www.iap.uni-jena.de

Optical Design with Zemax

for PhD - Advanced

Lecture 3: Physical modelling I

2014-12-03

Herbert Gross

Winter term 2014

2

Preliminary Schedule

No Date Subject Detailed content Examples

1 12.11. Repetition Correction, handling, multi-configuration

2 19.11. Illumination I Simple illumination problems 5-78,4-1,4-3

3 26.11. Illumination II Non-sequential raytrace 4-2 , RXI, TBD

4 03.12. Physical modeling I Gaussian beams, physical propagation 6-1, 6-2, resonator, anamorphot (2x)

5 10.12. Physical modeling II Polarization 6-3, POP, TBD

6 07.01. Physical modeling III Coatings 6-4, TBD

7 14.01. Tolerancing I Sensitivity, practical procedure 5-45, 5-58, 5-59

8 21.01. Tolerancing II Adjustment, thermal loading, ghosts 1-12, 2-15, (7-4)

9 28.01. Additional topics I Adaptive optics, stock lens matching, index fit 1-10, 1-20, 7-1

10 04.02. Additional topics II Macro language, coupling Zemax-Matlab 1-8, TBD, simple macro, raytrace,

Content

Gaussian beams

Gauss-Schell beams

Non-fundamental modes

Propagation methods

Numerical issues

POP in Zemax

2

2

)(

w

r

oeIrI

Gaussian Beams, Transverse Beam Profile

I(r) / I0

r / w

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-2 -1 0 1 2

0.135

0.0111.5

0.589

1.0

Transverse beam profile is gaussian

Beam radius w at 13.5% intensity

Expansion of the intensity distribution around the waist I(r,z)

Gaussian Beams

z

asymptotic

lines

x

hyperbolic

caustic curve

wo

w(z)

R(z)

o

zo

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

z / z

r / w

o

o

asymptotic

far field

waist

w(z)

o

intensity

13.5 %

Geometry of Gaussian Beams

2

2

2

1

2

2

2 1

2),(

o

To

z

zzw

r

o

To

e

z

zzw

PzrI

2

0 1)(

o

T

z

zzwzw

00000

zzw

o

o

Gaussian Beams, Definitions and Parameter

Paraxial TEM00 fundamental mode

Transverse intensity is gaussian

Axial isophotes are hyperbolic

Beam radius at 13.5% intensity

Only 2 independent beam parameter of the set:

1. waist radius wo

2. far field divergence angle o

3. Rayleigh range zo

4. Wavelength o

Relations

f

zfz

zzz

TT

oTT

1

1

1112'

Transform of Gaussian Beams

Diffraction effects are taken into account

Geometrical prediction corrected in the waist region

No singulare focal point: waist with finite width

Focal shift: waist located towards the system, intra focal shift

Transform of paraxial beam

propagation

z'T / f

zT / f

-6 -5 -4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

5

6

zo / f = 0.1

zo / f = 0.2

zo / f = 0.5

zo / f = 1

zo / f = 2

geometrical

limit zo / f = 0

Transform of Gaussian Beam

R

w

w'

R'

starting

plane

receiving

plane

paraxial

segment

A B

C D

incoming

Gaussian beam

z

outgoing

Gaussian beam

Transfer of a Gaussian beam by a paraxial ABCD system

w wB

wA

B

R'

2

2 2

R

AB

R

B

w

AB

RC

D

RD

B

w

'

2

2

2

2

2

-2

0

2-8

-6

-4

-2

0

2

4

6

8

0

0.5

1

z

intensity I

[a.u.]

x

Gaussian Beam Propagation

Paraxial transform of

a beam

Intensity I(x,z)

2

)(2

2 )(

2),(

zw

r

ezw

PzrI

2

21

2

1

21 )0,(

cL

rr

ezrr

2

1

2

1

c

o

L

w

2

2 11

c

o

L

wM

Gauß-Schell Beam: Definition

Partial coherent beams:

1. intensity profile gaussian

2. Coherence function gaussian

Extension of gaussian beams with similar description

Additional parameter: lateral coherence length Lc

Normalized degree of coherence

Beam quality depends on coherence

Approximate model do characterize multimode beams

-4 -3 -2 -1 0 1 2 3 4

1

2

3

4

0 z / zo

w / wo

0.25

0.50

1.0

-4 -3 -2 -1 0 1 2 3 4

1

2

3

4

0 z / zo

w / wo

1.0

0.50 0.25

Gauß-Schell Beams

Due to the additional parameter:

Waist radius and divergence angle are

independent

1. Fixed divergence:

waist radius decreases with

growing coherence

2. Fixed waist radius:

divergence angle decreases with

growing coherence

f

zfz

zzz

TT

oTT

1

1

1112'

s'/f'

1

1

s / f

= 0 geometrical

optic

0 < < 1 Gauss-

Schell beams

= 1 Gaussian

beams

Gauß-Schell Beam Transform

Similiar to gaussian beam propagation

Smooth transition between:

1. coherent gaussian beam

= 1

2. incoherent geometrical

optic = 0

Hermite Gaussian Modes

n = 0 n = 1 n = 2 n = 3 n = 4 n = 5

m = 5

m = 4

m = 3

m = 2

m = 1

m = 0

Mode profiles in circular

coordinates

Indices:

p: azimuthal

l: radial

p = 5p = 4p = 3p = 2

l = 5

p = 0 p = 1

l = 0

l = 1

l = 2

l = 3

l = 4

Laguerre Gaussian Modes

a/wo = 1

0 1 2 3 4 5 610

-12

10-10

10-8

10-6

10-4

10-2

100

a/wo = 2

a/wo = 3

a/wo = 4

x

Log |A|

Truncated Gaussian Beams

Untruncated gaussian beam: theoretical infinite extension

Real world: diameter D = 2a = 3w with 1% energy loss acceptable

Truncation: diffraction ripple occur, depending on ratio x = a / wo

Keplerian telecope with pinhole in focal plane

Higher modes with larger spatial extend are blocked:

only fundamental mode transmitted, beam clean up, mode filter

Approximate size of pinhole diameter: 20.637Pinhole

in in

f fD

w w

Beam Clean Up Filter

z

win

f

second lens ffirst lens f stop d

f wo

Focussed Gaussian beam with spherical aberration

Asymmetry intra - extra focal

depending on sign of spherical aberration

Gaussian profile perturbed

Gaussian Beam with Spherical Aberration

c9 = -0.25

c9 = 0.25

c9 = 0

Solution Methods of the Maxwell Equations

Maxwell-

equations

diffraction

integrals

asymptotic

approximation

Fresnel

approximation

Fraunhofer

approximation

finite

elements

finite

differences

exact/

numerical1st

approximation

direct

solutions of

the PDE

spectral

methods

plane wave

spectrum

vector

potentials

2nd

approximation

finite element

method

boundary

element

method

hybrid method

BEM + FEM

Debye

approximation

Kirchhoff-

integral

Rayleigh-

Sommerfeld

1st kind

Rayleigh-

Sommerfeld

2nd kind

mode

expansion

boundary

edge wave

Method Calculation Properties / Applications

Kirchhoff

diffraction integral E r

iE r

e

r rdF

i k r r

FAP

( ) ( ' )'

'

Small Fresnel numbers,

Numerical computation slow

Fourier method of

plane waves )(ˆˆ)'(21 xEFeFxE zvi

II

Large Fresnel numbers

Fast algorithm

Split step beam

propagation

Wave equation: derivatives approximated

En

yxnkE

z

Eik

z

E

o

1

),(2

2

222

2

2

Near field

Complex boundary geometries

Nonlinear effects

Raytracing Ray line law of refraction

r r sj j j j 1 sin '

'sini

n

ni

System components with a aberrations

Materials with index profile

Coherent mode

expansion

Field expansion into modes

n

nn xcxE )()( dxxxEc nn )()( *

Smooth intensity profiles

Fibers and waveguides

Incoherent mode

expansion

Intensity expansion into coherent modes

n

nn xcxI2

)()(

Partial coherent sources

Wave Optical Coherent Beam Propagation

Analytical approach according to Lommel (see Born/Wolf)

Two-argument Fresnel integrals

Lommel functions

Diffraction integral representation

21

Fresnel Diffraction

),(),()(2

1

0

20

2

vuSivuCdevJu

i

),(2/

2/cos),(

2/

2/sin

2sin

2

),(2/

2/sin),(

2/

2/cos

2

1cos)(2),(

10

2

21

1

0

2

0

vuVu

uvuV

u

u

u

v

v

vuUu

uvuU

u

u

duvJvuC

),(2/

2/sin),(

2/

2/cos

2cos

2

),(2/

2/cos),(

2/

2/sin

2

1sin)(2),(

10

2

21

1

0

2

0

vuVu

uvuV

u

u

u

v

u

vuUu

uvuU

u

u

duvJvuS

0

2

2

0

2

2

)()1(),(

)()1(),(

s

sn

sn

s

n

s

sn

sn

s

n

vJu

vvuV

vJv

uvuU

2

/sin),(2

2

/cos),(2),(),(1

2),(

),(),(2

),(

2

1

2

0

2

1

2

0

2

2

2

2

1

2

uvuvuV

uvuvuVvuVvuV

uIvuI

vuUvuUu

IvuI

obright

oshadow

Gaussian profile in the spatial domain

Fourier transform

Sampling theorem

N: number of discrete points

D: size of calculation domain

Zero padding with large D/w:

finer pixels in frequency space

22

FFT-Sampling of a Gaussian Profile

2

2

)( xw

x

exf

2

2

222

)( vx w

v

x

vw

x ewewvF

DvvN

D

NvxND

1,, max

x

vw

w

1

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nx = 25

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nv = 25

w = 0.1 w = 0.0353

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nx = 71

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nv = 9

w = 0.01

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nx = 7

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1N = 256 Nv = 91

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

Sampling of the Diffraction Integral

c9 = 3

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

= c9(6x4-6x2)

r

r

Re[ U ]

