Preface

This book discusses recent advances in vortex dynamics. It includes nine chapters written by twenty-two authors from Turkey, Czech Republic, Japan, China, Italy, Mexico, Sweden, India, and Spain.

Chapter 1 investigates the existence and stability properties of dipole solitons in a nonlocal nonlinear medium with self-focusing and self-defocusing quantic nonlinear responses. The second chapter is a review chapter in which the authors examine the existing literature on the propagation of coherent or partially coherent vortex beams through a random medium. Chapter 3 presents an algorithm developed to detect individual vortices via direct fitting of the measured velocity field. Chapter 4 discusses the existing literature on vortex dynamics in complex fluids by considering Taylor vortex flow. The fifth chapter discusses rotating fluid flows affected by a β-effect and blood flow through a natural or artificial valve in the left ventricle. The next chapter is a theoretical work that studies a system of partial differential equations that is related to point vortices that appear in fluid dynamics. Chapter 7 discusses the features of vortex structures that are shown to exist in the plasma wake of Venus and the momentum transport phenomena of the vortex motion. Chapter 8 is a discussion of the existence of vortex structures in a dusty plasma medium. Finally, Chapter 9 analyzes and compares the horseshoe vortex and some other known models applied to biomimetic Micro Aerial Vehicles (MAVs).

This book is a useful resource for researchers, scientists, and postgraduate students in academia as well as industry.

> **Dr. İlkay Bakırtaş and Dr. Nalan Antar** Faculty of Science and Letters, Mathematical Engineering Department, Istanbul Technical University, Istanbul, Turkey

Section 1

## Vortex Structures in Nonlinear Optics

#### **Chapter 1**

## Dipole Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing Quintic Nonlinear Responses

*Mahmut Bağcı, Melis Turgut, Nalan Antar and İlkay Bakırtaş*

#### **Abstract**

Stability dynamics of dipole solitons have been numerically investigated in a nonlocal nonlinear medium with self-focusing and self-defocusing quintic nonlinearity by the squared-operator method. It has been demonstrated that solitons can stay nonlinearly stable for a wide range of each parameter, and two nonlinearly stable regions have been found for dipole solitons in the gap domain. Moreover, it has been observed that instability of dipole solitons can be improved or suppressed by modification of the potential depth and strong anisotropy coefficient.

**Keywords:** dipole solitons, nonlinear response, nonlocal nonlinear medium, quintic nonlinearity

#### **1. Introduction**

Many phenomena in nature are modeled mathematically using nonlinear differential equations. Traveling wave solutions of nonlinear partial differential equations play a significant role in nonlinear wave propagation problems that are observed in various fields such as nonlinear optics, fluid dynamics, plasma physics, elastic media, and biology [1]. Some of the solutions to such nonlinear wave propagation problems are called solitons, which are localized wave solutions.

Optical solitons are formed because of the balance between the medium's diffraction and the self-phase modulation [2]. As a consequence of this, an optical field that does not change its shape occurs during propagation [3]. Recently, spatial solitons that can be used for optical switching and processing applications [4] have been extensively investigated in nonlinear optical systems with external optical lattices. There is a considerable amount of research about this subject in the literature. In 2003, Segev et al. experimentally observed spatial solitons in optically induced periodic potentials [5]. Fundamental and vortex solitons with real or complex lattices have been investigated in optical media with the cubic Kerr-type [6–10], the saturable [11], and competing nonlinearities [12]. Moreover, the existence of solitons has been observed in aperiodic or quasicrystal lattice structures [13–18] and the lattices that possess defects [19, 20] and dislocations [21, 22].

The dynamics of solitons are governed by nonlinear Schrödinger (NLS) type equations in optical media with nonlinearities and/or external potentials as in the referred studies. Additionally, the cubic nonlinear NLS equation needs to be modified to describe nonlinear optical materials that have both cubic and quadratic nonlinear responses [23–29], such as potassium niobate (KNbO3) [30] or lithium niobate (LiNbO3) [31]. These dynamics in quadratically polarized media are governed by the NLS equation with coupling to a mean term (d.c. field), which are denoted as NLSM systems and sometimes referred to as Benney-Roskes or Davey-Stewartson systems [32, 33].

NLSM equations were first studied by Benney and Roskes for water of finite depth in the free surface conditions in 1969 [32]. Later, in 1974, Davey and Stewartson derived the limiting integrable case, which is a reduced case of the Benney–Roske's system by studying the evolution of a 3D wave packet for water of finite depth [33]. In 1975, Ablowitz and Haberman [34] studied the integrability of NLSM systems in the shallow water limit. The effects of surface tension were included in the results of Benney and Roskes by Djordevic and Reddekopp [35] in 1977. From the first principles, Ablowitz et al. [23, 36, 37] discovered that NLSMtype equations describe the evolution of the electromagnetic field in a quadratic nonlinear media. The general NLSM system is given by [23, 36, 37]

$$
\rho \dot{u}\_x + \Delta u + |u|^2 u - \rho u \phi = 0,\\
\ \phi\_{\text{xx}} + v \phi\_{\text{yy}} = \left( |u|^2 \right)\_{\text{xx}},\tag{1}
$$

where *u*(*x, y, z*) corresponds to the normalized amplitude of the envelope of the static electric field propagating in the *z* direction, x and y are transverse spatial coordinates. Δ*u* � *uxx* þ *uyy* corresponds to diffraction, the cubic term in *u* originates from the Kerr-type nonlinear change of the refractive index. The parameter *ρ* is a coupling constant that comes from the combined optical rectification and electro-optic effects modeled by the *ϕ*ð Þ *x*, *y* field, and *v* is the coefficient that comes from the anisotropy of the material [37]. Such systems of equations arise due to the growth and depletion of the fundamental and second-harmonic fields at the moment that the phase velocity of the fundamental and the second-harmonic wave are not equal during propagation [38]. When the phase-matching condition is not satisfied, the equation of the second-harmonic field can be solved directly and generates an additional self-phase modulation contribution as a result of cascaded nonlinearity. Similarly, the NLSM systems describe the nonlocal–nonlinear coupling between the first harmonic with the cascading effect from the second harmonic and a static field that is related to the mean term [36, 37].

Wave collapses play a significant role in various branches of science. The peak amplitude of the wave solutions tends to infinity (blow-up) in finite time or finite propagation distance when a singularity occurs. This phenomenon is often called wave collapse [39]. In the NLS equation, it was first observed numerically by Kelley in 1965 [40]. In fact, this wave collapse phenomenon is similar for the NLSM systems. Wave collapse in the NLSM systems occurs with a modulated profile [41]. Merle and Raphael [42] analyzed the collapse behavior of the NLS equation and other related equations in detail. Moreover, Moll et al. investigated experimental observations of optical wave collapse in cubic nonlinearity and showed that the amplitude of the wave increases as the spatial extent decreases in a self-similar profile [43]. In Ref [39], Ablowitz and coworkers studied wave collapse that occurs with a quasi-self-similar profile in the NLSM system and found that collapse can be arrested by the small nonlinear saturation. Furthermore, in Ref [30], NLSM collapse was arrested by wave self-rectification. In this aforementioned study, they considered only the nonlinear evolution of beams with an initial Gaussian beam profile with several values of input power and/or beam ellipticity and found that the wave collapse can be arrested by increasing the coupling constant *ρ* or for an initially highly elliptic beam. Recently, the NLSM system collapse was arrested by adding a

*Dipole Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing… DOI: http://dx.doi.org/10.5772/intechopen.106207*

real periodic [24] and partially parity-time-symmetric [44] and azimuthal [45] external lattices (potential) to the governing system, and it was shown numerically that modification of potential depth provides great controllability on the stability of soliton.

More recently, Bağcı et al. [46] have numerically investigated stability dynamics of fundamental lattice solitons that are solutions of extended NLSM system in a nonlocal nonlinear medium with self-focusing and self-defocusing quintic nonlinearity. It has been shown that as the absolute value of *γ* increases for both self-focusing and self-defocusing cases, the obtained fundamental solitons become nonlinearly unstable. However, the stability of unstable fundamental solitons can be improved by modification of potential depth [46].

Dipole (two-phased) and higher-phase vortex solitons in the presence of an induced lattice have been studied analytically and experimentally in Bose-Einstein condensates (BECs) [47, 48] and in optical Kerr media [49–54]. In recent years, these types of solitons have attracted considerable interest because of their unique features and potential applications [55].

In this chapter, we numerically study the existence and stability of dipole soliton solutions of the NLSM system in a nonlocal nonlinear medium with the selfdefocusing quintic nonlinear response by adding an external lattice. In fact, this study is about the dynamics of dipole solitons instead of fundamental solitons in the problem that Bağcı and coworkers have addressed in recent book chapter [46]. The purpose of this study is to numerically investigate the effects of the strength of quintic nonlinearity that specify characteristics of the model and variation of potential depth on the existence and stability of dipole solitons. In several applications, many optical materials such as chalcogenide glasses are required quintic and seventh-order effects in addition to cubic nonlinear effects [56], and effective higher-order nonlinearities can reveal with pure Kerr materials in an inhomogeneous propagation media [57–59].

The chapter is outlined as follows: In Sec. 2, we present the model equations, and the squared-operator method is explained so that it is modified for the model. The dipole solitons are computed by this numerical method. Nonlinear evolution of the dipole solitons is examined to perform stability analysis, In Sec. 3. Finally in Sec. 4, results of this study are outlined.

#### **2. The model**

In this chapter, we modify the NLSM system (1) as follows to describe the dynamics of lattice solitons in a nonlocal nonlinear medium with cubic and quintic nonlinearity

$$\begin{aligned} i u\_x + \frac{1}{2} \Delta u + \beta |u|^2 u - \rho u \phi + \gamma |u|^4 u - V(x, y) u &= 0, \\ \phi\_{\text{xx}} + \nu \phi\_{\text{yy}} &= \left( |u|^2 \right)\_{\text{xx}} \end{aligned} \tag{2}$$

where *γ* is the coefficient of quintic nonlinearity and *V*(*x, y*) is the optical lattice. In this chapter, we consider lattices that can be written as the intensity of a sum of *N* phase-modulated plane waves [13]

$$V(\mathbf{x}, \mathbf{y}) = \frac{V\_0}{N^2} \left| \sum\_{\mathbf{z}=\mathbf{0}}^{N-1} e^{i \left(k\_\mathbf{x}^\mathbf{x} + k\_\mathbf{y}^\mathbf{z}\right)} \right|^2,\tag{3}$$

where *V*<sup>0</sup> *>* 0 is the peak depth of the potential and the wave vector *k<sup>n</sup> <sup>x</sup>*, *<sup>k</sup><sup>n</sup> y* <sup>¼</sup> ½ � *K* cos 2ð Þ *πn=N* , *K* sin 2ð Þ *πn=N :* The potential for *N* = 2*,* 3*,* 4*,* 6 yield crystal (periodic) lattices, while *N* = 5*,* 7 yield quasi-crystals (aperiodic) lattices. Contour image, contour plot, and diagonal cross-section of the lattice *V*(*x, y*) are plotted in **Figure 1** for *V*<sup>0</sup> ¼ 12*:*5, *N* = 4 and *kx* ¼ *ky* ¼ 2*π:* It can be seen that the lattice is periodic, and the center of lattice is a local maximum.

