1 Design and Correction of optical Systems Part 7: Geometrical aberrations Summer term 2012 Herbert Gross 2 Overview 2012-04-18 1. Basics 2012-04-18 2. Materials 2012-04-25 3.Components 2012-05-02 4. Paraxial optics 2012-05-09 5. Properties of optical systems 2012-05-16 6.Photometry 2012-05-23 7. Geometrical aberrations 2012-05-30 8. Wave optical aberrations 2012-06-06 9. Fourier optical image formation 2012-06-13 10.Performance criteria 1 2012-06-20 11.Performance criteria 2 2012-06-27 12.Measurement of system quality 2012-07-04 13.Correction of aberrations 1 2012-07-11 14.Optical system classification 2012-07-18 3 Part 7: Contents 2012-05-30 7.1 Representations 7.2 Power expansion of aberrations 7.3 Seidel aberrations 7.4 Surface contributions 7.5 Primary monochromatic aberrations 7.6 Chromatical aberrations 4 Optical Image Formation ° Perfect optical image: All rays coming from one object point intersect in one image point ° Real system with aberrations: 1. transverse aberrations in the image plane 2. longitudinal aberrations from the image plane 3. wave aberrations in the exit pupil 5 Representation of Geometrical Aberrations ° Longitudinal aberrations ∆s Gaussian image plane reference ray logitudinal aberration Gaussian along the reference ray image plane ∆∆∆l' ray ray reference U' point optical axis optical axis ∆∆∆l' o longitudinal aberration system projected on the axis system ∆ s' longitudinal aberration ° Transverse aberrations ∆y transverse reference ray ∆∆∆y' (real or ideal chief ray) aberration ray U' optical axis reference plane system 6 Representation of Geometrical Aberrations ° Angle aberrations ∆u ideal reference ray angular aberration ∆∆∆U' real ray optical axis system x reference sphere Gaussian wavefront reference plane ° Wave aberrations ∆W W > 0 paraxial ray real ray U' C z y' R ∆∆∆ ∆∆∆ s' < 0 7 Transverse Aberrations ° Typical low order polynomial contributions for: Defocus, coma, spherical, lateral color ° This allows a quick classification of real curves linear: quadratic: cubic: offset: defocus coma spherical lateral color 8 Transverse Aberrations ° Classical aberration curves ° Strong relation to spot diagram ° Usually only linear sampling along the x-, y-axis no information in the quadrant of the aperture ∆∆∆y ∆∆∆x 5 µµµm 5 µµµm tangential sagittal -1 y p xp 1 -1 1 Dx λλλ= 486 nm Dy λλλ= 588 nm λλλ= 656 nm 9 Spot Diagram ° 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 10 Spot Diagram ° Table with various values of: 1. Field size 486nm 546nm 656nm 2. Color ° Small circle: Airy diameter for axis comparison ° Large circle: Gaussian moment field zone full field 11 Aberrations of a Single Lens ° Single plane-convex lens, y BK7, f = 100 mm, λ = 500 nm ° Spot as a function of field position ° Coma shape rotates according to circular symmetry ° Decrease of performance with the distance to the axis x 12 Polynomial Expansion of the Aberrations ° Paraxial optics: small field and aperture angles Aberrations occur for larger angle values optical ° Two-dimensional