General Astigmatism Research Articles

Engineering Orbital Angular Momentum in Structured Beams in General Astigmatic Systems via Symplectic Matrix Approach

We studied theoretically and experimentally the propagation of structured Laguerre–Gaussian (sLG) beams through an optical system with general astigmatism based on symplectic ABCD transforms involving geometry of the second-order intensity moments symplectic matrices. The evolution of the coordinate submatrix ellipses accompanying the transformation of intensity patterns at different orientations of the cylindrical lens was studied. It was found that the coordinate submatrix W and the twistedness submatrix M of the symplectic matrix P degenerate in the astigmatic sLG beam with simple astigmatism, which sharply reduces the number of degrees of freedom, while general astigmatism removes the degeneracy. Nevertheless, degeneracy entails a simple relationship between the coordinate element Wxy and the twistedness elements Mxy and Myx of the submatrix M, which greatly simplifies the measurement of the total orbital angular momentum (OAM), reducing the full cycle of measurements of the Hermite–Gaussian (HG) mode spectrum (amplitudes and phases) of the structured beam to the only measurement of the intensity moment. Moreover, we have shown that Fourier transform by a spherical lens enables us to suppress the astigmatic OAM component and restore the original free-astigmatic sLG beam structure. However, with further propagation, the sLG beam restores its astigmatic structure while maintaining the maximum OAM.

Control of giant orbital angular momentum bursts of structured Laguerre-Gaussian beams in a medium with general astigmatism

We examined theoretically and experimentally the influence of astigmatic elements (for example, a cylindrical lens) on a structured Laguerre-Gaussian beam (sLG) when the lens axes are directed at an arbitrary angle to the laboratory coordinate axes (a general astigmatism). Although a structurally stable Laguerre-Gaussian (LG) beam contains a great number of axisymmetric modes (in the LG basis) with matching phases and amplitudes, their superposition already loses its original axial symmetry, but acquires new properties (for example, fast oscillations of the orbital angular momentum (OAM)), while the beam OAM cannot exceed the azimuthal number l of the original LG mode. The loss of axial symmetry occurs due to bringing phase and amplitude perturbations into each mode of the sLG beam, which destroy the ring dislocations. Since degenerate ring dislocations are formed by optical vortices with opposite topological charges, but with equal weights, their destruction is accompanied by appearing of vortex pairs in the form of topological dipoles (their number is equal to the radial number n). As a result, the mode spectrum of the sLG is extended to the value ±(2n + l). The astigmatic element (cylindrical lens) violates the equality of vortex weights in the dipoles, which leads to a sharp growing the OAM of the sLG beam. Moreover, the OAM can be controlled by changing the rotation of the cylindrical lens axes and the control parameters of the sLG beam. It is these processes that are discussed in detail in our article, both in theoretical and experimental aspects. We show that with a certain orientation of the cylindrical lens axes, the beam OAM can exceed the sum of the orbital and azimuthal numbers (OAM > n + l). Besides, we reveal that the intensity pattern of the astigmatic sLG beam can follow the rotation of the astigmatic element axes (the effect of the beam structure following the cylindrical lens axes) at certain relations between control parameters of the sLG beam and the astigmatic element.

Influence of thermal blooming on characterizations of general astigmatism laser beams propagating in the atmosphere

Influence of thermal blooming on characterizations of general astigmatism laser beams propagating in the atmosphere

General Astigmatism of Structured LG Beams: Evolution and Transformations of the OAM Super-Bursts

We have developed a structured beams theory, computer simulation and experiment that have lost rotational symmetry and propagate in a medium with a general astigmatism. It transpires that the astigmatic medium brings to light such hidden properties of the structured beams as a sharp growing of the orbital angular momentum exceeding the sum of the radial and azimuth numbers, as well as the effect of the beam structure following the cylindrical lens rotation.

Nonparaxial Tilted Waveobjects

Simple closed-form analytical expressions for tilted astigmatic wave beams and wavepackets that are exact solutions of the wave equation are constructed. They are obtained through two different but equivalent derivations, one is based on a complex shift in the Bateman-type solutions, the other employs their Lorentz transformation. Analytic expressions for the propagation invariants: energy, momentum and orbital angular momentum of tilted waveobjects with general astigmatism are presented.

