WAVEFRONT OPTICS FOR THE NON-MATHEMATICIAN
|
|
- Laureen Glenn
- 5 years ago
- Views:
Transcription
1 WAVEFRONT OPTICS FOR THE NON-MATHEMATICIAN Understanding its basics of and rationale can help clinicians broaden their horizons. Y GAURAV PRAKASH, MD, FRCS (GLASGOW); AND VISHAL JHANJI, MD, FRCOphth An understanding of wavefront optics is an important tool in the armamentarium of ophthalmologists. 1-4 This concept has been used to explain and detect conditions in refractive surgery, cataract surgery, amblyopia, wound healing, keratoconus, and ocular surface abnormalities such as pterygium and dry eye. A deeper understanding of the interpretations of wavefront optics involves the use of complex mathematics, which often prompts ophthalmologists to avoid the subject. The aim of this article is to summarize five fundamental points that can be used for a better understanding of wavefront optics. We have omitted advanced mathematics and calculus, and there is only a minimal need of basic mathematics in certain sections. First, it is important to understand why wavefront-based assessment of refractive error is crucial in certain situations. Figure 1 shows a fictional shape that represents an eye with an irregular cornea with an approximate refractive error of D sphere and D cylinder, plus a little extra error (to be discussed below). Let us assume that we have access only to spherical powers (3.00, 2.00, 1.00, -1.00, -2.00, D, etc) for correction of this error. In this scenario, our best guess fit would be the spherical equivalent, which is the spherical power plus half the cylinder power, or in this case -3 +(-2)/2= -4. So a D sphere correction would be a sufficient compromise but not accurate enough. If we add the inventory of cylinders, we realize that the actual error is much more accurately predicted by D sphere with the addition of D cylinder in the correct axis. A similar analogy between conventional refraction (retinoscopic/subjective) and wavefront-based refraction is helpful. Although we can get to a fair compromise with spherocylindrical correction, using and understanding the wavefront error of the eye helps us to measure that little extra mentioned above. This is useful in eyes in which this little extra is more than normal, such as those with pathologic corneas. With this knowledge basis, let us dive into the world of wavefront optics. A C Figure 1. A fictional shape representing an eye with an irregular cornea (A) corrected by spherical equivalent (), spherocylinder (C), and wavefront aberrometric refraction (D). THE ASICS OF THE MATHEMATICS AND PHYSICS INVOLVED Herein we will discuss four terms from mathematics and physics: (1) the x-y coordinate system; (2) its cousin, the polar coordinate system; (3) basic trigonometry; and (4) the wave nature of light. This is perhaps the minimum mathematics required to understand the concept of wavefront optics. However, this section can be skipped or revisited later by the reader without loss of continuity. The Cartesian or x-y coordinate system. The x-y coordinate system is based on the old-fashioned system of walking around the block (Figure 2A). Let us take the example of Jim and Tom. When Jim says, To get to Tom s house from mine, walk to the signal that is 4 km from here, then take a left turn and walk 3 km to reach your destination, Jim has essentially summarized the x-y (or the Cartesian) coordinate system. Jim s house is the origin, the 4 km to the signal is the distance on the x-axis, and the 3 km from the signal is the distance on the y-axis. A fancier way of saying this is T(x,y) = (4,3). What if you wanted to know the actual shortest distance between Jim s and Tom s D FERUARY 2016 CATARACT & REFRACTIVE SURGERY TODAY EUROPE 67
2 FUNDAMENTALS A C D Figure 2. asic mathematics required to understand wavefront optics, with Cartesian coordinates (A), polar coordinates (), their conversion (C), and the concept of a unit circle (D). houses, imagining you have magical powers to fly directly but are in fuel-conservation mode? Pythagoras and his famed equation come to the rescue. Just square the two terms, add the squares, and take the square root of the sum. In this case, the result is 5 km. This can be represented mathematically as (x 2 + y 2 ) 1/2. It is important to know this calculation, as it helps in understanding how wavefront terms are added and also elucidates our next topic, the polar coordinate system. Polar coordinate system. In principle, the polar coordinate system does the same work as the Cartesian system, ie, to locate a point. However, the origin (0,0), or Jim s house in the previous example, is called the pole (Figure 2). The direct distance between Jim s and Tom s houses that was calculated above using the Pythagorean theorem is called the radial coordinate, and the angle between the reference axis (horizontal) and the radial coordinate is called the polar coordinate, polar angle, or azimuth. The polar coordinate system makes it easier to express the Zernike polynomials (our real building blocks). AT A GLANCE Wavefront optics can be used to explain and detect optical conditions of the eye caused by refractive surgery, cataract surgery, amblyopia, wound healing, keratoconus, and ocular surface abnormalities. A comprehensive understanding of the interpretations of wavefront optics involves the use of complex mathematics, which often prompts clinicians to avoid the subject. Knowledge of a few key principles related to wavefront optics can be useful for ophthalmologists. As discussed above, both of these systems, Cartesian and polar, are convertible using the Pythagorean theorem and some trigonometry (Figure 2C). Given that R and Θ (angle theta) are the radial and polar coordinates, the Cartesian coordinates will be x= R*Cosine Θ and y= R*Sin Θ. This brings us to the next points, the well-known cosine and sine. asic trigonometry and unit circle. In a right triangle, where one of the three angles is 90, the longest side is the hypotenuse and the other two sides are called opposite (to our favored angle) and adjacent (Figure 2D). The sum of all angles in a triangle can only be 180. As the angle between the opposite angle and the adjacent angle is 90, this leaves only the remaining 90 for the other two angles. These angles interplay in a wavelike fashion, and the relationship can be given by the ratio of the sides. This is the origin of sine and cosine. Another way to visualize this fixed relationship between sine and cosine is to imagine them as two children playing on a seesaw. If one child goes up, the other comes down by a similar amount. Finally, tangent (tan) is the ratio of the opposite and adjacent sides. In terms of Cartesian coordinates, this can be written as tan Θ = (y/x). This right triangle can be inserted into a circle, with the hypotenuse being equal to the radius. When