Dual-mode varifocal Moiré metalens for quantitative phase and edge-enhanced imaging

2.1 Principle of the phase imaging based on Moiré metalens

Here, we proposed a dual-mode varifocal Moiré metalens for QPI and edge-enhanced imaging, as shown in Figure 1. This Moiré metalens is composed of two axially aligned metasurfaces M1 and M2. The metalens enables continuous optical zoom by rotating the metasurface M1. This varifocal metalens can obtain the required axial intensity images under x-polarization light incidence, with all metasurfaces maintain the axial position fixed. Combined with the TIE algorithm, the Moiré metalens can achieve high-precision phase retrieval. Additionally, this metalens generates a vortex-focusing phase profile under y-polarization light incidence. Therefore, edge-enhanced imaging can be achieved by convolving image with the point spread function (PSF) of the metalens as U out x , y = U in x , y ⊗ P r , φ . Here, ⊗ is the convolution symbol, U _out and U _in refer to the output and input light field, respectively, x , y are the spatial coordinates of the sampling plane, and r , φ are the polar coordinates. The P r , φ = F exp i φ corresponds to the vortex phase distribution, and here F ⋅ refers to the Fourier transform. The vortex phase metasurface can perform a two-dimensional spatial differentiation on the input light field.

Figure 1:

Schematics of the polarization-multiplexed Moiré metalens for quantitative phase imaging (left panel) and edge-enhanced imaging (right panel). The focal length varies continuously as the relative angle Δθ changes. The metalens enables the acquisition of axial intensity profiles for QPI in the x-polarization state and varifocal edge-enhanced imaging in the y-polarization state.

The quantitative relationship between the axial intensity derivative and the transverse phase distribution can be described through a second-order elliptic partial differential equation [54]:

where k refers to the wavenumber 2π/λ and the operating wavelength λ is 633 nm, r = x , y are the transverse coordinates, I r , z is the intensity distribution on the recorded plane at the propagation distance z, ϕ r refers to the phase distribution on the recorded plane, and ∇ is the two-dimensional gradient operator. For pure phase object (I ≈ constant), Eq. (1) can be simplified as:

Hence, the phase distribution ϕ r can be derived from the axial intensity derivative ∂I/∂z. Theoretically, phase retrieval requires only two intensity images at different propagation distances. However, in order to suppress noise in phase retrieval, multiple intensity profiles are typically required. Previous TIE systems generally capture the required intensity images by axially moving the detector. However, this approach limits systems’ compactness and response speed. Additionally, the impact of mechanical movement errors on retrieval accuracy is non-negligible. We obtained the required axial intensity images via the varifocal Moiré metalens zooming from 58.7 μm to 61.8 μm, corresponding to 3.1 μm defocus range.

2.2 Design and features of the Moiré metalens

The specially designed metasurfaces enable the metalens to exhibit Moiré and vortex-focusing phase profiles, respectively, in the x- and y-polarization states. The Moiré phase ϕ _f of the metalens can be expressed as:

The vortex phase profile ϕ _e is written as:

where f refers to the focal length, and r , φ are the polar coordinates of the metalens. The phase profile of the polarization-dependent metasurface M2 is:

Here, θ ₂ is the rotation angle of M2, ϕ _M2,x and ϕ _M2,y refer to the phase distributions, respectively, in the x- and y-polarization states. round ⋅ is the integer operator; thus, the discontinuity at the polar angle φ = π can be removed [55]. Additionally, the phase distribution of the polarization-independent metasurface M1 can be expressed as:

where θ ₁ refers to the rotation angle of M1. The phase profiles of the Moiré metalens can be written as the superposition of M1 and M2 phase distributions:

where ϕ _x and ϕ _y represent the phase profiles of the Moiré metalens, respectively, under x- and y-polarization light incidence, and Δθ = θ ₂ − θ ₁ refers to the relative angle between the two metasurfaces. Therefore, the relationship between the actual focal length f _θ and the relative angle Δθ can be expressed as:

Hence, the Moiré metalens enables flexible and continuous focal length adjustment via changing the relative angle Δθ.

The theoretical phase profiles of the Moiré metalens are demonstrated in Figure 2a. It’s illustrated that the phase distribution of metasurface M1 remains unchanged under different polarization states, corresponding to Eq. (7). In the y-polarization state, metasurface M2 introduces an additional vortex phase compared to the x-polarization state, as expressed in Eqs. (5) and (6). The metalens exhibits the Moiré and vortex phase profiles, respectively, under the two orthogonal polarization states, as shown in Eq. (8). Close agreement is achieved between theoretical and simulated phase distributions of the Moiré metalens, as illustrated in Figure 2a and b.

