DOCSLIB.ORG

Introduction to Aberrations OPTI 518 Lecture 13

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Aberrations OPTI 518 Lecture 13 Introduction to aberrations OPTI 518 Lecture 13 Prof. Jose Sasian OPTI 518 Topics • Aspheric surfaces • Stop shifting • Field curve concept Prof. Jose Sasian OPTI 518 Aspheric Surfaces • Meaning not spherical • Conic surfaces: Sphere, prolate ellipsoid, hyperboloid, paraboloid, oblate ellipsoid or spheroid • Cartesian Ovals • Polynomial surfaces • Infinite possibilities for an aspheric surface • Ray tracing for quadric surfaces uses closed formulas; for other surfaces iterative algorithms are used Prof. Jose Sasian OPTI 518 Aspheric surfaces The concept of the sag of a surface 2 cS 4 6 8 10 ZS ASASASAS4 6 8 10 ... 1 1 K ( c2 1 ) 2 S Sxy222 K 2 K is the conic constant K=0, sphere K=-1, parabola C is 1/r where r is the radius of curvature; K is the K<-1, hyperola conic constant (the eccentricity squared); -1<K<0, prolate ellipsoid A’s are aspheric coefficients K>0, oblate ellipsoid Prof. Jose Sasian OPTI 518 Conic surfaces focal properties • Focal points for Ellipsoid case mirrors Hyperboloid case Oblate ellipsoid • Focal points of lenses K n2 Hecht-Zajac Optics Prof. Jose Sasian OPTI 518 Refraction at a spherical surface Aspheric surface description 2 cS 4 6 8 10 ZS ASASASAS4 6 8 10 ... 1 1 K ( c2 1 ) 2 S 1122 2222 22 y x Aicrasphe y 1 x 4 K y Zx 28rr Prof. Jose Sasian OTI 518P Cartesian Ovals ln l'n' Cte . Prof. Jose Sasian OPTI 518 Aspheric cap Aspheric surface The aspheric surface can be thought of as comprising a base sphere and an aspheric cap Cap Spherical base surface Prof. Jose Sasian OPTI 518 Aspheric surface contributions Prof. Jose Sasian OPTI 518 2 1 y Wa220 4 y Prof. Jose Sasian OPTI 518 Stop at aspheric surface 4 2 WHcap , Ayn4 Prof. Jose Sasian OPTI 518 Upon stop shifting 4 2 WHcap , Ayn4 shift shift Prof. Jose Sasian OPTI 518 Aspheric cap contributions New stop y y Old stop Prof. Jose Sasian OPTI 518 Stop shifting 4 2 WHcap , Ayn4 shift shift shift SH yy y 0 y S new old new yyy Prof. Jose Sasian OPTI 518 2 Expansion of shift shift 2 2 2SH S H H shift shift 2 2SH S H H 2 2 2 44SH S H 2 234 24S HH S HH H S HH Prof. Jose Sasian OPTI 518 Aberration function upon stop shifting for an aspheric cap 4 2 WHcap , Ayn4 shift shift 442 Ay44 n 4 Ay n S H 2 42 42 42Ay44 n S H Ay n S H H 2 43 44 4Ay44 n S H H H Ay n S H H 112 12 aaSHaSH 2 82 2 11 1 2 aS23 H H aS H H H aS4 H H 42 8 4 aAyn8 4 Prof. Jose Sasian OPTI 518 Aspheric contributions Prof. Jose Sasian OPTI 518 Aberration function of base sphere and aspheric cap WH ,,, Wspherical H Wcap H Case of an aspherical surface with the stop at the surface: 2221 24u WAyAyn040 4 8 n 222 Sag Sphere A4 x y Prof. Jose Sasian OPTI 518 Stop shifting Prof. Jose Sasian OPTI 518 Stop shifting Prof. Jose Sasian OPTI 518 Stop shifting • How do the aberration coefficients change as we shift the stop? • Used to ease calculations • Most importantly, the formulas give insights • The concept of object shifting Prof. Jose Sasian OPTI 518 Stop shifting parameter Prof. Jose Sasian OPTI 518 Stop shifting formulas Prof. Jose Sasian OPTI 518 Derivation for A Ж Ay11 Ay Ж A22y Ay 0 A12AyAyy 12 yy y A AAAS21 yy AAAS Prof. Jose Sasian OPTI 518 Derivation of new Seidel sums upon stop shifting AASA 2 u SAyI n SSII And similarly for: uu SAAyAASAyII nn uu2 SSIV IV AAy S A y nn 23 SSSSSVV IVIII 33 SSSS III SSSSII II I 2 2 uu SAyASAyIII nn 222u A 2 AAS S A y n 2 SSIII III2 SSSS II I Prof. Jose Sasian OPTI 518 Field focus curve concept 2 w Provides limit for a given wavefront variance. Prof. Jose Sasian OPTI 518 Field focus curve concept Prof. Jose Sasian OPTI 518 Field focus curve concept •Must preserve DOF Prof. Jose Sasian OPTI 518 Summary • Variety in aspheric surfaces • Aspheric surface contributions to fourth- order aberrations • Stop shifting and how aberrations change • The concept of field focus curve Prof. Jose Sasian OPTI 518.
