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Frontiers in Physics ·

Volume 14 - 2026 | https://doi.org/10.3389/fphy.2026.1919249

Collapse dynamics of structured optical field in Kerr medium with chirality modulation

  • Yongjie Luo1
  • Cai-Xia Liu1,2*corresponding author
  • Qinglai Yan1
  • Xiao-Bo Hu2,3
  • Jie Zhao1
  • Yao Tang1
  1. 1Zhejiang Institute of Medical Device Supervision and Testing, Hangzhou, Zhejiang, China
  2. 2Key Laboratory of Optical Field Manipulation of Zhejiang Province, Department of Physics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, China
  3. 3Zhejiang-Mexico Joint Laboratory of Intelligent Optoelectronic Sensing, Zhejiang Sci-Tech University, Hangzhou, China
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Abstract

Based on the two-dimensional coupled nonlinear Schrödinger equation, we systematically investigate the collapse dynamics of vector optical fields (VOFs) in a chiral-Kerr medium, where chirality is induced by the unequal refractive indices of left- and right-circularly polarized components. The critical power for VOF collapse is derived analytically using the moment method. Numerical simulations reveal that the number, locations, propagation distances, and profiles of collapse events are determined by the real and imaginary parts of the chiral refractive indices, the initial power, and the polarization topological charge m. In an achiral Kerr medium, collapse occurs symmetrically. Circular birefringence (CB), i.e., a difference in the real parts of the refractive indices, induces rotation of the state of polarization (SoP) while preserving symmetric collapse positions about the coordinate axes. For a locally linearly polarized VOF, the number of collapse points equals 4×⌈ m /2⌉. For a hybrid polarized VOF under CB, the beam differentiates into 4 m collapse points. In contrast, circular dichroism (CD), arising from a disparity in the imaginary parts, triggers linear-to-elliptical/circular polarization conversion and shifts the collapse positions, with the rotation direction dictated by the relative magnitudes of n + and n −. Under low and high initial powers, the hybrid polarized VOF ultimately splits into 2 m beamlets concentrated at regions of dominant handedness under CD. The rotation angle of the collapse positions with respect to the Y-axis is π/(4 m). These findings establish chiral-Kerr media as a promising platform for controlling vector field collapse and polarization dynamics, with potential applications in chiral sensing, nonlinear optics, and structured light manipulation.

Keywords

1 Introduction

With the advent and development of lasers, flexible manipulation of optical fields across different media has attracted growing interest, aiming to deepen our understanding of light-matter interactions [1–3]. Structured optical fields, owing to their independently controllable degrees of freedom—including amplitude, phase, polarization, frequency, and time—have become a key enabler of advances in modern optics [4, 5]. In particular, vector optical fields (VOFs), formed by the orthogonal coherent superposition of left- and right-handed circular polarization components, represent a class of complex structured light fields with a spatially nonuniform polarization distribution [3, 6]. Owing to their unique spatial-polarization coupling structure, they have shown great potential in optical communication [7, 8], particle trapping [9, 10], optical metrology [11–16], imaging techniques [17–19], chiral sensing [20], and nonlinear optics [21, 22].

The Kerr medium, in which the refractive index change is linearly proportional to the intensity, is the most representative nonlinear medium. The evolution of a light beam in such a medium is governed by the intensity-dependent refractive index variation. When a beam interacts with a self-focusing Kerr medium, its spatial characteristics may be affected by modulation instability, potentially leading to filamentation [23]. Consequently, controlling the collapse and filamentation of structured optical fields—specifically, the collapse position, critical power, and suppression of collapse—represents an important topic in nonlinear optics [24, 25]. Chirality describes the geometric property where an object and its mirror image cannot perfectly coincide [26]. A chiral object and its mirror image are termed enantiomers [27–31]. A pair of enantiomers often exhibit markedly different physiological effects, pharmacological activities, and biological responses, particularly in the fields of pharmacy and toxicology [32–35]. The characterization of material chirality holds significant scientific and practical value in drug development, toxicological analysis, food safety, and environmental monitoring [36–38]. Researchers have proposed novel chiral optical fields for chiral sensing, including superchiral fields [39, 40], light fields carrying orbital angular momentum [41–43], and synthetic chiral light [44–47], which are used for efficient chiral detection. VOFs show great potential for chiral detection because their inherent chiral properties are similar to those of enantiomers [48–50]. Althrough the interaction between VOFs and nonlinear Kerr media has been explored [51–53]. It remains unknown whether the propagation of vector structured optical fields in chiral nonlinear media gives rise to new chirality-induced effects or phenomena. The ability to control VOFs in chiral nonlinear media provides a novel research approach, deepens the understanding of the underlying mechanisms, and may lead to the development of new optical measurement techniques.

