Unraveling the scattering of a circularly polarized Lommel beam by a dielectric sphere

1. Introduction

Non-diffracting laser beams hold a unique place in the class of well-known laser beams (Ren et al., 2021; Arfan et al., 2024). The capacity of non-diffracting beams to sustain distinctive intensity distributions over extended distances has garnered considerable interest in the domains of optics and photonics (Stearns 2016). In 1987, Durnin did groundbreaking work that led to the idea of non-diffracting beams (Durnin et al., 1987). Studies like these sparked a lot of interest in theory and experiment and paved the way for researchers to look for alternative approaches that don’t cause light to diffract (Durnin 1987). Following this, numerous non-diffracting beams have been suggested and studied, including Bessel beams (Arfan et al., 2023, Ambrosio and Hernández-Figueroa, 2011, Arfan et al., 2025), Tricomi beams (Zhu et al., 2021), Airy beams (Chen et al., 2023), Lommel beams (Cui et al., 2018; Asif et al., 2024), etc. In principle, there is no limit to the amount of energy and extent that may be carried by a non-diffracting beam. Consequently, the existence of a genuinely non-diffracting beam is unattainable.

Bessel beams and Lommel beams are known as non-diffracting beams. These beams, i.e., Bessel beams and Lommel beams, are described by Bessel functions and Lommel functions, respectively. Assuming specific parameters of the Lommel beam, i.e., asymmetry parameter $\left(c\right)$ and beam order $\left(v\right)$ zero, the Lommel modes transform to standard Bessel beams. Potentially far-reaching uses are being explored with each new finding of non-diffracting beams. The use of Bessel beams in optical micromanipulation, communication systems, and particle dynamics plays their role very well in this matter (Arfan et al., 2024). The ability to manipulate and drive microparticles by directing them along curved trajectories and allowing them to bypass obstacles is another benefit of Airy beams (Ivaškevičiūtė-Povilauskienė et al., 2022). It means Airy beams can guide microparticles around obstacles and help them move along curved trajectories, making them useful for precise and flexible control of particle manipulation. The scattering of Lommel beams by a spherical surface is a critical area of investigation within the optical research community, particularly with its pressing applications, such as remarkable depth of field and propagation stability. In contrast to planar electromagnetic wave irradiation, it provides supplementary information about the scatterer’s geometry and structure through interactions with diverse particles. Kovalev and Kotlyar in 2015 introduced the term “non-diffracting Lommel beam,” which refers to a beam represented as a linear combination of Bessel beam modes, characterized by complex amplitude factors dependent on two variables (Kovalev and Kotlyar 2015). These are defined as radial mode index $\left(n\right)$ and azimuthal mode index ($m$ or $l$). Here, $n$ specifies the number of concentric circles, while $m$ or $l$ specifies the beams’ orbital angular momentum (OAM) or phase variation. Owing to its unique spatial configuration, it is more efficient than other types of beams.

Beams with arbitrary shapes and polarizations contribute more significantly to scattering than plane waves do due to their distinctive properties. One of the main points of focus is always the interaction between particles and electromagnetic waves. Comprehending the optical scattering features of particles improves understanding of the light interaction processes and facilitates the examination of particle attributes through analysis of scattered light beams. Right here, we investigate the scattering of a dielectric sphere by a Lommel beam exhibiting CP. The CP Lommel beam can produce a focus with a spot size lower than the diffraction limit, resulting in fascinating effects in electromagnetic scattering, subwavelength focusing, and imaging. It possesses numerous characteristics of a conventional Bessel beam, including notable non-diffraction, self-accelerating, self-healing, spatial angular momentum, Bessel-like intensity profile, and polarization control (Kovalev and Kotlyar 2015; Yu and Zhang 2017).

