Based on the optical statistics theory, the spatial coherence is a second-order statistical property describing the correlation between two spatial points of the random light fields. The characteristics of spatial coherence are calculated by the degree of coherence (DOC) function

where $${\mathsf{\rho }\mathsf{\rho }}_1$$ and $${\mathsf{\rho }\mathsf{\rho }}_2$$ are two position vectors. $$J({\mathsf{\rho }\mathsf{\rho }}_1,{\mathsf{\rho }\mathsf{\rho }}_2)=⟨E^{\ast }({\mathsf{\rho }\mathsf{\rho }}_1)E({\mathsf{\rho }\mathsf{\rho }}_2)⟩$$ is the mutual intensity function, where E denotes the random electric fields, the asterisk and angle brackets represent the complex conjugate and ensemble average over random fluctuations, respectively. The upper and lower limits of the modulus of the DOC function are 1 and 0, describing fully coherent and incoherent beams. For spatially uniform coherence of light, the DOC function depends only on the separation of two spatial points, i.e., $$\mu ({\mathsf{\rho }\mathsf{\rho }}_1,{\mathsf{\rho }\mathsf{\rho }}_2)=\mu ({\mathsf{\rho }\mathsf{\rho }}_2-{\mathsf{\rho }\mathsf{\rho }}_1)$$. We consider a stochastic field generated by a deterministic beam passing through a random complex screen, given by $$E(\mathsf{\rho }\mathsf{\rho })=A(\mathsf{\rho }\mathsf{\rho })T(\mathsf{\rho }\mathsf{\rho })$$, where $$A(\mathsf{\rho }\mathsf{\rho })$$ and $$T(\mathsf{\rho }\mathsf{\rho })$$ represent the complex deterministic electric fields and a complex screen, respectively. Taking the autocorrelation of $$E(\mathsf{\rho }\mathsf{\rho })$$, we obtain the relation of the random phase screens and the DOC function as $$\mu ({\mathsf{\rho }\mathsf{\rho }}_2-{\mathsf{\rho }\mathsf{\rho }}_1)=⟨T^{\ast }({\mathsf{\rho }\mathsf{\rho }}_1)T({\mathsf{\rho }\mathsf{\rho }}_2)⟩$$

where $$\mathbf{f}=\hat{\mathbf{x}}{f}_x+\hat{\mathbf{y}}{f}_y$$ is the spatial frequency vector, r(f) is a delta-correlated function composed of zero-mean and unit-variance complex Gaussian random numbers, and P(f) is the power spectral density (PSD) of the light beams, which is the Fourier transform of the DOC function. In practical applications, it is challenging to simultaneously modulate the random amplitude and phase. However, even if we only extract the phase part from the complex screen, i.e., $$\phi (\mathsf{\rho }\mathsf{\rho })=\mathrm{\angle }T(\mathsf{\rho }\mathsf{\rho })$$, the autocorrelation of the pure phase screen is almost the same as that of the complex screen according to the simulations in Fig. S1 in Supplemental Material. Therefore, we use the random phase screen instead of the complex screen to customize the spatial coherence structure:

The phase fluctuation of $$\phi (\mathsf{\rho }\mathsf{\rho })$$ can be designed with a prescribed DOC $$\mu ({\mathsf{\rho }\mathsf{\rho }}_2-{\mathsf{\rho }\mathsf{\rho }}_1)$$ to manipulate the coherence length and coherence structure of the incident beams.

With the ability of metasurfaces to accurately engineer the wave-front of light beams

Fig 1  Design of the disordered metasurface and its operation principle.

a The incident light passing through the disordered metasurface with a modified coherence structure and generating source speckles. With a scattering barrier blocking out the optical transmission, the far-field speckles are obtained. b The unit cell of the designed nanofin with disorder-engineered orientation set {θ_dis}. c The coherence structure obtained from the source speckles in a. d The statistical intensity statistically obtained from the far-field speckles, containing the transmitted optical information with (top) and without (bottom) the barrier

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A proof-of-concept design to demonstrate the proposed strategy is shown in Fig. 2. To manipulate the spatial coherence structure and coherence length of the beams, we first investigate the mapping of the on-demand Schell-Model DOC to the phase distributions $$\phi (\mathsf{\rho }\mathsf{\rho })$$ of the metasurface (Fig. 2a). A PSD can be calculated from the Fourier transform of the desired DOC. The phase distribution of the metasurface can be further calculated by the PSD from Eq. (2). We proposed two series of designs to generate beams with different spatial coherence structures and coherence lengths, i.e., Hermite–Gaussian correlated structure with m = 2, n = 2 (HG₂₂) and Laguerre-Gaussian correlated structure with n = 5 (LG₅), as shown in Fig. 2b, c. The DOC of the related HGCSM beams

Fig 2  Correlation between the DOC and the disordered metasurface design.

