Optical Design with Zemax

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1 Optical Design with Zemax Lecture 3: Properties of optical sstems II Herbert Gross Sommer term 04

2 Preliminar Schedule.04. Introduction Properties of optical sstems I Properties of optical sstems II Aberrations I Aberrations II Optimiation I Introduction, Zemax interface, menues, file handling, preferences, Editors, updates, windows, coordinates, Sstem description, Component reversal, sstem insertion, scaling, 3D geometr, aperture, field, wavelength Diameters, stop and pupil, vignetting, Laouts, Materials, Glass catalogs, Ratrace, Ra fans and sampling, Footprints Tpes of surfaces, Aspheres, Gratings and diffractive surfaces, Gradient media, Cardinal elements, Lens properties, Imaging, magnification, paraxial approximation and modelling Representation of geometrical aberrations, Spot diagram, Transverse aberration diagrams, Aberration expansions, Primar aberrations, Wave aberrations, Zernike polnomials, Point spread function, Optical transfer function Principles of nonlinear optimiation, Optimiation in optical design, Global optimiation methods, Solves and pickups, variables, Sensitivit of variables in optical sstems Optimiation II Sstematic methods and optimiation process, Starting points, Optimiation in Zemax Advanced handling I Telecentricit, infinit object distance and afocal image, Local/global coordinates, Add fold mirror, Scale sstem, Make double pass, Vignetting, Diameter tpes, Ra aiming, Material index fit Advanced handling II Report graphics, Universal plot, Slider, Visual optimiation, IO of data, Multiconfiguration, Fiber coupling, Macro language, Lens catalogs Imaging Fundamentals of Fourier optics, Phsical optical image formation, Imaging in Zemax Correction I Correction II Smmetr principle, Lens bending, Correcting spherical aberration, Coma, stop position, Astigmatism, Field flattening, Chromatical correction, Retrofocus and telephoto setup, Design method Field lenses, Stop position influence, Aspheres and higher orders, Principles of glass selection, Sensitivit of a sstem correction Miscellaneous Microscopic objective lens, Zoom sstem, induced aberrations Phsical optical modelling Gaussian beams, POP propagation, polariation ratrace, illumination

3 3 Contents 3rd Lecture. Tpes of surfaces. Aspheres 3. Gratings and diffractive surfaces 4. Gradient media 5. Cardinal elements 6. Lens properties 7. Imaging 8. Magnification 9. Paraxial approximation and modelling

4 Surface properties and settings Setting of surface properties surface tpe additional drawing switches diameter local tilt and decenter operator and sampling for POP coating scattering options

5 Ideal lens Ideale lens Principal surfaces are spheres The marginal ra heights in the vortex plane are different for larger angles Inconsistencies in the laout drawings P P'

6 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 6 Aspherical surface tpes

7 7 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 = 0 sphere > 0 oblate ellipsoid (disc) 0 > > - prolate ellipsoid (cigar ) Relations with axis lengths a,b of conic sections a b c b a b c a c

8 Aspherical shape of conic sections Conic aspherical surface Variation of the conical parameter c c

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

10 Ellipsoid mirror Equation c: curvature /R : Eccentricit e c ( ) c b oblate vertex radius Rso F prolate vertex radius R sp a F' ellipsoid

11 Sag of a surface Sag at height for a spherical surface: parabola r r Paraxial approximation: quadratic term p r height p = /(r) sag sphere axis

12 Grating Diffraction Maximum intensit: constructive interference of the contributions of all periods grating Grating equation g sin sin m o grating constant g in-phase +. diffraction order s = incident light

13 Ideale diffraction grating Ideal diffraction grating: monochromatic incident collimated beam is decomposed into discrete sharp diffraction orders Constructive interference of the contributions of all periodic cells grating diffraction orders +. Onl two orders for sinusoidal +. incident collimated light g = / s grating constant

14 Finite width of real grating orders Interference function of a finite number N of periods Finite width of ever order depends on N Sharp order direction onl in the limit of g N sin sin I g sin sin sin / 4g N N N = 5 N = 5 N =

15 Diffractive Elements Original lens height profile h(x) Wrapping of the lens profile: h red (x) reduction on maximal height h Digitaliation of the reduced profile: h q (x) h 3 h h h(x) : continuous profile h h red (x) : wrapped reduced profile h q (x) : quantied profile

16 6 Diffracting surfaces Surface with grating structure: new ra direction follows the grating equation Local approximation in the case of space-varing grating width s' n s n' mg gˆ e n' d Ratrace onl into one desired diffraction order Notations: g : unit vector perpendicular to grooves d : local grating width m : diffraction order e : unit normal vector of surface s e g p p grooves s Applications: - diffractive elements - line gratings d - holographic components

