Design of Achromatic Corrector Plates for Schmidt Cameras and Related Optical Systems

Abstract

An analytical method for the design of achromatic corrector plates for either classical Schmidt cameras or related types of systems is presented. The method may be applied to the design of achromatic doublet correcting plates allowing for the attainment of any order of asphericity for the aspheric surfaces of the corrector plate.     

The Optical Effects of a Double Calcite Plate on a Converging Beam of Light

The aberrations and variation in transmission of a beam of light converging to a focus and passing through plates of calcite have been investigated, in order to estimate possible photometric errors due to these effects when a crossed calcite plate is used to calibrate photographic plates. The aberrations are calculated in terms of ray deviations whose magnitudes are derived, and the transmission coefficient is determined by solution of electromagnetic boundary conditions. The resultant formulae are evaluated numerically, and set limits on the photometric accuracy of images formed by a Schmidt telescope used with a calcite plate calibrator.     

Active optics and modified-Rumsey wide-field telescopes: MINITRUST demonstrators with vase- and tulip-form mirrors

Wide-field astronomy requires the development of larger aperture telescopes. The optical properties of a three-mirror modified-Rumsey design provide significant advantages when compared to other telescope designs: (i) at any wavelength, the design has a flat field and is anastigmatic; (ii) the system is extremely compact, i.e., it is almost four times shorter than a Schmidt. Compared to the equally compact flat-field Ritchey-Chrétien with a doublet-lens corrector, as developed for the Sloan digital sky survey-and which requires the polishing of six optical surfaces-the proposed modified-Rumsey design requires only a two-surface polishing and provides a better imaging quality. All the mirrors are spheroids of the hyperboloid type. Starting from the classical Rumsey design, it is shown that the use of all eight available free parameters allows the simultaneous aspherization of the primary and tertiary mirrors by active optics methods from a single deformable substrate. The continuity conditions between the primary and the tertiary hyperbolizations are achieved by an intermediate narrow ring of constant thickness that is not optically used. After the polishing of a double vase form in a spherical shape, the primary-tertiary hyperbolizations are achieved by in situ stressing. The tulip-form secondary is hyperbolized by stress polishing. Other active optics alternatives are possible for a space telescope. The modified-Rumsey design is of interest for developing large space- and ground-based survey telescopes in UV, visible, or IR ranges, such as currently demonstrated with the construction of identical telescopes MINITRUST-1 and -2, f/5-2 degrees field of view. Double-pass optical tests show diffraction-limited images.     

All-sky camera with a concave mirror

The characteristics of an all-sky camera with a concave mirror are analyzed. A differential equation for a concave aspheric mirror with constant angular magnification is derived for the general dependence of the camera image height on the camera field angle. This equation is solved in parametric form for the case of a concave mirror with a constant angular magnification. The explicit equations for the shape of the aspheric mirror are given for some particular values of the angular magnification. Parametric equations of the surface shape for sevenfold angular magnification are developed into a power series that is used to analyze the imaging performance of such a mirror. The performance of the concave aspheric mirror is compared with that of a spherical mirror. The minimal camera-to-mirror distance is determined as a function of the blur allowed and the camera lens aperture. Some characteristics of convex mirrors are also presented for comparison.     

Aplanatic corrector designs for the extremely large telescope

The next century is knocking on our door, bringing with it the possibility of telescopes even bigger than the 8-10-m-class instruments that have proliferated over the past decade. The fixed spherical reflector is the most economical and pragmatic way to construct an extremely large primary mirror (30-50 m in diameter). Although spherical mirrors have virtues such as manufacturability and identically figured segments, they also create great amounts of spherical aberration and coma. Here we show that there are several catoptric (all-reflecting) corrector designs that enable a fast telescope based on a spherical primary mirror.     

