Lens Design Fundamentals
`
`Second Edition
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`R U D O L F K I N G S L A K E
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`Chapter 4
`
`Aberration Theory
`
`4.1 INTRODUCTION
`
`In the preceding chapter, imaging was considered to be ideal or stigmatic. This
`means that rays from a point source P that pass through an optical system will
`converge to a point located at its Gaussian image P0
`. In a like manner, the portion
`of wavefronts from P passing through the optical system will converge as portions
`of spherical wavefronts toward P0
`. In other words, the point sources are mapped
`onto the image surface as point images according to the laws of Gaussian
`image formation presented in the prior chapter. Deviations from ideal image
`formation are the result of defects or aberrations inherent in the optical system.
`As will be discussed in this chapter, it is possible that the actual image ~P0
`is
`formed at a location other than at P0
`which can be caused by field curvature
`and distortion while still forming a stigmatic image. When an optical system
`fails to form a point image of a point source in the Gaussian image plane, the
`rays do not pass through the same location and the converging wavefront is
`no longer spherical as a consequence of the optical system suffering aberrations.
`In this chapter, a mathematical description of the aberrations for symmetrical
`optical systems will be presented primarily from the viewpoint of ray deviation
`errors rather than wavefront errors. In the following chapters, each of the aber-
`rations will be treated in significant detail in addition to their control during the
`optical design process.
`
`4.2 SYMMETRICAL OPTICAL SYSTEMS
`
`Figure 4.1 illustrates the basic elements of a symmetric optical system. This
`system is invariant under an arbitrary rotation about its optical axis (OA) and
`under reflection in any plane containing OA. Both of these symmetry character-
`istics are necessary properties of a symmetrical optical system.1 A right-hand
`
`Copyright # 2010, Elsevier Inc. All rights reserved.
`DOI: 10.1016/B978-0-12-374301-5.00008-5
`
`101
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`102
`
`Aberration Theory
`
`Y
`
`Hy
`
`X
`
`Y′
`
`X′
`
`Z
`Actual image
`
`x
`Image plane
`
`Ideal image
`
`Hx
`Point object
`
`Object plane
`
`Exit pupil
`Rotationally
`symmetric system
`
`Entrance pupil
`
`Figure 4.1 Basic elements of a symmetrical optical system.
`
`Cartesian coordinate system is used where the optical axis is always taken to lie
`along the z-axis.2 The ideal state of correction for a symmetrical optical system
`is when a system forms in the image plane (IP) normal to the optical axis a
`sharp and undistorted image of an object in the object plane (OP) orthogonal
`to the optical axis. These planes are designated as the image and object planes,
`respectively, and are conjugate since the optical system forms an image of one in
`the other. Unless otherwise specified, these planes should be considered to be
`orthogonal to the optical axis.
`Consider for the moment an arbitrary point P in the object space of a symmetric
`system. In general the family of rays from P traversing the optical system will fail to
`pass through a unique point in the image space and the image of P formed by the
`system is said to be astigmatic, that is, to suffer from aberrations. If, on the other
`hand, all rays from P do pass through a unique point P0
`in the image space, the
`image of point P is said to be stigmatic.3 From the definition of a symmetric system,
`it should be evident that if P0
`is the stigmatic image of some point P then the two
`points P and P0
`lie in a plane containing the optical axis. Now imagine that object
`points are constrained to lie in the object plane OP and that the images of all such
`points are stigmatic and that the object plane is stigmatically imaged by the system
`onto an image surface (in contrast to an image plane).
`Again relying on the definition of a symmetric system, it is obvious that the
`stigmatic image of a plane object surface OP, which is normal to the optical axis
`of a symmetric system, is a surface of revolution about the optical axis. When
`this image surface of revolution is not planar, the imagery is considered to suffer
`an aberration or image defect known as curvature of field although there is no
`blurring of the image. Since the optical system is considered to be rotationally
`symmetric, we can arbitrarily select a reference plane that contains the optical
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`4.2 Symmetrical Optical Systems
`
`103
`
`axis. Referring to Figure 4.1, this plane is the Y-Z plane and is generally called
`the tangential or meridional plane.
`Assume now that a stigmatic image of the object plane is formed in the image
`plane where the object has some geometrical shape. If the optical system forms an
`image having the same geometrical shape as the object to some scaling factor, the
`image is considered to be undistorted or be an accurate geometric representation
`of the object. Should the optical system form an image which is not geometrically
`similar to the object’s shape, then the image is said to suffer distortion. When the sys-
