Paraxial propagation of Hermite cosine-hyperbolic-Gaussian beams in a chiral medium

The propagation properties of the Hermite-cosine-hyperbolic-Gaussian beam (HChGB) in a chiral medium are investigated. The analytical formula for a HChGB beam propagating through a chiral medium is derived theoretically based on the Huygens-Fresnel integral, and the propagation properties are illustrated numerically and discussed. Results show that the evolution properties of the HChGB in a chiral medium are closely related to the beam order, the chirality factor, and the decentered parameter b.


Introduction
The propagation of light beams in optical systems is of great interest and has attracted the attention of many laser researchers over the last decades.Exact solutions to the wave equation have been investigated for certain initial conditions and symmetries.In particular, the Hermite-Sinusoidal-Gaussian beam (HSGB) was introduced for the rectangular symmetry by Tovar and coworkers (Casperson and Tovar 1998;Tovar and Casperson 1998) as a general solution to the paraxial equation.The amplitude of the HSGB is the product of Hermite polynomials, sinusoidal function, and Gaussian function with complex arguments.The Gaussian beams, sinusoidal-Gaussian beams, and Hermite-Gaussian beams can be regarded as particular cases of HSGB (Casperson et al. 1997;Lü et al. 1999;Hricha and Belafhal 2005a;Deng 2004).The HSGB can also be assumed as a superposition of two Hermite-decentered Gaussian beams (Tovar and Casperson 1998;Casperson et al. 1997;Lü et al. 1999;Hricha and Belafhal 2005a;Deng 2004Deng , 2006)).Among the HSGB family, the Hermite-hyperbolic cosine-Gaussian beam HChGB has gained much attention during the last years besides the cosh-Gaussian beams (ChGB) (Siegman 1986;Ibnchaikh et al. 2001;Hricha andBelafhal 2004, 2005b, c;Hricha et al. 1281 Page 2 of 12 2021a, 2021b, 2022a; Yaalou et al. 2020).In the waist plane, the HChGB cross-section is square-shaped, and its intensity distribution pattern is closely related to the parameters b and m, which are associated with the cosh part and mode index of the Hermite polynomials, respectively.For a small value of b, the HChGB is multi-petal-like with (m + 1) 2 lobes, and where the outer lobes intensity being greater than the axial ones.For large values of b, the beam profile is strictly four-petal-like.As b or m increases, the peak intensity of the beam becomes more acentric.In particular, by selecting the appropriate values of the beam parameters, a HChGB can describe the Gaussian beam, Hermite-Gaussian beam, and the cosh-Gaussian beam.The shape variability of the HChGB has stimulated various applications, e.g., micromanipulation of particles, optical trapping, telecommunications, high-power laser, and plasma physics (Li et al. 2007;Patil et al. 2012Patil et al. , 2009;;Kaur et al. 2017;Wani and Kant 2018;Konar et al. 2007;Mahajan et al. 2018;Gill 2010).So far, the propagation features of HChGB in different optical systems have been investigated.The propagation characteristics of this beam in a Kerr medium, a strongly nonlocal medium (SNNM), a fractional Fourier transform system (FrFT), a quadratic gradient media (GRIN), and uniaxial crystals have been examined (Chen and Dai 2010;Hricha et al. 2020Hricha et al. , 2021cHricha et al. , 2022bHricha et al. , 2023Hricha et al. , 2022c;;Du and Zhao 2008;Bayraktar 2021;Lazrek et al. 2022Lazrek et al. , 2023)).
On the other hand, the propagation properties of light beams in anisotropic optical media have received much attention because of their importance for optical devices and nonlinear optics.Indeed, many optical devices are designed based on the anisotropic properties of the medium on the electric field of light beams.Among the anisotropic systems, chiral media have extensively been researched for their chiral activity effect on the polarization and energy distribution of light beams.Chiral systems can give rise to many exotic properties such as negative refraction and circular dichroism, which can be explored in several applications, e.g., in biochemistry, medicine, and so on (Pendry 2004;Kwon et al. 2008;Sersic et al. 2012;Chern and Chang 2013;Baimuratov et al. 2017;Lee et al. 2013;Zhang et al. 2017;Beaulieu et al. 2018;Zhuang et al. 2011).In a chiral medium, an incident linearly polarized beam is split into two circularly polarized waves with right and left polarizations, and the two resulting waves have different phase velocities in the optical media.The interference between the two waves may lead to a more complex intensity distribution and will provide more richness in the resulting beam pattern.During the last years, the propagation properties of various light beams in a chiral medium have been examined such as stochastic electromagnetic beams (Jaggard et al. 1979), Airy beams (Zhuang et al. 2012;Wen and Chu 2014), Airy-Gaussian beams (Deng et al. 2016), Airy-Gauss-vortex beams (Liu and Zhao 2014), Chirped Airy beams (Hua et al. 2017), cosh-Airy 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).Recently, we have investigated the propagation properties of a ChGB passing through a chiral medium (Hricha et al. 2022d).The results have shown that the ChGB's intensity pattern can be controlled by adjusting the chirality factor and the decentered parameter b of the incident beam.The present paper aims to extend the previous study to the more general HChGB.
The remainder of this paper is structured as follows.In Sect.2, the theory model is described, and the analytical expression of a HChGB propagating in a chiral medium is derived based on the Huygens-Fresnel integral formula.Then, some numerical examples are presented to discuss the propagation characteristics of the output beam as functions of the chiral factor and the incident beam parameters in Sect.3. Finally, the main results of this study are highlighted in the conclusion part.

