Introduction

The phenomenon in which light incident on a small barrier or aperture can extend into a geometric shadow is called diffraction, which historically has helped us to understand the wave nature of light1. As the dimension of the barrier or aperture decreases to the scale of one wavelength, more light is extended into the shadow, thereby leading to high-order diffraction (HOD) in contrast to the light within the geometric projection (i.e., zero-order) of the barrier or aperture2. In optics, such HODs exist in scattered light from microstructures of diffractive optical elements3, such as spatial light modulators4 and micro-mirror devices5, leading to decreased efficiency and distorted patterns.

The solution to these HOD problems is twofold. The first strategy is to eliminate HOD by using optical components with subwavelength pixel pitches, e.g., well-developed metasurfaces with shape- or dimension-varying nanostructures6,7,8,9,10,11,12,13. Currently, the pixel pitch in most metasurfaces is chosen to be near half the wavelength14,15,16, limiting the spatial accuracy of optical modulation17. Smaller pixel pitches in metasurfaces lead to insufficient modulation of the phase or amplitude due to strong electromagnetic coupling between two neighboring nanostructures18, and challenge large-scale fabrication19.

The second method is to utilize HODs in some specialized fields. For example, optical spectroscopies employ large dispersions of 1st-order diffraction from blazed gratings to enhance spectral resolution20. However, little progress has been made in extending HOD to other territories. The challenge comes from the fundamental difficulty in independently manipulating HODs because they usually inherit similar but distorted physical properties from low-order diffraction2. A well-known case is that one can obtain much larger topologically charged vortices with distorted intensity profiles at the HOD of a fork grating even if this grating is designed to create a vortex with a topological charge of 1 at the first diffraction order21. Therefore, the key to using HODs in various applications is to manipulate HODs independently, which is in the infancy of optics nowadays. For soft X-ray, the HODs of zone plates have been used to enhance imaging resolution22, which, however, is still diffraction-limited due to the lack of independent manipulation of HODs.

Here, we manipulate the HOD of a flat lens through its analytical width of the zone so that the same-order diffraction of each zone can be modulated independently with a binary phase. Controllable interference from these selected orders of light is achieved after the phase is optimized, thereby creating a subdiffraction-limit focal spot with a lateral size of ~0.44 λ at NA = 0.9. Since all these HODs inherently originate from zones with non-subwavelength widths, HOD lenses facilitate low-cost manufacture of large-scale lenses. Using HOD lenses, we demonstrate confocal imaging with a center-to-center resolution of 190 nm that breaks the visible-light diffraction limit of 200 nm in a far-field, label-free, linear and dry circumstance, which behaves better than all previously reported confocal microscopies. Laser ablation-based lithography with an HOD lens is also demonstrated with a patterning resolution of 400 nm (0.385 λ) for the fabrication of various optical devices.

Results

HOD physics of the zones in a flat lens

We begin with optical diffraction from a zone (with its outermost and inner radii of rn and rn−1, respectively) in a flat lens (Fig. 1a). The incident light on this zone with a width of Δr = rn − rn−1 is redistributed into position (i.e., ρ)-dependent intensity at a given target plane z = f (where f refers to the focal length). With increasing Δr, the optical amplitude (Fig. 1b) at the designed focus of ρ = 0 oscillates with several valleys and peaks. Each valley implies the occurrence of destructive interference at ΔR = Rn − Rn−1 = m λ, where Rn and Rn-1 denote optical paths from the outermost (rn) and inner (rn−1) radial positions of the zone to the position ρ of interest (see Fig. 1a), m is an integer and λ is the wavelength of light. Such destructive interference extends to other radial positions (the dashed-white curves in Fig. 1b) when the width Δr changes. Due to the intensity valley at the destructive interference, the dashed-white curves define the boundaries between two neighboring diffraction orders. Thus, for a zone with a given rn and rn-1, we can derive the diffraction order where the light at the radial position ρ of the focal plane is located by using \(M=\left\lfloor \Delta R/\lambda \right\rfloor\) (where \(\left\lfloor a\right\rfloor\) denotes the integer part of a). In Fig. 1b, we have labeled the diffraction orders at the focal plane z = 75λ for an exemplified zone with its central position (rn + rn−1)/2 = 75 λ.

