As members of the thesis committee, we certify that we have read the thesis prepared by Zhamila Kulçukova with the title Final approval and acceptance of this thesis is conditional upon the submission of final copies of the thesis by the candidate to the Department of Physics. This study has the potential to define the specifics of the underlying mechanism that causes spontaneous nodulation.

In turbulence, one of the unsolved problems is the Navier-Stokes equation, which describes the flow of fluids; knots that appear in a turbulent flow can describe the transport properties of the plasma [3]. Uncovering the basic mechanisms of knotting is a difficult task due to the complexity of the system, so to understand it, we have to investigate mainly simpler cases.

## Lasers

To better understand the context of this problem, it is essential to first review the fundamental concepts. The laser beams can also have different shapes and configurations depending on the specific design of the optical resonator and the properties of the active medium [20]. The type of paraxial laser beam that we consider in our research has the Gaussian intensity profile and is therefore called the Gaussian beam [21–23].

The equation is derived through the paraxial wave equation (hence the name - paraxial laser beam) [24], and a more comprehensive description of the Gaussian beam can be found in Chapter 2. One of the properties of the Gaussian beam that makes it useful in our research is that it is also widely used to form optical vortices.

## Optical vortices

For this particular thesis, I focus on paraxial laser beams, which are beams that propagate at small angles with respect to the optical axis. Beams of this type are very well aligned, meaning they stay focused over long distances.

## Motivation

First, the paraxial wave equation is derived and the equation describing the Gaussian beam electric field is presented in more detail. In addition, this section provides an overview of the paper presenting vortex rings resulting from the interference of a radially symmetric beam and a plane wave, which serves as the basis of my research.

## Paraxial wave equation

Recalling the properties of paraxial laser beams, assuming propagation in the z-direction, the largest component of the wave vector must be k = kxxˆ+kyyˆ+kzˆz kz. As mentioned, for my thesis I am using a solution that specifically describes the Gaussian beam, which I will discuss in more detail in the next subsection.

## Gaussian beams

The shape of the beam is defined by the center w0 and wave number k. The mentioned mode is called the fundamental Gaussian beam and is given in the following form: 2.12) Now, using the scaling properties of the system, the dimensionless transverse variables (x, y, t) are introduced and derived from the dimensional variables (˜x,y,˜z) in the following˜ manner: x = ˜x /w0, y= ˜y/w0, t= ˜z/zR and κ=kzR. The amplitude E0 is normalized to 1. This will give the equations governing the shape of the beam in eq. 2.13b) Consequently, the Schr¨odinger-type paraxial wave equation in eq.

## Vortex rings

This equation is the dimensionless version of the fundamental Gaussian beam mode described in Eq. Looking back at Chapter 1, it is impossible to obtain vortices by examining only the zero-intensity lines in the field. To find the vortices, both the amplitude of the field and the phase must be analyzed. The solution to the equation is given in the form of spheroids: enclosed surfaces in three dimensions that resemble a deformed sphere. 2.18a) and by defining ρ(t) = r, the mathematical form of the spheroids is obtained:. 2.18b), the equation for the phase at the spheroids can be obtained:

Solving for a parameter ϕ, the values of t in eq. 2.20) indicate the locations of the rings along the propagation direction. 1 +t2 have an opposite topological charge to the rings outside them, meaning they spin in the opposite direction. The rings in this arrangement appear via two mechanisms: nucleation, in which a vortex ring collapses into a single point, and annihilation, in which two rings with opposite topological charges merge and cancel each other out.

3a, the single ring is nucleated by a single point along the optical axis, corresponding to one of the endpoints of the spheroid. 3b however, both mechanisms occur: the rings nucleate at endpoints ρ= 0 and annihilate at ρ=√. First, I derive the equation describing the elliptical Gaussian beam and consider the same interference problem as in Eq.

Although rotations are not unified within the scope of this thesis, they provide the basis for future research on knot formations, and therefore the equation is only mentioned here. 2.3, however, depending on the elliptical beam, the rings are deformed to different degrees. Furthermore, I suggest that for a very specific set of parameters P, ϕ and ellipticity ϵ, the rings can be reconnected and self-crossed.

