Marginal Ray Research Articles

Caustic and marginal ray curves obtained from images generated by means of a star test using digital image processing

Caustic and marginal ray curves obtained from images generated by means of a star test using digital image processing

Spot diagram borders are always caustic curves and/or marginal rays III: segmented mirror in hexagons and trapezoids

For annular (circular and elliptical) and hexagonal exit pupils with any optical path differences (described by Zernike Polynomials), it was proven [Appl. Opt. 61, 1876 (2022)APOPAI0003-693510.1364/AO.449270; Appl. Opt. 61, C20 (2022)APOPAI0003-693510.1364/AO.435850] that the border of any spot diagram is composed of the caustic surface and/or marginal rays. In this paper, the previous results are extended to segmented (by hexagons or trapezoids) exit pupils. Examples with exact spherical, coma, and astigmatism aberrations are included, while a mathematical demonstration is also included.

Optimum Optical Designs for Diffraction-Limited Terahertz Spectroscopy and Imaging Systems Using Off-Axis Parabolic Mirrors

Off-axis parabolic mirrors (OAPMs) are widely used in the THz and mm-wave communities for spectroscopy and imaging applications, as a result of their broadband, low-loss operation and high numerical apertures. However, the aspherical shape of an OAPM creates significant geometric aberrations: these make achieving diffraction-limited performance a challenge, and lower the peak electric field strength in the focal plane. Here, we quantify the impact of geometric aberrations on the performance of the most widely used spectrometer designs, by using ray tracing and physical optics calculations to investigate whether diffraction-limited performance can be achieved in both the sample and the detector plane. We identify simple rules, based on marginal ray propagation, that allow spectrometers to be designed that are more robust to misalignment errors, and which have minimal aberrations for THz beams. For a given source, this allows the design of optical paths that give the smallest THz beam focal spot, with the highest THz electric field strength possible. This is desirable for improved THz imaging, for better signal-to-noise ratios in linear THz spectroscopy and optical-pump THz-probe spectroscopy, and to achieve higher electric field strengths in non-linear THz spectroscopy.

Ontogeny of multiple variants in the stems of Phaseolus lunatus L. (Fabaceae)

Development of the cambial variant is characteristic to most of the climbing members of the Fabaceae and is considered as an alternative path to increase their stem diameter for efficient supply of minerals and photosynthate within limited stem diameter. These alternative forms of secondary growths are ascribed as an adaptive character to increase stem flexibility and most often they are found to be taxon-specific. In the present study, histological investigation on the main stems of Phaseolus lunatus L. (Fabaceae) showed multiple combinations of variant secondary growth like the formation of vascular cylinders external to the secondary phloem and proliferation of parenchyma (axial and ray) followed by the development of interxylary phloem, interxylary cambia (functionally regular and inverse) and ray cambia. The secondary xylem produced by regular cambium was diffuse-porous with indistinct growth rings and composed of an abundance of thin-walled unlignified parenchyma. With the increase in age, phloem rays underwent dilatation, subsequently became meristematic and formed oval to circular vascular cylinders/units with irregular orientation. Concomitantly, dedifferentiation xylem parenchyma not only formed small islands of interxylary phloem but also led to the origin of interxylary cambial segments that differentiated bidirectionally and deposited xylem and phloem having inverse, diagonal, tangential or radial orientations. Similar cell division activity was also observed in marginal xylem ray cells (ray cambium) that produced both the xylem and phloem on either side. All these cambial variants are described in detail and their possible significance in the climbing habit is discussed.

Intrinsic aberration coefficients for plane-symmetric optical systems consisting ofspherical surfaces.

This paper presents the analytical form of the intrinsic aberration coefficients for spherical plane-symmetric optical systems expressed as a function of first-order system parameters and the paraxial chief and marginal ray angles and heights. The derived aberration coefficients are in the third and fourth groups with the multiplication of two or three vector products of pupil and field vectors.

