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 - to distinguish: chief ray tangential and sagittal plane O 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'nsinu' 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' m p

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 and edge of pupil chief ray skew ray

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 xp

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 yu  n'y'u' . Derivation at a single refracting surface: 1. Common height h: h  su  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 . General expression: S z B A y yCR 1. Triangle SPB w' CR P s'ExP s'Exp s' 2. Triangle ABO' y'CR w's's'ExP 

y 3. Triangle SQA u' MR s' yMR 4. Gives L  n'u'y'CR n' w's's'ExP  n'yMRw'u'w's'ExP  s' 5. Final result for arbitrary z:

L n'w'yMR (z)u'yCR(z)

27 Nested Ray Path

Optical Image formation:

. Sequence of pupil and image planes . Matching of location and size of image planes necessary (trivial) . Matching of location and size of pupils necessary for invariance of energy density . In microscopy known as Köhler illumination

entrance 2nd intermediate pupil image exit object pupil 1st stop intermediate image

marginal ray chief ray 28 Uncertainty Relation in Optics

1. Slit diffraction q Diffraction angle inverse to slit q D width D D  q  D

2. Gaussian beam x

Constant product of waist size wo

and divergence angle qo  w q  0 0  qo

w o z

29 Helmholtz-Lagrange Invariant

. Laser optics: beam parameter product waist radius times far field divergence angle LGB  woqo

. Minimum value of L:  TEMoo - fundamental mode LGB   . Elementary area of phase space: Uncertainty relation in optics  L  w q  2n1 . Laser modes: discrete structure of phase space GB n n 

. Geometrical optics: quasi continuum

. L is a measure of quality of a beam small L corresponds to a good focussability

30

Phase Space: 90°-Rotation

. Transition pupil-image plane: 90° rotation in phase space . Planes Fourier inverse . Marginal ray: space coordinate x ---> angle q' . Chief ray: angle q ---> space coordinate x'

Fourier plane image pupil location

marginal ray

x' x q'

q chief ray

f

31 Phase Space

x

2' Direct phase space x'2 representation of raytrace: space domain spatial coordinate vs angle 2 x2 1' x'1 1 x1

z

x x phase space

2' x'2 2 x2

1' 1 x1 x'1 u u u1 u2 u1 u2 32 Phase Space

x

z

x x x

x

u I u I

33 Phase Space

x Direct phase space representation of raytrace: spatial coordinate vs angle lens 2 lens 1 4 3 3' 1 5

2 2' 6

z

x 1 grin 4 free lens transfer free 3 3' lens 2 transfer 5 free transfer 2 free lens 1 2' transfer 6

u

34 Phase Space

x . Grin lens with aberrations in phase space: - continuous bended curves - aberrations seen as nonlinear angle or spatial deviations z

x x x

u u u

35 Transfer of Energy in Optical Systems

. Conservation of energy d 2 d 2'

. Differential flux (L: radiance) d 2 LsinucosudAdu d

. No absorption T  1

. Sine condition nysinu  n' y'sinu'

y y'

F F'

n n'

u u' dA dA' s s'

EnP ExP

36 Helmholtz-Lagrange Invariant

D . Geometrical optic: L  field n sinu Etendue, light gathering capacity Geo 2

. Paraxial optic: invariant of Lagrange / Helmholtz L  n yu  n'y'u'

. General case: 2D

space y . Invariance corresponds to y'1 conservation of energy

u u . Interpretation in phase space: 1 2 y'3 u3 constant area, only shape is changed aperture y'2 at the transfer through an optical small

system 1 medium aperture large 2 aperture 3

angle u

37 Conservation of Energy

• Invariance of Energy: - constant area in phase space in the geometrical model - constant integral over density in the wave optical model

• Incoherent ensemble, quasi continuum: u Jacobian matrix of transformation relates the coordinate changes imaging in 1 phase space

L u L v J     x y y x invariant area Dx Du = const. 2

3 x

38 Example Phase Space of an Array

Simplified pictures to the changes of the phase space density. Etendue is enlarged, but no complete filling.

1) before array u 2) separation of u 3) lens effect u subapertures of subapertures

x x x

4) in focal plane u 5) in 2f-plane u 6) far away u

x x x

1 2 3 4 5 6

39 Pupil Sphere

. Generalization of paraxial picture: Principal surface works as effective location of ray bending for object points near the optical axis (isoplanatic patch)

. Paraxial approximation: plane

Can be used for all rays to find effective surface of the imaged ray P ray bending P'

. Real systems with corrected sine-condition (aplanatic): principal sphere y U'

. The pupil sphere can not be used to construct arbitrary ray paths

. If the sine correction is not fulfilled: more complicated shape of the arteficial surface, that represents the ray bending f' 40 Pupil Sphere

. Pupil sphere: object entrance exit image pupil pupil equidistant sine-

sampling yo y'

sin(U) sin(U') U U' z

object yo pupil sphere

equidistant sin(U)

angle U non- equidistant 41 Canonical Coordinates

Aplanatic system:

y p y' . Sine condition fulfilled y  p p y'p  . Pupil has spherical shape hEnP h'ExP . Normalized canonical coordinates n sinu n'sinu' for pupil and field y   y y'  y'  

y yp y'p y'

hEnP yp y'p U h'ExP U' O O'

entrance exit object pupil pupil image 42 Canonical Coordinates

x x' x  p x'  p . Normalized pupil coordinates p h p h' EnP ExP y y' p y'  p y p  p h h'ExP EnP . Special case: aplanatic imaging x'  x y'  y p p p p

nsin u n'sin u'sag . Normalized field coordinates x  sag  x x'  x'   nsin u n'sin u ' y  tan  y y' tan  y'  

. Special case: paraxial imaging x' x y' y

. Reference on chief ray NA n sinu  n sinuTCO  sinuCR 

nsinu n sinu  sinu  . Reduced image side coordinates x  sag  x y  CR tan  y   43 Analogy Optics - Mechanics

Mechanics Optics spatial variable x x Impuls variable p=mv angle u, direction cosine p,

spatial frequency sx, kx equation of motion spatial Lagrange Eikonal domain equation of motion phase space Hamilton Wigner transport potential mass m index of refraction n minimal principle Hamilton principle Fermat principle Liouville theorem constant number of constant energy particles system function impuls response point spread function wave equation Helmholtz Schrödinger propagation variable time t space z uncertainty h/2p l/2p continuum approximation classical mechanics geometrical optics exact description quantum mechanics wave optics