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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