This research work focuses on non-diffracting Lommel beam results by using the generalized Lorenz-Mie Theory (GLMT), which has been perpetually developed and broadened to accommodate the interaction of particles with arbitrarily-shaped light beam (Gouesbet and Gréhan 2011). GLMT has emerged as a vital theoretical paradigm for examining structured light interactions, incorporating fundamental Mie principles while broadening its capability to polarized and shaped light contexts (Gouesbet et al., 1998). The fundamentals of GLMT are presented, along with mathematical formulae relating to the scattered electromagnetic waves (Maheu et al., 1988). The GLMT theory and its potential applications connecting to the optical sizing are detailed in Gouesbet et al., 1991. GLMT addresses electromagnetic scattering properties and mechanical impacts (calculation of cross-sections) for spherical particles of any dimension in an elegant way (Gouesbet and Lock 2015). GLMT primarily aims to determine beam shape coefficients (BSCs) linked to specific structured light, offering significant understanding of the physical principles that regulate particle interactions (Gouesbet et al., 2021).

Studies on diffraction-free Lommel beams, such as the scattering of homogeneous spherical particles (Chafiq and Belafhal 2018) and non-spherical particles, have been investigated (Cui et al., 2018). The theory of Lommel beam modes is discussed in an elegant way (Kovalev and Kotlyar 2015). The scattering and propagation of Lommel-Gaussian beams toward atmospheric turbulence and anisotropic oceanic turbulence were studied (Ez-Zariy et al., 2016; Yu and Zhang 2017). Exploring the scattering response of spherical particles by using vector Lommel beams in the context of GLMT has been done (Ahmidi et al., 2024). Analyzing the interaction of diffraction-free Lommel beams with spherical surfaces, specifically dielectric spheres, is essential for understanding their optical scattering behavior. Consequently, motivated by the potential applications of the non-diffracting Lommel beams, the present work conducts a scattering response of a dielectric sphere using the GLMT in the Lommel beam field.

The organization of the work is given as follows: First, we discuss analytical formulae regarding scattering of a Lommel beam by a dielectric sphere utilizing the well-known optical theory such as GLMT in the section titled “The theoretical model.” The characteristics of Lommel beams and their interaction with a dielectric sphere are analyzed in the section titled “Numerical Analysis and Discussion.” The “Conclusions” section presents the findings.

2. The Theoretical Model

The optical researchers proposed the idea of diffraction-free Lommel beams as a superposition of Bessel beam modes (Kovalev and Kotlyar 2015). The complete distributions of the amplitude function of an ideal Lommel beam depending on its beam order $v$ in the context of cylindrical coordinates can be expressed as

On the Lommel beam, $c$ stands for the asymmetry parameter. In this context, $v$ denotes the beam’s topological charge. For Lommel beams, $v$ also represents the beam order. It defines the vortex characteristics. It is also commonly known as the beam topological charge. $\phi$ denotes the azimuthal angle, defined as $\phi ={\tan }^{-1}\left(\frac{y}{x}\right)$. Here, $k_z=kcos\alpha$ and $k_t=ksin\alpha$ represent the longitudinal and transverse wave vectors. $\alpha$ specifies the axicon or beam half-cone angle. It influences the central spot width of the Lommel beam. The axicone angle of a Lommel beam, also known as its half-cone angle, is created by the beam’s wave fronts with its optical axis of propagation, which also influences its optical characteristics. The full wave number can be written as $k=\sqrt{k_z^2+k_t^2}=\frac{2\pi }{\lambda }$ with $\lambda$ being the wavelength of the incident non-diffracting Lommel beam.

Fig. 1 shows the interaction of the Lommel beam field with a dielectric sphere, which is kept in free space. It propagates along the $z$ direction and is polarized along the $x$ direction. The radius of the dielectric sphere is $a$. Right here, $o$ indicates the center of the dielectric sphere with a coordinate system $xyz$, while the coordinate system symbolizes the propagation characteristics of the Lommel beam with center $o^{\prime }$. The beam center coordinates are$x_0{y}_0{z}_0$.

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Fig. 1.

