Paraxial Ray Tracing Research Articles

Evaluation of corneal power from an AS-OCT thick lens model and ray tracing: reliability of the keratometer index

Purpose: To investigate and compare different strategies of corneal power calculations using keratometry, paraxial thick lens calculations and ray tracing. Setting: Tertiary care center. Design: Retrospective single-center consecutive case series. Methods: Using a dataset with 9780 eyes of 9780 patients from a cataractous population the corneal front (Ra/Qa) and back (Rp/Qp) surface radius/asphericity, central corneal thickness (CCT), and entrance pupil size (PUP) were recorded using the Casia 2 tomographer. Beside keratometry with the Zeiss (PKZ) and Javal (PKJ) keratometer index, a thick lens paraxial formula (PG) and ray tracing (PR) was implemented to extract corneal power for pupil sizes from 2 mm to 5 mm in steps of 1 mm and PUP. Results: With PUP PKZ/PKJ overestimates the paraxial corneal power PG in around 97%/99% of cases and PR in around 80% to 85%/99%. PR is around 1/6 or 5/6 diopters (D) lower compared with PKZ or PKJ. For a 2 mm pupil PR is around 0.20/0.91 D lower compared with PKZ/PKJ and for a 5 mm pupil PR is comparable with PKZ (around 0.03 D lower) but around 0.70 to 0.75 D lower than PKJ. Conclusions: “True” values of corneal power are mostly required in lens power calculations before cataract surgery, and overestimation of corneal power could induce trend errors in refractive outcome with axial length and lens power if compensated with the effective lens position.

Calculation and Analysis of Key Parameters of Underwater Optical Imaging System.

When photographing objects underwater, it is important to utilize an optical window to isolate the imaging device from the water. The properties of the entire imaging system will change, and the imaging quality will decrease due to the refraction impact of the water and the window. The theoretical calculation method for air imaging is no longer relevant in this context. To analyze the unique rule, this research derives the formulas for key parameters of underwater imaging systems under paraxial circumstances. First, the optical window is modeled, then the formula for the optical window's focal length in the underwater environment is derived, and the change rule for the focal length of various window forms underwater is condensed. For the ideal imaging system using a domed optical window, the equivalent two-optical group model of the imaging system is established, and the formula for calculating the focal length, working distance, and depth of field of the underwater imaging system is derived through paraxial ray tracing. The accuracy of the formula is verified through the comparative analysis of the formula calculation results and the Zemax modeling simulation results. It provides an important theoretical basis for the in-depth study of underwater imaging technology.

A szemnyomás változásának hatása a szaruhártya és a szem törőerejére a Liou–Brennan-modellszem alapján

It is a clinical experience that with significantly higher intraocular pressure, glass ordering may be inaccurate, therefore, it should be performed after adjustment of intraocular pressure. To analyze the effect of intraocular pressure on the refractive power of the cornea and of the eye, using geometrical properties of the Liou-Brennan model eye and finite element modeling. Using corneal parameters of the Liou-Brennan model eye, a parametric geometrical model and with these data finite element modelling, using the Ansys (Ansys, Canonsburg, PA, USA) program, has been performed. Corneal tissue properties have been defined based on trilinear material model of Wollensak and boundary conditions based on Pandolfi. The effect of changes in intraocular pressure on geometrical corneal properties with optical impact (corneal curvature at the anterior and posterior corneal surface at the optical center and axial displacement of the anterior and posterior corneal surface points at the optical center) have been analyzed. Based on the geometric characteristics obtained, using the paraxial ray tracing method, changes in the optical properties of the cornea and the eye have been calculated. With increasing intraocular pressure load, anterior and posterior central corneal radius of curvature decreased and corneal surface points at the optical center at the anterior and posterior surface moved forwards, the cornea thinned. Axial displacement of corneal volume points was maximal for all pressure load values at the optical center of the posterior corneal surface. The refractive power of the cornea decreased as eye pressure increased, while the refractive power of the entire eye increased. We verified using the Liou-Brennan model eye and finite element modelling that a change of the intraocular pressure load between 10 and 30 mmHg may result in 0.31-0.86 mm decrease of the corneal radius of curvature, in 1.5-4.8 D increase in the keratometric values and in about 1.2-3.9 D increase in the total optical power of the eye. A decrease of 0.14 D to 0.69 D has been measured in the corneal optical power in the examined eye pressure range. Orv Hetil. 2022; 163(47): 1872-1879.

