Energy, Momentum, and Angular Momentum of Non-Diffracting Tricomi Beams

Abstract

In this work, we report a theoretical study of the energy, momentum, and angular momentum of non-diffracting Tricomi beams. By utilizing the vector potential in the Lorenz gauge, we derive the explicit analytical expressions for the electric and magnetic field components of non-diffracting Tricomi beams. A canonical theory is introduced to describe the energy, momentum, spin angular momentum (SAM), and orbital angular momentum (OAM) of the non-diffracting Tricomi beams. The effects of the asymmetry constants, topological charge, and half-cone angle on the energy, momentum, SAM, and OAM of the non-diffracting Tricomi beams are simulated and analyzed. This study provides fundamental physical insights into the dynamical characteristics of non-diffracting Tricomi beams relevant to potential optical manipulation applications.

1. Introduction

As is well known, diffraction is a fundamental wave phenomenon that unavoidably accompanies light propagation. Despite its unavoidable nature, there exist particular light beams that propagate freely without changing in shape or scale; these are thus called diffraction-free beams. Such beams, also known as non-diffracting beams, have been a subject of interest since they were first theoretically predicted and experimentally demonstrated by Durnin and his colleagues in 1987 [1,2]. Mathematically, an important group of non-diffracting beams are exact solutions to the Helmholtz wave equation, including the well-known plane waves in the Cartesian coordinate system, the Bessel beams in the circular cylindrical coordinate system [3,4,5], the Mathieu beams in the elliptic cylindrical coordinate system [6,7], and the parabolic beams in the parabolic cylindrical coordinate system [8,9]. Among these beams, particular interest has been focused on Bessel beams. Notably, ideal non-diffracting Bessel beams are not physically realizable because they carry infinite energy across any normal cross-section in their propagation direction. To describe non-diffracting Bessel beams more realistically, the Bessel–Gauss beam [10,11], as a realizable approximation of the ideal propagation-invariant Bessel field, is usually adopted. This beam carries finite power, maintains non-diffracting properties over a limited propagation distance, and can be understood as a finite-energy Bessel beam that has passed through a Gaussian aperture. Another group of non-diffracting beams are exact solutions to the potential-free Schrödinger equation, including the Airy beams [12,13] and Olver beams [14,15]. Additionally, many non-diffracting beams formed by the superposition of these fundamental non-diffracting beams have also been proposed [16,17,18,19,20,21,22,23,24]. Recently, Zhu et al. introduced a new class of non-diffracting beams referred to as Tricomi beams [25]. Such non-diffracting beams are exact solutions of the Helmholtz wave equation in the Cartesian coordinate system, expressed in terms of the Tricomi function [26,27,28]. Beyond its compact representation, the mathematical essence of the Tricomi function is deeply rooted in the operational framework of Laguerre-type derivatives introduced by Dattoli and colleagues [29]. As eigenfunctions of the Laguerre operator, these functions satisfy a second-order differential equation with a notably simpler structure compared to that of ordinary Bessel functions. Furthermore, Reference [30] demonstrated that the introduction of operational identities associated with generalized polynomials provides a more rigorous mathematical foundation for describing the evolution and symmetry properties of these non-diffracting fields. This operator-based perspective not only streamlines the analytical treatment of the Tricomi beams but also offers deeper physical insights into their structural stability and self-healing mechanisms within the paraxial regime. It is worth noting that the Tricomi beams can be reduced to conventional Bessel beams. However, compared to conventional non-diffracting Bessel beams, non-diffracting Tricomi beams exhibit more diverse and complex morphological characteristics [31]. By properly adjusting the asymmetry parameters, non-diffracting Tricomi beams can be transformed into symmetric, asymmetric, and off-axis Bessel beams [32,33,34,35,36,37,38]. Owing to their high versatility and good adjustability, non-diffracting Tricomi beams can be effectively applied in optical trapping and manipulation of particles. The interaction between non-diffracting Tricomi beams and particles, which involves the transfer of photon energy, momentum, and angular momentum, serves as the theoretical basis for such applications and constitutes an important research subject. The main dynamic quantities, energy, momentum, SAM, and OAM, are crucial for understanding the properties of non-diffracting Tricomi beams and their interactions with particles.

