Open Access
Add to BibSonomy
Abstract
We studied the super-resolution light modulation capability of Si nanodisks, a flat semi-transparent high index nanolens in the visible spectral range. A Laguerre-Gaussian beam-based optimization algorithm was developed to synthesize desired field distributions. Focused spots below 45 nm (< λ/9) were successfully achieved with 405 nm light over the whole center area of the nanolens. This superb light nano-focusing capability allows us to synthesize complex nano-patterns by simply superposing several focus spots together, making the Si nanolens a promising tool for super-resolution photolithography.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The optical resolution can be improved by using high refractive index materials. This method has been widely used with liquid, for instance oil immersion objective lenses [1–3] and water-immersion Deep Ultraviolet (DUV) lithography [4–6]. Recently, it was extended to solid dielectric materials, particularly micro-dielectric structures, such as polymer micro-droplets [7], dielectric microspheres [8] and TiO2 nanoparticle micro-droplets [9]. A variety of interesting results have been reported, such as super-resolution white-light imaging [8,10–15], fluorescence imaging [16–19], and lithography [20–22], to name a few. Despite the successes, the resolution of dielectric micro-lenses is fundamentally limited by their refractive index, which is normally lower than 2.0 in the visible spectral range. As a result, the smallest resolvable pitch size is limited to a value slightly better than 100 nm. Moreover, these micro-dielectric lenses are normally curved 3-dimensional structures, which cannot be produced using standard nano-fabrication techniques. Herein, we ask the following questions: “(1) Can we make a high index lens which can modulate light with spatial resolution well below 100 nm, and (2) can this lens be a flat structure which can be easily fabricated?”
To answer the above questions, we investigate the optical behaviors of a simple flat high index nanostructure, Si nanodisk, and study whether we can focus, scan and even generate any given pattern of light at the nanoscale using such a structure. Here, Si is chosen because high index materials commonly have a narrow bandgap which is associated with high material losses in the visible range but Si is an exception [23,24]. As an indirect band material, the absorption of Si is considerably lower than those of direct band materials in the visible and near-infrared spectral range. For example, the penetration depth of Si is approximately 200 nm at 405 nm, one of the most commonly used wavelengths for photolithography [25]. This is enough for building a Si nano-lens with a reasonable transmissivity. For example, we demonstrated that sub-50 nm focus can be achieved with a simple semispherical nanolens, but as a 3D structure, semispherical nanolens is extremely difficult to fabricate [26].
To check whether the flat Si lens can focus and further spatially modulate light at will, we need to solve an inverse problem. That is, for a target super-resolution field distribution, whether we can find corresponding incident fields, and what they are. Today, driven by the fast development of computational electrodynamics, this type of inverse problems have been attracting a lot of interests [27–31], and different algorithms were developed in various scenarios, for instance three-dimensional vectorial holography [29], surface plasmon [28], and Au nanostructure arrays [32]. In this work, we use a high-order Laguerre-Gaussian (LG) beam [33–35] based method together with a customized particle swarm optimization (PSO) algorithm [36,37] to solve the inverse problem.
2. Optimization algorithm for light manipulation
Figure 1 is the schematic of the system. The Si nanolens (SiNL) is made of a nanodisk (100 nm thick and 100 nm in radius) on a glass substrate. Its surrounding area is covered by a 40 nm Al film which blocks the unwanted light. Considering that it is difficult to obtain an ideal perpendicular wall in fabrication, the sidewall of the nanodisk is set at 10 degrees from the normal direction of the substrate. Here, the wavelength of light is set at 405 nm, which is widely used in lithography and fluorescence imaging.
