electron diffraction tube experiment

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Electron diffraction, what it shows:.

Louis de Broglie predicted that matter under certain circumstances would exhibit wave-like properties. A proof of this is the repeat of X-ray diffraction experiments using electrons, whose de Broglie wavelengths at high accelerating potentials are similar to X-ray wavelengths. Here we accelerate electrons into crystal targets and get diffraction patterns identical to those from X-ray diffraction.

How it works:

The apparatus is a Welch Scientific 2639 Electron Diffraction Tube mounted within its 2639A container and power supply. What we have then is a controllable electron gun with built in crystal targets. The samples provided are two dimensional hexagonal pyrolytic graphite which gives a monocrystal dot diffraction pattern; polycrystalline aluminum that gives a ring pattern and polycrystalline hexagonal pyrolytic graphite which gives an intermediate (beaded ring) diffraction pattern. By setting the anode voltage between 5kV and 10kV the electron beam can be steered onto a crystal sample by horizontal and vertical controls (details in Setting it Up ). Knowing the distance from the crystal target to the screen (18.16cm) an investigation into the crystal structure can be carried out using Bragg's law (examples of analyses are given in the tube handbook).

Setting it up:

Allow the tube to warm up for about five minutes with the anode voltage at around 5kV. Turn up the voltage to 7kV and scan the screen for crystals using the horizontal and vertical controls, but remember that anode voltage also affects deflection so is a third variable. Fine tune the beam with one control at a time. Fiddle with intensity and focus controls after you have located a crystal. To display, use a camera with zoom to fill a TV screen with the pattern. To cut glare on the front window from overhead lighting, use a sheet of 50cm × 75cm card as a visor. The camera also has to be protected from the central bright spot of the diffraction pattern; this can be done by sticking a (1cm diameter circular) piece of black tape over the spot when everything is aligned. Turn down the intensity until the demo is to be used.

If you are just going for one crystal, choose the polycrystalline graphite, although all are beautiful. Our electron diffraction tube is mounted on a Tektronix scope cart which is a perfect size. It is worth warning people unfamiliar with this setup that in-the-field crystal hunting is a tricky affair! A good companion demonstration would be the optical X-ray diffraction analog.

References:

H. F. Meiners: Instructions for Catalog No.2639 (Welch Scientific Company 1963) L. de Broglie: Phil Mag 47 , 446 (1924) C. J. Davisson and L. H. Germer: Phys. Rev. 30 , 705 (1927) W. L. Bragg: Proc. Camb. Phil. Soc. 17 , 43 (1913) G. P. Thompson: Proc. Roy. Soc. 117 , 600 (1928), 119 , 652 (1928)

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Newtonian Mechanics Fluid Mechanics Oscillations and Waves Electricity and Magnetism Light and Optics Quantum Physics and Relativity Thermal Physics Condensed Matter Astronomy and Astrophysics Geophysics Chemical Behavior of Matter Mathematical Topics

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Size : from small [S] (benchtop) to extra large [XL] (most of the hall)  Setup Time : <10 min [t], 10-15 min [t+], >15 min [t++] /span> Rating : from good [★] to wow! [★★★★] or not rated [—] 

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Electron Diffraction Apparatus

Figure 1: Electron diffraction demo in action

This apparatus shows the diffraction of electrons onto a white viewing screen as shown above in Figure 1.

Figure 2: Setup example

• Electron diffraction tube and stand (center of Figure 2) [Cabinet D2]
• Teltron power supply (right in Figure 2) [Cabinet K]
• BK Precision power supply (left in Figure 2) [Cabinet K]
• Banana cables
• Bar magnets

Figure 3: Electron diffraction circuit diagram

• Place the electron tube securely on the stand.

• The external bias refers to the BK Precision power supply connections.
• The small red lead gets grounded in the Teltron
• The 6.3 V gets applied across the large banana jack holes
• BK supply negative gets grounded with the Teltron
• BK supply positive goes to the large banana hole without the minus sign next to it
• To turn on the 6.3 V, flip the switch on the back of the Teltron.
• Tuning this voltage allows for sharper diffraction patterns.
• This portion is optional, useful diffraction patterns may be observed without this supply connected.
• Use the bar magnet to distort the shadow cast on the the screen to show that electrons are being shot out instead of light.
• Varying external bias will also distort the diffraction rings.
• An example of the desired diffraction pattern is provided below in Figure 4.
• Before turning off the apparatus, reduce HV to zero (otherwise the cathode can be damaged).

Figure 4: Electron diffraction pattern example

Explanation:

The diffraction rings are created by the electrons diffracting and interfering as they pass through the crystal structure of the carbon target. They behave like waves as they pass through the grating created by the graphite’s crystal structure. Deflecting the pattern with a magnet shows that the diffraction is produced by moving charged particles, rather than light or some other form of radiation. This shows both the particle and wave nature of electrons as they react to the magnetic field and pass through the graphite foil. When the voltage between cathode and anode is increased, the electrons diffract less and the rings get closer together. This shows that higher energy electrons have shorter wavelengths, which agrees with De Broglie’s formula.