Fig. 1: High-order diffraction in focusing.
figure 1

a Optical diffraction from a single zone that is sketched in a polar coordinate. b Simulated amplitude (normalized to the maximum) of light diffracted from the zones with different widths Δr. By changing Δr, the simulation is implemented by using Rayleigh-Sommerfeld diffraction with a fast Fourier transform17, where the pixel pitch of 0.02 λ × 0.02 λ is used to enhance the calculation accuracy. The zones have the same central position of (rn + rn-1)/2 = 75 λ for investigating the role of Δr. The diffraction orders of the zones are also labeled for better observation, where the dashed white curves (derived by using \(M=\left\lfloor \triangle R/\lambda \right\rfloor\)) denote the boundaries between two neighboring orders. For each Δr, the fast oscillations along the radial positions are not observed in current figure due to the large range in radial position. c On-axis (ρ = 0) intensity of diffracted light from axisymmetric zones with different widths Δr and the outermost radii rn (where the parameter \(\sin \alpha={r}_{n}/\sqrt{{f}^{2}+{r}_{n}^{2}}\) is used with f = 75 λ for better data display). The on-axis intensity of each order (labeled in different colors) has been normalized to the maximum of its own order. Two cases for the destructive (A with Δr = 4.69 λ and \(\sin \alpha\)=0.45) and constructive (B with Δr = 7.6 λ and\(\,\sin \alpha\) = 0.5) interferences have been exemplified to show its diffraction patterns (see Supplementary Fig. 1). The structural parameters of the 0th-order (dots) and 1st-order (triangles) FZPs are located at the region of constructive interference, implying diffraction-limited focusing similar to traditional objectives lenses. The destructive interference is indicated by white regions. d–e Simulated optical focusing via a standard 0th-order FZP (d) and high-order (e) FZP with binary phase. Optical paths from the nth radius to the focus are Rn = f + n λ/2 for standard FZP and Rn = f + n(2 M + 1)λ/2 for Mth-order FZP, which indicates their fundamental differences. f Analytical (gray bars) and simulated (blue-white-red bars) efficiencies of high-order FZPs with different NAs.

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By limiting the position of interest only at the focus (i.e., ρ = 0 and z = f) for a given zone, we have the analytical focal intensity

$$I\left(0,0,f\right)={f}^{2}\left[{R}_{n}^{-2}+{R}_{n-1}^{-2}-2{R}_{n}^{-1}{R}_{n-1}^{-1}\cos \left(k\Delta R\right)\right],$$
(1)

where k = 2π/λ (see its mathematical derivations in Supplementary Section 1). The focal intensity has multiple maxima with nearly identical values of \(I\left({\mathrm{0,0}},f\right)={f}^{2}{\left[{R}_{n}^{-1}+{R}_{n-1}^{-1}\right]}^{2}\) when kΔR = (2 M + 1)π, which coincides with the simulated on-axis peaks in Fig. 1b. This implies that, from different diffraction orders, one can obtain nearly the same focal spot by manipulating the width of a zone, which holds the fundamental basis for using HOD in optical superfocusing.

To show a detailed map of diffraction orders for different zones (with width Δr and outermost radius rn with an intermediate parameter \(\sin \alpha={r}_{n}/\sqrt{{f}^{2}+{r}_{n}^{2}}\)), we visualize the focal intensity (normalized to the maximum of its own order) of each order in different pseudocolors in Fig. 1c, where the brightest color denotes the maximum intensity (constructive interference) and the white color denotes the minimum intensity (destructive interference). Mathematically, constructive interference occurs at the focus when Rn − Rn−1 = (2 M + 1)λ/2 (labeled by the dashed-black curves in Fig. 1c). If all the zones in a flat lens satisfy such a straightforward relationship, we can derive optical path Rn from the nth zone (with its outermost radius rn) of the lens to the focus by using Rn = f + n(2 M + 1)λ/2 because the smallest Rn-1 (i.e., R0) equals f. Thus, we have the radius

$${r}_{n}=\sqrt{n\left(2M+1\right)\lambda f+{n}^{2}{\lambda }^{2}{(2M+1)}^{2}/4},$$
(2)

which analytically correlates the dimension of each zone with the diffraction order M. If M = 0 (see Fig. 1d), Eq. (2) gives the structural dimension of a standard Fresnel zone plate (FZP).

When M ≥ 1, Eq. (2) yields the proposed HOD FZP (Fig. 1e) where the smallest zone width of \(\Delta {r}_{\min }={{\mathrm{lim}}}_{n\to \infty }({r}_{n}-{r}_{n-1})=\left(2M+1\right)\lambda /2\) is (2 M + 1) times that of the standard FZP. This HOD FZP utilizes light from Mth-order diffraction of the zones, suggesting a fundamental difference from the traditional standard FZP. Because high diffraction orders offer light with larger spatial frequencies than the zero order, this feature enables the HOD FZP to realize super-focusing without any sub-wavelength feature. Such a non-subwavelength feature is extremely important for short-wavelength optics because it solves the challenging problem of fabricating finer structures for high-NA flat lenses at ultraviolet23,24 and X-ray spectra25.

The efficiency (Fig. 1f) of the HOD FZP decreases with increasing diffraction order M (see the simulation details in the Methods). The underlying reason is that the HODs of diffracted light carry high spatial frequencies, which occupy only a small fraction of the entire spatial-frequency spectrum2. Furthermore, the largest spatial frequencies provided by the HOD are always below 1/λ in an air environment, so the HOD FZPs still have diffraction-limited focal spots. Therefore, manipulating the phase or amplitude of each diffraction order via these selected zones (defined by Eq. (2)) is required to focus light beyond the diffraction limit.