#### **2.1 Numerical solution for the dipole solitons**

Yang and Lakoba developed an iterative numerical method called the squaredoperator method (SOM) [60]. The idea of this method is to iterate a modified differential equation whose linearization operator is square of the original equation together with a preconditioning (or acceleration) operator. To obtain the soliton solution of the (2+1)D NLSM model, this method is modified as follows:

Soliton solutions are sought in the form *u x*ð Þ¼ , *<sup>y</sup>*, *<sup>z</sup> U x*ð Þ , *<sup>y</sup> ei<sup>μ</sup><sup>z</sup>* where *<sup>U</sup>*(*x, y*) is real-valued function and *μ* is the propagation constant (or eigenvalue). Substituting the ansatz *u*(*x, y, z*), we get the following expressions:

$$\begin{aligned} \mu\_x &= i\mu U e^{i\mu x}, \\ \mu\_{xx} &= U\_{xx} e^{i\mu x}, \\ \mu\_{yy} &= U\_{yy} e^{i\mu x}, \\ \left| \mu \right|^2 &= U e^{i\mu x} U^\* e^{-i\mu x} = \left| U \right|^2, \\ \left| \mu \right|^4 &= \left| U \right|^2 \left| U \right|^2 = \left| U \right|^4 \end{aligned} \tag{4}$$

where *<sup>U</sup>* <sup>¼</sup> *<sup>U</sup>* <sup>∗</sup> in our case. Substituting the set of the terms in Eq. (4) into the (2+1) NLSM model, the following nonlinear equations for *U* are obtained

$$\begin{aligned} -\mu U + \frac{1}{2}\Delta U + \beta |U|^2 U - \rho \phi U + \gamma |U|^4 U - VU &= 0, \\ \phi\_{\text{xx}} + \nu \phi\_{\text{yy}} &= \left( |U|^2 \right)\_{\text{xx}}. \end{aligned} \tag{5}$$

**Figure 1.**

*(a) Contour image, (b) contour plot, and (c) diagonal cross-section of the lattice V(x, y) when V*<sup>0</sup> ¼ 12*:*5, *N = 4 and (x,y) [15,15].*

*Dipole Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing… DOI: http://dx.doi.org/10.5772/intechopen.106207*

Applying the Fourier transform to the eigenequations system (5) yields

$$\begin{split} -\mu \hat{U} - \frac{1}{2} \left( k\_x^2 + k\_y^2 \right) \hat{U} + \mathcal{F} \left\{ \beta |U|^2 U - \rho \phi U + \gamma |U|^4 U - VU \right\} &= 0, \\ k\_x^2 \hat{\phi} + \nu k\_y^2 \hat{\phi} &= k\_x^2 \mathcal{F} \left\{ |U|^2 \right\}, \end{split} \tag{6}$$

where <sup>F</sup> denotes the Fourier transform, *<sup>U</sup>*^ <sup>¼</sup> <sup>F</sup>f g *<sup>U</sup>* , *kx* and *ky* are the Fourier transform variables. Isolating *ϕ*^ from the second equation of Eq. (6) gives

$$
\hat{\phi} = \frac{k\_\times^2 \mathcal{F}\left\{|U|^2\right\}}{k\_\times^2 + \upsilon k\_\times^2}.\tag{7}
$$

Taking the inverse Fourier transform of Eq. (7), we get

$$\phi = \mathcal{F}^{-1}\left\{\frac{k\_\times^2 \mathcal{F}\left\{\left|U\right|^2\right\}}{k\_\times^2 + \nu k\_\times^2}\right\},\tag{8}$$

where F �<sup>1</sup> denotes the inverse Fourier transform and during iteration, the first element of *k*<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> *vk*<sup>2</sup> *<sup>y</sup>* is set to 1 in order to avoid division by zero error. By applying the inverse Fourier transform to first equation of Eq. (6) and substituting Eq. (8) into the obtained equation, we get

$$-\mu U + \mathcal{F}^{-1}\left\{-\frac{1}{2}\left(k\_x^2 + k\_y^2\right)\hat{U}\right\} + \beta |U|^2 U - \rho \mathcal{F}^{-1}\left\{\frac{k\_x^2 \mathcal{F}\left\{|U|^2\right\}}{k\_x^2 + \nu k\_y^2}\right\} U + \gamma |U|^4 U - VU = 0. \tag{9}$$

To obtain operator *L*0, Eq. (9) can be written as

$$L\_0 U = \mathcal{F}^{-1}\left\{-\frac{1}{2}\left(k\_x^2 + k\_y^2\right)\hat{U}\right\} + T\_0 U = 0,\tag{10}$$

where

$$T\_0 = -\mu + \beta |U|^2 - \rho \mathcal{F}^{-1} \left\{ \frac{k\_\text{\textpi}^2 \mathcal{F} \left\{ |U|^2 \right\}}{k\_\text{\textpi}^2 + \nu k\_\text{\textpi}^2} \right\} + \gamma |U|^4 - V. \tag{11}$$

Now, we should obtain operator *L*1, which denotes the linearized operator of *<sup>L</sup>*0*<sup>U</sup>* <sup>¼</sup> 0, with respect to the solution *<sup>U</sup>*, i.e., *<sup>L</sup>*<sup>0</sup> *<sup>U</sup>* <sup>þ</sup> *<sup>U</sup>*<sup>~</sup> � � <sup>¼</sup> *<sup>L</sup>*1*U*<sup>~</sup> <sup>þ</sup> *<sup>O</sup> <sup>U</sup>*<sup>~</sup> <sup>2</sup> � �, where *U*~ ≪ 1. However, it should be noted that we have obtained the operator *L*<sup>0</sup> by substituting the mean field term *ϕ*ð Þ *x*, *y* into the governing equation. Therefore, at this point, we have to perturb *ϕ*ð Þ *x*, *y* function as well. In accordance with this purpose, the soliton solution and the mean-field term should be perturbed as follows, respectively,

$$\begin{aligned} u(\mathbf{x}, \boldsymbol{y}, \boldsymbol{z}) &= \left[ U(\mathbf{x}, \ \boldsymbol{y}) + \tilde{U}(\mathbf{x}, \ \boldsymbol{y}) \right] \mathbf{e}^{i\mu\mathbf{x}} \\ \phi(\mathbf{x}, \boldsymbol{y}) &= \phi(\mathbf{x}, \boldsymbol{y}) + \tilde{\phi}(\mathbf{x}, \boldsymbol{y}) \end{aligned} \tag{12}$$

$$\begin{aligned} -\mu U + \frac{1}{2}\Delta U + F\left(|U|^2\right)U - \rho\phi U - VU &= 0, \\ \phi\_{\text{xx}} + \nu \phi\_{\text{yy}} &= \left(|U|^2\right)\_{\text{xx}}, \end{aligned} \tag{13}$$

$$F\left(|\boldsymbol{U}|^{2}\right) = F\left(\left|\boldsymbol{U} + \boldsymbol{\tilde{U}}\right|^{2}\right) = F\left(\boldsymbol{U}^{2} + 2\boldsymbol{U}\boldsymbol{\tilde{U}} + \boldsymbol{\tilde{U}}^{2}\right)$$

$$\approx F\left(\boldsymbol{U}^{2} + 2\boldsymbol{U}\boldsymbol{\tilde{U}}\right) \tag{14}$$

$$\approx F\left(|\boldsymbol{U}|^{2}\right) + 2\boldsymbol{U}\boldsymbol{\tilde{U}}\boldsymbol{F}'\_{|\boldsymbol{U}|^{2}}\left(|\boldsymbol{U}|^{2}\right)$$

$$\begin{split} F\left( |U|^2 \right) U &= F\left( |U+\bar{U}|^2 \right) \left( U+\bar{U} \right) \\ &\approx \left[ F\left( |U|^2 \right) + 2U\bar{U}F'\_{|U|^2} \left( |U|^2 \right) \right] \left( U+\bar{U} \right) \\ &\approx (U+\bar{U})F\left( |U|^2 \right) + 2U^2\bar{U}F'\_{|U|^2} \left( |U|^2 \right) + 2U\bar{U}^2 F'\_{|U|^2} \left( |U|^2 \right) \\ &\approx (U+\bar{U})F\left( |U|^2 \right) + 2U^2\bar{U}F'\_{|U|^2} \left( |U|^2 \right) + O\left( \bar{U}^2 \right). \end{split} \tag{15}$$

$$-\mu \left( U + \bar{U} \right) + \frac{1}{2} \Delta \left( U + \bar{U} \right) + \left( U + \bar{U} \right) F \left( |U|^2 \right) + 2U^2 \bar{U} F\_{|U|^2}' \left( |U|^2 \right)$$

$$-\rho \phi \left( U + \bar{U} \right) - \rho \bar{\phi} U - V \left( U + \bar{U} \right) = 0,\tag{16}$$

$$\phi\_{\text{xx}} + \nu \phi\_{\text{yy}} + \bar{\phi}\_{\text{xx}} + \nu \bar{\phi}\_{\text{yy}} = \left( |U|^2 \right)\_{\text{xx}} + \left( 2U \bar{U} \right)\_{\text{xx}}.$$

$$\begin{aligned} & -\mu \left( U + \bar{U} \right) + \frac{1}{2} \Delta \left( U + \bar{U} \right) + \left( U + \bar{U} \right) \left( \rho |U|^2 + \gamma |U|^4 \right) + 2U^2 \bar{U} \left( \beta + 2\gamma |U|^2 \right) \\ & -\rho \phi \left( U + \bar{U} \right) - \rho \bar{\phi} U - V \left( U + \bar{U} \right) = 0, \\ & \phi\_{\text{xx}} + \nu \phi\_{\text{yy}} + \bar{\phi}\_{\text{xx}} + \nu \bar{\phi}\_{\text{yy}} = \left( |U|^2 \right)\_{\text{xx}} + \left( 2U \bar{U} \right)\_{\text{xx}}. \end{aligned}$$

*Dipole Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing… DOI: http://dx.doi.org/10.5772/intechopen.106207*

Applying the Fourier transform to the first perturbed equation in Eq. (17) and the inverse Fourier transform to the obtained equation, we get

$$\begin{split} & -\mu \left( \mathcal{U} + \tilde{\mathcal{U}} \right) + \mathcal{F}^{-1} \left\{ -\frac{1}{2} \left( k\_x^2 + k\_y^2 \right) \left( \hat{\mathcal{U}} + \hat{\mathcal{U}} \right) \right\} + \left( \mathcal{U} + \tilde{\mathcal{U}} \right) \left( \boldsymbol{\rho} |\mathcal{U}|^2 + \boldsymbol{\gamma} |\mathcal{U}|^4 \right) \\ & + 2\boldsymbol{U}^2 \tilde{\mathcal{U}} \left( \boldsymbol{\rho} + 2\boldsymbol{\gamma} |\mathcal{U}|^2 \right) - \rho \boldsymbol{\phi} \left( \boldsymbol{U} + \tilde{\mathcal{U}} \right) - \rho \tilde{\boldsymbol{\phi}} \boldsymbol{U} - \boldsymbol{V} \left( \boldsymbol{U} + \tilde{\mathcal{U}} \right) = \mathbf{0}. \end{split} \tag{18}$$

Using Eq. (5) and the second equation in Eq. (17), following equation is obtained

$$
\tilde{\phi}\_{\infty} + \nu \tilde{\phi}\_{\mathcal{Y}} = \left( \mathbf{2} U \tilde{U} \right)\_{\infty}. \tag{19}
$$

