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Purpose To explain the concept behind the Castrop toric lens (tIOL) power calculation formula and demonstrate its application in clinical examples. Methods The Castrop vergence formula is based on a pseudophakic model eye with four refractive surfaces and three formula constants. All four surfaces (spectacle correction, corneal front and back surface, and toric lens implant) are expressed as spherocylindrical vergences. With tomographic data for the corneal front and back surface, these data are considered to define the thick lens model for the cornea exactly. With front surface data only, the back surface is defined from the front surface and a fixed ratio of radii and corneal thickness as preset. Spectacle correction can be predicted with an inverse calculation. Results Three clinical examples are presented to show the applicability of this calculation concept. In the 1st example, we derived the tIOL power for a spherocylindrical target refraction and corneal tomography data of corneal front and back surface. In the 2nd example, we calculated the tIOL power with keratometric data from corneal front surface measurements, and considered a surgically induced astigmatism and a correction for the corneal back surface astigmatism. In the 3rd example, we predicted the spherocylindrical power of spectacle refraction after implantation of any toric lens with an inverse calculation. Conclusions The Castrop formula for toric lenses is a generalization of the Castrop formula based on spherocylindrical vergences. The application in clinical studies is needed to prove the potential of this new concept.

This work presents a volumetric calibration method for Scheimpflug plenoptic camera in particle image velocimetry. To establish a prerequisite foundation for the Scheimpflug light-field PIV, the proposed technique makes use of the unique point-like pattern and plenoptic disc feature of particle light-field images to accurately determine affected pixels for a spatial voxel in a measurement volume. With Gaussian optics, an ideal projection model of the Scheimpflug plenoptic camera is derived to provide a basis for the calibration method. By taking lens defects, optical window’s imperfection, inaccurate tilted-angle and other practical application difficulties into account, the calibration method can establish the mapping relationship between a spatial voxel and its affected pixels correctly. The volumetric calibration method was validated by a classic vortex-ring PIV measurement where the correspondence between a voxel and its affected pixels was accurately determined and the volumetric velocity field was successfully acquired.

During the production of a new designed lens system, overweighted assembling and manufacturing errors can cause the degradation of the performance of the whole system. Therefore, optimizations to improve the assembling and manufacturing tolerances sensitivity are greatly required in lens design industry. Following the principle of Gaussian optics, there are some practical implications of the first order derivatives which are of great significance to analyse the sensitivity of a lens system with its structural parameters changing. It is demonstrated that manufacturing errors, such as tilting, decentering and so on, induce the real rays' height shifting on each surface of an imaging system, which subsequently induces the decline of image quality. Thus, we proposed a scheme to use the first order derivatives of the height of a real ray on the image surface in a lens system versus that on the pupil surface, which indicate the height variations of the real ray on the image surface of the lens system caused by the displacement of its corresponding height on the pupil surface, as a merit function to improve the tolerance sensitivity of the whole system. A test program using the ZPL language in Opticstudio has preliminarily confirmed the effectiveness of our proposed methods. And as it is aimed at controlling the real rays of a lens system to reduce its sensitivity according to assembling and manufacturing errors, our scheme is theoretically able to appropriate to all lens design systems.

