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Abstract
Planar metalenses are regarded as promising functional nanodevices because of their lightweight, nano-resolution properties, and, therefore, they can serve as versatile platforms for imaging and Fourier transforming. Here, we demonstrate a meta-device that functions as an isotropic bifocal all-dielectric Huygens' metalens to realize nanoscale real-time coaxial digital hologram generation. We design an isotropic bifocal metalens for micro/nano hologram recording, and the metalens utilizes the complete region compared to a previously reported interleaved multifocal metalens scheme. In addition, the hologram generation does not depend on complex polarization conversion, thereby improving the practical efficiency. For high-fidelity reconstruction, compressive reconstruction is utilized to remove twin-image and zero-order items and to suppress noise. Such concept would be extended to white-light achromatic meta-holography and three-dimensional micro/nano in vivo incoherent super-resolution imaging under subwavelength modulation.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces can be used to flexibly manipulate the phase, amplitude, polarization, and other properties of light, and they have the characteristics of miniaturization and lightweight [1–6]. Metasurfaces provide new opportunities to develop ultrathin devices that are easy to integrate into compact platforms. Metasrufaces have broad application in holography, Fourier optics, analog computing and so on [7–12]. Huygens’ metasurface is an important type of metasurface used to achieve almost uniform transmission efficiency and flexible phase manipulation. For simplicity, Huygens’ metasurface can function as a combination of orthogonal electric and magnetic dipoles with equal intensity. Huygens’s principle states that every point of wavefront can be considered as a second wavelet source and propagation forward under a certain condition, independent on input polarization [13]. As an important branch of metasurface, ultrathin metalens has received significant attention because of their planar, achromatic advantages. Metalenses can reduce the bulk size of diffractive lenses, and they can spatially modulate the phase discontinuities within ultrashort distances. Therefore, metalenses are used for white-light focusing and full-color imaging [14–18]. Specially, metalenses can be used to perform multi-foci/tunable focusing, spectral tomography, among others by applying different mechanisms [19–27]. In addition, such metalenses have wide applications to improve performance and open new avenues in imaging systems, optical tomography, information optics, and optical communications [28–33].
Digital holography (DH) refers to the technique for the acquisition of holograms using a digital sensor (CCD/CMOS). Usually, the object wave and reference wave interfere with each other to generate a hologram, which is recorded using an optical system. Subsequently, the hologram is processed and reconstructed using a computer [34]. Compared with computer-generated hologram, DH records in situ hologram of a real object by using an optical system in real time; therefore, the DH is generally used for microscopy, 3D holographic storage, and dynamic samples capture. The conventional coaxial holographic recording method (Fresnel incoherent correlation holography, FINCH) utilizes diffractive optical components and/or spatial light modulators (SLMs) to record digital holograms [35–42]. In the FINCH optical configuration, the object wave is split into two by using an encoding diffractive lens displayed on an SLM, and two images are created. An image sensor is placed between both the images, and the self-interference hologram is recorded. Because of the twin-image generated from coaxial hologram, many methods are used to eliminate it and enhance the reconstruction quality. In addition, three- or four-step phase-shifting technology, in which phase-shifting holograms are restored and superposed to produce a complex hologram and also reconstructed, is widely used as an effective approach to eliminating twin images, and these technologies are used for DH reconstruction. However, the commercial spatial light modulator has characteristic micrometer pixel size, and there exists dead zone that reflect or transmit light without modulation, which can cause zero background noise. Hence such method has some limitations for achieving micro-nano size holographic imaging. In addition, multi-holograms need more recording operations, thereby consuming significant amount of time. Moreover, the phase-shifting method for multi-holograms recording often faces the challenge of phase mismatch during the instances of mechanical movement or phase noise due to disturbances in system components [43–45], especially in meta-optical experimental systems.
