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www.iap.uni-jena.de

Lens Design II

Lecture 1: Aberrations and optimization

2019-10-16

Herbert Gross

Winter term 2019

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Preliminary Schedule Lens Design II 2018

1 16.10. Aberrations and optimization Repetition

2 23.10. Structural modificationsZero operands, lens splitting, lens addition, lens removal, material selection

3 30.10. Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres

4 06.11. FreeformsFreeform surfaces, general aspects, surface description, quality assessment, initial systems

5 13.11. Field flatteningAstigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses

6 20.11. Chromatical correction IAchromatization, axial versus transversal, glass selection rules, burried surfaces

7 27.11. Chromatical correction IISecondary spectrum, apochromatic correction, aplanatic achromates, spherochromatism

8 04.12. Special correction topics I Symmetry, wide field systems, stop position, vignetting

9 11.12. Special correction topics IITelecentricity, monocentric systems, anamorphotic lenses, Scheimpflug systems

10 18.18. Higher order aberrations High NA systems, broken achromates, induced aberrations

11 08.01. Further topics Sensitivity, scan systems, eyepieces

12 15.01. Mirror systems special aspects, double passes, catadioptric systems

13 22.01. Zoom systems Mechanical compensation, optical compensation

14 29.01. Diffractive elementsColor correction, ray equivalent model, straylight, third order aberrations, manufacturing

15 05.02.

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1. Geometrical aberrations

2. Wave aberrations and Zernikes

3. Point spread function

4. Modulation transfer function

5. Principles of optimization

Contents

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Optical Image Formation

optical

system

object

plane

image

plane

transverse

aberrations

longitudinal

aberrations

wave

aberrations

Perfect optical image:

All rays coming from one object point intersect in one image point

Real system with aberrations:

1. tranverse aberrations in the image plane

2. longitudinal aberrations from the image plane

3. wave aberrations in the exit pupil

4

5

x, y object coordinates, especially object height

x', y' image coordinates, especially image height

xp,yp coordinates of entrance pupil

x'p, y'p coordinates of exit pupil

s object distance form 1st surface

s' image distance form last surface

p entrance pupil distance from 1st surface

p' exit pupil distance from last surface

Dx' sagittal transverse aberration

Dy' meridional transverse aberration

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Notations for an Optical System

x

y

y

s

p

s'

p'

xP

yp

y'

x'

y'

x'P

y'p

object

plane

entrance

pupil

exit

pupil

image

plane

zDx'

Dy'

system

surfaces

xP

yp

x'P

y'p

P'P'0

P

6

Polynomial Expansion of Aberrations

Representation of 2-dimensional Taylor series vs field y and aperture r

Selection rules: checkerboard filling of the matrix

Constant sum of exponents according to the order

Field y

Spherical

y0 y 1 y 2 y3 y 4 y 5

Distortion

r

0

y

y3

primary

y5

secondary

r

1

r 1

Defocus

Aper-

ture

r

r

2

r2y Coma primary

r 3

r 3 Spherical

primary

r

4

r

5

r 5 Spherical

secondary

DistortionDistortionTilt

Coma Astigmatism

Image

location

Primary

aberrations /

Seidel

Astig./Curvat.

cos

cos

cos2

cos

Secondary

aberrations

cos

r 3y2

cos2

Coma

secondary

r4y cos

r2y

3 cos

3

r2y

3 cos

r1 y

4

r1 y

4 cos

2

r 3y2

r12 y

r12 y

6

7

Pupil Sampling

Ray plots

Spot

diagrams

sagittal ray fan

tangential ray fan

yp

Dy

tangential aberration

xp

Dy

xp

Dx

sagittal aberration

whole pupil area

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8

Typical chart of aberration representation

Reference: at paraxial focus

Spherical Aberration

Wave aberration

tangential sagittal

22

Primary spherical aberration at paraxial focus

Transverse ray aberration yD xD yD

Pupil: y-section x-section x-section

0.01 mm 0.01 mm 0.01 mm

Geometrical spot through

focus

0.0

2 m

m

Modulation Transfer Function MTF

MTF at paraxial focus MTF through focus for 100 cycles per mm

1

0

0.5

0 100 200cyc/mm z/mm

1

0

0.5

-0.02 0.020.0

z/mm-0.02 0.020.0-0.01 0.01

Ref: H. Zügge

8

9

Surface Contributions: Example

Seidel aberrations:

representation as sum of

surface contributions possible

Gives information on correction

of a system

Example: photographic lens

1

23 4

5

6 8 9

10

7

Retrofocus F/2.8

Field: w=37°

SI

Spherical Aberration

SII

Coma

-200

0

200

-1000

0

1000

-2000

0

2000

-1000

0

1000

-100

0

150

-400

0

600

-6000

0

6000

SIII

Astigmatism

SIVPetzval field curvature

SVDistortion

CIAxial color

CIILateral color

Surface 1 2 3 4 5 6 7 8 9 10 Sum

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10

Wave Aberration in Optical Systems

Definition of optical path length in an optical system:

Reference sphere around the ideal object point through the center of the pupil

Chief ray serves as reference

Difference of OPL : optical path difference OPD

Practical calculation: discrete sampling of the pupil area,

real wave surface represented as matrix

Exit plane

ExP

Image plane

Ip

Entrance pupil

EnP

Object plane

Op

chief

ray

w'

reference

sphere

wave

front

W

y yp y'p y'

z

chief

ray

wave

aberration

optical

systemupper

coma ray

lower coma

ray

image

point

object

point

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11

Pupil Sampling

y'p

x'p

yp

xp x'

y'

z

yo

xo

object plane

point

entrance pupil

equidistant grid

exit pupil

transferred grid

image plane

spot diagramoptical

system

All rays start in one point in the object plane

The entrance pupil is sampled equidistant

In the exit pupil, the transferred grid

may be distorted

In the image plane a spreaded spot

diagram is generated

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12

Zernike Polynomials

+ 6

+ 7

- 8

m = + 8

0 5 8764321n =

cos

sin

+ 5

+ 4

+ 3

+ 2

+ 1

0

- 1

- 2

- 3

- 4

- 5

- 6

- 7

Expansion of wave aberration surface

into elementary functions / shapes

Zernike functions are defined in circular

coordinates r,

Ordering of the Zernike polynomials by

indices:

n : radial

m : azimuthal, sin/cos

Mathematically orthonormal function

on unit circle for a constant weighting

function

Direct relation to primary aberration types

n

n

nm

m

nnm rZcrW ),(),(

01

0)(cos

0)(sin

)(),(

mfor

mform

mform

rRrZ mnm

n

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13

Airy distribution:

Gray scale picture

Zeros non-equidistant

Logarithmic scale

Encircled energy

Perfect Lateral Point Spread Function: Airy

DAiry

r / rAiry

Ecirc

(r)

0

1

2 3 4 5

1.831 2.655 3.477

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2. ring 2.79%

3. ring 1.48%

1. ring 7.26%

peak 83.8%

r

z

I(r,z)

13

14

0

2

12,0 Iv

vJvI

0

2

4/

4/sin0, I

u

uuI

-25 -20 -15 -10 -5 0 5 10 15 20 250,0

0,2

0,4

0,6

0,8

1,0

vertical

lateral

inte

nsity

u / v

Circular homogeneous illuminated aperture: intensity distribution

transversal: Airy

scale:

axial: sinc

scale

Resolution transversal better

than axial: Dx < Dz

Ref: M. Kempe

Scaled coordinates according to Wolf :

axial : u = 2 p z n / NA2

transversal : v = 2 p x / NA

Perfect Point Spread Function

NADAiry

22.1

2NA

nRE

r

z

lateral

aperture

cone

axial

image plane

optical

axis

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Spherical aberration Astigmatism Coma

c = 0.2

c = 0.3

c = 0.7

c = 0.5

c = 1.0

Psf with Aberrations

Zernike coefficients c in

Spherical aberration,

Circular symmetry

Astigmatism,

Split of two azimuths

Coma,

Asymmetric

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),(ˆ),( yxIFvvH PSFyxOTF

*

2

( ) ( )2 2

( )