Fig. 1. Simulated nanofocusing by a flat Si nanodisk lens. a. Schematic drawing of the model system, in which spatially modulated light is focused by the objective lens and Si nanodisk subsequentially. With optimization algorithm and numerical simulations, the minimum focus spot can be obtained. b. Principle of the Laguerre-Gaussian beam-based optimization process. First, focused LG beams are generated by an oil-immersion objective with spatially modulated beams as shown in the bottom panels in b. Then, the focused beams are further focused by the Si nanolens in the (xSiNL, ySiNL) plane. Finally, nano-focused spot is created in the (xSiNL, ySiNL) plane by linearly superposing the refocused LG beams with the help of PSO algorithm.
To focus light and further generate super-resolution patterns, spatially modulated beams generated by a spatial light modulator (SLM) are used as the incident light, which are first focused by an objective lens (NA = 1.4) and then further confined by the SiNL (Fig. 1(a)).
As a linear optical system, the focused field by the SiNL, ${{\mathbf E}_{SiNL}}$, can be written as
Here, ${{\mathbf E}_{in}}$ are the incident fields. ${{\mathbf M}_{obj}}$ and ${{\mathbf E}_{in}}$ are the optical transfer matrices of objective lens and Si nanolens, respectively. To calculate the fields, we discretize the focus and back focus planes of the objective into subwavelength meshes with N mesh points ri, $i \in [1,N]$ [32]. Fields ${{\mathbf E}_{SiNL}}$ and ${{\mathbf E}_{in}}$ can then be written as 3N-supervectors, ${{\mathbf E}_{SiNL}}\textrm{ = (}{{\mathbf E}_{SiNL}}({r_1}),{{\mathbf E}_{SiNL}}({r_2}),...,$ and ${{\mathbf E}_{in}}\textrm{ = (}{{\mathbf E}_{in}}({r_1}),{{\mathbf E}_{in}}({r_2}),...,{{\mathbf E}_{in}}({r_N})\textrm{)}$ ${{\mathbf M}_{obj}}$ and ${{\mathbf M}_{SiNL}}$ becomes 3N×3N matrices. Equation (1) can then be written as:
The goal of this work is to find the incident fields ${{\mathbf E}_{in}}$ for a given super-resolution field distribution ${{\mathbf E}_{SiNL}}$. To simplify this inverse problem, we write ${{\mathbf E}_{in}}$ into the coherent superimposition of LG beams
where ${C_{lp}}$ are the complex coefficients, andare the complex electric field distributions of LG beams. Substitute ${{\mathbf E}_{LG,lp}}$in Eq. (1) with Eq. (3), we haveHere, ${{\mathbf E}_{SiNL,lp}} = {{\mathbf M}_{SiNL}}{{\mathbf M}_{obj}}{{\mathbf E}_{LG,lp}}$, which presents the field distributions of the focused LG beams by the SiNL.${{\mathbf E}_{LG,lp}}$
Comparing to Eq. (1), Eq. (4) simplifies the problem significantly. Ideally, we only need to solve coefficient ${C_{lp}}$ for a given target field ${{\mathbf E}_{SiNL}}$. However, in the most cases, Eq. (4) is over determined, and does not have a solution. In fact, we even do not know what the best achievable spatial resolution for the SiNL is. To circumvent this issue, in this work, we first search for the narrowest synthesized focus spots at different positions by the LG beams, and then use these “points” as fundamental building blocks to construct more complex super-resolution patterns. By using this method, the inverse problem becomes an optimization problem, which can be solved using various algorithms. In this work, the PSO method, a global optimization algorithm, was used to search for the narrowest field distributions.
Figure 1(b) shows the detailed optimization process. LG beams, ${{\mathbf E}_{LG,lp}}$ with l and p ranging from 0 to 9 and -9 to 9 and polarization along both x and y directions were used as the incident fields. The corresponding focused fields ${{\mathbf M}_{obj}}{{\mathbf E}_{LG,lp}}$ at 405 nm by the objective and subsequentially ${{\mathbf E}_{SiNL,lp}}$ by the SiNL were calculated using the Kirchhoff angular spectrum diffraction method [38] and the finite-difference time-domain (FDTD) method (Lumerical FDTD), respectively [39].
