Optical Design with Zemax for PhD - Basics

www.iap.uni-jena.de

Optical Design with Zemax

for PhD - Basics

Seminar 5 : Physical Modelling II

2014-12-10

Herbert Gross

Winter term 2014

2

Preliminary Schedule

No Date Subject Detailed content

1 12.11. Repetition Correction, handling, multi-configuration

2 19.11. Illumination I Simple illumination problems

3 26.11. Illumination II Non-sequential raytrace

4 03.12. Physical modeling I Gaussian beams, physical propagation

5 10.12. Physical modeling II Polarization

6 07.01. Physical modeling III Coatings

7 14.01. Tolerancing I Sensitivity, practical procedure

8 21.01. Tolerancing II Adjustment, thermal loading, ghosts

9 28.01. Additional topics I Adaptive optics, stock lens matching, index fit

10 04.02. Additional topics II Macro language, coupling Zemax-Matlab

1. Introduction

2. Jones calculus

3. Stokes vector and Poincare sphere

4. Propagation of polarization

5. Illustration of polarization

6. Birefringence

7. Components

8. Polarization in Zemax

3

Contents

Scalar:

Helmholtz equation

Vectorial:

Maxwell equations

Scalar / vectorial Optics

0)(2 rEnko

k

E

H

k

0

Bk

iDk

BEk

jiDHk

EJ

Jk

MHB

PED

r

r

0

0

Description of electromagnetic fields:

- Maxwell equations

- vectorial nature of field strength

Decomposition of the field into components

Propagation plane wave:

- field vector rotates

- projection components are oscillating sinusoidal

yyxx etAetAE )cos(cos

z

x

y

Basic Notations of Polarization

1. Linear components in phase

2. circular phase difference of 90° between components

3. elliptical arbitrary but constant phase difference

x

y

z

E

E

x

y

z

EE

x

y

z

E

E

Basic Forms of Polarisation

Representation of the state of polarization by an ellipse

Field components

Axes of ellipse: a, b

Rotation angle of the field

Angle of eccentricity

a

btan

Polarization Ellipse

)sinsincos(cos xxxx AE

)sinsincos(cos yyyy AE

cos2

2tan22

yx

yx

AA

AA

x

y

x'

y'

a

b

Ax

Ay

Ex

Ey

Elimination of the time dependence:

Ellipse of the vector E

Different states of polarization:

- sense of rotation

- shape of ellipse

0° 45° 90° 135° 180°

225° 270° 315° 360°

2

2

2

2

2

sincos2

yx

yx

y

y

x

x

AA

EE

A

E

A

E

Polarization Ellipse

Descriptions of Polarization

E

Parameter Properties

1

Polarization ellipse

Ellipticity ,

orientation only complete polarization

2

Complex parameter

Parameter

only complete polarization

3

Jones vectors

Components of E

only complete polarization

4

Stokes vectors

Stokes parameter So ... S4

complete or partial

polarization

5

Poincare sphere

Points on or inside the

Poincare sphere only graphical representation

6

Coherence matrix

2x2 - matrix C

complete or partial

polarization

Decomposition of the field strength E

into two components in x/y or s/p

Relative phase angle between

components


Linear polarized light

Circular polarized light

y

x

i

y

i

x

y

x

eA

eA

E

EE

0

xy

0

10E

1

00E

sin

cos0E

iErz

1

2

1

iElz

1

2

1

Jones Vector

2

22

sincos2

yx

yx

y

y

x

x

AA

EE

A

E

A

E

Jones representation of full polarized field:

decomposition into 2 components

Cascading of system components:

Product of matrices

Transmission of intensity

Jones Calculus

1121 EJJJJE nnn

os

op

ppps

spss

s

p

oE

E

JJ

JJ

E

EEJE ,

System 1 :

J1

E1 System 2 :

J2

E2 System 3 :

J3

E3 System n :

Jn

En-1 EnE4

1

*

12 EJJEI

Three basic types of components, that change the polarization:

1. Change of amplitude: polarizer / analyzer

2. Change of phase: retarder

3. Change of orientation: rotator

)()0()()( DJDJ

Jones Matrices

p

s

trans t

tJ

10

01

2

2

0

0

i

i

ret

e

eJ

cossin

sincos)(rotJ

Decomposition of the field in 2

components x/y or p/s respectively

matrix representation with Jones vector

Phase angle, relativ


Linear polarized field

Circular polarized field

y

x

i

y

i

x

y

x

eA

eA

E

EE

0

xy

0

10E

1

00E

sin

cos0E

iErz

1

2

1

iElz

1

2

1

Jones Vector

2

22

sincos2

yx

yx

y

y

x

x

AA

EE

A

E

A

E

Jones representation of full polarized field:

decomposition into 2 components

Cascading of system components:

Product of matrices

In principle 3 types of components influencing polarization:

1. Change of amplitude polarizer, analyzer

2. Change of phase retarder

3. Rotation of field components rotator

Jones Calculus

1121 EJJJJE nnn

p

s

trans t

tJ

10

01

cossin

sincos)(rotJ

2

2

0

0

i

i

ret

e

eJ

os

op

ppps

spss

s

p

oE

E

JJ

JJ

E

EEJE ,

System 1 :

J1

E1 System 2 :

J2

E2 System 3 :

J3

E3 System n :

Jn

En-1 EnE4

Rotated component

Rotation matrix

Intensity

Three types of components to change the polarization:

1. amplitude: polarizer / analyzer

2. phase: retarder

3. orientation: rotator

Propagation:

1. free space

2. dielectric interface

3. mirror

)()0()()( DJDJ

cossin

sincos)(D

1

*

12 EJJEI

Jones Matrices

10

012OPLi

PRO eJ

p

s

TRA t

tJ

0

0

p

s

REF r

rJ

0

0

10

01rJ SP

Birefringence: index of refraction depends on field orientation

Uniaxial crystal:

ordinary index no perpendicular to crystal axis

extra-ordinary index ne along crystal axis

Difference of indices

Jones matrix

Relative phase angle

0

0

i n z

neo i n z

eJ

e

2 2

2 2

sin cos sin cos 1( , )

sin cos 1 cos sin

i i

neoi i

e eJ

e e

2

e o

zn n

oe nnn

Birefringence: Uniaxial Crystal

Description of polarization from the energetic point of view

- So total intensity

- S1 Difference of intensity in x-y linear

- S2 Difference of intensity linear under 45° / 135°

- S3 Difference of circular components

)0,90()0,0(0 IIS

)0,90()0,0(1 IIS

)0,135()0,45(2 IIS

)90,45()90,45(3 IIS

Stokes Vector

22

yxo EES

22

1 yx EES

cos22 yxEES

sin23 yxEES

Description of a polarization state with Stokes parameterInterpretation:

Components of the field on the Poincare sphere

Also partial polarization is taken into account

Relation

Unequal sign: partial polarization

Stokes vector 4x1

Propagation:

Müller matrix M

2

3

2

2

2

1

2

0 SSSS

3

2

1

0

S

S

S

S

S

Stokes Vector

S

mmmm

mmmm

mmmm

mmmm

SMS

33323130

23222120

13121110

03020100

'

Linear horizontal / vertical

LInear 45°

Circular clockwise / counter-clockwise

0

0

1

1

S

0

0

1

1

S

0

1

0

1

S

1

0

0

1

S

1

0

0

1

S

Examples of Stokes Vectors

Partial polarized light:

degree of polarization

p = 0 : un-polarized

p = 1 : fullypolarized

0 < p < 1 : partial polarized

Determination of Stokes parameter:

Unpolarized light

Fully polarized light

Partial Polarization

ges

pol

I

Ip

pS S S

S

1

2

2

2

3

2

0

pu SS

pSSpS

000 )1(

0321 SSS

2

3

2

2

2

1

2

0 SSSS

Fully polarized: point

Unpolarized: full surface

Partial polarized: probability distribution, points inside

y

x

z

P

y

x

z

y

x

z

fully polarized un-polarizedpartial

polarized

Partial POlarization on the Poincate Sphere

Every point on a uni sphere describes one state of polarization

In spherical coordinates:

Points on z-axis: circular polarized light

Meridian line: linear polarization

Points inside

partial polarization

Sphere of Poincare

2sin

2cos2cos

2sin2cos1

222

z

y

x

zyxr

y

x

z

right handed

circular polarized

left handed

circular polarized

elliptical

polarized

linear

polarized

2

2

Stokes parameter S1 , S2 , S3 :

Componenst along axis directions

Radius of sphere, length of vector: So

Projection into meridional plane:

angle 2 of polarization ellipse

Projection into meridian plane:

eccentricity angle 2

Poincare Sphere and Stokes Vector

So

S1

S2

S3

y

x

z

P

2

2

Description of a complex state of polarization as a time-averaged matrix of correlation of

the field

Matrix

Relation to Stokes vectors

Degree of polarization

Decomposition into Pauli-spin

matrices

RElation with Jones matrices in

case of fully polarization

dtEET

EEc

T

kit

kiik 0

** 1

yyxy

xyxx

yyyx

xyxx

cc

cc

cc

ccC *

* *

0x x y ydiag C E E E E S

2)(

det41

CSpur

Cp

1032

3210

21

SSiSS

iSSSSC

Coherence Matrix

3

0

21

i

iiSC

2

2

y

i

yx

i

yxx

pAeAA

eAAAC

Decomposition into unpolarized

and polarized part

Example matrices:

1. unpolarized light

2. linear polarized in x/y

3. circular polarized right / left-

handed

cd

dbaCCC pu *10

01

10

01

2

0SC

00

010SC

10

000SC

1

1

2

0

i

iSC

1

1

2

0

i

iSC

Coherence Matrix

Jones matrix changes Jones vector

Müller matrix changes Stokes vector

Change of coherence matrix

Procedure for real systems:

1. Raytracing

2. Definition of initial polarization

3. Jones vector or coherence matrix local on each ray

4. transport of ray and vector changes at all surfaces

5. 3D-effects of Fresnel equations on the field components

6. Coatings need a special treatment

7. Problems: ray splitting in case of birefringence

JCJC'

SMS

'

EJE

'

Propagation of Polarization

Raytracing for birefringence

splitting of the ray in two with different polarization states at the entrance into a

birefringent medium

For m lenses: 2m rays behind the system

Lateral walkoff of the rays

Effects on amplitude and phase

Re-decomposition of principal field components possible

In general two separated pupils for s/p, which van not interfere

Birefringence

1. 2. 3.

1

splitting

42

8

rays

Change of field strength:

calculation with polarization matrix,

transmission T

Diattenuation

Eigenvalues of Jones matrix

Retardation: phase difference

of complex eigenvalues

To be taken into account:

1. physical retardance due to refractive index: P

2. geometrical retardance due to geometrical ray bending: Q

Retardation matrix

Diattenuation and Retardation

EE

EPPE

E

EPT

T

*

*

2

2

minmax

minmax

TT

TTD

2/12/12/12/12/1 wewwJ

i

ret

21 argarg

totaltotalPQR

1

Rotation of plane of polarization with t / z

Phase angle 90°

Generation by /4 plate out of linear polarized light

Erzeugung :

Polarisator und / 4 -

Platte

b1

x

y

E

Er

El

t1

El

Er E

t2

b2

SA z

y

x

TA

45°

/ 4 - plate

linear

polarizer

LA

Circular polarized Light

Circular Polarization

Spiral curve of field vector

Superposition of left and right handed circular

polarized light:

resulting linear polarization

Ref: Manset

tx

y

E

Er

El

t

t1

t2

t3

Classical geometry of birefringent refraction refers on interface plane

Grandfathers method:

Calculation iterative due to non-linear equations in prism coordinates

History: formulas according to Muchel / Schöppe

Only plane setup considered, crystal axis in plane of incidence

Geometry of Raytrace at an Interface

incidentray

e

o

crystalaxis

wavenormal

'

o

''eo

air crystal

a

s'

s

n

3D calculus in global coordinates according to Chipman

Ray trace defines local coordinate system

Transform between local and

global coordinates

Polarization Raytrace

inLLinjL

in

inL szxsy

ss

ssx

1,1,,

1

11, ,,

'

