Generalized Ray Tracing With Basis Functions for Tomographic Projections

This thesis proposes a method to evaluate and quantify in a precise way the peripheral refraction induced by an ophthalmic lens. The motivation for this work stems from the progression of myopia and its possible causes; two of them are particularly key for this PhD: (a) Peripheral refraction of the eye may have an important role in the progression of myopia and (b) ophthalmic lenses are the compensating element more used in children and teenagers. These two elements fully justify the need for a reliable method to quantify the induced peripheral refraction by an ophthalmic lens. This method is based on two pillars: first, it must accurately assess the design of the ophthalmic lens and, second, it should consider what peripheral refractive pattern is acting, that is, without compensating element. The proposed method takes the advantages provided by the ray tracing strategies used in the classic design of ophthalmic lenses but applying them in parallel with amendments to evaluate the peripheral refraction. Thus, the simple scheme used in the classic design of ophthalmic lenses containing a remote sphere and a small aperture at the center of rotation of the eye becomes a scheme where the retina conjugate surface (RCS) and the nodal point of the eye play equivalent roles. In our case, the reference for ray tracing is the nodal point of the eye and the reference for measuring the induced peripheral refraction is the RCS. Ray tracing is based on a finite ray tracing (FRT) from the image space to the object space and on a generalized ray tracing (GRT) from object space to image space. Both have been implemented in a Matlab program and validated to provide a powerful tool for our purpose. GRT allows a quick and accurate assessment of the oblique astigmatism, ie the tangential and sagittal focal lens, in wide field of view considering accurately the lens design. This considers that each ray has a small wavefront associated traveling perpendicular to it. By GRT we are able to know how the wavefront shape changes when is propagated and refracted. Therefore, it is mandatory to have a locally description of the geometry of both the wavefront and the refractive surface at the point where the ray arrives to the refractive surface. This local description is determined by the normal and by the principal curvatures and directions of these surfaces at the point of interest; they can be obtained from a parametric description of the surface and then using Gaussian fundamental forms. This ray tracing procedure has been developed for the general case of any geometry to the surfaces of the ophthalmic lens and has been detailed for the case of an astigmatic lens. For calculating the induced peripheral refraction, a surface is modeled reflecting the peripheral refractive initial values before entering the lens; this is the aforementioned RCS. Two methods have been proposed to model this RCS. One is based on the trends observed in the different studies and uses three-dimensional surfaces power vectors associated with peripheral refraction. The second method uses experimental measurements obtained along four meridians of the retina to interpolate a surface. The expression of these surfaces by power vectors can easily be combined with the results obtained by tracing rays through the lens for the calculation of the induced peripheral refraction. We present in this manuscript some specific examples of how variations on the lens geometry modified the induced peripheral refraction. This opens up the possibility of custom designs ophthalmic lenses to prevent the progression of myopia.

A method for general implementation in any software platform of the generalized Coddington equations is presented, developed, and validated within a Matlab environment. The ophthalmic lens design strategy is presented thoroughly, and the basic concepts of generalized ray tracing are introduced. The methodology for ray tracing is shown to include two inter-related processes. Firstly, finite ray tracing is used to provide the main direction of propagation of the considered ray at the incidence point of interest. Afterwards, generalized ray tracing provides the principal curvatures of the local wavefront at that point, and its orientation after being refracted by the lens. The curvature values of the local wavefront are interpreted as the sagital and tangential powers of the lens at the point of interest. The proposed approach is validated using a double-check of the calculated lens performance in the spherical lens case: while finite ray tracing is validated using a commercial ray tracing software, generalized ray tracing is validated using a software application for ophthalmic lens design based on the classical version of Coddington equations. Equations of the complete tracing process are developed in detail for the case of generic astigmatic ophthalmic lenses as an example. Three-dimensional representation of the sagital and tangential powers of the ophthalmic lens at all directions of gaze then becomes possible, and results are presented for lenses with different geometries.

X - Generalized Ray Tracing

Generalized ray tracing is an algorithm for calculating the geometrical parameters of a wave front in the neighborhood of a traced ray. These calculations are applied, surface by surface, for each traced ray, to an optical system being designed. These calculations determine the two points of contact of each traced ray with the two sheets of a caustic surface. The caustic surfaces are, in fact, aberrated three-dimensional images of object points and therefore contain all information on the geometrical aberrations of the subject lens. Generalized bending is a procedure in which the curvature of a pair of adjacent spherical refracting surfaces, their separation, and the distance to the next succeeding or next preceding surface may be changed so that any paraxial ray is left invariant except at the two affected surfaces. In this study we show that the displacement of a caustic point caused by a generalized bending is in essentially a straight line, that the direction of the displacement is determined by which refracting surfaces are selected, and that the magnitude of the displacement is proportional to the logarithm of the bending parameter. This suggests that caustic surfaces can be used as a merit function in the optical design process, that the merit functions can be calculated by means of generalized ray tracing, and that generalized bending provides an effective means of optimizing the design when included in a feedback loop.

Generalized ray tracing allows for propagation in a slowly-varying medium changing in space and time. This is relevant in media which have velocities comparable with the wave velocity as is the case for solar wind and magnetosheath MHD waves. We show that the cumulative error in the phase is small. Even in a medium changing by 30% per wavelength, the phase error is less than 1/20 radian per wavelength. Thus the method is useful in considering Pc3 interference effects in the magnetosphere. Results of a number of computatuions are presented to illustrate the use of the method.