Applying Fourier transform to Eq. (19) and isolating *ϕ* ^~ from obtained equation, we get

$$k\_x^2 \hat{\phi} + vk\_y^2 \hat{\phi} = k\_x^2 \mathcal{F} \{ 2U\ddot{U} \}$$

$$\hat{\bar{\phi}} = \frac{k\_x^2 \mathcal{F} \{ 2U\ddot{U} \}}{k\_x^2 + vk\_y^2}. \tag{20}$$

Taking the inverse Fourier transform of *ϕ* ^~ yields

$$\tilde{\phi} = \mathcal{F}^{-1}\left\{\frac{k\_x^2 \mathcal{F}\{2U\tilde{U}\}}{k\_x^2 + vk\_y^2}\right\}.\tag{21}$$

Substituting Eq. (8) and Eq. (21) into Eq. (18) yields

$$-\mu \left( U + \bar{U} \right) + \mathcal{F}^{-1} \left\{ -\frac{1}{2} \left( k\_x^2 + k\_y^2 \right) \left( \bar{U} + \hat{\bar{U}} \right) \right\} + \left( U + \bar{U} \right) \left( \beta |U|^2 + \gamma |U|^4 \right)$$

$$-2U^2 \bar{U} \left( \beta + 2\gamma |U|^2 \right) - \rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ |U|^2 \right\}}{k\_x^2 + \nu k\_y^2} \right\} (U + \bar{U}) \tag{22}$$

$$-\rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ 2U \bar{U} \right\}}{k\_x^2 + \nu k\_y^2} \right\} U - V \left( U + \bar{U} \right) = 0.$$

After grouping the terms, Eq. (22) can be written as

$$\begin{aligned} & \left[ -\mu U + \mathcal{F}^{-1} \left\{ -\frac{1}{2} \left( k\_x^2 + k\_y^2 \right) \hat{U} \right\} + \beta |U|^2 U - \rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ |U|^2 \right\}}{k\_x^2 + \nu k\_y^2} \right\} U + \gamma |U|^4 U - VU \right] \\ & + \left[ -\mu \bar{U} + \mathcal{F}^{-1} \left\{ -\frac{1}{2} \left( k\_x^2 + k\_y^2 \right) \hat{U} \right\} + \beta |U|^2 \hat{U} + \gamma |U|^4 \hat{U} + 2\beta |U|^2 \hat{U} + 4\gamma |U|^4 \hat{U} \right] \\ & - \rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ |U|^2 \right\}}{k\_x^2 + \nu k\_y^2} \right\} \hat{U} - \rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ 2U \bar{U} \right\}}{k\_x^2 + \nu k\_y^2} \right\} U - V \hat{U} \right] = 0. \end{aligned} \tag{23}$$

Hence,

$$\begin{aligned} & \left[ -\mu U + \mathcal{F}^{-1} \left\{ -\frac{1}{2} \left( k\_x^2 + k\_y^2 \right) \dot{U} \right\} + \beta |U|^2 U - \rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ |U|^2 \right\}}{k\_x^2 + \nu k\_y^2} \right\} U + \gamma |U|^4 U - \mathbf{V}U \right] \\ & + \left[ -\mu \dot{U} + \mathcal{F}^{-1} \left\{ -\frac{1}{2} \left( k\_x^2 + k\_y^2 \right) \dot{U} \right\} + 3\beta |U|^2 \dot{U} - \rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ |U|^2 \right\}}{k\_x^2 + \nu k\_y^2} \right\} \dot{U} + 5\gamma |U|^4 \dot{U} \\ & - \rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ 2U \dot{U} \right\}}{k\_x^2 + \nu k\_y^2} \right\} U - V \dot{U} \right] = 0. \end{aligned} \tag{24}$$

From Eq. (9), we know that the first bracket is identically zero. Consequently, we obtain

$$\begin{split} & -\mu \ddot{U} + \mathcal{F}^{-1} \left\{ -\frac{1}{2} \left( k\_x^2 + k\_y^2 \right) \hat{\bar{U}} \right\} + 3\beta |U|^2 \ddot{U} - \rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ |U|^2 \right\}}{k\_x^2 + \nu k\_y^2} \right\} \ddot{U} + 5\gamma |U|^4 \ddot{U} \\\\ & -\rho \mathcal{F}^{-1} \left\{ \frac{k\_x^2 \mathcal{F} \left\{ 2U \bar{U} \right\}}{k\_x^2 + \nu k\_y^2} \right\} U - V \ddot{U} = 0. \end{split} \tag{25}$$

Moreover, Eq. (24) satisfied *<sup>L</sup>*<sup>0</sup> *<sup>U</sup>* <sup>þ</sup> *<sup>U</sup>*<sup>~</sup> � � <sup>¼</sup> *<sup>L</sup>*1*U*<sup>~</sup> <sup>þ</sup> *<sup>O</sup> <sup>U</sup>*<sup>~</sup> <sup>2</sup> � �. Therefore, to obtain a linearized operator *L*1L1, Eq. (25) can be written as

$$L\_1 \ddot{U} = \mathcal{F}^{-1} \left\{ -\frac{1}{2} \left( k\_\chi^2 + k\_\chi^2 \right) \hat{\dot{U}} \right\} + T\_1 \ddot{U} - \rho \mathcal{F}^{-1} \left\{ \frac{k\_\chi^2 \mathcal{F} \{ 2U \ddot{U} \}}{k\_\chi^2 + \nu k\_\chi^2} \right\} U = 0,\tag{26}$$

where

$$T\_1 = -\mu + 3\beta \left| U \right|^2 - \rho \mathcal{F}^{-1} \left\{ \frac{k\_\text{\textpi}^2 \mathcal{F} \left\{ \left| U \right|^2 \right\}}{k\_\text{\textpi}^2 + vk\_\text{\textpi}^2} \right\} + 5\gamma \left| U \right|^4 - V. \tag{27}$$

To obtain soliton solution *U*(*x*, *y*) in *L*0*U* ¼ 0, we numerically integrate the following distance-dependent squared-operator evolution equation

$$U\_x = -M^{-1}L\_1^\dagger M^{-1}L\_0U,\tag{28}$$

where ð Þ� † denotes the Hermitian of the operator and *<sup>M</sup>* is a real-valued positive definite Hermitian preconditioning operator that is introduced to accelerate the convergence. Since it is easily invertible to take the Fourier transform, we take the preconditioning operator *M* to be in the form of the following

$$M = \mathcal{c} - \left(\partial\_{\text{xx}} + \partial\_{\text{yy}}\right),\tag{29}$$

*Dipole Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing… DOI: http://dx.doi.org/10.5772/intechopen.106207*

where *c* > 0 is a parameter for parametrizing the numerical scheme. Applying the Fourier transform to Eq. (29) yields

$$\mathcal{F}\{\mathbf{M}\} = \mathfrak{c} + k\_{\mathfrak{x}}^2 + k\_{\mathfrak{y}}^2. \tag{30}$$

Consequently, in Eq. (28)

$$\mathbf{M}^{-1}\mathbf{L}\_1^\dagger \mathbf{M}^{-1} L\_0 \mathbf{U} = \mathcal{F}^{-1}\left\{\frac{\mathcal{F}\{L\_1 \mathbf{M}^{-1} L\_0 \mathbf{U}\}}{c + k\_\mathbf{x}^2 + k\_\mathbf{y}^2}\right\}.\tag{31}$$

Using the forward Euler method, steady-state solution *U* is computed by an iterative scheme as follows

$$U\_{n+1} = U\_n - \left[\mathbf{M}^{-1} \mathbf{L}\_1^\dagger \mathbf{M}^{-1} \mathbf{L}\_0 U\right]\_{U = U\_n} \Delta \mathbf{z},\tag{32}$$

where Δ*z* is an auxiliary distance-step parameter. It has been demonstrated that the SOM algorithm converges to a soliton solution for a wide range of nonlinear PDEs if the initial condition is sufficiently close to the exact solution and the distance-step Δ*z* in the iteration scheme is less than a specific threshold value [60, 61]. To obtain a convergent soliton solution, *c* and Δ*z* are chosen heuristically as positive real numbers. Moreover, our convergence criterion is that the obtained solution satisfies Eq. (10) with an absolute error less than 10�<sup>5</sup> .

In this chapter, to obtain dipole solitons, the initial condition of the SOM algorithm is chosen as a multi-humped Gaussian function which is given by

$$U\_0(\mathbf{x}, \boldsymbol{\mathcal{y}}, \mathbf{0}) = \sum\_{n=0}^{H-1} e^{-A\left[\left(\mathbf{x} + \mathbf{x}\_n\right)^2 + \left(\mathbf{y} + \mathbf{y}\_n\right)^2\right] + i\theta\_n},\tag{33}$$

where *xn* and *yn* represent the location of the solitons on the lattice, *H* corresponds to the number of humps, *A* is a positive integer, and *θ<sup>n</sup>* is the phase difference. Since we numerically investigate the dipole solitons, *H* is set to 2, thus Eq. (33) takes the following form:

$$U\_0(\mathbf{x}, \mathbf{y}, \mathbf{0}) = e^{-A\left[\left(\mathbf{x} + \mathbf{x}\_0\right)^2 + \left(\mathbf{y} + \mathbf{y}\_0\right)^2\right] + i\theta\_0} + e^{-A\left[\left(\mathbf{x} + \mathbf{x}\_1\right)^2 + \left(\mathbf{y} + \mathbf{y}\_1\right)^2\right] + i\theta\_1},\tag{34}$$

where (*x*0,*y*0) and (*x*1,*y*1) represent the locations of dipole solitons, *θ*<sup>0</sup> and *θ*<sup>1</sup> are the phase differences of dipole solitons. It was shown that the solitons located at the maximum of the lattices are unstable [13, 21, 24], due to this fact we will investigate the dipole solitons located on minima of the considered square lattice. A dipole (two-phased) localized soliton numerically found by

$$A = \mathbf{1}, \ \mathbf{x}\_n = r \cos \theta\_n, \ \mathcal{Y}\_n = r \sin \theta\_n, \ n = \mathbf{0}, \mathbf{1}. \tag{35}$$

Here *r* is set to be *π* and *θ<sup>n</sup>* ¼ *nπ*, so that the humps of the initial condition are located at the local minima of the lattice where *x*0, *y*<sup>0</sup> � � <sup>¼</sup> ð Þ *<sup>π</sup>*, 0 and *<sup>x</sup>*1, *<sup>y</sup>*<sup>1</sup> � � <sup>¼</sup> ð Þ �*π*, 0 .

Unless otherwise stated, parameters in the NLSM model (2) are fixed to

$$(\mu, \rho, v, \beta, \gamma, V\_0) = (-0.1, 0.5, 1.5, 2, -0.1, 12.5). \tag{36}$$

It is noted that *ρ* ¼ 0*:*5 and *v* ¼ 1*:*5 are especially chosen to simulate quadratic optical effects in potassium niobate (KNbO3) [30].