In this paper, we systematically investigate the manipulation of structured beam collapse and filamentation in a chiral-Kerr medium which refers to a nonlinear Kerr medium with chiral properties. Specifically, we compare the state of polarization (SoP) of VOFs before and after passing through purely Kerr media and chiral-Kerr media, as well as the collapse position under different topological charges. We analyze the interplay among VOFs, chirality, and the nonlinear Kerr effect, and explore the physical laws and mechanisms underlying these novel light-matter interactions.

2 Theoretical mothod

The vector optical field (VOF) can be expressed as a superposition of two arbitrary orthogonal polarization components at the source plane:where A 0 is the amplitude, and r is the radial coordinates normalized by the initial beam width w 0. φ = arctan(Y/X) is the azimuthal angle and φ 0 is the initial phase. The unit vectors in the X-direction and Y-direction are denoted by *e* ***X*** and *e* ***Y***, while *e* ***+*** and *e* − represent the left- and right-circular polarization basis vectors, respectively. The relationship between projections *e* ***X,Y*** and *e* ± is given by *e* ***+*** = *e* ***X*** + i *e* ***Y*** and *e* − = *e* ***X*** − i *e* ***Y***. The integer m denotes the polarization topological charge. Δθ determines the phase difference between the X and Y polarization components, where Δθ = 0 represents the locally linear polarization of VOF and Δθ = π/2 represents the hybrid polarization of VOF. Equation 1 reduces to a scalar beam when only one polarization component is considered. To model the propagation dynamics of a structured beam in a chiral-Kerr medium (i.e., a Kerr medium endowed with chirality), a chirality term is incorporated into the two-dimensional coupled nonlinear Schrödinger equation (NLSE) [54] ∇ ⊥ 2 U ± + 2 i k ∂ U ± / ∂ z + 4 n 2 k 2 U ± 2 + 2 U ∓ 2 U ± / 3 n 0 = 0. Specifically, we can get the normalized two-dimensional coupled NLSE ∇ ⊥ 2 E ± + i ∂ E ± / ∂ Z + 2 E ± E ± 2 + 2 E ∓ 2 / 3 = 0 by introducing the normalizations X = x/w 0, Y = y/w 0, Z = z/(2kw 0 2) and E(x,y) = kw 0(2n 2/n 0)1/2 U(x,y). Here X, Y and Z are the dimensionless coordinates, n 2 is the third-order nonlinear coefficient, n 0 is the linear refractive index, and k is the linear wave number.

E X , Y , Z = 0 = A 0 r m ⁡ exp − r 2 cos m φ + φ 0 e X + e i Δ θ ⁡ sin m φ + φ 0 e Y = A 0 r m ⁡ exp − r 2 e i m φ + φ 0 e + + e − i m φ + φ 0 e − Δ θ = 0 , = A 0 r m ⁡ exp − r 2 e i m φ + φ 0 e D + e − i m φ + φ 0 e A Δ θ = π / 2 , (1)

The propagation of the VOF in the chiral-Kerr medium can be described by normalized two-dimensional coupled NLSE with the chiral term as follows:where ∇ ⊥ 2 = ∂ 2 / ∂ X 2 + ∂ 2 / ∂ Y 2 is the transverse Laplace operator representing the diffraction of the light field. The second term on the left of the equation denotes the propagation term, and Z is the longitudinal coordinate along the beam propagation direction. The third term indicates the third-order Kerr nonlinear effect. The fourth term is refractive index modulation, which represents the chiral modulation. Where n + and n − denote the chiral refractive indices for the two circular polarization components. Here, the subscripts “+” and “−” represent the left- and right-handed circular polarization components, respectively. In the chiral-Kerr medium, the refractive indices for the two circular polarization components are complex, n ± = n R ± in I, where the real parts n R govern circular birefringence (CB) and the imaginary parts n I govern circular dichroism (CD, gain or loss).

