We study the formation of caustic surfaces produced by convex conic lenses, considering a linear set of point sources displayed on a plane, this linear array is placed at arbitrary position along the optical axis. The caustic surface can be defined as the envelope for either reflected or refracted rays propagating through an optical system. Implementing an exact ray trace, we have obtained an analytic equation that describes a parametric family of refracted rays propagating through a convex conic lens and by computing its envelope, we provide an exact equation for the caustic surface as a function of all the parameters involved in the process of refraction. Considering the geometric center of a conic surface, we have located the parabasal image for each point source produced by refraction, and by extending this concept for a set of point sources placed along a linear array, we obtain the image surface which is the locus where the succession of paraxial images are located. Finally, using a commercial 3D printer, we have fabricated a convex conic lens along with its image curved surface to implement a preliminary test to study the image formation for extended objects, with potential applications in uniform illumination systems.
We study different approaches to describe the evolution of wavefronts refracted through two simple lenses forming an achromat separated by finite distance (dialyte for short), assuming a plane wavefront incident propagated along the optical axis impinging on the optical system. This allows us to quantitatively evaluate the best optical design containing the minimum amount of spherical aberration produced by the optical system under test. Based on this study, we have implemented an interferometric array for testing a dialyte placing a reference mirror at predefined position along the optical axis, which permits to compensate the optical phase and properly produce a null interferogram to evaluate the performance for this kind of optical systems.
Traditionally, the Fresnel-type lens design often assumes thin lenses, since lenses of finite thickness cannot be completely described in analytical terms, then a numerical solution has to be found. We implement an exact ray trace considering a plane wavefront incident by reducing as much as possible the area of light concentration, while increasing the concentration ratio of energy. In such a way that we have mathematically determined a function to compute groove angles, to refract a bundle of rays from each planar echelon, which are propagated into a predetermined area of energy collection. In addition, we can evaluate the performance of the Fresnel-type lens design for a solar collector. Thus, an analytical formula to calculate the optical efficiency is obtained, in terms of the geometrical losses widely studied for nonimaging systems, such as blocking losses. Finally, the proposed aspherical Fresnel lens design is fabricated using a commercial 3D printer and subsequently polishing the surface for better performance. In this fashion we could test the performance of the manufactured prototype.
In this work we propose a procedure to divide a regular off-axis conic surface, which defines the parent surface represented by a parametric equation, in order to describe the optical surface for an off-axis Fresnel-type reflector, in such a way that this mathematical representation allows us to implement an exact ray trace considering a plane wavefront incident on the surface. Additionally, we can simulate the performance of a preliminary prototype proposed for designing a solar collector.
We design two different Hartmann type null screens based on an exact ray trace for testing a fast plano-convex aspherical condenser. The first null screen is designed for testing the external convex surface or periphery area for the condenser by reflection. We have implemented an exact ray trace assuming a point source placed along the optical axis, emitting a bundle of rays, which are reflected by the surface under test, to obtain a non-uniform array of spots, which are printed on paper sheet and wrapped on a plastic cylinder fabricated by using additive manufacture. Subsequently, by reversibility Principle’s after by reflection we obtain a uniform array pattern displayed at the detection plane. Alternatively, to evaluate the whole area for the condenser, the second null screen is designed for testing the central convex area for the condenser by refraction. Thus, we have implemented an exact ray trace assuming an incident plane wavefront, these rays are refracted through the lens under test, to obtain a non-uniform array of drop spots, which are printed on plastic sheet and placed in front of the lens under test. Finally, assuming the reversibility Principle’s after by refraction we obtain a uniform array pattern displayed at the detection plane. For this method, we have called Hartmann type hybrid null screens.
We have designed a Linear Fresnel-type Reflector (LFR) to reduce the area of light concentration based on the caustic surfaces produced by reflection. The LFR is designed by a set of planar mirrors, which appropriately have slopes in such a way that input energy can be focused at predefined absorber area. Also, losses due to riser steps were obtained from a geometrical point of view, to reduce and reconfigure the LFR shape in order to facilitate its manufacture. Finally, a LFR prototype will be fabricated on a single aluminum sheet where their grooves will be molded through CNC machine.
