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www.iap.uni-jena.de
Optical Engineering
Part 9: Point spread function
Herbert Gross
Summer term 2020
Huygens principle
Ideal point spread function
PSF for defocus
Point spread function with aberrations
Miscellaneous
2
Contents
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
object
point
spherical
wave
truncated
spherical
wave
image
plane
x = 1.22 / NA
point spread function
object plane
PSF by Huygens Principle
Huygens wavelets correspond to vectorial field components:
- represented by a small arrow
- the phase is represented by the direction
- the amplitude is represented by the length
Zeros in the diffraction pattern: destructive interference
Ideal point spread function:
pupil
stop
wave
front
point
spread
function
zero intensity
closed loop
side lobe peak
1 ½ round trips
central peak maximum
constructive interference
single wavelets
sum
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,)','(
rdrErr
erE
rrik
2
'
)('
)'(
0
2
12,0 Iv
vJvI
0
2
4/
4/sin0, I
u
uuI
Circular homogeneous illuminated aperture:
Transverse intensity:
Airy distribution
Dimension: DAirynormalized lateral
coordinate:
v = 2 x / NA
Axial intensity:
sinc-function
Dimension: Rayleigh unit Runormalized axial coordinate
u = 2 z n / NA2
Perfect Point Spread Function
NADAiry
22.1
2NA
nRu
r
z
Airy
lateral
aperture
cone
Rayleigh
axial
image plane
optical
axis
-25 -20 -15 -10 -5 0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
axial
lateral
u / v
Dairy
4Ru
I/I0
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%
log I(r)
r0 5 10 15 20 25 30
10
10
10
10
10
10
10
-6
-5
-4
-3
-2
-1
0
central
peak
1st diffraction
ring 2nd
diffraction
ring
Airy
radius
0.017
0.003
Abbe Resolution and Assumptions
Assumption Resolution enhancement
1 Circular pupil ring pupil, dipol, quadrupole
2 Perfect correction complex pupil masks
3 homogeneous illumination dipol, quadrupole
4 Illumination incoherent partial coherent illumination
5 no polarization special radiale polarization
6 Scalar approximation
7 stationary in time scanning, moving gratings
8 quasi monochromatic
9 circular symmetry oblique illumination
10 far field conditions near field conditions
11 linear emission/excitation non linear methods
Abbe resolution with scaling to /NA:
Assumptions for this estimation and possible changes
A resolution beyond the Abbe limit is only possible with violating of certain
assumptions
Defocussed Perfect Psf
Perfect point spread function with defocus
Representation with constant energy: extreme large dynamic changes
Fully symmetric around image plane
z = -2Ru z = +2Ruz = -1Ru z = +1Ru
normalized
intensity
constant
energy
focus
Imax = 5.1% Imax = 42%Imax = 9.8%
Normalized axial intensity
for uniform pupil amplitude
Decrease of intensity onto 80%:
Scaling measure: Rayleigh length
- geometrical optical definition
depth of focus: 1RE
- Gaussian beams: similar formula
22
'
'sin' NA
n
unRu
Depth of Focus: Diffraction Consideration
2
0
sin)(
u
uIuI
2' ou
nR
udiff Run
z
2
1
sin493.0
2
12
focalplane
beam
caustic
z
depth of focus
0.8
1
I(z)
z-Ru/2 0
r
intensity
at r = 0
+Ru/2
PSF by Huygens Principle
Apodization:
variable lengths
of arrows
Aberrations:
variable orientation
of arrows
pupil
stop
wave
front
point
spread
function
apodization:
decreasing length of arrows
homogeneous pupil:
same length of all arrows
rp
I(xp)
pupil
stop
ideal
wave
front
point
spread
function
ideal spherical wavefront
central peak maximum
real
wave
front
real wavefront
with aberrations
central peak reduced
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
12
Growing spherical aberration shows an asymmetric behavior around the nominal image
plane for defocussing
13
Caustic with Spherical aberration
c9 = 0 c9 = 0.7c9 = 0.3 c9 = 1
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