∇ ⊥ 2 E ± + i ∂ E ± ∂ Z + 2 3 E ± 2 + 2 E ∓ 2 E ± + n ± E ± = 0 . (2)

There are two important physical quantities: the total initial power P z = ∬ s E + 2 + E − 2 d X d Y and the Hamiltonian for the NLSE. Based on the moment method, the four corresponding integral quantities of the VOF in the chiral-Kerr medium are given as follows [54, 55]:where I 1 is the beam power, I 2 is the root-mean-square (RMS) beam width, I 3 denotes the momentum and I 4 represents the Hamiltonian, respectively. The above integral quantities satisfy the coupled ordinary differential equations in Equation 3 as [55]: dI 1(Z)/dZ = 0, dI 2(Z)/dZ = I 3(Z), dI 3(Z)/dZ = 4I 4(Z), dI 4(Z)/dZ = 0. There is an important invariant Q = 2I 2(Z)I 4(Z) - I 3 2(Z)/4 during the evolution of a VOF in a chiral-Kerr medium. The evolution of the quantity I 2 satisfies a differential equation [56]:

I 1 Z = ∬ s E + 2 + E − 2 d X d Y , I 2 Z = ∬ s X 2 + Y 2 E + 2 + E − 2 d X d Y , I 3 Z = i ∬ s X E + + E − ∂ E + * + E − * ∂ X − E + * + E − * ∂ E + + E − ∂ X d X d Y + i ∬ s Y E + + E − ∂ E + * + E − * ∂ Y − E + * + E − * ∂ E + + E − ∂ Y d X d Y , I 4 Z = ∬ s ∇ E + 2 + ∇ E − 2 − 1 3 E + 4 + E − 4 + 4 E + 2 E − 2 − n + E + 2 + n − E − 2 d X d Y , (3)

d 2 I 2 1 / 2 Z d Z 2 = Q I 2 3 / 2 Z . (4)

The general solution of Equation 4 for the initial VOF distribution can be given [56] as:

I 2 Z = I 2 Z = 0 + Q I 2 Z = 0 Z 2 . (5)

A total balance between diffraction effects and self-focusing effects occurs when Hamiltonian I 4 = 0, under which condition the beam width remains invariant [56, 57]. By setting I 4 = 0, the critical amplitude A 0 of the VOF in a chiral-Kerr medium can be obtained from Equation 5. Therefore, the critical power (P cr) can be obtained by the beam power I 1 as follows:

P cr = 4 m π 4 + 4 m − n + − n − Γ m + 1 2 Γ 2 m + 1 . (6)

For scalar beams, the P cr in a chiral-Kerr medium can be achieved by retaining only one orthogonal polarization component, as show in Equation 6. Equality between the input power P in and P cr leads to a constant root-mean-square beam width [58]. Exceeding this threshold triggers a global collapse, characterized by the beam width decaying to zero within a finite propagation range.

3 Numerical simulation and analysis

Numerical calculations are carried out using the split-step finite-difference based on Equation 2 to explore the dynamics of VOFs in chiral-Kerr media, with λ = 532 nm and w 0 = 10 μm. The evolution of the SoP of VOFs in the chiral-Kerr medium is visualized using colored and shaped markers: red and blue ellipses (or circles) denote left- and right-elliptical (or circular) polarizations, respectively, while green lines represent linear polarization. It should be clarified that these refractive index values in this paper are used for clear visualization of the underlying physics and are not intended to represent typical natural materials. All intensities are normalized to the initial incident peak intensity. Here, we take the P in where collapse occurs precisely at z = 2 as an illustrative example. In our simulations, all intensity values are scaled by the maximum intensity of the incident optical field; thus, when the color bar exceeds 1, it indicates that the local intensity has surpassed its initial value, which we use as a practical indicator that the beam is approaching or entering collapse. By employing the bisection method, we are able to identify the appropriate initial power that ensures collapse occurs at the fixed position z = 2 for each case.

