Design and Correction of optical Systems

Part 3: Components

Summer term 2012

Herbert Gross

1

Overview

1. Basics 2012-04-18

2. Materials 2012-04-25

3. Components 2012-05-02

4. Paraxial optics 2012-05-09

5. Properties of optical systems 2012-05-16

6. Photometry 2012-05-23

7. Geometrical aberrations 2012-05-30

8. Wave optical aberrations 2012-06-06

9. Fourier optical image formation 2012-06-13

10. Performance criteria 1 2012-06-20

11. Performance criteria 2 2012-06-27

12. Measurement of system quality 2012-07-04

13. Correction of aberrations 1 2012-07-11

14. Optical system classification 2012-07-18

2012-04-18

Part 3: Components

3.1 Lenses - description and parameters

- imaging formulas

3.2 Mirrors

3.3 Prisms - dispersion prims

- reflection prisms

- miscellaneous

3.4 Special components - gratings

- aspheres

- diffractive elements

- taper

- gardient lenses

Optical Imaging

� Optical image formation:

All rays starting in an object point meet in one image point

� Real image:

intersection length positive

� Virtual image:

intersection length negative

� Region near the optical axis:

ideal, paraxial, gaussian

linear, aberration-free

� Object and image:

conjugated

Single Surface

� 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=

−=−

−⋅=

−⋅=

'

11'

11

srn

srnQs

Sag of a Spherical Surface

� Sag z at height y for a spherical

surface:

� Paraxial approximation:

quadratic term

22yrrz −−=

r

yz p

2

2

≈

Cardinal Elements of a Lens

� Focal points:

1. incoming parallel ray

intersects the axis in F‘

2. ray through F is leaves the lens

parallel to the axis

� Principal plane P:

location of apparent ray bending

y

f '

u'P' F'

sBFL

sP'

principal

plane

focal plane

nodal planes

N N'

u

u'

� Nodal points:

Ray through N goes through N‘

and preserves the direction

Notations of a Lens

P principal point

S vertex of the surface

F focal point

s intersection point

of a ray with axis

f focal length PF

r radius of surface

curvature

d thickness SS‘

n refrative index

O

O'

y'

y

F F'

S

S'

P P'

N N'

n n n1 2

f'

a'

f'BFL

fBFL

a

f

s's

d

sP

s'P'

u'u

Lens Shapes

� Different shapes of singlet lenses:

1. bi-, symmetric

2. plane convex / concave, one surface plane

3. Meniscus, both surface radii with the same sign

� Convex: bending outside

Concave: hollow surface

� Principal planes P, P‘: outside for mesicus shaped lenses

P'P

bi-convex lens

P'P

plane-convex lens

P'P

positivemeniscus lens

P P'

bi-concave lens

P'P

plane-concave

lens

P P'

negativemeniscus lens

Bending of a Lens

� Bending: change of shape for

invariant focal length

� Parameter of bending

r1

r2

X = -1

X = 0.5 1 1.5 2 3 4 5 7 10 20

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

-3

-2

-1

0

1

2

3

4

X = -1.5

X = - 0.5

X = 0

X = - 2

X = - 3

X = - 4X = - 5X = - 7

X = - 10X = - 20

12

21

rr

rrX

−

+=

X=1

X>1

X=0

X=-1

meniscus lensX<-1

biconvex lens

biconcave lens

planconvex lens

planconcave lens

planconvex lens

planconcave lens

meniscus lens

Bending of a Lens and Principal Planes

� Ray path at a lens of constant focal length and different bending

� The ray angle inside the lens changes

� The ray incidence angles at the surfaces changes strongly

� The principal planes move

For invariant location of P, P‘ the position of the lens moves

P P'

F'

X = -4 X = -2 X = +2X = 0 X = +4

Properties of a Lens

� Main notations and properties of a lens:

- radii of curvature r1 , r2

curvatures c

sign: r > 0 : center of curvature

is located on the right side

- thickness d along the axis

- diameter D

- index of refraction of lens material n

� Focal length (paraxial)

� Optical power

� Back focal length

intersection length,

measured from the vertex point

2

2

1

1

11

rc

rc ==

'tan',

tan

'

u

yf

u

yf F ==

'

'

f

n

f

nF =−=

'' ' HF sfs +=

Formulas of Surface and Lens Imaging

� 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 ⋅

−+

−⋅−=

Thick Lens

� Definition of thick / non-thin:

1. geometrical:

thickness mach smaller than radius

2. physical:

significant difference of the ray height at front and rear surface

� Differences in bending point

and angle of exit ray

1,1 21 <<⋅<<⋅ dcdc

11

1

1

<<⋅⋅−

=∆

cdn

n

y

y

Conic Mirror: Paraboloid

Equation

c : curvature 1/Rs

κ : eccentricity ( = -1 )

radii of curvature :

22

2

)1(11 cy

ycz

κ+−+=

2

tan 1

+⋅=

s

sR

yRR

2

3

2

tan1

+⋅=

s

sR

yRR

vertex circle

parabolic

mirror

F

f

z

y

R s

C

Rsvertex circle

parabolic

mirror

F

y

z

y

ray

Rtan

x

Rsag

tangential circle

of curvature

sagittal circle of

curvature

Simple Asphere – Parabolic Mirror

sR

yz

2

2

=

axis w = 0° field w = 2° field w = 4°

� Equation

� Radius of curvature in vertex: Rs

� Perfect imaging on axis for object at infinity

� Strong coma aberration for finite field angles

� Applications:

1. Astronomical telescopes

2. Collector in illumination systems

Conic Mirror: Ellipsoid

Equation

c: curvature 1/R

κ: Eccentricity

22

2

)1(11 cy

ycz

κ+−+=

Simple Asphere – Elliptical Mirror

22

2

)1(11 cy

ycz

κ+−+=

F

s

s'

F'

� Equation

� Radius of curvature r in vertex, curvature c

eccentricity κ

� Two different shapes: oblate / prolate

� Perfect imaging on axis for finite object and image loaction

� Different magnifications depending on

used part of the mirror

� Applications:

Illumination systems

Modelling a Mirror Surface

� Problem in coordinate system based raytracing of mirror systems:

right-handed systems becomes left-handed

� Possible solutions:

1. Folding the mirror

- light propagation direction changed

z-component inverted

- tunnel diagram for prism

2. negative refractive index

3. inversion of the x-axis

r

sphericalmirror

F

f'

zC

P=P'

folded mirror

surface

Single Plane Dielectric Interface

� Refraction an single plane interface:

intersection length changed

� Optical denser medium:

length increased

� Example:

View into water, ground seems to

be nearer

n

nss

'' ⋅=

s

s'

n = 1 n' > 1

Medium

Plane Parallel Platte

� Plane parallel plate: image location changed,

Intersection length increased

� Reflection prisms works optically as plane parallel plate

inside optical systems

� Finite numerical aperture:

Generation of spherical aberration

Application: cover glass in microscopy

� Non-parallel ray path:

Generation of astigmatism

Application: Prism positions in

collimated beam path preferred

z

d

∆ s

u

3

1'

dd

n

ns ≈⋅

−=∆

( )3

22

2

sin1'

n

unds sph

−⋅=∆

( )3

22sin1

'n

wnds ast

⋅−⋅=∆

Dispersion Prism

� Parameters for characterization:

1. length of basis side: b

2. wedge angle α

� Angle of ray deviation:

