DOCSLIB.ORG

Zernike Polynomials: a Guide

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Zernike Polynomials: a Guide This article was downloaded by: [65.121.24.70] On: 02 November 2011, At: 12:16 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Modern Optics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tmop20 Zernike polynomials: a guide Vasudevan Lakshminarayanan a b & Andre Fleck a c a School of Optometry, University of Waterloo, Waterloo, Ontario, Canada b Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI, USA c Grand River Hospital, Kitchener, Ontario, Canada Available online: 04 Apr 2011 To cite this article: Vasudevan Lakshminarayanan & Andre Fleck (2011): Zernike polynomials: a guide, Journal of Modern Optics, 58:7, 545-561 To link to this article: http://dx.doi.org/10.1080/09500340.2011.554896 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Journal of Modern Optics Vol. 58, No. 7, 10 April 2011, 545–561 TUTORIAL REVIEW Zernike polynomials: a guide Vasudevan Lakshminarayanana,b*y and Andre Flecka,c aSchool of Optometry, University of Waterloo, Waterloo, Ontario, Canada; bMichigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI, USA; cGrand River Hospital, Kitchener, Ontario, Canada (Received 5 October 2010; final version received 9 January 2011) In this paper we review a special set of orthonormal functions, namely Zernike polynomials which are widely used in representing the aberrations of optical systems. We give the recurrence relations, relationship to other special functions, as well as scaling and other properties of these important polynomials. Mathematica code for certain operations are given in the Appendix. Keywords: optical aberrations; Zernike polynomials; special functions; aberrometry 1. Introduction the Zernikes will not be orthogonal over a discrete set The Zernike polynomials are a sequence of polyno- of points within a unit circle [11]. However, Zernike mials that are continuous and orthogonal over a unit polynomials offer distinct advantages over other poly- circle. A large fraction of optical systems in use today nomial sequences. Using the normalized Zernike employ imaging elements and pupils which are circu- expansion to describe aberrations offers the advantage lar. As a result, Zernike polynomials have been that the coefficient or value of each mode represents adopted as a mathematical description of optical the root mean square (RMS) wavefront error attrib- wavefronts propagating through such systems. An utable to that mode. The Zernike coefficients used to optical wavefront can be thought of as the surface of mathematically describe a wavefront are independent equivalent phase for radiation produced by a mono- of the number of polynomials used in the sequence. chromatic light source. For a point source at infinite This condition of independence or orthogonality, distance this surface is a plane wave. An example is means that any number of additional terms can be shown in Figure 1 with an aberrated wavefront for added without impact on those already computed. comparison. The mathematical description, offered by Coefficients of larger magnitude indicate greater con- Zernike polynomials, is useful in defining the magni- tribution of that particular mode to the total RMS tude and characteristics of the differences between the wavefront error of the system and thus greater image formed by an optical system and the original negative impact on the optical performance of the object. These optical aberrations can be a result of system. Downloaded by [65.121.24.70] at 12:16 02 November 2011 optical imperfections in the individual elements of an Having stated the advantages of Zernike polyno- optical system and/or the system as a whole. First mials in describing wavefront aberrations over a unit employed by F. Zernike, in his phase contrast circle it should also be clearly noted that Zernike method for testing circular mirrors [3], they have polynomials are not always the best polynomials for since gained widespread use due to their orthogonality fitting wavefront test data. In certain cases, Zernike and their balanced representation of classical aberra- polynomials may provide a poor representation of the tions yielding minimum variance over a circular pupil wavefront. Some of the effects of air turbulence in [4–9]. astronomy and