A Review: Phase Measurement Techniques Based on Metasurfaces

Abstract

Phase carries crucial information about the light propagation process, and the visualization and quantitative measurement of phase have important applications, ranging from ultra-precision metrology to biomedical imaging. Traditional phase measurement techniques typically require large and complex optical systems, limiting their applicability in various scenarios. Optical metasurfaces, as flat optical elements, offer a novel approach to phase measurement by manipulating light at the nanoscale through light-matter interactions. Metasurfaces are advantageous due to their lightweight, multifunctional, and easy-to-integrate nature, providing new possibilities for simplifying traditional phase measurement methods. This review categorizes phase measurement techniques into quantitative and non-quantitative methods and reviews the advancements in metasurface-based phase measurement technologies. Detailed discussions are provided on several methods, including vortex phase contrast, holographic interferometry, shearing interferometry, the Transport of Intensity Equation (TIE), and wavefront sensing. The advantages and limitations of metasurfaces in phase measurement are highlighted, and future research directions are explored.

1. Introduction

In physical optics, light is regarded as an electromagnetic wave whose characteristics can be described by a complex amplitude function. This complex amplitude encapsulates two key pieces of information: amplitude and phase. The amplitude reflects the energy intensity of the light wave, while the phase represents the position of the wave or the time delay in its propagation [1,2]. When light interacts with an object, the phase of the transmitted or reflected light carries crucial information about the object, such as its three-dimensional surface morphology, thickness, and refractive index. However, because the frequency of visible light is much higher than the response range of photodetectors, existing technology cannot directly record the phase information of light [3,4]. Typically, only the intensity of light is captured, leading to the loss of some critical information about the object. To address this issue, researchers have developed various phase measurement techniques. These techniques aim to reconstruct the phase information from the intensity data using specific physical or computational methods, a process commonly referred to as phase retrieval or phase reconstruction. Phase measurement techniques have significant applications across multiple fields. For instance, in ultra-precision measurement [5,6], phase information can be used to measure nanoscale surface roughness or thickness variations. In biological microscopy [7,8,9,10,11,12,13,14,15,16], phase information allows the visualization of internal cell structures without the need for staining. In adaptive optics [17,18,19,20], phase information helps in real-time correction of wavefront distortions, thereby improving imaging quality.

The earliest visualization of phase information was achieved through the invention of the Zernike phase contrast (ZPC) microscope, which transforms phase information of an object into intensity variations using spatial filtering in a 4f system and won the Nobel prize in 1953 [21,22]. This method has been particularly effective in high-contrast imaging of live biological cells without the need for labeling. Subsequently, Nomarski developed the Differential Interference Contrast (DIC) microscope, which operates on the principles of polarization beam splitting and shearing interference [23]. Compared with the Zernike phase contrast microscope, the DIC microscope provides higher-quality images with fewer artifacts. However, the intensity measurements obtained from these two methods do not have a strictly linear relationship with the object’s phase information, making it impossible to quantitatively recover phase information. To overcome this limitation, quantitative phase measurement methods, both interferometric and non-interferometric, have been developed. Interferometric methods, with digital holography as a prominent example [5], introduce a reference wavefront into the target light field to produce interference fringes, from which phase information can be accurately decoded. Other techniques in this category include speckle interferometry [24], microscopic interferometry [25] and quantum interference [26]. These methods offer high precision and are suitable for high-resolution imaging and nanoscale measurements, but they require extremely stable environmental conditions and are highly susceptible to external disturbances, limiting their practical applications. In contrast, non-interferometric methods, which do not require a reference wavefront, include techniques like the intensity transport equation [27,28,29] and wavefront sensing [19,20,30]. These methods recover phase information by directly analyzing intensity images, reducing dependence on the experimental environment and enhancing the system’s robustness. However, the phase recovery accuracy of non-interferometric methods is generally lower than that of interferometric methods, and they may face computational challenges when dealing with complex light fields.

Metasurfaces are two-dimensional materials composed of sub-wavelength-scale artificial structural units, demonstrating unprecedented potential in the control of electromagnetic waves [31]. Unlike traditional optical components, metasurfaces can flexibly manipulate the phase [32], amplitude [33], and polarization [34] of light on a nanoscale by precisely designing their sub-wavelength structural units. This flexibility enables metasurfaces to achieve complex optical functions within ultra-thin structures, such as metalens [35,36,37], holograms [38,39,40], complex beam generators [41,42], and optical computing [43,44,45]. Inspired by the principles of traditional phase measurement methods, metasurface technology not only allows for efficient, multifunctional phase measurement in a more compact form but also significantly simplifies the design of optical systems. By leveraging metasurfaces, researchers can enhance the performance and integration of optical devices without increasing system complexity, offering novel solutions for precision optical measurement and imaging applications.

This paper reviews recent advancements in the field of phase measurement using metasurfaces, with a focus on both interferometric and non-interferometric phase measurement techniques. By systematically categorizing and analyzing various types of metasurface-based phase measurement technologies, the review provides an in-depth examination of the strengths and limitations of each approach. Finally, the review discusses the challenges and future directions for the use of metasurfaces in phase measurement, offering insights that aim to guide further research.

2. Non-Quantitative Phase Measurement

The ZPC imaging technique utilizes the principle of spatial filtering by introducing a fixed phase shift to the zero-order light, converting subtle phase differences in the sample into intensity variations [21,46]. However, its limitation lies in the overlap of the zero-frequency spectrum and diffracted light, often resulting in a halo effect, particularly at the sample edges, which degrades image quality. In contrast, spiral phase contrast imaging employs a spiral phase modulation through a spiral phase plate, imparting orbital angular momentum to the light field [47,48]. This results in more uniform phase enhancement, making it particularly suitable for samples with symmetric or complex topological structures, effectively reducing the halo effect and providing higher resolution and contrast.