Practical technique for balancing the optical aberrations among conceptual thin-lens components of optical zoom lenses.

Solving conceptual thin-lens components with balanced optical aberrations among different optical configurations is essential for lens design from scratch. This paper presents a general and practical method for building conceptual thin-lens components that can be further optimized by Code V (a well-known optical design program) to approach specific aberration targets. The conceptual component consists of a Code V's lens module and a hologram with zero diffraction order. The latter provides spaces for storing the reference data of incident marginal and chief rays, spherical aberration, central coma, and longitudinal chromatic aberration without affecting the paraxial ray tracing. Those reference data are used to evaluate the real aberrations at different optical configurations by our earlier aberration variation algorithms [Appl. Opt.55, 10363 (2016)APOPAI0003-693510.1364/AO.55.010363]. The model helps investigate the properties of lenses and solve the balanced aberrations by the optimization process. An example of solving a zoom lens with different optimization goals is given to demonstrate the usage of the method. Detailed design data are listed for verification.

Effects of optic nerve head-related parameters on retinal vessel calibers measurement results on fundus photographs

BackgroundAlthough relationship between the retinal vessel caliber (RVC) and glaucoma is well known, there has been a paucity of information on the effects of glaucoma-related optic nerve head (ONH) structural factors on the RVC. Information on this relationship should be useful in studying possible roles of ocular circulation in the development and progression of glaucoma.MethodSubjects were from Kumejima Study participants aged 40 years and older in Kumejima, Japan. Normal subjects and eyes were defined according to standardized systemic and ocular examinations. The central retinal artery and vein equivalents (CRAE and CRVE) were determined on fundus photographs by correcting the magnification using the refractive error, corneal curvature, and axial length (AL) of an individual eye and paraxial ray tracing; the ONH structural parameters were determined by planimetry.ResultsIn a total of 558 right eyes (558 normal subjects), aged 49.0 ± 7.1 (standard deviation) years with gradable photographs and planimetric results, CRAE averaged 136.1 ± 12.3 μm and CRVE 216.9 ± 17.4 μm. After adjustment for the effects of confounding factors in multivariate analysis, the AL (P < 0.001), rim area (P = 0.019), disc area (P = 0.042), and smoking (P = 0.035–0.043) correlated positively, and the mean blood pressure (P < 0.001) negatively with CRAE; the AL (P < 0.001), rim area (P = 0.001), disc area (P = 0.005), smoking (P < 0.001), and male sex (P = 0.013) correlated positively, and the β-peripapillary atrophy (β-PPA) area (P = 0.044), vertical Cup/Disc ratio (v-C/D) (P = 0.035), and age (P < 0.001) negatively with CRVE.ConclusionThe current study showed significant effects of rim area, v-C/D or β-PPA area determined on the photographs on the RVC measurement results. Further, it showed a necessity to incorporate the glaucoma-related ONH structural parameters as co-variables to correctly estimate the effects of various factors on the RVC.

Analysis and calculation of the fill factor for microlens array scanning systems.

Microlens array (MLA) scanning systems are used in numerous applications, such as laser radar and optical communications. Excrescent light generated by overlapping divergent beams from adjacent microlenses increases with scanning angle, thereby reducing the relative aperture and affecting the MLA system resolution. The fill factor is hence proposed to characterize the relative aperture, and its effects on the modulation transfer function are analyzed. The fill factor is calculated from matrix optics paraxial ray tracing using the system parameters. The simulated results of the MLA system and theoretical calculations are in good agreement. This work can benefit the design and evaluation of MLA systems.