The focusing field ${{\mathbf E}_{obj}}({\mathbf r})$ at the object plane can be obtained from the ${{\mathbf E}_{LG,lp}}$ by Kirchhoff angular spectrum diffraction method [38]
We used the focused fields as the source in FDTD to illuminate the SiNL and calculated the field distributions in the optical near-filed. Perfect matched layer boundary conditions were used to eliminate unwanted reflections at the boundaries; the size of the simulated area was 5µm×5µm×1µm; the mesh size was 2 nm. Here, the refractive index of Si from Palik’s results (n = 5.425 + i0.331) was used in the simulation [25,26]. With the help of ${{\mathbf E}_{SiNL,lp}}$, we can then calculate the corresponding coefficients ${C_{lp}}$ in Eq. (4) for given target field distributions with the PSO algorithm.
In the PSO calculations, multiple parameters are considered in the optimization process to obtain a meaningful super-resolution focus with a clean background, symmetric shape and large focus depth. More detailed, we defined a certain number of particles in a high-dimensional parameter space spanned by Clp, the coefficients in Eq. (4). The particles moved in this parameter space via a stochastic optimization process to find the optimal value of a pre-defined merit function [35,36]. Here, the merit function includes all the key parameters for the nanofocusing process, namely the size, depth and position of the focus spots. We iterated 5000 times with a swarm of 60 particles in each calculation and we repeated this process until satisfactory results were found. Linear decreasing weight and contraction factor were used to accelerate the convergence rate and obtain high quality solutions. The detailed optimization strategy is discussed in the later section of this work.
3. Sub-50 nm focus of light
Using the above method, we first searched for the simplest case of super-resolution light modulation, namely the nano-focus at the center point of the SiNL by setting the target position at (0, 0) in the PSO algorithm. Figure 1(b) shows the result. A symmetric nano-focus spot was obtained without any strong sidelobes. The full width of half maximum (FWHM) reaches 45 nm at z = 0 nm, which is smaller than λ/9 and much smaller than diffraction limit (λ/2NA).
In addition to the extreme lateral confinement, the depth of the nano-focus spot reaches almost 100 nm in air. We plot the field distribution at different heights (10 nm, 40 nm, 70 nm and 100 nm), as shown in Fig. 2(a). The size of the nano-focus spot increases slowly along the z direction. At z = 100 nm, the spot size is still smaller than 100 nm. It is worth noting that although the lateral confinement of nano-focus spot is kept, the intensity of focus spot drops exponentially along the z direction (Fig. 2(c)). It indicates that the nano-focusing effect is essentially a near-field effect.
Fig. 2. Nanofocusing behavior of the Si nanolens at different height, z. a. Field distribution at z = 10, 40, 70 and 100 nm. b. Size of the focus spot as a function of z. c. Height-dependent peak intensity of the nano-focused fields.
4. Nano-focusing at any given position
After achieving nano-focusing at the center of the SiNL, a natural question is whether we can synthesize such a nano-spot at other positions. To answer this question, we changed the target position in the merit function and ran the optimization algorithm for each point. Since the SiNL is an axial symmetric structure, we only need to consider the focus behaviors along its radius (e.g., the x-axis in this work). If nano-focusing can be achieved at any position along the x-axis, we then will be able to focus light at a given position in the whole plane by simply rotating the system as shown in Fig. 3(a).
Fig. 3. Nanofocus at different positions of the Si nanolens. a. Focus spot at (xfoc, θ) can be obtained by rotating an optimized nano-focus spot (xfoc, 0) on x-axis. b. Synthesized nano-focusing spots at different positions on x-axis (x = 10, 20, 30, 40, 50 and 60 nm). c. Size of the focus spot as a function of z at different positions. d. Height-dependent peak intensity of the nano-focused fields at different positions.
