Lens Design II Lecture 3: Aspheres 6-- Herbert Gross Winter term 6 www.iap.uni-jena.de
Preliminar Schedule 9.. Aberrations and optimiation Repetition 6.. Structural modifications Zero operands, lens splitting, lens addition, lens removal, material selection 3.. Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres 4 9.. Freeforms Freeform surfaces 5 6.. Field flattening Astigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses 6 3.. Chromatical correction I Achromatiation, axial versus transversal, glass selection rules, burried surfaces 7 3.. Chromatical correction II secondar spectrum, apochromatic correction, spherochromatism 8 7.. Special correction topics I Smmetr, wide field sstems,stop position 9 4.. Special correction topics II Anamorphotic lenses, telecentricit.. Higher order aberrations high NA sstems, broken achromates, induced aberrations 4.. Further topics Sensitivit, scan sstems, eepieces.. Mirror sstems special aspects, double passes, catadioptric sstems 3 8.. Zoom sstems mechanical compensation, optical compensation 4 5.. Diffractive elements color correction, ra equivalent model, stralight, third order aberrations, manufacturing 5.. Realiation aspects Tolerancing, adjustment
3 Contents. Aspheres. Conic sections 3. Forbes aspheres 4. Sstem improvement b aspheres 5. Aspheres in Zemax
x x c x c x R R R R x x Conic section Special case spherical Cone Toroidal surface with radii R x and R in the two section planes Generalied onic section without circular smmetr Roof surface c x c c x c x x x tan 4 Aspherical Surface Tpes
Polnomial Aspherical Surface Other descriptions 6 6 4 4 4 4 x s s c s c s C s b s b B k A C B A M m N n n m ij M m m m h a h a h k h ) ( ) ( t g t f h Superconic (Grenolds ) Implicit -polnomial asphere (Lerner/Sasian ) Truncated parametric Talor (Lerner/Sasian ) 5 Ref: K. Uhlendorf 5
6 Conic Sections Explicite surface equation, resolved to Parameters: curvature c = / R conic parameter Influence of on the surface shape cx c x Parameter Surface shape = - paraboloid < - hperboloid = sphere > oblate ellipsoid (disc) > > - prolate ellipsoid (cigar ) Relations with axis lengths a,b of conic sections a b c b a b c a c
7 Aspherical Shape of Conic Sections Conic aspherical surface Variation of the conical parameter c c.8.6.4. - -.8 -.6 -.4 -...4.6.8
8 Simple Asphere Parabolic Mirror Equation Radius of curvature in vertex: R s Perfect imaging on axis for object at infinit Strong coma aberration for finite field angles Applications:. Astronomical telescopes. Collector in illumination sstems R s axis w = field w = field w = 4
9 Parabolic Mirror Equation c : curvature /R s : eccentricit ( = - ) c ( ) c C F ra R sag vertex circle sagittal circle of curvature tangential circle of curvature R tan vertex circle R s parabolic mirror F x R radii of curvature : tan Rs R s parabolic mirror R tan Rs R s R s f 3
Simple Asphere Elliptical Mirror Equation Radius of curvature r in vertex, curvature c eccentricit Two different shapes: oblate / prolate Perfect imaging on axis for finite object and image loaction Different magnifications depending on used part of the mirror Applications: Illumination sstems s c ( ) c s' F F'
Ellipsoid Mirror Equation c: curvature /R : Eccentricit e c ( ) c b oblate vertex radius Rso F prolate vertex radius R sp a F' ellipsoid
Asphere: Perfect Imaging on Axis Perfect stigmatic imaging on axis: Hperoloid rear surface r s n s n s r n n n s F Strong decrease of performance for finite field sie : dominant coma Alternative: ellipsoidal surface on front surface and concentric rear surface 5 D spot m] w in
3 Asphere: Perfect Imaging on Axis Perfect stigmatic imaging on axis: elliptical front surface elliptical concentric
Polnomial Aspherical Surface Standard rotational-smmetric description 4 Basic form of a conic section superimposed b a Talor expansion of ( h) h c h M m a m h m4 h... Radial distance to optical axis h x... Curvature c... Conic constant a m... Apherical coefficients,5,5 h^4 h^6 h^8 h^ h^ h^4 h^6,,4,6,8, h Ref: K. Uhlendorf
5 Aspheres - Geometr Reference: deviation from sphere Deviation along axis Better conditions: normal deviation r s () deviation height tangente () deviation along axis height sphere perpendicular deviation r s aspherical shape spherical surface aspherical contour
6 Aspherical Expansion Order Improvement b higher orders Generation of high gradients (r) 6. order 5 D rms [m] 3 4. order 8. order. order. order -5 -..4.6.8 r - 4 6 8 4 order k max
7 Aspheres: Correction of Higher Order Correction at discrete sampling Large deviations between sampling points Larger oscillations for higher orders Better description: slope, defines ra bending residual spherical transverse aberrations perfect correcting surface Corrected points with ' = corrected points residual angle deviation points with maximal angle error paraxial range ' = c d A /d real asphere with oscillations A