Huo et al. proposed a spin-multiplexed vortex phase imaging metasurface based on a 4f system [49]. By altering the spin state of the incident light, this system can switch between two imaging modes: bright-field imaging and vortex phase contrast imaging in Figure 1a. The metasurface leverages both the transmission phase and geometric phase by varying the length (D_x), width (D_y) and rotation angle (θ) of TiO₂ nanopillars, as shown in Figure 1b. The TiO₂ nanopillars have a fixed height of 600 nm, with a period of 450 nm, and act as individual nanoscale half-wave plates. This design allows for the generation of two independent phase distributions under the two spin states of the incident light. When left-handed circularly polarized (LCP) light is incident, the system produces bright-field imaging. Conversely, when right-handed circularly polarized (RCP) light is incident, the system generates vortex phase contrast imaging, which results in a two-dimensional edge image because spiral phase filtering operation is equivalent to two-dimensional spatial differentiation of the incident light field. Figure 1c shows the bright-field imaging and vortex phase contrast imaging of the onion epidermis at different wavelengths. This demonstrates the system’s capability to switch between imaging modes based on the incident light’s spin state, providing versatile imaging solutions for different applications.

The 4f system architecture, while effective, remains complex and bulky. To address this, Kim et al. combined the vortex phase and hyperbolic phase to design a single-layer metalens capable of isotropic vortex phase contrast imaging without the need for a 4f system [50], as shown in Figure 1d. The metasurface is composed of Si meta-atom, with a height of 400 nm and a period of 300 nm, and its diameter is 1 mm, NA is 0.8. Utilizing the geometric phase, the phase distribution of metalens is a superposition of the vortex phase and focusing phase. When circularly polarized light is incident, the metalens achieve a resolution as high as 0.78 µm within the visible broadband range. This design simplifies the optical setup, eliminating the need for the traditionally cumbersome 4f system while still delivering high-resolution imaging, demonstrating the potential for more compact and efficient phase contrast imaging systems. Further, Zhang et al. developed a single-layer metalens that simultaneously generates bright-field imaging and vortex phase contrast imaging within the same field of view [51], as shown in Figure 1g. The metalens is composed of Si nanopillars with a height of 600 nm, period of 300 nm, and diameter of 1 mm at a wavelength of 630 nm. The phase distribution of metalens is a superposition of two distinct phase profiles: one combines vortex phase and focusing phase distributions, while the other consists solely of focusing phase distribution. These two phase profiles are designed with opposite deflection phases to effectively separate the bright-field imaging from the vortex phase contrast imaging simultaneously without interference between them. The metasurface is composed of silicon nanopillars, which are designed to ensure polarization insensitivity. Figure 1i demonstrates the imaging performance of the metasurface, showcasing the distinct bright-field and vortex phase contrast images produced by the metalens.

In DIC microscopy, the optical transfer function (OTF) of the system exhibits asymmetry in the frequency domain. The introduction of asymmetry in the OTF can realize more complex spatial filtering to enhance phase visualization. Wesemann et al. developed a Nanophotonics Enhanced Coverslip (NEC) to improve the contrast and accuracy of traditional biological phase imaging [52] in Figure 2a. The NEC is composed of a 100 nm thick titanium dioxide (TiO₂) layer, 40 nm thick silver stripes, a PMMA layer, and a silicon dioxide substrate in Figure 2b. When light is incident on the metasurface, it is diffracted into three beams. The zero-order diffraction propagates in the same direction as the incident light, while the ±1 order diffraction beams couple into the TiO₂ waveguide layer. When these beams reach the metal stripes, they reflect and undergo a π phase shift, diffracting again to form new zero-order beams. These recombined beams propagate parallel to the incident light but out of phase, resulting in destructive interference and suppression of transmission. By changing the angle of incidence, the phase relationship between the beams is altered, leading to incomplete cancellation and, thus, transmission of light. The diffraction and waveguide effects enhance the phase contrast by selectively transmitting light with non-zero phase gradients. Effectively, the NEC acts as a high-pass filter, filtering out undiffracted light and only transmitting light that carries phase information, converting phase differences into intensity contrast and presenting the image in a pseudo-3D format. In Figure 2c, NEC was used to image human cancer cells (HeLa cells), demonstrating its capability to visualize internal cell structures without the need for staining. Compared with traditional DIC microscopy, the NEC simplifies the phase imaging system by eliminating the need for additional optical components while still generating high-contrast phase images.

The NEC requires a slight tilt (≈3°) relative to the optical axis of the imaging system to break the symmetry of its OTF, which introduces additional complexity. Further, Wesemann et al. proposed and demonstrated a real-time phase imaging method based on an asymmetric metasurface utilizing photon spin-orbit coupling [53]. The metasurface, based on the geometric phase principle, consists of silver nanorods with varying rotation angles placed on a Si₃N₄ waveguide, as shown in Figure 2f. When illuminated by circularly polarized light, the interaction between the light’s intrinsic spin and the differently oriented nanorods induces spin-orbit coupling, resulting in a phase shift and causing the asymmetric response of the metasurface. By switching between left- and right-handed circularly polarized light, the system enables contrast inversion in phase imaging, providing an additional degree of control for phase contrast modulation. This system can capture real-time changes in the optical wavefront’s phase without requiring post-processing. Figure 2g shows the phase imaging of the eggcrate pattern. However, the system’s numerical aperture (NA = 0.025) is relatively low, and its operating frequency is −0.025 < k_x/k₀ < 0.025, which limits its application in scenarios requiring high-NA imaging. In addition, the OTF of the devices exhibits directional dependence, which limits its ability to perform phase imaging uniformly in all directions.

