Optical field manipulation, a fundamental branch of modern optical research, focuses on the precise control of light propagation characteristics, including phase, amplitude, and polarization states^[1]. This technology has been applied extensively in various fields. In optical communications^[2], optical field manipulation significantly enhances bandwidth and transmission efficiency, driving advancements in information technology to meet the ever-growing data demands^[3]. In microscopy and imaging systems^[4], the manipulation of optical fields enhances resolution and contrast, driving progress in high-precision imaging technologies^[5]. Furthermore, the precise control of optical fields enables micromachining of materials and laser fabrication, which have substantial applications in the manufacturing sector^[6-7]. Optical field manipulation can also enhance the quality and characteristics of laser beams by enabling the generation of ultrashort pulse lasers^[8], thereby fostering innovation in laser technology^[9]. In terms of application prospects, research on optical field manipulation is evolving toward intelligent optical systems^[10], focusing on the development of adaptive optical devices that achieve real-time manipulation of optical fields, thus enhancing the flexibility and responsiveness of optical systems^[11-12]. In the field of nano-optics^[13-15], researchers are exploring the fine control of optical fields at the nanoscale for applications in optical information processing and sensor technologies, with the aim of enhancing system performance^[16-17]. The integration of optical field manipulation with quantum optical technology facilitates the preparation and detection of quantum states, thereby advancing the field of quantum information science^[18-19]. Moreover, the development of advanced biological imaging technologies enables non-invasive detection and diagnostics^[20-21], ultimately improving the precision and efficiency of medical imaging^[22-24]. In summary, advancements in optical field manipulation provide new directions for modern optical research as well as open up broader possibilities in related application fields.

In recent years, the arbitrary azimuthal manipulation of optical fields has emerged as a crucial technique in optics, demonstrating immense potential for applications involving complex optical fields and higher-order optical fields. Arbitrary directional manipulation of optical fields involves not only the adjustment of traditional optical characteristics but also holistic control of the spatial distribution of light, phase structures^[25], and their interactions with matter^[26]. Spatial light modulators (SLMs) are extensively employed in optical field manipulation. By designing optical elements, SLMs enable real-time control over the phase and amplitude of light, facilitating precise adjustments to the divergence angle and transmission modes of optical fields^[27-28]. For instance, liquid-crystal SLMs can generate various beam structures, such as vortices and honeycomb beams, thereby realizing multi-directional control. The optical phase array technology enables the arbitrary reconstruction of optical wavefronts^[29]. The desired optical field could be generated at any angle by adjusting the phase of each element in the array. The integration of this technology offers new design concepts for miniaturized optical devices, such as intelligent LiDAR and optical communication equipment^[30]. Computational optics further enhance the complex optical field structures through numerical calculations and algorithmic adjustments^[31-32]. Techniques such as digital holography theoretically enable arbitrary directional manipulation and offer significant applications in optical imaging and optic capturing. Furthermore, nonlinear optical effects, such as four-wave mixing and self-focusing, enable researchers to achieve arbitrary directional enhancement and control of optical fields^[33-34]. These nonlinear effects not only change the direction of light propagation but also modulate the properties of light, providing new avenues for optical field manipulation. The development of smart materials and adaptive optics technologies has enhanced the flexibility and immediacy of optical field manipulation. These technologies can automatically adjust the propagation direction and mode of light in response to changes in the external environment, thereby significantly improving the adaptability and functionality of optical fields. Despite significant advancements in the arbitrary azimuthal manipulation of optical fields, several challenges remain. The key issues include achieving more efficient spatial control over optical fields and integrating optical field manipulation technologies into compact devices to improve their practicality.

In this study, we propose dual-spiral arrays composed of two interleaved spiral arrays to achieve modulations of azimuthal optical fields. These arrays differed in their initial radii, whereas all other parameters remained unchanged. By precisely tuning the difference in radius between the two spiral structures, we achieved continuous control over the azimuthal field distribution. To gain deeper insight into the azimuthal distribution of the dual-spiral array structure, we performed numerical simulations. Through systematic analysis, we examined the relationship between the structural characteristics of the dual-spiral arrays and the resulting azimuthal optical field distribution. The simulation results demonstrate that, by accurately controlling the radius difference between the two arrays, the azimuth of the optical field can be smoothly varied across the entire range from 0 to 2π. Furthermore, we observed a nearly linear correlation between the radius difference and the azimuthal distribution, offering valuable insights for the design and optimization of optical devices based on this structure.

