Showing papers on "Paraxial approximation published in 2023"

Exact solutions for the electromagnetic fields of a flying focus

Abstract: The intensity peak of a "flying focus" travels at a programmable velocity over many Rayleigh ranges while maintaining a near-constant profile. Assessing the extent to which these features can enhance laser-based applications requires an accurate description of the electromagnetic fields. Here we present exact analytical solutions to Maxwell's equations for the electromagnetic fields of a constant-velocity flying focus, generalized for arbitrary polarization and orbital angular momentum. The approach combines the complex source-point method, which transforms multipole solutions into beam-like solutions, with the Lorentz invariance of Maxwell's equations. Propagating the fields backward in space reveals the space-time profile that an optical assembly must produce to realize these fields in the laboratory. Comparisons with simpler paraxial solutions provide conditions for their reliable use when modeling a flying focus.

X-ray dark-field and phase retrieval without optics, via the Fokker–Planck equation

Abstract: Emerging methods of x-ray imaging that capture phase and dark-field effects are equipping medicine with complementary sensitivity to conventional radiography. These methods are being applied over a wide range of scales, from virtual histology to clinical chest imaging, and typically require the introduction of optics such as gratings. Here, we consider extracting x-ray phase and dark-field signals from bright-field images collected using nothing more than a coherent x-ray source and a detector. Our approach is based on the Fokker-Planck equation for paraxial imaging, which is the diffusive generalization of the transport-of-intensity equation. Specifically, we utilize the Fokker-Planck equation in the context of propagation-based phase-contrast imaging, where we show that two intensity images are sufficient for successful retrieval of both the projected thickness and the dark-field signal associated with the sample. We show the results of our algorithm using both a simulated dataset and an experimental dataset. These demonstrate that the x-ray dark-field signal can be extracted from propagation-based images, and that sample thickness can be retrieved with better spatial resolution when dark-field effects are taken into account. We anticipate the proposed algorithm will be of benefit in biomedical imaging, industrial settings, and other non-invasive imaging applications.

Geometric descriptions for the polarization of nonparaxial light: a tutorial

Abstract: This tutorial provides an overview of the local description of polarization for nonparaxial light, for which all Cartesian components of the electric field are significant. The polarization of light at each point is characterized by a $3$ component vector in the case of full polarization or by a $3\times3$ polarization matrix for partial polarization. Standard concepts for paraxial polarization like the degree of polarization, the Stokes parameters and the Poincar\'e sphere then have generalizations for nonparaxial light that are either not unique or not trivial. This work aims to clarify some of these discrepancies, present some new concepts, and provide a framework that highlights the similarities and differences with the description for the paraxial regimes. Particular emphasis is placed on geometric interpretations.

XUV superfluorescence from helium gas in the paraxial three-dimensional approximation

Abstract: We present the results of a theoretical study of XUV superfluorescence from doubly excited states of helium resonantly pumped by free-electron laser (FEL) pulses. Our model allows us to predict both the spectrum and angular distribution of emitted XUV radiation in a wide range of experimentally accessible parameters. This is achieved by going beyond two key deficiencies of most previous models: The one-dimensional treatment in space is upgraded to three dimensions with electromagnetic fields treated in the paraxial approximation and spontaneous emission is modeled by a recently developed approach that avoids the unrealistic delayed response but preserves the expected characteristics of the emitted field in the spontaneous emission limit. The case study of $3a{\phantom{\rule{0.16em}{0ex}}}^{1}{P}^{o}$ resonance in helium with $63.66\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ excitation energy is presented for realistic parameters of seeded light pulses from the FERMI FEL facility and a recently developed high-pressure gas cell. Results of numerical simulations show that both the spectral width and angular divergence of emitted radiation vary with gas pressure and pump pulse intensity in a complex way.