3. Quantitative Phase Measurement

3.1. Interference Methods

Interference is a classical phase measurement technique [5]. The incident light is divided into two beams: one beam illuminates the sample, while the other serves as the reference. The interference between the object beam and the reference beam occurs when the peaks and troughs of the two waves align, producing constructive interference. When the peaks of one wave coincide with the troughs of the other, destructive interference occurs. The phase information of the object is recorded in the interference fringes, and the phase can be demodulated from the interference pattern through numerical calculation methods, offering extremely high precision and sensitivity. In 1948, Gabor proposed holography [54], successfully applying it to improve the resolution of electron microscopes, for which he was awarded the Nobel Prize in Physics in 1971. In 1967, Goodman recorded holograms using a camera and a computer for wavefront reconstruction, marking the development of digital holography [55]. Digital holography has emerged as a crucial tool in various fields due to its non-invasive nature and ability to capture high-resolution, label-free 3D information.

A novel incoherent holography method based on an all-dielectric metasurface was proposed, utilizing the polarization selectivity of the geometric phase to achieve high-resolution quantitative phase measurement of a plasmonic metasurface [56]. In Figure 3a, the metasurface acts as a geometric phase grating (GPG), which separates LCP light (scattered light) and RCP light (leaked light). The scattered light interacts with the object, carrying the object’s phase information, while the leaked light is simply reflected and serves as the reference phase. As the scattered and leaked light passes through the GPG and a 4f system, they generate self-interference, producing an off-axis phase hologram. The phase information can be recovered from the interference fringes. The imaging excels at the high spatial resolution that was demonstrated experimentally by the precise amplitude and phase restoration of vortex metalenses and a metasurface grating with 833 lines/mm. To simplify the optical system, Sardana J. et al. developed a compact single-shot digital holography system based on a single-layer dielectric metasurface, enabling quantitative phase imaging [57], as shown in Figure 3c. The system utilizes a birefringent dielectric metasurface designed to simultaneously achieve polarization beam splitting and focusing. The metasurface is composed of TiO₂ nanopillars with a height of 600 nm, period of 350 nm, focal length of 12 mm, and NA 0.1, which split the incident light beam into two orthogonally polarized beams and focused them at different spatial positions. By performing a Fourier transform on the captured hologram, the phase information of the object is extracted. The experiments demonstrated the system’s capability to measure the three-dimensional morphology of pure-phase objects. Compared with traditional interferometers, this system offers a more compact and stable setup that is less sensitive to vibrations, making it particularly suitable for dynamic measurements. However, the current design’s resolution is limited by the numerical aperture; in addition, the phase reconstruction algorithm requires further optimization to enhance accuracy and computational efficiency for complex three-dimensional morphology measurements.

Traditional interferometry relies on the interference between a measurement beam and a separate reference beam, leading to a complex optical setup. It is well-suited for measuring phase differences across an entire wavefront and is often used in precise measurements of static systems. However, traditional interferometry is highly susceptible to external environmental influences, which can degrade accuracy. In contrast, shearing interferometry eliminates the need for a separate reference beam by shearing and translating different parts of the same wavefront to produce interference, thereby simplifying the optical design and reducing sensitivity to environmental factors. Traditional differential interference contrast microscopy, however, is not a quantitative phase measurement method, and its optical components are complex and bulky, making it difficult to achieve compact and precise phase measurement systems.

Arbabi et al. developed a compact quantitative phase gradient microscope (QPGM) based on a multifunctional dielectric metasurface system [58], enabling quantitative phase imaging of transparent samples in a single shot, as shown in Figure 4a. The researchers designed a vertically integrated two-layer dielectric metasurface system. Each metasurface layer is composed of Si nanostructures with a height of 664 nm, a period of 380 nm, and a diameter of 600 μm, which have different functionalities. The first metasurface layer decomposes the incident light beam into two orthogonal polarization states (TE and TM) and generates a shearing focus. This layer also splits the light beam into three directions, corresponding to three different Differential Interference Contrast images. The second metasurface layer consists of three off-axis polarization-multiplexed metalenses, which receive the three beams from the first layer and form DIC images with different phase shifts on the imaging plane. Using the captured three DIC images, the phase gradient image of the sample is calculated through a three-step phase-shifting method. In Figure 4d, the system’s performance was demonstrated by measuring the thickness of phase targets, showing good agreement with thickness data obtained from Atomic Force Microscopy (AFM). Additionally, imaging biological samples, such as sea urchin cells, also yielded excellent results, as shown in Figure 4e. The system achieved a significant reduction in size while maintaining high sensitivity, capable of detecting a minimum phase gradient of 92.3 mrad/μm, with spatial resolutions of 2.76 μm along the x-axis and 3.48 μm along the y-axis. However, the method has some limitations. Since the system is configured close to a 4f setup, the focal lengths of the two metasurface layers and the distance between them must be matched, resulting in a relatively small field of view. Additionally, the system operates at a single wavelength and requires precise alignment of the two metasurface layers, which increases manufacturing complexity.

