Tight Focusing, Spin Textures, and Optical Forces of Amplitude–Phase-Asymmetric Hyperbolic-Chirped Double-Ring Airy–Gaussian Vortex Beams

Abstract

We propose an amplitude–phase-asymmetric hyperbolic-chirped double-ring Airy–Gaussian vortex beam (AHC-DRAGVB). Oppositely signed radial chirps are imposed on the inner and outer rings, respectively, while a hyperbolic-cosine amplitude modulation is applied only to the outer ring. A vectorial tight-focusing model based on Richards–Wolf diffraction is used to examine the effects of the scale factor G, vortex displacement, optical distribution factor b, hyperbolic modulation coefficient κ, and radial-chirp coefficients. The scale factor jointly controls the focal position, depth of focus, and longitudinal peak-intensity ratio. Off-axis vortex displacement modifies the focal distribution and reshapes the focal-plane spin density and polarization ellipses. Increasing b changes the relative Gouy phase and the associated local phase slopes and thereby reconstructs the axial spin texture. Increasing κ enhances longitudinal peak weighting, accompanied by changes in focal widths and competing transverse peaks. Radial chirp further shifts the focal region and modifies the local field, although excessive chirp degrades the intensity balance between foci. Rayleigh-force calculations are complemented by three-dimensional force directions and coordinate cuts to examine the local restoring response for b = 0.3.

1. Introduction

The optical manipulation of micro- and nanoparticles offers contact-free operation, low invasiveness, and high spatial precision, and has become an important tool in biomedicine, microfluidics, and precision fabrication. Conventional single-trap optical tweezers generally address one or a few particles locally and are less suited to parallel manipulation or long-range three-dimensional transport. These limitations have motivated the development of structured optical fields [1,2,3,4,5]. Airy beams are representative accelerating fields with quasi-nondiffracting, self-healing, and abruptly autofocusing behavior [6,7,8,9]. Circular Airy–Gaussian beams introduce a Gaussian envelope into a circular Airy field, retaining autofocusing while avoiding the infinite-energy requirement of an ideal Airy solution; they are therefore attractive for optical manipulation.

Coaxial superposition of two circular Airy–Gaussian fields with different ring radii produces a double-ring Airy–Gaussian vortex beam (DRAGVB). Coherent interference between the inner and outer rings can generate optical bottles, multiple foci, and helical propagation patterns, providing additional degrees of freedom for particle manipulation at multiple axial positions [10,11,12,13]. Previous studies have shown that the ring-radius ratio, Airy distribution factor, and topological charge can alter the bottle size, number of foci, and spacing between trapping sites. Recent work also studied dual-ring Airy–Gaussian vortex beams under tight focusing, using ring dimensions and topological-charge matching to tailor optical bottles and multiple spots [14]. The present work adds ring-dependent radial phase and amplitude modulation to this setting, rather than identifying double-ring focusing itself as a new mechanism.

Phase chirp provides an effective means of controlling the focusing evolution of structured light. A radial chirp changes the autofocusing distance of circular Airy-type beams, whereas chirps with opposite signs change the relative radial phase accumulated by the two rings [15,16]. Hyperbolic-cosine modulation can simultaneously reshape the outer-ring amplitude and alter the amplitude matching between the two rings. We therefore introduce an amplitude–phase-asymmetric hyperbolic-chirped double-ring Airy–Gaussian vortex beam (AHC-DRAGVB). Opposite quadratic radial phases are applied to the inner and outer rings, and only the outer ring is modulated by a hyperbolic-cosine amplitude factor. The resulting field combines distinct pupil phase and amplitude controls over the coherent superposition and autofocusing evolution of the two rings.

Distinct from existing demonstrations of autofocusing, vortex manipulation and tightly focused optical bottles [10,11,14], our modulation strategy uses opposite quadratic phases on two rings and a hyperbolic envelope only on the outer ring for coherent phase–amplitude tailoring, requiring no variation in ring radii or vortex orders. Specifically, the relative chirp phase is (C_in − C_out)r², while only the outer-to-inner amplitude ratio acquires cosh(κq_out). Relative phase and amplitude can therefore be varied without changing the ring radii or vortex orders. This adds a route to tailor focal position, longitudinal weighting and local polarization.