Manipulating optical phase of HOD for super-focusing

Because Eq. (2) defines the dimensions of the zones for a selected diffraction order, the phase modulation of HOD can be realized by introducing an additional phase delay at each zone in a binary-phase HOD FZP. We note that the additional phase also modulates other undersigned orders but does not influence optical focusing at the designed order M. Thus, by using the phase delay of each zone, we can realize the independent manipulation of the phase at the expected order. In practice, the additional phase delay is realized by a phase mask that can be integrated with an HOD FZP, forming a single high-order lens (Fig. 2a) with a phase profile of \(\varphi={\varphi }_{{mask}}+{\varphi }_{{HOD}-{FZP}}\) (where \({\varphi }_{{mask}}\) and \({\varphi }_{{HOD}-{FZP}}\) denote the phase profiles of the phase mask and the HOD FZP, respectively). Because each zone in the HOD FZP is modulated entirely, all the radii \({\rho }_{j}\) in the phase mask come from a certain radius rn in the HOD FZP. Consequently, no new finer structure is generated when combining the phase mask and the HOD FZP into a single device.

Fig. 2: Optical focusing with a high-order lens.
figure 2

a The high-order lens composed of a binary-phase high-order FZP and an additional phase mask. The phase mask is optimized to realize optical focusing beyond the diffraction limit. b Optical microscope image of the fabricated 2nd-order lens at the working wavelength of 405 nm. Insert: the photo of the entire lens and a ruler. c Simulated (left) and measured (right) intensity profiles at the focal plane of the 2nd-order lens. d Simulated (solid curve) and measured (forks and dots) line-scanning intensity profiles. e Measured cross-section (x-z plane) intensity profiles near the focal plane from Δz = −20 μm to Δz = 8 μm (where Δz = z-f and f = 1 mm). The on-axis intensity profile is presented in the upper panel, which reveals that the longitudinal FWHM of 10.86 μm. f Measured lateral FWHMs of the optical needle near the focal plane. RC: Rayleigh criterion (\(0.515\lambda /{NA}\)), SOC: Superoscillation criterion \((0.358\lambda /{NA})\).

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To design the high-order lens, we optimize ρj only in the phase mask due to the fixed dimension of an HOD FZP. The optimization of the radii ρj is implemented to minimize the size of the focal spot by using particle swarm algorithms26, the details of which can be found in our previous work27. A 2nd-order lens with a focal length f = 1 mm at a working wavelength of λ = 405 nm is designed with a relative 2nd-order FZP (with a total zone number N = 1274), resulting in a diameter of ~4.111 mm and an efficient NA = 0.9. The optimized phase mask consists of 33 rings (see their dimensions in Supplementary Fig. 2). Hence, the 2nd-order lens possesses 1242 concentric zones with the smallest feature size of ~1.1 μm, which enables low-cost fabrication (see the details in “Methods”). Figure 2b shows a microscopy image of our fabricated lens, indicating a high-fidelity profile.

To characterize its focusing properties, we measure the diffraction field near the focal region under the illumination of circularly polarized light. To collect all the diffracted light from the 2nd-order diffraction, an objective with a larger NA of 0.95 is utilized to project the focal spot onto a camera; see the experimental setup in Supplementary Fig. 3. From the simulated and experimental intensity profiles in Fig. 2c, we can derive their line intensity profiles (Fig. 2d), which show that the full width at half maximum (FWHM) of the measured focal spot agrees with the designed 180 nm (\(0.44\lambda\)). More importantly, such a well-confined subwavelength spot can be maintained over a longitudinal distance of 10.86 μm; see its cross-sectional (i.e., x-z plane) intensity profiles in Fig. 2e. The measured optical needle has a lateral FWHM of approximately 180 nm (Fig. 2f), which is smaller than the Rayleigh criterion (RC, calculated by \(0.515\lambda /{NA}=\)0.57 λ)1 but larger than the super-oscillation criterion (SOC, calculated by \(0.358\lambda /{NA}\)=0.4 λ)17,28,29,30. These properties reveal that the super-critical focusing is achieved in this 2nd-order lens31. Due to the diffraction effect17, a Gaussian beam with lateral size of w0 usually diverges beyond a Rayleigh distance of 0.5kw02. In comparison, our achieved optical needle exhibits diffraction-controlled propagation over nearly 43 times the Rayleigh distance, which is record-high among those experimental subwavelength needles31,32,33,34,35,36. The physical origin of such a long distance is attributed to the focal shift offered by the optimized phase mask in the 2nd-order lens. Theoretically, without changing the high-order zone plates (i.e., its focal length and NA are fixed), the longer distance is also possible if more rings (i.e., having a larger J in Fig. 2a) are used in the phase mask due to the greater spatial degrees of freedom available for light shaping37.

Because all the zones in the binary-phase HOD FZP contribute to constructive interference, the additional phase of π in the phase mask weakens such constructive interference, thus decreasing optical efficiency. Therefore, the efficiency of HOD FZPs is the theoretical limitation of optical efficiency for arbitrary high-order lenses. For example, the simulated efficiency of this 2nd-order lens is 0.17% (see the simulation details in “Methods”), which is less than the efficiency of 1.6% for the relative 2nd-order FZP.

