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Abstract
We proposed an alternative method to design a magnifying lens by optical conformal mapping. Different from previous hyperlens or superlens, the proposed lens needs no materials with negative or anisotropic refractive index. The lens has better photonic transporting efficiency than conventional a solid immersion lens due to impedance matching. The proposed lenses have many other advantages, such as broadband, low loss, and no need to redesign the sizes and material parameters when another magnifying ratio is required. Both numerical simulations and experimental demonstrations are implemented to verify the performance of the lens.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is well known that a conventional optical imaging system cannot resolve two objects with a distance smaller than about half a wavelength [1]. This is because the closer of the two objects, the more details are needed to distinguish them. These details usually carried by photons with large wave vectors can only exist as evanescent waves in the near field region and cannot propagate to the far field region to be captured by conventional optical imaging systems. The main reason is the surrounding medium (air or vacuum) cannot support these photons which occupy the extra available states. To obtain these details, one intuitive way is to capture them directly in the near field region. This method usually involves a near-field scanning system and an image reconstruction system, e.g., the scanning near-field optical microscopy [2] and stimulated emission depletion microscopy [3]. Another way is to transfer these near-field evanescent waves to the far field. In this case, materials with high photonic density of states are required, e.g., a dielectric slab (or semi-sphere) with high refractive index. Spherical shape dielectric (usually referred as the solid immersion lens [4–6]) has better performance than a slab because of its geometrical advantage, i.e., less rays will suffer from the total internal reflection. Although solid immersion lenses have the capability to improve the resolution, they usually have a low photonic transporting efficiency due to the impedance mismatching at the boundary. With the development of metamaterials, two different methods were proposed to capture the details in the far field region. One of them is called superlens [7,8], which magnifies the evanescent waves to compensate their exponential decay in air by using negative index materials. Another is called hyperlens [9–11], which uses fan-shaped highly anisotropic materials to transfer the evanescent waves to propagating waves. These two types of lenses are designed by transformation optics [12–14]: a mathematical tool relating material parameters and spatial geometries, which is usually used to design novel optical devices, such as invisibility cloaks [15,16], novel lens [17,18], optical wormholes [19], etc. From the perspective of transformation optics, superlens is the result of a spatial folding and hyperlens is the result of an extreme stretching. Although both two methods can get a better imaging resolution and a high photonic transporting efficiency, they require complex materials, i.e. negative or highly anisotropic. Some other improvements have been implemented based on these methods, such as a parameter simplified magnifying lens [20,21] and a magnifying glass for virtual imaging [22]. Among them, all dielectric lens is more attractive due to its low loss and wide bandwidth features. In Ref. [20], the authors proposed an all dielectric magnifying lens for far field virtual imaging, which is a better version of the conventional solid immersion lens due to its matched impedance to air and its high transmission efficiency. However, the magnifying ratio cannot be changed unless redesigning the lens with a different size or refractive index distribution. In Ref. [23], an all-dielectric subwavelength metasurface focusing lens is designed and fabricated, which is also interesting although it has different function with Ref. [20] and the proposed lens in this paper. Our lens and the lens in Ref. [20] are near field lens (have subwavelength distance from the objects or sources), which can only provide a virtual image and can be used as a front lens to provide magnifying effect for a following optical system. Reference [23] is a far field focusing lens with compressed volume, which provide real images and has the same function of a conventional focusing lens.
In the present work, we use optical conformal mapping (OCM) to design a lens with isotropic and positive refractive index, which can be made by ordinary dielectrics. Our lens has many advantages, such as broadband, low loss, impedance matched to air, and the magnifying ratio is linearly tunable with the height of the lens, i.e., by adding or removing layers of the lens to adjust the magnifying ratio. We experimentally realized the designed lens in microwave region and further discussed the extension to optical region, and we also provide the criterion of resolution capability of the proposed lens.
2. Theoretical method
We begin with the introduction of OCM. In complex analysis, conformal mapping denotes functions which can preserve orientation and angles locally; Riemann sheets are used to describe analytical multiple-valued functions: each Riemann sheet corresponds to one value domain. Lines or curves connecting different Riemann sheets are called branch cuts. OCM can be treated as a special case of general transformation optics. The major advantage of OCMs is their greatly simplified material parameters, i.e. only isotropic and non-magnetic materials are required. In this study, we use Zhukovski transformation, which can be written as:
Fig. 1. Basic design method for designing a magnifying lens. Two objects (two red dots) and the rays emitted from them (blue curves) in (a) the physical space and (b) the reference space of the Zhukovski transformation. The distance δ of the two objects is smaller than half a wavelength in the physical space, while it becomes large in the reference space (d > λ/2). The red circle and red line segments are the branch cuts in the physical space and the reference space, respectively.