this radius is 1 unit (cm/mm/m/any unit of distance), the circle is called a unit circle (Figure 2D). This is an important concept to understand, as Zernike polynomials are defined over a unit circle. The benefit of a unit circle is that there is no need to worry about the R from R Cosine Θ or R Sin Θ. Remember that R is the radius here, and the radius is 1. Anything multiplied by one is itself; thus, in a unit circle, the computations become generic. Wave nature of light. Finally, we come to the physics concept, the wave nature of light. Light can be interpreted as either a wave or a particle, but, for our purpose, its wave nature is more important. All of the points that originated at the same time from a point source, such as a small light bulb (or, analogously, a pebble dropped in still water), and that are in the same state of oscillation (phase), constitute the wavefront. UNDERSTANDING WAVEFRONT ERROR AND ITS MEASUREMENT An ideal wavefront of light traveling through a vacuum will have no additional imperfections, as there are no refracting media to cause refraction and scattering. However, this is not the case in the real world. Atmospheric interference distorts images of the celestial bodies, and to compensate for this astronomers came up with the concept of adaptive optics. To put it briefly, deformable mirrors were used to correct these 68 CATARACT & REFRACTIVE SURGERY TODAY EUROPE FERUARY 2016
3 distortions. Ophthalmology borrowed the concept. In the real-world scenario, a wavefront of parallel beams of light can be distorted by the refractive error of the eye. This wavefront error can be measured by aberrometers. There are multiple principles behind these devices, but, for the sake of simplicity, we will discuss the most commonly used method: the Hartmann-Shack principle. 2,4-6 Hartmann-Shack based devices have arrays of multiple small lenses, somewhat like the compound eye of an insect, rather than a single lens as in an autorefractor or a similar device (Figure 3). These small lenses measure deviations at multiple points on different parts of the emerging wavefront. ased on the relative position of the centroids, an image of the wavefront is reconstructed and then compared with the ideal wavefront. 2,4,6 This difference is the total wavefront error. It is typically measured in root-mean-square (RMS) microns and can be converted to equivalent defocus (in diopters). However, as the optical effects of 1.00 D of spherical power are not the same as those of 1.00 D of astigmatism, the equivalent effects of higher-order aberration (HOA) RMS and defocus are not same. Nonetheless, if a clinician does not want to use microns, he or she can use the following conversion formula to get the values in diopters: equivalent diopters = 4π 3 x (RMS in µm/pupil area in mm 2 ). 7 REAKING DOWN WAVEFRONT ERROR INTO ZERNIKE POLYNOMIALS Total wavefront error is a useful entity in itself. It gives an idea of total distortion in the eye; however, it is not of much use if we cannot break it into smaller parts. Imagine the total wavefront error to be a lump of dough. It would not make an attractive cookie at this stage. However, multiple cookie cutters can be used to shape smaller, more recognizable pieces. The benefit of using shapes such as coma, spherical aberration, and trefoil lies in the easier visualization of the data. One can break down the wavefront error into Zernike polynomials or shapes. A polynomial is an expression of two or more algebraic terms: for example, 4x 2 + 3x + y. Zernike polynomials are orthogonal over a unit circle, meaning that the integral (sum) of these terms will be equal to zero. These polynomials take forms such as P 3 Cos3Θ (in polar coordinate form) or x 3-3xy 2 (in Cartesian format). 6,8,9 The most intuitive and convenient way to write down the Zernike polynomials is the double-index scheme, Z m. In this, n the superscript index is for the meridional frequency (m), and the subscript index is of the order (n). 2,6,9,10 The index n is the highest power or order of the radial polynomial (P 3 ; therefore, 3 from the example above), and the index m describes the azimuthal frequency, or, in simpler terms, the complexity of the waveform regarding crests and troughs. For example, a meridional frequency of three in our example (3Θ) denotes three crests along the circumference of the waveform (for example, in trefoil). The polynomials with a cosine term are noted as horizontal, and the ones with a sine term are noted as vertical. Following this scheme, one can derive the common name of a polynomial from the given term. It may seem daunting to look at Zernike polynomials at first. However, it helps to write Zernike polynomials in a visually interesting and logical way, in the form of a Zernike pyramid (Figure 4). 2,6,9,10 In this scheme, the order of the polynomial increases as one moves from the apex to the base of the pyramid, and the meridional frequency changes (0, ±2, ±4 or, ±1, ±3, ± 5 ) as one moves laterally in the pyramid. We will discuss individual aberrations in the next section. efore that, let us consider the significance of the order. The zero order (piston) is the mean value of a wavefront or phase profile across the pupil of an optical system. The first order (tilt) aberration represents the image point location and not its quality. Thus, these do not represent or model the wavefront and are not important in our analysis of the refractive system. Our interest starts from the second order. The secondorder aberrations comprise the spherocylinder and are called defocus (Z 20 ) and astigmatism (Z -2 2 and Z 22 ). All orders larger than two are referred to as HOAs. In a clinical scenario, most wavefront errors can be sufficiently described up to the fifth order. In distorted corneas, such as in advanced keratoconus, further expansion can be performed to include more Zernike terms, or Fourier analysis may be needed to describe the error fully. 11,12 However, it should be noted that the quality of data capture in very distorted corneas is also compromised, and, therefore, the interpretation of this extra noise may be clinically distracting. FUNDAMENTALS Figure 3. Schematic diagram showing the principle of Hartmann-Shack aberrometry performed on the eye. INDIVIDUAL AERRATIONS It is helpful to understand refractive error as a 3-D shape (Figure 5). Among the lower-order aberrations, defocus (Z 20 ) is synonymous with spherical error. It is a symmetrical error, in that it is similar if you rotate the error along its axis. Astigmatism is described as oblique astigmatism (Z 2-2 ) and vertical astigmatism (Z 22 ); both are asymmetrical errors. Astigmatism has one principal meridian (axis/orientation) that corresponds to the steep shape of the 3-D refractive error. Third-order aberration: coma (vertical Z 3-1 and horizontal Z 31 ). In coma, aberration images appear to have a tail, FERUARY 2016 CATARACT & REFRACTIVE SURGERY TODAY EUROPE 69
