Chief Ray Research Articles

Variations of spherical aberration and central coma of conceptual thin lenses.

Based on our earlier matrix algorithms [J. Mod. Opt.55, 105 (2008)JMOPEW0950-034010.1080/09500340701308740] for evaluating the variations of Seidel aberrations, new concise equations are derived to directly evaluate the variations of spherical aberration and central coma of conceptual thin lenses when the incident marginal and chief rays are arbitrarily changed. In the earlier algorithms, four cases with distinct translation factors and matrix equations are required according to the relationships of the object and pupil positions; otherwise, division-by-zero errors or insufficient numerical accuracy will be encountered. Conversely, the new versatile equations are always accurate for all kinds of object and pupil positions; one example verifying this property is illustrated. Another example is given to explain why a symmetrical thin lens can provide only one, rather than two, degrees of freedom for correcting Seidel aberrations.

Quality of a Gaussian beam diffracted by a concave non-spherical grating

We examine the propagation of a complex-source-point spherical wave via a concave non-spherical grating. First we trace the paraxial paths of a chief ray propagating from a complex source point through the system to complex image points of different orders. Then we represent the wave function of the light in terms of Seidel-type aberrations, while the terms of up to fourth order in aperture variables are taken into account. Based on the derived formula, we also evaluate the quality of a Gaussian beam degraded by the system. We show how the beam qualities in the sagittal and tangential sections are influenced by the conic constant (or the aspheric coefficient of fourth order) and the positions of source and stop.

Analysis of the Dependence of the Viewing Zones on the Display Radius in Autostereoscopic 2 View Displays of Horizontally Concave Surface

In autostereoscopic display, ray trajectories through the optic element such as the parallax barrier or the lenticular lens are affected by the radius of horizontally concave display. Using the intersection of the calculated trajectories of the rays coming from the various positions of the autostereoscopic 2 view display of the curved surface, the viewing zones and area of the incomplete separation of the images for the left and right eyes were investigated. The area of incomplete separation of the images for the left and right eyes related to the triangular areas defined by three chief rays coming from the left edge, center, and the right edge. These triangular areas for the curved surface was smaller than those for the flat surface and showed the tendency of the decrease as the ratio of the radius of the curved and the designed viewing distance approached 1/2.

Anisotropic shadow effects with various pattern directions in an anamorphic high numerical aperture system

A high numerical aperture (NA) system with an NA larger than 0.5 is required to make patterns of 1X nm and below, even though extreme ultraviolet lithography uses a 13.5-nm wavelength source. To avoid the reflective efficiency loss and to avoid an increase in the chief ray angle of incident light, use of an anamorphic high-NA system is suggested. The suggested anamorphic NA system has nonisotropic magnification, x-magnification of 4× and y-magnification of 8×, and the mask NA shape is an ellipse due to the nonisotropic magnification distribution. Anamorphic NA systems have a nonconventional shadow effect due to nonisotropic incident angle distribution and magnification. These nonisotropic characteristics lead to the reduction of asymmetric shadow distribution and a reduction of horizontal–vertical bias. As a result, anamorphic NA systems can achieve balanced patterning results regardless of pattern direction and incident direction.

Flat-Concave Dual-Mirror Configuration Design for Upright Projection-Type Ultrashort Throw Ratio Projectors

An off-axis flat-concave dual-mirror (FCDM) configuration for an ultrashort focal length projection lens based on the digital light processor (DLP) upright projection-type projector is proposed in this paper. This projection lens is composed of two lens groups: the FCDM system and the relay optical system. The flat mirror and the contour of the concave mirror are determined according to the relationship between the chief ray from the exit pupil of each field of view (FOV) and the corresponding assigned position on the screen. A curved image is formed between exit pupil and concave mirror, aiming to correct field curvature. The second lens group consists of a set of conventional refractive lenses which convert a curved object onto the image panel. Based on this method, a projection lens with an ultrashort throw ratio (TR) of 0.34, including a full FOV 139.8 $^{\circ}$ and distortion of less than 1%, is designed. Additionally, the colorimetric performance, uniformity, optical efficiency and size of the designed projector are analyzed. Results show that the proposed projector system with high optical performance is compact and suitable for household or educational applications.