Subdiffraction-limit confocal microscopy with a high-order lens

In a scanning confocal microscope38,39,40 (SCM), a photomultiplier detector can record weak light so that the low efficiency of a high-order lens is not the key issue for scanning imaging. Hence, as a condenser, our 2nd-order lens is integrated into a transmission SCM (see its experimental setup in Fig. 3a). Due to the ultralong working distance, our 2nd-order lens allows focused light to pass through quartz first before illuminating the nano-objects (fixed on a 3-dimensional piezo stage for scanning). Thus, the light transmitted through nano-objects can be collected directly by using an ordinary objective without the need to correct optical aberrations caused by the quartz substrate in other SCMs with shorter working distances31,33,41,42.

Fig. 3: Scanning confocal microscopy with a 2nd-order lens.
figure 3

a Sketch for optical setup of SCM in a transmission mode. PMT: photomultiplier tube. b–e Double slits (b) and their images by using coherent bright-field microscope (c), traditional objective-based SCM (d) and our high-order-lens-based SCM (e). The double slits have the center-to-center distances of 180 nm (I), 190 nm (II), 200 nm (III), 210 nm (IV), 220 nm (V) and 230 nm (VI). f Line-scanning intensity profiles of images with different microscopies. g Measured valley-peak ratios for SCM images based on traditional objective (black dots) and our high-order lens (red asterisks). The valley-peak ratio γ is defined by using γ = Ix=0/I|x|=D/2, where Ix=0 and I|x|=D/2 denote the intensity values at the x = 0 (the symmetric axis of double slit) and |x | =D/2 (the center of each slit), respectively. The error bars are generated because the different I+|x| and I-|x| are achieved in experiment, as observed in (f). h–k Complex nano-object bird (h) and its images by using coherent bright-field microscope (i), traditional objective-based SCM (j) and our high-order lens-based SCM (k). The zoomed-in images at the wing region (labeled in the red boxes) are shown in the lower panels.

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Imaging resolution is tested by using double slits with varying center-to-center (CTC) distances from D = 180 nm to D = 230 nm (Fig. 3b), where each slit with a length of 2 μm and a width of 50 nm is etched through a 140 nm-thick chromium film on a 170 μm-thick quartz substrate. As a control, a coherent bright field microscope equipped with a 0.9 NA objective cannot resolve any double slit (Fig. 3c). Similarly, the conventional SCM with a 0.9 NA objective can resolve only double slits with the CTC distance of 230 nm (Fig. 3d). In Fig. 3e, the 2nd-order lens-based SCM yields an impressive CTC resolution of 190 nm (see the simulation results in Supplementary Fig. 4), which is doubly confirmed by the line-scanning intensity profiles in Fig. 3f. To quantitatively distinguish whether double slits are resolved, we use the ratio of the intensity Ix=0 at the symmetric axis (i.e., x = 0) of double slits to the intensity I|x|=D/2 at the center (x = ±D/2) of each slit, namely γ = Ix=0/I|x|=D/2. When γ < 1, an intensity valley appears in the scanning image, implying that the double slits are resolved; otherwise, the double slits are unsolved for γ ≥ 1. In our SCM, the experimental ratio γ < 1 for D ≥ 190 nm implies an imaging resolution of 190 nm, which, to the best of our knowledge, is the first to overcome the 200 nm limitation by using air-ambient visible-light SCMs31,33,35,41,42.

More complex, planar, and transparent nano-objects can also be imaged with high resolution by using our SCM. For a bird object with a size of 8 × 7 μm2 and the smallest feature of 200 nm (Fig. 3g), the experimental images obtained by using a coherent bright field microscope (Fig. 3i) and a traditional SCM (Fig. 3j) can only exhibit rough contours, leaving blurred details (see the lower panels in Figs. 3i–j). In contrast, our SCM successfully resolved all the contours and details at the bird’s wings (Fig. 3k), indicating greater resolving power.

Laser-ablation-based lithography with a high-order lens

To use a high-order lens in lithography (Fig. 4a), we design another 4 mm-diameter 1st-order lens (see its structural details in Supplementary Fig. 2) with a focal length of f = 1 mm at \(\lambda\)=1040 nm, yielding a simulated focal spot with a lateral size of ~450 nm (FWHM). For laser ablation43, the transient power of the focal spot must be sufficiently high to ablate the interacting materials (e.g., chromium used for demonstration). Hence, a circularly polarized femtosecond laser with a center wavelength of 1040 nm (with a nearly Gaussian shape and a spectral width of 10 nm) is used experimentally, creating an ~22 μm-long optical needle from Δz = −8 μm to Δz = 14 μm, where Δz = z-f (see its measured intensity profiles in Supplementary Figs. 5a–c). The measured optical needle has lateral sizes ranging from 550 nm to 450 nm (Supplementary Fig. 5d), which are slightly larger than the designed value of 450 nm. The reason mainly comes from the dispersion of the femtosecond light that broadens the spatial and temporal width of the focal spot, which has a time-dependent intensity profile (see Supplementary Figs. 5e–g).

