imaging and aberration theory
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Imaging and Aberration Theory
Lecture 2: Fourier- and Hamiltonian Optics
2017-10-23
Herbert Gross
Winter term 2017
2
Preliminary time schedule
1 16.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 23.10. Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 30.10. Eikonal Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 06.11. Aberration expansions single surface, general Taylor expansion, representations, various orders, stop shift formulas
5 13.11. Representation of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 20.11. Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders
7 27.11. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 04.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 11.12. Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum
10 18.12. Sine condition, aplanatism and isoplanatism
Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics
11 08.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations
12 15.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, measurement
13 22.01. Point spread function ideal psf, psf with aberrations, Strehl ratio
14 30.01. Transfer function transfer function, resolution and contrast
15 05.02. Additional topics Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability
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
3
Contents
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
Diaphragm in Optical Systems
y
y'
stop
pupil
coma ray
chief ray
marginal ray
aperture
angle
u
field anglew
object
image
4
Imaging on axis: circular / rotational symmetry
Only spherical aberration and chromatical aberrations
Finite field size, object point off-axis:
- chief ray as reference
- skew ray bundels:
coma and distortion
- Vignetting, cone of ray bundle
not circular symmetric
- to distinguish:
tangential and sagittal
plane
O
entrance
pupil
y yp
chief ray
exit
pupil
y' y'p
O'
w'
w
R'AP
u
chief ray
object
planeimage
plane
marginal/rim
ray
u'
Definition of Field of View and Aperture
5
Classical measure for the opening:
numerical aperture
In particular for camera lenses with
object at infinity:
F-number
Numerical aperture and F-number are to system properties, they are related to a conjugate
object/image location
Paraxial relation
Special case for small angles or sine-condition corrected systems
6
Numerical Aperture and F-number
'sin' unNA DEnP/2
f
image
plane
object in
infinity
u'
EnPD
fF #
'tan'2
1#
unF
'2
1#
NAF
More general definition
of the F-number for systems
with finite object location
Effective or working F-number
with
and
we get as a relation with the object-in-infinity-case
m is the system magnification, mp is the pupil magnification
7
Generalized F-Number
infinity
s'
f'
u'o u'
finite
objectDExP
mfs 1''
)1('2)1('2'2'sin
mfn
Dm
mf
D
s
Du
EnPpExPExP
p
eff
m
mF
unF
1
'sin'2
1##
Meridional rays:
in main cross section plane
Sagittal rays:
perpendicular to main cross
section plane
Coma rays:
Going through field point
and edge of pupil
Oblique rays:
without symmetry
Special rays in 3D
axis
y
x
p
p
pupil plane
object plane
x
y
axissagittal ray
meridional marginal ray
skew raychief ray
sagittal coma ray
upper meridional coma ray
lower meridional coma ray
field point
axis point
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
Ray-Fan Selection for Transverse Aberration Plots
x
y
pupil
tangential
ray fan
sagittal
ray fanobject
point
9
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
Pupil Sampling
y'p
x'p
yp
xp x'
y'
z
yo
xo
object
plane
entrance
pupil
exit
pupil
image
plane
tangential
sagittal
10
Pupil Sampling
Ray plots
Spot
diagrams
sagittal ray fan
tangential ray fan
yp
Dy
tangential aberration
xp
Dy
xp
Dx
sagittal aberration
whole pupil area
11
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
Pupil Sampling
y'p
x'p
yp
xp x'
y'
z
yo
xo
object
plane
entrance
pupil
exit
pupil
image
plane
12
Pupil Sampling
13
polar grid cartesian isoenergetic circular
hexagonal statistical pseudo-statistical (Halton)
Fibonacci spirals
Criteria: 1. iso energetic rays 2. good boundary description 3. good spatial resolution
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
Pupil Sampling Spot Artefacts
hexagonal statisticalcartesian
14
The physical stop defines
the aperture cone angle u
The real system may be
complex
The entrance pupil fixes the
acceptance cone in the
object space
The exit pupil fixes the
acceptance cone in the
image space
Diaphragm in Optical Systems
uobject
image
stop
EnP
ExP
object
image
black box
details complicated
real
system
? ?