`tem is free of distortion (undistorted), the ratio of image size to the corresponding
`object size is the magnification m, with the image for a positive lens being inverted
`
`
`and reverted with respect to the object. Let the object be a line extending from the
`origin of the object plane to the location denoted as point object in Figure 4.1 which
`has coordinates expressed as Hx; Hy
`. The image size can be computed by
`
`H 0
`
`x ¼ mHx and H 0y ¼ mHy
`
`since the line can be projected onto each axis and propagated independently
`without loss of generality since a paraxial skew ray is linearly separable into
`its orthogonal components.
`It is evident from the preceding discussion that an ideal image of the object
`plane requires three conditions to be satisfied, namely, stigmatic image forma-
`tion, no curvature of field, and no distortion. In contrast, an optical system hav-
`ing stigmatic image formation can still suffer the image defects of distortion and
`curvature of field.
`As explained, an ideal optical system forms a perfect or stigmatic image
`which essentially means that rays emanating from a point source will be con-
`verged by the optical system to a point image, although curvature of field and
`distortion may be present. At this juncture, image quality will be discussed in
`strictly geometric terms. In later chapters, the impact of diffraction on image
`quality will be discussed.
`The majority of this book addresses rotationally symmetric optical systems,
`their aberrations, and configurations. Figure 4.1 shows the generic geometry for
`such systems, which comprise five principal elements: the object plane, entrance
`pupil, lenses (including stop), exit pupil, and image plane.4 A ray propagating
`through this system is specified by its object coordinates ðHx; HyÞ and entrance
`pupil coordinates ðrx; ryÞ ¼ ~r, or in polar coordinates ðr; yÞ, as illustrated in
`Figure 4.2. This means that point P in the entrance pupil can be expressed by X ¼
`r cosðyÞ and Y ¼ r sinðyÞ where y is zero when ~r lies along the Y-axis.
`
`x; H 0yÞ and displaced or aberrant
`This ray is incident on the image plane at ðH 0
`from the ideal image location by ðex; eyÞ. Since the optical system is rotationally
`symmetric, the (point) object is assumed to always be located on the y-axis in the
`object plane, that is, H  ð0; HyÞ. This means the ideal image is located along the
`y-axis in the image plane, that is, h0 ¼ mH where m is the magnification. The actual
`
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`104
`
`Aberration Theory
`
`Hy
`
`H
`
`Hx
`
`P( X,Y )
`
`Y
`
`r
`
`q
`
` p l a n e
`
`O b j e c t
`
`E n t r a n c e p u p i
`
`l
`
`Figure 4.2 Entrance pupil coordinates of a ray.
`
`X
`
`Z
`
`image plane may be displaced a distance x from the ideal image plane. The ideal
`image plane is also called the Gaussian or paraxial image plane. The term image
`plane, as used in this book, means the planar surface where the image is formed
`which may be displaced from the ideal image plane by the defocus distance x.
`A ray exiting the exit pupil, as shown in Figure 4.3, intersects the image
`plane at ðX 0; Y0Þ which in general does not pass through the ideal
`image
`
`Optical
`axis
`
`Ray
`
`Y′
`
`Actual
`image
`
`X′
`
`Ideal image
`
`Z
`
`e
`
`y
`
`x
`
`e
`x
`
`I m a g e p l a n e
`
`Figure 4.3 Image plane coordinates of ray suffering aberrations.
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`4.2 Symmetrical Optical Systems
`
`105
`
`point shown in the figure as a consequence of aberrations. The object point is
`located at ~H ¼ Hx
`^i þ Hy
`^j (see Figure 4.1) with the ideal image point being
`
`x; h0yÞ and the actual image being located at ðX 0; Y0Þ.
`located at ðh0
`X 0ðr; y; ~H; xÞ  exðr; y; ~H; xÞ þ h0
`Y0ðr; y; ~H; xÞ  eyðr; y; ~H; xÞ þ h0
`
`x
`
`y
`
`Using vector notation and ignoring the defocus parameter for the moment, the
`ray aberration can be written as
`
`~eð~r; ~HÞ ¼ ~esð~r; ~HÞ þ~ecð~r; ~HÞ
`
`where ~es and ~ec are defined by
`
`~esð~r; ~HÞ ¼ 1
`~eð~r; ~HÞ ~eð~r; ~HÞ
`~ecð~r; ~HÞ ¼ 1
`~eð~r; ~HÞ þ~eð~r; ~HÞ
`
`
`
`
`2
`
`2
`
`and ~es and ~ec are called the symmetric and asymmetric aberrations as well as the
`astigmatic and the comatic aberrations, respectively, of the ray ð~r; ~HÞ.5 The
`
`importance of decomposing the ray aberration in this manner for our study of
`lens design will become evident. Consider first the symmetric term ~es which
`means that the ray error will be symmetric about the ideal image location
`assuming no distortion. Specifically this can be interpreted as ðex; eyÞ for
`ð~r; ~HÞand ðex; eyÞ for ð~r; ~HÞ. If a spot diagram of a point source is made
`
`for an optical system suffering only astigmatic aberration, the pattern formed
`will be symmetric.
`In contrast, the comatic or asymmetric aberration ~ec is invariant when the
`sign of ~r is changed. This means that rays ð~r; ~HÞ and ð~r; ~HÞ will suffer the
`identical image error ðex; eyÞ, that is, they each intercept the image plane at
`
`the same location. Consequently, the comatic aberration creates an asymmetry
`in the spot diagram. Further, it should be recognized that the astigmatic and
`comatic aberration components are decoupled and can not be used to balance
`one another. The importance of this knowledge in lens design will be explained
`in more detail in the following chapters.
