Analyzing the spreading properties of vortex beam in turbulent biological tissues

Presenting the intensity development of a circular Laguerre-cosh-Gaussian (CLChG) beam in turbulent mouse biological tissues is the major goal of the current work. Using the power spectrum refractive index from Schmitt's model and the extended Huygens-Fresnel integral, the propagation formula of the CLChG beam is produced. In order to determine the spreading properties of the studied beam, analytical expressions of the CLChG beam's effective beam size in turbulent mouse biological tissues are constructed. Some graphical representations have to be carried out in order to discover the impacts of beam and biological turbulence parameters on this sort of beam. The findings show that the transformation of the CLChG beam into a Gaussian-like beam in the far field occurs more quickly when the beam passes through the deep dermis of the mouse. The shape of the CLChG beam can also be changed by choosing a specific value for the parameter linked to the cosh-part. Because the effective beam spot radius along the x- and y-axis are equal, we also see that the beam spot in biological tissues takes on a circular shape.


Introduction
Recently, substantial interest has been carried out to examine the propagation properties of various aspects of laser beams through different turbulent mediums and optical systems. This attraction is a result of its wide applications in different fields, such as optical communication, information encoding, transmission, and lattice spectroscopy (Yadav et al. 2007a(Yadav et al. , b, 2008Han 2009). Several scientists, including our research group, have developed large-scale studies in this area (Cai 2006;Zhou and Chu 2009;Yang et al. 2009;Wang et al. 2010;Zhao and Cai 2011;Zhong et al. 2011;Khannous et al. 2015;Ez-Zariy et al. 2016;Boufalah et al. 2018;Ye et al. 2019;Chib et al. 2020Chib et al. , 2022Hricha et al. 2021a, b;Lazrek et al. 2021;Nossir et al. 2021;Nabil et al. 2022a, b, c). The progress of medicine and molecular biology had made of the turbulent biological tissues another important propagation environment. To understand the interactions between light beams and biological tissues, the spreading of optical beams through these mediums becomes an essential issue that is receiving intense trends from researchers (Jin et al. 2016;Wu et al. 2016;Baykal et al. 2017;Saad and Belafhal 2017;Gökçe et al. 2020;Ebrahim and Belafhal 2021). Especially since the average intensity and propagation of laser beams play crucial roles in biophotonics research as a way of collecting quantitative data for cancer diagnosis or screening (Yi and Backman 2012). Due to the spatial variability of its refractive index, biological tissue is an extremely complicated system in which light is widely distributed in propagation. For this reason, Schmitt and Kumar (Schmitt and Kumar 1996) examined and studied statistical features of refractive-index fluctuations in the biological tissue samples. Their research came up with statistically identical spatial correlations to those caused by atmospheric turbulence. Hence, they proved that the structural function of refractive-index inhomogeneities in mammalian tissues fulfills the traditional Kolmogorov turbulence model. According to this power spectrum model, considerable studies have been established to define the spreading characteristics of diverse optical beams in turbulent biological tissues.
On the other hand, it is well known that the biological environment contains diverse tissue components that destroy the transmission of optical signals. To solve this problem, it has been demonstrated that certain types of laser beams are less susceptible to turbulence-related deterioration. Vortex beams are one type of these beams. For these features, a considerable number of research has been conducted about the propagation of stochastic electromagnetic vortex beams (Luo et al. 2014;Duan et al. 2020), anomalous hollow Gaussian beams (Lu et al. 2016), and Circular and elliptical hollow Gaussian beams (Saad and Belafhal 2017) through turbulent biological tissues. Up to now, the Circular Laguerre-Gaussian beams have been successively introduced and analyzed by Ebrahim and Belafhal (2021). This paper, seeks to study Circular Laguerre-cosh-Gaussian (CLChG) beam traveling in turbulent biological tissues, especially those of mouse. The proposed model contains one additional control parameter compared to the Circular Laguerre-Gaussian beams. The CLChG beam is a generalized form, and the Circular Laguerre-Gaussian beams can be a limiting case of this field. The intensity of the central area of CLChG beam can be controlled by altering the beam parameters. The specificity of this field can be generated as vortex beams, particularly since the cosh function is considered to be a superposition of the Gaussian function. Additionally, the Laguerre polynomial modulated by a Gaussian envelope is already demonstrated and is practically realizable as vortex beams.
The fundamental principle supporting optical biomedical technology employed in optical imaging, diagnostics, and treatments is the relationship between optical and biological characteristics of tissue. Applications for light-scattering diagnostics are attractive in part because to the large number of parameters that may be measured. Information encoded in the polarization, wavevector, wavelength, and phase of scattered light may be diagnostically relevant, while optical phenomena such as interference and propagation (Rinehart et al. 2015). can reveal previously unmeasurable quantities. A wide range of technologies that measure one or more characteristics to extract useful information from tissue have been developed as a result of the abundance of information in the scattered field.
The remainder of the paper is organized as follows. In Sect. 2, we shall introduce the distribution of CLChG beam. The irradiance distribution of CLChG beam in turbulent biological tissues of mouse and the effective beam spot are established in Sect. 3. The impact of the beam and turbulent biological tissues parameters on the evolution comportment of CLChG beam are presented graphically. The conclusion is finally given in Sect. 4.