Design method for zoom systems based on tunable lenses

It is well known that tunable lenses, with refractive power that can be varied, e.g., by changing the curvature of a membrane, can replace the motion of lens groups in zoom systems. Similar to classical zoom systems, the performance of these systems is heavily influenced by the fundamental first-order layout. Moreover, the first-order layout sets the most important requirements for the employed tunable lenses. In this contribution, we present a method for the analysis of a large number of possible first-order solutions for typical requirements and for the selection of the most promising layouts. The first-order solution space is mapped, allowing the layouts to be automatically filtered and plotted depending on pre-defined characteristics. Ray tracing of the marginal and chief rays combined with the traditional thin lens aberration theory provide efficient estimations of the expected installation space requirements and performance for each first-order layout. Using an example, we demonstrate good agreement between these estimations and the corresponding real lens layout, optimized by commercial raytracing software. The presented design method for zoom systems based on tunable lenses is compared with similar approaches for classical zoom lenses.

Proof that spot diagram borders are always caustic curves and/or marginal rays II: annular (circular and elliptical) and single hexagonal exit pupils.

In a previous paper [Appl. Opt. 61, C20 (2022)] it was proven that for a circular exit pupil and any optical path differences, the border of any spot diagram is integrated by the caustic surface and/or marginal rays. In this paper, the previous results are extended to annular (circular an elliptical) as well as hexagonal (single and segmented) exit pupils. Several examples of wavefronts affected by linear combinations of orthonormal Zernike aberrations are shown.

Inferring the solution space of microscope objective lenses using deep learning

Lens design extrapolation (LDE) is a data-driven approach to optical design that aims to generate new optical systems inspired by reference designs. Here, we build on a deep learning-enabled LDE framework with the aim of generating a significant variety of microscope objective lenses (MOLs) that are similar in structure to the reference MOLs, but with varied sequences-defined as a particular arrangement of glass elements, air gaps, and aperture stop placement. We first formulate LDE as a one-to-many problem-specifically, generating varied lenses for any set of specifications and lens sequence. Next, by quantifying the structure of a MOL from the slopes of its marginal ray, we improve the training objective to capture the structures of the reference MOLs (e.g., Double-Gauss, Lister, retrofocus, etc.). From only 34 reference MOLs, we generate designs across 7432 lens sequences and show that the inferred designs accurately capture the structural diversity and performance of the dataset. Our contribution answers two current challenges of the LDE framework: incorporating a meaningful one-to-many mapping, and successfully extrapolating to lens sequences unseen in the dataset-a problem much harder than the one of extrapolating to new specifications.

Borders of spot diagrams are always caustic surfaces and/or marginal rays.

In this paper, it is proven that for any optical system with a circular exit pupil and wavefronts affected by any aberrations, the borders of all leaving rays are caustic surfaces and/or marginal rays. Several examples are shown for wavefronts affected by linear combinations of Zernike aberrations.

Sonic boom simulation for boom carpet determination

The Institute of Air Transportation Systems in Hamburg, Germany, researches methodologies for optimized supersonic flight routing that prevents the (primary) sonic boom from reaching the ground. The sonic boom shall either be cut off vertically or laterally. In initial trials, lateral cutoff positions of the sonic boom carpet were searched geometrically by ray tracing. Thereby, it was found that smooth interpolation of atmospheric data is necessary for geometrical convergence of different ray tracing methods, and that determining the marginal ray was oftentimes virtually impossible leeward. (These findings will be elaborated in more detail.) For those reasons, acoustic propagation has been introduced based on the Augmented Burgers equation, and implementing the Tricomi equation for covering flight maneuvers is planned in the short term. Our main research goal is finding numerical methods for determining the acoustic sonic boom cutoff that are first and foremost practical, i.e., fast, accurate, and robust. In this respect, we will discuss different termination conditions in signature propagation for the purpose of sonic boom carpet computation.

Two Cyc2CL transcripts (Cyc2CL-1 and Cyc2CL-2) may play key roles in the petal and stamen development of ray florets in chrysanthemum

