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Abstract
In this paper, a method to generate a sub-wavelength optical needle is proposed and demonstrated by wavelength shifting and center masking in the Fresnel zone plate (FZP). In theory, the vectorial angular spectrum (VAS) theory combined with genetic algorithm (GA) is used for the design of the center-masking FZP for generating optical needle, and finite-difference time-domain (FDTD) method is used for theoretical validation. In experiment, an amplitude-type center-masking FZP with a processing error of 5 nm is fabricated by focused ion beam etching (FIB), and the focusing intensity distribution of the optical needle is measured based on a self-made device. Finally, a sub-wavelength optical needle in far field is obtained, featuring a depth of focus of 7.16 µm, a central focal length of 26.87 µm, and a minimum full width at half maximum of 500 nm and 467 nm in x and y directions, respectively. This study provides both theoretical and experimental foundations for the practical application of FZPs.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical needles with sub-wavelength focusing spot and long depth-of-focus (DOF) are attractive in many applications such as optical trapping [1,2], optical data storage [3,4], super-resolution imaging [5–7], and lithography [8–10]. In recent years, optical needles have been demonstrated theoretically and experimentally. Optical needles can be generated by different methods, and the methods are categorized into three kinds: the reflective optical system using a paraboloid mirror [11–13], the refractive optical system using a regular lens [14–17], and the diffractive optical system using a microstructure [7,18–23]. For the reflective optical system, the implementation of the system requires a paraboloid mirror with high surface accuracy, which relies on ultra-precision aspherical machining methods, leading to a significant increase in costs. For the refractive optical system, the main drawbacks are the complexity of focusing optical systems and the significant limitation on the transverse beam width. Compared to the previous two systems, the diffractive optical systems are more appealing for achieving sub-wavelength optical needles in the far field using a microstructure, because they offer a simpler system, lightweight structure, and flexible design. Currently, common microstructures include metasurfaces, superlenses, super-oscillatory lenses, and plasmonic lenses, which can not only generate optical needles but also produce Airy optical beams and be used for super-resolution imaging [24–26]. However, the internal minimum units used in the microstructure of diffraction optical systems are on the scale of tens of nanometers, placing more stringent requirements on existing micro/nano fabrication methods, and the majority of research in this field currently leans towards simulation calculations, with a lack of experimental studies. Compared to commonly used microstructures such as metasurfaces, the structure of Fresnel zone plate (FZP) is simpler and has a larger minimum unit size. Based on this, a novel method is proposed in this paper for generating optical needles through wavelength shifting and center masking in FZP, which has been experimentally validated. This study provides an experimental foundation for the practical application of optical needles.
2. Design method of the focusing optical needle
Based on previous studies on the focusing performances of FZP under wavelength dispersion and center-masking, it is known that reducing the incident wavelength and applying center-masking can both elongate the axial focusing spot size [27–30]. Therefore, combining strong dispersion effects (wavelength shift) with a center masked FZP can provide a new method for generating optical needles. It can be achieved through two steps: Firstly, introducing a slight deviation between the actual illumination wavelength and the designed wavelength of the FZP, thus causing the focusing incident light to form a diffused light field distribution along the axis. At this point, the diffraction light field is no longer ideally focusing due to the violation of the equal optical path condition in theory. Therefore, the Debye-Wolf diffraction integral cannot accurately calculate the scattered light field distribution at this time and requires the use of the vectorial angular spectrum (VAS) theory for calculation [29]. Secondly, masking the annular bands in the near-axis region of the FZP (equivalent to central blocking) enables compression of the beam laterally. Simultaneously, increasing the area of the central masking region significantly compresses the radially polarization component, which is beneficial for expanding the focusing spot. Ultimately, this results in the compression of the focused beam laterally. This method can produce an optical needle with a length of several wavelengths behind the surface of the FZP.
Based on the above description, given the total annulus number N of a standard FZP and an illumination wavelength λ0, a single-objective, multi-variable, nonlinear constrained optimization model is established by optimizing the actual wavelength λd, center masking factor ε (expressed as the minimum transparent zone index Nd), and focal length f to minimize the target function, as shown in Eq. (1). The target function aims to minimize the intensity fluctuations within the axial focal depth range of the focusing spot, that is to ensure the excellent light uniformity of the optical needle.