Fig. 4: Laser-ablation-based lithography with a 1st-order lens.
figure 4

a Sketch for laser-ablation lithography on a chromium (Cr) film by focusing a femtosecond laser with a 1st-order lens with a focal length of 1 mm. b Optical microscope images of slits fabricated by using the different parts of optical needle. The out-of-focus distance Δz = z-f ranges from Δz = −8 μm to Δz = 13 μm. The exposure time for all the slits is 200 ms. c The widths of fabricated slits in (a) and their theoretical values. RC: Rayleigh criterion (\(0.515\lambda /{NA}\)). The error bars are formed by averaging the x- and y- width of half of measured peak intensity of optical needle. d Widths of slits fabricated by using traditional objective with exposure time of 200 ms. The incident pulse energy is decreased to enable the slit width close to those fabricated with high-order lens because traditional objectives have higher efficiency than our high-order lens. Insert: optical microscope image of fabricated slits with traditional objective. e Measured widths (triangles) of fabricated double slits at the different exposure time from 100 ms to 600 ms. The fitted data reveal a linear dependence between the width and the exposure time, which is obtained at a single pulse energy of 0.2 mJ with a repetition rate of 25 Hz. Insert: Optical microscope images of fabricated double slits by addressing the exposure time.

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To test the fabrication resolution, the optical needle is used to pattern slits on a chromium film through a laser ablation process (see the experimental setup in Supplementary Fig. 6). Since the line widths of fabricated slits are highly sensitive to pulse power and exposure time, we fix the power of each pulse to ~0.20 mJ with a repetition frequency of 25 Hz throughout this work and tune the exposure time to characterize the resolution. At an exposure time of 200 ms, some 5 μm-long slits in Fig. 4b are patterned on a 10 nm-thick chromium film on a quartz substrate by using the optical needle at different longitudinal positions Δz with a spatial interval of 1 μm. When the out-of-focus distance Δz ranges from -7 μm to 12 μm, the lines indicate the trajectories of the scanning spot where the chromium is ablated, thus leading to the dark lines in the reflective microscopic images (Fig. 4b). If the chromium is located beyond the region of the optical needle, the focused power decreases rapidly so that the chromium is maintained, leaving no pattern in the film. In Fig. 4c, these slits have measured widths ranging from 400 nm to 550 nm, where both ends of the optical needle create higher-resolution slits than the middle part. This tendency coincides with that of the predicted widths, which are calculated by using the lateral dimension of the measured optical needle above the ablation power threshold (evaluated by half of the measured peak intensity in the needle). Thus, we confirm that the proposed lithography has a depth-of-focus of 19 μm and a patterning resolution of 400 nm (0.38 λ), which is comparable to the resolution of 350 nm obtained by using a laser with a much-shorter wavelength of λ = 248 nm43,44. Compared with traditional objective (NA = 0.9)-based lithography, which yields a depth-of-focus of 0.6 μm and a resolution of 500 nm (Fig. 4d), our lithography shows better tolerance to the longitudinal misalignment of samples and higher patterning resolution. Furthermore, the resolution of our lithography system has a linear dependence on the exposure time (Fig. 4e), which is revealed experimentally by using the slit widths of the fabricated double-slits with different exposure time.

Using the proposed laser-ablation lithography with the 1st-order lens, we fabricate various optical devices, such as binary-amplitude FZP, fork gratings, and characters with directional gratings. Figure 5a shows the fabricated and designed FZPs with a diameter of 75 μm, where the experimental dimensions coincide with the designed ones (as doubly confirmed by the comparison between the measured and designed widths of zones in Fig. 5b). The optical performance of this fabricated FZP is also examined by comparing the simulated and measured focal spots, which exhibit nearly identical intensity profiles at the focal plane (Fig. 5c). The fabricated fork grating in Fig. 5d can generate multicolor vortices (Fig. 5e) under optical illumination with several discrete wavelengths from a supercontinuum laser; see the measurement details in Supplementary Fig. 7. Each vortex diffracted from the fork grating has a donut-shaped intensity profile with a dark center, where the expected phase singularity is located45,46. The fabricated characters “USTC” containing gratings with different periods and orientations (Fig. 5f) can be observed only at designed view angles of 0°, 45°, 90° and 135°(Fig. 5g), where the observation is recorded dynamically in Supplementary Movie 1. All these manufactured elements have exhibited the expected optical phenomena with good performance, validating high-quality fabrication by high-order-lens-based lithography. In addition to these elements, more complicated patterns (see Supplementary Fig. 6) can also be fabricated for various applications such as optical displays, inkless printing and high-resolution laser marking.

