Imaging and Aberration Theory

Home , Coma (optics), Petzval field curvature

Lecture 8: Astigmastism and field curvature 2013-12-19 Herbert Gross

Winter term 2013 www.iap.uni-jena.de 2 Preliminary time schedule

1 24.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems 2 07.11. Pupils, Fourier optics, pupil definition, basic Fourier relationship, phase space, analogy optics and Hamiltonian coordinates mechanics, Hamiltonian coordinates 3 14.11. Eikonal Fermat Principle, stationary phase, Eikonals, relation rays‐waves, geometrical approximation, inhomogeneous media 4 21.11. Aberration expansion single surface, general Taylor expansion, representations, various orders, stop shift formulas 5 28.11. Representations of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations 6 05.12. Spherical aberration phenomenology, sph‐free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders 7 12.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options 8 19.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options 9 09.01. Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum 10 16.01. Further reading on aberrations sensitivity in 3rd order, structure of a system, analysis of optical systems, lens contributions, Sine condition, isoplanatism, sine condition, Herschel condition, relation to coma and shift invariance, pupil aberrations, relation to Fourier optics 11 23.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations, relation to PSF and OTF 12 30.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, recalculation for offset, ellipticity, measurement 13 06.02. Miscellaneous Intrinsic and induced aberrations, Aldi theorem, vectorial aberrations, partial symmetric systems 3 Contents

1. Geometrical astigmatism 2. Point spread function for astigmatism 3. Field curvature 4. Petzval theorem 5. Correcting field curvature 6. Examples 4 Astigmatism

. Reason for astigmatism: chief ray passes a surface under an oblique angle, the refractive power in tangential and sagittal section are different . The astigmatism is influenced by the stop position . 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 5 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

x sagittal circle of least focus tangential confusion focus 6 Primary Aberration Spot Shape for Astigmatism

. Seidel formulas for field point only in y‘ considered 3 2 2 3 rSy  pp ryC p P ryPA Pp  yD ''cos'')''2()2cos2('''cos''' 3 2 2 rSx  pp ryC p P  ryP sin'''2sin'''sin''' Pp

2 2 . Field curvature:  ryPy Pp  ryPx sin'''',cos'''' Pp circle 24222  ryPyx ''''' p

. Astigmatism: 2 ryAy  x  0',cos'''2' focal line Pp

. General spot with defocus: 22 2 wave aberration ),( 20   22  ycyxcyxW

W W Transverse aberrations Rx  ,2 RyxRc 2  ccRy  x 20  y 2220

2 2 Circle/ring in the pupil rxy22 2 x y 2  2 1 delivers an elliptical spoit in the image Rrc20 )2( 2  ccRr 2220

Special case for c20 = - c22/2: x2 y2  1 circle of least confusion 2 2 Rrc22 Rrc22 )()( 7 Astigmatism

Primary astigmatism at sagittal focus Wave aberration tangential sagittal . At sagittal focus 2 2 . Defocus only in tangential cross section

Transverse ray aberration y x y 0.01 mm 0.01 mm 0.01 mm

Pupil: y-section x-section x-section Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 100 cycles per mm 1 1 tangential

sagittal sagittal

0.5 tangential 0.5

0 0 0100200 -0.04 0.0 0.04 cyc/mm z/mm

Geometrical spot through focus 0.02 mm 0.02 -0.04 -0.02 0.0 0.02 0.04 Ref: H. Zügge z/mm 8 Astigmatism

Primary astigmatism at medial focus Wave aberration tangential sagittal . At median focus 2 2 . Anti-symmetrical defocus in T-S-cross section

Transverse ray aberration y x y 0.01 mm 0.01 mm 0.01 mm

Pupil: y-section x-section x-section Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 100 cycles per mm 1 1

tangential sagittal tangential 0.5 sagittal 0.5

0 0 0100200 -0.04 0.0 0.04 cyc/mm z/mm

Geometrical spot through focus 0.02 mm -0.04 -0.02 0.0 0.02 0.04 Ref: H. Zügge z/mm 9 Astigmatism

. Imaging of a polar grid in different planes . Tangential focus: - blur in azumuthal direction - rings remain sharp . Sagittal focus: - blur in radial direction - spokes remain sharp 10 Astigmatism

