www.iap.uni-jena.de
Optical Design with Zemax
for PhD - Basics
Lecture 5: Aberrations II
2018-11-21
Herbert Gross
Speaker: Uwe Lippmann
Winter term 2018
Preliminary Schedule
No Date Subject Detailed content
1 17.10. Introduction
Zemax interface, menus, file handling, system description, editors, preferences, updates,
system reports, coordinate systems, aperture, field, wavelength, layouts, diameters, stop
and pupil, solves
2 24.10.Basic Zemax
handling
Raytrace, ray fans, paraxial optics, surface types, quick focus, catalogs, vignetting,
footprints, system insertion, scaling, component reversal
3 07.11.Properties of optical
systems
aspheres, gradient media, gratings and diffractive surfaces, special types of surfaces,
telecentricity, ray aiming, afocal systems
4 14.11. Aberrations I representations, spot, Seidel, transverse aberration curves, Zernike wave aberrations
5 21.11. Aberrations II Point spread function and transfer function
6 28.11. Optimization I algorithms, merit function, variables, pick up’s
7 05.12. Optimization II methodology, correction process, special requirements, examples
8 12.12. Advanced handling slider, universal plot, I/O of data, material index fit, multi configuration, macro language
9 09.01. Imaging Fourier imaging, geometrical images
10 16.01. Correction I Symmetry, field flattening, color correction
11 23.01. Correction II Higher orders, aspheres, freeforms, miscellaneous
12 30.01. Tolerancing I Practical tolerancing, sensitivity
13 06.02. Tolerancing II Adjustment, thermal loading, ghosts
14 13.02. Illumination I Photometry, light sources, non-sequential raytrace, homogenization, simple examples
15 20.02. Illumination II Examples, special components
16 27.02. Physical modeling I Gaussian beams, Gauss-Schell beams, general propagation, POP
17 06.03. Physical modeling II Polarization, Jones matrix, Stokes, propagation, birefringence, components
18 13.03. Physical modeling III Coatings, Fresnel formulas, matrix algorithm, types of coatings
19 20.03. Physical modeling IVScattering and straylight, PSD, calculation schemes, volume scattering, biomedical
applications
20 27.03. Additional topicsAdaptive optics, stock lens matching, index fit, Macro language, coupling Zemax-Matlab /
Python
1. Point spread function
2. Edge and line spread function
3. Optical transfer function
Contents
Diffraction at the System Aperture
Self luminous points: emission of spherical waves
Optical system: only a limited solid angle is propagated, the truncation of the spherical
wave results in a finite angle light cone
In the image space: uncomplete constructive interference of partial waves, the image
point is spread
The optical systems acts as a low pass filter
object
point
spherical
wave
truncated
spherical
wave
image
plane
x = 1.22 / NA
point
spread
function
object plane
Fraunhofer Point Spread Function
Rayleigh-Sommerfeld diffraction integral,
Mathematical formulation of the Huygens-principle
Fraunhofer approximation in the far field
for large Fresnel number
Optical systems:
numerical aperture NA in image space
Pupil amplitude/transmission/illumination T(xp,yp)
Wave aberration W(xp,yp)
complex pupil function A(xp,yp)
Transition from exit pupil to
image plane
Point spread function (PSF): Fourier transform of the complex
pupil function
1
2
z
rN
p
F
),(2),(),( pp yxWi
pppp eyxTyxA
pp
yyxxR
i
yxiW
pp
AP
dydxeeyxTyxEpp
APpp
''2
,2,)','(
''cos'
)'()('
dydxrr
erE
irE d
rrki
I
PSF by Huygens Principle
Huygens wavelets correspond to vectorial field components
The phase is represented by the direction
The amplitude is represented by the length
Zeros in the diffraction pattern: destructive interference
Aberrations from spherical wave: reduced conctructive superposition
pupil
stop
wave
front
ideal
reference
sphere
point
spread
function
zero
intensity
side lobe
peak
central peak maximum
constructive interference
reduced constructive
interference due to phase
aberration
0
2
12,0 I
v
vJvI
0
2
4/
4/sin0, I
u
uuI
-25 -20 -15 -10 -5 0 5 10 15 20 250,0
0,2
0,4
0,6
0,8
1,0
vertical
lateral
inte
nsity
u / v
Circular homogeneous illuminated
Aperture: intensity distribution
transversal: Airy
scale:
axial: sinc
scale
Resolution transversal better
than axial: x < z
Ref: M. Kempe
Scaled coordinates according to Wolf :
axial : u = 2 z n / NA2
transversal : v = 2 x / NA
Perfect Point Spread Function
NADAiry
22.1
2NA
nRE
log I(r)
r0 5 10 15 20 25 30
10
10
10
10
10
10
10
-6
-5
-4
-3
-2
-1
0
Airy distribution:
Gray scale picture
Zeros non-equidistant
Logarithmic scale
Encircled energy
Perfect Lateral Point Spread Function: Airy
DAiry
r / rAiry
Ecirc
(r)
0
1
2 3 4 5
1.831 2.655 3.477
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2. ring 2.79%
3. ring 1.48%
1. ring 7.26%
peak 83.8%
Axial distribution of intensity
Corresponds to defocus
Normalized axial coordinate
Scale for depth of focus :
Rayleigh length
Zero crossing points:
equidistant and symmetric,
Distance zeros around image plane 4RE
22
04/
4/sinsin)(
u
uI
z
zIzI o
42
2 uz
NAz
22
'
'sin' NA
n
unRE
Perfect Axial Point Spread Function
-4 -3 -2 -1 0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
I(z)
z/
RE
4RE
z = 2RE
Defocussed Perfect PSF
Perfect point spread function with defocus
Representation with constant energy: extreme large dynamic changes
z = -2RE z = +2REz = -1RE z = +1RE
normalized
intensity
constant
energy
focus
Imax = 5.1% Imax = 42%Imax = 9.8%
Comparison Geometrical Spot – Wave-Optical PSF
aberrations
spot
diameter
DAiry
exact
wave-optic
geometric-optic
approximated
diffraction limited,
failure of the
geometrical model
Fourier transform
ill conditioned
Large aberrations:
Waveoptical calculation shows bad conditioning
Wave aberrations small: diffraction limited,
geometrical spot too small and
wrong
Approximation for the
intermediate range:
22
GeoAirySpot DDD
0,0
0,0)(
)(
ideal
PSF
real
PSFS
I
ID
2
2),(2
),(
),(
dydxyxA
dydxeyxAD
yxWi
S
Important citerion for diffraction limited systems:
Strehl ratio (Strehl definition)
Ratio of real peak intensity (with aberrations) referenced on ideal peak intensity
DS takes values between 0...1
DS = 1 is perfect
Critical in use: the complete
information is reduced to only one
number
The criterion is useful for 'good'
systems with values Ds > 0.5
Strehl Ratio
r
1
peak reduced
Strehl ratio
distribution
broadened
ideal , without
aberrations
real with
aberrations
I ( x )
12
PSF with Aberrations
PSF for some low oder Zernike coefficients
The coefficients are changed between cj = 0...0.7
The peak intensities are renormalized
spherical
defocus
coma
astigmatism
trefoil
spherical
5. order
astigmatism
5. order
coma
5. order
c = 0.0
c = 0.1c = 0.2
c = 0.3c = 0.4
c = 0.5c = 0.7
13
Point Spread Function with Apodization
w
I(w)
1
0.8
0.6
0.4
0.2
00 1 2 3-2 -1
Airy
Bessel
Gauss
FWHM
w
E(w)
1
0.8
0.6
0.4
0.2
03 41 2
Airy
Bessel
Gauss
E95%
Apodisation of the pupil:
1. Homogeneous
2. Gaussian
3. Bessel
PSF in focus:
different convergence to zero forlarger radii
Encircled energy:
same behavior
Complicated:Definition of compactness of thecentral peak:
1. FWHM: Airy more compact as GaussBessel more compact as Airy
2. Energy 95%: Gauss more compact as AiryBessel extremly worse
Only far field model (Fraunhofer)
Two different algorithms available:
1. FFT-based
- fast
- equidistant exit pupil sampling assumed
- high resolution PSF needs many points
2. elementary integration (Huygens)
- slow (N4)
- independence of pupil and image sampling
- valid also for calculation of pupil distortion
- gives correct Strehl number
Different options for representation possible
15
PSF in Zemax
Logarithmic representation
16
PSF in Zemax
Line image: integral over point spread function
LSF: line spread function
Realization: narrow slit
convolution of slit width
But with deconvolution, the PSF can be reconstructed
dyyxIxI PSFLSF ),()(
Integration
intens
ity
x
Line spread function
PSF
dyyxIxI PSFLSF ),()(
Line Spread Function
Line image:
Fourier transform of pupil in one dimension
Line spreadfunction with aberrations
Here: defocussing
pppp
pp
xxR
i
pp
iLSF
dydxyxP
dydxeyxP
xI
pi
2
22
,
,
)(
x
ILSF
(x)
-10 -5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
W20
= 0.0
W20
= 0.1
W20
= 0.2
W20
= 0.3
W20
= 0.4
W20
= 0.5
W20
= 0.7
Line Spread Function
ESF with defocussing ESF with spherical aberration
x
IESF
(x)
-10 -5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
W20
= 0.0
W20
= 0.1
W20
= 0.2
W20
= 0.3
W20
= 0.4
W20
= 0.5
W20
= 0.7
x
IESF
(x)
-8 -6 -4 -2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
W40
= 0.0
W40
= 0.1
W40
= 0.2
W40
= 0.3
W40
= 0.4
W40
= 0.5
W40
= 0.7
Incoherent Edge Spread Function
Sampling of the Diffraction Integral
x-6 -4 -2 0 2 4
0
10
20
30
40
50
quadratic
phase
wrapped
phase
2
smallest sampling
intervall
phase
Oscillating exponent :
Fourier transform reduces on 2-
period
Most critical sampling usually
at boundary defines number
of sampling points
Steep phase gradients define the
