Phase-probability shaping for speckle-free holographic lithography

Abstract

Optical holography has undergone rapid development since its invention in 1948, but the accompanying speckles with alternating dark and bright spots of randomly varying shapes are still untamed now due to the intrinsic fluctuations from irregular complex-field superposition. Despite spatial, temporal and spectral averages for speckle reduction, holographic images cannot yet meet the requirement for high-homogeneity, edge-sharp and shape-unlimited features in optical display and lithography. Here we report that holographic speckles can be removed by narrowing the probability density distribution of encoded phase to homogenize optical superposition. Guided by this physical insight, an Adam-gradient-descent probability-shaping (APS) method is developed to prohibit the fluctuations of intensity in a computer-generated hologram (CGH), which empowers the experimental reconstruction of irregular images with ultralow speckle contrast (C = 0.08) and record-high edge sharpness (~1000 mm⁻¹). These well-behaved performances revitalize CGH for lensless lithography, enabling experimental fabrication of arbitrary-shape and edge-sharp patterns with spatial resolution of 0.54λ/NA.

The authors report lensless holography lithography with diffraction-limited resolution by proposing a phase-probability shaping mechanism to suppress speckle noise efficiently.

Introduction

Holography can reconstruct a predefined image from optical scattering of a well-designed mask with random phase or amplitude profile under the illumination of a high-coherence light source^1,2. The optical field at each position in the image plane can be taken as a coherent superposition of diffracted fields from all the pixelated sources at the holographic mask³. The random phase or amplitude encoded in the mask introduces the uncertainty between constructive and destructive interferences^4–6, leaving bright and dark spots with randomly varying shapes. Such fluctuations of intensity are named holographic speckles, which increase the noise level and destroy the uniformity, thereby constraining its applications in optical display^7–9 and lithography^10,11. Therefore, speckle reduction has become one of the central problems in optical holography towards practical usage.

Since randomness in the phase or amplitude mask is mandatorily required to reconstruct an irregular image, almost all the reported approaches for speckle reduction share the common origin of coherence control: the average of multiple incoherent speckles. For example, employing a degenerate-cavity laser with multiple orthogonal modes¹² and decreasing optical spatial coherence with a rotating diffusor¹³ are taken as the spatial average of the speckles^14,15, while the spectral average^16–18 is realized by broadening (or narrowing) the bandwidth of a laser⁵ (or a light-emitting diode¹⁹). In addition, the temporal average is also proposed by summarizing many speckles in a time sequence^20–22. These averaging techniques can reduce the speckles efficiently but at the cost of blurred edges, missing details or time-consuming exposure.

Subsequently, persistent efforts have been made by developing novel algorithms^19,23–27, but speckle reduction is achieved by introducing some additional approaches such as partially coherent sources¹⁹ and camera-in-the-loop training²³, or by decreasing the edge sharpness with spatial filtering of the image²⁴. Although some modified Gerchberg-Saxton (GS) algorithms^25–30 based on initial quadratic or random phase have been proposed to suppress the holographic speckles, they are usually reported with good performances for Fraunhofer holograms. When extended for Fresnel holograms, these algorithms cannot suppress the speckles satisfactorily (see Supplementary Section 3). Despite these advances in hologram design, the statistical property of the holographic phase is less considered. Since the random phase in holograms fundamentally governs the intensity fluctuations within speckles, its statistical properties may provide a fundamental yet hitherto unexploited avenue for the elimination of the holographic speckles.

Here, we demonstrate that the speckle contrast of a holographic image can be theoretically suppressed down to nearly zero by shaping the probability density of the encoded random phase even under high-coherence illumination. This finding guides us to develop a phase-probability shaping method based on Adam-gradient-descent algorithm for hologram design. Experimentally, we realize the designed phase by using dielectric geometric metasurfaces and reconstruct highly uniform images with sharp edges (such as a binary “bull”, a two-dimensional “barcode”, and an optical “fork grating”) to demonstrate the prototype of computer-generated-holography (CGH) lithography. To validate its feasibility in fabricating high-resolution structures, we also demonstrate binary-phase CGH lithography that could pattern integrated circuits with a feature size of 310 nm (0.54λ/NA, where NA is the numerical aperture of the hologram and the wavelength λ = 405 nm), which approaches the diffraction limit of light. Compared with other lithographic approaches, the CGH lithography is lensless without any complicated projection system and exhibits the comparable resolution at single exposure, thus offering a potential alternative for lithography technology.

Speckle suppression through phase-probability shaping

First, we derive the statistical properties of optical speckles generated by optical diffraction from a random phase mask (Fig. 1a). If each phase φ_n at the n^th pixel in the mask is valued by obeying a Gaussian-shape probability of $P\left(\phi \right)=\exp \left[-{\phi }^2/\left(2{\sigma }^2\right)\right]/\left(\sqrt{2\pi }\sigma \right)$ (where σ determines its probability distribution⁴), the diffraction field at the position (x, y, z) is expressed as E = u + iv = Ae^iθ, where u, v, A and θ are its real part, imaginary part, amplitude and phase, respectively. Because φ_n is random by obeying the Gaussian-shape probability, u and v are also Gaussian random variables. When the number of the pixels is large, we use the central limit theorem³¹ to calculate the u/v-joint probability density function, from which the intensity probability density P_I(x,y,z,I) (where I = A²) at position (x,y,z) can be derived after several variable transformations (see Supplementary Section 1). P_I(x,y,z,I) is the local intensity probability density due to its strong dependence on the spatial position. After integration with respect to x and y, we obtain the intensity probability density in the interested region Ʃ of the speckle at a z-cut plane

