1. Introduction

Vortex generation in superfluids and superconductors has been historically a matter of thorough study in condensed matter systems. Vortices also play an important role in Bose-Einstein condensates (BEC) so that their analysis in the crossover of these three disciplines is currently a rich field of exchanging ideas [1,2]. In all cases vortices appear as objects with quantized flux circulation, to which can be attributed quasi-particle properties. They have in common that they appear in interacting systems in which the fields describing the superfluid, superconductor or BEC is formally described by nonlinear equations of motion. Effective nontrivial interactions occur between vortices as quasi-particles which can be understood in terms of this nonlinear dynamics. The mechanisms for generation of vortices in all these cases are in this way intrinsically different from those associated to optical vortices in free propagation. Optical vortices in free propagation are instead described by the linear paraxial wave equation, which is the differential form of the classical Fresnel diffraction integral and it is formally analogous to the Schrödinger equation for a free particle in two transverse dimensions, with the axial variable z playing the role of time.

Apparently, we could expect trivial vortex dynamics in a free theory describing a non-interacting particle. However, as a panoply of results in the field of Singular Optics show this is not the case. Due to their special properties, optical vortices cores can be also considered as quasi-particles. They follow the trajectories of phase singularities, i.e., the points in space where the optical field vanishes as the optical beam propagate. A group of optical vortices can propagate inside a single beam or as a result of superposition of many waves. In both cases they move along complex paths. During propagation the optical vortices of opposite topological charges can be created or annihilated forming characteristic loops – vortex loops [3–6]. The problem of rules describing their trajectories was studied in several papers [7,8]. In simple cases the vortex trajectory can be easily determined as for example for three plane waves interference [9,10]. The other simple case is Gaussian beam with single off axis vortex [11,12]. However, the case of four or more plane waves interference or interference of more complex waves (like Gaussian or Bessel beams) results in complex trajectories which are difficult to determine [13,8]. The same problems occur in case of single beam with multiple vortices [14] or diffracted beams [15,16].

Creation and annihilation of vortex pairs with opposite charge can occur easily during free propagation since these processes do not violate the conservation of the topological charge. As compared to pair generation in superfluids, superconductors or BEC’s, generation of optical vortex pairs in free propagation are energetically favored. In the first case, the ground state is a field with a non-zero amplitude, so generating a solution with zero amplitude (the vortex core) requires some energy to deform spatially the ground state field to a point where the field vanishes. In optical free propagation the ground state is just the zero-field solution so introducing phase singularities (points of zero field) in a system tends do decrease its energy instead. This particularity can explain why is relatively easy to have a richness of vortex-anti-vortex pairs in free propagation by manipulating the boundary conditions of the optical field. The key point is then to what extent we can set the initial condition of the free propagating beam to predict and achieve full control of the creation and annihilation of vortex pairs inside the beam. It turns out that this process is surprisingly complex even in the simpler cases, as we will show in this paper by analyzing the phase singularity structure of a multisingular Gaussian beam with just four single-charged vortices at its waist.

Since the core of a vortex is a dark point, vortex trajectories are called by some authors dark rays. The theory describing the dark rays’ propagation is named dark optics. In this paper we want to contribute to dark optics by identifying some characteristic features describing the propagation of four vortices embedded into a Gaussian beam. We have started from the theory presented in [17,18] which opens the door for finding a deep insight to the behavior of many vortices seeded in a Gaussian beam. This four vortex case is simple enough for clear and short presentation and rich enough to exhibit the complex vortex behavior depending on a single parameter. The identified vortex dynamics can be split into separate classes which are structurally stable. That means that we can verify our results experimentally. Thus, we can show that the theory works in a desired way and it is a promising tool for systematic study of intricate order in case of Gaussian beams carrying more than four vortices. Additionally, we demonstrate both theoretically and experimentally a systematic procedure to control and manipulate the creation and annihilation of vortex-anti-vortex pairs using a few parameters easily accessible in standard diffraction experimental set-ups.