sampling
High order aberrations are a
problem
Propagation by Plane / Spherical Waves
Expansion field in simple-to-propagate waves
1. Spherical waves 2. Plane waves
Huygens principle spectral representation
rdrErr
erE
rrik
2
'
)('
)'(
x
x'
z
E(x)
eikr
r
)(ˆˆ)'( 1 rEFeFrE xy
zik
xyz
x
x'
z
E(x)
eik z z
Resolution of Fourier Components
Ref: D.Aronstein / J. Bentley
object
pointlow spatial
frequencies
high spatial
frequencies
high spatial
frequencies
numerical aperture
resolved
frequencies
object
object detail
decomposition
of Fourier
components
(sin waves)
image for
low NA
image for
high NA
object
sum
pppp
pp
vyvxi
pp
yxOTF
dydxyxg
dydxeyxg
vvH
ypxp
2
22
),(
),(
),(
),(ˆ),( yxIFvvH PSFyxOTF
pppp
pp
y
px
p
y
px
p
yxOTF
dydxyxP
dydxvf
yvf
xPvf
yvf
xP
vvH
2
*
),(
)2
,2
()2
,2
(
),(
Optical Transfer Function: Definition
Normalized optical transfer function
(OTF) in frequency space
Fourier transform of the PSF-
intensity
OTF: Autocorrelation of shifted pupil function, Duffieux-integral
Absolute value of OTF: modulation transfer function (MTF)
MTF is numerically identical to contrast of the image of a sine grating at the
corresponding spatial frequency
I Imax V
0.010 0.990 0.980
0.020 0.980 0.961
0.050 0.950 0.905
0.100 0.900 0.818
0.111 0.889 0.800
0.150 0.850 0.739
0.200 0.800 0.667
0.300 0.700 0.538
Contrast / Visibility
The MTF-value corresponds to the intensity contrast of an imaged sin grating
Visibility
The maximum value of the intensity
is not identical to the contrast value
since the minimal value is finite too
Concrete values:
minmax
minmax
II
IIV
I(x)
-2 -1.5 -1 -0.5 0 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Imax
Imin
object
image
peak
decreased
slope
decreased
minima
increased
Number of Supported Orders
A structure of the object is resolved, if the first diffraction order is propagated
through the optical imaging system
The fidelity of the image increases with the number of propagated diffracted orders
0. / +1. / -1. order
0. / +1. / -1.
+2. / -2.
order
0. / +1. -1. / +2. /
-2. / +3. / -3.
order
Optical Transfer Function of a Perfect System
Aberration free circular pupil:
Reference frequency
Maximum cut-off frequency:
Analytical representation
Separation of the complex OTF function into:
- absolute value: modulation transfer MTF
- phase value: phase transfer function PTF
'sinu
f
avo
'sin222 0max
un
f
navv
2
000 21
22arccos
2)(
v
v
v
v
v
vvHMTF
),(),(),( yxPTF vvHi
yxMTFyxOTF evvHvvH
/ max
00
1
0.5 1
0.5
gMTF
Due to the asymmetric geometry of the PSF for finite field sizes, the MTF depends on the
azimuthal orientation of the object structure
Generally, two MTF curves are considered for sagittal/tangential oriented object structures
Sagittal and Tangential MTF
y
tangential
plane
tangential sagittal
arbitrary
rotated
x sagittal
plane
tangential
sagittal
gMTF
tangential
ideal
sagittal
1
0
0.5
00.5 1
/ max
x p
y p
area of
integration
shifted pupil
areas
f x
y f
p
q
x
y
x
y
L
L
x
y
o
o
x'
y'
p
p
light
source
condenser
conjugate to object pupil
object
objective
pupil
direct
light
at object diffracted
light in 1st order
Interpretation of the Duffieux Iintegral
Interpretation of the Duffieux integral:
overlap area of 0th and 1st diffraction order,
interference between the two orders
The area of the overlap corresponds to the
information transfer of the structural details
Frequency limit of resolution:
areas completely separated
Test: Siemens Star
Determination of resolution and contrast
with Siemens star test chart:
Central segments b/w
Growing spatial frequency towards the
center
Gray ring zones: contrast zero
Calibrating spatial feature size by radial
diameter
Nested gray rings with finite contrast
in between:
contrast reversal, pseudo resolution
29
Contrast and Resolution
High frequent
structures :
contrast reduced
Low frequent structures:
resolution reduced
contrast
resolution
brillant
sharpblurred
milky
30
Optical Transfer Function of a Perfect System
Loss of contrast for higher spatial frequencies
contrast
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ideal
MTF
/max
/max
Various options:
1. FFT based calculation
2. representation as a function of
- field size
- defocus
3. Huygens PSF integral based
4. geometrical approximation via
spot calculation for not diffraction
limited systems
Different representation settings:
- maximun spatial frequency
- volume relief
- MTF / PTF
- changes over the field size
32
OTF in Zemax
Various MTF representations
33
OTF in Zemax