Ultra-wide angle lens design with relative illumination analysis

An ultra-wide angle lens design with relative illumination analysis is presented. The half field angle of 80°, the relative illumination of the image plane will be reduced. It is necessary to increase the image numerical aperture ratio for X and Y direction, and decrease the angle between the chief ray and optical axis in the image space. However the ultra-wide angle lens induces great barrel distortion causing image extrusion. Therefore, we use F-theta distortion to replace optical distortion. Seven lenses and two pieces of flat glass are used for a sensor 1/6” sensor with 2.1 million pixels. The final result for the lens is a half viewing angle of 80°, F/2.4, a focal length 1.1 mm, a length of 22.37 mm, F-theta distortion of less than 2%, and relative illumination greater than 83%.

Quality factors of an aberrant Gaussian beam diffracted through a convex grating spectrometer

We analyze the evolution of a complex-source-point spherical wave through a concentric catoptric system that is made up of one convex grating and two concave mirrors. First we trace the paraxial paths of a chief ray propagating from a complex source point through the system to complex image points of different orders. Then we find an expression of the fourth order-corrected Gaussian beam which includes Seidel-type aberrations. Finally, we analyze the quality of an aberrant Gaussian beam diffracted through the system. The results presented here will be useful in dealing with the propagation of a focused Gaussian beam through a diffractive optical system and the beam quality degradation due to aberration of the system.

Optical analysis of miniature lenses with curved imaging surfaces.

Miniature cameras for consumer electronics and mobile phones have been, and continue to be, in fast development. The system level requirements, such as manufacturing cost, packaging, and sensor characteristics, impose unique challenges for optical designers. In this paper, we discuss the potential optical benefits of having a curved image surface rather than a flat one. We show that curved sensor technology allows for optically faster lens solutions. We discuss trade-offs of several relevant characteristics, such as packaging, chief ray angle, image quality, and tolerance sensitivity. A comparison of a benchmark flat field lens, and an evaluation design imaging on a curved surface and working at f/1.6, provides useful specific insights. For a given image quality, departing from a flat imaging surface does not allow significantly reducing the total length of a lens.

Conditions for having identical aberration behaviors of various thick lenses

Two optical lenses with the same focal length but different lens thicknesses and principal plane locations can provide equivalent first-order optics on imagery if the marginal and chief ray heights and slope angles at their principal planes are equal, resulting in the same system magnification, object and image distances. In this paper, we will prove that if these lenses have the same primary aberrations (including Seidel aberrations, chromatic aberrations, and secondary spectrum) corresponding to the same marginal and chief rays then they must have the same new aberrations for any same new rays. This theorem is useful when designers would like to redesign a lens group in a system. The influence on the movements among lens groups, the special aberration properties of plane parallel plates, and afocal lens groups are also discussed.

Centroid shift analysis of microlens array detector in interference imaging system

Most CCD imaging detectors integrated microlens arrays (MLAs) to increase fill factor and sensitivity. However, they also introduce spot calibration issues with the inconsistency of spot geometry center and intensity distribution center. We setup theoretical and experimental models to research the problem of centroid shifting. According to the Seidel and Zernike coefficients of the optical model, we analyze main aberrations of microlens. In “Chief Ray” and “Centroid” reference frames, centroid shift numerical value is calculated with Geometric Ensquared Energy (GEE). Based on pentaprism test for 8.4m mirror segment, we conduct spot imaging experiment in interference system. Spots images are obtained, and two-dimensional centroid algorithm processing is performed on them to get the analog experiment values of centroid movements. The results show that the MLA placed in KAI‐16000 imaging detector causes the spot centroid to move. When there is a 14° (or −14°) angle of incident ray, the shifting values are about 1.46μm in simulation and 2.18μm in experiment. Our research makes a contribution to the compensation of calibrated error in metrology technology. We also prove that a significant portion of the shift comes from the low order aberration of microlens.

Centration axis in refractive surgery.

