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Medical Photonics Lecture
Optical Engineering
Lecture 5: Optical systems
2019-05-15
Herbert Gross
Speaker: Yi Zhong
Summer term 2019
2
Contents
No Subject Ref Date Detailed Content
1 Introduction Zhong 10.04.Materials, dispersion, ray picture, geometrical approach, paraxial approximation
2 Geometrical optics Zhong 17.04.Ray tracing, matrix approach, aberrations, imaging, Lagrange invariant
3 Diffraction Zhong 24.04.Basic phenomena, wave optics, interference, diffraction calculation, point spread function, transfer function
4 Components Kempe 08.05. Lenses, micro-optics, mirrors, prisms, gratings
5 Optical systems Zhong 15.05.Field, aperture, pupil, magnification, infinity cases, lens makers formula, etendue, vignetting
6 Aberrations Zhong 22.05. Introduction, primary aberrations, miscellaneous
7 Image quality Zhong 29.05. Spot, ray aberration curves, PSF and MTF, criteria
8 Instruments I Kempe 05.06.Human eye, loupe, eyepieces, photographic lenses, zoom lenses, telescopes
9 Instruments II Kempe 12.06.Microscopic systems, micro objectives, illumination, scanning microscopes, contrasts
10 Instruments III Kempe 19.06.Medical optical systems, endoscopes, ophthalmic devices, surgical microscopes
11 Photometry Zhong 26.06.Notations, fundamental laws, Lambert source, radiative transfer, photometry of optical systems, color theory
12 Illumination systems Gross 03.07.Light sources, basic systems, quality criteria, nonsequential raytrace
13 Metrology Gross 10.07. Measurement of basic parameters, quality measurements
1. Photo objective lens
2. Microscope objective lens
3. Binocular
4. Infrared afocal system
Typical Example Systems 1
5. Relay optics
6. Scan-objective lens
7. Collimator objective lens
Typical Example Systems 2
possible surfaces
under test
8. Projector lens
9. Telescope
10. Lithography projection
lens
Typical Example Systems 3
M1
M2
M3
11. Illumination collector system
12. Illumination condenser system
13. Head mounted display
Typical Example Systems 4
eye
pupil
image
total
internal
reflection
free formed
surface
free formed
surface
field angle 14°
14. Stereo microscope
15. Zoom system
Typical Example Systems 5
eyepiece
tube
system zoom
system object
plane
eye
common
axis
stereo
angle
common
objective
lens
f = 61
f = 113
f = 166
Lithographic Optics
H-Design
I-Design
X-Design
9
Model depth of Light Propagation
Different levels of modelling in optical propagation
Schematical illustration (not to scale)
Ref: A. Herkommer
accuracy
calculation
effort
paraxial optic
geometrical
optic
(raytrace)
scalar waveoptic
(high NA)
paraxial
waveoptic
vectorial waveoptic
rigorous waveoptic
Modelling of Optical Systems
Principal purpose of calculations:
1. Solving the direct problem of
understanding the properties:
analysis
2. Solving the inverse problem:
Finding the concret system data
for a required functionality:
synthesis
System, data of the structure(radii, distances, indices,...)
Function, data of properties,
quality performance(spot diameter, MTF, Strehl ratio,...)
Analysisimaging
aberration
theory
Synthesislens design
inverse
problem
Ref: W. Richter
10
Approximation of Optical Models
Imaging model with levels
of refinement
Paraxial model
(focal length, magnification, aperture,..)
approximation
l à 0
no description of
short pulses
Geometrical
optics
Analytical approximation
(3rd order aberrations,..)
exact geometry
Wave optics
no time dependence
Maxwell equations
Scalar approximation
Helmholtz equation
(PSF, OTF,...)
