OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-1 Section 25 Aberrations Supplemental Materials OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-2 fraction are not always small. always not fraction are ane. These rays are called ane. skew rays. es (tangents of angles). This removes the es (tangents of angles). of the images all point sources the surface of a sphere, not at the vertex the surface of a sphere, an analysis. The paraxial image is often an systems – spherical surfaces are true the d elements. Aberrations result d from the ered to be a collection of independently optical systems with perfect imagery. optical systems with real systems from this perfection. . Angles of incidence and re . ′

 sin ′ n =

 sin n linearity of raytraces. Real rays do not scale. plane. - Surfaces have sag, and refraction occurs at -is law Snell’s - angles must be used, not paraxial angl Real - rays may leave the meridional Real (y-z) pl First-order or paraxial systems are ideal Aberrations describe the deviation of An aberrated system will have image locations and magnifications approximately the same as those predicted by paraxial or Gaussi as a reference for the measurement of aberrations. used In geometrical optics, the object is consid image is the sum radiating point sources. The (independent irradiance patterns). There is no interference. exist for perfectly manufactured Aberrations spheres, and there are no tilted or decentere breakdown of the paraxial assumptions: Aberrations of the Rotationally Symmetric Optical System OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-3 P x , PP xy  P P r  y )  and  nt to all points in the pupil, through the nt to all points in the pupil, through from a given object point to all points in the or angle the image height e deviations of these rays from paraxial e ont approach may be used. The amount of may be used. The ont approach (or the polar pupil coordinates – normalized pupil coordinates the

P   cos sin

and y   P   H –H normalized object height the x . P P P y x is defined against the sign

 Note that by tradition, the azimuth that by Note angle convention. The physical pupil radius is r A ray analysis traces rays from the object poi A Th to the image plane. optical system, and are the aberrations. behavior pupil of the system. Either a ray or wavefr aberration will be a function of To analyze aberrations, light must propagate analyze To Aberration Analysis OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp ′ 25-4 z ′ x ′ y ′ y Image

  cos sin ′   x   P P y x P P y x P y Pupil P x z y h Object x Coordinate System OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-5 z Image Points are imaged to points. to a single image point. All rays converge All wavefronts are spherical. Object Perfect System OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-6 ) Y z  Focus Paraxial Aberration ( Transverse Ray Transverse ) Z  axial locations ( the pupil focus at different focus at different the pupil Rays from zones different in Not Spherical Measuring the errors: -errors – Ray relative measure to a reference image point - errors – Wave Sphere measure relative or Reference to a perfect wavefront Determine the deviations from perfection. Real System OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-7 are  Z  xy Reference z  Image Point YP,P  are measured in the W Z Paraxial  Image Plane R and longitudinal ray errors Y  , rsection the of the paraxial chief ray and X  rence sphere centered on the reference image are measured as a function of the pupil  P, P  Aberrated Wavefront Wxy Reference Wavefront P r XP is the radius of the reference sphere or the image distance. A positive wavefront or the image distance. A is the radius of reference sphere R Both the wavefront error and the ray errors coordinates for a particular object point. XP relative or refe XP to a reference wavefront point. error is shown. measured relative to this reference image point. Wavefront errors paraxial image plane. Transverse ray errors paraxial image plane. Transverse A reference image point is defined by the inte reference image point is defined by A Reference Image Point OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-8 W f 2/# 1    P R rnu  The transverse ray errors are related to the The ill change if the reference image point is ill change    P P y , , P  P     Wx y Wx    PP PP R rx R ry   PAPPRPP , ,,, , P xy xy    XPP YPP Wx y W x y W x y   are the image space index and marginal ray angle. and are the image space index ′ u and ′ moved. to the wavefront. rays are perpendicular The error: slope of the wavefront n The wavefront error is the OPL difference (or OPD) between the actual wavefront and the error w wavefront The reference wavefront. Wavefront Error and Ray Errors OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-9 z z P P y y Pupil Pupil P P x x   P P x0 y0 Object Object = 0. P x = 0. P y the meridional plane need be considered. A the meridional plane need be considered. tersects a general point in the pupil. Two rays intersect the pupil at Sagittal rays or transverse intersect the pupil at skew ray leaves the meridional plane and in for aberration analysis. special sets of rays are used Tangential rays or meridional By rotational By symmetry, only object points in Tangential and Sagittal Rays OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-10 P P y x 1 1 X Y -40 -40 40 40   -1 -1 Example: Spherical Aberration Example: Spherical = 0. P = 0. x P y for P for y P x r the tangential and sagittal r the tangential and sets of rays. versus Y