Изменение рефракции передней и задней поверхностей роговицы в раннем послеоперационном периоде после синустрабекулэктомии

Aim: to evaluate keratotopography of the anterior and posterior corneal surfaces in the early postoperative period following sinus trabeculectomy (STE). Patients and Methods: the study included 45 patients (45 eyes) with primary open angle glaucoma, stages II-III, who underwent trabeculectomy, as it was impossible to reduce intraocular pressure (IOP) by the prescribed medications. Corneal topographic parameters of the anterior and posterior corneal surfaces were evaluated with Scheimpflug camera before surgery, 1 week and 1 month after STE. Results: one week after surgery, general and corneal astigmatism increased (0.32 and 1.0 diopters, respectively). An increase in the anterior corneal curvature was usually associated with the vertical meridian and in the posterior corneal curvature — with the horizontal meridian. Topographically, there was an asymmetric steeping of individual corneal segments that cannot be explicitly classified. One month after the operation there was a trend to flattening of the anterior corneal surface and to increasing of the posterior curvature astigmatism by 0.5 diopters. Measurements taken with auto kerato-refractometer demonstrated that general astigmatism reduced by 0.69 diopters. However, the measurements revealed a 1.5-fold increase in mean corneal astigmatism compared to baseline. One week after surgery, the spherical component increased by 0.36 diopters, and following one month — by 0.62 diopters. The corneal thickness did not change during the entire follow-up period. Keratometric measurements of the anterior and posterior corneal surfaces strongly correlated at all follow-up evaluation points. There was a correlation between Scheimpflug imaging and measurements taken with auto kerato-refractometer. At different time points, a relationship was found between the corneal astigmatism and the mean curvature or the steep meridian of the anterior and posterior corneal surfaces. Conclusion: STE causes an induced astigmatism as it affects both the anterior and posterior corneal curvatures. Since the effect of induced astigmatism on the cornea is varying, it is not possible to find appropriate classification for such cases, and, as a rule, the astigmatism cannot be controlled with optical correction methods. Keywords: glaucoma, trabeculectomy, cornea, refraction, posterior corneal curvature. For citation: Averich V.V., Volzhanin A.V., Egorova G.B. Refraction changes in the anterior and posterior corneal surfaces during the early postoperative period following sinus trabeculectomy. Russian Journal of Clinical Ophthalmology. 2023;23(1):27–32 (in Russ.). DOI: 10.32364/2311-7729-2023-23-1-27-32.

Оптические аберрации глаза при кератоконусе

The review presents the basic concepts of the wavefront, ocular aberrations and their changes in keratoconus (KC) — the most common form of primary ectatic corneal disorder. It also considers the issues of the posterior corneal surface impact on the indicators of general astigmatism and the wavefront. The authors note that a possible reason for the increase in the aberration level in KC are concomitant signs of dry eye syndrome (DYS). Adding that, the review also discusses methods for correcting aberrations in KC. To date, contact lens vision correction with rigid gas permeable lenses remains the most effective method of visual rehabilitation. The prescription with a combination of artificial tears and reparative medicine allows to almost completely neutralize the phenomena of keratomalacia (conjunctival xerosis). If a patient with KC has initial signs of DYS, it is advisable to use preservative-free low-viscosity artificial tears based on hyaluronic acid of various concentrations. If the patient uses contact lens vision correction to prevent hypoxia complications, it is justified to prescribe the antioxidant agent, methylethylpiridinol, which has an antiaggregational and antihypoxic effect. Keywords: keratoconus, ocular aberrations, posterior corneal surface, wavefront, Zernike polynomials, dry eye syndrome, antioxidant. For citation: Averich V.V., Egorova G.B. Ocular aberrations in keratoconus. Russian Journal of Clinical Ophthalmology. 2022;22(3):168– 174 (in Russ.). DOI: 10.32364/2311-7729-2022-22-3-168-174.

The ABCD matrices for reflection and refraction for any incident angle and surface

A method to derive the ABCD matrices for reflection and refraction for any diopter under arbitrary incident angle is developed. The result is benchmarked on a commercial software for an optical system compound of two cylindrical lenses. It is used to simulate the general astigmatism and calculate the resulting Orbital Angular Momentum of a laser beam that propagate through a pair of cylindrical lenses. It more generally useful to ease and simplify such kind of simulations in any optical system by avoiding use of complicated commercial softwares.

