Flexible manipulation of optical angular momentum using a spherical wavefront

Shuaichao Li; Hongjie Huang; Yili Lu; Pei Wang; Yonghua Lu

doi:10.3788/COL202523.093601

1. Introduction

Light carries two types of optical angular momentum (AM): spin angular momentum (SAM) and orbital angular momentum (OAM). SAM is linked to circular polarization, valued at $σ ℏ$ ( $ℏ$ , reduced Planck’s constant), where $σ = 1$ corresponds to left-handed circular polarization (LCP) and $σ = - 1$ to right-handed circular polarization (RCP). OAM arises from the wavefront’s helical phase winding, valued at $ℓ ℏ$ , where $ℓ$ is the topological charge that represents the number of $2 π$ phase winding^[1]. Counterclockwise winding corresponds to $ℓ > 0$ , while clockwise winding corresponds to $ℓ < 0.$ Optical AM is applied in areas such as particle trapping^[2,3], microscopy^[4,5], holography^[6,7], and secure information technologies^[8–11]. SAM is routinely controlled using polarizers or waveplates, while OAM is manipulated via phase plates^[12] or holographic gratings^[13]. However, these methods are often limited by narrow spectral ranges and bulky setups, making flexible AM manipulation a challenge.

A promising solution lies in the intrinsic coupling between SAM and OAM, governed by the conservation of total AM, $L ℏ$ , where $L = σ + ℓ$ ^[14]. This coupling is realized through processes like tight focusing^[15], particle scattering^[16], and diffraction^[17], where changes in SAM are compensated for by modifications in OAM. For example, converting LCP ( $σ = 1$ ) to RCP ( $σ = - 1$ ) induces an OAM change of $+ 2 ℏ$ ^[18]; decomposing radially polarized light into RCP with SAM of $- ℏ$ results in an OAM change from zero to $+ ℏ$ ^[19]. Such coupling enhances the flexibility of AM manipulation. Additionally, the geometric topological charge ( $g$ ) of structures like plasmonic spirals^[20–23] and meta-surfaces^[24–26] can further influence optical AM. Here, the total AM of output light depends on both the incident light’s AM and the structure’s geometric topological charge, significantly expanding the information carried by light. It is noted that all the above studies on optical spin-orbital interactions employ plane wavefronts, with AM adjustments achieved by altering structural parameters or polarization states. Investigating AM manipulation under spherical wavefront illumination could offer new possibilities. Convergent or divergent spherical wavefronts, easily achieved by defocusing a light beam, provide a promising avenue for controlling SAM and OAM. However, this approach remains largely unexplored.

In this Letter, we investigate the optical spin and orbital interaction when a spiral nanostructure is illuminated by a spherical wavefront and propose a novel approach to flexibly control the optical SAM and OAM states. The spiral structure consisting of the Λ-shaped rectangular aperture pair is elaborately designed. The scattering and diffraction of a focusing radially polarized incident light by the spiral structure is theoretically investigated and numerically simulated. The optical spin and orbital interaction during the scattering and diffraction is particularly investigated. Different SAM and OAM are achieved for convergent and divergent spherical wavefront illumination. The flexible manipulation of both SAM and OAM states is experimentally demonstrated, which is potentially applicable in lithography, particle manipulation, and information encryption.

2. Design of the Λ-shaped Apertures

Figure 1 schematically illustrates the optical field manipulation achieved by the spiral nanostructure under spherical wavefront illumination. As shown in Fig. 1(a), the incident light is converged by a lens, and the spiral nanoapertures are illuminated by a spherical wavefront when the sample is shifted away from the focal plane of the lens. The inset displays a unit of paired Λ-shaped apertures, which are etched into $h = 200 nm$ -thick silver film. Considering both the polarization and the spherical wavefront, the illuminating optical field is described as^[27] $| E_{in} ⟩ = A e^{i δ} = A e^{i \frac{n_{g} π}{d λ} r^{2}},$ (1)where $d$ is the defocus position determining the curvature of the spherical wavefront ( $d > 0$ , diverging spherical wavefront; $d < 0$ , converging spherical wavefront), $r$ is the lateral distance from the beam center, and $n_{g}$ is the refractive index of the incident medium. $λ$ is the free-space wavelength of the incident light, and $A$ is the Jones vector describing the polarization of the incident light.

Figure 1.(a) Schematic showing the phase difference introduced at the defocus position where d > 0. A plane wave is transformed to a spherical wave after being focused by a lens; the thickness of the silver film is h = 200 nm, and the inset displays a unit of paired Λ-shaped apertures. (b) Details of the Λ-shaped apertures on sliver film with parameters l = 150 nm, w = 60 nm, and θ = π/4; green arrows indicate the polarization direction of the scattered light.