Nowadays the prevention of dementia is a challenge for humanity. There are some preventive intervention programs for dementia, which are mainly based in the modification of multicomponent lifestyles such as: physical and cognitive activity, weight control, metabolic-comorbidity control and social support. Recently, Mind and Movement Program to have Cognitive Health is a collaborative methodological proposal between the countries Mexico, Japan and Canada, which consists of three components: aerobic exercise; aerobic and cognitive exercises, as well as a motivation program. For performing aerobic and cognitive exercises, the monitoring of vital signs in real time is necessary through a statistical analysis of the data of each patient, in such a way that the doctor knows the state of health of the patient. As a consequence of the COVID-19 pandemic, the original program to acquire experimental data underwent modifications. Since the older adults were isolated, they were required to do their physical exercises at home, implementing a remote monitoring system based on a wearable smart band, which was properly developed to monitor the vital signs for each patient. Hence, a personalized quantification of the oxygen saturation and cardiac pressure based on light sensors and pressure sensors, respectively, was measured and monitored in real time. On the other hand, predefined programming based on Artificial intelligence, provides certain advantages for easy handling by the older adults. Currently, we are working along with a hospital, where doctors involved in the program are testing the prototype for the validation of the wearable smart bands.
Nowadays, optical systems commonly use either aspheric or free-form surfaces to improve their performance; however, to ensure that the manufactured surfaces become successful in concordance with to the nominal design, some geometrical parameters such as radius of curvature, conic constant, aspheric coefficients, etc., must be measured, including the surface shape under test. In this work, we propose a simple method to evaluate the optical quality of a plano-convex aspheric lens, where the convex face is modeled as an aspheric or free-form surface. We design a non-uniform pattern on the plane face of the plano-convex lens, to obtain a uniform pattern on a predefined detection plane by using the law of refraction in vector form. Additionally, implementing numerical simulations, we calculate the synthetic images produced through a predefined optical surface that we will use as if they were obtained from an experimental test. Finally, we apply an iterative method to retrieve the shape of the surface by using the normal vector field to demonstrate the feasibility of our proposal.
We present an iterative procedure to retrieve the wavefront using a Shack–Hartmann sensor. Traditionally, a uniform array of microlens is used as a domain to reconstruct the wavefront under test; however, this properly works if the wavefront differs slightly from a plane. But generally in optical tests, astronomy, and ophthalmology the wavefronts under test can have appreciable deviations with respect to a plane wavefront. The proposed method considers the reconstruction of the wavefront deformations with respect to a known reference wavefront. At each iteration, the wavefront deformation is used to find a reference wavefront closer to the wavefront under test and a domain closer to the actual domain. When the values of the wavefront deformations are small enough, we can take the reference as the wavefront under test. In addition, we simulate the centroid positions of the spot pattern used to retrieve the wavefront under test using the proposed method. We compare our results with those obtained by three other different approaching methods described in the literature (Modal, Trapezoidal Rule, and Southwell). For the simulations used in this work, our method retrieves wavefronts closer to the real wavefront than the other methods. Also, we apply the proposed method to an experimental case to reconstruct the wavefront under test using a Shack–Hartmann sensor.
One of the most important components of a wind turbine are the blades, the evaluation of their manufacturing quality and aerodynamic capabilities can be very costly, for this reason a 3D reconstruction by stereo vision is proposed. This technique consists of projecting a laser line in each face of the blade. Using a linear stage, two cameras will scan simultaneously, considering bidirectional disparities and feature correspondences between the two pictures. Two symmetric airfoils of the NACA 0012 family are evaluated. The expected precision is 0.1mm.
We study the propagation of wavefronts refracted through separated doublet lenses (SDL), considering a plane wavefront propagating parallel to the optical axis. We provide formulas for the zero-distance phase front refracted through SDL by using Huygens’s principle. Additionally, we obtain formulae to represent the shape of refracted wavefronts propagated at arbitrary distances along the optical axis, as a function of all parameters involved in the process of refraction. Finally, some examples for commercial SDL showing the evolution of the wavefronts arbitrary distances are presented, assuming different wavelengths for the refractive indices of the lenses, displaying dispersion effects produced through SDL.
We have designed a Linear Fresnel Reflector (LFR), with potential applications for solar concentration, by using an exact ray tracing. We have mathematically parameterized the slopes of LFR to provide predefined areas of light concentration. LFR planar mirrors were calculated in such a way that an incident plane wavefront can be focused at minimum absorber area. Finally, prototypes of LFR were manufactured by using a 3D printer, considering a set of small sized mirrors to join up with the aim of producing a linear focus.