 versus X  plot the transverse ray error.  Ray x Actual Position X  Y  Reference Image Point  y The tangential ray fan plots The sagittal ray fan plots Ray fans (or ray intercept curves) Wave fans are plots of the wavefront error fo Wave fans are plots of the wavefront Wave Fans and Ray Fans Wave Fans and Ray OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-11 X  for each ray. for each . Y ......  ...... Y ...... , ......  ...... X ......  ...... Spot Diagram , rays are traced through a uniform grid in , rays are traced through ount of energy. The spot diagram plots all The ount of energy. the spot diagram must be symmetric with and spot diagrams) will with the and vary timate of the image blur produced by that by timate of the image blur produced is anti-symmetric. All of the aberration is anti-symmetric. ce image point. The common grids are square, common ce image point. The . In a similar fashion, the sagittal fan wave 0  P x X  placing a dot at the value of by . H P y Square Grid system aberration. From a single object point to the same am ray corresponds the EP. Each the ray intersections relative to the referen hexapolar and dithered. the ray intercepts Graphically show The spot diagram provides a geometrical spot diagram provides es The Spot Diagrams For a rotationally symmetric optical system, a For and respect to the meridional plane, must be symmetric, and the sagittal ray fan fans, ray fans measures (including the wave or FOV object/image height OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-12 . The ray errors are 1/2    1/2    RMS   1  N 1 i   N i  N N       Xi  1/2 1  1/2 N i e location is found by averaging the ray errors:  through the transverse ray errors to find through 1 N  X  22 22 the root mean squared spot size

     

 

 00 00   21 1 21  N i  11 11         1 N .      y  222 RY X YYY YiY XXX XiX YYi  and MS RMS RMS x RMS d d R RMS d d A better measure of the spot size is A The spot size (max to min) is found by sorting by spot size (max to min) is found The in total range The spot centroid relative to the reference imag The integrated or summed over the pupil: radial RMS spot size can also be determined: A Spot Size OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-13 = 1). H

 cos

 IJ K Defocus Tilt Wavefront Spherical Aberration (SA) Coma Astigmatism Field Curvature Distortion  IJK WH ves the value of wavefront aberration stem. These aberrations are inherent to the stem. These expansion for the wavefront aberrations for the wavefront expansion = 1) and the edge of the field ( the requirements of rotational symmetry, the P y . The coefficient subscript encodes the powers of  is the amount of wavefront error associated with this is the amount of wavefront cos

 IJK H W

 2 

and 

2   cos cos 2 2 Third-Order Terms Third-Order cos 3 ,   cos  2 2 2  3  4 2 H H H H  H

 H 040 131 222 220 311 111 020 W W W W W W W + + + + + + = W the corresponding polynomial term: the corresponding inherent to a rotationally symmetric optical sy design of the system. In order to satisfy expansion terms are The wavefront expansion is a power series Wavefront Expansion Using normalized field and pupil coordinates gi meaning. physical coefficients aberration term at the edge of the pupil ( aberration term at the edge OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-14 ) and 000 W Fifth-Order SA Fifth-Order Linear Coma Fifth-Order Astigmatism Fifth-Order Field Curvature Fifth-Order Distortion SA Sagittal Oblique Tangential Oblique SA Cubic ComaLine Coma Eliptical Coma , etc.), are usually ignored. usually , etc.), are 400 W ,

   2 2 3

 200   W cos cos cos cos cos 2 2 cos 4 4 3 3 Fifth-Order Terms

5       

 4 4 5 2 2 3 3 6

 H H H H H H H H 060 151 422 420 511 250 252 331 333 W W W W W W W W W + + + + + + + + + Higher Order Terms + field-dependent phase ( The wavefront terms are denoted by the order of their ray aberration, which is one less piston ( pupil dependence, with no Terms aberration order. than the wavefront Wavefront Expansion –Wavefront Expansion Order Terms Higher OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-15 the paraxial magnification and actual P 111 WH P R r