For the single-objective, multi-variable, nonlinear constrained optimization model, the genetic algorithm (GA) is used to solve it, and the FZP structural parameters (center masking factor ε and focal length f) and actual wavelength λd can be obtained for achieving the focusing optical needle. Due to the circular symmetric structure of FZP, the fast Hankel transform algorithm is utilized during the solving process to improve computational efficiency. The optimization procedure of a FZP for a sub-wavelength optical needle is shown in Fig. 1. And the steps for optimizing the model using GA are as follows:
- (1) According to the constraint intervals, λd, Nd, and f are binary encoded with corresponding bit lengths of 12, 6, and 6, respectively. Consequently, the length of an individual’s encoding is 24. On this basis, 30 different individuals are randomly generated as the initial population.
- (2) Calculate the axial focusing optical field distribution for each individual, and then obtain the fitness value of each individual based on the target function in Eq. (1).
- (3) Check if the optimization condition is satisfied. If it is satisfied, output the results directly. Otherwise, proceed to the next step. Here, the optimization condition is whether the maximum number of generations (the total generations is 150) has been reached.
- (4) Perform genetic operations among the individuals in the population: elite selection strategy (the selection probability is 0.2), two-point crossover (the crossover probability is 0.8), and two-point mutation (the mutation probability is 0.12), to form the new generation of the population.
- (5) Continue with the second step and perform genetic operations repeatedly until the optimization condition is met. When the optimization process is finished, output the optimal solution (λd, Nd, and f).
3. Design results and FDTD simulation verification
To demonstrate the efficiency of the aforementioned method, the GA is employed to optimize the design of two different FZP microstructures for achieving a focusing optical needle. Additionally, the correctness of the design is validated through finite-difference time-domain (FDTD) simulations.
3.1 Design example 1
Given an illumination wavelength λ0 = 633 nm (linearly polarized beam, polarized along x direction), the total number of zones N = 40, and a medium of air, the optimized generated optical needle has a length of 5.16 µm (8.15λ0). The center of the optical needle is located at a distance of 17.83 µm behind FZPD1, and the axial intensity distribution of the focused optical needle is shown by the solid black line in Fig. 2(a). The FZPD1 structure parameters obtained by genetic algorithm optimization are λd = 478 nm, Nd = 6, f = 10 µm, and the numerical aperture (NA) is 0.897. Furthermore, the radius of each ring rn of FZPD1 can be calculated using Eq. (2), as shown in Table 1. The first ring is the opaque annulus, while the second ring is the transparent annulus. Subsequently, the opaque annulus and transparent annulus alternate in sequence.
Fig. 2. Focusing intensity distribution of FZPD1. (a) comparison of axial focal intensity distribution between VAS theory and FDTD; (b) x-y plane intensity distribution at z = 17.83 µm; (c) and (d) intensity distribution of optical needle in the x-z plane and y-z plane.
Table 1. Annulus radii of FZPD1
To further validate the performance of the optimized design structure, FDTD simulations are conducted and compared with VAS. The results, as shown in Fig. 2(a) and (b), depict the axial intensity distribution and the intensity distribution of the optical needle in the transverse x-y plane (z = 17.83 µm), respectively. Figure 2(c) and (d) display the calculated intensity distributions of the optical needle in the x-z and y-z planes, respectively, obtained through rigorous FDTD computations. It can be observed that the optimized design results exhibit excellent consistency with the FDTD simulations, confirming that the optimized FZPD1 structure successfully achieves the desired focal effect of the optical needle.
3.2 Design example 2
Given an illumination wavelength λ0 = 633 nm (linearly polarized beam, polarized along x direction), the total number of zones N = 60, and a medium of air, the optimized generated optical needle has a length of 4.96 µm (7.84 λ0). The center of the optical needle is located at a distance of 27.51 µm behind FZPD2, and the axial intensity distribution of the focused optical needle is shown by the solid black line in Fig. 3(a). The FZPD2 structure parameters obtained by genetic algorithm optimization are λd = 533 nm, Nd = 4, f = 20 µm, and NA = 0.858. Furthermore, the radius of each ring rn of FZPD2 can be calculated using Eq. (2), as shown in Table 2. The comparison results between FDTD and VAS for further simulation calculations are shown in Fig. 3(a) and (b), which represent the axial intensity distribution and the transverse x-y plane optical needle field intensity distribution (z = 27.51 µm), respectively. Figure 3(c) and (d) show the accurately calculated optical needle field intensity distribution in the x-z and y-z planes by FDTD, respectively. The calculation results indicate that FZPD2 also produces the expected focused optical needle. However, there are some differences between the calculation results by using VAS theory and FDTD method in Fig. 3. The reason is that there is a certain approximation in calculating the intensity distribution of the focusing light field using VAS, and the electromagnetic properties of the material are not taken into account [31–34]. The FDTD method involves solving Maxwell’s equations to describe how the electromagnetic field in space changes over time and provides a direct approach to modeling the evolution process of the electromagnetic field, which is more accurate and realistic [28].