Propagation of HChGB in a chiral medium
It is assumed that a HChGB is propagating toward a chiral medium (z > 0), and the z-axis is taken to be the propagation direction (see Fig. 1).
The electric field of a HChGB at the source plane z = 0 takes the form (Tovar and Casperson 1998;Ibnchaikh et al. 2001).
where In these equations, x 0 , y 0 are the transverse Cartesian coordinates at arbitrary point in the source plane and 0 is the beam waist radius of the Gaussian part.b x , b y are the decentered parameter associated with the x-and y-directions, respectively.H p is the Hermite polynomials of p th -order.
As the amplitude expression of HChG field is separable, we will use a one-dimensional (1D) description to study the beam propagation in a paraxial optical system.
The (1D) HChGB can expressed (in the x-direction) as By using the cosh function definition, i.e. cosh(u) = e u +e −u 2 , Eq. ( 2) can be written as This last equation indicates that a HChGB can be produced by a superposition of two Hermite-decentered Gaussian beams with the same waist and in phase.The beam shape is determined by the decentered parameter b and the mode index p.For a small value of b, the beam is multi-lobe-like with (p + 1) lobes, and for a large value of b, the beam profile is two-lobe-like.As b or p increases, the peak intensity of the beam becomes more abaxial.
(1a) E p x 0 , y 0 = E p x 0 , 0 E p y 0 , 0 , (2) Now, let us assume that a HChGB is propagating through a paraxial ABCD system along the z-axis.Accordingly, the output beam E(x, z) can be described by the Huygens- Fresnel integral formula as (Collins 1970) where E p x 0 , 0 is the electric field at the source plane z = 0. z is the distance from the ini- tial plane to the output plane.A, B, and D are the elements of the transfer matrix associated with the optical system, and k = 2 the wave number with λ is the wavelength of the light beam in vacuum.
By substituting Eqs. ( 2) into (4), and using the expansion form of the Hermite function H p (.) (Gradshteyn and Ryzhik 1994).and recalling the following integral formula (Belafhal et al. 2020;Failed 1986).( 4) x n e −px 2 +2qx dx = e q 2 p � p where with α is the auxiliary parameter defined as After some algebraic calculations, one can obtain the propagation formula of a HChGB through a paraxial optical system as where Next, we will apply the above formalism to investigate the evolution of HChGB in a chiral medium (see Fig. 1).As is well-known, the transfer matrix of a chiral medium can be expressed according to the direction of circular polarization of the incident waves as where n L = n 0 1−n 0 k and n R = n 0 1+n 0 k are the refractive indices associated with the left circularly polarized (LCP) wave and right circularly polarized (RCP) wave, with γ as the chiral parameter, and n 0 as the original refractive index.
On substituting from Eqs. (9) into Eq.( 8), one can directly obtain the closed-form expression of the LCP and RCP waves as and ( 9) where f pL (x) and f pR (x) are related to n L and n R , respectively.The (2D) electric field in the chiral medium is then given by where The beam intensity can be expressed as with I L (x, y, z) and I R (x, y, z) are the intensity of the individual components LCP and RLP, respectively, and I int (x, y, z) is the interference term which is given as where the asterisk '*' denotes the complex conjugation.
(10b) 6 The contour graph of a HChGB for different chirality factors, with p = q = 2 and 3 (with b = 0.1, 1.5, and 4) propagating in a chiral medium at z = 3z r , and z = 10z r Equations ( 12) and ( 13) indicate that the intensity of the output beam depends on the chirality parameter γ, and the beam parameters p and b.The interference term I int (x, y, z) is the additional contribution due to the chirality of the medium.Thus, the HChGB propagating in a chiral medium is expected to allow more richness in the intensity distribution pattern in comparison to the free space propagation (or homogeneous media propagation).