Load more

Recommended publications
  • Chapter 11. Three Dimensional Analytic Geometry and Vectors
    Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1.
    [Show full text]
  • Radar Back-Scattering from Non-Spherical Scatterers
    REPORT OF INVESTIGATION NO. 28 STATE OF ILLINOIS WILLIAM G. STRATION, Governor DEPARTMENT OF REGISTRATION AND EDUCATION VERA M. BINKS, Director RADAR BACK-SCATTERING FROM NON-SPHERICAL SCATTERERS PART 1 CROSS-SECTIONS OF CONDUCTING PROLATES AND SPHEROIDAL FUNCTIONS PART 11 CROSS-SECTIONS FROM NON-SPHERICAL RAINDROPS BY Prem N. Mathur and Eugam A. Mueller STATE WATER SURVEY DIVISION A. M. BUSWELL, Chief URBANA. ILLINOIS (Printed by authority of State of Illinois} REPORT OF INVESTIGATION NO. 28 1955 STATE OF ILLINOIS WILLIAM G. STRATTON, Governor DEPARTMENT OF REGISTRATION AND EDUCATION VERA M. BINKS, Director RADAR BACK-SCATTERING FROM NON-SPHERICAL SCATTERERS PART 1 CROSS-SECTIONS OF CONDUCTING PROLATES AND SPHEROIDAL FUNCTIONS PART 11 CROSS-SECTIONS FROM NON-SPHERICAL RAINDROPS BY Prem N. Mathur and Eugene A. Mueller STATE WATER SURVEY DIVISION A. M. BUSWELL, Chief URBANA, ILLINOIS (Printed by authority of State of Illinois) Definitions of Terms Part I semi minor axis of spheroid semi major axis of spheroid wavelength of incident field = a measure of size of particle prolate spheroidal coordinates = eccentricity of ellipse = angular spheroidal functions = Legendse polynomials = expansion coefficients = radial spheroidal functions = spherical Bessel functions = electric field vector = magnetic field vector = back scattering cross section = geometric back scattering cross section Part II = semi minor axis of spheroid = semi major axis of spheroid = Poynting vector = measure of size of spheroid = wavelength of the radiation = back scattering
    [Show full text]
  • John Ellipsoid 5.1 John Ellipsoid
    CSE 599: Interplay between Convex Optimization and Geometry Winter 2018 Lecture 5: John Ellipsoid Lecturer: Yin Tat Lee Disclaimer: Please tell me any mistake you noticed. The algorithm at the end is unpublished. Feel free to contact me for collaboration. 5.1 John Ellipsoid In the last lecture, we discussed that any convex set is very close to anp ellipsoid in probabilistic sense. More precisely, after renormalization by covariance matrix, we have kxk2 = n ± Θ(1) with high probability. In this lecture, we will talk about how convex set is close to an ellipsoid in a strict sense. If the convex set is isotropic, it is close to a sphere as follows: Theorem 5.1.1. Let K be a convex body in Rn in isotropic position. Then, rn + 1 B ⊆ K ⊆ pn(n + 1)B : n n n Roughly speaking, this says that any convex set can be approximated by an ellipsoid by a n factor. This result has a lot of applications. Although the bound is tight, making a body isotropic is pretty time- consuming. In fact, making a body isotropic is the current bottleneck for obtaining faster algorithm for sampling in convex sets. Currently, it can only be done in O∗(n4) membership oracle plus O∗(n5) total time. Problem 5.1.2. Find a faster algorithm to approximate the covariance matrix of a convex set. In this lecture, we consider another popular position of a convex set called John position and its correspond- ing ellipsoid is called John ellipsoid. Definition 5.1.3. Given a convex set K.
