Optical singularities, a rapidly growing field in contemporary optics, have garnered considerable attention due to their unique physical properties [1–3]. The related research has opened up new possibilities in various domains, including optical communication, super-resolution imaging, optical tweezers, and so on [4–8]. Two distinct types of optical singularities exist: phasic and polarized manifestations. The phase singularity typically occurs in optical vortex, with an indeterminate phase [9]. As for the polarization singularities, they can be broadly classified into two types such as V-point and C-point [10]. The V-point occurs in a local linearly polarized inhomogeneous field, and it is an intensity null point with indeterminate polarization orientation [11]. The C-point exists in an inhomogeneous localized elliptical polarization field, and it refers to a complete circular polarization whose major axis orientation is undefined [12].

Sharp changes of phase and polarization usually exist within a tiny region around the optical singularities; thus, it provides alternative schemes for ultra-sensitivity mutation detection and metrology [13]. Physically, the V-point singularity is a zero-intensity point that lies in an ultra-dark region, making it an incredibly hard task to be detected [14]. The C-point singularity, in contrast, can occur at arbitrary intensities theoretically with chiral polarization distribution properties. It thus offers intriguing application possibilities for detectable chiral light–matter interactions [12]. Interestingly, when a V-point is subjected to an asymmetrical perturbation, it can disintegrate into a C-points pair with opposite chirality [15]. However, although the division from V-point to C-point has been achieved, the introduced asymmetric factors are usually unquantifiable [16–18]. It is still a great challenge currently to modulate the C-points precisely, which severely limits its further applications [19].

In this work, we propose a method that splits a V-point into a controllable C-point pair under a highly focusing condition, and further demonstrate its potential in precise displacement sensing. It is realized by quantifying the asymmetric parameter, i.e., the relative offset factor β, of an incident azimuthal polarized beam (APB) that is a typical vector structured beam. For a nonzero β, the optical spin Hall effect undergoes in the focal plane, and a C-points pair with opposite chirality is obtained [15,16,18,20]. The β value, particularly, impacts deterministically the splitting distance between the two C-points, and the offset angle α controls the direction of division. Then, an approximate linear sharp chiral reversal between the opposite chiral singularities is achievable within a sub-wavelength range, which can be harnessed by varying the offset. As the reversal distance between the two C-points is controllable at the nanometric scale, it is possible for sensitive displacement sensing. Consequently, far-field scattering schemes of sweeping chiral helices and spherical particles in such a C-points pair field are conducted, and a Q index is defined to evaluate the scattering properties. The results show that the Q value has a strong specific response on chirality of the structure, and the distinct tilt variation curves validate its feasibility for iota displacement sensing. This finding creates a novel methodology for ultra-precise displacement sensing by controlling the split polarization singularities in the focusing field, as well as efficient detection and characterization of chiral structures. It is expected to hold significant contribution to many frontier domains including biomedicine, material science, and nanotechnology.

The APB is composed of spatially anisotropic line polarizations varying with azimuth ϕ and centred at the V-point. As a general and typical vector structured light field that contains a V-point, the tight focusing field of the APB still maintains a V-point due to its symmetry. However, by introducing an offset of the incident beam, the original V-point in the tightly focused field is transformed into a C-point pair. Figure 1 depicts the schematic of controllable splitting C-points by shifting the APB from a co-axial to an off-axial scenario.

When the V-point of the incident APB coincides with the center of the objective, as shown in Fig. 1(b), it is named the co-axial condition; while the off-axial condition occurs when the incident APB is offset laterally where the V-point mismatches the objective center, as shown in Fig. 1(c). For quantitative analysis, δ is defined as the distance of the offset V-point relative to the objective center, and R is the radius of objective pupil. The offset factor β is then written as β=δR.(1)

Thus, β can quantify the asymmetric break of the offset APB. Naturally, β ranges in [0, ∞). For β=0, it refers to the coaxial situation, as shown in Fig. 1(b). For this scenario, the incident beam is the standard APB with δ=0, and a V-point occurs at the focal field center, as shown in Fig. 1(d). Figure 1(c) depicts the off-axial cases with β>0. It should be noted that the singularity point will exceed the objective entrance pupil and cannot be focused when β>1. Therefore, we focus on the case of β locating in [0, 1) in the following works, to ensure that the incident V-point locates within the pupil of the objective.