Recent vortex studies link vortex-pair control to photonic spin Hall symmetry [17], relate longitudinal spin splitting to the Goos–Hänchen shift [18], demonstrate reflection-induced C-point polarization-structure transitions [19], and show reconfigurable free-space optical skyrmion topology [20]. These studies motivate examining relative focal-component phase and spin orientation; the present spin trajectories do not by themselves establish a spin Hall displacement or skyrmion invariant.

Using the Richards–Wolf vector diffraction theory, we establish a tight-focusing model of the AHC-DRAGVB and investigate how the scale factor G, vortex-pair displacement, optical distribution factor b, hyperbolic modulation coefficient κ, and inner- and outer-ring chirp coefficients affect the focal electric field, spin density, polarization state, and optical force on Rayleigh particles [3,21,22,23,24,25,26,27,28]. The x–z intensity distribution, focal-plane vector field, focal locations, axial interaction range, threshold-defined transverse dimensions, interference sidelobes, and longitudinal peak-intensity ratio are analyzed to clarify the available parameter-control mechanisms.

2. Theoretical Model and Computational Methods

Figure 1 illustrates the tight-focusing geometry of the AHC-DRAGVB. The incident field consists of two concentric Airy–Gaussian annular fields with different orbit radii. The origin of the Cartesian coordinate system is placed at the geometrical focus. The lens is assumed to be aplanatic and to satisfy the sine condition, so that the radial pupil coordinate is r = f sin θ, where f is the focal length and θ is the polar angle. The inner- and outer-ring fields in the input plane are denoted by E_in(r, φ) and E_out(r, φ), respectively, and are written as follows [21].

$$E_{\mathrm{i}\mathrm{n}}=\mathrm{A}\mathrm{i}\left(q_{\mathrm{i}\mathrm{n}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(aG{q}_{\mathrm{i}\mathrm{n}}\right)\mathrm{e}\mathrm{x}\mathrm{p}(-b^2{q}_{\mathrm{i}\mathrm{n}}^2)\mathrm{e}\mathrm{x}\mathrm{p}\left(i{C}_{\mathrm{i}\mathrm{n}}{r}^2\right)V(r,\phi )$$

(1)

$$E_{\mathrm{o}\mathrm{u}\mathrm{t}}=\mathrm{A}\mathrm{i}\left(q_{\mathrm{o}\mathrm{u}\mathrm{t}}\right)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\kappa q_{\mathrm{o}\mathrm{u}\mathrm{t}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(aG{q}_{\mathrm{o}\mathrm{u}\mathrm{t}}\right)\mathrm{e}\mathrm{x}\mathrm{p}(-b^2{q}_{\mathrm{o}\mathrm{u}\mathrm{t}}^2)\mathrm{e}\mathrm{x}\mathrm{p}\left(i{C}_{\mathrm{o}\mathrm{u}\mathrm{t}}{r}^2\right)V(r,\phi )$$

(2)

Figure 1. Schematic of the tightly focused asymmetric double-ring Airy–Gaussian vortex beam. The inner and outer rings have different radii and radial chirp coefficients C_in and C_out; f is the focal length, and θ and φ denote the polar and azimuthal angles, respectively.

Here, q_j = (G r_j − r)/(G b w₀), j = in or out. The factor exp(aGq_j) follows the supplied numerical definition. Ai is the Airy function; G is the scale factor; a is the truncation factor; b is the optical distribution factor; w₀ is the transverse length scale; and r_in and r_out are reference ring radii. The shared displaced vortex product is

$$V(r,\phi )={\left({\displaystyle \frac{r{e}^{i\phi }-r_{k1}}{w_0}}\right)}^m{\left({\displaystyle \frac{r{e}^{i\phi }+r_{k2}}{w_0}}\right)}^n$$

(3)

The vortex orders are m = n = 1; their positions are x = r_k1 and x = −r_k2. The dimensionless hyperbolic coefficient is κ; C_in and C_out have units m⁻². The total incident field is

$$E_0(r,\phi )=A(E_{\mathrm{i}\mathrm{n}}-E_{\mathrm{o}\mathrm{u}\mathrm{t}})$$

(4)

