www.iap.uni-jena.de

Design and Correction of Optical

Systems

Lecture 3: Paraxial optics

2017-04-28

Herbert Gross

Summer term 2017

2

Preliminary Schedule - DCS 2017

1 07.04. Basics Law of refraction, Fresnel formulas, optical system model, raytrace, calculation

approaches

2 21.04. Materials and Components Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,

aspheres, diffractive elements

3 28.04. Paraxial Optics Paraxial approximation, basic notations, imaging equation, multi-component

systems, matrix calculation, Lagrange invariant, phase space visualization

4 05.05. Optical Systems Pupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,

photometry

5 12.05. Geometrical Aberrations Longitudinal and transverse aberrations, spot diagram, polynomial expansion,

primary aberrations, chromatical aberrations, Seidels surface contributions

6 19.05. Wave Aberrations Fermat principle and Eikonal, wave aberrations, expansion and higher orders,

Zernike polynomials, measurement of system quality

7 26.05. PSF and Transfer function Diffraction, point spread function, PSF with aberrations, optical transfer function,

Fourier imaging model

8 02.06. Further Performance Criteria Rayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based

criteria, further options

9 09.06. Optimization and Correction Principles of optimization, initial setups, constraints, sensitivity, optimization of

optical systems, global approaches

10 16.06. Correction Principles I Symmetry, lens bending, lens splitting, special options for spherical aberration,

astigmatism, coma and distortion, aspheres

11 23.06. Correction Principles II Field flattening and Petzval theorem, chromatical correction, achromate,

apochromate, sensitivity analysis, diffractive elements

12 30.06. Optical System Classification Overview, photographic lenses, microscopic objectives, lithographic systems,

eyepieces, scan systems, telescopes, endoscopes

13 07.07. Special System Examples Zoom systems, confocal systems

1. Paraxial approximation

2. Ideal surfaces and lenses

3. Imaging equation

4. Matrix formalism

5. Lagrange invariant

6. Phase space considerations

7. Delano diagram

3

Contents

Paraxiality is given for small angles

relative to the optical axis for all rays

Large numerical aperture angle u

violates the paraxiality,

spherical aberration occurs

Large field angles w violates the

paraxiality,

coma, astigmatism, distortion, field

curvature occurs

Paraxial Approximation

4

Paraxial approximation:

Law of refraction for finite angles I, I‘

sin-expansion

Small incidence angles allows for a linearization of the law of refraction

Small angles of rays at every surface

All optical imaging conditions become linear (Gaussian optics),

calculation with ABCD matrix calculus is possible

No aberrations occur in optical systems

There are no truncation effects due to transverse finite sized components

Serves as a reference for ideal system conditions

Is the fundament for many system properties (focal length, principal plane, magnification,...)

The sag of optical surfaces (difference in z between vertex plane and real surface

intersection point) can be neglected

All waves are plane of spherical (parabolic)

The phase factor of spherical waves is quadratic

Paraxial approximation

'' inin

R

xi

eExE

2

0)(

...36288050401206

sin9753

iiii

ii

'sin'sin InIn

5

Taylor expansion of the

sin-function

Definition of allowed error

10-4

Deviation of the various

approximations:

- linear: 5°

- cubic: 24°

- 5th order: 542°

6

Paraxial approximation

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

sin(x)

x [°]

exact sin(x)

linear

cubic

5th order

x = 5° x = 24° x = 52° deviation 10-4

Law of refraction

Expansioin of the sine-function:

Linearized approximation of the

law 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' = 1.9

n' = 1.7

n' = 1.5

...!5!3

sin53

xx

xx

'sin'sin InIn

1

'

sinarcsin

'

'

''

n

inn

in

I

Ii

'' inin

7

8

Generalized Paraxiality

Pitfalls in the classical definition of paraxiality:

1. Central obscurartion and ring-shaped pupil:

- Paraxial marginal ray of no relevance

- Reference on centroid ray

2. General 3D system without straight axis:

Central ray as reference, calculated finite

parabasal rays in the neighborhood of the

real chief ray

Distortion information is lost

General quasi-parabasal rays:

- macroscopic astigmatism

- aberrations reference definition more complicated

- separated view on cheif ray / marginal ray

M

F

M1

circular

symmetric

asphere

M2

pupil

freeform M3

freeform

image

Optical Image formation:

All ray emerging from one object point meet in the perfect image point

Region near axis:

gaussian imaging

ideal, paraxial

Image field size:

Chief ray

Aperture/size of

light cone:

marginal ray

defined by pupil

stop

image

object

optical

system

O2field

point

axis

pupil

stop

marginal

ray

O1 O'1

O'2

chief

ray

Optical imaging

9

Single surface

imaging equation

Thin lens in air

focal length

Thin lens in air with one plane

surface, focal length

Thin symmetrical bi-lens

Thick lens in air

focal length

'

1'

'

'

fr

nn

s

n

s

n

21

111

'

1

rrn

f

1'

n

rf

12'

n

rf

21

2

21

1111

'

1

rrn

dn

rrn

f

Formulas for surface and lens imaging

10

Single surface between two media

Radius r, refractive indices n, n‘

Imaging condition, paraxial

Abbe invariant

alternative representation of the

imaging equation

'

1'

'

'

fr

nn

s

n

s

n

y

n'ny'

r

C

ray through center of curvature C principal

plane

vertex S

s

s'

object

surface

image

arbitrary ray

'

11'

11

srn

srnQs

Single Surface

11

Imaging with a lens

Location of the image:

lens equation

Size of the image:

Magnification

Imaging by a Lens

object image

-sf

+s'

system

lens

y

y'

fss

11

'

1

s

s

y

ym

''

12

Ranges of imaging

Location of the image for a single

lens system

Change of object loaction

Image could be:

1. real / virtual

2. enlarged/reduced

3. in finite/infinite distance

Imaging by a Lens

|s| < f'

image virtual

magnified FObjekt

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 at

infinity

image real

magnified

image real

1 : 1

image real

reduced

13

Imaging by a lens in air:

lens makers formula

Magnification

Real imaging:

s < 0 , s' > 0

Intersection lengths s, s'

measured with respective to the

principal planes P, P'

fss

11

'

1

s'

2f'

4f'

2f' 4f'

s-2f'- 4f'

-2f'

- 4f'

real object

real image

real object

virtual object

virtual image

virtual image

real image

virtual image

Imaging equation

s

sm

'

14

Distance object-image:

(transfer length)

L = |s| + |s‘|

Two solution for a given L with

different magnifications

No real imaging for L < 4f

Transfer Length / Total Track L

mmfL

12'

''21

'2

2

f

L

f

L

f

Lm

| 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

Lateral magnification for finite imaging

Scaling of image size

'tan'

tan'

uf

uf

y

ym

z f f' z'

y

P P'

principal planes

object

imagefocal pointfocal point

s

s'

y'

F F'

Magnification

16

Afocal systems with object/image in infinity

Definition with field angle w

angular magnification

Relation with finite-distance magnification

''tan

'tan

hn

nh

w

w

'f

fm

Angle Magnification

h

h'w'

w

17

Imaging equation according to Newton:

distances z, z' measured relative to the

focal points

'' ffzz

Newton Formula

-z -f f' z'

y

P P'

principal planes

image

focal

point F'focal

point F

-s s'

object

18

Graphical image construction

according to Listing by

3 special rays:

1. First parallel through axis,

through focal point in image

space F‘

2. First through focal point F,

then parallel to optical axis

3. Through nodal points,

leaves the lens with the same

angle

Procedure work for positive

and negative lenses

For negative lenses the F / F‘ sequence is

reversed

Graphical Image Construction after Listing

F

F'

y

y'

P'P

1

2

3

F'

F

y

y'

P'P

1

2

3

19

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

F'F

P'P

Qarbitrary ray

desired output ray

parallel ray

through F

f'