Figure 2:

Phase distributions of the polarization-multiplexed Moiré metalens. (a) Theoretical phase distributions of metasurfaces M1, M2 and the superimposed Moiré phase profiles of the Moiré metalens under x-polarized and y-polarized light incidence, respectively. (b) Simulated phase distributions of the Moiré metalens under x-polarized and y-polarized light incidence, respectively.

High refractive index material TiO₂ (n = 2.58) is used for the metalens structure, with SiO₂ (n = 1.45) as the substrate. In the bilayer metasurfaces, M1 is composed of the nanocylinders, while M2 consists of the elliptical nanopillars. In order to endow the metalens with polarization-multiplexing capability for dual-mode imaging, we construct M2 with the polarization-sensitive units. We compose M1 with the polarization-independent units; thus, the superposition phase of M1 and M2 can exhibit the corresponding diverse phase profiles in different polarization states, as shown in Eq. (8). Besides, the phase distribution of M1 can remain constant during rotation. The structure of the elliptical nanopillar in M2 is shown in Figure 3a, where the periodicity of the meta-atoms is p = 450 nm, the height of the nanopillars is h ₂ = 600 nm. The long and short semi-axes of the ellipse a and b vary from 50 to 200 nm, respectively. The highest aspect ratio of the TiO₂ is 1:6, satisfying the fabrication constraints [56]. Considering the fabrication tolerances of the elliptical cylinder structures, we verify the feasibility of the design through simulation. The parameter sweep was implemented through the commercial software FDTD (Ansys Lumerical FDTD). In the simulation, the periodic boundary condition is set in the x and y direction, while the perfectly matched layer (PML) boundary condition is set in the z direction. The mesh size is set as 10 nm. As h ₂ is close to the working wavelength λ = 633 nm, the meta-atoms can act as truncated waveguides and thus feature enhanced phase control capability. The proposed metalens can also be tuned to the infrared wavelength range. For an infrared-band device, crystalline silicon (c-Si) can serve as a constituent material for the metasurface, which can be compatible with semiconductor processing [57]. The phase distributions of the meta-atoms cover 0 to 2π, satisfying the needs of the metalens design, as shown in Figure 3b. As for the meta-atoms of M1, the height of the nanocylinders is h ₁ = 650 nm with the radius r ranging from 50 nm to 120 nm, and the corresponding phase distributions also satisfy the 2π phase range. We select the meta-atoms covering the full 2π phase range with the transmittance higher than 80 % to construct the Moiré metalens, as illustrated in Figure 3c. The maximum error between the real and target phase is 0.4 rad.

Figure 3:

Schematics and characteristics of the meta-atoms. (a) Three-dimensional view, side view, and top view of the meta-atom. (b) The phase distribution chart corresponding to different elliptic parameters a and b of the nanopillars under x-polarization light incidence. (c) The transmittance distribution chart corresponding to different elliptic parameters a and b of the nanopillars under x-polarization light incidence.

To verify the structural design, a Moiré metalens with an aperture of 58 μm is constructed in the commercial software FDTD. The simulation area is set to a three-dimension domain of 58 × 58 × 5 μm³, and the mesh size is set as 10 nm. We set the PML boundary conditions in the x, y, and z directions. The two axially aligned metasurfaces M1, M2 constitute the compact metalens, with an axial distance of 200 nm. The focusing effect can be calculated based on Fresnel diffraction theory. The simulated intensity distributions in the x–z plane and focal plane under y-polarization light incidence are demonstrated in Figure 4. The corresponding focal length f _θ increases from 56.4 μm to 64.7 μm with the relative angle Δθ decreasing from 6° to −6°, as demonstrated in Figure 4a and b. The measured relationship between f _θ and Δθ shows excellent agreement with the theoretical function, and the maximum deviation doesn’t exceed 0.23 μm. The slight discrepancy between the theoretical and simulated focal length f _θ stems from phase matching error. The periodic boundary condition is applied in the simulation of a single meta-atom. However, in the simulation of the entire metalens, the adjacent meta-atoms may not be identical. The mutual interaction between meta-atoms can cause a local deviation in the Moiré phase, resulting in a small random shift in the focal length. Due to the inclusion of vortex phase, the intensity profile in the focal plane shows a donut-shaped distribution, as shown in Figure 4c and d.

Figure 4:

Simulated results of the Moiré metalens in the y-polarization state. (a) Normalized intensity distributions on the x–z plane corresponding to different relative angles Δθ. (b) Theoretical and simulated focal length f _θ variation curves under different relative angles Δθ. Normalized intensity distribution on the focal plane (c) and x-direction tangent (d) at the relative angle Δθ of 0°.

