Thus far, the vector field of light probed by a nanostructured super-oscillatory lens (SOL) has mostly been studied by approximate theoretical means. Here the first rigorous electromagnetic (EM) test has been presented through an established electromagnetic model solved by the three-dimensional (3D) finite-difference time-domain (FDTD) method. It is found through comparisons that scalar/vectorial theories currently used for designing the metal-film-coated SOL can effectively predict the on-axis intensity behind a SOL simulated by FDTD for both linearly and circularly polarized beams; however, they cannot reflect the true 3D EM vector field distribution particularly for the linearly polarized beam and imprecise results for the total electric energy density have appeared in certain transverse planes, e.g. a relative error as high as 26% is produced for the size of the main focus behind a SOL of 14μm large in diameter. Besides, it is found that current theories cannot be used for designing the glass-etched phase-type SOL.
© 2015 Optical Society of America
Super-oscillatory lens (SOL) is one type of planar, multi-annular, nanostructured metasurface proposed for subwavelength focusing and nanoscopic imaging . Based on SOLs, highly suppressed subwavelength hotspots [1–4] and subdiffraction optical needles [5,6] have been modulated with three major advantages. First, it obtains far-field optical subdiffraction and super-resolution without near-field evanescent waves , which essentially differs from the superlens  or the plasmonic lens . Second, it is a planar, lens-free, ultra-high numerical aperture (NA) focusing plate, in contrast to a traditional lens-based optical system with pupil filters [10–12]. Third, compared with a Fresnel zone plate of nanoscale outermost annulus [13,14], the smallest annulus width of a SOL can be prescribed from 100 nm to micrometers if the ring number can be sufficiently large [1–4]; meanwhile, widths of the transparent annuli are certain times of that for the narrowest annulus, which facilitates the practical fabrication [1,6].
So far, approximate theoretical methods have been used in the design of subwavelength focusing binary amplitude-type SOLs (metal-film-coated). These methods have been based on the scalar angular spectrum theory [1,5], the vectorial angular spectrum theory [2,4], or the vectorial Rayleigh-Sommerfeld diffraction integral [3,6], respectively. All these methods have assumed a polarization-insensitive scalar function to model the physical process of light transmitting through a micro-/nanostructured SOL. It is necessary to implement a rigorous electromagnetic test to examine the validity of this basic assumption and the design theories [1–6]. There are three main reasons. First of all, although the minimum annulus width (or the feature size) of a metal-film-coated SOL can be designed in micrometers, it is basically in a subwavelength scale to facilitate the optimization process (more annuli for a give SOL diameter) [1–6]. In this situation, the polarization-dependent transmission property, a phenomenon known as extraordinary optical transmission, becomes very pronounced  and has been verified through nanohole arrays . In , this extraordinary transmission property and the vectorial nature of light have not been considered. Although vectorial design methods have effectively considered the illumination of vector beams [2,3], the validity of these theoretical generalizations is fundamentally limited to the transmission assumption of a SOL. Second, it is difficult to explicitly establish a physical model to effectively describe the electromagnetic transmission behavior of a metal-film-coated nanostructured SOL, which requires riogously solving an electromagnetic boundary value problem. Third, the phase-type SOL has recently been proposed with an advantage of remarkably enhancing the light throughput [4,17], but it has not yet been validated. Thus, implementing a rigorous electromagnetic test of super-oscillatory lenses is imperative. In this paper, the developed vectorial design theory is briefly described and used to design typical SOLs with the linearly and circularly polarized beams. An electromagnetic simulation model is then established and rigorously solved by the three-dimensional (3D) finite-difference time-domain (FDTD) method. Finally, comparisons and discussions are presented for both the metal-film-coated amplitude-type SOL and the glass-etched phase-type SOL.
2. Vectorial design theory and electromagnetic simulation model
2.1 Vectorial design theory
SOL is composed of many concentric rings either opaque or transparent and the amplitude distribution of the incident vector beam is assumed to be rotationally symmetric. The schematic diagram of the focusing geometry of a SOL is shown in Fig. 1.
The developed vectorial design theory of SOLs is based on the vectorial angular spectrum (VAS) representation and uses the genetic algorithm for optimization. This approach generalizes the scalar angular spectrum method  and uses a fast Hankel transform algorithm to accelerate the optimization . According to the VAS theory, the integral representations for the electric field behind a SOL are briefly outlined for a linearly, circularly, or radially polarized vector beam. The specific derivations are routine and hence omitted here. For the radially polarized beam, the derivations in detail can be found in .