In addition to the offset factor, the offset direction will also affect the focusing field distribution. In Fig. 1(c), the offset angle α represents the offset direction of the V-point offset relative to the objective center. These two offset parameters, i.e., α and β, then together describe the state of offset APB relative to the objective. Due to the asymmetry introduced by the offset, as depicted in Fig. 1(c), the V-point splits into two orthogonal ellipse-field singularities (C-points) in the focal plane, as shown in Fig. 1(e). For a typical APB, it can be expressed as [21] Ein=−sinϕe→x+cosϕe→y,(2)where ex→ and e→y are orthogonal linearly polarized basis vectors, and ϕ is the azimuth angle. For the coaxial scenario, the relationship between ϕ and the position coordinates is ϕ=arctan(yx),(3)where (x, y) are the Cartesian coordinates. When the APB is offset, the beam incident into the objective is no longer a standard APB and the expression should be related to δ (δ=βR) and α, where a coordinate transformation is needed. After the coordinate transformation, a new azimuth ϕ′ to describe the polarization distribution should be modified as ϕ′=arctan(y+βRsinαx+βRcosα),(4)where βRsinα and βRcosα represent the y- and x-direction offset distances, respectively. As ϕ′ is a function of the offset parameters, it is no longer uniformly varying from 0 to 2π; so, the rearrangements of azimuthal and radial polarization directions are achieved at the same time. It is worth mentioning that the implementation of this process needs a precondition where the incident beam radius is greater than the objective lens aperture in physical settings. Then, by substituting ϕ′ into Eq. (2), the new offset APB can be expressed as E′in=−sinϕ′e→x+cosϕ′e→y.(5)

After tightly focusing the incident beam, the focus field distribution can be calculated by the Richard-Wolf vector diffraction integral using linear polarization as the basis vector [22], yielding E(r,φ,z=0)=−ikf2π∫0θ∫02πcosθ[(−sinϕ′(cos2ϕcosθ+sin2ϕ)+cosϕ′cosϕsinϕ(cosθ−1))e→x(−sinϕ′sinϕcosϕ(cosθ−1)+cosϕ′(cos2ϕcosθ+sin2ϕ))e→y(cosϕ′sinϕ−cosϕsinϕ′)sinθe→z]×exp(ikrsinθcos(φ−ϕ))sinθdθdϕ,(6)where (r, φ, z) are the coordinates to describe the focal field. Here, z=0 denotes the focal plane, and the variations of polarization singularities in such a plane are discussed in detail in the following.

The Stokes parameter is a method to characterize the polarization state, and S0, S1, S2, and S3 are capable to characterize the arbitrary polarization distribution of the beam. To precisely characterize the position of polarization singularity, the Stokes phase is a commonly used method, where a complex Stokes field can be constructed as [23] S1+iS2=Aijexp(iϕ12),(7)where S1 and S2 are normalized Stokes parameters, and ϕ12 is the Stokes phase. In such a Stokes field, the polarization singularity is embodied as a phase singularity. Similar to the topological charge of phase singularity, the corresponding polarization singularity Stokes index σ12 is defined using the Stokes phase, being as σ12=12π∮cdϕ12,(8)where c is a circuit enclosing the Stokes phase singularity. Since the Stokes phase has the limitation of being unable to distinguish between the integer charged C-points and V-points, two parameters connected to the Stokes index σ12 are defined to make the distinction: the polarization V-point can be characterized by the Poincaré–Hopf index η=σij/2, and the C-point by the index Ic=σij/2 [1]. Moreover, the Stokes phase distribution potentially looks identical and fails to distinguish the C-points and V-points with the same index. For better analysis of the polarization singularity, it is necessary to couple the Stokes phase with the polarization ellipses distribution. For a quantitative analysis of the spatial distribution property of the split C-points, the Stokes phase and localized polarization characteristics of the focus field with various off-axial shifts are drawn in Fig. 2. To match the tight focusing condition, the NA is set as 0.9.