Here, we set a series of target positions of the nano-focus spots along the x-axis with a 10 nm step (i.e., xfoc = 10 nm, 20 nm, 30 nm, 40 nm, 50 nm and 60 nm). Figure 3(b) shows the optimized results. Light can be tightly focused into a nano-spot (FWHM < 60 nm) at all the 6 positions with intensity of the sidelobes kept below half of the intensity of the main focus spot. In addition, the depths of all the nano-focus spots exceed 100 nm similar to the case of the focus spot at (0, 0).
The above results show that the field of view (FOV) of the SiNL can reach 120 nm (a circular area with radius of 60 nm). Within the FOV, light can always be focused into a deep subwavelength nano-spot with a depth of focus (DOF) over several tens of nanometers (Fig. 3(c) and 3(d)). In other words, the spot size is kept below 100 nm within the DOF. We also test the cases with xfoc > 60 nm. Focus spots can still be obtained, but their shapes become asymmetric and the sizes are considerable larger than 50 nm.
5. Super-resolution spatial light modulation
Once we can obtain nano-focus spots at any given position in the FOV, we can generate any super-resolution patterns by using the linear superposition of several different nano-spots. To demonstrate this capacity, we here tested two different types of patterns, point arrays and lines.
As shown in Fig. 4(a) and 4(b), two different point array patterns are synthesized, namely point pair and 2-by-2 array. The patterns are composted of focused point 60 nm away from the center point of the FOV. After the supposition, all the spots are still well separated. In the process, the size of the each spot slightly increases due to the background induced by the rest nano-focus spots. Here, a further step of optimization was implemented based on the supposition results to further rectify the shape and suppress the undesired background.
Fig. 4. Nanopattern generation by the Si nanolens. a and b. Synthesized point pair and 2-by-2 point arrays. c and d. Synthesized super-resolution lines along the x-axis and diagnostics.
In addition to the point array, super-resolution lines can also be constructed by the SiNL. Here, we take lines along the x-axis and diagnostics as examples. As shown in Fig. 4(c) and 4(d), the lengths of the lines exceed 100 nm, and the width is below 60 nm. Similar to the case of point arrays, an additional step of optimization was used to perfect the shapes of the lines.
6. Multi-step optimization strategy
To achieve above demonstrated super-resolution patterns, the key is to choose a proper optimization strategy. Particularly, in this work, multiple independent parameters are required to quantify the quality of a single nano-focus spot (namely, position, size, symmetry, size of sidelobes and depth of focus). It makes the choice of the detailed optimization process, as well as the merit function, a challenging task. For example, if all the parameters are considered in the same time, we will not be able to find satisfactory results in the most case. By setting optimization strategy as merit function, the optimization process of PSO can be constrained. For example, if the center region is set as the range of the strongest point of light intensity, the results of the non-strongest point of the center can be discarded. This method reduces the computation and saves the convergence time. Based on the above consideration, here, we use a multi-step optimization strategy, in which the lateral profile and DOF of the focused fields are considered separately in subsequent steps.
One of the major challenges in the lateral optimization process is how to quantitively evaluate the shape and size of the focused spots, which can be in any irregular shapes. Here, the isoline at 50% of the maximum intensity is used to describe the shape of focus spot. More detailed, we use the radius of the smallest excircle of the 50% isoline, Rmax, as the spot size, and define $\mathrm{\eta}$ = Rmin/Rmax, the ratio between radius of the smallest excircle and largest incircle of the 50% isoline to evaluate the symmetry of the spot.
After the optimal lateral profiles (i.e., focus spots with the smallest Rmax and $\mathrm{\eta}$) are found at the z = 20 nm plane, we further optimize the DOF of the focus spots to avoid the cases of extremely small DOF. More detailed, we use $\sum\nolimits_i {{z_i}{R_{\max ,i}}}$, the sum of the product between height zi (ranging from 10 nm to 200 nm) and lateral size Rmax,i at zi as the merit function to search for the best results. This will finally lead to the nano-focus spot with a large DOF.