Numerical geometric acoustics

We present a new approach to offline room acoustic modeling and simulation based on solving the eikonal equation, a nonlinear partial differential equation describing the arrival times of a high-frequency wavefront. We call the approach numerical geometric acoustics, since it combines some of the advantages of both numerical acoustics, where the acoustic wave equation or the Helmholtz equation are solved, and geometric acoustics, typically based on a variety of raytracing techniques. In our approach, reflection and diffraction effects are determined from boundary conditions connecting different eikonal problems generated by a recursion. This allows us to compute the multipath eikonal, a multivalued function parametrizing the arrival times of all acoustic rays propagating throughout a scene. Each branch of the multipath eikonal is computed by numerically solving the eikonal equation using a recently developed high-order semi-Lagrangian direct solver. This approach naturally encompasses spatially varying sound speeds. Our solver is compact and sufficiently high-order to allow us to transport the amplitude prefactor directly using paraxial raytracing. We discuss how to compute the acoustic parameter fields typical of precomputed room acoustic simulations used for virtual reality or games.

Development of a new intraocular lens power calculation method based on lens position estimated with optical coherence tomography

A new method is developed and validated for intraocular lens (IOL) power calculation based on paraxial ray tracing of the postoperative IOL positions, which are obtained with the use of anterior segment optical coherence tomography. Of the 474 eyes studied, 137 and 337 were grouped into training and validation sets, respectively. The positions of the implanted IOLs of the training datasets were characterized with multiple linear regression analyses one month after the operations. A new regression formula was developed to predict the postoperative anterior chamber depth with the use of the stepwise analysis results. In the validation dataset, postoperative refractive values were calculated according to the paraxial ray tracing of the cornea and lens based on the assumption of finite structural thicknesses with separate surface curvatures. The predicted refraction error was calculated as the difference of the expected postoperative refraction from the spherical-equivalent objective refraction values. The percentage error (within ±0.50 diopters) of the new formula was 84.3%. This was not significantly correlated to the axial length or keratometry. The developed formula yielded excellent postoperative refraction predictions and could be applicable to eyes with abnormal proportions, such as steep or flat corneal curvatures and short and long axial lengths.

Roadmap for geometrical optics based on Taylor series expansion of skew ray.

The skew ray R¯n on the image plane of an optical system possessing n boundary surfaces has the form of an n-layered deep composite function. It is hence difficult to evaluate the system performance using ray tracing alone. The present study therefore uses the Taylor series expansion to expand R¯n with respect to the source ray variable vector. It is shown that the paraxial ray tracing equations, point spread function, caustic surfaces and modulation transfer function can all be explored using the first-order expansion. Furthermore, the primary and secondary ray aberrations of an axis-symmetrical system can be determined from the third- and fifth-order expansions, respectively. It is thus proposed that the Taylor series expansion of the skew ray serves as a useful basis for exploring a wide variety of problems in geometrical optics.

Non-orthogonal beam coordinate system wave propagation and reverse time migration

Abstract Grid size has a significant influence on the computation efficiency and accuracy of finite-difference seismic modeling and can change the workload of reverse time migration (RTM) remarkably. This paper proposes a non-orthogonal analytical coordinate system, beam coordinate system (BCS), for the solution of seismic wave propagation and RTM. Starting with an optical Gaussian beam width equation, we expand the representation on vertically variable velocity media, which is the most common scenario in seismic exploration. The BCS based on this representation can be used to implement an irregular-grid increment finite-difference that improves the efficiency of RTM. Based on the Laplacian expression in Riemannian space, we derive the wave equation in the BCS. The new coordinate system can generate an irregular grid with increment increasing vertically along depth. Through paraxial ray tracing, it can be extended to non-analytical beam coordinate system (NBCS). Experiments for the RTM on the Marmousi model with the BCS demonstrate that the proposed method improves the efficiency about 52% while maintaining good image quality.

Derivation of conventional formula of the third-order aberration for off-axial optical system

A formula of the third-order aberration based on the primary aberration expansion for an off-axial optical system is investigated. The point eikonal analysis and the coordinate system using the surface normal vector lead to a simple wavefront-based expression. Three-ray system employing the central, the principal and an arbitrary ray is examined for the derivation. The rays are forced to reach from the last surface to a virtual image point. Therefore, the contradiction of the optical path lengths brings the wavefront aberration. The extracted third-order aberrations are not strictly the same expressions as the conventional formula but show clearly the features of the five kinds of the primary aberrations along with the term originated by the off-axial layout. The relation between the second-order term and the expanded gaussian matrix for paraxial ray tracing and a term related to field curvature are discussed for off-axial systems.