Geometrical diagram of Lommel beam and dielectric sphere used in studying the present interaction model. The intensity distribution of Lommel with beam wavelength 632.8 nm in the source plane has been generated. The dielectric sphere has a radius designated as a=1 µm.

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Based on the well-established GLMT (Gouesbet and Gréhan 2011), the incident electric and magnetic fields are represented by the BSCs and expansion of the vector spherical wave functions (VSWFs) as (Gong et al., 2018)

here $c_n^{pw}={\left(-i\right)}^{n+1}\frac{2n+1}{n\left(n+1\right)}$

$g_{n,TM}^m$ and $g_{n,TE}^m$ represent the BSCs of order $m$ of CP Lommel beam and these can be expressed as (Wang et al., 2016; Ahmidi et al., 2024)

with ${\rm T}={\sum }_{p=0}^{\infty }{\left(-1\right)}^p{c}^{2p}{J}_{m-\nu -2p-1}\left(k_t{\rho }_0\right)e^{-i\left(m-\nu -2p-1\right){\phi }_0}$.

and

with ${\rm K}={\sum }_{p=0}^{\infty }{\left(-1\right)}^p{c}^{2p}{J}_{m-\nu -2p+1}\left(k_t{\rho }_0\right)e^{-i\left(m-\nu -2p+1\right){\phi }_0}$

Similar expressions are used for the scattered and internal electromagnetic fields:

${\text{M}}_{\text{mn}}^{\left(1,3\right)}$ and ${\text{N}}_{mn}^{\left(1,3\right)}$ denote VSWFs (Frezza et al., 2018). Following are the boundary conditions (BCs) that can be used to obtain these unknown expansion coefficients from the boundary of the dielectric sphere.

By applying the electromagnetic field equations (2)-(3) and (8)-(11) to the BCs outlined in Eqs. (12)-(13), the relationships among the unknown field coefficients (incident, scattered, and internal fields) can be achieved.

The coefficients $a_n$, $b_n$, $c_n$, and $d_n$ define the classical scattering coefficients within the Lorenz-Mie theory (LMT). The generalized scattering field coefficients can be expressed as the product of LMT-scattering coefficients and BSCs (Gouesbet and Gréhan 2011).

$\mu$ and ${\mu }_1$ are the permeabilities of the free space and the dielectric sphere, respectively. Here, $x=ka$ symbolizes the size parameter, $a$ be the radius of the dielectric sphere, while $m_r$ symbolizes the refractive index of the dielectric sphere with $m_r=\frac{k_1}{k}=\frac{N_1}{N}$. Here, $N_1$ and $N$ specify the refractive indices of the dielectric sphere and the background medium. When light travels through a material, its angular distribution and scattering intensity are both affected by the presence of a dielectric constant. The dielectric constant helps track how the nondiffracting beams interact with the material during scattering. Enhanced scattering occurs when there is a difference in the dielectric characteristics between the particles and the surrounding material due to changes in the refractive index. Now, by introducing the Riccati–Bessel functions, i.e., ${\psi }_n\left(.\right)=x*j_n\left(.\right)$ and ${\xi }_n\left(.\right)=x*h_n^{\left(1\right)}\left(.\right)$, the scattering coefficients can be written as

When the spherical Hankel functions are approximated in the far field, NDFSI can be expressed as (Tang et al., 2022)

The far field, or far zone, denotes the area distant from a scatterer where the angular distribution of the field is clearly defined. In the context of far field or far zone approximation, $\left(kr\gg 1\right)$ holds very well.

where $S_1\left(\theta ,\phi \right)$ and $S_2\left(\theta ,\phi \right)$ denote amplitude functions (Gouesbet et al., 1988)

