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Abstract
Due to the wave nature of light, the resolution achieved in conventional imaging systems is limited to around half of the wavelength. The reason behind this limitation, called diffraction limit, is that part of the information of the object carried by the evanescent waves scattered from an abject. Although retrieving information from propagating waves is not difficult in the far-field region, it is very challenging in the case of evanescent waves, which decay exponentially as travel and lose their power in the far-field region. In this paper, we design a high-order continuous dielectric metasurface to convert evanescent waves into propagating modes and subsequently to reconstruct super-resolution images in the far field. The designed metasurface is characterized and its performance for sub-wavelength imaging is verified using full wave numerical simulations. Simulation results show that the designed continuous high-order metasurface can convert a large group of evanescent waves into propagating ones. The designed metasurface is then used to reconstruct the image of objects with sub-wavelength features, and an image with the resolution of λ/5.5 is achieved.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In many research fields, achieving high-resolution images can lead to a better understanding of the subject under study. However, owing to the diffraction limit, the resolution of conventional imaging systems is restricted to λ/2NA, where λ is the working wavelength, and NA is the numerical aperture of the imaging system [1–3]. The reason behind this limit is that the evanescent waves, scattered from the object, exponentially decay and finally disappear in the far-field region [1–3]. Therefore, developing methods to conquer such a limitation would be of pivotal significance.
Different methods have been introduced to reconstruct super-resolution images [4–21]. Generally, the previously developed methods can be categorized as near-field [4–10], and far-field techniques [11–21]. Some of near-field methods are based on scanning the near-field area of an object, where the evanescent waves have not yet been decayed [4–6], using a probe with subwavelength dimensions. The requirement to scan the object in these methods makes them time consuming and limits the observation area. In another category of near-field techniques, flat lenses made of very thin layer of noble metals, such as silver or gold, are used to enhance the evanescent waves resulting in a resolution higher than diffraction limit [7–10]. However, those enhanced evanescent waves start to decay just after leaving the flat lens, and, therefore, the image should be reconstructed in the near field area very close to the lens and object (in nano-meter scales in practical works) which is impractical for many applications [7–10]. To overcome the drawbacks of near-field imaging techniques, different far-field imaging methods have been developed up to now [11–21]. In the category of far-filed imaging, we can name hyperlenses, which exploit curved hyperbolic metamaterials to convert the evanescent waves scattered from the object to the propagating ones, which can be detected in the far-field region [11–15]. However, the fabrication of a curved structure in nanometer scales is very challenging [13]. Another far-field imaging technique is using far-field superlenses, which use periodic structures to convert evanescent waves to propagating modes [16–18]. However, in this process, some undesired diffraction orders will be produced, and eliminating them requires a filter to be placed before the grating structure so as to obviate the undesired orders [16–18].
To eliminate the need for a filter in far-field superlenses, in our previous work [19], we proposed to use gradient metasurfaces instead of grating structures and showed that gradient metasurfaces can be used to convert evanescent modes to propagating ones leading to a resolution higher than diffraction limit [19]. However, the designed structure suffered from low efficiency, polarization conversion, and limited accuracy provided by discrete gradient metasurfaces. Metasurfaces, due to their ultrathin dimension in the direction of propagation, have been replaced 3D bulky metamaterials in many applications such as invisibility cloaks [22–25], flat lenses [26–28], holograms [29,30], optical beam steering [31–33], polarization control [34–36], plasmonic solar cells [37,38], photonic topological insulators [39], subwavelength imaging [19] and perfect absorbers [40,41]. Plasmonic metasurfaces [42] suffer from low efficiency, which can limit their application in imaging. To increase the efficiency of transmissive metasurfaces, several methods such as using metascreens [43] and metamaterial Huygens’ Surfaces [44] have been introduced. However, a more practical solution to increase efficiency of metasurfaces is using dielectric unit-cells [45–52]. The efficiency of dielectric metasurfaces can be dramatically higher than plasmonic metasurfaces since they use dielectric resonators with negligible loss at the optical wavelengths [53,54]. On the other hand, using several separate unit-cells to form a metasurface results in a discrete spatial phase profile for reflection and transmission wavefronts. In this regard, continuous metasurfaces [55–58] can be used to reconstruct the wavefronts more accurately. In other words, due to the spatial continuity of phase response in continuous metasurfaces, they have higher efficiency than the ones with discrete unit-cells [56–58].