To date, there have been some studies on the energy, momentum, SAM, and OAM of fundamental non-diffracting beams, including the Bessel beams and Airy beams [39,40,41,42,43,44,45,46,47,48,49]. Specifically, Surzhykov et al. investigated the energy flow of Bessel beams [39]. Volke-Sepulveda et al. studied the OAM density of high-order Bessel beams [40]. Litvin et al. further studied the OAM density of superpositions of Bessel beams [41]. Schulze et al. presented an approach for measuring the OAM density of Bessel beams [42]. Belyi et al. analyzed the spin-to-orbital angular momentum conversion in Bessel beams within crystals [43,44]. Sztul and Alfano examined the evolution of the energy flow and angular momentum of Airy beams [45]. Deng et al. further examined the energy flow and angular momentum of nonparaxial Airy beams [46]. Kim explored the transverse spin angular momentum of Airy beams [47]. Hui et al. reported a study of the canonical momentum and angular momentum of Airy beams [48]. In a recent work [49], Kotlyar et al. studied the canonical energy backflow in Airy beams. These existing studies have revealed the influences of the parameters of non-diffracting Bessel beams and Airy beams on their dynamical characteristics, including the energy, momentum, SAM, and OAM. Now, a question arises: for versatile and adjustable non-diffracting beams characterized by multiple parameters, which include the asymmetry constants, topological charge, and half-cone angle, how do these parameters affect the dynamical characteristics of Tricomi beams? To the best of our knowledge, this issue has not been addressed before. It is the purpose of this article to report a theoretical study of the energy, momentum, and angular momentum of non-diffracting Tricomi beams. These quantities are essential for understanding their optical interactions.

This paper is organized as follows. In Section 2, we first carry out a full-vector wave analysis of the non-diffracting Tricomi beams. Then, we briefly recall the canonical approach for describing the energy, momentum, SAM, and OAM of the non-diffracting Tricomi beams. Some numerical simulations are performed and analyzed in Section 3. Finally, we draw the conclusions in Section 4.

2. Theoretical Formulae

2.1. Vector Wave Analysis of the Non-Diffracting Tricomi Beams

As a starting point of our analysis, let us consider the scalar field of the non-diffracting Tricomi beams. As stated earlier, the non-diffracting Tricomi beams are exact solutions of the Helmholtz wave equation. Following the work of Zhu et al. [25], in Cartesian coordinates $\left(x,y,z\right)$, the scalar field of the Tricomi beams propagating in the $+z$ direction is given by

$$E_n\left(x,y,z\right)=C_n\left[{\displaystyle \frac{k_t^2{\left(x+\alpha \right)}^2+k_t^2{\left(y+\beta \right)}^2}{4}}\right]{\left[{\displaystyle \frac{i{k}_t\left(x+\alpha \right)+k_t\left(y+\beta \right)}{2}}\right]}^n\exp \left(i{k}_{z}z\right)$$

(1)

where

$$C_n\left(x\right)={\displaystyle \sum _{r=0}^{\infty }\frac{{\left(-1\right)}^r}{r!\left(n+r\right)!}}{x}^r$$

(2)

is the so-called Tricomi function; $n$ is the order of the beams, also known as the topological charge; $\alpha$ and $\beta$ are complex constants for describing the asymmetry of the Tricomi beams; and $k_t=k\sin {\theta }_0$ and $k_z=k\cos {\theta }_0$ are the transversal and longitudinal wavenumbers, respectively. The parameters $k=2\pi /\lambda$ and ${\theta }_0$ are the wavenumber and half-cone angle of the Tricomi beams, respectively, with $\lambda$ being the wavelength of the beams.

