Imaging and Aberration Theory Lecture 2: Fourier- and Hamiltonian Optics 2016-10-26 Herbert Gross Winter term 2016 www.iap.uni-jena.de 2 Preliminary time schedule 1 19.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems Pupils, Fourier optics, pupil definition, basic Fourier relationship, phase space, analogy optics and 2 26.11. Hamiltonian coordinates mechanics, Hamiltonian coordinates Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical 3 02.11. Eikonal approximation, inhomogeneous media single surface, general Taylor expansion, representations, various orders, stop 4 09.11. Aberration expansions shift formulas different types of representations, fields of application, limitations and pitfalls, 5 16.11. Representation of aberrations measurement of aberrations phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical 6 23.11. Spherical aberration surfaces, higher orders phenomenology, relation to sine condition, aplanatic sytems, effect of stop 7 30.11. Distortion and coma position, various topics, correction options 8 07.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options Dispersion, axial chromatical aberration, transverse chromatical aberration, 9 14.12. Chromatical aberrations spherochromatism, secondary spoectrum Sine condition, aplanatism and Sine condition, isoplanatism, relation to coma and shift invariance, pupil 10 21.12. isoplanatism aberrations, Herschel condition, relation to Fourier optics 11 04.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations special expansion for circular symmetry, problems, calculation, optimal balancing, 12 11.01. Zernike polynomials influence of normalization, measurement 13 18.01. Point spread function ideal psf, psf with aberrations, Strehl ratio 14 25.01. Transfer function transfer function, resolution and contrast Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic 15 01.02. Additional topics and induced aberrations, revertability 3 Contents 1. Definition of aperture and pupil 2. Special ray sets 3. Vignetting 4. Helmholtz-Lagrange invariant 5. Phase space 6. Resolution and uncertainty relation 7. Hamiltonian coordinates 8. Analogy mechanics – optics 4 Diaphragm in Optical Systems . Pupil stop defines: 1. chief ray angle w 2. aperture cone angle u . The chief ray gives the center line of the oblique ray cone of an off-axis object point . The coma rays limit the off-axis ray cone . The marginal rays limit the axial ray cone stop pupil coma ray marginal ray y field angle image w u object aperture angle y' chief ray 5 Definition of Field of View and Aperture . Imaging on axis: circular / rotational symmetry Only spherical aberration and chromatical aberrations . Finite field size, object point off-axis: y y' - chief ray as reference y p y'p - skew ray bundels: coma and distortion O' marginal/rim ray - Vignetting, cone of ray bundle RA' P chief ray not circular symmetric u w' u' w chief ray - to distinguish: tangential and sagittal O plane entrance object exit image pupil plane pupil plane 6 Numerical Aperture and F-number image . Classical measure for the opening: plane numerical aperture object in infinity NA'nsinu' DEnP/2 u' . In particular for camera lenses with object at infinity: F-number f f F # DEnP . Numerical aperture and F-number are to system properties, they are related to a conjugate object/image location 1 . Paraxial relation F # 2n'tanu' . Special case for small angles or sine-condition corrected systems 1 F # 2NA' 7 Generalized F-Number infinity . More general definition finite object of the F-number for systems DExP u' with finite object location u'o . Effective or working F-number with s' f '1 m and f' D D m D sinu' ExP ExP p EnP s' 2s' 2 f '(1 m) 2n' f (1 m) we get as a relation with the object-in-infinity-case eff 1 1 m F# F# 2n'sinu' mp m is the system magnification, mp is the pupil magnification Special rays in 3D . Meridional rays: in main cross section plane yp . Sagittal rays: upper perpendicular to main cross meridional coma ray axis section plane sagittal coma . Coma rays: ray Going through field point chief ray skew ray and edge of pupil x meridional p . Oblique rays: marginal ray without symmetry y pupil plane field point lower meridional coma ray sagittal ray axis point axis x object plane 9 Ray-Fan Selection for Transverse Aberration Plots . Transverse aberrations: Ray deviation form ideal image point in meridional and sagittal plane respectively . The sampling of the pupil is only filled in two perpendicular directions along the axes . No information on the performance of rays in the quadrants of the pupil y x tangential ray fan pupil object sagittal point ray fan 10 Pupil Sampling . Pupil sampling for calculation of tranverse aberrations: all rays from one object point to all pupil points on x- and y-axis . Two planes with 1-dimensional ray fans . No complete information: no skew rays object entrance exit image plane pupil pupil plane yo yp y'p y' tangential xp x'p x' xo z sagittal 11 Pupil Sampling . Ray plots tangential aberration sagittal aberration Dy Dx . Spot sagittal ray fan diagrams xp x p tangential ray fan Dy yp whole pupil area 12 Pupil Sampling . Pupil sampling in 3D for spot diagram: all rays from one object point through all pupil points in 2D . Light cone completly filled with rays object entrance exit image plane pupil pupil plane yo yp y'p y' xo xp x'p x' z 13 Pupil Sampling . Criteria: 1. iso energetic rays 2. good boundary description 3. good spatial resolution polar grid cartesian isoenergetic circular Fibonacci spirals hexagonal statistical pseudo-statistical (Halton) 14 Pupil Sampling Spot Artefacts . Artefacts due to regular gridding of the pupil of the spot in the image plane . In reality a smooth density of the spot is true . The line structures are discretization effects of the sampling cartesian hexagonal statistical 15 Diaphragm in Optical Systems black box . The physical stop defines details complicated the aperture cone angle u ? ? . The real system may be image complex object real system . The entrance pupil fixes the acceptance cone in the object space object u image . The exit pupil fixes the acceptance cone in the image space stop ExP EnP Ref: Julie Bentley 16 Properties of the Pupil Relevance of the system pupil : . Brightness of the image Transfer of energy . Resolution of details Information transfer . Image quality Aberrations due to aperture . Image perspective Perception of depth . Compound systems: matching of pupils is necessary, location and size 17 Entrance and Exit Pupil field point of image upper object marginal ray point U on axis U' W on axis lower marginal chief point of ray ray image upper coma ray lower coma ray stop outer field point of object exit entrance pupil pupil 18 Parameter of Excentricity . Relative position of stop inside system . Quantitative measure: h h Parameter of excentricity c CR MR (h absolut values) hCR hMR . Special cases: c = 1 image plane c = -1 pupil plane c = 0 same effective distance from image and pupil pupil image surface no j marginal ray hMR (ima) hCR hCR chief ray c -1 0 +1 19 Parameter of Excentricity 50 . Example: 40 30 excentricity for all surfaces marginal ray height 20 pupil . Change: 10 c = +1 ...-1 ...+1 image 0 object chief ray height -10 surface 0 5 10 15 20 25 30 35 index 1 0.5 excentricity 0 -0.5 -1 surface index 0 5 10 15 20 25 30 35 9 11 13 15 17 1 2 3 4 19 5 7 22 23 26 28 30 32 34 object stop image 20 Pupil Mismatch . Telescopic observation with different f-numbers . Bad match of pupil location: key hole effect F# = 2.8 F# = 8 F# = 22 a) pupil adapted b) pupil location mismatch Ref: H. Schlemmer 21 Vignetting field . Artificial vignetting: stop truncation Truncation of the free area of the aperture light cone D axis 0.8 D truncation AExp . Natural Vignetting: Decrease of brightness according to cos w 4 due to oblique projection of areas and changed photometric distances field angle w imaging with imaging without vignetting imaging with vignetting vignetting complete field of view 22 Vignetting . 3D-effects due to vignetting . Truncation of the at different surfaces for the upper and the lower part of the cone object lens 1 aperture lens 2 image stop upper truncation chief ray lower sagittal coma truncation trauncation rays 23 Vignetting projection of the . Truncation of the light cone free area of the rim of the 1st lens with asymmetric ray path aperture for off-axis field points meridional . Intensity decrease towards chief coma rays ray the edge of the image . Definition of the chief ray: sagittal ray through energetic centroid coma rays . Vignetting can be used to avoid projection of uncorrectable coma aberrations aperture stop in the outer field . Effective free area with extrem aspect ratio: projection of the anamorphic resolution rim of the 2nd lens 24 Vignetting . Illumination fall off in the image due to vignetting at the field boundary 25 Helmholtz-Lagrange Invariant . Product of field size y and numercial aperture is invariant in a paraxial system L n yu n'y'u' . Derivation at a single refracting surface: 1. Common height h: h su s'u' 2. Triangles y y' w , w' s s' 3. Refraction: nw n'w' 4. Elimination of s, s',w,w' . The invariance corresponds to: 1. Energy conservation 2. Liouville theorem 3. Invariant phase space volume (area) 4. Constant transfer of information n n' marginal ray u h u' w' w y surface chief ray s' s 26 Helmholtz-Lagrange Invariant arbitrary z pupil image . Basic formulation of the Lagrange y yp y' invariant: O' Uses image height, Q chief ray only valid in field planes y'CR yMR marginal ray .