Relativistic transformations of quasi-monochromatic paraxial optical beams

Abstract: A monochromatic plane wave recorded by an observer moving with respect to the source undergoes a Doppler shift and spatial aberration. We investigate here the transformation undergone by a generic, paraxial, spectrally coherent quasimonochromatic optical beam (of finite transverse width) when recorded by a moving detector. Because of the space-time coupling intrinsic to the Lorentz transformation, the monochromatic beam is converted into a propagation-invariant pulsed beam traveling at a group velocity equal to that of the relative motion and which belongs to the recently studied class of ``space-time wave packets.'' We show that the predicted transformation from a quasimonochromatic beam to a pulsed wave packet can be observed even at terrestrial speeds.

Diffractive Lens Design

Abstract: Lens systems can incorporate diffractive optical elements or surfaces. A diffractive surface provides unique advantages for the optical designer, enabling lens systems to be simplified. This book covers all aspects of the design of diffractive lens systems, and is an essential reference for incorporating diffractive optical elements into imaging systems. Topics include paraxial imaging theory, aberration theory, ray tracing, image formation theory, tolerancing, image quality modelling by scalar diffraction theory and diffractive lens design in commercial lens design software. Design examples are used to illustrate key aspects of the design processes, with files available in CODEV and Zemax. The key audiences for this text include optical designers, optical system engineers and postgraduate students in optical science and engineering. Part of IOP Series in Emerging Technologies in Optics and Photonics. Key features • A unique and comprehensive treatment of the design of optical systems incorporating diffractive surfaces. • An essential text for the optical designer contemplating the use of diffractive surfaces to improve their optical system. • Presents a new and consistent approach to the development of aberration and image formation theory applied to diffractive lenses. • Provides detailed knowledge, software tools and design examples to reduce the optical engineer’s learning curve in the understanding of the unique properties of these optical elements. • Brings all the relevant material associated with diffractive lens design into a single volume, including a comprehensive set of references.

Partially coherent sources whose coherent modes are spatiotemporal optical vortex beams

Abstract: We analyze three non-stationary partially coherent sources whose coherent modes are spatiotemporal optical vortex (STOV) beams. Using spatiotemporal (ST) Bessel–Gauss and Laguerre–Gauss beams (STOV-carrying solutions to the space-time paraxial wave equation) as eigenfunctions in the coherent-modes representation of the mutual coherence function, we derive the ST versions of J 0-Bessel-correlated, I n -Bessel-correlated, and twisted Gaussian Schell-model beams. We model, in simulation, these ST random beams via their coherent-modes expansions, compare and contrast the simulated results to theory, and analyze/discuss their free-space propagation characteristics. The work presented in this paper will be useful for simulating or physically generating these ST beams for use in applications or future studies.

“Capillary” Structures in Transversely Trapped Nonlinear Optical Beams

Abstract: A mathematical analogy between paraxial optics with two circular polarizations of light in a defocusing Kerr medium with positive dispersion, binary Bose-Einstein condensates of cold atoms in the phase separation regime, and hydrodynamics of two immiscible compressible liquids can help in theoretical search for unknown three-dimensional coherent optical structures. In this work, transversely trapped (by a smooth profile of the refractive index) light beams are considered and new numerical examples are presented, including a ``floating drop'', a precessing longitudinal optical vortex with an inhomogeneous profile of filling with the second component, and the combination of a drop and a vortex filament. Filled vortices that are perpendicular to the beam axis and propagate at large distances have also been simulated. V. P. Ruban, JETP Lett. 117(4), 292 (2023); DOI: 10.1134/S0021364022603311

Toward Arbitrary Spin‐Orbit Flat Optics Via Structured Geometric Phase Gratings

Abstract: Reciprocal spin‐orbit coupling (SOC) via geometric phase with flat optics provides a promising platform for shaping and controlling paraxial structured light. Current devices, from the pioneering q‐plates to the recent J‐plates, provide only spin‐dependent wavefront modulation without amplitude control. However, achieving control over all the spatial dimensions of paraxial SOC states requires spin‐dependent control of corresponding complex amplitude, which remains challenging for flat optics. Here, to address this issue, a new type of flat‐optics elements termed structured geometric phase gratings is presented, that is capable of conjugated complex‐amplitude control for orthogonal input circular polarizations. By using a microstructured liquid crystal photoalignment technique, a series of flat‐optics elements is engineered and their excellent precision in arbitrary SOC control is shown. This principle unlocks the full‐field control of paraxial structured light via flat optics, providing a promising way to develop an information exchange and processing units for general photonic SOC states, as well as extra‐/intracavity mode convertors for high‐precision laser beam shaping.

