Aberrations of a Spherical Mirror

The position and the size of the Gaussian image of an object formed by an optical imaging system is determined by using its Gaussian imaging equations. We have stated in Chapter 1 that the quality of the diffraction image depends on the aberrations of the system. A spherical wave originating at a point object is incident on the system. The image formed by the system is aberration free and perfect if the wave exiting from the system is also spherical. In this case, the rays originating at the point object and traced through the system all pass through the Gaussian image point. If the optical wavefront exiting from the exit pupil is not spherical, its optical deviations from a spherical form represent its wave aberrations. These wave aberrations play a fundamental role in determining the quality of the aberrated image. The rays traced from the object point through the system, instead of passing through the Gaussian image point, intersect the image plane in its vicinity. The distance of the point of intersection of a ray in the image plane from the Gaussian image point is called the transverse ray aberration, and the distribution of the rays is referred to as the spot diagram. In this chapter, we define the wave and ray aberrations and give a relationship between them. We relate the longitudinal defocus of an image to the defocus wave aberration, and its wavefront tilt to the wavefront tilt aberration. Next, the possible aberrations of an imaging system that is rotationally symmetric about its optical axis are described. The aberration function of the system is expanded in a power series of the object and pupil coordinates, and primary (or Seidel), secondary (or Schwarzschild), and tertiary aberrations are introduced. We also discusss briefly how the aberrations may be observed using a Twyman–Green interferometer and what the fringe pattern of a primary or Seidel aberration looks like. A short summary of the chapter is given at the end.

4.1 Definition of a Thin Lens A thin lens is a theoretical lens whose thickness is assumed to be zero and, therefore, negligible during thin-lens calculations. The thin lens is a design tool used to simulate a real lens during the preliminary stages of optical system design and analysis. The concept is particularly valuable because it enables the optical engineer to quickly establish many basic characteristics of a lens or optical system design. By assuming a lens form where the thickness is zero, the formulas used to calculate object and image relationships are greatly simplified. The drawback to the thin-lens approach is that it is not possible to determine image quality without including actual lens thickness (and other lens data) in the calculations. As a result, while it is possible to establish many valuable facts about an optical system by the application of thin-lens theory and formulas, the ultimate quality of the final image can, at best, only be estimated. 4.2 Properties of a Thin Lens Figure 4.1 is an illustration of a positive thin lens. Any lens system analysis must start with several known factors that will generally be provided by the end user, i.e., the customer. From these given starting factors it will be possible, using thin-lens theory and formulas, to generate the missing information required to completely describe the final lens system. In the case shown in Fig. 4.1, for example, it is given that the system detector (image size) will be 25 mm in diameter and that the full field of view of the lens will be 10 deg. Other system considerations dealing with detector sensitivity - thus required image brightness - dictate that a lens speed (f/#) of f/2.0 will be required. Since we know the image size and the field of view, applying the formulas shown in Fig. 4.1, we can calculate the effective focal length (EFL) of the lens. The image size dimension used for this calculation is measured from the optical axis and is designated as y1. In this case, the maximum value for y1 is 12.5 mm.

Offner imaging spectrometer consists of a convex grating and two concave mirrors. The concentric characteristics of the optical structure make it have the advantages of large relative aperture, small distortion and compact structure. In order to reduce the alignment difficulty of the Offner imaging spectrometer and improve the efficiency, a fast alignment method of Offner imaging spectrometer is presented in this paper based on the concentric characteristic and spherical autostigmatic method. Firstly, the spherical autostigmatic device is built, which can generate point source, and when the point source is located at the spherical mirror's center of curvature (CoC), its reflection image point and the point source coincide. By measuring the distance between the reflection image point and the point source, the position deviation of the spherical mirror's CoC can be determined. The Offner imaging spectrometer is completed with this device by locating the CoC of its primary mirror, convex grating and tertiary mirrors. The results show that location error of the two off-axis concave mirrors's CoC can be controlled within 10μm, and the imaging performance of the imaging spectral system is satisfied. Compared with the existing method, this method has the advantages such as easy to operate, low cost and fast alignment.

The Marshall Grazing Incidence X-Ray Spectrometer (MaGIXS) is an X-ray imaging spectrometer that will observe the solar corona in the soft X-ray regime with both spatial and spectral resolution. The science goal of MaGIXS is to better understand the problem of coronal heating by measuring the temperature distribution, composition, and temporal variability of hot plasmas (>4 MK) in active regions. In order to do this, the instrument will observe the corona with a fast cadence (~5 seconds) in wavelengths between 6-24 A with a 6” spatial resolution and a 0.1 A spectral resolution. To ensure that this instrument can achieve such a resolution, it is crucial to have exact measurements of the focal lengths of the mirrors. The mirrors will be aligned and mounted using the Centroid Detector Assembly (CDA) (a steerable laser originally developed for aligning the AXAF mirrors), a CMM Romer arm, and Hartmann aperture masks to perform the focal length measurements. We have designed metrology supports that elevate the aperture mask and mirror up to the height of the optical axis defined by the CDA of the laser, allows the aperture mask 3 translational degrees of freedom, and the allows the mirror 3 translational and 3 rotational degrees of freedom needed for alignment. The measured and verified focal lengths will then be used to carry out the alignment of the mirrors as the MaGIXS instrument is assembled for launch.