Aniseikonia as one of the major risk factors for asthenopic problems is mostly overlooked in modern cataract surgery. The purpose of this study was to develop a simple calculation scheme for clinicians to predict the object to image magnification in a pseudophakic eye with biometric data.The calculation scheme for object to image magnification in the pseudophakic eye is based on a vergence calculation of the lens power with theoretical optical formulae. From the biometric data, which are typically derived from both eyes during lens power calculation, the vergences in front of and behind the 3 or 4 refractive surfaces of the pseudophakic eye model are used to predict the magnification for objects at infinity or objects located at a finite measurement distance (e.g. 5 m).With a formula-based lens power calculation a pseudophakic eye model is set up with 3 or 4 refractive surfaces (postoperative spectacle refraction; thick cornea described by anterior surface or thick cornea characterized by anterior and posterior surfaces; intraocular lens). The vergence in front of and behind each refractive surface is derived by means of linear Gaussian optics. The quotient of the product of all vergences in front of the surfaces and the product of all vergences behind the respective surfaces describes the object to image magnification of the eye. A comparison of the object to image magnification of both eyes yields the retinal image size disparity or aniseikonia. This calculation strategy is shown in a step-by-step approach exemplarily for the Haigis and Hoffer‑Q formulae (3 surfaces) and the Castrop formula (4 surfaces).If during planning and lens power calculation biometry is performed for both eyes, ocular magnification of both eyes can be easily derived with this calculation scheme and aniseikonia can be extracted from a comparison of magnification of both eyes. Such a simple prediction should be established as a standard for precataract biometry and lens power calculation for early detection and avoidance of asthenopic complaints after cataract surgery.HINTERGRUND UND ZIELSETZUNG: Die Aniseikonie als mögliche Ursache asthenopischer Beschwerden tritt bei der modernen Kataraktchirurgie oft in den Hintergrund. Ziel der vorliegenden Arbeit ist es, dem Kliniker ein einfaches Berechnungsmodell an die Hand zu geben, mit dem der Abbildungsmaßstab des pseudophaken Auges abgeschätzt werden kann.Das Berechnungsschema für den Abbildungsmaßstab des pseudophaken Auges bezieht sich auf die formelbasierte (vergenzbasierte) Berechnung der Intraokularlinse mit theoretisch-optischen Formeln. Aus den biometrischen Größen, die in der Regel für beide Augen bei der Linsenberechnung vorliegen, kann aus den Vergenzen vor und hinter den 3 oder 4 refraktiven Grenzflächen im pseudophaken Augenmodell der Abbildungsmaßstab für Objekte im Unendlichen oder in einer endlichen Messdistanz ermittelt werden.Bei der formelbasierten Berechnung wird ein pseudophakes Augenmodell mit 3 bzw. 4 refraktiven Grenzflächen (postoperative Brillenrefraktion; dünne Hornhaut, beschrieben durch die Vorderfläche, bzw. dicke Hornhaut, beschrieben durch die Vorder- und Rückfläche; Intraokularlinse) definiert und mit den Methoden der linearen Optik die Vergenz vor und hinter jeder Grenzfläche bestimmt. Der Quotient aus dem Produkt der Vergenzen vor den Grenzflächen und dem Produkt der Vergenzen unmittelbar hinter den Grenzflächen beschreibt direkt den Abbildungsmaßstab des Auges. Aus dem Vergleich des Abbildungsmaßstabs beider Augen kann unmittelbar der retinale Bildgrößenunterschied ermittelt werden. Exemplarisch wird dies anhand der Haigis- und Hoffer-Q-Formel (3 Flächen) und der Castrop-Formel (4 Flächen) gezeigt.Wird bei der Planung der Kataraktoperation die Biometrie und Linsenberechnung an beiden Augen durchgeführt, so kann mit einfachen Mitteln der Abbildungsmaßstab bei beiden Augen und aus dem Vergleich beider Augen die Aniseikonie des pseudophaken Patienten ermittelt werden. Eine derartige Abschätzung sollte fester Bestandteil der Linsenberechnung werden, um mögliche asthenopische Beschwerden nach der Kataraktoperation früh zu erkennen.

Trapped atomic ensembles are convenient systems for quantum information storage in the long-lived sublevels of the electronic ground state and its conversion to propagating optical photons via stimulated Raman processes. Here we investigate a phase-matched emission of photons from a coherently prepared atomic ensemble. We consider an ensemble of cold atoms in an elongated harmonic trap with normal density distribution, and determine the parameters of paraxial optics to match the mode geometry of the emitted radiation and optimally collect it into an optical waveguide.

In the ophthalmic Shack-Hartmann wavefront sensor beacon light originates from various depths within the retina. We model the retina of the human and mouse eyes as formed by two backscattering planes, showing that the use of lenslets with low Fresnel number and/or too small search boxes in the centroiding algorithm will produce artifactual aberrations. We evaluate the impact of these errors for four common beacon illumination strategies: full circular, annular, small circular on-axis and small circular off-axis. We find that artifactual aberrations are larger for annular and off-axis beacon illumination, dominated by defocus plus spherical aberration and defocus plus coma, respectively. These artifactual aberrations can be almost completely eliminated by selecting the minimum centroid search box size based on a simple Gaussian optics model and ocular biometry, provided the lenslet Fresnel number is sufficiently large to introduce minimal cross-talk between images of adjacent lenslets.