Figure 1 presents the intensity and SoP distributions of a locally linearly polarized VOF propagating in an achiral Kerr medium compared with a chiral-Kerr medium, illustrating the influence of the disparity in the real parts of the refractive indices for left- and right-circular polarizations on the resulting SoP. When a VOF propagates in a chiral-Kerr medium with an initial power exceeding the critical threshold, the self-focusing effect overcomes the diffraction effect, leading to collapse, as illustrated in Figures 1b, c. Notably, the SoP of the VOF rotates during propagation in a chiral-Kerr medium which is different from that in an achiral Kerr medium, as depicted in Figure 1. The distance at which collapse occurs is strongly dependent on the initial power. When the real parts of the refractive indices for left-circularly polarized (LCP) and right-circularly polarized (RCP) components are different (n + ≠ n −), they experience distinct phase velocities, and hence different propagation speeds. This phenomenon arises from CB. The relative magnitude of n + and n − determines the rotation direction of the SoP. Specifically, a clockwise rotation occurs when Re (n + − n −) > 0, whereas a counterclockwise rotation takes place for Re (n +) < Re(n −), as shown in Figures 1b, c, respectively.

The normalized three-dimensional intensity and the cross-sectional intensity and SoP distributions at different propagation distances for a VOF with *m* = 1: (a) *n* <sub>+</sub> = *n* <sub>−</sub> = 1.3, *P* <sub>in</sub> = 1.7*P* <sub>cr</sub>; (b) *n* <sub>+</sub> = 1.3, *n* <sub>−</sub> = 2.3, *P* <sub>in</sub> = 2.1*P* <sub>cr</sub>; (c) *n* <sub>+</sub> = 2.3, *n* <sub>−</sub> = 1.3, *P* <sub>in</sub> = 2.1*P* <sub>cr</sub> in an achiral Kerr medium and a chiral-Kerr medium. Green lines represent linear polarization, *n* <sub>+</sub> and *n* <sub>−</sub> are schematic values.
Figure 1. The normalized three-dimensional intensity and the cross-sectional intensity and SoP distributions at different propagation distances for a VOF with m = 1: (a) n + = n − = 1.3, P in = 1.7P cr; (b) n + = 1.3, n − = 2.3, P in = 2.1P cr; (c) n + = 2.3, n − = 1.3, P in = 2.1P cr in an achiral Kerr medium and a chiral-Kerr medium. Green lines represent linear polarization, n + and n − are schematic values.

Figure 2 shows the intensity and SoP of a locally linearly polarized VOF during propagation in a chiral-Kerr medium, illustrating how m influences both the collapse positions and the evolution of the SoP. Owing to the faster collapse dynamics of the VOF at linearly polarized locations compared to left- and right-elliptical (or circular) SoP, the collapse events predominantly occur at positions exhibiting linear polarization. It is noteworthy, however, that the total number of collapse points is not 4m. This discrepancy arises because the collapse rates differ for linear polarizations oriented along distinct directions, preventing all such positions from undergoing collapse simultaneously. Specifically, for adjacent odd and even topological charges (e.g., m = 3 and m = 4), although the total count of collapse points remains the same, their intensity distributions are spatially complementary. In general, the number of collapse points equals 4×⌈m/2⌉for locally linear polarized VOF, where ⌈x⌉ denotes the ceiling function. The collapse positions are symmetric about the coordinate axes.

The normalized three-dimensional intensity distribution,cross-sectional intensity (at *z* = 2) and SoP evolution of a locally linearly polarized VOF in the chiral-Kerr medium with *n* <sub>+</sub> = 1.3, *n* <sub>−</sub> = 2.3 for different topological charges *m*: **(a)** *m* = 2, *P* <sub>in</sub> = 1.44*P* <sub>cr</sub>; **(b)** *m* = 3, *P* <sub>in</sub> = 1.18*P* <sub>cr</sub>; **(c)** *m* = 4, *P* <sub>in</sub> = 1.03*P* <sub>cr</sub>; **(d)** *m* = 5, *P* <sub>in</sub> = 0.92*P* <sub>cr</sub>. (Red and blue ellipses (or circles) denote left- and right-elliptical (or circular) polarizations, green lines represent linear polarization. *n* <sub>+</sub> and *n* <sub>−</sub> are schematic values.).
Figure 2. The normalized three-dimensional intensity distribution,cross-sectional intensity (at z = 2) and SoP evolution of a locally linearly polarized VOF in the chiral-Kerr medium with n + = 1.3, n − = 2.3 for different topological charges m: (a) m = 2, P in = 1.44P cr; (b) m = 3, P in = 1.18P cr; (c) m = 4, P in = 1.03P cr; (d) m = 5, P in = 0.92P cr. (Red and blue ellipses (or circles) denote left- and right-elliptical (or circular) polarizations, green lines represent linear polarization. n + and n − are schematic values.).