 0   X cos Y    111 111  WWH WHy magnification of the system. Wavefront tilt describes a difference between Wavefront Tilt OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp P P y x 25-16 f/5 1 1 X Y    -20 -20 20 20  20 W = 1 W =  -1 -1 P is used. is used. 20 W   P P , the actual image plane is lance and better image quality. 20 20 20 20  020 WWx WWy W PP PP RR RR rr rr 2sin2 2cos2 222

 X YP 20 20       WW Wxy    Defocus is not really an aberration. In a system with defocus displaced from the paraxial image plane. More importantly, allows the image plane or reference point defocus to be shifted for aberration ba with spherical aberration, a better image is For example Recognizing inside paraxial focus. an image plane formed on the notation decision, user that this shift is a Defocus OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-17 20 , an image 2 X  z , 2  Y  f   8/#  8/# 20  W zf W

    XX

     X  YY Y nce sphere, not the actual wavefront in   z and transverse ray aberrations W Focus Paraxial z  Plane Image Shifted   WW W the XP of the system. the XP Moving the image plane changes the refere Moving the image plane changes In a system that has a wavefront error In a system that has wavefront plane shift the measured apparent aberration: changes Defocus – Image Plane Shift OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-18 Defocus – Focus Spot Through Diagrams OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp P P y x 25-19 1 1 X Y -40 -40 40 40   040 W 1 = f/5 -1 -1 2 power or focal length of the power 3 3 3 3 P   PP 040 040 040 040  W Wx Wy W P P P P R R r R r R r r 4sin 4 4 4cos 422      X YP X Y

040 040      WW W x y Ray fans: Spherochromatism is SA that varies with wavelength. Spherochromatism is SA system to vary with pupil radius. Spherical aberration causes the Spherical Aberration OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp Z  P 1 y 25-20 LA z TA Focus Z Paraxial  LA Focus Marginal r from the top of pupil: . The real marginal P y 2 P inal ray crosses the axis). The 040 Wy  2 is the distance from paraxial focus to distance from paraxial focus is the 1 is the transverse ray erro LA f tan TA  for SA are quadratic with for SA  z YP  16 /#     Z TA LA U TA y  . ′ U The transverse aberration ray angle is longitudinal ray errors marginal focus (where the real marg marginal focus (where The longitudinal aberration Transverse and Longitudinal SA OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp z 25-21 TA MID Focus RMS z Paraxial z  MIN  z  LA MARGINAL z  Focus Marginal Caustic .5 .67 .75     zLA zLA zLA zLA ch as stars. This condition is ch as stars. used for

    wavefront variance condition which is 040 1.33 1.5 2 2 20 040 20 040 20 040 20 040  WW WW WW WW     16 /#  LA f W Mid focus: Min RMS: Min circle: focus: Marginal Mid focus corresponds to the minimum optimum for viewing isolated point sources su sources isolated point optimum for viewing designing telescopes. Spherical Aberration and Defocus Spherical Aberration and The image plane can be shifted from paraxial focus to obtain better image quality in the presence of SA. Focus criteria include mid focus, minimum RMS spot size, and minimum circle the marginal ray crosses (where caustic). OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-22 Spherical Aberration –Focus Spot Diagrams Through OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-23 Square Grid Variance Minimum Wavefront Wavefront Mid Focus Spot Size Minimum RMS Paraxial Focus Minimum Circle Marginal Focus Spherical Aberration –Positions Diagrams at Focus Spot OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-24 No field dependence Spherical Aberration – Plot Field OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-25 > 0), independent of lens bending. > 0), independent of lens bending. 040 W This plot is for an object at infinity in visible The stop is light at (n about 1.5). spherical aberration is The the lens. that is bending a minimized for coma The approximately convex-plano. is also zero at this bending. positive thin lens has positive SA A ( Bending can only change the magnitude This is called undercorrected of the SA. in all is the situation the and shown SA figures. . However, the aberrations of lens do However, .  rence in the curvatures of its two surfaces: 12   1  nCC  