2. Pair generation in Gaussian beams

Multisingular Gaussian beams are Gaussian beams embedding a set of positive and negative single-charged phase singularities. They are monochromatic optical beams given by the scalar field ${\phi _\omega }({x,y,z,t} )= \phi ({x,y,z} ){e^{i\omega t}}$ that represents the electric component of the optical field in the scalar approximation. They verify the paraxial wave equation

Physically, the initial condition (2) corresponds to a set of ${N_ + }$ positive (at complex positions $\{{{a_i}} \}_{i = 1}^{{N_ + }}$) and ${N_ - }$ negative (at complex positions $\{{{b_i}} \}_{i = 1}^{{N_ - }}$) single-charged phase singularities inserted at the waist of a fundamental Gaussian beam. The singularities are canonical, which means that the phase of the field in the neighborhood of every singularity grows linearly with the polar angle around the singularity $\textrm{arg}(\phi )\sim{\pm} \theta $.

Multi-singular Gaussian beam functions have the form of a polynomial times the Gaussian function ${\phi _{00}}$. They can be derived analytically in a simple manner using the so-called Scattering Modes [19]. In this manuscript, we will adopt this technique to calculate them. A general multi-singular Gaussian beam verifying the initial condition (2) has then the following form:

The complex zeros of this polynomial at every z determine the trajectories of the phase singularities of the beam. These trajectories are dark rays since the intensity vanishes ($\phi = 0$) along them. Conservation of the topological charge is preserved in this type of beams. However, generation and annihilation of pairs of a positive and negative charge is not forbidden by the conservation of charge and, in fact, it is possible for proper initial conditions. The number of phase singularities is not a constant value in the beam [20].

A necessary condition for pair generation is that at $z = 0$ the beam contains a mixture of positive and negative charges. In other words, Gaussian beams with all singularities equally charged at z = 0 do not generate pairs in forward or backward evolution. According to the initial condition (2) this type of beams are generated by polynomial depending exclusively on w (positive charges) or ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w}}$ (negative charges). Using the expansion in Scattering Modes [19] one can see that the general complex polynomial (5) for this type of beam also fulfills this property:

An explicit example of conformal Gaussian beam showing this property is a beam with initial condition given by four identical single charges $({ + , + , + , + } )$ symmetrically distributed around the origin at a distance a and at the vertices of a square of side $\sqrt 2 a$. This is the counterpart of the quadrupole configuration $({ + , - , + , - } )$ that will be analyzed in the next section. This conformal beam has positive single charges at $z = 0$ located at the complex points ${a_j} = \sqrt 2 a{e^{i\left( {\frac{\pi }{2}j + \frac{\pi }{4}} \right)}},\,\,\,j = 1,2,3,4$ and it fulfills the initial condition (2) with the conformal polynomial ${P_{4 + }}({w,0} )= 4{a^4} + {w^4}$. As expected from the previous discussion, the polynomial of the global solution is also conformal and given by

In this way, since $q(z )= z + i{z_{R\; }}$, the dark rays of the four singularities follow straight lines for all values of z, as shown in Fig. 1. Thus, there is no generation of pairs and phase singularities act as a set of non-interacting particles with trivial dynamics.

 

Fig. 1. Dark rays’ structure of a conformal Gaussian beam with four singularities with charge $q ={+} 1$ (green trajectory indicates positive $q$): (a) general view, (b) top view, (c) near field central region.

Download Full Size | PDF

3. Quadrupole Gaussian beams

The situation changes dramatically when we introduce positive and negative charges in the initial condition. Now instead of four identical charges $({ + , + , + , + } )$ at the vertices of a square of size $\sqrt 2 a$ we consider a quadrupole arrangement of alternating charges $({ + , - , + , - } )$. We will show next that this simple arrangement presents a highly nontrivial and rich singularity structure, completely different from the previous case. We perform a complete analysis of the dark beam structure of this type of beam by taking advantage of its polynomial nature and the fact that they admit a closed analytical form.