Distribution of the circular Laguerre-cosh-Gaussian beam
In the cylindrical coordinates system, the electric field distribution of CLChG beam at the source plane (z = 0) is defined as where and are radial and azimuthal coordinates, A 0 is a constant (to simplify, we will take it equal to 1), L l m (.) denotes the Laguerre polynomial with m and l are the radial and angular mode orders, 0 is the waist width of the Gaussian part of CLChG beam, e il is the phase term for the beam, and Ω is the displacement parameter associated with the cosh-part.
Applying the following expression formulae (Abramowitz and Stegun 1970;Kimel and Elias 1993) and where H j (.) is the jth order Hermite polynomial and m n denotes the binomial coefficient, (1) can be written as The last equation represents the input field of the CLChG beam, which is introduced as a finite sum of Hermite-Gaussian modes in Cartesian coordinates x and y with 2 = x 2 + y 2 . First, we will perform the evolution of the intensity for CLChG beam in the input plane for different values of the beam orders l and m with 0x = 0y = 2 μm and Ω = 0.15 μm −1 (see Fig. 1).
From this figure, it is observed that for m = l = 0, the beam becomes identical to the original Gaussian beam and when both m and l are different to zero the shape of CLChG beam has a central dark region. Additionally, we can find that when the beam orders m and l increase, the considered beam does not satisfy Hermite distribution due to the influence of Ω . The effect of the parameter associated with the cosh-part on the intensity distribution is presented in Fig. 2.
The other parameters are set to be: m = l = 2 and 0x = 0y = 2 μm. From this illustration, we can observe that the principal lobe disappears gradually with the increase of the parameter Ω , the intensity of the side lobes becomes more brilliant and the dark spot size widens. We can thus conclude that the variation of this parameter eliminates the law of the Hermite distribution.
In the following Section, we shall investigate the propagation of the CLChG beam through the turbulent biological tissues by means of the generalized Huygens-Fresnel diffraction integral in order to study their propagation characteristics.