`Since the optical system is rotationally symmetric, the object can be placed in
`the meridional plane, or y-axis of the object plane, without the loss of generality
`and the advantage of simplifying the computation and interpretation of the
`resulting aberrations. Consequently, since the x-component is zero the object
`is denoted by H and the ideal image by h0
`. The actual image coordinates now
`become
`
`X 0ðr; y; H; xÞ  exðr; y; H; xÞ
`Y 0ðr; y; H; xÞ  eyðr; y; H; xÞ þ h0
`
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`106
`
`Aberration Theory
`
`for the specific ray coordinates r; y; and H, and the image plane defocus x.
`It has been found useful to decompose the aberration into two elements with
`respect to how the aberrations transform under a change in the sign of r. These
`elements are called symmetric and asymmetric components and are orthogonal
`to one another. Since the object and image are located in the meridional plane,
`all rays emanating from the object point having entrance pupil coordinates
`
`~r ¼ ðr; 0oÞ necessarily lie in the meridional plane.6 Consequently, ex ¼ 0. It is
`common to plot the ray aberration for the meridional fan of rays with r being
`normalized (1 to þ1). The ordinate of the plot is the ray error measured from
`Figure 4.4 provides an example of such a plot. In this case, H 6¼ 0 to allow
`
`intercept of the principal ray.
`
`illustration of the symmetric and asymmetric components of the ray aberration.
`As explained above, ~es and ~ec represent these components. In this figure, the
`comatic and the stigmatic contributions for the total aberration are shown.
`Notice that the comatic aberration is symmetric about the r ¼ 0 axis. In other
`words, any ray pair having entrance pupil coordinates of ðr; 0oÞ and ðr; 0oÞ
`will have the same ray error, that is, eyðr; 0; HÞ ¼ eyðr; 0; HÞ. In contrast,
`any ray pair having entrance pupil coordinates of ðr; 0oÞ and ðr; 0oÞ will suffer
`ray errors of equal and opposite sign, that is, eyðr; 0; HÞ ¼ eyðr; 0; HÞ.
`
`the astigmatic aberration is asymmetric about the same axis. This means that
`
`Examination of the total aberration curve illustrates that it can be neither sym-
`metric nor asymmetric. In this particular case, both the comatic and astigmatic
`aberrations comprise third- and fifth-order terms of opposite signs. The total
`aberration curve is simply the sum of the comatic and astigmatic values.
`
`1.0
`
`0.5
`
`0.0
`
`y
`e
`
`TOTAL
`
`Astigmatic/Symmetric
`
`Comatic/Asymmetric
`
`–0.5
`–1.0
`
`–0.5
`
`0.0
`r
`
`0.5
`
`1.0
`
`Figure 4.4 Ray aberration for a meridional fan of rays.
`
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`4.2 Symmetrical Optical Systems
`
`107
`
`The interpretation of such plots for use in lens design will become evident in the
`following material.
`Plots, such as that shown in Figure 4.4, are very useful during a lens design
`process; however, the meridional plots provide only a portion of the insight into
`the complete aberrations suffered by a particular lens design. Additional plots
`can be generated using non-meridional rays, which are generally called skew
`rays. The most common skew rays utilized have entrance pupil coordinates of
`
`ðr; 90oÞ and are commonly called sagittal rays. The name sagittal is generally
`
`
`skew rays that lie in a plane perpendicular to the
`and 270
`given to the 90
`meridional plane, containing the principal ray.
`The sagittal plane is not one single plane throughout a lens, but it changes its
`tilt after each surface refraction/reflection. The point of intersection of a sagittal
`ray with the paraxial image plane may have both a vertical error and a horizon-
`tal error relative to the point of intersection of the principal ray, and both these
`errors can be plotted separately against some suitable ray parameter. This
`parameter is often the horizontal distance from the meridional plane to the
`point where the entering ray pierces the entrance pupil. The meridional plot,
`of course, has no symmetry, but the two sagittal ray plots do have symmetry.
`As a consequence, sagittal ray plots are often shown for only positive values
`of r since it is realized that
`exðr; 90o; H; xÞ ¼ exðr; 90o; H; xÞ and eyðr; 90o; H; xÞ ¼ eyðr; 90o; H; xÞ:
`
`It has been shown that the ray aberration can be decomposed into astigmatic
`and comatic components, which are orthogonal. These two components can be
`further decomposed. For the astigmatic component, it comprises spherical aber-
`ration, astigmatism, and defocus. In a like manner, the comatic component
`comprises coma and distortion. The following two equations for the ray errors
`ex and ey show this decomposition. The abbreviations for the various compo-
`nents will be utilized extensively in the following material.
`
`ASTIGMATIC COMPONENTS
`
`COMATIC COMPONENTS
`
`(4-1)
`
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`exðr; y; H; xÞ ¼ SPHxðr; y; 0Þ þ ASTxðr; y; HÞ þ DFxðr; y; xÞ
`|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
`þ CMAxðr; y; HÞ
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`eyðr; y; H; xÞ ¼ SPHyðr; y; 0Þ þ ASTyðr; y; HÞ þ DFyðr; y; xÞ
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ CMAyðr; y; HÞ þ DISTðHÞ
`
`ASTIGMATIC COMPONENTS
`
`COMATIC COMPONENTS
`
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`