$$P_I\left(z,I\right)=\frac{1}{S}{\int }_{\mathrm{\Sigma }}{P}_I\left(x,y,z,I\right)dxdy=\frac{1}{S}{\int }_{\mathrm{\Sigma }}\left\{\frac{1}{4\pi {\sigma }_u{\sigma }_v\sqrt{1-{\rho }_{u,v}^2}}{\int }_{-\pi }^{\pi }\exp \left[-\frac{{\left(\frac{\sqrt{I}\cos \theta -ū}{{\sigma }_u}\right)}^2+{\left(\frac{\sqrt{I}\sin \theta -\overline{v}}{{\sigma }_v}\right)}^2-2{\rho }_{u,v}\cdot \frac{\left(\sqrt{I}\cos \theta -ū\right)\left(\sqrt{I}\sin \theta -\overline{v}\right)}{{\sigma }_u{\sigma }_v}}{2\left(1-{\rho }_{u,v}^2\right)}\right]d\theta \right\}dxdy$$ 1

where S is the area of the region Ʃ, and the average of the variable X is defined as $\overline{X}=\int X\cdot P\left(\phi \right)d\phi$, ${\sigma }_X^2=\overline{{\left(X-\overline{X}\right)}^2}$ and ${\rho }_{u,v}=\overline{\left(u-ū\right)\left(v-\overline{v}\right)}/\left({\sigma }_u{\sigma }_v\right)$. Once z and Ʃ are fixed, the parameters$ū$, $\overline{v}$, ${\sigma }_u$, ${\sigma }_v$ and ${\rho }_{u,v}$ determine the relative probability density of intensity in the speckle. Since these parameters are the statistical average under the Gaussian probability P(φ), Eq. (1) bridges the link between the statistical properties (i.e., $P_I\left(z,I\right)$) of the speckle and the phase probability P(φ) of the mask.

Fig. 1. Working principle of speckle suppression through phase-probability shaping.

a Sketch for optical diffraction from a random phase mask that is located at z = 0. The size of the phase mask is exemplified to be 128 μm × 128 μm, resulting in the pixel number N = 512 × 512 because ∆x = ∆y = 0.25 μm in our simulations. The propagation distance between the phase mask and the target plane (where the speckle is evaluated in this work) is chosen at z = 250 μm for the proof-of-concept demonstration. b Probability density distributions of the phase mask. The red curves show the probability shapes that are labeled by different standard deviations from σ = 0.1π to σ = 0.7π with an interval of 0.1π. The cyan parts exhibit equivalent shapes after phase wrapping for |φ | >π. c Simulated diffraction patterns from the phase masks with different σ. The region of interest Ʃ is a square of 100 μm × 100 μm centered on the position (x = 0, y = 0, z). For a better comparison, all intensity profiles are normalized to the maximum intensity of the σ = 0.1π case. d Theoretical (solid curves) and simulated (black hollow circles) probability density distributions of the diffraction patterns at z = 250 μm. The theoretical results are obtained by using Eq. (1). e Speckle contrasts. The solid curves denote the theoretical prediction by Eq. (1), while the diamonds are obtained by the statistical analysis of the diffraction patterns in (c).

Now, we investigate the role of the mask’s phase probability in determining the properties of the speckle. By using the Rayleigh-Sommerfeld integral³², the simulated diffraction patterns of the masks with different phase-probability shapes (denoted by σ in Fig. 1b) indicate stronger fluctuations of intensity for larger σ (Fig. 1c). When σ approaches infinity and P(φ) is considered a uniform probability distribution, Eq. (1) is simplified into $P_I\left(z,I\right)=\frac{1}{S}{\iint }_{\mathrm{\Sigma }}{e}^{-I/\left(2{\sigma }_u^2\right)}/\left(2{\sigma }_u^2\right)dxdy$ (where ${\sigma }_u^2={\sum }_{n=1}^N{a}_n^2/2$), which behaves like the Rayleigh statistics for a fully developed speckle⁴. Such an exponentially decaying probability density also exists in the diffracted patterns for σ = 0.6π and 0.7π (Fig. 1d) and exhibits nearly perfect agreement with the theoretical prediction by Eq. (1). The speckle contrasts^4,33,34 ($C=\sqrt{\overline{I^2}-{\left(Ī\right)}^2}/Ī$, where the average variable $\overline{X}=\int X\cdot P_I\left(z,I\right)dI$) is close to 1, implying poor uniformity. In contrast, for the cases of small σ (e.g., σ = 0.1π), the diffraction patterns are homogeneous with reduced fluctuations of intensity (Fig. 1c), leading to the narrowly distributed probability density (Fig. 1d) and low speckle contrast C (Fig. 1e). These results suggest that optical speckles can be suppressed by reducing the width of the phase probability distribution in the mask, which holds the physical origin of speckle-free holograms. Namely, the uniform probability distribution in the phase mask should be excluded during the hologram design.

