts fuse and complete the rings at the rims of the toroids in Fig. 7(b,d).
Fig. 7. Vortex rings in counter-propagating vortex and plane waves. Vortex beam charge $l=1$ in (a,b) and $l=3$ in (c,d); plane wave amplitude $P=0.6$. Gray-colored surfaces show the iso-intensity surfaces $|\mathcal {E}|^2=1$; vortex lines are colored as in Fig. 5. In all panels the propagation distance is one wavelength: in (a,c) centered at focal plane $t=0$, and in (b,d) at the right rim $t=t_{\rm max}$ of corresponding toroid in Fig. 4(b). The later panels show the fusion of the two lines of each vortex ring at $\rho _1(t_{\rm max})=\rho _2(t_{\rm max})$.
The twist angle per period of interference grating is defined by the vortex charge $l$: a single vortex ring makes two full turns per wavelength in Fig. 7(a) while the three vortex rings in Fig. 7(c) twist by the angle $4\pi /3$ per wavelength. Similar picture we observe for smaller values of $P$ (not shown): although this parameter has strong influence on the size and shape of the toroids in Fig. 4(b), the $P$-dependence is suppressed in Eq. (14).
6. Conclusions
Simple model, presented here, provides an accessible way to study vortex rings in virtually any optics laboratory. The interference of paraxial laser beams with a plane wave is governed by the phases of the Gaussian beams: dynamic phase, forming phase fronts, and geometric Gouy phase, dominating near the optical axis. Therefore, the vortex rings, being the result of destructive interference, elucidate the phase structure of Gaussian beams. In particular, the inter- and counter-play between dynamic and geometric phases determine peculiar twists of rings born in vortex beams in Fig. 5(c,d). Furthermore, the control over relative phase of two beams allows to move and nucleate vortex rings in fundamental Gaussian beam at will in real time, as in Visualization 1 and Visualization 2. Setting up a scheme with fast relative phase modulation, e.g. using reflection from a mirror mounted on a piezoelectric transducer driven by an ac voltage at 1 kHz frequency, would produce the vortex rings moving back and forth with speeds of the order of $10^3 z_d$ per second.
The techniques to measure the three-dimensional configurations of optical vortices are elaborate but well established [71–73]. The difficulty of measuring the complex vortex rings here, at small $P$, is that they are mostly located in the darker regions of small intensity. However, they still challenge the intuitive notion that, as $P$ vanishes, the rings disappear “at infinity”. Although this is true in the limit $P\to 0$, the axial extent of the “ring domain” grows as $t_{\max }\sim 1/P$, e.g. for $P=0.1$ (intensity of the plane wave is 1% of the peak Gaussian intensity) the 2 or 3 vortex rings can be found within 10 diffraction lengths. The radial positions also do not exceed 2-3 local beam sizes, especially near focal region.
It is interesting to further study accessible optical vortex rings by varying the structure of the Gaussian beams, namely how breaking the symmetry of the beam changes vortex ring structure. Preliminary results for Gaussian beams with elliptic cross-sections suggest that, instead of pairwise ring nucleation discussed above, the deformed rings undergo series of ring reconnections, as the control parameters vary. Furthermore, we expect that, similar to the self-trapped beams in nonlinear media [84], the ring — anti-rings pairs will be born from the cylindrical edge dislocations in higher-order radial Gaussian modes. It remains to be explored what kind of topological transformations vortex rings undergo in twisted Gaussian beams carrying orbital angular momentum [81].
Funding
Nazarbayev University (021220FD3551).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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