S'S

20

Two lenses with distance d

Focal length

distance of inner focal points e

Sequence of thin lenses close

together

Sequence of surfaces with relative

ray heights hj, paraxial

Magnification

n

FFdFFF 21

21

e

ff

dff

fff 21

21

21

k

kFF

k k

kkk

rnn

h

hF

1'

1

kk

k

n

n

s

s

s

s

s

sm

'

''' 1

2

2

1

1

Multi-Surface Systems

21

System of two separated thin lenses

Variation of the back principal plane

as a function of the distribution of

refractive power

22

Principal Planes

P' L1 L2P F

plate

plate

Imaging with tilted object plane

If principal plane, object and image plane meet in a common point:

Scheimpflug condition,

sharp imaging possible

Scheimpflug equation

tan'tan

tantan

'

s

s

Scheimpflug Imaging

'

y

s

s'

h

y'

h'

tilted

object

tilted

image

system

optical axis

principal

plane

23

General :

1. Image plane is tilted

2. Magnification is anamorphic

Example :

Scheimpflug-Imaging

Scheimpflug System

object plane

lens

image plane

'

s

s'

'sin

sin2

oy mm

tan

'tan'

s

smm ox

24

Linear relation of ray transport

Simple case: free space

propagation

Advantages of matrix calculus:

1. simple calculation of component

combinations

2. Automatic correct signs of

properties

3. Easy to implement

General case:

paraxial segment with matrix

ABCD-matrix :

u

xM

u

x

DC

BA

u

x

'

'

z

x x'

ray

x'

u'

u

x

B

Matrix Formulation of Paraxial Optics

A B

C D

z

x x'

ray x'

u'u

x

25

Matrix Calculus

Paraxial raytrace transfer

Matrix formulation

Matrix formalism for finite angles

Paraxial raytrace refraction

Inserted

Matrix formulation

111 jjjj Udyy

1 jjjj Uyi in

nij

j

j

j''

1' jj UU

1 jj yy

1

'

''

j

j

j

j

j

jjj

j Un

ny

n

nnU

'' 1 jjjj iiUU

j

jj

j

j

U

yd

U

y

10

1

'

'1

j

j

j

j

j

jjj

j

j

U

y

n

n

n

nnU

y

'

'01

'

'

j

j

j

j

u

y

DC

BA

u

y

tan'tan

'

26

Linear transfer of spation coordinate x

and 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

uDxCu

uBxAx

'

'

u

xM

u

x

DC

BA

u

x

'

'

mxxA /'

uuD /'

xuC /'

121 ... MMMMM kk

'det

n

nCBDAM

Matrix Formulation of Paraxial Optics

27

System inversion

Transition over distance L

Thin lens with focal length f

Dielectric plane interface

Afocal telescope

AC

BDM

1

10

1 LM

11

01

f

M

'0

01

n

nM

0

1L

M

Matrix Formulation of Paraxial Optics

28

Calculation of intersection length

Magnifications:

1. lateral

2. angle

3. axial, depth

Principal planes

Focal points

Matrix Formulation of Paraxial Optics

DsC

BsAs

'

DsC

BCADm

2'

DsC

BCAD

ds

ds

'sCA

BCADDsC

C

DBCADaH

C

AaH

1'

C

AaF '

C

DaF

29

Decomposition of ABCD-Matrix

2x2 ABCD-matrix of a system in air: 3 arbitrary parameters

Every arbitrary ABCD-setup can be decomposed into a simple system

Decomposition in 3 elementary partitions is alway possible

Case 1: C # 0 one lens, 2 transitions

System data

MA B

C D

L

f

L

1

0 1

1 01

1

1

0 1

1 2

LD

C1

1

LA

C2

1

fC

1

Output

xo

Input

xi

Lens

f

L2

L1

Decomposition of ABCD-Matrix

Case 2: B # 0 two lenses, one transition

System data:

MA B

C D f

L

f

1 01

11

0 1

1 01

12 1

fB

A1

1

L B

fB

D2

1

OutputInput

Lens 1

f1

L

Lens 2

f2

Product of field size y and numercial aperture is

invariant in a paraxial system

Derivation at a single refracting surface:

1. Common height h:

2. Triangles

3. Refraction:

4. Elimination of s, s',w,w'

The invariance corresponds to:

1. Energy conservation

2. Liouville theorem

3. Invariant phase space volume (area)

4. Constant transfer of information

''' uynuynL

Helmholtz-Lagrange Invariant

y

u u'

chief ray

marginal ray

w

h

ss'

w'

n n'

surface

''usush

'

'',

s

yw

s

yw

''wnnw

32

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 to

1. Energy conservation

2. Liouville theorem

3. Constant transfer of information

y

y'

u u'

marginal ray

chief ray

object

image

system

and stop

''' uynuynL

Helmholtz-Lagrange Invariant

33

Basic formulation of the Lagrange

invariant:

Uses image heigth,

only valid in field planes

General expression:

1. Triangle SPB

2. Triangle ABO'

3. Triangle SQA

4. Gives

5. Final result for arbitrary z:

Helmholtz-Lagrange Invariant

ExPCR sswy ''''

ExP

CR

s

yw

''

''

s

yu MR

z

y'y yp

pupil imagearbitrary z

chief

ray

marginal

ray

s's'Exp

y'CR

yMR

yCR

B A

O'

Q

S

P

ExPMRExPMR

CR swuwynssws

ynyunL ''''''''

'''''

)(')('' zyuzywnL CRMR

34

Simple example:

- A microscope is a 4f-system with

objective lens (fobj = 3 mm) and tube lens (fobj = 180 mm)

- the numerical aperture is NA = 0.9 and the intermediate Image size D = 2yima = 25 mm

- magnification

- image sided aperture

- pupil size

- object field

60/ objTL ffm

Helmholtz-Lagrange Invariant

yobj

yimauima

image diameter

25 mm

tube lens:

focal length 180 mm

uobj

objective lens

focal length 3 mm

NA = 0.9

pupil

015.0/ muu objima

mmNAfD objpup 7.2

mmuuyy objimaimaobj 42.0/

35

Geometrical optic:

Etendue, light gathering capacity

Paraxial optic: invariant of Lagrange / Helmholtz

General case: 2D

Invariance corresponds to

conservation of energy

Interpretation in phase space:

constant area, only shape is changed

at the transfer through an optical

system

unD

Lfield

Geo sin2

''' uynuynL

Helmholtz-Lagrange Invariant

space y

angle u

large

aperture

aperture

small

1

23

medium

aperture

y'1

y'2

y'3u1 u2

u3

36

Direct phase space

representation of raytrace:

spatial coordinate vs angle

Phase Space

z

x

x

u u

x

x1

x'1

x2

x'2

x1 x'1

x2

x'2

u1 u2 u1 u2

11'

2

2'

1

1'

2

2'

space

domain

phase

space

37

Phase Space

z

x

I

x

I

x

u

x

u

x

38

Phase Space

z

x

x

u

1

1

2 2'

2 2'

3 3' 4

3

3'

5

4

5

6

6

free

transfer

lens 1

lens 2

grin

lens

lens 1

lens 2

free

transfer

free

transfer

free

transfer

Direct phase space

representation of raytrace:

spatial coordinate vs angle

39

Grin lens with aberrations in

phase space:

- continuous bended curves

- aberrations seen as nonlinear

angle or spatial deviations

Phase Space

z

x

x

u u

x

u

x

40

1. Slit diffraction

Diffraction angle inverse to slit

width D

2. Gaussian beam

Constant product of waist size wo

and divergence angle o

00w

D

D D

o

wo

x

z

Uncertainty Relation in Optics

41

• Angle u is limited

• Typical shapes:

Ray : point (delta function)

Coherent plane wave: horizonthal line

Extended source : area

Isotropic point source: vertical line

Gaussian beam: elliptical area with

minimal size

• Range of small etendues: modes, discrete

structure

• Range of large etendues: quasi continuum

Phase Space in Optics

plane wave

(laser)

x

u

ray

spherical

wave

gaussian

beam

LED

point

source

x

u

discrete

mode

points

quasi continuum

42

Simplified pictures to the changes of the phase space density.