`108
`
`Aberration Theory
`
`where SPH  spherical aberration, AST  astigmatism, CMA  coma, DIST 
`distortion, and DF  defocus. It should be recognized that the comatic compo-
`nent of ex does not contain a distortion term since it is assumed that the object
`lies in the meridional plane.
`Being that the ray intercept error can be described as the linear combination
`of the astigmatic and comatic contributions, these contributions can be written
`as a power series in terms of H and r. Several conventions exist for expansion
`nomenclature; however, most follow that given by Buchdahl. Specifically, an
`aberration depending on r and H in the combination rnsH s is said to be of
`the type
`
`. nth order, sth degree coma if (n-s) is even, or
`. nth order, (n-s)th degree astigmatism if (n-s) is odd.
`
`For simplicity, the arguments of ex and ey are not explicitly shown unless
`needed for clarity, defocus is assumed zero, and recalling that the expansions
`are a function of y for the general skew ray, the expansion of the ray errors
`are given by
`
`1
`
`SPHERICAL
`
`3
`
`LINEAR or CIRCULAR COMA
`
`CUBIC COMA
`
`QUINTIC COMA
`
`(4-2)
`
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`r5 þ t1r7 þ . . .Þ sinðyÞ
`ex ¼ ðs1r3 þ m
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`r4 þ t3r6 þ . . .Þ sinð2yÞH
`þ ðs2r2 þ m
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ ðm9 sinð2yÞr2 þ ðt9 sinð2yÞ þ t10 sinð4yÞÞr4 þ . . .ÞH 3
`
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ ðt17 sinð2yÞr2 þ . . .ÞH 5
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`H 4 þ t19H 6 þ . . .Þ sinðyÞr
`þ ððs3 þ s4ÞH 2 þ m
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`6 cos2ðyÞÞH 2 þ ðt13 þ t14 cos2ðyÞÞH 4 þ . . .Þ sinðyÞr3
`þ ððm
`þ m
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ ððt5 þ t6 cos2ðyÞÞH 2 þ . . .Þ sinðyÞr5
`
`11
`
`LINEAR ASTIGMATISM
`
`5
`
`CUBIC ASTIGMATISM
`
`QUINTIC ASTIGMATISM
`
`þ . . . HIGHER ORDER ABERRATIONS IN TERMS OF r AND H:
`
`Apple v. Corephotonics
`IPR2019-00030
`Exhibit 2024 Page 10 of 37
`
`