17 7 Diffracting surfaces Local micro-structured surface Location of ra bending : macroscopic carrier surface Direction of ra bending : local grating micro-structure Independent degrees of freedom:. shape of substrate determines the point of the ra bending. local grating constant determines the direction of the bended ra lens local grating g(x,) thin laer bending angle macroscopic surface curvature m-th order

18 Modellierung diffractive elements Discrete topograph on the surface Phase unwrapped to get a smooth surface Onl one desired order can be calculated with ratrace numerical optimied DOE Model approximation according to Sweatt: same refraction b small height and high index phase unwrapped Sweatt-model ( thin lens with high index ) ERL-model ( equivalent refractive lens ) black box model

19 Diffractive Optics: Sweatt Model Phase function redistributed: large index / small height tpical : n = 0000 ( x, ) n ( x, ) n* *( x, ) Calculation in conventional software with ratrace possible ra bending equivalent refractive aspherical lens the same ra bending Sweatt lens * real index n snthetic large index n*

20 0 Ratracing in GRIN media Ra: in general curved line Numerical solution of Eikonal equation Step-based Runge-Kutta algorithm 4th order expansion, adaptive step width Large computational times necessar for high accurac d r dt nn D n n x n n n n x Brechahl : n(x,,) b b ' c s c s x' Strahl

21 Description of GRIN media Analtical description of grin media b Talor expansions of the function n(x,,) Separation of coordinates nn c o, c hc 0 xc h x c 3 c Circular smmetr, nested expansion with mixed terms nn c o, Circular smmetr onl radial h 4 c x 3 4 h c 6 3 c 5 c h 8 4 c 6 c c ch c3h c4h c5c 6h c7h c8h c9h c c h c h c h c h c c h c h c h c h h n n o, c ( c h) c 3 ( c h) c 4 ( c h) c 5 ( c h) c 6 ( c h) c 8 3 c Onl axial gradients 4 6 n n o, c ( c ) c 3 ( c ) c 4 ( c ) c 5 ( c ) 8 Circular smmetr, separated, wavelength dependent nn c h c h c h c h c c c o,,, 3, 4, 5, 6, 7, 3

22 Gradient Lens Tpes Curved ra path in inhomogeneous media Different tpes of profiles n entrance () n exit () radial gradient rod lens axial gradient rod lens radial and axial gradient rod lens n i n o n(x,,) radial gradient lens axial gradient lens radial and axial gradient lens

23 Gradium Lenses Axial profile 'Gradium' k Focussing effect onl for oblique ras Combined effect of front surface curvature n( ) n k k max and index gradient Special clindrical blanks with given profile allows choice of individual -interval for the lens n.75 lens.70 max

24 Collecting radial selfoc lens F L F' Thick Wood lens with parabolic index profile Principal planes at /3 and /3 of thickness P P' L n( r) n n r 0 n > 0 : collecting lens n < 0 : negative lens F' P P'

25 Gradient Lenses marginal Tpes of lenses with parabolic profile n( r) n 0 n 0 n r n Ar 0 n r r coma Pitch length 0.5 Pitch Object at infinit p n0 n n r 0.50 Pitch Object at front surface 0.75 Pitch Object at infinit.0 Pitch Object at front surface Pitch

26 Cardinal elements of a lens Focal points:. incoming parallel ra intersects the axis in F. ra through F is leaves the lens parallel to the axis Principal plane P: location of apparent ra bending principal plane P' f ' u' s BFL F' focal plane nodal planes s P' u N N' u' Nodal points: Ra through N goes through N and preserves the direction

27 Notations of a lens P principal point S vertex of the surface F focal point O n n n s f intersection point of a ra with axis focal length PF u F S P P' N N' S' u' F' r radius of surface curvature ' O' d thickness SS s f f' s' n refrative index f BFL s P s' P' f' BFL a d a'

28 Main properties of a lens Main notations and properties of a lens: - radii of curvature r, r curvatures c sign: r > 0 : center of curvature is located on the right side - thickness d along the axis - diameter D - index of refraction of lens material n Focal length (paraxial) Optical power Back focal length intersection length, measured from the vertex point c r c r F ' f, f ' tan u F s n f f s n' f ' F ' ' H ' tan u'

29 Lens shape Different shapes of singlet lenses:. bi-, smmetric. plane convex / concave, one surface plane 3. Meniscus, both surface radii with the same sign Convex: bending outside Concave: hollow surface Principal planes P, P : outside for mesicus shaped lenses P P' P P' P P' P P' P P' P P' bi-convex lens plane-convex lens positive meniscus lens bi-concave lens plane-concave lens negative meniscus lens

30 Lens bending und shift of principal plane Ra path at a lens of constant focal length and different bending Quantitative parameter of description X: The ra angle inside the lens changes X R R R R The ra incidence angles at the surfaces changes strongl The principal planes move For invariant location of P, P the position of the lens moves P P' F' X = -4 X = - X = 0 X = + X = +4

