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Lens Design II

Lecture 14: Diffractive elements

2019-02-06

Herbert Gross

Winter term 2018

2

Preliminary Schedule Lens Design II 2018

1 17.10. Aberrations and optimization Repetition

2 24.10. Structural modificationsZero operands, lens splitting, lens addition, lens removal, material selection

3 07.11. Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres

4 14.11. FreeformsFreeform surfaces, general aspects, surface description, quality assessment, initial systems

5 21.11. Field flatteningAstigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses

6 28.11. Chromatical correction IAchromatization, axial versus transversal, glass selection rules, burried surfaces

7 05.12. Chromatical correction IISecondary spectrum, apochromatic correction, aplanatic achromates, spherochromatism

8 12.12. Special correction topics I Symmetry, wide field systems, stop position, vignetting

9 19.12. Special correction topics IITelecentricity, monocentric systems, anamorphotic lenses, Scheimpflug systems

10 09.01. Higher order aberrations High NA systems, broken achromates, induced aberrations

11 16.01. Further topics Sensitivity, scan systems, eyepieces

12 23.01. Mirror systems special aspects, double passes, catadioptric systems

13 30.01. Zoom systems Mechanical compensation, optical compensation

14 06.02. Diffractive elementsColor 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

Comparison: Lens - Asphere - Diffractive Element

Location of ray bending: r,z

New ray direction:

Substrate: z(r)

DOE phase function (r)

Spherical lens:

- z(r) has only 1 parameter

- r,z and coupled over complete diameter

Aspherical lens:

- z(r) has many parameters

- r,z and locally coupled

- r,z and coupling changes over diameter

Diffractive element on curved aspherical

substrate:

- r,z and completly decoupled

aspherical

profile

bending

angle

ray

z

r

P(r,z)

a) Spherical lens

P(r,z)

spherical

profile

bending

angle

ray

z

r

P(r,z)

P(r,z) fix arbitrary

b) Asphere

substrate

aspherical

profile

local

grating

g(x,y)

c) DOE on aspherical substrate

bending

angle

arbitrary

ray

thin

layer

z

r

P(r,z)

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)

8

Grating: constant period g

Diffractive Lens

Grating with varying period g

g

Arbitrary modification of

light possible

Focussing of light

Fresnel-Zone-plate

Diffractive optical element

k /2

f

Equation of circle: r1² + f ² = ( f + /2)²

r2² + f ² = ( f + 2 /2)²

r3² + f ² = ( f + 3 /2)²

rk² + f ² = f ² + 2 f k/2 + (k/2)²

Then rk² = k f because << f

With rk =

r1

𝑘 r1 𝑓 =𝑟1²

𝜆

𝑓 =𝑟1²

𝜆

gmm

)sin( max

Ref: B. Böhme

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

Single Blazed DOE

Length difference between

neighbouring cells in order k

Corresponding phase difference

Corresponding sawtooth height

Blaze angle of teeth

Transmission function of profile

Product of cell function and interference function gives amplitude/intensity

hmax

b

L

p

ray

period

p

rays

period

path difference

D = k /nmed

nmed

nlen

( )blaze

OPD

med

k

n

D

( )

max( )

blaze

med len med

kh

n n n

( )

max 2blaze

med

k

n

( )( ) max

( )

blazeblaze

med len med

h k

p p n n n

b

( ) sinc 1 sinc

len med

jlen med

n njt v k v L v

n n k

b

b

real cell

function

imaginary cell

function

grating

function

interference

function

total

transfer

function

amount

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)

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

kkk

k

k rrn

mh

tan

cos1

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

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

14

Diffracting Surfaces

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)()/( )()( diff

periodic

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

Transition Diffractiv-Refractiv

Ideal

- design

0

0.5

1

0

0.5

1

- change

3 %

- change

15 %

prism / interference total

-20 0 20 40 60 80 100

-20 0 20 40 60 80 100

-20 0 20 40 60 80 100-20 0 20 40 60 80 100

-20 0 20 40 60 80 100

-20 0 20 40 60 80 100

0.5

1

0.

0.5

1

0.

0.5

1

0.

0.5

1

0.

Changed wavelength:

- perfect blaze condition perturbed

- no ideal suppression of higher orders

- parasitic orders occur

19

Grating Equation Validity

Simple grating equation

Several hidden assumption:

infinite number of periods

In case of a finite number of periods N accross the beam cross section:

width of the side lobes changing

with

-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

N = 10

N = 50

N = 200

angle [a.u.]

0sin sinm

g

2 2

sin ( ) / sin sin / sin

( ) / sin sin / sinN

m g NgI gN

m g N g

1)(

1)()(

01

1

0

n

n

Intensity I

N=1

2

5

10

50

20

Grating Equation Validity

Real optical systems:

the number of periods per

local ray bundle cross section

is essential

The formula is almost stable for

approximately Nbundle > 5

ray bundle

area

surface

N grooves

1bundlebundle

groove

DN

g

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

0Ds'

Dy'

yp

yp

-0.5 mm50 m

0Ds'

Dy'

yp

yp

-0.5 mm50 m

0Ds'

Dy'

yp

yp

-0.5 mm50 m

0Ds'

Dy'

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

Dy' Dy'

yp

yp

Dx'

xp

axis fieldfield

Dy' Dy'

yp

yp

Dx'

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

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

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

,DD

D

f

sFF

sphref

refdif

ref

refdif

dif

dif

,

nn

n

nn

n

D

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

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