Here, we demonstrate a high-efficiency isotropic bifocal Huygens’ metalens, which is used for polarization-insensitive nanoscale coaxial digital hologram generation and imaging-system minimization. Such an optimal all-dielectric bifocal metalens has two foci along the optical axis in the forward transmission space. Therefore, the real object will form two on-axis focused images along the forward direction. The coaxial hologram is generated because of the interference of two identical defocused images on a fixed position between two focused planes. Compared with the traditional hologram recording process that uses diffractive optical elements or chemical films, such real time generation provides a significantly simpler and more efficient method. In addition, such bifocal metalenses can minimize the imaging system, improve the resolution, and provide an easy approach for micro/nano scale high resolution DH. In addition, we adopt the compressive sensing (CS) algorithm to reconstruct the coaxial hologram to eliminate the twin-image and zero-order items. Compressive sensing is a method that can retrieval the original signal through less sampling points than the Nyquist sampling theorem [46]. Because there is larger difference between the reconstructed image and the twin image in the coaxial holographic reconstruction. CS can greatly suppress the twin image and obtain a high-quality image through sparse sampling and reconstruction methods. Such holographic metalens configuration enables flexible and non-mismatched hologram recording, in contrast to a previously reported static metasurface hologram based on preparatory calculation. Furthermore, through delicate design and fabrication, such a bifocal metalens can even be expanded to broadband achromatic range applied for 3D phase or biomedical object recording and reconstruction under incoherent source/fluorescence illumination or in vivo imaging [15–16,47–48].
2. Methods and results
We demonstrated the real-time holographic recording performed using a bifocal isotropic metalens. Such a meta-device minimizes the holographic imaging system. In addition, the reconstruction is realized on the basis of the CS algorithm, as depicted in Fig. 1. To obtain a high-quality hologram, arbitrary linearly polarized light is used as the illumination. When linearly polarized light passes through the object (USAF1951, negative resolution test chart) and impinges on the bifocal metalens, two images with different magnifications are generated on two focused axial planes. While in the overlap region between the two focal planes, there always exists one fixed plane (maximum section of overlap) with approximately same magnification and intensity, and the plane can be used for recording the interference hologram by camera. Because the recorded hologram is attributed to the coaxial interference, the real image and its twin image will simultaneously appear at the center of the reconstructed plane and become indistinguishable. An effective and time-saving CS algorithm is used for removing the twin-image in the reconstruction process. Consequently, high-quality reconstruction results are obtained.
The principle of the real-time hologram generation by using a bifocal metalens is based on two sphere wave interference along the coaxial propagation direction, as depicted in Fig. 2. One on-axis point of the object is propagated to the bifocal lens with object distance Ds and is imaged to two different on-axis points, which are located at D1 and D2 planes, respectively. Hence, the two coherent beams can interfere with each other to form a point-source hologram between both the focus planes. Similarly, for the point cloud of the object, all the interference point-sources (hologel) are superimposed to generate the entire hologram. Notably, for simplicity, only the case of on-axis points is considered. However, the principle and results are available for all the off-axis points in the linear space invariant system.
Fig. 2. Hologram recording diagram, (a)Coaxial holography principle scheme; (b) Bifocal metalens wrap phase profile. Ds: distance between object voxel and bifocal metalens; Dh: recording distance; D1 /D2: two beams converge distance from bifocal metalens.
The intensity distribution of the on-axis point-hologram E obeys Fresnel paraxial approximation, as follows [49]:
The above-mentioned hologram E can also be derived as follows:
For designing the polarization-independent bifocal metalens, cylindrical silicon nanodisks with various diameters based on Huygens’ metasurface principle are arranged according to the bifocal metalens phase distribution:
The 45° side views of the scanning electric microscopy images are depicted in Figs. 4(a) and (b). Before performing the imaging experiment, we first characterized both the foci of the metalens. The propagation along the longitudinal direction was simulated according to Fresnel diffraction, as depicted in Fig. 4(c). We observed two foci located at 1400 and 1600 µm respectively, showing consistency with the theoretical design. In addition, clear interference fringes were observed between the two foci. To experimentally measure the focal lengths, a plane wave was directly illuminated on the metalens without using the resolution chart in the setup. We changed the distance between the metalens and objective via the step size of 10 µm by using a precision 3D stage; therefore, different propagation planes could be captured using the CCD. The normalized diffraction intensity along the axial direction is depicted in Fig. 4(d). Both the experimentally characterized foci are located at the distances of 1340 and 1580 µm, respectively, showing satisfactory correspondence to the simulation. Note that there is slightly deviation of the measured foci from the simulation results, which may due to the finite aperture of metalens as well as fabrication accuracy. We provide normalized intensity distribution and its line profile at both foci plane in Fig. 4(e). Note there are some fringes around the focal points, which may result from the diffraction of the bifocal metalens. With the optimization of the metalens and the improvement of the fabrication accuracy, the resolution can be further enhanced. Similarly, we observed a clear interference fringe between both the foci, and this is the precondition for hologram generation. In addition, we performed the Fourier transform infrared spectroscopy (FTIR) of the metalens in the range from 500 to 1100 nm, as depicted in Fig. 4(f). The linear polarization transmission efficiency reached up to 77.6%. Such high efficiency can guarantee the high-quality recording of coaxial holograms.