The human eye is an asymmetric optical system and the real cornea is not a rotationally symmetrical volume. Each optical element in the eye has its own optical and neural axes. Defining the optimum center for laser ablation is difficult with many available approaches. We explain the various centration approaches (based on these reference axes) in refractive surgery and review their clinical outcomes. The line-of-sight (LOS) (the line joining the entrance pupil center with the fixation point) is often the recommended reference axis for representing wavefront aberrations of the whole eye (derived from the definition of chief ray in geometrical optics); however pupil centration can be unstable and change with the pupil size. The corneal vertex (CV) represents a stable preferable morphologic reference which is the best approximate for alignment to the visual axis. However, the corneal light reflex can be considered as non-constant, but dependent on the direction of gaze of the eye with respect to the light source. A compromise between the pupil and CV centered ablations is seen in the form of an asymmetric offset where the manifest refraction is referenced to the CV while the higher order aberrations are referenced to the pupil center. There is a need for a flexible choice of centration in excimer laser systems to design customized and non-customized treatments optimally.

Defect mitigation considerations for EUV photomasks

The introduction of extreme ultraviolet (EUV) lithography into manufacturing requires changes in all aspects of the infrastructure, including the photomask. EUV reflective masks consist of a sophisticated multilayer (ML) mirror, capping layer, absorber layer, and anti-reflective coating thereby dramatically increasing the complexity of the photomask. In addition to absorber type defects similar to those the industry was forced to contend with for deep ultraviolet lithography, the complexity of the mask leads to new classes of ML defects. Furthermore, these approaches are complicated not only by the mask itself but also by unique aspects associated with the exposure of the photomask by the EUV scanner. This paper focuses on the challenges for handling defects associated with inspection, review, and repair for EUV photomasks. Blank inspection and pattern shifting, two completely new steps within the mask manufacturing process that arise from these considerations, and their relationship to mask review and repair are discussed. The impact of shadowing effects on absorber defect repair height is taken into account. The effect of mask biasing and the chief ray angle rotation due to the scanner slit arc shape will be discussed along with the implications of obtaining die-to-die references for inspection and repair. The success criteria for compensational repair of ML defects will be reviewed.

Hybrid approach for the design of mirror array to produce freeform illumination sources in immersion lithography

Source and mask optimization (SMO) has emerged as a key resolution enhancement technique (RET) for 45nm technology node and below in lithography. The design method of freeform illumination sources predicted by SMO is significant for the scanner development. We present a hybrid approach combining simultaneous and sequential approaches to optimize the tilt angles of the mirrors to produce multi target freeform illumination sources accurately and quickly. The size of spot reflect by plane mirror can be easily controlled by changing different microlens arrays with appropriate focal length, which reduce the complexity of the system and makes it more flexible to produce the specific freeform sources compared to curving the mirrors used in previous work. The relationship between the tilt angles of plane mirrors and the positions of the spots in the pupil is obtained by chief ray tracing. Using the hybrid approach the freeform illumination sources required by SMO can be designed by merely adjusting the tilt angles of mirrors without changing other parameters of optical elements, which is most effective for both lithography tool manufacture and its applications. The real ray tracing results demonstrate that our design method is capable of creating multi freeform illumination sources with high transmittance, and confirm that the effectiveness of the hybrid approach for optimized design and control of mirror array in immersion lithography system.

Field curvature correction method for ultrashort throw ratio projection optics design using an odd polynomial mirror surface.

This paper presents a field curvature correction method of designing an ultrashort throw ratio (TR) projection lens for an imaging system. The projection lens is composed of several refractive optical elements and an odd polynomial mirror surface. A curved image is formed in a direction away from the odd polynomial mirror surface by the refractive optical elements from the image formed on the digital micromirror device (DMD) panel, and the curved image formed is its virtual image. Then the odd polynomial mirror surface enlarges the curved image and a plane image is formed on the screen. Based on the relationship between the chief ray from the exit pupil of each field of view (FOV) and the corresponding predescribed position on the screen, the initial profile of the freeform mirror surface is calculated by using segments of the hyperbolic according to the laws of reflection. For further optimization, the value of the high-order odd polynomial surface is used to express the freeform mirror surface through a least-squares fitting method. As an example, an ultrashort TR projection lens that realizes projection onto a large 50 in. screen at a distance of only 510 mm is presented. The optical performance for the designed projection lens is analyzed by ray tracing method. Results show that an ultrashort TR projection lens modulation transfer function of over 60% at 0.5 cycles/mm for all optimization fields is achievable with f-number of 2.0, 126° full FOV, <1% distortion, and 0.46 TR. Moreover, in comparing the proposed projection lens' optical specifications to that of traditional projection lenses, aspheric mirror projection lenses, and conventional short TR projection lenses, results indicate that this projection lens has the advantages of ultrashort TR, low f-number, wide full FOV, and small distortion.