**Figure 2.**

*3D dipole profiles centered at the lattice minima (first column), the phase structures of the dipole (second column), and the contour plot of the dipole solitons superimposed on the underlying lattice (third column), which are obtained for (a) <sup>γ</sup>* ¼ �0*:*3,*<sup>c</sup>* <sup>¼</sup> <sup>2</sup>*:*1,Δ*<sup>z</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>4</sup> *and error is order of* <sup>10</sup>�<sup>6</sup>*, (b) <sup>γ</sup>* ¼ �0*:*1,*<sup>c</sup>* <sup>¼</sup> <sup>2</sup>*:*5,Δ*<sup>z</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>4</sup> *and error is order of* <sup>10</sup>�<sup>8</sup>*, and (c) <sup>γ</sup>* <sup>¼</sup> <sup>0</sup>*:*3,*<sup>c</sup>* <sup>¼</sup> <sup>2</sup>*:*5,Δ*<sup>z</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>4</sup> *and error is order of* <sup>10</sup>�<sup>8</sup>*. All other parameters are fixed to the values in Eq. (36).*

Dipole solitons of the NLSM model (2) are calculated by the SOM method. In **Figure 2**, 3D views (first column), phase structures (second column), and contour plots of dipole solitons on the underlying lattice (third column) are displayed for selfdefocusing (*γ* <0) and self-focusing (*γ* > 0) quintic nonlinearities. *γ* is set to be �0*.*3, �0*.*1 and 0*.*3 in **Figure 2(a)**–**(c)**, respectively, and all other parameters are fixed to the values given in Eq. (36). **Figure 2** shows that the dipole solitons can be generated on the lattice minima (see the third column), and the amplitudes of dipole solitons are decreased as *γ* increased (from �0*.*3 to 0*.*3) (see the first column).

#### **3. Stability analysis**

The stability dynamics of dipole solitons obtained by the SOM method are studied by the power analysis and direct simulation of the nonlinear evolution.

The power of solitons plays an important role in the stability analysis and it is calculated by

*Dipole Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing… DOI: http://dx.doi.org/10.5772/intechopen.106207*

$$P(\mu) = \int\_{-\infty}^{+\infty} \int\_{-\infty}^{+\infty} |U(\infty, y; \mu)|^2 d\mathbf{x} dy. \tag{37}$$

Vakhitov and Kolokolov proved a necessary condition for the linear stability of solitons in Ref [62]. They demonstrated that a soliton is linearly stable only if its power increases as propagation constant (or eigenvalue) *μ* increases. In other words, a necessary condition for the stability of solitons is

$$\frac{dP}{d\mu} > 0.\tag{38}$$

Moreover, Weinstein and Rose [63, 64] proved that a necessary condition for the nonlinear stability of solitons is also the slope condition given in Eq. (38).

To analyze nonlinear stability of the NLSM model (2), we examine the direct simulation of dipole solitons obtained by the SOM method. A finite-difference discretization scheme is used in the spatial domain (*x, y*) and the dipole solitons are advanced in the *z* direction with a fourth-order Runge-Kutta method. The initial condition of the nonlinear evolution is taken to be a dipole soliton, and 1% random noise is inserted into the amplitude of the initial condition.

The power diagrams of dipole solitons are displayed for varied *μ*, *γ*, *β* and *ρ* values in **Figure 3(a)**–**(d)**, respectively. It is noted that the domain of existence for the varied parameter is shown on the *x*-axis of each panel when other parameters are fixed to the values in Eq. (36). **Figure 3** shows that the power of dipole solitons increases as *μ* and *ρ* increase, whereas the power of dipole solitons decreases as *γ* and *β* increase. Moreover, the stability (solid blue) and instability (red dotted) regions of parameters are determined by the nonlinear evolution of dipole solitons for each point on the power curves.

The dipole solitons are found to be nonlinearly stable for self-defocusing quintic nonlinearity (*γ* ¼ �0*:*1) when the power P ∈ [0.99, 1.87] and propagation constant *μ*∈½ � �0*:*75, �0*:*6 . Also, the dipole solitons are nonlinearly stable when P ∈ [3.12, 4.26] and *μ*∈½ � �0*:*35, �0*:*06 , which is the second nonlinearly stable gap (see **Figure 3(a)**). These results are consistent with key analytical results on nonlinear stability, which Weinstein and Rose proved in Ref [63, 64], since slope of the power-eigenvalue (P � μ) diagram is positive. As can be seen from **Figure 3(b)**, the dipole solitons are obtained for *γ* ∈½ � �0*:*7, 25 , when other parameters are fixed, and dipole solitons are nonlinearly stable for *γ* ∈½ � �0*:*21, 0*:*25 . Zoom-in view of this stability domain is depicted in **Figure 3(b)**. Furthermore, it is observed that dipole solitons are nonlinearly stable for *β* ∈½ � 1*:*6, 18*:*9 (see **Figure 3(c)**), and dipole solitons are stable for *ρ*∈½ � 0, 0*:*8 (see **Figure 3(d)**) in their existence domains when other parameters are fixed.

In **Figure 4**, nonlinear evolution of peak amplitudes, 3D views of the evolved dipole solitons, and the phase structures of evolved dipole solitons are plotted for the dipole solitons that are shown in **Figure 2**. The effect of quintic nonlinearity (*γ*) on nonlinear stability is investigated by fixing other parameters as in Eq. (36).

**Figure 4(b)** shows that peak amplitudes of dipole solitons oscillate mildly (first column), and the 3D profile (second column) and phase structure (third column) of dipole solitons are preserved for *γ* ¼ �0*:*1. Thus, the dipole solitons are nonlinearly stable for the defocusing quintic nonlinearity for the considered parameter regime. On the other hand, as shown in **Figure 4(a)** and **(c)**, when the quintic nonlinearity is strong (*γ* ¼ �0*:*3 and *γ* ¼ þ0*:*3), peak amplitudes of dipole solitons increase significantly in a short propagation distance *z* (first column), dipole profiles (second column) cannot be preserved, and phase structures of dipole solitons (third column) break up after evolution. Comparing **Figure 4(a)** and **(c)**, it is observed

**Figure 3.**

*Power of dipole solitons (a) for varying eigenvalue μ, (b) for varying quintic nonlinearity coefficient γ, (c) for varying cubic nonlinearity coefficient β, and (d) for varying quadratic nonlinear response ρ. The nonlinear stability and instability regions are shown by solid blue and red dotted lines, respectively.*

that the propagation distance of dipole solitons in a medium with strong selffocusing nonlinearity (*γ* ¼ 0*:*3) is longer than that of a medium with strong selfdefocusing nonlinearity (*γ* ¼ �0*:*3). Considering these evolution results in **Figure 4** and the existing domain for *γ* in **Figure 3(b)**, it is demonstrated that both strong self-focusing and self-defocusing quintic nonlinearities have a negative effect on the nonlinear stability of dipole solitons.

In previous studies [6, 14, 24], it is found that modification of the depth of potential can suppress nonlinear instabilities. More recently, Bağcı and coworkers [46] have demonstrated that nonlinear stability of fundamental solitons in an NLSM system (2) with quintic nonlinearity can be improved by the modification of lattice depth *V*0. They showed that increased lattice depth supports the stability of fundamental solitons in a medium with strong self-focusing (*γ* ¼ 0*:*3) quintic nonlinearity, and the stability of solitons in a medium with strong self-defocusing (*γ* ¼ �0*:*3) quintic nonlinearity can be improved by decreasing lattice depth. For the dipole solitons, evolution of peak amplitudes is depicted for varying potential depths, when *γ* ¼ �0*:*3 and *γ* ¼ 0*:*3 in **Figure 5(a)** and **(b)**, respectively. **Figure 5 (a)** shows that the stability of dipole solitons is improved by decreasing lattice depth (from 25 to 5) for strong self-defocusing nonlinearity (*γ* ¼ �0*:*3), and collapse can be arrested when *V*<sup>0</sup> ¼ 5. In contrast, as shown in **Figure 5(b)**, the propagation distance of dipole solitons in a medium with strong self-focusing

*Dipole Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing… DOI: http://dx.doi.org/10.5772/intechopen.106207*

#### **Figure 4.**

*Nonlinear evolution of maximum amplitudes as a function of propagation distance z (first column), 3D views of the dipole solitons after evolution (second column), and the phase structures of dipole solitons after evolution (third column) for (a) γ* ¼ �0*:*3*, (b) γ* ¼ �0*:*1*, and (c) γ* ¼ 0*:*3*. All other parameters are taken as in Eq. (36).*

nonlinearity (*γ* ¼ 0*:*3) is extended by increasing lattice depth (from 5 to 50). It should be noted that these results are in agreement with the findings of the aforementioned studies. Thus modification of the lattice depth can be utilized to improve the nonlinear stability of dipole solitons.

It is also known that when the quadratic [15, 24, 44] and quintic [46] electrooptic effects are strong, the instability of fundamental solitons can be improved by increasing the anisotropy parameter. To examine the effect of anisotropy coefficient *v* on the nonlinear stability of dipole solitons in a medium with strong quintic nonlinearity (*γ* ¼ �0*:*3 and *γ* ¼ 0*:*3), evolution of the peak amplitudes is displayed for varied *v* values in **Figure 6**. **Figure 6** shows that increasing the anisotropy coefficient *v* from 0.001 to 10 stabilizes the dipole solitons in a medium with strong self-defocusing nonlinearity (*γ* ¼ �0*:*3), and increasing *v* from 0.001 to 1000 extends the propagation distance of dipole solitons in a medium with strong selffocusing nonlinearity (*γ* ¼ 0*:*3). Thus, larger anisotropy coefficient supports the nonlinear stability of the dipole solitons, and this result complies with the results of the previous studies [46]. It is important to note that the parameters *ρ* and *v* are predetermined coefficients that depend on the type of optical materials; larger

#### **Figure 5.**

*Maximum amplitudes of the evolved dipole solitons for varying depth of potential V0, when the dipole soliton is obtained for (a) γ* ¼ �0*:*3 *and (b) γ* ¼ 0*:*3*.*

values of *v* cannot be applied to real optical systems. In this chapter, the effect of extremely large *v* values on the stability of dipole solitons is explored numerically.

#### **4. Conclusions**

In this chapter, the existence and nonlinear stability dynamics of dipole solitons have been investigated for a nonlocal nonlinear medium with quintic nonlinear response. This medium was characterized by the (2+1)D NLSM system with a periodic external lattice. Dipole solitons were obtained for self-defocusing (*γ* <0) and self-focusing (*γ* >0) quintic nonlinearities by the SOM method, and the nonlinear stability of these dipole structures has been investigated by the direct simulation of the model equations. Power of dipole solitons was determined for varying *μ, γ, β,* and *ρ* parameters and it has shown that the power of dipole solitons increases as the eigenvalue *μ* and quadratic nonlinear response *ρ* increase, whereas the power of dipole solitons decreases as quintic nonlinearity coefficient *γ* and cubic nonlinearity coefficient *β* increase.

Nonlinear evolution of the dipole solitons showed that the dipole solitons are stable for the weak self-focusing and self-defocusing quintic nonlinearity. In other words, as an absolute value of *γ* increases, the obtained dipole solitons become nonlinearly unstable in both self-focusing and self-defocusing media. It has been demonstrated that the collapse of dipole solitons can be arrested by decreased

*Dipole Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing… DOI: http://dx.doi.org/10.5772/intechopen.106207*

#### **Figure 6.**

*Maximum amplitudes of the evolved dipole solitons for varying anisotropy coefficients v, when the dipole soliton is obtained for (a) γ* ¼ �0*:*3*, and (b) γ* ¼ 0*:*3*.*

potential depth in a medium with strong self-defocusing quintic nonlinearity (*γ* ¼ �0*:*3), while the deeper lattice extends the propagation distance of dipole solitons in a medium with strong self-focusing quintic nonlinearity (*γ* ¼ 0*:*3). Furthermore, it has been observed that increasing the anisotropy coefficient (*v*) extends the propagation distance of the dipole solitons for strong self-focusing quintic nonlinearity, and it stabilizes the dipole solitons for strong self-defocusing quintic nonlinearity.