4 FUNDAMENTALS certain conditions can change the spherical aberration profile of the eye. One such condition is excimer laser ablation, especially with a small optical zone and a small transition zone for a higher power (Figure 8). Fourth-order aberration: quadrafoil. This and subsequent foils are similar to trefoil. Instead of three steep zones, qudrafoil has four, and pentafoil has five. Secondary fourth-order aberrations: (secondary coma, secondary astigmatism, etc). These are more complex shapes based on the primary shapes described above. Figure 4. Schematic diagram of the Zernike pyramid. like a comet. Coma can be described three dimensionally as an aberration in which there is an area of more positive power and another area of more negative power in the same axis. Regarding ophthalmic optics, another way to understand coma is that it occurs when there is a difference in the power in the refracting area (cornea or lens) through which the light ray passes (in the zone of the entrance pupil). Note that this difference must be rotationally nonsymmetrical. In ophthalmology, the most common causes for the exaggerated difference in corneal power over the pupillary area are a decentered ablation, an eccentric corneal graft, and keratoconus. In a mirror and lens system, coma occurs when the incident light rays, especially the peripheral ones, are not perpendicular to the refracting surface (Figure 6A). This can also be the case when the object plane or the image plane is not parallel to the lens plane. The visual complaint voiced by patients with coma is increased tailing around light sources (Figure 6). Third-order aberration: trefoil (vertical Z 3-3 and oblique Z 33 ). Trefoil means three leaves. This aberration looks like three streaks of light originating from a point source. The simplest way to understand trefoil is to look at it as astigmatism with three difference axes, or triangular astigmatism. This is evident when one looks at the 3-D image of trefoil aberration (Figure 5). Fourth-order aberration: spherical (Z 40 ). In a curved refracting surface, the light rays falling on the periphery tend to deviate more compared with the ones falling on the center (paraxial rays). This occurs because of the additional prismatic effect of the periphery of the curved refractive surface (Figure 7). In positive spherical aberration, peripheral rays bend more than paraxial rays and meet before the expected focus. In negative spherical aberration, peripheral rays bend less and meet beyond the expected focus. This aberration is over and above defocus, and these two should not be confused, despite their similar shapes. The most common visual complaints of patients with increased spherical aberration are a loss of crispness and halos around images. In a normal human eye, multiple methods such as pupillary aperture and asphericity of the cornea reduce the effective spherical aberration of light entering the eye. However, INTERPRETING WAVEFRONT MAPS AND CLINICAL EXAMPLES Wavefront maps and outputs vary from device to device. The basic principles remain the same. A systematic approach can help the clinician derive the most useful data from the maps. The first thing to look for should be the quality of the scan. Some devices give an option of looking at the raw centroid image; however, most automatically decide the quality of acquisition. The next step is to look at the pupil size and the wavefront diameter. The RMS values are dependent on the pupillary diameter, and, therefore, as an attempt toward standardization, most studies report data at 4 or 6 mm for whole-eye aberrations. Next, one should look at the total wavefront error and higher-order wavefront error. The percentage of the wavefront error governed by the HOAs can provide two important interpretations: (1) how much of the refractive error can be corrected by glasses, and (2) whether a patient s symptoms are due to HOAs, especially in postoperative situations. The absolute amount of total wavefront error is also useful, as it gives an overall picture of the wavefront error and the level of distortion in the eye. However, all aberrations are not equal, and different aberrations have varying effects on visual function. 13,14 Thus, there is value in looking at the aberration modes separately. Individual aberrations are often demonstrated in a bar chart. Either signed or polar coefficients may be Figure 5. 3-D surface plots representing the shapes of common aberrations. 70 CATARACT & REFRACTIVE SURGERY TODAY EUROPE FERUARY 2016
5 A FUNDAMENTALS Figure 6. Optical principle of induction of coma (A). Visual simulation of coma (). The left column shows the 2-D plots, the central column shows the simulation on a vision chart, and the right column shows the point spread function. Figure 7. Optical principle of induction of spherical aberration. displayed, depending on the device and user selection, and one must be careful when comparing these values. A review of the aberration bar chart can give insights into the pathology, as discussed above in the sections on coma and spherical aberration. Finally, a word about point spread function (PSF): The PSF corresponding to a specific aberration profile provides an approximation of how a point source of light (as practical examples, a distant street light or bulb) appears to a person with that particular aberration profile. It is a useful tool for understanding patient symptoms and can be used in patient counseling. CONCLUSION Wavefront optics involves complicated mathematics. However, understanding the basics of and rationale behind this concept can help clinicians broaden their horizons. For clinical purposes, the aforementioned principles and points are sufficiently useful for most ophthalmologists. n 1. Mello GR, Rocha KM, Santhiago MR, Smadja D, Krueger RR. Applications of wavefront technology. J Cataract Refract Surg. 2012;38: Krueger RR, Applegate RA, MacRae S. Wavefront Customized Visual Correction: The Quest for Super Vision II. Thorofare, NJ: Slack, Ruttig NJ, Jancevski M, Shah SA. Evaluating wavefront analysis application in intraocular lens placement. Curr Opin Ophthalmol. 2008;19: ruce AS, Catania LJ. Clinical applications of wavefront refraction. Optom Vis Sci. 2014;91: Platt C, Shack R. History and principles of Shack-Hartmann wavefront sensing. J Refract Surg. 2001;17:S Jagerman LS. Ophthalmologists, Meet Zernike and Fourier! Victoria, C, Canada: Trafford, Thibos LN, Himebaugh N, Coe CD. Wavefront refraction. In: enjamin WJ, ed. orish s Clinical Refraction. 2nd ed. St. Louis, MO: utterworth Heinemann Elsevier, van rug HH. Efficient Cartesian representation of Zernike polynomials in computer memory. Proc SPIE. 1997; McAlinden C, McCartney M, Moore J. Mathematics of Zernike polynomials: a review. Clin Experiment Ophthalmol. 