Freeform optical design for a nonscanning corneal imaging system with a convexly curved image

Design Method of Freeform Surface Optical Systems with Low Coupling Position Error Sensitivity

Freeform off-axis reflective systems are significantly more difficult to align and assemble owing to their asymmetric surface shapes and system structures. In this study, a freeform surface system design method with low coupling position error sensitivity (FCPESM) was proposed. First, we established a mathematical model of a reflective system when it was perturbed by coupling position errors and used the clustering-microelement method to establish the coupling error sensitivity evaluation function. The evaluation function was then applied to the design process of a freeform surface off-axis three-mirror optical system. The results showed that the FCPESM optical design method can significantly relax the assembly tolerance requirements of optical systems on the basis of ensuring image performance. In this study, the reflective system was perturbed by tilt and decenter simultaneously, and the disturbance mechanism of position errors on optical systems was further improved. Through this research, freeform surface systems with both image performance and error sensitivity can be obtained, which makes freeform off-axis reflective systems with better engineering realizability.

Design of freeform mirrors using the concentric rings method

Through the methodology of optical surface design based on concentric rings, this paper proposes the design of freeform mirrors, initially by employing segmented rings, each of them with different spherical radii of curvature, and then by employing segmented conic rings with different conic constants in each of the segments. These surfaces will then produce the desired images. For the case of segmented spherical rings, mathematical expressions were deduced to obtain the image points as a function of the radii of curvature. Furthermore, it is shown that in the case where conic rings were used, there is a decrease in spherical aberration, which allows the manipulation of the generated image. Finally, several proposals are presented for the design of mirrors to generate both the desired size of the image and the desired distribution of energy, together with their analyses.

Full circumferential morphological analysis of Schlemm's canal in human eyes using megahertz swept source OCT

We performed full circumferential imaging of the Schlemm's canal (SC) of two human eyes using a Fourier domain mode-lock laser (FDML) based 1.66-MHz SS-OCT prototype at 1060 nm. Eight volumes with overlapping margins were acquired around the limbal area with customized raster scanning patterns designed to fully cover the SC while minimizing motion artifacts. The SC was segmented from the volumes using a semi-automated active contour segmentation algorithm, whose mean dice similarity coefficient was 0.76 compared to the manual segmentation results. We also reconstructed three-dimensional (3D) renderings of the 360° SC by stitching the segmented SCs from the volumetric datasets. Quantitative metrics of the full circumferential SC were provided, including the mean and standard deviation (SD) of the cross-sectional area (CSA), the maximum CSA, the minimum and maximum SC opening width, and the number of collector channels (CC) stemming from the SC.

Automated freeform imaging system design with generalized ray tracing and simultaneous multi-surface analytic calculation

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.

Freeform imaging systems: Fermat's principle unlocks "first time right" design

For more than 150 years, scientists have advanced aberration theory to describe, analyze and eliminate imperfections that disturb the imaging quality of optical components and systems. Simultaneously, they have developed optical design methods for and manufacturing techniques of imaging systems with ever-increasing complexity and performance up to the point where they are now including optical elements that are unrestricted in their surface shape. These so-called optical freeform elements offer degrees of freedom that can greatly extend the functionalities and further boost the specifications of state-of-the-art imaging systems. However, the drastically increased number of surface coefficients of these freeform surfaces poses severe challenges for the optical design process, such that the deployment of freeform optics remained limited until today. In this paper, we present a deterministic direct optical design method for freeform imaging systems based on differential equations derived from Fermat's principle and solved using power series. The method allows calculating the optical surface coefficients that ensure minimal image blurring for each individual order of aberrations. We demonstrate the systematic, deterministic, scalable, and holistic character of our method with catoptric and catadioptric design examples. As such we introduce a disruptive methodology to design optical imaging systems from scratch, we largely reduce the "trial-and-error" approach in present-day optical design, and we pave the way to a fast-track uptake of freeform elements to create the next-generation high-end optics. We include a user application that allows users to experience this unique design method hands-on.

Correction of 2D-telecentric scan systems with freeform surfaces

For scanning systems the resolution, distortion as well as the telecentricity are important performance criteria. For two-dimensional scanning systems, scan mirrors deflecting in only one transverse direction are not allowing for telecentricity in x and y simultaneously in case of an axisymmetric system. It is possible to achieve two-dimensional telecentricity by splitting the pupils in x- and y-direction and shifting the principal planes in one dimension by changing the focal power using an anamorphic setup. However, for higher specifications concerning a large aperture and wide scanning angle, using cylindrical lenses are not enough to achieve a good system quality. It has been proved in many researches that freeform surfaces are effective to improve the resolution of systems without rotational symmetry. In this work, a systematic case study is presented to investigate the potential of freeform surfaces to improve the resolution, telecentricity, and distortion simultaneously. It is shown as a result that freeform surfaces offer large correction ability in all the three aspects concerning high specifications of 2D-telecentric anamorphic scan systems. This contribution provides the insight into the application of freeform surfaces in non-rotationally symmetric optical systems with refractive components.