Assuming two objects with a distance of δ (< λ/2) exist in the physical space, which are optical indistinguishable when they are placed in air. Here, we place them in a proper position inside the red circle (transformed from the branch cut) in Fig. 1(a) to ensure they have a larger distance d (>λ/2) in the reference space in Fig. 1(b), which can be fulfilled as the Zhukovski transformation has a magnifying effect when some region inside the red circle is transformed to the lower Riemann sheet. Detectors (or human eyes) in the physical space can distinguish these two objects because from their view the two objects have a visual distance d not their real distance δ (see the optical path in the reference space).
The chosen of the positions of the two object is important. For convenience, we assume they have the same height (i.e., the same y coordinate). Their coordinates can be written as: (δ /2, h) and (-δ /2, h) (see Fig. 1(a)), where δ is the distance between the two objects and h is their distance from the x-axis, we can deduce their visual distance d (distance in the reference space) from Eq. (1):
and the magnifying ratio M defined as d/δ can be written as:3. Numerical simulations and discussions
Two-dimensional (2D) numerical simulations are used to verify the performance of the lens. All the following 2D simulations are performed by a finite-element-method based software COMSOL Multiphysics, the boundary is set as perfect matched layers, and the mesh is generated smaller than 1/5 of the effective wavelength. The configurations are shown in Fig. 2(a), where the radius of the branch cut a = 30 mm (red circle) and the wavelength λ0 = a. The lens (light blue region) is above the x-axis with a height of h = 2/3a. The lens has been truncated with an outer radius of 3a since the refractive index outside is very close to the background. With the theory of OCM, the refractive index can be written as:
where z = x + i y, n0 = 1 is the refractive index of the background. Two out-of-plane current sources (yellow dots) with a distance of δ = 1/3a = 1/3λ0 are placed at the bottom of the lens. The electric field distributions Ez is plotted in Figs. 2(b)–2(d). Note our lens can only provide virtual image, and for virtual image, a direct way to show the resolution capability of the lens is comparing the near field or far field patterns with other sources without lens (with different distances). Without the lens, these two sources with a subwavelength distance cannot be resolved as shown in Fig. 2(c). When covered by the lens, the two sources with a distance of δ = 1/3λ0 can be resolved because they have a visual distance of d = M·δ = 1.04λ0, whose far field distribution (see Fig. 2(d)) is very similar to the far field distributions of two sources (in the air) with a real distance of d = 1.04λ0 (see Fig. 2(a)). Although the original refractive index of Eq. (4) has three singularities (two points with n = 0 and one with n = ∞), these points (intersections of the circular branch cut and the x-axis, and the coordinate origin) are excluded from the lens. Almost all the regions of the lens have a refractive index above 1, and the other regions are very close to 1, where we have replaced by the air (we have done simulations to confirm this assumption is valid, no shown here). The refractive index of the lens ranges from 1 to 3.25, which can be realized in a broadband in microwave frequency. We also shown the broadband performance of the lens, which is shown in Fig. 3. We use the same two sources with distance of δ = 1/3a. Five groups of field distributions at different frequencies, i.e. (a-b) 8 GHz, (c-d) 9 GHz, (e-f) 10 GHz, (g-h) 11 GHz and (i-j) 12 GHz, are plotted in Fig. 3 with (right) and without (left) the lens.Fig. 2. (a) Configuration; simulated electric field distributions when the distance between the two sources is (b) δ = 1.04λ0; (c) δ = 1/3λ0; (d) δ = 1/3λ0 (covered by a lens).
Fig. 3. Electric field distributions at different frequencies, i.e. 8GHz-12 GHz from top to bottom. The right and left column are the field patterns with and without the lens, respectively.
