High-precision positioning of nanoantennas is conducive to accurate excitation and integration of nanoantennas and is of great significance in the fields of surface-enhanced Raman spectroscopy [1–3], optical superresolution [4], optical switching [5], biosensing [6,7], enhancement of light-emitting diodes [8], and quantum technology [9–11]. Since single-element antennas are subwavelength nanoparticles, whose size is beyond the diffraction limit, it is quite challenging to precisely position and excite these antennas using optical approaches [12,13]. Previous works have utilized focused cylindrical vector beams for positioning nanoantennas [14–22]. The swift change in polarization of the focusing field near the vector point (V-point) [23] in the transverse electric field alters the far-field scattering of the nanoantenna. By analyzing the scattering field, precise positioning of the nanoantenna near the V-point is achieved. However, the transverse electric field near the V-point acts as a dark field, which limits the signal-to-noise ratio of this method. Consequently, the current limit of accuracy is challenging to improve beyond 100 pm. Another advanced technique involves placing antennas on a metal film [24]. This approach improves the measurement of signal-to-noise ratio by profiting the existence of surface plasmon polaritons to localize and enhance the light field. However, this method necessitates the presence of a metal film, making it unsuitable for conventional antenna structures. Additionally, a V-point refers to a unique line within a three-dimensional space. In the technique described earlier, this unique line aligns with the longitudinal direction. Within this specific direction, the light field’s polarization remains relatively constant, inhibiting the ability to accurately detect the nanoantenna’s position along its length.

In this research, a liquid crystal plate is employed to manipulate the relative phase and position of left-handed and right-handed circularly polarized light fields. This control enables the creation of an interfered light field, whose characteristics are adjustable through the liquid crystal plate, allowing for spatial superposition within the wavelength scale in a tightly focused field. By choosing appropriate parameters, a novel light field can be designed for accurate positioning of nanoantennas, as shown in Fig. 1. This innovative light field is characterized by the gradual reversal of its polarized chiral structure along a predefined axis, while maintaining uniform intensity throughout the entire path, including along the longitudinal direction, at either the wavelength or subwavelength level. As the spherical nanoantenna follows the predefined trajectory, the polarization of the light field undergoes rapid changes. This, in turn, causes the polarization component of the scattered light field to adjust correspondingly. Based on this principle, the position of the nanoantenna can be identified with an accuracy close to 20 pm. Given that the trajectory of the polarization transformation in the excitation field can be specified, it becomes possible to measure not just the transverse but also the longitudinal position of the nanoantenna. This capability is advantageous for stacking multiple layers of antenna arrays along the longitudinal direction. Moreover, more precise positioning of the nanoantennas facilitates improved excitation, which in turn enhances the intensity and selectivity of the signal. This characteristic is especially significant for applications such as biosensors and the enhancement of Raman scattering signals.

To obtain the excitation light field whose polarization transforms along the designated three-dimensional trajectory in subwavelength scale, tight focus technology is used to compress the focus spot size. In the framework of Richards–Wolf theory [25], we design three transformation factors (α, β, λ) to control the position of the focal spot in the x-, y-, and z-directions, respectively. The complex expression of the transformation factor can be expressed as E=exp(−ikαρcosϕf)exp(−ikβρsinϕf)⋅exp(−ikγf2−ρ2f),(1)where (ρ, ϕ) denote the radial and angular coordinates on the incident plane, k stands for the wavenumber, and f indicates the focal length (for the detailed modulation process, see Section 4 and Appendix A.1).

Then loading the conjugated transformation factors to the left-handed and right-handed circular polarization components can realize the separation of the focal spots of the two components. The separated distance affects the polarization of the light field. If the separated distance is too small, the light field is approximately linearly polarized, and if the separated distance is too large, the light spot is completely separated. As the separated distance of the focal spots of the two components reaches about half-wavelength to several wavelengths, the structure of the excitation light field whose polarization changes rapidly from right-handed to left-handed circular polarization components is generated.

In the experiment, a customized liquid crystal chip loads the conjugated transformation factors in left-handed and right-handed circular polarization components. Through the high numerical aperture (NA) objective, the excitation light field whose polarization transformation from right-handed to left-handed circular polarization components is generated (for details, see Appendix A.2).