Purpose To analyse corneal power based on a large optical coherence tomography dataset using raytracing, and to evaluate corneal power with respect to the corneal front apex plane for different definitions of best focus. Methods A large OCT dataset (10,218 eyes of 8,430 patients) from the Casia 2 (Tomey, Japan) was post-processed in MATLAB (MathWorks, USA). Using radius of curvature, corneal front and back surface asphericity, central corneal thickness, and pupil size (aperture) a bundle of rays was traced through the cornea. Various best focus definitions were tested: a) minimum wavefront error, b) root mean squared ray scatter, c) mean absolute ray scatter, and d) total spot diameter. All 4 target optimisation criteria were tested with each best focus plane. With the best-fit keratometer index the difference of corneal power and keratometric power was evaluated using a multivariate linear model. Results The mean corneal powers for a/b/c/d were 43.02±1.61/42.92±1.58/42.91±1.58/42.94±1.59 dpt respectively. The root mean squared deviations of corneal power from keratometric power (nK = 1.3317/1.3309/1.3308/1.3311 for a/b/c/d) were 0.308/0.185/0.171/0.209 dpt. With the multivariate linear model the respective RMS error was reduced to 0.110/0.052/0.043/0.065 dpt (R² = 0.872/0.921/0.935/0.904). Conclusions Raytracing improves on linear Gaussian optics by considering the asphericity of both refracting surfaces and using Snell’s law of refraction in preference to paraxial simplifications. However, there is no unique definition of best focus, and therefore the calculated corneal power varies depending on the definition of best focus. The multivariate linear model enabled more precise estimation of corneal power compared to the simple keratometer equation.

We analyze the use of two varifocal lenses, with fixed interlens separation, for achieving tunable magnification at a specific throw. Our discussion extends the Hopkins procedure circumscribed to the determination of fixed optical powers in a multilens system. We illustrate our results by presenting the Gaussian optics design of surgical spectacles, which have tunable magnification while generating virtual images with zero throw. We also report novel formulas describing this type of two-lens zoom system, which works without any mechanical compensation.

An incoherent optical detection sensor (often referred to as energy or direct detection sensor) used for remote detection and ranging purposes is a useful tool. While the accuracy and robustness of an incoherent sensor relative to a coherent sensor may be lacking particularly in cluttered environments, it has a place in the world due to its simplicity and performance. With this, a best design approach is sought to meet requirements in a stochastic fashion. In developing the design approach, motivations are borrowed from decades of research in radar systems. This article provides a sensor- or top-level design approach for an incoherent optical detection sensor based mainly on paths developed in radar.

In Gaussian optics all observers experience the same magnification, the instrument’s angular magnification, when viewing distant objects though a telescope or other afocal instruments. However, ana...

The latest generation of Cassegrain radio astronomy antennas is designed for multiple frequency bands with receivers for individual bands offset from the antenna axis. The offset feed arrangement typically has two focusing elements in the form of ellipsoidal mirrors in the optical path between the feed horn and the antenna focus. This arrangement aligns the beam from the offset feed horn to illuminate the subreflector. The additional focusing elements increase the number of design variables, namely the distances between the horn aperture and the first mirror and that between the two mirrors, and their focal lengths. There are a huge number of possible combinations of these four variables in which the optics system can take on. The design aim is to seek the combination that will give the optimum antenna efficiency, not only at the centre frequency of the particular band but also across its bandwidth. To pick the optimum combination of the variables, it requires working through, by computational mean, a continuum range of variable values at different frequencies which will fit the optics system within the allocated physical space. Physical optics (PO) is a common technique used in optics design. However, due to the repeated iteration of the huge number of computation involved, the use of PO is not feasible. We present a procedure based on using multimode Gaussian optics to pick the optimum design and using PO for final verification of the system performance. The best antenna efficiency is achieved when the beam illuminating the subreflector is truncated with the optimum edge taper. The optimization procedure uses the beam’s edge taper at the subreflector as the iteration target. The band 6 receiver optics design for the Atacama Large Millimetre Array (ALMA) antenna is used to illustrate the optimization procedure.

