Superoscillations are band-limited functions with the counterintuitive property that they can vary arbitrarily faster than their fastest Fourier component, over arbitrarily long intervals. Modern studies originated in quantum theory, but there were anticipations in radar and optics. The mathematical understanding—still being explored—recognises that functions are extremely small where they superoscillate; this has implications for information theory. Applications to optical vortices, sub-wavelength microscopy and related areas of nanoscience are now moving from the theoretical and the demonstrative to the practical. This Roadmap surveys all these areas, providing background, current research, and anticipating future developments.

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Roadmap on superoscillations

The resolution of conventional optical elements and systems has long been perceived to satisfy the classic Rayleigh criterion. Paramount efforts have been made to develop different types of superresolution techniques to achieve optical resolution down to several nanometres, such as by using evanescent waves, fluorescence labelling, and postprocessing. Superresolution imaging techniques, which are noncontact, far field and label free, are highly desirable but challenging to implement. The concept of superoscillation offers an alternative route to optical superresolution and enables the engineering of focal spots and point-spread functions of arbitrarily small size without theoretical limitations. This paper reviews recent developments in optical superoscillation technologies, design approaches, methods of characterizing superoscillatory optical fields, and applications in noncontact, far-field and label-free superresolution microscopy. This work may promote the wider adoption and application of optical superresolution across different wave types and application domains.

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Superoscillation: from physics to optical applications.

We demonstrate a metamaterial superlens: a planar array of discrete subwavelength metamolecules with individual scattering characteristics tailored to vary spatially to create subdiﬀraction superoscillatory focus of, in principle, arbitrary shape and size. Metamaterial free-space lenses with previously unattainable eﬀective numerical apertures – as high as 1.52 – and foci as small as 0.33λ in size are demonstrated. Super-resolution imaging with such lenses is experimentally veriﬁed breaking the conventional diﬀraction limit of resolution and exhibiting resolution close to the size of the focus. Our approach will enable far-ﬁeld label-free super-resolution nonalgorithmic microscopies at harmless levels of intensity, including imaging inside cells, nanostructures, and silicon chips, without impregnating them with ﬂuorescent materials.

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Far-field superoscillatory metamaterial superlens

The generation of a sub-diffraction longitudinally polarized spot is of great interest in various applications, such as optical tweezers, super-resolution microscopy, high-resolution Raman spectroscopy, and high-density optical data storage. Many theoretical investigations have been conducted into the tight focusing of a longitudinally polarized spot with high-numerical-aperture aplanatic lenses in combination with optical filters. Optical super-oscillation provides a new approach to focusing light beyond the diffraction limit. Here, we propose a planar binary phase lens and experimentally demonstrate the generation of a longitudinally polarized sub-diffraction focal spot by focusing radially polarized light. The lens has a numerical aperture of 0.93 and a long focal length of 200λ for wavelength λ = 632.8 nm, and the generated focal spot has a full-width-at-half-maximum of about 0.456λ, which is smaller than the diffraction limit, 0.54λ. A 5λ-long longitudinally polarized optical needle with sub-diffraction size is also observed near the designed focal point.

Creation of Sub-diffraction Longitudinally Polarized Spot by Focusing Radially Polarized Light with Binary Phase Lens.

This review focuses on the imaging applications of metasurfaces. These optical elements provide a unique platform to control light; not only do they have a reduced size and complexity compared to conventional imaging systems but they also enable novel imaging modalities, such as functional-imaging techniques. This review highlights the development of metalenses, from their basic principles, to the achievement of achromatic and tunable lenses, and metasurfaces implemented in functional optical imaging applications.

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Metasurfaces-based imaging and applications: from miniaturized optical components to functional imaging platforms

In this paper, we numerically demonstrate the advantage of utilizing continuous amplitude and phase modulation in super-oscillation focusing lens design. Numerical results show that compared with simple binary amplitude modulation, continuous amplitude and phase modulation can greatly improve the super-oscillation focusing performance by increasing the central lobe intensity and the ratio of its energy to the total energy, reducing the sidelobe intensity, and substantially extending the field of view. Our study also reveals the role of phase distribution in reducing the spatial frequency bandwidth of the super-oscillation optical field on the focal plane. Based on continuous amplitude and binary phase modulation, a lens was designed with double layer metal slit array for wavelength of 4.6 µm. COMSOL is used to carry out the 2D simulation. The lens focal length is 40.18λ and the focal spot FWHM is 0.308λ. Two largest sidelobes are located right next to the central lobe with intensity about 40% of the central lobe intensity. Except for the two sidelobes, other sidelobes have intensity less than 25% of the central lobe intensity, which leads to a clear field of view on the whole focal plane.