Taylor expansion shows field yp axis entrance and aperture dependence pupil xp ° Expansion for one meridional field point y xp coma rays chief ° Pupil: cartesian or polar grid in x p / y p outer rays of ray aperture cone r θθθ ray yp O field sagittal point plane object height y meridional plane axis point object plane 13 Polynomial Expansion of Aberrations ° Taylor expansion of the deviation: k l m y' Image height index k ∆y(y ,' rp ,θ) = ∑ aklm ⋅ y' ⋅rp⋅cos θ lk ,, m rp Pupil height index l θ Pupil azimuth angle index m ° Symmetry invariance: selection of special combinations of exponent terms ° Number of terms: sum of isum number Type of aberration indices in the exponent isum of terms 2 2 image location ° The order of the aperture function depends on the aberration type used: 4 5 primary aberrations, 3rd/4th order primary aberrations: 6 9 secondary aberrations, 5th/6th order - 3rd order in transverse aberration ∆y 8 14 higher order - 4th order in wave aberration W Since the coupling relation 1 ∂W ∆y = − ⋅ R ∂x p changes the order by 1 14 Power Series Expansion of Aberrations ° General case : two coordinates in object plane and pupil ° Rotational symmetry: 3 invariants 1. Scalar product of field vector and pupil vector r r yp P ⋅ F = P ⋅ F ⋅ cos( ϕ F −ϕ P ) = x p ⋅ x + y p ⋅ y z 2. Square of field height xp P yp r r xp P F ⋅ F = F 2 = x 2 + y 2 y 3. Square of pupil height r r 2 2 2 F P ⋅ P = P = x + y x P p p upil F ° y Therefore: x Only special power combinations are physically meaningful Ob ject 15 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 Primary location aberrations / Seidel Field y Secondary Spherical Coma Astigmatism aberrations y0 y 1 y 2 y 3 y 4 y 5 y cos y 3 cos y 5 cos 0 Distortion r Tilt Distortion Distortion primary secondary 2 1 r 1 y r cos 2 y 4 r 1 2 1 2 1 cos r Defocus y r y 4 r 1 Aper- Astig./Curvat. 2 y 3 2 ture r cos y r cos 3 r 2 Coma r 3 2 primary y r cos r 3 y 2 r 3 cos 2 3 r Spherical 2 3 primary y r y r 4 cos r 4 Coma secondary r 5 r 5 Spherical secondary 16 Primary Aberrations ° Expansion of the transverse aberration ∆y on image height y and pupil height r ° Lowest order 3 of real aberrations: primary or Seidel aberrations ° Spherical aberration: S 3 2 2 - no dependence on field, valid on axis ∆y = r ⋅ S + r ⋅ y ⋅r ⋅cos θ ⋅C - depends in 3rd order on apertur + y 2 ⋅r ⋅cos 2 θ ⋅ A + y 2 ⋅r ⋅ P ° Coma: C - linear function of field y + y3 ⋅ D - depends in 2rd order on apertur with azimuthal variation ° Astigmatism: A - linear function of apertur with azimuthal variation - quadratic function of field size ° Image curvature (Petzval): P - linear dependence on apertur - quadratic function of field size ° Distortion: D - No dependence on apertur - depends in 3rd order on the field size 17 Transverse Aberrations of Seidel 2 2 4 2 2 3 ° Transverse deviations x' (x' + y' )s' [2x' (x' x' + y' y' )+ x'(x' + y' )]s' s' ∆x'= p p p S'− p p p p p p C' 2n' R'3 2n' R'3 ° Sum of surface p p 2 2 2 2 2 2 contributions x'()x' x' + y' y' s' s' x' ()x' + y' s' s' + p