optical design with zemax · 2012. 10. 22. · 3 30.10. properties of optical systems ii types of...

$: Optical Design with Zemax · 2012. 10. 22. · 3 30.10. Properties of optical systems II Types of surfaces, Aspheres, Gratings and diffractive surfaces, Gradient media, Cardinal elements,$
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Optical Design with Zemax

Lecture 2: Properties of optical systems II

2012-10-30

Herbert Gross

Winter term 2012

2 2 Properties of Optical Systems II

Preliminary time schedule

1 16.10. Introduction

Introduction, Zemax interface, menues, file handling, preferences, Editors, updates, windows,

Coordinate systems and notations, System description, Component reversal, system insertion,

scaling, 3D geometry, aperture, field, wavelength

2 23.10. Properties of optical systems I Diameters, stop and pupil, vignetting, Layouts, Materials, Glass catalogs, Raytrace, Ray fans

and sampling, Footprints

3 30.10. Properties of optical systems II Types of surfaces, Aspheres, Gratings and diffractive surfaces, Gradient media, Cardinal

elements, Lens properties, Imaging, magnification, paraxial approximation and modelling

4 06.11. Aberrations I Representation of geometrical aberrations, Spot diagram, Transverse aberration diagrams,

Aberration expansions, Primary aberrations,

5 13.11. Aberrations II Wave aberrations, Zernike polynomials, Point spread function, Optical transfer function

6 20.11. Optimization I Principles of nonlinear optimization, Optimization in optical design, Global optimization

methods, Solves and pickups, variables, Sensitivity of variables in optical systems

7 27.11. Optimization II Systematic methods and optimization process, Starting points, Optimization in Zemax

8 04.12 Imaging Fundamentals of Fourier optics, Physical optical image formation, Imaging in Zemax

9 11.12. Illumination Introduction in illumination, Simple photometry of optical systems, Non-sequential raytrace,

Illumination in Zemax

10 18.12. Advanced handling I

Telecentricity, infinity object distance and afocal image, Local/global coordinates, Add fold

mirror, Scale system, Make double pass, Vignetting, Diameter types, Ray aiming, Material

index fit

11 08.01. Advanced handling II Report graphics, Universal plot, Slider, Visual optimization, IO of data, Multiconfiguration,

Fiber coupling, Macro language, Lens catalogs

12 15.01. Correction I

Symmetry principle, Lens bending, Correcting spherical aberration, Coma, stop position,

Astigmatism, Field flattening, Chromatical correction, Retrofocus and telephoto setup, Design

method

13 22.01. Correction II Field lenses, Stop position influence, Aspheres and higher orders, Principles of glass

selection, Sensitivity of a system correction, Microscopic objective lens, Zoom system

14 29.01. Physical optical modelling I Gaussian beams, POP propagation, polarization raytrace, polarization transmission,

polarization aberrations

15 05.02. Physical optical modelling II coatings, representations, transmission and phase effects, ghost imaging, general straylight

with BRDF

1. Types of surfaces

2. Aspheres

3. Gratings and diffractive surfaces

4. Gradient media

5. Cardinal elements

6. Lens properties

7. Imaging

8. Magnification

9. Paraxial approximation and modelling

3 2 Properties of Optical Systems II

Contents 3rd Lecture

Setting of surface properties

local tilt

and

decenterdiameter

surface type

additional drawing

switches

coatingoperator and

sampling for POP

scattering

options

2 Properties of optical systems II

Surface properties and settings

Ideale lens

Principal surfaces are spheres

The marginal ray heights in the vortex plane are different for larger angles

Inconsistencies in the layout drawings


Ideal lens

P P'

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 tan

6 2 Properties of optical systems II

Aspherical surface types

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 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 )


Conic sections

Conic aspherical surface

Variation of the conical parameter


Aspherical shape of conic sections

z

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y

22

2

111 yc

cyz

Equation

c : curvature 1/Rs

: eccentricity ( = -1 )

radii of curvature :

22

2

)1(11 cy

ycz

2

tan 1

s

sR

yRR

2

32

tan 1

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


Parabolic mirror

Equation

c: curvature 1/R

: Eccentricity

22

2

)1(11 cy

ycz

ellipsoid

F'

F

e

a

b

oblate

vertex

radius Rso

prolate

vertex

radius Rsp


Ellipsoid mirror

Sag z at height y for a spherical

surface:

Paraxial approximation:

quadratic term

22 yrrz

r

yz p

2

2

z

y

height

y

z

zp = y2/(2r)

sphere

parabola

sag

axis


Sag of a surface

Maximum intensity:

constructive interference of the contributions

of all periods

Grating equation


Grating Diffraction

g mo sin sin

grating

g

incident

light

+ 1.

diffraction

order

s =

in-phase

grating

constant

Ideal diffraction grating:

monochromatic incident collimated

beam is decomposed into

discrete sharp diffraction orders

Constructive interference of the

contributions of all periodic cells

Only two orders for sinusoidal


Ideale diffraction grating

grating

g = 1 / s

incidentcollimated

light

grating constant

-1.