Super-oscillation focusing lens based on continuous amplitude and binary phase modulation.

A super-oscillation far-field focusing micro-lens based on continuous amplitude modulation is experimentally demonstrated with 40-nm thick width-varied sub-wavelength metallic slit array. The $228\times 200~\mu \text{m}^{\mathrm {\mathbf {2}}}$ micro-lens is designed and fabricated with numerical aperture 0.976 and focal length $40.1\lambda $ for wavelength $\lambda =632.8$ nm. Experimental results show that the focal length is $\sim 26.5\pm 1~\mu \text{m}$ , and the focal line full width at half maximum is $0.379\lambda $ , which is smaller than the corresponding diffraction limit $0.512\lambda $ and the super-oscillation criterion $0.389\lambda $ . A great suppression of sidelobes was observed in the measured focal plane area, and the largest sidelobe intensity was found only 10.6% of the central lobe intensity, leading to a wide field of view.

Super-Oscillation Far-Field Focusing Lens Based on Ultra-Thin Width-Varied Metallic Slit Array

Optical hollow beams are suitable for materials processing, optical micromanipulation, microscopy, and optical lithography. However, conventional optical hollow beams are diffraction-limited. The generation of sub-wavelength optical hollow beams using a high numerical aperture objective lens and pupil filters has been theoretically proposed. Although sub-diffraction hollow spot has been reported, nondiffracting hollow beams of sub-diffraction transverse dimensions have not yet been experimentally demonstrated. Here, a planar lens based on binary-phase modulation is proposed to overcome these constraints. The lens has an ultra-long focal length of 300λ. An azimuthally polarized optical hollow needle is experimentally demonstrated with a super-oscillatory transverse size (less than 0.38λ/NA) of 0.34λ to 0.42λ, where λ is the working wavelength and NA is the lens numerical aperture, and a large depth of focus of 6.5λ. For a sub-diffraction transverse size of 0.34λ to 0.52λ, the nondiffracting propagation distance of the proposed optical hollow needle is greater than 10λ. Numerical simulation also reveals a good penetrability of the proposed optical hollow needle at an air-water interface, where the needle propagates through water with a doubled propagation distance and without loss of its super-oscillatory property. The proposed lens is suitable for nanofabrication, optical nanomanipulation, super-resolution imaging, and nanolithography applications.

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Planar binary-phase lens for super-oscillatory optical hollow needles.

Due to its unique properties of nondiffracting propagation, highly-localized intensity distribution, small beam cross-section, and self-healing, a nondiffracting beam is attractive for materials processing, microscopy, and optical research. Various methods have been investigated to generate such beams with conventional optics. However, the transverse size of those nondiffracting beams is restricted by the diffraction-limit. To overcome the diffraction limit, we use the concepts of super-oscillation and the vectorial angular spectrum method to design a phase mask mirror with a focal length of 1 m, radius of 5 mm, and numerical aperture of about 0.005 for a wavelength of 632.8 nm. The phase mask mirror was created with a phase spatial light modulator. Under the illumination of a linearly polarized Gaussian wave, a nondiffracting beam was created with sub-diffraction transverse size. The maximum transverse size of the beam is smaller than the diffraction limit of 0.5λ/NA for a propagation distance greater than 43.3 mm. A nondiffracting beam with smaller transverse size can be realized by further increasing the NA value.

Creating a nondiffracting beam with sub-diffraction size by a phase spatial light modulator

We have demonstrated numerically a lens design based on continuous modulation of both phase and amplitude utilizing array of double-layer metallic aluminum holes filled with silicon dioxide. The tuning range is 0-π and 0-0.3 for phase and amplitude, respectively, at wavelength 365 nm, allowing flexible lens design. Two lenses are designed and compared. At wavelength 365 nm, the lens based on pure phase modulation generates a 600-nm hot spot at a distance of 6.5 μm, while the phase-amplitude-modulation-based lens leads to an improved focusing spot size of 450 nm, which is beyond the diffraction limit 608 nm and the superoscillation criterion 462 nm. An example of subwavelength superoscillation focusing (0.32λ) is also demonstrated with a continuous amplitude modulated lens, which shows small sidelobe and broad view field. Our approach shows great potential in superoscillation lens design and can be applied to other optical spectrum ranges as well.

Yinghu He

Papers

Super-oscillation focusing lens based on continuous amplitude and binary phase modulation.

Super-Oscillation Far-Field Focusing Lens Based on Ultra-Thin Width-Varied Metallic Slit Array

Planar binary-phase lens for super-oscillatory optical hollow needles.

Creating a nondiffracting beam with sub-diffraction size by a phase spatial light modulator

Double-Layer Metallic Holes Lens Based on Continuous Modulation of Phase and Amplitude