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 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 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, , 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

° 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 r 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 ° : 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

° Transverse deviations x' (x'2 + y'2 )s'4 [2x' (x' x' + y' y' )+ x'(x'2 + y'2 )]s'3 s' ∆x'= p p p S'− p p p p p p C' 2n' R'3 2n' R'3 ° Sum of surface p p contributions x'()x' x' + y' y' s'2 s'2 x' ()x'2 + y'2 s'2 s'2 + 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'= P ∑ j y'()x' x' + y' y' s'2 s'2 y' ()x'2 + y'2 s'2 s'2 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 n s jj 

  n ω Q   ° Coma 4 2  1 1  1 pj pj  1 1  C j = ω j Q j  − ⋅ ⋅ −   n' j s' j n s jj  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 n s jj   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 n s jj   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 = ⋅ω ωj 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 = − νn''nω n f ⋅

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 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 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  2( )1 ° Coma Clens = 2 ⋅  X − n + 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 focus ° Real marginal rays: shorter intersection length (for single positive lens) ° Optimal image plane: circle of least rms value

plane of the medium smallest image waist plane

As

2 A s

marginal plane of the paraxial ray focus smallest focus rms-value 25 Spherical Aberration

° Single positive lens

° Paraxial focal plane near axis, Largest intersection length

° Shorter intersection length for rim ray and outer aperture zones 26 Spherical Aberration: Lens Bending

object principal image ° Spherical aberration and focal spot diameter plane plane plane as a function of the lens bending (for n=1.5) ° Optimal bending for incidence averaged incidence angles ° Minimum larger than zero: usually no complete correction possible 27 Aplanatic Surfaces

° Aplanatic surfaces: zero spherical aberration: 1. Ray through vertex s'= s = 0

2. concentric s''= s und u= u 3. Aplanatic ns = n' s'

° Condition for aplanatic surface: ns n'' s ss ' r = = = n + n' n + n' s + s'

° Virtual image location

° Applications: 1. Microscopic objective lens 2. Interferometer objective lens 28 Aplanatic Lenses

A-A : ° Aplanatic lenses parallel offset

° Combination of one concentric and one aplanatic surface: zero contribution of the whole lens to spherical aberration A-C : convergence enhanced

° Not useful: 1. aplanatic-aplanatic 2. concentric-concentric bended plane parallel plate, C-A : nearly vanishing effect on rays convergence reduced

C-C : no effect 29 Astigmatism

° Reason for astigmatism: chief ray passes a surface under an oblique angle, the refractive power in tangential and sagittal section are different ° A tangential and a sagittal focal line is found in different distances ° Tangential rays meets closer to the surface ° In the midpoint between both focal lines: circle of least confusion 30 Astigmatism

° Beam cross section in the case of astigmatism: - Elliptical shape transforms its aspect ratio - degenerate into focal lines in the focal plane distances - special case of a circle in the midpoint: smallest spot

y tangential aperture

sagittal aperture

z

x sagittal circle of least focus tangential confusion focus 31 Astigmatism

Imaging of a polar grid in different planes image space circle sagittal line sagittal tangential line focus exit pupil

entrance best focus pupil

tangential focus

object 32 Field Curvature and Image Shells

° Imaging with astigmatism: Tangential and sagittal image shell depending on the azimuth ° Difference between the image shells: astigmatism ° Astigmatism corrected: It remains a curved image shell, Bended field: also called Petzval curvature ° System with astigmatism: Petzval sphere is not an optimal surface with good imaging resolution ° Law of Petzval: curvature given by: 1 1 = −n' ⋅∑ rpk n k⋅ f k

° No effect of bending on curvature, important: distribution of lens powers and indices 33 Astigmatisms and Curvature of Field

° Image surfaces: y' 1. Gaussian image plane 2. tangential and sagittal image shells (curved) ∆∆∆ ∆∆∆ 3. image shell of best sharpness ∆∆∆ 4. Petzval shell, arteficial, not a good image ∆∆∆s' ° Seidel theory: pet ∆∆∆s' ∆s''''− ∆ s =3 ⋅ ∆ s − ∆ s sag tan pet( sag pet ) tangential ∆∆∆s' 3∆s'sag −∆s'tan surface tan ∆s' pet = 2 medium surface ° Astigmatism is difference sagittal surface Petzval ∆ s'''ast= ∆ stan −∆ s sag surface Gaussian ° Best image shell image ∆ s''+∆ s ∆ s' = sag tan plane best 2

z 34 Field Curvature

° Focussing into different planes of a system with field curvature ° Sharp imaged zone changes from centre to margin of the image field

focused at field boundary focused in field zone focused in center (mean image plane) (paraxial image plane)

y'

receiving planes

z

image sphere 35 Blurred Coma Spot

° Coma aberration: for oblique bundels and finite aperture due to asymmetry ° Primary effect: coma grows linear with field size y ° Systems with large field of view: coma hard to correct ° Relation of spot circles and pupil zones as shown chief ray zone 1 zone 2 coma zone 3 blur

lens / pupil

axis 36 Distortion Example: 10%

° What is the type of degradation of this image ? ° Sharpness good everywhere !

