We propose and analyze an innovative device called “Hybrid-Super-Hyperlens”. This lens is made of two hyperbolic metamaterials with different signs in their dielectric tensor and different isofrequency dispersion curves. The ability of the proposed lens to break the optical diffraction limit is demonstrated using numerical simulations (with the resolution power of about λ/6). Both a pair of nano-slits and a nano-ring can be imaged and resolved by the proposed lens using the radially polarized light source. Such a lens has great potential applications in photolithography and real-time nanoscale imaging.
©2013 Optical Society of America
Ernst Abbe first introduced so-called diffraction limit , whereλis the wavelength, is the distance between two objects, and is the numerical aperture (NA) . Generally, features smaller than half wavelength of the light are permanently lost in the image because the waves which carry information of the sub-wavelength details are transverse waves with their wave vectors larger than free space wave vector. Those wave vectors decay exponentially from the surface of the object in free space . This is the key reason for conventional optical microscopy cannot capture the minuscule details of the object in the far field region. In order to improve the optical imaging resolution, and make good use of the optical evanescent wave or near field, photon scanning tunneling microscopy [3–5] and near-field scanning optical microscopy [6–8] are the major paths to achieve near-field super-resolution image. However, low throughput, poor compatibility with various environment/samples, and inability to obtain the whole image at one scan are the drawbacks need to be overcome.
According to Pendry’s conceptual model, using a slab with the refractive index , both propagating and evanescent waves excited from the objects can contribute to the resolution image . The perfect lens should be realized by tuning the parameters of constituent elements (thin metal wires and split ring) which provide the potential to fulfill the super-resolution condition as mentioned above [10, 11]. However, it is difficult to materialize at present due to the feasibility of simultaneously reaching negative permittivity and permeability, as well as the impedance mismatch between the perfect lens and surrounding medium . Some other lenses which are similar to perfect lens named superlens have shown the ability of breaking diffraction limit theoretically  and experimentally . However, in these so-called superlenses, all the fine features (evanescent waves) cannot be brought to focus by conventional optical devices and instruments.
Recently, the multi-layered metal-dielectric structures [15–17] and the metal nanorod array structures [18–20] have been proposed to have the ability of delivering evanescent information to far-field space optical microscopy. Due to their unusual optical property, the fine features with evanescent mode can be changed into propagation mode . They can be understood by considering the isofrequency curve (dispersion relation between the frequency and the wave vector) with hyperbolic forms which are known as hyperbolic metamaterials. Especially, the hyperbolic metamaterials with multi-layered metal-dielectric shape used for super-resolution are named hyperlens and have caught the eye and drawn attention to their potential applications since its first demonstration in 2007 [21, 22].
The hyperlens with cylindrically or spherically curved multilayer stacks uses an approach of magnifying the sub-wavelength features . The evanescent waves excited from the objects (placed near or on the curved hyperlens) are magnified and transformed into the propagating waves in such anisotropic medium with a hyperbolic dispersion. However, the cylindrical structure has shortcomings [2, 21, 22], since it is inconvenient to put the objects on the curved platform for practical applications. Further, the semicircle space that is constructed by metal will form a cavity and affect the resolution ability at specific operation wavelength.
In this investigation, the proposed hybrid-super-hyperlens [23, 24] with linearly and radially polarized incident light is theoretically investigated. The capability of this lens to break optical diffraction limit is proved by finite element method (FEM) and finite-difference time-domain (FDTD) simulation. The challenge relative to resolve complicated nano pattern is also investigated. Basically, the polarization of light source is an important factor which affects the integrity of the resolution image results from exciting surface plasmon polarions (SPPs) near the patterned region. We demonstrate that the whole magnifying far-field images can be obtained at one scan procedure by using radially polarized light source. That is, superposition of the images under incident light with different polarized directions is unnecessary. The applicability and superiority of hybrid-super-hyperlens for real applications such as photolithography and planar integrated optical devices [25, 26] will also be discussed.
2. Analytical model structure and simulation method
Figure 1(a) shows the conceptual objective lens which is composed of two hyperbolic metamaterials i.e., superlens and hyperlens. For TM polarized wave, both of the optical dispersion relations of the superlens and hyperlens can be obtained by the transfer matrix method 
Considering the long-wavelength approximation (λ>>Λ) and using Taylor expansion to the first-order term, the terms , , and in Eq. (1) will be replaced by , , and ,respectively. Then Eq. (1) is transformed to the following form:Equation (2) denotes an anisotropic metamaterial with extraordinary relative permittivity in different propagating directions.
Based on the superresolution conditions of the hybrid-super-hyperlens that is shown in Ref. 23 and 24, the calculated isofrequency dispersion curves of the uper planar-superlens for the incident light of 405 nm are shown in Fig. 2. Figure 2 exhibits that the isofrequency dispersion curves obtained from Eq. (1) and Eq. (2) are essentially the same in their characters. Hence, under the long-wavelength approximation, it is plausible to apply Eq. (2) to design the multi-layered metal-dielectric shape superlens and hyperlens.
The numerical results shown in this paper are obtained by computational electromagnetism program Lumerical and COMSOL MultiphysicsTM 3.5a which are based on the FDTD and FEM numerical methods respectively. The silver in the visible region is described by the Lorentz -Drude model Fig. 1) is adopted. Since the SPPs only exist for TM polarization, only the grooves that aligned to x axis can be resolved (the corresponding simulation results are shown in Section 3). In contrary, for the beams with radially polarized mode, every position in the beam has the polarization vector (electric field) pointing towards the center of the beam. In cylindrical coordinates system, the electric field distributions (on xy-plane) of the radially beam can be expressed by Fig. 3, in which the arrows indicate the polarization direction of the electric fields on xy-plane.
