Imaging and Aberration Theory Lecture 10 Sine condition

Imaging and Aberration Theory Lecture 10: Sine condition, aplanatism and isplanatism 2017 -12 -18 Herbert Gross Winter term 2017 www. iap. uni-jena. de

2 Preliminary time schedule 1 2 16. 10. Paraxial imaging Pupils, Fourier optics, 23. 10. Hamiltonian coordinates paraxial optics, fundamental laws of geometrical imaging, compound systems pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates 7 Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media single surface, general Taylor expansion, representations, various orders, stop 06. 11. Aberration expansions shift formulas different types of representations, fields of application, limitations and pitfalls, 13. 11. Representation of aberrations measurement of aberrations phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical 20. 11. Spherical aberration surfaces, higher orders phenomenology, relation to sine condition, aplanatic sytems, effect of stop 27. 11. Distortion and coma position, various topics, correction options 8 04. 12. Astigmatism and curvature 3 4 5 6 30. 10. Eikonal 11 phenomenology, Coddington equations, Petzval law, correction options Dispersion, axial chromatical aberration, transverse chromatical aberration, 11. 12. Chromatical aberrations spherochromatism, secondary spoectrum Sine condition, aplanatism and Sine condition, isoplanatism, relation to coma and shift invariance, pupil 18. 12. aberrations, Herschel condition, relation to Fourier optics isoplanatism definition, various expansion forms, propagation of wave aberrations 08. 01. Wave aberrations 12 15. 01. Zernike polynomials 13 14 22. 01. Point spread function 30. 01. Transfer function 15 05. 02. Additional topics 9 10 special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, measurement ideal psf, psf with aberrations, Strehl ratio transfer function, resolution and contrast Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability

3 Contents 1. Sine condition 2. Pupil aberrations 3. Aplanatism 4. Isoplanatism 5. Miscellaneous

4 Sine Condition § Lagrange invariante for paraxial angles U, U‘ § sin-condition: extension for finite aperture angle u § Corresponds to energy conservation in the system § Constant magnification for alle aperture zones § Pupil shape for finite aperture is a sphere § Definition of violation of the sine condition: OSC (offense against sine condition) § OSC = 0 means correction of sagittal coma (aplanatic system)

5 Optical Sine Condition § Condition for finite angles § Condition for object at infinity § Condition for afocal system § In the formulation the sagittal magnification is used

6 Abbe Sine Condition § If for example a small field area and a widespread ray bundle is considered, a perfect imaging is possible § The eikonal with the expression can be written for d. L=0 as § In the special case of an angle 90°we get with cos(q)=sin(u) the Abbe sine condition with the lateral magnification

7 Derivation of the Sine Condition § From geometry P 0 CQ, Mo'QC § Refraction § Division and substitution

8 Pupil Sphere § Pupil sphere: Relation between pupil height and aperture angle according to sine condition § Can only be fulfilled by spherical shaped pupil surface

9 Principal Sphere and Aplanatism § If a point A is imaged aplanatic: it is impossible, that a second point B is also aplanatic with the principal sphere construction

10 Principal Sphere § The principal sphere can only image on axis points § Off axis points gives wrong image location § Conclusion: Aplanatism is only locally valid (axial and lateral)

11 Vectorial Sine Condition § General vectorial sine condition: spatial frequencies / direction cosines are linear related from entrance to exit pupil § Generalization can be applied for anamorphic systems § General mapping between pupil surfaces gives the condition

12 Transfer of Energy in Optical Systems § Conservation of energy § Invariant local differential flux § Assumption: no absorption § Delivers the sine condition

13 Pupil Sphere § Sine condition fulfilled: linear scaling from entrance to exit pupil § Pupil surface must be sperical § The pupil height scales with the sine of the angle

14 Pupil Distortion § Sine condition fulfilled: linear scaling from entrance to exit pupil § Offence against the sine condition (OSC): Exit pupil grid is distorted § Consequences: 1. Photometric effect causes apodization 2. Wave aberration could be calculated wrong 3. Spatial filtering on warped grid

15 Pupil Distortion § Afocal system: Lagrange invariant (classical Lagrange invariant for pupil imaging) § Magnification § Sine condition § Fulfillment of the sine condition: linear scaling of entrance to exit pupil

16 Sine Condition in Microscopic Objective Lens § Typical high-NA system § Virtual pupil located inside § Typical grid distortion

17 OSC and Apodization § Photometric effect of pupil distortion: illumination changes at pupil boundary § Effect induces apodization § Sign of distortion determines the effect: outer zone of pupil brighter / darker § Additional effect: absolute diameter of pupil changes