3. Numerical Analysis and Discussion

The numerical study is conducted using GLMT, concentrating on scattering characteristics of the Lommel beam. To investigate the analytical calculations, the normalized dimensionless far-field scattering intensity (NDFSI) is analyzed for Lommel beam configuration parameters. Here, the impact of varying the scattering angle $\theta$ on NDFSI is emphasized. In light of the theoretical formulation given in Section 2, the CP Lommel beam with an incident wavelength of $\lambda = 632.8 nm$ illuminating the dielectric sphere is considered. Furthermore, the effects of the Lommel beam configuration parameters, i.e., beam order $\left(v\right)$, asymmetry parameter $\left(c\right)$, beam half-cone angle $\left(\alpha \right)$, and beam center coordinate positions $\left(x_0, y_0, z_0\right)$ for the dielectric sphere were analyzed and discussed for NDFSI.

For the validation of the numerical results, the scattering efficiency $\left(Q_{sca}\right)$ of the Lommel beam for the dielectric sphere was plotted. The configuration parameters for the Lommel beam, such as beam order besides beam asymmetry, were adopted as $v=0=c$. As the Lommel beam is considered a linear combination of the Bessel beam modes, the results of the Bessel vortex beam of zeroth order, i.e., $\left(l=0\right)$ for a dielectric sphere, are also the same. The size parameter is unitless. It is varied from $0-5$. Incident beam wavelength is $\lambda = 632.8 nm$. Following this, the wave number can be computed as $k=\frac{2\pi }{\lambda }$. As $\phi ={\tan }^{-1}\left(\frac{y}{x}\right)$ with $x=1*\lambda$ and $y=1*\lambda$. The axicon angle was set as $\alpha =2^0$. The comparison shows that convergence has been achieved for the limiting case of the Lommel beam for $(v=0=c)$ and the Bessel vortex beam for $(l=0)$ toward the dielectric sphere. The result of this comparison has been plotted in Fig. 2. The plot shows integral scattering efficiency.

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Fig. 2.

Scattering efficiency for Bessel beam of zeroth order, (l = 0) and Lommel beam for v=c=0. The other parameters are, beam wavelength=632.8 nm, axicone angle= 2, a=1 µm.

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Fig. 3 illustrates the NDFSI of the Lommel beam for different beam orders. Increasing by heightened by increase the beam order $\left(v\right)$ of a Lommel beam leads to heightened scattered intensity upon its interaction with a dielectric sphere, attributable to various interrelated physical phenomena and mathematical factors. A more intricate transverse structure of the Lommel beam, which concentrates energy in an off-axis ring pattern, becomes apparent with increasing beam order. As a result of improved interaction with the sphere’s higher-order Mie modes, the scattering is amplified. Based on the Mie theory, dielectric spheres have the ability to scatter light as a summation of multipole modes. Lommel beams with lower order predominantly stimulate multipoles of lower order. Increasing the order of the Lommel beams results in stimulating the higher-order multipoles with various enhanced field distributions that are identical to and corresponding to their field strengths. Higher-order Lommel beams exhibit an enhanced pattern of intensity gradients and more tangible interference patterns in proximity to the spherical particle. These gradients can improve the local field interactions, resulting in stronger induced dipoles and amplified scattered electromagnetic intensity.

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Fig. 3.

Scattered intensity for (a-b) LCP and (c-d) RCP Lommel beams plotted vs scattering angles for various values of beam order, v=2,4,6,8, with c=0.2, size parameter=10, axicone angle= 2, and mr=1.01.

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Fig. 4 displays the effects of the beam asymmetry parameter on the NDFSI for both the left CP and the right CP Lommel beams. In Lommel beams, the parameter $c$ regulates the degree to which the beam’s intensity distribution diverges from cylindrical symmetry. With the increase in $c$, the beam exhibits shifting, tilting, or enhanced azimuthal structuring, resulting in the emergence of energy lobes displaced from the beam center. It means intensity is not focused at its center, and the beam propagation sets off about the optical axis, and the formation of more ring or petal structures. Increased $c$ directs intensity towards lateral scattering lobes or arcs. The asymmetry parameter $c$ in a Lommel beam alters the amplitude and phase are altered, resulting in an intensity pattern that is spatially asymmetrical. For lower values of this parameter, a symmetric Lommel beam becomes more centered. However, for higher $c$, the energy pattern shifts into various forms such as side lobes, twisting arms, or asymmetrical rings. The other beam configuration parameters are also contributing to it, such as beam order, etc. This impacts the energy distribution and its contribution to the dielectric sphere. This, in turn, affects optical scattering.