Fig. 5: Complex optical devices fabricated by using high-order lens-based lithography with laser ablation.
figure 5

a Fabricated (left) and designed (right) binary-amplitude zone plates with a focal length of 200 μm at a working wavelength of 405 nm. Image of the fabricated zone plates is taken by using reflective microscopy, which reveals darkness in the etched region and brightness in the non-etched Cr film. b Designed and measured widths of zones in the FZP. c Simulated (solid curves) and measured (dots and forks) intensity profiles at the focal plane of the FZP. Inserts: 2-dimensional intensity profiles of the focal spots. d Microscope image of fabricated fork grating for generation of optical vortex. e Measured optical vortices at first-order diffraction of the fork grating under the illumination of a supercontinuum laser with the selected (using an acousto-optic modulator) wavelengths: 477 nm, 500 nm, 550 nm, 600 nm and 650 nm. f Fabricated “USTC” letters composed of gratings with different periods and orientations. The periods and orientations are labeled in the different groups of dashed rectangles. The inserts sketch the orientation of the gratings in each group. The grating periods are 1 μm, 2 μm, 3 μm 4 μm for the characters from the upper to the lower rows. g Observed images of the fabricated letters by rotating the sample that is illuminated by a white light. The letters appear at the designed angles, which confirms the validity of the fabricated orientations in the gratings. The color overlaying of the letters in each view angle is different, which is attributed to the dispersion of gratings with different periods. All the samples are fabricated on a 10-nm-thick Cr film at \(\Delta z=0\,{{\rm{\mu }}}{{\rm{m}}}\) with a pulse power of 0.2 mJ at a repetition rate of 25 Hz and an exposure time of 200 ms.

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Discussion

Although non-subwavelength zones for super-focusing were first proposed in 201328 and subsequently demonstrated in experiment after 201531,32,35,47, the role of HOD in optical focusing has not been revealed; thus, all those diffractive lenses with zones of fixed non-subwavelength widths must block the central parts of these lenses to decrease the low spatial-frequency light. The accompanying cost is the decrease in both the optical efficiency and spatial degree of freedom for focusing. In contrast, guided by the physics of HOD, our high-order lenses fully use the entire region inside the diameter without any spatial blocking, which offers sufficient space for confining light with improved efficiency at arbitrary NAs. Benefiting from this advantage, our high-order lens can simultaneously realize subdiffraction-limit focusing, a high NA ( ≥ 0.9), a millimeter-scale focal length, and a non-subwavelength feature, which cannot be achieved in any other reported flat lenses17,48,49,50,51 (see the detailed comparison in Supplementary Section 10).

For our high-order lenses, the ratios of the achieved focal size to the smallest feature of the lens are 1/6.1 (λ = 405 nm) and 1/3.78 (λ = 1040 nm), which are much better than the ratio of 1/1.2 in soft X-ray focusing52. If this technique is used to design a high-order lens with a feature size of 15 nm (the same as that in ref.52) for X-rays, the expected focal size or imaging resolution can be scaled down to 2.5 ~ 4.2 nm, which is highly desirable in soft X-ray optics and imaging53,54,55,56. Therefore, a high-order lens is a viable approach for realizing subdiffraction-limit focusing especially for ultraviolet and soft X-ray spectra, where traditional refractive lenses are unavailable.

Low efficiency from HODs is obtained under the assumption of ultra-thin zone structures. In fact, when the height of the zone structures increases, the waveguide effect in the thick zone can be used to enhance the efficiency with the optimized thickness and line-to-space ratio57. Moreover, the shrunken efficiency from HODs can be compensated by using higher-power laser sources or high-sensitivity detector in practical applications. For example, although the high-order lens in our confocal microscopy has low efficiency, the transmitted light is detected by using a photomultiplier tube that can record weak optical signals. In the laser-ablation lithography, high transient power of the incident ultrafast pulses can also compensate low efficiency of the high-order lens so that the focused light is sufficiently strong to pattern various structures on the materials.

In summary, we have demonstrated optical subdiffraction-limit focusing with HOD of non-subwavelength zones in a flat lens of arbitrary NA. HODs carrying high spatial frequencies provide a theoretical basis for super-focusing with a micrometer-feature, large-scale and low-cost lenses, which significantly reduces the threshold of fabrication tools. Using high-order lenses, we demonstrated confocal imaging with a record-high CTC resolution of 190 nm (λ = 405 nm) in air and laser-ablation lithography with a patterning resolution of 400 nm (λ = 1040 nm) on a metal film. This work could impact applications in imaging17, lithography58, laser surgery, additive manufacturing, and material processing and could also benefit especially short-wavelength optics.

