Design and Correction of optical Systemsand+correction+of... · Graphical image construction...

Design and Correction of optical Systems

Part 4: Paraxial optics

Summer term 2012Herbert Gross

1

Overview

1. Basics 2012-04-182. Materials 2012-04-253. Components 2012-05-024. Paraxial optics 2012-05-095. Properties of optical systems 2012-05-166. Photometry 2012-05-237. Geometrical aberrations 2012-05-308. Wave optical aberrations 2012-06-069. Fourier optical image formation 2012-06-1310. Performance criteria 1 2012-06-2011. Performance criteria 2 2012-06-2712. Measurement of system quality 2012-07-0413. Correction of aberrations 1 2012-07-1114. Optical system classification 2012-07-18

2012-04-18

4.1 Imaging - basic notations

- paraxial approximation

- linear collineation

- graphical image construction

- lens makers formula

4.2 Optical system properties - special aspects

- imaging

- multiple components

4.3 Matrix calculus - simple matrices

- relations

4.4 Phase space - basic idea

- invariant

Part 4: Paraxial Optics

Optical Image formation:All ray emerging from one object point meet in the perfect image point

Region near axis:gaussian imagingideal, paraxial

Image field size:Chief ray

Aperture/size of light cone:marginal raydefined by pupil stop

Optical Imaging

image

object

optical system

O2fieldpoint

axis

pupilstop

marginal ray

O1 O'1

O'2

chiefray

Single surface between two mediaRadius r, refractive indices n, n‘

Imaging condition, paraxial

Abbe invariantalternative representation of theimaging equation

'1'

''

frnn

sn

sn

'11'11sr

nsr

nQs

Single Surface

Law of refraction

Expansioin of the sine-function:

Linearized approximation of thelaw of refraction: I ----> i

Relative error of the approximation

Paraxial Approximation

i0 5 10 15 20 25 30 35 40

0

0.01

0.02

0.03

0.04

0.05

i'- I') / I'

n' = 1.9n' = 1.7n' = 1.5

...!5!3

sin53

xxxx

'sin'sin InIn

1

'sinarcsin

''

''

nin

nin

IIi

'' inin

Linear Collineation

General rational transformation

Linear expression

Describes linear collinear transform x,y,z ---> x',y',z'

Analog in the image space

Inserted in only 2 dimensions

Focal lengths

Principal planes

0

3

0

2

0

1 ',','FFz

FFy

FFx

3,2,1,0, jdzcybxaF jjjjj

0

3

0

2

0

1

'',

'',

''

FFz

FFy

FFx

3,2,1,0,'''''''' jdzcybxaF jjjjj

0

1

0

33 ','dzc

yaydzcdzcz

oo

01

0303

0

1 ',ca

cddcfcaf

01

030313'

0

01 ,ca

cddcaczc

daz PP

Linear Collineation

Special choice of origin of coordinate systems: Newton imaging equations

Finite angles: tan(u) must be taken:Magnification:

Focal length:

Invariant:

O

O'

y'

y

F F'

P P'

N N'

f'

a'a

f

z'z

u'u

h

uum

tan'tan

huu

ftan'tan

'1

'tan''tan uynuny

Positive lensReal image for s > f

Negative lensvirtual image

F'Fy

f f

y'

s's

F'Fy

f f

y'

s'

s

Image Construction of a Lens by simple Rays

Graphical image constructionaccording to Listing by3 special rays:

1. First parallel through axis,through focal point in imagespace F‘

2. First through focal point F,then parallel to optical axis

3. Through nodal points,leaves the lens with the sameangle

Procedure work for positiveand negative lensesFor negative lenses the F / F‘ sequence isreversed

Graphical Image Construction after Listing

First ray parallel to arbitrary ray through focal point, becomes parallel to optical axis

Arbitrary ray:- constant height in principal planes S S‘- meets the first ray in the back focal plane,

desired ray is S‘Q

General Graphical Ray Construction

Principle setups of the image of apositive lens

Object and image can be real/virtual

Positive Lens

from infinity

real image

virtual image

F F'