a Diagram of the synthesis of the transmitted function T of the disordered metasurface. b, c Intensity (top) and phase (bottom) distributions on the metasurfaces to manipulate spatial coherence with various prescribed DOC (HG₂₂ in b, LG₅ in c). The coherence lengths are 25 μm (left) and 10 μm (right). d An illustration of the spatial coherence manipulation with the metasurface. The spatial correlation of the instantaneous intensity distributions at the source plane determines the DOC of the average intensity transmitted from the metasurface

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where $${\delta }_0$$ denotes the coherence length of the partially coherent beams, H_m denotes the Hermite polynomial of order m, $$L_n^0$$ denotes the Laguerre polynomial of mode order n and 0. Due to the statistical properties of the partially coherent beams and the random phase distributions, the resulting phase distributions shown in Fig. 2b, c are not unique under the abovementioned generation condition of PSD. Figure 2d shows an illustration of the spatial coherence manipulation with the disordered metasurface. When illuminating the metasurface at different instants of time and different positions, different instantaneous speckle distributions are acquired containing the spatial correlation information of the light beam. The coherence length and coherence structure can be obtained from the statistical superposition of the instantaneous intensity distributions at the source plane. The instantaneous speckle distributions can be obtained through different methods, such as mechanical scanning, piezoelectric effect, structured illumination.

Figure 3a illustrates the detailed experimental process for capturing the intensity distributions and measuring the coherence structure. The experimental setup is shown in Fig. S5 in Supplementary Material. The disordered metasurface is controlled by an electric translation stage to uniformly move along the x-direction for loading a phase change to the wave-front correlated with a designed DOC. A right-handed circularly polarized Gaussian beam with a beam size of 15 μm is illuminated on the sample. An imaging system combined with an objective ×100, a tube lens (TL), and a CCD camera is used to capture the transmitted cross-polarized light and record the light distributions to verify the spatial coherence manipulation of the incident fields. The disordered metasurface is fabricated with the standard atomic layer deposition method through single-step lithography

Fig 3  Spatial coherence structures manipulated by the disordered metasurface and propagation properties of the generated beams.

a A right-handed circularly polarized incident light is focused on the metasurface, and the transmitted left-handed circularly polarized light can be collected by an objective paired with a tube lens. The propagation planes at different z_n are imaged by an imaging system to capture instantaneous intensity distributions at the source plane and far-fields. b SEM images of the fabricated metasurface. c, d The spatial coherence structure and far-field intensity distributions of the HG₂₂-correlated (c) and LG₅-correlated (d) beams customized in theory (top) and their measured counterparts (bottom)

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The spatial coherence structure determines the light distribution of the beams in the far-field. Such spatially distributed fields can carry transportable optical information of the metasurfaces. As shown in the right columns of Fig. 3c, d, the partially coherent beam with HG₂₂-correlated structure splits into four light spots, while the beam with LG₅-correlated structure possesses a dark hollow in the center. Both of the coherence structure and the far-field optical information can be arbitrarily shaped owing to the one-to-one correspondence between the DOC and far-field light distribution. The coherence structure can decide the beam profile in the far field. The coherence length can also affect the intensity distributions of these beams in the far field and the robustness of these beams during propagation. The detailed propagation properties of these beams with different coherence structures and coherence lengths are discussed in Supplementary Note S1.

We further experimentally demonstrate the self-reconstruction of the HG₀₁-correlated Schell-model beams with coherence lengths of δ₀ = 10 μm and δ₀ = 25 μm (Fig. 4). The schematic diagram of the beam self-reconstruction is shown in Fig. 4a. An obstacle placed away from the metasurface is illuminated by the generated partially coherent beam with a specially correlated coherence structure. A lens is used to perform the Fourier transform of the partially coherent beams to obtain the intensity at an infinite distance according to the Fraunhofer diffraction theorem. To better characterize the optical information carried by the beams, we normalized the light intensity to its maximum for both of the obstructed and unobstructed beams to exclude the effect of energy loss. The calculated intensity distributions of the obstructed and unobstructed beams are consistent with each other (Fig. 4b). The slight deviation is caused by the loss of a small portion of the coherence structure. Such information loss can be further decreased if decreasing the coherence length. The spatial coherence structure induces a re-arrangement of the energy distribution of the partially coherent beam as the beam propagates. Since the coherence structure is a global effect that is encoded to the whole beam, the partially coherent beam possesses self-similarity even for large obstructed angle α. We employed the self-reconstruction similarity degree (D_p) to characterize the similarity of the two intensity distributions and the self-reconstruction capability, which can be expressed as$$D_p=\frac{{[\iint ⟨I_{wt}(\mathsf{\rho }\mathsf{\rho })⟩⟨I_{ob}(\mathsf{\rho }\mathsf{\rho })⟩d^2\mathsf{\rho }\mathsf{\rho }]}^2}{\iint ⟨I_{wt}(\mathsf{\rho }\mathsf{\rho })⟩d^2\mathsf{\rho }\mathsf{\rho }\iint ⟨I_{ob}(\mathsf{\rho }\mathsf{\rho })⟩d^2\mathsf{\rho }\mathsf{\rho }}$$. We theoretically calculated D_p as a function of the coherence length of the HG₀₁-correlated Schell-model beams obstructed by an α = 3/2π and 7/4π sector-shaped opaque object, respectively (Fig. 4c). It is demonstrated that the self-reconstruction capability is independent of the obstacle shape in Fig. S8 in Supplemental Material. The D_p increases as the coherence length decreasing and reaches almost the same when $${\delta }_0\to 0$$. Figure 4d shows the measured beams without and with an α = 3/2π sector-shaped opaque object at the z = 0 plane. And more measured self-reconstructed beams can be found in Fig. S9 in Supplemental Material. The optical information at $$z\to \mathrm{\infty }$$ is measured at the focal plane with different coherence lengths (Fig. 4e, f). Compared with the unobstructed beam intensities, the obstructed ones rampantly excluding 3/4 areas of the beam can still provide sufficient optical information. The self-construction performance with δ₀ = 10 μm is better than that with δ₀ = 25 μm, and the calculated D_p are 0.9485 and 0.9024, respectively. Consequently, the coherence length and coherence structure of the partially coherent beams both determine the self-reconstruction ability. The presented self-reconstruction effect of the partially coherent beams can be used for robust information transfer and optical communications.

