Imaging and Aberration Theory

Transcription

1 Imaging and Aberration Theory Lecture 8: Astigmatism and field curvature 0--4 Herbert Gross Winter term 0

2 Preliminary time schedule 9.0. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems 6.0. Pupils, Fourier optics, Hamiltonian coordinates pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates Eikonal Fermat Principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media Aberration expansion single surface, general Taylor expansion, representations, various orders, stop shift formulas Representations of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options 9.. Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum 0.0. Further reading on aberrations sensitivity in 3rd order, structure of a system, superposition and induced aberrations, analysis of optical systems, lens contributions, Sine condition, isoplanatism, sine condition, Herschel condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, relation to Fourier optics and phase space 8.0. Wave aberrations definition, various expansion forms, propagation of wave aberrations, relation to PSF and OTF 5.0. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, recalculation for offset, ellipticity, measurement Miscellaneous Aldi theorem, telecentric case, afocal case, aberration balancing, Delano diagram, Scheimpflug imaging, Fresnel lenses, statistical aberrations Vectorial aberrations Introduction, special cases, actual research, anamorphotic, partial symmetric

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

4 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 tangential differential ray y chief ray image points x O sag O tan O circle sagittal differential ray optical axis

5 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 z x tangential focus circle of least confusion sagittal focus

6 Primary Aberration Spot Shape for Astigmatism Seidel formulas for field point only in y considered Field curvature: circle Astigmatism: focal line General spot with defocus: wave aberration Transverse aberrations Circle/ring in the pupil delivers an elliptical spoit in the image Special case for c 0 = - c /: circle of least confusion P p P p p p P p P p p p r y P r y C r S x y D r y P A r y C r S y sin sin sin cos ) ( ) cos ( cos , cos x r y A y P p 4 sin, cos p P p P p r y P y x r y P x r y P y 0 ), ( y c y x c y x W 0 0, c c Ry y W R y x Rc x W R x r x y ) ( 0 0 c c Rr y Rrc x ) ( ) ( Rrc y Rrc x

7 0.0 mm Astigmatism At sagittal focus Defocus only in tangentail cross section Primary astigmatism at sagittal focus Wave aberration tangential sagittal y Transverse ray aberration x y 0.0 mm 0.0 mm 0.0 mm Pupil: y-section x-section x-section Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 00 cycles per mm tangential sagittal sagittal 0.5 tangential cyc/mm z/mm Geometrical spot through focus Ref: H. Zügge z/mm

8 0.0 mm Astigmatism Primary astigmatism at medial focus At median focus Anti-symmetrical defocus in T-S-cross section Wave aberration tangential sagittal Transverse ray aberration y x y 0.0 mm 0.0 mm 0.0 mm Pupil: y-section x-section x-section Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 00 cycles per mm 0.5 tangential sagittal 0.5 tangential sagittal cyc/mm z/mm Geometrical spot through focus Ref: H. Zügge z/mm

9 9 Astigmatism Imaging of a polar grid in different planes Tangential focus: - blur in azumuthal direction - rings remain sharp image space Sagittal focus: - blur in radial direction - spokes remain sharp exit pupil circle tangential line sagittal line sagittal focus entrance pupil best focus tangential focus object

10 Astigmatism Behavior of a local position of the field:. good resolution of horizonthal structures. 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 ncos l tan i ncos l tan i ncos i ncos i R n l sag n l sag ncos i ncos i R Right side: oblique power of the spherical interface The sagittal image must be located on the auxiliary axis by symmetry surface stop lsag i i chief ray l tan object R l tan,sag auxiliary ray through the center of curvature C tangential image sagittal image

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 cylindrical x c 4 = -.0 c 4 = -0.5 toroidal c 7 only c 4 = -0.5 c 4 = 0 cylindrical y c 4 = 0.5 c 4 = 0.5 c 4 =.0

13 Psf for Astigmatism Zernike coefficients c in Pure astigmatism Shape is not circular symmetric due to finite width of focal lines c 5 = 0. c 5 = 0.5 c 5 = 0.7 c 5 =.0

14 Psf bei Astigmatismus Ref: Francon, Atlas of optical phenomena

15 Astigmatism: Lens Bending Bending effects astigmatism For a single lens bending with zero astigmatism, but remaining field curvature Astigmatism Seidel coefficients in [] Surface Sum Surface Curvature of surface T S T S S T ST T S Ref : H. Zügge

16 6 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 image surfaces sagittal shell tangential shell y ideal image plane

17 7 Astigmatisms and Curvature of Field Image surfaces:. Gaussian image plane. tangential and sagittal image shells (curved) 3. mean image shell of best sharpness 4. Petzval shell, arteficial, not a good image y Seidel theory: s tan s 3 s s 3s sags tan s pet s s s s ast best s sag s pet sag pet Astigmatism is difference tan Best image shell tan sag tangential surface medium surface sagittal surface s pet s sag s tan Petzval surface Gaussian image plane z

