Optical Design with Zemax

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1 Otical Design with Zema Lecture 4: Aberrations I Herbert Gross Summer term 013

2 Preliminar Schedule Introduction Proerties of otical sstems I Proerties of otical sstems II Aberrations I Aberrations II Otimization I Introduction, Zema interface, menues, file handling, references, Editors, udates, windows, coordinates, Sstem descrition, Comonent reversal, sstem insertion, scaling, 3D geometr, aerture, field, wavelength Diameters, sto and uil, vignetting, Laouts, Materials, Glass catalogs, Ratrace, Ra fans and samling, Footrints Tes of surfaces, Asheres, Gratings and diffractive surfaces, Gradient media, Cardinal elements, Lens roerties, Imaging, magnification, araial aroimation and modelling Reresentation of geometrical aberrations, Sot diagram, Transverse aberration diagrams, Aberration eansions, Primar aberrations, Wave aberrations, Zernike olnomials, Point sread function, Otical transfer function Princiles of nonlinear otimization, Otimization in otical design, Global otimization methods, Solves and ickus, variables, Sensitivit of variables in otical sstems Otimization II Sstematic methods and otimization rocess, Starting oints, Otimization in Zema Imaging Fundamentals of Fourier otics, Phsical otical image formation, Imaging in Zema Illumination Advanced handling I Advanced handling II Correction I Correction II Phsical otical modelling Introduction in illumination, Simle hotometr of otical sstems, Non-sequential ratrace, Illumination in Zema Telecentricit, infinit object distance and afocal image, Local/global coordinates, Add fold mirror, Scale sstem, Make double ass, Vignetting, Diameter tes, Ra aiming, Material inde fit Reort grahics, Universal lot, Slider, Visual otimization, IO of data, Multiconfiguration, Fiber couling, Macro language, Lens catalogs Smmetr rincile, Lens bending, Correcting sherical aberration, Coma, sto osition, Astigmatism, Field flattening, Chromatical correction, Retrofocus and telehoto setu, Design method Field lenses, Sto osition influence, Asheres and higher orders, Princiles of glass selection, Sensitivit of a sstem correction, Microscoic objective lens, Zoom sstem Gaussian beams, POP roagation, olarization ratrace, olarization transmission, olarization aberrations

3 3 Contents 1. Reresentation of geometrical aberrations. Sot diagram 3. Transverse aberration diagrams 4. Aberration eansions 5. Primar aberrations

4 4 Otical Image Formation Perfect otical image: All ras coming from one object oint intersect in one image oint Real sstem with aberrations: 1. transverse aberrations in the image lane. longitudinal aberrations from the image lane 3. wave aberrations in the eit uil object lane wave aberrations image lane otical sstem transverse aberrations longitudinal aberrations

5 5 Reresentation of Geometrical Aberrations Longitudinal aberrations Ds reference ra ra logitudinal aberration along the reference ra Dl Gaussian image lane ra Gaussian image lane U reference oint otical ais Dl o otical ais sstem longitudinal aberration rojected on the ais sstem D s Transverse aberrations D longitudinal aberration reference ra (real or ideal chief ra) D transverse aberration ra U otical ais reference lane sstem

6 6 Reresentation of Geometrical Aberrations Angle aberrations Du ideal reference ra DU angular aberration real ra otical ais sstem Wave aberrations DW reference shere wavefront W > 0 Gaussian reference lane araial ra real ra U C z R D D s < 0

7 Sot Diagram All ras start in one oint in the object lane The entrance uil is samled equidistant In the eit uil, the transferred grid ma be distorted In the

7 7 Sot Diagram All ras start in one oint in the object lane The entrance uil is samled equidistant In the eit uil, the transferred grid ma be distorted In the image lane a sreaded sot diagram is generated object lane oint entrance uil equidistant grid otical sstem eit uil transferred grid image lane sot diagram o o z

8 8 Sot Diagram Table with various values of: 1. Field size. Color Small circle: Air diameter for comarison Large circle: Gaussian moment ais field zone 486 nm 546 nm 656 nm full field

9 9 Aberrations of a Single Lens Single lane-conve lens, BK7, f = 100 mm, l = 500 nm Sot as a function of field osition Coma shae rotates according to circular smmetr Decrease of erformance with the distance to the ais

10 10 Transverse Aberrations Tical low order olnomial contributions for: Defocus, coma, sherical, lateral color This allows a quick classification of real curves linear: defocus quadratic: coma cubic: sherical offset: lateral color

11 11 Transverse Aberrations Classical aberration curves Strong relation to sot diagram Usuall onl linear samling along the -, -ais no information in the quadrant of the aerture tangential D 5 m sagittal D 5 m D D l= 486 nm l= 588 nm l= 656 nm