• Warm up the cathode for five to ten seconds before applying the high voltage
• Reduce the high voltage to zero before turning off the cathode heater

Written by Noah Peake

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Electron Diffraction Tube

In this video, produced by the National STEM Learning Centre and Network and the Institute of Physics, Alom Shaha explains the elements of an electron diffraction tube and how to use one to demonstrate a diffraction pattern. The video is aimed at teachers but it could also be used with older students to illustrate the working of an electron diffraction tube.

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• Open access
• Published: 06 September 2024

High-order diffraction for optical superfocusing

• Jun He 1   na1 ,
• Hong Liu   ORCID: orcid.org/0000-0002-3560-9401 2   na1 ,
• Dong Zhao   ORCID: orcid.org/0000-0001-7173-0500 1   na1 ,
• Jodhbir S. Mehta   ORCID: orcid.org/0000-0001-6362-4898 3 ,
• Cheng-Wei Qiu 4 ,
• Fangwen Sun   ORCID: orcid.org/0000-0002-9625-7390 1 ,
• Jinghua Teng   ORCID: orcid.org/0000-0001-5331-3092 2 &
• Kun Huang   ORCID: orcid.org/0000-0002-9391-149X 1  

Nature Communications volume  15 , Article number:  7819 ( 2024 ) Cite this article

Metrics details

• Confocal microscopy
• Micro-optics

High-order diffraction (HOD) from optical microstructures is undesirable in many applications because of the accompanying ghosting patterns and loss of efficiency. In contrast to suppressing HOD with subwavelength structures that challenge the fabrication of large-scale devices, managing HOD is less developed due to the lack of an efficient method for independently manipulating HOD. Here, we report independent manipulation of HODs, which are unexploited for subdiffraction-limit focusing in diffractive lenses, through an analytical formula that correlates the diffraction order and the width of each zone. The large spatial frequencies offered by the HODs enable our lenses to reduce the lateral focal size down to 0.44  λ even without any subwavelength feature (indispensable in most high- NA diffractive lenses), facilitating large-scale manufacture. Experimentally, we demonstrate high-order lens-based confocal imaging with a center-to-center dry resolution of 190 nm, the highest among visible-light confocal microscopies, and laser-ablation lithography with achieved direct-writing resolution of 400 nm (0.385  λ ).

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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 light 1 . 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 aperture 2 . In optics, such HODs exist in scattered light from microstructures of diffractive optical elements 3 , such as spatial light modulators 4 and micro-mirror devices 5 , 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 nanostructures 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 . Currently, the pixel pitch in most metasurfaces is chosen to be near half the wavelength 14 , 15 , 16 , limiting the spatial accuracy of optical modulation 17 . Smaller pixel pitches in metasurfaces lead to insufficient modulation of the phase or amplitude due to strong electromagnetic coupling between two neighboring nanostructures 18 , and challenge large-scale fabrication 19 .

The second method is to utilize HODs in some specialized fields. For example, optical spectroscopies employ large dispersions of 1 st -order diffraction from blazed gratings to enhance spectral resolution 20 . 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 diffraction 2 . 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 order 21 . 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 resolution 22 , 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.

HOD physics of the zones in a flat lens

We begin with optical diffraction from a zone (with its outermost and inner radii of r n and r n −1 , respectively) in a flat lens (Fig.  1a ). The incident light on this zone with a width of Δr  =  r n   −  r n −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  =  R n   −  R n −1  =  m   λ , where R n and R n-1 denote optical paths from the outermost ( r n ) and inner ( r n −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 r n and r n-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  = 75 λ for an exemplified zone with its central position ( r n  +  r n−1 )/2 = 75  λ .

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 transform 17 , 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 ( r n  +  r n-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 r n (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 0 th -order (dots) and 1 st -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 0 th -order FZP ( d ) and high-order ( e ) FZP with binary phase. Optical paths from the n th radius to the focus are R n  =  f  +  n   λ /2 for standard FZP and R n  =  f  +  n (2  M  + 1) λ /2 for M th -order FZP, which indicates their fundamental differences. f Analytical (gray bars) and simulated (blue-white-red bars) efficiencies of high-order FZPs with different NA s.

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

where k  = 2 π / 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 r n 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 R n  −  R n−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 R n from the n th zone (with its outermost radius r n ) of the lens to the focus by using R n  =  f  +  n (2  M  + 1) λ /2 because the smallest R n-1 (i.e., R 0 ) equals f . Thus, we have the radius

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 M th -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 ultraviolet 23 , 24 and X-ray spectra 25 .

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 spectrum 2 . 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 r n in the HOD FZP. Consequently, no new finer structure is generated when combining the phase mask and the HOD FZP into a single device.

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 2 nd -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 2 nd -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 \((0.358\lambda /{NA})\) .

