www.iap.uni-jena.de
Optical Design with Zemaxfor PhD - Basics
Lecture 5: Aberrations II
2014-05-22
Herbert Gross
Summer term 2014
2
Preliminary Schedule
No Date Subject Detailed content
1 16.04. IntroductionZemax interface, menus, file handling, system description, editors, preferences, updates, system reports, coordinate systems, aperture, field, wavelength, layouts, diameters, stop and pupil, solves
2 24.04. Basic Zemax handling Raytrace, ray fans, paraxial optics, surface types, quick focus, catalogs, vignetting, footprints, system insertion, scaling, component reversal
3 08.05. Properties of optical systems aspheres, gradient media, gratings and diffractive surfaces, special types of surfaces, telecentricity, ray aiming, afocal systems
4 15.05. Aberrations I representations, spot, Seidel, transverse aberration curves, Zernike wave aberrations
5 22.05. Aberrations II PSF, MTF, ESF
6 05.06. Optimization I algorithms, merit function, variables, pick up’s
7 12.06. Optimization II methodology, correction process, special requirements, examples
8 19.06. Advanced handling slider, universal plot, I/O of data, material index fit, multi configuration, macro language
9 25.06. Imaging Fourier imaging, geometrical images
10 03.07. Correction I simple and medium examples
11 10.07. Correction II advanced examples
12 xx.07. Illumination simple illumination calculations, non-sequential option
13 xx.07. Physical optical modelling I Gaussian beams, POP propagation
14 xx.07. Physical optical modelling II polarization raytrace, polarization transmission, polarization aberrations, coatings, representations, transmission and phase effects
15 xx.07. Tolerancing Sensitivities, Tolerancing, Adjustment
Diffraction at the System Aperture
Self luminous points: emission of spherical waves Optical system: only a limited solid angle is propagated, the truncaton 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 spreaded The optical systems works as a low pass filter
objectpoint
sphericalwave
truncatedspherical
wave
imageplane
x = 1.22 / NA
pointspread
function
object plane
Fraunhofer Point Spread Function
Rayleigh-Sommerfeld diffraction integral, Mathematical formulation of the Huygens-principle
Fraunhofer approximation in the far fieldfor large Fresnel number
Optical systems: numerical aperture NA in image spacePupil 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
12
z
rN p
F
),(2),(),( pp yxWipppp eyxTyxA
pp
yyxxR
iyxiW
ppAP
dydxeeyxTyxEpp
APpp''2
,2,)','(
''cos'
)'()('
dydxrr
erEirE 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
0
212,0 Iv
vJvI
0
2
4/4/sin0, I
uuuI
-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 illuminatedAperture: intensity distribution transversal: Airy
scale:
axial: sincscale
Resolution transversal betterthan 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
2NAnRE
log I(r)
r0 5 10 15 20 25 3010
10
10
10
10
10
10
-6
-5
-4
-3
-2
-1
0Airy 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 intensityCorresponds 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
0 4/4/sinsin)(
uuI
zzIzI o
42
2 uzNAz
22
''sin' NA
nun
RE
Perfect Axial Point Spread Function
Defocussed Perfect Psf
Perfect point spread function with defocus Representation with constant energy: extreme large dynamic changes
z = -2RE z = +2REz = -1RE z = +1RE
normalizedintensity
constantenergy
focus
Imax = 5.1% Imax = 42%Imax = 9.8%
Comparison Geometrical Spot – Wave-Optical Psf
aberrations
spotdiameter
DAiry
exactwave-optic
geometric-opticapproximated
diffraction limited,failure of the
geometrical model
Fourier transformill conditioned
Large aberrations:Waveoptical calculation shows bad conditioning
Wave aberrations small: diffraction limited, geometrical spot too small and wrong
Approximation for the intermediate range:
22GeoAirySpot DDD
0,0
0,0)(
)(
idealPSF
realPSF
S IID
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...1DS = 1 is perfect
Critical in use: the completeinformation is reduced to only one number
