Download ODZ_PhD_Optical design with Zemax PhD Basics 5 Aberrations II

Optical Design with Zemax
for PhD - Basics
Lecture 5: Aberrations II
2014-05-22
Herbert Gross
Summer term 2014
www.iap.uni-jena.de
2
Preliminary Schedule
No
1
2
3
Date
Subject
Detailed content
Zemax interface, menus, file handling, system description, editors, preferences,
updates, system reports, coordinate systems, aperture, field, wavelength, layouts,
diameters, stop and pupil, solves
Raytrace, ray fans, paraxial optics, surface types, quick focus, catalogs, vignetting,
24.04. Basic Zemax handling
footprints, system insertion, scaling, component reversal
aspheres, gradient media, gratings and diffractive surfaces, special types of
08.05. Properties of optical systems
surfaces, telecentricity, ray aiming, afocal systems
16.04. Introduction
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
3
Contents
1. Point spread function
2. Edge and line spread function
3. Optical transfer function
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
spherical
wave
image
plane
truncated
spherical
wave
object
point
point
spread
function
object plane
x = 1.22  / NA
Fraunhofer Point Spread Function
 Rayleigh-Sommerfeld diffraction integral,
Mathematical formulation of the Huygens-principle
 Fraunhofer approximation in the far field
for large Fresnel number
 
ik r  r '

i
EI (r )  
 e
E
(
r
' )    cos d dx' dy'
 
r  r'
NF 
 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
E ( x' , y ' )  T x p , y p   e
rp2
 z
1

2 iW x p , y p
AP
 Point spread function (PSF): Fourier transform of the complex pupil
function
A( x p , y p )  T ( x p , y p )  e
2 iW ( x p , y p )

e
2 i
x x' y p y'
 RAP p


dx p dy p
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
Perfect Point Spread Function
Circular homogeneous illuminated
Aperture: intensity distribution
 transversal: Airy
scale:
1.22  
 axial: sinc
scale
vertical
lateral
0,8
NA
n
RE 
NA2
 Resolution transversal better
than axial: x < z
intensity
DAiry 
1,0
0,6
0,4
0,2
0,0
-25
-20
-15
-10
-5
0
5
10
15
20
25
u/v
 2 J v 
I 0,v    1  I0
 v 
2
 sin u / 4
I u,0  
I0

u
/
4


Scaled coordinates according to Wolf :
axial :
u = 2  z n /  NA2
transversal : v = 2  x /  NA
Ref: M. Kempe
2
Perfect Lateral Point Spread Function: Airy
Airy distribution:




Gray scale picture
Zeros non-equidistant
Logarithmic scale
Encircled energy
log I(r)
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
0
5
10
15
20
Ecirc(r)
1
0.9
3. ring 1.48%
0.8
2. ring 2.79%
0.7
1. ring 7.26%
0.6
peak 83.8%
DAiry
0.5
0.4
0.3
0.2
0.1
0
2
0
1
1.831
3
2.655
4
3.477
5
r / rAiry
25
30
r
Perfect Axial Point Spread Function
 Axial distribution of intensity
Corresponds to defocus
 Normalized axial coordinate
u
 NA2
z
z
2
4
 Scale for depth of focus :
Rayleigh length
RE 

n' sin 2 u '

n' 
NA2
 Zero crossing points:
equidistant and symmetric,
Distance zeros around image plane 4RE
 sin z  
 sin u / 4 
I ( z)  I 0  
I




o 
u
z
/
4




2
2
Defocussed Perfect Psf
 Perfect point spread function with defocus
 Representation with constant energy: extreme large dynamic changes
z = -2RE
z = -1RE
focus
normalized
intensity
Imax = 5.1% Imax = 9.8%
Imax = 42%
constant
energy
z = +1RE
z = +2RE
Comparison Geometrical Spot – Wave-Optical Psf
 Large aberrations:
Waveoptical calculation shows bad conditioning
 Wave aberrations small: diffraction limited,
geometrical spot too small and
wrong
spot
diameter
 Approximation for the
intermediate range:
2
2
DSpot  DAiry
 DGeo
exact
wave-optic
DAiry
geometric-optic
approximated
aberrations
diffraction limited,
failure of the
geometrical model
Fourier transform
ill conditioned
12
Strehl Ratio
 Important citerion for diffraction limited systems:
Strehl ratio (Strehl definition)
Ratio of real peak intensity (with aberrations) referenced on ideal peak intensity
( real )
PSF
( ideal )
PSF
I
DS 
I
0,0
0,0
A( x, y )e
dxdy