For a linearly polarized beam (LPB, polarized along the x direction) normally illuminating a SOL, as shown in Fig. 1, components of the electric field E for any point in the observation plane (z>0) are derived asEq. (1) is expressed as . represents the scalar polarization-insensitive transmission function for a SOL, which is a basic assumption for all current design theories [1,2,6]. For a sufficiently thin mask (thin element approximation), the transverse electric field immediately behind the aperture plane is approximated by the multiplication of the illumination transverse electric field in the mask plane and the transmission function of the SOL [1–6]. denotes the amplitude distribution of the illumination vector beam at the mask plane of the SOL. has been assumed in this paper. It can be seen from Eq. (1) that the longitudinally polarized component vanishes along the y direction (). The total electric energy density (or light intensity) is calculated by .
It should be noted that the validity of Eq. (1) depends on the scalar transmission assumption of a SOL, i.e., a scalar function of is used. It is found that Eq. (1) is in essence consistent with Eqs. (12) and (13) in . One can find detailed derivations directly from Maxwell’s equations in ; however, Eq. (1) is derived from the vectorial angular spectrum theory here. Using the above assumptions, Ey = 0 and Ex is independent of the observation angle φ. However, in practice, it is found by the rigorous electromagnetic simulation that a minute Ey will appear and Ex will also slightly depend on φ. Thus, the difference of electric fields between the VAS prediction and FDTD simulation shows that it is one of the limitations of current theories used in the design of SOLs compared with the rigorous FDTD electromagnetic calculations. In addition, it should be indicated that the difference of Ey is not the main reason to account for the overall error since it is quite small.
For a left-handed circularly polarized beam (CPB), the electric field is derived as
For a radially polarized vector beam (RPB), radially, azimuthally, and longitudinally polarized components of the electric field E are expressed as2,10] or a subwavelength optical needle [11,19].
For a scalar light beam (neglecting the polarization effect), according to the scalar angular spectrum theory, the light field is described as1], SOL is designed based on Eq. (4). Compared with Eqs. (1) and (2), Eq. (4) represents the transversely polarized electric field component without the longitudinally polarized component Ez.
Based on the above integrals, subwavelength focusing SOLs can be optimized with the prescribed optimization targets. The optimization solver uses a purposely configured genetic algorithm and more importantly implements a fast Hankel transform algorithm to accelerate the optimization. The annulus widths for the initial structure of a SOL have been assumed to be the same. The main idea is to minimize the width of the subwavelength light spot in a post-evanescent transverse plane at a restricted (not fixed) distance away from the SOL surface, while retaining a certain dark ring surrounding the central main lobe. The structure of a SOL is produced for a given maximum iteration number. More details can be found in the previous researches [2,4]. It is found by many examples that the developed optimization method is quite efficient and versatile, which can be used to design binary amplitude and binary phase SOLs of arbitrary size illuminated with various vector beams [2,4].
2.2 Electromagnetic simulation model
Basically, the electromagnetic focusing process of SOL can be divided into two fundamental steps: light transmitting microstructured rings or grooves (area I) and light propagating in a homogeneous, dielectric medium (area II), as shown in Fig. 2 (a cross section view).
If the electric field immediately behind the front surface of the SOL (Fig. 2) is known, the vectorial angular spectrum (VAS) theory is valid for describing the light propagation in area II. However, it is very complicated and even challenging to explicitly determine the electric field distribution. So assumptions have to be applied, which mostly uses a scalar function and neglects the polarization-dependant transmission property of micro-/nanostructured SOLs. This assumption may induce imprecise results  and thus the design theories need to be tested. An electromagnetic simulation model is described in Fig. 2. This can be physically solved using the three-dimensional FDTD method, which rigorously solves Maxwell’s equations . The total-field scattered-field (TFSF) boundary and perfectly matched layer (PML) absorbing boundary condition are applied in the 3D FDTD. Material index parameters are modeled based on . The illuminating wavelength is λ0 = 640 nm. SOL is placed in air (refractive index n = 1) or the oil immersion medium (n = 1.514). The 3D simulation area is set as x, y: , and z: . The front surface of the SOL is initially locating at z = 0. The mesh size (Yee cell) is (x, y, and z), less than λ0/(20n). This grid setting has considered the stability and convergence condition for the FDTD method , as well as the available workstation configuration. For binary amplitude SOLs, a 100nm-thick metal film is coated on the glass substrate to attenuate the incident light. For binary phase SOLs (in air), the groove height is so as to induce a phase shift of π. is the refractive index of the glass substrate. For λ0 = 640 nm, and h = 700.48 nm.