Evaluation of least-squares phase-diversity technique for space telescope wave-front sensing

Because of mechanical aspects of fabrication, launch, and operational environment, space telescope optics can suffer from unforeseen aberrations, detracting from their intended diffraction-limited performance goals. We give the results of simulation studies designed to explore how wave-front aberration information for such near-diffraction-limited telescopes can be estimated through a regularized, low-pass filtered version of the Gonsalves (least-squares) phase-diversity technique. We numerically simulate models of both monolithic and segmented space telescope mirrors; the segmented case is a simplified model of the proposed next generation space telescope. The simulation results quantify the accuracy of phase diversity as a wave-front sensing (WFS) technique in estimating the pupil phase map. The pupil phase is estimated from pairs of conventional and out-of-focus photon-limited point-source images. Image photon statistics are simulated for three different average light levels. Simulation results give an indication of the minimum light level required for reliable estimation of a large number of aberration parameters under the least-squares paradigm. For weak aberrations that average a 0.10lambda pupil rms, the average WFS estimation errors obtained here range from a worst case of 0.057lambda pupil rms to a best case of only 0.002lambda pupil rms, depending on the light level as well as on the types and degrees of freedom of the aberrations present.     

Special configuration of a very large Schmidt telescope for extensive astronomical spectroscopic observation

A special reflecting Schmidt telescope is used to observe celestial objects. The telescope has an aperture of 4m, f ratio of 5, and a 5° field of view. Its optical axis is fixed and tilted 25° to the horizontal that runs from south to north. The celestial objects were observed for 1.5 h as they passed through the meridian. The shape of the reflecting Schmidt plate has to be changed with each different declination δ and in the tracking process. This is achieved with active optics. The sky area to be observed is -10° ≤ δ ≤ +90°. There are plans to place ~4000 optical fibers on the telescope focal surface that will lead to a dozen spectrographs.     

Grazing-incidence hyperboloid-hyperboloid designs for wide-field x-ray imaging applications

The classical Wolter type I grazing-incidence x-ray telescope consists of a paraboloidal primary mirror and a confocal hyperboloidal secondary mirror. This design exhibits stigmatic imaging on-axis but suffers from coma, astigmatism, field curvature, and higher-order aberrations such as oblique spherical aberration. Wolter-Schwarzschild designs have been developed that strictly satisfy the Abbe sine condition and thus exhibit no spherical aberration or coma. However, for wide-field applications such as the solar x-ray imager (SXI), there is little merit in a design with stigmatic imaging on-axis. Instead, one needs to optimize some area-weighted-average measure of resolution over the desired operational field of view. This has traditionally been accomplished by mere despacing of the focal plane of the classical Wolter type I telescope. Here we present and evaluate in detail a family of hyperboloid-hyperboloid grazing-incidence x-ray telescope designs whose wide-field performance is much improved over that of an optimally despaced Wolter type I and even somewhat improved over that of an optimally despaced Wolter-Schwarzschild design.     

Design example for the use of hybrid optical elements in the infrared

The Petzval objective is an effective design form for IR camera systems. It has excellent performance characteristics for high-aperture applications. Its field coverage is limited to a few degrees by the uncorrected Petzval sum.

Using a hybrid aspheric/diffractive front element, it is possible to achieve broadband color correction over the 3-5 μm or 8-12 μm region using only one lens material. For the long wavelength region, Germanium is typically used (except when the application environment is expected to exceed 60° C where Ge becomes opaque). This hybrid combination allows for the correction of the spherical and chromatic aberrations.

The advantage of the high precision obtainable with diamond turning makes it possible to generate simultaneously the diffractive phase profile superimposed on the aspheric surface. To protect the diffractive profile from the environment, it is placed on the rear surface of the first element.

     

Modeling technique for the Hubble Space Telescope wave-front deformation

Images from the Hubble Space Telescope suffer from an overcorrected spherical aberration that is due to a conic-constant error in the primary mirror. Within the program known as the corrective optics space telescope axial replacement (COSTAR) simulators have been built to provide the point-spread function (PSF) of the telescope alone and of the telescope with the faint-object camera F/96. It was found that the experimental PSF's were identical to those in orbit, which was not the case when the PSF's were calculated with commonly used optical software. We explain this discrepancy and propose a modeling method that is based on the determination of the wave-front error at the exit-pupil level that gives results that are consistent with observations.     

Off-axis systems for 4-m class telescopes

We describe here an off-axis design for a 4.0-m astronomical telescope. We show that the geometric optical performance of this configuration can equal that of an on-axis conventional configuration while the diffractive performance fundamentally surpasses conventional telescopes because of the absence of pupil obstruction. The specific optical design described here uses a single off-axis primary mirror to obtain three distinct final focus ports: an f/10 port (with corrector) for wide-field imaging and spectroscopy with a field of view (FOV) of 15 arc min; a small-field, 2-reflection f/10 port suitable for polarimetry and coronagraphy; and a slower, f/16(3-reflection) port with a 7 arc min FOV. For general astronomical observations requiring high optical throughput and low scattered light, this design is superior to conventional Ritchey-Chretien optical configurations.     