2011;39: Figure 8. Schematic diagram showing clinical scenarios that can induce spherical aberration and coma. (Not drawn to scale. In real-world scenario, multiple aberrations will be induced.) 10. Thibos LN, Applegate RA, Schwiegerling JT, Webb R. Appendix A: Optical Society of America s standards for reporting optical aberrations. In: Porter J, Queener HM, Lin JE, Thorn K, Awwal A, eds. Adaptive Optics for Vision Science: Principles, Practices, Design, and Applications. Hoboken, NJ: John Wiley & Sons, Dai GM. Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms. J Refract Surg. 2006;22: Yoon G, Pantanelli S, MacRae S. Comparison of Zernike and Fourier wavefront reconstruction algorithms in representing corneal aberration of normal and abnormal eyes. J Refract Surg. 2008;24: Applegate RA, Sarver EJ, Khemsara V. Are all aberrations equal? J Refract Surg. 2002;18: S Applegate RA, allentine C, Gross H, Sarver EJ, Sarver CA. Visual acuity as a function of Zernike mode and level of root mean square error. Optom Vis Sci. 2003;80: Suggested further reading: Krueger RR, Applegate RA, MacRae S. Wavefront Customized Visual Correction: The Quest for Super Vision II. Thorofare, NJ: Slack, Jagerman LS. Ophthalmologists, Meet Zernike and Fourier! Victoria, C, Canada: Trafford, Vishal Jhanji, MD, FRCOphth n Assistant Professor of Ophthalmology, The Chinese University of Hong Kong n Honorary fellow, Centre for Eye Research Australia, University of Melbourne, Royal Victorian Eye and Ear Hospital, Australia n vishaljhanji@cuhk.edu.hk n Financial disclosure: None Gaurav Prakash, MD, FRCS (Glasgow) n Cornea and refractive surgeon, NMC Specialty Hospital, United Arab Emirates n drgauravprakash@gmail.com n Financial disclosure: None FERUARY 2016 CATARACT & REFRACTIVE SURGERY TODAY EUROPE 71
Ocular Wavefront Error Representation (ANSI Standard) Jim Schwiegerling, PhD Ophthalmology & Vision Sciences Optical Sciences University of Arizona
Ocular Wavefront Error Representation (ANSI Standard) Jim Schwiegerling, PhD Ophthalmology & Vision Sciences Optical Sciences University of Arizona People Ray Applegate, David Atchison, Arthur Bradley,
More informationWavefront Optics for Vision Correction
Wavefront Optics for Vision Correction Guang-mingDai SPIE PRESS Bellingham, Washington USA Contents Preface Symbols, Notations, and Abbreviations xiii xv 1 Introduction 1 1.1 Wavefront Optics and Vision
More informationAOL Spring Wavefront Sensing. Figure 1: Principle of operation of the Shack-Hartmann wavefront sensor
AOL Spring Wavefront Sensing The Shack Hartmann Wavefront Sensor system provides accurate, high-speed measurements of the wavefront shape and intensity distribution of beams by analyzing the location and
More informationAN INTERPOLATION METHOD FOR RIGID CONTACT LENSES
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 4, Number, Pages 167 178 13 Institute for Scientific Computing and Information AN INTERPOLATION METHOD FOR RIGID CONTACT LENSES
More informationSystematic errors analysis for a large dynamic range aberrometer based on aberration theory
Systematic errors analysis for a large dynamic range aberrometer based on aberration theory Peng Wu,,2 Sheng Liu, Edward DeHoog, 3 and Jim Schwiegerling,3, * College of Optical Sciences, University of
More informationWorking with Zernike Polynomials
AN06 Working with Zernike Polynomials Author: Justin D. Mansell, Ph.D. Active Optical Systems, LLC Revision: 6/6/0 In this application note we are documenting some of the properties of Zernike polynomials
More informationWavefront aberration analysis with a multi-order diffractive optical element
Wavefront aberration analysis with a multi-order diffractive optical element P.A. Khorin 1, S.A. Degtyarev 1,2 1 Samara National Research University, 34 Moskovskoe Shosse, 443086, Samara, Russia 2 Image
More informationLens Design II. Lecture 1: Aberrations and optimization Herbert Gross. Winter term
Lens Design II Lecture 1: Aberrations and optimization 18-1-17 Herbert Gross Winter term 18 www.iap.uni-jena.de Preliminary Schedule Lens Design II 18 1 17.1. Aberrations and optimization Repetition 4.1.
More informationWavefront Sensing using Polarization Shearing Interferometer. A report on the work done for my Ph.D. J.P.Lancelot
Wavefront Sensing using Polarization Shearing Interferometer A report on the work done for my Ph.D J.P.Lancelot CONTENTS 1. Introduction 2. Imaging Through Atmospheric turbulence 2.1 The statistics of
More informationWave-front generation of Zernike polynomial modes with a micromachined membrane deformable mirror
Wave-front generation of Zernike polynomial modes with a micromachined membrane deformable mirror Lijun Zhu, Pang-Chen Sun, Dirk-Uwe Bartsch, William R. Freeman, and Yeshaiahu Fainman We investigate the
More informationAVEFRONT ANALYSIS BASED ON ZERNIKE POLYNOMIALS
AVEFRONT ANALYSIS BASED ON ZERNIKE POLYNOMIALS M.S. Kirilenko 1,2, P.A. Khorin 1, A.P. Porfirev 1,2 1 Samara National Research University, Samara, Russia 2 Image Processing Systems Institute - Branch of
More informationModeling corneal surfaces with radial polynomials
Modeling corneal surfaces with radial polynomials Author Iskander, Robert, R. Morelande, Mark, J. Collins, Michael, Davis, Brett Published 2002 Journal Title IEEE Transactions on Biomedical Engineering
More informationLaurent Dumas 1, Damien Gatinel 2 and Jacques Malet 3
ESAIM: PROCEEDINGS AND SURVEYS, September 2018, Vol. 62, p. 43-55 Muhammad DAUHOO, Laurent DUMAS, Pierre GABRIEL and Pauline LAFITTE A NEW DECOMPOSITION BASIS FOR THE CLASSIFICATION OF ABERRATIONS OF THE
More informationOptical/IR Observational Astronomy Telescopes I: Optical Principles. David Buckley, SAAO. 24 Feb 2012 NASSP OT1: Telescopes I-1
David Buckley, SAAO 24 Feb 2012 NASSP OT1: Telescopes I-1 1 What Do Telescopes Do? They collect light They form images of distant objects The images are analyzed by instruments The human eye Photographic
More informationThe Accuracy of Tower' Maps to Display Curvature Data in Corneal Topography Systems
The Accuracy of Tower' Maps to Display Curvature Data in Corneal Topography Systems Cynthia Roberts Purpose. To quantify the error introduced by videokeratographic corneal topography devices in using a
More information10 Lecture, 5 October 1999
10 Lecture, 5 October 1999 10.1 Aberration compensation for spherical primaries: the Schmidt camera All-reflecting optical systems are called catoptric; all-refracting systems are called dioptric. Mixed
More informationPhase Retrieval for the Hubble Space Telescope and other Applications Abstract: Introduction: Theory:
Phase Retrieval for the Hubble Space Telescope and other Applications Stephanie Barnes College of Optical Sciences, University of Arizona, Tucson, Arizona 85721 sab3@email.arizona.edu Abstract: James R.