Fig. 3. Focusing intensity distribution of FZPD2. (a) comparison of axial focal intensity distribution between VAS theory and FDTD; (b) x-y plane intensity distribution at z = 27.51 µm; (c) and (d) intensity distribution of optical needle in the x-z plane and y-z plane.
Table 2. Annulus radii of FZPD2
4. Experimental validation
4.1 Fabrication and characterization of FZPD2
For the fabrication of micro-ring structures, focused ion beam etching (FIB), electron beam lithography (EBL) and two-photon polymerization (2PP) are the main techniques used [35–40]. Among them, 2PP is mainly used for the fabrication of phase-type micro-ring structures, while EBL and FIB can be used for both amplitude and phase-type micro-ring structures. Compared to EBL, FIB processing has simpler procedures and better fabrication results. Based on this, the FZPD2 is fabricated using the dual-beam system of a focused ion beam and scanning electron microscope (FIB-SEM, Crossbeam 540, ZEISS, Germany). The ion beam of this equipment can achieve a maximum resolution of 3 nm (at an acceleration voltage of 30 kV). The system is equipped with five types of injection gases to reduce the re-deposition of sputtered particles, ensuring the quality of processed samples and etching efficiency.
The specific process of fabricating the amplitude-type FZPD2 using FIB is as follows: First, prepare a quartz glass substrate with dimensions of 20 mm × 20 mm × 0.5 mm. Clean the surface with acetone, alcohol, and deionized water using ultrasonic cleaning to remove surface impurities, and then dry it. Next, deposit a 100 nm thick layer of aluminum (Al) on the substrate surface using electron beam evaporation (TF500, HHV, UK). Import the designed FZPD2 structure dimensions into the FIB system and select appropriate FIB processing parameters to etch the sample. Since the minimum line width of FZPD2 is greater than 300 nm, an ion beam dose of 93 mC/cm2 and an ion beam current of 300 pA are selected [35]. Additionally, to reduce the impact of material sputtering deposition around the annular structure during the fabrication process, a radially outward circular processing method is employed [35]. The characterization results of the fabricated sample using scanning electron microscopy (SEM) are shown in Fig. 4. The widths of the outermost transparent ring and the innermost transparent ring are 367.1 nm and 882.5 nm, respectively. The measured values are compared with the designed values, and the error is less than 5 nm, which meets the requirements for subsequent experiments on focusing optical needles.
Fig. 4. SEM images of FZPD2. The left and right red boxes in (b) represent (a) and (c) respectively.
4.2 Building of focusing optical needle field measurement device
Methods for detecting the micro-nano focusing optical field distribution of planar optical elements mainly include three types: knife-edge detection method [41,42], near-field optical microscopy detection method [43–45], and wide-field microscopy imaging amplification method based on high numerical aperture objective [28,34,46]. Among them, the knife-edge detection method and near-field optical microscopy detection method are contact measurement methods that require direct contact between the detection unit and the detected optical field, making it easy to affect the detected field. This is a difficult disadvantage to overcome. However, the wide-field microscopy imaging amplification method is a non-contact method for detecting the distribution of optical fields and is currently the most commonly used detection method. The basic principle is to use a high-magnification and large numerical aperture microscopic objective and tube lens to form an infinite optical imaging system to detect the optical field of the observation plane. In this study, the wide-field microscopy imaging amplification method will be used to measure the focusing optical needle field generated by FZPD2.