In conclusion, the existence and stability properties of dipole solitons have been numerically explored in a nonlocal nonlinear medium with quintic nonlinear response, and it has been demonstrated that the instability of dipole solitons can be suppressed by modification of the lattice depth and increased anisotropy coefficient.

#### **Conflict of interest**

The authors declare no conflict of interest.

#### **Author details**

Mahmut Bağcı 1 \*†, Melis Turgut2†, Nalan Antar2† and İlkay Bakırtaş 2†

1 Department of Management Information Systems, Marmara University, Istanbul, Türkiye

2 Department of Mathematical Engineering, Istanbul Technical University, Istanbul, Türkiye

\*Address all correspondence to: bagcimahmut@gmail.com

† These authors contributed equally.

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Dipole Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing… DOI: http://dx.doi.org/10.5772/intechopen.106207*

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**Chapter 2**

## The Propagation of Vortex Beams in Random Mediums

*Sekip Dalgac and Kholoud Elmabruk*

#### **Abstract**

Vortex beams acquire increasing attention due to their unique properties. These beams have an annular spatial profile with a dark spot at the center, the so-called phase singularity. This singularity defines the helical phase structure which is related to the topological charge value. Topological charge value allows vortex beams to carry orbital angular momentum. The existence of orbital angular momentum offers a large capacity and high dimensional information processing which make vortex beams very attractive for free-space optical communications. Besides that, these beams are well capable of reducing turbulence-induced scintillation which leads to better system performance. This chapter introduces the research conducted up to date either theoretically or experimentally regarding vortex beam irradiance, scintillation, and other properties while propagating in turbulent mediums.

**Keywords:** vortex beams, random medium, turbulence, scintillation, optical communications

#### **1. Introduction**

Wave front dislocations, in other words, phase defects which consist of edge dislocations, screw dislocations and mixed edge-screw dislocations are firstly proposed by Nye and Berry as a new type of light field principle [1]. The screw dislocation most prevalently known as front dislocation which presents a phase singularity at the center of the beam with zero amplitude and indefinite phase. Also, when both the real and imaginary parts of the wave function (ψ) equal zero the phase singularity is observed. Due to the fact that light field possesses unique properties such as phase singularly or dislocations, it paves the way for modern optics which called singular optics. Optical vortices are the primary topic of the singular optics [2]. Allen in 1992 realized that a beam of photons can hold singularity with azimuthally phase structure e*jl<sup>Ɵ</sup>* and carry an orbital angular momentum (OAM) where *l* is the topological charge and *Ɵ is the* azimuth angle [3]. Vortex beams possess distinct optical properties compared to the other beam types since they carry OAM. These beams have introduced a great diversity in a wide range of applications namely optical manipulation, biomedical applications, microfabrication, imaging, and micro-mechanics. Furthermore, they played an important role in the new generation of optical communication where OAM is employed as a new modulation technique in the optical communication systems [4, 5]. Such a feature makes the beams carrying OAM a perfect solution for the increasing demand of larger bandwidth and higher data rates in a diversity of applications such as 5G and 6G communication links, laser satellite communications, and remote sensing. However, in these applications the propagation of the laser beam in a random medium, which represents the channel of the system, degrades the probability of error performance of the system [6–12].

Actually, the propagation of laser beams through a random medium is governed fundamentally by three main phenomena namely absorption, scattering and refractive-index fluctuations. While absorption, scattering, which are caused by constituent gas and particles in the medium, resulted in the energy dissipation [13, 14]. The refractive-index fluctuations named turbulence originate from the temperature differences and cause intensity fluctuations (scintillation) that degrade the probability of error performance of the wireless optical communication system. In case that turbulence presence, the beams involve in extra beam spreading, beam wander, and scintillation that greatly hamper the performance of the communication system. Consequently, understanding the effects of turbulent medium on the propagating beam is an important issue for the researchers that paves the way towards mitigating the limitations caused by turbulence [15, 16].

This chapter presents a detailed review of the conducted work up to date on the propagation of vortex beams through random mediums. Accordingly, Section 2 starts with the representation of different types of vortex beams. Then, followed by the theory of the propagation in a random medium in Section 3. Subsequently, Section 4 discusses the atmospheric turbulence effect on the fully and partially coherent vortex beams. In addition, it represents the scintillation properties of vortex beams. In Section 5, we evaluate coherent and partially coherent vortex beam properties in oceanic turbulence. Furthermore, it covers the scintillation effects on the vortex beams propagating oceanic turbulence medium. Finally, Section 6 sums up the chapter by concluding the advantages that vortex beams offer for optical communication systems through the degradation of turbulence effects.

#### **2. Representations of vortex beams**

In this part of the chapter, expressions of different vortex beams are given at the source plane on the fundamental coordinate systems, neither Cartesian sx, sy or radial s, ð Þ φ [17, 18]. Firstly, the source field expression of Gaussian vortex beam is;

$$E(s,\rho) = \left(\frac{s}{a\_{\mathfrak{s}}}\right)^l \exp\left(-\frac{s^2}{a\_{\mathfrak{s}}^2}\right) \exp\left(jl\rho\right) \tag{1}$$

where αs, and l represent the source size and topological charge respectively.

Besides that, the source field expression of elliptical Gaussian vortex beam [19] can be written with the Cartesian coordinate as follows;

$$E(\mathbf{s}\_{\mathbf{x}}, \mathbf{s}\_{\mathbf{y}}) = \left(\frac{\mathbf{s}\_{\mathbf{x}} + j\varepsilon\_{\mathbf{s}}\mathbf{s}\_{\mathbf{y}}}{a\_{\mathbf{t}}}\right)^{l} \exp\left(-\frac{\mathbf{s}\_{\mathbf{x}}^{2} + \varepsilon\_{\mathbf{s}}^{2}\mathbf{s}\_{\mathbf{y}}^{2}}{a\_{\mathbf{t}}}\right) \exp\left[jl\tan^{-1}\left(\frac{\varepsilon\_{\mathbf{s}}\mathbf{s}\_{\mathbf{y}}}{\varepsilon\_{\mathbf{x}}}\right)\right] \tag{2}$$

ε<sup>s</sup> is the degree of ellipticity. Another widely investigated beam type is the Laguerre Gaussian vortex beam which can be expressed as [20, 21];

$$E(s,\rho) = \left(\frac{s}{a\_s}\right)^l \exp\left(-\frac{s^2}{a\_s^2}\right) L\_n{}^m \left(\frac{s^2}{a\_s^2}\right) \exp\left(jl\rho\right) \tag{3}$$

*The Propagation of Vortex Beams in Random Mediums DOI: http://dx.doi.org/10.5772/intechopen.101061*

Ln <sup>m</sup> is the Laguerre polynomial, with a polynomial degree of n. If n>0, Bessel function Jn <sup>m</sup> with orders can also generate vortex beams [22]. Thus, Bessel– Gaussian vortex beam can be written as;

$$E(s,\rho) = \exp\left(-\frac{s^2}{a\_t^2}\right) I\_m\left(\frac{s}{a\_t}\right) \exp\left(jl\rho\right) \tag{4}$$

Jm is the Bessel function order with m. Finally, Flat topped Gaussian vortex beam expressed with the related source field expression [23] as given;

$$E(\mathbf{s}\_{\mathbf{x}}, \mathbf{s}\_{\mathbf{y}}) = \frac{1}{N} \left(\frac{\mathbf{s}\_{\mathbf{x}} + j\mathbf{s}\_{\mathbf{y}}}{a\_{\mathbf{s}}}\right)^{m} \sum\_{n=1}^{N} (-1)^{n-1} (Nn) \exp\left(-n\frac{\mathbf{s}\_{\mathbf{x}}^{2} + \mathbf{s}\_{\mathbf{y}}^{2}}{a\_{\mathbf{s}}^{2}}\right) \tag{5}$$

N indicates the order of flat-topped Gaussian vortex beam. Moreover, Hermite– Gaussian vortex beam can be written as the superposition of two orthogonally polarized components under paraxial approximation [24]. The equation of Hermite–Gaussian vortex beam can be written as;

$$\mathbf{E}(\mathbf{s}\_{\mathbf{x}},\mathbf{s}\_{\mathbf{y}}) = \exp\left(-\frac{\mathbf{s}\_{\mathbf{x}}^{2}+\mathbf{s}\_{\mathbf{y}}^{2}}{\mathbf{a}\_{\mathbf{s}}\,^{2}}\right) \left[\mathbf{H}\_{\text{mx}}\left(\frac{\mathbf{s}\_{\mathbf{x}}}{\mathbf{a}\_{\mathbf{s}}}\right)\mathbf{H}\_{\text{ny}}\left(\frac{\mathbf{s}\_{\mathbf{y}}}{\mathbf{a}\_{\mathbf{s}}}\right)\overrightarrow{\mathbf{s}\_{\mathbf{x}}} + \mathbf{H}\_{\text{mx}}\left(\frac{\mathbf{s}\_{\mathbf{x}}}{\mathbf{a}\_{\mathbf{s}}}\right)\mathbf{H}\_{\text{my}}\left(\frac{\mathbf{s}\_{\mathbf{y}}}{\mathbf{a}\_{\mathbf{s}}}\right)\overrightarrow{\mathbf{s}\_{\mathbf{y}}}\right] \tag{6}$$

where the orders of the Hermite polynomials such as nx, ny, mx, my in HnxðÞ, HnyðÞ, HmxðÞ, HmyðÞ can be introduced as odd integers to create the desired zero on-axis field amplitude, this way such combinations can be regarded as vortex beams.

The optical field of the sinh-Gaussian vortex beam in the source plane can be specified as given in [25];

$$E(\overrightarrow{\mathbf{s}}, \mathbf{0}) = \sinh\left[\mathcal{Q}(\mathbf{x}\_0 + \mathbf{y}\_0)\right] \exp\left(-\frac{\mathbf{x}\_0^2 + \mathbf{y}\_0^2}{w\_0^2}\right) \left[\mathbf{x}\_0 + \mathbf{j}\cdot\text{sgn}(l)\mathbf{y}\_0\right]^{|l|}\tag{7}$$

s ! is the position vector, Ω denote the constant parameter of the hyperbolic sinusoidal part, where sgn lð Þ can be introduced as a symbolic function.