PURPOSE Cardinal points are used for ray tracing through Gaussian systems. Anti-principal and anti-nodal points (which we shall refer to as the anti-cardinal points), along with the six familiar cardinal points, belong to a much larger set of special points. The purpose of this paper is to obtain a set of relationships and resulting equalities among the cardinal and anti-cardinal points and to illustrate them using Pascal's ring. METHODS The methodology used relies on Gaussian optics and the transference T. We make use of two equations, obtained via the transference, which give the locations of the six cardinal and four anti-cardinal points with respect to the system. We obtain equalities among the cardinal and anti-cardinal points. We utilise Pascal's ring to illustrate which points depend on frequency and their displacement with change in frequency. RESULTS Pascal described a memory schema in the shape of a hexagon for remembering equalities among the points and illustrating shifts in these points when an aspect of the system changes. We modify and extend Pascal's ring to include the anti-cardinal points. We make use of Pascal's ring extended to illustrate which points are dependent on the frequency of light and the direction of shift of the equalities with change in frequency. For the reduced eye the principal and nodal points are independent of frequency, but the focal points and the anti-cardinal points depend on frequency. For Le Grand's four-surface model eye all six cardinal and four anti-cardinal points depend on frequency. This has implications for definitions, particularly of chromatic aberrations of the eye, that make use of cardinal points and that themselves depend on frequency. CONCLUSIONS Pascal's ring and Pascal's ring extended are novel memory schema for remembering the equalities among the cardinal and anti-cardinal points. The rings are useful for illustrating changes among the equalities and direction of shift of points when an aspect of a system changes. Care should be taken when defining concepts that rely on cardinal points that depend on frequency.

PURPOSE To show that 14-dimensional spaces of augmented point P and angle Q characteristics, matrices obtained from the ray transference, are suitable for quantitative analysis although only the latter define an inner-product space and only on it can one define distances and angles. The paper examines the nature of the spaces and their relationships to other spaces including symmetric dioptric power space. METHODS The paper makes use of linear optics, a three-dimensional generalization of Gaussian optics. Symmetric 2 × 2 dioptric power matrices F define a three-dimensional inner-product space which provides a sound basis for quantitative analysis (calculation of changes, arithmetic means, etc.) of refractive errors and thin systems. For general systems the optical character is defined by the dimensionally-heterogeneous 4 × 4 symplectic matrix S, the transference, or if explicit allowance is made for heterocentricity, the 5 × 5 augmented symplectic matrix T. Ordinary quantitative analysis cannot be performed on them because matrices of neither of these types constitute vector spaces. Suitable transformations have been proposed but because the transforms are dimensionally heterogeneous the spaces are not naturally inner-product spaces. RESULTS The paper obtains 14-dimensional spaces of augmented point P and angle Q characteristics. The 14-dimensional space defined by the augmented angle characteristics Q is dimensionally homogenous and an inner-product space. A 10-dimensional subspace of the space of augmented point characteristics P is also an inner-product space. CONCLUSIONS The spaces are suitable for quantitative analysis of the optical character of eyes and many other systems. Distances and angles can be defined in the inner-product spaces. The optical systems may have multiple separated astigmatic and decentred refracting elements.

The light–energy coupling during femtosecond laser processing of glass is mediated by non-linear ionization mechanisms through the formation of an electron plasma. Its transient optical properties provide information about the density, temperature and scattering rate of the excited electrons. In turn, these properties strongly condition the features and size of the permanent optical modification near the focal volume that are desirable to fabricate photonic devices, such as optical waveguides inside transparent materials. Here, we report on the spectral response of a fs-laser-induced electron plasma inside fused silica by measuring its transient transmission using a broadband probe. We model the interaction of the probe beam with the plasma by combining Drude–Sommerfeld model with Gaussian optics. In this manner, we take into account both the laser–plasma interaction and the influence of the chromatic aberration inherent to a broadband-based system. We find good agreement between experiments at several processing energies and simulations and provide an estimate of the dielectric function of the excited material.

