HTML
-
High transverse resolution and high longitudinal resolution are both indispensable for tomographic imaging. As we know, high NA is exceedingly beneficial to high resolution in both the transverse direction (0.61λ/NA by Rayleigh criterion44) and the longitudinal direction (${\rm{DOF}} = \lambda {\rm{/}}(1 - \sqrt {1 - {\rm{N}}{{\rm{A}}^2}} )$45). For a large-NA lens, spherical aberration also plays an important role in both the transverse and longitudinal resolutions. In common cases, the phase profile of a focal metalens is defined by
$$ \varphi = \frac{{2\pi }}{\lambda }(f - \sqrt {{R^2} + {f^2}} ) $$ (1) where R is the position in the radius dimension and f is the focal length. It is derived from the plane wave incidence, and there is no spherical aberration for the flat metalens46. However, in cases of microscopic imaging, the object is quite close to the lens, and the objective field tends to be a spherical wave instead of a plane wave. Therefore, using a normal metalens will cause large spherical aberrations in microscopic imaging, which definitely degrades the image quality. To correct the spherical aberration with respect to the high-NA metalens, we introduced an aplanatic phase profile as a function of the working distances based on the generalized laws of refraction and imaging1 as
$$ \varphi = \frac{{2\pi s}}{{\lambda \left( {f - s} \right)}}\left( { - \frac{{\left( {f - s} \right)\sqrt {{s^2} + {R^2}} }}{s}{\mkern 1mu} + {\mkern 1mu} \frac{{\sqrt {\left[ {2{f^2} - 2fs + {s^2} + \left( {2f - s} \right)\left( {\frac{{2{s^2}}}{{{s^2} + {R^2}}} - 1} \right)} \right]\frac{{{s^2} + {R^2}}}{{{s^2}}}} }}{{\sqrt 2 }}} \right) $$ (2) where s is the object distance and λ is the working wavelength. This is a generalized aplanatic phase profile; when s approaches infinity, it is simplified to Eq. (1). Here, to capture tomographic images as accurately as possible, we choose s = 2 f; then, the aplanatic phase profile becomes
$$ \varphi = \frac{{4\pi }}{\lambda }\left( {2f - \sqrt {{R^2} + 4{f^2}} } \right) $$ (3) The phase profiles of normal and aplanatic metalenses are shown in Fig. 1a, where the parameters are set as f = 80 μm at λ = 532 nm. Figure 1b shows the comparison of spherical aberration (Δs′) between these two kinds of metalens when the object distance s = 2 f = 160 μm with respect to different NA. It is apparent that as the NA increases, the corresponding Δs′ of the normal metalens (blue curve) increases dramatically, while that of the aplanatic metalens (red line) remains at zero. This phenomenon can be clearly manifested by the ray traces of light with a Zemax simulation, as shown in the inset of Fig. 1b (with NA = 0.78), in which all light rays from different incident angles converge at the same point for the aplanatic lens, while the normal lens shows obvious divergence. Although the aplanatic phase profile is designed for a fixed wavelength (532 nm), it still works well in a broadband spectrum. Figure 1c illustrates the theoretical results of normalized spherical aberration (Δs′/s′) as a function of the wavelength λ and lens NA. The region inside the dotted lines indicates that the change in Δs′/s′ is lower than 10%, corresponding to a good imaging performance.
Fig. 1 Aplanatic metalens design.
a Phase profiles designed for a normal (blue) and an aplanatic (red) metalens. b Spherical aberration (Δs′) of the normal and aplanatic metalenses as a function of the numerical aperture (NA), where the inset figures are the corresponding ray tracing results with a lens NA = 0.78 as an example. c Normalized spherical aberration (Δs′/s′) as a function of the NA and wavelength, showing the broadband performance. d, e Full-wave simulation results of the point-source imaging with the normal and aplanatic metalenses, respectivelyTo confirm the theoretical analyses, full-wave simulations were performed by a commercial solution (CST Microwave Studio). Here, f is reduced to 10 μm, and s is set as 20 μm in the simulations to avoid an unnecessary large computation time. Figure 1d shows the 2D axial point imaging with a normal metalens, where to the left is the point source and to the right is the point image, with its transverse and longitudinal full width at half maximum (FWHM) being 968 nm and 11.93 μm, respectively. However, those of the aplanatic metalens are 366 nm and 1.23 μm, respectively, as shown in Fig. 1e. It is evident that the aplanatic lens exhibits greatly improved focusing capability for a certain point source in the 2 f distance, which indicates that a high resolution can be achieved as it is utilized for tomographic imaging.