We have known that when the refractive indices n + and n − are unequal, rotation of the SoP in the cross-section occurs, as seen in Figures 1, 2. Unlike CB, CD arises from the differential absorption of LCP and RCP, i.e., CD ∝ Im (n + − n −). Figure 3 displays the intensity and SoP of a locally linearly polarized VOF in a chiral-Kerr medium, demonstrating the effect of the difference in the imaginary parts of the refractive indices for the left- and right-circular polarizations on the SoP. The differential absorption causes an amplitude imbalance between the LCP and RCP components, transforming the initially linear SoP into an elliptical one. When one component is completely absorbed, a circular SoP is obtained. As illustrated in Figure 3, during the propagation of a locally linearly polarized vector field in a chiral-Kerr medium, the SoP transitions from linear to a uniform distribution of either left- or right-handed elliptical (circular) polarization. The handedness of the resulting elliptical (circular) polarization is dictated by the relative magnitudes of the imaginary parts of the refractive indices. If Im (n + − n −) > 0, the LCP component is absorbed more strongly, so the RCP component dominates, resulting in a right-handed elliptical (or circular) polarization distribution. Conversely, if Im (n + − n −) < 0, the LCP component dominates, leading to a left-handed distribution. If only the real parts differ, the SoP rotates, while the collapse positions remain symmetrically distributed about the coordinate axes under CB. In contrast, the CD effect is not merely a rotation of the SoP but a conversion from linear to circular polarization. Furthermore, such polarization conversion leads to a rotation in the collapse positions with the rotation direction, as shown by the solid white line in Figure 3. The direction of rotation and the angle are determined by the relative magnitudes of n + and n −.

The evolution of normalized three-dimensional intensity distribution, intensity and SoP distribution for the locally linearly polarized VOF in a chiral-Kerr medium under circular dichroism effect during propagation: **(a)** *m* = 1, *n* <sub>+</sub> = 1.3 + *i*, *n* <sub>−</sub> = 1.3 − *i, P* <sub>in</sub> = 1.62*P* <sub>cr</sub>; **(b)** *m* = 2, *n* <sub>+</sub> = 1.3 − *i*, *n* <sub>−</sub> = 1.3 + *i*; *P* <sub>in</sub> = 1.22 *P* <sub>cr.</sub> (The red and blue ellipses (or circles) denote left- and right-elliptical (or circular) polarizations. The solid white lines represent the *Y*-axis direction under CB. *n* <sub>+</sub> and *n* <sub>−</sub> are schematic values.).
Figure 3. The evolution of normalized three-dimensional intensity distribution, intensity and SoP distribution for the locally linearly polarized VOF in a chiral-Kerr medium under circular dichroism effect during propagation: (a) m = 1, n + = 1.3 + i, n − = 1.3 − i, P in = 1.62P cr; (b) m = 2, n + = 1.3 − i, n − = 1.3 + i; P in = 1.22 P cr. (The red and blue ellipses (or circles) denote left- and right-elliptical (or circular) polarizations. The solid white lines represent the Y-axis direction under CB. n + and n − are schematic values.).

In particular, the number of collapse points does not increase with m, and only four collapse points ultimately survive, as the other four exhibit weaker intensities in Figure 3b. The topological charge m not only affects the number and positions of collapse but also influences the size of the beam and the optical singularity. The precise SoP distribution is intricately linked to m. Moreover, the magnitude of the imaginary parts of the refractive indices affects the collapse positions. As depicted in Figures 1, 2, 3, the beam width evolves during propagation, exhibiting phenomena such as an initial broadening followed by focusing. This behavior originates from the interplay among diffraction, self-focusing, and the chiral modulation term.

A more complex optical field with a hybrid polarization distribution (simultaneously containing linear, elliptical, and circular polarizations) is shown in Figure 4. Figure 4 illustrates the influence of CB on the 3D evolution of a hybrid VOF in a chiral-Kerr medium for different polarization topological charges, as well as the intensity and SoP distributions at various transverse cross-sections. Specifically, Figures 4a-c corresponds to m = 1, 2, and 3, respectively. For the locally linearly polarized VOF (as in Figures 1, 2, 3), the beam first expands and then contracts, such that the cross-sectional area at the final collapse points is smaller than that of the initial field. For a hybrid polarized optical field, the beam initially undergoes a slight contraction before expanding, resulting in a final collapse cross-sectional area larger than that of the initial field. Furthermore, when a hybrid VOF propagates in a chiral-Kerr medium, circular birefringence—arising from the disparity in the real parts of the refractive indices for the LCP and RCP components—causes all SoP to evolve with propagation distance. For instance, the orientation of linear SoP rotates, while the axial direction of elliptical (or circular) SoP changes. More importantly, the topological charge m governs the number and locations of collapse events as shown in Figure 4. Compared to the circularly polarized state, linear SoP collapses earlier; consequently, the positions of linear SoP dictate the number of collapse points, which remains 4m. Note that the relative magnitudes of the refractive indices for the LCP and RCP components determine both the rotation of the SoP and the orientation of the elliptical/circular polarization axes.