More informationSky demonstration of potential for ground layer adaptive optics correction
Sky demonstration of potential for ground layer adaptive optics correction Christoph J. Baranec, Michael Lloyd-Hart, Johanan L. Codona, N. Mark Milton Center for Astronomical Adaptive Optics, Steward Observatory,
More informationDesigning a Computer Generated Hologram for Testing an Aspheric Surface
Nasrin Ghanbari OPTI 521 Graduate Report 2 Designing a Computer Generated Hologram for Testing an Aspheric Surface 1. Introduction Aspheric surfaces offer numerous advantages in designing optical systems.
More informationMonochromatic aberrations of the human eye in a large population
Porter et al. Vol. 18, No. 8/August 2001/J. Opt. Soc. Am. A 1793 Monochromatic aberrations of the human eye in a large population Jason Porter The Institute of Optics, University of Rochester, Rochester,
More informationOrthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of Zernike polynomials
Orthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of ernike polynomials Chunyu hao and James H. Burge College of Optical ciences, the University of Arizona 630 E.
More informationn The visual examination of the image of a point source is one of the most basic and important tests that can be performed.
8.2.11 Star Test n The visual examination of the image of a point source is one of the most basic and important tests that can be performed. Interpretation of the image is to a large degree a matter of
More informationProblems with wavefront aberrations applied to refractive surgery: Developing standards
Problems with wavefront aberrations applied to refractive surgery: Developing standards Stanley A. Klein* School of Optometry, University of California, Berkeley ABSTRACT Refractive surgery is evolving
More informationVISUAL OPTICS LABORATORY POWER MEASUREMENTS. Prof.Dr.A.Necmeddin YAZICI. GAZİANTEP UNIVERSITY OPTİCAL and ACOUSTICAL ENGINEERING DEPARTMENT
VISUAL OPTICS LABORATORY POWER MEASUREMENTS Prof.Dr.A.Necmeddin YAZICI GAZİANTEP UNIVERSITY OPTİCAL and ACOUSTICAL ENGINEERING DEPARTMENT http://opac.gantep.edu.tr/index.php/tr/ 1 SURFACE GEOMETRY 2 The
More informationTelescopes and Optics II. Observational Astronomy 2017 Part 4 Prof. S.C. Trager
Telescopes and Optics II Observational Astronomy 2017 Part 4 Prof. S.C. Trager Fermat s principle Optics using Fermat s principle Fermat s principle The path a (light) ray takes is such that the time of
More informationDesign and Correction of optical Systems
Design and Correction of optical Systems Part 10: Performance criteria 1 Summer term 01 Herbert Gross Overview 1. Basics 01-04-18. Materials 01-04-5 3. Components 01-05-0 4. Paraxial optics 01-05-09 5.
More informationPHYS 102 Exams. PHYS 102 Exam 3 PRINT (A)
PHYS 102 Exams PHYS 102 Exam 3 PRINT (A) The next two questions pertain to the situation described below. A metal ring, in the page, is in a region of uniform magnetic field pointing out of the page as
More informationOverview of Aberrations
Overview of Aberrations Lens Design OPTI 57 Aberration From the Latin, aberrare, to wander from; Latin, ab, away, errare, to wander. Symmetry properties Overview of Aberrations (Departures from ideal behavior)
More informationPRINCIPLES OF PHYSICAL OPTICS
PRINCIPLES OF PHYSICAL OPTICS C. A. Bennett University of North Carolina At Asheville WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION CONTENTS Preface 1 The Physics of Waves 1 1.1 Introduction
More informationMore Accommodating to your Presbyopic Patients
1 2 Designed to provide enhanced near, intermediate and distance vision with fewer halos and glare and no loss of contrast, The Tetraflex delivers clearer, safer vision without compromise... a true evolution
More informationcorrelated to the Indiana Academic Standards for Precalculus CC2
correlated to the Indiana Academic Standards for Precalculus CC2 6/2003 2003 Introduction to Advanced Mathematics 2003 by Richard G. Brown Advanced Mathematics offers comprehensive coverage of precalculus
More informationTrigonometric ratios:
0 Trigonometric ratios: The six trigonometric ratios of A are: Sine Cosine Tangent sin A = opposite leg hypotenuse adjacent leg cos A = hypotenuse tan A = opposite adjacent leg leg and their inverses:
More informationApplications of the Abbe Sine Condition in Multi-Channel Imaging Systems
Applications of the Abbe Sine Condition in Multi-Channel Imaging Systems Barbara Kruse, University of Arizona, College of Optical Sciences May 6, 2016 Abstract Background In multi-channel imaging systems,
More informationPhysics 3312 Lecture 7 February 6, 2019
Physics 3312 Lecture 7 February 6, 2019 LAST TIME: Reviewed thick lenses and lens systems, examples, chromatic aberration and its reduction, aberration function, spherical aberration How do we reduce spherical
More informationAstro 500 A500/L-7 1
Astro 500 1 Telescopes & Optics Outline Defining the telescope & observatory Mounts Foci Optical designs Geometric optics Aberrations Conceptually separate Critical for understanding telescope and instrument
More informationResearch Article Corneal Topographic and Aberrometric Measurements Obtained with a Multidiagnostic Device in Healthy Eyes: Intrasession Repeatability
Journal of Ophthalmology Volume 2017, Article ID 2149145, 9 pages https://doi.org/10.1155/2017/2149145 Research Article Corneal Topographic and Aberrometric Measurements Obtained with a Multidiagnostic
More informationInternational Journal of Scientific & Engineering Research, Volume 4, Issue 7, July ISSN
International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 96 Performance and Evaluation of Interferometric based Wavefront Sensors M.Mohamed Ismail1, M.Mohamed Sathik2 Research
More informationDownloaded from
Question 10.1: Monochromatic light of wavelength 589 nm is incident from air on a water surface. What are the wavelength, frequency and speed of (a) reflected, and (b) refracted light? Refractive index
More informationPupil matching of Zernike aberrations
Pupil matching of Zernike aberrations C. E. Leroux, A. Tzschachmann, and J. C. Dainty Applied Optics Group, School of Physics, National University of Ireland, Galway charleleroux@yahoo.fr Abstract: The
More informationThe Use of Performance-based Metrics to Design Corrections and Predict Outcomes
The Use of Performance-based Metrics to Design Corrections and Predict Outcomes Jason D. Marsack Visual Optics Institute, College of Optometry, University of Houston 1/28/6 1: 1:2 am Visual Optics Institute