7. Role of polarization in nano-focusing
In addition to the optimization strategy, the polarization of LG beams can also influence the quality of the nano-focuses. It requires both LG beams with x-polarization and y-polarization to achieve a nano-focus spot below 45 nm. When only x-polarized LG beams are used, the optimized results are considerably larger (always > 55 nm), as shown in Fig. 5. Moreover, the relative intensity of the sidelobes also becomes larger (reaches 0.7) and the DOF becomes smaller than 50nm.
Fig. 5. Polarization effect in nano-focusing. a. Polarization maps of the nano-focusing spot by both x- and y-polarized LG beams at different z. The contour maps are shows the intensity distribution. b. Polarization maps of the nano-focusing spot x-polarized LG beams.
In order to understand the behind mechanism, we plotted the polarization of the electric fields at different height, as shown in Fig. 5(a) and 5(b). In the case that both x- and y-polarized LG beam are used, there are two vortices formed in the close vicinity of the central axis of the SiNL, and these two vortices form low intensity areas which pinch the fields into a tightly focused spot along the central axis. More importantly, due to the vortices, the structure of the focused fields stay unchanged along z, and this leads to the large DOF. On the contrary, there is not such parallel vortices in the case of focused field synthesized by only x-polarized LG-beams. As a result, the structure of electric fields varies along z-axis, and the focus spot diverges rapidly, as shown in Fig. 5(b). It worth noting that the coexistence of vortices and field localization is a common phenomenon, which has been found in other types of super-resolution focus spots [40].
8. Size effect of the SiNL
Finally, we would like to emphasize that the performance of the SiNL is not sensitive to its size. When the thickness of SiNL is comparable to or less than the penetration depth of the Si (195 nm at 405 nm) and the radius is comparable to the thickness, we can always achieve a nano-focus spot approximately 50 nm. More detailed, we tested SiNL with R ranging from 50 nm to 250 nm, and thickness from 100 nm to 200 nm. The size of the optimized focus spot is always below 60 nm, and no correlation between the size of nano-focus spots and size of SiNL is found. It means that SiNLs are able to tolerate shape discrepancies induced in fabrication processes.
9. Summary
In summary, we studied the super-resolution light modulation capability of a flat Si nanolens at 405 nm, which is semi-transparent and has a refractive index > 5. A Laguerre-Gaussian beam expansion-based optimization method was established to synthesize nano-focus spots on the upper-surface of the SiNL. With this method, we realized a nano-focus at the center of the nanolens with a FWHM of 45 nm and DOF larger than 100nm. Moreover, we can obtain such a nano-focus spot at any positions in the whole center area of the nanolens (120 nm in diameter) and further utilize these nano-spots as building blocks for constructing more complex super-resolution patterns, including point arrays and lines by linearly combining several separated nano-focus spots together. Finally, we investigated the nano-focusing capability of Si nanolens with different sizes, and found that their performance is not sensitive to the geometrical parameters. We believe that this flat Si nanodisk lenses, with its superb nano-focusing capability and fabrication-friendly design, can have important applications in super-resolution imaging and lithography.
Funding
Key Technologies Research and Development Program (2016YFA0201104, 2021YFA1400803).
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.
References
1. H. Kawata, J. M. Carter, A. Yen, and H. I. Smith, “Optical projection lithography using lenses with numerical apertures greater than unity,” Microelectron. Eng. 9(1-4), 31–36 (1989). [CrossRef]
2. G. J. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light-microscopy with high aperture immersion lenses,” Journal of Microscopy-Oxford 117(2), 219–232 (1979). [CrossRef]
3. H. E. Keller, “Objective Lenses for Confocal Microscopy,” in Handbook Of Biological Confocal Microscopy, J. B. Pawley, ed. (Springer US, Boston, MA, 2006), pp. 145–161.