For a fixed size of the branch cut a, we can adjust the parameter h, (the thickness of the lens is a-h) to obtain different magnifying ratio. For example, when we choose h = 2/3a, 1/2a, and 1/3a, the magnifying ratios are M = 3.1, 4.6, and 8.2. We can use the lens with M = 3.1 as a fundamental lens, and then we could just add an additional thin layer (or two layers) to change M to 4.6 (or 8.2), i.e. we can change the magnifying ratio just by adding/removing the additional slab accessories. Therefore, we don’t need to redesign the size and materials as previous hyperlens (or solid immersion lens) does when another magnifying ratio is required [20–22]. Figure 4 shows the electric field distributions of two sources (δ = 1/3λ0) (a) with the fundamental lens, (b) added by one-layer slab, and (c) added by two-layer slabs. We can see the magnifying ratio is increased when additional thin slabs are added. The corresponding field distributions of two sources with magnified distance (d = M·δ = 1.03λ, 1.53λ and 2.73λ) in the air are shown in Figs. 4(d)–4(f). To get a clearer picture in mind, we have drawn the corresponding position of the imaginary branch cut (red line) although we don’t take into consideration of the branch cut. Actually, waves in the lower Riemann sheet cannot radiate out from the white line, i.e., some waves cannot exist in the upper Riemann sheet, e.g. where the black arrows point, and that is the reason the far field patterns in Figs. 4(d)–4(f) have some small difference from the ones in Figs. 4(a)–4(c).
Fig. 4. Field distributions of the two-point sources covered by lenses with different magnifying ratios, (a) M = 3.1, (b) M = 4.6, and (c) M = 8.2. (d-f) are the corresponding field distributions in the air.
4. Design with natural materials
The permittivity distribution of the proposed lens with magnifying ratio M = 3.1 is shown in Fig. 5(a), which ranges from 1 to 10. We can use subwavelength structures of alumina ceramics (with ɛ = 9.9) to realize the required permittivity in microwave region. The size length of the square unit cell is chosen to be p = 5 mm, and the side length s of each alumina ceramic square is dependent on its local permittivity, which can be calculated using effective medium theory [24]:
We can use only three types of ceramic blocks to achieve the distribution of discretized permittivity, which is shown in Fig. 5(b) with real structures. The yellow squares are alumina ceramics and the light blue region is air. Figure 5(c) is the three-dimensional (3D) simulation (top view) of the electric field distributions at 10 GHz (λ0 = 30 mm) with material parameters in Fig. 5(b). The top and bottom boundaries are perfect electric conductors. We can see the lens can resolve two objects with a distance of δ = 1/3λ0. For high frequency band, some novel experimental methods to implement all-dielectric lenses have been proposed recently for on-chip applications, such as subwavelength grating waveguide [25], varying the thickness of guiding layer [26], and graded photonic crystals [27].Fig. 5. (a) Permittivity distribution of the lens without discretization. (b) Real structure of the lens with alumina ceramic blocks. (c) Electric field distributions of two subwavelength sources (δ = 1/3λ0) covered by the lens with real 3D structures.
5. Experimental sections
A schematic of the experimental setup is shown in Fig. 6. The sample includes 48 alumina ceramic blocks, which are the same as in the 3D simulation in Fig. 5(c). A photograph of the sample is shown in the upper right corner of Fig. 6. Two dipole sources with a distance of δ = 1/3λ0 = 10 mm are set at the bottom of the sample. The whole sample is placed between two metal plates (yellow planes) with a distance of 6 mm. The top plate is fixed while the bottom plate is bound to one arm of a 2D motorized linear stage. Therefore, the bottom plate together with the sample on it can move freely in the x-y plane. One end of the source cable is connected to the output port (Port 1) of a R&S ZVA 40 vector network analyzer (VNA), and the other end connects to a power divider. Two channel outputs from the power divider are followed by a phase shifter and an attenuator, respectively. Signals from these two channels are carefully modulated to have the same phase and amplitude. A detecting probe with fixed position is connected to the input port (Port 2) of the VNA by a detecting cable. The 2D motorized linear stage is connected to its controlling computer by data cables (blue bidirectional arrowlines). Therefore, we can use a LabView program to give a precise control on the position of the sample. The whole sample is surrounded by a pyramidal microwave absorber (colored blue) so as to reduce reflection and minimize the external electromagnetic interference. The detecting region we choose has a size of 180 mm×140 mm.
Fig. 6. Schematic diagram of the basic experiment setup. A photograph of the sample is drawn in the upper right corner.