    [Show full text]
  • Geodetic Position Computations
    GEODETIC POSITION COMPUTATIONS E. J. KRAKIWSKY D. B. THOMSON February 1974 TECHNICALLECTURE NOTES REPORT NO.NO. 21739 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. GEODETIC POSITION COMPUTATIONS E.J. Krakiwsky D.B. Thomson Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton. N .B. Canada E3B5A3 February 197 4 Latest Reprinting December 1995 PREFACE The purpose of these notes is to give the theory and use of some methods of computing the geodetic positions of points on a reference ellipsoid and on the terrain. Justification for the first three sections o{ these lecture notes, which are concerned with the classical problem of "cCDputation of geodetic positions on the surface of an ellipsoid" is not easy to come by. It can onl.y be stated that the attempt has been to produce a self contained package , cont8.i.ning the complete development of same representative methods that exist in the literature. The last section is an introduction to three dimensional computation methods , and is offered as an alternative to the classical approach. Several problems, and their respective solutions, are presented. The approach t~en herein is to perform complete derivations, thus stqing awrq f'rcm the practice of giving a list of for11111lae to use in the solution of' a problem.
    [Show full text]
  • Models for Earth and Maps
    Earth Models and Maps James R. Clynch, Naval Postgraduate School, 2002 I. Earth Models Maps are just a model of the world, or a small part of it. This is true if the model is a globe of the entire world, a paper chart of a harbor or a digital database of streets in San Francisco. A model of the earth is needed to convert measurements made on the curved earth to maps or databases. Each model has advantages and disadvantages. Each is usually in error at some level of accuracy. Some of these error are due to the nature of the model, not the measurements used to make the model. Three are three common models of the earth, the spherical (or globe) model, the ellipsoidal model, and the real earth model. The spherical model is the form encountered in elementary discussions. It is quite good for some approximations. The world is approximately a sphere. The sphere is the shape that minimizes the potential energy of the gravitational attraction of all the little mass elements for each other. The direction of gravity is toward the center of the earth. This is how we define down. It is the direction that a string takes when a weight is at one end - that is a plumb bob. A spirit level will define the horizontal which is perpendicular to up-down. The ellipsoidal model is a better representation of the earth because the earth rotates. This generates other forces on the mass elements and distorts the shape. The minimum energy form is now an ellipse rotated about the polar axis.
    [Show full text]
  • World Geodetic System 1984
    World Geodetic System 1984 Responsible Organization: National Geospatial-Intelligence Agency Abbreviated Frame Name: WGS 84 Associated TRS: WGS 84 Coverage of Frame: Global Type of Frame: 3-Dimensional Last Version: WGS 84 (G1674) Reference Epoch: 2005.0 Brief Description: WGS 84 is an Earth-centered, Earth-fixed terrestrial reference system and geodetic datum. WGS 84 is based on a consistent set of constants and model parameters that describe the Earth's size, shape, and gravity and geomagnetic fields. WGS 84 is the standard U.S. Department of Defense definition of a global reference system for geospatial information and is the reference system for the Global Positioning System (GPS). It is compatible with the International Terrestrial Reference System (ITRS). Definition of Frame • Origin: Earth’s center of mass being defined for the whole Earth including oceans and atmosphere • Axes: o Z-Axis = The direction of the IERS Reference Pole (IRP). This direction corresponds to the direction of the BIH Conventional Terrestrial Pole (CTP) (epoch 1984.0) with an uncertainty of 0.005″ o X-Axis = Intersection of the IERS Reference Meridian (IRM) and the plane passing through the origin and normal to the Z-axis. The IRM is coincident with the BIH Zero Meridian (epoch 1984.0) with an uncertainty of 0.005″ o Y-Axis = Completes a right-handed, Earth-Centered Earth-Fixed (ECEF) orthogonal coordinate system • Scale: Its scale is that of the local Earth frame, in the meaning of a relativistic theory of gravitation. Aligns with ITRS • Orientation: Given by the Bureau International de l’Heure (BIH) orientation of 1984.0 • Time Evolution: Its time evolution in orientation will create no residual global rotation with regards to the crust Coordinate System: Cartesian Coordinates (X, Y, Z).
    [Show full text]
  • Ductile Deformation - Concepts of Finite Strain
    327 Ductile deformation - Concepts of finite strain Deformation includes any process that results in a change in shape, size or location of a body. A solid body subjected to external forces tends to move or change its displacement. These displacements can involve four distinct component patterns: - 1) A body is forced to change its position; it undergoes translation. - 2) A body is forced to change its orientation; it undergoes rotation. - 3) A body is forced to change size; it undergoes dilation. - 4) A body is forced to change shape; it undergoes distortion. These movement components are often described in terms of slip or flow. The distinction is scale- dependent, slip describing movement on a discrete plane, whereas flow is a penetrative movement that involves the whole of the rock. The four basic movements may be combined. - During rigid body deformation, rocks are translated and/or rotated but the original size and shape are preserved. - If instead of moving, the body absorbs some or all the forces, it becomes stressed. The forces then cause particle displacement within the body so that the body changes its shape and/or size; it becomes deformed. Deformation describes the complete transformation from the initial to the final geometry and location of a body. Deformation produces discontinuities in brittle rocks. In ductile rocks, deformation is macroscopically continuous, distributed within the mass of the rock. Instead, brittle deformation essentially involves relative movements between undeformed (but displaced) blocks. Finite strain jpb, 2019 328 Strain describes the non-rigid body deformation, i.e. the amount of movement caused by stresses between parts of a body.
