Advanced Lens Design Lecture 3: Aberrations I 214-11-4 Herbert Gross Winter term 214 www.iap.uni-jena.de
2 Preliminary Schedule 1 21.1. Basics Paraxial optics, imaging, Zemax handling 2 28.1. Optical systems Optical systems, raytrace, advanced handling 3 4.11. Aberrations I Geometrical aberrations, wave aberrations 4 11.11. Aberrations II PSF, OTF, sine condition, aplanatism, isoplanatism 5 18.11. Optimization I Basic principles, merit function setup, optimization in Zemax 6 25.11. Optimization II Initial systems, solves, bending, AC meniscus lenses, constraints, effectiveness of variables 7 2.12. Structural modifications Zero operands, lens splitting, lens addition, lens removal, material selection 8 9.12. Aspheres and freeforms Correction with aspheres, Forbes approach, freeform surfaces, optimal location of aspheres, several aspheres 9 16.12. Field flattening Astigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses 1 6.1. Chromatical correction Achromatization, apochromatic correction, axial versus transversal, glass selection rules, burried surfaces 11 13.1. Special correction topics Symmetry, sensitivity, anamorphotic lenses, field lens, telecentricity, stop position 12 2.1. Higher order aberrations high NA systems, broken achromates, induced aberrations 13 27.1. Advanced optimization Local vs global optimization, control of iteration, growing strategies requirements 14 3.2. Mirror systems special aspects, double passes, catadioptric systems 15 1.2. Diffractive elements color correction, ray equivalent model, straylight, third order aberrations, manufacturing
3 Contents 1. Geometrical aberrations 2. Seidel theory 3. Wave aberrations 4. Zernike coefficients
Optical Image Formation Perfect optical image: All rays coming from one object point intersect in one image point Real system with aberrations: 1. tranverse aberrations in the image plane 2. longitudinal aberrations from the image plane 3. wave aberrations in the exit pupil object plane wave aberrations image plane optical system transverse aberrations longitudinal aberrations
Representation of Geometrical Aberrations Longitudinal aberrations Ds reference ray ray logitudinal aberration along the reference ray Dl' Gaussian image plane ray Gaussian image plane U' reference point optical axis Dl' o optical axis system longitudinal aberration projected on the axis system D s' Transverse aberrations Dy longitudinal aberration reference ray (real or ideal chief ray) Dy' transverse aberration ray U' optical axis reference plane system
Representation of Geometrical Aberrations Angle aberrations Du ideal reference ray DU' angular aberration real ray optical axis system Wave aberrations DW x reference sphere wavefront W > Gaussian reference plane paraxial ray real ray U' C z R Dy' D s' <
Transverse Aberrations Typical low order polynomial contributions for: defocus, coma, spherical aberration, lateral color This allows a quick classification of real curves Dy' K' r' p cos p Dy' S' r' 3 p cos p Dy' C' y' r' 2 p (2 cos2 ) P linear: defocus quadratic: coma cubic: spherical offset: lateral color
Pupil Sampling Ray plots Spot sagittal ray fan tangential aberration Dy sagittal aberration Dx diagrams x p x p tangential ray fan Dy y p whole pupil area
.2 mm Spherical Aberration Typical chart of aberration representation Reference: at paraxial focus Primary spherical aberration at paraxial focus Wave aberration tangential sagittal 2 2 Transverse ray aberration Dy Dx Dy.1 mm.1 mm.1 mm Pupil: y-section x-section x-section Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 1 cycles per mm 1 1.5.5 1 2 cyc/mm -.2..2 z/mm Geometrical spot through focus Ref: H. Zügge -.2 -.1..1.2 z/mm