18 Correction of Astigmatism and Field Curvature Different possibilities for the correction of astigmatism and field curvature Two independend aberrations allow 4 scenarious a) bended image plane residual astigmatism b) bended image plane corrected astigmatism c) flattened image plane residual astigmatism d) flattened image plane corrected astigmatism T S y T S y S y T S T z z z z

19 9 Petzval Shell The Petzval shell is not a desirable image surface It lies outside the S- and T-shell: s pet 3s sag s tan The Petzval curvature is a result of the Seidel aberration theory P S T TSP P S T P S T P S T (a) s 0 ast s s sag tan s s pet sag (b) s 0 ast s s sag tan s s pet pet (c) s ast s s sag tan ( 3) s ( 3) s 0 pet pet (d) s ast s s sag tan s ( s pet ) s sag pet (e) s ast s s sag tan s 0 s pet pet

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:. sag and tan image shell. difference (astigmatism) and mean (medial image shell) of sag and tan Longitudinal Astigmatism, sagittal and tangential field curvature Paraxial focus Sagittal focus Medial focus field field field Ref: H. Zügge

21 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 (mean image plane) focused in center (paraxial image plane) y receiving planes z image sphere

22 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 rp n n For a system of several lenses, the Petzval curvature is given by n r n f p k k k object plane n surface n stop C curved image shell

23 Petzval Theorem for Field Curvature Petzval theorem for field curvature:. formulation for surfaces. formulation for thin lenses (in air) Important: no dependence on bending R ptz R ptz n m k n f j j nk nk n n r j k k k Natural behavior: image curved towards system object plane Problem: collecting systems with f > 0: If only positive lenses: R ptz always negative R optical system real image shell ideal image plane

24 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 a s r, a s r Imaging condition n n n s ns nn r For the special case of a flat object gives R p n n nr surface a r a auxiliary axis C image s object s

25 Petzval Theorem for Field Curvature Goal: vanishing Petzval curvature and positive total refractive power for multi-component systems R f ptz n f j h h j f j j j Solution: General principle for correction of curvature of image field:. Positive lenses with: - high refractive index - large marginal ray heights - gives large contribution to power and low weighting in Petzval sum. Negative lenses with: - low refractive index - samll marginal ray heights - gives small negative contribution to power and high weighting in Petzval sum

26 Flattening Meniscus Lenses Possible lenses / lens groups for correcting field curvature Interesting candidates: thick meniscus shaped lenses r nk nk n n r Rptz k k k k n f n n d r r r d. Hoeghs mensicus: identical radii - Petzval sum zero - remaining positive refractive power F ( n ) nr d. Concentric meniscus, - Petzval sum negative - weak negative focal length - refractive power for thickness d: 3. Thick meniscus without refractive power Relation between radii r R ptz r d ( n ) d n r r d r r d n n R ptz ( n ) d F nr ( r d) n r ( n ) d nr d ( n ) 0

27 Correcting Petzval Curvature Group of meniscus lenses n n d collimated r r Effect of distance and refractive indices /R pet [/mm] 0 - K5 / d=5 mm 0 - K5 / d=5 mm SF66 / d=5 mm r [mm] Ref : H. Zügge

28 Correcting Petzval Curvature Triplet group with r r r 3 d/ collimated n n Effect of distance and refractive indices 0 - /R pet [/mm] SF66 / FK3 / SF BK7 0-3 Ref: H. Zügge r [mm]

29 Flattening Field Lens Effect of a field lens for flattening the image surface. Without field lens. With field lens curved image surface image plane image shell flat image field lens

30 Conic Sections Explicite surface equation, resolved to z Parameters: curvature c = / R conic parameter Influence of k on the surface shape cx y c x z y Parameter Surface shape = - paraboloid < - hyperboloid = 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 Radii of curvature R T c 3/ c x, R c x / S c

31 Local Radii of Curvature Principal radii of curvature in and perpendicular to the plane of incidence R h Rcosq, Rh R cosq Projection of the principal radii of curvatur into an arbitrary plane with azimuthal angle q: ray tangential plane normal e h plane of incidence R cos q sin R h R h q R h R h h R cos q sin R h R h q st plane of principal curvature nd plane of principal curvature

32 Plane Parallel Platte Plane parallel plate: image location changed, Intersection length increased n s d n d 3 Reflection prisms works optically as plane parallel plate inside optical systems Finite numerical aperture: Generation of spherical aberration Application: cover glass in microscopy d s sph n sin 3 n u d s Non-parallel ray path: Generation of astigmatism Application: Prism positions in collimated beam path preferred d s ast n sin 3 n w u z

33 Astigmatism of Oblique Mirrors Mirror with finite incidence angle: effective focal lengths mirror f tan R cos i f sag R cos i s i Mirror introduces astigmatism s ast s R sin i R cos i R cos i s s cos i C focal line L R s sag s ast s / R Parametric behavior of scales astigmatism s / R = s / R = s / R = s / R = s / R = i

34 Field Curvature of a Mirror Mirror: opposite sign of curvature than lens Correction principle: field flattening by mirror f > 0 / R > 0 f > 0 / R < 0 mirror Gaussian image plane Petzval surface lens Petzval surface Gaussian image plane