 Smith –Smith Optical Engineering Modern The lens can be bent without changing its power change with the bending. The power of a thin lens depends on the diffe on of a thin lens depends power The Spherical Aberration and Lens Bending OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-26 Previously Covered Plano-Convex the surfaces. This minimizes the angles of the surfaces. as close to paraxial as possible. This is as close to paraxial possible. pherical aberration. It can only be minimized. the light is bent same amount at each ndex, as is often found in the IR, best shape as is often found ndex, ent bendings to minimize spherical aberration. ent bendings y bending occurs at one surface. In the convex- occurs at one y bending nimum deviation for a refracting prism. for a nimum deviation Convex-Plano There is no bending that completely eliminates There is no bending s In the plano-convex configuration, all of the ra In the plano-convex is split configuration, the ray bending between plano makes the situation incidence at the surfaces and to the angle of mi directly analogous require differ object/image conjugates Different At high i The optimum shape varies with index. meniscus. is a of the lens. other aberrations also vary with the bending The lens surface. Object at Infinity: The minimum The spherical aberration occurs when Lens Bending and Minimum SA OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-27 z z z Previously Covered At 1:1 imaging, an equiconvex lens is At 1:1 imaging, an equiconvex is to split the biconvex lens into two plano- is to split the biconvex o lenses in their infinite conjugate minimum spherical aberration results than in the bi- is required to minimize spherical aberration. o lenses can be changed to match the object be changed o lenses can This final vertex-to-vertex arrangement uses tw uses This final vertex-to-vertex arrangement spherical aberration configuration. Much less The focal lengths of the tw solution. convex lenses. in fast condenser This solution is also sometimes used image conjugates. and convex lenses and then flip each of the lenses: then flip each lenses and convex A trick to further minimize spherical aberration A The particular shape depends on the magnification. used. At finite singlet image conjugates, a bi-convex Spherical Aberration and Finite Spherical Aberration and Congugates OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp P P y x f/5 25-28 1 1 X Y H = 1 -30 -30 30 30   131 W = 1 W = -1 -1 . H 2cos2 sin 2  2 2 2   131 131 131

 WHy WH WH P P R r R P r R r cos 3 3 0       X X Y YP     131  WWH Ray fans: Coma results when the magnification of the system varies with system the magnification of the results when Coma pupil position. An asymmetric as the entire blur is produced side of the paraxial image location. The image blur is to one image blur increases linearly with height Coma OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-29 Point Image T is assumed to is assumed Paraxial C H < 0). S 60° C 131 W S C e pupil maps to a displaced circle of light Depending on the sign of coma, pattern Depending and about 55% of a the light 60 degree wedge, S coma. This is the natural stop position. coma. bending and the stop position. For any bending, the stop position. For any and bending C 131 > 0) or away from the optical axis ( > 0) or away 131 P 131 R and sagittal coma r P R r T 3 W C   S T CW CW For a thin lens, coma varies with lens there is a stop location that eliminates Tangential coma Tangential measures of coma: are two in the image blur. The blur is contained in blur is contained in the image blur. The in the first is contained third of the pattern. can flare towards ( For a given object point, each annular zone in th annular zone object point, each For a given represent a positive image height. Coma – Diagram Spot OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-30 Coma – Focus Spot Diagrams Through OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-31 Coma – Field Plot OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp P P y x 25-32 1 1 X Y H = 1 -20 -20 20 20   222 W = 1W = f/5 -1 -1 With positive astigmatism, light from a With closer to the vertical meridian is focused the horizontal lens than light through two object point produces meridian. Each are the These perpendicular line images. tangential focus and the sagittal focus. P ans is different as a function ans is different as a 2 222 WHy P R cos r 2

