Propagation properties of higher-order cosine-hyperbolic-Gaussian beams in a chiral medium

The paraxial propagation of a higher-order cosine-hyperbolic-Gaussian beam (HOChGB) in a chiral medium is investigated theoretically. Analytical expression of the HOChGB passing through a chiral medium is derived based on the Huygens-Fresnel Integral and the ABCD transfer matrix. From the obtained formula, the propagation properties of this beam in a chiral medium are analyzed with numerical illustrative examples. The obtained results show that the evolution of the intensity pattern of HOChGB depends strongly on the chiral factor, the beam order, and the decentered parameter .


Introduction
The sinusoidal-Gaussian beams are exact set solutions of the paraxial wave equation in the Cartesian coordinate system and represent a generalized form of light beams propagating in a complex paraxial system (Casperson et al. 1997).Among the sinusoidal-Gaussian beams, the cosine hyperbolic-Gaussian beam (ChGB) has a flexible intensity profile and highly efficient power extraction (Lü et al. 1999;Eyyuboglu and Baykal 2004;Hricha and Belafhal 2005;Chu 2007; Li et al. 2007).The ChGBs have been largely researched for their potential application in many fields, e.g.micromanipulation of particles, optical trapping, telecommunications, highpower laser, and plasma physics (Patil et al. 2012;Kaur et al. 2017;Wani and Kant 2014;Konar et al. 2007;Mahajan et al. 2018;Patil et al. 2009;Gill 2010;Chen and Dai 2009) .A more generalized expression of the ChGB called the higher-order cosh-Gaussian beam (HOChGB) was introduced by Zhou and Zheng (Zhou and Zheng 2009) in 2009.The HOChGB is defined as a higher-order cosh function multiplied by a Gaussian function and characterized by two parameters, named the beam order N and the decentered parameter b.By selecting appropriate parameters of the source, the HOChGB can take on three different intensity profiles, a Gaussian-like, a flattened, and dark-hollow-like.The cosh-Gaussian and the Gaussian beams can be regarded as special cases of the HOChGB when N=1 and N=0, respectively.The propagation characteristics of the HOChGB through various optical systems, including a paraxial ABCD system, turbulent atmosphere, Airy Transform system, uniaxial crystal, and also the vectorial structure properties of the beam have been investigated (Zhou 2009;Chen et al 2009;Li et al. 2010a, b;Zhou 2011;Kang et al. 2015;Yaalou et al. 2020;Hricha et al. 2021).
On the other hand, chiral mediums have many unique properties compared with ordinary ones; e.g., negative refraction, circular dichroism, polarization rotation, and so on.In the past years, research on materials with chiral properties has received much attention due to their applications, e.g. in the fields of biochemistry and medicine (Pendry 2004;Chern and Chang 2013;Sersic et al. 2012;Baimuratov et al 2017;Kwon et al 2008;Lee et al 2013;Zhang et al 2017;Beaulieu et al. 2018).When a linearly polarized light beam is incident upon a chiral medium, it splits into right circularly polarized (RCP) and left circularly polarized (LCP) filed components, with different phase velocities.The combination of both components results in the interference phenomenon, and a new intensity structure can be produced.Recently, there has been extensive research on the propagation characteristics of various laser beams in chiral media.Among them, one can cite stochastic electromagnetic beams (Zhuang et al. 2011), Airy beams (Zhuang et al. 2012), Vortex Airy beam (Liu and Zhao 2014), Airy-Gaussian beams (Deng et al. 2016), Airy-Gaussian-vortex beams (Hua et al 2017), first order chirped Airy-vortex beams (Xie et al. 2018), Cosh-Airy vortex beams (Yang et al 2020), Bessel-Gaussian beams (Hui et al. 2018), and Pearcey-Gaussian beams (Zeng and Deng 2020), cosine-hyperbolic-Gaussian (Hricha et al. 2022) and Hermite-cosh-Gaussian beam (Yaalou et al. 2023).However, to the best of our knowledge, there has been no report on the properties of a HOChGB when it passes through a chiral medium.Therefore, the present work aims to investigate this research subject.The remainder of the manuscript is organized as follows.In Section 2, the analytical formula of a HOChGB propagating in a chiral medium is derived by using the Huygens-Fresnel diffraction integral.
The evolution of the intensity distribution of the HOChGB in a chiral is analyzed numerically as a function of the chiral factor and the initial beam parameters in Section 3. Finally, the main results of this study are highlighted in the conclusion part.