In the x-polarization state, the Moiré metalens realizes continuous zoom. As illustrated in Figure 5a and b, the focal length f _θ increases by 3.1 μm (58.7 μm–61.8 μm) with the relative angle Δθ decreasing by 4° (2° to −2°), matching the theoretical calculations. Figure 5c depicts the full width at half maximum (FWHM) of the metalens at the relative angle Δθ of 0°. The measured FWHM is 0.74 μm, which is close to the diffraction limit of λ/2NA. The simulated results prove that the designed Moiré metalens can achieve flexible and continuous adjustment of the focal length f _θ by regulating the relative angle Δθ. Moreover, we also simulated potential axial misalignments during the experiment by slightly offsetting the two metasurfaces M1 and M2. The different focal spots under different axial misalignment distances are demonstrated in Figure 5d, and within the 2 µm (−1 µm–1 µm) offset range, the metalens can maintain consistent focusing performance. In practical application, it’s necessary to maintain the two metasurfaces center-aligned. However, the impact of minor axial misalignment on focusing performance of the proposed Moiré metalens is controllable.

Figure 5:

Simulated results of the Moiré metalens in the x-polarization state. (a) Normalized intensity distributions on the x–z plane corresponding to different relative angles Δθ. (b) Theoretical and simulated focal length f _θ variation curves under different relative angles Δθ. (c) The FWHM of the metalens at the relative angle Δθ of 0°, the inset illustrates the focal spot profile. (d) Diagrams of the metasurfaces misalignment and focal spot maps under different axial misalignment distances at the relative angle Δθ of 0°.

3.1 Quantitative phase imaging performance

We combine iterative angular spectrum and the TIE method (AS-TIE method) to achieve high-precision QPI. In comparison with other phase retrieval methods, AS-TIE has advantages of low computational complexity and phase-unwrapping-free. The flowchart is demonstrated in Figure 6.

Figure 6:

Flowchart of the phase retrieval method.

First, we extract the axial intensity derivative for phase retrieval, based on the zoom capability of the Moiré metalens. The selection of the defocus distances has a significant impact on the results’ accuracy. Too short defocus distance can’t filter the noise, whereas too long defocus distance will lead to the loss of high-frequency information. The phase profile can be retrieved from unequally spaced intensity distributions. The axial intensity derivative is represented by a linear combination of the intensity distributions at different defocus distances. The extraction of the axial intensity derivative ∂I/∂z is based on the higher-order Taylor expansion of the measured intensity. The calculations of ∂I/∂z and the reconstructed phase ϕ r are written as [58]:

Here, n is the serial number of the defocus plane, I z n r refers to the intensity distribution of the n-th defocus plane, and I 0 r is the intensity distribution of the in-focus plane. C _n is the corresponding factor, and z _n refers to the defocus distance of the n-th defocus plane. The intensity distributions corresponding to different axial distances z _n can be obtained based on the Moiré metalens, as demonstrated in Figure 7c. The axial intensity derivative ∂I/∂z can be obtained based on Eq. (10). According to Eq. (2), the phase profile can be obtained by solving the Poisson equation, as shown in Eq. (10). The calculation can be simplified in Fourier domain, and the Laplace operator is ∇ − 2 = − 1 / 4 π 2 f x 2 + f x 2 , where f _x and f _x are the spatial frequency coordinates.

Figure 7:

Quantitative phase imaging of the Moiré metalens. (a) The incident phase profile and the reconstructed phase profiles from different frames. The input image exhibits uniform intensity (I = 1) and the phase satisfies the three-peak Gauss distribution with different peak intensities. (b) Comparison of the target profile and the reconstructed phase profiles from different frames. (c) The intensity distributions of different relative angles Δθ.

Second, the previously obtained phase profile will be input as the initial value into the iterative angular spectrum algorithm. According to the angular spectrum theory, the forward and backward propagation process between the object and image plane (the focal plane) can be represented by the angular spectral transfer function H. The light field relation can be expressed as:

where u ₀ and u _i refer to the optical field distribution of the object and image plane, respectively. The initial incident complex amplitude in this iteration is:

Here, φ ₀ is the previously extracted phase profile, and I ₀ refers to the intensity distribution of the incident plane. As for pure phase object, the incident intensity distribution is uniform, i.e., I ₀ ≈ 1. The light field distribution of the focal plane can be calculated based on the forward propagation Eq. (11). According to the calculated focal plane complex amplitude u f = I n f exp i φ n f , the corresponding focal intensity distribution I _nf of the n-th cycle can be obtained. The iteration value I _nf should be compared to the measured focal intensity I _f , and I f − I n f is taken as the iterative evaluation function. If the evaluation value satisfies the criteria, the iteration terminates; otherwise, the iteration continues. Then, the focal amplitude I n f should be replaced by the measured value I f . Furthermore, the derived incident complex amplitude u n 0 = I n 0 exp i φ n 0 is obtained based on backward propagation. Moreover, we perform the amplitude substitution again, replacing I n 0 with the actual value I 0 . Repeat the process until the iteration converges or the maximum iteration count is met.