 0 Smith –Smith Optical Engineering Modern 22 2 22   X YP   222 222  WWH WHy of image height. In a system with astigmatism, of the optical system the power vertical in horizontal and meridi Astigmatism OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-33 z 2 Plane 222 2 Paraxial Image T-M-S No Field Curvature 222 /2 4( /#) 222 222 8( /#)   DL f WH LfWH 22 22 8( /#) 4( /#) 0    L zfHW zfHW z

   and a line image in the meridional plane is and ntial or meridional rays focus, and a line and ntial or meridional rays focus, onal plane. Located between these two line 2 Sagittal 2 zero image heights. On axis, image heights. On zero te curved image plane. In the te curved .5 0 D 20 20 222 20 222 W WWH WWH Medial    L Tangential Sagittal focus: Medial focus: Tangential focus: Tangential presence of astigmatism only: This aberrational astigmatism is not astigmatism. there is no caused by manufacturing errors. Each of these foci lies on a separa The field dependence of astigmatism is due to apparent foreshortening of the pupil at non- Tangential Focus and Sagittal Focus Sagittal focus is where the sagittal rays focus, sagittal rays the is where Sagittal focus formed. Tangential focus is where the tange image is formed perpendicular to the meridi foci is a circular focus called the medial focus. OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-34 Sagittal Medial Tangential Astigmatism –Diagrams Spot Focus Through OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-35 Astigmatism – Plot – Field Focal Surface Sagittal OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-36 Astigmatism – Plot – Field Surface Focal Medial OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-37 Astigmatism – Plot – Field Focal Surface Tangential OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-38 tric optical system does not exist on axis not exist on tric optical system does s not vary with field of view. Imperfectly not vary with field of view. s from imperfections of either the cornea or from imperfections of either the cornea the circular pupil is viewed at an oblique at an the circular pupil is viewed ve a cylindrical component to their shape. ese two different distinguish ese two conditions. To ries with meridional cross section. Line power results power from the apparent foreshortening xhibit this same type of error. - Aberrational astigmatism - Visual astigmatism or axial astigmatism or at zero field height. This asymmetry in fields When of the pupil at non-zero of view. angle, it will appear elliptical in shape. results Visual astigmatism, the other hand, on or both ha the crystalline lens of the eye. One va of the eye As a result, the paraxial power images result on axis, and this type of error doe manufactured optical systems may e term The “astigmatism” is used to describe th them, adjectives can be used: The astigmatism The present in a rotationally symme Visual Astigmatism OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp P P y x f/5 25-39 1 1 X Y -20 -20 20 20 H = 1   220 W = 1 W = -1 -1 z Blur z Image . A compromise A .  2 H  P P P e natural tendency of optical e natural tendency  2 2 220 220 z WHx WHy P Blur R P r R Image r 2 2  22 2 2 2   X YP   220 220   WWH WHx y Field curvature characterizes th Field curvature systems to have curved image planes. A positive singlet image planes. A has curved systems to have image surface. an inward bending Field Curvature flat image plane that reduces the average image blur occurs flat image plane that reduces the average inside paraxial focus. A perfect image is formed on a curved surface, and the image A blur at the paraxial image plane increases as OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-40 nd occur in the same relative order: T-M-S- in the same relative occur order: nd at medial focus. An artificially An flattened at medial focus. surface represents the fundamental field the construction parameters of system: .5 .5 balancing astigmatism and field curvature. 22 2 22 22 ure for the astigmatic image surfaces. 20 220 222 20 220 20 220 222 20 220 222 WWHWH WWH WWHWH WWHWH     The field curvature is a bias curvat Sagittal surface: Medial surface: Tangential surface: Petzval surface: image surface, the Petzval While not a good curvature of the system. only on It depends element indices of refraction. surface curvatures and Field Curvature –Field Curvature Surface Petzval P or P-S-M-T. The best image quality occurs The or P-S-M-T. P field or medial surface can be obtained by These four image surfaces are equally spaced a four image surfaces are equally spaced These OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-41 Field Curvature –Field Curvature Plot – Field Focus Paraxial OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-42 Field Curvature –Field Curvature Plot – Field Paraxial Focus Inside OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-43 H . Straight H the image. Points still map to points, so and the image point position is displaced and stinguished by their different field or their different stinguished by nd distortion both are constant with respect tion varies with the image height tion varies with P 3 311 WH P R cos r 0  33   X Y 