18 Pupil Aberration § Interlinked imaging of field and pupil § Distortion of object imaging corresponds to spherical aberration of the pupil imaging § Corrected spherical pupil aberration: tangent condition

19 Pupil Aberrations § Pupil imaging: interchange of marginal and chief ray § Spherical aberration and coma of pupil imaging: - variable exit pupil position - distortion of the image § Field and pupil aberrations connected

20 Pupil Aberrations § Wave aberration of pupil imaging § Wave aberrations of field and pupil imaging related L: Lagrange invariant Pupil imaging Spherical aberration Coma Field imaging Piston Astigmatism Distortion Field curvature Distortion Coma Piston Spherical aberration Relation

21 Pupil Aberrations § Transverse aberrations Pupil imaging Effect Grid change Spherical Kidney bean lateral shift aberration Pupil walking Coma Slyusarev Vignetting Astigmatism Field curvature Quadratic distortion I Distortion Apodization Loss of telecentricity Cubic (classical) distortion Piston Spherical no effect Relation Shape of new grid Magnification and anamorphism Quadratic distortion II

22 Pupil Imaging § Assumed ideal imaging of field planes, OO' § Fermat principle for constant OPD in case of pupil imaging: - axis point SS' as reference - same OPD only valid for BB' - point AA' not imaged perfect - rim of the stop only imaged ideal, if pupil spherical

23 Pupil Imaging

24 Pupil Imaging § General case: system stop is imaged in object and image domain § Distorted imaging in entrance and exit pupil § Practical consequences for intensity, stop boundary and wave optical calculations Equidistant sampling in: Stop correct interpreted Lambert source correct sampled Stop boundary correct without aiming PSF needs no apodization W calculation needs on a regular grid PSF calculation with FFT possible Entrance Stop pupil Y N N N Exit pupil N N Y Y

25 Pupil Aberrations § Spherical aberration of the chief ray / pupil imaging § Exit pupil location depends on the field height

26 Pupil Aberration § Eyepiece with pupil aberration § Illumination for decentered pupil : dark zones due to vignetting

27 Pupil-Surface § High numerical aperture microscopic lens § Numerical calculated pupil surface by starting narrow parallel ray bundes in the object plane § Strongly curved surface is obtained

28 Pupil Aberrations § Telecentric system: Pupil aberrations means loss of telecentricity § Special extreme case of pupil aberrations: walk offset of pupil in Fisheye lenses

29 Sine Condition and Aplanatism § Geometry same for l' ratio § Ratio § With magnification § Sine condition fulfilled for aplanatism

30 Sine Condition and Coma § Seidel contribution spherical aberration § Seidel contribution coma for arbitrary stop location Interpretation: 1. term: aplanatism 2. term: thickness dependend 3. term: residual spherical aberration § If spherical corrected and thin lenses: Coma corrected if aplanatism fulfilled

Wavefront and Spot for Coma § Coma Seidel trans verse aberrations § Wavefront for coma with § Relationship § Here

Wavefront and Spot for Coma § Schematic geometry: Notice the doubled revolution in the image plane due to combined effect of azimuthal rotation and tilt of wavefront

Tangential and Sagittal Coma § 2 terms of tangential transverse aberration: - Sagittal coma depends on xp, describes the asymmetry - Tangential coma depends on yp, corresponds to spherical aberration under skew conditions larger by a factor of 3 § Only asymmetry removed with sine condition: sagittal coma vanishes

34 Sine Condition and Coma § Linear coma (all orders) § Transverse aberrations § Sagittal coma yp = 0, xp = a § Sine condition fulfilled: linear sagittal coma vanishes § If in addition spherical aberration is corrected (aplanatic): also tangential coma vanishes

35 Skew Spherical aberration § Decomposition of coma: 1. part symmetrical around chief ray: skew spherical aberration 2. asymmetrical part: tangential coma § Skew spherical aberration: - higher order aberration - caustic symmetric around chief ray

36 Aplanatic and Perfect Imaging § Perfect imaging on axis due to conic section - not aplanatic: linear growth of coma with field size § Aplanatic: - Perfect stigmatic imaging on axis, spherical corrected - linear coma vanishes: good correction off-axis but near to axis - quadratic grows of spot size due to astigmatism - aplanatic and perfect marginal ray quite different

37 Aplanatic vs Non-aplanatic Lens § Same focal length § Same NA = 0. 75 § Principal surfaces completly different § Incoming diameter different § Field correction different

38 Spherical Corrected Surface § Seidel contribution of spherical aberration with § Result § Vanishing contribution: 1. first bracket: vertex ray 2. second bracket: concentric 3. bracket: aplanatic surface § Discussion with the Delano formula 2. concentric corresponds to i' = i 3. aplanatic condition corresponds to i' = u

