www.iap.uni-jena.de

Optical Design with Zemax

Lecture 1: Introduction

2012-07-17

Herbert Gross

Summer term 2012

Planned dates: 17.07. 24.07. 14.08. 28.08. 11.09. 25.09. 09.10. 23.10. 06.11. 20.11.

Shift of some days can not be avoided

Tuesday, 8.30 – 12.00

Location: Computerpool, Helmholtzweg 4

Web page on IAP homepage under ‚learning‘ provides slides and exercise Zemax files

Content(stype of the lecture:

Not: pure Zemax handling

But: - optical design with Zemax as tool

- understanding of simulation opportunities and limits

- learning by doing

- mix of theory/principles, presented examples and own exercises

- questions and dialog welcome

The content is adapted and is changed depending on progress

2 1 Introduction

Lecture data

3 1 Introduction

Preliminary time schedule

1. Kingslake Lens design fundamentals, SPIE Press, 2010 (x)

2. Mouroulis / McDonald Geometrical Optics and Optical Design, Oxford, 1997 (x)

3. Fischer / Tadic-Galeb Optical System Design, McGraw Hill, 2000 (x)

4. Malacara / Malacara Handbook of Lens Design, Dekker, 1994 (x)

5. Laikin Lens Design, Dekker, 2007 (x)

6. W. Smith Modern Optical Engineering, Graw Hill, 2000 (x)

7. W. Smith Modern lens design, McGraw Hill, 2005 (x)

8. Geary Lens Design with practical Examples, Willmann-Bell, 2002 (x)

9. Gross (Ed.) Handbook of optical systems, Vol 1-5, Wiley, 2005

10. Shannon The art and science of optical design,

Cambridge Univ. Press, 1997

11. G. Smith Practical computer-aided lens design, Willman Bell, 1998

4 1 Introduction

Literature on optical design

1. Introduction

2. Zemax interface, menues, file handling, preferences

3. Editors, updates, windows

4. Coordinate systems and notations

5. System description

6. Component reversal, system insertion, scaling

7. Solves and pickups, variables

8. 3D geometry

9. Aperture, field, wavelength

10. Diameters, stop and pupil, vignetting

11. Layouts

12. Glass catalogs

13. Raytrace

14. Ray fans and sampling

15. Footprints

5 1 Introduction

Contents

There are 4 types of windows in Zemax:

1. Editors for data input:

lens data, extra data, multiconfiguration, tolerances

2. Output windows for graphical representation of results

Here mostly setting-windowss are supported to optimize the layout

3. Text windows for output in ASCII numerical numbers (can be exported)

4. Dialog boxes for data input, error reports and more

There are several files associates with Zemax

1. Data files (.ZMX)

2. Session files (.SES) for system settings (can be de-activated)

3. Glass catalogs, lens catalogs, coating catalogs, BRDF catalogs, macros,

images, POP data, refractive index files,...

There are in general two working modes of Zemax

1. Sequential raytrace (or partial non-sequencial)

2. Non-sequential

1 Introduction

Zemax interface

Helpful shortcuts:

1. F3 undo

2. F2 edit a cell in the editor

3. cntr A multiconfiguration toggle

4. cntr V variable toggle

5. F6 merit function editor

6. cntr U update

7. shift cntr Q quick focus

Window options:

1. several export options

2. fixed aspect ratios

3. clone

4. adding comments or graphics

1 Introduction

Zemax interface

Coordinate systems

2D sections: y-z shown

Sign of lengths, radii, angles:

z / optical axis

y / meridional section

tangential plane

x / sagittal plane

+ s- s

negative:

to the left

positive:

to the right+ R

+ j

reference

angle positive:

counterclockwise

+ R1

negative:

C to the leftpositive:

C to the right

- R2

C1C2

1 Introduction

Coordinate systems and sign of quantities

Interface surfaces

- mathematical modelled surfaces

- planes, spheres, aspheres, conics, free shaped surfaces,…

Size of components

- thickness and distances along the axis

- transversal size,circular diameter, complicated contours

Geometry of the setup

- special case: rotational symmetry

- general case: 3D, tilt angles, offsets and decentrations, needs vectorial approach

Materials

- refractive indices for all used wavelengths

- other properties: absorption, birefringence, nonlinear coefficients, index gradients,…

Special surfaces

- gratings, diffractive elements

- arrays, scattering surfaces

9 1 Introduction

Description of optical systems

Single step:

- surface and transition

- parameters: radius, diameter, thickness,

refractive index, aspherical constants,

conic parameter, decenter, tilt,...