3. Rigorous electromagnetic calculations and discussions
3.1 Test samples
Based on the vectorial design theory described in Section 2.1, SOL1~6 are designed (Table 1). For the optimization, one can find more details in . SOL1~3 are designed with the LPB, and SOL4~6 with the CPB. There are 35 concentric rings with an equidistant ring width of 200nm. The diameter of SOL1~6 is 14 μm. SOL2 and SOL5 are inserted into the oil immersed medium to increase the equivalent NA, which can sharpen the diffracted beam as well as the subwavelength focus. The binary amplitude-type SOL is coded with digits , while the binary phase-type SOL is coded with digits (replacing ‘-1’ with ‘b’ for the sake of clarity in Table 1). The electromagnetic simulation result by FDTD is used to examine the accuracy of the electric field distribution behind the SOL predicted by the scalar/vectorial angular spectrum theory [1,2,4].
3.2 Influence of metal-film material
Three kind of typical metal film materials, aluminum (Al), silver (Ag), and gold (Au), are compared with the same film thickness of 100 nm. For SOL1 and SOL2, the electric field intensity distributions (|E|2) along the axial direction are particularly compared in Fig. 3. The black dashed lines correspond to the theoretical results predicted by the scalar or vectorial angular spectrum theory (the same for the on-axis intensity distributions). The intensity distributions in Figs. 3(a) and 3(c) are further normalized by their respective maximum values as shown in Figs. 3(b) and 3(d). It can be seen that the on-axis intensity distribution predicted by the scalar/vectorial angular spectrum theory broadly coincides with the rigorous electromagnetic simulation result by FDTD for these three metal materials, although remarkable differences can be found (e.g. in the axial range of 5~7 μm in Fig. 3(b) and 3~4 μm in Fig. 3(d)). It should be noted that the axial position of z = 0 is better to locate at the back surface of the SOL (Fig. 2), which corrects the usual assumption of the front surface in previous publications [1–6], especially for a linearly polarized beam. The induced axial displacement is the thickness of the coated metal film (100 nm). At the wavelength of 640 nm, the theoretical curve (black dashed line) using the Au film agrees better with the electromagnetic simulation result. The light throughput of the SOL depends on the coated metal material. For example, the Al film produces a brighter focus for SOL1, while the peak intensity is highest with the Ag film for SOL2. As a result, it will generally have a certain axial shift for a practical metal-film-coated SOL compared with the VAS prediction. Although the electric field distributions are similar for different metal films, there are still obvious differences for the details.
3.3 Metal-film-coated amplitude-type SOL
So far, it has been reported that a circular aperture with a diameter of 5λ0 has been studied by a 3D FDTD method and compared with the vectorial Rayleigh-Sommerfeld diffraction integral in . Besides, a simple 2D FDTD simulation at a mid-infared wavelength (4.6 μm) has been reported in  to investigate a 1D multi-amplitude and binary-phase metallic slit (i.e., a complex amplitude metallic slit). However, the widely used binary amplitude-type SOL has not been examined by rigorous electromagnetic approaches [1–6]. So SOL1,2 (x-polarized LPB) and SOL4,5 (left-handed CPB) in Table 1 are particularly used for comparison (100 nm-thick Al film is used). SOL2 and SOL5 are oil immersed. The electric field intensity distributions in the x-z plane calculated by the VAS theory (Figs. 4(a), 4(c), 4(e)) and FDTD (Figs. 4(b), 4(d), 4(f)) are compared. The plotted area is 4 μm × 7.8 μm (x: −2~2 μm, z: 0~7.8 μm). The x-polarized (|Ex|2) and z-polarized (|Ez|2) electric field intensity components are plotted in Figs. 4(c), 4(d) and Figs. 4(e), 4(f), respectively. The total electric energy density (|E|2) distributions are compared in Figs. 4(a) and 4(b). The significant differences have been observed though the on-axis intensity distributions are similar as shown in Fig. 3(b). The dumbbell shape of the main focus as predicted by the VAS theory in Fig. 4(a) does not appear according to the rigorous FDTD simulation in Fig. 4(b). The longitudinally polarized component |Ez|2 