For the longitudinal positioning of nanoantennas, the polarization components of the excitation light field are separated in the z-direction by assigning values to parameter γ. In this case, the intensity distributions of the right-handed and left-handed circular polarization components of the excitation light field in the x–z plane are shown in Figs. 2(a) and 2(b). Notably, the focal positions of these two light field components exhibit a separation along the z-axis. For transformational factors γ=0.5λ and λ, the light spots retained a distinctly high Gaussian waveform. The peak separation between the left and right circularly polarized components measured 1.1λ and 2.0λ, respectively. These findings closely align with the theoretical prediction of 2γ, as shown in Figs. 2(c) and 2(d).

Then, the above field was used to excite the spherical gold nanoantenna with a diameter of 200 nm. The intensity distributions of the left- and right-handed circular polarization components of the scattered light field in the far field calculated by finite-difference time-domain (FDTD) algorithm are shown in Figs. 3(a)–3(c) (see Appendix A.3). The polarization component of the scattering field from the gold nanoantenna varies as the nanoantenna moves along the z-axis. To quantify these changes in the scattering field, we introduced a component factor “g” as a metric, defined as follows: g=IL−IRIL+IR.(2)

In this context, IL and IR denote the total intensities of the left-handed and right-handed circular polarization components of the scattered field, respectively. Different from the description of molecular properties in the previous works [26], the asymmetric factor in this work is employed to characterize the polarization composition of the scattered light field, thus eliminating the need for coefficients.

The experimental results, obtained with nanoantenna movement step sizes of 5 nm and 1 nm, are displayed in Figs. 3(d) and 3(e). As the gold nanoantenna moves, the component factor of the backscattering exhibits a linear change, resulting in a measurement standard error of approximately 20 pm (for error analysis, see Appendix A.4). These findings suggest that the spherical single-element nanoantennas can be accurately distinguished in the z-direction, which also provides a reference for identifying more complex antenna structures.

The potential of the above technique is not only to identify the nanoantenna position along the z-axis, but it also can identify the nanoantenna position along the discretionarily designated three-dimensional trajectory. By adjusting α, β, and γ simultaneously, the polarization transformation direction of the excitation light field can point in an arbitrary direction in three-dimensional space (see Visualization 1). Taking the transformation in the x-direction as an example, the detailed demonstration of the structure of the light field and the excitation of the nanoantenna is also studied. For α=0.36λ, the intensity and the ellipse degree of the excitation light field are shown in Figs. 4(a) and 4(b) (for the experimental demonstration, see Appendix A.2; for the calculation of ellipse degree, see Appendix A.5). Along the x-axis, the intensity is approximately constant, and the ellipse degree is changed from negative to positive, which means that the polarization chirality is reversed. Notably, the polarization transformation along the x-axis occurs more rapidly compared to the z-axis, attributable to the smaller size of the light spot in the transverse direction. The current excitation light field is selected to excite the nanoantenna, and the scattered field will become very sensitive to the position of the nanoantenna in the x-direction. Moreover, due to the hot spot distribution of the excitation light field in the process of polarization transformation, the signal-to-noise ratio of the measurement is greatly improved. The nanoantenna is controlled to move with steps of 1 nm and 5 nm, the component factor of the scattered field is used for measurement, and the measurement standard error approaches 20 pm. This measurement standard error is not very different from the z-direction measurement, mainly due to the impact of environmental noise, such as the accuracy of the piezoelectric displacement table. In addition, the measurement along the y-direction is the same as the situation along the x-direction, so we will not prove it.

In this paper, the high-precision positioning of the spherical gold nanoantenna is realized by designing a novel excitation light field whose polarization transforms from right-hand to left-hand along the designated trajectory at the wavelength or subwavelength scale. Compared with previous similar technologies using the singularity light field to excite nanoantennas, this technique can not only measure the transverse position of the nanoantenna, but also realize the longitudinal measurement, which is of significance for the precise excitation and integration of nanoantennas. Furthermore, since the excitation light field of the measurement region is always in the hot spot region, the measurement of the nanoantenna has a higher signal-to-noise ratio, resulting in higher measurement accuracy (approaching 20 pm), and may be suitable for the measurement of smaller antennas.