12 1 Polnomial Eansion of the Aberrations Paraial otics: small field and aerture angles Aberrations occur for larger angle values Two-dimensional Talor eansion shows field and aerture deendence Eansion for one meridional field oint Puil: cartesian or olar grid in / coma ras outer ras of aerture cone r entrance uil otical ais chief ra ra field oint O sagittal lane object height meridional lane ais oint object lane

13 13 Polnomial Eansion of Aberrations Talor eansion of the deviation: Image height inde k r Puil height inde l Puil azimuth angle inde m k l m D(, r, ) a r cos k, l, m klm Smmetr invariance: selection of secial combinations of eonent terms Number of terms: sum of indices in the eonent isum The order of the aerture function deends on the aberration te used: rimar aberrations: - 3rd order in transverse aberration D - 4th order in wave aberration W Since the couling relation W D R changes the order b 1 i sum number of terms Te of aberration image location 4 5 rimar aberrations, 3rd/4th order 6 9 secondar aberrations, 5th/6th order 8 14 higher order

14 14 Polnomial Eansion of Aberrations Reresentation of -dimensional Talor series vs field and aerture r Selection rules: checkerboard filling of the matri Constant sum of eonents according to the order Image location Primar aberrations / Seidel r 5 Field Sherical Coma Astigmatism cos 3 cos 5 cos Distortion r 0 Tilt Distortion Distortion rimar secondar r 1 r 1 cos 4 r 1 cos r 1 Defocus r 1 Aerture Astig./Curvat. 4 r 1 r cos 3 r r r cos 3 Coma rimar 3 r cos r 3 r 3 cos r 3 Sherical rimar r 3 r 4 cos r 4 Coma secondar Sherical r 5 secondar Secondar aberrations

15 15 Primar Aberrations Eansion of the transverse aberration D on image height and uil height r Lowest order 3 of real aberrations: rimar or Seidel aberrations Sherical aberration: S - no deendence on field, valid on ais - deends in 3rd order on aertur D Coma: C - linear function of field - deends in rd order on aertur with azimuthal variation Astigmatism: A - linear function of aertur with azimuthal variation - quadratic function of field size Image curvature (Petzval): P - linear deendence on aertur - quadratic function of field size Distortion: D - No deendence on aertur - deends in 3rd order on the field size r 3 S r 3 r D cos r A cos C r P

16 Transverse Aberrations of Seidel Transverse deviations Sum of surface contributions D R n s s P R n s s A R n s s C R n s s S R n s D D R n s s P R n s s A R n s s C R n s s S R n s D k j S j S 1 k j C j C 1 k j A j A 1 k j P j P 1 k j D j D 1 16

17 17 Surface Contributions: Eamle 00 Seidel aberrations: reresentation as sum of surface contributions ossible Gives information on correction of a sstem Eamle: hotograhic lens Retrofocus F/.8 Field: w= S I Sherical Aberratio S II Coma S III Astigmatism S IV Petzval field curvatu S V Distortion C I Aial color Surface Sum C II Lateral color

18 18 Sherical Aberration Sherical aberration: On ais, circular smmetr Perfect focussing near ais: araial focus Real marginal ras: shorter intersection length (for single ositive lens) Otimal image lane: circle of least rms value lane of the smallest waist medium image lane A s A s marginal ra focus lane of the smallest rms-value araial focus

19 19 Sherical Aberration Single ositive lens lane of best focus Paraial focal lane near ais, Largest intersection length araial Shorter intersection length for rim ra and outer aerture zones zone rim

20 0 Sherical Aberration: Lens Bending Sherical aberration and focal sot diameter as a function of the lens bending (for n=1.5) Otimal bending for incidence averaged incidence angles Minimum larger than zero: usuall no comlete correction ossible object lane rincial lane image lane diameter bending X

21 1 Alanatic Surfaces Alanatic surfaces: zero sherical aberration: 1. Ra through verte. concentric 3. Alanatic Condition for alanatic surface: r ns n s ss n n n n s s Virtual image location Alications: 1. Microscoic objective lens. Interferometer objective lens s s 0 s s und u u ns n s herboloid oblate ellisoid oblate ellisoid rolate ellisoid + ower series + ower series + ower series + ower series Ds verte shere concentric shere alanatic S

22 Alanatic Lenses Alanatic lenses Combination of one concentric and one alanatic surface: zero contribution of the whole lens to sherical aberration Not useful: 1. alanatic-alanatic. concentric-concentric bended lane arallel late, nearl vanishing effect on ras A-A : arallel offset A-C : convergence enhanced C-A : convergence reduced C-C : no effect

