Advanced Lens Design

Lecture 9: Chromatical correction 2013-12-10 Herbert Gross

Winter term 2013 www.iap.uni-jena.de

2 Contents

1. Dispersion 2. Glass diagram 3. Relative partial dispersion 4. Achromate 5. Apochromate 6. Schupman principle 7. Dialyte system 8. Burried surfaces 9. Transverse color aberratios 3 Overview on Chromatical Aberrations

1. Third order chromatical aberrations: - axial chromatical aberration error of the marginal ray by dispersion - transverse chromatical aberration error of the chief ray by dispersion 2. Higher order chromatical aberrations: - secondary spectrum residual axial error, if only selected wavelength are coinciding - spherochromatism chromatical variation of the spherical aberration, observed in an achromate - chromatical variation of all monochromatic aberrations e.g. astigmatism, coma, pupil location,... 4 Dispersion and Abbe number

. Description of dispersion:

Abbe number n 1 n  n  n F ' C' refractive index n . Visual range of wavelengths: 1.8 typically d,F,C or e,F’,C’ used 1.75 ne 1 ne  nF '  nC' 1.7

SF1 flint . Typical range of glasses 1.65 ne = 20 ...100 1.6 . Two fundamental types of glass: Crown glasses: 1.55 n small, n large, dispersion low BK7 Flint glasses: 1.5 n large, n small, dispersion high crown 1.45  0.5 0.75 1.0 1.25 1.5 1.75 2.0 5 Dispersion

Material with different dispersion values: - Different slope and curvature of the dispersion curve - Stronger change of index over wavelength for large dispersion - Inversion of index sequence at the boundaries of the spectrum possible

refractive index n

1.7

flint F6 n small slope large 1.675

1.65 crown n large slope small

1.625 SK18A

VIS 1.6  0.5 0.75 1.0 1.25 1.5 1.75 2.0 6 Glass Diagram

. Usual representation of glasses: diagram of refractive index vs dispersion n(n)

. Left to right: Increasing dispersion decreasing Abbe number

7 Relative Partial Dispersion

. Long crown and short flint as special realizations of large P

Long crown

Short flint

Crown Flint Ref.:H. Zuegge 8 Anomalous Partial Dispersion

N-SF57 P . There are some special glasses with g,F SF1 a large deviation from the normal line . Of special interest:

long crowns and short flints line of normal dispersion KZFSN4

PSK53A LASFN30 FK51 FK5 LAK8 ZKN7 n

Pg,F n heavy flints with character of long crowns

flint long crowns

long crown short flint short flints

crown normal line

n  9 Anomalous Partial Dispersion

. Arrows in the glass map:

indication of the deviation from the PhF' normal line

. Vertical component: at the red

horizontal: at the blue

end of the spectrum

P12  a12 n d b12  P12 normal line

Glass nd

PhF' n

arrow of deviation nd PtC'

red side glass location PhF' blue side

nd 10 Axial Colour: Achromate

(a) (b) (c) . Compensation of axial colour by appropriate glass choice

. Chromatical variation of the spherical aberrations: spherochromatism (Gaussian aberration)

. Therefore perfect axial color correction BK7 BK7 F2 BK7 N-SSK8F2 (on axis) are often not feasable n = 1.5168 n = 1.5168 1.6200 n = 1.5168 1.6177 n = 64.17 n = 64.17 36.37 n = 64.17 49.83 F= 1 F = 2.31 -1.31 F = 4.47 -3.47

r p r p r p 1 1 1

486 nm 588 nm 656 nm

z z z -2.5 0 -0.20 0 0.20 -100 0 100

Ref : H. Zügge

Achromate

. Achromate

. Longitudinal aberration . Transverse aberration . Spot diagram 486 nm 587 nm 656 nm y'

 = 486 nm axis

rp 1

 = 656 nm sinu'  = 587 nm 1.4° 486 nm 587 nm 656 nm

2° s' [mm] 0 0.1 0.2

Achromate

. Achromate: - Axial colour correction by cementing two Crown in different glasses front - Bending: correction of spherical aberration at the full aperture - Aplanatic coma correction possible be clever choice of materials