F'F

F'F

virtual objectlocation

Ranges of imagingLocation of the image for a singlelens system

Change of object loaction

Image could be:1. real / virtual2. enlarged/reduced3. in finite/infinite distance

Imaging by a Lens

|s| < f'image virtual

magnified F Objekt

s

F object

s

Fobject

s

F

object

s

F'

F

object

image

s

|s| = f'

2f' > |s| > f'

|s| = 2f'

|s| > 2f'

F'

F'

F'

F'

image

image

image

image

image atinfinity

image realmagnified

image real1 : 1

image realreduced

Lateral magnification for finite imaging Scaling of image size 'tan'

tan'ufuf

yym

Magnification

z f f' z'

y

P P'

principal planes

objectimage

focal pointfocal point

s

s'

y'

F F'

Imaging on axis: circular / rotational symmetryOnly 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 bundlenot circular symmetric

- to distinguish:tangential and sagittalplane

Definition of Field of View and Aperture

Afocal systems with object/image in infinity Definition with field angle w

angular amgnification

Relation with finite-distance magnification

''tan'tan

hnnh

ww

Magnification

w

w'

'ff

Axial magnification

Approximation for small z and n = n‘

fzf

fzz

1

1'' 2

'tantan

2

22

uu

Axial Magnification

z z'

Distance object-image:(transfer length)

Two solution for a given L withdifferent magnifications

No real imaging for L < 4f

Transfer Length / Total Track L

mmfL 12'

''21

'2

2

fL

fL

fLm

| m |

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

L / f'

magnified

reduced

Lmin = 4f' mmax = L / f' - 2

4f-imaging

Magnification parameter M:defines ray path through the lens

Special cases:1. M = 0 : symmetrical 4f-imaging setup2. M = -1: object in front focal plane 3. M = +1: object in infinity

The parameter M strongly influences the aberrations

1'

21211

''

sf

sf

UUUUM

Magnification Parameter

M=0

M=-1

M<-1

M=+1

M>+1

Image in infinity:- collimated exit ray bundle- realized in binoculars

Object in infinity- input ray bundle collimated- realized in telescopes- aperture defined by diameter

not by angle

Special Setups of Object / Image

object atinfinity

image infocalplane

lens acts asaperture stop

collimatedentrance bundle

image atinfinity

stop

image

eye lensfield lens

Imaging by a lens in air:lens makers formula

Magnification

Real imaging:s < 0 , s' > 0

Intersection lengths s, s'measured with respective to theprincipal planes P, P'

fss11

'1

ss'

Imaging Equation

s'

2f'

4f'

2f' 4f's

-2f'- 4f'

-2f'

- 4f'

real objectreal image

real objectvirtual objectvirtual image

virtual imagereal image

virtual image

Imaging equation in inverserepresentation with refractivepower and vergence 1/s:linear behavior

Imaging Equation

1/s'

2

4

2 41/s

-2- 4

-2

-4

real objectreal image

real objectvirtaul image

virtual objectvirtual image

virtual objectreal image

-

Imaging equation according to Newton:distances z, z' measured relative to the focal points

'' ffzz

Newton Formula

Two lenses with distance d

Focal lengthdistance of inner focal points e

Sequence of thin lenses closetogether

Sequence of surfaces with relativeray heights hj, paraxial

Magnification

nFFdFFF 21

21

eff

dfffff 21

21

21

k

kFF

k k

kkk

rnn

hhF 1'

1

kk

k

nn

ss

ss

ss

'''' 1

2

2

1

1

Multi-Surface Systems

Focal lengthe: tube length

Image location

Two-Lens System

eff

dfffff 21

21

21 '''''''

1

1

21

212 '

')'(''

')'('f

fdfdfffdfs

Matrix Calculus

Paraxial raytrace transfer

Matrix formulation

Paraxial raytrace refraction

Inserted

Matrix formulation

111 jjjj Udyy

1 jjjj Uyi inn

ijj

jj'

'

1' jj UU

1 jj yy

1'