. Behavior of a local position of the field: 1. good resolution of horizonthal structures 2. bad resolution of vertical structures . Imaging of polar diagram shows not the classical behaviour of the complete field 11 Coddington Equations

. For an oblique ray, the effective curvatures of the spherical surface depend on azimuth . There are two focal points for sagittal / tangential aperture rays . This splitting occurs already for infinitesimal aperture angles around the chief ray . Intersection lengths along the chief ray: Coddington equations

2 in 'cos' 2 in  cos'cos'cos inin n' n  cos'cos' inin   l'sag lsag R l'tan ltan R . Right side: oblique power of the spherical interface . The sagittal image must be located on the auxiliary axis by symmetry 12 Wavefront for Astigmatiusm with Defocus

. Astigmatic wavefront with defocus . Purely cylindrical for focal lines in x/y . Purely toroidal without defocus: circle of least confusion 13 Psf for Astigmatism

. Zernike coefficients c in  . Pure astigmatism . Shape is not circular symmetric due to finite width of focal lines 14 Psf bei Astigmatismus

Ref: Francon, Atlas of optical phenomena 15 Astigmatism: Lens Bending

. Bending effects astigmatism . For a single lens 2 bending with zero astigmatism, but remaining field curvature

Ref : H. Zügge 16 Field Curvature and Image Shells

. Imaging with astigmatism: Tangential and sagittal image shell sharp depending on the azimuth . Difference between the image shells: astigmatism . Astigmatism corrected: It remains one curved image shell, Bended field: also called Petzval curvature . System with astigmatism: Petzval sphere is not an optimal surface with good imaging resolution . No effect of lens bending on curvature, important: distribution of lens powers and indices 17 Astigmatisms and Curvature of Field

. Image surfaces: 1. Gaussian image plane 2. tangential and sagittal image shells (curved) 3. mean image shell of best sharpness 4. Petzval shell, arteficial, not a good image

. Seidel theory:

ss''tan pet 3   ss ''sag pet  ss ''3 s'  sag tan pet 2

. Astigmatism is difference

 sss'''ast   tan  sag

. Best image shell  ss''  s'  sag tan best 2 18 Correction of Astigmatism and Field Curvature

. Different possibilities for the correction of astigmatism and field curvature . Two independend aberrations allow 4 scenarious 19 Petzval Shell

. The Petzval shell is not a desirable image surface . It lies outside the S- and T-shell:  ss ''3 s'  sag tan pet 2 . The Petzval curvature is a result of the Seidel aberration theory 20 Field Curvature

. The image splits into two curved shells in the field . The two shells belong to tangential / sagittal aperture rays . There are two different possibilities for description: 1. sag and tan image shell 2. difference (astigmatism) and mean (medial image shell) of sag and tan

Ref: H. Zügge 21 Field curvature

. Basic observation: A plane object gives a curved image 22 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)

receiving planes

image sphere 23 Field for Single Surface

. The image is generated on a curved shell . In 3rd order, this is a sphere . For a single refracting surface, the Petzval radius is given by nr r  p nn' . For a system of several lenses, the Petzval curvature is given by 11 n'  rpkk nf k 24 Petzval Theorem for Field Curvature

. Petzval theorem for field curvature: 1 'nn kk 1. formulation for surfaces  nm ' Rptz k ' rnn kkk 1 1 2. formulation for thin lenses (in air)   ptz j  fnR jj . Important: no dependence on bending

. Natural behavior: image curved towards system

. Problem: collecting systems with f > 0: If only positive lenses:

Rptz always negative 25 Petzval Theorem

. Elementary derivation by a momocentric system of three surfaces: interface surface with r, object and image surface . Consideration of a skew auxiliary axis    '', rsarsa

n' n 'nn . Imaging condition  s' s r

. For the special case of a flat object gives with p ,' aRa  

'1 nn  Rp nr 26 Petzval Theorem for Field Curvature

. Goal: vanishing Petzval curvature 1 1   ptz j  fnR jj h 11 and positive total refractive power  j  f j 1 fh j for multi-component systems