Hologram design with the APS method

To design speckle-free holograms in a framework of phase-probability shaping, we develop the APS algorithm, as sketched in Fig. 2a and b and introduced in detail in the Methods. Our APS algorithm is built with three key features: 1) an iteration method based on the Adam gradient descent³⁵ to build up the phase-probability configuration by using the independent phase feedback of each pixel (Fig. 2b); 2) a hybrid loss function including RMSE (the root-mean-square error between the simulated and ideal images), SD (standard deviation of the image) and optical efficiency η to fully evaluate the image quality; 3) the dynamic weighted coefficients w₁, w₂ and w₃ with their maximums controlled by the constants w_RMSE, w_SD and w_Eff (Fig. 2c) to control the singularity feature of the image.

Fig. 2. Hologram design via phase-probability shaping.

a Sketch for holographic imaging using geometric metasurfaces. b, c Flowchart (b) for the phase updating strategy in each iteration. The loss function is a weighted superposition of RMSE, standard deviation and optical efficiency, where their weights w₁, w₂ and w₃ depend tightly on the index (labeled by i) of the iterations, as shown in (c). The mathematical expressions of these weights are provided in the Methods. d Map of phase singularity in the optimized images for different w_SD and w_Eff. Depending on both parameters (because w_RMSE = 1), the phase singularity disappears (cyan) or exists (orange) in the images. Both parts have the boundary of a white dashed line (w_SD = 0.2956w_Eff-0.008096), which is obtained by fitting our simulated boundary (black dots). In our simulations, a metasurface hologram with a pixel pitch of 250 nm × 250 nm is illuminated by a laser light of λ = 633 nm and then reconstructs a “bull”-pattern image at a propagating distance of z = 250 μm. e Simulated images without (A) or with (B) phase singularities. f Probability evolution of the designed phases for the singularity-free (top) and singularity (middle) cases. Their weights with increasing iterations (bottom) are also plotted for a better observation of the phase-probability change. g Parametric (i.e., optical efficiency, SD and RMSE) evolution of the optimized image with the increment of the iteration number (distinguished by the line color). Cases A and B are the evolution processes for a holographic image with or without phase singularity. Their parametric evolutions go along two completely different paths, finally arriving at (RMSE = 0.025, SD = 0.079, η = 0.296) for A and (RMSE = 0.038, SD = 0.197, η = 0.396) for B. h Phase probability distributions of designed masks for the singularity-free (triangles) and singularity (squares) cases after 1500 iterations. For case A, the fitted probability density (red curve) has the form of $P\left(\phi \right)={\sum }_{m=1}^4{a}_m\exp \left[-{\left(\frac{\phi -b_n}{C_m}\right)}^2\right]$, where a₁ = 0.2141, a₂ = 0.1658, a₃ = 0.1181, a₄ = 0.1249, b₁ = 1.067, b₂ = 2.15, b₃ = −1.169, b₄ = 0.15, c₁ = 0.5, c₂ = 0.45, c₃ = 0.57 and c₄ = 2.55. For case B, the fitted probability density (black curve) is $P\left(\phi \right)=0.1106+0.171\times \exp \left[-{\left(\frac{\phi -0.935}{1.007}\right)}^2\right]$. In the fitted results, the phase probability for |φ | >π is wrapped into the region |φ | ≤π.

By carrying out the APS algorithm with 1500 iterations, we designed different holograms by scanning the constants w_SD and w_Eff from 0 to 1 (w_RMSE = 1 is used for all the simulations due to its dominating role in converging the diffraction pattern to the ideal image). To highlight the speckle features in Fig. 2d, we divide all the simulated images into two types: the patterns with (orange part) or without (cyan part) phase singularities (the simulated efficiency, RMSE and SD are discussed in Supplementary Section 2). Both parts are separated by a linear boundary, which is the fitting of the discrete positions at the simulated boundary. In the parameter space of (w_SD, w_Eff), the cyan part has a larger area than the orange part, revealing that our APS algorithm can generate more singularity-free solutions than the singularity ones. To show their differences, two simulated diffraction patterns, labeled A (w_SD = 1, w_Eff = 0.1) and B (w_SD = 0.1, w_Eff = 1), are exemplified in Fig. 2e, revealing the good uniformity for case A and the dark spots with phase singularities for case B. These results have theoretically confirmed the validity of our APS algorithm in tailoring holographic speckles via the parametric operations of w_SD and w_Eff.

Next, we reveal the fundamental importance of the phase probability in achieving the speckle-free holograms with our APS algorithm. Due to the different w_SD and w_Eff in both cases (A and B), their phase possibilities undergo completely distinguished evolution (Fig. 2f). For case A, the weight w₂ (because w_SD = 1) dominates the loss function when i < 175, where its phase probability, RMSE and efficiency (Fig. 2g) are unchanged but its SD decreases quickly down to zero to enhance the uniformity of the image. When i > 175 (where the weight w₁ dominates), the phase probability density extends linearly with three peaks and then becomes stable for i > 400 (where RMSE decreases monotonously for approaching the target but both the SD and efficiency go through one peak). In comparison, for case B, the initial w₃-dominated optimization expands the phase probability to enhance the optical efficiency, but the accompanying large SD leads to undesirable phase singularities that always exist in the following w₁-dominated optimization (where the phase probability is nearly unchanged). As a result, the phase probability density (Fig. 2h) for case B can be taken as a superposition of the Gaussian and uniform distributions. According to the prediction in Eq. (1), its uniformly distributed part leads to the creation of optical speckle, which is the underlying reason for the phase singularities in the relative image (Fig. 2e, bottom). Therefore, for case A, the narrowly distributed phase probability as a summation of four Gaussian shapes enables the reconstruction of the speckle-free images. The physical origin of such speckle-free reconstruction is discussed in Supplementary Section 2.