To design the high-order lens, we optimize ρ j only in the phase mask due to the fixed dimension of an HOD FZP. The optimization of the radii ρ j is implemented to minimize the size of the focal spot by using particle swarm algorithms 26 , the details of which can be found in our previous work 27 . A 2 nd -order lens with a focal length f  = 1 mm at a working wavelength of λ  = 405 nm is designed with a relative 2 nd -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 2 nd -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 2 nd -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 2 nd -order lens 31 . Due to the diffraction effect 17 , a Gaussian beam with lateral size of w 0 usually diverges beyond a Rayleigh distance of 0.5 kw 0 2 . 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 needles 31 , 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 2 nd -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 shaping 37 .

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 2 nd -order lens is 0.17% (see the simulation details in “Methods”), which is less than the efficiency of 1.6% for the relative 2 nd -order FZP.

Subdiffraction-limit confocal microscopy with a high-order lens

In a scanning confocal microscope 38 , 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 2 nd -order lens is integrated into a transmission SCM (see its experimental setup in Fig.  3a ). Due to the ultralong working distance, our 2 nd -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 distances 31 , 33 , 41 , 42 .

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  =  I x =0 / I |x|=D/2 , where I x =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.

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 2 nd -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 I x =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 =  I x =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 SCMs 31 , 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 μm 2 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.  3 i– 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 1 st -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 ablation 43 , 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 ).

a Sketch for laser-ablation lithography on a chromium (Cr) film by focusing a femtosecond laser with a 1 st -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.

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 nm 43 , 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 1 st -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 located 45 , 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.

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.

Although non-subwavelength zones for super-focusing were first proposed in 2013 28 and subsequently demonstrated in experiment after 2015 31 , 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 lenses 17 , 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 focusing 52 . 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 imaging 53 , 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 ratio 57 . 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 imaging 17 , lithography 58 , laser surgery, additive manufacturing, and material processing and could also benefit especially short-wavelength optics.

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:

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 r N 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:

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

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

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

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 field 17 :

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:

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:

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

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.  3 d- 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.  3 j- 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 1 st -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.

Data availability

The all data generated in this study have been deposited in the Zenodo database 61 .

Code availability

The codes for designing higher order lens and simulating focus profile and imaging have been deposited in the Zenodo database 61 .

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Acknowledgements

K.H. thanks the CAS Project for Young Scientists in Basic Research (Grant No.YSBR-049), the National Natural Science Foundation of China (Grant No. 12134013 and 62322512), the National Key Research and Development Program of China (No. 2022YFB3607300), and support from the University of Science and Technology of China’s Center for Micro and Nanoscale Research and Fabrication. D.Z. thanks for the support from the China Postdoctoral Science Foundation under Grant Number 2023M743364. The numerical calculations were performed on the supercomputing system at Hefei Advanced Computing Center and the Supercomputing Center of the University of Science and Technology of China. J.T. thanks the A*STAR AME IRG program (Grant No. A2083c0058). This work was partially supported by the A*STAR BEP Grant No. 1521480031.

Author information

These authors contributed equally: Jun He, Hong Liu, Dong Zhao.

Authors and Affiliations

Department of Optics and Optical Engineering, University of Science and Technology of China, Hefei, Anhui, China

Jun He, Dong Zhao, Fangwen Sun & Kun Huang

Institute of Materials Research and Engineering, Agency for Science Technology and Research (A*STAR), 2 Fusionopolis Way, #08-03, Innovis, Singapore, Singapore

Hong Liu & Jinghua Teng

Tissue Engineering and Cell Therapy Group, Singapore Eye Research Institute, Singapore, Singapore

Jodhbir S. Mehta

Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore, Singapore

Cheng-Wei Qiu

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Contributions

K.H. and J.T. conceived the idea. K.H., J.H., F.S., and C.Q. performed the simulations and the lens designs. H.L. and D.Z. prepared and fabricated optical samples. J.H., D.Z., and J.M. built up the experimental setup and performed the characterization. J.H., H.L., D.Z., K.H., C.Q., and J.T. wrote the manuscript. K.H. and J.T. supervised the overall project. All authors discussed the results and commented on the manuscript.

Corresponding authors

Correspondence to Jinghua Teng or Kun Huang .

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The authors J.H., D.Z., J.M., C.Q., and F.S. declare no competing interests. The authors K.H., H.L., and J.T. declare the following competing interests. K.H., H.L., and J.T. have filed one patent application related to this work through the Institute of Materials Research and Engineering. This patent (K.H., H.L., and J.T., “Optical Lens and Method of Forming the Same,” Singapore patent 10201901616 R(2019)) has been applied by the Institute of Materials Research and Engineering. It refers to the design and fabrication of high-order lenses for the realization of super-focusing.

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He, J., Liu, H., Zhao, D. et al. High-order diffraction for optical superfocusing. Nat Commun 15 , 7819 (2024). https://doi.org/10.1038/s41467-024-52256-y

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Received : 31 January 2024

Accepted : 02 September 2024

Published : 06 September 2024

DOI : https://doi.org/10.1038/s41467-024-52256-y

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