. Solution: General principle for correction of curvature of image field: 1. Positive lenses with: - high refractive index - large marginal ray heights - gives large contribution to power and low weighting in Petzval sum 2. Negative lenses with: - low refractive index - samll marginal ray heights - gives small negative contribution to power and high weighting in Petzval sum 27 Flattening Meniscus Lenses

. Possible lenses / lens groups for correcting field curvature . Interesting candidates: thick meniscus shaped lenses

r2 2 1 'nn  n 11  d   kk     d Rptz k ' kkk  fnrnn  n  rr 21 r1

1. Hoeghs mensicus: identical radii 2  )1( dn - Petzval sum zero F' rn 2 - remaining positive refractive power

2. Concentric meniscus,   drr - Petzval sum negative 12 - weak negative focal length  )1(1 dn ()nd1  F' - refractive power for thickness d: R  drrn ptz 11 nr11() r d

3. Thick meniscus without refractive power n 1 )1(1 2  dn Relation between radii  0 rrd21 n Rptz 11 ndnrrn  )1( 28 Correcting Petzval Curvature

. Group of meniscus lenses n n

d collimated

r 1 r 2

. Effect of distance and refractive indices

Ref : H. Zügge 29 Correcting Petzval Curvature

r r . Triplet group with + - + 1 2

r 3

d/2 collimated

. Effect of distance and refractive indices

Ref: H. Zügge 30 Flattening Field Lens

Effect of a field lens for flattening the image surface

1. Without field lens 2. With field lens

curved image surface image plane 31 Conic Sections

. Explicite surface equation, resolved to z   yxc 22  z  Parameters: curvature c = 1 / R 111   yxc 222 conic parameter 

. Influence of k on the surface shape Parameter Surface shape  = - 1 paraboloid  < - 1 hyperboloid  = 0 sphere  > 0 oblate ellipsoid (disc) 0 >  > - 1 prolate ellipsoid (cigar )

. Relations with axis lengths a,b of conic sections

2  a  b 1 1   1   c  2 b  a   b  a c1 c 1

. Radii of curvature 1 2/3 1 2/1 R 1  xc 22  , R 1  xc 22 T c S c 32 Astigmatism of Oblique Mirrors

mirror . Mirror with finite incidence angle: effective focal lengths cosiR R s f  f  i tan 2 sag cos2 i

. Mirror introduces astigmatism R C s'sag 2  sin 2 iRs s'  ast  cosiR   R  cos2  si    s   cos22 i focal     line L

s'ast

s' / R

1 . Parametric behavior of scales astigmatism 0.9 s / R = 2 s / R = 0.6 s / R = 1 0.8

0.7

0.6 s / R = 0.4 0.5

0.4

0.3

0.2

0.1 s / R = 0.2

0 i 0 10 20 30 40 50 60 33 Field Curvature of a Mirror

. Mirror: opposite sign of curvature than lens . Correction principle: field flattening by mirror 34 Microscope Objective Lens

a) . Possible setups for flattening the field single meniscus . Goal: lense - reduction of Petzval sum b) - keeping astigmatism corrected two meniscus . Three different classes: lenses 1. No effort c) 2. Semi-flat symmetrical triplet 3. Completely flat

D S d) achromatized 1 meniscus lens not plane plane 0.8 diffraction limit e) two 0.6 meniscus lenses achromatized 0.4 semi plane f) 0.2 modIfied achromatized rel. triplet solution 0 field 00.50.707 1 35 New Achromate

. An achromate is typically corrected for axial chromatical aberration

. The achromatization condition for two thin Petzval y' shell lenses close together reads

FF mean 1  2 0 image shell 1 2

. The Petzval sum usually is negative RP and the field is curved

f perfect image 11 plane   P j fnR jj . A flat field is obtained, if the following condition is fulfilled

F F 1 2  0 n1 n2

. This gives the special condition of simultaneous correction of achromatization and flatness of field  n 1  1 2 n2 36 New Achromate

. This condition correponds to the requirement to find two glasses on one straight line in the glass map

. The solution is well known as simple photographic lens (landscape lens) stop K5 F2 37 Field Curvature

. Correction of Petzval curvature in photographic lens Tessar

. Positive lenses: green nj small

. Negative lenses : blue nj large . Correction principle: special choice of refractive indices 1 F   j R j n j