Figure 1(a) (color online) schematically illustrates the modulation model of the dual-spiral arrays. Defined in polar coordinates (r, θ), the dual-spiral arrays can be expressed as

$ \begin{split} &{r_1} = - \left[ {{r_{10}} + \frac{{l\textit{z}\lambda }}{{2{\text{π}} {r_{10}}}}\left( {{\text{π}} - \theta } \right)} \right] \\ &{r_2} = {r_{20}} + \frac{{l\textit{z}\lambda }}{{2{\text{π}} {r_{20}}}}\theta \quad, \end{split} $ (1)

where r₁₀ and r₂₀ are the initial radii, l indicates the topological charge, z denotes the diffraction distance, and λ indicates the wavelength. Owing to the counter rotational structure and mirror-symmetric configuration of the dual-spiral arrays, the spiral r₁ is assigned a negative coordinate accompanied by an azimuthal transformation of π–θ. The modulation of the transmitted optical field is fundamentally governed by the parametric dependence of the spiral radius r variation with respect to the azimuthal angle θ. As the beam passes through this dual-spiral structure, the spirals introduce a specific phase delay in the light perpendicular to the wavefront. Various phase distributions can be obtained by adjusting the topological charges of the spirals. The combination of these distinct phase shifts and amplitude variations collectively formed a helical phase structure. The interaction of the two helical phase fronts from the spirals, along with the modulation effects of the constructive and destructive interference, resulted in an optical field that exhibited two focal spots surrounding a central zero-intensity distribution, as shown in Fig. 1(c).

Figure 1(b) shows a front view of the dual-spiral arrays. These arrays consisted of two interleaved spiral structures with both the inner and outer spirals composed of multiple pinholes. The inner spiral exhibited a right-handed orientation, whereas the outer spiral was left-handed, with the rotational directions demonstrating mirror symmetry. The mirror-symmetric configuration of the dual-spiral arrays facilitated complementary phase modulations within the optical field, resulting in petal-like intensity distributions through the interplay between the constructive and destructive interference effects. The modulation mechanism of the dual-spiral arrays involves precise control over both the phase and amplitude of the light, enabling the modulation of the optical field to achieve specific azimuthal characteristics.

For this operational mode, we set the wavelength to 1550 nm, the propagation distance to 1 m, the initial radius of each spiral array to 5 mm, the radius of pinholes to 30 μm, and the number of pinholes to 100. Once the number of pinholes reached a threshold sufficient to form a uniform spiral, any further increase in the number of pinholes had a negligible effect on the optical field characteristics and phase distribution. The structural parameters of the dual-spiral configuration were strategically optimized to ensure practical feasibility for future experimental implementations. The initial radius of the spiral arrays was chosen to accommodate both the number and size of the pinholes to achieve an optimal spatial distribution within the azimuthal periodicity of the spiral. The pinhole dimensions were determined based on the achievable fabrication precision and tolerance in femtosecond laser micromachining. In addition, the operational wavelength is strategically aligned with standard telecommunications to facilitate future practical applications in optical communication. Keeping the aforementioned parameters fixed, we varied the radius difference and topological charge of the dual spirals to investigate their influence on the optical field characteristics and phase distribution of multiple beams.

Figure 2(a) (color online) shows dual-spiral arrays with varying radial differences between the inner and outer spirals, denoted by the variable ∆r₃ at the center of the figure. Specifically, the radius difference is defined as the arithmetic difference between their respective initial radii, mathematically expressed as ∆r = r₂₀ − r₁₀. The gray arrows on the outer circle indicate the direction of rotation, whereas the numbers within the arrows correspond to the radial differences of the dual-spiral arrays. The radius differences are selected uniformly within the range of 10 μm to 250 μm. As the radius difference increases from (a1) to (a12) in Fig. 2, the inner spiral contracts progressively inward. Initially, in (a1), the inner spiral nearly coincides with the outer spiral. The corresponding intensity distributions of the dual-spiral arrays in Fig. 2(b) (color online) exhibit two focal spots surrounding the central zero-intensity region. Central phase distribution displays uniform and singularity-free equiphase lines with phase jumps forming a pattern of two semicircles. By continuously adjusting the radius difference between the two spirals, the resulting intensity and phase distributions can be smoothly rotated. However, despite the changes in the radius difference, the overall profiles of the intensity and phase distributions remained consistent, as clearly illustrated in Fig. 2.