The theoretical model for a HOChGB propagating in a chiral medium
The electric field distribution of a (HOChGB at the source plane (z=0) in the rectangular coordinate system can be expressed as (Patil et al. 2009) where ( ) , with p=N or M. (2) The integers N and M stand for the beam order along the x-and y-directions, respectively.x0 and y0 are the transverse coordinates at the source plane, cosh (.) is the hyperbolic-cosine function,  is the waist size of the Gaussian part, and b is the decentered parameter.
As the HOChG field in the initial plane is separable in the x-and y-directions, we will use for the sake of convenience the one-dimensional form (1D) of the beam to derive the output field in the chiral medium.So, in the following, we assume a (1D) HOChGB of the form ( ) By using the explicit form of cosh (.) and the well-known binomial formula, Eq. ( 2) can be rearranged as (Casperson et al 1997;Li et al. 2010b) where , and (5) Eq. ( 4) indicates that a (1D) HOChGB can be regarded as a superposition of (N+1) decentered Gaussian beams with different weight coefficients and the same waist.This also means that the HOChGB can be produced by a superposition of (N+1) 2 decentered Gaussian beams (Lü et al. 1999;Eyyuboglu and Baykal 2004;Hricha and Belafhal 2005;Chu 2007).
In the special case b=0, the HOChGB turns to the fundamental Gaussian beam.
Within the framework of the paraxial approximation, the output electric field ( ) propagating through a paraxial ABCD optical system along the z-axis obeys the Huygens-Fresnel diffraction integral, which is given by (Collins 1970) ( ) ( ) where z is the distance from the initial to the output plane.A, B, and D are the matrix elements of the paraxial ABCD optical system,  The refractive indices, n (L) and n (R)  , which are related to LCP and RCP fields (respectively), are given as where  is the chiral factor, and n0 is the original refractive index.
Hence, the ABCD matrix associated with each component (i.e., LCP or RCP fields) is then given as J=L or R.
Substituting from Eq. ( 4) and Eq. ( 8) into Eq.( 4), one can obtain where ) and Now, recalling the following equality (Belafhal et al. 2020) after doing some algebraic calculations, Eq. ( 7a) can be expressed as where with R 0 iz q = , and The propagation of the beam components (i.e., LCP or RCP fields) in a chiral medium can be regarded as a combination of decentered Gaussian modes.
The total output field is given as -In the special case of b=0, or N=0, one obtains the intensity formula for the Gaussian beam in the chiral medium.
-By substituting =0 in Eq. ( 14), one can obtain the expression of HOChGB propagating in homogeneous optical media of index n0, which is consistent with the result reported in Ref. (Zhou 2009;Hricha et al 2021).The superposition of the two beam components propagating in the chiral medium will lead to the interference effect, as a result, the intensity distribution of the output beam will be more complicated compared to the initial beam.
The intensity can be expressed as are the intensity of LCP and RLP components, respectively, ( ) is the interference term, and the asterisk *denotes the complex conjugation.By substituting Eq. ( 12) into Eq.( 15), one can directly obtain the intensity equation of the HOChGB in the chiral medium.From the derived expressions, one can see that the intensity of the output beam depends on the chirality factor , propagation distance z, and the initial HOChGB parameters (N, b, , and 0).
The field expression for the tridimensional HOChGB (i.e., with two transverse coordinates) in the chiral medium can be written as where with J=L or R.
The total intensity is then given as with and (18d)