23 3 Astigmatism Reason for astigmatism: chief ra asses a surface under an oblique angle, the refractive ower in tangential and sagittal section are different A tangential and a sagittal focal line is found in different distances Tangential ras meets closer to the surface In the midoint between both focal lines: circle of least confusion chief ra image oints O sag O tan O circle tangential differential ra sagittal differential ra otical ais

24 4 Astigmatism Beam cross section in the case of astigmatism: - Ellitical shae transforms its asect ratio - degenerate into focal lines in the focal lane distances - secial case of a circle in the midoint: smallest sot tangential aerture sagittal aerture z tangential focus circle of least confusion sagittal focus

25 Astigmatism 5 Imaging of a olar grid in different lanes eit uil image sace circle sagittal line tangential line sagittal focus entrance uil best focus tangential focus object

26 6 Field Curvature and Image Shells Imaging with astigmatism: Tangential and sagittal image shell deending on the azimuth Difference between the image shells: astigmatism Astigmatism corrected: It remains a curved image shell, Bended field: also called Petzval curvature Sstem with astigmatism: Petzval shere is not an otimal surface with good imaging resolution Law of Petzval: curvature given b: 1 1 n r n f k k k No effect of bending on curvature, imortant: distribution of lens owers and indices image surfaces sagittal shell tangential shell ideal image lane

Field Curvature 7 Focussing into different

27 Field Curvature 7 Focussing into different lanes of a sstem with field curvature Shar imaged zone changes from centre to margin of the image field focused at field boundar focused in field zone (mean image lane) focused in center (araial image lane) receiving lanes z image shere

28 8 Blurred Coma Sot Coma aberration: for oblique bundels and finite aerture due to asmmetr Primar effect: coma grows linear with field size Sstems with large field of view: coma hard to correct Relation of sot circles and uil zones as shown coma blur chief ra zone 1 zone zone 3 lens / uil ais

29 9 Distortion Eamle: 10% What is the te of degradation of this image? Sharness good everwhere! Ref : H. Zügge

30 30 Distortion Eamle: 10% Image with shar but bended edges/lines No distortion along central directions Ref : H. Zügge

31 31 Distortion Purel geometrical deviations without an blurr Distortion corresonds to sherical aberration of the chief ra Imortant is the location of the sto: defines the chief ra ath Two rimar tes with different sign: 1. barrel, D < 0 front sto. incushion, D > 0 rear sto Definition of local magnification changes lens rear sto image incussion distortion D > 0 D real ideal ideal object front sto barrel distortion D < 0

32 3 Aial Chromatical Aberration Aial chromatical aberration: Higher refractive inde in the blue results in a shorter intersection length for a single lens The colored images are defocussed along the ais Definition of the error: change in image location / intersection length Correction needs several glasses with different disersion white H D s s s CHL F C s F s e blue s C green red

33 33 Aial Chromatical Aberration Ds Simle achromatization / first order correction: - two glasses with different disersion - equal intersection length for outer wavelengths (blue F, red C) - residual deviation for middle wavelength (green e) F e C l Residual erros in image location: secondar sectrum white H Aochromat: - coincidence of the image location for at least 3 wavelengths - three glasses necessar, onl with anomal artial disersion (ecetions ossible) s F s e s C secondar sectrum blue red green

34 34 Aial Chromatical Aberration Longitudinal chromatical aberration for a single lens Best image lane changes with wavelength l best image lane l= 648 nm l= 546 nm l= 480 nm defocus z Ref : H. Zügge

35 35 Chromatic Variation of Magnification Lateral chromatical aberration: Higher refractive inde in the blue results in a stronger ra bending of the chief ra for a single lens The colored images have different size, the magnification is wavelength deendent Definition of the error: change in image height/magnification Correction needs several glasses with different disersion The aberration strongl deends on the sto osition D D CHV CHV F F e C C sto red D CHV blue reference image lane

sequence deends on sign of CHV original without lateral chromatic

36 36 Chromatic Variation of Magnification Imression of CHV in real images Tical colored fringes blue/red at edges visible Color sequence deends on sign of CHV original without lateral chromatic aberration 0.5 % lateral chromatic aberration 1 % lateral chromatic aberration

Imaging and Aberration Theory

Imaging and Aberration Theory Imaging and Aberration Theor Lecture 5: Aberration Reresentations 1-11-16 Herbert Gross Winter term 1 www.ia.uni-jena.de Preliminar time schedule 1 19.1. Paraxial imaging araxial otics, fundamental laws