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Figure 1(b) provides a detailed view of the paired Λ-shaped apertures. The apertures, with a length of $l = 150 nm$ , a width of $w = 60 nm$ , and an orientation angle $θ = π / 4$ , are perpendicular to each other. The scattering field radiated from each rectangular aperture can be regarded as equivalent to the radiation from a dipole^[28,29] oriented along its short axis (green arrows in the figure), more details are provided in the Supplement 1, Sec. 1.1. By adjusting the spacing between the two apertures, a relative phase difference $Δ δ = δ_{2} - δ_{1}$ can be introduced between the two dipoles. For a diverging spherical wavefront ( $d > 0$ ), $Δ δ = π / 2$ , resulting in LCP scattered light. Conversely, for a converging spherical wavefront ( $d < 0$ ), $Δ δ = - π / 2$ , resulting in RCP scattered light. When the paired Λ-shaped apertures are arranged in a varying spiral configuration and illuminated by focusing radially polarized light, each Λ-shaped aperture scatters circularly polarized light. Due to the varying positions of those apertures, the initial phase of the scattered light from each pair of apertures differs, specifically, when the spiral structure parameters—inner curve radius, orientation angle, and spacing—vary as ${\begin{matrix} r_{1}^{2} = r_{0}^{2} + m \frac{φ λ | d |}{n_{g} π} \\ s = \sqrt{r_{1}^{2} + \frac{| d | λ}{2 n_{g}}} - r_{1} \\ θ = π / 4 + φ \end{matrix},$ (2)where $r_{0}$ is the starting radius, $φ$ is the azimuth angle (positive in the counterclockwise direction), and $m$ is the arm number (with $m > 0$ corresponding to a counterclockwise spiral). At different defocus positions, the output scattered light carrying different spin SAM and OAM can be generated, which is $| E_{out} ⟩ = {\begin{matrix} J_{| m - 1 |} e^{i (m - 1) φ_{f}} | L ⟩, & d > 0 \\ J_{| 1 - m |} e^{i (1 - m) φ_{f}} | R ⟩ & d < 0 \end{matrix},$ (3)where $J$ denotes the Bessel function, $φ_{f}$ is the azimuthal angle in the observation plane, and $| L ⟩$ and $| R ⟩$ represent the LCP light and RCP light. Details are provided in the Supplement 1, Sec. 1.2.

Table 1 summarizes the AM states of the scattered light derived directly from Eq. (3). The SAM changes sign with the defocus position: it is $σ_{out} = + 1$ for diverging spherical wavefronts ( $d > 0$ ) and $σ_{out} = - 1$ for converging spherical wavefronts ( $d < 0$ ). The OAM depends on the arm number $m$ of the spiral structure and the defocus position, with $ℓ_{out} = m - 1$ for $d > 0$ and $ℓ_{out} = 1 - m$ for $d < 0$ . The total AM ( $L_{out}$ ) equals to the geometrical topological charge of the spiral, defined as $g = m | d | / d$ . Specifically, $g = m$ for $d > 0$ and $g = - m$ for $d < 0$ , reflecting the reversal of the phase accumulation direction along the spiral.

Table 1. Scattered Light’s AM Illuminated by Focusing Radially Polarized Light^a

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Table 1. Scattered Light’s AM Illuminated by Focusing Radially Polarized Light^a

Arm number Defocus σ_out ℓ_out L_out g
m = +1 d > 0 +1 0 +1 +1
d < 0 −1 0 −1 −1
m = 0 d > 0 +1 −1 0 0
d < 0 −1 +1 0 0
m = −1 d > 0 +1 −2 −1 −1
d < 0 −1 +2 +1 +1