D
 A( x, y)dxdy
2 iW ( x , y )
S
2
2
 DS takes values between 0...1
DS = 1 is perfect
 Critical in use: the complete
information is reduced to only one
number
I( x )
1
peak reduced
Strehl ratio
ideal , without
aberrations
 The criterion is useful for 'good'
systems with values Ds > 0.5
distribution
broadened
real with
aberrations
r
13
Psf with Aberrations
 Psf for some low oder Zernike coefficients
 The coefficients are changed between cj = 0...0.7 
 The peak intensities are renormalized
coma
5. order
astigmatism
5. order
spherical
5. order
trefoil
c =0.0
coma
c =0.1
astigmatism
c =0.2
c =0.3
spherical
c =0.4
c =0.5
c =0.7
defocus
Point Spread Function with Apodization
 Apodisation of the pupil:
1. Homogeneous
2. Gaussian
3. Bessel
 Psf in focus:
different convergence to zero for
larger radii
I(w)
1
Airy
Bessel
Gauss
0.8
0.6
FWHM
0.4
0.2
 Encircled energy:
same behavior
 Complicated:
Definition of compactness of the
central peak:
1. FWHM:
Airy more compact as Gauss
Bessel more compact as Airy
2. Energy 95%:
Gauss more compact as Airy
Bessel extremly worse
0
-2
-1
0
1
2
3
w
PSF in Zemax
 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
Fresnel Edge Diffraction
 Diffraction at an edge in Fresnel
approximation
I(t)
1.5
 Intensity distribution,
Fresnel integrals C(x) and S(x)
1
2
2
1  1
 
 1
I (t )     C (t )    S (t )  
2  2
 2
 
scaled argument
t
k
2
x
 x  2N F
z
z 
 Intensity:
- at the geometrical shadow edge: 0.25
- shadow region: smooth profile
- bright region: oscillations
0.5
0
-4
-2
0
2
4
6
t
Incoherent Edge Spread Function
 ESF with defocussing
ESF with spherical aberration
IESF(x)
IESF(x)
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
W20 = 0.0
0.5
0.5
W20 = 0.1
W20 = 0.2
W20 = 0.3
W20 = 0.4
W20 = 0.5
0.4
0.3
0.2
0.4
0.3
0.2
W20 = 0.7
0.1
0.1
0
-10
-5
0
5
W40 = 0.0
W40 = 0.1
W40 = 0.2
W40 = 0.3
W40 = 0.4
W40 = 0.5
W40 = 0.7
10
x
x
0
-8
-6
-4
-2
0
2
4
6
8
Linienbild
 Line image: integral over point sptread function
LSF: line spread function
I LSF ( x)   I PSF ( x, y )dy
 Realization: narrow slit
convolution of slit width
 But with deconvolution, the PSF can be reconstructed
PSF
intens
ity
Integration
I LSF ( x) I PSF ( x, y )dy
x
Line spread function
Line Spread Function
 Line image:
Fourier transform of pupil in one dimension
I LSF ( xi ) 
  Px
p
, y p  e
  Px