effects of fabrication errors in the The Zernike polynomials are but one of infinite production of optical elements may not be well number of complete sets of polynomials, with two represented by even a large expansion of the Zernike variables, that are orthogonal and continuous over the sequence. In the testing of conical optical elements interior of a unit circle [10]. The condition of being or irregular ocular optics, such as a condition known continuous is important to note because, in general, as keratoconus (literally a cone shaped cornea), *Corresponding author. Email: [email protected] yVL is also with the departments of Physics and Electrical Engineering at the University of Waterloo. ISSN 0950–0340 print/ISSN 1362–3044 online ß 2011 Taylor & Francis DOI: 10.1080/09500340.2011.554896 http://www.informaworld.com 546 V. Lakshminarayanan and A. Fleck Figure 1. A plane wave (perfect wavefront) and an aberrated wavefront subdivided into smaller wavefronts where the local slope determines the location of a ray tracing projection of the sublet (subdivided wavelet). This is the operating principle of wavefront sensors [1]. (The color version of this figure is included in the online version of the journal.) additional terms must be added to Zernike polyno- mials to accurately represent alignment errors and even in so doing, the description of the optical quality may be incomplete [12]. In any system with noncircular pupils, the Zernike circle polynomials will not be orthogonal over the whole of the pupil area and thus may not be the ideal polynomial sequence. 2. Mathematical basis In general, the function describing an arbitrary wavefront in polar coordinates (r, ), denoted by W(r, ), can be expanded in terms of a sequence of polynomials Z that are orthonormal over the entire surface of the circular pupil:X m m Wðr, Þ¼ Cn Zn ðr, Þ, ð1Þ Downloaded by [65.121.24.70] at 12:16 02 November 2011 n, m where C denotes the Zernike amplitudes or coefficients Figure 2. Cartesian (x, y) and polar (r, ) coordinates of a and Z the polynomials. The coordinate system is point Q in the plane of a unit circle representing the circular shown in Figure 2. exit pupil of an imaging system [6]. The Zernike polynomials expressed in polar coor- dinates (X ¼ r sin , Y ¼ r cos ) are given by the complex combination: clockwise from the y-axis. This is consistent with Zmðr, ÞiZÀmðr, Þ¼VÀmðr cos , r sin Þ n n n aberration theory definitions, but different from the m ¼ Rn ðrÞ expðimÞ, conventional mathematical definition of polar coordi- which leads to nates. The convention employed is at the discretion of Zmðr, Þ¼RmðrÞ cos m for m 0, the author and may differ depending on the applica- n n m Àm m tion. The radial function, Rn ðrÞ, is described by: Zn ðr, Þ¼Rn ðrÞ sin m for m 5 0, ð2Þ where r is restricted to the unit circle (0 r 1), ðnXÀmÞ=2 ð1Þl ðn À lÞ! RmðrÞ¼ ÂÃÂÃrnÀ2l: ð3Þ meaning that the radial coordinate is normalized by n l! 1 ðn þ mÞl ! 1 ðn À mÞl ! the semi-diameter of the pupil, and is measured l¼0 2 2 Journal of Modern Optics 547 visual/ophthalmic optics; it is nothing but the ordinary spectacle correction that is commonly prescribed by optometrists. Furthermore, the radial order n ¼ 1is equivalent to prism terms that are prescribed in spectacles as well. The normalization has been chosen to satisfy Æm Rn ð1Þ¼1 for all values of n and m. This gives a normalization constant described by: 1=2 m 2ðn þ 1Þ Nn ¼ , ð4Þ 1 þ m0 where m0 is the Kronecker delta (m0 ¼ 0 for m 6¼ 0). The algebraic form of the first 10 orders of Zernike polynomials which correspond to the surface plots of Figure 4 are shown in Tables 1 and 2, in both polar and Cartesian form, for comparison. In Cartesian coordi- nates, the 0 r 1 limit becomes x2 þ y2 1, and the recursion relationship for the sequence in Cartesian form is given by [13]: ! ! Xq XM MXÀJ n À 2M M À j m iþj Zn ðx, yÞ¼ ð1Þ i¼0 j¼0 k¼0 2i þ p k ðn À j Þ!  xy, ð5Þ j!ðM À j Þ!ðn À M À j Þ! where n n jsj M ¼ À m À , p ¼ ðs þ 1Þ, 2 2 2 s q ¼ðd À sMod½n,2Þ , s ¼ sgnðd Þ, d ¼ n À 2m, 2 ¼ 2ði þ kÞþp, ¼ n À 2ði þ j þ kÞp: As can be seen from the relationship above and the expanded form shown in Tables 1 and 2, representa- Downloaded by [65.121.24.70] at 12:16 02 November 2011 tion of the polynomials in polar form offers some advantage in efficiency by nature of its circular symmetry.