Design method of an astronomical telescope with reduced sensitivity to misalignment

Approximate formulas are derived for the axial coma resulting from tilt and decenter of a surface and for the spherical aberration resulting from a change in its axial position. These expressions include terms that represent aberrations induced by the subsystem preceding the surface in addition to other terms that are intrinsic contributions from the misaligned surface itself. This separation of the terms gives a simple method of designing a system that is insensitive to a misalignment at a given surface. The method is illustrated by applying it to a two-mirror astronomical telescope with corrector. Two examples are given-one for tilt and the other for despace. In both examples an appreciable reduction in the sensitivity is obtained. The limitations of these solutions and the problem of simultaneous correction for two types of misalignment are examined.     

Parabola-doublet aplanat becomes anastigmatic when second doublet is inserted near focus

A doublet of choice glasses may be located in the converging focal cone of the infinity-focused parabola to yield an aplanatic telescope or camera. The resulting angular field is limited by high astigmatism but is significantly larger than that of the coma-limited parabola. The spherical and chromatic aberrations are so well corrected and the coma so well balanced that the doublet may be used unaltered with a parabola of arbitrary focal length and speed with excellent results for the unvignetted rays. A second doublet nearer to the focus and designed independently of the first corrects the system's astigmatism while preserving its aplanaticism. It may also be designed for flattening the field. This arrangement may allow for greater flexibility in the placing of optical elements than does Wynne's triplet for modest-aperture systems. Equations are presented for choosing candidate glasses for the first doublet from the very limited manifold of solving glasses.     

Application of the see-saw method to all refracting optical systems

The optical see-saw diagram is a method that describes image correction to third-order approximation over a finite field of view in rotationally symmetric systems that employ aspheric surfaces. The aim of this paper is to describe the correction of aberrations caused by plane surfaces in all refracting optical systems in terms of the see-saw diagram. A lens correction algorithm based on the see-saw method is described to correct analytically the Seidel aberrations, primary spherical aberration, coma, astigmatism, and distortion, in such systems. We then apply this lens correction algorithm to the design of equivalent configurations by aspherizing different surfaces of the system, and the high-order aberrations of the equivalent configurations are evaluated by means of transverse-ray-aberration plots. Results indicate that this method gives information on what the contribution must be to the third-order aberrations that each component should provide to the system to give a better balance of high-order aberrations. Examples of the lens correction algorithm applied to lenses with six refracting surfaces and working for both finite and infinite object conjugates are given.     

Efficient testing of segmented aspherical mirrors by use of reference plate and computer-generated holograms. I. Theory and system optimization

Telescopes with large aspherical primary mirrors collect more light and are therefore sought after by astronomers. Instead of large mirrors as a single piece, they can be made by use of numerous smaller segments. Because the segments must fit together to create the effect of a single mirror, segmented optics present unique challenges to fabrication and testing that are absent for monolithic optics. We have developed a new method for measuring large quantities of segments accurately, quickly, and economically using an interferometric test plate and computer-generated hologram (CGH). In this test, the aspheric mirror segments are interferometrically measured by use of a test plate with a best-fit spherical surface. The aspherical departure is accommodated with a small CGH that is imaged onto the test plates. The radius of curvature is tightly controlled by maintaining the gap between the test plate and the segment. We present a summary of the test and give the basic design tradeoffs for using a single system to measure all of the segments of a large aspheric mirror.     

Optimization of sensing parameters for a confocal signal-based wavefront corrector in microscopy

The accuracy of a confocal signal-based wavefront corrector depends on several parameters such as spatial variation of optical properties within the specimen, aberration magnitude and composition, time required for the correction, etc. Here, a numerical analysis has been performed with the aim to improve system performance. The goal of the search algorithm in a confocal signal-based wavefront corrector is to estimate the Zernike coefficients of the aberrations. High-magnitude aberrations show low Strehl ratios. Repeating the correction process results in higher Strehl ratios, but at the cost of increased time. An in-focus on-axis specimen results in higher Strehl ratio compared to an out-of-focus and off-optical-axis specimen. For all cases, the wavefront correction accuracy is better, when the diameter of the pinhole is chosen to be equal to that of the Airy disk. The lower limit on the pinhole size for detecting small magnitude aberrations is set by noise.     