Figure 2(a) illustrates the Stokes phase with β=0, and Fig. 2(e) shows the corresponding localized polarization states. For such a scenario, the incident beam is a standard APB, thus resulting in an inhomogeneous locally linearly polarized field at the focal plane. The Stokes phase and localized polarized states indicate the existence of a V-point polarization singularity at the center, whose Poincaré–Hopf index is η=1. In Figs. 2(b)–2(d), the offset appears along the x-axis, y-axis, and 45° direction (α=0, π/2, and π/4), respectively. Taking the x-axial shift as an example (α=0), as depicted in Fig. 2(b), it can be observed that the phase singularity in the Stokes phase splits along the y-direction, which corresponds to two C-points in Fig. 2(f). Here, the localized polarization states at the strict C-points are complete circular polarization, and general elliptical polarizations elsewhere. The split occurs perpendicular to the offset direction and stays symmetrically with equal distances from the field center, which refers to the symmetry breaking in this direction [20].

Likewise, in Figs. 2(c) and 2(d), the split direction keeps perpendicular to the offset angle. With this property, the direction of singularity splitting can be controlled by the offset angle α. Moreover, the two C-points hold opposite circular polarizations but possess the same singularity index of Ic=1/2 to keep the sum as a constant of 1, being equal to the singularity index of the original V-point [15]. This is mainly because the singularity index remains conserved during the splitting process. In addition, there forms an L-line at the demarcation line between the two coexisting and neighboring C-points [15].

As has been demonstrated, with a nonzero β, the polarization states of the focal plane change from a global localized linear polarization into a double localized elliptical polarization composed of equal left- and right-chirality. Since splitting occurs in the perpendicular direction of symmetry breaking, it provides a degree of freedom to control the emergence orientation of the C-points. To employ the singularity splitting effect in wider ultra-sensitive sensing scenes, the splitting distance between the two C-points should be further modulated. Consequently, its dependence manner with the offset factor β is investigated in detail.

Figures 3(a)–3(c) depict the modulation results of splitting distance with various β of 0.1, 0.2, and 0.3, respectively. Figures 3(d)–3(f) are the corresponding polarization ellipse states in the focal plane. The results indicate that the splitting distance between the two C-points increases gradually with an increased β, and its spatial polarization chirality stays consistent. The red dots in Fig. 3(g) plot the relationship between the splitting distances and offset index β∈ [0, 0.5], and an excellent linear fitting is achieved. This provides an additional degree of freedom to precisely modulate the split distance of the C-points. Moreover, the focusing degree determines the size of the focal field, as well as the spatial distribution of the polarization status. Figure 3(g) shows the dependence of splitting distances on β under different NAs. It suggests that the NA does not impact the linear association between them but exerts a discernible influence on the slope of the fitting line. For a larger NA, a fixed β leads to a compressed splitting interval, providing an improved precision for modulation. Consequently, it is feasible to position the C-point singularities at any desired location in the focus field within a certain range by this method.