Using the mathematical relation between the $n\mathrm{th}$-order Tricomi function $C_n\left(x\right)$ and the $n\mathrm{th}$-Bessel function $J_n\left(x\right)$ [23,24,25,26,27,28]

$$C_n\left(x\right)=x^{-{\displaystyle \frac{n}{2}}}{J}_n\left(2\sqrt{x}\right)$$

(3)

We can represent Equation (1) as

$$E_n\left(x,y,z\right)=J_n\left[k_t\sqrt{{\left(x+\alpha \right)}^2+{\left(y+\beta \right)}^2}\right]{\left[{\displaystyle \frac{i\left(x+\alpha \right)+\left(y+\beta \right)}{i\left(x+\alpha \right)-\left(y+\beta \right)}}\right]}^{n/2}{\left(-1\right)}^{{\displaystyle \frac{n}{2}}}\exp \left(i{k}_{z}z\right)$$

(4)

Obviously, the features of non-diffracting Tricomi beams depend on four adjustable parameters: the asymmetry constants $\alpha$ and $\beta$, the topological charge $n$, and the half- cone angle ${\theta }_0$, which determines the transversal and longitudinal wavenumbers $k_t$ and $k_z$. By selecting appropriate asymmetry constants $\alpha$ and $\beta$, the non-diffracting Tricomi beams can be transformed into the symmetric, asymmetric, and off-axis Bessel beams [32,33,34,35,36,37,38]. In particular, for the case of $\alpha =\beta =0$, the field distribution of non-diffracting Tricomi beams reduces to that of conventional non-diffracting Bessel beams [25]. The detailed mathematical derivation is given in Appendix A.

Before proceeding further, note that to study the energy, momentum, SAM, and OAM of non-diffracting Tricomi beams, the analytical expressions for their electric and magnetic field components must first be known. Usually, for this purpose, there are two approaches that can be used to derive these expressions for structured light beams: the vector potential method [50,51,52] and the vector angular spectrum representation [53,54,55]. Here, we adopt the vector potential in the Lorenz gauge to derive the explicit analytical expressions for the corresponding electric and magnetic field components of these beams. The basic idea of this approach is to define a vector potential based on the scalar field of the propagating non-diffracting Tricomi beams and then apply the Lorenz gauge condition. Specifically, the vector potential of non-diffracting Tricomi beams can be defined as

$$\mathbf{A}=\left(p_x\widehat{\mathbf{x}}+p_y\widehat{\mathbf{y}}\right)E_n\left(x,y,z\right)$$

(5)

where $p_x$ and $p_y$ are the polarization parameters, which determine the states of polarization of the beams.

Stemming from the vector Maxwell’s equations and using the Lorenz gauge condition, the electric and magnetic field vectors of the non-diffracting Tricomi beams can be expressed in terms of $\mathbf{A}$ as

$$\mathbf{E}=ikZ\left[\mathbf{A}+{\displaystyle \frac{1}{k^2}}\nabla \left(\nabla \cdot \mathbf{A}\right)\right]$$

(6)

$$\mathit{H}=\nabla \times \mathbf{A}$$

(7)

where $k$ and $Z$ are the wave number and wave impedance of the beams, respectively.

Substituting Equations (4) and (5) into Equations (6) and (7), and performing a lengthy but straightforward derivation, we obtain the explicit expressions for the electric and magnetic field components of the non-diffracting Tricomi beams, as follows

$$E_x=ikZ\left\{p_x\left[A^{\prime }+{\displaystyle \frac{1}{k^2}}\nabla {\left(\nabla \cdot {\mathbf{A}}^{\prime }\right)}_x\right]+p_y\left[{\displaystyle \frac{1}{k^2}}\nabla {\left(\nabla \cdot {\mathbf{A}}^{″}\right)}_x\right]\right\}$$

(8)

$$E_y=ikZ\left\{p_x\left[{\displaystyle \frac{1}{k^2}}\nabla {\left(\nabla \cdot {\mathbf{A}}^{\prime }\right)}_y\right]+p_y\left[A^{″}+{\displaystyle \frac{1}{k^2}}\nabla {\left(\nabla \cdot {\mathbf{A}}^{″}\right)}_y\right]\right\}$$

(9)

$$E_z=ikZ\left\{p_x\left[{\displaystyle \frac{1}{k^2}}\nabla {\left(\nabla \cdot {\mathbf{A}}^{\prime }\right)}_z\right]+p_y\left[{\displaystyle \frac{1}{k^2}}\nabla {\left(\nabla \cdot {\mathbf{A}}^{″}\right)}_z\right]\right\}$$