To quantitatively evaluate the SAM of the scattered light, we define the purity of circular polarization (PCP) of the scattered light from the Λ-shaped apertures, with $n_{g} = 1.46$ . The PCP is defined as $PCP = \frac{I_{LCP} (k_{x}, k_{y}) - I_{RCP} (k_{x}, k_{y})}{I_{LCP} (k_{x}, k_{y}) + I_{RCP} (k_{x}, k_{y})},$ (4)where $I_{LCP} and I_{RCP}$ represent the intensities of scattered light’s LCP and RCP components, and $k_{x} and k_{y}$ are the transverse components of the wave vector. It is evident that the $PCP = 1$ corresponds to LCP ( $σ_{out} = 1$ ), while $PCP = - 1$ corresponds to RCP ( $σ_{out} = - 1$ ). The PCP is computed using the finite-difference time-domain (FDTD) method, taking the extremum values within the range $- 0.05 \leq (k_{x}^{2} + k_{y}^{2}) / k_{0}^{2} \leq 0.05$ , to minimize the influence of FDTD mesh refinement and wavefront distortions. The PCPs of the scattered light at $d > 0$ and $d < 0$ defocus positions are shown in Figs. 2(a) and 2(b), where the corresponding system is indicated in Fig. 1(a). Here, the transverse distance $r_{1} = 2000 nm$ and the apertures spacing $s = 250 nm$ are fixed. The PCP reaches a maximum extreme value of $+ 1$ near defocus position $d = + 5000 nm$ , and a minimum extreme value of $- 1$ near $d = - 5000 nm$ . The black dashed lines indicate the defocus positions predicted by Eq. (2). This discrepancy between the theoretical prediction ( $d = \pm 5800 nm$ ) and the numerical results ( $d = \pm 5000 nm$ ) arises from the simplified assumption of an ideal spherical wave in the theoretical model. In practice, the focused field deviates due to factors such as lens aberrations and higher-order phase corrections (e.g., $r^{4}$ terms). Nevertheless, PCP values remain very high ( $PCP = \pm 0.98$ ) at $d = \pm 5800 nm$ , demonstrating that the theoretical framework provides reliable guidance. After determining the optimal defocusing positions, we further examined the effect of varying the transverse distance $r_{1}$ and aperture spacing $s$ on the polarization of the scattered light; the variation of $r_{1}$ is taken from the values on the horizontal axis and spacing $s$ varies according to Eq. (2). The defocus positions are fixed at $d = + 5000 nm$ and $d = - 5000 nm$ , respectively. The corresponding results are shown in Figs. 2(c) and 2(d). At those defocusing positions, the scattered light maintains high circular polarization purity ( $| PCP | \geq 0.9$ ), indicating circularly polarized light can be preserved when the apertures are arranged in a spiral form by Eq. (2).

Figure 2.(a) Variation of the scattered light’s PCP on d > 0 defocus positions for r₁ = 2000 nm and s = 250 nm. (b) Variation of the scattered light’s PCP on d < 0 defocus positions for r₁ = 2000 nm and s = 250 nm. The black dashed lines indicate the defocus positions predicted by Eq. (2). (c) Variation of the scattered light’s PCP with r₁ and s at the defocus position of d = 5000 nm. (d) Variation of the scattered light’s PCP with r₁ and s at the defocus position of d = −5000 nm. The PCP is computed by the FDTD method, taking the extremum values within the range −0.05 ≤ (k_x² + k_y²) / k₀² ≤ 0.05.

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Using the designed apertures, we configure the spirals to generate scattered light with specific SAM and OAM states. Figure 3(a) shows three spiral curves with arm number $m = 1$ , 0, and $- 1$ , and a starting radius $r_{0} = 2000 nm$ . The apertures vary according to Eq. (2). The performance of the structures at different defocus positions of $d = + 5000 nm$ and $d = - 5000 nm$ is investigated using FDTD simulations. The scattered light’s intensity, PCP, and phase distribution are shown in Figs. 3(b)–3(d), where the phase distribution corresponds to the $E_{x}$ component, and the numerical aperture (NA) of the collection objective is set as NA = 0.2. For the spiral curve arrangement with $m = 1$ , the scattered light’s intensity exhibits a form of a solid distribution at a defocused position of $d = + 5000 nm$ , with polarization of LCP and no helical phase winding ( $σ_{out} = 1, ℓ_{out} = 0$ ). Upon switching the defocused position to $d = - 5000 nm$ , the polarization state is reversed and there is still no helical phase winding ( $σ_{out} = - 1, ℓ_{out} = 0$ ). For $m = 0$ , the scattered light’s intensity exhibits a donut-shaped distribution at the defocused position of $d = + 5000 nm$ , with the polarization of LCP and a clockwise helical phase winding ( $σ_{out} = 1, ℓ_{out} = - 1$ ). After switching the defocused position to $d = - 5000 nm$ , the intensity distribution remains, but the polarization state switches to RCP, and the helical phase winding becomes counterclockwise ( $σ_{out} = - 1, ℓ_{out} = 1$ ). For $m = - 1$ , the empty center of the intensity distribution is further expanded. At the defocused position of $d = + 5000 nm$ , the scattered light is locked to LCP with two clockwise helical phase windings ( $σ_{out} = 1, ℓ_{out} = - 2$ ). After switching its defocused position to $d = - 5000 nm$ , the polarization switches to RCP with two counterclockwise helical phase windings ( $σ_{out} = - 1, ℓ_{out} = 2$ ). Meanwhile, it is noted that at $d > 0 (d < 0)$ defocusing positions, the scattered light does not perfectly correspond to LCP (RCP), as reflected in the PCP diagrams; both LCP and RCP components are present, which can be attributed to imperfect polarization transitions. In Figs. 2(c) and 2(d), we showed that variations in the spacing $s$ and distance $r_{1}$ cause fluctuations in the PCP, manifested as $| PCP | \leq 1$ . Although the target polarization state is still dominant, an orthogonal polarization state is mixed in. After propagating, their intensity distributions become separated; more details are provided in the Supplement 1, Sect. 2. To understand the influence of that imperfect polarization transition, we fix the defocus position and extract the spatial overlap of the RCP and LCP intensities at $k_{y} = 0$ . The results are shown in Fig. 3(e), where the overlap reaches a maximum value of 10% for $m = - 1$ , which can be considered negligible.