More informationDeviations of Lambert-Beer s law affect corneal refractive parameters after refractive surgery
Deviations of Lambert-Beer s law affect corneal refractive parameters after refractive surgery José Ramón Jiménez, Francisco Rodríguez-Marín, Rosario González Anera and Luis Jiménez del Barco Departamento
More informationComparison of the Adaptive Optics Vision Analyzer and the KR-1W for measuring ocular wave aberrations
Comparison of the Adaptive Optics Vision Analyzer and the KR-W for measuring ocular wave aberrations Journal: Clinical and Experimental Optometry Manuscript ID CEOptom---OP.R Manuscript Type: Original
More informationResponse of DIMM turbulence sensor
Response of DIMM turbulence sensor A. Tokovinin Version 1. December 20, 2006 [tdimm/doc/dimmsensor.tex] 1 Introduction Differential Image Motion Monitor (DIMM) is an instrument destined to measure optical
More informationPolarization Shearing Interferometer (PSI) Based Wavefront Sensor for Adaptive Optics Application. A.K.Saxena and J.P.Lancelot
Polarization Shearing Interferometer (PSI) Based Wavefront Sensor for Adaptive Optics Application A.K.Saxena and J.P.Lancelot Adaptive Optics A Closed loop Optical system to compensate atmospheric turbulence
More information2. FUNCTIONS AND ALGEBRA
2. FUNCTIONS AND ALGEBRA You might think of this chapter as an icebreaker. Functions are the primary participants in the game of calculus, so before we play the game we ought to get to know a few functions.
More informationPhysics 142 Mathematical Notes Page 1. Mathematical Notes
Physics 142 Mathematical Notes Page 1 Mathematical Notes This set of notes contains: a review of vector algebra, emphasizing products of two vectors; some material on the mathematics of vector fields;
More informationPhysics 142 Wave Optics 1 Page 1. Wave Optics 1. For every complex problem there is one solution that is simple, neat, and wrong. H.L.
Physics 142 Wave Optics 1 Page 1 Wave Optics 1 For every complex problem there is one solution that is simple, neat, and wrong. H.L. Mencken Interference and diffraction of waves The essential characteristic
More informationIndiana Academic Standards for Precalculus
PRECALCULUS correlated to the Indiana Academic Standards for Precalculus CC2 6/2003 2004 Introduction to Precalculus 2004 by Roland E. Larson and Robert P. Hostetler Precalculus thoroughly explores topics
More informationPhysics 214 Course Overview
Physics 214 Course Overview Lecturer: Mike Kagan Course topics Electromagnetic waves Optics Thin lenses Interference Diffraction Relativity Photons Matter waves Black Holes EM waves Intensity Polarization
More informationPrecalculus Summer Assignment 2015
Precalculus Summer Assignment 2015 The following packet contains topics and definitions that you will be required to know in order to succeed in CP Pre-calculus this year. You are advised to be familiar
More informationVECTORS. Given two vectors! and! we can express the law of vector addition geometrically. + = Fig. 1 Geometrical definition of vector addition
VECTORS Vectors in 2- D and 3- D in Euclidean space or flatland are easy compared to vectors in non- Euclidean space. In Cartesian coordinates we write a component of a vector as where the index i stands
More informationPopulation distribution of wavefront aberrations in the peripheral human eye
19 J. Opt. Soc. Am. A/ Vol. 6, No. 1/ October 9 Lundström et al. Population distribution of wavefront aberrations in the peripheral human eye Linda Lundström, 1,, * Jörgen Gustafsson, 3 and Peter Unsbo
More informationFORCE TABLE INTRODUCTION
FORCE TABLE INTRODUCTION All measurable quantities can be classified as either a scalar 1 or a vector 2. A scalar has only magnitude while a vector has both magnitude and direction. Examples of scalar
More informationUsing a Membrane DM to Generate Zernike Modes
Using a Membrane DM to Generate Zernike Modes Author: Justin D. Mansell, Ph.D. Active Optical Systems, LLC Revision: 12/23/08 Membrane DMs have been used quite extensively to impose a known phase onto
More informationIntegers, Fractions, Decimals and Percentages. Equations and Inequations
Integers, Fractions, Decimals and Percentages Round a whole number to a specified number of significant figures Round a decimal number to a specified number of decimal places or significant figures Perform
More informationAnswer Explanations for: ACT June 2012, Form 70C
Answer Explanations for: ACT June 2012, Form 70C Mathematics 1) C) A mean is a regular average and can be found using the following formula: (average of set) = (sum of items in set)/(number of items in
More informationPerformance Enhancement of 157 nm Newtonian Catadioptric Objectives
Performance Enhancement of 157 nm Newtonian Catadioptric Objectives James Webb, Timothy Rich, Anthony Phillips and Jim Cornell Corning Tropel Corporation, 60 O Connor Rd, Fairport, NY 14450, 585-377-3200
More informationSurgical correction of hyperopia with the excimer laser involves
Corneal Asphericity Change after Excimer Laser Hyperopic Surgery: Theoretical Effects on Corneal Profiles and Corresponding Zernike Expansions Damien Gatinel, 1 Jacques Malet, 2 Thanh Hoang-Xuan, 1 and
More informationYEAR 10 PROGRAM TERM 1 TERM 2 TERM 3 TERM 4
YEAR 10 PROGRAM TERM 1 1. Revision of number operations 3 + T wk 2 2. Expansion 3 + T wk 4 3. Factorisation 7 + T wk 6 4. Algebraic Fractions 4 + T wk 7 5. Formulae 5 + T wk 9 6. Linear Equations 10 +T
More informationMaking FEA Results Useful in Optical Analysis Victor Genberg, Gregory Michels Sigmadyne, Inc. Rochester, NY
Making FEA Results Useful in Optical Analysis Victor Genberg, Gregory Michels Sigmadyne, Inc. Rochester, NY Keith Doyle Optical Research Associates,Westborough, MA ABSTRACT Thermal and structural output
More informationVECTORS. 3-1 What is Physics? 3-2 Vectors and Scalars CHAPTER
CHAPTER 3 VECTORS 3-1 What is Physics? Physics deals with a great many quantities that have both size and direction, and it needs a special mathematical language the language of vectors to describe those
More informationObjectives and Essential Questions
VECTORS Objectives and Essential Questions Objectives Distinguish between basic trigonometric functions (SOH CAH TOA) Distinguish between vector and scalar quantities Add vectors using graphical and analytical
More information2014 Summer Review for Students Entering Algebra 2. TI-84 Plus Graphing Calculator is required for this course.