'

'',',' 11,1,1,1,1, szxsyxx LLinLLL

zzLzL

yyLyL

xxLxL

out

inzinyinx

zLyLxL

xLyLxL

in

syx

syx

syx

T

sss

yyy

xxx

T

11,1,

11,1,

11,1,

,1

,,,

1,1,1,

1,1,1,

,11

''

''

''

,

plane of

incidence

interface

plane

incoming

ray

outgoing

ray

kj-1 kj

xj

yj

y'j

Embedded local 2x2 Jones matrix

Matrices of refracting surface

and reflection

Field propagation

Cascading of operator matrices

Transfer properties

1. Physical changes

2. Geometrical bending effects

Polarization Raytrace

1,

1,

1,

,

,

,

1

jz

jy

jx

zzyzxz

zxyyxy

zxyxxx

jz

jy

jx

jjj

E

E

E

ppp

ppp

ppp

E

E

E

EPE

121 .... PPPPP MMtotal

100

00

00

,

100

00

00

s

p

rs

p

t r

r

Jt

t

J

100

0

0

2221

1211

,1 jj

jj

J refr

1

,1,1,11

inrefrout TJTP

1

,1,1,11

inbendout TJTQ

Change of field strength:

calculation with polarization matrix,

transmission T

Diattenuation

Eigenvalues of Jones matrix

Retardation: phase difference

of complex eigenvalues

To be taken into account:

1. physical retardance due to refractive index: P

2. geometrical retardance due to geometrical ray bending: Q

Retardation matrix

Diattenuation and Retardation

EE

EPPE

E

EPT

T

*

*

2

2

minmax

minmax

TT

TTD

2/12/12/12/12/1 wewwJ

i

ret

21 argarg

totaltotalPQR

1

Pupil

Vectorial Diffraction at high NA

Linear initial polarization

NA = 0.95

z-component orders of magnitute larger than cross coupling

High NA and Vectorial Diffraction

Relative size of vectorial effects as a function of the numerical aperture

Characteristic size of errors:

I / Io

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

-6

10-5

10-4

10-3

10-2

10-1

100

NA

axial

lateral

error axial lateral

0.01 0.52 0.98

0.001 0.18 0.68

Polarization

Polarization of a donat mode in the focal region:

1. In focal plane 2. In defocussed plane

Ref: F. Wyrowski

Change of incoming linear polarization

in the pupil area

Total or specific decomposition

Polarization Performance Evaluation

negative

positive

piston defocustilt

Müller matrix visualization

Interpretation not trivial

Retardation and diattenuation map

across the pupil

Polarization Zernike pupil aberration

according to M. Totzeck (complicated)

Polarization Performance Evaluation

Relationship between electric field and

displacement vector

Linear relationship of tensor-equation

First term, coefficient :

local direction, birefringence

Second term, coefficient g:

gradient of field, third order, optical activity, polarization rotated

Third term, :

forth order, intrinsic birefringence, spatial dispersion of birefringence

General:

Due to the tensor properties of the coefficients, all these effects are anisotropic

The field E and the displacement D are no longer aligned

Anisotropic Media

ED r

0

qml

lqmjlmq

ml

lmjlm

l

ljlj EEED,,,

g

EEssnEssnD

22

Birefringence

Different direction of E and D

Ray splitting not identical to wave splitting

Ey

electric

field

constant

energy

density

Ex

E

D

displacement

vector

Vanishing solution determinante of the

wave equation

Value of speed of ligth depending on the ray

direction (phase velocity)

Alternative: axis 1/n

Fresnel or Ray Ellipsoid

022

2

22

2

22

2

z

z

y

y

x

x

cc

k

cc

k

cc

k

1/nx

x

y

z

1/ny

1/nz

Ey

electric

field

constant

energy

density

Ex

E

D

displacement

vector

Special case uniaxial crystal:

Two cases:

ce < co : positive ( prolate, cigar )

ce > co : negativ ( oblate, disc )

Ray Ellipsoid for Uniaxial Crystals

optical axis

c

c0

c0

c()

c0ce

Effective refractive index for ray direction :

Intersection point of ray with index ellipsoid

k2

k3

k

n0k

0

n0k

0

n() k0

n0k

0n

ek

0

2

2

2

2

2

sincos1

eooeff nnn

Ray Ellipsoid: Index for Arbitrary Direction

optical

axis

c

c0

c0

c()

c0ce

Inverse matrix of the dielectric tensor:

index or normal ellipsoid

Gives the refractive index as a function of

the orientation

Also possible: ellipsoid of k-values

Index or Normal Ellipsoid

constn

z

n

y

n

x

zyx

2

2

2

2

2

2

z

y

x

r

100

01

0

001

1iin

x

y

z

ny

nz

nx

022

22

22

22

22

22

z

zz

y

yy

x

xx

nn

nk

nn

nk

nn

nk

Dy

electric

fieldconstant

energy

density

Dx

D

E

displacement

vector

Three cases of symmetry:

1. Isotropic 2.. uniaxial 3. biaxial

Index Ellipsoid

k1/k0

isotropic

n

n

nk2/k0

k3/k0

k1/k0

uniaxial

no

k2/k0

k3/k0

no

no

ne

ne

k1/k0

biaxial

k2/k0

k3/k0

n1

n2

n3

n1

n2

n3

Special case uniaxial crystal:

ellipsoid rotational symmetry

no ordinary direction valid for two directions

ne extra ordinary valid for only one direction

Two cases:

ne > no : positive ( prolate, cigar )

ne < no : negativ ( oblate, disc )

Arbitrary orientation : intersection points

Index Ellipsoid for Uniaxial Crystals

optical

axis

a) positive birefringence ne > no

o-ray

e-ray

optical

axis

b) negative birefringence ne < no

o-ray

e-ray

x

y

z

no

ne

no

k2

k3

kn0k0

n0k0

n() k0

n0k0 nek0

Ray direction: k

Normal plane on k intersects the ellipsoid

in an ellipse

The axis of the ellipse are the principal

axis of polarization

The effective index along a field direction

E is given by the corresponding intersection

point

Effective Refractive Indices

n1

x

y

z

n2

n3

kna

nb

Da

Db

index

ellipse

index

ellipsoid

propagation

direction

Orientation of field vectors in the uniaxial crystal for different input orientation

Index Ellipsoid for Uniaxial Crystals

x

a) propagation along z

field E in x-y-plane: o-o

ne

y

crystal

axis z

ne

ne

no

no

kz

Eox

Eoy

x

b) propagation along y

field E in x-z-plane: e-o

ne

y

crystal

axis z

ne

ne

no

no

ky

Eox

Eez

x

c) propagation along x

field E in y-z-plane: e-o

ne

y

crystal

axis z

ne

ne

no

no

kx

Eez

Eoy

Normal ellipsoid can be written as a quartic surface

Index ellipsoid as osculating surface

Special case of sy = 0:

Normal Surface

x

z

index

ellipsoid

ordinary

extra

ordinary

normal

surface

ordinary

extra

ordinary

0222222222222222 yxzzxyzyx ccccsccccsccccs

0222222222 ozeyxo ccsccsscc

Classical physical consideration:

Due to anisotropic refractive index the incoming rays splits into

1. ordinary ray, Snellius law with no

2. extraordinary ray

Poynting vector:

- energy flow

- perpendicular to

wave envelope

Ray Splitting for Refraction at Birefringent Interface

projection of

crystal axis

i

o

e

extraordinary

wave

ordinary

ray

incident

ray

a

se

si

so

te

extraordinary

ray

ordinary

wave

Poynting

direction

z

z

x

k S

phase energy

Incident plane wave

Osculating tangential plane at the ordinary index-sphere:

defines normal to o-ray direction

Osculating tangential plane at the index o/e-index ellipsoid:

normal to e-direction

Wave-Optical Construction of the o/e-direction

crystal

axis

incident ray

o - rayeo-ray

velocity

ellipsoid

wavefronts

Incident ray direction: k

Osculating tangential plane at ellipsoid perpendicular to k:

wavefront, with normal wave vector

Parallel line through center defines effective index n()