Fig. 3. Nanostructure and optical setup: (a) Nanodisk structure; (b) The phase and amplitude of the nanodisks at wavelength 800 nm; (c) Partial phase profile of the double-focal metalens, and it corresponds to nanodisks with various radii; (d) Nanodisk electromagnetic energy distribution; (e) The optical setup of coaxial holography (USAF1951: resolution chart; objective: 20×/0.45; CCD: camera).
Next, we included the resolution chart to the setup. The real-time in situ hologram was generated on the fixed plane of approximately 1500 µm away from metalens. In addition, we used the CCD to record a point-source hologram and a series of holograms with Arabic numbers holograms 1 − 6 of group 5 (measured linewidth: 4.32, 3.51, 3.24, 2.97, 2.7, and 2.16 µm) in the standard resolution chart(USAF 1951) in Figs. 5(a)–(g). Because these holograms were generated from the coaxial optical setup configuration, twin images and zero-order items were unavoidable during the traditional reconstruction process upon using the Fresnel back-diffraction method, as depicted in Figs. 5(h)–(n). Specifically, the reconstructed point is surrounded by a strong diffraction halo, as depicted in Fig. 5(h). In addition, the reconstructed Arabic numbers (holograms 1 − 6) are disturbed because of twin image diffraction.
Fig. 4. Scanning electric microscopy images and bifocal metalens axial intensity. (a) Top-view and (b) 45°-side-view SEM images. : (c) Simulated and (d) Experimental results of intensity distribution along propagation direction of such bifocal metalens; simulated focal lengths: 1400 and 1600 µm; measured focal lengths: 1340 and 1580 µm; (e) Normalized intensity distribution at the two foci and their line cross-section; (f) FTIR-measurement of different polarization channels.
To remove the twin images and enhance the reconstruction quality, the CS algorithm was adopted in the reconstruction process [46,49–50]. Assuming the measurement intensity Ig and the objective signal f satisfy the relationship as follows:
where Φ denotes sensing matrix (M × N), and Ig represents the measurement on the detector (reform to a M × 1 column vector; Ig is the hologram). Because natural input signals are not sparse, sparse sampling and transform are needed. While f is the object wave (N × 1 column vector). Using the CS algorithm, the object can be retrieved using a two-step thresholding by solving the following equation: where $\mathop {\arg \min }\limits_f \{{\cdot} \}$ denotes the minimum operator and $||\cdot ||_2^2$ means the l-2 Euclidean norm. The term τ denotes the relative weight between l-2 Euclidean norm of residual. It is important to improve estimation accuracy by adjusting its value, until obtaining the best reconstruction. In addition, ${||f ||_{TV}}\textrm{ = }\sum\limits_i {\sqrt {{{|{\Delta _i^xf} |}^2} + {{|{\Delta _i^yf} |}^2}} }$, where $\Delta _i^x$ and $\Delta _i^y$ denote the horizontal and vertical first-order local difference operations, respectively. TV-norm is a function of the total variation (TV) regularization, which is adopted as our sparsity function [51]. The sharper edge cause by twin image, the larger TV norm.Fig. 5. Experimental results: (a-g) Coaxial holograms; (h-n) Fresnel back-propagation reconstructions; (o-u) CS reconstructions of point-source and USAF1951 resolution chart Group 5; scalebar: 16.6 µm.