On the importance of the longitudinal component of paraxial optical vortices in the interaction with atoms

Abstract: Experimental evidence and theory on the head-on excitation of atoms by paraxial Laguerre–Gaussian beams revealed that the longitudinal component of the field has to be taken into account. Optical vortices are in fact a large family of fields, Laguerre–Gaussian being only one particular case. Here, we extend the previous study to a broader set of vortex fields. We demonstrate that, in general, paraxial optical vortices that have opposite orbital and spin angular momenta exhibit a longitudinal component that cannot be disregarded in the light–matter interaction.

Multimodal intrinsic speckle-tracking (MIST) to extract images of rapidly-varying diffuse X-ray dark-field

Abstract: Abstract Speckle-based phase-contrast X-ray imaging (SB-PCXI) can reconstruct high-resolution images of weakly-attenuating materials that would otherwise be indistinguishable in conventional attenuation-based X-ray imaging. The experimental setup of SB-PCXI requires only a sufficiently coherent X-ray source and spatially random mask, positioned between the source and detector. The technique can extract sample information at length scales smaller than the imaging system’s spatial resolution; this enables multimodal signal reconstruction. “Multimodal Intrinsic Speckle-Tracking” (MIST) is a rapid and deterministic formalism derived from the paraxial-optics form of the Fokker–Planck equation. MIST simultaneously extracts attenuation, refraction, and small-angle scattering (diffusive dark-field) signals from a sample and is more computationally efficient compared to alternative speckle-tracking approaches. Hitherto, variants of MIST have assumed the diffusive dark-field signal to be spatially slowly varying. Although successful, these approaches have been unable to well-describe unresolved sample microstructure whose statistical form is not spatially slowly varying. Here, we extend the MIST formalism such that this restriction is removed, in terms of a sample’s rotationally-isotropic diffusive dark-field signal. We reconstruct multimodal signals of two samples, each with distinct X-ray attenuation and scattering properties. The reconstructed diffusive dark-field signals have superior image quality—as measured by the naturalness image quality evaluator, signal-to-noise ratio, and azimuthally averaged power-spectrum—compared to our previous approaches which assume the diffusive dark-field to be a slowly varying function of transverse position. Our generalisation may assist increased adoption of SB-PCXI in applications such as engineering and biomedical disciplines, forestry, and palaeontology, and is anticipated to aid the development of speckle-based diffusive dark-field tensor tomography.

Average paraxial power of a lens and visual acuity

Abstract: Abstract To provide a solution for average paraxial lens power (A p P) of a lens. Orthogonal and oblique sections through a lens of power $$F$$ F were reduced to a paraxial representation of lens power followed by integration. Visual acuity was measured using lenses of different powers (cylinders of − 1.0 and − 2.0D) and axes, mean spherical equivalent (MSE) of S + C/2, A p P and a toric correction, with the order of correction randomised. A digital screen at 6 m was used on which a Landolt C with crowding bars was displayed for 0.3 s before vanishing. The general equation for a symmetrical lens of refractive index (n), radius of curvature R, in medium of refractive index n1, through orthogonal ( $$\theta$$ θ ) and oblique meridians ( $$\gamma$$ γ ) as a function of the angle of incidence ( $$\alpha$$ α ) reduces for paraxial rays ( $$\alpha \sim 0$$ α ∼ 0 ) to $$F_{n,R} \left( {\alpha ,\theta ,\gamma } \right)\left. \right|_{\alpha \sim 0} = \frac{{n - n_{1} }}{R}\cos^{2} \theta \cos^{2} \gamma$$ F n , R α , θ , γ α ∼ 0 = n - n 1 R cos 2 θ cos 2 γ . The average of this function is $$F_{n,R} \left( {\alpha ,\theta ,\gamma } \right)\left. \right|_{\alpha \sim 0} = \frac{{n - n_{1} }}{4R} $$ F n , R α , θ , γ α ∼ 0 = n - n 1 4 R providing a solution of $$\frac{F}{4}$$ F 4 for A p P.For central ( p = 0.04), but not peripheral ( p = 0.17) viewing, correction with A p P was associated with better visual acuity than a MSE across all tested refractive errors ( p = 0.04). These findings suggest that $$\frac{F}{4}$$ F 4 may be a more inclusive representation of the average paraxial power of a cylindrical lens than the MSE.