3.1 Analytical form

The initial condition of a quadrupole Gaussian beam (QGB) is given by ${\phi _Q}\left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w} ,0} \right) = {P_Q}\left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w} } \right){\phi _{00}}\left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w} ,0} \right)$, where the polynomial ${P_Q}$ describes a set of two canonical phase singularities with charge $q ={-} 1$ at positions $1/\sqrt 2 ({ + a, + a} )$, $1/\sqrt 2 ({ - a, - a} )$, and two with $q ={+} 1$ at $1/\sqrt 2 ({ - a, + a} )$, $1/\sqrt 2 ({ + a, - a} )$:

By means of the Scattering Modes method (see Ref. [19]), the solution of the paraxial wave equation (1) with initial condition determined by the polynomial (4) can be easily found (see Supplement 1 for the explicit calculation). It has a polynomial form at every $z$

3.2 Classification

The nature of zeroes of the quadrupole polynomial ${P_Q}\left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w} ,z} \right) = 0$ at every z determines the classification of the dark beam structure of QGB. The coefficients of the P_Q polynomial (7) depend only on two parameters: a and ${z_R}$. Equivalently, we can use alternatively the beam width at its waist ${W_0} = \sqrt {{z_R}/\pi } $ and the vortex-anti-vortex distance at z = 0, $d = \sqrt 2 a$. These parameters are fixed by the initial condition (Eq. (9)) and thus they can be easily controlled experimentally. Unlike conformal Gaussian beams, where pairs are absent, QGB present a highly non-trivial dark beam 3D structure characterized by a different number of pairs. By analyzing the number of vortex pairs ${N_p}$ that are created or annihilated for $z > 0$ (from the beam waist to the far field) in terms of W₀ and the dimensionless ratio $\sigma = {W_0}/d$ we can classify QGB in only three equivalence classes:

 

Fig. 2. Class I dark beam structure with ${N_p} = 6$. (a) General view. (b) Top view. (c) Central loop in the near field. Dark rays created by vortices of opposite topological charges are indicated by different colors (green and red).

Download Full Size | PDF

 

Fig. 3. Class II dark beam structure with ${N_p} = 2$. (a) General view. (b) Top view. (c) No central loop in the near field.

Download Full Size | PDF

 

Fig. 4. Class III dark beam structure with ${N_p} = 4$. (a) General view. (b) Top view. (c) Central loop in the near field.

Download Full Size | PDF

The total number of points where pairs are created or annihilated for the whole dark beam is just $2{N_p}$ due to the symmetries of the quadrupole polynomial (see next section).

Remarkably, these classes depend only on the dimensionless parameter $\sigma = {W_0}/d$ and not on d as shown in the phase diagram in Fig. 5. In terms of forward propagation from $z = 0$ class I and III beams exhibit annihilation of the original two pairs at an axial distance z_loop, followed by propagation with no singularities until generation of one (class III) or two (class I) new pairs at further axial positions. Contrarily, in class II QGB the original two pairs never annihilates and only two new outer pairs are created at further z.

 

Fig. 5. Phase diagram of the classes of QGB structures in terms of σ and d. The vertical dashed white line corresponds to the critical value ${\sigma _{crit}} = 1/{2^{3/4}}$.

Download Full Size | PDF

4. Properties of the quadrupole Gaussian beam

Some features of QGB can be understood using their symmetry properties and asymptotic behavior.

