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The Zernike polynomials are a complete set of continuous functions orthogonal over a unit circle. Since first developed by Zernike in 1934, they have been in widespread use in many fields ranging from optics, vision sciences, to image processing. However, due to the lack of a unified definition, many confusing indices have been used in the past decades and mathematical properties are scattered in the literature. This review provides a comprehensive account of Zernike circle polynomials and their noncircular derivatives, including history, definitions, mathematical properties, roles in wavefront fitting, relationships with optical aberrations, and connections with other polynomials. We also survey state-of-the-art applications of Zernike polynomials in a range of fields, including the diffraction theory of aberrations, optical design, optical testing, ophthalmic optics, adaptive optics, and image analysis. Owing to their elegant and rigorous mathematical properties, the range of scientific and industrial applications of Zernike polynomials is likely to expand. This review is expected to clear up the confusion of different indices, provide a self-contained reference guide for beginners as well as specialists, and facilitate further developments and applications of the Zernike polynomials.

A circularly polarized focussed Gaussian beam carries total angular momentum of $\hbar$ per photon about the beam axis, but less than $\hbar$ spin per photon, due to the focussing of the beam. The remainder of the angular momentum is carried as orbital angular momentum. When such beams are used to rotate microscopic birefringent particles in optical tweezers, the change in angular momentum can be optically measured. However, this measurement is made using the collimated transmitted beam, rather than the focussed beam. Therefore, the conversion of spin to orbital angular momentum by focussing or collimating the beam is expected to affect the measurement. We show that for the typical cases where rotating optical tweezers are used for such measurements, the error due to spin–orbit conversion is unimportant, but there exist cases where a spin-only torque measurement would be completely erroneous.

Structured electromagnetic waves carrying orbital angular momentum (OAM) have been explored in various frequency regimes to enhance the data capacity of communication systems by multiplexing multiple co-propagating orthogonal OAM beams (i.e. mode-division multiplexing (MDM)). Terahertz (THz) communications in free space have gained interest as THz waves tend to have: (a) larger bandwidth and lower beam divergence than millimeter-waves, and (b) lower interaction with matter conditions than optical waves. In this paper, we review recent experimental demonstrations of OAM-based THz MDM communication systems, including (a) THz MDM system with two multiplexed OAM beams; (b) THz OAM multiplexing together with frequency-division-multiplexing and polarization-division-multiplexing; (c) multiplexing a full set of two-dimensional Laguerre–Gaussian (${\text{L}}{{\text{G}}_{\ell ,{\text{p}}}}$) beams; and (d) THz integrated OAM emitter for OAM mode generation and multiplexing. System performance of THz OAM links with the effect of turbulence, divergence, and multipath is also simulated and analyzed.

In this work, the density matrix formalism that describes any standard polarization state (fully or partially polarized) is applied to describe vector beams and spatial modes with orbital angular momentum (OAM). Within this framework, we provide a comprehensive description of the mapping between the corresponding Poincaré spheres (PSs); namely: the polarization PS, the higher-order PS (HOPS) and the orbital angular momentum PS (OAMPS). Whereas previous works focus on states located on the surface of these spheres, here we study vector and scalar modes lying inside the corresponding PS. We show that they can be obtained as the incoherent superposition of two orthogonal vector (or scalar) modes lying on the corresponding sphere surface. The degree of polarization (DoP) of a classical polarization state is thus extended to vector beams and OAM modes. Experimental results validate the theoretical physical interpretation, where we used a q-plate to map any state in the polarization PS onto the HOPS, and a linear polarizer to finally project onto the OAMPS. Three input states to such q-plate-polarizer system are considered: totally unpolarized, partially polarized, and fully polarized light. For that purpose, we design a new polarization state generator, based on two geometric phase gratings and a randomly polarized laser, which generates partially polarized light in an efficient and controlled way. We believe that the extension of the DoP concept to vector and OAM beams introduces a degree of freedom to describe spatially polarization and phase variant light beams.