Load more

Recommended publications
  • Quaternion Zernike Spherical Polynomials
    MATHEMATICS OF COMPUTATION Volume 84, Number 293, May 2015, Pages 1317–1337 S 0025-5718(2014)02888-3 Article electronically published on August 29, 2014 QUATERNION ZERNIKE SPHERICAL POLYNOMIALS J. MORAIS AND I. CAC¸ AO˜ Abstract. Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs) in many different fields of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this develop- ment. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In the present paper we introduce the Zernike spherical polynomials within quater- nionic analysis ((R)QZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with R3 and R4). (R)QZSPs are orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second or- der homogeneous differential equation are also discussed. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. (R)QZSPs are new in literature and have some consequences that are now under investigation. 1. Introduction 1.1. The Zernike spherical polynomials. The complex Zernike polynomials (ZPs) have long been successfully used in many different fields of optics.
    [Show full text]
  • Wavefront Parameters Recovering by Using Point Spread Function
    Wavefront Parameters Recovering by Using Point Spread Function Olga Kalinkina[0000-0002-2522-8496], Tatyana Ivanova[0000-0001-8564-243X] and Julia Kushtyseva[0000-0003-1101-6641] ITMO University, 197101 Kronverksky pr. 49, bldg. A, Saint-Petersburg, Russia [email protected], [email protected], [email protected] Abstract. At various stages of the life cycle of optical systems, one of the most important tasks is quality of optical system elements assembly and alignment control. The different wavefront reconstruction algorithms have already proven themselves to be excellent assistants in this. Every year increasing technical ca- pacities opens access to the new algorithms and the possibilities of their appli- cation. The paper considers an iterative algorithm for recovering the wavefront parameters. The parameters of the wavefront are the Zernike polynomials coef- ficients. The method involves using a previously known point spread function to recover Zernike polynomials coefficients. This work is devoted to the re- search of the defocusing influence on the convergence of the algorithm. The method is designed to control the manufacturing quality of optical systems by point image. A substantial part of the optical systems can use this method with- out additional equipment. It can help automate the controlled optical system ad- justment process. Keywords: Point Spread Function, Wavefront, Zernike Polynomials, Optimiza- tion, Aberrations. 1 Introduction At the stage of manufacturing optical systems, one of the most important tasks is quality of optical system elements assembly and alignment control. There are various methods for solving this problem, for example, interference methods. However, in some cases, one of which is the alignment of the telescope during its operation, other control methods are required [1, 2], for example control by point image (point spread function) or the image of another known object.
    [Show full text]
  • Wavefront Aberrations
    11 Wavefront Aberrations Mirko Resan, Miroslav Vukosavljević and Milorad Milivojević Eye Clinic, Military Medical Academy, Belgrade, Serbia 1. Introduction The eye is an optical system having several optical elements that focus light rays representing images onto the retina. Imperfections in the components and materials in the eye may cause light rays to deviate from the desired path. These deviations, referred to as optical or wavefront aberrations, result in blurred images and decreased visual performance (1). Wavefront aberrations are optical imperfections of the eye that prevent light from focusing perfectly on the retina, resulting in defects in the visual image. There are two kinds of aberrations: 1. Lower order aberrations (0, 1st and 2nd order) 2. Higher order aberrations (3rd, 4th, … order) Lower order aberrations are another way to describe refractive errors: myopia, hyperopia and astigmatism, correctible with glasses, contact lenses or refractive surgery. Lower order aberrations is a term used in wavefront technology to describe second-order Zernike polynomials. Second-order Zernike terms represent the conventional aberrations defocus (myopia, hyperopia and astigmatism). Lower order aberrations make up about 85 per cent of all aberrations in the eye. Higher order aberrations are optical imperfections which cannot be corrected by any reliable means of present technology. All eyes have at least some degree of higher order aberrations. These aberrations are now more recognized because technology has been developed to diagnose them properly. Wavefront aberrometer is actually used to diagnose and measure higher order aberrations. Higher order aberrations is a term used to describe Zernike aberrations above second-order. Third-order Zernike terms are coma and trefoil.