Intraocular lens power calculation using a Placido disk–Scheimpflug tomographer in eyes that had previous myopic corneal excimer laser surgery

To assess the accuracy of different methods to calculate the intraocular lens (IOL) power from measurements obtained with an optical biometer combined with a Placido disk-Scheimpflug tomographer (Galilei G6) in eyes that had previous myopic excimer laser surgery. G.B. Bietti Foundation, Rome, Italy. Prospective case series. When prerefractive surgery data were available, the following methods were evaluated: the Barrett True-K, total corneal power 1 (TCP 1) with double-K SRK/T, TCP with double-K Holladay 1, as well as the Masket and Savini methods. In all eyes, these "no-history" methods were studied: Barrett True-K No-History, TCP 1 using the American Society of Cataract and Refractive Surgery calculator, a ray-tracing program, and the Shammas No-History formula. In eyes with historical data (n=15), the Masket method, Savini method, and TCP 1 with double-K SRK/T produced the highest percentage of cases with an absolute prediction error in refraction of 0.50 diopters (D) or less and the lowest median absolute error (0.32 D, 0.28 D, and 0.25 D, respectively). In the whole cohort (N=22), the most accurate no-history method was paraxial ray tracing, which had the highest percentage of eyes with an absolute prediction error of 0.50 D or less and the lowest median absolute error of 0.31 D. Different solutions can provide accurate IOL power calculations when using the measurements obtained with the optical biometer combined with a Placido-dual Scheimpflug tomographer.

Extending the depth of field with chromatic aberration for dual-wavelength iris imaging.

We propose a method of extending the depth of field to twice that achievable by conventional lenses for the purpose of a low cost iris recognition front-facing camera in mobile phones. By introducing intrinsic primary chromatic aberration in the lens, the depth of field is doubled by means of dual wavelength illumination. The lens parameters (radius of curvature, optical power) can be found analytically by using paraxial raytracing. The effective range of distances covered increases with dispersion of the glass chosen and with larger distance for the near object point.

Design and implementation of optical system for Placido-disc topography

Corneal topography provides powerful support in the diagnosis and treatment of corneal disease by displaying the corneal surface topography in data or image format. To realize the precise detection of corneal surface topography, an optical system for the corneal topography that is based on a Placido disc is designed, which includes a ring distribution on a Placido disc, an imaging system and a collimating illumination system. First, a mathematical model that is based on the corneal topography working principles is established with MATLAB to determine the distribution of white-and-black rings on the Placido disc, in which the ellipsoid facial rings-target of the Placido disc is utilized. Second, the imaging lens structure is designed and optimized by Zemax software. Last, the collimating illumination lens structure is designed by paraxial ray trace equations. The quality of the corneal topography, which is based on our designed optical system, is evaluated. The high-contrast image of uniformly distributed white-and-black rings is observed through the CCD camera. Our optical system for the corneal topography has high precision, with a measuring region of the cornea with a diameter of approximately 10 mm. Therefore, the creation of this optical system offers guidance for designing and improving the optical system of Placido-disc topography.

Enhancement in Seismic Imaging using Diffraction Studies and Hybrid Traveltime Technique for PSDM

The accurate migration of seismic data is conditional on the parameters which are nominated. The effective velocity used in residual processing for migration is small compared to the original migration velocity. Considering traveltime computation is a significant part of seismic imaging algorithms. Conventional implementation of Kirchhoff migration is essential for precomputing a traveltime table from the categories involving traditional ray-tracing methods and finite difference eikonal solvers. In this paper, we examine the accuracy using, the eikonal solver and paraxial ray tracing traveltime computation in pre-stack Kirchhoff depth migration. This hybrid traveltime technique can be applied to a variety of problems related to faults, fractures, and complex region. To evaluate the relevance of this identical traveltime technique, we applied on a Marmousi data set.

Relative location of microseismicity

Applying the two-point paraxial ray tracing, we develop a technique for relative location of microseismic events. Our technique assumes the availability of a perforation shot or an already located microseismic event, termed the master, for which the paraxial ray tracing has been performed. The ray-tracing output for the master makes it possible to compute the relative locations of adjacent microseismic events, as many as a data set contains, with an efficient algorithm that requires no additional ray tracing and reduces to solving a series of simple, low-dimensional and well-behaved optimization problems. The relative event-location approach discussed in our paper is especially well suited for surface microseismic monitoring because the high accuracy of the paraxial ray approximation in the directions orthogonal to the reference rays, typically spanning the stimulated horizons for surface microseismic geometries, ensures the calculation of precise event hypocenters at appreciable distances from the master. Also, since our computations operate with differences of the recorded times of microseismic events rather than with the times themselves, inaccuracies in static corrections for surface receiver stations are largely eliminated. We test the relative location technique on synthetic and field microseismic data to demonstrate its accuracy, computational efficiency, and insensitivity to velocity errors.