We study the free space propagation of arrays of N × N point vortices with equal unitary topological charge embedded in a Gaussian beam, initially distributed into either a square lattice or a disordered array with the same mean vortex density and separation. The far-field patterns and the near-field evolution are analyzed as a function of different control parameters, such as the size of the carrier beam, the number of vortices, the mean separation distance among them P, and a geometrical parameter S_p, defined as the ratio of the size of the array to the size of the host beam. The value of S_p turns out to be determinant for the final spatial distribution of the optical vortices in the far-field pattern and also plays a very relevant role in the near-field evolution. While the initial array structure is basically preserved, rotated and scaled up when $S_p\gt\,\sim\!0.75$, the vortices redistribute forming a circumference around the propagation axis for $S_p\lt\,\sim\!0.5$, depleting most of the central intensity, even if the vortex density is far lower than the previously predicted limit for depletion. In the near field, we find an expression for the rotation angle of the whole structure when preserved, as a function of the propagation distance, which depends on P, but it is around eight-fold larger than the Gouy phase shift of the host. Also, we observe the formation of a high intensity ring pattern, whose radius is independent of the size of the carrier and the initial distribution of vortices, but it is only determined by the number of vortices. We verified our numerical results with experiments, finding a very good agreement between them.

Orbital angular momentum (OAM) has been known and understood in mechanical systems for centuries, but far less venerable in optical systems. It was only 30 years ago that OAM was directly associated with the spatial structure of light, specifically its phase structure, allowing OAM carrying light to be routinely created in optical laboratories. The explosion in activity since then has been startling, with OAM finding applications in microscopy, imaging, metrology and sensing, optical trapping and tweezing, communication and quantum science. Many of these advances have been reported in this very journal, and so it is fitting that the Journal of Optics should have a special issue dedicated to the topic, celebrating 30 years of advances with a collection that includes original work, reviews and tutorials, covering the past, present while pointing to an exciting future.

Point source excitation and point detection in the near-field provides new perspective to study the near-field optical phenomena of plasmonic nanostructures. Using the automated dual-tip scanning near-field optical microscope (SNOM), we have measured the optical near-field response of a dipolar emission near the edge of a monocrystalline gold platelet. The image dipole method was used to analytically calculate the interference pattern due to surface plasmon polaritons excited at the position of aperture tip and those reflected from edges of the gold platelet. The near-field enhancement was observed on the edges of the gold platelet. Our results verify that automated dual-tip SNOM is an intriguing technique for quantum plasmonic studies where deterministic coupling of quantum emitters and the detection of the near-field enhancement are of great interest.

Evanescent waves are central to many technologies such as near-field imaging that beats the diffraction limit and plasmonic devices. Frustrated total internal reflection (FTIR) is an experimental method commonly used to study evanescent waves. In this paper, we shape the incident beam of the FTIR process with a Mach–Zehnder interferometer and measure light transmittance while varying the path length difference and interferometric visibility. Our results show that the transmittance varies with the path length difference and, thus, the intensity distribution of the shaped beam. Experiment and finite element method simulation produce results that agree. We also show, through simulations, that the transmittance can be controlled via other methods of beam shaping. Our work provides a proof-of-concept demonstration of the coherent control of the FTIR process, which could lead to advancements in numerous applications of evanescent waves and FTIR.

An efficient and straightforward method to obtain all-fiber pulsed sources at 2 μm is presented and experimentally demonstrated. It is based on the soliton self-frequency shift effect in a highly nonlinear fiber. The output power of a supercontinuum source is previously increased by an optimized homemade thulium-doped fiber amplifier. By coupling the amplified output power in a highly nonlinear fiber, the spectrum is shifted toward 130 nm and the spectral peak is located at 2014 nm. The power conversion factor reaches values as high as 0.93, without employing additional amplifiers. The mean spectral power of the 2 μm source reaches −4.6 dBm nm⁻¹ (0.35 mW nm⁻¹), its output power is 38 mW and the peak power of each pulse is higher than 27 kW.