The excitation light field is generated through the superposition of tightly focused left- and right-handed circularly polarized light at very small scales, as shown in Figs. 5(a)–5(c). Mathematically, the incident light in the tight focusing system can be expressed as Ein(ρ)=[ERe→RELe→L]=[Ee→RE′e→L],(3)where Ein represents the incident light, comprising two key components, ER and EL, which correspond to the right-handed and left-handed circular polarization, respectively. ρ denotes the radial coordinate on the incident plane, and E is the complex expression of the transformation factor, described as Eq. (1).

In the condition of tightly focusing, the complex expression of the transformation factor can be simplified as {exp(−ikαρcosϕf)=exp(−ikαcosϕsinθ)exp(−ikβρsinϕf)=exp(−ikβsinϕsinθ)exp(−ikγf2−ρ2f)=exp(−ikγcosθ).(4)

Equation (4) can be further integrated with the formula for tight focusing [25,27], resulting in exp{iksinθ[(x−α)cosϕ+(y−β)sinϕ]+ik(z−γ)cosθ} (for details, see Appendix A.1). In this context, the transformational factors α, β, and γ correspond to the displacements of the light field along the x-, y-, and z-axes at the focus point, respectively. The left- and right-handed circularly polarized light each has a conjugate modulation term E′, which leads to a counteracting displacement for each polarization component within the focus field. This theoretical approach facilitates the separation of left- and right-handed circular polarization components at a fine scale and sets the stage for their superposition within wavelength-scale light fields.

Given that the left- and right-handed circularly polarized lights are orthogonal, the intensity distribution of the excitation light field is equivalent to the cumulative intensity of these two polarization components. Therefore, when the transformational factors are sufficiently minimized, the focal field retains a singular, focused spot distribution. In this scenario, we generate the excitation light field, characterized by a rapid polarization transition along a linear trajectory that connects the left- and right-handed circular polarization components.

To visually represent the polarization transformation, the polarized ellipse degree at each point is determined using the following formula: ε=atan(A2A1)sign(Δψ),(5)where A1 and A2 are the major and minor axes of the polarization ellipse, while Δψ denotes the phase difference between the x- and y-polarization components of the light field, and sign(·) is used to determine the sign of the variable. These physical quantities can be derived from the electric field expression of the excitation light field (for the derivation details, see Appendix A.5). It can be obtained from Eq. (5) that ε∈[−π/4,π/4], and for ε>0, the light field is right-handed polarization; for ε<0, the light field is left-handed polarization. ε=±π/4 represents full circular polarization.

Based on the aforementioned theory and considering a system with an NA of 0.8, the characteristics of the excitation light field, when the direction of polarization transformation is along the z-axis, are depicted in the second row of Fig. 5. Along the z-axis, the excitation light field’s elliptical degree gradually inverts while maintaining a single hot spot-type intensity distribution.

In Fig. 5(f), the effect of varying transformational factors on the elliptical degree of polarization in the z-direction is quantitatively presented. The elliptical degree transitions from positive to negative across the focus point, moving from −0.44λ to 0.44λ along the z-axis. Contrary to what might be expected, the rate of change in the elliptical degree of the excitation light field increases as the separation between the left- and right-handed circularly polarized light grows. This is due to the mutual cancellation effect of the right- and left-handed circular polarizations when they overlap. A smaller separation distance leads to a more pronounced cancellation effect between the two polarization components. In the instance of perfect overlap, the left- and right-handed components completely negate each other, resulting in linear polarization. As such, larger transformational factors facilitate quicker polarization transformations. However, there is a limit imposed by the intensity distribution, which prevents indefinite increases in transformational factors. Excessively large factors cause the intensity to split into two spots, disrupting the three-dimensional structure of the excitation light field. In our analysis, the z-polarization component is disregarded due to the dark core structure in the area of interest, a result of the spin–orbit transformation [28].