(10)

$$H_x=p_x{\left(\nabla \times {\mathbf{A}}^{\prime }\right)}_x+p_y{\left(\nabla \times {\mathbf{A}}^{″}\right)}_x$$

(11)

$$H_y=p_x{\left(\nabla \times {\mathbf{A}}^{\prime }\right)}_y+p_y{\left(\nabla \times {\mathbf{A}}^{″}\right)}_y$$

(12)

$$H_z=p_x{\left(\nabla \times {\mathbf{A}}^{\prime }\right)}_z+p_y{\left(\nabla \times {\mathbf{A}}^{″}\right)}_z$$

(13)

where ${\mathbf{A}}^{\prime }=\widehat{\mathbf{x}}{E}_n\left(x,y,z\right)$ and ${\mathbf{A}}^{″}=\widehat{\mathbf{y}}{E}_n\left(x,y,z\right)$. It is obvious that the determination of the electric and magnetic field components necessitates differential operations on the vector potentials ${\mathbf{A}}^{\prime }$ and ${\mathbf{A}}^{″}$. Since the explicit expansion of the gradient of the divergence and the curl is algebraically complex, the full derivation, along with the component-wise identities, is presented in Appendix B.

Once the analytical expressions for the electric and magnetic field components of the non-diffracting Tricomi beams are obtained, Equations (1) and (2) can be explicitly written as

$$\mathbf{E}=E_x\widehat{\mathbf{x}}+E_y\widehat{\mathbf{y}}+E_z\widehat{\mathbf{z}},\mathbf{H}=H_x\widehat{\mathbf{x}}+H_y\widehat{\mathbf{y}}+H_z\widehat{\mathbf{z}}$$

(14)

2.2. Description of the Energy, Momentum, SAM, and OAM

We now proceed to describe the energy, momentum, SAM, and OAM of the non-diffracting Tricomi beams by adopting a canonical approach proposed by Bliokh et al. [56]. With this approach, the energy density is described by the Brillouin formula, the canonical momentum density corresponds to the local gradient of the phase of the field, the SAM density is proportional to the local ellipticity of the field polarization, and the OAM density is defined via the canonical momentum density [57]. In particular, the energy, canonical momentum, SAM, and OAM densities of the non-diffracting Tricomi beams can be expressed as

$$W={\displaystyle \frac{1}{4}}\left(\epsilon {\left|\mathbf{E}\right|}^2+\mu {\left|\mathbf{H}\right|}^2\right)$$

(15)

$$\mathbf{P}={\displaystyle \frac{1}{4\omega }}\mathrm{Im}\left[\epsilon {\mathbf{E}}^{\ast }\cdot \left(\nabla \right)\mathbf{E}+\mu {\mathbf{H}}^{\ast }\cdot \left(\nabla \right)\mathbf{H}\right]$$

(16)

$$\mathbf{S}={\displaystyle \frac{1}{4\omega }}\mathrm{Im}\left[\epsilon {\mathbf{E}}^{\ast }\times \mathbf{E}+\mu {\mathbf{H}}^{\ast }\times \mathbf{H}\right]$$

(17)

$$\mathbf{L}=\mathbf{r}\times \mathbf{P}$$

(18)

where $\epsilon$ and $\mu$ are the permittivity and the permeability of the medium in which the Tricomi beams propagate, $\omega$ is the angular frequency of the beams, $\mathrm{Im}\left[\cdot \right]$ denotes the imaginary parts, the superscript “$\ast$” represents the complex conjugate, and the notation $\mathbf{A}\cdot \left(\nabla \right)\mathbf{B}=A_x\nabla B_x+A_y\nabla B_y+A_z\nabla B_z$ is used.

Before concluding this section, we should point out that, in a recent work [28], the derivation of the mathematical form of vector Tricomi beams is presented based on a linear combination of three fundamental vector solutions, which is unlike the vector potential in the Lorenz gauge adopted in our work. Furthermore, the aforementioned work focused on the beams’ intensity, polarization, singularities, and Stokes parameters, whereas our work investigates their energy, momentum, SAM, and OAM. These differences in both methodological frameworks and research objectives make the two works fundamentally different.