4.1 Symmetries

The symmetry of a QGB is determined by the symmetry of its initial condition $\phi \left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w} ,0} \right) = {P_Q}\left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w} } \right){\phi _{00}}\left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w} ,0} \right)$. The function ${\phi _{00}}\left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w} ,0} \right) = {e^{ - \pi |w{|^2}/{z_R}}}$ is real and invariant under any rotation $w \to {e^{i\theta }}w$. On the other hand, the quadrupole polynomial P_Q at $z = 0$ as can be checked in Eq. (9), is invariant under $\pi $ rotations in the $xy$ plane but also under the combined effect of complex conjugation and a $\pi /2$ rotation. Physically, this means that we recover the same structure at $z = 0$ either by rotating the beam around the z axis by $\pi $ or, equivalently, by rotating it by $\pi /2$ and conjugating at the same time (thus converting a positive charge into a negative one and vice versa) as seen in Figs. 2–4. This implies that (we use C for the conjugation operator and ${\mathrm{{\cal C}}_N}$ for a discrete rotation of angle $2\pi /N$)

Due to the invariance of ${H_0}$ under rotations, the ${\mathrm{{\cal C}}_2}$ invariance of the initial condition is also preserved under propagation:

On the other hand, the evolution operator ${e^{i{H_0}z}}$ changes under the action of the $C{\mathrm{{\cal C}}_4}$ operator as the “time” reversal operator $T:z \to - z$ (mirror reflection with respect the $z = 0$ plane) changes the sign of propagation:

Using the definition of the evolution operator $|{{\phi_Q}(z )\rangle} = {e^{i{H_0}z}}|{{\phi_Q}(0 )\rangle} $ and the commutation relation (Eq. (14)) we find

Or equivalently,

These properties can be directly checked on the explicit expressions for ${\phi _{00}}\left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w} ,z} \right)$ and ${P_Q}\left( {w,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over w},z} \right)$ taking into account that the beam parameter q verifies $q({ - z} )={-} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over q} (z )$. For the ${P_Q}$ polynomial, it is straightforwardly checked from Eq. (11) that

The physical consequences of these symmetries are apparent in all figures of beam classes Figs. 2–4. All dark beam structures are invariant under $\pi $ rotations around z. Looking at the top views, which are projections on the $({0,0,\infty } )$ plane, one can appreciate the $\pi /2$ symmetry of the dark beam provided we change the sign of its charge (color) under rotation. Note that the $\pi /2$-rotated counterpart of a dark ray at z is located at $- z$. That is, $C{\mathrm{{\cal C}}_4}$ is equivalent to a mirror reflection as expressed in Eq. (17).

4.2 Asymptotic dark beam structure

Far field properties of the Gaussian beams are particularly useful since they have simpler experimental access. For QGB these properties can be easily obtained by studying the asymptotic behavior of the ${P_Q}$ polynomial for $z \gg {z_R}$. In this regime all dark ray trajectories follow straight lines. So, we can study the zeroes of ${P_Q}$ in the limit $w \to \sigma z$. At leading order in an $1/z$ asymptotic expansion around $z = \infty $, we obtain a complex polynomial for the complex slope $\sigma $:

Using the modulus and argument decomposition for the complex slope $\sigma = {\sigma _\infty }{e^{i{\theta _\infty }}}$, we get the binomial form of the complex function $P_Q^{\textrm{ass}}$

Its imaginary part provides the angular asymptotic positions of the dark beams in the far field: $\textrm{Im}({P_Q^{\textrm{ass}}} )= 0 \Rightarrow \textrm{cos}({2{\theta_\infty }} )= 0$, which correspond to the polar angles ${\theta _\infty } ={-} 3\pi /4,{\; \; } - \pi /4,{\; }\pi /4,3\pi /4$, i.e., the same angular positions as in the initial condition. This asymptotic distribution can be appreciated in the top views of Figs. 2–4. On the other hand, $\textrm{Re}({P_Q^{\textrm{ass}}} )= 0$ is a second order polynomial in $\sigma _\infty ^2$ (since ${\sigma _\infty } \ge 0$ we ignore negative roots), which provides one solution for the radial asymptotic position $|{{w_{\textrm{ass}}}} |\sim {\sigma _\infty }z$ if $\sqrt {{z_R}} /a < \sqrt \pi /{2^{1/4}}$ or two solutions if $\sqrt {{z_R}} /a > \sqrt \pi /{2^{1/4}}$. Equivalently, using the dimensionless parameter $\sigma = {W_0}/d = \left( {1/\sqrt {2\pi } } \right)\sqrt {{z_R}} /a\; $ as in the phase diagram in Fig. 5, we can define the critical parameter