For the off-axial scenario, the localized general elliptical polarization possesses spin properties. Many studies have demonstrated that when the symmetry of a tightly focused beam is disrupted, an optical spin Hall effect occurs in the focused field [17,20]. Due to the optical spin Hall effect, the focus field is divided into two parts perpendicular to the split orientation. Moreover, the polarization ellipses hold different spin angular momenta (SAM). The SAM density plays a crucial role in studying the mechanisms of interaction between photons and matters [24]. To analyze the SAM density and its associated optical spin Hall effect, the local SAM density is determined [25]: S=Im[E*×E]E*·E,(9)where E is the electric field at the focal field. Figure 4(a) shows the local SAM density (Sz) of the focus field under various β, where the two parts hold opposite nature. The positive and negative signs give chirality of the spin status. For a complete circular polarization at the C-point, the absolute value of longitudinal spin Sz reaches the maximum of 1. Meanwhile, as it is linear polarization along the middle dividing line, there exists no spin property and thus holds a zero Sz. The general elliptic polarizations elsewhere, its Sz lies in a general intermediate state. Figure 4(b) shows the variation characteristic of the Sz along the dashed line in Fig. 4(a). As β increases, however, the maximum absolute value of Sz in Fig. 4(b) becomes smaller than 1. This is because when the offset is large, the symmetry of the azimuthal polarization is more broken, and a minor longitudinal component appears in the incident field, resulting in a certain amount of transverse spin in the focal field. Whatever, the transverse spin is very small within the scope of the study in this paper and has few effects on the position of C-points.

Meanwhile, it shows that the Sz undergoes a rapid change within an ultra-short distance in Fig. 4(b), indicating a high spin gradient. The sharp gradient of spin always has enormous potential for application in precision probing [26]. In accordance with Fig. 4(a), the results in Fig. 4(b) verify that value of β can also be used to regulate the spin gradient. With a decreased value of β, the position of the extreme value moves toward the central divided line, signifying an enhanced spin gradient. Taking β=0.1, for instance, the linkage change of Sz from +1 to −1 between the two C-points spans only 0.080λ in Fig. 4(b), and the full width at half-maximum (FWHM) of the spin gradient is only 0.040λ, as shown in Fig. 4(c). For an incident wavelength of 532 nm, it signifies that the polarization state changes from right to left circular within ∼42nm, which is far beyond the diffraction limitation [27]. Typically, the distance between the two positive and negative extrema narrows with decreased β, indicating a higher spin gradient accordingly. The presence of such a huge spin gradient is of paramount importance for high-precision sensing and chiral detection applications [28].

To deepen the prospective applications of this polarization singularity in precise sensing, chiral metal nanostructures are employed to sweep the region along the linkage line of two C-points by using a 3D finite-difference time-domain (3D FDTD) method (Lumerical FDTD Solutions 2020 R2.4) [29], corresponding to the yellow line in Fig. 4(a). Typically, a gold nanohelix with three spiral cycles is selected as the chiral sample, with a width of 30 nm, length of 70 nm, and wire diameter of 10 nm. For comparison of the specific response of spin distribution around the C-points to chiral structures, an achiral gold nanosphere (radius of 50 nm) is employed within the same scheme. The far-field intensity distribution of the scattering field, which is easy to be detected, is retrieved. These far-field scattering results are obtainable through a Fraunhofer diffraction of the near-field signals, as shown in Fig. 5(a). It enables chiral recognition at the microstructural level, and facilitates high-precision displacement sensing.

The left column in Fig. 5(a) depicts the schematic of the nanostructures, while the right panels show the far-field intensity distribution of the scattered signal, when the nanostructure is positioned at focal plane, and their coordinates in the x- and y-axes are 0 and 0.075λ, respectively. It indicates that the far-field intensity distribution of the scattering field shapes as a notched ring, both for chiral helix structures and gold particles. When sweeping the gold helix, the notches exhibit evident rotation along the ring profile. This is mainly because, in the sweeping process, spin characteristic of the field interacting with the structure changes gradually. Indeed, for a certain field, the glide keeps the same direction regardless of the chirality of the structures. This glide behavior is attributed to the inherent chirality and complex interactions between the spin field and chiral structure, highlighting the intricate nature of light–matter interactions at the nanoscale.

As the scattered field carries specific spin characteristics, it thus leads to the variation in scattering angles and intensity patterns in the far field. Remarkably, the far-field scattering reveals distinct responses for both the chiral and achiral structures. To quantitatively analyze the changes of the scattering intensity field in the sweeping process, a split detection approach is employed here. By dividing the far-field intensity map into four quadrants A, B, C, and D, as shown in Fig. 5(a), a quality index Q is defined as (IB+IC)/(IA+ID) to quantify the scattering intensity field distribution. Through simulations we find that the intensity of the far scattered field is typically the 10−6 order of the incident light. When the incident field power is 1 mW, the scattered field distribution achieves about 10−9W. As has been demonstrated, a lower β leads to a sharper reversal Sz, and the range with linear spin density gradient narrows accordingly. Consequently, the sweeping range is more limited for a smaller value of β.

