www.iap.uni-jena.de

Design and Correction of Optical

Systems

Lecture 1: Basics

2019-04-08

Herbert Gross

Summer term 2019

2

Preliminary Schedule - DCS 2019

1 08.04. BasicsLaw of refraction, Fresnel formulas, optical system model, raytrace, calculation

approaches

2 15.04. Materials and ComponentsDispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,

aspheres, diffractive elements

3 29.04. Paraxial OpticsParaxial approximation, basic notations, imaging equation, multi-component

systems, matrix calculation, Lagrange invariant, phase space visualization

4 06.05. Optical SystemsPupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,

photometry

5 13.05. Geometrical AberrationsLongitudinal and transverse aberrations, spot diagram, polynomial expansion,

primary aberrations, chromatical aberrations, Seidels surface contributions

6 20.05. Wave AberrationsFermat principle and Eikonal, wave aberrations, expansion and higher orders,

Zernike polynomials, measurement of system quality

7 27.05. PSF and Transfer functionDiffraction, point spread function, PSF with aberrations, optical transfer function,

Fourier imaging model

8 03.06. Further Performance CriteriaRayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based

criteria, further options

9 17.06. Optimization and CorrectionPrinciples of optimization, initial setups, constraints, sensitivity, optimization of

optical systems, global approaches

10 24.06. Correction Principles ISymmetry, lens bending, lens splitting, special options for spherical aberration,

astigmatism, coma and distortion, aspheres

11 01.07. Correction Principles IIField flattening and Petzval theorem, chromatical correction, achromate,

apochromate, sensitivity analysis, diffractive elements

12 08.07. Optical System ClassificationOverview, photographic lenses, microscopic objectives, lithographic systems,

eyepieces, scan systems, telescopes, endoscopes

1. Refraction

2. Fresnel formulas

3. Optical systems

4. Raytrace

5. Calculation approaches

3

Contents

Law of Refraction

Angle deviation at skew incidence

Change of magnification at curved surfaces, lensing effect

surface

normal

interface

plane

n

Q

n'

P

Oi

i'

e

ray

s'

s

4

Law of Refraction

Scalar law of refraction

(Snells law)

Vectorial form

Special case reflection

All vectors in the plane of incidence

Fundamental basis:

Principle of Fermat

Invariance of field components

'sin'sin inin

seess 2'

esen

nse

n

ns

n

ns

2

2

1'

1''

'

n n'incidence

reflection

refraction

interface

normal direction

i

i'i

s

s's'

e

5

Law of Refraction

i

i'

n

n'

a

a'

d

'' anan

'sin'

sin

ida

ida

'sin'sin inin

Simple derivation of the law of refraction:

constant optical path length for two rays,

optical path length:

product of index of refraction times geometrical path length

Geometrical condition of triangles

Insertion delivers the law of

refraction

6

Law of Refraction

n i n i sin ' sin ' Scalar law of refraction

Sine is limited by -1…+1:

1. gracing incidence at i=90°

2. total internal reflection with

i‘ = 90°

for

i0 10 20 30 40 50 60 70 80 90

0

10

20

30

40

50

60

70

80

90

i'

n/n' = 1.9

n/n' = 1.5

n/n' = 1.2

n/n' = 1

n/n' = 1/1.2

n/n' = 1/1.5

n/n' = 1/1.9

total internal reflection

grazing

incidence

nni /'sin

7

Fresnel Formulas

n n'

incidence

reflection transmission

interface

E

B

i

i

E

B

r

r

E

Bt

t

normal to the interface

i

i'i

a) s-polarization

n n'

incidence

reflection

transmission

interface

B

E

i

i

B

E

r

r

B

E t

t

normal to the interface

i'i

i

b) p-polarization

Schematical illustration of the ray refraction ( reflection at an interface

The cases of s- and p-polarization must be distinguished

8

Fresnel Formulas

Electrical transverse polarization

TE, s- or s-polarization, E perpendicular to incidence plane

Magnetical transverse polarization

TM, p- or p-polarization, E in incidence plane

Boundary condition of Maxwell equations

at a dielectric interface:

continuous tangential component of E-field

Amplitude coefficients for

reflected field

transmitted field

Reflectivity and transmission

of light power

TEe

rTE

E

Er

1 TE

TEe

tTE r

E

Et 1

' TMTM r

n

nt

nn EE 2211

tt EE 21

TMe

rTM

E

Er

2rP

PR

e

r 2

cos

'cos't

in

in

P

PT

e

t

||

9

Fresnel Formulas: Stokes Relations

1 rt

1cos

'cos|||| r

i

it

Relation between the amplitude coefficients for reflection/transmission:

1. s-components:

field components additive

minus sign due to phase jump

2. p-components:

energy preservation but change of

area size due to projection,

correction factor, no additivity of

intensities

field

amplitudes

incidence reflection

transmission

Ei

Er

Et

cross section

area

continuous

tangential

component

10

Fresnel Formulas

Coefficients of amplitude for reflected rays, s and p

Coefficients of amplitude for transmitted rays, s and p

tzez

tzezE

kk

kk

inin

inin

innin

innin

ii

iir

'cos'cos

'cos'cos

sin'cos

sin'cos

)'sin(

)'sin(

222

222

tzez

tzezE

knkn

knkn

inin

inin

innnin

innnin

ii

iir

22

22

2222

2222

||'

'

'coscos'

'coscos'

sin'cos'

sin'cos'

)'tan(

)'tan(

tzez

ezE

kk

k

inin

in

innin

in

inin

int

2

'cos'cos

cos2

sin'cos

cos2

'cos'cos

cos2

222

tzez

ezE

knkn

kn

inin

in

innnin

inn

inin

int

22

2

2222||

'

'2

'coscos'

cos2

sin'cos'

cos'2

'coscos'

cos2

11

Fresnel Formulas

Typical behavior of the Fresnel amplitude coefficients as a function of the incidence angle

for a fixed combination of refractive indices

i = 0

Transmission independent

on polarization

Reflected p-rays without

phase jump

Reflected s-rays with

phase jump of p

(corresponds to r<0)

i = 90°No transmission possible

Reflected light independent

on polarization

Brewster angle:

completely s-polarized

reflected light

r

r

t

t

i

Brewster

12

Fresnel Formulas: Energy vs. Intensity

Fresnel formulas, different representations:

1. Amplitude coefficients, with sign

2. Intensity cefficients: no additivity due to area projection

3. Power coefficients: additivity due to energy preservation

r , t

0 10 20 30 40 50 60 70 80 90-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

i

t

r

r

t

13

R , T

i0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

T

R

R

R , T

i0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

T

R

R

Fresnel Formulas

Reflectivity and transmittivity of power

Arbitrary azimuthal angle t of polarization: decompositioin of components

In case of vanishing absorption:

Energy preservation

Special case of normal incidence

Typical values for some glasses and optical materials in air

)'(sin

)'(sin2

2

ii

iiR

)'(tan

)'(tan2

2

||ii

iiR

)'(sin

2cos'2sin2 ii

iiT

)'(cos)'(sin

'2sin2cos22||

iiii

iiT

ee RRR tt 22

|| sincos ee TTT tt 22

|| sincos

1TR

2

'

'

nn

nnR

2'

'4

nn

nnT

n R

1.4 2.778 %

1.5 4.0 %

1.8 8.16 %

2.4 16.96 %

14

Transmission in Optical Systems

Residual reflectivity of the (identical) surfaces in an optical system with n surfaces:

Overall transmission of energy:

Transmission decreases

nonlinear

Practical consequences:

1. loss of signal energy

2. contrast reduction in

case of imaging

3.occurence of ghost images

n0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

R = 1 %

R = 2 %

R = 4 %

R = 6 %R = 10 %

nges RT 1

15

Brewster Angle

Brewster case of reflection:

reflected and transmitted ray are perpendicular

Condition of Brewster angle

The reflected light is completely

s polarized

Application:

stack of plates under Brewster angle

as polarizer

n

niB

'tan

90'ii

n n'

incidence

reflectiontransmission

interface

normal to the interface

Brewster case

i

90°

i'

i

16

Total Internal Reflection in Fibers

Total internal reflection between core and cladding in a step index fiber

Ref: M. Kaschke

17

Total Internal Reflection

Limiting angle ic of total internal

reflection:

no light leaves the medium with

the higher index

Condition:

n

nic

'sin

1|| RR

n

n'

i

i'