`

`4.2 Symmetrical Optical Systems
`
`109
`
`and
`
`1
`
`SPHERICAL
`
`2
`
`LINEAR or CIRCULAR COMA
`
`CUBIC COMA
`
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`r5 þ t1r7 þ . . .Þ cosðyÞ
`ey ¼ðs1r3 þ m
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`
`þ ðs2ð2þ cosð2yÞÞr2 þðm
`þ m3 cosð2yÞÞr4 þðt2 þ t3 cosð2yÞÞr6 þ . . .ÞH
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ ððm
`
`þ m8 cosð2yÞÞr2 þðt7 þ t8 cosð2yÞþ t10 cosð4yÞÞr4 þ . . .ÞH 3
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ ððt15 þ t16Þ cosð2yÞr2 þ . . .ÞH 5
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ ðð3s3 þ s4ÞH 2 þ m
`H 4 þ t18H 6 þ . . .Þ cosðyÞr
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ ððm
`
`þ m6 cos2ðyÞÞH 2 þðt11 þ t12 cos2ðyÞÞH 4 þ . . .Þ cosðyÞr3
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ ððt4 þ t6 cos2ðyÞÞH 2 þ . . .Þ cosðyÞr5
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ s5H 3 þ m
`H 5 þ t20H 7 þ . . .
`þ . . . HIGHER ORDER ABERRATIONS IN TERMS OF r AND H:
`
`7
`
`4
`
`QUINTIC COMA
`
`10
`
`LINEAR ASTIGMATISM
`
`CUBIC ASTIGMATISM
`
`QUINTIC ASTIGMATISM
`
`12
`
`DISTORTION
`
`(4-3)
`The five s, twelve m, and twenty t coefficients represent the third-, fifth-, and
`seventh-order terms, respectively. Even-order terms do not appear as a conse-
`quence of the rotational symmetry of the optical system. Further, there are
`actually five, nine, and 14 independent coefficients for the third-, fifth-, and
`seventh-order terms, respectively.7
`There exist three identities between the m coefficients, and six identities
`between the t coefficients. These identities take the form of a linear combination
`of the nth-order coefficients being equal to combinations of products of the lower-
`order coefficients. If, for example, all of the third-order coefficients have been
`corrected to zero, then the following identities for the fifth-order coefficients exist:
` 2
` m
` m
` m
` m
`m
`m
`9
`8
`7
`6
`5
`4
`2
`3
`3
`ficients is straightforward, although tedious, using the iterative process developed
`by Hans Buchdahl.1 The third-order terms were first popularized by the publication
`of Seidel and are often referred to as the Seidel aberrations.8 The fifth-order terms
`were first computed in the early twentieth century.9
`
`¼ 0; and m
`
`¼ 0. Calculation of these coef-
`
`¼ 0; m
`
`Apple v. Corephotonics
`IPR2019-00030
`Exhibit 2024 Page 11 of 37
`
`