We study the formation of caustic produced by refraction through conic surfaces, considering a point source placed along the optical axis at arbitrary distances from the vertex of refracting surface. We demand that the optical surface is represented by a mathematical function, which is smooth, continuous, and derivable. We implement an exact ray trace to obtain a monoparametric equation that describes a family of refracted rays, which are propagated as a function of the angle of emission from the point source for each ray. Subsequently, by using the envelope’s method, we provide an analytical equation for the caustic surface as a function of all the parameters involved in the process of refraction. We analyze the paraxial approximation assuming a very small angle about the axis of the system. Additionally, we provide a formula, which describes the conditions for total internal reflection. Finally, we present the formation of caustic surfaces by considering a liquid polymer in a rotating vessel forming a paraboloid surface, also we present the caustic surface produced by the package of a Light Emitting Diode, to produce an uniform illumination pattern.
A method for designing afocal achromatic doublet is presented. We have implemented an exact ray trace through a separated doublet lens considering a plane wavefront propagating along the optical axis. The analytic equation of both the caustic surface and the back focal length for separated doublet lenses are provided. Demanding that the back focal length tends to infinity, we impose the conditions to design afocal optical systems, obtaining sixth and fourth degree polynomials as a function of the radii of curvature. In order to produce an afocal achromatic optical system, we solve numerically a set of two nonlinear equations assuming two spectral lines. Therefore, we have two unknowns which are the curvature radii for both the front surface and the rear surface. The contribution of this work is to provide simple formulas for designing optical beam expander or reducer devices based on separated doublets.
We study the caustic surfaces produced by a separated doublet assuming arbitrary distances between two simple lenses in order to design achromatic doublets. We have implemented an exact ray tracing considering a plane wavefront propagating along the optical axis, impinging on the first lens and being refracted through it, subsequently these rays are propagated up to the second lens and finally are refracted outside of the last one obtaining their respective caustic surface. The contribution in this work it is to provide simple formulas of the caustic surface caused by the refraction process through two separated lenses having arbitrary distances.
We study the formation of caustic surfaces produced by simple lenses, considering a point source placed at arbitrary position along the optical axis. It is well known that the shape of the caustic is directly related to the image errors introduced by an optical system, and it can be modified by changing the parameters involved in the process of refraction. Implementing an exact ray tracing, we have obtained an analytic equation that describes a family of rays that are propagated radially from the point source placed at arbitrary distances along the optical axis as a function of an angle of emission of the source. Once obtained the parametric equation that describes the family of all the rays refracted through the singlet lens, we provide an analytical equation for the caustic surface as a function of all the parameters involved in the process of refraction. Additionally, assuming a paraxial approximation, we obain Gauss equation of image forming and some examples are presented.
We study the propagation of wavefronts produced through cemented doublet lenses, considering a plane wavefront propagating parallel to the optical axis. We provide formulas for the zero-distance phase front by using Huygens's principle, also we provide formulae to represent the shape of refracted wavefronts propagated arbitrary distances along the optical axis, which are function of all parameters involved in the process of refraction. We present examples of doublet lenses showing the evolution of the wavefronts arbitrary distances, assuming different wavelengths for the refractive indices of the lenses, for this purpose we compare the dispersion effects produced through this particular kind of lenses.
We design Fresnel mirrors by using an exact ray tracing considering an incident plane wavefront propagating along the optical axis, impinging at arbitrary reflective surfaces, in order to efficiently redirect the light at a predefined area where will be placed the absorber. The solar concentrator consists of a set of planar mirrors, each one has its own slope in such a way that all the rays impinging on it will be focused at a predefined area as a CPC does. Finally, we provide a qualitative test for a commercial FM based on the null screen method.
A method to design Ronchi-Hartmann screens for testing a fast plano-convex aspherical Fresnel lens is presented. We design null screens that produce either aligned straight fringes or quasi-angular spots arrays for observed patterns. The designs of these null screens are based on knowledge of the caustic by refraction through arbitrary curves. A qualitative test for a Fresnel lens is presented.
To measure the quality of optical surfaces, one of the most used methods is the deflectometry. To implement this technique, a screen is used to choose some incident rays on the surface under test. Subsequently, the intersection of the rays is measured, after having passed through the surface, in a detection plane perpendicular to the optical axis. With the coordinates of the points in the detection plane, the normal vectors are determined in each point of the surface under test. The process is simple if the incident rays are chosen in a configuration called null, that is, in the detection plane the measured points will be distributed in a uniform configuration, rectangular, circular, radial, etc. In this work we present the numerical simulations, considering an incident spherical wavefront in the null screen that is placed at an arbitrary position between the source and the flat surface of an aspheric lens that was used in the experimental arrangement. In the simulations it is expected to obtain a uniformly distributed arrangement of spots, which will be compared with the experimental results.