BackgroundChrysanthemum morifolium is one of the most popular ornamental crops. The capitulum, which is the main ornamental part of chrysanthemum plants, consists of ligulate marginal ray florets, an attractive corolla (petals), and radially hermaphroditic disc florets, but no stamens. In Asteraceae species, the zygomorphic ray florets evolved from the actinomorphic disc florets. During this process, the zygomorphic ligulate corolla arose and the stamens were aborted. Although molecular genetic research has clarified ray floret development to some extent, the precise molecular mechanism underlying ray floret development in chrysanthemum remained unclear.ResultsA CYC2-like gene, Cyc2CL, was cloned from C. morifolium ‘Fenditan’. Subsequent analyses revealed that the alternative splicing of Cyc2CL, which occurred in the flower differentiation stage, resulted in the production of Cyc2CL-1 and Cyc2CL-2 in the apical buds. Prior to this stage, only Cyc2CL-1 was produced in the apical buds. A fluorescence in situ hybridization analysis of labeled Cyc2CL-1 and Cyc2CL-2 RNA indicated that Cyc2CL-2 was first expressed in the involucre tissue during the final involucre differentiation stage, but was subsequently expressed in the receptacle and floret primordia as the floral bud differentiation stage progressed. Moreover, Cyc2CL-2 was highly expressed in the inflorescence tissue during the corolla formation stage, and the expression remained high until the end of the floral bud differentiation stage. Furthermore, the overexpression of Cyc2CL-1 and Cyc2CL-2 in transgenic Arabidopsis inhibited stamen and petal development. Therefore, both Cyc2CL-1 and Cyc2CL-2 encode candidate regulators of petal development and stamen abortion and are important for the ray floret development in chrysanthemum.ConclusionIn this study, we characterized the alternatively spliced transcripts of the CYC2-like gene that differ subtly regarding expression and function. The data presented herein will be useful for clarifying the regulatory mechanisms associated with the CYC2-like gene and may also be important for identifying the key genes and molecular mechanisms controlling the development of ray florets in chrysanthemum.

Second Fin Ray Shows Promise for Estimating Ages of Juvenile but Not Adult Lake Sturgeon

AbstractThe first marginal pectoral fin ray (fin spine) is the most common structure used for estimating the age of sturgeons, including Lake Sturgeon Acipenser fulvescens. However, conflicting results from studies on the effects of fin spine removal have made some managers hesitant about the practice. We investigated whether the second pectoral fin ray, which can be removed in a less invasive procedure, could be used for estimating ages of Lake Sturgeon. Ages estimated from fin spine and second fin ray samples were compared for 53 wild (470 to 1,981 mm TL) and 16 stocked, known‐age (ages 8–18) Lake Sturgeon. Mean coefficient of variation for all samples was 12.4% for the fin spine and 17.5% for the second fin ray. In known‐age fish, 17% of estimated ages for the fin spine and the second fin ray matched true age. For the remaining estimates, the difference between the second fin ray and true age was greater than the difference between the fin spine and true age (P < 0.05, Wilcoxon’s signed rank test). In juvenile fish (n = 24), 75% of ages estimated from fin spines and second fin rays were within ±4 annuli, which was similar to differences in reader agreement for the same fin spine. Age estimates for adult Lake Sturgeon (n = 45) were less when using the second fin ray relative to the fin spine (up to –34 years). Additionally, poor annulus clarity was observed in >70% of the second fin rays sampled from adult fish. Our results suggest that the second fin ray does not provide reliable age estimates for adult Lake Sturgeon but may have some utility for estimating age of juvenile Lake Sturgeon. Additional research with a larger sample size would be required to provide more conclusive results.

Genome-wide analysis of the MADS-Box gene family in Chrysanthemum

MADS-box family transcription factors play key roles in various developmental processes in plants. Here, we identified 108 MADS-box genes in the genome of chrysanthemum (Chrysanthemum nankingense). We classified these genes based on their phylogenetic relationships with MADS-box genes in Arabidopsis thaliana and lettuce (Lactuca sativa). Type I genes were subdivided into classes Mα (19 genes), Mβ (12 genes), and Mγ (10 genes), and type II genes were subdivided into classes MIKCC (64 genes) and MIKC* (3 genes). The MIKCC class genes were further divided into 16 subclasses that included genes described in the ABCDE flower development model. Each group of MADS-box genes showed a specific pattern of conserved protein motifs and exon–intron structure. We analyzed the expression levels of each MADS-box gene in root, stem, leaf, flower bud, disc floret, and ray floret tissues. Subfamilies AGL18, FLC, and SVP contained more members in chrysanthemum. The asterid-specific TM8 subfamily and eleven Asteraceae Specific-MADS CnMADS genes were present in chrysanthemum. Chrysanthemum is the lacking members of the AGL15 subfamily. Among the genes responsible for the ABCDE model, B-class genes were expanded in chrysanthemum with three AP3 and four PI genes. One AP3 homolog functions in marginal ray floret development, whereas the two other AP3 homologs function in the development of the central disc floret. Two of the four PI genes are expressed in chrysanthemum, specifically in both types of florets. The results of this study lay the foundation for further studies of the roles of MADS-box genes in flower development in chrysanthemum and of the evolution of MADS-box genes in plants.