311 311  WWH WHy . These two aberration terms . can be di P y lines in the object are mapped to curved lines in to curved lines in the object are mapped image blur associated with distortion. there is no Distortion occurs when image magnifica when Distortion occurs Distortion Distortion is a quadratic magnification error, Distortion is a in a radial direction. tilt transverse ray fans for wavefront The a to dependence: linear for wavefront tilt and cubic for distortion. OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-44   x x   y y . H . The . H H H represents a positive image height. H 0 0 for for 00 00     311 311 Y Y

W 

W  Barrel and Pincushion Distortion Barrel and Pincushion Barrel distortion results when the actual the distortion results when Barrel the paraxial less than magnification becomes magnification with increasing in towards the corners of a square are pushed optical axis. actual the distortion results when Pincushion the than larger magnification becomes paraxial magnification with increasing from The corners of a square are pulled away the optical axis. The figures assume OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-45 = 0 ray H 3 2 2 H H H H H H None None . A similar chart P H x 3 P P P 0 0 x and x x vs. P X y  , P x . A positive slope of the A . H and P y P x 2 3 P P raxial focus, as a ray from the top of pupil P P P y y y y y vs. aberrations, and the ray fans encode the the ray fans encode aberrations, and e slope is proportional to the separation. Y > 0). The image plane is outside paraxial focus The > 0).

 ConstantConstant 0 0 are especially important for deciphering the are especially important for deciphering P y > 0 for Y  Aberration Defocus Astigmatism Coma Field Curvature Wavefront tilt Wavefront Distortion SA fan indicates that the image plane is inside pa has not yet crossed the axis ( for a negative slope. The magnitude of th The magnitude for a negative slope. The slopes of the ray fans at origins The field astigmatism a non-zero curvature and defocus, produce aberration content. Only on dependence slope, but each has a different aberration content in the dependence of the ray errors on aberration content in the dependence A real system will be degraded by multiple by degraded real system will be A exists for wave fans. exists for wave Combinations of Aberrations OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-46 100% Astigmatism 75% Astigmatism 50% Astigmatism 25% Astigmatism 25% Coma 50% Coma 75% Coma 100% SA 75% SA 50% SA 100% Coma 25% SA Defocus has been added to each to produce the minimum RMS spot size. Combined Aberrations – Spot Diagrams OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-47 Stop Shift – the Stop With Moving SA Introduces Off-Axis Aberrations OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-48 is the k e flat or spherical. is the radial coordinate and is the radial coordinate most optical surfaces ar most optical surfaces r ted by rotation of a conic section about the ci. A source placed at one focus will image The sag of a conic is given by 221/2 Cr 2 ten used as reflecting surfaces. Cr  1(1(1) )  () sr is the base curvature of surface, C where conic constant. Conics are of The introduction of aspheric surfaces provides more optimization introduction of aspheric surfaces provides variables for aberration The correction. Rotational symmetry is maintained. The first class of aspheric surfaces is genera optical axis. Conics are defined by two fo without aberration to the other focus. Because of the ease of fabrication and testing, of the ease fabrication and Because Conics and Aspheres OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-49 2 F 2 F 1 2 F 2 F F 1 F 1 F .... 2468 1234 virtual. Hyperbolas are used as are used virtual. Hyperbolas 1/2 r is at the focal point of 22 Cr 2  Cr  1 1  0     10

      111   sr Ar Ar Ar Ar negative reflecting elements. to conic the to added be rotationally symmetric terms can Other obtain a generalized asphere: reflecting for imaging distant objects. surface. Parabolas are used Ellipse: foci are real. Elliptical for relaying Both surfaces are used images. Hyperbola: other is the is real, and focus One Circle: foci are at the center of curvature. Both Parabola: is at infinity, the othe focus One Conic Sections OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-50 Parabola Parabola Parabola to correct spherical aberration. The to correct spherical aberration. Hyperbola yout, except that it uses two hyperbolic mirrors Ellipse Newtonian telescope: a parabola with a fold flat. It is directly to a Keplerian telescope. analogous The imaging properties of conic surfaces are telescopes. in the design of mirror-based used Gregorian telescope: the parabola is followed by an ellipse to relay As with the intermediate image. a relayed Keplerian telescope, for terrestrial is good this design an applications as it produces erect image. Ritchey-Chretien telescope is identical in la Ritchey-Chretien to correct coma as well spherical aberration. Cassegrain telescope: the parabola is Cassegrain telescope: the parabola combined with a hyperbolic secondary mirror the system length. to reduce combination of the primary and The is the mirror equivalent of a secondary telephoto objective. The Cassegrain design uses two conic surfaces Mirror Based Telescopes Mirror Based OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-51