Ref: Julie Bentley
15
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
Properties of the Pupil
16
Entrance and Exit Pupil
exit
pupil
upper
marginal ray
chief
ray
lower coma
raystop
field point
of image
UU'
W
lower marginal
ray
upper coma
ray
on axis
point of
image
outer field
point of
object
object
point
on axis
entrance
pupil
17
Relative position of stop inside system
Quantitative measure:
Parameter of excentricity
(h absolut values)
Special cases: c = 1 image plane
c = -1 pupil plane
c = 0 same effective distance from image and pupil
MRCR
MRCR
hh
hh
c
Parameter of Excentricity
hCR(ima)
chief ray
pupil image
marginal ray hMR
surface
no j
hCR
c
-1 0 +1
18
Example:
excentricity for all surfaces
Change:
c = +1 ...-1 ...+1
19
Parameter of Excentricity
1 2 3 45 7
9 11 13 15 1719
22 23 26 28 30 3234
stop imageobject
pupil
0 5 10 15 20 25 30 35
-10
0
10
20
30
40
50
0 5 10 15 20 25 30 35
-1
-0.5
0
0.5
1
object
image
chief ray height
marginal ray
height
excentricity
surface
index
surface
index
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
20
Artificial vignetting:
Truncation of the free area
of the aperture light cone
Natural Vignetting:
Decrease of brightness
according to cos w 4 due
to oblique projection of areas
and changed photometric
distances
Vignetting
w
AExp
imaging without vignetting
complete field of view
imaging with
vignetting
imaging with
vignetting
field
angle
D
0.8 Daxis
field
truncation
truncation
stop
21
3D-effects due to vignetting
Truncation of the at different surfaces for the upper and the lower part
of the cone
Vignetting
object lens 1 lens 2 imageaperture
stop
lower
truncation
upper
truncation
sagittal
trauncation
chief
ray
coma
rays
22
Truncation of the light cone
with asymmetric ray path
for off-axis field points
Intensity decrease towards
the edge of the image
Definition of the chief ray:
ray through energetic centroid
Vignetting can be used to avoid
uncorrectable coma aberrations
in the outer field
Effective free area with extrem
aspect ratio:
anamorphic resolution
Vignetting
projection of the
rim of the 2nd lens
projection of the
rim of the 1st lens
projection of
aperture stop
free area of the
aperture
sagittal
coma rays
meridional
coma rayschief
ray
23
Vignetting
Illumination fall off in the image due to vignetting at the field boundary
24
Product of field size y and numercial aperture is
invariant in a paraxial system
Derivation at a single refracting surface:
1. Common height h:
2. Triangles
3. Refraction:
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
''' uynuynL
Helmholtz-Lagrange Invariant
y
u u'
chief ray
marginal ray
w
h
ss'
w'
n n'
surface
''usush
'
'',
s
yw
s
yw
''wnnw
25
Basic formulation of the Lagrange
invariant:
Uses image height,
only valid in field planes
General expression:
1. Triangle SPB
2. Triangle ABO'
3. Triangle SQA
4. Gives
5. Final result for arbitrary z:
Helmholtz-Lagrange Invariant
ExPCR sswy ''''
ExP
CR
s
yw
''
''
s
yu MR
z
y'y yp
pupil imagearbitrary z
chief
ray
marginal
ray
s's'Exp
y'CR
yMR
yCR
B A
O'
Q
S
P
ExPMRExPMR
CR swuwynssws
ynyunL ''''''''
'''''
)(')('' zyuzywnL CRMR
26
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
Nested Ray Path
object
marginal
ray chief ray
1st
intermediate
image
entrance
pupil
stop
exit
pupil
2nd intermediate
image
27
1. Slit diffraction
Diffraction angle inverse to slit
width D
2. Gaussian beam
Constant product of waist size wo
and divergence angle qo
q 00w
D
q
q
D D
q
qo
wo
x
z
Uncertainty Relation in Optics
28
Laser optics: beam parameter product
waist radius times far field divergence angle
Minimum value of L:
TEMoo - fundamental mode
Elementary area of phase space:
Uncertainty relation in optics
Laser modes: discrete structure of phase space
Geometrical optics: quasi continuum
L is a measure of quality of a beam
small L corresponds to a good focussability
ooGB wL q
GBL
12 nwL nnGB
q
Helmholtz-Lagrange Invariant
29
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'
f
xx'
q
q'
Fourier plane
pupil
image
location
marginal ray
chief
ray
30
Direct phase space
representation of raytrace:
spatial coordinate vs angle
Phase Space
z
x
x
u u
x
x1
x'1
x2
x'2
x1 x'1
x2
x'2
u1 u2 u1 u2
11'
2
2'
1
1'
2
2'
space
domain
phase
space
31
Phase Space
z
x
I
x
I
x
u
x
u
x
32
Phase Space
z
x
x
u
1
1
2 2'
2 2'
3 3' 4
3
3'
5
4
5
6
6
free
transfer
lens 1
lens 2
grin
lens
lens 1
lens 2
free
transfer
free
transfer
free
transfer
Direct phase space
representation of raytrace:
spatial coordinate vs angle
33
Grin lens with aberrations in
phase space:
- continuous bended curves
- aberrations seen as nonlinear
angle or spatial deviations
Phase Space
z
x
x
u u
x
u
x
34
Transfer of Energy in Optical Systems
Conservation of energy
Differential flux (L: radiance)
No absorption
Sine condition
d d2 2 '
ddudAuuLd cossin2
T 1
y
dA dA's's
EnP ExP
n n'
F'F
y'
u u'
'sin''sin uynuyn
35
Geometrical optic:
Etendue, light gathering capacity
Paraxial optic: invariant of Lagrange / Helmholtz
General case: 2D
Invariance corresponds to
conservation of energy
Interpretation in phase space:
constant area, only shape is changed
at the transfer through an optical
system
unD
Lfield
Geo sin2
''' uynuynL
Helmholtz-Lagrange Invariant
space y
angle u
large
aperture
aperture
small
1
23
medium
aperture
y'1
y'2
y'3u1 u2
u3
36
• 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:
Jacobian matrix of transformation relates
the coordinate changes
x
u
imaging in
phase space1
2
3
invariant area
Dx Du = const.
x
v
y
L
y
u
x
LJ
Conservation of Energy
37
Simplified pictures to the changes of the phase space density.
Etendue is enlarged, but no complete filling.
x
u
x
u
x
u1) before array
x
u
x
u
x
u
2) separation of
subapertures
3) lens effect
of subapertures
4) in focal plane 5) in 2f-plane 6) far away
1 2 3 4 5 6
Example Phase Space of an Array
38
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
the imaged ray
Real systems with corrected
sine-condition (aplanatic):
principal sphere
The principal 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
Principal Sphere
effective surface of
ray bending P'
y
f'
U'
P
39
Pupil sphere:
equidistant sine-
sampling
Pupil Sphere
z
object entrance
pupil
image
yo y'
U U'sin(U) sin(U')
exit
pupil
objectyo
equidistant
sin(U)
angle U non-
equidistant
pupil
sphere
40
Aplanatic system:
Sine condition fulfilled
Pupil has spherical shape
Normalized canonical coordinates
for pupil and field
Canonical Coordinates
O'O
y
yphEnP
y'
entrance
pupilexit
pupilobject image
U U'h'ExP
y'p
yp y'p
''sin'
' yun
y
EnP
p
ph
yy
ExP
p
ph
yy
'
''
41
yun
y
sin
Normalized pupil coordinates
Special case: aplanatic imaging
Normalized field coordinates
Special case: paraxial imaging
Reference on chief ray
Reduced image side coordinates
EnP
p
ph
xx
ExP
p
ph
xx
'
''
pp yy '
xx '
CRTCO uununNA sinsinsin
xun
xsag
sin y
uuny CR
tansinsin
Canonical Coordinates
42
EnP
p
ph
yy
ExP
p
ph
yy
'
''
pp xx '
yy '
yun
y
tansin'
'sin'' tan y
uny
xun
xsag
sin'
'sin'' x
unx
sag
Analogy Optics - Mechanics
43
Mechanics Optics
spatial variable x x
Impuls variable p=mv angle u, direction cosine p,
spatial frequency sx, kx
equation of motion spatial
domain
Lagrange Eikonal
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
particles
constant energy
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