Experimental verification of speckle-free holograms

To verify these predictions, two exemplified phase profiles are designed by using the APS (Fig. 3a) and GS (Fig. 3b) algorithms³⁶, respectively. For our APS case, the phase probability density has a narrow distribution (featured by the zero probability at −0.86π in Fig. 3c), which allows less fluctuations of intensity than that (taken as a superposition of the uniform and Gaussian shapes) for the GS case. As expected, the simulated image (Fig. 3d) for the APS case reveals better uniformity (with the speckle contrast $C=SD/Ī=0.07$) than that (C = 0.46) of the GS case (Fig. 3e).

Fig. 3. Experimental characterization of speckle-free holograms.

a, b Designed phase profiles by using APS (a) and GS (b) algorithms. Without the loss of generality, we use the parameters w_SD = 0 and w_Eff = 0 (near the boundary but still in the singularity-free region of Fig. 3(d)), so that the designed holographic images have no speckle. All the other parameters in both the APS algorithm and optical system are identical to those in Fig. 2. c Probability density distributions of the designed phases in (a) and (b). The APS case has a fitted probability density of $P\left(\phi \right)=0.2504\times \exp \left[-{\left(\frac{\phi -4.067}{0.3872}\right)}^2\right]+0.1714\times \exp \left[-{\left(\frac{\phi -3.55}{2.557}\right)}^8\right]$(red curve), while $P\left(\phi \right)=0.0949+0.0875\times \exp \left[-{\left(\frac{\phi -3.233}{1.003}\right)}^2\right]+0.2557\times \exp \left[-{\left(\frac{\phi -4.028}{0.5478}\right)}^2\right]$ for the GS case (blue curve). (d-e) Simulated intensity profiles for the APS (d) and GS (e) cases. The dashed lines (AA’ and BB’) denote the position for line-scanning intensity for a comparison between the simulated and experimental results. Scale bars: 20 μm. f, g SEM images of the fabricated silicon metasurfaces for the APS (f) and GS (g) cases. Scale bars: 300 nm. h, i Measured intensity profiles for the APS (h) and GS (i) cases. Scale bars: 20 μm. j Comparison of the experimental and simulated line-scanning intensity profiles for the APS (left panel, along AA’ in (d) and (h)) and GS (right panel, along BB’ in (e) and (i)) cases. (k) Probability density distributions of measured intensity profiles for the APS (triangles) and GS (squares) cases. The Gaussian shape is used to fit the experimental probability density. (l-m) The FWHMs (l) of the intensity probability density and speckle contrasts (m) at the different wavelengths. The simulated data are shown as solid lines and the experimental data are labeled by triangles for the APS and squares for the GS case. n Measured (curve) and fitted (line) line-scanning intensity profile at the edge of the holographic image. The insert shows the edge position of the experimental image. The sharpness is calculated by using the formula in the insert. o Experimental (triangles) and simulated (curve) sharpness of the images at the different wavelengths.

Experimentally, we utilize geometric metasurfaces^37–42 composed of rotating dielectric nanobricks (see the design details in Supplementary Section 4) to realize phase masks, which are fabricated via standard electron-beam lithography with a dry-etching process (see the fabrication steps in Supplementary Section 5). The well-fabricated samples (see their scanning-electron-microscopy (SEM) images in Fig. 3f, g) enable us to achieve high-fidelity holographic images in Fig. 3h and 3i by using a self-built measurement setup (Supplementary Section 6). By comparing their experimental speckle contrasts (C = 0.08 and C = 0.50 for the APS and GS cases respectively) and line-scanning intensity profiles (Fig. 3j), we confirm the expected speckle reduction by using our APS algorithm. To ensure this further, the probability density of its normalized intensity is calculated with a FWHM (i.e., full width at half maximum, Fig. 3k) of 0.122, which is approximately one order of magnitude smaller than that (FWHM = 1.14) for the GS image. A further discussion about the experimental factors influencing the uniformity of the holographic images is provided in Supplementary Section 7.

Due to the dispersion-less properties of geometric metasurfaces⁴³, the speckle-free features are valid over a wide spectrum (see the measured images in Supplementary Fig. 7). Nearly constant values near λ = 633 nm are observed in the simulated and experimental FWHMs (Fig. 3l) and speckle contrasts (Fig. 3m). For λ < 608 nm, the absorption of silicon in geometric metasurfaces decreases the conversion efficiency (Supplementary Fig. 8) from the incident circular polarization to its cross polarization, thus increasing the background that leads to fluctuations of intensity (featured by larger FWHMs in Fig. 3l and speckle contrasts in Fig. 3m). Nevertheless, the APS case still has better uniformity over the broadband spectrum than the GS case.