Complete system:

- sequence of surfaces

- object has index 0

- image has index N

- tN does not exist

Ray path has fixed

sequence

0-1-2-...-(N-1)-N surface

index

object

plane

1

0 2

image

plane

N-23 j N-1.... .... N

0

1

2 3 j N-2 N-1 (N)

thickness

index

surfaces

surface j

medium j

tj / nj

radius rj

diameter Dj

1 Introduction

System model

Menu: Reports / Prescription

data

Menu:

reports / prescription data

1 Introduction

System data tables

1 Introduction

System data tables

Necessary data for system calculation:

1. system surfaces with parameters (radius)

2. distances with parameters (length, material)

3. stop surface

4. wavelength(s)

5. aperture

6. field point(s)

Optional inputs:

1. finite diameters

2. vignetting factors

3. decenter and tilt

4. coordinate reference

5. weighting factors

6. multi configurations

7. ...

1 Introduction

System data

Useful commands for system changes:

1. Scaling (e.g. patents)

2. Insert system

with other system file

File - Insert Lens

2. Reverse system

14 1. Introduction

System changes

Setting of surface properties

local tilt

and

decenterdiameter

surface type

additional drawing

switches

coatingoperator and

sampling for POP

scattering

options

1 Introduction

Surface properties and settings

Value of the parameter dependents on other requirement

Pickup of radius/thickness: linear dependence on other system parameter

Determined to have fixed: - marginal ray height

- chief ray angle

- marginal ray normal

- chief ray normal

- aplanatic surface

- element power

- concentric surface

- concentric radius

- F number

- marginal ray height

- chief ray height

- edge thickness

- optical path difference

- position

- compensator

- center of curvature

- pupil position

1 Introduction

Solves

Examples for solves:

1. last radius forces given image aperture

2. get symmetry of system parts

3. multiple used system parts

4. moving lenses with constant system length

5. bending of a lens with constant focal length

6. non-negative edge thickness of a lens

7. bending angle of a mirror (i'=i)

8. decenter/tilt of a component with return

1 Introduction

Solves

Open different menus with a right-mouse-click in the corresponding editor cell

Solves can be chosen individually

Individual data for every surface in this menu

1 Introduction

Solves

General input of tilt and decenter:

Coordinate break surface

Change of coordinate system with lateral translation and 3 rotations angles

Direct listing in lens editor

Not shown in layout drawing

19 1 Introduction

3D geometry

Auxiliary menus:

1. Tilt/Decenter element

2. Folding mirror

20 1 Introduction

3D geometry

Local tilt and decenter of a surface

1. no direct visibility in lens editor

only + near surface index

2. input in surface properties

3. with effect on following system surfaces

21 1 Introduction

3D geometry

Imaging on axis: circular / rotational symmetry

Only spherical aberration and chromatical aberrations

Finite field size, object point off-axis:

- chief ray as reference

- skew ray bundels:

coma and distortion

- Vignetting, cone of ray bundle

not circular symmetric

- to distinguish:

tangential and sagittal

plane

O

entrance

pupil

y yp

chief ray

exit

pupil

y' y'p

O'

w'

w

R'AP

u

chief ray

object

planeimage

plane

marginal/rim

ray

u'

1 Introduction

Definition of aperture and field

Quantitative measures of relative opening / size of accepted light cone

Numerical aperture

F-number

Approximation for small

apertures:

'sin unNA

EXD

fF

'#

NAF

2

1#

image

plane

marginal ray

exit

pupil

chief ray

U'W'

DEX

f'

1 Introduction

Aperture definition

Pupil stop defines:

1. chief ray angle w

2. aperture cone angle u

The chief ray gives the center line of the oblique ray cone of an off-axis object point

The coma rays limit the off-axis ray cone

The marginal rays limit the axial ray cone

y

y'

stop

pupil

coma ray

chief ray

marginal ray

aperture

angle

u

field anglew

object

image

1 Introduction

Optical system stop

The physical stop defines

the aperture cone angle u

The real system may be complex

The entrance pupil fixes the

acceptance cone in the

object space

The exit pupil fixes the

acceptance cone in the

image space

uobject

image

stop

EnP

ExP

object

image

black box

details complicated

real

system

? ?