39 Aplanatic Surfaces with Vanishing Spherical Aberration § Aplanatic surfaces: zero spherical aberration: 1. Ray through vertex 2. concentric 3. Aplanatic § Condition for aplanatic surface: § Virtual image location § Applications: 1. Microscopic objective lens 2. Interferometer objective lens

40 Aplanatic Lenses § Aplanatic lenses § Combination of two spherical corrected surfaces: one concentric and one aplanatic surface: zero contribution of the whole lens to spherical aberration § Not useful: 1. aplanatic-aplanatic 2. concentric-concentric bended plane parallel plate, nearly vanishing effect on rays

41 Aplanatic Lenses § Impact of aplanatic lenses in microscopy on magnification § Three cases of typical combinations

42 Aplanatic Surface § Simple construction of general aplanatic surface § Equations § No general solution for arbitrary refractive index

43 General Aplanatic Surface § General approach of Fermat principle: aplanatic surface § Cartesian oval, 4 th order § Special case OPD = 0: Solution is spherical aplanatic surface

44 General Aplanatic Surface § In general ellipsoidal surface § Only one value with degenerated ellipsoid: sphere

45 Isoplanatism § General definition of isoplanatism: - Invariance of performance for small lateral shifts of the field position - spherical aberration not necessarily corrected § Usual simple case: near to axis § Consequences: - vanishing linear growing coma - caustic symmetrical around chief ray

46 Isoplanatism § General definition of isoplanatism: - Invariance of performance for small lateral shifts of the field position - spherical aberration not necessarily corrected § Usual simple case: near to axis § Consequences: - vanishing linear growing coma - caustic symmetrical around chief ray § Example: Microscopic lens 46

47 Isoplanatism § Simple derivation for symmetric off-axis ray cone § Geometry relations § Finally Staeble-Lihotzky condition is obtained

48 Isoplanatism Condition of Staeble-Lihotzky § Sagittal coma aberration: from the geometry of the figure and Lagrange invariant § Condition of Staeble-Lihotzky § Problems: - no quantitative measure - only tangential rays are considered - integral criterion

49 Isoplanatism from Wave Aberrations § Lateral shift of object point § Change in image § Change of wave aberration § Isoplanatism: change is equal d. W' = d. W

50 Isoplanatism § Berek's condition of proportionality § Berek's coincidence condition § Isoplanatism in case of defocussing: can only be fulfilled in one plane or for telecentricity

51 Piecewise Isoplanatism § Invariance of PSF: to be defined System § Possible options: 1. relative change of Strehl 2. correlation of PSF's § Examples for microscopic lenses with and without flattening correction § In medium field size: small isoplanatic patches § On axis: large isoplanatic area § Criteria not useful at the edge for low performance MO plane 100 x 1. 25 isoplanatic patch size in mm MO not plane 40 x 0. 85 isoplanatic patch size in mm Strehl 1% Psf correlation 0. 5% on axis 70 72 81 100 half field 3. 8 27 3. 1 field zone 2. 5 29 39 full field 45 3. 8 117 62

52 Offence Against the Sine Condition § Conradys OSC (offense against sine condition): - measurement of deviation of sagittal coma - quantitative validation of the sine condition § Only sagittal coma considered in case of OSC=0 the Staeble-Lihotzky condition is automatically fulfilled § OSC allows for the definition of surface contribution

53 OSC § Coma and isoplanatism are strongly connected § Vectorial OSC: linear scaling of spatial frequencies: perturbation of the linearity

54 Skew and Symmetrical Ray bundles § Three possibilities for symmetry and correction of skew bundles: 1. no correction coma, bundle asymmetric 2. isoplanatic coma corrected, bundle symmetric oblique spherical remains 3. aplanatic symmetric and corrected, also spherical on axis corrected

55 General Invariant of Welford § Rotation around axis for small angle calculation of change in wave aberration § Welfords condition (scalar triple product) § All other conditions can be obtained as special cases: 1. sine condition 2. off axis isoplanatism 3. Herrschel condition 4. Smith cos-invariant

56 Cos-Condition of Smith § From Eikonal theory: General condition of Smith: Invariance of the scalar product § Special case: P on axis, q = 90°: Abbe sine condition, invariant transverse magnification § Special case: P on axis, q = 0°: Herschel condition, invariant axial magnification

57 Herschel Condition § Herschel condition: Invariance of the depth magnification § In principle not compatible with the sine condition § Therefore a perfect imaging of a volume is impossible

58 Overview Aplanatism-Isoplanatism § Overview on conditions for aberrations and aplanatism-isoplanatism

59 Overview § Overview on invariants and conditions