In order to achieve wide-band operation, easy to build an optical system, Zhou et al. proposed a Fourier optical spin splitting microscopy technique [59]. In this method, a geometric phase metasurface is placed at the Fourier plane of a 4f imaging system, as shown in Figure 5a. The phase delay introduced by the metasurface is determined by the rotation direction of nanoholes, with a rotational angle distribution given by φ(x,y) = πx/Λ, where Λ represents the period of the metasurface. The incident linearly polarized light is decomposed into LCP and RCP components. The metasurface imparts opposite phase delays (±2φ) to these components, resulting in the formation of two identical images on the imaging plane, with a certain shear distance between them. This shear distance is given by ∆ = λf/Λ, where f is the focal length of the second lens. When the metasurface is placed between a pair of orthogonal polarizers, it behaves similarly to a sinusoidal amplitude grating. By laterally shifting the metasurface along the x-axis in the Fourier plane, the phase delay between LCP and RCP components is altered. Moving the metasurface along the z-axis results in angular and spatial displacements of the object’s angular spectrum in the imaging plane. For a pure-phase object, the intensity distribution on the imaging plane can be approximately expressed as follows:

$$I_{out}\left(x_3,y_3\right)\approx {\displaystyle \frac{1}{2}}\left\{1-cos\left[2∆{\displaystyle \frac{d\mathsf{\Phi }\left(x_3,y_3\right)}{d{x}_3}}-2\beta \left(x_3\right)-2\theta \right]\right\},$$

(1)

where Δ = λf/Λ represents the shear distance between the two identical images, $d\mathsf{\Phi }(x_3,y_3)/d{x}_3$ represents the phase gradient. $\beta \left(x_3\right)=2\pi \epsilon /\left(\mathsf{\Lambda }f\right)$, where $\epsilon$ is the longitudinal movement distance of the metasurface. $\theta =2\pi s/\mathsf{\Lambda }$, where s is the lateral movement distance of the metasurface. The indices (0, 1, 2, 3) refer to different planes in the system: 0 represents the input plane, 1 is the metasurface plane, 2 is the Fourier plane, and 3 is the imaging plane. By varying the lateral displacement s of the metasurface, multiple images with different phase delays can be obtained. These images can be used to calculate the phase gradient $G_x$ using the three-step phase-shifting algorithm as follows:

$$G_x={\displaystyle \frac{d\mathsf{\Phi }(x_3,y_3)}{d{x}_3}}={\displaystyle \frac{arctan\frac{\sqrt{3}(I_1-I_3)}{2I_2-I_1-I_3}}{2\mathsf{\Delta }}}$$

(2)

Once the phase gradient is determined, a two-dimensional integration along the x and y directions is performed to reconstruct the phase distribution. Figure 5c shows the quantitative phase imaging of NH3T3 cells obtained when the phase delay is set to −120°, 0°, and 120° in the horizontal and vertical directions, respectively. In this method, the metasurface replaces the polarization beam-splitting prism used in traditional DIC microscopy, while the geometric phase allows for a broad bandwidth. The phase shift is introduced by laterally moving the metasurface. However, this approach requires multiple lateral movements of the metasurface to capture the different phase-shifted images.

Wu et al. demonstrated that placing a pair of dielectric geometric phase metasurfaces near the conjugate plane of a traditional microscope’s optical path enables single-shot quantitative amplitude and phase imaging [60], as shown in Figure 5d. The working principle involves the metasurfaces first deflecting and separating the left- and right-handed components of a linearly polarized light beam and then recombining them. The lateral shear distance between the two identical images is determined by the distance between the two metasurfaces, while the phase bias delay is influenced by the metasurface period, relative displacement, and the rotation angle of the linear analyzer. Subsequently, the deflected camera captures four images with different phase delays in a single shot. These images are then processed using the four-step phase-shifting algorithm to recover the phase and amplitude information of the object. The rotation angle distribution of the meta-atoms is given by ϕ(x,y) = πx/Λ. When the metasurface is laterally shifted by ξ along the x-axis, the rotation angle distribution becomes ϕ(x,y) = π(x−ξ)/Λ. For incident left-handed circularly polarized and right-handed circularly polarized light, the phase distribution is ±2ϕ(x,y), introducing opposite phase gradients. When linearly polarized light is incident perpendicularly on the first metasurface, the LCP and RCP components are deflected in different directions and converted into RCP and LCP, respectively, with deflection angles of ±arcsin(λ/Λ). Similarly, after passing through the second metasurface, the RCP and LCP components are deflected by ±arcsin(λ/Λ) and are again converted into LCP and RCP, returning to their original propagation direction. According to the scalar angular spectrum diffraction theory and under the Fresnel approximation, the output intensity distribution when the relative displacement between the two metasurfaces is set to zero can be expressed as follows:

$$I_{out}(x_3,y_{3,}\mathsf{\Theta })~{\left|E_{in}\left(x_3-{∆}_0,y_3\right)e^{j\mathsf{\Theta }}+E_{in}\left(x_3+{∆}_0,y_3\right)e^{-j\mathsf{\Theta }}\right|}^2$$

(3)

${∆}_0$ represents the shear distance, and $\mathsf{\Theta }$ is the phase delay. The polarization camera’s analyzer A is rotated to different angles Θ to introduce four distinct phase delays. The camera captures intensity images corresponding to four polarization directions: 0°, 45°, 90°, and 135°, and the phase gradient $G_x$ is calculated by a four-step phase shift algorithm as follows:

$$G_x\approx {\displaystyle \frac{1}{2{∆}_0}}[\varphi \left(x_3+{∆}_0,y_3\right)-\varphi \left(x_3-{∆}_0,y_3\right)]={\displaystyle \frac{1}{2{∆}_0}}\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{I_2-I_4)}{I_1-I_3}}\right)$$

(4)

Additionally, phase reconstruction obtained from unidirectional Quantitative Phase Gradient Imaging (QPGI) images captured in simple integrated experiments often results in unwanted linear artifacts along the shear direction. The authors employ the Alternating Direction Method of Multipliers (ADMM) algorithm to solve the phase reconstruction problem. In the experiment, two pairs of metasurfaces, each measuring 8 × 8 mm with periods of 8 mm and 1 mm, were fabricated. These metasurfaces were used to demonstrate single-shot phase imaging of transparent cells, as shown in Figure 5g. Compared with the Fourier beam splitting method, using two metasurfaces allows the cosine background to cancel out, which facilitates direct observation of the phase gradient in the image plane. However, this method cannot overcome the anisotropic nature of phase imaging due to lateral shear, meaning that it can only perform phase gradient imaging in a single direction.