4. T. Ito and S. J. N. Okazaki, “Pushing the limits of lithography,” Nature 406(6799), 1027–1031 (2000). [CrossRef]
5. A. Raub and S. Brueck, Deep UV immersion interferometric lithography, Microlithography 2003 (SPIE, 2003), Vol. 5040.
6. M. Totzeck, W. Ulrich, A. Göhnermeier, and W. Kaiser, “Pushing deep ultraviolet lithography to its limits,” Nat. Photonics 1(11), 629–631 (2007). [CrossRef]
7. J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I. C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature 460(7254), 498–501 (2009). [CrossRef]
8. Z. Wang, W. Guo, L. Li, B. Luk’yanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at 50 nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2(1), 218 (2011). [CrossRef]
9. W. Fan, B. Yan, Z. B. Wang, and L. M. Wu, “Three-dimensional all-dielectric metamaterial solid immersion lens for subwavelength imaging at visible frequencies,” Sci. Adv. 2(8), e1600901 (2016). [CrossRef]
10. A. Darafsheh, G. F. Walsh, L. Dal Negro, and V. N. Astratov, “Optical super-resolution by high-index liquid-immersed microspheres,” Appl. Phys. Lett. 101(14), 141128 (2012). [CrossRef]
11. H. Yang, N. Moullan, J. Auwerx, and M. A. M. Gijs, “Super-Resolution Biological Microscopy Using Virtual Imaging by a Microsphere Nanoscope,” Small 10(9), 1712–1718 (2014). [CrossRef]
12. F. Wang, L. Liu, H. Yu, Y. Wen, P. Yu, Z. Liu, Y. Wang, and W. J. Li, “Scanning superlens microscopy for non-invasive large field-of-view visible light nanoscale imaging,” Nat. Commun. 7(1), 13748 (2016). [CrossRef]
13. F. G. Wang, S. L. Yang, H. F. Ma, P. Shen, N. Wei, M. Wang, Y. Xia, Y. Deng, and Y. H. Ye, “Microsphere-assisted super-resolution imaging with enlarged numerical aperture by semi-immersion,” Appl. Phys. Lett. 112(2), 023101 (2018). [CrossRef]
14. N. Franch, J. Canals, V. Moro, A. Vila, A. Romano-Rodriguez, J. D. Prades, J. Gulink, D. Bezshlyakh, A. Waag, K. Kluczyk-Korch, M. Auf der Maur, A. di Carlo, and A. Dieguez, “Nano illumination microscopy: a technique based on scanning with an array of individually addressable nanoLEDs,” Opt. Express 28(13), 19044–19057 (2020). [CrossRef]
15. J. Haug, M. Palei, J. D. Shrout, E. Narimanov, P. W. Bohn, and A. J. Hoffman, “Confined hyperbolic metasurface modes for structured illumination microscopy,” Opt. Express 29(26), 42331–42342 (2021). [CrossRef]
16. K. Koyama, M. Yoshita, M. Baba, T. Suemoto, and H. Akiyama, “High collection efficiency in fluorescence microscopy with a solid immersion lens,” Appl. Phys. Lett. 75(12), 1667–1669 (1999). [CrossRef]
17. Z. H. Liu, B. B. Goldberg, S. B. Ippolito, A. N. Vamivakas, M. S. Unlu, and R. Mirin, “High resolution, high collection efficiency in numerical aperture increasing lens microscopy of individual quantum dots,” Appl. Phys. Lett. 87(7), 071905 (2005). [CrossRef]
18. D. Wildanger, B. R. Patton, H. Schill, L. Marseglia, J. P. Hadden, S. Knauer, A. Schoenle, J. G. Rarity, J. L. O’Brien, S. W. Hell, and J. M. Smith, “Solid Immersion Facilitates Fluorescence Microscopy with Nanometer Resolution and Sub-Angstrom Emitter Localization,” Adv. Mater. 24(44), OP309–OP313 (2012). [CrossRef]