3. Results and Discussion

In this section, we perform some numerical simulations to explore the effects of the asymmetry constants, topological charge, and half-cone angle on the energy, momentum, SAM, and OAM of non-diffracting Tricomi beams. In the simulations, the beams are assumed to propagate in the free space with $\epsilon ={\epsilon }_0$ and $\mu ={\mu }_0$. Besides the parameters given below for every figure, the common parameters are set as the wavelength $\lambda =632.8 \mathrm{nm}$, the polarization parameters $\left(p_x,p_y\right)=\left(1,0\right)$, the topological charge $n=3$, the half-cone angle ${\theta }_0=15°$, the asymmetry constants $\alpha =0$ and $\beta =i\times {10}^{-7}$, and the position of the observed plane $z=0$.

To start, we consider the effects of the asymmetry constants, topological charge, and half-cone angle on the energy density of non-diffracting Tricomi beams. As shown in Figure 1, the normalized energy density distributions of the non-diffracting Tricomi beams for different asymmetry constants $\alpha$ and $\beta$ are explored. From Figure 1a, we can easily find that when $\alpha =\beta =0$, the Tricomi beam degenerates into a standard Bessel beam, exhibiting a perfect concentric ring structure in its energy density distribution. From Figure 1b–d, it can be seen that when either or both of the parameters $\alpha$ and $\beta$ take finite real values, the Tricomi beam transforms into an off-axis Bessel beam. Its energy density distribution consists of a set of concentric rings that deviate from the propagation axis in different directions, depending on the values of $\alpha$ and $\beta$. As observed in Figure 1e–h, when one or both of $\alpha$ and $\beta$ take imaginary values, the Tricomi beam transforms into an asymmetric Bessel beam. In this case, the energy density distribution loses its cylindrical symmetry about the propagation axis, and its main lobe takes on a crescent shape, the bending direction of which varies with the values of $\alpha$ and $\beta$. A comparison of Figure 1g–j reveals that as $\left|\alpha \right|$ and $\left|\beta \right|$ increase, the beam energy shifts toward one side and the asymmetry of the normalized energy density distribution becomes more pronounced. Further comparison of Figure 1i–l shows that as $\left|\alpha \right|$ and $\left|\beta \right|$ continue to increase, the original concentric ring structure gradually breaks apart into an arc-shaped fringe structure, eventually forming a highly stretched, single sheet-like intensity distribution whose spatial scale expands with the asymmetry parameters. The above results indicate that the asymmetry constants $\alpha$ and $\beta$ have a significant effect on the energy density distribution of the Tricomi beam. The introduction of the imaginary part of these constants breaks the axial symmetry of the beam, gradually transforming its energy density distribution from a ring structure into a distorted distribution with pronounced directionality. Without loss of generality, in the following numerical simulations, we focus only on cases where either $\alpha$ or $\beta$ (or both) take an imaginary value.

Figure 2 shows the normalized energy density distributions of the non-diffracting Tricomi beams for different topological charges $n$ and half-cone angles ${\theta }_0$. It can be seen from Figure 2(a1–a4) that, as the topological charge increases, the radial position of the main energy ring shifts outward, the spatial extent of the energy distribution expands significantly, the central low-intensity region enlarges, and the high-energy-density lobes gradually shift rightward. In contrast, as the half-cone angle increases, the radius of the main energy ring gradually decreases, the overall spatial extent of the energy distribution shrinks, the central dark region continuously diminishes, and the high-energy-density lobes progressively shift leftward, as illustrated in Figure 2(b1–b4). In addition, clearly asymmetric arc-shaped high-intensity regions indicate localized energy enhancement along specific orientations. Outside the main energy ring, gradually decaying oscillatory fringes reveal the diffractive oscillatory structure that accompanies beam propagation. Based on the above analysis, we conclude that the topological charge primarily governs the radial extent of the energy density distribution through a positive scaling relationship with the peak position, thereby redistributing energy over a broader radial range by an enhanced centrifugal tendency. The half-cone angle, on the other hand, modulates the spatial concentration of energy through its inverse relationship with the beam’s characteristic scale, enabling fine control over the extent of beam energy localization.