Ref: Julie Bentley

1 Introduction

Optical system stop

Relevance of the system pupil :

Brightness of the image

Transfer of energy

Resolution of details

Information transfer

Image quality

Aberrations due to aperture

Image perspective

Perception of depth

Compound systems:

matching of pupils is necessary, location and size

1 Introduction

Properties of the pupil

exit

pupil

upper

marginal ray

chief

ray

lower coma

raystop

field point

of image

UU'

W

lower marginal

ray

upper coma

ray

on axis

point of

image

outer field

point of

object

object

point

on axis

entrance

pupil

1 Introduction

Entrance and exit pupil

Generalization of paraxial picture:

Principal surface works as effective location of ray bending

Paraxial approximation: plane

Real systems with corrected

sine-condition (aplanatic):

principal sphere

effektive surface of

ray bending P'

y

f'

U'

P

1 Introduction

Pupil sphere

Pupil sphere:

equidistant sine-

sampling

z

object entrance

pupil

image

yo y'

U U'sin(U) sin(U')

exit

pupil

objectyo

equidistant

sin(U)

angle U non-

equidistant

pupil

sphere

1 Introduction

Pupil sphere

Different possible options for specification of the aperture in Zemax:

1. Entrance pupil diameter

2. Image space F#

3. Object space NA

4. Paraxial working F#

5. Object cone angle

6. Floating by stop size

Stop location:

1. Fixes the chief ray intersection point

2. input not necessary for telecentric object space

3. is used for aperture determination in case of aiming

Special cases:

1. Object in infinity (NA, cone angle input impossible)

2. Image in infinity (afocal)

3. Object space telecentric

1 Introduction

Aperture data in Zemax

1. Determination of one surface as system stop:

- Fixes the chief ray intersection point with axis

- can be set in surface properties menu

- indicated by STO in lens data editor

- determines the aperture for the option 'float by stop size'

2. Diameters in lens data editor

- indicated U for user defined

- only circular shape

- effects drawing

- effects ray vignetting

- can be used to draw 'nice lenses' with

overflow of diameter

3. Diameters as surface properties:

- effects on rays in drawing (vignetting)

- no effect on lens shapes in drawing

- also complicated shapes and decenter

possible

- indicated in lens data editor by a star

31 1 Introduction

Diameters and stop sizes

4. Individual aperture sizes for every field point can be set by the vignetting factors of the

Field menu

- real diameters at surfaces must be set

- reduces light cones are drawn in the layout

32 1 Introduction

Diameters and stop sizes

Graphical control of system

and ray path

Principal options in Zemax:

1. 2D section for circular symmetry

2. 3D general drawing

Several options in settings

Zooming with mouse

33 1. Introduction

Layout options

Different options for 3D case

Multiconfiguration plot possible

Rayfan can be chosen

34 1. Introduction

Layout options

Professional graphic

Many layout options

Rotation with mouse or arrow buttons

35 1. Introduction

Layout options

Important types of optical materials:

1. Glasses

2. Crystals

3. Liquids

4. Plastics, cement

5. Gases

6. Metals

Optical parameters for characterization of materials

1. Refractive index, spectral resolved n(l)

2. Spectral transmission T(l)

3. Reflectivity R

4. Absorption

5. Anisotropy, index gradient, eigenfluorescence,…

Important non-optical parameters

1. Thermal expansion coefficient

2. Hardness

3. Chemical properties (resistence,…)

1 Introduction

Optical materials

l in [nm] Name Color Element

248.3 UV Hg

280.4 UV Hg

296.7278 UV Hg

312.5663 UV Hg

334.1478 UV Hg

365.0146 i UV Hg

404.6561 h violett Hg

435.8343 g blau Hg

479.9914 F' blau Cd

486.1327 F blau H

546.0740 e grün Hg

587.5618 d gelb He

589.2938 D gelb Na

632.8 HeNe-Laser

643.8469 C' rot Cd

656.2725 C rot H

706.5188 r rot He

852.11 s IR Cä

1013.98 t IR Hg

1060.0 Nd:YAG-Laser

1 Introduction

Test wavelengths

refractive

index n

1.65

1.6

1.5

1.8

1.55

1.75

1.7

BK7

SF1

l0.5 0.75 1.0 1.25 1.751.5 2.0

1.45

Flint

Kron

Description of dispersion:

Abbe number

Visual range of wavelengths:

Typical range of glasses

ne = 20 ...120

Two fundamental types of glass:

Crone glasses:

n small, n large

Flint glasses

n large, n small

n ll

n

n nF C

1

' '

ne

e

F C

n

n n

1

' '

1 Introduction

Dispersion and Abbe number

Material with different dispersion values:

- Different curvature of the dispersion curve

- Stronger change of index over wavelength for large dispersion

- Inversion of index sequence at the boundaries of the spectrum possible

refractive index n

F6

SK18A

l

1.675

1.7

1.65

1.625

1.60.5 0.75 1.0 1.25 1.751.5 2.0

1 Introduction

Dispersion

Schott formula

empirical

Sellmeier

Based on oscillator model

Bausch-Lomb

empirical

Herzberger

Based on oscillator model

Hartmann

Based on oscillator model

n a a a a a ao + + + + +

1

2

2

2

3

4

4

6

5

8l l l l l

n A B C( )ll

l l

l

l l +

+

2

212

2

222

n A B CD E

Fo

o

( )

( )

l l ll

l

l ll

l l

+ + + +

+

2 4

2

2

2 22

2 2

mmit

aaaan

o

oo

o

l

llllll

168.0

)(222

3

22

22

1

+

++

lll

+

+

5

4

3

1)(a

a

a

aan o

1 Introduction

Dispersion formulas

Relative partial dispersion :

Change of dispersion slope with l

Definition of local slope for selected

wavelengths relative to secondary

colors

Special selections for characteristic

ranges of the visible spectrum

l = 656 / 1014 nm far IR

l = 656 / 852 nm near IR

l = 486 / 546 nm blue edge of VIS

l = 435 / 486 nm near UV

l = 365 / 435 nm far UV

P

n n

n nF C

l l

l l

1 2

1 2

' '

l

n

400 600 800 1000700500 900 1100

e : 546 nm

main color

F' : 480 nm

1. secondary

color

g : 435 nmUV edge

C' : 644 nm

s : 852 nm

IR edge

t : 1014 nm

IR edge

C : 656 nmF : 486 nm

d : 588 nm

i : 365 nm

UV edge

i - g

F - C

C - s

C - t

F - e

g - F

2. secondary

color

1.48

1.49

1.5

1.51

1.52

1.53

1.54

n(l)

1 Introduction

Relative partial dispersion

Usual representation of

glasses:

diagram of refractive index

vs dispersion n(n)

Left to right:

Increasing dispersion

decreasing Abbe number

1 Introduction

Glass diagram

Glass:

LAFN7

Part 1

Main data Refractive

indices

Internal transmission

data

relative partial

dispersion

Anomalous

partial

dispersion

l

[nm] n

l

[nm]

i [10

mm]

i [25

mm]

LAFN7 750350.4

38 n2325,4 2325,4 1,70211 2500,0 0,380 0,090 Ps,t 0,2360 Pc,t 0,0174

nd 1,74950 n1970,1 1970,1 1,70934 2325,4 0,700 0,410 PC,s 0,4921 Pc,s 0,0078

ne 1,75458 n1529,6 1529,6 1,71726 1970,1 0,940 0,850 Pd,C 0,2941 PF,e - 0,0011

nd 34,95 n1060,0 1060,0 1,72642 1529,6 0,984 0,960 Pe,d 0,2369 Pg,F - 0,0025

ne 34,72 nt 1014,0 1,72758 1060,0 0,998 0,996 Pg,F 0,5825 Pi,g - 0,0093

nF-nC 0,02145 ns 852,1 1,73264 700,0 0,998 0,996 Pi,h 0,9160

nF'-nC' 0,021735 nr 706,5 1,73970 660,0 0,998 0,995

nC 656,3 1,74319 620,0 0,998 0,995 P's,t 0,2329

nC' 643,8 1,74418 580,0 0,998 0,995 P'C',s 0,5311

n632,8 632,8 1,74511 546,1 0,998 0,994 P'd,C' 0,2446

nD 589,3 1,74931 500,0 0,998 0,994 P'e,d 0,2338

nd 587,6 1,74950 460,0 0,993 0,982 P'g,F' 0,5158

ne 546,1 1,75458 435,8 0,986 0,965 P'i,h 0,9037

nF 486,1 1,76464 420,0 0,976 0,940

nF' 480,0 1,76592 404,7 0,950 0,880

ng 435,8 1,77713 400,0 0,940 0,850

nh 404,7 1,78798 390,0 0,910 0,780

ni 365,0 1,80762 380,0 0,840 0,650

n3 34,1 334,1 0,00000 370,0 0,690 0,400

n312,6 312,6 0,00000 365,0 0,550 0,220

n296,7 296,7 0,00000 350,0 0,130 0,010

n280,4 280,4 0,00000 334,1 0,000 0,000

n248,3 248,3 0,00000 320,0 0,000 0,000

310,0 0,000 0,000

1 Introduction

Glass data sheet

Sellmeier

dispersion

constants

Constants of

dn/dT

Other properties Temperature coefficients

dn rel / dT [10 -6/K]