''

j

j

jj

j

jjjj U

nn

yn

nnU

'' 1 jjjj iiUU

j

jj

j

j

Uyd

Uy

101

'' 1

j

j

j

j

j

jjj

j

j

Uy

nn

nnn

Uy

''

01

''

Linear Collineation

Matrix formalism for finite angles

j

j

j

j

uy

DCBA

uy

tan'tan'

Linear relation of ray transport

Simple case: free space propagation

General case:paraxial segment with matrixABCD-matrix :

ux

Mux

DCBA

ux''

z

x x'

ray

x'

u'

u

xB

Matrix Formulation of Paraxial Optics

A BC D

z

x x'

ray x'

u'u

x

Linear transfer of spation coordinate xand angle u

Matrix representation

Lateral magnification for u=0

Angle magnification of conjugated planes

Refractive power for u=0

Composition of systems

Determinant, only 3 variables

uDxCuuBxAx

''

ux

Mux

DCBA

ux''

xxA /'

uuD /'

xuC /'

121 ... MMMMM kk

'det

nnCBDAM

Matrix Formulation of Paraxial Optics

System inversion

Transition over distance L

Thin lens with focal length f

Dielectric plane interface

Afocal telescope

ACBD

M 1

101 L

M

1101

fM

'0

01

nnM

0

1 LM

Matrix Formulation of Paraxial Optics

Calculation of intersection length

Magnifications:1. lateral

2. angle

3. axial, depth

Principal planes

Focal points

Matrix Formulation of Paraxial Optics

DsCBsAs

'

DsCBCAD

2'

DsCBCAD

dsds

'sCABCADDsC

CDBCADaH

CAaH

1'

CAaF ' C

DaF

Direct phase space representation of raytrace:spatial coordinate vs angle

Phase Space

Phase Space

z

x

I

x

I

x

u

x

u

x

Phase Space

z

x

x

u

1

1

2 2'

2 2'

3 3' 4

33'

5

4

5

6

6

free transfer

lens 1

lens 2

grin lens

lens 1lens 2

free transfer

free transfer

free transfer

Direct phase space representation of raytrace:spatial coordinate vs angle

Grin lens with aberrations inphase space:- continuous bended curves- aberrations seen as nonlinear

angle or spatial deviations

Phase Space

z

x

x

u u

x

u

x

Phase Space: 90°-Rotation

Transition pupil-image plane: 90° rotation in phase space Planes Fourier inverse Marginal ray: space coordinate x ---> angle ' Chief ray: angle ---> space coordinate x'

Product of field size y and numercial aperture is invariant in a paraxial system

The invariant L describes to the phase space volume (area)

The invariance corresponds to1. Energy conservation2. Liouville theorem3. Constant transfer of information

y

y'u u'

marginal ray

chief ray

object

image

system and stop

''' uynuynL

Helmholtz-Lagrange Invariant

Geometrical optic:Etendue, light gathering capacity

Paraxial optic: invariant of Lagrange / Helmholtz

Invariance corresponds to conservation of energy

Interpretation in phase space:constant area, only shape is changedat the transfer through an optical system

uDLGeo sin2

''' uynuynL

Helmholtz-Lagrange Invariant

Laser optics: beam parameter product waist radius times far field divergence angle

Minimum value of L: TEMoo - fundamental mode

Elementary area of phase space:Uncertainty relation in optics

Laser modes: discrete structure of phase space

Geometrical optics: quasi continuum

L is a measure of quality of a beamsmall L corresponds to a good focussability

ooGB wL

GBL

12 nwL nnGB

Helmholtz-Lagrange Invariant

Outlook

Next lecture: Part 5 – Properties of optical systemsDate: Wednesday, 2012-05-16

Content: 5.1 Pupil - basic notations- pupil- special rays- vignetting- ray sets

5.2 Special imaging setups - telecentricity- anamorphotic imaging- Scheimpflug condition

5.3 Canonical coordinates - normalized properties- pupil sphere

5.4 Delano diagram - basic idea- examples