Methods

Optical efficiency of HOD FZPs

To investigate the efficiency of binary-phase HOD FZPs, we can make a physically reasonable approximation of Eq. (2) when the diffraction order M is small or the focal length f is large, which means that the second term in the right part of Eq. (2) can be ignored due to relatively small value (compared with the first item). We can rewrite Eq.(2) as:

$${r}_{n}=\sqrt{n\left(2M+1\right)\lambda f}\sqrt{1+\frac{n\lambda \left(2M+1\right)}{4f}}.$$
(3)

According to Eq. (3), \({r}_{n}\) can be approximately as \(\sqrt{n\left(2M+1\right)\lambda f}\) when n ≪ 4 f/[λ(2 M + 1)]. Because the zone index n is valued between 0 to N, it requires N ≪ 4 f/[λ(2 M + 1)], which yields the diffraction order M ≪ 2 f/(Nλ)-0.5 and \(f\gg \sqrt{2}{r}_{N}/4\). This approximation indicates that the radius rN is small compared with the focal length f, thus leading to small NA of the lens. Therefore, optical efficiency derived under the approximation matches the realistic efficiency only for a small-NA lens. By using the approximation, we have \({{r}_{n}}^{2}=n\left(2M+1\right)\lambda f\). Thus, the structural parameter \({{r}_{n}}^{2}\) of the HOD FZP has a period of (2 M + 1)λf, so that it can be taken as a binary phase grating with a periodicity of (2 M + 1)λf with respect to the variable \({r}^{2}\)2,59,60. Mathematically, the transmission function of the binary phase grating is:

$$t\left({r}^{2}\right)=\left\{\begin{array}{cc}1,& {r}^{2}\le \frac{(2M+1)\lambda f}{2}\\ -1,& \frac{(2M+1)\lambda f}{2}\le {r}^{2}\le (2M+1)\lambda f,\end{array}\right.$$
(4)

Because the transmission function \(t({r}^{2})\) is periodic, it can be expressed by using the Fourier series:

$$t\left({r}^{2}\right)={\sum }_{-{{\infty }}}^{{{\infty }}}{c}_{q}{e}^{-i2\pi q{f}_{0}{r}^{2}},$$
(5)

where \({f}_{0}=1/\left[(2M+1)\lambda f\right]\). According to Eqs. (4) and (5), the Fourier coefficient \({c}_{q}\) is

$${c}_{2M+1}={sinc}\left(\frac{2M+1}{2}\right)-{sinc}\left(2M+1\right).$$
(6)

Therefore, the theoretical diffraction efficiency of HOD FZPs at diffraction order M in the focal plane \(z=f\) has the following analytical form:

$${\eta }_{{M}_{{theoretical}}}={{c}_{2M+1}}^{2}={\left[{sinc}\left(\frac{2M+1}{2}\right)-{sinc}\left(2M+1\right)\right]}^{2}.$$
(7)

By using Eq. (7), Fig. 1f shows the theoretical efficiency for different diffractive orders M. The analytical η depends on only the diffraction order M. It is important to emphasize that Eq. (7) is obtained under the assumption of N ≪ 4 f/[λ(2 M + 1)], which means that Eq. (7) coincides better with the efficiency for small NA than that for large NA, as observed in Fig. 1f.

To numerically confirm the efficiency of HOD FZPs, we can employ Rayleigh-Sommerfeld diffraction to obtain the focal electric field17:

$$E\left(x,y,z=f\right)=-\frac{1}{2\pi }\iint _{-{{\infty }}}^{{{\infty }}}{E}_{0}\left({x}_{0},{y}_{0}\right)\cdot \frac{{e}^{{ikR}}}{{R}^{2}}\left({ik}-\frac{1}{R}\right)\cdot z{{dx}}_{0}{{dy}}_{0},$$
(8)

where \(R=\sqrt{{(x-{x}_{0})}^{2}+{(y-{y}_{0})}^{2}+{z}^{2}}\), \({x}_{0}\) and \({y}_{0}\) are the coordinates of the incident plane and x and y are the coordinates at the plane of interest. Assuming that a plane wave with uniform electric field \(E\left({x}_{0},{y}_{0}\right)\) = 1 is incident on the HOD FZPs, we can obtain the total power in a circle (with a radius of 3 times the FWHM of the focal spot) at the focal plane:

$${W}_{{{\rm{out}}}}=\int _{0}^{3{FWHM}}\int _{0}^{2\pi }{I}_{{{\rm{out}}}}\left(r\right){rdrd}\varphi,$$
(9)

where \({I}_{{{\rm{out}}}}\left(r\right)={\left|E\left(r,z=f\right)\right|}^{2}={\left|E\left(x,y,z=f\right)\right|}^{2}\), \(r=\sqrt{{x}^{2}+{y}^{2}}\). For the simulations in this work, the employed pixel size is 10 nm for high accuracy in the calculations. The incident power is evaluated by:

$${W}_{{in}}=\int _{0}^{{r}_{n}^{\max }}\int _{0}^{2\pi }{{{|}}{E}_{0}{{|}}}^{2}{rdrd}\varphi,$$
(10)

where \({r}_{n}^{\max }\) is the maximum radius of the HOD FZP. Furthermore, the diffraction efficiency is:

$${\eta }_{{M}_{{numerical}}}=\frac{{W}_{{out}}}{{W}_{{in}}}.$$
(11)

In the simulations in Fig. 1f, the HOD FZPs have an NA ranging from 0.9 to 0.3 with an interval of 0.2, a focal length of 200 μm, and a working wavelength of 405 nm. Because the small window leads to insufficient calculation of the focal power, the simulated efficiency is lower than the analytical efficiency for all these diffraction orders from M = 0 to M = 3. Extending the calculation window at the focal region might overestimate the efficiency by involving the power from other diffraction orders. Therefore, the frequently used definition by using the circle with its radius of 3 times FWHM of the focal spot is a good balance in numerically calculating the efficiency.