dn abs / dT

[10-6/K]

B1 1,66842615E+00 D0 7,27E -06 - 30/+70°C [10 -6 /K] 5,3 [°C] 1060,0 e g 1060,0 e g

B2 2,98512803E -01 D1 1,31E -08 +20/+300°C [10 -6 /K] 6,4 -40/ -20 6,00 7,80 9,70 3,70 5,40 7,20

B3 1,07743760E+00 D2 - 3,32E -11 Tg[°C] 500 +20/+40 6,30 8,30 10,40 4,80 6,70 8,90

C1 1,03159999E -02 E0 8,88E -07 T10 13,0[°C] 481 +60/+80 6,50 8,60 10,90 5,30 7,40 9,70

C2 4,69216348E -02 E1 9,32E -10 T10 7,6[°C] 573

C3 8,25078509E+01

l TK [

m] 2,48E -01 cp [J/(g·K)] 0,000

l [W/(m·K)] 0,770

[g/cm 3] 4,38

E[10 3 N/mm 2] 80

0,280

K[10 -6 mm 2 /N] 1,77

HK0,1/20 520

HG 3

B 0

CR 3

FR 1

SR 53

AR 2,2

PR 4,3

Glass:

LAFN7

Part 2

1 Introduction

Glass data sheet

Crown glasses

n

n

e e

e e

für n

für n

54 7 16028

49 7 16028

. .

. .

Flint glasses

n

n

e e

e e

für n

für n

54 7 16028

49 7 16028

. .

. .

n

n

80 70 60 50 40 30 2025354555657585

1.45

1.50

1.55

1.60

1.65

1.70

1.75

1.80

1.85

1.90

1.95

2.00

LaK

LaSF

SF

TiSF

TiF

BaSF

F

LFLLF

BaLF

LaF

PSK

PK

FK TiK

BKK

SK

BaK

SSK

KF

crown

glass

flint

glass

BaF

1 Introduction

Ranges of glass map

n

n

20 30 40 50 60 70 80 90 1001.4

1.5

1.6

1.7

1.8

1.9

2

20 30 40 50 60 70 80 90 1001.4

1.5

1.6

1.7

1.8

1.9

2

Schott Ohara

Nearly equal area in the glass diagramm with available materials for all vendors

No options with Index / dispersion

1. high / high

2. low / low

1 Introduction

Ranges of glass map - availability

zoptical

axis

y j

u'j-1

ij

dj-1

ds j-1

ds j

i'j

u'j

n j

nj-1

mediummedium

surface j-1

surface j

ray

dj

vertex distance

oblique thickness

rr

Ray: straight line between two intersection points

System: sequence of spherical surfaces

Data: - radii, curvature c=1/r

- vertex distances

- refractive indices

- transverse diameter

Surfaces of 2nd order:

Calculation of intersection points

analytically possible: fast

computation

47 1 Introduction

Scheme of raytrace

yj

z

Pj+1

sj

xj

yj+1

xj+1

Pj

surface

No j

surface

No j+1

dj sj+1

intersection

point

ej

normal

vector

ej+1

ray

distance

intersection

point

normal

vector

General 3D geometry

Tilt and decenter of surfaces

General shaped free form surfaces

Full description with 3 components

Global and local coordinate systems

48 1 Introduction

Vectorial raytrace

r D R F R D rH H V V'

Single surface:

- tilt and decenter before refraction

- decenter and tilt after refraction

General setup for position and orientation in 3D

z

x

y

z

xF

yF

zF

j-1

j+1global

coordinates

surface

No j

VV

VH

KV

KH

shift before

shift after

tilt after

tilt before

49 1 Introduction

Surfaces in 3D

Restrictions:

- surfaces of second order, fast analytical calculation of intersection point possible

