www.iap.uni-jena.de

Optical Design with Zemax

for PhD

Lecture 9: Imaging

2016-02-03

Herbert Gross

Winter term 2015

2

Preliminary Schedule

No Date Subject Detailed content

1 11.11. Introduction

Zemax interface, menus, file handling, system description, editors, preferences,

updates, system reports, coordinate systems, aperture, field, wavelength, layouts,

raytrace, diameters, stop and pupil, solves, ray fans, paraxial optics

2 02.12. Basic Zemax handling surface types, quick focus, catalogs, vignetting, footprints, system insertion, scaling,

component reversal

3 09.12. Properties of optical systems aspheres, gradient media, gratings and diffractive surfaces, special types of

surfaces, telecentricity, ray aiming, afocal systems

4 16.12. Aberrations I representations, spot, Seidel, transverse aberration curves, Zernike wave

aberrations

5 06.01. Aberrations II PSF, MTF, ESF

6 13.01. Optimization I algorithms, merit function, variables, pick up’s

7 20.01. Optimization II methodology, correction process, special requirements, examples

8 27.01. Advanced handling slider, universal plot, I/O of data, material index fit, multi configuration, macro

language

9 03.02. Imaging Fourier imaging, geometrical images

10 10.02. Correction I simple and medium examples

11 17.02. Correction II advanced examples

12 24.02. Illumination simple illumination calculations, non-sequential option

13 02.03. Physical optical modelling Gaussian beams, POP propagation

14 07.03. Tolerancing Sensitivities, Tolerancing, Adjustment

1. Fourier imaging

2. Coherence

3. Phase imaging

4. Imaging in Zemax

3

Contents

diffracted ray

direction

object

structure

g = 1 /

/ g

k

kT

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

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

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

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.

+3.+3.

6

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

Abbe Theorie of the Microscopic Resolution

Diffraction of the illumination wave at the object structure

Occurence of the diffraction orders in the pupil

Image generation by constructive interference of the supported orders

Object details with high spatial frequency are blocked by the system aperture and can not

be resolved

-1

1

object planesource

pupil plane

image plane

0

imaginglens

Ref: W. Singer

Imaging of a crossed grating object

Spatial frequency filtering by a slit:

Case 1:

- pupil open

- Cross grating imaged

Case 2:

- truncation of the pupil by a split

- only one direction of the grating is

resolved

Fourier Filtering

pupil complete open

pupil truncated by slit

image

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

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

(isoplanasy):

incoherent convolution

Intensities are additive

dydxyxIyyxxgyxI psfinc

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

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

dydxyxIyyxxgyxI psfinc

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

Fourier Theory of Incoherent Image Formation

object

intensity

I(x,y)

squared PSF,

intensity-

response

Ipsf (x',y')

image

intensity

I'(x',y')

convolution

result

object

intensity

spectrum

I(vx,vy)

optical

transfer

function

HOTF (vx,vy)

image

intensity

spectrum

I'(vx,vy)

product

result

Fourier

transform

Fourier

transform

Fourier

transform

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

Comparison Coherent – Incoherent Image Formation

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

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

Partial Coherent Imaging

Complete description of an optical system:

1. Light source

2. Illumination system, amplitude response hill

3. Transmission object

4. Observation / imaging system with amplitude response hobs

sourceillumination

system

observation

system

pupil Psensor

object

plane

image

planeillumination

field

xs , ys

Is hill Iihobs

Io

xp , yp xi , yi

obs

ill

u

u

sin

sin

Coherence Parameter

Finite size of source : aperture cone with angle uill

Observation system: aperture angle uobs

Definition of coherence parameter :

Ratio of numerical apertures

Limiting cases:

coherent = 0

uill << uobs

incoherent

= 1

uill >> uobs

object imagesource

xi , yixo , yo

lens

uobs

uill

illumination observation

Fourier optical steps and effect of pupil mask

Example images for system with different aberrations

Phase Contrast Simulation

original phase DIC image perfect system DIC image defocussed system DIC image aberrated system

phase object pupil spectrum without pupil mask with pupil mask

mask

)()()( xiexBxP

)(1)( xixE

)'()'(' xiiaxE

)'()'(' xaxI

condenserimage

Bertrand

lens

objective

lens

ring shaped

stopobject

phase

plate

Zernike Phase Contrast

Pure phase object is not visible

Pupil modified: 0th order damped (a)

and phase rotated (90°)

Approximation small phase:

Intensity proportional to phase angle

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

20

Imaging in Zemax

Field size definitions

Total field size in data (angle or length)

Selected field index

Relative size of structure in the total field

Shown part of the field

Not completly consistent in the different imaging tools

21

Imaging in Zemax

plotted image size

/ pixel x Npix

selected field (index)

field heigth

relative field size

of object structure

y

real maximum size

of the field

x

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

- zero padding included (not: size = diameter)

- image size shown 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

22

General Image Simulation

Geometrical imaging by raytrace

Binary IMA-files with geometrical shapes

Choice of:

- field size

- image size,

- wavelengths

- number of rays

Interpolation possible

23

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

24

Geometric Imaging II

Different types of partial coherent model algorithms possible

Only IMA-Files can be used as objects

a describes the coherence factor (relative pupil filling)

Control and algorithms not clear, not stable

25

Partial Coherent Imaging

Classical convolution of psf

with pixels of IMA-File

Coherent and incoherent

model possible

PSF may vary over field

position

26

Extended Diffraction

Geometric Raytrace

27

object image

7x7 pixel IMA file 10 000 raytraces

From random position inside object pixel

To random position in entrance pupil

Spot diagram

Ref.: M. Eßlinger

28

10 rays per pixel 1 ray per pixel

100 rays per pixel 1000 rays per pixel

Geometric Raytrace

Ref.: M. Eßlinger

Imaging Objects

29

Ref.: M. Eßlinger

IMA (Image file)

- Illumination brightness in each point of the object

- Zemax provides basic shapes like the letter „F“

- ASCII format with 10 different grey values or binary with 256 grey values

BIM (binary image)

- like IMA, but 64bit (double precision) float values

ZBF (Zemax beam file)

- for sophisticated illumination optics

- many features only available in Premium Version of Zemax

BMP (bmp, jpg or png)

- 3 x 8 bit RGB values ( raytrace with FdC: 656 nm, 587 nm and 486 nm)

- for greyscale detector: raytrace with FdC, averaging on detector plane

Imaging: Summary

30

Ref.: M. Eßlinger

Advantages and Disadvantages of Geometric Raytracing

+ Easy to understand

+ Field dependent errors are considered automatically

- Does not include Diffraction Limit

- Requires large number of rays (slow)

- Coherent imaging is difficult (not possible with Zemax)

Object File type

conv. raytrace diffraction

spatial variant aberr.

pupil aberr.

Coherence

BMP IMA BIM ZBF coh. part. coh. incoh.

Image Simulation X X X

Geometric X X X X X

Geometric Bitmap X X X X X X

Partially Coherent X X X

Extended Diffraction X X X X

Imaging: Summary

31

Ref.: M. Eßlinger

Advantages and Disadvantages of Convolution based algorithms

+ Easy to implement Diffraction Limit

+ Fast Fourier Transform based (fast)

+ Possibility to calculate also coherent imaging

- Although Diffraction is considered, aberrations are calculated from tracing

geometrical rays (result is not a rigorous wave-optical solution)

- FFT-PSF in systems with pupil-aberration creates wrong results,

( Only Hugens-PSF method works correctly )

- Field dependent errors and spatial variant aberrations

are not trivial to implement (possible with Zemax, but it‘s not always clear

how this is performed)