. Four possible solutions: - Crown in front, two different bendings - Flint in front, two different bendings Flint in front . Typical: - Correction for object in infinity - spherical correction at center wavelength with zone - diffraction limited for NA < 0.1 - only very small field corrected

solution 1 solution 2

Achromate: Realization Versions

. Advantage of cementing: solid state setup is stable at sensitive middle surface with large curvature

. Disadvantage: a) flint in front loss of one degree of freedom

. Different possible realization forms in practice

edge contact cemented

b) crown in front

edge contact cemented contact on axis broken, Gaussian setup

Achromate : Basic Formulas

. Idea: 1. Two thin lenses close together with different materials 2. Total power F  F1  F2

F F 3. Achromatic correction condition 1  2  0 n1 n 2 . Individual power values 1 1 F1   F F   F n 2 1 2 n1 1 n1 n . Properties: 2 1. One positive and one negative lens necessary 2. Two different sequences of plus (crown) / minus (flint) 3. Large n-difference relaxes the bendings 4. Achromatic correction indipendent from bending 5. Bending corrects spherical aberration at the margin 6. Aplanatic coma correction for special glass choices 7. Further optimization of materials reduces the spherical zonal aberration

Achromate: Correction

. Cemented achromate: 6 degrees of freedom: 3 radii, 2 indices, ratio n1/n2

. Correction of spherical aberration: diverging cemented surface with positive spherical contribution for nneg > npos  s'rim . Choice of glass: possible goals 1. aplanatic coma correction 2. minimization of spherochromatism case with 2 solutions 3. minimization of secondary spectrum

R . Bending has no impact on chromatical 1 correction: is used to correct spherical aberration case with at the edge one solution and coma case without solution, correction . Three solution regions for bending only sperical minimum 1. no spherical correction 2. two equivalent solutions 3. one aplanatic solution, very stable

16 Optimization of Achromatic Glasses

. For one given flint a line indicates the usefull crown glasses and vice versa

. Perfect aplanatic line of corresponding glasses (corrected for coma)

Condition:  2  .  1 1  n1 1 n1 n2 1 n 2 n1  n1 n2 n2           n1 1      r s'  n n n n n n  n n 1  2 2   1 2 2  1 2   2  2 2 

n n

fixed flint glass line of minimal spherical aberration line of minimal spherical aberration

fixed crown glass n n 17 Achromate Spherical Correction

Seidel surface contributions, same scale n2 n1 =70 / n2=30 / n1/n2 = 2.33

. The correction of the spherical aberration in an achromate for the marginal ray is performed by the undercorrecting effect of

n1 the cemented surface n2

n1 =65 / n2=40 / n1/n2 = 1.63

. A necessary condition for this is n1 < n2

. For large values of e n1 /n 2 n must be considerably larger then n since 2 1 n1 n2 the cemented surface is rather weak bended n1 =55 / n2=45 / n1/n2 = 1.43

. For values of e close to 1, the radii are strongly bended, the undercorrection of the cemented surface can only compensate the dominating n1 n2 first surface for a small difference in the n n1 =55 / n2=45 / n1/n2 = 1.22

. The figure shows the corresponding layouts and Seidel contributions for the case

n1 = 1.5 and n2 =1.8 n1 n2 n1 =55 / n2=47 / n1/n2 = 1.17

n1 18 Achromate Spherical Correction

. Spherical aberration for n1 = 1.5

Log n2 = 1.5 Sph n /n = 2.33 1 1 2 10 n1/n2 = 1.86 n1/n2 = 1.63 n1/n2 = 1.43 n1/n2 = 1.22 n1/n2 = 1.20 n1/n2 = 1.17 0 10