. Cemented component: New Achromate . Spherical aberration not correctable in the New Achromate 38 Asymmetrical Anastigmatic Doublets

. Antiplanet

. Protar

. Dagor

. Orthostigmat 39 Field Curvature

1 F . Correction of Petzval field curvature in lithographic lens  j R  n for flat wafer j j h F j  F . Positive lenses: green hj large, nj large  j j h1 . Negative lenses : blue hj small , nj small

. Correction principle: certain number of bulges 40 Field Flatness

one waist two waists . Principle of multi-bulges to reduce Petzval sum

11 n'  rpkk nf k

. Seidel contributions show principle

Petz Petz 1. bulge 1. waist 2. bulge 2. waist 3. bulge

0.2

-0.2

10 20 30 40 50 60 41 Field Flatness

. Effect of mirror on Petzval 1 sum 0 -1 . Flatness of field for cata- 1. intermediate concave 2. intermediate dioptric lenses image mirror image 42 Field Curvature

. Symmetrical system . Astigmatism corrected symmetrical system field . Field curvature remains curvature

T S y'

s' -2 -1 0 +1 +2 43 Astigmatism of Eyeglasses with Rotating Eye

object . Rotating eye: Astigmatism image . Coddington equations: pupil Elliptical line with vanishing astigmatism: Tscherning ellipses

a s' s 44 Semiconductor Laser Beams

. Semiconductor laser sources show two types of astigmatism: 1. elliptical anamorphotic aperture (not a real astigmatism) 2. z-variation of the internal source points in case of index guiding) . In laser physics, the quadratic astigmatic beam shape is not considered to be a degradation, the M2 is constant, it can be corrected by a simple cylindrical optic

perpendicular

Q Q junction z plane parallel

semiconductor 45 Anamorphotic Systems

. Anamorphotic or cylindrical/toroidal system are used to get a circular profile form a semiconductor laser x . Example: laser beam collimator

axi-symmetric cylindrical

x-z-section NA = 0.1 z

y-z-section NA = 0.5 z 46 45° Effects of Anamorphotic Systems

. Example of two aspherical cylindrical lenses with different focal lengths . For high numerical apertures: no longer additivity and decoupling of x-and y-cross section . The wavefront shows the deviations in the 45° directions 47 Decentered System

. Real systems with centering errors: - non-circular symmetric surface on axis - astigmatism on axis - point spread function no longer circular symmetric

decenter x

surface local astigmatic 48 Astigmatism due to Fabrication Errors

. Real surfaces with varying radius of curvature

. Toroidal surface shape with Rmax, Rmin . Astigmatic effects on axis . Irregularity errors in tolerancing measured by interferometry as ring difference

circular symmetric surface with surface irregularity error

Rmax

Rmin 49 Astigmatic Correction by Clocking

. Two lens surfaces nearby with equal astigmatism due to manufacturing errors . Azimuthal rotation of one lens against the other compensates astigmatism in first approximation . This clocking procedure is used in practice for adjusting sensitive systems 50 Adjustment and Compensation

. Example Microscopic lens

On axis coma . Adjusting: compensator Spherical aberration

1. Axial shifting lens : focus astigmatism On axis 1 Nr. compensator compensator Defocus compensator 2. Clocking: astigmatism 3. Lateral shifting lens: coma

On axis astigmatism Stop . Ideal : Strehl DS = 99.62 % compensator Nr. 2 Object plane Object With tolerances : DS = 0.1 %

After adjusting : DS = 99.3 %

PSF PSF (energy normalized) System with tolerances (intensity normalized) Strehl: 0,2%

Ref.: M. Peschka 51 Adjustment and Compensation

100 99.32% . Sucessive steps of improvements 97.2% 80 [%]

o 60 l rati PSF h 40 tre

(intensity S 20 25.48% normalized) 21.77% PSF 0.2% (energy 0 normalized) Not Step 1 Step 2 Step 3 Step 4 adjusted Z,49 Z Z,78 Z Z,56 Z ~ Step 2 Z,78 Z

With Tolerances

Step 1 (Z49 , Z )

Step 2 (Z78 , Z )

Step 3 (Z56 , Z )

Step 4 ~ Step 2 (Z78 , Z )

Ref.: M. Peschka