Polarization-independent distortion corrector fabricated using polymer-dispersed liquid crystals

We demonstrate a polarization-independent distortion corrector fabricated using a polymer-dispersed liquid crystal (PDLC) cell placed on the intermediate image plane of an optical system. At low voltage, a hazy PDLC cell scatters the incident rays and redirects the off-axis propagated chief ray. The chief ray approaches the principal point of the lens element, thereby decreasing image distortion. At high voltage, the PDLC cell becomes transparent, thereby restoring the image distortion. The PDLC-based distortion corrector is an easy-to-fabricate universal device that can be applied to various optical systems. With a large lens diameter, the distortion of a PDLC-corrected image is approximately 1/5 of that of an uncorrected image.

Comparison of Design and Spatial Stability of Two Kinds of Optical Systems for Stereoscopic Mapping Camera

To achieve a certain precision when mapping in accord with a particular proportion or scale with stereo mapping camera, the change of chief ray height of the edge field due to the fluctuation of working temperature is required to be within the range of microns, and at the meantime the size and structure layout of three-linear array stereo mapping camera are determined directly by the configuration of optical system. Based on the requirements of refractive optical system with long focal length, academic calculation and actual optical designs based on two typical configurations for stereoscopic mapping camera are made,the actual working temperature and performance requirements are also taken into consideration, the results including the modulation transfer function, distortion and stability comparison are also given.

Method to design two aspheric surfaces for imaging system

This paper proposes a method of designing two aspheric surfaces simultaneously for an imaging system. By coupling the three rays (two marginal rays and one chief ray from the exit pupil) of each field of view to their corresponding image point, microspherical lens corresponding to each field of view can be obtained by the refraction law. Profiles of the aspheric surfaces can be obtained by employing the position and normal vector of midpoint of the microsphere corresponding to each field of view. Theoretically, through this method, we can realize any mapping relationship between the object points and their image points in 2D. Combining an original projector with the aspheric lens designed in this paper, an example that realizes projection onto a spherical surface with both low distortion and high image quality is presented.

Distortion aberration correction device fabricated with liquid crystal lens array

A distortion aberration (DA) correction device is fabricated using a liquid crystal lens array (LCLA), which is placed at the intermediate image plane of the optical system. Without voltage, the LCLA does not work, the image is distorted due to the aberration from the optical system; with voltage, the incident light is focused by the LCLA and then the distorted image is corrected. The correction of distorted image by LCLA is attributed to the redirection of the off-axis propagated chief ray approaches the principal point of the lens element.

Actinic microscope for extreme ultraviolet lithography photomask inspection and review

Two dual-configuration extreme ultraviolet (EUV, 13.5nm wavelength) optical designs are described as a means to overcome principal EUV photomask metrology challenges. Semiconductor industry-wide efforts to define performance requirements and create standalone tools that can be used to discover, review, and accurately locate phase, amplitude, and mask pattern defects are described. The reference designs co-optimize low and high magnification configurations for orthogonal chief ray planes to avoid inspection and review trade-offs and emulate the aerial image of a lithography scanner.

Skew ray tracing and sensitivity analysis of hyperboloid optical boundary surfaces

One of the most popular mathematical tools in fields of robotics, mechanisms and computer graphics is the 4×4 homogeneous transformation matrix. Our group's previous application of the homogeneous transformation matrix to flat and spherical optical boundaries has been extended to hyperboloid surfaces for: (1) skew ray tracing to determine the paths of reflected/refracted skew rays; (2) sensitivity analysis for direct mathematical expression of differential changes of incident points and reflected/refracted vectors with respect to changes in incident light sources and boundary geometric parameters; (3) a sensitivity analysis-based merit function derived directly from mathematical expression of catadioptric imaging system components. The presented methodology is highly suited to digital implementation and offers direct and rapid analytical statement of ray path, chief ray, marginal rays and merit functions of optical systems.