-1 10

-2 n2 10 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 19 Achromate

. Crown glass locations for M flint aplanatic achromates glass

crown glass n location

n n flint 1.8 aplanatic crown lines

1.7

1.6

available glasses 1.5

n 80 70 60 50 40 30 20 Achromatic Solutions in the Glass Diagram

large n-differences give relaxed bendings

flint crown negative lens positive lens

Achromat

Coma Correction: Achromate

Image height: y’ = 0 mm y’ = 2 mm Achromat . Bending of an achromate Pupil section: meridional meridional sagittal bending Transverse y' y' y' - optimal choice: small residual spherical Aberration: 0.05 mm 0.05 mm 0.05 mm aberration (a) - remaining coma for finite field size . Splitting achromate: additional degree of freedom: - better total correction possible (b) - high sensitivity of thin air space . Aplanatic glass choice: vanishing coma (c) . Cases: a) simple achromate, Achromat, splitting sph corrected, with coma b) simple achromate, (d) coma corrected by bending, with sph Achromat, aplanatic glass choice c) other glass choice: sph better, coma reversed d) splitted achromate: all corrected (e) e) aplanatic glass choice: all corrected Wave length:

Ref : H. Zügge

Achromate

. Residual aberrations of an achromate . Clearly seen: 1. Distortion 2. Chromatical magnification 3. Astigmatism

23 Spherochromatism: Achromate

. Residual spherochromatism of an achromate . Representation as function of apeture or wavelength

longitudinal aberration defocus variation rp  1 656

pupil height :

rp = 1 r = 0.707 587 p rp = 0.4 rp = 0

486 nm 587 nm 656 nm 486 z  z 0 0.04 0.08 0.12 -0.04 0 0.04 0.08 24 Axial Colour: Achromate and Apochromate

. Effect of different materials . Axial chromatical aberration changes with wavelength . Different levels of correction: apochromate achromate 1.No correction: lens,  one zero crossing point 644 2.Achromatic correction: C' - coincidence of outer colors - remaining error for center residual error apochromate wavelength - two zero crossing points 3. Apochromatic correction: lens - coincidence of at least three e 546 colors residual error achromate - small residual aberrations singlet - at least 3 zero crossing points - special choice of glass types F'  s' with anomalous partial 480 dispertion necessery -100 0 100 25 Axial Colour : Apochromate

P gF . Choice of at least one special glass

0,62 N-FS6 . Correction of secondary spectrum: (2) anomalous partial dispersion 0,60

. At least one glass should deviate 0,58 significantly form the normal glass line (1)+(2) T 0,56 N-KZFS11 (3) 656nm 0,54 (1) N-FK51 n 588nm 90 80 70 60 50 40 30 20

486nm

z 436nm z -0.2mm -0.2mm 0 1mm 26 Apochromate

. Focal power condition F  F1  F2  F3

F F F . Achromatic condition 1  2  3  0 n1 n 2 n 3 P  F P  F P  F . Secondary spectrum 1 1  2 2  3 3  0 n1 n 2 n 3

1 Pb  Pc . Curvatures of lenses ca   f  E n a n c  na,1  na,3 1 1 c   1 P  P r r c   c a 1 2 b f  E n n  n  n a c b,1 b,3 1 P  P c   a b c f  E n n  n  n a c c,1 c,3 . Parameter E 1 E  n P  P n P  P n P  P  n n a b c b c a c a b a c . The 3 materials are not allowed to be on the normal line

. The triangle of the 3 points should be large: small cj give relaxed design 27

Correction of Spherochromatism: Splitted Achromate

. Broken-contact achromate: different ray heights allow fof y red

correcting spherochromatism blue

rp

SK11 SF12

486 nm 587 nm 656 nm

 z in [mm] 0 0.2 0.4 28 Axial Color Correction with Schupman Lens

wavelength . Non-compact system in mm 0.656 . Generalized achromatic condition Schupman lens with marginal ray hieghts yj achromate y2 y2 1  F  2  F 0 v 1 v 2 1 2 0.571 . Use of a long distance and

negative F2 for correction . Only possible for virtual imaging

0.486 s -50 mm 0 50 mm

first lens second lens positive positive

intermediate virtual image image

Axial Color Correction with Schupman Lens

. Special layout with mirror for real imaging

f = -100 mm 2 mirror

f1 = 300 mm real image

30 Two-Lens Apochromate

. Special glasses . with anormal relative partial dispersion

. High refractive powers in the two components result in large spherical zonal aberration