Propagation of CLChG beam through the turbulent biological tissues of mouse
Based on the extended Huygens-Fresnel integral formula, the propagation of a laser beam in the turbulent biological tissues of mouse can be presented as (Andrews and Phillips 1998) where ⟨ I x , y , z ⟩ refers to the irradiance intensity distribution at the receiver plan z, k = 2 ∕ is the wavenumber with denotes the wavelength of the source radiation and ⟨⋅⟩ is the ensemble average over the turbulent media; which describes the influence of the turbulence on the propagation of laser beam. It can be expressed as In the last equations 0 is the coherence length of a spherical wave passing through biological tissues. The parameter C 2 n is the constant of refractive index structure of biological tissues indicates the ensemble-averaged variance of the refractive index, L 0 being the outer scale of the refractive index size and d f means the fractal dimension of the tissue.
In Eq. (4), (x i , y i ) with (i = 1 or 2) and x , y as the components of the position factor at the input plane (z = 0) and observation plane (z), respectively.
By substituting Eqs. (3) and (5) into Eq. (4), we obtain where (6) and with (j = x or y). Applying the following formulae (Abramowitz and Stegun 1970;Erdelyi et al. 1954;Belafhal et al. 2020) and Equation (6) can be written in the below form  The expressions of S 2x and S 2y are obtained from S 1x and S 1y if b x , b y in S 1x , S 1y are replaced by u x and u y , respectively. In a similar way, S 4x and S 4y are achieved by the replacement of b x , b y in S 3x , S 3y by u x , u y , respectively, with u j = a * j − 1 a j 4 0 .

and
In the following Section, referring to the results obtained in the previous part, we investigate numerically the evolution behavior of CLChG beam after spreading through turbulent biological tissues of mouse focusing on intestinal epithelium and deep dermis of mouse.

Graphical simulation results and discussion
Firstly, we will focus on the effect of the beam width, the propagation distance z, the beam orders (m, l), the structure constant of the refractive-index C 2 n , and the parameter associated with the cosh-part Ω on the average irradiance distribution for CLChG beam. (1, 2, 3, 4), j = j or a j , j = b j or u j , j 1 = n 1 , f 1 , g 1 , j 2 = n 2 , f 2 , g 2 , Furthermore, we examine the impact of C 2 n and wavelength on effective beam size. In our simulation, the wavelength is 632.8 nm. Figure 3 illustrates the variation of the normalized irradiation distribution of CLChG beam through the intestinal epithelium of mouse for different propagation distances and beam widths. The other parameters are set to be: C 2 n = 0.6 × 10 −4 μm −1 , Ω = 10 −2 μm −1 and m = l = 1. From these curves, we can observe that, in the source plane, the beam is composed of the main lobe and a side one with a dark spot at the center. As we move away from the source, the CLChG beam undergoes two main changes: in the first the beam begins to have a certain brightness in the center while in the second the beam turns to Gaussian one. Besides, the small width of the beam accelerates the process of the form change. In this term, the CLChG beam, with a larger beam width, can resist more against the turbulence created in intestinal epithelium tissues of mouse. Figure 4 shows the various forms of the CLChG beam with several beam orders and two tissue types. The influence of the parameter l is identical in both intestinal epithelium mouse C 2 n = 0.6 × 10 −4 μm −1 and deep dermis of mouse C 2 n = 0.22 × 10 −3 μm −1 (Schmitt and Kumar 1996). For l = 0, we notice that the maximum intensity is located in the center and by increasing the value of the beam order l, the dark region of the CLChG beam widens to obtain hollow ones (see Fig. 4a and b at z = 0). It should be Z Fig. 3 Evolution of the normalized irradiance distribution of CLChG beam propagating in intestinal epithelium of mouse for the different beam widths with C 2 n = 0.6 × 10 −4 μm −1 and Ω = 10 −2 μm −1 1 3 98 Page 12 of 18 noted this characteristic is very important because it applies to a confinement of atoms (Ashkin 2006). At z = 0 the average intensity distribution of the CLChG beam is transformed into a Gaussian beam like, with a small value of the beam order, and for l = 6 the CLChG beam has a flattened shape. As we compare Fig. 4a and b, we can infer that the evolution of the transformation of the CLChG beam into a Gaussian beam is carried out rapidly in the deep dermis of the mouse and also the CLChG beam with bigger l has a better capacity to resist the turbulence in biological tissues. Figure 5 describes the evolution of the normalized