    [Show full text]
  • Zernike Polynomials: a Guide
    See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/241585467 Zernike polynomials: A guide Article in Journal of Modern Optics · April 2011 DOI: 10.1080/09500340.2011.633763 CITATIONS READS 41 12,492 2 authors: Vasudevan Lakshminarayanan Andre Fleck University of Waterloo Grand River Hospital 399 PUBLICATIONS 1,806 CITATIONS 40 PUBLICATIONS 1,210 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Theoretical and bench color and optics work View project Quantum Relational Databases View project All content following this page was uploaded by Vasudevan Lakshminarayanan on 13 May 2014. The user has requested enhancement of the downloaded file. Journal of Modern Optics Vol. 58, No. 7, 10 April 2011, 545–561 TUTORIAL REVIEW Zernike polynomials: a guide Vasudevan Lakshminarayanana,b*y and Andre Flecka,c aSchool of Optometry, University of Waterloo, Waterloo, Ontario, Canada; bMichigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI, USA; cGrand River Hospital, Kitchener, Ontario, Canada (Received 5 October 2010; final version received 9 January 2011) In this paper we review a special set of orthonormal functions, namely Zernike polynomials which are widely used in representing the aberrations of optical systems. We give the recurrence relations, relationship to other special functions, as well as scaling and other properties of these important polynomials. Mathematica code for certain operations are given in the Appendix. Keywords: optical aberrations; Zernike polynomials; special functions; aberrometry 1. Introduction the Zernikes will not be orthogonal over a discrete set The Zernike polynomials are a sequence of polyno- of points within a unit circle [11].
    [Show full text]
  • New Separated Polynomial Solutions to the Zernike System on the Unit Disk and Interbasis Expansion
    1844 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America A Research Article New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion 1,2,3 4, 1 GEORGE S. POGOSYAN, KURT BERNARDO WOLF, * AND ALEXANDER YAKHNO 1Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico 2Yerevan State University, Yerevan, Armenia 3Joint Institute for Nuclear Research, Dubna, Russia 4Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Cuernavaca, Morelos 62251, Mexico *Corresponding author: [email protected] Received 23 May 2017; revised 7 August 2017; accepted 22 August 2017; posted 23 August 2017 (Doc. ID 296673); published 19 September 2017 The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the unit disk to classify wavefront aberrations in circular pupils is shown to have a set of new orthonormal solution bases involving Legendre and Gegenbauer polynomials in nonorthogonal coordinates, close to Cartesian ones. We find the overlaps between the original Zernike basis and a representative of the new set, which turn out to be Clebsch–Gordan coefficients. © 2017 Optical Society of America OCIS codes: (000.3860) Mathematical methods in physics; (050.1220) Apertures; (050.5080) Phase shift; (080.1010) Aberrations (global); (350.7420) Waves. https://doi.org/10.1364/JOSAA.34.001844 1. INTRODUCTION: THE ZERNIKE SYSTEM to explain the symmetry hidden in the equal spacing of n familiar from the oscillator model. In 1934, Frits Zernike published a paper that gave rise to phase- ’ contrast microscopy [1].
    [Show full text]
  • Zernike Polynomials Background
    Zernike Polynomials • Fitting irregular and non-rotationally symmetric surfaces over a circular region. • Atmospheric Turbulence. • Corneal Topography • Interferometer measurements. • Ocular Aberrometry Background • The mathematical functions were originally described by Frits Zernike in 1934. • They were developed to describe the diffracted wavefront in phase contrast imaging. • Zernike won the 1953 Nobel Prize in Physics for developing Phase Contrast Microscopy. 1 Phase Contrast Microscopy Transparent specimens leave the amplitude of the illumination virtually unchanged, but introduces a change in phase. Applications • Typically used to fit a wavefront or surface sag over a circular aperture. • Astronomy - fitting the wavefront entering a telescope that has been distorted by atmospheric turbulence. • Diffraction Theory - fitting the wavefront in the exit pupil of a system and using Fourier transform properties to determine the Point Spread Function. Source: http://salzgeber.at/astro/moon/seeing.html 2 Applications • Ophthalmic Optics - fitting corneal topography and ocular wavefront data. • Optical Testing - fitting reflected and transmitted wavefront data measured interferometically. Surface Fitting • Reoccurring Theme: Fitting a complex, non-rotationally symmetric surfaces (phase fronts) over a circular domain. • Possible goals of fitting a surface: – Exact fit to measured data points? – Minimize “Error” between fit and data points? – Extract Features from the data? 3 1D Curve Fitting 25 20 15 10 5 0 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 Low-order Polynomial Fit 25 y = 9.9146x + 2.3839 R2 = 0.9383 20 15 10 5 0 -0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 In this case, the error is the vertical distance between the line and the data point.