Numerical results and discussion
In this Section, we will investigate numerically the properties of a HOChGB propagating in a chiral medium based on the formulae derived above.In the simulations, the calculation parameters are set as 0=0.1mm,=632.8nm, n0=3, 2 0 / 2 4.96 The typical 1D-normalized intensity distributions and the contour graphs of HOChGBs propagating in the chiral medium, for the beam orders N=2 and 3, and three decentered parameters, b=0.1, 1.5, and 4 are illustrated in Figs. 2 and 3.
From the plots, one can see that the HOChGB gradually spreads upon propagation and its  When b has a small value (typically b<1), the beam propagating in the near field region (i.e., z=zR) presents a main central bright spot with some secondary concentric ringed spots around.
The beam profile is preserved within the far-field region although the relative intensity of the ringed spots changes slightly.When b is large (saying b=4), the beam exhibits a four-lobe-like pattern in the near field, and each intensity lobe includes many interference fringes.The parameters are the same as in Fig. 2.
One can notice that the inter-fringe increases gradually as the propagation distance increases.
In the far-field region, the interference fringes overlap, and the beam evolves into a complex structured profile, with a bright central spot.The structure of the output intensity pattern depends on the value of N (see the third column in Figs. 2 and 3).
To compare the above beam behavior in the chiral medium with the one in free space, we present in Figs. 4 and 5, the intensity distribution of the HOChGB in free space under the same conditions as in Figs. 2 and 4.
One can easily see that, the beam with a small value of b retains its initial Gaussian profile upon propagation in free space (see the first column in Fig. 4 and 5) in contrast with the chiral medium case where the initial beam profile is transformed into a ringed beam-like.In addition, the beam spreading is slower in the chiral medium than in free space propagation.6) shows the effect of increasing the chiral factor  ( is increased from 0.1/k0 to 0.22/k0) on the output intensity distribution in the near-field and far-field zones.One can observe from the plots that in the near field zone if  is augmented, the number of peak intensity oscillations increases and their interspace decreases, i.e., bright ringed spot become thinner.In the far-field region (saying z=10zR), as  is increased the peak intensity oscillations become thinner, and the beam width decreases.From the numerical results, one can conclude that a HOChGB propagating in chiral medium can generate rich optical intensity patterns by adjusting the optical field structure and the chiral medium conditions.

Conclusion
The propagation properties of a linearly polarized HOChGB in a chiral medium are investigated by using the Huygens-Fresnel diffraction integral and the ABCD matrix method.
Based on the obtained field expression, the intensity distribution of the beam is calculated with numerical examples and analyzed as a function of the chiral factor and the initial beam parameters.It is shown that the output intensity distribution depends sensitively on the chirality factor, the beam order, and the decentered parameters.Different beam shapes can be generated by adjusting the parameters of the optical field structure.HOChGB brings more possibilities for particle manipulation and trapping, due to their rich controllability under chiral medium conditions.This study could be beneficial for applying HOChGB in beam shaping and optical micromanipulation.
and  is the wavelength of laser light in a vacuum.Let us assume an incident HOChGB with a linear polarization propagating through a chiral medium along the z-axis as schematized in Fig.1.Because of the chirality of the medium, the initial beam will produce right circularly polarized (RCP) and left circularly polarized (LCP) components.The two beam components will have different phase velocities due to their different refractive indices(Zhuang and Zhao 2011)

Figure 1 :
Figure 1: Schematic of a HOChGB passing through a chiral medium.
intensity profile undergoes sensitive changes, which depend on the initial beam parameters N and b.Based on numerous numerical simulations, we have discovered that the evolution of the HOChGB in the chiral medium depends strongly on the decentered parameter b.There are two types of beam comportment, which are determined by the value of b.

Figure 2 :
Figure 2: The normalized intensity (1D) in the x-direction of HOChGB in a chiral medium (with

Figure 3 :
Figure 3: The intensity contour graph of a (2D) HOChGB in a chiral medium.
One can notice from the same plots that a HOChGB with large b can exhibit structured intensity patterns with rich interference fringes, and which can be controlled by adjusting the values of N and b.The above beam dynamics in the chiral medium is physically the results of the interference of different initial Gaussian components and their splitting into LCP and RCP modes.

Figure 4 :
Figure 4: The intensity distribution in the x-direction of a HOChGB in free space at different propagation distances z and different parameters b.First row (A) N=2 , second row (B) N=3.The parameters are the same as in Fig. 2.

Figure 5 :
Figure 5: The normalized intensity contour graph of a (2D) HOChGB in free space.Parameters are the same as in Fig. 2.

Figure 6 :
Figure 6: The normalized intensity in the x-direction of a HOChGB in a chiral medium with different parameters  at the plane z= zR (near field) and z=10 zR (far-field).