According to the Richards–Wolf vector diffraction integral, the electric field of an x-linearly polarized AHC-DRAGVB at the observation point is

$$\mathit{E}(\rho ,\Phi ,z)=-{\displaystyle \frac{i{k}_{0}f}{2\pi }}{\int }_0^{\alpha }{\int }_0^{2\pi }{E}_0(f\mathrm{s}\mathrm{i}\mathrm{n}\theta ,\phi )\sqrt{\mathrm{c}\mathrm{o}\mathrm{s}\theta }\mathrm{s}\mathrm{i}\mathrm{n}\theta e^{i{k}_0\left[\rho \mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{c}\mathrm{o}\mathrm{s}\right(\phi -\Phi )+z\mathrm{c}\mathrm{o}\mathrm{s}\theta ]}\mathit{p}d\phi d\theta$$

(5)

The polarization vector is p = (cosθ cos²φ + sin²φ, (cosθ − 1)cosφ sinφ, −sinθ cosφ)ᵀ. We use exp(−iωt); the chirp signs specify opposite quadratic pupil phases.

Here, k₀ = 2πn_m/λ, λ = 532 nm, and α = arcsin(NA/n_m). The medium index is n_m, whereas n denotes the second vortex order. The angular frequency is ω = 2πc₀/λ. The incident field is linearly polarized along x. Numerical integration yields E_x, E_y and E_z. After removing the common propagation phase, their component and relative phases are

$${\delta }_i\left(z\right)=\mathrm{A}\mathrm{r}\mathrm{g}\left[E_i\right(z\left)\right]-k_{0}z$$

(6)

$${\delta }_{ij}\left(z\right)={\delta }_i\left(z\right)-{\delta }_j\left(z\right),i,j\in \{x,y,z\}$$

(7)

The vectors α(r), β(r) and γ(r) describe the major axis, minor axis and normal of the three-dimensional electric polarization ellipse [22,23,24,25,26]. The following expressions apply away from the circular-polarization degeneracy E·E = 0; at a degeneracy, the ellipse axes require a limiting construction.

$$\alpha ={\displaystyle \frac{\mathrm{R}\mathrm{e}\left\{{\mathit{E}}^{\ast }\sqrt{\mathit{E}·\mathit{E}}\right\}}{\left|\sqrt{\mathit{E}·\mathit{E}}\right|}}$$

(8)

$$\beta ={\displaystyle \frac{\mathrm{I}\mathrm{m}\left\{{\mathit{E}}^{\ast }\sqrt{\mathit{E}·\mathit{E}}\right\}}{\left|\sqrt{\mathit{E}·\mathit{E}}\right|}}$$

(9)

$$\mathit{\gamma }=\mathrm{I}\mathrm{m}({\mathit{E}}^{\ast }\times \mathit{E})$$

(10)

The polarization ellipse lies in the plane spanned by α and β. For peak-amplitude phasors in air, its normal is parallel to the electric contribution to the time-averaged spin density [22,23,24,25],

$${\mathit{S}}_{\mathrm{E}}={\displaystyle \frac{{\epsilon }_0}{4\omega }}\mathrm{I}\mathrm{m}({\mathit{E}}^{\ast }\times \mathit{E})$$

(11)

Here, ε₀ is the vacuum permittivity and ω is the angular frequency. We use a unit vector to describe spin orientation, rather than an energy-normalized spin magnitude:

$${\hat{\mathit{S}}}_{\mathrm{E}}={\displaystyle \frac{{\mathit{S}}_{\mathrm{E}}}{\left|{\mathit{S}}_{\mathrm{E}}\right|}}={\displaystyle \frac{\mathrm{I}\mathrm{m}({\mathit{E}}^{\ast }\times \mathit{E})}{2\sqrt{{\xi }_{yx}^2+{\xi }_{zx}^2+{\xi }_{zy}^2}}}$$

(12)

The unit vector is undefined at zero spin. The auxiliary quantities are

$${\xi }_{yx}=\left|E_x\right|\left|E_y\right|\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{yx},{\xi }_{zx}=\left|E_x\right|\left|E_z\right|\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{zx},{\xi }_{zy}=\left|E_y\right|\left|E_z\right|\mathrm{s}\mathrm{i}\mathrm{n}{\delta }_{zy}$$