Scaling law for computational imaging using spherical optics

The resolution of a camera system determines the fidelity of visual features in captured images. Higher resolution implies greater fidelity and, thus, greater accuracy when performing automated vision tasks, such as object detection, recognition, and tracking. However, the resolution of any camera is fundamentally limited by geometric aberrations. In the past, it has generally been accepted that the resolution of lenses with geometric aberrations cannot be increased beyond a certain threshold. We derive an analytic scaling law showing that, for lenses with spherical aberrations, resolution can be increased beyond the aberration limit by applying a postcapture deblurring step. We then show that resolution can be further increased when image priors are introduced. Based on our analysis, we advocate for computational camera designs consisting of a spherical lens shared by several small planar sensors. We show example images captured with a proof-of-concept gigapixel camera, demonstrating that high resolution can be achieved with a compact form factor and low complexity. We conclude with an analysis on the trade-off between performance and complexity for computational imaging systems with spherical lenses.     

Field-of-view limitations of phased telescope arrays

The optical performance of imaging phased telescope arrays is degraded by various design, manufacturing, and operational errors. Perhaps the most basic and fundamental of these error sources are the residual aberrations of the optical design chosen for the individual telescopes. We show that third-order field curvature and distortion, which are rather benign aberrations in a conventional telescope, result in relative phase and tilt errors between the individual telescopes making up the array. The field-dependent image degradation caused by these relative phase and tilt errors is then predicted for different subaperture configurations and telescope design parameters. For phased arrays made up of simple two-mirror telescopes, distortion limits the field of view to less than 5 arcmin for small subapertures (D < 0.5 m), and field curvature limits the field of view to less than 1 arcmin for subaperture diameters greater than 2 m. Quantitative parametric results yielding tolerances for residual field curvature as the phased array is scaled up in size are presented graphically. If a 0.5-deg field of view is desired for telescope diameters greater than 2 m, complex telescope configurations are necessary to satisfy the rather tight tolerances on both field curvature and distortion.     

Statistical variation of aberration structure and image quality in a normal population of healthy eyes

A Shack-Hartmann aberrometer was used to measure the monochromatic aberration structure along the primary line of sight of 200 cyclopleged, normal, healthy eyes from 100 individuals. Sphero-cylindrical refractive errors were corrected with ophthalmic spectacle lenses based on the results of a subjective refraction performed immediately prior to experimentation. Zernike expansions of the experimental wave-front aberration functions were used to determine aberration coefficients for a series of pupil diameters. The residual Zernike coefficients for defocus were not zero but varied systematically with pupil diameter and with the Zernike coefficient for spherical aberration in a way that maximizes visual acuity. We infer from these results that subjective best focus occurs when the area of the central, aberration-free region of the pupil is maximized. We found that the population averages of Zernike coefficients were nearly zero for all of the higher-order modes except spherical aberration. This result indicates that a hypothetical average eye representing the central tendency of the population is nearly free of aberrations, suggesting the possible influence of an emmetropization process or evolutionary pressure. However, for any individual eye the aberration coefficients were rarely zero for any Zernike mode. To first approximation, wave-front error fell exponentially with Zernike order and increased linearly with pupil area. On average, the total wave-front variance produced by higher-order aberrations was less than the wave-front variance of residual defocus and astigmatism. For example, the average amount of higher-order aberrations present for a 7.5-mm pupil was equivalent to the wave-front error produced by less than 1/4 diopter (D) of defocus. The largest pupil for which an eye may be considered diffraction-limited was 1.22 mm on average. Correlation of aberrations from the left and right eyes indicated the presence of significant bilateral symmetry. No evidence was found of a universal anatomical feature responsible for third-order optical aberrations. Using the Marechal criterion, we conclude that correction of the 12 largest principal components, or 14 largest Zernike modes, would be required to achieve diffraction-limited performance on average for a 6-mm pupil. Different methods of computing population averages provided upper and lower limits to the mean optical transfer function and mean point-spread function for our population of eyes.