We present generation of asymmetric aberration laser beams (aALBs) with controlled intensity distribution, using a diffractive optical element (DOE) involving phase asymmetry. The asymmetry in the phase distribution is introduced by shifting the coordinates in a complex plane. The results show that autofocusing properties of aALBs remain invariant with respect to the asymmetry parameters. However, a controlled variation in the phase asymmetry allows to control the spatial intensity distribution of aALBs. In an ideal ALB containing equal intensity bright lobes, by introducing asymmetry most of the intensity can be transferred to any one of single bright lobe, and forms a high-power density lobe. For a given beam parameter m, the precise spatial position of high-power density lobe can be controlled by the asymmetry parameter β, and we have determined the empirical relations for them. We have found that for the specific values of β, the intensity in the high-power density lobe can be enhanced by several times the intensity in other lobes. The experimental results show a good agreement with the numerical simulations. The findings can be suitable for applications such as in optical trapping and manipulation as well as material processing.

When light propagates through aberrated optical systems, the resulting degradation in amplitude and phase has deleterious effects, for example, on resolution in imaging, spot sizes in focussing, and the beam quality factor of the output beam. Traditionally, this is either pre- or post-corrected by adaptive optics or phase conjugation. Here, we consider the medium as a complex channel and determine the corresponding eigenmodes which are impervious of the channel perturbation. We employ a quantum-inspired approach and apply it to the tilted lens as our example channel, a highly astigmatic system that is routinely used as a measure of orbital angular momentum. We find the eigenmodes analytically, show their robustness in a practical experiment, and outline how this approach may be extended to arbitrary astigmatic systems.

In the field of industrial metrology, the 3D nondestructive imaging of reflective metallic surfaces is a delicate task. In this work, we propose a novel application of the electrically tunable lens (ETL) in digital holography for imaging specularly reflecting objects. The precise surface profile of the microscopic step height at different axial depths is obtained by tuning the liquid lens at different currents. Initially, the ETL's focal length is set by tuning its control current to image the specular reflection observed from the surface of the reflecting sample. The current of the ETL is tuned accordingly as the sample is moved to different z-positions. In order to demonstrate the efficacy of the proposed setup, the object is kept at multiple axial distances within the depth of field of the ETL. The step height measurements are carried out and a measurement uncertainty of 0.083 µm is calculated for step height measurements at different axial positions ranging from 2 cm–21 cm. The axial range of the setup is validated by keeping two specularly reflecting samples in the field of view of the ETL. The experimental results demonstrate the ETL's efficiency in a digital holographic system for accurately imaging specularly reflecting objects present at multiple axial depths. The setup is useful for precise step height measurements and for obtaining surface profiles of microstructures.

We have demonstrated using higher-order Stokes correlations that the retrieved amplitude and phase information of the optical vortex (OV) beam and its orbital angular momentum spectrum when it is propagating through a scattering medium are insensitive to the external aberration from the optical system. A theoretical framework of the proposed technique is described and validated by considering an aberration in the propagation channel by optical system. The usefulness of the technique is demonstrated in the recovery of amplitude and phase information of an OV beam even in presence of aberration. Comparisons between aberrated and non-aberrated cases are discussed to examine and evaluate the performance of the technique.

We investigate the applicability of the circular arrays of coupled single-mode optical waveguides in transferring the non-classical state of light for quantum information processing. We study the nonclassical states of light, such as a single-photon Fock state, a two-photon NOON state, a single-mode squeezed state and a two-mode squeezed state as inputs to the lattice, which are key resources for various applications in the field of quantum information science. In addition, for comparison, we also examine a coherent state. We investigate the transport of non-classical features and quantum states of light from one waveguide mode to another. For the single and two-mode squeezed states, we perform a detailed study of the evolution of the squeezing. Our work highlights the potential of the circular arrays of optical waveguides platform for the transport of non-classical features and quantum states of light. We expect our results should have applications in the physical implementation of photonic quantum technologies.