`

`110
`
`Aberration Theory
`
`In the late 1940s, Buchdahl published his work on how to calculate the coef-
`ficients to any arbitrary order. However, recent investigation into the historical
`work of Joseph Petzval, a Hungarian professor of mathematics at Vienna, has
`lead to the belief that he had developed in the late 1830s a computational
`scheme through fifth-order and perhaps to seventh-order for spherical aberra-
`tion.10 Conrady was well aware of the Petzval sum in addition to Petzval’s
`greater contributions to optics as evidenced when he wrote:
`
`[Petzval] who investigated the aberrations of oblique pencils about 1840, and
`apparently arrived at a complete theory not only of the primary, but also of the
`secondary oblique aberrations; but he never published his methods in any complete
`form, he lost the priority which undoubtedly would have been his. It is, however,
`perfectly clear from his occasional brief publications that he had a more accurate
`knowledge of the profound significance of the Petzval theorem than any of his
`successors in the investigation of the oblique aberrations for some eighty years
`after his original discovery.11
`
`1
`
`SPHERICAL
`
`5
`
`11
`
`r
`
`are given by
`
`and
`
`Regrettably, the preponderance of his work was lost to posterity. The design
`and development for today’s optical systems were made possible by theoretical
`understanding of optical aberrations through the contributions of numerous
`individuals. Although the subject is still evolving, serious research spans over
`four centuries.12
`As an example, consider a meridional ray intersecting the paraxial image
`plane, and having entrance pupil coordinates of ðr; 900; H; 0Þ: The ex and ey
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`r5 þ t1r7 þ . . .Þ
`ex ¼ ðs1r3 þ m
`
`
`
`
`þ ðs3 þ s4ÞH 2 þ m
`H 4 þ t19H 6 þ . . .
`
`
`þ m
`H 2 þ t13H 4 þ . . .
`r3
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ t5H 2 þ . . .
`r5
`þ . . . HIGHER ORDER ABERRATIONS IN TERMS OF r AND H:
`
`
`Þr4 þ ðt2 t3Þr6 þ . . .ÞH
` m
`ey ¼ ðs2r2 þ ðm
`
`
`þ ðm
`Þr2 þ ðt7 t8 þ t10Þr4 þ . . .
` m
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ ðt15 t16Þr2 þ . . .
`|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
`þ s5H 3 þ m
`H 5 þ t20H 7 þ . . .
`þ . . . HIGHER ORDER ABERRATIONS IN TERMS OF r AND H:
`
`ASTIGMATISM
`
`3
`
`2
`
`8
`
`7
`
`H 5
`
`COMA
`
`12
`
`DISTORTION
`
`H 3
`
`Apple v. Corephotonics
`IPR2019-00030
`Exhibit 2024 Page 12 of 37
`
`

`

`4.2 Symmetrical Optical Systems
`
`111
`
`Observe that the sagittal term ex comprises only astigmatic contributions,
`while the meridional term ey contains only comatic contributions. The ability
`to isolate specific contributions of the ray error by proper selection of one or
`more rays will be exploited in the remainder of this chapter.
`As previously explained, the actual ray height in the paraxial image plane
`can be considered to comprise two principal elements: the Gaussian ray height
`and the ray aberration, as illustrated in the aberration map shown in Figure 4.5
`on the next page. The total aberration for a rotationally symmetric optical
`system comprises two orthogonal components, astigmatic and comatic. The
`astigmatic aberration is segmented into field independent and dependent com-
`ponents while the comatic aberration is divided into aperture independent and
`dependent components.
`The field-independent astigmatic aberration has two contributions, which are
`defocus and spherical aberration. The defocus x is linearly dependent on the
`entrance pupil radius r while the spherical aberration is dependent on the
`odd orders of third and above of the entrance pupil radius, namely, r3; r5; . . ..
`The field-independent astigmatic aberration introduces a uniform aberration or
`blur over the optical system’s field-of-view.
`
`Field-dependent astigmatic aberrations comprise two contributions which
`are linear astigmatism and oblique spherical aberration. Both of these aberra-
`tions are dependent on even orders of H, namely, H 2; H 4; . . .. Linear astigma-
`tism is linearly dependent on the entrance pupil radius r while the oblique
`spherical aberration is dependent on the odd orders of third and above of the
`entrance pupil radius. It should be noted that the defocus and linear astigma-
`tism comprise the linearly-dependent entrance-pupil-radius components of the
`astigmatic aberration (r; H 0; H 2; H 4; . . .). In a like manner, spherical and
`oblique spherical aberrations comprise the higher-order terms in entrance-
`pupil-radius (r3; r5; . . . ;H 0; H 2; H 4; . . .).
`Aperture-independent comatic aberration has two contributions, which are
`the Gaussian image height and distortion. Although the Gaussian image height
`is not considered an actual aberration, it is shown in the aberration map in a
`dashed box since the Gaussian image height is linearly proportional to H and
`aperture independent. Distortion is also aperture independent, but is dependent
`on the odd orders of third and above of H, namely, H 3; H 5; . . . .
`Aperture-dependent comatic aberration also has two contributions, which
`are linear coma and nonlinear coma. Linear coma is linearly dependent on the
`field angle and on even orders of the entrance pupil radius (r2; r4; . . . ;H ). Non-
`linear coma has the same entrance pupil radius dependence as does linear coma,
`but is dependent on the odd orders of third and above of H in the same manner
`as distortion. Perhaps the most common element of nonlinear coma is referred
`to as elliptical coma; however, there are many other contributions to the nonlin-
`ear comatic aberration.
`
`Apple v. Corephotonics
`IPR2019-00030
`Exhibit 2024 Page 13 of 37
`
`