Next, we explore the effects of the asymmetry constants, topological charge, and half-cone angle on the momentum density of non-diffracting Tricomi beams. Figure 3a,b illustrates normalized momentum density distributions of the non-diffracting Tricomi beams for different asymmetry constants $\alpha$ and $\beta$. As we can see, when $\alpha$ and $\beta$ are $i\times {10}^{-7}$, the momentum density distribution exhibits pronounced oscillations, with numerous side lobes of large amplitude. As the values of $\alpha$ and $\beta$ increase to $5i\times {10}^{-7}$ and $10i\times {10}^{-7}$, the main peak of the momentum density distribution gradually broadens, and the side-lobe amplitudes decrease. At $20i\times {10}^{-7}$, the distribution exhibits a single peak, and the side lobes largely disappear. A comparison of Figure 3a,b shows that the momentum density distributions along the $y$-and $x$-axes are symmetric. Without loss of generality, the plane defined by $x=y$ is used as the observation plane to explore the effects of the topological charge, and half-cone angle on the momentum density of non-diffracting Tricomi beams, as illustrated in Figure 3c,d. It is noted that as the topological charge increases, the maximum value of the momentum density decreases, and main peak gradually shifts outward from the center. We also note that as the half-cone angle increases, the maximum value of the momentum density increases, main peak gradually shifts toward the center and becomes sharper, while the side lobe structure becomes denser and is accompanied by high-frequency oscillations.

Finally, we explore the effects of the asymmetry constants, topological charge, and half-cone angle on the SAM and OAM densities of non-diffracting Tricomi beams. In Figure 4, the normalized SAM and OAM density distributions of the non-diffracting Tricomi beams for different asymmetry constants $\alpha$ and $\beta$ are illustrated. We can easily find that, as the values of $\alpha$ and $\beta$ increase, the regular four-lobe pattern at the center of the longitudinal SAM becomes distorted and fragmented. Meanwhile, the pattern of the transverse SAM exhibits a pronounced shift and localization, gradually stretching from an initial double-arc crescent shape into diagonal strips on both sides. The multiple concentric rings of the longitudinal and transverse OAM density distributions gradually collapse along the diagonal direction. Under strongly asymmetric conditions, both the longitudinal and transverse OAM evolve into a distinct “butterfly” pattern. Notably, the distributions of the OAM density show a more pronounced sensitivity to asymmetry than that of the SAM density. This phenomenon can be attributed to the fundamental difference in the physical origins of SAM and OAM. While the SAM density is primarily governed by the local spin density (polarization state), the OAM density is intrinsically linked to the spatial phase gradient and the transverse coordinate. Any deviation from circular symmetry in the beam profile introduces significant perturbations in the spatial phase structure and the resulting azimuthal phase gradient. Furthermore, because OAM involves the cross product of the position vector and linear momentum $(r\times p)$, structural deformations—particularly those occurring at the beam’s periphery—are effectively scaled and amplified by the radial distance $r$. This coordinate-dependent amplification mechanism makes the OAM density distribution far more responsive to beam shape asymmetries than the SAM density, which remains relatively robust as it lacks this radial scaling and phase-dependency. Consequently, orbital momentum serves as a more sensitive physical indicator for diagnosing and characterizing beam shape deformation. In addition, a comparison of the colorbar ranges of the transverse and longitudinal components of the SAM and OAM densities reveals that the transverse part dominates the angular momentum density of the linearly polarized non-diffracting Tricomi beams. Overall, the above results demonstrate that increasing asymmetry gradually breaks the rotational symmetry of non-diffracting Tricomi beams, leading to a localized redistribution of both SAM and OAM. Such behaviors highlight the theoretical value of the momentum and energy distribution laws for understanding complex field–matter interactions, establishing a critical foundation for characterizing momentum transfer mechanisms.