## Elliptical Gaussian beams

The vortex rings are similar to fig. 2.3, but depending on the elliptical shape of the beam, the rings are deformed to different degrees. To do so, I solve the same Schr¨odinger-type paraxial wave equation as in Eq. 3.6) Variable θ introduces an additional degree of freedom; at fixed θ the beam is non-rotating, but by varying θ in time the rotation of the beam can be introduced.

To reiterate, this equation describes the rotating elliptic radius, although not used, it is important for future investigation.

## Vortex rings and deformations

Now solve for the real and imaginary parts in eq. 2.19), describes the intensity isosurfaces (spheroids) on which the rings are located. 2.19), where it was possible to easily solve for r, this equation will remain in an implicit form. However, this still allows me to numerically solve for the intersection of the two surfaces cf. Due to the symmetry of sinϵ and cosϵ functions, the spheroids of intensity P, phase ϕ and ellipticityϵ=π/4±ϵ0 will have the same shape and behave in the same way despite being elongated along different axes.

The circumference of the spheroids can be calculated by setting x= 0 and y = 0 and evaluating the resulting equation (t2−sin22ϵ)2+ 4t2 =P−4sin42ϵ for the set of parameters {ϵ, P}. The same can be seen from fig. 4b, the shape of the ϵ = 0.456 curve has a visible peak along the x -axis compared to other curves with milder ellipticity. It may appear as if the hoop is bending toward the center, but in fact it is bending outwards, as the greater degree of deflection on one of the axes "pulls" the hoops outward.

In this case, ϵ= 0.628<0.785 (ellipticity of a radially symmetric beam) means that the degree of diffraction is higher along the y-axis, so the ring is curved outwards at x= 0. Looking more closely at the ring shapes, at lower ϕ the two rings on the right side of the spheroid deform towards the point between them on the oni = 0 plane. Although I say it is "bent towards the origin", in reality the high ellipticity of ϵ= 0.571 means that the deformation of the annulus more severe, causing the edges of the annulus to be pulled outwards more than appears to be the case with milder ellipticities.

Self-crossing of the ring is a phenomenon that was not expected to be rediscovered. Moreover, there is not a single justification for this behavior of the ring. Taking all of the above into account, it is possible to predict vortex ring reconnections and self-transitions by tracking a very specific set of parameters {ϵ, P, ϕ}.

## Reconnections and self-crossings

Further observing the loops for different parameters, with a slightly higher ellipticity and ϕ close to 0, the loop on the left side of the beam appears to cross itself, bending towards the origin, as shown in the figure below: The blue line represents the curve along the extreme points; the red line ε=π/4≈0.785 is the ellipticity of the symmetric radial radius given for. For a fixed ϵ, the curve is anti-symmetric about the ψ0 axis, meaning that a specific ϕcandtc will give the same result, but on the other side of the beam compared to -ϕc and -tc, as if the image were mirrored .

They are divided into four: the lowest unmarked region is where no reconnection occurs beyond t = 0; the area where self-transition can occur, labeled as "self-transition" on the diagram;. region where reconnections occur only outside oft= 0; and the uppermost unmarked. The lower limit of the region is clearly the lower limit of the extremal curve, while the upper limit of the region is located by finding ϵ, where one set of extreme points is at ψ0 = 0. So, intuitively, it may seem like the upper limit for the self-transition region is at ϵ= 0.591, but observations suggest otherwise.

This is the only feasible explanation why this relation makes ϵ= 0.578 the upper limit for the self-intersecting region. The shape and locations of the vortex rings formed as a result of the interference of the elliptical Gaussian beam and the plane wave are determined by three parameters: the relative amplitude P, the phase difference ϕ and the elliptical beam ϵ. The two types of recombinations have been identified: recombinations that fuse the rings and the self-crossing of a single ring.

It was revealed that for a fixed ϵ, the reconnections occur at the extremum points of the phase ψ(t). In the astigmatic beam, the focal points of the x and y components of the beam do not coincide, causing the asymmetry that is not present in the current setup. Previous research has indicated that the presence of beam rotation leads to the appearance of vortices, so after observing the effects of astigmatism, the next step is to introduce the case of the rotating beam.

Dennis, "Both and coupled phase singularities in monochromatic waves", Proceedings of the Royal Society of London A. Prokhorov, "Theory of molecular generator and molecular power amplifier", Soviet Journal of Experimental and Theoretical Physics.