Wang et al. proposed a computational metasurface system [61], which operates similarly to the Fourier optical spin splitting microscope but with an improved experimental setup. In Figure 6a, a pair of orthogonal geometric phase metasurfaces are placed in the two arms of a Mach-Zehnder interferometer, introducing phase gradients in the x and y directions, respectively. The incident light is 45° linearly polarized to ensure that the intensity of the images in both the x and y directions is equal. After passing through Polarizing Beam Splitter 1 (PBS1), the light containing the target phase information undergoes shearing interference in the x and y directions within Path 1 and Path 2 of the interferometer, respectively. This process results in the acquisition of phase gradient information in two orthogonal directions. After the recombination of the two beams by Polarizing Beam Splitter 2 (PBS2), an isotropic phase gradient image is captured by the CCD camera. This setup allows for the simultaneous measurement of phase gradients in both the x and y directions, leading to a more complete and accurate reconstruction of the phase distribution across the sample, overcoming the anisotropy limitation of traditional single-directional phase gradient imaging methods. Figure 6b shows the imaging results of fishtail cross-cut cells with broadband vectorial DIC microscopy. However, the complexity of the optical path limits the application.

Li et al. developed an imaging system based on single-layer metalens and a polarization camera [62], which utilizes spatial and polarization multiplexing techniques to simultaneously record the shearing interference patterns of LCP and RCP light. The system captures four unique phase-shifted shearing interference patterns in a single shot, and the phase gradients in the x and y directions are calculated using the four-step phase-shifting method. The final two-dimensional complex amplitude image is then reconstructed through higher-order finite difference least-squares integration, as illustrated in Figure 6c. The metalens is composed of rectangular silicon nanofins with a height of 600 nm and a period of 350 nm, designed to achieve polarization and spatial multiplexing through the geometric phase principle. The focal length of the metalens is set to 1.5 cm, with a numerical aperture (NA) of 0.066, ensuring effective separation and focusing of LCP and RCP light at the operating wavelength of 800 nm. Figure 6d shows the phase gradients along the x and y axes calculated from the shearing interference images. Additionally, the system’s ability to reconstruct the two-dimensional surface topography of a UV glue sample is demonstrated in Figure 5e, comparing it with the surface topography of the same sample measured using a commercial white-light interferometer (WLI). The system’s relative average deviation (RAD) is 3.79%, indicating a high degree of consistency with the WLI measurements while offering significantly faster measurement speeds (completed within 10 ms). This highlights the system’s advantages in surface topography measurement applications.

To achieve isotropic shear interference, Zhou et al. proposed a broadband two-dimensional optical spatial differentiator based on a dielectric metasurface [63], utilizing the principle of radial shear interferometry, as shown in Figure 7a. By designing a dielectric metasurface with a radially symmetric phase gradient, the system efficiently achieves edge detection and high-contrast imaging. The diameter of the metasurface is 4 mm, and it is composed of Si nanostructures. The metasurface separates LCP and RCP light, generating a shear interference effect in the Fourier plane, which extracts the edge information of the object. The system demonstrates high transmission efficiency (95%) and a broadband response, covering the entire visible spectrum. In Figure 7b, experimental results using cellular samples for edge detection showcase the potential of this technique in biomedical imaging, providing a powerful tool for high-contrast edge detection in various applications. Further, Wang et al. proposed a single-shot isotropic DIC microscope based on metasurfaces [64]. By employing orthogonal polarization multiplexing with metasurfaces, this work addresses the anisotropy issue inherent in traditional DIC microscopes, which results from linear shear. The metasurface is composed of Si nanopillars with a height of 360 nm, period of 300 nm, diameter of 200 μm, and focal length of 1 mm. The single-layer metasurface integrates focusing, beam splitting, and phase shifting functions, significantly simplifying the optical design of traditional DIC microscopes, as shown in Figure 7c. When a plane wave illuminates the object, it passes through a 45° polarizer, decomposing into x- and y-polarized components. After interacting with the metasurface, these components undergo radial displacement and phase biasing. Following passage through a −45° polarizer, an isotropic phase gradient map is produced. The metasurface phase distribution is as follows:

$$\begin{array}{}{\mathsf{\varphi }}_x\left(\xi ,\eta \right)=-{\displaystyle \frac{\pi }{\lambda f}}\left({\xi }^2+{\eta }^2\right)+{\displaystyle \frac{2\pi \mathsf{\Delta }s}{\lambda f}}\sqrt{{\xi }^2{+\eta }^2}-\mathsf{\Delta }\phi \\ {\mathsf{\varphi }}_y\left(\xi ,\eta \right)=-{\displaystyle \frac{\pi }{\lambda f}}\left({\xi }^2+{\eta }^2\right)-{\displaystyle \frac{2\pi \mathsf{\Delta }s}{\lambda f}}\sqrt{{\xi }^2{+\eta }^2}+\mathsf{\Delta }\phi \end{array}$$

(5)

In order, these represent the focus term, the shear term, and the phase bias term. Figure 6d shows the Imaging results with a-DIC and i-DIC microscopy.