In addition to coherent vortex beams, there exist various important types of partially coherent vortex beams in the literature. The cross spectral density (CSD) of partially coherent beams in the source plane can be expressed by the following general form [17];

$$\mathcal{W}(\mathbf{s}\_1, \mathbf{s}\_2) = \left< E\_{(\mathfrak{s}\_1)} E^\* \right>\_{(\mathfrak{s}\_2)} \rangle = \mathcal{A}\_{(\mathfrak{s}\_1)} \mathcal{A}\_{(\mathfrak{s}\_2)} \exp\left[j l (\mathfrak{\rho}\_1 - \mathfrak{\rho}\_2) \right] \mathfrak{g}(\mathfrak{s}\_1 - \mathfrak{s}\_2) \tag{8}$$

Es and As are electric field and its amplitude, respectively. The angular bracket and the asterisk denote ensemble average and complex conjugate. s1 and s2 are the two arbitrary points in the source plane, g sð Þ <sup>1</sup> � s2 is the correlation function between two arbitrary points. As the amplitude (As) of the beam changes, different kind of partially coherent beams are obtainable. The correlation function for the Gaussian distribution is [26];

$$\lg(\mathbf{s}\_1 - \mathbf{s}\_2) = \exp\left[-\frac{(\mathbf{s}\_1 - \mathbf{s}\_2)}{2\delta\_0^2}\right] \tag{9}$$

That δ<sup>0</sup> indicates initial coherence width. In case of δ<sup>0</sup> ! ∞, Eq. (9) tends to a fully coherent vortex beam, However, when δ<sup>0</sup> ! 0, Eq. (9) reduces to an

incoherent vortex beam. On the other hand the CSD function of Gaussian Schellmodel (GSM) vortex beam in the source plane is written as [27];

$$E(\mathbf{s}\_1, \mathbf{s}\_2, \boldsymbol{\varphi}\_1, \boldsymbol{\varphi}\_2) = \exp\left[ -\frac{\mathbf{s}\_1^2 + \mathbf{s}\_2^2}{4\sigma\_0^2} - \frac{\mathbf{s}\_1^2 + \mathbf{s}^2 - 2\mathbf{s}\mathbf{s}\_2\cos\left(\varphi\_1 - \varphi\_2\right)}{2\delta\_0^2} + jl(\varphi\_1 - \varphi\_2) \right] \tag{10}$$

Where σ<sup>0</sup> indicates transverse beam width. Furthermore, the CSD of the partially coherent LG beam in the source plane is obtained as [28];

$$\mathbf{E}(\mathbf{s}\_{1},\mathbf{s}\_{2},\boldsymbol{\uprho}\_{1},\boldsymbol{\uprho}\_{2}) = \left(\frac{\sqrt{2}\mathbf{s}\_{1}}{\boldsymbol{\upalpha}\_{0}}\right)^{\mathbf{m}} \left(\frac{\sqrt{2}\mathbf{s}\_{2}}{\boldsymbol{\upalpha}\_{0}}\right)^{\mathbf{m}} \mathbf{L}\_{\mathbf{n}}^{\mathbf{m}} \left(\frac{\mathbf{2s}\_{1}{}^{2}}{\boldsymbol{\upalpha}\_{0}}\right) \mathbf{L}\_{\mathbf{p}}^{\mathbf{m}} \left(\frac{\mathbf{2s}\_{2}{}^{2}}{\boldsymbol{\upalpha}\_{0}}\right) \exp\ - \left(\frac{\mathbf{s}\_{1}^{2} + \mathbf{s}\_{2}^{2}}{4\boldsymbol{\upalpha}\_{0}}\right) \tag{11}$$

$$\times \exp\left[-\frac{\mathbf{s}\_{1}^{2} + \mathbf{s}\_{2}^{2} - 2\mathbf{s}\_{1}\mathbf{s}\_{2}\cos\left(\boldsymbol{\uprho}\_{1} - \boldsymbol{\uprho}\_{2}\right)}{2\boldsymbol{\updelta}\_{0}}\right] \exp\left[\mathbf{j}!(\boldsymbol{\uprho}\_{1} - \boldsymbol{\uprho}\_{2})\right] \tag{12}$$

Ln <sup>m</sup> is Laguerre polynomial with mode orders n and m. In the case that n <sup>¼</sup> 0, Eq. (11) becomes a partially coherent LG0l beam. However, having the both mode ordersn and m set to zero, the beam turns to the well-known GSM beam. Furthermore, Laguerre Gaussian correlated Schell-model vortex (LGCSMV) beam in the source plane as special kind of the correlated partially coherent vortex beams, can be expressed as [29].

$$E(\mathbf{s}\_1, \mathbf{s}\_2) = \exp\left[ -\frac{\mathfrak{s}\_1^2 + \mathfrak{s}\_2^2}{4\sigma\_0^2} - \frac{(\mathfrak{s}\_1 - \mathfrak{s}\_2)^2}{2\delta\_0^2} \right] L\_n^{-0} \frac{(\mathfrak{s}\_1 - \mathfrak{s}\_2)^2}{2\delta\_0^2} \exp\left[jl(\theta\_1 - \theta\_2)\right] \tag{12}$$

#### **3. Theoretical background of beam propagation through random medium**

In this part of the chapter, atmospheric and oceanic turbulence phenomena that influence the optical laser beam propagation are explained. Also, the theoretical background regarding the laser beam propagation is provided.

#### **3.1 Atmospheric turbulence**

Atmosphere is a medium that surrounds the Earth which mainly consists of gaseous such as nitrogen, oxygen, water vapor, carbon dioxide, methane, nitrous oxide, and ozone. As the beam propagates through atmospheric medium, the change of atmosphere temperature and wind velocity results in variation of the atmosphere's refractive index. These changes simply called atmospheric turbulence. Atmospheric turbulence is a non-linear process that is governed by Navier–Stokes equations. Since solving such kind of equations is challenging, the statistical approaches are developed. One of the widely used approaches is Kolmogrov power spectrum model that is given below [30];

$$\Phi\_{\mathbf{n}}(\mathbf{k}) = \mathbf{0}.\mathbf{0}\mathbf{3}\mathbf{3}\mathbf{C}\_{\mathbf{n}}^{2}\mathbf{x}^{-11/3};\quad \mathbb{1}\_{\mathrm{L}\_{0}}\ll\mathbf{x}\ll\mathbb{1}\boldsymbol{\zeta}\_{0}\tag{13}$$

Cn <sup>2</sup> indicates the refractive index structure and <sup>κ</sup> <sup>¼</sup> j j <sup>Κ</sup> is the scalar wave number. Kolmogrov power spectrum does ignore the effects of the inner (l0) and outer scales (L0) of the turbulence since outer scale is infinity and the inner scale is so small. However, more elaborated power spectrums are suggested by Tatarski and Von-Karman. Tatarski power spectrum is defined as [31];

$$\Phi\_{\mathbf{n}}(\mathbf{k}) = \mathbf{0}.033 \mathbf{C}\_{\mathbf{n}}^2 \mathbf{x}^{-11/3} \exp\left(-\frac{\mathbf{x}^2}{\kappa\_{\mathbf{m}}^2}\right); \quad \mathbf{x} \gg 1 \rangle\_{\mathcal{L}\_0} \tag{14}$$

where κ<sup>m</sup> ¼ 5*:*92*=*l0. If the limit 1*=*L0 ! 0 Lð Þ <sup>0</sup> ! ∞ then this spectrum has a singularity at κ ¼ 0.If the inner and outer scales of the turbulence are considered, Von-Karman spectrum can be defined to model the turbulence as follows [32];

$$\Phi\_{\mathbf{n}}(\kappa) = 0.033 \mathbf{C}\_{\mathbf{n}}^2 \kappa^{-11/3} \frac{\exp\left(-\kappa^2/\kappa\_{\mathbf{m}}^2\right)}{\left(\kappa^2 + \kappa\_0^2\right)^{11/6}}; \quad 0 \le \mathbf{x} < \infty \tag{15}$$

#### **3.2 Oceanic turbulence**

As it is stated above, optical turbulence refers to the index of refraction fluctuations, which is one of the most significant features of optical wave propagation. Depending on the medium type, external and internal effect, there are some distinctions among the index of refraction fluctuations. For instance, while temperature fluctuation is fundamental reason for atmospheric turbulence, refraction index variation in seawater is caused by not only temperature fluctuations but also fluctuations of salinity. For that reason, power spectrum of ocean that considers both temperature and salinity fluctuations was firstly proposed in 2000 [33]. Power spectrum of oceanic turbulence is given for homogeneous and isotropic underwater media as follows;

$$\begin{split} \Phi\_{\mathbf{n}}(\kappa) &= 0.388 \\ &\times 10^{-8} \varepsilon^{-11/3} \left[ + 2.35 (\kappa \mathfrak{q})^{2/3} \right] \frac{\chi\_{\rm T}}{\mathfrak{s}^2} \left[ \mathfrak{s}^2 \exp \left( -\mathbf{A}\_{\rm T} \mathfrak{d} \right) + \exp \left( -\mathbf{A}\_{\rm S} \mathfrak{d} \right) - 2 \mathfrak{g} \exp(-\mathbf{A}\_{\rm TS} \mathfrak{d}) \right] \end{split} \tag{16}$$

ε is the rate of dissipation for turbulent kinetic energy per unit mass of fluid, χ<sup>T</sup> is the rate of dissipation of mean square temperature. AT <sup>¼</sup> <sup>1</sup>*:*<sup>863</sup> � <sup>10</sup>�<sup>2</sup> , ATS ¼ <sup>9</sup>*:*<sup>41</sup> � <sup>10</sup>�<sup>3</sup> δ = 8.284ð Þ κƞ 4*=*3 +12.987ð Þ κƞ 2 . ς is the relative strength of temperature and salinity fluctuations, and finally, ƞ represents the Kolmogrov inner scale.

#### **3.3 Turbulence Modeling**

The behavior of optical beams propagating in random medium can be understood by characterizing the medium qualitatively and quantitatively. Huygens–Fresnel principle is one of the most important modeling types to characterize beam propagation in turbulent medium [34]. The average intensity distribution at the observation plane can be expressed via Huygens–Fresnel principle as Eq. (17);

$$
\begin{split}
\left<\mathbf{J}\left(\overrightarrow{\mathbf{R}},\mathbf{L}\right)\right> &= \frac{\mathbf{k}^{2}}{(2\pi\mathbf{L})^{2}} \iint \mathbf{E}\_{0}\left(\overrightarrow{\mathbf{r}}\_{1},\mathbf{0}\right) \mathbf{E}\_{0} \,^{\ast}\left(\overrightarrow{\mathbf{r}}\_{2},\mathbf{0}\right) \\
&\times \exp\left\{\frac{\mathbf{ik}}{2\mathbf{L}} \left[\left(\mathbf{R}-\overrightarrow{\mathbf{r}}\_{1}\right)^{2} - \left(\mathbf{R}-\overrightarrow{\mathbf{r}}\_{2}\right)^{2}\right]\right\} \\
&\times \left<\mathbf{exp}\left[\left\|\mathbf{q}\left(\mathbf{R},\overrightarrow{\mathbf{r}}\_{1}\right) + \boldsymbol{\Psi}^{\ast}\left(\mathbf{R},\overrightarrow{\mathbf{r}}\_{2}\right)\right]\right] \mathbf{d}\,\overrightarrow{\mathbf{r}}\_{1}\mathbf{d}\,\overrightarrow{\mathbf{r}}\_{2} \right>\tag{17}
\end{split}
\tag{17}
$$

R ! denotes the position vector at the observation plane, r! <sup>1</sup> and r! <sup>2</sup> represent the position vectors at the source plane, k is the wave number, the asterisk denotes the complex conjugation, the < > indicates the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere, ψ R, r! 1 � � is the random part complex phase of a spherical wave. Moreover, Rytov approximation is another type of turbulence modeling for weak atmospheric turbulence [35]. The beam at the receiver plane can be written in terms of Rytov approximation as given Eq. (18);

$$\mathbf{E}(\rho, \mathbf{z}) = -\frac{\mathbf{i}}{\lambda \mathbf{z}} \exp\left(i \mathbf{k}z\right) \int\_{0}^{2\pi} \mathbf{E}(\mathbf{r}, \theta) \exp\left[\frac{i\mathbf{k}}{2\mathbf{z}}(\rho - \mathbf{r})^{2}\right] \exp\left[\Psi(\mathbf{r}, \rho, \mathbf{z})\right] \mathbf{r} \, d\mathbf{r} \, d\theta \,\tag{18}$$

ψð Þ r, ρ, z denotes the random part due to the turbulence and ρ is the position vector. Furthermore, Random Phase Screen method can be employed to modeling turbulence [36]. It is described by the spatial spectrum of phase fluctuations. The spectrum of random phase can be expressed as;

$$
\Phi(\mathbf{q}\_x, \mathbf{q}\_y, \mathbf{q}\_z) = 2\pi \mathbf{k} \Delta z \,\Phi(\mathbf{q}\_x, \mathbf{q}\_y) \tag{19}
$$

#### **4. Coherent vortex beams in turbulent media**

Fluctuation of electric field between two or more points can be considered as coherence of the light beams. The effect of coherence parameters is analyzed by different research groups [37, 38]. The term of coherent vortex beam firstly was revealed by Coullet in 1989, then Allen found that vortex beams can carry OAM [39, 40]. Since the first exploration of the vortex beams, many studies utilized in numerous fields such as quantum information [41], optical processing [42], optical manipulation [43] and optical communication systems [44]. Thus, the propagation of fully coherent vortex beams in turbulent medium has been investigated intensively in the literature.