    [Show full text]
  • Computing Human Optical Point Spread Functions
    Journal of Vision (2015) 15(2):26, 1–25 http://www.journalofvision.org/content/15/2/26 1 Computing human optical point spread functions Andrew B. Watson NASA Ames Research Center, Moffett Field, CA, USA $ There is renewed interest in the role of optics in human The wavefront aberrations can be used to compute vision. At the same time there have been advances that the optical point spread function (PSF; Dai, 2008; allow for routine standardized measurement of the Goodman, 2005; Mahajan, 2013). To do this we first wavefront aberrations of the human eye. Computational define the complex-valued generalized pupil function g, methods have been developed to convert these measurements to a description of the human visual i2p gðx0; y0Þ¼pðx0; y0Þexp wðx0; y0Þ ð2Þ optical point spread function (PSF), and to thereby k calculate the retinal image. However, tools to implement these calculations for vision science are not The real-valued function p is the pupil function that widely available or widely understood. In this report we describes transmission through the pupil. It may be describe software to compute the human optical PSF, defined as 1 within the pupil area and 0 elsewhere, but and we discuss constraints and limitations. may also be used to describe variable transmission as a function of position (apodization), such as due to the Stiles-Crawford effect (Atchison & Scott, 2002; Met- calf, 1965; Stiles & Crawford, 1933). Introduction The PSF for incoherent light is then given by the squared modulus of the Fourier transform of the Vision begins with a retinal image formed by the generalized pupil function, optics of the eye.
    [Show full text]
  • Zernike Polynomials
    ZernikePolynomialsForTheWeb.nb James C. Wyant, 2003 1 Zernike Polynomials 1 Introduction Often, to aid in the interpretation of optical test results it is convenient to express wavefront data in polynomial form. Zernike polynomials are often used for this purpose since they are made up of terms that are of the same form as the types of aberrations often observed in optical tests (Zernike, 1934). This is not to say that Zernike polynomials are the best polynomials for fitting test data. Sometimes Zernike polynomials give a poor representation of the wavefront data. For example, Zernikes have little value when air turbulence is present. Likewise, fabrication errors in the single point diamond turning process cannot be represented using a reasonable number of terms in the Zernike polynomial. In the testing of conical optical elements, additional terms must be added to Zernike polynomials to accurately represent alignment errors. The blind use of Zernike polynomials to represent test results can lead to disastrous results. Zernike polynomials are one of an infinite number of complete sets of polynomials in two variables, r and q, that are orthogonal in a continuous fashion over the interior of a unit circle. It is important to note that the Zernikes are orthogonal only in a continuous fashion over the interior of a unit circle, and in general they will not be orthogonal over a discrete set of data points within a unit circle. Zernike polynomials have three properties that distinguish them from other sets of orthogonal polynomials. First, they have simple rotational symmetry properties that lead to a polynomial product of the form r ρ g θ , where g[q] is a continuous function that repeats self every 2p radians and satisfies the requirement that rotating the coordinate system by an angle a does not change the form of the polynomial.
    [Show full text]
  • Computation of 2D Fourier Transforms and Diffraction Integrals Using Gaussian Radial Basis Functions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repositorio Institucional de la Universidad de Almería (Spain) Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions A. Martínez-Finkelshtein, D. Ramos-López and D.R. Iskander Applied and Computational Harmonic Analysis 2016 This is not the published version of the paper, but a pre-print. Please follow the links below for the final version and cite this paper as: A. Martínez-Finkelshtein, D. Ramos-López, D.R. Iskander. Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions. Applied and Computational Harmonic Analysis, ISSN 1063-5203 (2016). http://dx.doi.org/10.1016/j.acha.2016.01.007 http://www.sciencedirect.com/science/article/pii/S1063520316000087 Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions A. Martínez-Finkelshtein, D. Ramos-López and D.R. Iskander This is not the published version of the paper, but a pre-print. Please follow the links below for the final version and cite this paper as: A. Martínez-Finkelshtein, D. Ramos-López, D.R. Iskander. Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions. Applied and Computational Harmonic Analysis, ISSN 1063-5203 (2016). http://dx.doi.org/10.1016/j.acha.2016.01.007 http://www.sciencedirect.com/science/article/pii/S1063520316000087 Computation of 2D Fourier transforms and diffraction integrals using Gaussian
    [Show full text]
  • IAT14 Imaging and Aberration Theory Lecture 12 Zernike Polynomials.Pdf
    Imaging and Aberration Theory Lecture 12: Zernike polynomials 2015-01-29 Herbert Gross Winter term 2014 www.iap.uni-jena.de 2 Preliminary time schedule 1 30.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems Pupils, Fourier optics, pupil definition, basic Fourier relationship, phase space, analogy optics and 2 06.11. Hamiltonian coordinates mechanics, Hamiltonian coordinates Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical 3 13.11. Eikonal approximation, inhomogeneous media single surface, general Taylor expansion, representations, various orders, stop 4 20.11. Aberration expansions shift formulas different types of representations, fields of application, limitations and pitfalls, 5 27.11. Representation of aberrations measurement of aberrations phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical 6 04.12. Spherical aberration surfaces, higher orders phenomenology, relation to sine condition, aplanatic sytems, effect of stop 7 11.12. Distortion and coma position, various topics, correction options 8 18.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options Dispersion, axial chromatical aberration, transverse chromatical aberration, 9 08.01. Chromatical aberrations spherochromatism, secondary spoectrum Sine condition, aplanatism and Sine condition, isoplanatism, relation to coma and shift invariance, pupil 10 15.01. isoplanatism aberrations, Herschel condition, relation to Fourier optics 11 22.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations special expansion for circular symmetry, problems, calculation, optimal balancing, 12 29.01. Zernike polynomials influence of normalization, measurement ideal psf, psf with aberrations, Strehl ratio, transfer function, resolution and 13 05.02. PSF and transfer function contrast Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic 14 12.02.