Direction from center to tangential point: direction of energy flow

Ray Splitting : Huygens Construction

optical

crystal axis

ne

no ray

energy

wave

vector k

wavefrontn()

tangential

point

Classical uniaxial media used in polarization components:

1. Quartz, positive birefringent, small difference

2. Calcite, negative birefringent, larger difference

Birefringent Uniaxial Media

oe

e

o

quartz calcite

material sign no neo

Calcite negative 1.6584 1.4864

Quarz, SiO2 positive 1.5443 1.5534

Optical symmetry axis of crystal

material breaks symmetry

Split of rays depends on the

axis orientation

Split of field into two orthogonal

polarisation components

Energy propagation (ray,

Poynting) in general not

perpendicular to wavefront

Refraction with Birefringence

divergent rays

crystal

axis

parallel rays

with phase difference

crystal

axiscrystal

axis

parallel rays

in phase

o-ray e-ray

Polarizer with attenuation cs/p

Rotated polarizer

Polarizer in y-direction

p

s

LIN c

cJ

10

01

z

y

x

TA

2

2

sincossin

cossincos)(PJ

10

00)0(PJ

Polarizer

Polarizer and analyzer with rotation

angle

Law of Malus:

Energy transmission

TA

z

y

x

TA

linear

polarizer y

linear

polarizer

E

E cos

2cos)( oII

I

0 90° 180° 270° 360°

Pair Polarizer-Analyzer

parallel

polarizer

analyzer

perpendicular

Phase difference between field

components

Retarder plate with rotation angle

Special value:

/ 4 - plate generates circular polarized light

1. fast axis y

2. fast axis 45°

2

2

0

0

i

i

RET

e

eJ

z

y

x

SA

LA

ii

ii

Vee

eeJ

22

22

cossin1cossin

1cossinsincos),(

iJ V

0

01)2/,0(

1

1

2

1)2/,4/(

i

iiJ V

Retarder

Rotate the of plane of polarization

Realization with magnetic field:

Farady effect

Verdet constant V

bb

bb

cossin

sincosROTJ

z

y

x

b

VLB

b

Rotator

Polarization-prism setups for DIC microscopy:

1. Original Wollaston geometry

2. modified geometries according to Nomarski

Both configurations can be realized in two cases with slightly different ray path

of the o- and e-rays

Advantage: virtual source point is accessible outside prism and can be used

with microscopic objective lens in pupil plane

Possible Configurations for DIC Prisms

o

e o

e

a) case 1 b) case 2

standard setup / Wollaston

o

eo

e

c) case 3 d) case 4

modified setup / Nomarski

Model:

1. definition of a starting polarization

2. every ray carries a Jones vector of polarization, therefore a spatial variation of polarization

is obtained.

3. at any interface, the field is decomposed into s- and p-component in the local system

4. changes of the polarization component due to Fresnel formulas or coatings:

- amplitude, diattenuation

- phase, retardance

Spatial variations of the polarization phase accross the pupil are aberrations,

the interference is influenced and Psf, MTF, Strehl,... are changed

Polarization in Zemax

Starting polarization

Polarization influences:

1. surfaces, by Fresnel formulas or coatings

2. direct input of Jones matrix surfaces with


EJE

'

y

x

imreimre

imreimre

y

x

y

x

E

E

DiDCiC

BiBAiA

E

E

DC

BA

E

E

'

'

Analysis of system polarization:

1. pupil map shows the spatial variant

polarization ellipse

2. The transmission fan shows the variation of

the transmission with the pupil height

3. the transmission table showes the mean

values of every surface


Single ray polarization raytrace:

detailed numbers of

- angles

- field components

- transmission

- reflection

at all surfaces


Detailed polarization analyses are possible at the individual surfaces by using the coating

menue options


Optical Design with Zemax for PhD - Basics

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