average intensity of the CLChG beam with several m in turbulent biological tissues of mouse. At the input plane, it is seen that the evolution of the intensity for the considered beam fulfills the Hermite distribution with (m + 1) lobes. After propagation, it can be found that the form of the beam takes different shapes and at a large propagation distance the CLChG beam turn into Gaussian. The reason that CLChG beam converts into Gaussian-like beam is because at long-distance propagation the output beams become the completely incoherent light due to the effect of the turbulent medium ( 0 → 0 as z increases) (Eyyuboğlu et al. 2006). We can also observe that the propagation of the CLChG beam in the deep dermis of the mouse accelerates the process of evolution toward Gaussian beam compared with intestinal epithelium mouse.
To illustrate the influence of the cosh-part parameter on the propagation properties of CLChG beam propagating in turbulent biological tissues of mouse is presented in Fig. 6. Three different values of Ω are chosen as Ω = 0.1 μm −1 ; Ω = 0.3 μm −1 , and Ω = 0.6 μm −1 , and the other parameters are = 632.8 nm, m = l = 1, and z = 6.5 μm. From this figure, we can observe that in intestinal epithelium mouse the CLChG beam has a peak intensity in the center, for Ω = 0.3 µm −1 the intensity profile presents a central bright spot with side lobe located sideways. When the increase of Ω up to 0.6 µm −1 , it can be seen that the produced beam has a dark-centered distribution surrounded by three peaks of bright. On the other hand, in the deep dermis of the mouse when Ω = 0.1 μm −1 , the CLChG beam has a bright spot in the center. By varying this parameter the main peak divides into two peaks. Finally, we observe that the CLChG beam takes a flattened profile. We can say that the type of tissue has a considerably great impact in the form of the beam and the intensity of the central area of the beam can be controlled by adjusting the value of Ω.
Based on our analytical formulas provided by Eqs. (16-20), we investigate the square root beam width by numerical computations. To understand the propagation characteristics of CLChG beam, we numerically analyze the effective beam sizes W x (z) and W y (z) versus the propagation distance z in turbulent biological tissues of mouse for different values of the structure constant C 2 n in Fig. 7. In this figure, the calculation parameters are chosen as: m = l = 1, = 632.8 nm , Ω = 10 −2 μm −1 , 0x = 5 μm and 0y = 2 μm.
As observed from this illustration, at a long distance the quantity W x (z) becomes equivalent to W y (z) in turbulent biological tissues of mouse. As a result of the effect of turbulent biological tissues, the effective beam spot radius gradually becomes circular. Furthermore, we find that in the absence of turbulence (free space) the beam pot size clearly decreases and the effective beam spot radius in intestinal epithelium mouse is different from the deep dermis of the mouse. So, we can conclude that the quality of the beam depends on the tissue types.
Finally, we elaborate Fig. 8 to investigate more about the influence of wavelength on the effective beam spot size of the CLChG beam in terms of the propagation distance.

Conclusion
The propagation properties of a CLChG beam in turbulent biological tissues especially those of mouse are investigated in detail. The analytical formula of the average intensity of the analyzed beam in turbulent biological tissues is established within the turbulence theory and the power spectrum refractive-index model. Graphical representations performing the effects of the beam parameters and the turbulence strength on the beam spreading are illustrated. Also, we have evaluated the effective beam size of the CLChG beam in order to qualify the spreading properties of the considered beam. It is found that in the far-field, the CLChG beam turns into a Gaussian-like beam and the process is made faster when the beam propagates in the deep dermis of the mouse. Furthermore, the shape of the CLChG beam is adjustable by choosing a specific value of the cosh-part parameter. We find also that the beam spot has a circular shape due to the equivalent between W x (z) and W y (z) in turbulent biological tissues of mouse. Our study could be important for imaging and medical diagnosis. Effective beam sizes W x (z) and W y (z) of CLChG beam versus the propagation distance z in turbulent biological tissues of mouse for different values of C 2 n Fig. 8 The effect of wavelength on the effective beam size W x (z) of CLChG beam in the deep dermis of the mouse with 0x = 0y = 2 μm , Ω = 10 −2 μm −1 , and C 2 n = 0.22 × 10 −3 μm −1