Note that the asymptotic polynomial (Eq. (18)) recovers the original $C{\mathrm{{\cal C}}_4}$ symmetry of its $z = 0$ counterpart: $P_Q^{\textrm{ass}}\left( {\sigma ,\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over \sigma } } \right) = \bar{P}_Q^{\textrm{ass}}\left( {{e^{i\pi /2}}\sigma ,{e^{ - i\pi /2}}\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftharpoonup$}} \over \sigma } } \right)$.

4.3 Stability of dark beams

As in superfluidity or superconductivity vortex singularities can be studied in terms of effective vortex-vortex interactions. From this point of view, phase singularities in the case of conformal Gaussian beams behave as a set of non-interacting particles since all trajectories are straight lines no matter their relative position. This feature seems compatible with the fact that the conformal Gaussian beams fulfill Eq. (1), which is formally identical to the Schrödinger equation of a free particle.

However, phase singularities of the QGB are a completely different matter. They follow nontrivial trajectories except at long distances, which indicates they possess intricate dynamics when considered as particles. This effective non-triviality of their dynamics poses the question of whether the different classes of dark beam structures previously reported are stable or not under perturbation of initial conditions. This question makes sense despite the original equation for the beam field being linear.

From an experimental point of view, dark beam stability is closely linked to the tolerance of the optical beam features under misalignments and other inherent symmetry-breaking processes unavoidable in realistic measurements. For this reason, we chose to study the evolution of beam trajectories when the initial condition is perturbed by randomly moving the four phase singularities from their symmetric position for a given class of QGB. The dark beam structure will be stable if the number of points ${N_p}$ where pairs are created or annihilated for $z > 0$ remains unchanged under perturbations of reasonable strength. It is clear that if the perturbation is as big as to make the ratio σ=W₀/d change enough to become larger or smaller than some of the critical values in the phase diagram (Fig. 5), we would observe a change in N_p.

We show in Fig. 6 the results obtained by a perturbation of the initial condition of the class I QGB depicted previously in Fig. 2 The perturbation consists in modifying the positions of the singularities of the quadrupole configuration at $z = 0$ with respect their original symmetric locations as shown in Fig. 6(a). This is done by adding a random complex number of range $\epsilon = 0.2$ at every singularity location and analytically calculating the resulting perturbed QGB by means of the same technique previously used. The result obtained shows that the curves $\textrm{Re}(\phi )= 0$ and $\textrm{Im}(\phi )= 0$ are deformed with respect the symmetric case for every value of $z > 0$ but pairs are annihilated and created at exactly 6 points, as in the symmetric case. The dark beam structure is no longer symmetric but ${N_p}$ remains invariant. Thus, we obtain a deformed version of a class I QGB perfectly recognizable in the experiment.

 

Fig. 6. Stability of class I QGB’s under a random complex displacement of range $\epsilon = 0.2$ of symmetric charge positions at $z = 0$ (breaking the original $C{\mathrm{{\cal C}}_4}$ rotational symmetry). Curves for $Re(\phi )= 0$ (solid) and $Im(\phi )= 0$ (dashed) at different relevant axial positions are shown (same parameters as in Fig. 2). Circle and square crossing points indicate phase singularities for the original and perturbed cases, respectively. Curves are shown in the following cases: (a) initial condition at $z = 0$. (b)-(c) below and above the annihilation of the original two pairs in the symmetric case at $z = {z_{loop}}$. (d) right above the generation of the first two pairs in the symmetric case at $z = {z_{pair,1}}$. (e) right above the generation of the two extra pairs in the symmetric case at $z = {z_{pair,2}}$. Only one extra pair is generated in the perturbed case. (f) further above $z = {z_{pair,2}}$ a second pair is generated in the perturbed case. The total number of space points where pairs are generated or annihilated for all values of $z > 0$ is 6 in both cases.