1. exact

2. approximation for small wedge angles

αϕ ⋅−= )1(n

]sincossinsinarcsin[ 11

22

1 iini ⋅−⋅−+−= αααϕ

αααα

ϕϕϕϕ

b

I1

I'1 I

2

I'2

Dispersion Prism

� Angle deviation ϕ changes with tilt of prism

� Minimum value of deviation,

approximately realized for a symmetrical ray path

� The deviations allows to

measure the refractive

index

Dispersion Prism

� White light dispersion by a prism wedge

Dispersion Prism

� Separation of white light into the

spectral components due to

dispersion

� Normal dispersion:

blue color stronger bending than red color

� Application.

spectral spreading of the wavelengths

in spectrometer

λα

αϕ

d

dn

ndn

d⋅

⋅−

⋅=

2sin1

2sin2

22

ϕϕϕϕ

ϕϕϕϕ∆∆∆∆

red

green

blue

white

Prism Magnification

� A beam with diameter Din changes ist width for a non-symmetrical path through a

dispersion prism

� The magnification of the prism defines the change of diameter (in the main section) as

� The prism magnification depends on

angle of incidence and refractive index

� Application:

Anamorphotic prism pairs,

change of ellipticity of beam cross

section in laser beam guiding

2

2

1

1

'cos

cos

'cos

cos

i

i

i

iM ⋅=

Din D

out

Dprism

i1

i'1

i'2

i2

Anamorphotic Prism Pair

� Appropriate combination of two dispersion prisms:

Change of cross section ellipticity of a collimated beam without angle deviation

‚prism compressor‘

� Application: transform the profile of semiconductor laser beams in a circular cross section

D out

D in

beam cross-section

at the entrance

beam cross-sectionat the exit

Din

θθθθ

Dout

αααα

( )2tansincos αθθ ⋅−=

in

out

D

D

Reflection Prisms

Properties of reflection prims:

� Bending of the beam path, deflection of the axial ray direction

Application in instrumental optics and folded ray paths

� Parallel off-set, lateral displacement of the axial ray

� Modification of the image orientation with four options:

1. Invariant image orientation

2. Reverted image ( side reversal )

3. Inverted image ( upside down )

4. Complete image inversion (inverted-reverted image)

� The number of mirrors is important

Every mirror generates a complete inversion,

No change for even numbers

l/r and u/d separation by roof-edge prisms

� Off-set of the image position, shift of image position forwards in the propagation direction.

� Aberrations introduced

1. Astigmatism

2. Chromatic aberration

3. Spherical aberration in non-collimated beams

Comparison: Mirror vs. Prism Systems

Prisms Mirrors

Transmission utilizing total internal reflection +

Chromatic properties, dispersion +

Weight +

Centering sensitivity , monolithic components +

Complexity, number of mechanical holders +

Coatings +

Material absorption and inhomogeneities +

Aberrations in a non-parallel beam path +

Ghost images +

Complexity of alignment +

Separately adjustable reflecting surfaces +

Tunnel Diagram

� Tunnel diagram:

Unfoldung the ray path with invariant sign of the z-component of the optical axis

� Optical effect of prisms corresponds to plane parallel plates

� More rigorous model:

Exact geometry of various prisms can cause vignetting

3

1 2

2

3

Transformation of Image Orientation

� Modification of the image orientation with four options:

1. Invariant image orientation

2. Reverted image ( side reversal )

3. Inverted image ( upside down )

4. Complete image inversion

(inverted-reverted image)

� Image side reversal in the

principal plane of one mirror

� Inversion for an odd number

of reflections

� Special case roof prims:

Corresponds to one reflection

in the edge plane,

Corresponds to two reflections

perpendicular to the edge plane

y

x

y

x

y

x

mirror 1

mirror 2

y - z- foldingplane

z

z

Transformation of Image Orientation

image reversion in the

folding plane(upside down)

image

unchanged

imageinversion

original

folding planeimage reversion

perpendicular to the

folding plane

Transform of Image Orientation

� Rotatable Dove prism:

Azimutal angle: image rotates by the double angle

� Application: periscopes

object

Bild

0° 45° 90°angle of prism

rotation

angle of image

rotation 0° 90° 180°

Roof-Edge Prism

� Roof edge:

- two reflecting surfaces with 90°

- change of lateral coordinate in one section

� Critical in practice:

Precision of 90°angle,

typical tolerance 1‘‘

errors cause image split

� Coatings critical due to

polarization effects

sA

D

B

C

roof edge

intersection planewith angle of 90°ββββ

intersectionplanes withangles of 2ββββ

ϕϕϕϕ

Types of Reflection Prisms: 90° Prism

� Classical 90°prism

� Version with roof edge

� Version with arbitrary deviation angle

(Amici prism)

90°

D

δδδδ

αααα

b

D

h

ββββ

Types of Reflection Prisms: Porro Prism

� Porrro Prism

� Incoming ray direction inverted in one section

� Version with roof-edge:

Ray direction inverted in 3D (retro reflector, cats eye reflector)

D

90°

a

v

Types of Reflection Prisms: Penta Prism

� Classical penta prism

� Penta prism with roof edge

� Penta prism with arbitrary deviation angle

D

90°

b

d

22.5°

D

90°

a

67.5°

D

αααα

δδδδ

ββββ

Types of Reflection Prisms: Bauernfeind Prism

� Classical Bauernfeind prism

� One surface used for entrance and

in reflection

� Prism with roof-edge

D

ββββ

D

a

ααααδδδδ

αααα/2 αααα

Deviation of Light

Mechanisms of light deviation and ray bending

� Refraction

� Reflection

� Diffraction according to the grating equation

� Scattering ( non-deterministic)

'sin'sin θθ ⋅=⋅ nn

'θθ −=

( )g mo⋅ − = ⋅sin sinθ θ λ

Reflection Grating

� Geometry of grating diffraction

� Generation of diffraction orders

y

x

z

incidence

diffracted

orders

g

h

0.

+1.

-1.

+2.

-2.

Grating Diffraction

� Maximum intensity:

constructive interference of the contributions

of all periods

� Grating equation

( )g mo⋅ − = ⋅sin sinθ θ λ

grating

g

incidentlight

+ 1.

diffraction

order

∆∆∆∆s = λλλλ

in-phase

θθθθ

θθθθοοοο

grating

constant

Grating Equation

� Intensity of grating diffraction pattern

(scalar approximation g >> λ)

� Product of slit-diffraction and

interference function

� Maxima of pattern:

coincidence of peaks of both

functions: grating equation

� Angle spread of an order decreases

with growing number od periods N

� Oblique phase gradient:

- relative shift of both functions

- selection of peaks/order

- basic principle of blazing

2

22

sin

sinsin

⋅

⋅

⋅⋅=

λ

π

λ

π

λ

π

λ

π

ugN

ugN

ug

ug

gNI

( ) λθθ ⋅=−⋅ mg osinsin

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u = π/λπ/λπ/λπ/λ sinθθθθ

Real Diffraction Grating

Real diffraction grating:

1. Finite number of periods

2. Finite width of diffraction orders

grating

incidentlight

- 1.

0.

diffractionorders :

finite width

+ 1.∆λ∆λ∆λ∆λ : spectral width

∆θ∆θ∆θ∆θ finite divergenceN : finite

number ofperiods

∆θ∆θ∆θ∆θ

Spectral Resolution of a Grating

� Angle dispersion of a grating

� Separation of two spectral lines

� Complete setup with all orders:

Overlap of spectra possible at higher orders

m

m

d

dD

θλ

θθ

λ

θ

cos

sinsin 0

⋅

−==

NmLA m ⋅=⋅−

=∆

=λ

θθ

λ

λ 0sinsin

0.+1. +2.

+3.+4.-4.

-3.-2. -1.