31 Optical imaging Optical Image formation: All ra emerging from one object point meet in the perfect image point Region near axis: gaussian imaging ideal, paraxial Image field sie: Chief ra field point O chief ra pupil stop Aperture/sie of light cone: marginal ra defined b pupil stop object axis marginal ra optical sstem O O' image O'

32 Single surface imaging equation Thin lens in air focal length Thin lens in air with one plane surface, focal length Thin smmetrical bi-lens Thick lens in air focal length ' ' ' ' f r n n s n s n ' r r n f ' n r f ' n r f ' r r n d n r r n f Formulas for surface and lens imaging

33 Magnification Lateral magnification for finite imaging Scaling of image sie m ' f tan u f ' tan u' principal planes object focal point focal point F P P' F' f f' ' image ' s s'

34 Object or field at infinit Image in infinit: - collimated exit ra bundle - realied in binoculars image image at infinit Object in infinit - input ra bundle collimated - realied in telescopes - aperture defined b diameter not b angle lens acts as aperture stop field lens stop ee lens object at infinit collimated entrance bundle image in focal plane

35 Imaging equation s' Imaging b a lens in air: lens makers formula s' s f real object real image 4f' f' virtual image real image Magnification s m ' s - 4f' -f' f' 4f' s Real imaging: s < 0, s' > 0 Intersection lengths s, s' measured with respective to the principal planes P, P' real object virtual image -f' virtual object virtual image - 4f'

36 Angle Magnification Afocal sstems with object/image in infinit Definition with field angle w angular magnification tan w' tan w nh n' h' w' w Relation with finite-distance magnification m f f '

37 Paraxial Approximation Paraxialit is given for small angles relative to the optical axis for all ras Large numerical aperture angle u violates the paraxialit, spherical aberration occurs Large field angles w violates the paraxialit, coma, astigmatism, distortion, field curvature occurs

38 Paraxial approximation Paraxial approximation: ni n' i' Small angles of ras at ever surface Small incidence angles allows for a lineariation of the law of refraction All optical imaging conditions become linear (Gaussian optics), calculation with ABCD matrix calculus is possible No aberrations occur in optical sstems There are no truncation effects due to transverse finite sied components Serves as a reference for ideal sstem conditions Is the fundament for man sstem properties (focal length, principal plane, magnification,...) The sag of optical surfaces (difference in between vertex plane and real surface intersection point) can be neglected i x All waves are plane of spherical (parabolic) R E( x) E0 e The phase factor of spherical waves is quadratic

39 Paraxial approximation Law of refraction nsin I n' sin I' Talor expansion 3 5 x x sin x x... 3! 5! Linear formulation of the law of refraction ni n' i' Error of the paraxial approximation i'- I') / I' n' =.9 n' =.7 n' =.5 ni i' I' n' I' nsin i arcsin n' i

40 Modelling of Optical Sstems Principal purpose of calculations: Imaging model with levels of refinement Sstem, data of the structure (radii, distances, indices,...) Analsis imaging aberration theorie Snthesis lens design Function, data of properties, qualit performance (spot diameter, MTF, Strehl ratio,...) Paraxial model (focal length, magnification, aperture,..) linear approximation Analtical approximation and classificationl (aberrations,..) Talor expansion Geometrical optics (transverse aberrations, wave aberration, distortion,...) with diffraction approximation --> 0 Wave optics (point spread function, OTF,...) Ref: W. Richter

41 Modelling of Optical Sstems Five levels of modelling:. Geometrical ratrace with analsis. Equivalent geometrical quantities, classification 3. Phsical model: complex pupil function 4. Primar phsical quantities 5. Secondar phsical quantities Blue arrows: conversion of quantities Geometrical ratrace with Snells law Geometrical equivalents classification Phsical model Primar phsical quantities Secondar phsical quantities ra tracing intersection points optical path length reference sphere wave aberration W exponential function of the phase pupil function Kirchhoff integral integration autocorrelation Duffieux integral orthogonal expansion optical transfer function approximation diameter of the spot Zernike coefficients Fourier transform Luneburg integral ( far field ) point spread function (PSF) maximum of the squared amplitude Fourier transform squared amplitude Raleigh unit equivalence tpes of aberrations longitudinal aberrations final analsis reference ra in the image space Strehl number approximation threshold value spatial frequenc resolution analsis sum of coefficients Marechal approximation sum of squares Marechal approximation rms value integration of spatial frequencies geometrical optical transfer function transverse aberration differen tiation definition single tpes of aberrations Marechal approximation final analsis reference ra in the image plane full aperture geometrical spot diagramm Fourier transform threshold value spatial frequenc approximation spot diameter

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