Etendue is enlarged, but no complete filling.

x

u

x

u

x

u1) before array

x

u

x

u

x

u

2) separation of

subapertures

3) lens effect

of subapertures

4) in focal plane 5) in 2f-plane 6) far away

1 2 3 4 5 6

Example Phase Space of an Array

43

Analogy Optics - Mechanics

44

Mechanics Optics

spatial variable x x

Impuls variable p=mv angle u, direction cosine p,

spatial frequency sx, kx

equation of motion spatial

domain

Lagrange Eikonal

equation of motion phase space Hamilton Wigner transport

potential mass m index of refraction n

minimal principle Hamilton principle Fermat principle

Liouville theorem constant number of

particles

constant energy

system function impuls response point spread function

wave equation Schrödinger Helmholtz

propagation variable time t space z

uncertainty h/2p l/2p

continuum approximation classical mechanics geometrical optics

exact description quantum mechanics wave optics

Delano Diagram

Special representation of ray bundles in

optical systems:

marginal ray height

vs.

chief ray height

Delano digram gives useful insight into

system layout

Every z-position in the system corresponds

to a point on the line of the diagram

Interpretation needs experience

CRyy

lens

y

field lens collimatormarginal ray

chief ray

y

y

y

lens at

pupil

position

field lens

in the focal

plane

collimator

lens

MRyy

46

Delano Diagram

Delano’s

skew ray

Image

d2 d1

yC yM

(yC,yM) yM

a a

b

c c

d d

yC Delano ray (blue)=

Chief ray (red) in x +

Marginal ray (green) in y

Delano Diagram =

Delano ray projected

into the xy-Plane

Substitution

x -->

y = Pupil coordinate

= yc Field coordinate

Stop

Lens

a b

c

d

y (or yM)

y

y

y

Ref.: M. Schwab / M. Geiser

chief ray

marginal ray

Delano diagram:

projection

along z

b

x

y

marginal

ray

chief

ray

skew ray

y

y

y

y

diagram

image

object

Pupil locations:

intersection points with y-axis

Field planes/object/image:

intersectioin points with y-bar axis

Construction of focal points by

parallel lines to initial and final line

through origin

y

y

object

plane

lens

image

plane

stop and

entrance pupil

exit pupil

y

y

object

space

image

space

front focal

point Frear focal

point F'

Delano Diagram

Delano Diagram

Influence of lenses:

diagram line bended

Location of principal planes

y

y

strong positive

refractive power

weak positive

refractive power

weak negative

refractive power

y

y

object space image space

principal

plane

yP

yP

Delano Diagram

Afocal Kepler-type telescope

Effect of a field lens

y

y

lens 1

objective

lens 2

eyepiece

intermediate

focal point

y

y

lens 1

objective

lens 2

eyepiece

field lensintermediate

focal point

Delano Diagram

Microscopic system

y

y

eyepiece

microscope

objective tube lens

object

image at infinity

aperture

stop

intermediate

image

exit pupil

telecentric

Conjugated point are located on a

straight line through the origin

Distance of a system point from

origin gives the systems half

diameter

Delano Diagram

y

y

object

space image

space

conjugate

line

conjugate line

with m = 1

principal point

conjugate

points

y

y

maximum height of

the coma ray at

lens 2

curve of

the system

lens 1lens 2

lens 3

D/2

51

Location of principal planes in the Delano diagram

Triplet Effect of stop shift

Delano Diagram

y

y

object

plane

lens L1

lens L2

lens L3

image

plane

stop shift

y

y

object

spaceimage

space

principal plane

yP

yP

Vignetting :

ray heigth from axis

Marginal and chief ray considered

Line parallel to -45° maximum diameter

yya

Delano Diagram

object

pupil

chief ray

marginal ray

coma ray

yyy + y

y

y

maximum height at

lens 2

system polygon line

lens 1

lens 2

lens 3

D/2