Focal length, EFL, and the eye.

Abstract: The focal length is often called the effective focal length, or efl instead, and although this is acceptable for a lens in air, it is not otherwise correct. The eye is used as an example here for an optical system where the object is in air and the image is in fluid. Welford, Aberrations of Optical Systems (1986) has paraxial equations that are consistent with historical use while also clearly defining efl. These are based on power at a surface having to be the same for light traveling in both directions (n '/f '). The focal length f ' is the actual physical distance from the 2nd principal point to the paraxial focus, and the equivalent focal length, or efl, is the focal length divided by the image index (f '/n '). Separately, when the object is in air, the efl is shown to act at the nodal point, with the lens system represented by either an equivalent thin lens at the principal point with a focal length or a different equivalent thin lens in air at the nodal point with an efl. The rationale for using effective instead of equivalent for efl is unclear, but efl is used more as a symbol than as an acronym.

Optical singularity dynamics and spin-orbit interaction due to a normal-incident optical beam reflected at a plane dielectric interface

Abstract: The degenerate case of normal incidence and reflection of an optical beam (both paraxial and nonparaxial) at a plane isotropic dielectric interface, which is azimuthally symmetric in terms of the momentum-spatial variation of Fresnel coefficients but not in terms of the fundamental polarization inhomogeneity of the incident field, requires in-depth analyses. In this paper, we use the reflection and transmission coefficient matrix formalism to derive an exact field expression of a normal-reflected diverging beam. The availability of the exact field information allows controlled variations of the system parameters, leading to significant dynamics of phase and polarization singularities hitherto unanticipated in the literature. We carry out a detailed exploration of these dynamics in our simulated system, and also verify them experimentally by using an appropriate setup. We then use Barnett's formalism to determine the associated orbital angular momentum (OAM) fluxes, leading to a subtle interpretation and mathematical characterization of spin-orbit interaction (SOI) in the system. Our paper thus represents a nontrivial unification of the most fundamental electromagnetic reflection and transmission problem at a plane dielectric interface and the emerging areas of optical singularity dynamics with their understanding in terms of OAM flux and SOI. The normal-incidence--retroreflection geometry being especially amenable to applications, these beam-field phenomena are anticipated to have applications in interface characterization, particle rotation and manipulation, and other nano-optical processes.

Paraxial beams in fluctuating fusion plasmas: Diffusive limit and beyond

Abstract: A paraxial expansion of the (ensemble-averaged) Wigner function in the relevant wave kinetic equation for electron cyclotron waves in fluctuating plasmas allows the derivation of phase-space equations similar to the equations for the Gaussian beam parameters in the paraxial WKB method [G.V. Pereverzev, Phys. Plasmas 5, 3529 (1998)]. This is relatively straightforward when the scattering of the wave field by density fluctuations can be described by a diffusion operator in refractive-index space. The general case is rather more complicated, yet we could find a heuristic construction of a paraxial Wigner function. Here we use a simple model, which has an analytical solution, to test both the theoretical validity of the diffusion approximation and the heuristic paraxial approach beyond the diffusion approximation.

Transcending shift-invariance in the paraxial regime via end-to-end inverse design of freeform nanophotonics

Abstract: Traditional optical elements and conventional metasurfaces obey shift-invariance in the paraxial regime. For imaging systems obeying paraxial shift-invariance, a small shift in input angle causes a corresponding shift in the sensor image. Shift-invariance has deep implications for the design and functionality of optical devices, such as the necessity of free space between components (as in compound objectives made of several curved surfaces). We present a method for nanophotonic inverse design of compact imaging systems whose resolution is not constrained by paraxial shift-invariance. Our method is end-to-end, in that it integrates density-based full-Maxwell topology optimization with a fully iterative elastic-net reconstruction algorithm. By the design of nanophotonic structures that scatter light in a non-shift-invariant manner, our optimized nanophotonic imaging system overcomes the limitations of paraxial shift-invariance, achieving accurate, noise-robust image reconstruction beyond shift-invariant resolution.