Observation and analysis of single and multiple high-order Laguerre-Gaussian beams generated from a hemi-cylindrical cavity with general astigmatism

We experimentally verified that anisotropic Hermite-Gaussian modes can be generated from a hemi-cylindrical laser cavity and can be transformed into high-order Laguerre-Gaussian modes using an extra-cavity cylindrical lens. We further combined the Huygens integral and the ABCD law to clearly demonstrate the transformation along the propagation direction. By controlling the pump offset and the pump size in hemi-cylindrical cavities, we experimentally observed the unique laser patterns that displayed the optical waves related to the coherent superposition of Laguerre-Gaussian modes.

Rays and fields in general astigmatic resonators

General astigmatic (GA) resonators are discussed in detail. Eigenrays, eigenmodes and eigenvalues (Gouy-factors) of this resonator are evaluated. A stability diagram for such resonators is introduced, which clearly depicts the stable and unstable regions for rays as well as for fields. Eigenrays and their stability regions are evaluated using the ABCD-law. For the beam propagation Collins' integral and the second moment method are applied. The eigenfunctions for rectangular symmetry are the generalized Hermite polynomials multiplied by the Gaussian exponential factor. It is shown that for general astigmatic resonators these polynomials are the product of normal Hermite polynomials. The generating function of the eigenfunctions depends on the special resonator. It is a useful tool for all calculations and it is determined. Furthermore it is shown that the second moment characterization of the modes is a useful and easy to handle procedure to evaluate beam width, beam divergence, radius of curvature and twist of the generalized Gauss–Hermite functions.

Nonparaxial wave beams and packets with general astigmatism

We present exact solutions of the wave equation involving an arbitrary wave form with a phase closely similar to the general astigmatic phase of paraxial wave optics. Special choices of the wave form allow general astigmatic beamlike and pulselike waves with a Gaussian-type unrestricted localization in space and time. These solutions are generalizations of the known Bateman-type waves obtained from the connection existing between beamlike solutions of the paraxial parabolic equation and relatively undistorted wave solutions of the wave equation. As a technical tool, we present a full description of parametrizations of $2\ifmmode\times\else\texttimes\fi{}2$ symmetric matrices with positive imaginary part, which arise in the theory of Gaussian beams.

Analytical description of a Gaussian beam in a ring resonator with a nonplanar axial contour and an odd number of mirrors

Stability conditions for a ring resonator with an odd number of mirrors and a nonplanar axial contour are studied analytically. New explicit expressions are derived to describe the transverse field distribution of the Gaussian mode with general astigmatism produced in this resonator. Field characteristics for a resonator with the specified parameters are calculated.

General astigmatic transform of Hermite–Laguerre–Gaussian beams

The general astigmatic transform, or two-dimensional non-separable linear canonical transform of a Hermite-Laguerre-Gaussian beam, is investigated by theoretical means. Some corollaries that apply to Hermite-Gaussian and Laguerre-Gaussian beam propagation are presented and discussed.

Simple formula for a Gaussian beam with general astigmatism in a homogeneous medium

A simple analytical formula that describes the evolution of a Gaussian beam with general astigmatism in a homogeneous medium is obtained. The quadratic form matrix, which determines the transverse distribution of the field in the beam for an arbitrary value of the longitudinal coordinate, is represented in the form of a linear combination of such matrix in a certain initial cross section and a unit matrix. The coefficients of the linear combination are complex-valued and are expressed via the longitudinal coordinate of the considered cross section, the refractive index of the medium, and the determinant and the trace of the initial matrix.

Analytical description of a Gaussian beam in a ring resonatorwith a nonplanar axial contour and an even number of mirrors

Stability conditions for a ring resonator with aneven number of mirrors and a nonplanar axial contour arestudied analytically. New explicit expressions are derived todescribe the transverse field distribution of the Gaussian modewith general astigmatism produced in this resonator. Fieldcharacteristics for a resonator with the specified parametersare calculated.