may remarkably influence the total electric energy density distribution, e.g. in the axial range of 2.5~3.5 μm. It even dominates in the near-field range (0~500 nm). However, according to the scalar angular spectrum theory, this pronounced electric energy component has not been considered . The reason for the enhancement of Ez is that the equivalent NA at the observation x-y plane is increased to be as high as ~0.9 (higher when z<3.5 μm). The electric field intensity distributions in the x-y plane at z = 2.62 μm are plotted in Figs. 4(g)~4(m). |Ex|2, |Ez|2, and |E|2 calculated by the VAS theory are shown in Figs. 4(g), 4(i), and 4(k), respectively, compared with the FDTD simulation result in Figs. 4(h), 4(j), and 4(m). The total electric energy density distribution obviously takes a dumbbell shape in Fig. 4(k), which does not agree with the FDTD result, as shown in Fig. 4(m). It should also be noted that the x-polarized component |Ex|2 calculated by FDTD is not circularly symmetric (slightly dependent on the observation angle φ) as shown in Fig. 4(h), in contrast to the VAS prediction in Fig. 4(g). It is wider in the x direction than that in the y direction. Intensity distributions are further compared in Fig. 4(n). The full width at half maximum (FWHM) for |E|2 in the x direction predicted by the VAS theory is 593 nm (blue dashed line), which is 1.35 times as wide as that calculated by FDTD (blue solid line). This implies a 26% relative error of FWHM for the VAS theory. Again in the x direction for |Ex|2, FWHM = 312 nm (0.49λ0) by the VAS theory (red dashed line), which is slightly sharper than that by the FDTD simulation. It can be seen that the intensity distribution is sharpest in the y direction with FWHM = 278 nm (0.43λ0) by the FDTD simulation (black solid line), in contrast to 0.49λ0 evaluated from the VAS theory (black dotted line). For SOL2, it is also observed that the total electric energy density distribution calculated by FDTD largely deviates from what is predicted by the VAS theory. For example, in the axial range of 1~2 μm, a pronounced dumbbell shape appears again according the VAS theory due to the enhanced |Ez|2 component. However, this does not agree with the FDTD simulation. It can be concluded that current scalar/vectorial theories have inherent limitations in the design of metal-film-coated SOLs with a LPB, and imprecise results may be produced in certain transverse planes.
For the circularly polarized beam (CPB), SOL4 and SOL5 are tested particularly. The diffracted light field will be circularly symmetric along the axial direction for the CPB in contrast the LPB. The electric field intensity distributions for SOL5 are compared in Fig. 5. In the x-z plane, the total electric energy density (|E|2) distributions agree with each other between the VAS theory (Fig. 5(a)) and FDTD (Fig. 5(b)). The on-axis intensity distributions are further compared in Fig. 5(c), and the axial FWHM of the focus is ~λ0. Near the peak intensity plane, the electric field intensity distributions are compared in the x-y plane at z = 2.90 μm, as plotted in Fig. 5(d). It can be seen that the light density (|E|2) distributions have been broadened (FWHM≈270 nm, 0.42λ0) due to the enhanced |Ez|2 component. The focus is sharper when neglecting the |Ez|2 component, which are compared in red lines in Fig. 5(d). The transverse FWHM of the focus is reduced to be 218 nm (0.34λ0). Again, for SOL4, the transversely polarized electric components (|Ex|2 + |Ey|2) are agreed; however, the total electric field intensity (|E|2) distributions have slight differences due to the |Ez|2 component. It can be concluded that current design theories are able to predict the 3D electric field intensity distribution behind a metal-film-coated SOL with the CPB in contrast to the LPB.
It should also be indicated that for the vectorial angular spectrum theory, as a fast Hankel transform algorithm with sufficient high accuracy and computation efficiency has been used [2,4], the theoretical results are accurate and reliable if a suitable cutoff frequency is selected. On the other hand, for the 3D FDTD simulations, a fine setting of the grid size will generally lead to a more accurate result; however, there will be no significant differences if the grid size is less than λ0/(20n) .