1. Solving Linear Equations 2. Solving Linear Systems of Equations 3. Multiplying Polynomials and Solving Quadratics 4. Writing the Equation of a Line 5. Laws of Exponents and Scientific Notation 6. Solving
More informationChapter 2. Altitude Measurement
Chapter Altitude Measurement Although altitudes and zenith distances are equally suitable for navigational calculations, most formulas are traditionally based upon altitudes which are easily accessible
More informationThe Not-Formula Book for C2 Everything you need to know for Core 2 that won t be in the formula book Examination Board: AQA
Not The Not-Formula Book for C Everything you need to know for Core that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationCrash Course in Trigonometry
Crash Course in Trigonometry Dr. Don Spickler September 5, 003 Contents 1 Trigonometric Functions 1 1.1 Introduction.................................... 1 1. Right Triangle Trigonometry...........................
More informationSection 6.1 Sinusoidal Graphs
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a right triangle, and related to points on a circle We noticed how the x and y values
More informationMathematics for Graphics and Vision
Mathematics for Graphics and Vision Steven Mills March 3, 06 Contents Introduction 5 Scalars 6. Visualising Scalars........................ 6. Operations on Scalars...................... 6.3 A Note on
More informationTrigonometry.notebook. March 16, Trigonometry. hypotenuse opposite. Recall: adjacent
Trigonometry Recall: hypotenuse opposite adjacent 1 There are 3 other ratios: the reciprocals of sine, cosine and tangent. Secant: Cosecant: (cosec θ) Cotangent: 2 Example: Determine the value of x. a)
More informationIntroduction Assignment
PRE-CALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationES 111 Mathematical Methods in the Earth Sciences Lecture Outline 3 - Thurs 5th Oct 2017 Vectors and 3D geometry
ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 3 - Thurs 5th Oct 2017 Vectors and 3D geometry So far, all our calculus has been two-dimensional, involving only x and y. Nature is threedimensional,
More informationIntroduction to aberrations OPTI518 Lecture 5
Introduction to aberrations OPTI518 Lecture 5 Second-order terms 1 Second-order terms W H W W H W H W, cos 2 2 000 200 111 020 Piston Change of image location Change of magnification 2 Reference for OPD
More informationIMPROVING BEAM QUALITY NEW TELESCOPE ALIGNMENT PROCEDURE
IMPROVING BEAM QUALITY NEW TELESCOPE ALIGNMENT PROCEDURE by Laszlo Sturmann Fiber coupled combiners and visual band observations are more sensitive to telescope alignment problems than bulk combiners in
More informationProgressive Addition Lenses
Characterizing the Optics of Progressive Addition Lenses Thomas Raasch, OD, PhD Ching Yao Huang, PhD, MS Mark Bullimore, MCOptom, PhD Allen Yi, PhD Lijuan Su, PhD T. Raasch 6/2/2011 1 Outline Background
More informationAlignment aberrations of the New Solar Telescope
Alignment aberrations of the New Solar Telescope Anastacia M. Manuel and James H. Burge College of Optical Sciences/University of Arizona 1630 East University Blvd., Tucson, AZ 85721, USA ABSTRACT The
More informationMathematics Teachers Enrichment Program MTEP 2012 Trigonometry and Bearings
Mathematics Teachers Enrichment Program MTEP 2012 Trigonometry and Bearings Trigonometry in Right Triangles A In right ABC, AC is called the hypotenuse. The vertices are labelled using capital letters.
More informationLIGHT. A beam is made up of several rays. It maybe parallel, diverging (spreading out) or converging (getting narrower). Parallel Diverging Converging
LIGHT Light is a form of energy. It stimulates the retina of the eye and produces the sensation of sight. We see an object when light leaves it and enters the eye. Objects such as flames, the sum and stars
More informationMATH 109 TOPIC 3 RIGHT TRIANGLE TRIGONOMETRY. 3a. Right Triangle Definitions of the Trigonometric Functions
Math 09 Ta-Right Triangle Trigonometry Review Page MTH 09 TOPIC RIGHT TRINGLE TRIGONOMETRY a. Right Triangle Definitions of the Trigonometric Functions a. Practice Problems b. 5 5 90 and 0 60 90 Triangles
More informationElectro Magnetic Field Dr. Harishankar Ramachandran Department of Electrical Engineering Indian Institute of Technology Madras
Electro Magnetic Field Dr. Harishankar Ramachandran Department of Electrical Engineering Indian Institute of Technology Madras Lecture - 7 Gauss s Law Good morning. Today, I want to discuss two or three
More informationTriangles and Vectors
Chapter 3 Triangles and Vectors As was stated at the start of Chapter 1, trigonometry had its origins in the study of triangles. In fact, the word trigonometry comes from the Greek words for triangle measurement.