(13)

Under the Rayleigh approximation, a nonabsorbing particle with r_p = 20 nm < λ/20 is treated as an electric dipole. We use n_p = 1.59, n_m = 1 and n_r = n_p/n_m, with c₀ = 299,792,458 m s⁻¹ and ε₀ = 8.85,418,781,28 × 10⁻¹² F m⁻¹. The static polarizability and radiation-reaction correction are [28,29,30]

$${\alpha }_{\mathrm{C}\mathrm{M}}=4\pi {\epsilon }_0{n}_{\mathrm{m}}^2{r}_{\mathrm{p}}^3{\displaystyle \frac{n_{\mathrm{r}}^2-1}{n_{\mathrm{r}}^2+2}},\alpha ={\displaystyle \frac{{\alpha }_{\mathrm{C}\mathrm{M}}}{1-i{k}_0^3{\alpha }_{\mathrm{C}\mathrm{M}}/\left(6\pi {\epsilon }_0{n}_{\mathrm{m}}^2\right)}}$$

(14)

$$F_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d},i}={\displaystyle \frac{\mathrm{R}\mathrm{e}\alpha }{4}}{\partial }_i{\sum }_{j=x,y,z}|E_j{|}^2$$

(15)

$$F_{\mathrm{m}\mathrm{o}\mathrm{m},i}={\displaystyle \frac{\mathrm{I}\mathrm{m}\alpha }{2}}\mathrm{I}\mathrm{m}{\sum }_{j=x,y,z}{E}_j^{\ast }{\partial }_i{E}_j$$

(16)

The momentum term generalizes the scattering contribution to local phase gradients in all three directions. It is not constrained to point along z. Spatial derivatives are evaluated analytically inside the diffraction integral at the requested coordinates. The complete time-averaged force is

$$F_i=F_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d},i}+F_{\mathrm{m}\mathrm{o}\mathrm{m},i}={\displaystyle \frac{1}{2}}\mathrm{R}\mathrm{e}\left[{\alpha }^{\ast }{\sum }_{j=x,y,z}{E}_j^{\ast }{\partial }_i{E}_j\right]$$

(17)

All simulations use NA = 0.95, n_m = 1, λ = 532 nm, w₀ = 3 mm, a = 0.1, m = n = 1 and x-linearly polarized input. The second vortex order is called l in the source programs. The transverse scale w₀ uses italic Latin w, not the angular frequency ω. The pupil radius is R = f sinα = 0.95f; r_0max in the displacement notation denotes R. Original x–z field calculations sample −4 ≤ x/λ ≤ 4 and −50 ≤ z/λ ≤ 50 on a 200 × 200 grid; the polar-angle integral uses 200 samples over 0 ≤ θ ≤ α. Figure-specific parameters are stated with the corresponding results and captions.

The original calculations set the pupil-field amplitude factor M₀ = 1 and retain their plotted intensity normalization. This amplitude convention does not specify a physical incident power of 1 W. The additional total-force calculations use a transmitted-pupil power reference of 1 W, with forces reported per watt. The corresponding optical and particle parameters are specified in the optical-force analysis.

The longitudinal peak ratio is η = max_Ω|E_z|²/max_Ω(|E_x|² + |E_y|² + |E_z|²), where Ω is the sampled x–z domain. It is a ratio of maxima, not an integrated energy fraction. On the transverse slice selected at the E_z intensity maximum, R_in and R_out are half the spans between the outermost samples above 0.5 and exp(−2) of the corresponding slice maximum. These threshold radii measure bright-field extent rather than a geometrical dark-cavity radius. The transverse and axial FWHM, T_s and A_s, use x and z cuts through the selected component maximum; the nearest half-height boundaries delimit the selected peak. Widths are reported in units of λ.

A peak-based sidelobe ratio compares the strongest sidelobe peak with the principal peak. The original SLR curves instead use max{I_j: I_j < 0.7 I_j,max}/I_j,max over the sampled x–z field, where I_j = |E_j|². This threshold statistic is distinguished from a physical peak-based SLR and is not used to claim systematic sidelobe suppression.