1 Notations for an Optical System x, y object coordinates, especially object height x', y' image coordinates, especially image height x p,y p coordinates of entrance pupil x' p, y' p coordinates of exit pupil s object distance form 1st surface s' image distance form last surface p entrance pupil distance from 1st surface p' exit pupil distance from last surface Dx' sagittal transverse aberration Dy' meridional transverse aberration y p x' P y' p p' x' s' y' P' y' Dx' P' Dy' image plane z x P y' p x y system surfaces s p x P y p entrance pupil x' P exit pupil object plane y P
Polynomial Expansion of Aberrations Representation of 2-dimensional Taylor series vs field y and aperture r Selection rules: checkerboard filling of the matrix Constant sum of exponents according to the order Image location Primary aberrations / Seidel y r 5 Field y Spherical Coma Astigmatism y y 1 y 2 y 3 y 4 y 5 y cos 3 cos y 5 cos Distortion r Tilt Distortion Distortion primary secondary r 1 y 2 r 1 cos 2 y 4 r 1 cos 2 r 1 Defocus y 2 r 1 Aperture Astig./Curvat. y 4 r 1 y r 2 cos y 3 r r 2 r 2 cos 3 Coma primary y 3 r 2 cos r 3 y 2 r 3 cos 2 r 3 Spherical primary y 2 r 3 y r 4 cos r 4 Coma secondary Spherical r 5 secondary Secondary aberrations
Surface Contributions: Example Seidel aberrations: representation as sum of surface contributions possible Gives information on correction of a system Example: photographic lens Retrofocus F/2.8 Field: w=37 2-2 1-1 2-2 1-1 6 S I Spherical Aberration S II Coma S III Astigmatism S IV Petzval field curvature S V Distortion -6 1 2 3 4 5 1 6 7 8 9 15-1 6 C I Axial color -4 Surface 1 2 3 4 5 6 7 8 9 1 Sum C II Lateral color
Wave Aberration in Optical Systems Definition of optical path length in an optical system: Reference sphere around the ideal object point through the center of the pupil Chief ray serves as reference Difference of OPL : optical path difference OPD Practical calculation: discrete sampling of the pupil area, real wave surface represented as matrix y y p y' p y' upper coma ray optical system W wave aberration chief ray w' image point z chief ray wave front object point lower coma ray reference sphere Object plane Op Entrance pupil EnP Exit plane ExP Image plane Ip
14 Pupil Sampling All rays start in one point in the object plane The entrance pupil is sampled equidistant In the exit pupil, the transferred grid may be distorted In the image plane a spreaded spot diagram is generated object plane point entrance pupil equidistant grid optical system exit pupil transferred grid image plane spot diagram y o y p y' p y' x o x p x' p x' z
15 Zernike Polynomials Expansion of wave aberration surface into elementary functions / shapes m W ( r, ) c Z ( r, ) n n m n nm n m = + 8 + 7 + 6 + 5 cos Zernike functions are defined in circular coordinates r, Z m n ( r, ) R m n sin ( m ) for ( r) cos( m ) for 1 for m m m Ordering of the Zernike polynomials by indices: n : radial m : azimuthal, sin/cos + 4 + 3 + 2 + 1-1 - 2-3 - 4 Mathematically orthonormal function on unit circle for a constant weighting function Direct relation to primary aberration types - 5-6 - 7-8 n = sin 1 2 3 4 5 6 7 8
16 Zernike Polynomials: Fringe Convention Nr Cartesian representation Circular representation 1 1 1 2 x r sin 3 y r cos 4 2 x 2 + 2 y 2-1 2 r² - 1 5 2 x y r² sin 2 6 y 2 - x 2 r² cos 2 7 ( 3x 2 + 3 y 2-2 ) x ( 3r 3-2r ) sin 8 ( 3x 2 + 3 y 2-2 ) y ( 3r 3-2r ) cos 9 6 (x 2 +y 2 ) 2-6 (x 2 +y 2 ) +1 6r 4-6r² + 1 1 ( 3y 2 -x 2 ) x r³ sin 3 11 ( y 2-3x 2 ) y r³ cos 3 12 (4x 2 +4y 2-3) 2xy ( 4r 4-3r² ) sin 2 13 (4x 2 +4y 2-3) (y 2 - x 2 ) ( 4r 4-3r² ) cos 2 14 [1(x 2 +y 2 ) 2-12(x 2 +y 2 )+3] x ( 1r 5-12r³ + 3r ) sin 15 [1(x 2 +y 2 ) 2-12(x 2 +y 2 )+3] y ( 1r 5-12r³ + 3r ) cos 16 2 (x 2 +y 2 ) 3-3 (x 2 +y 2 ) 2 + 12 (x 2 +y 2 ) - 1 2r 6-3r 4 + 12r² - 1 17 (y 2 -x 2 ) 4xy R 4 sin 4 18 y 4 +x 4-6x 2 y 2 R 4 cos 4
Indices of Zernike Fringe Polynomials 17