- homogeneous media

Direction unit vector of the straight ray

Vector of intersection point on a surface

Ray equation with skew thickness dsj

index j of the surface and the space behind

Equation of the surface 2.order

The coefficients H, F, G contains the surface shape parameters

s j

j

j

j

r

x

y

z

j

j

j

j

11,1 + jjsjj sdrr

02 1,

2

1, + jjsjjsj GdFdH

50 1 Introduction

Vectorial raytrace formulas

Special case spherical surface with

curvature c = 1/R

Coefficients H, G, F

Unit vector normal to the surface

Insertion of the ray equation into surface equation:

skew thickness

Angle of incidence

Refraction

or

reflection

jj cH

jjjjjj zzyxcG 2222 ++

jjjjjjjjj zyxcF ++

jj

jj

jj

j

zc

yc

xc

e

1

jjjj

j

js

GHFF

Gd

++

21,

cosi s ej j j

cos ' cosin

nij

j

j

j

+

1 11

2

2

cos ' cosi ij j

51 1 Introduction

Vectorial raytrace formulas

Auxiliary parameter

New ray direction vector

j j j j jn i n i +1 cos ' cos

s

n

ns

nej

j

j

j

j

j

j+

+ +

+1

1 1

52 1 Introduction

Vectorial raytrace formulas

Vignetting/truncation of ray at finite sized diameter:

can or can not considered (optional)

No physical intersection point of ray with surface

Total internal reflection

Negative edge thickness of lenses

Negative thickness without mirror-reflection

Diffraction at boundaries

index

j

index

j+1

axis

negative

un-physical

regular

irregular

axis

no intersection

point

axis

intersection:

- mathematical possible

- physical not realized

axis

total

internal

reflection

1 Introduction

Raytrace errors

Meridional rays:

in main cross section plane

Sagittal rays:

perpendicular to main cross

section plane

Coma rays:

Going through field point

and edge of pupil

Oblique rays:

without symmetry

axis

y

x

p

p

pupil plane

object plane

x

y

axissagittal ray

meridional

marginal ray

skew raychief ray

sagittal coma

ray

upper

meridional

coma ray

lower

meridional

coma ray

field

point

axis point

1 Introduction

Special rays in 3D

Off-axis object point:

1. Meridional plane / tangential plane / main cross section plane

contains object point and optical axis

2. Sagittal plane:

perpendicular to meridional plane through object point

x

y

x'

y'

z

lens

meridional

plane

sagittal

plane

object

planeimage

plane

1 Introduction

Tangential and sagittal plane

Ray fan:

2-dimensional plane set of rays

Ray cone:

3-dimensional filled ray cone

object

point

pupil

grid

1 Introduction

Ray fans and ray cones

Transverse aberrations:

Ray deviation form ideal image point in meridional and sagittal plane respectively

The sampling of the pupil is only filled in two perpendicular directions along the axes

No information on the performance of rays in the quadrants of the pupil

x

y

pupil

tangential

ray fan

sagittal

ray fanobject

point

1 Introduction

Ray fan selection for transverse aberrations plots

Pupil sampling for calculation of transverse aberrations:

all rays from one object point to all pupil points on x- and y-axis

Two planes with 1-dimensional ray fans

No complete information: no skew rays

y'p

x'p

yp

xp x'

y'

z

yo

xo

object

plane

entrance

pupil

exit

pupil

image

plane

tangential

sagittal

1 Introduction

Pupil sampling

Pupil sampling in 3D for spot diagram:

all rays from one object point through all pupil points in 2D

Light cone completly filled with rays

y'p

x'p

yp

xp x'

y'

z

yo

xo

object

plane

entrance

pupil

exit

pupil

image

plane

1 Introduction

Sampling of pupil area

Different types of sampling with pro and con‘s:

1. Polar grid: not isoenergetic

2. Cartesian: good for FFT, boundary discretization bad

3. Isoenergetic circular: good

4. Hexagonal: good

5. Statistical: good non-regularity,

holes ?

polar grid cartesian isoenergetic circular

hexagonal statistical

1 Introduction

Pupil sampling

Artefacts due to regular gridding of the pupil of the spot in the image plane

In reality a smooth density of the spot is true

The line structures are discretization effects of the sampling

hexagonal statisticalcartesian

1 Introduction

Artefacts of pupil sampling

Looking for the ray bundle cross sections

62 1. Introduction

Footprints