How to calculate toric intraocular lenses in clinical practice?

SummaryThere are several options for calculation of toric intraocular lenses, some are based on the classical intraocular lens calculation formulas and others utilize paraxial or numerical ray tracing techniques. Most online calculators use a two‐step system based on precalculated data for the equivalent monofocal lens. Other methods calculate intraocular lens power separately for both cardinal meridians. Advanced methods directly calculate spherocylindrical power by matrix or ray tracing techniques. Especially when considering superposition of regular and irregular astigmatic components, postoperative implant rotation or anisometropia and aniseikonia, the classical formula approach might be inadequate for estimating toric intraocular lens power. Therefore surgeons dealing with toric intraocular lenses require deeper knowledge on advanced calculation techniques to improve patient satisfaction.

Modelling high resolution Echelle spectrographs for calibrations: Hanle Echelle spectrograph, a case study

We present a modelling scheme that predicts the centroids of spectral line features for a high resolution Echelle spectrograph to a high accuracy. Towards this, a computing scheme is used, whereby any astronomical spectrograph can be modelled and controlled without recourse to a ray tracing program. The computations are based on paraxial ray trace and exact corrections added for certain surface types and Buchdahl aberration coefficients for complex modules. The resultant chain of paraxial ray traces and corrections for all relevant components is used to calculate the location of any spectral line on the detector under all normal operating conditions with a high degree of certainty. This will allow a semi-autonomous control using simple in-house, programming modules. The scheme is simple enough to be implemented even in a spreadsheet or in any scripting language. Such a model along with an optimization routine can represent the real time behaviour of the instrument. We present here a case study for Hanle Echelle Spectrograph. We show that our results match well with a popular commercial ray tracing software. The model is further optimized using Thorium Argon calibration lamp exposures taken during the preliminary alignment of the instrument. The model predictions matched the calibration frames at a level of 0.08 pixel. Monte Carlo simulations were performed to show the photon noise effect on the model predictions.

Evaluation of Geometrical Modulation Transfer Function in Optical Lens System

This paper presents ray tracing algorithms to evaluate the geometrical modulation transfer function (GMTF) of optical lens system. There are two kinds of ray tracings methods that can be applied to help simulate the point spread function (PSF) in the image plane, for example, paraxial optics and real ray tracings. The paraxial optics ray tracing is used to calculate the first-order properties such as the effective focal length (EFL) and the entrance pupil position through less cost of computation. However, the PSF could have a large tolerance by only using paraxial optics ray tracing for simulation. Some formulas for real ray tracing are applied in the sagittal and tangential line spread function (LSF). The algorithms are developed to demonstrate the simulation of LSF. Finally, the GMTF is evaluated after the fast Fourier transform (FFT) of the LSF.

Geometry-invariant GRIN lens: finite ray tracing.

The refractive index distribution of the geometry-invariant gradient refractive index lens (GIGL) model is derived as a function of Cartesian coordinates. The adjustable external geometry of the GIGL model aims to mimic the shape of the human and animal crystalline lens. The refractive index distribution is based on an adjustable power-law profile, which provides additional flexibility of the model. An analytical method for layer-by-layer finite ray tracing through the GIGL model is developed and used to calculate aberrations of the GIGL model. The result of the finite ray tracing aberrations of the GIGL model are compared to those obtained with paraxial ray tracing. The derived analytical expression for the refractive index distribution can be employed in the reconstruction processes of the eye using the conventional ray tracing methods. The layer-by-layer finite ray tracing approach would be an asset in ray tracing through a modified GIGL model, where the refractive index distribution cannot be described analytically. Using the layer-by-layer finite ray-tracing method, the potential of the GIGL model in representing continuous as well as shell-like layered structures is illustrated and the results for both cases are presented and analysed.