Here, to overcome the limitations of previously developed works, we have designed a high-order dielectric continuous metasurface to efficiently convert a large group of evanescent waves to propagating ones, resulting in a sub-wavelength image with resolution of much higher than the diffraction limit. In order to increase the resultant resolution, in the proposed technique, we use both first order and second order metasurfaces in the imaging process, and then we combine the information obtained using them. Finally, based on the derived information, images with subwavelength resolutions are reconstructed. Unlike previously reported works, the designed metasurface does not change the polarization of the incident waves and therefore eliminate the need for a polarizer in the imaging setup. In order to improve the efficiency of the method, the utilized metasurfaces are designed based on all-dielectric unit cells to avoid plasmonic loss. Moreover, the designed metasurfaces have continuous structures to reconstruct the transmitted wavefronts more accurately. It should be noted that the technique we propose here, is different from using metalenses [26,28]. Metalenses are flat lenses that can reconstruct images by themselves. However, their resolution is limited by diffraction limit [19]. Here we use phase gradient metasurfaces for evanescent-to-propagating wave conversion. A capability that can be used as part of an imaging system to reconstruct sub-wavelength images.
The paper is organized as follows. First in section 2, the method to design the high order continuous metasurface is explained and the designed structure is introduced and discussed. Then, in section 3, the performance of the designed metasurface in converting evanescent waves to propagating ones is investigated using full-wave numerical simulations. After that, in section 4, the designed metasurface is used for subwavelength imaging and the reconstructed images are presented and discussed, and the resolution limit of the proposed structure is calculated through numerical simulations. Finally, section 5 concludes the paper.
2. Converting evanescent waves to propagating ones using a continuous high order metasurface
The structure of the designed metasurface is shown in Fig. 1. Figure 1(A) illustrates the unit-cell used here as the building block of the designed metasurface. As shown in this figure, the unit cell is designed to be an all-dielectric resonator to avoid plasmonic loss. Figures 1(B) and (C) show the top view of the designed continuous metasurfaces. Figure 1(B) illustrates the designed continuous first order metasurface while Fig. 1(C) shows the second order one. In this work, we have designed continuous metasurfaces to avoid step-wise spatial phase profile in the transmitted wavefront provided by discrete metasurfaces.
Fig. 1. A: 3D view of the unit-cell, where t1 = 900 nm is the height of the silicon resonator, t2 = 200 nm is the height of the silica substrate, l = 230 nm is the length of the resonator, p = 230 nm is the dimension of the unit-cell, and w is the width of the resonator. B: Top view of the first order supercell with the length of Γ1 = 1610 nm. C: Top view of the second order supercell with the length of Γ2 = 1610 nm.
To explain the behavior of our designed metasurfaces, we use generalized Snell’s law [42]. Assuming that our metasurface is illuminated by an incident plane wave with the transverse wavenumber of kt,i, according to the generalized Snell’s law, the transmitted wave will have a transverse wavenumber of kt,t, which can be written as:
where $\nabla {\varphi _m}$ is the gradient of the phase produced by the metasurface, m is the order of metasurface, and $\mathrm{\Gamma }$ is the period of the supercell of metasurface (see Fig. 1(B) and Fig. 1(C)). According to Eq. (1) and Eq. (2), a shift can be applied to the incident transverse wavenumber using metasurfaces. Therefore, by properly designing the metasurfaces, a desired range of evanescent waves (waves with transverse wave numbers larger than free space wavenumber, ${k_0}\; ) $ can be transferred into the propagating range (waves with transverse wave numbers smaller than ${k_0}\; )$. Moreover, Eq. (1) and (2) show that as the phase gradient (m2$\pi $ /Γ) increases, larger values of incident transverse wavenumbers can be transferred into propagating range. This is equivalent to achieve information about smaller sub-wavelength features of the object, leading to a higher resolution in the imaging process. In this regard, we have designed both the first (m = 1) and second order (m = 2) metasurfaces (see Fig. 1) to transfer a wide range of large transverse wavenumbers into propagating range.As shown in Fig. 1, in our designed unit-cells (see Fig. 1), we have used Silicon on Silica structure, in order to benefit from the high refractive index of Silicon at optical frequencies to reduce the height of the unit-cells. It should be noted that, owing to the exponential decaying of evanescent waves, the thickness of resonators should be as less as possible so that evanescent waves would be able to interact with the resonators efficiently. Here, the thickness of the Silicon resonators are selected to be t1 = 900 nm = 0.6λ0, at the wavelength of λ0 = 1500 nm. Moreover, to design a continuous metasurface for evanescent-to-propagating wave conversion, the length of nano-brick resonators, l, is selected equal to the periodicity of the unit-cells, p = 230 nm. Here, without loss of generality, the thickness of Silica substrate is chosen to be 200 nm. However, in practice, this thickness can be larger and its effect should be considered in the computation of transfer functions, as will be explained in details in the next sections.