#### **4.1 Atmospheric turbulence**

Despite the great advantages of free-space optical communication (FSO) systems, the propagation of the laser beam in atmosphere limits the performance of these systems. Mitigating these effects can be achieved through understanding the behavior of the propagating beam under different atmosphere circumstances. Accordingly, the literature has significant investigations on this topic. Recently, vortex beams have become one of the beam types under concentration. Considering Laguerre Gaussian (LG) vortex beams, it is proved that, as the topological charge increases, LG beam undergoes less broadening as given **Figure 1a**. Also, it is obtained that LG vortex beam is less affected by the turbulence than Gaussian beam as a result of the numerical analysis in [17, 18]. Thus, Gaussian beam suffers from more broadening than LG vortex beam. Moreover, Algebraic sum of the topological charges of LG beam is determined. Accordingly, the phase singularities existing in test aperture is approximately equal to the topological charge of the input LG vortex beam [48]. Fiber coupling of LG vortex beam in turbulent atmosphere is investigated by a theoretical model. LG beam that have small OAM number, low radial

*The Propagation of Vortex Beams in Random Mediums DOI: http://dx.doi.org/10.5772/intechopen.101061*

#### **Figure 1.**

*Influence of (a) topological charge and (b) mode probability density (MPD) and crosstalk probability density of low-order LG beams for the various propagation distance for angular mode = 1. (c) the capacity of wireless optical links using AV beams versus LG beams, and (d) effect of turbulence on the intensity and phase distributions of Bessel vortex beams versus LG vortex beams [17, 45–47].*

index and long wavelength gives higher coupling efficiency [49]. Mode probability density (MPD) of LG beam propagating in atmospheric turbulence is analyzed. MPD of LG vortex beam decreases while the distance increases as given **Figure 1b**. Additionally, MPD is increases by lower radial and waist radius, lower refractive index constant and shorter propagation distance [45]. The propagation properties of synthesized vortex beams compared with LG beams in free-space and in atmosphere is explored numerically. Propagation properties of LG beam shows the same characteristics with those of the synthesized vortex beams [50]. Furthermore, spiral spectrum of LG vortex beam and Anomalous vortex beam (AVB) is studied in details. It is achieved that; effects of atmospheric turbulence on LG vortex beam are more than those on Anomalous vortex beams as illustrated **Figure 1c**. Also, the spiral spectrum of the AVB is less affected by the turbulent atmosphere compared with LG vortex beam, in the case that AVB has larger beam order, longer wavelength, smaller topological charge, and at smaller refractive index structure constant, also propagating shorter distances [46]. Different kinds of vortex beams, including LG vortex beam and Bessel vortex beam were analyzed under the same turbulence conditions as given **Figure 1d**. Bessel vortex beams are more affected by the turbulence than LG vortex beams under the same circumstances [47]. It was experimentally demonstrated that, LG vortex beam exhibit enhanced backscatter (EBS) when only having even topological charge and LG beam may convert into corresponding Hermite Gaussian (HG) mode [51].

In addition researchers have analyzed Bessel Vortex Beams (BVB) in atmospheric turbulence. The degree of coherence of Bessel vortex beam decreases much faster under higher levels of fluctuation in the atmosphere [52]. The mean intensity of BVB versus dimensionless parameter ðξ) is given **Figure 2a**. It is obvious from the figure that increasing the topological charge results in decreasing the mean intensity of BVB [22]. Also, Bessel-Gaussian Vortex beams (BGV) have been studied numerically and experimentally, where it is observed that the OAM mean value does not show any variation during the propagation in atmospheric turbulence [56]. The mean intensity of BGV beams possessing phase singularities versus wavelength is given **Figure 2b**. It is clear that the central hole and the dark ring of the beams are gradually filled with the decrease of wavelength. Also, mean intensity of BGV beam decreases faster as the beam operating at a shorter wavelength or having either a narrower beam width, or a smaller topological charge [53]. Finally, a comparison between LG beam and BGV beam is conducted in terms of transmission quality and stability. According to the study given in [57], transmission quality and stability of BGV beam were observed to be better than those of the LG beams. Gaussian Vortex (GV) beam is another beam type that investigated frequently by the researchers.

#### **Figure 2.**

*(a) The average intensity of vortex Bessel beam versus dimensionless parameter and (b) BesselGaussian vortex beam with different wavelength, (c) illustration of the radius of a ring dislocation of vortex beam as a function of structure constant, and (d) beam order effect on the intensity distributions for four petal GV beam [22, 53–55].*

*The Propagation of Vortex Beams in Random Mediums DOI: http://dx.doi.org/10.5772/intechopen.101061*

GV beam enables us to calculate atmospheric turbulence strength by measuring radius of ring dislocations with different beam width as given **Figure 2c** [54]. Four Petal GV beam possessing high beam order undergoes transformation into more petals in the far field as achieved in **Figure 2d** [55].

The laser wavelength effect on the annular vortex beam is investigated when propagating in atmospheric turbulence [58]. It is observed that, operating at higher wavelengths causes lowering the central relative intensity and the central dark hollow is more achievable as stated in **Figure 3a**. Furthermore, beam width of a collimated vortex beam increases with the decrease of the wavelength [61]. Elliptically polarized (EP) vortex beams in turbulent atmosphere evolve into a Gaussian beam shape when the propagation distance is long enough and also flat-topped profile is obtained at a longer propagation distance as the topological charge increases [62]. Initial dark hollow profile of flat-topped vortex hollow beams remains the same in the short propagation distance then the beam evolves into a Gaussian-like beam under the strong turbulence [44]. Rectangular vortex beam array with arbitrary topological charge through atmospheric turbulence is analyzed and the obtained results clarify that beam array transform into a fan structure under moderate turbulence after propagating 1000 m, then turns to a single vortex beam after propagating 5000 m as given in **Figure 3b** and **c**. [59]. Also, optical vortex beams with higher topological charge are able to propagate longer distances in weak turbulent atmosphere. However, when the particular distance exceeds 500 km the output beam finally loses the vortex property and gradually becomes a Gaussianshaped beam as illustrated in **Figure 3d** [60].

**Figure 3.**

*(a) Average intensity of annular vortex beam with different beam wavelengths and rectangular vortex beam when distance (b) 1000 m and (c) 5000 m and optical vortex beam with distance of 500 km [58–60].*

Furthermore, the influence of topological charge, wavelength, zenith, receiver aperture, waist radius, radial index and inner scale on spiral spectrum is investigated on the LG vortex beam propagating in slant atmospheric medium. It is achieved that, when propagation distance, topological charge, zenith and receive aperture increases, the spiral spectrum becomes wider. However, with the increase of wavelength and turbulence inner scale, the spiral spectrum spread less [63].

#### **4.2 Scintillation properties**

Optical wave propagating through a random medium such as the atmosphere, ocean and tissue etc. encounters fluctuations of beam intensity during the short and long propagation paths. This mechanism briefly explained by the scintillation of medium. Scintillation is caused by the external effect which is temperature variations in the random medium, resulting in index-of-refraction fluctuations (i.e., optical turbulence). Theoretical and experimental studies of scintillation have become more important nowadays since optical communication system adopts many types of beams. Accordingly, the scintillation index of LG beams is investigated in [17, 64]. The scintillation index of LG beam having different topological charges is demonstrated in **Figure 4a**. It is shown that, as propagation distance increases scintillation index increases as well. Also, it is obvious that the scintillation

#### **Figure 4.**

*Scintillation index of (a) LG beam with different beam orders (b) vector versus scalar vortex beams (c) LG vortex against Gaussian beam, (d) single and double vortex beam [21, 64–66].*

*The Propagation of Vortex Beams in Random Mediums DOI: http://dx.doi.org/10.5772/intechopen.101061*

of non-vortex beams is higher than that of the vortex beams since having a higher topological charge results in lower scintillation levels [64]. Also, it is obtained that, Gaussian beams are much more affected by the scintillation than LG vortex beams [17]. Additionally, the scintillation properties of vectorel and scalar vortex beam are analyzed both numerically and experimentally as shown in **Figure 4b**. This study realized that vectorel vortex beam provides an advantage over the scalar vortex beam since it has lower scintillation index for long propagation distances [21]. Furthermore, scintillation performance of various vortex beams (flat-topped Gaussian vortex, elliptical Gaussian vortex beam, Gaussian vortex beam) in strong turbulence region is investigated in [67]. It is achieved that, higher topological charges uniformly leading to lower scintillation [67]. The scintillation performance of Sinh Gaussian (SH-G) vortex beam has derived and investigated in [68]. This study has discovered that scintillation index of SH-G beam is higher than that of SH-G vortex beam under the same propagation circumstances. Comparison between Gauss and LG vortex beam in terms of scintillation index with different radius of targets is given in **Figure 4c**. The scintillation indices of the two beams decrease while weak turbulence effect exists. However, in case of strong turbulence, the scintillation indices increase. Moreover, the scintillation indices of Gaussian beam are higher than those of LG vortex beams [65]. Finally, **Figure 4d** shows the scintillation indices of single (beam 1 and beam 2) and double vortex beam (beam 3 and beam 4). All the beams have a similar scintillation levels at short propagation distance. On the other hand, the scintillation indices of the single vortex beams increase gradually at longer propagation distance [66]. Flat-topped Gaussian vortex beam propagating in a weakly turbulent atmosphere is investigated and scintillation properties are observed. It is found that flat-topped Gaussian vortex beam with high topological charges has less scintillation than the fundamental Gaussian beam [69].