3.2. Non-Interference Methods

The quantitative relationship between the axial intensity derivative and the phase of the transverse optical field was first derived by Teague in a second-order elliptic partial differential equation [27], later termed transport-of-intensity equation (TIE), which is written as follows:

$$-k{\displaystyle \frac{\partial I\left(x,y\right)}{\partial z}}=\nabla \bullet \left[I\left(x,y\right)\nabla \phi (x,y)\right],$$

(6)

where k is the wavenumber 2π = λ, λ is the wavelength, z is the coordinate along the optical axis, I and φ are the intensity and phase distributions at the recorded plane, $\nabla$ is the two-dimensional gradient operator over the transverse coordinates x and y. According to the TIE method, at least two out-of-focus intensity images are required to perform finite difference estimation in order to compute the axial intensity derivative on the left side of the equation. This derivative is then used to numerically solve for the phase distribution of the object at the in-focus plane without the need for any iterative process. TIE bypasses the need for complex optical setups or iterative algorithms, relying instead on simple intensity measurements at different focal planes to reconstruct the phase information. Traditional phase measurement methods based on the TIE often require manually moving the camera to different positions and capturing multiple images. This process places high demands on alignment and system stability, as even small misalignments or environmental vibrations can affect the accuracy of the phase reconstruction.

Engay et al. designed a 4f system that utilizes a metasurface to control the orthogonal polarization states of incident light, capturing two images with different propagation distances on the same plane [65]. This enables single-shot phase recovery using the Transport of Intensity Equation (TIE). As shown in Figure 8a, a polarizer is placed at a 45° angle in front of the 4f system, and the metasurface is positioned at the Fourier plane. The metasurface applies distinct phase shifts to the TE and TM components of the incident light, as follow:

$$\begin{array}{}{\varphi }_{\mathrm{T}\mathrm{E}}\left(\eta ,\nu \right)={\displaystyle \frac{2\pi }{\lambda }}\eta tan\theta \\ {\varphi }_{\mathrm{T}\mathrm{M}}\left(\eta ,\nu \right)={\displaystyle \frac{2\pi }{\lambda }}\left(-\eta tan\theta +\mathsf{\Delta }z\sqrt{1-{\displaystyle \frac{1}{f^2}}\left({\eta }^2+{\nu }^2\right)}\right)\end{array}$$

(7)

The TE and TM component is deflected along the x-axis by angle θ and −θ, simultaneously introducing an axial displacement Δz. After passing through lens2, a lateral separation d = 2ftanθ is generated between the TE and TM polarized light beams, and both images are recorded on the same plane. Due to the axial offset Δz between the two images, there is no need to move the camera, allowing for direct phase retrieval using the Transport of Intensity Equation. The metasurface, as shown in Figure 8b, consists of a silicon dioxide substrate and elliptical amorphous silicon nanopillars with a period of 350 nm and a height of 411 nm. By varying the major and minor axes of the elliptical nanopillars, the phase and transmission responses for the two orthogonal polarizations can be independently controlled. Figure 8c shows the optical thickness profile obtained from the reconstructed phase distribution. This approach eliminates the traditional need to move the camera or sample to acquire multiple images, greatly simplifying the experimental setup and improving the speed of data acquisition. However, recording two images simultaneously comes at the cost of halving the effective field of view compared with capturing a single image across the full width of the camera sensor. Additionally, this method only captures two images at once, relying on first-order finite difference approximations for the intensity derivative, which can limit the accuracy of TIE-based phase retrieval to some extent.

Zhou et al. addressed the issue of insufficient accuracy in intensity derivative approximation by proposing a single-shot phase retrieval method based on an anisotropic metasurface [66]. Through the careful design of the metasurface, they achieved phase gradient control for orthogonal circularly polarized light, enabling the acquisition of three diffraction images in a single exposure. The phase retrieval is then performed using an improved three-image Transport of Intensity Equation (TTIE) algorithm. As shown in Figure 8d, the incident linearly polarized light can be considered a superposition of two orthogonal circularly polarized components. The metasurface introduces different phase gradients for LCP and RCP components. After passing through the metasurface, the incident light is split into three beams, including a central zero-order diffraction image and two off-focus ±1 order diffraction images. These images correspond to different focal planes, forming three images with distinct phase information on the imaging plane. The metasurface, based on geometric principles, is composed of rectangular silicon nanopillars with a height of 600 nm and a period of 300 nm, as shown in Figure 8e. The phase shift is twice the rotation angle of the nanopillars, and the phase distribution follows the following formula:

$$M\left(f_x,f_y\right)=\mathrm{e}\mathrm{x}\mathrm{p}(j\sigma k∆d\sqrt{1-{\left(\lambda f_x\right)}^2-{\left(\lambda f_y\right)}^2})\mathrm{e}\mathrm{x}\mathrm{p}(-j\sigma k{f}_{x}sin\alpha ),$$

(8)

where $M\left(f_x,f_y\right)$ represents the phase distribution function of the metasurface, $\sigma$ indicates the handedness of the incident light (with +1 for LCP and −1 for RCP), k is the wavevector, Δd is the pre-set propagation distance, $f_x$ and $f_y$ are the frequency coordinates, and α represents the phase gradient angle along the x-axis. The experimental results in Figure 8f demonstrate that, when combined with the iterative TIE algorithm, the phase retrieval performance is significantly superior to that obtained using the back-propagation method. The phase maps derived from this approach show much higher accuracy and less noise, particularly when addressing complex phase distributions. This improvement is attributed to the enhanced precision of the three-image Transport of Intensity Equation (TTIE) algorithm, facilitated by the metasurface’s ability to simultaneously capture multiple diffraction images with varying phase shifts.

In addition to polarization, frequency is another degree of freedom that light can utilize. Wang et al. developed a compact quantitative phase imaging system based on dispersion metalens [67], achieving multi-focus imaging without mechanical movement by adjusting the illumination wavelength, as shown in Figure 9. The system uses a single-layer dispersion metalens, where different illumination wavelengths result in varying focal lengths. The metalens is composed of SiN_x nanofins with a height of 1000 nm, a period of 300 nm, a diameter of 2 mm, and an NA of 0.58. By capturing seven images at different focal planes, the system applies the Transport of Intensity Equation method for phase retrieval, ultimately acquiring precise phase information. The phase recovery accuracy of the system reached 0.03λ when applied to microlens arrays and biological samples, enabling high-precision phase measurements.