iic

2. limiting case of total internal reflection

3. total internal reflection1. refraction

refracted rays

total internal

reflection

medium n

source point

plane of interface

total internal

reflection

18

Total Internal Reflection

In case of total internal reflection, the Fresnel formulae reads

The wave penetrates the boundary

and generates an evanescent wave

which propagates

along the boundary

22223

22223

||

'sin1'cos

'sin1'cos

ninnin

ninninr

222

222

'sin1cos

'sin1cos

ninin

nininr

x

phase surfaces

evanescent :

parallel

incident

reflected

z

19

20

Evanescent Field

Visualization of the evanescent wave

Evanescent field: finite penetration depth

propagation parallel to the interface plane

Data: n = 1.5 , n‘ = 1.0 ; q = 45°

Ref: Peatros

Description of Optical Systems

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

21

Modelling of Optical Systems

Principal purpose of calculations:

1. Solving the direct problem of

understanding the properties:

analysis

2. Solving the inverse problem:

Finding the concret system data

for a required functionality:

synthesis

System, data of the structure(radii, distances, indices,...)

Function, data of properties,

quality performance(spot diameter, MTF, Strehl ratio,...)

Analysisimaging

aberration

theory

Synthesislens design

inverse

problem

Ref: W. Richter

22

Approximation of Optical Models

Imaging model with levels

of refinement

Paraxial model

(focal length, magnification, aperture,..)

approximation

l à 0

no description of

short pulses

Geometrical

optics

Analytical approximation

(3rd order aberrations,..)

exact geometry

Wave optics

no time dependence

Maxwell equations

Scalar approximation

Helmholtz equation

(PSF, OTF,...)

linear

approximation

no description of

small structures

and polarization

effects

no diffraction

no higher order

aberrations

no aberrations

exact

23

Five levels of modelling:

1. Geometrical raytrace with analysis

2. Equivalent geometrical quantities,

classification

3. Physical model:

complex pupil function

4. Primary physical quantities

5. Secondary physical quantities

Blue arrows: conversion of quantities

Modelling of Optical Systems

ray

tracing

optical path

length

wave

aberration W

transverse

aberrationlongitudinal aberrations

Zernike

coefficients

pupil

function

point spread

function (PSF)

Strehlnumber

optical

transfer function

geometricalspot diagramm

rms

value

intersectionpoints

final analysis reference ray in

the image space

referencesphere

orthogonalexpansion

analysis

sum of

coefficientsMarechal

approxima-

tion

exponentialfunction

of the

phase

Fourier

transformLuneburg integral

( far field )

Kirchhoffintegral

maximum

of the squared

amplitude

Fouriertransformsquared amplitude

sum of

squaresMarechalapproxima-

tion

integration ofspatial

frequencies

Rayleigh unit

equivalencetypes of

aberrationsdifferen

tiationinte-

gration

full

aperture

single types of aberrations

definition

geometricaloptical

transfer function

Fouriertransform

approximation

auto-correlationDuffieux

integral

resolution

threshold value spatial frequency

threshold value spatial

frequency approximationspot diameter

approximation diameter of the

spot

Marechalapproximation

final analysis reference ray in the image planeGeometrical

raytrace

with Snells law

Geometrical

equivalents

classification

Physical

model

Primary

physical

quantities

Secondary

physical

quantities

24

Scheme of Raytrace

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

25

Workflow Raytrace

transport of the ray

over a distance in the

medium

homogeneous medium

equation of a straight line

inhomogeneous medium

solution of the eikonal equation

e.g. with Runge-Kutta method

calculation of the

intersection of the ray

with the next surface

sequential raytrace :

next surface index

non-sequential raytrace : correct

index from the smallest distance of all

alternatives

surface of second order :

analytical solution for intersection

point with the ray

aspherical surface :

numerical iterative calculation of the

intersection point

check if intersection

point is correct

vignetting of the ray due to the size of

the surface

ray does not hit the surface

calculation of the new

ray direction

transition of the ray

into the new medium

refraction of the ray

reflection of the ray

new direction due to diffraction at a

diffractive surface

change of the direction due to

scattering at the surface

special case : total reflection

Step

No j

Step

No j+1

Two step process:

1. Transition to next surface

intersection point,

test of plausibility

2. Dielectric interface

refraction and new direction

test of plausibility

26

Raytrace Formulas

Different sets of formulas:

1. Paraxial formulas, for reference of ideal imaging

2. Meridional formulas, for circular symmetry

3. Vectorial formulas, 3D systems, state of the art today

4. Differential formulas, transfer of neighbourhood, ray density, astigmatism

Special aspects:

1. Aspherical surfaces, numerical iterative calculation of intersection points

2. Gradient media, eikonal differentail equation, Runge-Kutta numerical stepwise

3. Diffractive elements, local grating equation

4. Non-sequential raytrace, illumination and straylight

5. Scattering surfaces, Monte-Carlo decision for new direction

6. Photometric correct raytracing, transfer of relative weighting factor

7. Polarization raytrace, transfer of Jones vector on a ray

8. Geometrical approximated edge diffraction, ray deviation depends on edge distance

27

Paraxial y-U-Method

Paraxial ray trace

formulas

Parameters of ray

description:

1. ray height y

2. ray angle U

Transfer to next surface

index j -à j+1

height

angle of incidence

refraction

new direction angle

y y d Uj j j j 1 1 1

1 jjjj Uyci

in

nij

j

j

j''

''' 1 jjjjjjj icyiiUU

i

i'

rsinu-U

-U'-U

QQ'

-rsinur

L'

L

M

28

Normal Distance of Rays to Surface Vertex

Projection of ray to surface vertex point

length: Q

In the paraxial regime identical with ray intersection height y

yo

z

object

plane

starting

plane

entrance

pupil

marginal

raychief ray

yp

U1

W1

QRS1

QHS1

normals to

the ray

29

Exact Meridional Q-U-Raytrace Method

Exact raytrace scheme in the meridional plane

Ray description parameters:

- angle u with optical axis

- distance Q between ray and surface vertex point

Set of formuals:

(1) angle of incidence

(2) refraction

(3) new angle of ray

(4) auxiliary parameter

(5) distance to vertex

(6) intersection length

(7) final distance

uQci sinsin cos sini i 1 2

sin ''sini

n

ni cos ' sin 'i i 1 2

u u i i' '

GQ

u i

cos cos

'cos'coscos

'cos'sin'sin'

iu

G

u

uQ

c

uiQ

Q Q d uj j 1 sin '

L L dj j 1

Q L u1 sin

30

Exact Meridional Q-U-Raytrace Method (2)

Set of formuals (2):

(8) final intersection length

(9) intersection point

(10) failure condition 1:

ray don‘t hit surface

(11) failure condition 2:

total internal reflection

LQ

u'

'

sin '

)cos(1)sin(

iuGc

iuy

)sin()cos(1

iuGc

iuz

sin i 1

sin 'i 1

31

Vectorial 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

32

Vectorial Raytrace Formulas

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

33

Vectorial Raytrace Formulas (2)

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

34

Vectorial Raytrace Formulas (3)

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

35

Conic Sections

222

22

111 yxc

yxcz

1

2

b

a

2a

bc

1

1

cb

1

1

ca

Explicite surface equation, resolved to z

Parameters: curvature c = 1 / R

conic parameter

Influence of k on the surface shape

Relations with axis lengths a,b of conic sections

Parameter Surface shape

= - 1 paraboloid

< - 1 hyperboloid

= 0 sphere

> 0 oblate ellipsoid (disc)

0 > > - 1 prolate ellipsoid (cigar )

36

Aspherical Surface Types

q

22 yxz

222

22

111 yxc

yxcz

22

22 yxRRRRz xxyy

Conic section

Special case spherical

Cone

Toroidal surface with

radii Rx and Ry in the two

section planes

Generalized onic section without

circular symmetry

Roof surface

2222

22

1111 ycxc

ycxcz

yyxx

yx

z y tanq

37

Aspherical Surfaces with Rotational Symmetry

22

122

2

111)(

k

k

k ycyc

ycyz

z

y

height

y

z(y)

deviationz

aspherical

surface

spherical

surface

Classical representation of an aspheric surface with rotational symmetry:

- basic shape of a conic section

- correction of real sag height with Taylor expansion

Aspherical constants cj

Usually only even orders, no cusp on axis

Problems with this representation:

- deviation z not perpendicular to surface

- Taylor expansion terms not orthogonal

- oscillations of higher order terms

38

Numerical Iterative Raytrace at Aspheres

Calculation of the intersection point with an aspherical surface:

- no fast analytical calculation possible

- iterative numerical computation is used

- often problems with stability for

steep aspheres

General scheme:

- intersection point with vertex plane Q0

- projection onto surface, point Q1

- determine the tangential plane in Q1

- intersection with tangential plane Q‘1

- projection onto surface, point Q2

- …

Qo

y

z

tangential plane in Q1

surface No j

ray

e2

unit vector in Q

Q1

Q2

Q'1

tangential plane in Q2

Q

sj

z-projection

vertex plane

39

egdn

gms

n

ns

l ˆ''

'

Diffracting Surfaces

grooves

s

s

e

d

gp

p

Surface with grating structure:

new ray direction follows the grating equation

Local approximation in the case of space-varying

grating width

Raytrace only into one desired diffraction order

Notations:

g : unit vector perpendicular to grooves

d : local grating width

m : diffraction order

e : unit normal vector of surface

Applications:

- diffractive elements

- line gratings

- holographic components

40

Raytracing in Grin Media

s

y

x

Strahl

Brechzahl :

n(x,y,z)

b

c s

b

c

y'

x'

z

nn

y

nn

x

nn

Dnndt

rd

2

2

Ray: in general curved line

Numerical solution of Eikonal equation

Step-based Runge-Kutta algorithm

4th order expansion, adaptive step width

Large computational times necessary for high accuracy

41

Description of Grin Media

3

15

2

1413

3

12

2

1110

4

9

3

8

2

76

8

5

6

4

4

3

2

21,

ycycycxcxcxc

zczczczchchchchchcnn o

l

Analytical description of grin media by Taylor expansions of the function n(x,y,z)

Separation of coordinates

Circular symmetry, nested expansion with mixed terms

Circular symmetry only radial

Only axial gradients

Circular symmetry, separated, wavelength dependent

8

19

6

18

4

17

2

1615

38

14

6

13

4

12

2

1110

2

8

9

6

8

4

7

2

65

8

4

6

3

4

2

2

1,

hchchchcczhchchchccz

hchchchcczhchchchcnn o

l

n n c c h c c h c c h c c h c c ho , ( ) ( ) ( ) ( ) ( )l 1 2 1

2

3 1

4

4 1

6

5 1

8

6 1

10

n n c c z c c z c c z c c zo , ( ) ( ) ( ) ( )l 1 2 1

2

3 1

4

4 1

6

5 1

8

n n c h c h c h c h c z c z c zo , , , , , , , ,l l l l l l l l1

2

2

4

3

6

4

8

5 6

2

7

3

42

Gradient Lenses

Refocusing in parabolic profile

Helical ray path in 3 dimensions

axis ray bundle

off axis ray bundle

waist

points

view

along z

perspectivic viewy

x

y

x

y'

x'

z

43

Non-Sequential Raytrace

Conventional raytrace:

- the sequence of surface hits of a ray is pre-given and is defined by the index vector

- simple and fast programming of the surface-loop of the raytrace

Non-sequential raytrace:

- the sequence of surface hits is not fixed

- every ray gets ist individual path

- the logic of the raytrace algorithm determines the next surface hit at run-time

- surface with several new directions of the ray are allowed:

1. partial reflection, especially Fresnel-formulas

2. statistical scattering surfaces

3. diffraction with several grating orders or ranges of deviation angles

Many generalizations possible:

several light sources, segmented surfaces, absorption, …

Applications:

1. illumination modelling

2. statistical components (scatter plates)

3. straylight calculation

44

Nonsequential Raytrace: Examples

Signal

1 2 3 4

Reflex 1 - 2

Reflex 3 - 2

1

2

3

1. Prism with total internal

reflection

2. Ghost images in optical systems

with imperfect coatings

45

Non-Sequential Raytrace: Examples

3. Illumination systems, here:

- cylindrical pump-tube of a solid state laser

- two flash lamps (A, B) with cooling flow tubes (C, D)

- laser rod (E) with flow tube (F, G)

- double-elliptical mirror

for refocussing (H)

Different ray paths

possible

A: flash lamp gas

H

4

B: glass tube of

lamp

C: water cooling

D: glass tube of cooling

5

6

3

2

1

7

E: laser rod

F: water cooling

G: glass tube of cooling

46