3. Results and Discussion

3.1. Effect of the Optical Distribution Factor on Focal-Plane Intensity and Phase

Figure 2 retains the original spatial illustrations for b = 0.1, 0.3 and 0.5, with f = 6 mm, G = 0.8, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm, r_out = 1.6 mm and r_k1 = r_k2 = 0. The distribution factor changes the Airy–Gaussian pupil weighting and focal ring structure [6,7,8,9,10,13]. The on-axis vortex pair preserves the scalar pupil symmetry, whereas x-polarized vector focusing produces distinct E_x, E_y and E_z patterns. Individually normalized spatial panels illustrate morphology. The original absolute-phase display is retained with its explicit plotting convention; it is not used to infer an additional physical vortex order.

Figure 2. Original spatial illustrations. (a) Three-dimensional E_x intensity for (a1) b = 0.1, (a2) b =0.3 and (a3) b =0.5; (b) original phase display following the original plotting convention for (b1) b = 0.1, (b2) b =0.3 and (b3) b =0.5; and (c) Two-dimensional distributions of the transverse electric-field intensity E_y for (c1) b = 0.1, (c2) b =0.3 and (c3) b =0.5; (d) the longitudinal electric-field intensity E_z for (d1) b = 0.1, (d2) b =0.3 and (d3) b =0. The spatial intensity color scales are normalized display values. The original phase routine displays Arg[(E_x + iEy)exp(i3φ)]; this plotting phase is retained for consistency with the original figure and is not interpreted as an additional physical pupil vortex. f = 6 mm, G = 0.8, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm, r_out = 1.6 mm and r_k1 = r_k2 = 0. Common wavelength, NA, polarization and vortex orders are stated in Section 2.

3.2. Effect of the Scale Factor on the Focal Field

Figure 3a–c compares the x–z component maps for G = 0.8, 1.0 and 1.2. This family uses f = 6 mm, b = 0.2, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.3R = 1.71 mm, r_out = 0.6R = 3.42 mm and r_k1 = r_k2 = 0. Changing G modifies pupil weighting and radial interference [10,11,12,13]. Figure 3d,e gives the transverse threshold radii for E_x and E_z, respectively; Figure 3f gives the E_z peak and longitudinal peak ratio. These measures describe the redistribution of the focal field without establishing a universal critical value at G = 1.1.

Figure 3. Scale-factor dependence. (a) Intensity distribution of E_x in the x–z plane for (a1) G = 0.8, (a2) G = 1.0 and (a3) G =1.2; (b) Intensity distribution of E_y in the x–z plane for (b1) G = 0.8, (b2) G = 1.0 and (b3) G =1.2; (c) Intensity distribution of E_y in the x–z plane for (c1) G = 0.8, (c2) G = 1.0 and (c3) G =1.2; (d,e) E_x and E_z transverse threshold radii R_in and R_out; (f) E_z peak I_max and longitudinal peak ratio η. Curves sample G = 0.7–1.2 in steps of 0.1 and retain the original amplitude normalization. f = 6 mm, b = 0.2, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 1.71 mm, r_out = 3.42 mm and r_k1 = r_k2 = 0.

3.3. Effect of Vortex Displacement on the Focal Field

Figure 4 retains the original component, spin and polarization illustrations for f = 6 mm, b = 0.5, G = 0.8, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm and r_out = 1.6 mm. In Figure 4a, r_k1 = r_k2 changes from 0 to 0.3R; in Figure 4b, it changes from 0 to 0.5R. The two singularities move in opposite transverse directions and reshape the focal branches. Figure 4c compares spin orientation for r_k1 = 0 and 0.3R with r_k2 = 0. Figure 4d shows polarization ellipses for (r_k1,r_k2) = (0.3R,0) and (0,0.5R). The arrows project a three-dimensional spin direction, and the ellipses illustrate local orientation and ellipticity [22,23,24,25,26,31,32]. These vector-field changes do not by themselves establish a measurable particle torque.