    [Show full text]
  • Standards for Reporting the Optical Aberrations of Eyes
    Standards for Reporting the Optical Aberrations of Eyes Larry N. Thibos, PhD; Raymond A. Applegate, OD, PhD; James T. Schwiegerling, PhD; Robert Webb, PhD; VSIA Standards Taskforce Members ABSTRACT tional standards for reporting aberration data and In response to a perceived need in the vision to specify test procedures for evaluating the accura- community, an OSA taskforce was formed at the cy of data collection and data analysis methods. 1999 topical meeting on vision science and its appli- 1 cations (VSIA-99) and charged with developing con- Following a call for participation , approximately sensus recommendations on definitions, conven- 20 people met at VSIA-99 to discuss the proposal to tions, and standards for reporting of optical aber- form a taskforce that would recommend standards rations of human eyes. Progress reports were pre- for reporting optical aberrations of eyes. The group sented at the 1999 OSA annual meeting and at VSIA- agreed to form three working parties that would 2000 by the chairs of three taskforce subcommittees on (1) reference axes, (2) describing functions, and take responsibility for developing consensus recom- (3) model eyes. [J Refract Surg 2002;18:S652-S660] mendations on definitions, conventions and stan- dards for the following three topics: (1) reference axes, (2) describing functions, and (3) model eyes. It he recent resurgence of activity in visual was decided that the strategy for Phase I of this pro- optics research and related clinical disciplines ject would be to concentrate on articulating defini- T(eg, refractive surgery, ophthalmic lens tions, conventions, and standards for those issues design, ametropia diagnosis) demands that the which are not empirical in nature.
    [Show full text]
  • Aberration Correction Using Adaptive Optics in an Optical Trap
    Aberration Correction Using Adaptive Optics in an Optical Trap Carrie Voycheck, 2006 Advised by Daniel G. Cole, Research Assistant Department of Mechanical Engineering Introduction trap is created when the gradient forces are greater than the The processing of optical trapping is extremely useful in scattering forces. the biological sciences for manipulating microscopic particles The optical trapping system consists of a laser propagating and organisms using laser light. Imperfections in the elements through four lenses and into an inverted microscope in order comprising the optical trap cause deformations in the laser to trap a specimen on the stage. Here the beam partially beam, which decrease the effectiveness of the trap. The overfills the microscope objective. Moving from the laser to the wavefront distortion on the beam can be described by Zernike objective, the focal length of the lenses are: 500mm, (SLM, polynomials or disk harmonic functions. The coefficients of see below), 750mm, 300mm, and 150mm. The microscope these functions can be manipulated to compensate for the objective is 100x (focal length of 1.64mm). When constructing aberration making a diffraction limited spot, and therefore, a single beam optical trap, an objective with a high numerical a more effective trap. The aberration correction is performed aperture is necessary in order to bring the beam to a tight focus. using an adaptive simulation in the programming language Two telescopes, created by L1 and L2, and L3 and L4 (Figure MATLAB. Three elements are required in an adaptive 2), are used to match the size of the beam at the entrance pupil optics system: a wavefront sensor, a corrective element, and of the objective to the spatial light modulator (SLM).2 A spatial a feedback or control loop.
    [Show full text]