Wavefront sensing technology is a non-interference method of wavefront reconstruction. Liu et al. developed a novel image sensor at near-infrared wavelengths based on an asymmetric metasurface photodetector for single-shot quantitative phase imaging [68]. The plasmonic metasurface, composed of gold nanostripes, exhibits asymmetry and responds differently to incident light from various angles. The surface of the photodetector is coated with the plasmonic metasurface, as shown in Figure 10a, where the phase information of the incident light is encoded into the photodetector’s photocurrent signal through the metasurface’s angular response. This signal is then converted into an electrical signal, allowing for the measurement of the local phase gradient of the light wavefront. Using an array of photodetectors and an integrated computational imaging algorithm, the system can directly capture phase images from a camera or microscope without the need for complex optical components. The system was optimized for use with monochromatic light sources in experiments, demonstrating excellent phase imaging performance. However, its applicability across a broader spectrum range remains to be further explored.

In a similar idea, Ji et al. proposed a novel phase contrast imaging method based on a non-local angular-selective metasurface in the visible range [69] in Figure 10b, achieving quantitative phase contrast imaging with a phase precision of 0.02π. The metasurface consists of a surface relief grating with a period of 390 nm etched onto a 240 nm thick single-mode silicon nitride (Si₃N₄) slab waveguide placed on a fused silica substrate. This metasurface exhibits high-Q resonances. At the resonance wavelength of 630 nm, near-axis incident light experiences significant transmission attenuation, while off-axis light intensity remains nearly unaffected. Additionally, the near-axis light undergoes a π/2 phase shift relative to the off-axis light. By adjusting the illumination wavelength away from the resonance, the amplitude of near-axis light can be controlled, allowing modulation of the relative intensity and phase between background and foreground light, thereby achieving maximum contrast imaging. This method eliminates the need for placing optical elements in the Fourier plane of a 4f system and does not require complex multi-step optical alignment.

The Shack–Hartmann wavefront sensor (SHWFS) is one of the most widely used wavefront sensing methods. Its simple design, consisting of a lenslet array and an image sensor, allows for single-shot operation. However, in traditional SHWFS, the lenslet array is typically fabricated using MEMS technology, which limits the minimum feature size and the maximum curvature of the microstructures. As a result, lenslet sizes are typically in the range of ~100 µm, and the numerical aperture (NA) is relatively low [70]. To overcome these limitations, Go et al. developed a metasurface-based Shack–Hartmann wavefront sensor (meta SHWFS), which improves the sampling density and the maximum measurable angle of traditional sensors [71], as shown in Figure 11a. By designing and fabricating a metasurface SHWFS composed of 100 × 100 metalenses, they achieved a sampling density of 5963 per mm² and a maximum acceptance angle of 8°, far exceeding that of conventional SHWFS. Metalenses consisted of silicon nitride (SiN_x) rectangular cuboids arranged on a subwavelength square lattice with a period of 350 nm. Each lens is designed at a wavelength of 532 nm, with a diameter of 12.95 µm and a focal length of 30 µm, corresponding to a numerical aperture of 0.21. The system captures the displacement of focal spots, and numerical integration is then used to reconstruct the two-dimensional phase information of the optical wavefront. Compared with traditional Shack–Hartmann sensors, the meta SHWFS offers higher spatial resolution and a larger angular acceptance range, significantly enhancing its applicability for high-precision phase measurements in complex environments.

To enhance the adaptability of the Shack–Hartmann wavefront sensor to different observing conditions, Hu et al. [72] proposed a polarization-dependent pitch-switchable metalens array based on non-interleaved silicon metasurfaces, capable of working at multiple wavelengths in Figure 11c. The metasurface array, utilizing rectangular silicon nanostructures with a height of 600 nm and a period of 600 nm, features polarization tri-channel functionality and achromatic properties. By harnessing the full degrees of freedom in the Jones matrix of a single birefringent meta-atom, the metasurface array (MA) was designed with three adjustable pitches. This design allows for free switching between different polarization states of the incident and outgoing light, depending on the required conditions. Additionally, the metasurface exploits structural dispersion freedom, enabling operation at two distinct wavelengths: 950 nm and 1030 nm. This provides adaptability to different wavefront sensing conditions, optimizing detection performance under varying scenarios. Yang et al. [73] proposed a generalized Hartmann sensor based on a metalens array. The metalens array consists of elliptical silicon nanostructures with a height of 340 nm and a period of 1500 nm, and six distinct metalenses within each sub-array allow for the complete determination of the Stokes parameters at every pixel in the array. This innovative device can not only measure the phase distribution but also simultaneously capture the spatial polarization distribution at a wavelength of 1550 nm.

Weak measurement has been widely applied in precision metrology, as it allows the amplification of small phase changes by manipulating the near-orthogonality of the preselected and postselected states, enabling the extraction of phase information [74,75]. Luo et al. proposed a phase and amplitude reconstruction method based on weak measurement [76], as shown in Figure 12a, using a Pancharatnam–Berry (PB) phase metasurface to introduce weak coupling, controlling both the phase and amplitude of the light wave. By selecting appropriate preselected and postselected states, the real or imaginary parts of the weak value can be obtained, which are used to reconstruct the amplitude and phase, respectively.

Speckle-based computational imaging is a lensless imaging technique that can also be employed for quantitative phase imaging [77], as shown in Figure 12c. Kwon et al. developed a novel computational imaging system by replacing traditional scattering media with a dielectric metasurface scatterer. The system generates speckle patterns using the metasurface scatterer, and by combining precomputed scattering matrices with captured speckle intensity, it reconstructs the complex optical field of the object through the Speckle Scattering Matrix (SSM) method. The metasurface consists of a high-contrast rectangular silicon array, 652 nm in height and with a periodicity of 500 nm, designed as a scattering mask with uniform phase response. It effectively scatters light at an 850 nm wavelength, enhancing imaging resolution. By utilizing a metasurface with well-characterized optical properties, the system overcomes the stability issues associated with traditional scattering media and simplifies the complex, time-consuming experimental characterization process.