Evolution of normalized three-dimensional intensity, cross-sectional intensity, and SoP distribution of the hybrid VOF in a chiral-Kerr medium under circular birefringence with different topological charges during propagation: **(a)** *n* <sub>+</sub> = 1.3, *n* <sub>−</sub> = 2.3, m = 1, *P* <sub>in</sub> = 3.4*P* <sub>cr</sub>; **(b)** *n* <sub>+</sub> = 1.3, *n* <sub>−</sub> = 2.3, *m* = 2, *P* <sub>in</sub> = 2.59*P* <sub>cr</sub>; **(c)** *n* <sub>+</sub> = 2.3, *n* <sub>−</sub> = 1.3, m = 3, *P* <sub>in</sub> = 2.16*P* <sub>cr</sub>. (Red and blue ellipses (or circles) denote left- and right-elliptical (or circular) polarizations, green lines represent linear polarization. *n* <sub>+</sub> and *n* <sub>−</sub> are schematic values.).
Figure 4. Evolution of normalized three-dimensional intensity, cross-sectional intensity, and SoP distribution of the hybrid VOF in a chiral-Kerr medium under circular birefringence with different topological charges during propagation: (a) n + = 1.3, n − = 2.3, m = 1, P in = 3.4P cr; (b) n + = 1.3, n − = 2.3, m = 2, P in = 2.59P cr; (c) n + = 2.3, n − = 1.3, m = 3, P in = 2.16P cr. (Red and blue ellipses (or circles) denote left- and right-elliptical (or circular) polarizations, green lines represent linear polarization. n + and n − are schematic values.).

Figure 5 illustrates the propagation dynamics of a hybrid VOF in a chiral-Kerr medium under different initial powers and polarization topological charges, along with the corresponding cross-sectional intensity and SoP distributions at various positions and 3D intensity distributions. Specifically, the initial VOF distributions for m = 2 and m = 3 are shown in the left-top panels of Figures 4b, c, respectively. For a low initial power, Figure 5a, c present the intensity and SoP distributions in the transverse plane at propagation distances z = 0.5 and z = 2, as well as the three-dimensional intensity profile up to z = 2. Figures 5b, d depict analogous results for a higher initial power compared to Figures 5a, c. It can be observed from Figure 5 that, owing to the differential absorption between left- and right-circular polarizations in the chiral medium, the initially symmetric and uniform distribution of left- and right-elliptical/circular polarizations in the hybrid VOF becomes uneven, leading to a competitive dominance relationship between the two components. At a low incident power, as shown in Figures 5a, c, although the annular beam has not yet undergone collapse during propagation, it still splits into multiple beamlets. The intensity becomes concentrated at locations where either left- or right-elliptical/circular polarization dominates, and the number of such beamlets is 2m. If Im (n + - n −) > 0, the right-circular component experiences less absorption and thus dominates, collapse occurs earlier in these dominated regions Figures 5a, b. Conversely, if Im (n + - n −) < 0, the left-circular component dominates and collapses first in those areas Figures 5c, d.