Both the twin image and zero-order I1 could be suppressed using the CS algorithm by estimating the correlation between the measured hologram and the reconstructed image at the reconstructed plane. The zero-order item I1 and twin image can result in a strong background and blurry the reconstructed patterns; nevertheless, such blurry speckle has a low correlation with the recorded hologram. In addition, the blurry pattern due to the zero-order item I1 is not sparse. Therefore, most of the estimated error due to zero-order would be eliminated. Regarding the twin image G[1/zr] and other noises, they can be removed using the TV-norm constraint. The TV-norm acts as a sparse filter to suppress the diffused twin image. Finally, high-quality reconstructions of point-source (full width half maximum, Δ = 2.58 µm) and holograms 1 − 6 of group 5 (reconstructed linewidth: 4.83, 4.19, 3.54, 3.87, 3.22, and 2.9 µm) are obtained, as depicted in Figs. 5(i)–(u).
Comparing the two reconstruction methods, i.e., Fresnel diffraction and CS algorithm, for the generation of such an in situ hologram, the latter performs better. For example, the reconstruction image of the point-source hologram possesses a bright circle point with the full width half maximum of 2.58 µm by using the CS algorithm. In addition, the theoretical effective width of the point spread function of such a system ($\Delta = \frac{{0.61\lambda }}{{NA}}$) is 1.08 µm. The experimental obtained image resolution is lower than the theoretical prediction, which may due to the fabrication accuracy and deviation of material property. In addition, because of the inherent twin images resulted from the coaxial hologram recording process, the reconstruction performed using Fresnel back-propagation is not satisfactory. Nevertheless, the reconstructed image qualities can be significantly improved using the CS algorithm. We utilized a single hologram to realize high-quality performance by using such a compressive reconstruction. Compared with the three- or four-step phase-shifting algorithm, the mismatch of multi-hologram recording is avoided, and such method is time efficient as well. In addition, one hologram recording significantly avoids cumulative uncertainty due to system errors and sample fabrication errors.
3. Discussion and conclusions
Notably, our method combined the techniques of FINCH and metasurface elements. FINCH is an enhanced resolution imaging technique that exhibits a non-classical modulation transfer function than that of a regular incoherent imager. Our metalens offered the possibility for continuous 0–2π Huygens’ phase modulation via the spatial arrangement of sub-wavelength meta atoms with constant thickness, unlike classical diffractive elements, which achieve 0–2π phase modulation with thickness variation. Such real-time in situ hologram generation and processing performed using a bifocal metalens can simplify and minimize the traditional double diffractive lenses/SLMs system of FINCH. Because of the integrated compact imaging meta-device, the recording system has higher stability than that of a traditional DH imaging system. In addition, we used the CS algorithm to remove the twin image and bias terms without requiring multiple camera shots or phase shift. Such a technology becomes fast, efficient, and accurate upon integration with the CS algorithm.
In summary, a bifocal metalens was designed and fabricated for real-time in situ hologram generation based on Huygens’ metasurface. Because bifocal metalenses have advantages of being ultra-thin and micro/nano scale complex-amplitude manipulation, multi-plane imaging with high efficiency could be achieved. We further demonstrated the minimization of coaxial hologram generation and reconstruction based on such bifocal metalenses, followed by the CS algorithm. High-fidelity reconstruction images were obtained using the CS algorithm rather than Fresnel diffraction or phase-shifting technology. The recording process time was significantly compressed. Simultaneously, such an optical system can improve the efficiency of recording holograms, and is especially appropriate for the imaging of biological tissues or in vivo cells. Furthermore, our design can expand from the narrowband to broadband achromatic metalenses in the visible and near-infrared range. Such techniques might be promising for many practical applications, such as in vivo fluorescence microscopy, 3D dynamic holography, and rainbow holography and so on.
Funding
National Key Research and Development Program of China 2017YFB1002900, Ministry of Science and Technology, China; Fok Ying-Tong Education Foundation of China (161009); Beijing Nova Program (Z171100001117047); Natural Science Foundation of Beijing Municipality (4172057); National Natural Science Foundation of China (61775019); Beijing Outstanding Young Scientist Program (BJJWZYJH01201910007022).
Disclosures
All the authors declare no conflict of interest.
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