Ray flower initiation in the Helianthus radula inflorescence is influenced by a functional allele of the HrCYC2c gene.

The radiate pseudanthium, with actinomorphic disk flowers surrounded by showy marginal zygomorphic ray flowers, is the most common inflorescence in the Helianthus genus. In Helianthus radula, ray flower primordia are normally absent at the dorsal domain of the inner phyllaries (discoid heads) while the occurrence of radiate inflorescences is uncommon. In Helianthus spp., flower symmetry and inflorescence architecture are mainly controlled by CYCLOIDEA (CYC)-like genes but the putative role of these genes in the development of discoid inflorescences has not been investigate. Three CYC genes of H. radula with a role in ray flower identity (HrCYC2c, HrCYC2d, and HrCYC2e) were isolated. The phylogenetic analysis placed these genes within the CYC2 subclade. We identified two different alleles for the HrCYC2c gene. A mutant allele, designed HrCYC2c-m, shows a thymine to adenine transversion, which generates a TGA stop codon after a translation of 14 amino acids. We established homozygous dominant (HrCYC2c/HrCYC2c) and recessive (HrCYC2c-m/HrCYC2c-m) plants for this nonsense mutation. Inflorescences of both HrCYC2c/HrCYC2c and HrCYC2c/HrCYC2c-m plants initiated ray flowers, despite at low frequency. By contrast, plants homozygous for the mutant allele (HrCYC2c-m/HrCYC2c-m) failed at all to develop ray flowers. The results support, for the first time, a role of the HrCYC2c gene on the initiation of ray flower primordia. However, also in the two dominant phenotypes, discoid heads are the prevalent architecture suggesting that this gene is required but not sufficient to initiate ray flowers in pseudanthia. Other unknown major genes are most likely required in the shift from discoid to radiate inflorescence.

Calculation of the different focus positions for meridional and marginal rays of a plano-convex thick lens

In this paper, we describe the equations for the calculation of the different focus positions for the meridional and marginal rays of a plano-convex thick lens. The equations are exact and were obtained from the trigonometric ray trace analysis of a single meridional ray parallel to the optical axis of the lens. We apply these equations to derive the expressions of the paraxial focal lengths for a plano-convex thick lens.

TCP and MADS-Box Transcription Factor Networks Regulate Heteromorphic Flower Type Identity in Gerbera hybrida.

The large sunflower family, Asteraceae, is characterized by compressed, flower-like inflorescences that may bear phenotypically distinct flower types. The CYCLOIDEA (CYC)/TEOSINTE BRANCHED1-like transcription factors (TFs) belonging to the TEOSINTE BRANCHED1/CYCLOIDEA/PROLIFERATING CELL FACTOR (TCP) protein family are known to regulate bilateral symmetry in single flowers. In Asteraceae, they function at the inflorescence level, and were recruited to define differential flower type identities. Here, we identified upstream regulators of GhCYC3, a gene that specifies ray flower identity at the flower head margin in the model plant Gerbera hybrida We discovered a previously unidentified expression domain and functional role for the paralogous CINCINNATA-like TCP proteins. They function upstream of GhCYC3 and affect the developmental delay of marginal ray primordia during their early ontogeny. At the level of single flowers, the Asteraceae CYC genes show a unique function in regulating the elongation of showy ventral ligules that play a major role in pollinator attraction. We discovered that during ligule development, the E class MADS-box TF GRCD5 activates GhCYC3 expression. We propose that the C class MADS-box TF GAGA1 contributes to stamen development upstream of GhCYC3 Our data demonstrate how interactions among and between the conserved floral regulators, TCP and MADS-box TFs, contribute to the evolution of the elaborate inflorescence architecture of Asteraceae.

Optical system optimization method for as-built performance based on nodal aberration theory.