In general, the radially-polarized source has been used to produce a smaller focused spot and in other applications such as optical trapping . Here, we employ the characteristics of the radially polarized beam that possesses electric components with arbitrary polarization directions. Using the radially-polarized incident wave, the SPPs along all directions of the pattern carved on the sample plate can be excited. Therefore, the whole pattern is expected to be resolved on the image plane. (The corresponding simulation results are shown below in Section 3).
3. Numerical simulation results
The abilities of the hybrid-super-hyperlens to break optical diffraction limit are demonstrated first. Here the simulated object is a pair of nano-slits and the two-dimensional FEM (COMSOL) is utilized in the simulation. The metal in the hybrid-super-hyperlens is silver. The other simulation parameters are listed in Table 1. The relative permittivity values are designed based on Eq. (2), and the thickness ratio () is 1 in the simulation. The corresponding material can be prepared by using available nano-fabrication techniques. (Note that the constituent parameters of the planar superlens and cylindrical (sphere) hyperlens fulfilled the requirement given in ).
Figures 4(a)-4(c) plot the simulated time-averaged power flows (left) and the normalized power intensities as a function of x position measured along the dotted line (right) with the incident wavelengths of 532 nm, 632.8 nm and 650 nm, respectively. The figures show that the pair of slit pair whose center-to-center distance smaller than can be resolved by the hybrid-super-hyperlens. The resolution powers for various incident wavelengths are also listed in Table 1. The figures also display that the signals extracted from the slits can propagate in the planar superlens and in the cylindrical hyperlens, and finally transfer into the far field. This is the reason why the hybrid-super-hyperlens can break the optical diffraction limit.
Next, the images of three-dimensional nano-patterns formed by the proposed hybrid-super-hyperlens are examined by FDTD (Lumerical) simulations. Figures 5(a) and 5(b) plot the simulated time-averaged power flows images below the bottom facet of the spherical hyperlens for a pair of nano-slits and a nano-ring, respectively, carved on the chromium sample stage and illuminated by linearly polarized light with the incident wavelength of 405 nm. (Note that, the sample stage design such as the thickness (width) of slits and type of material can be found in Ref [31, 32].) The materials and geometry parameters of the proposed lens are also listed in Table 1. Both the nano-slits and nano-ring have the widths of 50 nm and are filled with a dielectric of refractive index 1. (The structures are depicted in the insets of Figs. 5). Figure 5(a) displays that the far-field magnified image of the nano-slits pair can be obtained via the hybrid-super-hyperlens. Since the z component of electric field of the incident linearly light is perpendicular to the metal (chromium) surface, SPPs in the vicinity of the pair (nano-slits) are excited . The Fabry-Perot-like resonance inside the slits would induce the SPPs coupling between the upper and lower surface of chromium stage. When the multi-layered superlens is close to the chromium, the excited higher order signals (fine features) with evanescent form can be transferred to propagation mode . After the signals pass through the superlens and arrive at the interface between superlens and hyperlens, they can be received and delivered by spherical hyperlens from the interface to the outer surface. Because they become a propagating mode, the energy can be detected in the far field through a conventional optical microscope with a magnification factor () . Conversely, Fig. 5(b) shows that the resolvable image of the nano-ring cannot be obtained by the lens with a linearly polarized incident light. This result is attributed to the fact that SPPs are excited at parts of the nano-ring only. That is, the polarization direction of the incident light plays an important role in the ability of the lens to resolve a complicated geometrical pattern. To resolve the nano-ring pattern, all the radial directions of this structure should be “seen”. SPPs along the groove of the nano-ring should be excited and all the corresponding fine feature signals should be delivered to the superlens at the same time.
Figures 6(a) and 6(b) depict the time-averaged power flow images of a pair of nano-slits and a nano-ring that are irradiated with radially polarization source (using Lumerical). The images are also extracted at 20 nm below the hybrid-super-hyperlen outer facet. Figures 6(a) and 6(b) exhibit that both of the nano-slits and the nano-ring are resolvable by the radially-polarized incident light, which are significantly different from those in Figs. 5. As we have mentioned, the formation of the images are caused by excitation of SPPs along the groove and transfer of fine-structure signals into the far-field. Notably, an unbroken and completed image can be obtained at one scan procedure by using the proposed hybrib-super-hyperlens. To image more complex nano patterns, the unpolarized light source should be required. Furthermore, all parameters of material presented here are currently existent and available in industry, so that they can be applied to design real and operable devices. The choice of parameters of the material in the hybrid-super-hyperlens is flexible; basically, any values that suit the requirement shown in section 2 will also work well.
The hybrid-super-hyperlens is proposed and numerically investigated. In this proposed lens, by using the anisotropic feature of superlens and hyperlens, the high order signals that reveal the fine feature of the resolved object can be transferred and delivered into far field region. The proposed lens has demonstrated the resolution aboutλ/6 that breaks the diffraction limit. Both the nano-slit pair and the nano-ring are successfully resolved using radially incident source. The feasibility of the proposed lens is assured since all the materials are available.
This work is supported by the National Science Council of Taiwan under grants NSC100-2923-M-002-007-MY3, 101-2112-M-006-002-MY3, 101-3113-P-002-021, 101-2911-I-002-107 and 101-2112-M-002-023. We also thank National Center for High-Performance Computing, Research Center for Applied Sciences, Academia Sinica, Taiwan, National Center for Theoretical Sciences, Taipei Office, and Molecular Imaging Center of National Taiwan University for their kind support.
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