    [Show full text]
  • Lecture 19 1 the Ellipsoid Method for LP
    ORIE 6300 Mathematical Programming I October 31, 2014 Lecture 19 Lecturer: David P. Williamson Scribe: Nozomi Hitomi 1 The Ellipsoid Method for LP Recall we discussed the ellipsoid method last time: Given some bounded polyhedron P = fx 2 n R : Cx ≤ Dg, either finds x 2 P or states that P = ;; that is, P is infeasible. How can we use this feasibility detector to solve an optimization problem such as min cT x : Ax ≤ b; x ≥ 0? We claim that we can do this by making three calls to the ellipsoid method. 1.1 Idea of Ellipsoid method Let's now give the basic idea of how the ellipsoid method will work. • Start with a sphere large enough to contain all feasible points. Call the sphere E0, center a0. • If ak 2 P , done (i.e. obeys constraints). Return ak. • If not, ak 2= P , since Cjak > dj for some j. • Divide ellipsoid Ek in half through center ak with a hyperplane parallel to the constraint Cjx = dj. • Compute new ellipsoid Ek+1, center ak+1, containing the "good" half of Ek that contains P . Repeat. 1.2 Showing progress using the Ellipsoid method Recall: L ≡ number of bits to represent C; d. For any vertex x; xj needs at most nU + n log(n) bits to represent n ≡ numbers of vars, U ≡ largest entry in C; d. nU+n log(n) • =) jxjj ≤ 2 • =) sphere of radius 2L contains all vertices and has volume 2O(nL). Note, our initial ellipsoid is a sphere centered at origin and we know for any vertex x; jxjj ≤ 2nU+n log n, so sphere of radius 2L, volume 2O(nL), will contain the feasible region.
    [Show full text]
  • Inferring 3D Ellipsoids Based on Cross-Sectional Images with Applications to Porosity Control of Additive Manufacturing
    Inferring 3D Ellipsoids based on Cross-Sectional Images with Applications to Porosity Control of Additive Manufacturing Jianguo Wu1*, Yuan Yuan2, Haijun Gong3, Tzu-Liang (Bill) Tseng4 1* Department of Industrial Engineering and Management, Peking University, Beijing, China, 100871 E-mail: [email protected] 2 IBM Research, Singapore 18983 3 Department of Manufacturing Engineering, Georgia Southern University, Statesboro, GA 30458, USA 4 Department of Industrial, Manufacturing and Systems Engineering, University of Texas at El Paso, TX 79968, USA Abstract This paper develops a series of statistical approaches to inferring size distribution, volume number density and volume fraction of 3D ellipsoidal particles based on 2D cross-sectional images. Specifically, this paper first establishes an explicit linkage between the size of ellipsoidal particles and the size of cross-sectional elliptical contours. Then an efficient Quasi-Monte Carlo EM algorithm is developed to overcome the challenge of 3D size distribution estimation based on the established complex linkage. The relationship between the 3D and 2D particle number densities is also identified to estimate the volume number density and volume fraction. The effectiveness of the proposed method is demonstrated through simulation and case studies. KEY WORDS: Ellipsoid Intersection; Quasi-Monte Carlo; Expectation Maximization; Additive manufacturing; Porosity. 1. INTRODUCTION This paper develops a novel inspection method to infer the three-dimensional (3D) size distribution, volume number density (number per unit volume), and volume fraction (volume percentage) of ellipsoidal particles in materials based on 2D cross-sectional microscopic images. Although recent development of measurement technology makes the direct measurement of 3D particles possible, indirect measurement from lower dimension are still widely used for practical reasons.