4. Conclusions

Overall, this review summarizes the advancements and challenges in phase measurement techniques based on metasurfaces. Currently, the core of phase measurement lies in recovering phase information from easily accessible intensity data. Common phase retrieval methods include phase contrast, holographic interferometry, shearing interferometry, the Transport of Intensity Equation and wavefront sensing.

Phase contrast techniques can leverage vortex phase contrast metasurfaces and non-local metasurfaces, with the primary advantage being real-time, rapid imaging. Vortex phase contrast metasurfaces can be integrated into a 4f system or used as a single-layer system to achieve isotropic phase imaging, making them compatible with traditional microscope systems and reducing the complexity of optical system construction. However, they cannot perform quantitative phase measurements. On the other hand, non-local metasurfaces, which break the symmetry of the optical transfer function through resonance, can improve resolution compared with traditional DIC microscopy. Combined with phase retrieval algorithms, they can achieve quantitative phase measurement, although they only work within a narrow bandwidth around their resonance wavelength.

Holographic interferometry requires the interference of a reference beam with a measurement beam. Metasurfaces with polarization multiplexing can simplify the co-axial optical path, but the setup is still affected by environmental factors, the size of the metasurface, and fabrication errors. Shearing interferometry, which avoids the use of a reference beam, relies on introducing small shear and phase shifts in the incident light. Traditionally, this shear is generated using Wollaston prisms, which are complex to manufacture and align. Metasurfaces offer a more flexible and straightforward approach by leveraging transmission phase and geometric phase principles. Combined with lateral shearing interferometry, metasurfaces have enabled single-shot quantitative phase imaging systems and Fourier spin-splitting microscopy, eliminating the need for capturing multiple phase-shifted images and enabling high-precision phase measurements. To address the issue of anisotropic phase gradients, metasurfaces can perform radially symmetric shearing of the incident light, achieving real-time isotropic phase measurements.

The Transport of Intensity Equation method requires mechanical movement to capture intensity information at different positions, and phase information is recovered using iterative algorithms. Metasurfaces, with appropriate design, can present information from different positions on the same plane, simplifying the optical path and eliminating the need for mechanical movement. However, the resolution of phase measurements is still limited by the field of view and the performance of iterative algorithms, leaving room for significant improvement. In traditional Shack-Hartmann Wavefront Sensor, lenslet arrays are constrained by MEMS manufacturing, resulting in large size and low numerical aperture. Metasurfaces, composed of subwavelength structures, can overcome these limitations by offering higher sampling density and larger measurable angles, achieving far higher resolution than traditional sensors.

The phase measurement techniques based on metasurfaces are briefly summarized in

Table 1. Summary of phase measurement techniques based on metasurfaces.

Technique	Quantitative (Yes or No)	Isotropic (Yes or No)	Optical System	Accuracy/Resolution	Ref.
Interferometry
Holography interferometry	✓	-	4f	0.15 rad	[56]
	✓	-	Single layer	-	[57]
Shearing interferometry	✓	O	Two layer	2.76 μm	[58]
	✓	O	4f	-	[59]
	✓	O	Two layer	-	[60]
	✓	✓	Mach-Zehnder	2 μm	[61]
	✓	✓	Single layer	5.52 μm	[62]
	✓	✓	Single layer	0.775 μm	[64]
Non-interferometry
Vortex phase	O	✓	4f	3.11 μm	[49]
	O	✓	Single layer	0.78 μm	[50]
	O	✓	Single layer	2.2 μm	[51]
Asymmetric OTF	O	O	Single layer	-	[52,53]
	✓	O	Single layer	10 mrad	[68]
	✓	O	Single layer	0.063 rad	[69]
TIE	✓	–	Single layer	0.1 rad	[67]
Shack–Hartmann wavefront sensor	✓	–	Single layer	0.1λ	[71]
Weak measurement	✓	–	4f	-	[76]
SSM	✓	–	Single layer	-	[77]

. Due to the metasurface’s ability to flexibly manipulate the light field, combined with its small size and multifunctionality, the optical system design becomes simplified when applying different phase retrieval methods. This paves the way for compact system designs that enable real-time, portable, high-precision, and non-destructive measurements. The metasurface design varies depending on the phase retrieval method used. The performance of measurement systems is limited not only by the phase retrieval principles but also by the traditional metasurface design and fabrication processes. Despite significant progress, achieving truly isotropic, high-precision, and quantitative metasurface-based phase measurement systems remains a challenge.

The challenges also point to possible future directions for the development of phase measurement based on metasurfaces. Several examples, though not exhaustive, are as follows: (i) New theoretical and experimental strategies to improve existing phase retrieval methods. (ii) Optimization algorithms for metasurface design to improve the performance of metasurface, such as field of view. (iii) Efficient phase recovery algorithms to accelerate phase reconstruction and enable high-sensitivity, real-time dynamic phase measurements. (iv) The use of tunable metasurfaces instead of static ones to develop multifunctional phase measurement systems, such as breaking the OTF symmetry dynamically. (v) Explore the combination of metasurface and quantum technology, quantum phase imaging offers high-resolution phase profiling [26].

Compared with traditional phase measurement techniques, metasurface-based phase measurement technology offers significant advantages in terms of miniaturization, integration, and multifunctionality. We believe that metasurface-based phase measurement systems will become increasingly compact, with trends toward higher resolution and isotropy, and will see widespread application in future technological developments.