The multiple regression formulas and correlation of ocular components with refractive errors are presented by Gaussian optics. The refractive error changing rate for the cornea and lens power, the axial length, anterior chamber depth (ACD) and vitreous chamber depth (VCD) are calculated, including nonlinear terms for more accurate rate functions than the linear theory. Our theory, consistent with the empirical data, shows that the Pearson correlation coefficients for spherical equivalent (SE) and ocular components are highest for SE with axial length, ACD and VCD and weakest for corneal power, lens power and lens thickness. Moreover, our regression formulas show the asymmetric feature of the correlation that the axial length, ACD and VCD are more strongly correlated (with higher negative regression constants) with refractive errors in eyes with hyperopia than in eyes with myopia, particularly for severe hyperopia.

Light-field particle image velocimetry (LF-PIV) was recently introduced to measure three-dimensional, three-component velocity field with just a single light-field camera. One of the major challenges lies in the small viewing aperture affecting the depth resolution of such a single-camera based LF-PIV approach. In the present study, we show that this limitation may be mitigated by a dual light-field camera framework, one which includes a novel volumetric calibration model derived from Gaussian optics, a particle intensity reconstruction algorithm based on the multiplicative algebraic reconstruction technique and a post-processing technique for the reconstructed particle intensity field. The proposed approach was firstly validated with synthetic light-field particle images as well as experimental light-field images of five tiny glass beads imitating tracer particles. Secondly, parametric studies were conducted to analyze the influence of the viewing angle and seeding particle density on the reconstruction quality and spatial resolution. In particular, synthetic light-field particle images of a direct numerical simulation jet data set were utilized to compare the performance of single- and dual-camera LF-PIV techniques. Finally, experimental volumetric flow field results of a circular vortex-ring were also measured by single- and dual-camera LF-PIV techniques and compared. It is determined here that an additional light-field camera can mitigate the elongation effects of reconstructed particles and improve the measurement resolution in the depth direction. The authors acknowledge the financial support provided by the National Natural Science Foundation of China (Grant no. 11772197) for the present study. The support of Australian Research Council (ARC) through Discovery and LIEF Grant is also greatly acknowledged.

This work presents a volumetric calibration method for single-camera light-field particle image velocimetry (light-field PIV or LF-PIV). The proposed technique makes use of the unique point-like feature of particle light-field images to accurately determine affected pixels for a spatial voxel over a relative large measurement volume. A calibration model is derived based on Gaussian optics, which relates a spatial point light source with its confusion circle produced on microlens array (MLA), and optical distortions are accounted for by introducing five calibration parameters. By taking lens defects and misalignment between MLA and image sensor into account, the calibration method can calculate weighting coefficient for particle image reconstruction more accurately than the theoretical ray-tracing method, especially for regions further away from focal plane where light ray deflections are significant due to optical distortions. The volumetric calibration method was validated by simulation tests using synthetic light-field images, and has been successfully applied to a classic vortex-ring LF-PIV measurement, where the measurable range in depth direction was successfully extended and the quality of reconstructed volumetric velocity field was greatly improved. MOE (Min. of Education, S’pore) Accepted version