`

`Actual
`ray
`height
`
`Total
`aberration
`
`Gaussian
`ray
`height
`
`Astigmatic
`
`Comatic
`
`Field
`independent
`H 0
`
`Field
`dependent
`H 2+H 4+...
`
`Aperture
`independent
`r0
`
`Aperture
`dependent
`r2 + r4 + ...
`
`Defocus
`r
`
`Spherical
`r3 + r5 +...
`
`Linear
`astigmatism
`r
`
`Oblique
`spherical
`r3 + r5+...
`
`Gaussian
`image height
`H
`
`Distortion
`
`H 3 + H 5+...
`
`Linear
`coma
`H
`
`Nonlinear
`coma
`H 3+ H 5+...
`
`Figure 4.5 Ray aberration map showing the astigmatic and comatic elements comprising the total ray aberration.
`
`Apple v. Corephotonics
`IPR2019-00030
`Exhibit 2024 Page 14 of 37
`
`

`

`4.2 Symmetrical Optical Systems
`
`113
`
`The aperture-dependent comatic aberration can be viewed as a variation
`of magnification from one zone to another zone of the entrance pupil. It is
`also noted that because the astigmatic and comatic contributions are orthog-
`onal, changing the location of the image plane from the paraxial location
`can impact the resultant astigmatic aberration while having no effect on the
`comatic contribution of the total aberration. In other words, the defocus can
`change the astigmatic contribution to the total aberration while having no effect
`on the comatic contribution. This will be discussed in more detail later in this
`chapter.
`An interesting aspect of the Buchdahl aberration expansion is that the contri-
`bution for each coefficient is computed surface by surface and then summed to
`determine the value of the coefficient at the image plane. For example, s1 is the
`third-order spherical aberration coefficient. Its value for an optical system com-
`prising n surfaces is computed as
`
`s1 ¼
`
`Xn
`
`i¼1
`
`is1
`
`Although there will be no attempt to compute the general set of Buchdahl
`aberration coefficients in this study,
`it is important to understand certain
`aspects of their relationship to the design process. It can be shown that these
`aberration coefficients have intrinsic and extrinsic contributions. The third-
`order aberration coefficients have only intrinsic contributions, which mean that
`the value of the aberration coefficients computed for any arbitrary surface are
`not dependent on the aberration coefficient values for any other surface. For
`the higher-order aberration coefficients, extrinsic contributions exist in addition
`to the intrinsic contributions. This means that aberration coefficients for the kth
`surface are to some extent dependent on the preceding surfaces while not at all
`dependent on the subsequent surfaces.
`Two other characteristics of aberration coefficients are valuable for the lens
`designer to understand. The first is that lower-order aberration coefficients
`affect similar high-order aberration coefficients. An alternative way to express
`this behavior is that higher-order aberration coefficients do not affect the
`value of lower-order aberration coefficients; that is, adjustment of say t1 does
`not change the third- and fifth-order contributions. The second characteristic
`is that higher-order aberration coefficients move or change their values slowly
`with changes in constructional parameters (radii, thickness, etc.) compared to
`the movement of lower-order aberration coefficients. In short, this means that
`higher-order aberrations, be they astigmatic or comatic, are far more stable
`than lower-order aberrations.
`
`Apple v. Corephotonics
`IPR2019-00030
`Exhibit 2024 Page 15 of 37
`
`

`

`114
`
`Aberration Theory
`
`4.3 ABERRATION DETERMINATION
`USING RAY TRACE DATA
`
`The elements comprising the total ray aberration can be computed directly
`from specific ray trace data. How this is done and the relationship to the aber-
`ration coefficients are presented in this section. It should be noted that the
`method discussed decouples the defocus element from the astigmatic elements
`there