The general astigmatic matter wave

A new sort of matter wave with the general astigmatism is proposed, whose orientation of the phase front and the density profile are crossed. A generalized ABCD law is developed to treat the evolution of the general astigmatic matter wave. It is revealed that two "cylindrical lens" pulses with oblique orientations can bring general astigmatism to the matter wave. The evolution characteristics of the general astigmatic matter wave in the gravitational field is discussed and illustrated numerically in detail. It is found that the orientations of the density profile and the phase front of the matter wave rotate continuously during evolution.

General Astigmatic Beam Propagation in Complex Quasi-Optical Launchers: Application to the ITER ECRH Remote Steering Upper Launcher

Launching systems that use in sequence more than one mirror to direct and focus electron cyclotron (EC) waves with a sufficient steering capability in relevant absorption regions of fusion plasmas may produce general astigmatic beams. The double curvature of a generic reflecting surface, even induced by deformation effects in quasi-optical systems that handle high power, is an additional source of general astigmatism. Describing the propagation of general astigmatic Gaussian beams is a necessary step in the optimization phase of a complex EC resonance heating (ECRH) launcher, since simple astigmatism treatment does not reproduce the main feature of these beams, whose spot and phase ellipse orientation changes with propagation even in free space. The correct orientation for both spot ellipse and phase ellipse is one of the input key parameters to perform realistic calculations with beam-tracing codes, which aim to characterize a launching system in terms of localized heating and current drive efficiency. In this work we describe the influence of double-curvature effects and deformations on beam propagation in terms of beam dimensions and directions. In particular, we present an application of the theory of general astigmatic Gaussian beam propagation in vacuum to the case of the remote steering option for the ITER ECRH upper launcher. In this option beams are found to be strongly astigmatic, with a major/minor axis ratio in relevant absorption regions ranging from 2.3 to 4.4 in the cases examined. Furthermore, the major axis of the resulting spot ellipses presents an orientation angle variation (from the last mirror to the expected absorption regions) ranging from 9.1 to 22.8 deg in the cases investigated. The final orientation is close to a vertical direction with respect to the equatorial plane of ITER.

Reflection and refraction of narrow Gaussian beams with general astigmatism at tilted optical surfaces: a derivation oriented toward lens design

The formulas for the reflection and refraction of a narrow Gaussian beam with general astigmatism at a tilted optical surface are derived by ray-tracing techniques. The propagation direction of the reflected and refracted beams is computed by tracing the central ray of the incident beam, and the characteristic parameters of the respective wavefronts are worked out by applying the formulas developed for the generalized ray tracing. Moreover, the Gaussian form of the reflected and refracted amplitude distributions along the transverse coordinates is determined by requiring the matching of the incident, reflected, and refracted light spots on the optical surface. No limiting assumptions are made regarding the form of the optical interface or the orientation of the incident astigmatic wavefront. In the end, to illustrate a simple application of these formulas, the reflection of a Gaussian beam at a conicoid is considered, and a simple property of the conicoidal mirrors is reported.

Analytical formulas for a circular or non-circular flat-topped beam propagating through an apertured paraxial optical system

Propagation of a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry through an apertured optical system is investigated. By expanding the hard aperture function as a finite sum of complex Gaussian functions, some approximate analytical propagation formulas are derived for a flat-topped beam of circular or non-circular (rectangular or elliptical) symmetry propagating through an apertured paraxial general astigmatic (GA) optical system or an apertured paraxial misaligned stigmatic (ST) optical system. The derived formulas are very fast to compute. The results obtained by using the approximate analytical expressions are in a good agreement with those obtained by direct numerical integration. The present analytical formulas provide a convenient and effective way for studying the propagation and transformation of a circular or non-circular flat-topped beam through an apertured general optical system.

Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, we derived an approximate analytical formula for a partially coherent twisted anisotropic Gaussian Schell-model (AGSM) beam propagating through an apertured paraxial general astigmatic (GA) optical system by use of a tensor method. The results obtained by using the approximate analytical formula are in good agreement with those obtained by using the numerical integral calculation. Our formulas avoid time-consuming numerical integration and provide a convenient and effective way for studying the propagation and transformation of a partially coherent twisted AGSM beam through an apertured paraxial GA optical system.