First Page

It is possible to change the figure of reflecting surfaces quantitatively by controlled deposition of metals, evaporated in vacuum. The theory for calculating screens and the experimental technique are described. A spherical 12-inch mirror has been repeatedly parabolized, obtaining a surface perfect within 0.05 wave-length. A 558‐inch spherical mirror has been parabolized with the optical axis outside of its periphery. A defective parabolic mirror was corrected for turned up and down edges. A convex spherical secondary mirror was hyperbolized. The limitations and possible applications of the method are discussed.

A collimator was required to qualify the Advanced Land Imager (ALI) instrument of the EO-1 new millennium satellite for focus, MTF measurements, and distortion. It was used during assembly of the instrument and during thermal cycling while the instrument was in a vacuum tank. It had to be diffraction-limited over a 3 X 3 degree field of view and over a waveband of 400 to 2500 nm, have no obstruction, and have a virtual exit pupil that could be imaged onto the entrance pupil of the telescope. To satisfy these requirements the collimator, external to the vacuum tank, was built comprising a spherical mirror with exit pupil at its center of curvature, a full-aperture beamsplitter in collimated space, and a single lens to flatten the field. The optical layout and method of verifying collimation will be presented as well as optical performance, of interest since no corrector plate could be used as in the usual Schmidt camera configuration.

We analytically, experimentally and computationally explore the solar radiation flux distribution in the interior region of a spherical mirror and compare it to that of a paraboloidal one with the same aperture area. Our investigation has been performed in the framework of geometrical optics. It is shown that despite one can assign a quasi focus, at half the radius, to a spherical mirror, the light concentration occurs as well on an extended line region which starts at half-radius on the optical axis. In contrast to a paraboloidal concentrator, a spherical mirror can concentrate the radiation parallel to its optical axis both in a point-focus and in a line-focus manner. The envelope of the reflected rays is also obtained. It is shown that the flux distribution has an axial symmetry. The radial dependence of the flux on a flat circular receiver is obtained. The flux longitudinal dependence is shown to exhibit three distinctive regions in the interval [0, R] (R is mirror radius). We obtain the radiational (optical) concentration ratio characteristics and find the optimal location of the flat receiver of a given size at which the concentration ratio is maximised. In contrast to a parabolic mirror, it is shown that this location depends on the receiver size. Our findings offers that in spherical mirrors one can alternatively use a line receiver and gains a considerable thermal energy harvest. Our results are supported by Monte Carlo ray tracing performed by Zemax optical software. Experimental validation has been performed in lab with a silver-coated lens as the spherical mirror.