The numerically calculated phase and amplitude of the transmitted wave versus dimensions of unit cell, w and l (see Fig. 1), are shown in Fig. 2. The results are achieved through full wave numerical simulation in COMSOL Multiphysics software (a commercial solver operating based on finite element method). In this simulation, the periodic boundary condition is applied around the unit cell, and the structure is excited using a normally incident plane wave. As the results of Fig. 2 shows, a full phase coverage from 0 to 360° can be achieved for the designed unit cells, and also, the amplitude of the transmitted wave has a high value (more than 80%) for a large number of unit cells with different values of w and l. Therefore, according to the numerical analysis results, the designed unit cell has a proper performance for the application of sub-wavelength imaging.
Fig. 2. 2D phase graph (A) and 2D amplitude graph (B) versus different values of l ranging from 80 nm to 230 nm and w ranging from 80 nm to 210 nm (with 5 nm steps for both l and w).
As a basic principle in designing the high-order metasurfaces, the number of resonators in a super-cell must not be a factor of the order of the correspondence metasurface [59,60]. Therefore, for the first and second order metasurfaces, we have used 7 unit-cells in a super-cell. To design a second order metasurface (which linearly covers 0-4π phase range), first, the first order metasurface (which linearly covers 0-2π phase range), is designed. Then, by rearranging its unit-cells, the second order metasurface is obtained (see Fig. 1(C)). The width of the unit-cells used in the first and second order metasurfaces and their corresponding phase and amplitude graphs, are shown in Table 1, and Fig. 3, respectively.
Fig. 3. Phase (A), and amplitude (B) graphs of the first order (blue) and second order (red) metasurfaces versus the longitudinal direction (x-direction).
Table 1. The width of resonators selected in the design of first and second order metasurfaces
3. Numerical results for evanescent to propagating wave conversion
Here, we numerically show that using the designed metasurfaces, evanescent waves can be converted to propagating ones. To do so, full wave numerical simulations are done using COMSOL Multiphysics software. In these simulations, the metasurfaces are excited using evanescent waves with incident transverse wavenumbers, kt,i, larger than k0. For the boundary conditions, we have used perfect electric conductor (PEC) boundary condition in the y direction and periodic boundary condition in the x direction (see Fig. 1(A)). The results of these simulations are shown in Figs. 4 and 5, for the first and second order metasurfaces, respectively.
Fig. 4. A, C, and E: Evanescent waves with transverse wave numbers of 1.2k0, 1.5k0, and 1.7k0, respectively, in the absence of metasurface. B, D, and F: The corresponding A, C, and E evanescent waves, respectively, have been converted to propagating plane waves in the presence of the first order metasurface.
According to (2), the first-order metasurface with Γ1 = 1610 nm (see Fig. 1(B)), and negative phase gradient (negative sign in (2)), reduces the incident transverse wavenumbers with the value of −1500k0/1610 = -0.93k0. In other words, by using this metasurface, transverse wavenumbers in the range of {-0.07k0 < kt,i < 1.93k0} can be transferred into propagating range, {-k0 < kt,t < k0}. As shown in Figs. 4(A), (C), (E), where the first order metasurface does not exist, the incident evanescent waves with transverse wavenumbers of 1.2k0, 1.5k0, and 1.7k0 decay exponentially in the z-direction. However, by using the designed first-order metasurface, these waves have been converted into propagating waves (see Figs. 4(B), (D), (F)). It should be noted that for the positive phase gradient metasurface (which is obtained by simply rotating the metasurface around the z-axis by 180 degrees), the evanescent waves with transverse wavenumbers in the range of {-1.93k0 < kt,i < -0.07k0} can be transferred into the propagating range.
For the second order metasurface, the value of m in (2) is 2. In this regard, the rate of phase gradient for the second-order metasurface is 2 × 0.93k0 = -1.86k0, and therefore transverse wavenumbers in the range of {0.86k0 < kt,i < 2.86k0} can be transferred into propagating range. Therefore, as the order of metasurface increases, evanescent waves with larger values of transverse wave number can be converted to propagating waves. The performance of the second-order metasurface in evanescent-to-propagating wave conversion is shown in Fig. 5, where incident evanescent waves with large transverse wavenumbers are illuminated on the metasurface and then converted to propagating waves. Similarly, by the help of the positive phase gradient metasurface, the evanescent waves with transverse wavenumbers in the range of {-2.86k0 < kt,i < -0.86k0} can be transferred into the propagating range.