35 Microscope Objective Lens Possible setups for flattening the field Goal: - reduction of Petzval sum - keeping astigmatism corrected Three different classes:. No effort. Semi-flat 3. Completely flat a) single meniscus lense b) two meniscus lenses c) symmetrical triplet D S not plane plane diffraction limit semi plane rel. field d) achromatized meniscus lens e) two meniscus lenses achromatized f) modified achromatized triplet solution

36 New Achromate An achromate is typically corrected for axial chromatical aberration The achromatization condition for two thin lenses close together reads F F 0 Petzval shell mean image shell y The Petzval sum usually is negative and the field is curved R P R P j n f j j f perfect image plane A flat field is obtained, if the following condition is fulfilled F n n F 0 This gives the special condition of simultaneous correction of achromatization and flatness of field n n

37 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) K5 F stop

38 Field Curvature Correction of Petzval curvature in photographic lens Tessar Positive lenses: green n j small Negative lenses : blue n j large Correction principle: special choice of refractive indices R j F n j j Cemented component: New Achromate Spherical aberration not correctable in the New Achromate

39 Asymmetrical Anastigmatic Doublets Antiplanet Protar Dagor Orthostigmat

40 Field Curvature Correction of Petzval field curvature in lithographic lens for flat wafer Positive lenses: green h j large, n j small Negative lenses : blue h j small, n j large R F j j h h j F n j j F j Correction principle: certain number of bulges

41 Field Flatness one waist two waists Principle of multi-bulges to reduce Petzval sum n r n f p k k k Seidel contributions show principle. bulge. waist. bulge. waist 3. bulge Petz Petz

42 Field Flatness Effect of mirror on Petzval sum Flatness of field for catadioptric lenses 0 -. intermediate image concave mirror. intermediate image

43 Field Curvature Symmetrical system Astigmatism corrected Field curvature remains symmetrical system field curvature T S y s

44 Astigmatism of Eyeglasses with Rotating Eye Rotating eye: Astigmatism Coddington equations: Elliptical line with vanishing astigmatism: Tscherning ellipses object pupil image s a s front refractive power F [dpt] 0 5 Wollastonlens Ostwaldlens no astigmatism refractive power F [dpt]

45 Semiconductor Laser Beams Semiconductor laser sources show two types of astigmatism:. elliptical anamorphotic aperture (not a real astigmatism). z-variation of the internal source points in case of index guiding) In laser physics, the quadratic astigmatic beam shape is not semiconductor considered to be a q degradation, the M is constant, Q it can be corrected by a simple junction plane cylindrical optic q perpendicular parallel z perpendicular Q junction plane Q q parallel z semiconductor

46 Anamorphotic Systems Anamorphotic or cylindrical/toroidal system are used to get a circular profile form a semiconductor laser x Example: laser beam collimator y axi-symmetric cylindrical x-z-section NA = 0. z y-z-section NA = 0.5 z

47 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 y-z-section NA = 0.4 x-z-section NA =

48 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 x [mm] surface local astigmatic

49 Astigmatism due to Fabrication Errors Real surfaces with varying radius of curvature Toroidal surface shape with R max, R min Astigmatic effects on axis Irregularity errors in tolerancing measured by interferometry as ring difference R max R min R spec circular symmetric surface surface with irregularity error R max R min

50 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 c 5 [a.u.] t = 90 t = 67.5 t = 45 t =.5 t = 0 cylindrical power fixed orientation cylindrical power rotated by t astigmatism corrected astigmatic caustic

51 Object plane Defocus compensator On compensator axis astigmatism Nr. Adjustment and Compensation Example Microscopic lens Adjusting:. Axial shifting lens : focus. Clocking: astigmatism 3. Lateral shifting lens: coma Onaxiscoma compensator Spherical aberrati compensator Ideal : Strehl D S = 99.6 % With tolerances : D S = 0. % After adjusting : D S = 99.3 % Onaxisastig compensator N PSF (energynormalized) Systemwithtolerances Strehl: 0,% Ref.: M. Peschka

52 Strehl ratio [%] Adjustment and Compensation Sucessive steps of improvements PSF (intensity PSF normalized) (energy normalized) % Not adjusted.77% 5.48% 97.% Step Step Step 3 Z, 4 Z 9 Z, 7 Z 8 Z, 5 Z 6 WithTolerances Step(Z, Z) 4 9 Ref.: M. Peschka Step(Z, Z) 7 8

53 Summary Astigmatism is caused by the different effective radii of curvature in the cross sections of a skew chief ray Crossed sagittal / tangential focal lines forms two sharp image shells The image locations are given by the Coddington equations and are already obtained in an infinitesimal environment of the chief ray Oblique mirror and prisms in converging beams generate astigmatism The mean image shell between S and T image shell defines a curved image The flatness of the image shell is given in third order, if the Petzval theorem is fulfilled The flattening can be obtained by thick meniscus lenses In particular oblique mirrors show a positive image curvature In real systems with manufacturing deviations, astigmatism can be observed already on axis