= c9(6x4)

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

r

= c9(6x4-9x2)

a

b

c

Wave with spherical aberration

Real part of the electric field for different values of defocussing

Optimal defocussing means minimal slope of the difference between wavefront and reference sphere

In optimal defocussing the sampling requirements are strongly reduced

The important measure is not the absolut aberration value, but the slope

a

NAaRRD

pxN

RRvx

'

Numerical Computation of Psf by FFT

Diffraction angle at a stop with radius a

Focal diameter in the range (if diffraction limited)

R radius of reference sphere

Sampling theorem for FFT

pixel size in pupil: xp

sampling interval of spatial frequency: n

number of sampling points: N

Pixel size in image plane

Good resolution in pupil gives very large x‘:

only few significant points in Psf

Resolution enhancement in the image plane:

1. zero padding, N large

2. Direct integration method, high numerical effort

3. chirp-z-transform, preferred, slow ba factor 2.2

Nvxp /1

Basis: 1. superposition of solutions

2. separability of coordinates

Plane wave

Spectral expansion

A(k): plane wave spectrum

Dispersion relation: Ewald sphere

Transverse expansion

Main idea:

- Field decomposition in plane waves

- Switch into Fourier space of spatial frequencies

- Propagation of plane wave as simple phase factor

- Back transform into spatial domain

- Superposition of plane wave with modified phase

Plane Wave Expansion

kdekArE rki

)(

2

13

trkietrE

,

2

2222

o

zyxc

nkkkk

E x y z A k k z e dk dkx y

i xk yk

x y

x y, , ( , , )

1

22

k k k k k kT z x y z

A k k z E x y z e dxdyx y

i xk ykx y, , ( , , )

Propagation of plane waves:

pure phase factor

1. exact sphere

2. Fresnel quadratic approximation

Evanescent waves

components damped in z

important only for near field setups

Propagation algorithm

x-y-sections are coupled

Paraxial approximation

x-y-section decoupled

Plane Wave Expansion

A k k z A k k A k k ex y x y

ik z

x y

ik zk

k

k

kz

x x

, , , , , ,

0 01

2 2

22

22

0,,

0,,,, 2

yx

yx

vvzizik

yx

kkk

iz

zik

yxyx

eevvA

eekkAzkkA

evanescentkkforkkkik yxyxz ,0222

0

22

),(ˆˆ

),(ˆˆ)','(

222/121

1

yxEFeF

yxEFeFyxE

xy

vviz

xy

xy

zik

xy

yx

z

),(ˆˆ)','(22

1 yxEFeFyxE xy

vvzi

xyyx

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

Kirchhoff diffraction integral in Fresnel approximation

Fourier transform: plan wave expansion

Equivalent form

Curvature removed

Calculation in spherical coordinates

Fresnel Propagation with Equivalence Transform

222 )1(

1'

)1(

)(ˆˆ)'(x

z

Miv

M

zix

Mz

MiM

zik

OO exEFeFeM

exE

x

R<0

z1

z2

starting

plane

observation

plane 1

focus

observation

plane 2

R'<0

R2'>0

x''

x x'

a

a

xxz

i

dxexECxE

2'

)()'(

Optimal Conditioning of the Fresnel-Propagator

Four different cases of propagation in a caustic

Starting plane / final plane inside/outside the focal region

Flattening transform only necessary outside focal range

z

y

inside waist region:

weak curvature

outside waist region:

strong curvature

case 1 : I - I

case 42 : O - O

case 41 : O - O

case 2 : O - I case 3 : I - O

waist

plane

R < 0

R > 0

Sampling and Phase Space

x

u

with

curvature

plane

f

radius of

curvatureR

xp xsumax << 1

umax large

Wave front with spherical curvature:

large angle interval to be sampled

Quasi collimated beam:

very small angle interval

Phase space consideration:

smaller number of sampling points

necessary

Setting of initial beam and

sampling parameters

Beam Propagation in Zemax

Individual control of parameters

at every surface

Model of calculation:

1. propagator from surface to

surface

2. estimation of sampling by pilot

gaussian beam

3. mostly Fresnel propagator with

near-far-selection

4. re-sampling possible

5. polarization, finite transmission,...

possible

Beam Propagation in Zemax

Example:

Focussing of a tophat beam with

one-sided truncation

Beam Propagation in Zemax

1. Geometrical with raytrace:

image of circular object

only geometrical truncation on the dia-

meter is considered

2. Geometrical with raytrace:

footprint

only geometrical truncation on the dia-

meter is considered

3. Monomode fiber:

special menue entry:

Calculations / Fiber Coupling Efficiency

Transmission, apodization, vignetting

are taken into account

Angle and spatial acceptance is

considered simultaneously

Huygens integral PSF is calculated

4. With physical optical propagation code

Most general tool

35

Fiber Coupling

Monomode fiber coupling example

36

Fiber Coupling