3.4 Glass-etched phase-type SOL
The prospective advantage of a binary phase-type SOL is the enhanced light throughput. As indicated by the theoretical prediction [4,17], a much brighter hotspot can be focused by a phase SOL compared with the amplitude counterpart. For example, the peak intensity of the focus probed by a phase SOL can be over 5 times as bright as that by an amplitude SOL without increasing the size of the hotspot [4,17]. It is however to see that the theoretical VAS predictions of the glass-etched phase-type SOLs are not confirmed by the rigorous electromagnetic simulations by FDTD. The on-axis intensity distributions for SOL3 (x-polarized LPB) and SOL6 (left-handed CPB) are particularly plotted in Fig. 6. The zero position is chosen to coincide with the front surface of the SOL (the glass-air interface), in contrast to the back surface for the metal-film-coated SOL. The reason is to induce a phase shift of π. For SOL3 at z = 0.94 μm, the transverse FWHM of the main focus along the y direction is 262 nm (0.41λ0) while extending to 400 nm (0.63λ0) in the x direction. For SOL6 at z = 3.02 μm, the transverse FWHM of the main focus is as large as 356 nm (0.56λ0) while sharpening to 315 nm (0.49λ0) when neglecting the Ez component. It can be concluded that current theories cannot be used to design the glass-etched phase-type SOL though it is attractive in improving the light efficiency.
The electromagnetic focusing properties of the SOL have been thoroughly investigated using the established electromagnetic simulation model rigorously solved by the 3D FDTD method. Both the metal-film-coated amplitude-type SOL and the glass-etched phase-type SOL (in air or oil immersion medium) have been examined. For the metal-film-coated SOL with the LPB, it is found that, although the vectorial design theory can be effectively used to predict the on-axis intensity distribution, it cannot reflect the true 3D electric field intensity distribution. In certain transverse planes, imprecise results may be produced, e.g. a relative error as high as 26% for the FWHM of the focus behind a SOL of 14μm large in diameter. The vectorial theory can however be valid in the design of the metal-film-coated SOL with the CPB. In addition, current theories cannot be used in the design of the glass-etched phase-type SOL, although it is attractive in the enhanced light throughput. As a result, current scalar/vectorial theories have inherent limitations and it is necessary to develop more accurate design theories when applying SOLs in the fields of subwavelength focusing, subdiffraction optical microlithography, and far-field nanoscopic imaging.
The authors are grateful to the financial supports from National Natural Science Foundation of China (NSFC) (No. 61505158, No. 51175418) and China Postdoctoral Science Foundation (No. 2014M560767). The authors thank Prof. Chunfang Li (Shanghai University) for the discussion on the vectorial angular spectrum theory. It is also a pleasure to thank Dr. Lu Zhang (Xi’an Jiaotong University) and Dr. Guilin Sun (Lumerical Solutions, Inc., Canada) for beneficial discussions on the FDTD simulation.
References and links
1. E. T. F. 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] [PubMed]
3. H. Ye, C.-W. Qiu, K. Huang, J. Teng, B. Luk’yanchuk, and S. P. Yeo, “Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh-Sommerfeld method,” Laser Phys. Lett. 10(6), 065004 (2013). [CrossRef]
4. T. Liu, T. Shen, S. Yang, and Z. Jiang, “Subwavelength focusing by binary multi-annular plates: design theory and experiment,” J. Opt. 17(3), 035610 (2015). [CrossRef]
5. E. T. F. Rogers, S. Savo, J. Lindberg, T. Roy, M. R. Dennis, and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. 102(3), 031108 (2013). [CrossRef]
6. G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4, 6333 (2014). [CrossRef] [PubMed]
11. H. Wang, L. Shi, B. Luk’yanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]
12. T. Liu, J. Tan, and J. Liu, “Tighter focusing of amplitude modulated radially polarized vector beams in ultra-high numerical aperture lens systems,” Opt. Commun. 294, 21–23 (2013). [CrossRef]
13. W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005). [CrossRef] [PubMed]
15. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]
16. R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong polarization in the optical transmission through elliptical nanohole arrays,” Phys. Rev. Lett. 92(3), 037401 (2004). [CrossRef] [PubMed]
17. Z. Chen, Y. Zhang, and M. Xiao, “Design of a superoscillator lens for a polarized beam,” J. Opt. Soc. Am. B 32(8), 1731–1735 (2015). [CrossRef]
18. V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 52(3), 330–339 (2013). [CrossRef] [PubMed]
19. T. Liu, J. Tan, J. Liu, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60(5), 378–381 (2013). [CrossRef]
20. A. Talflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. ( Artech House, 2005).
21. E. D. Palik, Handbook of Optical Constants of Solids II (Academic, 1991).
22. Z. Wen, Y. He, Y. Li, L. Chen, and G. Chen, “Super-oscillation focusing lens based on continuous amplitude and binary phase modulation,” Opt. Express 22(18), 22163–22171 (2014). [CrossRef] [PubMed]