The traditional optical design process isolates the two steps of system performance optimization and tolerance allocation, making it difficult to achieve optimal design of as-built performance. To solve this problem, this paper proposes an analytical method for optimizing the as-built performance of optical systems. The method uses the nodal aberration theory to derive the wavefront aberration estimated value under the given surface decenter and tilt tolerance, and establishes the optical system as-built performance evaluation model. The as-built performance evaluation does not require a large amount of ray tracing, which can be completed only by tracking the paraxial marginal ray and the principal ray, and the calculation amount is small. The as-built performance evaluation model can be directly used as error function in optical design software for optical system optimization. A Cooke triplet system is taken as an example to compare the as-built performance optimization method, Code V and Zemax OpticStudio's built-in optimization methods and the traditional method which optimizes only nominal performance by Monte-Carlo tolerance analysis.

Comparative anatomy of oleoresin producing and non-producting trees of Copaifera multijuga Hayne in primary forests and plantations

The objective of this study was to identify and describe differences in the anatomical structure of productive and non-productive trees of Copaifera multijuga Hayne, and to understand the anatomical basis for production and non-production of oleoresin. We also studied how oleoresin is produced, transported and stored. The study was carried out on a primary forest area (2° 57′ 43′′ South, 59° 55′ 38′′ West) and at a 33 year-old plantation (2° 35′ 51.28′′ South and 60° 02′ 10.57′′ West), both located near Manaus, Amazonas, Brazil. A total of 27 trees were selected, including productive and non-productive trees from both areas. Wood was sampled using destructive methods (3 trees in plantations) and non-destructivly (with 2 bores from 24 trees, each giving a 5 mm radial sample). Macroscopic analysis: samples were polished until all cellular elements could be seen, then examined under a stereoscopic microscope. Microscopic analysis: Semipermanent histological sections (15 μm thick) were made in the transverse, tangential and radial longitudinal planes. Sections were also made of macerated material. In general, anatomical observations made agree with previous published studies, and show similarities to other species in the genus Copaifera. With the exception of vessel diameter, which was always higher in trees with oleoresin production (Kruskal-Wallis: p < 0.001), no qualitative or quantitative anatomical differences were observed between productive and non-productive trees. Both types of trees had oil-containing cellular elements, including the marginal parenchyma cells, rays and vessels. However, the quantity of oil-bearing cells was much higher in productive trees. Secretory canals, present in the marginal parenchyma, were mostly unobstructed (without oleoresin). However, ray cells and vessels were always filled with oleoresin, suggesting that, although oil synthesis occurs in the secretory cannals epithelial cells, its transport occurs via the ray cells to the vessels, in the heartwood region, where it is stored.

Alignment accuracy of gradient-index rod lens in an array

Lens arrays of gradient-index (GRIN) rods that project an erected unit magnification image are used in light sources of LED printers. It is shown how alignment accuracy of the GRIN rod affects imaging performance to maintain the image uniformity. The coordinates and inclinations of the chief and marginal rays of the GRIN rod that has a different object distance or is tilted are analyzed with the ray matrix. We compare the ray coordinate shift with the object distance difference with that with the GRIN rod optical axis tilt. The image shift with the tilted GRIN rod is several times larger than that with the different object distance of the GRIN rod. However, the image by the tilted GRIN rod shifts parallel on the image plane without being almost blurred. Consequently, the GRIN rod tilt should be minimized sufficiently.

Design of two spherical mirror unobscured relay telescopes using nodal aberration theory.

Nodal aberration theory is used to calculate the third-order aberrations that result in image blur for an unobscured modified 4f relay (2f1 + 2f2) formed by two tilted spherical mirrors for objects at infinity (infinite conjugate) and near the front focal plane of the first mirror (finite conjugate). The field-averaged wavefront variance containing only non-rotationally symmetric aberration coefficients is then proposed as an optimization metric. Analytical and ray tracing optimization are demonstrated through sample designs. The particular cases of in-plane and orthogonal folding of the optical axis ray are discussed, followed by an analysis of a modified 2f1 + 2f2 relay in which the distance of the first mirror to the object or pupil is allowed to vary for aberration correction. The sensitivity of the infinite conjugate 2f1 + 2f2 relay to the input marginal ray angle is also examined. Finally, the optimization of multiple conjugate systems through a weighted combination of wavefront variances is proposed.