Fig 4  Experimental self-reconstruction of the HG₀₁-correlated beam blocked by a sector-shaped opaque object.

a Schematic of the self-reconstruction and intensity distributions of the partially coherent beams manipulated by the metasurface. b The theoretical cut-lines of the intensity distributions of the obstructed and unobstructed beams. c D_p as a function of the coherence length of the HG₀₁-correlated Schell-model beams obstructed by different sector-shaped opaque objects. d Intensity distributions for the unobstructed and α = 3/2π obstructed optical transmission at the z = 0 plane. e, f Intensity distributions of the beams at the focal plane for unobstructed and α = 3/2π obstructed optical transmission with a coherence length of 10 μm (e) and 25 μm (f)

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We further demonstrate the proposed strategy also enables self-reconstructed imaging with the designed disordered metasurfaces (Fig. 5). We fabricated two disordered metasurfaces with different DOCs related to two target images 'Panda' and 'Meta', as shown in the top left panels of Fig. 5a, b. The measured coherence structure information stored in the metasurfaces is shown in the top right panels. Compared with the images without any obstacles [(ⅰ) of Fig. 5a, b], all of the setups with different obstructed angles provides significant far-field self-reconstruction of the images [(ⅱ)-(ⅳ) of Fig. 5a, b]. The bright dot in the center is caused by the zero-order light resulting from the imperfection of fabrication and measurement, which more or less incorporates unwished polarized light. Surprisingly, even if the angle of the sector-shaped opaque object is as large as 15/8π, the details of the images are still well preserved. Especially shown in Fig. 5a, the characteristic information of the small-sized eyes and nose of the panda can still be retained. Meanwhile, the overall field-of-view (FOV) information of the patterns is also completely preserved, demonstrating the robustness of both the detailed and global optical information. The intensity loss near the boundary of the FOV is caused by the experimental error such as deviation of the FOV measured in the experiment, which is caused by the imperfect modulation of the collimation of the optical devices.

Fig 5  Self-reconstructed imaging by metasurface based on spatial coherence structures engineering.

The target image 'Panda' (a) and 'Meta' (b) to design the metasurface and the corresponding DOC distribution (top). (ⅰ)-(ⅳ): the intensity distributions at the focal plane with different obstacles

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The dielectric disordered TiO₂ metasurfaces were manufactured on a meticulously prepared fused silica substrate. Initially, a 550 nm layer of electron-beam resist (specifically, ZEP 520 A by Zeon) was uniformly applied to the substrate using a spin-coating process. This coated substrate was then subjected to a hot plate treatment at 180 ℃ for a duration of 1 min. To prevent any electric charge buildup during the subsequent electron-beam writing step, a thin layer of E-spacer 300Z (Showa Denko) was applied atop the resist layer. The precise nanostructures were defined through electron-beam lithography, specifically employing a Raith150 instrument operating at 30 kV with a current of 20 pA and a dose of 80 μC/cm². Subsequently, the exposed structures were developed in an n-amyl acetate solvent at room temperature for a period of 60 s. A conformal layer of TiO₂, with an approximate thickness of 70 nm, was deposited onto the substrate using an atomic layer deposition system (Picosun). This deposition process involved the use of titanium tetrachloride and H₂O as precursor materials within a reactor maintained at a temperature of 130 ℃. Following the deposition, a blank-etching procedure of the TiO₂ layer was carried out using CHF₃ plasma within an inductively coupled plasma-reactive ion etching (ICP-RIE) system. Under controlled conditions, the TiO₂ layer was etched until the underlying ZEP 520 A resist was exposed. These etching conditions involved the use of 20 sccm of CHF₃, with a bias power of 20 W and an induction power of 500 W, all within an operating pressure of 10 mTorr. This process resulted in an etching rate of approximately 40 nm per minute for the TiO₂ layer. Finally, O₂ plasma with a minor addition of CHF₃ was used to fully remove the remaining ZEP resist.