On the paraxial approximation in quantum optics II: Henochromatic modes of a Maxwell field

Abstract: A companion paper has argued that the best way to associate single-particle quantum states of a scalar field to the modes of a narrowly collimated beam of classical radiation modeled in the paraxial approximation uses the ``henochromatic'' states previously introduced by Sudarshan, Simon and Mukunda. This paper extends that result to Maxwell fields, again emphasizing the central role of unitarity in defining the association. The principal new technical element in the present discussion has to do with the intertwining of polarization and spatial degrees of freedom in the resulting single-photon states.

Efficient post-processing of electromagnetic plane wave simulations to model arbitrary structured beams incident on axisymmetric structures

Abstract: The study of an optical beam interacting with material structures is a fundamental of nanophotonics. Computational electromagnetic solvers facilitate the rapid calculation of the scattering from material structures with arbitrary geometry and complexity, but have limited efficiency when employing structured excitation fields. We have developed a post-processing method and package that can efficiently calculate the full three-dimensional electric and magnetic fields for any optical beam incident on a particle or structure with at least one axis of continuous rotational symmetry, called an axisymmetric body (such as a sphere, cylinder, cone, torus or surface). Provided an initial batch of plane wave simulations is computed, this open-source package combines data from computational electromagnetic solvers in a post-processing fashion using the angular spectrum representation to create arbitrarily structured beams, including vector vortex beams. Any and all possible incident beams can be generated from the initial batch of plane wave simulations, without the need for further simulations. This allows for efficiently performing parameter sweeps such as changing the angle of illumination or translating the particle position relative to the beam, all in post-processing, with no need for additional time-consuming simulations. We demonstrate some applications by numerically calculating optical force and torque maps for a spherical plasmonic nanoparticle in a tightly focused Gaussian beam, a plasmonic nanocone in an azimuthally polarised beam and compute the fields of a non-paraxial Laguerre-Gaussian vortex beam reflecting on a multilayered surface. We believe this package, called BEAMS, is a valuable tool for rapidly quantifying electromagnetic systems that are beyond traditional analytical methods.

Propagation characteristics of high power beam in weakly relativistic-ponderomotive thermal quantum plasma

Abstract: The present work explores the propagation characteristics of high power beam in weakly relativistic-ponderomotive thermal quantum plasma (TQP). Q-gaussian laser beam is taken in present investigation. The quasi optics equation obtained in present study is solved through well established WKB approximation and paraxial theory approach for obtaining 2nd order differential equation describing the behavior of beam width of laser beam. Further, numerical simulation of this 2nd order differential equation is carried out for determining the behavior of beam width with dimensionless distance for established laser-plasma parameters. The comparison of present study is made with ordinary quantum plasma and classical relativistic plasma cases

Axial correlation revivals and number factorization with structured random waves

Abstract: We advance a general theory of field correlation revivals of structured random wave packets, composed of superpositions of propagation-invariant modes, at pairs of planes transverse to the packet propagation direction. We derive an elegant analytical relation between the normalized intensity autocorrelation function of thus structured paraxial light fields at a pair of points on an optical axis of the system and a Gauss sum, thereby establishing a fundamental link between statistical optics and number theory. We propose and experimentally implement a simple, robust analog random wave computer that can efficiently decompose numbers into prime factors.

3D Modeling of Hermite-Gaussian Modes Propagation

Abstract: The Fresnel transform is used to simulate the propagation of paraxial optical beams in free space, and the results are usually displayed in two-dimensional form. In this paper, we consider a three-dimensional model of Hermite-Gaussian modes propagation, as well as their superposition over a given propagation interval.