p p A'+ p p p p P' k 3 3 n' R' p 2n' R' p S'= ∑ S j j=1 2 2 3 x'()x' + y' s s'' p k − 3 D' 2n' R' p C'= ∑C j j=1 k 2 2 4 2 2 3 A'= ∑ Aj y' p (x' p + y' p )s' [2y' p (x' x' p + y' y' p )+ y'(x' p + y' p )]s' s' p j=1 ∆y'= 3 S'− 3 C' k 2n'R' p 2n' R' p P'= ∑ Pj 2 2 2 2 2 2 y'()x' x' + y' y' s' s' y' ()x' + y' s' s' j=1 + p p p A'+ p p p p P' k 3 3 n' R' p 2n'R' p D'= ∑ D j 2 2 3 j=1 y'()x' + y' s s'' p − 3 D' 2n' R' p 18 Surface Contributions ° Spherical aberration 1 1 S 4Q2 j = ω j j − n' j s' j njs j n ω Q ° Coma 4 2 1 1 1 pj pj 1 1 C j = ω j Q j − ⋅ ⋅ − n' j s' j njs j ω jQ j s1 s p1 2 2 n ω Q ° Astigmatisms 4 2 1 1 1 pj pj 1 1 Aj = ω j Q j − ⋅ ⋅ − n' j s' j njs j ω jQ j s1 s p1 2 2 ° Field curvature 1 1 n ω Q 1 1 1 1 1 P 4Q2 1 pj pj j = ω j j − ⋅ ⋅ − − ⋅ − n' j s' j njs j ω jQ j s1 s p1 rr n' j n j ° Distortion 2 2 1 1 n ω Q 1 1 1 1 1 n ω Q 1 1 D = ω 4Q2 − ⋅ 1 pj pj ⋅ − − ⋅ − ⋅ 1 pj pj − j j j n' s' n s ω Q s s r n' n ω Q s s j j j j j j 1 p1 r j j j j 1 p1 19 Surface Contributions of Chromatical Aberrations ° Axial chromatical aberration (CHL ) 2 Φ j K j = ω j ν j ° Transverse chromatical aberration s⋅ s n −1 CHV 1p 1 j H j = ⋅ωj ω pjQ pj∆ F sp1− s 1 n jν j ° Total chromatical errors of a system 2 2 s'n CHL h ∆ s'CHL = −2 ⋅KLinse = − n''nω n f ⋅ ν n CHV ∆ y''CHV= y n ⋅∑ H j j=1 20 Surface Contributions ° Abbreviations: hj 1. height ratio of marginal rays ω j = h1 hpj 2. height ratio of chief rays ω pj = hp1 1 1 3. Surface invariant of Abbe Qj= n j ⋅ − rj s j 4. Abbe invariant of the pupil imaging 1 1 Qpj= n j ⋅ − rj s pj 21 Surface Contributions of Seidel ° In 3rd order (Seidel) : Additive contributions of all surfaces of a system to the total aberration 4 (' ' ) 1 1 2 ' ' ' ° Spherical aberration n n −n n n + n s S = − 2 ⋅ − ⋅ − 8n R s' R s' p' R − s'+ p' ° Coma C = 4⋅ ⋅ S s'−R 2 R − s'+ p' A = 4⋅ ⋅ S ° Astigmatisms s'−R 2 R − s'+ p' n (' n'−n) ° Field curvature P = 2⋅ ⋅ S − s'−R 4nRp '2 3 R − s'+ p' n (' n'−n) R − s'+ p' ° Distortion E = 4⋅ ⋅ S − ⋅ s'−R 2nRp '2 s'−R 22 Surface Contributions: Example 200 SI 0 ° Seidel aberrations: Spherical Aberration representation as sum of -200 1000 surface contributions possible SII ° Gives information on correction 0 Coma of a system -1000 ° Example: photographic lens 2000 SIII 0 Astigmatism -2000 1000 SIV 0 Retrofocus F/2.8 Petzval field curvature Field: w=37° -1000 6000 SV 0 Distortion -6000 5 10 150 3 4 2 CI 1 67 8 9 0 Axial color -100 600 CII 0 Lateral color -400 Surface 1 2 3 4 5 6 7 8 9 10 Sum 23 Lens Contributions of Seidel ° In 3rd order (Seidel) : Additive contributions of lenses to the total aberration value (stop at lens position) 2 1 n3 n 2 (2 n2 )1 n2 (n )1 + − − 2 ° Spherical aberration Slens = 3 + ⋅ X − ⋅ M − ⋅ M 32 n(n − )1 f n −1 n −1 n + 2 n + 2 1 n +1 ° Coma Clens = 2 ⋅ X − 2( n + )1 M 4ns ' f n −1 1 ° Astigmatisms A = − lens 2 f ⋅ s'2 n +1 ° Field curvature P = − lens 4nf ⋅ s'2 ° Distortion Dlens = 0 24 Spherical Aberration ° Spherical aberration: On axis, circular symmetry ° Perfect focussing near axis: paraxial