656nm

588nm

486nm

z 436nm -2mm 5m 0 z

Ref.:H. Zuegge

Dialyt-Achromat

. Dialyt approch: one glass, but separated lenses no real imaging possible 2 . Data: kf k fa  fb  k  f (k 1) d   f k 1 k 1 . Setup

lens a

image lens b plane

y a y b

s'

d f a

Dialyt-Achromat

. Dialyt approach: achromatic correction by two lenses a/b in distance d d . Calculation: k  1. scaling parameter k f 'a

1 1 1 k   f fa fb

y2 y2 2. achromatic condition a  b 0 with ray heights y fa n a fb n b

3. focal lengths  n b  fa  f  1   n a 1 k

 n 1 k  a   fb  f 1 k 1   n b 

 n b  4. distance of lenses d  k  1   f  n a (1 k)

33 Principles of Glass Selection in Optimization

. Design rules for glass selection

. Different design goals: 1. Color correction: index n large dispersion positive lens field flattening difference desired Petzval curvature 2. Field flattening: large index difference + + desired

negative lens color correction + -

availability of glasses - - dispersion n

Ref : H. Zügge 34 Burried Surfaces

. Nearly equal refractive indices n

2 . Difference in Abbe number not larger than 30

1.9

1.8 Glass 1 Glass 2 n n n n  25 KZFN2 N-PK51 1.53188 0.00169 25.4 KZFN1 PSK3 1.55379 0.00061 13.9 1.7 N-LLF1 Ultran30 1.55097 0.00016 28.4 n  30 KZFSN2 LF8 1.56082 0.00667 10.4 1.6 F2 N-PSK53 1.62408 0.00010 27.1 n  30 F2 SK16 1.62408 0.00037 24.0 F2 SK55 1.62408 0.00037 23.8 1.5 F2 N-SSK2 1.62408 0.00099 16.9 F3 SK4 1.61685 0.00021 21.6 1.4 n F4 PSK52 1.62058 0.00031 22.6 20 30 40 50 60 70 80 90 100 F4 SSK3 1.62058 0.00288 14.6 F4 SSK4A 1.62058 0.00026 18.7 F6 LAKL21 1.64062 0.00242 24.4 F9 N-PSK53 1.62431 0.00031 25.4 F9 N-SK16 1.62431 0.00004 22.2 F9 SK55 1.62431 0.00004 22.0 F9 SK51 1.62431 0.00045 22.2 F13 N-SK10 1.62646 0.00041 20.9 N-SF4 N-LAK33 1.76164 0.00013 24.9 SFL4 N-LAK33 1.76175 0.00020 25.1 SFL56 LAFN10 1.79179 0.00312 17.8 SF1 LAF3 1.72310 0.00255 18.5 35 Burried Surfaces

. Cemented component with plane outer surfaces

. For center wavelength only plane parallel plate, not seen in collimated light

. Curved cemeneted surface: - dispersion for outer spectral weavelengths - color correction without disturbing the main wavelength

n n green undeflected . Example n1 n2

a) singlet 

0.630

0.580 singlet b) color corrected

0.530

corrected

0.480 z -320 -160 0 160 320 36 Lateral Color Aberration

. Dispersion of the chief ray deviation in the lens . Effect ressembles the disperion of a prism in the upper part of the lens . In the image plane, the differences in the colored ray angles cause changes in the ray height . The lateral color aberrations corresponds to a change of magnification with the wavelength

dispersion y prism effect yCHV chief ray

z

image stop plane 37 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 38 Chromatic Variation of Magnification

Representation of CHV: spot 1. Spot diagram diagram chromatical field 2. Magnification m() magnification height 3. Transverse aberration: difference Y‘ offset of chief ray reference

CHV 0.047 0.108

transverse aberration curves field field axis tangential sagital y y x

y

yp xp yp 39 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 chromatic aberration

0.5 % lateral chromatic aberration

1 % lateral chromatic aberration 40 Lateral Color Correction: Principle of Symmetry

. Perfect symmetrical system: magnification m = -1 . Stop in centre of symmetry . Symmetrical contributions of wave aberrations are doubled (spherical) . Asymmetrical contributions of wave aberration vanishes W(-x) = -W(x) . Easy correction of: coma, distortion, chromatical change of magnification

front part rear part

  2 3

 1