-
As a proof of concept, an aplanatic metalens based on Eq. (3) with NA = 0.78 at λ = 532 nm with the Pancharatnam–Berry (PB) phase design47, 48 was fabricated with gallium nitride (GaN) nanopillars on a sapphire substrate25 (see Methods). GaN was selected here because it is a low-loss semiconductor material over the entire visible spectrum (the band gap is approximately 3.4 eV, equal to the wavelength of 364.67 nm49). The period of the unit cell is chosen as p = 240 nm, which is smaller than the equivalent wavelength range (450–660 nm) in the substrate (450 nm/nAl2O3 = 253 nm) but greater than the diffraction condition (660 nm/2nAl2O3 = 187 nm) to suppress higher-ordered diffractions. A nanopillar with a height of 800 nm is designed and fabricated after careful optimizations, and the length and width of the nanopillar are 200 nm and 100 nm, respectively, to maintain the anisotropy allowed by fabrication. Figure 2a depicts the calculated conversion efficiency within the band of 450–660 nm with an average efficiency of 79%. Figure 2b shows the optical image and scanning electron microscope (SEM) images of this metalens with a diameter of 200 μm.
Fig. 2 Experimental characterization of the aplanatic metalens.
a Calculated conversion efficiency of the optical field by the unit cell in the working wavelength range. b Optical and SEM images of the fabricated metalens with NA = 0.78. c Experimental longitudinal cross section of the focusing light intensity by a metalens at different wavelengths in the plane wave incidence. d Images of the USAF resolution test chart with the metalens at different wavelengths. e SEM image of a nanoslit sample. f Longitudinal cross section of the moving slit as a 2D point source imaged with the aplanatic (left) and normal (right) metalensesFirst, the chromatic dispersion of the aplanatic metalens was demonstrated as it was illuminated by a white-light laser (Fianium Super-continuum, 4 W) with a wavelength ranging from 450 to 660 nm. A linear polarizer (LP) and quarter-wave plate (QWP) were employed to generate the circular polarization incidence, and another LP and QWP, for cross-polarization analyses after the metalens. To capture the light intensity profile, an achromatic objective (×100, NA = 0.70) and a coupled charge device (CCD) were placed on a motorized stage and moved together along the propagation direction. The measured cross-sectional intensities of the focusing light are partially displayed in Fig. 2c. It is observed that the focal length changes from 102.7 to 69.2 μm as the incident wavelength sweeps from 450 to 660 nm, showing a large focal length change of 42% compared with that of the center wavelength (λ = 532 nm). Due to the limited NA of the collecting objective (×100, NA = 0.70), we did not precisely measure the efficiency in the experiments.
Next, the 1951 United States Air Force (USAF) resolution test chart was employed to test the lens resolution. Instead of the super-continuous laser, a halogen light (incoherent) source is used for the illumination, with filters of 10 nm bandwidth to acquire clear images and avoid speckle noises. The sample was directly mounted approximately s = 2 f = 160 μm in front of the metalens, corresponding to a 4 f optical configuration without the image zoom (here, f = 80 μm at λ = 530 nm). Figure 2d shows the microscopic images of the USAF resolution test chart with filters of λ = 480, 530, and 630 nm from left to right, manifesting clear resolution of Element 3 and Group 9 (i.e., a resolution of 775 nm). As illustrated in Fig. 1c, this aplanatic design has a considerable bandwidth that guarantees this high transverse resolution (≤775 nm) in the whole concerned wavelength range (450–660 nm) (more detailed data are provided in the Supporting Information, section S1), indicating the capability of working in the broadband for the spectral tomography function. As expected, this high resolution obtained from the aplanatic lens design also shows its advantage compared with the normal metalens, as illustrated in the Supporting Information, section S2.