Freeform geometrical optics I: principles

Lens design uses a calculation of the lens' surfaces that permits us to obtain an image from a given object. A set of general rules and laws permits us to calculate the essential points of the optical system, such as distances, thickness, pupils, and focal distances, among others. Now, the theory on which classical lens design is based has changed radically, as our theoretical foundations do not rely on the classical ray-tracing rules. We show that with the rules expressed in a reduced vector analytical solution set of equations, we can take into account all optical elements, i.e., refractive, reflective, and catadioptric. These foundations permit us to keep under control the system aberration budget in every surface. It reduces the computation time dramatically. The examples presented here were possible because of the versatility of this theoretical approach.

Vectorial aberrations of biconic surfaces

For non-rotationally symmetric systems, spherical surfaces provide a non-sufficient effect in correcting aberrations. In anamorphic systems, biconic surfaces are used to provide different focal power in tangential and sagittal directions. The biconic surface shape is also used as the basic shape of the freeform surface representation. The different focal power in tangential and sagittal planes generates large field-constant astigmatism, which is decoupled from a coma generated by the biconic surface. In this paper, the biconic surface sag is expanded up to the fourth order and decomposed into the spherical part, the aspherical part, and the freeform part. The vectorial aberrations of the biconic surface are derived based on the extended nodal aberration theory. When biconic surfaces are used to obtain the initial setup of off-axis systems, it is beneficial due to the decoupling of astigmatism and coma. For instance, in the initial system design of reflective camera systems, biconic surfaces provide the possibility to obtain the nodal point in the center of the field of view, which is beneficial for systems with large field of view. This paper provides the insight of aberration analysis of biconic surfaces and the conversion of biconic surfaces into Zernike fringe freeform surfaces.

Design and fabrication of a compact off-axis see-through head-mounted display using a freeform surface

Head-mounted display (HMD) has been widely used in many fields, and most existing HMDs are complex and typically non-aesthetically pleasing. In this paper, a novel compact, lightweight and wearable off-axis HMD with freeform surface is reported. It is challenging to achieve large field of view (FOV) and maintain compact structure simultaneously for this type system. In this design, the compact off-axis HMD consists of a tilted ellipsoid combiner and a four pieces relay lenses. It offers a diagonal FOV of 30°, and an exit pupil diameter of 7 mm. No diffractive surfaces are used, thus avoiding the effect of stray light and ghost image in previous designs. The x-y polynomial freeform surface is adopted in the relay lens for improving the image quality and minimizing the structure size. Analytical expressions to determine the initial structure of HMD has been given, while structure constrains, optimization strategy and tolerance analysis are also described in details. The optical system is easy to manufacture by ordinary method which is beneficial for mass production. Further, a prototype of this compact HMD is successfully implemented with good imaging performance. The compact structure of HMD makes it well suited for integrating a normal glass, significantly advancing the application of HMD in people's daily life.

Aberration analysis for freeform surface terms overlay on general decentered and tilted optical surfaces

Aberration theory helps designers to better understand the nature of imaging systems. However, the existing aberration theory of freeform surfaces has many limitations. For example, it only works in the special case when the central area of the freeform surface is used. In addition, the light footprint is limited to a circle, which does not match the case of an elliptical footprint for general systems. In this paper, aberrations generated by freeform surface term overlay on general decentered and tilted optical surfaces are analyzed. For the case when the off-axis section of a freeform surface is used, the aberration equation for using stop and nonstop surfaces is discussed, and the aberrations generated by Zernike terms up to Z_17/18 are analyzed in detail. To solve the problem of the elliptical light footprint for tilted freeform surfaces, the scaled pupil vector is used in the aberration analysis. The mechanism of aberration transformation is discovered, and the aberrations generated by different Zernike terms in this case are calculated. Finally we proposed aberration equations for freeform terms on general decentered and tilted freeform surfaces. The research result given in this paper offers an important reference for optical designers and engineers, and it is of great importance in developing analytical methods for general freeform system design, tolerance analysis, and system assembly.

Multiple surface expansion method for design of freeform imaging systems

The relative aperture size and the field-of-view (FOV) are two significant parameters for optical imaging systems. However, it is difficult to improve relative aperture size and FOV simultaneously. In this paper, a freeform design method is proposed that is particularly effective for high performance systems. In this step-by-step method, the FOV is enlarged from a small initial value in equal-length steps until it reaches the full FOV; in each step, part of the area of one system surface is constructed. A freeform off-axis three-mirror imaging system with large relative aperture size and a wide FOV is designed as an example. The system operates at F/2.5 with 150 mm effective focal length and a 60° × 1° FOV. The average root-mean-square wavefront error of the system is 0.089λ (working wavelength λ = 530.5 nm).

Multi-element direct design using a freeform surface for a compact illumination system

An iterative optimization algorithm is introduced to address the surface iterative errors as well as source extension issues in a freeform illumination system for producing satisfactory illumination distribution. A unique two-parameter coordinate system is utilized to represent the emitted ray directions. Then, the direction vector for the incident rays, which propagate through several surfaces, is obtained using ray-tracing techniques. Based on the mapping between the incoming rays and a target grid, a freeform surface is generated as a good starting design. An iterative optimization strategy is further employed to alleviate the deterioration of illumination distribution on the target region, and the uniformity of the illumination system is evaluated during optimization. Very few variables are demanded, and more flexibility in the design of the freeform surface is offered. Successive iterations can be performed until the desired result is attained. An optical system is used as an example to demonstrate the validity of the proposed method, and numerical simulations are carried out to evaluate the optical performance. The simulation results show that a small angular intensity distribution and prescribed rectangular illumination pattern can be achieved simultaneously.