Fig. 5. A, C, and E: Evanescent waves with transverse wave numbers of 2k0, 2.5k0, and 2.6k0, respectively, in the absence of metasurfaces. B, D, and F: The corresponding A, C, and E evanescent waves, respectively, have been converted to propagating plane waves in the presence of the second order metasurface.
For a more detailed study of the evanescent-to-propagating wave conversion, the transmitted transverse wavenumber versus the incident transverse wavenumber is shown for both first and second order metasurfaces, in Fig. 6. Multiple transmitted modes can be observed in these figures, which their existence was predictable according to relations (1) and (2). According to Fig. 6(A), most of the power is concentrated in “m = 1 mode,” which shows the domination of the first order mode in the first order metasurface. On the other hand, for the case of second order metasurface, the “m = 2 mode” is dominant (see Fig. 6(B)). There are also undesired modes (m = -1, 0, and 2 for the first order metasurface, and m = 0, 1, and -1 for the second order metasurface) in both of the designed metasurfaces. Undesired modes other than m = 0, are due to the periodicity of the super-cells. However, in the imaging process, the interaction between light and metasurfaces is restricted to only few unit-cells, therefore these modes will not have the opportunity to be excited [19]. Unlike these modes, m = 0 mode is due to the ordinary refraction and exists at far-field in the imaging process. To eliminate this mode in the imaging process, field distribution in the absence of the metasurfaces is recorded at far-field and then is subtracted from the total field.
Fig. 6. The 2D intensity graph of different potentially existing modes for the first order metasurface (A), and for the second order metasurface (B).
Moreover, it should be noted that the m = 0 mode (propagating waves scattered from the sample in the absence of the metasurface), should be added to the information obtained by using metasurfaces. The reason behind this is that due to the shift in the transverse wavenumbers applied by the proposed metasurface, part of previously propagating waves is converted to the evanescent waves and will be lost in the far-field. Imaging without metasurface can be used to retrieve this part.
4. Subwavelength imaging using the designed high order metasurfaces
The first step for reconstructing subwavelength images with this method is calculating the transfer functions (T-functions) for the two metasurfaces, designed in the previous sections. In this regard, we have put an aperture with a subwavelength width (wa = λ0/15), acting like a delta function, in front of each of the two metasurfaces (see Fig. 7). Then, a y-polarized plane wave is illuminated to the aperture. After the illumination, on the other side of the aperture, many evanescent and propagating waves with different transverse wave numbers as well as different polarizations will be produced (see Fig. 8). Here, the distance between the metasurface and the sample, d, is 150 nm. This is due to the fact that before extinction of evanescent waves scattered from the sample, they must interact with the metasurface. In practice, this distance can be realized using a suitable spacer between the sample and the metasurface [17].
Fig. 7. The simulation setup used to retrieve the transfer function of the metasurface in which wa = λ0/15 is the width of the aperture, ta = 500 nm is the height of the aperture, d = 150 nm is the distance between aperture and the metasurface, r = 3000 nm is the distance between substrate to image plane.
Fig. 8. A: The numerically calculated field distribution for the first order metasurface (A), and the second order metasurface (B).
In the far-field region, the amplitude and phase of Ey is recorded on the image plane (see Fig. 7), and the m = 0 mode is removed for both the first order metasurface, Ey1 (see Fig. 9(A)), and second order metasurface, Ey2 (see Fig. 9(B)), and their corresponding Fourier transforms (F{Ey1}| Image plane and F{Ey2}| Image plane), are computed. Then, these Fourier transforms must be shifted by $\nabla {\varphi _1}$ (F{Ey1}| Shifted), for the first-order metasurface and $\nabla {\varphi _2}$ (F{Ey2}| Shifted) for the second order metasurface, to compensate the phase gradients applied by the metasurfaces [19]. Finally, F{Ey1}| Shifted and F{Ey2}| Shifted, are divided by the Fourier transform of Ey on the input plane and in the absence of the metasurfaces (F{Ey}|Obj. plane). The results will be the T-functions of the first-order and second-order metasurfaces, which are shown in Figs. 10(A) and 10(B).
Fig. 9. A: Recorded electric field on the image plane in the presence of the first order metasurface (A), and in the presence of the second order metasurface (B).
Fig. 10. A: The amplitude (blue) and phase (red) graphs of the transfer function of the first order metasurface (A) and the second order metasurface (B).