Optrodes are neural probes for delivering light at neurons, at least with an irradiance of 1 mW/mm2. MicroLED (microlight-emitting diode) optrodes with a microlens to collimate the light can improve the irradiance without increasing the microLED power consumption, minimizing the brain overheating. This paper presents FEM (finite element method) simulations of a PDMS (polydimethylsiloxane) microlens capable of collimating the light from a microLED, ensuring an average irradiance of 310 mW/mm2. Based on the simulated microlens, a microlens array was fabricated, using microfabrication processes. One of the PDMS microlens from the fabricated array was experimentally characterized, using AFM (atomic force microscopy) and SEM (scanning electron microscopy). The average surface roughness, base diameter and height of the microlens is 2.3 nm, 80 μm and 55 μm, respectively. It was also possible to determine the microlens focal length (96 μm), numerical aperture (0.42), f-number (1.2) and depth of focus (2.66 μm), using Gaussian optics. The fabrication process was validated, even using PDMS/PDMS demolding without surface treatment of the negative mold., This work is supported by FCT: OpticalBrain, PTDC/CTM-REF/28406/2017 operation code NORTE-01-0145-FEDER- 028406; OCT-RAMAN, PTDC/FIS- OTI/28296/2017 operation code NORTE-01-0145-FEDER-028296; CMEMS-UMinho strategic project UIDB/04436/2020; project of Infrastructures Micro&NanoFabs@PTNORTE- 01-0145-FEDER-022090, PORNorte, Portugal 2020 by FEDER funds through the COMPETE 2020 Programa Operacional Competitividade e Internacionalização (POCI). The work is also supported by RFF Forskningsfond Oslofjordfondet (Project No. 285575 and 296654), National Natural Science Foundation of China (Project No. 61950410621, 61531008 and 11702045). The support from Chongqing Research Program of Basic Research and Frontier Technology (Project No. cstc2016jcyjA0292, cstc2019jcyj-msxmX0776, cstc2017jcyjAX0191, cstc2017jcyjA1842 and cstc2018jcyjA4046), and Chongqing Education Commission – Science and Technology Research Program (Proj. nos. KJZD-K201800802, KJZD-K201900802) are also acknowledged.

Quantifying the microcirculation system is helpful for a more accurate diagnosis of blood flow related diseases. In this work, we report a depth-resolved fractal parameter analysis of human microvasculature obtained with a swept-source optical coherence tomography (SS-OCT) system after depth compensation. In our analysis, the characteristics of Gaussian optics and the attenuation of light intensity with the propagation distance into the tissue were taken into account. To obtain accurate depth-resolved fractal parameter of microvasculature in human skin, a swept-source optical coherence tomography (SS-OCT) system was built. First, three-dimension blood flow datasets were acquired. Then the signals from multiple depth layers of human skin were compensated based on our model, and depth-resolved fractal parameters were extracted finally. The en-face images of the microvasculature of different areas were presented. To the best of our knowledge, for the first time, it was found that the characteristic change of the fractal parameters of the microvasculature of in vivo human skin may be missed due to the light attenuation and the features of Gaussian optics. The depth-resolved fractal analysis of human microvasculature after depth-dependent compensation of imaging signals allows more accurate blood flow related diagnose diseases or monitoring renormalization of the microvasculature after treatment.

Optics technology is being increasingly used in mainstream industrial and research domains such as terrestrial telescopes, biomedical imaging and optical communication. One of the most widely used modeling approaches for such systems is Gaussian optics, which describes light as a beam. In this paper, we propose to use higher-order-logic theorem proving for the analysis of Gaussian optical systems. In particular, we present the formalization of Gaussian beams and verify the corresponding properties such as beam transformation, beam waist radius and location. Consequently, we build formal reasoning support for the analysis of quasi-optical systems. In order to demonstrate the effectiveness of our approach, we present a case study about the receiver module of a real-world Atacama Pathfinder Experiment (APEX) telescope.

The way to generalize scalar to wave optics, thus including polarization in the treatment consistent with the Maxwell equations was shown by Mukunda, Simon and Sudarshan for paraxial systems, based on a group theoretical analysis. Here, the Mukunda–Simon–Sudarshan (MSS) theory for the passage from scalar to vector optics is derived by casting the basic formalism in a framework very similar to the Dirac electron theory. The resulting formalism is suitable for extending the MSS-theory beyond the paraxial approximation.

This work deals with a problem of paraxial analysis of optical systems for transformation of Gaussian beams. General equations describing the paraxial parameters of a two-element zoom system for transformation of a Gaussian beam are derived. It is shown that such a zoom lens has different kinematics with respect to classical zoom system designed for homocentric beams, which depends on the parameters of the transformed Gaussian beam.

This paper reports on the discovery of the most unambiguous generalized vectorial laws of reflection and refraction along with the launching of a new world of geometrical optics with a lot of novel interesting physical insights to optical phenomena.