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Principles Of Optics Electromagnetic Theory Of Propagation Interference And Diffraction Of Light

Light-sheet microscopy facilitates rapid, high-contrast, volumetric imaging with minimal sample exposure. However, the rapid divergence of a traditional Gaussian light sheet restricts the field of view (FOV) that provides innate subcellular resolution. We show that the Airy beam innately yields high contrast and resolution up to a tenfold larger FOV. In contrast to the Bessel beam, which also provides an increased FOV, the Airy beam's characteristic asymmetric excitation pattern results in all fluorescence contributing positively to the contrast, enabling a step change for light-sheet microscopy.

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Light-sheet microscopy using an Airy beam

The diffraction of electrons through a nanoscale hologram that imprints a certain phase modulation on the electrons’ wavefunction produces a non-spreading electron Airy beam that follows a parabolic trajectory and can reconstruct its original shape after passing an obstacle. Light, as is widely known, travels in straight lines. Yet a few years ago it was shown that specially tailored light beams can follow a curved trajectory, without spreading. Such beams follow a waveform known from quantum mechanics, called the Airy function, a concept originally developed by the astronomer Sir George Biddell Airy in work on the trajectories of light in rainbows. Now, with the demonstration of Airy beams consisting of free electrons, new possibilities for manipulating electrons are in prospect. Airy electron beam arcs were generated by the diffraction of electrons through a nanoscale hologram, which imprints a specific phase modulation on the electrons' wavefunction. These beams can bend in space without any external force, stay localized over distances of up to 100 metres and self-heal after passing an obstacle. Possible applications include use in high-performance electron microscopes and as a basis for a new type of electron interferometer. Within the framework of quantum mechanics, a unique particle wave packet exists1 in the form of the Airy function2,3. Its counterintuitive properties are revealed as it propagates in time or space: the quantum probability wave packet preserves its shape despite dispersion or diffraction and propagates along a parabolic caustic trajectory, even though no force is applied. This does not contradict Newton’s laws of motion, because the wave packet centroid propagates along a straight line. Nearly 30 years later, this wave packet, known as an accelerating Airy beam, was realized4 in the optical domain; later it was generalized to an orthogonal and complete family of beams5 that propagate along parabolic trajectories, as well as to beams that propagate along arbitrary convex trajectories6. Here we report the experimental generation and observation of the Airy beams of free electrons. These electron Airy beams were generated by diffraction of electrons through a nanoscale hologram7,8,9, which imprinted on the electrons’ wavefunction a cubic phase modulation in the transverse plane. The highest-intensity lobes of the generated beams indeed followed parabolic trajectories. We directly observed a non-spreading electron wavefunction that self-heals10, restoring its original shape after passing an obstacle. This holographic generation of electron Airy beams opens up new avenues for steering electronic wave packets like their photonic counterparts, because the wave packets can be imprinted with arbitrary shapes5 or trajectories6.

Generation of electron Airy beams

In quantum mechanics, the space-fractional Schrodinger equation provides a natural extension of the standard Schrodinger equation when the Brownian trajectories in Feynman path integrals are replaced by Levy flights. Here an optical realization of the fractional Schrodinger equation, based on transverse light dynamics in aspherical optical cavities, is proposed. As an example, a laser implementation of the fractional quantum harmonic oscillator is presented in which dual Airy beams can be selectively generated under off-axis longitudinal pumping.

Fractional Schrödinger equation in optics.

Customizing the output beam shape from a laser invariably involves specialized optical elements in the form of apertures, diffractive optics and free-form mirrors. Such optics require considerable design and fabrication effort and suffer from the further disadvantage of being immutably connected to the selection of a particular spatial mode. Here we overcome these limitations with the first digital laser comprising an electrically addressed reflective phase-only spatial light modulator as an intra-cavity digitally addressed holographic mirror. The phase and amplitude of the holographic mirror may be controlled simply by writing a computer-generated hologram in the form of a grey-scale image to the device, for on-demand laser modes. We show that we can digitally control the laser modes with ease, and demonstrate real-time switching between spatial modes in an otherwise standard solid-state laser resonator. Our work opens new possibilities for the customizing of laser modes at source.

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A digital laser for on-demand laser modes

A method to design lasers that emit an arbitrary beam profile is studied. In these lasers, output-coupling is performed by a diffraction grating that imposes a phase and amplitude distribution onto the diffracted light. A solid-state laser emitting beams with a two-dimensional Airy intensity profile is demonstrated both theoretically and experimentally. In this case, the diffraction grating adds a transverse cubic phase to the diffracted light. An Airy beam is obtained by performing optical Fourier transform of the out-coupled light. The laser beam profile and power characteristics are shown to agree with theory.

Airy beam laser.

A set of diffractive optical elements for multiple-stripe structured illumination was designed, fabricated and characterized. Each of these elements with a single layer of binary surface relief combines functions of a diffractive lens, Gaussian-to-tophat beam shaper, and Dammann beam splitter. The optical investigations of laser light patterns at 20° fanout angle reveal up to 88% diffraction efficiency, high contrast, and nearly diffraction limited resolution. The developed technology has the potential for reducing complexity, number of optical components, power consumption and costs of structured light projectors in mobile and stationary 3D sensors.

Multifunctional binary diffractive optical elements for structured light projectors.

Surface-relief resonance-domain diffraction gratings with deep and dense grooves provide considerable changes in light propagation direction, wavefront curvature, and nearly 100% Bragg diffraction efficiency usually attributed only to volume optical holograms. In this paper, we present design, computer simulation, fabrication, and experimental results of binary resonance-domain diffraction gratings in the visible spectral region. Performance of imperfectly fabricated diffraction groove profiles was optimized by controlling the DC and the depth of the grooves. Indeed, more than 97% absolute Bragg diffraction efficiency was measured at the 635 nm wavelength with binary gratings having periods of 520 nm and groove depths of about 1000 nm, fabricated by direct electron-beam lithography and reactive ion etching.

Design and experimental investigation of highly efficient resonance-domain diffraction gratings in the visible spectral region.

Early expectations for a role of diffractive lenses were dramatically lessened by their high order overlapping foci, low optical powers, and competing advances in refractive micro-optics. By bringing the Bragg properties of volume holograms to diffractive lenses we got rid of ghost diffractive orders and the critical trade-off between diffraction efficiency, number of phase levels, and spatial feature-size. Binary off-axis resonance domain diffractive lens with high numerical aperture of 0.16 was designed with analytical effective grating theory, fabricated by direct e-beam writing, etched in fused silica and experimentally investigated. More than 81% measured diffraction efficiency exceeds twice the limits of thin binary optics.

Resonance domain surface relief diffractive lens for the visible spectral region

Inherent strong lateral and longitudinal chromatic dispersion of a transmission resonance domain off-axis diffractive lens were studied theoretically and experimentally. It is shown that a 4 mm diameter and 0.14 NA diffractive lens provides both focusing and dispersion with a spectral resolution of up to 0.09 nm, which is suitable for laser line spectral measurements. Experimental results for measured spectra of a mercury–argon source, a helium–neon laser, and RGB laser diodes pave a technological path to compact spectral sensors and microspectrometers.

Omri Barlev

Papers

Airy beam laser.

Multifunctional binary diffractive optical elements for structured light projectors.

Design and experimental investigation of highly efficient resonance-domain diffraction gratings in the visible spectral region.

Resonance domain surface relief diffractive lens for the visible spectral region

Chromatic dispersion of a high-efficiency resonance domain diffractive lens.