Download Full Size | PDF

5. Experiment

The scheme of experimental setup is shown in Fig. 7. We used a Spatial Light Modulator (SLM, LCR 720 Holoeye pixel size 20µm, dimension 1280 × 768 pixel) to imprint a vortex structure into a laser beam. He-Ne laser beam (633 nm) was expanded so the Gaussian beam of a diameter 5.4 mm illuminated SLM. The quadrupole of the side size 2a = 0.8 mm was displayed on SLM. To have access to both arms of the Gaussian beam (in front of and behind the beam waist) additionally a phase mask of the Fresnel lens was added to the vortex quadrupole mask and the beam divergence angle was 43 mrad. CCD camera (pixel size 5.86µm, dimension 1920 x 1200 pixel) was moved along the optical axis. For each camera position the intensity distribution and interferogram have been detected. At the interferograms we looked for the fork fringes to verify whether the vortices are present.

 

Fig. 7. Scheme of the experimental set-up. To focus the beam reflected from the SLM, a Fresnel lens phase mask was added to the quadrupole phase mask.

Download Full Size | PDF

To determine the quadrupole trajectory we needed to find vortex position within the beam. To do this we adopted the vortex filtering method [21–23] together with the specially trained neural network [24,25]. The vortex filtering generates the pseudo-phase map on which the spiral phase geometry of the vortex is visible. The procedure is schematically shown in Fig. 8. Figure 8(a) presents the detected intensity image with the preliminary identified regions of low intensity. Then image is divided into four parts, each containing the separate vortex (Fig. 8(b) upper row). Then the numerical vortex filtering is applied to these images and the extracted spiral pseudophase is shown in Fig. 8(b) (bottom row). The specially trained neural network localizes the end of the spiral showing the vortex position. Vortices found in this way (indicated on the intensity image by white circles) are shown in Fig. 8(c). The black cross in Fig. 8(c) is their centroid position, that allow us to adjust the beam center along the optical axis.

 

Fig. 8. (a) Detected intensity image with the marked dark regions, where the vortices are located, (b) upper row -four regions to which the vortex filtering was applied, bottom row- the extracted pseudo phase images, red circle indicate vortex position).

Download Full Size | PDF

As was stated in the theoretical section, quadrupole vortex configurations split into classes depending on the parameter σ value. We verified experimentally the main features of these structures.

Figure 9 shows the exemplary results for the vortex quadrupole annihilation during propagation (class 3). Vortex forming quadrupole were indicated there by the black spots. They were connected by the red solid lines to show the shape of the quadrangle (of sides a1, a2, b1, b2). The dashed-line circle is the diameter of the Gaussian beam. A vortex quadrupole (confirmed by the four fork interference fringes) is formed far before and behind the focusing distance. When the observation plane z approaches the focus, the vortices along sides a1 and a2 of the quadrangle become closer and finally annihilate (fork fringes disappear). And then on the opposite site of the focus vortex quadrupole is recovered. This behavior is clearly seen in the xz and yz projections shown in Fig. 10(a), (b). Figure 11(a) shows the xy projections at different z planes. We can see here the evolution of the quadrangle defined by quadrupole vortex configuration. We observe the stable quadrangle orientation and expansion of its sides a and b. The evolution of the lengths a and b along the z-axis are shown in Fig. 11(b). The length of the side a increases much faster than the length of side b. At the longer z distance they become equal. The near field loop was not possible to detect, as the focus spot was too small and too bright.

 

Fig. 9. Quadrupole trajectory within the Gaussian beam. Black spots indicate vortex position. Vortex positions are connected by the red quadrangle (sides a1, a2, b1, b2) at each observation plane. Dashed red circle is the Gaussian beam diameter.

Download Full Size | PDF

 

Fig. 10. XZ (a) and YZ (b) projections of the trajectory shown in Fig. 9. Here the red spots indicate vortices.