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Fig. 4.

Scattered intensity for (a-b) LCP and (c-d) RCP Lommel beams plotted vs scattering angles for various values of beam asymmetry parameters, c=0.2,0.4,0.6,0.8 with v=2, size parameter=10, axicone angle= 2, and mr=1.01.

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Fig. 5 shows the effect of the half-cone angle $\left(\alpha \right)$of the beam on the NDFSI distributions. Increasing $\alpha$ results in decreasing the far-field scattering intensity. A greater $\alpha$ indicates that the beam’s wave vectors scatter over a wider angular spectrum, deviating further from the propagation axis of the Lommel beam. The energy disperses (spreads) transversely instead of focusing in a forward direction. For a non-diffracting Lommel beam with a larger $\alpha$, there is a reduced forward momentum and an increased lateral momentum, resulting in diminished energy available for far-field electromagnetic scattering. A beam with a large $\alpha$ tends to excite higher-order, more spatially complex modes that radiate less energy into straightforward far-field patterns, resulting in a reduction in far-field scattering intensity when normalized. A non-diffracting Lommel beam with a large half-cone angle $\alpha$inclines to stimulate higher-order, spatially complex beam modes that radiate a reduced amount of energy into far-field/far-zone scattering patterns, yielding a reduced NDFSI.

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Fig. 5.

Scattered intensity for (a-b) LCP and (c-d) RCP Lommel beams plotted vs. scattering angles for different beam axicon angles, (2, 3, 4, 5) with azimuthal angle=0 for (a-c) and 90 for (b-d). The other parameters are c=0.2, size parameter=10, v=2, and mr=1.01.

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In Fig. 6, the impact of beam center coordinates on the NDFSI has been plotted. Increasing the beam center locations results in the beam being offset from the center of the dielectric sphere rather than striking it directly. Shifting the beam center results in a reduced overlap of the high-intensity region with the dielectric sphere, causing the dielectric sphere to perceive a weaker section of the Lommel beam. As the beam recedes, the dielectric sphere experiences a diminished interaction with the beam’s field. Despite the total interaction of the incident beam, the effective power experienced and scattered by the dielectric sphere is diminished, leading to the repression of the NDFSI. The diffraction-free Lommel beam exhibits a complex phase constant and amplitude profile. Shifting the center of the dielectric sphere relative to the beam center results in asymmetric irradiation, leading to destructive interference in the far-field/far-zone scattered field and a subsequent reduction in the scattered intensity.

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Fig. 6.

Scattered intensity for (a-b) LCP and (c-d) RCP Lommel beams plotted vs. scattering angles for different positions of beam center coordinates (x0, y0, z0). The other parameters are c=0.2, size parameter=10, v=2, axicon angle=2, and mr=1.01.

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For investigating the influences of beam parameters and scattering angle, Figs. 7 and 8 depict the NDFSI for a dielectric sphere. Besides electromagnetic scattering, interference phenomena also contribute to NDFSI. The impact of variation for $\left(0\le \theta \le \pi \right)$ and $\left(0\le \phi \le 2\pi \right)$ is quite obvious in Fig. 7. For Figs. 7 and 8, left and right circularly polarized (CP) Lommel beams are considered in the E-plane. In Fig. 8, besides the scattering angle, the beam asymmetry parameter is also varied. Graphical behavior shows that NDFSI is increasing up to a maximum and then decaying. The parameters for configuring the CP Lommel beam play their role in tuning the scattering characteristics. The selection of the Lommel beam configuration parameters other than the type of dielectric sphere results in augmentation of its non-diffraction characteristics over longer distances. The introduction of increased asymmetry$c$ results in angular phase differences within the Lommel beam, which induce interference patterns among various scattering trajectories. This phenomenon leads to oscillations in the angular distribution of scattered light, particularly evident when plotting scattered intensity against scattering angle.

