www.iap.uni-jena.de

Optical Design with Zemax

Lecture 9: Imaging

2013-01-08

Herbert Gross

Winter term 2012

2 9 Imaging

Time schedule

1 16.10. Introduction

Introduction, Zemax interface, menues, file handling, preferences, Editors, updates, windows,

Coordinate systems and notations, System description, Component reversal, system insertion,

scaling, 3D geometry, aperture, field, wavelength

2 23.10. Properties of optical systems I Diameters, stop and pupil, vignetting, Layouts, Materials, Glass catalogs, Raytrace, Ray fans

and sampling, Footprints

3 30.10. Properties of optical systems II Types of surfaces, Aspheres, Gratings and diffractive surfaces, Gradient media, Cardinal

elements, Lens properties, Imaging, magnification, paraxial approximation and modelling

4 06.11. Aberrations I Representation of geometrical aberrations, Spot diagram, Transverse aberration diagrams,

Aberration expansions, Primary aberrations,

5 13.+27.11. Aberrations II Wave aberrations, Zernike polynomials, Point spread function, Optical transfer function

6 04.12. Advanced handling

Telecentricity, infinity object distance and afocal image, Local/global coordinates, Add fold

mirror, Vignetting, Diameter types, Ray aiming, Material index fit, Universal plot, Slider,IO of

data, Multiconfiguration, Macro language, Lens catalogs

7 11.12. Optimization I Principles of nonlinear optimization, Optimization in optical design, Global optimization

methods, Solves and pickups, variables, Sensitivity of variables in optical systems

8 18.12. Optimization II Systematic methods and optimization process, Starting points, Optimization in Zemax

9 08.01 Imaging Fundamentals of Fourier optics, Physical optical image formation, Imaging in Zemax

10 15.01. Illumination Introduction in illumination, Simple photometry of optical systems, Non-sequential raytrace,

Illumination in Zemax

11 22.01. Correction I

Symmetry principle, Lens bending, Correcting spherical aberration, Coma, stop position,

Astigmatism, Field flattening, Chromatical correction, Retrofocus and telephoto setup, Design

method

12 29.01. Correction II Field lenses, Stop position influence, Aspheres and higher orders, Principles of glass

selection, Sensitivity of a system correction, Microscopic objective lens, Zoom system

13 05.02. Physical optical modelling Gaussian beams, POP propagation, polarization raytrace, coatings

1. Fundamentals of Fourier optics

2. Physical optical image formation

3. Imaging in Zemax

3 9 Imaging

Contents

diffracted ray

direction

object

structure

g = 1 /

/ g

k

kT

9 Imaging

Definitions of Fourier Optics

Phase space with spatial coordinate x and

1. angle

2. spatial frequency in mm-1

3. transverse wavenumber kx

Fourier spectrum

corresponds to a plane wave expansion

Diffraction at a grating with period g:

deviation angle of first diffraction order varies linear with = 1/g

0k

kv x

xx

),(ˆ),( yxEFvvA yx

A k k z E x y z e dxdyx y

i xk ykx y, , ( , , )

vk 2

vg

1

sin

4

Arbitrary object expaneded into a

spatial

frequency spectrum by Fourier

transform

Every frequency component is

considered separately

To resolve a spatial detail, at least

two orders must be supported by the

system

NAg

mg

sin

sin

off-axis

illumination

NAg

2

Ref: M. Kempe

9 Imaging

Grating Diffraction and Resolution

optical

system

object

diffracted orders

a)

resolved

b) not

resolved

+1.

+1.

+2.

+2.

0.

-2.

-1.

0.

-2.

-1.

incident

light

+1.

0.

+2.

+1.

0.

+2.

-2.

-2. -1.

-1.

5

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. order

0. / 1. / 2.

order

0. / 1. / 2. / 3.

order

9 Imaging

Resolution: Number of Supported Orders

6

Improved resolution by oblique illumination in microscopy

Enhancement only in one direction

centered illumination:

supported orders in y : 0 , +1 , -1

oblique illumination :

supported orders in y : 0 , +1 , + 2

0th order y

1st order in y

2nd order in y

- 1st order in y

no change

in x direction

9 Imaging

Oblique Illumination in Microscopy

7

9 Imaging

Fourier Optical Fundamentals

Helmholtz wave equation:

Propagation with Green‘s function g,

Amplitude transfer function, impulse response

For shift-invariance:

convolution

Green‘s function of a spherical wave

Fresnel approximation

Calculation in frequency space: product

Optical systems:

Impulse response g(x,y) is coherent transfer function, point spread function (PSF).

G(x,y) corresponds to the complex pupil function

Fourier transform: corresponds to a plane wave expansion

),(*),','(),','( yxEzyyxxgzyxE

g r rr r

ei k r r

( , ' )'

'

1

4

g x y zi

ze eikz

i

zx y

( , , )( )

2 2

),(),,(),,( yxyxyx vvEzvvGzvE

)( 22

),,( yx vvziikz

yx ezvvG

dydxyxEyxyxgzyxE

),(),,','(),','(

8

9 Imaging

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

9

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

(

),(

9 Imaging

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

10

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

9 Imaging

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

11

9 Imaging

Optical Transfer Function of a Perfect System

Aberration free circular pupil:

Reference frequency

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 0

un

f

navvG

2

000 21

22arccos

2)(

v

v

v

v

v

vvHMTF

),(),(),( yxPTF vvHi

yxMTFyxOTF evvHvvH

/ max

00

1

0.5 1

0.5

gMTF

12

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

9 Imaging

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

13

9 Imaging

Fourier Optics – Point Spread Function

Optical system with magnification m

Pupil function P,

Pupil coordinates xp,yp

PSF is Fourier transform

of the pupil function

(scaled coordinates)

pp

myyymxxxz

ik

pppsf dydxeyxPNyxyxgpp ''