To test the phase retrieval accuracy, we choose the phase profile containing three different-peak Gauss distributions as the input phase profile, as illustrated in Figure 7a. We select this target phase to verify the proposed meta-device’s capability for phase reconstruction across the varying peak intensities and spatial coordinates. The system’s focal length is defined as 60 μm of 0° relative angle. As demonstrated in Figure 7a and b, the proposed Moiré metalens can achieve a high-precision phase retrieval. The reconstructed phase based on three images shows a larger gap from the target phase, and the background noise introduces notable interference. The retrieval accuracy can be improved by increasing the number of sample images. It can be observed that the reconstructed phase distributions with 5 and 7 frames closely match the target. The RMSE reach 0.029 rad and 0.015 rad, corresponding to the wavefront errors of 0.0046λ and 0.0024λ, respectively, and the calculation formula of RMSE is:

where φ o i , j and φ r i , j refer to the input and reconstructed phase, respectively, of each pixel, and N is the pixel number. As mentioned above, except for the intensity image frames, the defocus distance also has a significant impact on the phase retrieval accuracy. To simplify the operation and calculation process, the five-frame phase retrieval is a better choice, and the corresponding imaging distances are −1.3 μm, −0.4 μm, 0 μm, 1.0 μm, and 1.8 μm, corresponding to the rotation angles of 2°, 1°, 0°, −1°, and −2°.

3.2 Edge-enhanced imaging performance

In the y-polarization state, the Moiré metalens exhibits vortex-focusing phase distribution. Thus, the metalens enables isotropic edge-enhanced imaging of the input image based on the radial Hilbert transform [59]. Compared with brightfield imaging, the edge-enhanced imaging enables higher sensitive detection of objects with a large difference in the refractive index from the background. To test the edge-enhanced imaging capability of the Moiré metalens, we choose the resolution-chart-like panel as the incident image, as illustrated in Figure 8a and c. We characterize both the amplitude and phase images to assess the edge-enhanced imaging performance. As for the incident amplitude image, the phase remains constant (φ = 0) and the input image is encoded by binary pixel intensities (0 or 1). The incident pure phase image exhibits a uniform intensity distribution (I = 1), with its light field expressed as u 0 = exp i φ , where the phase φ is binary-encoded by pixels with φ = 0 or π/2. The edge-enhanced images of the varifocal Moiré metalens for the amplitude and phase objects are shown in Figure 8a and c, respectively. With the relative angle changing from −6° to 6°, the metalens can achieve varifocal edge-enhanced imaging from 56.4 μm to 64.7 μm. Figure 8b and d demonstrates the normalized intensity of the edge-enhanced images in the y direction. It’s illustrated that the peaks of output images match well with the incident image. The resolution of the metalens can be extracted from the FWHMs of the edge response. For the amplitude object, the resolution shifts from 1.84 μm to 1.6 μm, within the 12° range of rotation. For the phase object, the resolution changes from 1.95 μm to 1.3 μm within 12° angular variation. Moreover, by employing the tree-shaped pattern and the cell microscopy image as the incident images, we further tested the edge-enhanced capability of the Moiré metalens. As illustrated in Figure 8e and f, the proposed metalens maintains high-quality edge-enhanced imaging even for complex and irregular input images. The designed Moiré metalens proves practically applicable and shows potential in biomedical imaging and related fields.

Figure 8:

Schematics of the varifocal edge-enhanced imaging. (a) The intensity distribution of the incident amplitude image and the corresponding edge-enhanced images obtained by the metalens at different relative angles Δθ. (b) Normalized intensity curves along the dotted lines in (a). (c) The phase distribution of the incident pure phase image and the corresponding edge-enhanced images obtained by the metalens at different relative angles Δθ. (d) The phase curve and normalized intensity curves along the dotted lines in (c). (e) Edge-enhanced images of the tree-shaped pattern. (f) Edge-enhanced images of the cell microscopy image, the red solid-line boxes highlight the same corresponding cells in different image panels.