Figure 3.(a) Top view of the spiral curve with arm numbers of m = +1, 0, −1 from top to bottom. (b)–(d) Simulated scattered light’s intensity, PCP, and phase distribution at d = +5000 nm and d = −5000 nm defocus positions. The phase distribution corresponds to the E_x component, and the NA of the collection lens is set to 0.2. (e) Simulated RCP and LCP intensity profiles and their spatial overlap at k_y = 0. The incident light is radially polarized, which is required to excite the desired scattering characteristics of the spiral structure.

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3. Experimental Results

Based on the design, we fabricated the corresponding structures. Fabrication details are provided in the Supplement 1, Sec. 3. The top views of the structures are shown in Fig. 4(a). A spherical protrusion is placed at the center of the structure to calibrate its lateral position relative to the focal point, and it is opaque; the defocus positions are calibrated using a high-precision piezoelectric translation stage (COREMORROW P79.XYZ50S). Figure 4(b) shows the intensity of the scattered light, which exhibits a form of solid distribution for the spiral with $m = 1$ , and a donut-like shape for $m = 0$ . For the spiral with $m = - 1$ , the empty center of the intensity distribution is further expanded, which is consistent with our previous prediction. For the extraction of the LCP and RCP polarization states, the scattered light is passed through a quarter-wave plate and a linear polarizer, projecting it into different circular polarizations. The corresponding PCP maps are shown in Fig. 4(c). For those three structures, LCP dominates at the $d > 0$ defocus position, while the RCP dominates at the $d < 0$ defocus position, in agreement with our previous prediction. To determine the number of OAM topological charge, we perform self-interference of the scattered light. Typically, interference between an OAM beam and a plane wave produces fork-shaped interference fringes, with the number and direction of the forks indicating the sign and magnitude of OAM topological charge^[30]. The corresponding results are shown in Fig. 4(d). For the spiral with $m = 1$ , no fork interference pattern is observed at either $d > 0$ or $d < 0$ defocus positions, indicating OAM topological charge $ℓ_{out} = 0$ . For the spiral with $m = 0$ , one single fork-shaped stripe appears in the interference stripes at both $d > 0$ and $d < 0$ defocused positions, pointing in opposite directions, indicating a sign reversal of the OAM topological charge ( $ℓ_{out} = - 1 \to ℓ_{out} = + 1$ ). For the spiral with $m = - 1$ , two forked stripes appear in the interference stripes at both $d > 0$ and $< 0$ defocused positions, pointing in opposite directions, indicating a sign reversal of the OAM topological charge ( $ℓ_{out} = - 2 \to ℓ_{out} = + 2$ ), which is also in good agreement with our previous predictions. Details of the experimental setup are provided in the Supplement 1, Sec. 4.

Figure 4.(a) Scanning electron microscope (SEM) images of the spiral curve with arm numbers of m = +1, 0, and −1 from top to bottom. (b), (c) Measured scattered light’s intensity and PCP distribution; NA = 0.2. (d) Self-interference pattern of the scattered light.

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4. Conclusion

In conclusion, we have demonstrated an approach for the flexible manipulation of optical SAM and OAM under illumination of a spherical wavefront. The sign of SAM can be easily reversed by switching between convergent and divergent spherical wavefronts of the incident spherical wave, while the OAM is correspondingly modified. Moreover, the optical OAM can be further modulated through the spiral’s geometric topological charge because of the conservation of total angular momentum. Due to the ultra-thin and compact nature of the spiral nanostructure, we foresee its integration into chip-scale applications, which could benefit fields such as lithography, particle manipulation, and information encryption.

Category: Nanophotonics, Metamaterials, and Plasmonics

Received: Mar. 7, 2025

Accepted: Apr. 28, 2025

Published Online: Aug. 13, 2025

The Author Email: Yonghua Lu (yhlu@ustc.edu.cn)

DOI:10.3788/COL202523.093601

CSTR:32184.14.COL202523.093601

Table 1. Scattered Light’s AM Illuminated by Focusing Radially Polarized Lighta

Table 1. Scattered Light’s AM Illuminated by Focusing Radially Polarized Lighta

Table 1. Scattered Light’s AM Illuminated by Focusing Radially Polarized Light^a

Table 1. Scattered Light’s AM Illuminated by Focusing Radially Polarized Light^a