More informationKinematics in Two Dimensions; 2D- Vectors
Kinematics in Two Dimensions; 2D- Vectors Addition of Vectors Graphical Methods Below are two example vector additions of 1-D displacement vectors. For vectors in one dimension, simple addition and subtraction
More informationFor information: Fred W. Duckworth, Jr. c/o Jewels Educational Services 1560 East Vernon Avenue Los Angeles, CA
THAT S TRIGONOMETRY For information: Fred W. Duckworth, Jr. c/o Jewels Educational Services 1560 East Vernon Avenue Los Angeles, CA 90011-3839 E-mail: admin@trinitytutors.com Website: www.trinitytutors.com
More information2. Pythagorean Theorem:
Chapter 4 Applications of Trigonometric Functions 4.1 Right triangle trigonometry; Applications 1. A triangle in which one angle is a right angle (90 0 ) is called a. The side opposite the right angle
More informationDEGREE OF POLARIZATION VS. POINCARÉ SPHERE COVERAGE - WHICH IS NECESSARY TO MEASURE PDL ACCURATELY?
DEGREE OF POLARIZATION VS. POINCARÉ SPHERE COVERAGE - WHICH IS NECESSARY TO MEASURE PDL ACCURATELY? DEGREE OF POLARIZATION VS. POINCARE SPHERE COVERAGE - WHICH IS NECESSARY TO MEASURE PDL ACCURATELY? Introduction
More informationBowling Balls and Binary Switches
Andrew York April 11, 2016 Back in the old programming days, my favorite line of code was "if-then." If some condition is met, then do something specific - incredibly useful for making things happen exactly
More informationEngineering Physics 1 Prof. G.D. Vermaa Department of Physics Indian Institute of Technology-Roorkee
Engineering Physics 1 Prof. G.D. Vermaa Department of Physics Indian Institute of Technology-Roorkee Module-04 Lecture-02 Diffraction Part - 02 In the previous lecture I discussed single slit and double
More informationDeformable mirror fitting error by correcting the segmented wavefronts
1st AO4ELT conference, 06008 (2010) DOI:10.1051/ao4elt/201006008 Owned by the authors, published by EDP Sciences, 2010 Deformable mirror fitting error by correcting the segmented wavefronts Natalia Yaitskova
More informationMeasurement of specimen-induced aberrations of biological. samples using phase stepping interferometry
Journal of Microscopy, Vol. 213, Pt 1 January 2004, pp. 11 19 Received 10 June 2003; accepted 2 September 2003 Measurement of specimen-induced aberrations of biological Blackwell Publishing Ltd. samples
More informationPhysics 101 Final Exam Problem Guide
Physics 101 Final Exam Problem Guide Liam Brown, Physics 101 Tutor C.Liam.Brown@gmail.com General Advice Focus on one step at a time don t try to imagine the whole solution at once. Draw a lot of diagrams:
More informationPrism Applications 2/11/2011. Copyright 2006, Phernell Walker, II, AS, NCLC, ABOM 1. Contact Information: Ophthalmic Prism.
Prism Applications Contact Information: Phernell Walker, II, AS, NCLC, ABOM Master in Ophthalmic Optics Phernell Walker, II, AS, NCLC, ABOM Email: pureoptics@earthlink.net www.pureoptics.com (254) 338-7946
More informationPart 1 - Basic Interferometers for Optical Testing
Part 1 - Basic Interferometers for Optical Testing Two Beam Interference Fizeau and Twyman-Green interferometers Basic techniques for testing flat and spherical surfaces Mach-Zehnder Zehnder,, Scatterplate
More information2. Geometry of lens systems
Lens geometry 1 2. Geometry of lens systems Deriving the geometry of a lens system - Method 1 (the Lens Equation) Consider the idealised system shown in Figure 2.1, where the symbols D s, D l and D ls
More informationAstronomy 203/403, Fall 1999
Astronom 0/40 all 999 8 ecture 8 September 999 8 ff-axis aberrations of a paraboloidal mirror Since paraboloids have no spherical aberration the provide the cleanest wa to stud the other primar aberrations
More informationTECHNICAL REPORT. Paraxial Optics of Astigmatic Systems: Relations Between the Wavefront and the Ray Picture Approaches
1040-5488/07/8401-0072/0 VOL. 84, NO. 1, PP. E72 E78 OPTOMETRY AND VISION SCIENCE Copyright 2007 American Academy of Optometry TECHNICAL REPORT Paraxial Optics of Astigmatic Systems: Relations Between
More informationAlignment metrology for the Antarctica Kunlun Dark Universe Survey Telescope
doi:10.1093/mnras/stv268 Alignment metrology for the Antarctica Kunlun Dark Universe Survey Telescope Zhengyang Li, 1,2,3 Xiangyan Yuan 1,2 and Xiangqun Cui 1,2 1 National Astronomical Observatories/Nanjing
More informationPhys 531 Lecture 27 6 December 2005
Phys 531 Lecture 27 6 December 2005 Final Review Last time: introduction to quantum field theory Like QM, but field is quantum variable rather than x, p for particle Understand photons, noise, weird quantum
More informationROINN NA FISICE Department of Physics
ROINN NA FISICE Department of 1.1 Astrophysics Telescopes Profs Gabuzda & Callanan 1.2 Astrophysics Faraday Rotation Prof. Gabuzda 1.3 Laser Spectroscopy Cavity Enhanced Absorption Spectroscopy Prof. Ruth
More informationKinematics in Two Dimensions; Vectors
Kinematics in Two Dimensions; Vectors Vectors & Scalars!! Scalars They are specified only by a number and units and have no direction associated with them, such as time, mass, and temperature.!! Vectors
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University Abstract This handout defines the trigonometric function of angles and discusses the relationship between trigonometric
More information5. Aberration Theory
5. Aberration Theory Last lecture Matrix methods in paraxial optics matrix for a two-lens system, principal planes This lecture Wavefront aberrations Chromatic Aberration Third-order (Seidel) aberration
More information