www.iap.uni-jena.de

Advanced Lens Design

Lecture 15: Diffractive elements

2015-02-10

Herbert Gross

Winter term 2014

2

Preliminary Schedule

1 21.10. Basics Paraxial optics, imaging, Zemax handling

2 28.10. Optical systems Optical systems, raytrace, advanced handling

3 04.11. Aberrations I Geometrical aberrations, wave aberrations

4 11.11. Aberrations II PSF, OTF, sine condition, aplanatism, isoplanatism

5 18.11. Optimization I Basic principles, merit function setup, optimization in Zemax

6 25.11. Optimization II Initial systems, solves, bending, AC meniscus lenses, constraints, effectiveness of variables

7 02.12. Structural modifications Zero operands, lens splitting, lens addition, lens removal, material selection

8 09.12. Aspheres and freeforms Correction with aspheres, Forbes approach, freeform surfaces, optimal location of aspheres, several aspheres

9 16.12. Field flattening Astigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses

10 06.01. Chromatical correction Achromatization, apochromatic correction, axial versus transversal, glass selection rules, burried surfaces

11 13.01. Special correction topics Symmetry, sensitivity, anamorphotic lenses, field lens, telecentricity, stop position

12 20.01. Higher order aberrations high NA systems, broken achromates, induced aberrations

13 27.01. Advanced optimization strategies

Local vs global optimization, control of iteration, growing requirements

14 03.02. Mirror systems special aspects, double passes, catadioptric systems

15 10.02. Diffractive elements color correction, ray equivalent model, straylight, third order aberrations, manufacturing

1. Basic principle

2. Diffraction orders and efficiency

3. Calculation models

4. Miscellaneous

5. Correction

6. Broadband solutions

7. Manufacturing

3

Contents

Deviation of Light

reflection

mirror

scattering

scatter plate

refraction

lens

diffraction

grating

Mechanisms of light deviation and ray bending

Refraction

Reflection

Diffraction according to the grating equation

Scattering ( non-deterministic)

'sin'sin nn

'

g mo sin sin

Diffractive Optics:

Local micro-structured surface

Location of ray bending :

macroscopic carrier surface

Direction of ray bending :

local grating micro-structure

Local aspherical phase effect local grating constant, q ist the wrapping factor

macroscopic

surface

curvature

local

grating

g(x,y)

lens

bending

angle

m-th

order

thin

layer

mj

mj

jm yxckyx,

),(

),(

2),(

yx

qyxg

Diffractive Elements

z

h2

hred(x) : wrapped

reduced profile

h(x) : continuous

profile

3 h2

2 h2

1 h2

hq(x) : quantized

profile

ray refracted at

smooth asphereray refracted at

Fresnel lens

wrapping

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)

Diffraction Orders

diffractive

structure

diffraction orders

mm-1

m-2m-3

m+2m+1

m+3

desired

order

Usually all diffraction orders are obtained simultaneously

Blazed structure: suppression of perturbing orders

Unwanted orders: false light, contrast and efficiency reduced

Efficiency of Diffractive Elements

diff

0

0.2

0.4

0.6

0.8

1

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

m = 2

m = 1

m = 0

m = 3

design

Efficiency optimized for one desired central wavelength

Strong dispersion for other wavelengths

Reduction of efficiency for deviating wavelengths

(figure: scalar approximation)

95.0

Nc

1sin 2

0 2 4 6 8 10 12 14 160.4

0.5

0.6

0.7

0.8

0.9

1

N

0.4053

0.8106

0.9496

0.9872

real

Efficiency of Binary DOE's

Binarized diffractive elements have a reduced efficiency

The efficiency grows theoretically with the number of quantization levels

For 8 levels the efficiency is

In reality for a larger number of levels the efficiency shrinks due to tolerances

Estimation of N levels

in scalar approximation

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 at the rings,

improved power transmission

fr r r r r12345 F

f+f+2

f+3

f+4

f+5

fmrm 2

Diffractive Lens

Diffractive Fresnel lens

Zone rings with radii

Blaze in every zone (surface slope)

k

rk

Zone No k

f

hk

k

fkmrk 2

kkkk

k rrn

mh

tan

cos1

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

12

Diffracting Surfaces

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

Fresnel zone plate: works as lens

p-phase jumps at Fresnel zones, - only odd zones are considered - no destructive interference

Fresnel Zone Plate

phase of lens intensity I(x,z)intensity I(x,y)

I(x) I(z)