 = 0, the diffraction- (Kathy Creath) W

 /, /  xffyf  XY f 2 , PP 

 appears as a phase factor in the pupil of as a phase appears 2/ iWxy W on PSF equals the squared modulus of the modulus on PSF equals the squared The result is scaled to the image plane The t amounts of defocus (in OPD): waves XP P r D    cyl         (, ) ). The wavefront error wavefront ). The ′ y , ′ PSF x y e x Degradation of the Airy disk by differen Degradation of the Fourier transform of the pupil function. coordinates ( The diffraction-based point spread functi limited Airy disk results. disk limited Airy system. The cylinder function defines the pupil diameter. When system. The Diffraction and Aberrations Diffraction and OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-52 (Kathy Creath) /4. This is /4. the Rayleigh  These improved spot patterns result the image plane to from refocusing find a “best” focus location. Spherical aberration balanced with an Spherical aberration balanced equal and opposite amount of defocus (both in waves of OPD): Note that for both SA and defocus, the and defocus, Note that for both SA limited when system is nearly diffraction the introduced wavefront aberration is less than criterion for image quality. criterion for image quality. Spherical Aberration and Diffraction Spherical Aberration and Different Amounts of SA (in waves OPD): OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-53 r 0.7  Airy Disk Aberrated W Airy Disk SR  1.22 f/# E 0  0 E E ure of image quality for systems with small  0 0 E E  SR The SR has a maximum value of one, and Airy it of the measures the degradation reduction of the SR is directly Any disk. variance. proportional to the wavefront is correlates well with image SR The to values of about 0.5. quality down amounts of aberration. The Strehl The ratio SR is a single-number meas Strehl Ratio OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-54 ), c , Popular Photography ns defects, spherical aberration. line or a sharp vertical a line. Coma, for the strong horizontal line in image ( d slightly either forward or back, the st of the light energy within a very small st of the light energy lly the image should be a point (a), but the lly the image should when the light source is in line with axis when the lens. In the photomicrographs on the on the lens. In photomicrographs the light source is off axis, is depicted in is depicted is off axis, the light source rration. Astigmatism shows up in purer form rration. Astigmatism up shows off-axis images produced when fast, modern by magnifying the images formed at focal by rcle with a diameter of .03 millimeter. ), which combines a complex mixture of coma, astigmatism mixture of coma, combines a complex ), which f Popular Photography Magazine. Popular Photography ) exhibits one of the most common le of the most common ) exhibits one b ). If the focus of the lens were move ). If the focus of lens were d ). The final image ( ). The e astigmatism would produce either a sharp horizontal familiar type of aberration that arises when image ( which also exhibits coma and spherical abe also exhibits coma and which in image ( Another common defect, astigmatism, accounts Images from William H. Price, “The Photographic Lens,” Scientific American, 72-83 (August 1976). The following photomicrographs were taken by Norman Goldberg who was Technical Director of Caption: Lens aberrations can be studied plane when a point light source is beamed at plane when technical director of Goldberg, made by Norman opposite page, and chromatic aberration, is typical of the chromatic aberration, and at full for the great magnification of image, lenses are used aperture. After allowance it however, is evident that the lens focuses mo in this case is a ci “blur disk,” which the images are magnified 600 diameters. Idea the images are magnified 600 as in this case, only ideal is usually achieved, of the lens. Image ( Aberration Images With Diffraction OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 25-55 Chromatic Aberration Chromatic Aberration and Chromatic Aberration Images With Diffraction D) Astigmatism D) Astigmatism E) Coma Astigmatism F) Combination of Coma, A) Diffraction-limited point image A) Diffraction-limited Aberration B) Spherical and Coma with Astigmatism C)