We study the formation of caustic surfaces produced by cemented doublet lenses, considering a plane wavefront propagating parallel to the optical axis and neglecting the refractive index of the optical cement between two simple lenses which are in contact. We have already seen that the shape of caustic surfaces can represent the monochromatic aberrations that we call image errors; furthermore the shape of the caustic can be modified by changing the parameters of design for two conic lenses in contact, in such a way that if we are able to vanish the caustic, the optical system produces either an image without spherical aberration or longitudinal chromatic aberration reduced, commonly known as spherochromatism. The shape of the caustic surface is a function of the indices of refraction involved in the process of refraction, and all the parameters of two simple lenses. We provide an analytic equation for the caustic surfaces in a meridional plane and an example is presented.
One of the most used methods to measure the quality of optical surfaces by light transmission is the de ectometry. To implement this technique, an arbitrary ray selector is used to choose some incident rays. Measuring the intersection of the rays refracted by the surface, in a posterior detection plane perpendicular to the optical axis, the normal at the surface under test, it is determined. Whith measurement points, the shape of the surface is determined by their integration. The process is simple if the incident rays are chosen in a called null configuration, since it is expected that all refracted rays will strike a predetermined ordered array in the detection plane. To numerical calculation we using an ideal surface on this test. In order to full measuare of the surface on simulation, we used vector form of exact ray tracing. To carry out the necessary numerical simulations for a null screen, we considering a point light source and the selector placed in arbitrary position between the point source lens of the test. Some numerical simulations are shown and are compared with experimental results.
KEYWORDS: Solar energy, Prototyping, Current controlled current source, Solar concentrators, Photocatalysis, Ray tracing, Compound parabolic concentrators, Glasses, Optical components, Freeform optics
We study the propagation of light in order to efficiently redirect the reflected light on photocatalytic samples placed inside a commercial solar simulator, and we have designed a small-scale prototype of Cycloidal Collectors (CCs), resembling a compound parabolic collector. The prototype consists of either cycloidal trough or cycloidal collector having symmetry of rotation, which has been designed considering an exact ray tracing assuming a bundle of rays propagating parallel to the optical axis and impinging on a curate cycloidal surface, obtaining its caustic surface produced by reflection.
We study the formation of caustic surfaces produced by bi-conic lenses, considering a plane wavefront propagating parallel to the optical axis. We have already seen that the shape of caustic surfaces can represent the monochromatic aberrations that we call image errors, furthermore the shape of the caustic can be modified by changing the parameters of the lens in such a way that if we are able to vanish the caustic, the optical system produces an image without spherical aberration, alternatively caustic surfaces having a large area could be applied to design non-imaging optical systems, such as diffusers of light. The shape of the caustic surface is a function of the indices of refraction involved in the process of refraction, and all the parameters of the bi-conic lens. We provide an analytic equation for the caustic surface in a meridional plane and some examples are presented.
We have obtained a formula to represent the wavefront produced by a plano-convex aspheric lens with symmetry of revolution considering a plane wavefront propagating parallel to the optical axis and impinging on the refracting surface, it is called a zero-distance phase front, being it the first wavefront to be out of the optical system. Using a concept of differential geometry called parallel curves it is possible to obtain an analytic formula to represent the wavefront propagated at arbitrary distances through the optical axis. In order to evaluate qualitatively a plano-convex aspheric lens, we have modified slightly an interferometer Tywman-Green as follow: In the reference beam we use a plane mirror and the beam of test we have used a spatial light modulator (SLM) to compensate the phase produced by the lens under test. It will be called a null phase interferometer. The main idea is to recombine both wavefronts in order to get a null interferogram, otherwise we will associate the patterns of the interferogram to deformations of the lens under test. The null phase screens are formed with concentric circumferences assuming different gray levels printed on SLM.