For β=0.1, the sweeping ranges from −0.0375λ to 0.0375λ (e.g., −20nm to +20nm for λ=532nm) with a step size of 4 nm, as shown in Fig. 5(b). Similarly, it expands from −0.094λ to +0.094λ with a larger step size of 10 nm for β=0.3 in Fig. 5(c). Figures 5(b) and 5(c) plot the Q values of the scattering fields by the three structures, respectively. Ideal linear fitting of Q index and the structure displacement is achieved for each group of data, holding similar variation tendency. The nanosphere, as a symmetric structure, results in a less pronounced response to chirality, so its Q values show a slow and gentle change around 1.0 in the sweeping process. This variation is induced by the chirality reversal of the localized spin states in the optical field. In contrast, the slope of the fitted line for chiral structures is apparently larger. It signifies that this exquisite spin structure associated with C-points is suitable and promising for high-precision sensing of chiral structures. It should be noted that, although a helix with opposite chirality can exhibit significantly different Q and is enlarged for a smaller β, both of their fitted lines exhibit similar slopes. Moreover, Fig. 5 shows only the results of scanning along the y-axis (x=0) at the focal plane. Additional simulations in Ref. [34] verify that the method still works when the helix has a small deviation along the z-axis or the x-axis. The main reason is that the spin density distribution along the y-axis within the focus region possesses good continuity, and thus a good linear relationship between the helix position and the Q value can be maintained.

The Q index has a huge advantage in precise sensing, as it shows the dependence of iota displacement on the far-field intensity distribution. Meanwhile, the differential form of Q index can significantly reduce the error caused by random noise. In exact experiments, the calculated Q index depends on the detection accuracy of the scattering field intensity. Taking the right-chirality helix sweeping result at β=0.3 as an example, there is a significant change of 0.507 within the range of 100 nm, equivalent to an average change of 0.00507 for Q index per nanometer. The accuracy of the Q comes from the measured light intensity. For photoelectric detection technology, the smallest change in light intensity that can be detected is determined by the noise equivalent power (NEP), and picowatt (10−12W) NEP has become possible for current photodetectors [30–32]. Considering the realistic experimental conditions, the displacement detection precision is promising up to a sub-nanometric level with advanced photoelectric detection technology [33]. This pivotal finding validates the immense potential in customizing polarization singularity for advancements in the realm of optical sensing.

In conclusion, we proposed a method to accurately control the position of split C-point polarization singularities at the focal plane by off-axis shifting the APB. Here, the relative offset index β leads to the splitting of the original V-point into two C-points, due to the optical spin Hall effect in the focal plane. The C-point pair constitutes a vector structured field with special anisotropic polarization distribution. Excitingly, the offset index determines the splitting distance between the two C-points, which can be manipulated at a deep sub-wavelength scale. In a broad sense, the offset method is also available for other vector-structured beams that contain V-points, to generate flexibly controllable C-points. As the C-points possess opposite chiral spin state, the spin density sharply reverses along their link-line. This signifies a high linear spin gradient within a narrow range, offering possibilities for high-precision displacement sensing. Therefore, its far-field scattering scheme is implemented by sweeping gold nanohelices and nanoparticles along the link-line. A defined Q index of these scattering fields indicates highly linear dependence on the position of the target in such singularity optical field, and it is more sensitive to chiral nanostructures. It has also proved that the good linear relationship works within the focus region. With the existing measurement accuracy for far-field detection, a sub-nanometric precision for displacement sensing is achievable by combining tailored polarization singularities with chiral structures. Thus, it provides an excellent alternative for applications requiring high precision, as well as opportunities for studying phenomena at nanoscales.