linear
approximation
no description of
small structures
and polarization
effects
no diffraction
no higher order
aberrations
no aberrations
exact
11
Five levels of modelling:
1. Geometrical raytrace with analysis
2. Equivalent geometrical quantities,
classification
3. Physical model:
complex pupil function
4. Primary physical quantities
5. Secondary physical quantities
Blue arrows: conversion of quantities
Modelling of Optical Systems
ray
tracing
optical path
length
wave
aberration W
transverse
aberrationlongitudinal aberrations
Zernike
coefficients
pupil
function
point spread
function (PSF)
Strehlnumber
optical
transfer function
geometricalspot diagramm
rms
value
intersectionpoints
final analysis reference ray in
the image space
referencesphere
orthogonalexpansion
analysis
sum of
coefficientsMarechal
approxima-
tion
exponentialfunction
of the
phase
Fourier
transformLuneburg integral
( far field )
Kirchhoffintegral
maximum
of the squared
amplitude
Fouriertransformsquared amplitude
sum of
squaresMarechalapproxima-
tion
integration ofspatial
frequencies
Rayleigh unit
equivalencetypes of
aberrationsdifferen
tiationinte-
gration
full
aperture
single types of aberrations
definition
geometricaloptical
transfer function
Fouriertransform
approximation
auto-correlationDuffieux
integral
resolution
threshold value spatial frequency
threshold value spatial
frequency approximationspot diameter
approximation diameter of the
spot
Marechalapproximation
final analysis reference ray in the image planeGeometrical
raytrace
with Snells law
Geometrical
equivalents
classification
Physical
model
Primary
physical
quantities
Secondary
physical
quantities
13
Aberrations
Non-perfect imaging through a real optical system
Only in paraxial approximation of small angles the systems are perfect
A perturbation theory for larger angles of
1. the chief ray for the field size description
2. the marginal ray for the aperture size description
explains the aberrations as non-linear effects
The monochromatic primary aberrations are of 3rd order in transverse representation,
they are:
1. spherical aberration: only on axis, circular symmetric
2. coma, grows linear with the field of view, asymmetric
3. astigmatism
4. field curvature, the image is sharp on a curved surface
5. distortion, geometrical deformation of the image, full resolution
Furthermore there are two types of primary chromatical aberrations as a result of the
material dispersion with wavelength:
1. axial color aberration, dispersion of the marginal ray
2. lateral color, dispersion of the chief ray
14
Spherical aberration
Ray intersection length with the optical axis depends on the ray height
plane of
best focus
zone
paraxial
rim
15
Astigmatism and Field Curvature
Astigmatism:
different focal lengths in sag and tan cross
section
Field curvature:
image is sharp on a curved surface
ideal
image
plane
tangential
shell
sagittal
shell
image surfacesy'
focused in center
(paraxial image plane)focused in field zone
(mean image plane)
focused at field boundary
z
y'
receiving
planes
image
sphere
16
Coma
Asymmetrical shape of the
spot
Centroid no longer peak
intensity
saggital
marginal rays
central
ray
angle 60°
coma
blur
lens / pupil
axis
tangential
marginal
rays
circle
radius ~ r² Tangential
coma CT
Saggital
coma CS
CS ~ CT / 3
c7 = 0.3 c7 = 0.5 c7 = 1
centroid
17
Distortion
Deformation of the geometry
No blurrly
Described by chief ray error on
pincussion
distortion
barrel
distortion
object
D < 0
D > 0
lens
rear
stop
imagex
x
y
y
y'
x'
y'
x'
front
stop
18
Chromatical aberrations
Reason of chromatical aberrations is the dispersion of the materials
Two primary aberrations:
1. marginal ray aberration:
axial color, images of different colors at different z-positions
2. chief ray aberration:
transverse color, images of different colors have different size
object
chief ray
marginal
ray
chief rays
marginal rays
l2
l1
ExP
l2
ExP
l1
transverse
chromatical
aberration
axial
chromatical
aberration
ideal
image
l1
ideal
image
l2
19
Chromatical aberrations
Axial color aberration
Transverse color aberration z
l = 648 nm
defocus
-2 -1 0 1 2
l = 546 nm
l = 480 nm
best image planel
z
y
stop
dispersion
prism effect
image
plane
DyCHV
chief
ray
Achromate
Residual aberrations of an achromate
Clearly seen:
1. Distortion
2. Chromatical magnification
3. Astigmatism
20
System performance:
• Aberrations
• Spot diameter
• Wavefront
• Zernike coefficients
• Contrast (MTF)
• Point spread function
Depends on:
• aperture
• field position
• wavelength
• object distance
10:15:47
ORA 24-Apr-07
40X POWER, 0.70 NA
PLAN-ACHROMAT
RAY ABERRATIONS ( MILLIMETERS )
656.3000 NM
587.6000 NM
546.1000 NM
486.1000 NM
435.8000 NM
-0.005
0.005
-0.005
0.005
0.00 RELATIVE
FIELD HEIGHT
( 0.000 )O
-0.005
0.005
-0.005
0.005
0.75 RELATIVE
FIELD HEIGHT
( -3.64 )O
-0.005
0.005
-0.005
0.005
TANGENTIAL 1.00 RELATIVE SAGITTAL