Transition Refractive - Diffractive

refractive :

one direction

n

diffractive blazed :

one direction

g

g

+1. order

+2. order

-2. order

-1. order

0. order

diffractive binary :

all directions

Refractive: no surface structure

Blazed diffractive: only one diffraction order,

for one wavelength and one angle of incidence

Binary diffractive: several orders simultaneously

qh

L

g

q = 1 q = 2 q = 4 q = 8)()/( )()( diffperiodic

dr

prism TxTxT

m

ni

mgxL

xrecte

g

xrectxT

1

2

)(

Transition Refractive - Diffractive

Phase of refracting blaze grating: convolution of prism transmission with periodic

interference function

Notation:

L total width

g length of period

blaze angle

m index of periods

q order of phase jump with

height h

Complete transmission:

1. one period with linear phase

2. total width

3. periodicity of cell

L / d = 8

prism / interference total

L / d = 4

L / d = 1

-20 0 20 40 60 80 100

0

0.5

1

-20 0 20 40 60 80 100

0

0.5

1

-20 0 20 40 60 80 100

0

0.5

1

-20 0 20 40 60 80 100

0

0.5

1

-20 0 20 40 60 80 100

0

0.5

1

-20 0 20 40 60 80 100

0

0.5

1

Transition Refractive - Diffractive

Higher number of

periods

Increasing order q

Widths of orders

decrease

Limiting case:

one prism,

slit diffraction

Lens with diffractive structured

surface: hybrid lens

Refractive lens: dispersion with

Abbe number n = 25...90

Diffractive lens: equivalent Abbe

number

Combination of refractive and

diffractive surfaces:

achromatic correction for compensated

dispersion

Usually remains a residual high

secondary spectrum

Broadband color correction is possible

but complicated

refractive

lens

red

blue

green

blue

red

green

red

bluegreen

diffractive

lens

hybrid

lens

R D

453.3

CF

dd

n

Achromatic Hybrid Lens

Dispersion by grating diffraction:

Abbe number

Relative partial dispersion

Consequence :

Large secondary spectrum

n-P-diagram

Diffractive Optics: Dispersion

330.3''

CF

ee

n

2695.0''

'

',

CF

Fg

FgP

normal

line

ne

0.8

0.6

0.4

0.2-20 0 4020 60 80

DOE

SF59SF10

F4KzFS1

BK7

PgF'

Combination of DOE

and aspherical carrier

Diffractive Optics: Singlet Solutions

50 m

0s'

y'

yp

yp

-0.5 mm50 m

0s'

y'

yp

yp

-0.5 mm50 m

0s'

y'

yp

yp

-0.5 mm50 m

0s'

y'

yp

yp

-0.5 mm

diffractive

surface,

carrier

aspherical

refractive

surface,

aspherical

diffractive

surface, phase

aspherical

all 2. order

d

d

d

d,a

a

Upper part:

conventional lens

Lower part : lens with

diffractive surface

Diffractive Optics: Example Lens

0

Wrms

0 5 10w

486

588

656

poly

0.5

0.25

0

Wrms

0 5 10w

486

588

656

poly

0.5

0.25

y' y'

yp

yp

x'

xp

axis fieldfield

y' y'

yp

yp

x'

xp

axis fieldfield

all refractive lens

lens with one

diffractive surface

diffractive

Data :

- = 193 nm

- NA = 0.65

- b = 50

- sfree = 7.8 mm

Properties:

- short total track

- extreme large free working distance

- few lenses

Hybrid - Microscope Objective Lens

diffractive

element

Straylight suppression

Diffractive Optics: Hybrid Lens

0. order

1. order

2. order

transmitted

light

blocked light

DOE

Nearly index matched materials

Height of second layer

Efficiency

Broad Band DOE: Nearly Index Matching

x

z

ho

h(x)

n1

n2

)()(1)(1)(

21

2

2

1

1

''

eeee

CF

e nnnn

n

n

)()( 21 ee

e

nnh

m

nn

nn

oo

onim

)()(

)()(sinc

21

212

dif

0

0.2

0.4

0.6

0.8

1

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

air

nim

[m]

Principle of achromatic correction

Ratio of Abbe numbers defines refractive power distribution

Diffractive element: Abbe number n = -3.45

Diffractive element gets only

approx. 5% of the refractive

power

diffglas

glas

refr FFnn

n

diffglas

diffs

refr FFnn

n

Color Correction of a Hybrid System

first refractive

power

short 1

long 2

image plane

corrected

second powerrefractive /diffractive

refractivesolution

diffractivesolution

1

2'1

'2

bending angles

Dispersion by grating diffraction:

Abbe number: small and negative !