Ref : H. Zügge 37 Distortion Example: 10%

° Image with sharp but bended edges/lines ° No distortion along central directions

Ref : H. Zügge 38 Distortion

° Purely geometrical deviations without any blurr ° Distortion corresponds to spherical aberration of the chief ray ° Important is the location of the stop: defines the chief ray path y' ° Two primary types with different sign: 1. barrel, D < 0 rear stop pincussion front stop y lens x' distortion 2. pincushion, D > 0 rear stop D > 0 ° Definition of local x magnification image changes y'

y' −y' front D= real ideal object y stop barrel x' distortion y'ideal D < 0 x 39 General Distortion Types

original ° Non-symmetrical systems: Generalized distortion types ° Correction complicated

anamorphism, a 10 x shear, a 01 y

1. order linear

2 2 a20 x keystone, a 11 xy line bowing, a 02 y

2. order quadratic

3 2 2 3 a30 x a21 x y a12 xy a03 y

3. order cubic 40 Axial Chromatical Aberration

° Axial chromatical aberration: Higher refractive index in the blue results in a shorter intersection length for a single lens ° The colored images are defocussed along the axis ° Definition of the error: change in image location / ∆ s'''= s− s intersection length CHLFC'' ° Correction needs several glasses with different dispersion

H'

white

s' F'

blue s'e

green s'C'

red 41 Axial Chromatical Aberration

∆∆∆s ° Simple achromatization / first order correction: - two glasses with different dispersion λλλ - equal intersection length for outer F' e C' wavelengths (blue F', red C') - residual deviation for middle wavelength (green e)

H' ° Residual erros in image location: white secondary spectrum

secondary spectrum ° Apochromat: - coincidence of the image location for at least 3 wavelengths

- three glasses necessary, only with s' s' anomal partial dispersion F' C' blue (exceptions possible) s' e red

green 42 Axial Chromatical Aberration

° Longitudinal chromatical aberration for a single lens ° Best image plane changes with wavelength

Ref : H. Zügge 43 Chromatic Variation of Magnification

° Lateral chromatical aberration: Higher refractive index in the blue results in a stronger ∆y'CHV = y'F ' −y'C' ray bending of the chief ray for a single lens ' ' ° The colored images have different size, y F ' −y C' ∆y'CHV = the magnification is wavelength dependent y'e ° Definition of the error: change in image height/magnification ° Correction needs several glasses with different dispersion ° The aberration strongly depends on the stop position

stop y' CHV red

blue

reference image plane 44 Chromatic Variation of Magnification

° Impression of CHV in real images ° Typical colored fringes blue/red at edges visible ° Color sequence depends on sign of CHV

original

without lateral

0.5 % lateral chromatic aberration

1 % lateral chromatic aberration 45 Summary of Important Topics

° There are different representations for aberrations ° Transverse aberrations are the most useful type ° Aberrations can be expanded into a Taylor series for field and aperture coordinate ° There are 7 primary aberrations (Seidel) of 3rd order; spherical aberration, coma, astigmatism, field curvature, distortion, axial chromatical, lateral chromatical ° In the Seidel definition, surface and lens contributions are additive ° Lens bending strongly changes aberration contribution due to variation of the incidence angles ° There are special setups with vanishing spherical aberration, important are aplanatic surfaces ° The image surface has two different equivalent representations: 1. tangential and sagittal image shell 2. astigmatism (difference) and field curvature (mean) ° Chromatic aberrations: axial and lateral, same size as monochromatic primary aberrations 46 Outlook

Next lecture: Part 8 – Wave optical aberrations Date: Wednesday, 2012-04-25 Contents: 8.1 Introduction, phase and optical path length 8.2 Definition of wave aberrations 8.3 Power expansion of the wave aberration 8.4 8.5 Measurement of wave aberrations 8.6 Special aspects 47 Outlook

Next lecture: Part 7 – Geometrical aberrations Date: Wednesday, 2012-06-06 Contents: 7.1 Representations 7.2 Power expansion of aberrations 7.3 Surface contributions 7.4 Primary monochromatic aberrations 7.5 Chromatical aberrations