The criterion is useful for 'good'systems with values Ds > 0.5
Strehl Ratio
r
1
peak reducedStrehl ratio
distributionbroadened
ideal , withoutaberrations
real withaberrations
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 = 0.0c = 0.1
c = 0.2c = 0.3
c = 0.4c = 0.5
c = 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
AiryBesselGauss
FWHM
Apodisation of the pupil:1. Homogeneous2. Gaussian3. 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
Diffraction at an edge in Fresnelapproximation
Intensity distribution,Fresnel integrals C(x) and S(x)
scaled argument
Intensity:- at the geometrical shadow edge: 0.25- shadow region: smooth profile- bright region: oscillations
22
)(21)(
21
21)( tStCtI
FNxz
xzkt 22
t
-4 -2 0 2 4 60
0.5
1
1.5
I(t)
Fresnel Edge Diffraction
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 = 0.0W20 = 0.1W20 = 0.2W20 = 0.3W20 = 0.4W20 = 0.5W20 = 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 = 0.0W40 = 0.1W40 = 0.2W40 = 0.3W40 = 0.4W40 = 0.5W40 = 0.7
Incoherent Edge Spread Function
Line image: integral over point sptread functionLSF: line spread function
Realization: narrow slitconvolution of slit width
But with deconvolution, the PSF can be reconstructed
dyyxIxI PSFLSF ),()(
Integration
intensity
x
Line spread function
PSF
dyyxIxI PSFLSF ),()(
Linienbild
Line image:Fourier transform of pupil in one dimension
Line spreadfunction with aberrationsHere: defocussing
pppp
pp
xxRi
pp
iLSFdydxyxP
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 = 0.0W20 = 0.1W20 = 0.2W20 = 0.3W20 = 0.4W20 = 0.5W20 = 0.7
Line Spread Function
Sampling of the Diffraction Integral
x-6 -4 -2 0 2 4
0
10
20
30
40
50quadratic
phase
wrappedphase
2
smallest samplingintervall
phase
Oscillating exponent :Fourier transform reduces on 2-period Most critical sampling usually
at boundary defines numberof 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 wavesHuygens principle spectral representation
rdrErr
erErrik
2'
)('
)'(
)(ˆˆ)'( 1 rEFeFrE xy
zikxy
z
pppp
ppvyvxi
pp
yxOTF
dydxyxg
dydxeyxgvvH
ypxp
2
22
),(
),(),(
),(ˆ),( yxIFvvH PSFyxOTF
pppp
ppy
px
py
px
p
yxOTF
dydxyxP
dydxvf
yvfxPvf
yvfxPvvH
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 thecorresponding 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 intensityis not identical to the contrast valuesince the minimal value is finite too
Concrete values:
minmax
minmax
IIIIV
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
peakdecreased
slopedecreased
minimaincreased
Number of Supported Orders
A structure of the object is resolved, if the first diffraction order is propagatedthrough the optical imaging system
The fidelity of the image increases with the number of propagated diffracted orders
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
favo
'sin222 0max
unf
navv
2
000 21
22arccos2)(
vv
vv
vvvHMTF
),(),(),( yxPTF vvHiyxMTFyxOTF evvHvvH
Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on theazimuthal orientation of the object structure
Generally, two MTF curves are considered for sagittal/tangential oriented object structures
Sagittal and Tangential MTF
y
tangentialplane
tangential sagittal
arbitraryrotated
x sagittalplane
tangential
sagittal
gMTF
tangential
ideal
sagittal
1
0
0.5
0 0.5 1 / max
x
y
x
y
L
L
x
y
o
o
x'
y'
p
p
light source
condenserconjugate to object pupil
object
objectivepupil
directlight
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 theinformation 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
30
Contrast and Resolution
High frequent structures :contrast reduced
Low frequent structures:resolution reduced
contrast
resolution
brillant
sharpblurred
milky
31
Various options:1. FFT based calculation2. representation as a function of
- field size- defocus
3. Huygens PSF integral based4. geometrical approximation via
spot calculation for not diffractionlimited systems
Different representation settings:- maximun spatial frequency- volume relief- MTF / PTF- changes over the field size
33
OTF in Zemax