In order to evaluate either qualitative or quantitatively the shape of fast plano-convex aspheric lenses, a method to design null screens type Hartmann is proposed. The null screens are formed with non-uniform spots, which allows to have uniform images at detection's plane. The screens are printed on a foil sheet and placed in front of the lens under test, they are illuminated with a collimated monochromatic beam propagating along the optical axis, in such a way that through the process of refraction will form a uniform spot patterns which are recorded at a predefined plane of detection. Finally, processing properly its image recorded we could be able to get a quantitative evaluation of the lens under test. The designs of these null screens are based on the equations of the caustic surface produced by refraction. A preliminary test for a fast plano-convex aspheric lens with F=# = 0:8 is presented in this work. This method could also be applied to alignment of optical systems.
We study the formation of wavefronts produced by smooth arbitrary surfaces with symmetry of revolution considering a plane wavefront propagating parallel to the optical axis and impinging on the refracting surface. The wavefronts are obtained by using the Malus-Dupin theorem and they represent the monochromatic aberrations which can be called image errors, furthermore their shapes could be modified by changing the parameters of the lens in such a way that if a caustic surface is vanished the optical system produces a perfect image, on the other hand for a caustic possessing a large area it could be applied to design non-imaging optical systems. The shape of the wavefront depends only on the indices of refraction and geometrical properties of the refracting surface such as the first derivative and their parameters associated. This analytic formula has potential applications in the microscopy field, illumination or corrector plates.
We study the formation of caustic produced by smooth arbitrary surfaces considering a plane wavefront propagating parallel to the optical axis and impinging on the refracting surface. We have already seen that the shape of the caustic surface can represent the monochromatic aberrations that we call image errors, furthermore the shape of the caustic can be modified by changing the parameters of the lens in such a way that if the caustic surfaces is vanished the optical system produces a perfect image, on the other hand for a caustic possessing a large area it could be applied to design no-imaging optical systems. The shape of the caustic depends only on the indices of refraction involved in the process of refraction, the refracting surface which is formed by smooth arbitrary plano-convex lens. We provide an analytic equation for the caustic surface after refraction of a plane wave from every rotationally symmetric surface..
In order to evaluate either qualitative or quantitatively the shape of fast plano-convex aspheric lenses, a method to design null screens type sub-structured Ronchi is proposed. The null screens are formed with nonuniform curves which allows us to have both thin and thick monochrome strips between contiguous curves. The screens are printed on a light transmission modulator and placed in front of the lens under test, they are illuminated with a collimated monochromatic beam propagating along the optical axis, in such a way that through the process of refraction will form a uniform straight fringes pattern which are recorded at a predefined plane of detection, finally processing its image recorded we could be able to get a quantitative evaluation of the lens under test. The designs of these null screens are based on the equations of the caustic surface produced by refraction. The null screens can be printed in gray levels on a light transmission modulator depending on the applied voltage on it. A preliminary test for a fast plano-convex aspheric lens with F=# = 0:8 is presented in this work. This method also could be applied to alignment of optical systems.
We report the qualitative testing of a spherical concave surface with deformation coefficients based on the null-screen principles. The design of cylindrical null screen with curved grid is described; its image, which is formed by reflection on the test surface, becomes an exact square grid if the surface is perfect. Any departure from this geometry is indicative of defects on the surface. In contrast to others tests, here the whole surface is tested at once. The surface under test is 140 mm in diameter and a radius of curvature of 97 mm. The surface is testing during the manufacturing process; departures from the design surface will be analyzed and discussed.
In order to obtain a clearer interpretation of the Intensity Transport Equation (ITE), in this work, we propose an algorithm to solve it for some particular wavefronts and its corresponding intensity distributions. By simulating intensity distributions in some planes, the ITE is turns into a Poisson equation with Neumann boundary conditions. The Poisson equation is solved by means of the iterative algorithm SOR (Simultaneous Over-Relaxation).
A method to design Ronchi-Hartmann null screens for improved alignment in the testing of fast concave conic mirrors is presented. The designs of these null screens are based on knowledge of the caustic by reflection.
We provide analytic formulas for fews aspheric terms either plano-convex or convex-plano aspheric lenses. These
formulas are obtained considering an expansion in Taylor's series from exact caustic equation produced by
aspheric lenses. A comparison between our method and numerical methods of design are presented, showing a
well agreement in order to reduce the spherical aberration.
KEYWORDS: Microwave radiation, Near field, Resonators, Dielectrics, Gold, Microscopy, Near field optics, Signal detection, Statistical analysis, Capacitive coupling
In this work we present near field microwave images of microelectronic circuits and their interpretation to
complement the conventional optical analysis. We show a highly simplified design of a resonant probe with
dynamically tunable capacitive coupling and with high sensitivity. Images were obtained by measuring the
microwave reflection coefficient operating a 7 GHz. This design represents a simplified and highly effective
approach to implementing near field microwave microscopy.