To characterize the edge sharpness, we employ the inverse of the spatial range between the 10% and 90% of the maximum intensity at the edge¹² (Fig. 3n). Figure 3o shows the simulated and measured sharpness of ~1000 mm⁻¹, which is enhanced by nearly ~100 times compared with the latest reported sharpness via the spatial-mode average¹². Such a giant enhancement originates fundamentally from our high-coherence and single-mode reconstruction via phase-probability shaping, where none of the previous averaging or filtering methods is included. Moreover, by using larger-size metasurfaces that offer much higher spatial frequencies^3,44 for holographic reconstruction, we can enhance the edge sharpness further to fabricate finer patterns in optical lithography.

Prototype of holographic lithography

To demonstrate proof-of-concept holographic lithography, we design a phase mask at λ = 405 nm and fabricated relative geometric metasurfaces made of less-absorption silicon-nitride nanobricks (see the SEM image in Fig. 4a). To show the capacity of patterning complex structures, two holographic images (i.e., a two-dimensional random barcode and a fork grating) are exemplified with their measured intensity profiles shown in Fig. 4b, c, respectively. Due to the limited imaging distance of 390 μm, the holographic images are magnified first by using a couple of objective lenses and then projected onto the photoresist (Microposit S1813, Shipley) with an exposure time of 10 seconds (see the details in Supplementary Section 8). The developed photoresist patterns (Fig. 4d, e) show the expected details with clear edges, implying highly homogeneous exposure. In the exposure area, one cannot see random speckle dots or broken lines, which appear in previous attempts^10,11,45,46 to demonstrate holographic lithography due to the un-eliminated speckles.

Fig. 4. Demonstration of CGH lithography.

a Sketch for holographic lithography by using silicon nitride (Si₃N₄) metasurfaces with the SEM image shown in the insert. b, c Measured intensity profiles of a two-dimensional barcode (b) and a fork grating (c). The design and characterization details are provided in Supplementary Section 8. Scale bars: 10 μm. d, e Microscopy images of the developed photoresists with the barcode (d) and fork grating (e) patterns. The smallest feature sizes are ~25 μm in the patterned structures for both cases due to their different magnifications of ~7.3 (barcode) and ~10.3 (fork grating). Scale bars: 100 μm. f Diffraction patterns of the patterned fork grating at different wavelengths from 410 nm to 590 nm with an interval of 20 nm. Only three orders (0 and ±1) are shown here due to the limited detection region of our camera. g Measured height of the patterned photoresist along the dashed line CC’ in (e). h Simulated (curve) and measured (green diamond) ratio of the 1-order diffracted intensity to the total intensity of three (0 and ±1) orders. The predicted phase delay between the patterned or unpatterned region in the developed fork grating is shown on the right axis.

Such holographic lithography is doubly examined by characterizing the optical performance of the patterned fork grating. Under the illumination of laser light (see Supplementary Section 9), the fork grating diffracts light into different orders, where the dark centers in ±1-orders (Fig. 4f) indicate the broadband creation of the expected vortex phase^47,48. To quantitatively evaluate its performance, the photoresist in the fork grating is measured with a height of h = 950 nm (Fig. 4g) by using a profilometer, yielding a wavelength-dependent phase delay (Fig. 4h) of ψ = 2π(n-1) h/λ (n is the refractive index of the photoresist). When such phase delay is odd times of π, the ratio of the 1^st-order intensity to the total intensity of 0 and ±1 orders reach its maximum of 0.5, which is verified by the simulated and experimental results with high consistency (Fig. 4h). These results confirm that, this fork grating becomes the first optics-level and high-quality element delivered by using CGH lithography.

To explore additional characteristics of CGH lithography, such as phase-modulation depth, working wavelength, and resolution, a binary phase plate is employed here instead of metasurfaces for the following demonstration. The use of a binary phase plate is justified for three key reasons. Firstly, the multi-level phase modulation provided by dielectric metasurfaces for speckle-free holograms is only effective at long-wavelength spectra. When the working wavelength falls below 200 nm, high-efficiency metasurfaces with multi-level phase modulation are unattainable due to the lack of materials with both high refractive indices and low absorption. Consequently, it is natural to investigate whether fewer phase levels (e.g., binary) are sufficient to achieve speckle-free holograms in CGH lithography. Secondly, binary-phase modulation for speckle-free holograms determines whether CGH lithography can be extended into short-wavelength spectra, such as vacuum and extreme ultraviolet wavelengths. Thirdly, binary-phase-based CGH lithography provides a generalized platform to demonstrate its resolution limit, as binary-phase modulation is easily achievable across a broader spectrum.

Figure. 5a illustrates the designed phase with a probability of ~0.5 for each phase level of 0 and π (see Fig. 5b), maintaining an equal density for the binary phase to ensure sufficient modulation for the reconstruction of speckle-free images. Using a sample (see microscopy images in Fig. 5c) fabricated via electron-beam lithography, we measured the experimental image with an “integrated circuits” pattern (its total size is ~27 μm × 27 μm, see Fig. 5d) at a distance of z = 300 μm. Given that the size of the binary phase plate (600 μm × 600 μm) is significantly larger than the reconstructed images, this hologram functions similarly to a lensing system with an efficient NA of 0.71 (=sin[tan⁻¹(R/z)] = sin(π/4)), implying a diffraction limit of ~294 nm (0.515λ/NA). Therefore, the linewidth of the “integrated circuits” pattern is designed to be 300 nm, approaching the diffraction limit. The reconstructed pattern in Fig. 5d shows the uniform lines without speckles, confirming that binary phase modulation provides sufficient depth to achieve speckle-free holograms with features close to the diffraction limit of light.