19. L. Liang, D. B. L. Teh, N.-D. Dinh, W. Chen, Q. Chen, Y. Wu, S. Chowdhury, A. Yamanaka, T. C. Sum, C.-H. Chen, N. V. Thakor, A. H. All, and X. Liu, “Upconversion amplification through dielectric superlensing modulation,” Nat. Commun. 10(1), 1391 (2019). [CrossRef]
20. L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, “Near-field photolithography with a solid immersion lens,” Appl. Phys. Lett. 74(4), 501–503 (1999). [CrossRef]
21. M. M. Alkaisi, R. J. Blaikie, S. J. McNab, R. Cheung, and D. R. S. Cumming, “Sub-diffraction-limited patterning using evanescent near-field optical lithography,” Appl. Phys. Lett. 75(22), 3560–3562 (1999). [CrossRef]
22. M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009). [CrossRef]
23. D. E. Aspnes and A. A. Studna, “Dielectric Functions and Optical-Parameters of Si, Ge, Gap, Gaas, Gasb, Inp, Inas, and Insb from 1.5 to 6.0 Ev,” Phys. Rev. B 27(2), 985–1009 (1983). [CrossRef]
24. H. Ehrenreich and F. Spaepen, Solid state physics (Academic Press, New York, 2001), Vol. 56.
25. E. D. Palik, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, New York, 1998).
26. Z. Wang and W. Zhang, “Sub-50 nm focusing of a 405 nm laser by hemispherical silicon nanolens,” J. Opt. Soc. Am. B 38(1), 44–50 (2021). [CrossRef]
27. K. Huang, H. P. Ye, J. H. Teng, S. P. Yeo, B. Luk’yanchuk, and C. W. Qiu, “Optimization-free superoscillatory lens using phase and amplitude masks,” Laser & Photonics Reviews 8(1), 152–157 (2014). [CrossRef]
28. I. Malkiel, M. Mrejen, A. Nagler, U. Arieli, L. Wolf, and H. Suchowski, “Plasmonic nanostructure design and characterization via Deep Learning,” Light: Sci. Appl. 7(1), 60 (2018). [CrossRef]
29. H. Ren, W. Shao, Y. Li, F. Salim, and M. Gu, “Three-dimensional vectorial holography based on machine learning inverse design,” Sci. Adv. 6(16), 4261 (2020). [CrossRef]
30. M. Bancerek, K. M. Czajkowski, and R. Kotynski, “Far-field signature of sub-wavelength microscopic objects,” Opt. Express 28(24), 36206–36218 (2020). [CrossRef]
31. L. H. Yeh, L. Tian, and L. Waller, “Structured illumination microscopy with unknown patterns and a statistical prior,” Biomed. Opt. Express 8(2), 695–711 (2017). [CrossRef]
32. G. Volpe, G. Molina-Terriza, and R. Quidant, “Deterministic Subwavelength Control of Light Confinement in Nanostructures,” Phys. Rev. Lett. 105(21), 216802 (2010). [CrossRef]
33. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
34. C. Rosales-Guzman, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20(12), 123001 (2018). [CrossRef]
35. Y. J. Shen, X. J. Wang, Z. W. Xie, C. J. Min, X. Fu, Q. Liu, M. L. Gong, and X. C. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019). [CrossRef]
36. J. Robinson and Y. Rahmat-Samii, “Particle Swarm Optimization in Electromagnetics,” IEEE Trans. Antennas Propag. 52(2), 397–407 (2004). [CrossRef]
37. E. T. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11(5), 432–435 (2012). [CrossRef]
38. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, New York, 2006).
39. A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method (Artech house, 2005).
40. G. Yuan, E. T. F. Rogers, and N. I. Zheludev, ““Plasmonics” in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields,” Light: Sci. Appl. 8(1), 2 (2019). [CrossRef]