-2.

-3.

0.

-4.

+1.

+2.

+3.

+4.

diffraction orders

Interference function of a finite number

N of periods

Finite width of every order depends on N

Sharp order direction only in the limit of


Finite width of real grating orders

sinsin

sinsin

2

2

g

Ng

I

N = 15 N = 50

-30 -20 -10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-30 -20 -10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-30 -20 -10 0 10 20 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N = 5

Ng

42sin 2/1

N

z

h2

hred(x) : wrapped

reduced profile

h(x) :

continuous

profile

3 h2

2 h2

1 h2

hq(x) : quantized

profile


Diffractive Elements

Original lens height profile h(x)

Wrapping of the lens profile: hred(x) reduction on

maximal height h2

Digitalization of the reduced profile: hq(x)

egdn

gms

n

ns

ˆ''

'

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


Diffracting surfaces


Diffracting surfaces

Local micro-structured surface

Location of ray bending :

macroscopic carrier surface

Direction of ray bending :

local grating micro-structure

Independent degrees of freedom:

1. shape of substrate determines the

point of the ray bending

2. local grating constant determines

the direction of the bended ray

macroscopic

surface

curvature

local

grating

g(x,y)

lens

bending

angle

m-th

order

thin

layer

Discrete topography on the surface

Phase unwrapped to get a smooth surface

Only one desired order can be

calculated with raytrace

Model approximation

according to Sweatt:

same refraction by small height

and high index

numerical optimized DOE

phase unwrapped

ERL-model( equivalent refractive lens )

black box modelSweatt-model( thin lens with high index )


Modellierung diffractive elements

Phase function redistributed:

large index / small height

typical : n = 10000

Calculation in conventional software with raytrace possible


Diffractive Optics: Sweatt Model

),(**2),(2),( yxznyxznyx

equivalent refractive

aspherical lens

real index n synthetic

large index n*

Sweatt lens

zz*

the same ray

bending

ray bending

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


Raytracing in 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

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

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

2

2

4

3

6

4

8

5 6

2

7

3


Description of GRIN media

Curved ray path in inhomogeneous media

Different types of profiles


Gradient Lens Types

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


Gradium Lenses

Axial profile 'Gradium'

Focussing effect only for oblique rays

Combined effect of front surface curvature

and index gradient

Special cylindrical blanks with given profile allows choice of individual z-interval for the lens

k

k

kz

znzn

max

)(

n

1.75

1.70z

zmaxz

lens


Collecting radial selfoc lens

L

P P'

F

F'

L

P P'

F'

Thick Wood lens with parabolic index

profile

Principal planes at 1/3 and 2/3 of

thickness

n2 > 0 : collecting lens

n2 < 0 : negative lens

2

20)( rnnrn

Types of lenses with parabolic profile

Pitch length


Gradient Lenses

ymarginal

ycoma

2

0

2

0

2

20

2

11

1)(

rAn

rnnrnnrn r

rnn

np

2

2

22

2

0

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

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


Cardinal elements 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


Notations 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


Main properties of a lens

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

positive

meniscus lens

P P'

bi-concave lens

P'P

plane-concave

lens

P P'

negative

meniscus lens


Lens shape

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


Lens bending und shift of principal plane

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

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

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

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

object at

infinity

image in

focal

plane

lens acts as

aperture stop

collimated

entrance bundle

image at

infinity

stop

image

eye lens

field lens


Object or field at infinity

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

'

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

w

w'

'f

fm

Angle Magnification

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

Paraxial approximation:

Small angles of rays at every surface

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

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)(

Law of refraction

Taylor expansion

Linear formulation of the law of refraction

Error of the paraxial 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


Modelling of Optical Systems

Principal purpose of calculations:

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

Function, data of properties,

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

Analysisimaging

aberration

theorie

Synthesislens design

Ref: W. Richter

Imaging model with levels of refinement

Analytical approximation and classificationl

(aberrations,..)

Paraxial model

(focal length, magnification, aperture,..)

Geometrical optics

(transverse aberrations, wave aberration,

distortion,...)

Wave optics

(point spread function, OTF,...)

with

diffractionapproximation

--> 0

Taylor

expansion

linear

approximation

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

optical design with zemax · 2012. 10. 22. · 3 30.10. properties of optical systems ii types of...

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