Relative partial dispersion

Consequence :

Large secondary spectrum

n-P-diagram

Diffractive Optics: Dispersion

330.3''

CF

ee

n

2695.0''

'

',

CF

Fg

FgP

normal

line

ne

0.8

0.6

0.4

0.2-20 0 4020 60 80

DOE

SF59SF10

F4KzFS1

BK7

PgF'

Achromatic doublet

Ratio of Abbe numbers and refractive powers

Visible : 5% refractive power of DOE

Infrared : strong variations depending on substrat material

Diffractive Optics: Dispersion

nrefr/ndiff

m

BK7

SF6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

5

10

15

20

25

30

35

40

silica

CaF2

poly-carbonat

LAF2

0 2 4 6 8 10 12

20

40

60

80

100

120

m

nrefr/ndiff

ZnS

ZnSe

sapphireCaF2

germanium

a) VIS b) IR

Spherical Hybrid Achromate

Classical achromate:

- two lenses, different glasses

- strong curved cemented surface

Hybride achromate:

- one lens

- one surface spherical with

diffractive structure

- tolerances relaxed

1 2 3

refractive

solution

1 2

diffractive

solution

1

2

3

tilt

sensitivity

1 2

tilt

sensitivity

Types of achromates

a) Spherical surfaces

uncemented

b) Hybrid with aspherical and

diffractive surface

Relaxed tolerances for

hybrid solution

classical

solution

1

2

3 4

aspherical

diffractive

solution

coma for

surface tilt

a) classical

solution

1 2 3 4

b) aspherical

diffractive

solution

1 2

all surfaces

spherical

apherical diffractive

Aspherical Hybrid Achromate

Expansion of the optical pathlength for one field point:

Primary Seidel aberrations:

No field curvature

No distortion (stop at lens)

Ray bending in a plane corresponds to linear collineation

Equivalent bending of lens

Primary Aberrations of a Diffractive Lens

f

r

w

DOE

chief

rayimage

plane

f

rw

f

wr

f

r

f

rrW

4

3

282

2)(

22

2

3

3

42

2

2am

ccfX

diff

diff

Seidel spherical aberration of a hybrid lens (A4: aspherical equivalent asphericity)

Optimal bending: choice of A4

Large spherochromatism

Optimization of refractive power of the DOE (10% instead of 5%)

Optimization of a Hybrid Lens

4

4

2

2

34

823

)1(

)1(4

1)1(

2

4yAm

n

n

nn

n

n

n

nn

nFySSS

ref

difrefsph

n

n

nn

n

n

n

nn

n

m

FA

ref 23

)1(

)1(4

1)1(

2

32

2

2

3

4

dif

sphref

gauss

ss

n

,

f

sFF

sphref

refdif

ref

refdif

dif

dif

,

nn

n

nn

n

f

sFF

sphref

refdif

ref

refdif

ref

ref

,

nn

n

nn

n

Realizations of Diffractive Elements

grating of index

example: HOEsurface contour multi phase level

binary grating

phase grating amplitude grating

DOE's

blazed

DOE's

quantized

DOE's

Phase reduced by 2m m: wrapping height perhaps rather small feature sizes at the boundary (large slope)

Changed wrapping number m from center to boundary to prevent from small sizes

Discrete binarization, constant lateral feature size

Discretisation and Wrapping

phase

x

x

reduced

phase

phase 2m

near boundarym

phase

x

x

reduced

phase

phase 2m

phase

x

x

discrete

phase

spatial constant transverse

feature size

Scaling error in mask

Typical Tolerances

x

Maske

1. Schritt

x

Maske

2. Schritt

Tolerances of Diffractive Elements

surface

gradient

wrongideal profile

edge

profile

rounded

profile

discrete

approximated

Round edges in a Fresnel lens:

enlarged diffraction fringes

Encircled energy broadened

Diffractive Optics: Rounding of Edges

a

b

c

d

-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

-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

-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

-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

Log Ipsf

(x) E(r)

x r0 0.1 0.2 0.3 0.4 0.5

0.8

0.85

0.9

0.95

1

-0.5 0 0.5

10-4

10-3

10-2

10-1

100

ab

c

d

Different profile shapes of linear gratings

General diffractive elements

Grating and Diffractive Structures

37 Ref: R. Steiner

7200 lines/mm (139 nm period)

profile depth 70 nm

multi level DOE 5 mRef: Brenner/Jahns, Kley

Ref: Axetris Corp.