( )

x xp p p

OTF x

p p

f v f vP x P x dx

H v

P x dx

Optical Transfer Function: Definition

Normalized optical transfer function (OTF) in frequency space:

Fourier transform of the Psf- intensity

Absolute value of OTF: modulation transfer function MTF

Gives the contrast at a special spatial frequency of a

sine grating

OTF: Autocorrelation of shifted pupil function, Duffieux-integral

Interpretation: interference of 0th and 1st diffraction of

the light in the pupil

x

y

x

y

L

L

x

y

o

o

x'

y'

p

p

light

source

condenser

conjugate to object pupil

object

objective

pupil

direct

light

at object diffracted

light in 1st order

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Optical Transfer Function of a Perfect System

Aberration free circular pupil:

Reference frequency

Maximum cut-off frequency:

Analytical representation

Separation of the complex OTF function into:

- absolute value: modulation transfer MTF

- phase value: phase transfer function PTF

'sinu

f

avo

'sin222 0max

un

f

navv

2

000 21

22arccos

2)(

v

v

v

v

v

vvHMTF

p

),(),(),( yxPTF

vvHi

yxMTFyxOTF evvHvvH

/ max0

0

1

0.5 1

0.5

gMTF

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18

Basic Idea of Optimization

iteration

path

topology of

meritfunction F

x1

x2

start

Topology of the merit function in 2 dimensions

Iterative down climbing in the topology

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19

Mathematical description of the problem:

n variable parameters

m target values

Jacobi system matrix of derivatives,

Influence of a parameter change on the

various target values,

sensitivity function

Scalar merit function

Gradient vector of topology

Hesse matrix of 2nd derivatives

Nonlinear Optimization

x

)(xf

j

iji

x

fJ

21

)()(

m

i

ii xfywxF

j

jx

Fg

kj

kjxx

FH

2

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20

Merit function:

Weighted sum of deviations from target values

Formulation of target values:

1. fixed numbers

2. one-sided interval (e.g. maximum value)

3. interval

Problems:

1. linear dependence of variables

2. internal contradiction of requirements

3. initail value far off from final solution

Types of constraints:

1. exact condition (hard requirements)

2. soft constraints: weighted target

Finding initial system setup:

1. modification of similar known solution

2. Literature and patents

3. Intuition and experience

Optimization in Optical Design

g f fj jist

j

soll

j m

2

1,

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Goal of optimization:

Find the system layout which meets the required performance targets according of the

specification

Formulation of performance criteria must be done for:

- Apertur rays

- Field points

- Wavelengths

- Optional several zoom or scan positions

Selection of performance criteria depends on the application:

- Ray aberrations

- Spot diameter

- Wavefornt description by Zernike coefficients, rms value

- Strehl ratio, Point spread function

- Contrast values for selected spatial frequencies

- Uniformity of illumination

Usual scenario:

Number of requirements and targets quite larger than degrees od freedom,

Therefore only solution with compromize possible

Optimization Merit Function in Optical Design

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22

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

Quantitative parameter of description X:

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

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21

RR

RRX

22

23

Correction of spherical aberration:

Splitting of lenses

Distribution of ray bending on several

surfaces:

- smaller incidence angles reduces the

effect of nonlinearity

- decreasing of contributions at every

surface, but same sign

Last example (e): one surface with

compensating effect

Correcting Spherical Aberration: Lens Splitting

Transverse aberration

5 mm

5 mm

5 mm

(a)

(b)

(c)

(d)

Improvement

(a)à(b) : 1/4

(c)à(d) : 1/4

(b)à(c) : 1/2

Improvement

Improvement

(e)

0.005 mm

(d)à(e) : 1/75

Improvement

5 mm

Ref : H. Zügge23