Fig. 5. Binary-phase-based CGH lithography.

a, b Designed phase profiles (a) and its probability density distribution (b). The total size of the phase plate is 600 μm with a pixel pitch of 300 nm × 300 nm. c Optical image of the fabricated binary-phase plate. The inset shows its zoomed-in image for a better observation. d Measured “integrated circuit” pattern at an imaging distance of z = 300 μm under the illumination of a 405 nm-wavelength laser with a coherent length of 1 meter. e SEM images of the developed photoresist by using the pattern in (d) at the exposure time of 11 seconds. f Statistical counts at different widths of lines in the developed pattern. The widths are extracted from the SEM image in (e). g SEM images of the developed photoresists under different exposure times. The red and blue circles denote the disappearance of two adhesive points when the exposure time increases. h Average widths of the patterns in (g). i, j SEM images (i) and adhesive points (j) of the developed photoresist patterns under different defocusing distances. The defocusing distance Δz = z-f ranges from Δz = −0.3 μm to Δz = 0.4 μm. The exposure time is 11 s.

To demonstrate the potential in lithography, the reconstructed “integrated circuits” is used to directly expose the photoresist for 11 seconds using a homemade setup (see Supplementary Fig. 13a). The developed photoresist exhibits the expected pattern (see its SEM image in Fig. 5e) with uniform lines and sharp edges, verifying the feasibility of binary-phase-based CGH lithography. To assess resolution, randomly selected lines from the developed photoresist are measured (without the loss of generality). Fig. 5f shows an experimentally etched linewidth of around 310 nm, slightly larger than the expected 300 nm due to photoresist exposure control and defocusing distance. By varying only the exposure time T, the profiles of the developed patterns in Fig. 5g (see full version in Supplementary Fig. 13b) reveal adhesive points (denoted by red and blue circles) for T ≤ 10 seconds, which disappear with longer exposure time. This suggests an optimal exposure time of around 11 seconds in our current lithographic demonstration. Near the optimal exposure, the experimental linewidths are ~300 nm (see Fig. 5h).

Additionally, the depth of focus (DOF) is characterized using defocusing distance Δz with a longitudinal interval of 100 nm, resulting in different exposed patterns in Fig. 5i. As |Δz| increases, the exposed patterns exhibit more adhesive points (Fig. 5j) due to non-uniform lines in the reconstructed patterns, as the designed coherent superposition for speckle-free images cannot be maintained at large defocusing distances. Simulated images (Supplementary Fig. 14) with different defocusing distances provide theoretical evidence for these adhesive points caused by low-intensity parts of non-uniform lines. Nevertheless, due to the slow variation of the light field along the propagation direction, a DOF of ~200 nm is achieved (extracted from the exposed patterns in Fig. 5j and simulated images in Supplementary Fig. 14), smaller than that of a focusing lens with the same NA ( ~0.5λ/(1-cosθ_max), where θ_max is the maximum convergent angle). This shorter DOF indicates that coherent control in speckle-free holograms is more stringent than constructive interference in a lensing system.

Discussion

Although CGH lithography has demonstrated good feasibility, further optimization is needed in resolution limits, exposure control, alignment, and DOF extension. To reduce the linewidth further, we designed the “integrated circuits” patterns with subdiffraction-limit linewidths of 258 nm (Supplementary Fig. 15) and 279 nm (Supplementary Fig. 16). Despite the good profile of the simulated patterns, the resulting lithographic patterns exhibit broken lines and adhesive points due to non-uniform experimental lines and imperfect control of exposure, see the detailed discussions in Supplementary Section 10. Achieving uniform lines with subdiffraction-limit widths via CGH lithography remains an open question nowadays. To explore the theoretical boundaries of linewidth, we offer a concise analysis of holographic patterns grounded in Fourier optics, proposing a theoretical resolution limit of 0.25λz/NA (see Supplementary Section 11) for CGH lithography.

In addition to exposure time, photoresist thickness is another critical parameter for optimization. For instance, when the photoresist thickness (e.g., 400 nm, the minimum achieved experimentally in this work) exceeds the DOF (~200 nm in our binary-phase CGH lithography), non-uniform lines at defocusing distances are recorded on the photoresist, potentially causing broken lines and adhesive points in the etched patterns. For alignment, industry-level high-precision stages with defocusing feedback are necessary to position the photoresist at the focus. Finally, to extend the DOF, modified algorithms should be developed to reduce speckles at defocusing planes through multi-plane holographic reconstruction³⁹.

Compared to other approaches, such as mixed region amplitude freedom (MRAF) lithography⁴⁹, complex amplitude modulation holographic femtosecond laser printing (CAM-HFLP) method⁵⁰, maskless optical projection nanolithography (MLOP-NL)⁵¹, patterned pulse laser lithography (PPLL)¹⁰, and DUV/EUV lithography^52–55, CGH lithography is the only lensless tool (see detailed comparison in Supplementary Table 1) that combines the lensing and holography functionalities simultaneously, eliminating the need for costly projection systems (indispensable in other approaches). Additionally, CGH lithography does not need high-power sources like femtosecond lasers used in other approaches, although it necessitates sufficient coherent length to ensure optical coherent superposition at the target plane. Although CGH lithography theoretically allows large-area (e.g., centimeter-scale) single-shot exposure, this poses challenges in the fabrication and design of larger phase mask.

