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