Figure 5 illustrates normalized SAM and OAM density distributions of the non-diffracting Tricomi beams for different topological charges $n$. From the longitudinal SAM distributions shown in Figure 5(a1–d1), as the topological charge increases, the multipole lobe structure at the center expands radially, while its angular distribution exhibits pronounced azimuthal non-uniformity. From Figure 5(a2–d2), a series of distinct crescent-shaped structures is observed in the transverse SAM. When $n$ is small, the transverse SAM density is mainly concentrated in the inner crescent structures. As $n$ increases, the transverse SAM gradually shifts outward, causing the density in the outer crescents to exceed that in the inner ones—a clear indication of a significant radial redistribution. As shown in Figure 5(a3–d3), with increasing topological charge, the dark core radius of the longitudinal OAM density distribution increases markedly, leading to a clear outward radial shift in the distribution. The spatial evolution of the longitudinal OAM density reveals the substantial modulation effect of higher-order topological charges on the transverse non-uniformity of the optical field and the spin–orbit coupling strength. Figure 5(a4–d4) further confirm this evolutionary behavior for the transverse OAM. Overall, for small $n$, the angular momentum density distribution is relatively dispersed, spreading over concentric crescent-shaped side lobes at different radial orders. As $n$ increases, the transverse component of the angular momentum density distribution becomes more concentrated, gradually converging into high-density crescent structures with well-defined contours. These observations demonstrate that the topological charge is a key parameter governing the spatial distribution pattern and concentration of angular momentum density.

Figure 6 illustrates normalized SAM and OAM density distributions of the non-diffracting Tricomi beams for half-cone angles ${\theta }_0$. As shown in Figure 6(a1–d1), with increasing half-cone angle, the lobed structure of the longitudinal SAM density narrows sharply toward the propagation axis, accompanied by a significant increase in the main peak of the longitudinal SAM density distribution, indicating local angular momentum enhancement under strong focusing conditions. Figure 6(a2–d2) reveal that as the half-cone angle increases, the crescent-shaped distribution of the transverse SAM density becomes increasingly narrow and sharp. Meanwhile, the radial distance between the main peak and the side lobes decreases substantially, resulting in an extremely high spin angular momentum density gradient. Figure 6(a3–d3) show a clear centripetal contraction trend of the longitudinal OAM density as the half-cone angle increases. The originally wide crescent arc band transforms into a compact linear crescent structure, exhibiting high spatial clustering. A similar deformation is observed in the transverse OAM density distributions shown in Figure 6(a4–d4), where the crescent-shaped structures exhibit rapid oscillations over a very small spatial scale. These findings suggest that increasing the half-cone angle enables the high-precision compression and local enhancement of the beam’s angular momentum density, thereby providing a theoretical foundation for understanding the momentum and energy distribution laws that are essential for complex field–matter interactions.

Based on the above results, we conclude that the asymmetry constants, topological charge, and half-cone angle collectively enable the flexible manipulation of the energy, momentum, SAM, and OAM of non-diffracting Tricomi beams, which offers crucial guidance for optical trapping and particle manipulation applications.

4. Conclusions

In conclusion, a theoretical study of the energy, momentum, SAM, and OAM of the non-diffracting Tricomi beams is reported. A full-vector wave analysis of the non-diffracting Tricomi beams is carried out and the corresponding electric and magnetic field components are derived in detail. With the aid of Brillouin energy density and the canonical momentum density together with the corresponding SAM and OAM densities, we perform some numerical simulations and the results show that the asymmetry constants, topological charge, and half-cone angle have a significant effect on the energy, momentum, SAM, and OAM of the non-diffracting Tricomi beams. In particular, the asymmetry constants break the axial symmetry of non-diffracting Tricomi beams, which in turn distorts their energy, momentum, SAM, and OAM densities. The topological charge primarily governs the radial expansion of the energy density via an intrinsic scaling relationship and shifts the main peak of the momentum density outward, while also radially redistributing the angular momentum density and enhancing its concentration. The half-cone angle controls the spatial confinement of energy density through transverse wavenumber scaling, increases the maximum of momentum density, and enables high-precision localization of both SAM and OAM densities along the propagation axis. These findings provide fundamental physical insights into the dynamical characteristics of non-diffracting Tricomi beams, laying the groundwork for future studies on their optical mechanical properties and potential applications in particle manipulation and trapping.