The influence of the circular dichroism on the evolution of the normalized three-dimensional intensity, cross-sectional intensity, and SoP distribution of a hybrid VOF in a chiral-Kerr medium for different topological charges and initial power during propagation: **(a)** *m* = 2, *n* <sub>+</sub> = 1.3 + *i*, *n* <sub>−</sub> = 1.3 - *i, P* <sub>in</sub> = 0.1*P* <sub>cr</sub>; **(b)** *m* = 2, *n* <sub>+</sub> = 1.3 + *i*, *n* <sub>−</sub> = 1.3 - *i, P* <sub>in</sub> = 1.7*P* <sub>cr</sub>; **(c)** *m* = 3, *n* <sub>+</sub> = 1.3 - *i*, *n* <sub>−</sub> = 1.3 + *i*; *P* <sub>in</sub> = 0.1*P* <sub>cr</sub>; **(d)** *m* = 3, *n* <sub>+</sub> = 1.3 - *i*, *n* <sub>−</sub> = 1.3 + *i*; *P* <sub>in</sub> = 1.47*P* <sub>cr</sub> (The red and blue ellipses (or circles) denote left- and right-elliptical (or circular) polarizations, while the green lines represent linear polarization. The dashed green lines represent the *Y*-axis direction with CD, and the solid white lineindicates the reference *Y*-axis direction in the absence of chirality. *n* <sub>+</sub> and *n* <sub>−</sub> are schematic values.).
Figure 5. The influence of the circular dichroism on the evolution of the normalized three-dimensional intensity, cross-sectional intensity, and SoP distribution of a hybrid VOF in a chiral-Kerr medium for different topological charges and initial power during propagation: (a) m = 2, n + = 1.3 + i, n − = 1.3 - i, P in = 0.1P cr; (b) m = 2, n + = 1.3 + i, n − = 1.3 - i, P in = 1.7P cr; (c) m = 3, n + = 1.3 - i, n − = 1.3 + i; P in = 0.1P cr; (d) m = 3, n + = 1.3 - i, n − = 1.3 + i; P in = 1.47P cr (The red and blue ellipses (or circles) denote left- and right-elliptical (or circular) polarizations, while the green lines represent linear polarization. The dashed green lines represent the Y-axis direction with CD, and the solid white lineindicates the reference Y-axis direction in the absence of chirality. n + and n − are schematic values.).

For a sufficiently high incident power, the hybrid VOF first differentiates into 4m beamlets at positions dominated by either left- or right-elliptical/circular polarization. As propagation proceeds further, the differential absorption causes one of the two circular handednesses to collapse earlier, gradually reducing the number of beamlets by half, i.e., to 2m. At this stage, the SoP of each beamlet corresponds entirely to the dominant handedness, either left- or right-elliptical/circular polarization. The angle between the solid white line and the dashed green line indicates the rotation direction relative to the Y-axis, and its value is given by π/(4m).

4 Conclusion

In this work, we have systematically studied the collapse dynamics of VOFs in a chiral-Kerr medium. The critical power for collapse is derived analytically via the moment method. Key findings reveal that circular birefringence (CB) induces polarization rotation without altering symmetric collapse positions, whereas circular dichroism (CD) triggers linear-to-elliptical/circular polarization conversion and shifts collapse locations. The topological charge m governs the number of collapse points. Under CB, the number of collapse points follows 4×⌈m/2⌉ for a locally linearly polarized VOF and 4m for a hybrid polarized VOF. Under the CD effect, the hybrid polarized VOF initially splits into 4m beamlets before reducing to 2m due to differential absorption. The resulting collapse positions are rotated by an angle of π/(4m) relative to the Y-axis. These findings establish chiral-Kerr media as a promising platform for manipulating vector field collapse, with potential implications for chiral sensing, nonlinear optics, and structured light manipulation.

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Footnotes

Edited by: Peng Gao, Xidian University, China

Reviewed by: Qiang Wang, Shantou University, China

Dan Wang, Taiyuan University of Technology, China

Statements

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Funding

The author(s) declared that financial support was not received for this work and/or its publication.

Data availability

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

YL: Conceptualization, Data curation, Funding acquisition, Methodology, Validation, Visualization, Writing – original draft, Writing – review and editing. C-XL: Conceptualization, Data curation, Methodology, Validation, Visualization, Writing – original draft, Writing – review and editing. QY: Formal Analysis, Investigation, Resources, Supervision, Writing – review and editing. X-BH: Investigation, Supervision, Writing – review and editing. JZ: Data curation, Project administration, Software, Validation, Writing – review and editing. YT: Data curation, Project administration, Software, Validation, Writing – review and editing.

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Citation

Luo, Y., et al. (2026). Collapse dynamics of structured optical field in Kerr medium with chirality modulation. Frontiers in Physics, Volume 14. https://doi.org/
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14 - 2026

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Copyright © 2026 Luo, Liu, Yan, Hu, Zhao and Tang.

License: https://creativecommons.org/licenses/by/4.0/

*Correspondence: Cai-Xia Liu

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