Scanning MeV-ion microprobe for light and heavy ions

A computational camera uses a combination of optics and processing to produce images that cannot be captured with traditional cameras. Over the last decade, a range of computational cameras have been proposed, which use various optics designs to encode and use computation to decode useful visual information. What is often missing, however, is the quantitative analysis of the relation between camera design and the captured visual information, and little systematic work has been done to evaluate and optimize these computational camera designs. While the optics of computational cameras may be quite complicated, many of them can be effectively characterized by their point spread functions (PSFs): the intensity distribution on an image sensor as a response to a point light source in a scene. This thesis explores the techniques to characterize, evaluate and optimize computational cameras via PSF engineering for computer vision tasks, including image recovery, 3D reconstruction, and image refocusing. I first explore PSF engineering techniques to recover image details from blurry images. Image blur is a problem commonly seen in photography due to a number of reasons, including defocus, lens aberration, diffraction, atmospheric turbulence, object motion, and etc. Image blur is often formulated as a convolution of the latent blur-free image and a PSF, and deconvolution (or deblurring) techniques have to be used to recover image details from a blurry region. Here, I propose a comprehensive framework of PSF evaluation for the purpose of image deblurring, in which the effects of image noise, deblurring algorithm, and the structure of natural images are all taken into account. In defocus blur, the shape of defocus PSF is largely determined by the aperture pattern of the camera lens. By using an evaluation criterion derived from the comprehensive framework, I optimize the aperture pattern to preserve many more image details when defocus occurs. Both through simulations and experiments, I demonstrate the significant improvement gained by using optimized aperture patterns. While defocus blur causes a loss in image detail, it also encodes depth information of scenes in images. A typical depth from defocus (DFD) technique computes depth from two or more images that are captured with circular aperture lenses of different focus settings. Circular apertures produce circular defocus PSFs. In this thesis, I show that the use of a circular aperture severely restricts the accuracy of DFD, and propose a comprehensive framework of PSF evaluation for depth recovery. With this framework, we can derive a criterion for evaluating a pair of apertures with respect to the precision of depth recovery. This criterion is optimized using a genetic algorithm and gradient descent search to arrive at a pair of high resolution apertures. The two coded apertures are found to complement each other in the scene frequencies they preserve. With this property it becomes possible to not only recover depth with greater fidelity but also to obtain a high quality all-focused image from the two defocused images. While depth recovery can significantly benefit from optimized aperture patterns, its overall performance is rigidly limited by the lens aperture's physical size. To transcend this limitation, I propose a novel depth recovery technique using an optical diffuser - referred to as depth from diffusion (DFDiff). I show that DFDiff is analogous to conventional DFD, in which the scatter angle of the diffuser determines the system's effective aperture. High precision depth estimation can be achieved by choosing a proper diffuser and no longer requires the large lenses that DFD requires. Even a consumer camera with a low-end small lens can be used to do high-precision depth estimation when coupled with an optical diffuser. In my detailed analysis of the image formation properties of a DFDiff system, I show a number of examples demonstrating greater precision in depth estimation when using DFDiff. While the finite depth of field (DOF) of a lens camera leads to defocus blur, it also produces artistic visual experience. Many of today's displays are interactive in nature, which opens up a possibility for new kind of visual representations. Users could, for example, interactively refocus images to different depths, so that they can experience the artistic narrow DOF images while simultaneously making available the image detail for the entire image. To enable image refocusing, one typical approach is to capture the entire light field. But this method has the drawback of a significant sacrifice in spatial resolution due to the dimensionality gap: the captured information (light field) is 4D, while the required information (focal stack) is only 3D.In this thesis, I present an imaging system that directly captures focal stacks by a sweeping focal plane. First, I describe how to synchronize focus sweeping with image capturing so that the summed DOF of a focal stack efficiently covers the entire depth range. Then, I take a customized algorithm to enable a seamless refocusing experience, even in textureless regions or with moving objects. Prototype cameras are presented to capture real space-time focal stacks. There is also an interactive refocusing viewer available online at www.focalsweep.com.

The BPT-4500 and method of tuning the acceleration by focusing a plasma lens is described. Test results show that overall thruster efficiency and plume shape is improved by adjusting the curvature of magnetic field lines in the acceleration zone. Thrust efficiency is improved to 50% from 21% at 150 V and from 58% to 62.5% at 350 V using this technique. At 125 V, the BPT-4500 operates at 45% thrust efficiency and 1030 s specific impulse yielding a thrust/power ratio only 10% less than a 600 s hydrazine arcjet. Introduction Multi-mode Hall thruster capability enables one thruster to serve constant power applications needing the highest possible thrust for orbit transfer and high specific impulse for station keeping. Towards this objective, we have evaluated a GENERAL DYNAMICS (formerly PRIMEX Aerospace Company) approach for tuning the BPT-4500 for the best possible performance over a range of accelerating potentials. This paper describes our initial exploratory efforts to improve thruster efficiency by actively changing magnetic field shape. Thruster Description The BPT-4500 is a 4.5 kW Hall field plasma accelerator based on an earlier 4000 watt device''. These thrusters employ a short discharge to minimize contact with the insulators and pole pieces to shape the curvature of the magnetic field in the ion creation zone. As is well known in Hall type accelerators, the magnetic field lines become nearly equipotential lines defining the acceleration vector of the ions. Also, the technique of adjusting electric potential shape by magnetized electrons in a magnetic field is commonly called a plasma lens. By virtue of the discharge properties in the BPT and shaping these fields, we have attempted to Copyright (c) 2001 by GENERAL DYNAMICS. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. create a plasma lens with adjustable focus. Plasma Lens Focusing Morozov in the Soviet Union has shown that using two sets of electromagnets to change the gradient of the radial magnetic field can have a significant effect on the behavior of ion acceleration in Hall type discharges. Gavryushin and Kim have shown that the axial gradient of the radial magnetic field regulated by varying the degree of screening of the accelerator channel. These accelerators are characterized by relatively long acceleration lengths in channels that typically have a length to width of 2:1. Insofar as these accelerators had a long discharge, the curvature of the field was not emphasized in the description of this early work. Instead, the researchers discuss primarily the axial gradient of Curvature of ion path changed by Anode focal length Anode potential contour Cathode potential contour Figure 1 Hall Acceleration with Plasma Lens Concept (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. the magnetic field. The BPT discharge is configured to take place in a very short channel, typically with a length to width of 1:2. As such, physical evidence from a variety of geometry studies suggests that the BPT discharge creates ions in a thin, -1-2 mm layer, occurring approximately 3-4 mm downstream of the anode, schematically illustrated in Fig 1. As such, we hypothesize the BPT discharge embodies Hall acceleration within a plasma lens. Our innovation is to exploit the features of this short discharge by adjusting the focal length of a plasma lens using a trim or focusing coil. Variation of the curvature of the fields in this layer has the effect of changing the initial trajectory of ions. Changes in the shape downstream of the ionization zone are minimized by the fact that magnetic fields are manipulated inside the primary magnetic structure. The BPT-4500 thruster shown in Fig 2 is composed of a primary magnetic structure with four outer coils and an inner coil. The BPT-4500 has a 13 cm mid-diameter and a 2 cm annulus providing slightly more discharge area than the BPT-4000. The schematic cross section of the thruster is shown in Fig. 3. This shows the anode is surrounded by a secondary magnetic structure to add or subtract from the magnetic field radius of curvature in the annulus between insulator rings. Figure 2 BPT-4500 Laboratory Style Thruster Flight Version A flight capable version of the thruster is under development in the Lockheed Martin funded Hall Thruster Propulsion System (HTPS) program. This flight thruster is referred to as the BPT-4000. This thruster has a fixed magnetic field and plasma lens taken from the optimal configuration found here. GENERAL DYNAMICS is on track to deliver a flight qualified thruster, power processor and xenon flow control system in 2003.