Because the analytical efficiency in Eq. (7) is obtained under the paraxial approximation that holds well for low-NA FZPs, the simulated efficiencies for low-NA cases are more consistent with the analytical efficiency than those for high-NA cases. Therefore, the theoretical efficiency for a high-NA FZP is lower than the analytical efficiency predicted by Eq. (7).

Fabrication details

The designed high-order lenses are fabricated through a standard top-down process. The quartz substrate is first deposited with a chromium film using the sputtering system. After coating and baking, the photoresist is patterned by using I-line lithography (Canon stepper). Using a photoresist as a hard mask, the patterns are transferred into the chromium film after the exposed chromium film is etched by an inductively coupled plasma (ICP) etching system. Then, the residual photoresist is removed, and the quartz substrate is etched to a designated thickness by an inductively coupled plasma-reactive ion etching (ICP-RIE) system. Finally, the chromium film is removed by a chromium etchant, yielding the expected binary-phase high-order lenses.

In addition, the double slits and bird objects used for scanning confocal microscopy are fabricated through focused-ion-beam (FIB) lithography.

Experimental details of scanning confocal imaging

In Fig. 3a, we integrate a 3-dimensional piezo stage (PI-545.3R8S) and its controller (PI-E727) into a single device, providing a scanning resolution of approximately 1 nm for high-accuracy imaging. The movement of the 3-dimensional piezo stage and the signals collected by the photomultiplier tube are correlated via a computer. We employ a 10 μm-diameter fiber (Thorlabs M64L01, 10 μm, 0.1 NA) as the pinhole in our setup. Additionally, we used a DAQ card (NI USB-6000) with 12-bit resolution and a sampling rate of 10 Ks/s to realize the analog-to-digital conversion of signals.

In the characterization experiment of imaging resolution in Figs. 3d-3e, we use a scanning range of 3 μm × 1 μm with a 100 × 100 pixel resolution, which requires ~15 minutes to complete one scan. The scanning speed can be further improved by utilizing a high-speed stage and digital-to-analog converter. In Figs. 3j-3k, a scanning range of 2.4 μm × 0.6 μm is employed to fully cover the objects. In the experiment involving complex nano-objects, the scanning ranges are 9 μm × 8 μm with 150 × 150 sampling points, which takes ~34 minutes to obtain one image.

Experimental details of laser-ablation lithography

The optical setup for laser-ablation-based lithography is sketched in Supplementary Fig. 6, which shows three key elements, namely, a mechanical shutter, our high-order lens and a 3-dimensional piezo scanning stage. A DAQ card (NI USB-6000) is used to create and send the signals to the shutter controller (ThorLabs SC10, shutter: SH05R/M), which switches the femtosecond laser pulse. The 1st-order lens working at a wavelength of 1040 nm is used to focus the light pulse into the subdiffraction-limit focal spot, which carries high power for laser ablation. The piezo stage and its controller are the same as those used in scanning imaging. Both the shutter and the piezo stage are correlated with the same time sequence in a computer for the fabrication of various patterns.

As a proof-of-concept demonstration of laser-ablation lithography, all patterns are fabricated on a sample composed of a 10 nm-thick Cr film and a 500 \({{\rm{\mu }}}{{\rm{m}}}\)-thick SiO2 substrate by a point-to-point scanning method during fabrication, i.e., first, the shutter is closed to move the sample to the target position; then, the shutter is opened with an optimized etching or exposure time; and finally, the shutter is closed. Except for the fabricated characters in Fig. 5f where the distance between two neighboring fabrication points is 0.5 μm, the distance between two neighboring fabrication points of all the other patterns is 0.2 μm, which is covered by a 550 nm-FWHM focal spot. The delay of the closing time for the shutter is 50 ms. Under the conditions of an exposure time of 200 ms and the closing time of 50 ms, it takes approximately 34 s to fabricate 100 points. To fabricate the “USTC” letters in Fig. 5f, we mount the 3D piezo stage on a mechanical 3D translation stage with an accuracy of 10 μm. When one single letter with a size of \(200\times 200{{\rm{\mu }}}{{\rm{m}}}\) is finished, the sample is moved to the next position by the mechanical 3D translation stage. To reduce the fabrication time of the “USTC” letters, the filling factors for these gratings with their periods of 1 μm, 2 μm, 3 μm and 4 μm (see. Figure 5f) are 1:1, 1:3, 1:5 and 1:7, respectively.