In summary, we have reported high-uniformity, edge-sharp and shape-unlimited holographic images under the illumination of single-mode, high-coherence lasers by shaping the phase-probability with the physics-guided APS algorithm. Benefiting from these high-quality images, we have presented a prototype of CGH lithography that exhibits the capability of patterning optics-level complex micro/nanostructures with a diffraction-limited resolution. This technique offers the possibilities to develop low-power and lensless lithography for fabricating large-size nanophotonic devices efficiently.

Methods

Adam-gradient-descent probability-shaping algorithm

This algorithm is developed for designing the phase-type compute-generated hologram with the controllable probability-shaping method. Here, the geometric metasurfaces are used to realize an arbitrarily designed phase (twice the rotating angle of the dielectric nanobricks⁴⁰) for the incident circular-polarization light because their subwavelength pixel pitches can exclude the undesired twin images and high diffraction orders^56,57. Figure 2a sketches the reconstruction of the holographic image through optical diffraction from phase-encoded geometric metasurfaces. The design algorithm contains the following key steps:

I. Simulating the diffraction pattern. The rigorous Rayleigh-Sommerfeld diffraction integral³² is employed to numerically simulate the diffraction patterns without any approximation. In each iteration, the simulated intensity pattern at the target plane is employed to calculate the key parameters such as RMSE, SD and optical efficiency.

II. Updating the key parameters. To avoid any confusion, we provide their definitions:$RMSE=\sqrt{{\sum }_{n=1}^N{\left(I_n/Ī-I_{ideal}\right)}^2/N}$, $SD=\sqrt{{\sum }_{m=1}^M{\left(I_m-Ī\right)}^2/M}$, $\eta =P_{image}/P_{incident}$, where I_n is the intensity at the n^th pixel located in the entire simulated pattern (containing N pixels), $Ī$ is the average intensity only in the bright image region, I_m is the intensity at the m^th pixel located only in the entire image (containing the bright ‘bull’ image and the surrounding dark region), I_ideal is the intensity of the ideal target, P_image is the total power encircled within the image and P_incident is the total power of incidence. By using the above definitions, we can update the RMSE, SD and optical efficiency in one iteration. Note that, these three parameters are highly dependent on the simulated patterns and hence become the key parameters to evaluate the image quality of the hologram. The RMSE can evaluate the convergence of the simulated image to the ideal one, the SD is critical in removing the phase singularities with dark intensity and enhancing the uniformity of the holographic image, and the parameter η is employed to improve the optical efficiency of the hologram.

III. Constructing the loss function. To obtain holographic images with high quality such as good uniformity without any speckle and high efficiency, we adopt a multi-parameter loss function with the form of ${LF}_i=w_1\left(i\right)\cdot {RMSE}_i+w_2\left(i\right)\cdot {SD}_i+w_3\left(i\right)\cdot \left({1-\eta }_i\right)$, where i is the index of the iterations, and w₁(i), w₂(i) and w₃(i) are the i-dependent weights of RMSE, SD and η, respectively. The reason we choose these three parameters with i-dependent weights is threefold. First, the commonly used RMSE in various holographic algorithms is responsible for only the similarity between the simulated and ideal images, so the uniformity and efficiency issues concerned here are not well-solved by using a single RMSE parameter. This issue has been observed in many reported algorithms. Second, both the SD and efficiency also have more serious singleness of functionality so that they are seldom used in most algorithms. If the weight of SD is too large in a fixed loss function, the designed image might have low efficiency despite good uniformity; a similar issue occurs for the parameter η of efficiency. Third, a dynamic loss function is employed by using the dynamic weight with a strong dependence on the iteration. To realize the dynamic loss function, our strategy of arranging these three parameters is that, the SD and efficiency parts dominate only the beginning of the iterations while the RMSE is the only parameter in the loss function for the remaining iterations. Following this strategy, we suggest $w_1\left(i\right)=w_{RMSE}\left\{\tanh \left[\left(i-i_0\right)/D_0\right]+1\right\}/2$, $w_2\left(i\right)=w_{SD}\left\{\tanh \left[-\left(i-i_0\right)/D_0\right]+1\right\}/2$, and $w_3\left(i\right)=w_{Eff}\left\{\tanh \left[-\left(i-i_0\right)/D_0\right]+1\right\}/2$, where w_RMSE, w_SD and w_Eff are used to control the weight of each parameter in the loss function, i₀ is the transition position between two states in the iterations, and D₀ determines the length of the transition process. Here, i₀ = 175 and D₀ = 40 are used in this work to offer a nearly null SD for i < i₀ (see case A in Fig. 2g) so that we can exclude any solutions that lead to the phase singularities in the holographic images. Their dependences on the iterations from 1 to 500 are shown in Fig. 2c for a better understanding of its working principle. The constants w_RMSE, w_SD and w_Eff are important in investigating their roles in phase-probability shaping and enhancing the image quality. Note that, w_RMSE = 1 is employed throughout our optimization because of the RMSE’s good performance in optimizing the hologram. However, w_SD and w_Eff are valued between 0 and 1 to investigate the tradeoff between the uniformity and efficiency for the hologram design.