FIELD HEIGHT
( -4.85 )O
11:16:28
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MODULATION
300 600 900 1200 1500 1800 2100 2400
SPATIAL FREQUENCY (CYCLES/MM)
40X POWER, 0.70 NA
PLAN-ACHROMAT
DIFFRACTION MTF
ORA 29-Apr-07
DIFFRACTION LIMIT
AXIS
T
R0.5 FIELD ( )-2.43 O
T
R0.8 FIELD ( )-3.64 O
T
R1.0 FIELD ( )-4.85 O
WAVELENGTH WEIGHT
656.3 NM 14
587.6 NM 71
546.1 NM 90
486.1 NM 26
435.8 NM 2
DEFOCUSING 0.00000
Ref : C. Menke
Performance Criteria
Raytrace Through a Lens
R1R2
plane
t
n
D
Tracing of rays throug a lens
Application of the law of refraction at every interface
surface between different indices of refraction
Simple geometrical parameter:
- radii of curvature: R1, R2
- thickness t
- diameter D
- refractive index n
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
23
System of two separated thin lenses
Variation of the back principal plane
as a function of the distribution of
refractive power
24
Principal Planes
P' L1 L2P F
plate
plate
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 bundles:
coma and distortion
- Vignetting, cone of ray bundle
not circular symmetric
- to distinguish:
tangential and sagittal
planeO
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 Field of View and Aperture
25
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
26
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
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
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
28
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
29
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
30
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
31
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
32
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 must be fulfilled
O O’
stop and
entrance pupil
optical system
exit pupil
objectimage
Object imaging Pupil imaging
Blue rays
Red rays
Marginal rays
Marginal raysChief rays
Chief rays
.tan
'tanconst
w
w
33
Sine condition not fulfilled:
- nonlinear scaling from entrance to exit pupil
- spatial filtering on warped grid, nonlinear sampling of spatial frequencies
- pupil size changes
- apodization due to distortion
- wave aberration could be calculated wrong
- quantitative mesaure of offence against the sine condition (OSC):
distortion of exit pupil grid
Sine Condition
xo xp
sphere distorted exit
pupil surface
object
plane
exit
pupil
optical
system
sx
u
xp
x'o
u'
x'p
x'p
image
plane
entrance
pupil
grid
distortion
1sin
unf
xD
ap
p
34
Spherical aberration of the chief ray / pupil imaging
Exit pupil location depends on the field height
Pupil Aberrations
yobject
sP
chief rays
pupil position
pupil
location
35
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
36
Eyepiece with pupil aberration
Illumination for decentered pupil :
dark zones due to vignetting
Kidney beam effect
Pupil Aberration
eyepiecelens and
pupil of
the eye
retina
caustic of the pupil
image enlarged
instrument
pupil
37
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
38
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
Projektion der
Aperturblende
free area of the
aperture
sagittal
coma rays
meridional
coma rayschief
ray
39
Vignetting
Illumination fall off in the image due to vignetting at the field boundary
40
Special stop positions:
1. stop in back focal plane: object sided telecentricity
2. stop in front focal plane: image sided telecentricity
3. stop in intermediate focal plane: both-sided telecentricity
Telecentricity:
1. pupil in infinity
2. chief ray parallel to the optical axis
Telecentricity
telecentric
stopobject imageobject sides chief rays
parallel to the optical axis
41
Double telecentric system: stop in intermediate focus
Realization in lithographic projection systems
Telecentricity
telecentric
stopobject imagelens f1 lens f2
f1
f1
f2
f2
42
43
Infinity cases
sample layoutexit pupilentrance
pupilimageobjectcase
finitefinitefinitefinite1
infinity
image
telecentric
finitefinitefinite2
infinity
image
telecentric
infinity
object
telecentric
finitefinite3
finitefiniteinfinityinfinity4
finiteinfinityfinitefinite5
finitefinitefiniteinfinity6
finitefiniteinfinityfinite7
finite
infinity
object
telecentric
infinityfinite8
infinity
image
telecentric
finitefiniteinfinity9
example
relay
metrology lens
lithographic
projection lens
4f-system
afocal zoom
telescopes
beam expander
metrology lens
camera lens
focussing lens
eyepiece
collimator
microscopic lens
infinity metrology
lens
finiteinfinityfiniteinfinity10
infinityfiniteinfinityfinite11
impossible
impossible
finiteinfinityinfinityinfinity12
infinityinfinityfiniteinfinity13
impossible
impossible
infinityfiniteinfinityinfinity14
infinityinfinityinfinityfinite15
impossible
impossible
infinityinfinityinfinityinfinity16 impossible
Systematic of all
infinity cases
Physically impossible:
1. object and entrance
pupil in infinity
2. image and exit
pupil in infinity