The longitudinal resolution is another important index in our spectral imaging process. Thus, a single nanoslit of 800 nm in width fabricated by a focus ion beam (FIB, dual-beam FEI Helios 600i) was used as a 2D point source to measure the DOF (here, the image distance is fixed at s′ = 2 f = 160 μm). The SEM image of the slit is shown in Fig. 2e. This slit sample was directly placed in front of the metalens and mounted onto an electric translation stage to carefully adjust its position, and then a series of images obtained using the metalens were captured with different object distances. Figure 2f (left) displays the longitudinal map of the middle of the slit imaged by the aplanatic metalens with respect to the translation object distance (Δs), which shows a relatively small DOF of approximately 6.7 μm (532 nm filter with a bandwidth of 3 nm). The clear and blurred inset figures correspond to the focused and defocused cases, respectively. However, for a normal metalens without an aplanatic design, a much larger DOF of 20.7 μm was observed, as shown in Fig. 2f (right). It qualitatively agrees well with our theoretical prediction in Fig. 1d, e (more data at other wavelengths can be found in the Supporting Information, section S1 and S2). This considerably high longitudinal resolution significantly enables this aplanatic metalens to work in a microscopic tomography.
-
This aplanatic metalens based on the geometrical PB phase has demonstrated a large dispersed object distance with respect to wavelengths in the above resolution analyses; therefore, it is capable of resolving different DOF information of a 3D structured object. To clearly demonstrate this process, we prepared four slides with printed rectangular holes of different sizes and orientations, which are particularly arranged along the optical axis. Here, an additional object (O2) (×40, NA = 0.75) is used to translate the macroscopic slide objects to a series of microscopic second-order objects for the metalens imaging with the optical setup shown in Fig. 3a, where the translated object distances are indicated with respect to each slide microscopic image through O2. With the obtained knowledge of the focal length at each wavelength, it is time to obtain image information at different object depths without any moving elements. Figure 3b shows the images of the recorded rotated objects through the aplanatic metalens with corresponding wavelengths of 510 nm, 540 nm, 580 nm, and 640 nm from left to right, and the calculated object distances are 267.0 μm, 224.7 μm, 185.5 μm, and 147.0 μm, respectively, in which the errors are less than 4.5% (see Supporting Information Table S1). From these images, it is easy to distinguish each layer of the orientated holes, confirming the function of tomography. For comparison, we also measured a normal metalens without the aplanatic design for tomographic imaging. The corresponding results are shown in Fig. 3c (detailed comparison results within the wavelength range of 510–640 nm are provided in Supporting Information Fig. S7). It is quite evident that the images from different layers overlap, which indicates loss of the function of tomography. Obviously, the aplanatic design not only endows the metalens with much better imaging quality (i.e., higher resolution) but also shows its necessity in tomographic imaging.
Fig. 3 Spectral 3D tomographic imaging.
a Schematics of the imaging setup. Lighting source is a halogen lamp. The four pictures in the inset are images through an achromatic objective O2 (×40, NA = 0.75) and are used as objects with different depths for the metalens to verify the tomographic imaging. The scale bar is 10 μm. The marked distances are measured distances from the metalens. The experiment captured images through an achromatic objective O1 (×100, NA = 0.70) and a CCD for the cases of (b) the aplanatic metalens and (c) the normal metalens for the same numerical aperture (NA = 0.78) -
With the ability of the DOF resolution being confirmed, this dispersive aplanatic metalens is highly expected to image microscopic biological specimens via tomography. Here, we placed a specimen of frog egg cells directly in front of the metalens with a certain object distance. Figure 4a shows a group of microscopic images obtained by the metalens with different wavelengths from 500 to 560 nm, the colors of which have been removed; the sizes are normalized according to the zoom scaling to show the relatively realistic morphology and inner structures of the frog cells. The direct white-light image obtained by the metalens is given in Fig. 4b, showing a colorful picture due to the large dispersion. It is clear that the images of the cell membrane and nucleus evolve from blurry to clear and back to blurry again as the wavelength increases. Significantly, it is found that the cell nucleus changes from a large dark appearance to a small bright one, the change contrast of which is much stronger than that of the membrane images. This indeed indicates that the cell membrane and nucleus have different depths of field according to their different sizes. By a more careful comparison of these images, one may find that the clearest image of the cell membrane is at λ = 520 nm, while that is at λ = 530 nm for the nucleus, implying that there would be a small location distance between the layer centers of the membrane and nucleus. The derived imaging data of the layer position and imaging scaling with respect to the wavelengths are plotted in Fig. 4c, according to which the depth of the frog cell is roughly estimated to be 35 μm, while that of the nucleus is 5 μm. The detailed spectral images can be switched to another group by modulating the specimen position, which is compared with the results of Fig. 4a without dropping colors, as shown in the Supporting Information, section S3. From this information, we can also deduce the same morphologic information of the cell samples. Note that the frog egg cells are from a specimen that has already been flattened to some extent during preparation; thus, their morphologies are not kept spherical as was the case for the living ones.