Similar to Fig. 2(a), Fig. 3(a) (color online) depicts dual-spiral arrays with varying radius differences, where the spirals are assigned a topological charge of 3. The radius difference between the inner and outer spirals is denoted by the variable ∆r₄. The gray arrows on the outer circle indicate the direction of rotation, with the numbers inside the arrows corresponding to the respective radius differences of the dual-spiral arrays. As the radius difference increases from (a1) to (a12) in Fig. 3, the inner spiral contracts progressively inward. Fig. 3(b) (color online) shows that the modulation of the dual-spiral arrays, which carry a third-order topological charge, results in an intensity distribution characterized by a six-petal flower-like symmetry surrounding a central dark spot. The central phase distribution rotates continuously from 0 to 6π over the full azimuthal cycle. Radius differences are selected uniformly, ranging from 10 μm to 850 μm, facilitating a continuous rotation of both intensity and phase distributions. Moreover, modifying the radius difference alone does not affect the shapes of the intensity and phase distributions, which is a feature similar to the observations shown in Fig. 2.

In summary, the numerical simulation analyses presented in Figs. 2 and 3 indicate that when the inner and outer spirals share the same order of topological charge, adjusting the radius difference between the two spirals enables precise control of the azimuthal variation of the generated optical field. A comparison between Figs. 2 and 3 further reveals that, as the topological charge of the dual-spiral arrays increases from the first to the third order, a greater radius difference is required for the azimuthal angle of the optical field to complete a full rotation. Notably, the petal-like intensity profile of the generated optical field was deformed and eventually deteriorated as the topological charge increased to higher orders. This phenomenon arises from the inherent structural asymmetry of a single-spiral configuration. As the topological charge l increases, the expanding radial variation of the spiral induces a deviation from circular symmetry, preventing the formation of the expected intensity profile^[35-36].

However, parameter adjustments can be applied to achieve the desired modulation effect. Specifically, as the topological charge l increases, reducing the diffraction distance z aids in maintaining the modulation effect of the spiral structure. The supplementary file provides additional data and figures illustrating the azimuthal optical modulations for topological charges l = 6 and l = 9.

Based on the above analysis, we extracted data on azimuthal angles and spiral radius differences to investigate their specific relationships. Figure 4(a) (color online) presents a comparative analysis of the azimuthal optical field as a function of the difference in radii. The purple and blue linear fit lines, corresponding to spiral topological charges of l = 1 and l = 3, respectively, exhibit a quasi-linear relationship between Δr and azimuthal angle. Figures 4(b) and 4(c) (color online) display the corresponding sector plots. The results reveal the following principles: First, varying the radius difference of the dual-spiral arrays enables azimuthal manipulation of the optical field over a full 0 to 2π range. Second, the relationship between the radius differences and azimuth exhibited a quasi-linear variation. Third, for the same radius difference in the dual-spiral arrays, different topological charges lead to varying amplitudes of the azimuthal variation in the optical field, with the amplitude decreasing as the topological charge increases.

This study reveals that the radius disparity between the dual-spiral arrays plays a crucial role in the azimuthal manipulation of the optical field. The stabilities of the intensity and phase distributions indicate that the generated optical field maintained a consistent profile. These findings provide a theoretical foundation for the precise control of azimuthal optical field manipulation, which is of considerable importance for flexible steering of optical fields. Drawing on the modulation principle of dual-spiral arrays, which induce a phase gradient through radial differences, any phase-modulated approach that conforms to spiral phase profile encoding can be employed to achieve precise azimuthal control of the optical field. For example, liquid crystals, which are excellent phase-encoded modulators owing to their exceptional electro-optic tunability and high-resolution phase modulation, play a vital role in optical control. Recent advancements in the dynamic control of optical vortices and complex wavefronts have further demonstrated their significance in optical field manipulation^[37-39].

This paper introduces the concept of dual-spiral arrays, comprising two interleaved spiral arrays designed to manipulate the azimuthal optical field. These arrays were distinguished solely by their initial radii, with all the other parameters remaining constant. By precisely controlling the radius difference between the two spiral configurations, the azimuthal distribution of the optical field can be continuously adjusted. To better understand how the dual-spiral array structure influences the azimuthal distribution, we conducted numerical simulations. Through comprehensive analysis, we investigated the relationship between the structural characteristics of the dual-spiral arrays and the resulting azimuthal optical field distribution. The simulation results reveal that by accurately controlling the radius difference, the azimuth of the optical field can be continuously varied across the full range from 0 to 2π. Furthermore, we observed a quasi-linear correlation between the radius difference and the azimuthal distribution, indicating that changes in the radius difference produced nearly proportional adjustments to the azimuthal distribution of the optical field. These findings provide valuable insights into the manipulation of optical fields, potentially paving the way for new applications in fields such as optical communication, microscopy, and optical trapping, where the precise control of the azimuthal optical field is crucial.