Figure 4. Original vortex-displacement illustrations. (a) Spatial distribution of E_x for (a1) r_k1 = r_k2 = 0 and (a2) r_k1 = r_k2 = 0.3R; (b) Spatial distribution of E_z for (b1) r_k1 = r_k2 = 0 and (b2) r_k1 = r_k2 =0.5R. (c) Spin-direction projections for (c1) r_k1 = 0 and (c2) r_k1 =0.3R with r_k2 = 0. (d) Polarization ellipses for (d1) (r_k1,r_k2) = (0.3R,0) and (d2) (r_k1,r_k2) = (0,0.5R). R = f sinα = 5.7 mm. f = 6 mm, b = 0.5, G = 0.8, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm and r_out = 1.6 mm. Spatial maps retain their original normalization and appearance; spin and ellipse panels illustrate local orientation.

3.4. Effect of the Optical Distribution Factor on Spin Density and Relative Gouy Phase

Figure 5 retains the original spin-direction trajectories and relative-phase curves for b = 0.3, 0.5 and 0.7. The settings are f = 6 mm, G = 0.8, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm, r_out = 1.6 mm, r_k1 = 0.3R and r_k2 = 0. The transverse spin components vary along z as E_x, E_y and E_z change in amplitude and relative phase. The b = 0.3 phase is not globally monotonic. The displayed phase is wrapped modulo 2π, so a branch jump is a representation change rather than necessarily a physical discontinuity. Relative phase can be evaluated as Arg(E_z E_x*) modulo 2π and is undefined when either component vanishes. Accordingly, the curves support local phase variation and spin-texture control [22,23,24,25], rather than a claim of global monotonicity or a phase winding inferred across zeros.

Figure 5. Original spin trajectories and relative-phase curves for b = 0.3, 0.5 and 0.7, left to right. (a–c) Three-dimensional spin-direction trajectories and transverse arrows; the vertical coordinate is position z/λ. (d–f) Relative phase δ_zx displayed modulo 2π. Branch jumps do not by themselves indicate field discontinuities; relative phase is undefined at a zero of either component. f = 6 mm, G = 0.8, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm, r_out = 1.6 mm, r_k1 = 0.3R and r_k2 = 0. All original spatial and phase panels are retained.

3.5. Effect of the Hyperbolic Modulation Coefficient on the Focal Field

Figure 6 shows the hyperbolic-modulation dependence for f = 6 mm, b = 0.1, G = 0.8, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm, r_out = 1.6 mm and r_k1 = r_k2 = 0. The maps use κ = 0.2, 0.5 and 0.8; the curves sample 0.2–0.8 in steps of 0.1. Outer-ring amplitude weighting redistributes the focal peaks and increases the longitudinal peak ratio. The original threshold-based SLR statistic is described in Section 2 and is not interpreted as proof of physical sidelobe suppression [12,13,15,16,33].

Figure 6. Hyperbolic-modulation dependence. Columns (a): Spatial distribution of E_x for (a1) κ = 0.2, (a2) κ = 0.5 and (a3) κ = 0.8; (b) Spatial distribution of E_y for (b1) κ = 0.2, (b2) κ = 0.5 and (b3) κ = 0.8; (c) Spatial distribution of E_z for (c1) κ = 0.2, (c2) κ = 0.5 and (c3) κ = 0.8; (d) E_x transverse and axial widths T_s and A_s, in λ; (e) threshold-based E_z SLR statistic defined in Section 2; (f) E_z peak I_max and longitudinal peak ratio η. Curves sample κ = 0.2–0.8 in steps of 0.1 and retain the original amplitude normalization. f = 6 mm, b = 0.1, G = 0.8, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm, r_out = 1.6 mm and r_k1 = r_k2 = 0.

3.6. Effect of the Radial-Chirp Coefficients on the Focal Field

Figure 7 compares opposite radial chirps for f = 9 mm = 3w₀, b = 0.1, G = 0.8, κ = 0.8, r_in = 0.8 mm, r_out = 1.6 mm and r_k1 = r_k2 = 0. The maps use |C| = 1100 and 550,000 m⁻²; the six curve samples use 1100, 5500, 11,000, 55,000, 110,000 and 550,000 m⁻², with C_in = C and C_out = −C. The radial phases reshape the focal distribution and component balance [10,12,15,33]. The focal-coordinate, threshold-radius and peak-intensity plots demonstrate this redistribution. The SLR panels use the threshold statistic defined in Section 2 rather than a physical peak-sidelobe ratio.