2 i
x x
R i p
2
dx p dy p

, y p  dx p dy p
2
p
ILSF(x)
1
 Line spreadfunction with aberrations
Here: defocussing
0.9
W20 = 0.0
W20 = 0.1
W20 = 0.2
W20 = 0.3
W20 = 0.4
W20 = 0.5
W20 = 0.7
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-10
-5
0
5
10
x
Sampling of the Diffraction Integral
 Oscillating exponent :
Fourier transform reduces on 2period
phase
50
 Most critical sampling usually
at boundary defines number
quadratic
phase
40
of sampling points
 Steep phase gradients define the
sampling
30
 High order aberrations are a
problem
20
wrapped
phase
10
2
0
-6
-4
-2
0
2
4
x
smallest sampling
intervall
Propagation by Plane / Spherical Waves
 Expansion field in simple-to-propagate waves
 1. Spherical waves
Huygens principle
 
ik r  r '



e
E (r ' )      E (r ) d 2 r
r  r'
2. Plane waves
spectral representation




E (r ' )  Fˆxy1 eik z z  Fˆxy E (r )
Resolution of Fourier Components
Ref: D.Aronstein / J. Bentley
Optical Transfer Function: Definition
 Normalized optical transfer function
(OTF) in frequency space
 
H OTF ( v x , v y ) 

2
g( xp , y p )  e
  
 


 2 i x p v x  y p v y

dx p dy p
2
g ( x p , y p ) dx p dy p
  
 Fourier transform of the Psfintensity
H OTF ( v x , v y )Fˆ I PSF ( x, y )
 OTF: Autocorrelation of shifted pupil function, Duffieux-integral
 
H OTF ( v x , v y ) 

 P( x p 
  
 f vx
2
, yp 
 f vy
2
)  P* ( x p 
 

 f vx
2
, yp 
2
P( x p , y p ) dx p dy p
  
 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
 f vy
2
) dx p dy p
Contrast / Visibility
 The MTF-value corresponds to the intensity contrast of an imaged sin grating
 Visibility
V
 The maximum value of the intensity
is not identical to the contrast value
since the minimal value is finite too
I max  I min
I max  I min
I(x)
peak
decreased
slope
decreased
1
object
 Concrete values:
Imax
0.9
0.8
I
0.010
0.020
0.050
0.100
0.111
0.150
0.200
0.300
Imax
0.990
0.980
0.950
0.900
0.889
0.850
0.800
0.700
V
0.980
0.961
0.905
0.818
0.800
0.739
0.667
0.538
0.7
0.6
0.5
0.4
0.3
0.2
image
Imin
0.1
0
-2
-1.5
-1
-0.5
0
minima
increased
1
1.5
2
x
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
Optical Transfer Function of a Perfect System
 Aberration free circular pupil:
Reference frequency
vo 
a sin u'

f

 Maximum cut-off frequency:
vmax  2v0 
2na 2n sin u '

f

 Analytical representation
2

 v   v 
 v  
2
  
 1  

H MTF (v )  arccos

2
v
2
v
2
v
0 
0 
0  





 Separation of the complex OTF function into:
- absolute value: modulation transfer MTF
- phase value: phase transfer function PTF
H OTF (vx , v y )  H MTF (vx , v y ) e
i H PTF ( v x ,v y )
Sagittal and Tangential MTF
 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
gMTF
1
tangential
plane
y
ideal
0.5
sagittal
tangential
tangential
sagittal
 /  max
0

0
0.5
arbitrary
rotated
tangential
x
sagittal
sagittal
plane
1
Interpretation of the Duffieux Iintegral
y'
p
 Interpretation of the Duffieux integral:
overlap area of 0th and 1st diffraction order,
interference between the two orders
objective
pupil
 The area of the overlap corresponds to the
information transfer of the structural details
x'p
y
o
 Frequency limit of resolution:
areas completely separated
at object diffracted
light in 1st order
xo
y
L
object
xL
y
condenser
conjugate to object pupil
x
light
source
direct
light
30
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
31
Contrast and Resolution
 High frequent
contrast
brillant
structures :
contrast reduced
 Low frequent structures:
resolution reduced
sharp
blurred
milky
resolution
Optical Transfer Function of a Perfect System
 Loss of contrast for higher spatial frequencies
OTF in Zemax
 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
33
OTF in Zemax
 Various MTF representations
34