#### **4.3 Oceanic turbulence**

Underwater Optical communication has attracted much attention due to its ability to provide the required large capacity and high-speed communication. Accordingly, many scientific research and exploration regarding the underwater environment are on progress. Among these, studying the propagation of laser beams under the effect of oceanic parameters namely spatial correlation length (σ), dissipation rate of temperature (χt), kinetic energy per unit mass of fluid (ε), relative strength of temperature, salinity fluctuations (ζ) and wavelength (*λ*). In this context, the detection probability characteristics of a Hyper geometric-Gaussian (HyGG) vortex beam propagating in oceanic turbulence are analyzed with different wavelengths as in **Figure 5a**. Beams operating at higher wavelengths have higher detection probability [70]. Further, HyGG beam with smaller topological charge is more resistant to oceanic turbulence. Furthermore, detection probability of Hermite Gaussian vortex beam tends to increase by the increase of ε [74]. Scintillation index of Gaussian vortex beam in oceanic turbulence is investigated for different waist widths as given in [75]. Besides that, Flat-topped vortex hollow beam is analyzed, where it recognized that this beam keeps its original intensity pattern in short propagation distances. Yet, it evolves into Gaussian like beam in far-field. Also, flat-topped vortex beam transforms into a Gaussian beam with decreasing of σ, ζ and ε as well as increasing of χ<sup>t</sup> [76]. As given in **Figure 5b**, the detection ratio of Airy vortex beam is higher than that of LG beam when topological charge is higher than 5. Otherwise, it is the opposite when the topological charge is less than or equal 4. Likewise, the interference of Airy vortex beam becomes stronger when χt, ζ and the propagation distance increases [71]. Stochastic

**Figure 5.**

*(a) Detection probability of HyGG vortex beam with different wavelength and (b) detection ratio of LG versus airy vortex beam, (c) average intensity of stochastic electromagnetic vortex beam with different topological charges and (d) elliptical chirped Gaussian vortex with different value of χ<sup>t</sup> [70–73].*

electromagnetic vortex beam depending on the topological charge is analyzed in oceanic turbulence. While the topological charge increases, larger dark vortex core is obtained as in **Figure 5c** [72]. The effect of χ<sup>t</sup> on rotating elliptical chirped Gaussian vortex beam (RECGVB) is illustrated in **Figure 5d**. It is obvious that, while χ<sup>t</sup> increases, the minimum normalized intensity distribution of RECGVB increases and the spreading of the beam becomes wider which turn into Gaussianlike distribution at the receiver [73]. Finally, Lorentz-Gauss vortex beam propagating through oceanic turbulence is studied. As a result, it is obtained that beams with higher order topological charges have larger dark center and the beam can protect these properties as the distance increases [77].

#### **4.4 Other mediums**

Besides the oceanic and atmospheric medium, propagation of vortex beams in other mediums is important for the optical communication system. The propagation properties of Gaussian vortex beam in gradient index medium are investigated where the phase distribution of the beam is calculated by the Gradient index parameter. While the gradient index parameter increases, periodical cycles become shorter. The topological charge can also influence the period of the phase distributions [78]. Finally, anamalous vortex beam is investigated in strongly nonlocal nonlinear medium. The results present that, the input power plays a key role in the beam evolution. By selecting a proper input power, the beam width can be controlled [79].

#### **5. Propagation properties of partially coherent vortex beams**

A partially coherent beam is the beam with a low coherence length which was first demonstrated by Gori et al. [80]. This beam types have some unique properties, such as the cross-spectral density, and correlation function which is different than that of fully coherent beams. On the other hand, partially coherent beams are able to reduce the scintillation induced by the turbulence, the beam spreading, and the image noise when compared with the fully coherent beams [81, 82]. Recently, many research groups have conducted a wide range of studies regarding the propagation of partially coherent vortex beams either in atmosphere, ocean or other mediums.

#### **5.1 Atmospheric turbulence**

GSM vortex beam can be introduced as a partially coherent vortex (PCV) beam and many studies have inquired into this beam type. The influence of structure constant, spatial correlation length and beam index on GSM beam is investigated in details. As given in **Figure 6a** as the structure constant increases, the normalized propagation factor increases as well. Additionally, the beam width increases likewise [83, 86]. Similarly, multi GSM vortex beam with smaller correlation length tends to lose its dark hollow center and evolve into a Gaussian beam as obtained in **Figure 6b** [84, 87]. **Figure 6c** illustrates the scintillation index of GSM beam against

#### **Figure 6.**

*(a) Average intensity of GSM vortex beam for different structure constant values, (b) average intensity of multi GSM vortex beam at different correlation lengths, (c) scintillation index of GSM beam against GSM vortex beams and (d) beam spreading of fully and partially coherent beam types [9, 83–85].*

GSM vortex beam. It is clear that, the scintillation index of the two beams increases as the coherence length increases. Also, for the coherence length being larger than 0.35 mm, GSM vortex beam is less affected by the turbulence than the GSM beam [9]. Moreover, beam index is another important parameter that affect the GSM vortex beam. While the beam index increases, the focused beam profile becomes flatter [88, 89]. Finally, **Figure 6d** explains that, the PCV beam is obviously suffers from less beam spreading than the fully coherent vortex beam as expected [85].

Besides that, the propagation of partially coherent double-vortex beams in turbulent atmosphere is investigated deeply. Accordingly, it is observed that the topological charge, source beam width, degree of coherence at the source plane and the propagation distance are effective parameters on the intensity distributions. Consequently, as the propagation distance increases, beam profile changes to a Gaussian beam shape [90]. Moreover, the spreading of partially coherent flat-topped and Gaussian vortex beams in atmospheric turbulence is analyzed. It is achieved that the beam width of partially coherent beams increases as the distance increases and vortex beams are less affected by the atmospheric turbulence than the non-vortex ones. [91, 92]. Another study analyzes the partially four-petal elliptic Gaussian vortex beams propagating in turbulent atmosphere. It is achieved that partially coherent four-petal elliptic Gaussian beams with larger topological charge, smaller beam order, and larger ellipticity factor are less influenced by atmospheric turbulence. Moreover, vortex beams spread faster with the decreasing of the coherence length [93]. Scintillation index of partially coherent radially polarized vortex (PCRPV) beams, and PCV are analyzed as well. According to the obtained numerical results, scintillation index of PCRPV beams is lower than that of the PCV beams [94]. The propagation of partially coherent electromagnetic rotating elliptical Gaussian vortex (PCEREGV) beam through non-Kolmogorov turbulence is investigated numerically. Thus, it is realized that the normalized spectrum density of PCEREGV beam is slightly affected by the inner scale, while the operating wavelength greatly influences the spectrum density. Normalized spectrum density distributes more dispersedly and its minimum becomes larger when operating at higher wavelengths [95]. Finally, partially coherent twisted elliptical and circular vortex beams are analyzed and it is obtained that, elliptical vortex structure beam has advantage over the circular vortex with twisted phase modulation [96].

On the other hand, partially coherent LG and GSM vortex beams in slant atmospheric medium are analyzed by the researchers in [97, 98]. The beam wandering of GSM vortex beams along a slant path is lower than the horizontal path in case of long propagation distances [97]. Also, when partially coherent LG vortex beam is propagating in a slant path, bigger source coherence parameter causes a smaller transverse coherence length. A large zenith angle results in a small transverse coherence length of the beam [98].

#### **5.2 Oceanic turbulence**

Cross-spectral density and average intensity of GSM vortex beams propagating in oceanic turbulence are discussed and their analytical expressions are obtained using extended Huygens–Fresnel principle. The intensity equals zero at the center then as the distance increases, flat-topped beam takes place and, consequently, evolves into a Gaussian beam shape [99]. Not only the increase in χt, and ζ but also the decrease in ε lead the partially coherent GSM vortex beams to lose their dark hollow center pattern and evolve into a flat-topped beam and Gaussian-like beams as the propagation distance increases under the strong oceanic turbulence [87, 100]. In addition, Lorentz–Gauss vortex beam generated by a Schell-model source becomes wider with the increase of the oceanic turbulence parameters namely χt,

*The Propagation of Vortex Beams in Random Mediums DOI: http://dx.doi.org/10.5772/intechopen.101061*

and ζ [101]. Furthermore, partially coherent flat-topped vortex hollow beam in oceanic turbulence with higher beam order loses its initial dark hollow center slower compared to the beam with lower beam order [23]. Partially coherent four-petal Gaussian vortex, anomalous hollow vortex beams are also discussed under the effect of oceanic turbulence. It is found that the partially coherent four-petal Gaussian vortex that has four petals profile in near field propagation, then turns into a Gauss-like beam rapidly with either decreasing σ, ς and ε, or increasing the oceanic parameter χ<sup>t</sup> in the far field [102]. For partially coherent anomalous hollow vortex beam, the parameters χ<sup>t</sup> and ς give rise to larger spreading of beam rather than ε [103].

#### **5.3 Other mediums**

The propagation of PCV beams in other mediums is also investigated. Consequently, the propagation properties of PCV beams in gain media are investigated. For longer propagation distances, PCV beams keep their original dark hollow intensity profile when having a higher topological charge value and larger coherence length. As the coherence length increases, the effective transmission distance of PCV beams with hollow distribution increases. However, fully coherent vortex beams always keep the hollow distribution while propagating in the gain medium [104].

#### **6. Conclusion**

The increasing importance of underwater and atmosphere wireless optical communication in a wide range of applications, has shaded the light on understanding the laser beam propagation in random media. In this context vortex beams play a role as one of the attractive laser beams which have become a widely investigated beam. The interest that these beams gained is due to their phase distribution that can be modulated to transmit the message signals. This way, they pose an alternative to the classical intensity or phase modulations that wireless optical communication links use. Thus, vortex beams are able to increase the ability of optical communication systems through mode multiplexing and high ratio terabit freespace data transmission. On the other hand, vortex beams are able to reduce the turbulence-induced scintillation, that leads to a better system performance.

In this context, this chapter introduces the research conducted up to date regarding the propagation of different vortex beam types in random medium. Besides summaries the effects of a variety of parameters such as the beam order, topological charge, coherence length, wavelength, source size, relative strength of temperature and salinity fluctuations on the beam properties. It observed that both Gaussian–Schell model vortex and elliptical vortex beams are able to improve the system performance through the reduction of scintillation that is induced by the atmospheric turbulence. Besides that, Laguerre–Gaussian vortex beam as an information carrier in the free-space optical link decreases the aperture averaged scintillation when increasing the topological charge value. The Laguerre–Gaussian vortex and combined Gaussian-vortex beams provides a room for the system performance improvement which is originated from the effective reduction of the scintillation index especially with the increase of the topological charge. Therefore, vortex beams are capable to propagate longer distances. In addition, beams with OAM mode provide another degree of freedom for multiplexing applications, especially space-division-multiplexing (SDM) systems which is sufficient for higher communication capacity. On the other hand, the double vortex beams offer advantages

over the single vortex beams for long communication links. Moreover, a comparative study investigated the propagation of different of vortex beam types in strong turbulence, and revealed that as the values of topological charge increases the scintillation level decreases. Partially coherent vortex beams are able to reduce the scintillation, and beam spreading when compared to the fully coherent beams.

This chapter sets the models of optical wave propagating in random medium such as atmosphere, ocean and gain media. Then, focuses on the propagation of different vortex beams, either fully coherent or partially coherent, in different turbulent mediums. The presented results serve as an adequate database for understanding the propagation of vortex beams in random medium. Thus, provides an essential aid for further investigations in utilizing vortex beams in a wide range of application namely not only underwater optical communication, laser satellite communication systems but also sensing systems.

#### **Author details**

Sekip Dalgac and Kholoud Elmabruk\* Electrical Electronics Engineering Department, Sivas University of Science and Technology, Sivas, Turkey

\*Address all correspondence to: elmabruk@sivas.edu.tr

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*The Propagation of Vortex Beams in Random Mediums DOI: http://dx.doi.org/10.5772/intechopen.101061*

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