Figure 7. Opposite radial chirps. (a) Spatial distribution of E_x for (a1) |C| = 1100 and (a2) 550,000 m⁻²; (b) Spatial distribution of E_y for (b1) |C| = 1100 and (b2) 550,000 m⁻²; (c) Spatial distribution of E_y for (c1) |C| = 1100 and (c2) 550,000 m⁻²; (d) E_x peak coordinates and the original inset; (e) E_x threshold radii R_in and R_out; (f) E_x and E_z peaks; (g) component threshold-based SLR statistics. Six samples use |C| = 1100, 5500, 11,000, 55,000, 110,000 and 550,000 m⁻², with C_in = C and C_out = −C. The original intensity normalization is retained. f = 9 mm, b = 0.1, G = 0.8, κ = 0.8, r_in = 0.8 mm, r_out = 1.6 mm and r_k1 = r_k2 = 0.

4. Optical Forces on Rayleigh Particles

Figure 8a–f presents the original transverse gradient-force maps, transverse force cuts and axial gradient and scattering contributions for f = 3 mm = w₀, G = 0.8, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm, r_out = 1.6 mm and r_k1 = r_k2 = 0. Figure 8g shows the three-dimensional total-force directions near the selected b = 0.3 equilibrium, marked in red. Arrow lengths are normalized to emphasize direction. Figure 8h shows the corresponding F_x, F_y and F_z cuts: one coordinate is varied at a time while the other two remain fixed at the same equilibrium. Each component vanishes at zero displacement and reverses sign across it, consistent with a local restoring response along the three coordinate cuts. These two panels complement the separate force distributions in Figure 8a–f.

Figure 8. Optical forces. (a,b) Original transverse gradient-force maps; (c,d) original x and y gradient-force cuts; (e,f) original axial gradient and scattering contributions for b = 0.1 and 0.3. Original panels a–f retain their display scales and do not by themselves establish a common three-dimensional equilibrium. (g) Three-dimensional total-force directions near the b = 0.3 equilibrium, with normalized arrows and a red equilibrium marker. (h) Three total-force component cuts through the same equilibrium, with the other coordinates held fixed; offsets are measured from that point. Only the new curves in h use the corrected force normalization, in pN W⁻¹. f = 3 mm, G = 0.8, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm, r_out = 1.6 mm, r_k1 = r_k2 = 0, r_p = 20 nm, n_p = 1.59 and n_m = 1.

Figure 8. Optical forces. (a,b) Original transverse gradient-force maps; (c,d) original x and y gradient-force cuts; (e,f) original axial gradient and scattering contributions for b = 0.1 and 0.3. Original panels a–f retain their display scales and do not by themselves establish a common three-dimensional equilibrium. (g) Three-dimensional total-force directions near the b = 0.3 equilibrium, with normalized arrows and a red equilibrium marker. (h) Three total-force component cuts through the same equilibrium, with the other coordinates held fixed; offsets are measured from that point. Only the new curves in h use the corrected force normalization, in pN W⁻¹. f = 3 mm, G = 0.8, κ = 0.5, C_in = −C_out = 1100 m⁻², r_in = 0.8 mm, r_out = 1.6 mm, r_k1 = r_k2 = 0, r_p = 20 nm, n_p = 1.59 and n_m = 1.

5. Conclusions

We introduced an AHC-DRAGVB with asymmetric radial phase and amplitude modulation and analyzed its tightly focused field, spin texture and Rayleigh-particle forces using the Richards–Wolf formalism. G modifies focal interference and longitudinal weighting; vortex displacement reshapes the focal pattern, spin orientation and polarization; and b changes local relative-phase evolution. The hyperbolic coefficient κ changes longitudinal peak weighting, while opposite chirps tune focal position and component balance. Additional three-dimensional total-force directions and coordinate cuts illustrate the local restoring response near a common equilibrium for b = 0.3. These results provide a basis for further study of structured-light manipulation.