Download Full Size | PDF

 

Fig. 11. (a) XY projection from Fig. 9 (for several z-positions behind the focus, larger quadrangles correspond to further distance); (b) change of the quadrupole size with propagation.

Download Full Size | PDF

Another characteristic feature that appears during quadrupole propagation is formation of the additional branch of vortex-antivortex pair (class 1 and 2). To observe it, the laser beam was less expanded (the beam divergence angle was 4 mrad), but the quadrupole size displayed on SLM was unchanged (0.8 mm). The results are presented in Fig. 12 where one can see intensity distribution (in a log scale), interferogram and recovered phase map (to do this we applied Fourier method [26] to interferogram). Black dashed circles indicate four fork fringes in the central part of the beam. The additional fork fringes are also visible (indicated by the yellow dashed circles). The sign of the optical vortex can be recognized by the fork orientation, or directly from the recovered phase map (here the black color corresponds to phase equal to zero, and white color to phase 2π respectively).

 

Fig. 12. The detected additional vortex anti-vortex pair, (a) intensity picture (log scale), (b) interferogram, (c) recovered phase map (black color correspond to phase equal to 0 and white to 2π).

Download Full Size | PDF

Similar experiment has been performed for the conformal beam, where all vortices had the same topological sign. As was pointed in the section 2, its propagation is different than in the case of quadrupole configuration. First of all, there is no vortex annihilation at the focal range. The interferogram shown on the left side of Fig. 13 is detected for the same conditions as that one from Fig. 9. We can see here four fork fringes indicating the presence of the vortices in the beam. We can also observe the rotation of the whole pattern which is also in agreement with the theoretical part.

 

Fig. 13. Propagation of the optical vortex structure with the same topological sign. At the focal region where the vortices in the quadrupole annihilate, the vortices of the same topological sign do not disappear (see interferogram on the left). Additionally, we observe the quadrangle rotation when moving along the z-axis which is not observed in the case of vortex quadrupole propagation.

Download Full Size | PDF

6. Conclusions

The general theory describing the propagation of multivortex system embedded into a Gaussian beam was presented in papers [19,20]. The theory reduces the problem to the analysis of complex polynomials. In general, the problem is split into conformal case with vortices having the same topological charge and non-conformal one. In the latter case the vortices behavior is complex. In this work the system of four optical vortices, forming a square at the beam waist, was investigated. The dark rays (vortex trajectories) were investigated by analyzing the zeros of polynomial (11). When the vortices have the same sign of topological charge (conformal case) the trajectories are of simple geometry. The vortices preserve their square arrangement, however it rotates and changes its size with the beam divergence. They behave like non-interacting particles. This was confirmed experimentally (Fig. 13). In case of vortex pairs of positive and negative charges (QGB case) the system exhibits a rich behavior- the vortex pairs can annihilate or create. We have shown that the creation and annihilation of vortex pairs during the beam propagation can be controlled by a single parameter σ that is related to the vortex quadrupole versus the Gaussian beam waist. This parameter defines the initial condition of vortex system. Three different classes of vortex trajectories can be identified on the base of number of created or annihilated vortex pairs Np (section 3.2). These classes are strictly related to the system symmetries as discussed in section (4.1). We have also shown that the vortex trajectory pattern is resistant against small deviation from perfect initial parameters (section 4.3). Moving from one class to another can be considered as a phase transition (Fig. 5). Thus, we can expect that they preserve their character in real optical system, which enables experimental verification as presented in section 5. It also means that our results have practical meaning as we can control in the systematic way this rich internal beam arrangement. Our analysis of the QGB case shows that this relatively simple case exhibits rich behavior including the vortex pairs’ creation and annihilation. The findings presented in the manuscript contribute to a deeper understanding of the complex behavior of vortices in laser beams and their potential applications in modern optics. The study opens up new possibilities for controlling and manipulating vortices in laser beams, which could have practical applications in various fields of optics and photonics.