Numerical calculations and analyses
Based on the analytical formulas obtained in the previous section, we will investigate numerically the propagation properties of a HChGB in a chiral medium.In the following numerical examples, the calculation parameters are set to be (otherwise stated) as ω 0 = 0.1 mm, λ = 632.8nm, n 0 = 3, γ = 0.16/k.It is to be noted that the intensity is normalized to its maximum value in the transversal plane, and z r is the Rayleigh length given by z r = k 2 0 ∕2.Figures 2 and 3 show the 1D-normalized intensity distribution and the contour graph of a HChGB propagating in a chiral medium at four propagation distances (z = 0, 0.5z r , 3z r , and 10z r ), and for different incident beam parameters.From the plots of these figures, one can see that the output beam evolves upon propagation, and the intensity distribution depends strongly on the values of the beam order p and decentered parameter b.One can point out that in the near field region, the initial peak intensity widens, and in the far field region, many secondary weak peak intensities appear as p becomes larger (see the fourth column and line in Figs. 2 and 3).Besides, one can observe that the parameter b affects the outgoing beam in a short propagation range.For a small value of b (saying b < 1.5), many lobes appear, and their widths widen as p increases.When b has large values (saying b > 1.5), the obtained field is strictly four-petal-like in the near field, but many new peaks appear in the far field (when p is greater than the unity).One can note that the beam has zero intensity at the center when the beam order is odd, and a bright lobe when the beam order is even.
The plots of the intensity distributions in Fig. 4 show that the RCP beam's component widens more quickly compared to the LCP one.The output beam evolves into multiple lobes, and the beam spreading decreases in the far-field region.The beam evolution in the chiral medium can be explained physically by the interference term, i.e., the overlapping of the LCP and RCP components.
Figures 5 and 6 illustrate the influence of the chirality factor (γ = 0, γ = 0.15/k, 0.2/k, and 0.25/k) on the output beam pattern in near-field and far-field zones (z = 3z r and z = 10z r ).From these figures, one can see that many peak intensities appear as γ increases, which means that the outgoing beam shape undergoes deterioration of intensity distributions compared to the free space propagation (i.e. the case γ = 0).The intensity distribution depends strongly on the values of the beam order, the factors γ and b.From the above results, it can be deduced that the profile of the output beam can be broadly controlled, for instance by varying the size of the central bright region and the number of the peak intensities.

Conclusion
In summary, a closed-form expression for a linearly polarized HChGB propagating in a chiral medium has been derived by using the Huygens-Fresnel integral formula and the ABCD matrix method.The propagation characteristics of the HChG beam in a chiral medium are illustrated numerically as functions of the chiral and the initial beam parameters.It is found that the output beam shape can be controlled by adjusting the chirality factor, the beam order, and the decentered parameter b of the incident of the HChGB.The interference between the two waves created by a linearly polarized beam in the chiral medium may lead to a more complex intensity distribution, and this may provide more richness in the resulting beam pattern.

Fig. 1
Fig. 1 Schematic illustration of a HChGB propagating through a chiral medium

Fig. 3 1281
Fig.3The contour graph of the normalized intensity of a HChGB in a chiral medium.The calculation parameters are the same as in Fig.2

Fig. 4
Fig.4Contour graphs of the normalized intensity of a HChGB in a chiral medium for p = q = 1 and 4 at z = 3z r and z = 10z r. .The calculation parameters are the same as in Fig.2

Fig. 5
Fig. 5 Normalized intensity (in the x-direction) of a HChGB with p = 2 and 3 (with b = 0.1, 1.5, and 4) in a chiral medium at z = 3z r , and z = 10z r for different chirality factors