The schematic diagram of the detection optical path and the physical image of the experimental setup are shown in Fig. 5. The laser light source consists of a supercontinuum white light source (wavelength range 400-2400 nm, Rock 400, LEUKOS, France) and a tunable filter (wavelength adjustment range 400-850 nm, Bebop, LEUKOS, France), forming a tunable laser light source (the spectral width of the wavelength is 5 nm). The incident laser beam passes through an optical fiber, a collimating lens (F220APC-633, Thorlabs, USA), and a reflection mirror to vertically illuminate the surface of the FZPD2, forming a focusing optical needle field at a certain distance. The distribution of the focusing optical field is recorded by a CMOS camera through an objective lens (100x, NA = 0.95, Nikon, Japan) and a tube lens (focal length 200 mm). The axial distribution of focusing optical needle intensity is measured by axial scanning using a piezoelectric transducer (PZT) nano-positioning stage (P-611.ZS, PI, Germany) with a scanning step size of 100 nm. Simultaneously, the precise distance from the observation plane to the FZPD2 is precisely controlled by the PZT nano-positioning stage.
Fig. 5. Schematic diagram of non-contact detection method experimental plan. (a) experimental setup diagram; (b) optical path diagram.
4.3 Focusing optical needle field measurement experiment and results
To demonstrate the accuracy of the theoretical simulation of optical needle generation by occlusion and wavelength shift, the setup shown in Fig. 5 is employed using an incident wavelength of 533 nm (linearly polarized beam, polarized along x direction). The FIB fabricated FZPD2 is utilized for the optical field detection. The x-z plane intensity distribution of the optical field is obtained through PZT axial scanning (with a scanning step size of 100 nm) and data reconstruction, as shown in Fig. 6(a). The focal plane optical field distribution is illustrated in Fig. 6(b). Furthermore, the optical field distribution along the light axis in the x-z plane and the intensity distribution in the x and y directions at the focal plane is extracted, as depicted in Fig. 6(c) and (d), respectively. According to the measurements illustrated in Fig. 6, it is evident that occlusion at the center and wavelength shift of the standard FZP can generate a sub-wavelength focusing optical needle in far field. The depth of focus is the length of the optical needle along the axial direction, and the central focal length is the distance from the rear surface center of a FZP to the midpoint of optical needle along the axial direction. This optical needle showcases a depth of focus of 7.16 µm, a central focal length of 26.87 µm, and a minimum full width at half maximum (FWHM) of 500 nm and 467 nm in x and y directions, respectively.
Fig. 6. Experimental detection result of focusing optical needle of FZPD2. (a) optical field distribution in the x-z plane; (b) focal plane optical field distribution (x-y plane); (c) axial optical field distribution; (d) intensity distributions of the focal plane in the x and y directions.
Comparing the experimental measurement data with the theoretical calculations based on VAS, as shown in Table 3, it can be observed that the focal spot elongates and the size of the spot increases. It means that the theoretical and experimental results for depths of focus and focal spot sizes in the x and y directions are different dramatically. The main reason is that the laser source used in the experiment has a relatively large spectral width (5 nm) compared to the ideal single-frequency incident light source used in the simulation. In other words, when the spectral width of the incident wavelength increases, it leads to the elongation of the focal spot and the broadening of the spot size. Meanwhile, it can be seen that the focal spot size in x direction is wider than that along y direction. The reason is that a linearly polarized beam (LPB, polarized along x direction) is used as the incident light [29]. If a circularly polarized beam (CPB) or a radially polarized beam (RPB) is adopted, the focal spot is circularly symmetric [31–33]. In this instance, the polarimetric techniques can be employed to measure the Stokes parameters of optical needles [47,48].
Table 3. Comparison between the theoretical calculations and the experimental measurements of FZPD2
5. Conclusions
In summary, a novel method of generating a sub-wavelength optical needle in far field is proposed and experimentally verified. This sub-wavelength optical needle is realized using wavelength shifting and center masking of FZP. Firstly, this study establishes a model for optimizing the FZP structural parameters and actual incident wavelength based on the VAS theory, and applies genetic algorithm to solve the model. Secondly, the designed center-masking FZP capable of generating sub-wavelength optical needle is theoretically validated using FDTD method. Furthermore, the amplitude-type center-masking FZP is fabricated using FIB, and the experimental validation of the optical needle field is performed using a self-made device and wide-field microscopy imaging amplification method. Finally, a sub-wavelength optical needle with a minimum full width at half maximum of 500 nm and 467 nm in x and y directions is observed at a distance of 26.87 µm away from the rear surface of the center-masking FZP.
Funding
Shaanxi University of Technology Science Foundation of China (SLGRCQD2132); Key Research and Development Program of Shaanxi Province (2023-YBGY-385); General project of Shaanxi Natural Science Basic Research Plan (2022JM-131, 2023-JC-YB-018); Project of the College Young Talents Promotion Program of Shaanxi Association of Science and Technology (20210423).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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