Recently, freeform optics has been widely used due to its unprecedented compactness and high performance, especially in the reflective designs for broad-wavelength imaging applications. Here, we present a generalized differentiable ray tracing approach suitable for most optical surfaces. The established automated freeform design framework simultaneously calculates multi-surface coefficients with merely the system geometry known, very fast for generating abundant feasible starting points. In addition, we provide a "double-pass surface" strategy with desired overlap (not mutually centered) that enables a component reduction for very compact yet high-performing designs. The effectiveness of the method is firstly demonstrated by designing a wide field-of-view, fast f-number, four-mirror freeform telescope. Another example shows a two-freeform, three-mirror, four-reflection design with high compactness and cost-friendly considerations with a double-pass spherical mirror. The present work provides a robust design scheme for reflective freeform imaging systems in general, and it unlocks a series of new 'double-pass surface' designs for very compact, high-performing freeform imaging systems.

Oblate drops of water can produce caustics where, unlike a simple Airy caustic, more than two rays merge. We extend previous treatments of generalized primary rainbows based on catastrophe optics [Opt. Lett. 10, 588 (1985); Proc. R. Soc. (London) A 438, 397 (1992)] to rays having (p - 1) = 2 to 5 internal reflections. The analysis is for a horizontally illuminated ellipsoid with a vertical symmetry axis. Aspect ratios causing a vanishing of the vertical curvature at the equator for the outgoing wave front are found from generalized ray tracing. In response to infinitesimal deformation, the axial caustic of real glory rays unfolds producing cusps. Laboratory observations with laser illumination demonstrate that cusps resulting from rays with five internal reflections extend into Alexander's dark band when the drop's aspect ratio is near 1.08. The evolution of this p = 6 scattering pattern as cusps meet the quinary rainbow is suggestive of an E(6) catastrophe. For ellipsoids of varying aspect ratio and refractive index N, there is an organizing singularity associated with an exceptionally flat outgoing wave front from spheres with N = p.

An imaging design approach for optical systems consisting of two aspheres which is free of astigmatism is presented in this paper. A set of implicit differential equations is derived from generalized ray tracing. The solution of the derived equations provides the profiles of the two aspheres as well as the object to image mapping. The obtained design can be used as a good starting point for optimization. Particular examples are given.

Glare points associated with the Airy caustics of once and twice internally reflected rays are visible in the scattering by sunlit icicles. Supporting color photographs include an image of the far-field scattering. Relevant rays are analogous to the Descartes rays of primary and secondary rainbows of drops; however, the caustic conditions for the icicle are predicted to be affected by tilt of the illumination relative to the axis of the icicle. A model for the caustic evolution, given for a circular dielectric cylinder, manifests a transition in which the Airy caustic (and associated glare points) merge in the meridional plane at a critical tilt. At this critical tilt the merged glare point is predicted to be very bright. The calculations use the Bravais effective refractive index and generalized ray tracing.

Generalized ray tracing and the caustic surface

Supplementary document for Automated freeform imaging system design with generalized ray tracing and simultaneous multi-surface analytic calculation - 5236973.pdf

An imaging design approach which is free of third-order astigmatism for one freeform optical surface and the image is presented in this paper. A set of differential equations is derived from generalized ray tracing. The solution of the above derived equations provides the anastigmatic freeform optical surface, the image surface and the object to image mapping. The obtained design can be used as a good starting point for optimization. As an example, a reflective freeform surface is designed for a single reflective Head Mounted Display (HMD). This example has a 3 mm pupil, 15mm eye clearance, 24-degree diagonal full field of view, and the final design yields an average MTF of 62.6% across 17 field points.

Light scattered from spheroidal drops produces a variety of complicated diffraction patterns which are examples of optical catastrophes. These patterns decorate caustics in the outgoing rays which have been refracted by (and internally reflected from) the drop’s surface. Near the rainbow scattering angle, hyperbolic umbilic, cusp, and other catastrophes have been observed with laser and white light illumination. Lips caustics in the backscattering pattern have also been obseved. Calculations of caustic parameters (from generalized ray tracing) and wavefields are summarized.

Raytracing is used in many applications to simulate physical phenomena. In optical applications, the concepts of sequential and non-sequential raytrace are commonly used is optical design and analysis software. A broader Generalized Raytrace technique will be described as well as how it is used to solve imaging and non-imaging optical analysis tasks. This technique performs the typical reflection and refraction calculation that occur due to surface interactions and includes physical models to simulate surface and bulk scattering, polarization and birefringence and permits analysis on arbitrarily complex geometric systems.

Sound scattered by an oblate penetrable spheroid should produce a transverse cusp caustic in the region associated with the rainbow in optics (for relative speed of sound cscatt/c < 1). The principal curvatures of the generic local wave front that produces the far‐field transverse cusp are examined. This wave front is shown to generate a caustic curve (U − Uc)3 = d∝ V3, where U and V are horizontal and vertical scattering angles, and Uc is the cusp point direction. The far‐field opening rate d∝ is calculated for the transverse cusp. It is shown that d∝ has a simple dependence on the parameters of the generic wave front. Define the aspect ratio q = D/H, where H is the height and D is the equatorial width of the penetrable spheroid. Generalized ray tracing is used to relate q to principal curvatures and shape parameters of the outgoing wave front and hence to d∝. Measurements of d∝ in the optically analogous problem appear to support the calculation. As q goes to q14 ≈ 1.31, the critical value for the gener...