,)',',,(

pppsf yxPFNyxg ,ˆ),(

object

planeimage

plane

source

point

point

image

distribution

14

9 Imaging

Fourier Theory of Coherent Image Formation

Transfer of an extended

object distribution I(x,y)

In the case of shift invariance

(isoplanasie):

coherent convolution of fields

Complex fields are

additive

object

plane image

plane

object

amplitude

distribution

single point

image

image

amplitude

distribution

dydxyxEyxyxgyxE psf ),(',',,)','(

dydxyxEyyxxgyxE psf ),(',')','(

),(,)','( yxEyxgyxE psf

15

9 Imaging

Fourier Theory of Coherent Image Formation

object

amplitude

U(x,y)

PSF

amplitude-

response

Hpsf (xp,yp)

image

amplitude

U'(x',y')

convolution

result

object

amplitude

spectrum

u(vx,vy)

coherent

transfer

function

hCTF (vx,vy)

image

amplitude

spectrum

u'(v'x,v'y)

product

result

Fourier

transform

Fourier

transform

Fourier

transform

16

9 Imaging

Fourier Theory of Incoherent Image Formation

objectintensity image

intensity

single

psf

object

planeimage

plane

Transfer of an extended

object distribution I(x,y)

In the case of shift invariance

(isoplanasie):

incoherent convolution

Intensities are additive

dydxyxIyyxxgyxI psfinc

),()','()','(2

),(*),()','( yxIyxIyxI objpsfimage

dydxyxIyyxxgyxI psfinc

),(),',,'()','(2

17

9 Imaging

Fourier Theory of Incoherent Image Formation

object

intensity

I(x,y)

squared PSF,

intensity-

response

Ipsf

(xp,y

p)

image

intensity

I'(x',y')

convolution

result

object

intensity

spectrum

I(vx,v

y)

optical

transfer

function

HOTF

(vx,v

y)

image

intensity

spectrum

I'(vx',v

y')

produkt

result

Fourier

transform

Fourier

transform

Fourier

transform

18

9 Imaging

Incoherent Image Formation in Frequency Space

Incoherent illumination:

No correlation between neighbouring object points

Superposition of intensity in the image

In the case of shift invariance

(isoplanasie):

Incoherent imaging with convolution

In frequency space:

Product of spectra, linear transfer theory

The spectrum of the psf works as low pass filter onto the object spectrum

Optical transfer function

dydxyxIyyxxgyxI psfinc

),()','()','(2

),(*),()','( yxIyxIyxI objpsfimage

dydxyxIyyxxgyxI psfinc

),(),',,'()','(2

),(),( yxIFTvvH PSFyxotf

),(),(),( yxobjyxotfyximage vvIvvHvvI

19

Circular disc with diameter

D = d x Dairy

Small d << 1 : Airy disc

Increasing d:

Diffraction ripple disappear

0 0,5 1 1,5 210 -4

10 -3

10 -2

10 -1

10 0

d = 0.1

d = 0.2

d = 0.5

d = 1.0

d = 1.5

d = 2.0

Log I(r)

rsinu /

d = 0.1 d = 0.2 d = 0.5 d = 1.0 d = 1.5 d = 2.0

9 Imaging

Incoherent Image of a Circular Disc

20

9 Imaging

Comparison Coherent – Incoherent Images

object

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

incoherent coherent

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

bars resolved bars not resolved bars resolved bars not resolved

21

Possible options in Zemax:

Convolution of image with Psf

1. geometrical

2. with diffraction

Geometrical raytrace analysis

1. simple geometrical shapes (IMA-files)

2. bitmaps

Diffraction imaging:

1. partial coherent

2. extended with variable PSF

Structure of options in Zemax not clear

Redundance

Field definition and size scaling not good

Numerical conditions and algorithms partially not stable

22 9 Imaging

Imaging in Zemax

Field height: location of object in the specific coordinates of the system

- zero padding included (not: size = diameter)

- image size shon is product of pixel number x pixel size

- can be full field or centre of local extracted part of the field

PSF-X/Y points: number of field points to incorporate the changes of the PSF,

interpolation between this

coarse grid

Object: bitmap

PSF: geometrical or diffraction

23 9 Imaging

General Image Simulation

Total field size: defined by system

Field height/size: reduced field corresponding to the structure as considered in the

imaging calculation

Field position: reference point of the considered reduced field (center) in the total field

Image size: size of the represented field size, should be a little larger as field size

to clearly see the boundary

In some tools calculated as product of pixel number and pixel size

24 9 Imaging

General Image Simulation

total field of

the system

field height/

field size

image size /

pixel size times

pixel number

field

position /

field index

x

y

Geometrical imaging by raytrace

Binary IMA-files with geometrical shapes

Choice of:

- field size

- image size,

- wavelengths

- number of rays

Interpolation possible

25 9 Imaging

Geometric Imaging I

Geometrical imaging by raytrace

of bitmaps

Extension of 1st option:

can be calculated at any surface

If full field is used, this

corresponds to a footprint with

all rays

Example: light distribution

in pupil, at last surface, in image

26 9 Imaging

Geometric Imaging II

Different types of partial coherent model algorithms possible

Only IMA-Files can be used as objects

describes the coherence factor (relative pupil filling)

Control and algorithms not clear, not stable

27 9 Imaging

Patial Coherent Imaging

Classical convolution of psf

with pixels of IMA-File

Coherent and incoherent

model possible

PSF may vary over field

position

28 9 Imaging

Extended Diffraction