We derive simple formulas for the caustic produced by a positive convex-plano and plano-convex conic lens
by considering a plane wave incident on the lens along the optical axis. By using these equations a paraxial
approximation for the caustics are provided in both configurations. Also, by using these equations it is possible
to obtain the third order coefficient of spherical aberration. Changing the parameters of the lens (refraction
index, conic constant, radius of curvature, thickness of the lens, etc.) we can modify the shape of the caustic,
furthermore there are cases where the spherical aberration changes from positive to negative when we vary
exclusively the conic constant. A formula for the Principal Surface as a function of the height also is given. We
believe that the method to obtain the caustic that we report is straightforward, obtaining a relationship between
caustics, wavefronts, and measurements of the spherical aberrations.
A new design of a null Hartman's screen to test quantitatively a fast plano-convex conic lens is presented. The
design of the null screen is based on the caustic produced by refraction through the lens. Additionally, the null
screen can be used to improve the alignment in optical systems. A quantitative evaluation of medium precision
by using a trapezoidal integration method is presented.
In this work we report a method for testing a parabolic trough solar collector (PTSC) based on the null screen
principles. For surfaces with symmetry of revolution a cylindrical null screen is used, now, for testing the PTSC we
use a flat null screen. The design of the null screen with ellipsoidal spots is described; its image, which is formed by
reflection on the test surface, becomes an exact square array of circular spots if the surface is perfect. Any departure
from this geometry is indicative of defects on the surface. The flat null screen design and the surface evaluation
algorithm are presented. Here the surface is tested in sections and the evaluation of the shape of the surface is
performed with stitching method. Results of the evaluation for a square PTSC with 1000 mm by side (F/0.49) are
shown.
We extend the principles of the null screen method for testing a fast ellipsoidal concave mirror by designing a null cylindrical screen, located around the near focal point of the ellipsoid (being parallel to the optical axis) and the observing CCD camera is near the far focal point. We present the formulae to design the screen in such a way that the image on the CCD is a perfect square grid; the departures of the surface from a perfect ellipsoidal shape are observed as deformations of the grid in the image. We show qualitative experimental results.
The generalized ray tracing for the extraordinary ray through uniaxial crystals developed by M. Avendanyo-Alejo and O. Stavroudis, is applied to investigate the optical path difference between the ordinary and the extraordinary rays in a plane parallel uniaxial plate. When a ray of light from a monochromatic source S is incident on the surface of a plane parallel uniaxial plate, two rays: an ordinary ray and an extraordinary ray will propagate inside the plate until they are refracted at the second interface of the plate. These two rays are orthogonally polarized so they do not interfere unless a polarizer is placed after the plate to make the parallel components of their respective electric fields interfere. In the present work we analyze the optical path difference traversed by the ordinary ray and the extraordinary ray.
We propose the design of tilted null screens in order to test the off-axis segments of conic surfaces. Furthermore we reduce the size of the screen in order to increase the performance of the test. The sensitivity is increased while the size of the screen is reduced in the saggital caustic region and vice versa in the tangential caustic region. Further analysis and experimental results are presented, for an off-axis concave parabolic mirror which has an elliptical aperture, with a distance offset Xc=25.4mm, the radius of curvature at the vertex R= 20.4mm, major axis of the mirror DM=49.4mm and minor axis Dm=29.5mm.
In early published contributions we obtain ray tracing formulas for uniaxial crystals. In the present contribution we show the ray tracing for an uniaxial cartesian oval, when the crystal axis lies in the incident plane. The uniaxial crystals have two refracted rays (ordinary and extraordinary rays). When the object is placed at infinity, the ordinary rays have a common focus called the distal focus. For the extraordinary ray the uniaxial cartesian oval show spherical aberration produced by the medium itself. In order to reduce the spherical aberration we analyze several cases for the orientation of the crystal axis, this improve the performance on this optical design.
Ray tracing formulas in a plane-parallel uniaxial plate bounded by an isotropic medium is analyzed when the crystal axis lies in the incident plane, and when its orientation is arbitrary. We present the behavior of the critical angle for the extraordinary ray as a function of the crystal axis position with respect to the normal to the refracting surface.
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