    [Show full text]
  • A Astronomical Terminology
    A Astronomical Terminology A:1 Introduction When we discover a new type of astronomical entity on an optical image of the sky or in a radio-astronomical record, we refer to it as a new object. It need not be a star. It might be a galaxy, a planet, or perhaps a cloud of interstellar matter. The word “object” is convenient because it allows us to discuss the entity before its true character is established. Astronomy seeks to provide an accurate description of all natural objects beyond the Earth’s atmosphere. From time to time the brightness of an object may change, or its color might become altered, or else it might go through some other kind of transition. We then talk about the occurrence of an event. Astrophysics attempts to explain the sequence of events that mark the evolution of astronomical objects. A great variety of different objects populate the Universe. Three of these concern us most immediately in everyday life: the Sun that lights our atmosphere during the day and establishes the moderate temperatures needed for the existence of life, the Earth that forms our habitat, and the Moon that occasionally lights the night sky. Fainter, but far more numerous, are the stars that we can only see after the Sun has set. The objects nearest to us in space comprise the Solar System. They form a grav- itationally bound group orbiting a common center of mass. The Sun is the one star that we can study in great detail and at close range. Ultimately it may reveal pre- cisely what nuclear processes take place in its center and just how a star derives its energy.
    [Show full text]
  • Demo of Some Simple Cylinders and Quadratic Surfaces
    Demo of some simple cylinders and quadratic surfaces Yunkai Zhou Department of Mathematics Southern Methodist University (Prepared for Calculus-III, Math 2339) Acknowledgement: The very nice free software K3dSurf was used for the plots. Math 2339, SMU – p. 1/22 Left: Cylinder x = cos(z); Right: Cylinder y = sin(z) Math 2339, SMU – p. 2/22 Left: Cylinder x2 + y2 =1; Right: Three cylinders x = cos(z), y = sin(z), x2 + y2 =1 intersecting each other. Notice the intersection of the three cylinders is the well-known space curve helix −→r (t)= cos(t), sin(t), t Math 2339, SMU – p. 3/22 x2 y2 z2 Ellipsoid + + =1 a2 b2 c2 Math 2339, SMU – p. 4/22 x2 y2 z2 Ellipsoid + + =1; y = b , (|b | < |b|) a2 b2 c2 1 1 (Notice the intersection is an ellipse. In fact the intersection of an ellipsoid with any plane that intersects with it is an ellipse.) Math 2339, SMU – p. 5/22 x2 y2 z2 Ellipsoid + + = 1; x = c , y = c , z = c a2 b2 c2 1 2 3 (The intersection of an ellipsoid with any plane (not necessarily parallel to the coordinate planes) is an ellipse.) Math 2339, SMU – p. 6/22 x2 y2 Elliptic paraboloid z = + + c a2 b2 Math 2339, SMU – p. 7/22 x2 y2 Elliptic paraboloid z = + + c; x = c , y = c a2 b2 1 2 The intersection of an elliptic paraboloid (1) with any plane parallel to the z-axis is a parabola; (2) with any plane not parallel to the z-axis but intersects the paraboloid is an ellipse.
    [Show full text]
  • 11. Quadratic Forms and Ellipsoids
    CS/ECE/ISyE 524 Introduction to Optimization Spring 2017{18 11. Quadratic forms and ellipsoids Quadratic forms Orthogonal decomposition Positive definite matrices Ellipsoids Laurent Lessard (www.laurentlessard.com) Quadratic forms Linear functions: sum of terms of the form ci xi where the ci are parameters and xi are variables. General form: T c1x1 + ··· + cnxn = c x Quadratic functions: sum of terms of the form qij xi xj where qij are parameters and xi are variables. General form: 2 2 2 q11x1 + q12x1x2 + ··· + qnnxn (n terms) 2 3T 2 3 2 3 x1 q11 ::: q1n x1 6 . 7 6 . .. 7 6 . 7 T = 4 . 5 4 . 5 4 . 5 = x Qx xn qn1 ::: qnn xn 11-2 Quadratic forms Example: 4x 2 + 6xy − 2yz + y 2 − z 2 2x3T 2460 3 2x3 4y5 4010 5 4y5 z 0 −2 −1 z 2 3T 2 3 2 3 8 x 4 p2 q2 x < p1 + p2 = 6 In general: 4y5 4p1 1 r2 5 4y5 q1 + q2 = 0 z q1 r1 −1 z : r1 + r2 = −2 2x3T 24 3 0 3 2x3 Symmetric: 4y5 43 1 −15 4y5 z 0 −1 −1 z 11-3 Quadratic forms Any quadratic function f (x1;:::; xn) can be written in the form x TQx where Q is a symmetric matrix (Q = QT). T Proof: Suppose f (x1;:::; xn) = x Rx where R is not symmetric. Since it is a scalar, we can take the transpose: T x TRx = x TRx = x TRTx Therefore: T 1 T T T T 1 T x Rx = 2 x Rx + x R x = x 2 (R + R )x 1 T So we're done, because 2 (R + R ) is symmetric! 11-4 Orthogonal decomposition Theorem.
    [Show full text]