After calculating the T-functions, in the next step, we want to reconstruct the images of two sub-wavelength apertures placed next to each other with a subwavelength gap of s, (see Fig. 11). In this regard, the sample is put behind the first and second order metasurfaces each time, as shown in Fig. 11(a) and y-polarized plane wave is incident on it. The field distributions (see Fig. 12), on the image plane is recorded in both cases of the first and second order metasurfaces and the m = 0 mode is eliminated as demonstrated in the previous section. By taking Fourier transform of the derived fields and dividing them by their corresponding T-function, and finally putting the two calculated spectrums next to each other, the Fourier spectrum of images will be achieved. The information obtained using positive phase gradients also can be added to these spectrums. The final combined Fourier transforms of the images are shown for cases of s = λ0/3, λ0/4, λ0/5, and λ0/5.5 in Figs. 13(A), 14(A), 15(A), and 16(A), respectively. Eventually, by getting the inverse Fourier transform, the final images will be reconstructed (W MS1 & MS2), which are shown in Figs. 13(B)–16(B), and the results have been compared with the case that only the first order metasurface has been used in the imaging process (W MS1), no metasurface has been used (WO MS), and the sample itself. In the case of s = λ0/3 (see Fig. 13(B)), without using the designed metasurfaces, the aperture cannot be resolved in the reconstructed image. However, only by using the first order metasurface these apertures are resolvable. In other words, the first order metasurface converts a part of evanescent waves into propagating modes, and using the information included in these modes is enough to resolve the apertures. Moreover, Fig. 13(B) shows that using both first and second order metasurfaces, a deeper depth is created between the images of the apertures, which is due to using information of the second order metasurface in addition to the first one. In the cases of s = λ0/4 and s = λ0/5 (see Figs. 14(B) and 15(B)), again without the metasurfaces, the apertures are not resolved. For these two cases, unlike the s = λ0/3 case, using only the information obtained by the first order metasurface is not enough to resolve the apertures. This is evident in Figs. 14(B) and 15(B), where an erroneous peak is appeared between the images of the two apertures. This is due to the loss of information contained in larger transverse wavenumbers, which have not been retrieved by the first order metasurface. However, when the information extracted from the second order metasurface is also used, the erroneous peak disappears and the apertures become resolvable. As the gap between the apertures reduces, the erroneous peak will be appeared even if the second order metasurface is used. This is evident in Fig. 16(B), for the case of s = λ0/5.5. According to the Rayleigh criterion, for gaps smaller than λ0/5.5, the erroneous peak will be observed as an object, which does not really exist. Therefore, according to these figures, a resolution of λ0/5.5 is obtained by the proposed technique.
Fig. 11. The imaging setup in which da = λ0/4 is the width of the apertures, ta = 500 nm is the height of the aperture, d = 150 nm is the distance between aperture and the metasurface, r = 3000nm is the distance between substrate to the image plane, and s is the distance between two apertures.
Fig. 12. A: The field distribution of the apertures with s = λ0/4, in the presence of the first order metasurface (A), and in the presence of the second order metasurface (B).
Fig. 13. A: Combined Fourier transform after doing the imaging process for s = λ0/3. B: The results of different imaging processes for s = λ0/3. Here, WO, MS, W, MS1, and MS2 stand for “without”, “metasurface”, “with”, “first order metasurface”, and “second order metasurface”, respectively.
Fig. 14. A:Combined Fourier transform after doing the imaging process for s = λ0/4. B: The results of different imaging processes for s = λ0/4. Here, WO, MS, W, MS1, and MS2 stand for “without”, “metasurface”, “with”, “first order metasurface”, and “second order metasurface”, respectively.
Fig. 15. A: Combined Fourier transform after doing the imaging process for s = λ0/5. B: The results of different imaging processes for s = λ0/5. Here, WO, MS, W, MS1, and MS2 stand for “without”, “metasurface”, “with”, “first order metasurface”, and “second order metasurface”, respectively.
Fig. 16. A: Combined Fourier transform after doing the imaging process for s = λ0/5.5. B: The results of different imaging processes for s = λ0/5.5. Here, WO, MS, W, MS1, and MS2 stand for “without”, “metasurface”, “with”, “first order metasurface”, and “second order metasurface”, respectively.
5. Conclusion
In this work, we introduced a technique for far-field subwavelength imaging by using high-order metasurfaces. In this regard, the first and second order continuous dielectric metasurfaces were designed and used for sub-wavelength image reconstruction and a resolution up to λ0/5.5 was obtained. For making this method more efficient, the use of tunable metasurfaces [61,62] can be considered in order to reduce the number of steps in the imaging process. Also, for achieving better image resolutions, we need to design supercells with smaller resonators. This is due to the fact that smaller resonators result in larger phase gradients and therefore larger transverse wavenumbers can be transferred into propagating range. Moreover, by rearranging these resonators we can achieve higher-order metasurfaces with even larger phase gradients and, as a result, broader Fourier spectrums will be obtained, and consequently, the resolution will be improved even further.
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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