"Analytical Continuation'' of Flattened Gaussian Beams

Abstract: A purely analytical extension of the flattened Gaussian beams [Opt. Commun. \textbf{107,} 335 (1994)] to any values of the beam order, is here proposed. Thanks to it, the paraxial propagation problem of axially symmetric, coherent flat-top beams through arbitrary $ABCD$ optical systems can definitely be closed in terms of a particular bivariate confluent hypergeometric function.

Non-Diffracting Polarisation Features around Far-Field Zeros of Electromagnetic Radiation

Abstract: Light from any physical source diffracts and becomes paraxial in the far field region, where polarisation is virtually transverse to the local propagation direction. A longitudinal polarisation component remains and is insignificant unless the transverse field components vanish. Maxwell's equations show that any such transverse field zero, where the longitudinal component dominates, develops non-paraxial features which do not diffract. Non-diffracting structures, independent of the distance to the source, include a zero-enclosing intensity ratio tube, and parallel, non-diverging polarisation singularities. Remarkably, the polarisation's spatial profile appears to be detached from the radiation's intensity profile. While the intensity spreads out in space at larger distances from the source, the polarisation profile maintains a fixed transverse spatial extent. Numerical examples presented for multipole radiation and phased antenna arrays confirm our findings.

Multimodal Intrinsic Speckle-Tracking (MIST) to extract rapidly-varying diffuse X-ray scatter

Abstract: Speckle-based phase-contrast X-ray imaging (SB-PCXI) can reconstruct high-resolution images of weakly-attenuating materials that would otherwise be indistinguishable in conventional attenuation-based imaging. The experimental setup of SB-PCXI requires only a sufficiently coherent source and spatially random mask, positioned between the source and detector. The technique can extract sample information at length scales smaller than the imaging system's spatial resolution; this enables multimodal signal reconstruction. ``Multimodal Intrinsic Speckle-Tracking'' (MIST) is a rapid and deterministic formalism derived from the paraxial-optics form of the Fokker-Planck equation. MIST simultaneously extracts attenuation, refraction, and small-angle scattering (diffusive-dark-field) signals from a sample and is more computationally efficient compared to alternative speckle-tracking approaches. Hitherto, variants of MIST have assumed the diffusive-dark-field signal to be spatially slowly varying. Although successful, these approaches have been unable to well-describe unresolved sample microstructure whose statistical form is not spatially slowly varying. Here, we extend the MIST formalism such that there is no such restriction, in terms of a sample's rotationally-isotropic diffusive-dark-field signal. We reconstruct multimodal signals of two samples, each with distinct X-ray attenuation and scattering properties. The reconstructed diffusive-dark-field signals have superior image quality compared to our previous approaches which assume the diffusive-dark-field to be a slowly varying function of transverse position. Our generalisation may assist increased adoption of SB-PCXI in applications such as engineering and biomedical disciplines, forestry, and palaeontology, and is anticipated to aid the development of speckle-based diffusive-dark-field tensor tomography.

Design and implementation of plenoptic imaging system with reduced aberrations

Abstract: Industrial inspection is critical for 3D semiconductor manufacturing process and automated optical inspection for 3D ICs has attracted a lot of attention in these years. Plenoptic imaging systems, based on a micro-lens array, acquire light field from parallax and computes 3D information with lower costs. To reduce aberrations from the optical design with microlens array, especially for off-axial micro lenses, the design flow for plenoptic imaging systems is proposed. Based on the parameters designed from paraxial approximation, lateral image quality is optimized by a commercial optical design software, and then depth-related performances are estimated from the simulated images of the optical system. The experimental system for validation is tested quantitatively with modulation transfer function (MTF), by the slanted-edge method of ISO 12233. The difference of MTF between the paraxial and off-axial regions is approximately 0.02, which is within the repeatability error 0.03. Moreover, the synthesized images of a PCIe card refocused on the chip and the board clearly show the elements at the refocusing depth only. The depth map and the all-in-focus image are estimated to build a 3D model. However, significant artifacts appear on depth maps when lighting is not uniform. With combination of the ring light and coaxial light, the depth maps of objects with different surface properties can be estimated with less artifacts. Furthermore, accuracy and resolution can be enhanced by deep-learning technologies, which will be implemented in the future.