The floating wire method for determining the useful aperture of a magnetic quadrupole lens

In this report the induced surface current method is used to investigate the spatial structure of the radiated electric field for a number of paraboloidal antennas. The paraboloids are excited by three different types of feeds, namely, a small electric dipole, an elemental plane wave source, and a rectangular horn. For the case of electric dipole excitation, formulas are derived that show the following characteristics: (i) For a reflector of constant ratio of focal length to the aperture diameter the magnitude of the cross-polarized lobe nearest to the antenna axis (paraboloid axis) remains constant relative to the maximum of the main lobe of the principally-polarized wave and is independent of the aperture size. (ii) For a given aperture size the magnitude of the cross-polarized component relative to its own principally-polarized maximum decreases with the focal length. (iii) The position of the maximum of the cross-polarized lobe depends only on the aperture size and is independent of the focal length. The problem of cross-polarization is also solved by using a simple model which gives results that are in surprisingly close agreement with those obtained by the more complete expressions. In addition this crude model explains the angular variation of the amplitude of cross-polarization component at angles not necessarily small from the paraboloid axis. For a paraboloid excited by an elemental Huyghens source the cross-polarization in the forward direction is reduced but the component in the laterally-directed radiation is increased relative to that of an electric dipole. In the case of the horn-excited paraboloid we obtain a formula that explains the experimentally-observed large crosspolarization. Finally, we show that the problems of the paraboloids excited by a small electric dipole and a plane (Huyghens) source are merely particular cases of the horn excitation problem.

We analyze the dispersion effect of the focal length of low-index thin lenses by using scalar diffraction and finite difference time domain (FDTD) methods. We compare the dispersion results obtained by using these methods with reported experimental results, and the well-known analytical formula for focal length (f) of diffractive lenses as a function of wavelength ( λ ) , f ( λ ) = f 0 λ 0 λ where f0 is the designed focal length for wavelength λ0. We show that when the analytical formula is applied to thin flat lenses with low-refractive index, the results are accurate for small numerical aperture (NA) up to 0.2. For larger NA, the error between the analytical approximation and the FDTD analysis remains around 8% over a wide range of NA.

To ensure the performance of the optical system, the machining accuracy of lens with long focal lengths is required to ensure the image quality. A new method for lens transmission wavefront power spectral density (PSD) in mid-frequency domain measurement using binary phase computer-generated hologram (CGH) is presented. This technique is widely applicable and is particularly useful for measuring large-size lenses with long focal lengths. A comparison experiment of the CGH measurement with results from a Fizeau sphere interferometry method is carried out to verify the accuracy and convenience of the measurement. Furthermore, measurement uncertainty due to CGH fabrication process is analysed. Analysis of the CGH test showed the overall accuracy of less than 1 nm RMS for a sphere lens with over 30 m focal length and Φ410 mm clear aperture. CGH can provide reference spheres with high precision, in the meantime greatly shorten air space, thus reducing the effect of vibration and air turbulence, therefore is of great importance for lens transmission wavefront PSD measurement. The realization of high precision, high efficiency and nondestructive testing of long focal-lens wavefront PSD ensure the ultra-precision and certainty level of machining, hence improving the comprehensive performance of the optical system.