Aberrationsin
`Optical Imaging
`Systems
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`JOSE SASIAN
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`CAMBRIDGE
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`APPLE V COREPHOTONICS
`IPR2020-00906
`Exhibit 2030
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`Published in the United States of America by Cambridge University Press, New York
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`www.cambridge.org
`Informationonthis title: www.cambridge.org/9781 107006331
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`© J. Sasian 2013
`
`This publication is in copyright. Subject to statutory exception
`and to the provisions of relevant collective licensing agreements,
`no reproduction of any part may take place without the written
`permission of Cambridge University Press.
`
`First published 2013
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`A catalog recordforthis publication is available fromthe British Library
`
`Library of Congress Cataloging in Publication data
`Sasian, José M.
`Introduction to aberrations in optical imaging systems / José Sasidn.
`p.
`cm.
`Includes bibliographical references and index.
`ISBN 978-1-107-00633-1 (hardback)
`2. Imaging systems — Image quality.
`3. Optical engineering.
`QC671.S27
`2012
`621.36'7 — de23
`2012027121
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`1. Aberration.
`
`. Title.
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`ISBN 978-1-107-00633-1 Hardback
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`Cambridge University Press has no responsibility for the persistence or
`accuracy of URLsfor externalorthird-party internet websites referred to
`in this publication and does not guarantee that any content on such
`websites is, or will remain, accurate or appropriate.
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`5.6 Parity of the aberrations
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`73
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`5.5 Determination of the wavefront deformation
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`‘sesine of the angle @ between these vectors. Table 5.1 summarizes thefirst four
`ders of aberrations using both vector and algebraic expressions. The fourth-order
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`=mms are often called the primary aberrations. The ten sixth-order terms can be
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`ided into two groups. The first group (first six terms) can be considered as an
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`“smprovementupontheprimary aberrationsby their increased field dependence, and
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`= second group (last four terms) represents new wavefront deformation forms.
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`re 5.1 showsthe shape (aperture dependenceonly) of the zero, second,fourth,
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`d the new wavefront shapesof the sixth-orderaberrations.
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`In Table 5.1 the piston terms represent a uniform phase change across the
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`‘perture that does not degrade the image quality. Physically piston terms represent
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`time delay or advancein the time of arrival of the wavefront as it propagates
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`om
`the object to the exit pupil. The second-order term magnification represents a
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`ange of magnification and the focus term represents a changein the axial location
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`the image. The coefficients for magnification and focus are set to zero given
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`at Gaussian and Newtonian optics accurately predict the size and location of an
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`‘mage. However,a focusterm is usually added to minimizeaberrationsorto select
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`observation plane other than the ideal imageplane. In addition, the change of
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`2enification and focus with respectto the wavelength are knownasthe transverse
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`d longitudinal chromatic aberrations respectively.
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`when rays of light do not pass through an ideal image point, the wavefront must
`deformed. The wavefront deformation is measured with the aid of a reference
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`sere. The reference sphere for a given field point passes through the on-axis exit
`oil point and its center coincides with the ideal image. As shownin Figure 5.3
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`wavefront deformation multiplied by the index of refraction is the optical path
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`etween the wavefront and the reference sphere measured alongtheray.
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`By convention the wavefront deformation is negative if the wavefrontlags the
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`“ference sphere and positive if it leads the reference sphere. The units of the
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`svefront deformation are linear dimensions of millimeters, micrometers, etc.
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`Sowever, often the wavefront deformationis divided by the wavelength oflight A,
`d then the deformation is expressed in waves. The reference sphere is centeredat
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`pint Vy, H inthe imageplane. Note thatthe tip of the aperture vector defines where
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`= ray intersects the exit pupil plane. In this mannerthe aperture vector designates
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`= samepupil pointforall field points. This definition eventually makeseasier the
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`ssiculation of sixth-order coefficients that are coordinate-system dependent.
`5.6 Parity of the aberrations
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`aberrationscan beclassified as even or odd aberrations. For example, spherical
`he
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`aberration,astigmatism,field curvature, and the chromatic change of focusare even
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`APPLE V COREPHOTONICS
`IPR2020-00906
`Exhibit 2030
`Page 3
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`APPLE V COREPHOTONICS
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`74
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`The wave aberration function
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`aberrations. Coma,distortion, and the chromatic change of magnification are odd
`aberrations. The parity is found by observation of the algebraic powerparity of
`the field and aperture vectors in the aberration coefficients. The odd aberrations
`have the important property that they cancel, or tend to cancel, in a system that
`has symmetry aboutthe stop. That is, each half of the system contributes the same
`amountof aberration but with opposite algebraic sign. In contrast, in a symmetrical
`system the even aberrations from each half of the system add,rather than cancel.
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`5.7 Note on the choice of coordinates
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`The aberration theory developed in this book uses polar coordinates with the field
`vector H serving as a reference to define the polar angle @ and the aperture vector
`p. Given the system’s axial symmetry,inherently only three variables are necessary,
`|\H |, ||, and cos(#), and eventually this leads to many simplifications. The other
`obvious choice is the use of Cartesian coordinates, which for historical reasons,
`previous works on wave aberration theory, and simplicity, are little used in the
`present treatment.
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`5.8 Summary
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`In this chapter we have introducedthe aberration function as a polynomial depend-
`ing on the field and aperture of the system. The terms in the aberration function
`representaberrationsas a wavefront deformation with respect to a reference sphere.
`The aberration coefficients provide the maximum amplitude of the deformation as
`an optical path. The aberration function provides a wealth of insight into the nature
`of an optical system andits aberrations. Symmetry considerations are important in
`developing the aberration function.
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`Exercises
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`5.1. Using symmetry considerations, explain why the sine of the angle between
`the field and aperture vector does not appearin the aberration function.
`5.2. Determine the aberration function up to fourth order of a system that has two
`orthogonal planes of symmetry. The intersection of these planes defines the
`optical axis. Use the unit vector i to specify the direction of one of the planes
`of symmetry, andthefield H and aperture p vectors.
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`References
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`[1] W. R. Hamilton, “Theory of systemsof rays,” Trans. R. Irish Acad. 15(1828), 69-174.
`[2] W. R. Hamilton, “Supplementto an essay on the theory of systemsof rays,” Trans. R.
`Trish Acad. 16(1830), 1-61.
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`APPLE V COREPHOTONICS
`IPR2020-00906
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`APPLE V COREPHOTONICS
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