8 Forbes Aspheres New representation of aspherical expansions according to Forbes (7) ( r) c r c r k max k Special polnomials Q k (r ):. Contributions are orthogonal slope. tolerancing is easil measurable 3. optimiation has better performance 4. usuall fewer coefficients are necessar 5. use of normalied radial coordinate makes coefficients independent on diameter a k Q k ( r ) Two different versions possible: a) strong aspheres: deviation defined along b) mild aspheres: deviation defined perpendicular to the surface
Polnomial Aspherical Surface Forbes Aspheres - Q con 9 New orthogonaliation and normaliation using Jacobi-polnomials Q m ( h) h c h M 4 h / h a Q h / h max m m m max requires normaliation radius h max (: conversion to standard aspheres possible),5 Mean square slope M m a m / m 5,5 -,5,,4,6,8, h^4*q h^4*q h^4*q h^4*q3 h^4*q4 h^4*q5 - h Ref: K. Uhlendorf 9
Polnomial Aspherical Surface Forbes Aspheres - Q bfs Limit gradients b special choice of the scalar product (: conversion to standard aspheres not possible) h u u ( h) h M a B / m m h h m u mit u : h,5 max Mean square slope / h max M m a m,,4,6,8, u(-u)b u(-u)b u(-u)b u(-u)b3 u(-u)b4 u(-u)b5 -,5 h Ref: K. Uhlendorf
Forbes Aspheres Strong asphere Q con sag along -axis slope orthogonal true polnom tpe Q in Zemax cr r r a Q r cr kmax 4 ( ) ( ) k k k direct tolerancing of coefficients Mild asphere Q bfs difference to best fit sphere sag along local surface normal not slope orthogonal not a polnomial tpe Q in Zemax cr (r) cr r r c cr M m a B no direct relation of coefficients to slope m m r,5,5,5,,4,6,8, h^4*q h^4*q h^4*q h^4*q3 h^4*q4 h^4*q5,,4,6,8, u(-u)b u(-u)b u(-u)b u(-u)b3 u(-u)b4 u(-u)b5 -,5 - h -,5 h
Aplanatic Aspherical Sstems Aplanatic Telescope with two aspheres primar mirror D P f S secondar mirror u P d u' f P image F Point-b-point determination of aplanatic imaging conditions asphere asphere ra ra
3 Aspheres Correcting Residual Wave Aberrations Special correcting free shaped aspheres: Inversion of incoming wave front Application: final correction of lithographic sstems conventional lens lens with correcting surface
4 Aspherical Single Lens Correction on axis and field point Field correction: two aspheres spherical axis field, tangential field, sagittal 5 m 5 m 5 m a one aspherical 5 m 5 m 5 m a a double aspherical 5 m 5 m 5 m
5 Reducing the Number of Lenses with Aspheres Example photographic oom lens Equivalent performance 9 lenses reduced to 6 lenses Overall length reduced a) all spherical 9 lenses Vario Sonnar 3.5-6.5 / f = 8-56 b) with 3 aspheres 6 lenses length reduced aspherical surfaces Ref: H. Zügge
6 Reducing the Number of Lenses with Aspheres Example photographic oom lens Equivalent performance 9 lenses reduced to 6 lenses Overall length reduced Photographic lens f = 53 mm, F# = 6.5 a) all spherical, 9 lenses axis field x x 436 nm 588 nm 656 nm p x p p x p b) 3 aspheres, 6 lenses, shorter, better performance axis field x x A A 3 A p x p p x p Ref: H. Zügge
7 Reducing the Number of Lenses with Aspheres Binocular Lenses.5x Nearl equivalent performance Distortion, Field curvature and pupil aberration slightl improved lens removed Better ee relief distance a) Binocular.5x, all spherical field curvature in dptr distortion in % tan sag b) Binocular.5x, aspherical surface - + -5 +5 tan sag - + -5 +5
8 Lithographic Projection: Improvement b Aspheres Considerable reduction of length and diameter b aspherical surfaces Performance equivalent a) NA =.8 spherical 3 lenses lenses removable b) NA =.8, 8 aspherical surfaces -9% -3% 9 lenses Ref: W. Ulrich
9 Best Position of Aspheres Location depending on correction target: spherical : pupil plane coma and astigmatism: field plane No effect on Petval curvature aspherical effect.4.3. spherical coma astigmatism distortion. -. -. -.5.5 d/p'
3 Aspherical Sensitivit 3 spherical aberration Example: Lithographic lens Sensitivities for aspherical correction.8.6 5 5 5 3 35 coma surface index.4..4.3 5 5 5 3 35 astigmatism surface index....5 5 5 5 3 35 distortion surface index..5 S S4 S5 S S6 stop S3 S8 5 5 5 3 35 surface index
3 Aspheriation of a Camera Lens Selection of one aspherical surface in a photographic lens S S 5 S 9 S S4..5 spherical aberration..5..5. 3 4 5 6 7 8 9 3 4 coma surface index spherical sstem: 97 nm surface : 96 nm surface 5: 85 nm surface 9: 87 nm surface : 78 nm surface 4: 78 nm.5.4.3 3 4 5 6 7 8 9 3 4 astigmatism surface index...5 3 4 5 6 7 8 9 3 4 distortion surface index.5 3 4 5 6 7 8 9 3 4 surface index
3 Realiation Aspects for Aspheres Strong asphere : Turning points ''= Deviation from sphere asphere profile (r) - - 4 6 8 4 6 r.. derivative '(r) 3 profile deviation (r) -. 4 6 8 4 6.5 r 4 6 8 4 6 r.5. derivative '(r). derivative ''(r) -.5 4 6 8 4 6 r 4 6 8 4 6 r
33 Surface properties and settings Setting of surface properties
Important Surface Tpes 34 34 Standard Even asphere Paraxial Paraxial XY Coordinate break Diffraction grating Gradient Toroidal Zernike Fringe sag Extended polnomial Black Box Lens ABCD spherical and conic sections classical asphere ideal lens ideal toric lens change of coordinate sstem line grating gradient medium clindrical lens surface as superposition of Zernike functions generalied asphere hidden sstem, from vendors paraxial segment