IV. Independent phase feedback via the gradient descent method of Adam. The Adam method³⁵ is used here to offer an independent increment of each phase so that we can shape the phase probability density by tuning the dynamic loss function. At the position (x_n, y_n) in the i^th iteration, the gradient of the loss function with respect to the local phase φ(x_n, y_n) is defined as $g_i=\partial \left({LF}_{i-1}\right)/\partial {\phi }_{i-1}\left(x_n,y_n\right)$, which is realized numerically by calculating the changing value of LF under a small variation of φ(x_n, y_n) (keeping all other phase values unchanged). To update all the phase values in one iteration, N (i.e., the total number of all the pixels) gradient calculations are needed so that the cost in both time and computing is too high for a large N. Therefore, our algorithms are implemented in an open-access TensorFlow for accelerating the optimization. Based on the calculated gradient g_i, we can update the phase (see Fig. 2b) ${\phi }_i\left(x_n,y_n\right)={\phi }_{i-1}\left(x_n,y_n\right)+f\left(g_i\right)$, where $f\left(g_i\right)=-\alpha {\widehat{m}}_i/\left(\sqrt{{\widehat{v}}_i}+\epsilon \right)$, ${\widehat{m}}_i=\left[{\beta }_1{m}_{i-1}+\left(1-{\beta }_1\right)g_i\right]/\left(1-{\beta }_1^i\right)$, ${\widehat{v}}_i=\left[{\beta }_2{v}_{i-1}+\left(1-{\beta }_2\right)g_i^2\right]/\left(1-{\beta }_2^i\right)$, β₁ = 0.9, β₂ = 0.999, ε = 10⁻⁸, m₀ = v₀ = 0 and α = 0.01. Note that, each phase${\phi }_i\left(x_n,y_n\right)$ is updated independently, which enables this algorithm to shape the phase probability. Such phase-probability shaping can be observed in different examples, as shown in Fig. 2f and 2g. This feature in our algorithm is not available in other algorithms that are based on the inverse propagation of the target image, such as the Gerchberg-Saxton algorithm³⁶ and YangGu algorithm⁵⁸.

The above four steps are the core of our hologram design method. In fact, the Adam approach in our algorithm offers the entire configuration of phase-probability shaping, while the weights w_RMSE, w_SD and w_Eff are the key to controlling the image quality of the hologram. To test its performance, we run the code on a personal computer (CPU Intel Core i5-7500, 32 G RAM). The details of the parameters are the pixel pitch p_x = p_y = 250 nm, the hologram size of 128 μm × 128 μm, the propagation distance of z = 250 μm between the metasurfaces and the image plane, and the operating wavelength λ = 633 nm. Under such conditions, it takes ~42 min to run 1500 iterations, while GS algorithm takes ~5 min. Moreover, in one computer with GPU (CPU: Intel Core i7-13700KF, 32 G RAM; GPU RTX 4090, 24 G RAM), the APS and GS algorithms take ~34 seconds and ~91 seconds to run 1500 iterations respectively. Further acceleration of optimization speed for our APS algorithm can be realized by calculating the gradient with Wirtinger derivatives⁵⁹.

Due to the discontinuous nature of binary phase modulation, its design for the CGH lithography is implemented using a modified genetic algorithm^56,60–62, which enables the independent updating of the phase for each pixel. This approach demonstrates a phase-probability shaping mechanism similar to that of the APS algorithm. Despite slight difference in the algorithms used for designing binary and multi-level phase modulation, both approaches are grounded in the same principle of phase-probability shaping to achieve speckle-free holograms.

Author contributions

K.H. conceived the idea and developed the theory. W.F., J.H., and K.H. performed the simulations. D.Z. and Z.L. prepared and fabricated optical samples. W.F. and D.Z. built up the experimental setup and performed the characterization. J.H., W.F., D.Z., and F.S. demonstrated the lithography. K.H. and D.Z. wrote the manuscript. K.H. supervised the overall project. All authors discussed the results, carried out the data analysis, and commented on the manuscript.

Peer review

Peer review information

Nature Communications thanks anonymous reviewer(s) for their contribution to the peer review of this work. [A peer review file is available].

Data availability

Data supporting the findings of this study are available from the manuscript and its Supplementary Information. The data is available from the corresponding author upon request.

Code availability

The code for the APS algorithm used in this study is available at Code Ocean (DOI: 10.24433/CO.4463535.v1).

Competing interests

The authors J.H., Z.L., and F.S. declare no competing financial interests. The authors D.Z., W.F., and K.H. declare the following competing interests. D.Z., K.H., and W.F. have filed one patent application related to this work through the University of Science and Technology of China. This patent (D.Z., K.H. and W.F., “A method of phase-probability shaping for eliminating speckles in holography”, 2022113559812(2022), under review) applied by University of Science and Technology of China refers to the design algorithms of Adam-gradient-descent phase-probability shaping for speckle-free holograms. D.Z. and K.H. have filed two patents application related to this work through University of Science and Technology of China. The first patent applied (D.Z. and K.H., “A phase-type diffractive optical element and system for computer-generated-holography lithography in dry circumstance”, 2023100401398 (2023), under review) by University of Science and Technology of China refers to the binary-phase/amplitude CGH lithography in air. The second patent applied (D.Z. and K.H., “A diffractive optical element and system for computer-generated-holography lithography in liquid circumstance”, 2023100400821 (2023), under review) by University of Science and Technology of China refers to the binary-phase/amplitude CGH lithography in liquid.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-64554-0.