0

0.2

0.4

0.6

0.8

1

I(x)

mλ λ λ λ /g

sinθθθθ

∆θ∆θ∆θ∆θm(λ+∆λ) λ+∆λ) λ+∆λ) λ+∆λ) /g

Diffractive Optics:

� Local micro-structured surface

� Location of ray bending :

macroscopic carrier surface

� Direction of ray bending :

local grating micro-structure

macroscopicsurface

curvature

local

gratingg(x,y)

lens

bendingangle

ϕϕϕϕ

m-th

order

thinlayer

Diffractive Elements

� Original lens height profile h(x)

� Wrapping of the lens profile: hred(x) Reduction on

maximal height h2π

� Digitalization of the reduced profile: hq(x)

Diffraction Orders

diffractivestructure

diffraction orders

mm-1

m-2m-3

m+2m+1

m+3

desiredorder

� Usually all diffraction orders are obtained simultaneously

� Blazed structure: suppression of perturbing orders

� Unwanted orders: false light, contrast and efficiency reduced

Fresnel Zone Plate

� Circular rings at radii

� Classical Fresnel zone lens:

only rings with same sign of phase have tranmission 1

� Modern zone plate (Wood):

phase steps of π at the rings,

improved power transmission

fr r r r r12345 F

f+λf+2

f+3

f+4

f+5

λλ

λ

λ

λ⋅⋅= fmrm 2

Diffractive Lens

� Diffractive Fresnel lens

� Zone rings with radii

� Blaze in every zone (surface slope)

λπ ⋅⋅⋅⋅= fkmrk 2

( ) kkk

k

k rrn

mh ψ

θ

λtan

cos1 ⋅−=

−

⋅= +

Gradient Lens Types

� Curved ray path in inhomogeneous media

� Different types of profiles

n(x,y,z)

non

i

nentrance

(y)

y

z

nexit

(y)

radial gradient

rod lens

axial gradient

rod lens

radial and axial

gradient

rod lens

radial gradient

lens

axial gradient

lens

radial and axial

gradient lens

Selfoc Lens

L

P P'

F

F'

� Transverse parabolic profile of refractive index:

Rod works as a periodical focussing lens

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

Gradient Lenses

� Types of lenses with parabolic profile

� Pitch length

ymarginal

ycoma

( )

⋅−⋅=

⋅−⋅=

⋅−=

2

0

2

0

2

20

2

11

1

)(

rAn

rnn

rnnrn

r

rnn

np

2

2

22

2

0 ππ ====⋅⋅⋅⋅====

0.25 Pitch

Object at infinity

0.50 Pitch

Object at front surface

0.75 Pitch

Object at infinity

1.0 Pitch

Object at front surface

Pitch 0.25 0.50 0.75 1.0

Summary of Important Topics

� Single lens, paraxial imaging

� Cardinal points: focal point, principal plane, nodal points

� Simple formulas for focal length of thin and thick lenses

� Thick lens: lateral changes of ray height inside the lens is not neglectable

� Important for correction: bending of lenses: shape changed, focal length preserved

� Mirrors: mainly conic sections are of interest

� Plane parallel plate: image in z-direction shifted

� Dispersion prisms: spectral spreading of white light, spectroscopic applications

� Dispersion prisms: anamorphotic magnification

� Reflection prisms: use for beam deflection, change of image orientation

� Of special interest: roof prisms with one-sided image flip

� Gratings: overlay of diffraction effects of single period and interference function

separation of the light into discrete diffraction orders

� Generalized: diffractive elements, local grating structures, problems with efficiencies and

flase light of unwanted orders

� Gradient lenses: spatiually variant refractive index causes bended ray paths,

can be used for imaging or beam profiling

54

Next lecture: Part 4 – Paraxial optics

Date: Wednesday, 2012-05-09

Contents: 4.1 Imaging - basic notations

- paraxial approximation

- linear collineation

- graphical image construction

- lens makers formula

4.2 Optical system properties - pupil

- special rays

- special configurations

4.3 Matrix calculus - simple matrices

- non-centered systems

4.4 Phase space - basic idea

- invariants

Part 4: Paraxial Optics