A Matlab program was written to process the original scanning data (the measured S21 from VNA) and plot the field distributions. Figure 7 shows the normalized measured electric field distributions 7(d) with the magnifying lens and 7(f) without the magnifying lens. Figures 7(a) and 7(c) are the corresponding 3D simulation results. We can see the measured results fit well with the simulation results. Figure 2(b) and Fig. 7(a) have similar scattering patterns, showing the magnifying effect, and Figs. 7(a) and 7(d) have similar scattering patterns, experimentally verifying the magnifying effect. The interference exists in Figs. 7(a) and 7(d), and we can distinguish the two dipoles by a following imaging system, e.g. human eyes. While in Fig. 7(c) and 7(f), they appear like one dipole, i.e., we cannot distinguish them. Note the beam in Figs. 7(a) and 7(d) is due to the interference effect instead of a directional emission effect. To confirm this, the two dipoles in Figs. 7(a) and 7(d) are replaced by a single dipole, and the resulting field distribution is shown Figs. 7(b) and 7(e). The field distribution is quite different from those in Figs. 7(a) and 7(d). Therefore, the field distributions in Figs. 7(a) and 7(d) are caused by the magnifying effect instead of directional emission effect. Note the sidelobe in the experimental results is weak due to the discretization and the fabrication error, and the scattering pattern could be more obvious with finer discretization and finer fabrication precision.
Fig. 7. Measured electric field distribution of two subwavelength sources (d) with and (f) without lens. (a) and (c) are the corresponding 3D simulation results. (b) and (e) are for the case of a single dipole with lens.
6. Discussion
For optical region, the realization is still challenging as the requirement of the inhomogeneous refractive index. We use ray tracing to show the validity of the lens in optical region (i.e., a≫λ), and further to give the criterion of resolution capability with the lens. Figure 8(a) shows the ray tracing result of two-point sources. Note we only show the rays from one source to see clearly. As mentioned above, the lens can only give a virtual image, and the virtual object (red dots in Fig. 8(a)) is in the convergence point of the reverse extension of the rays. The virtual objects have a larger distance than the real objects (blue dots), which shows the magnifying effect for optical region. The Rayleigh criterion can be written as ${\sigma _R} = \frac{{0.61\lambda }}{{n\sin u}}$, which implies a perfect optical system (with an aperture angle of u = π/2) can resolve two sources with a minimum distance of 0.61λ in free space. We use σphy to denote the minimum distance of two resolvable sources in the physical space covered by our lens (with height of h). In the reference space, the distance between the source and the branch cut can be written as $h^{\prime} = {a^2}/h - h$. The branch cut with finite length of 4a blocks some rays and restricts the aperture angle, i.e., $\sin {u_{ref}} = \frac{{2a}}{{\sqrt {{{({2a} )}^2} + {{\left( {h - \frac{{{a^2}}}{h}} \right)}^2}} }} = \frac{2}{{h/a + a/h}}$, see Fig. 8(b). Therefore, the minimum distance of two resolvable sources in the reference space is ${\sigma _{ref}} = \frac{{0.61\lambda }}{{\sin {u_{ref}}}}$ . With the magnifying ratio M in Eq. (3), the criterion of resolution capability in physical space can be written as:
When the aperture angle u of the following optical system is smaller than uref, the criterion of resolution capability in physical space can be written directly as the multiply of 1/M and Rayleigh criterion σR:Fig. 8. (a) The virtual image obtained by ray tracing method. (b) Branch cut in the reference space acts like an aperture. (c) Relative minimum resolvable distance in free space with (σphy) and without (σR) the lens.
Figure 8(c) shows the comparison of the criterions of resolution capability with and without lens, i.e., σphy, andσR, where we choose sin u = 0.87. A better resolution capability is obtained with the lens, and the minimum resolvable distance is linearly tunable by the relative height h/a. Therefore, we can choose h as small as possible to obtain a better resolution. Note a better resolution is originated from a higher refractive index, which is restricted by the available natural materials or available fabrication technique for artificial materials.
7. Summary
We have proposed a method to achieve a magnifying lens with high resolutions using OCM. Different from previous hyperlens or superlens, the proposed lens has no negative index materials or anisotropic materials, and can be easily realized in microwave region. The lens also has some other advantages, including (1) impedance matched to air, (2) broadband and low loss, (3) can adjust the magnifying ratio by simply adding or removing additional slab accessories. The lens can be used to real time far-field imaging with high resolution, and may pave the way to the development of supperresolution optical microscope systems.
Funding
National Natural Science Foundation of China (61905208, 61971300); National Key Research and Development Program of China (2017YFA0205700).
Disclosures
The authors declare no conflicts of interest.
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