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Fig. 7.

Scattered intensity for (a) LCP and (b) RCP Lommel beams plotted for various scattering angles and azimuthal angles. Here, v = 5, c = 0.5, size parameter=10, axicon angle=5, and mr=1.01.

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Fig. 8.

Scattered intensity for (a) LCP and (b) RCP Lommel beams plotted for various scattering angles and beam orders (v) with azimuthal angle=0. Here, c = 0.5, size parameter=10, axicon angle=5, and mr=1.01.

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Fig. 9 displays density plots that vary with the beam asymmetry parameter and scattering angle. As $c$ increases, the amplitude and phase of the Lommel beam vary both radially and azimuthally. Light from various segments of the beam interacts with the dielectric sphere at distinct phases and orientations. The various scattering paths interact either constructively or destructively at distinct angles. On increasing $c$, an increase is observed in the incident wave’s angular frequency. The result is accelerated oscillations in the electromagnetic scattered field as a function of scattering angle. If $c$ is small, the scattering intensity vs. scattering angle graph is smooth; but, as $c$ increases, the pattern becomes more rippling or oscillatory. When $c=0$, the non-diffracting Lommel beam is nearly symmetric. Optical scattering response is smooth in its nature. With the augmentation of the $c$ parameter, the asymmetric characteristics arise in the Lommel beam field. This aid yields interference effects in the scattered field. As a result, the discontinuous bright and dark bands appear across the scattering angle. For larger $c$, the density plot displays rapid oscillations characterized by numerous bright and dark fringes closely arranged along the angle axis. Summing up, the following Table 1 shows the characteristics of CP Lommel beam for the asymmetry parameter with the scattering angle.

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Fig. 9.

Density plot for various scattering angles and asymmetry parameters (a) LCP (b) RCP. The other parameters are v = 2, size parameter=10, axicon angle=5, and mr=1.01.

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4. Conclusions

This work analyzes the electromagnetic scattering of the CP Lommel beam with the dielectric sphere, based on the GLMT. The NDFSI is numerically analyzed and discussed. The influence of the beam order $v$, asymmetry parameter $c$, beam axicone angle $\alpha$, and beam center coordinate positions $x_0, y_0, z_0$, of the dielectric sphere has been discussed. It is important to understand that when a dielectric sphere interacts with a CP Lommel beam that has a large half-cone angle $\alpha$, it is predicted to create large scattering lobes. It is decided by the beam features of the scattered electromagnetic field, which have something to do with the internal structure of the incident beam field of the CP Lommel beam. Besides the internal structure of the non-diffracting Lommel beam, the interference as well as diffraction phenomena between the incident and the scattered CP Lommel beams are introduced by the dielectric sphere. Furthermore, it is worth noting that the beam order plays a key role in optical scattering. As $v$ increases, the NDFSI of the dielectric sphere augments. Enhanced beam scattering is achieved by carefully selecting the configuration parameters of the Lommel beam, i.e., beam order, asymmetry parameter, beam axicone angle, and beam center coordinates. By doing this, the non-diffracting characteristics are maintained over considerable distances. The meticulous selection of beam parameters can also enhance the stability and resistance to diffraction characteristics in Lommel beams. The beam order and beam asymmetry parameter influence the energy distribution of the Lommel beam. By optimizing the configuration parameters, the competence of the beam for lowering the distortion and minimizing spreading is also enhanced. This work is expected to provide a better understanding of how non-diffracting light beams interact with different materials, which is helpful for controlling light, trapping objects with light, and directing light.