www.iap.uni-jena.de

Metrology and Sensing

Lecture 2: Wave optics

2016-10-19

Herbert Gross

Winter term 2016

2

Preliminary Schedule

No Date Subject Detailed Content

1 18.10. Introduction Introduction, optical measurements, shape measurements, errors,

definition of the meter, sampling theorem

2 19.10. Wave optics (ACP) Basics, polarization, wave aberrations, PSF, OTF

3 01.11. Sensors Introduction, basic properties, CCDs, filtering, noise

4 08.11. Fringe projection Moire principle, illumination coding, fringe projection, deflectometry

5 09.11. Interferometry I (ACP) Introduction, interference, types of interferometers, miscellaneous

6 22.11. Interferometry II Examples, interferogram interpretation, fringe evaluation methods

7 29.11. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods

8 06.12. Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight,

Scheimpflug setup

9 13.12. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology

10 20.12. Holography Introduction, holographic interferometry, applications, miscellaneous

11 03.01. Measurement of basic

system properties Bssic properties, knife edge, slit scan, MTF measurement

12 10.01. Phase retrieval Introduction, algorithms, practical aspects, accuracy

13 17.01. Metrology of aspheres

and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms

14 24.01. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous

15 31.01. Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method

3

Content

Basic wave optics

Polarization

Wave aberrations

Point spread function

Transfer function

4

Basic Wave Optics

Scalar wave

phase function

Phase surface:

- fixed phase for one time

- phase surface perpendicular to

unit vektor e

( , )( ) ( ) e i rA r a r

( , )r

A0

2k r r e const

Ref: W. Osten

5

Plane and Spherical Waves

Plane wave

wave vector k

Spherical wave

)(),( trkiAetrE

)(),( trkier

AtrE

Ref.: B. Dörband

6

Basic Wave Optics

Scalar wave

Different types of waves:

phase function

amplitude function

Plane wave

Phase

Amplitude in 2D

Spherical wave

Parabolic wave

( , )( ) ( ) e i rA r a r

( ) ( ) e

oi krA r a r

( ) or k r

2

cos sin

( , ) (x, z) ei x z

A x z a

( , )r

( )a r

( ) ( ) e oi k r

A r a r

( ) or k r

22sin

2(x) (x) e

xi x

RA a

7

Basic Wave Optics

Spherical wave interference

( ) ( ) e oi k r

A r a r

sensor

spherical

wave 1

spherical

wave 2

maxima

hyperbola

8

Interference of Waves

The main property is the phase difference

between two waves

Interference of two waves

special case of equal intensites

Maxima of intensity at even phase differences

Minima of intensity at odd phase differences

Interference of plane waves

Interference of spherical waves:

1. outgoing waves

rotational hyperboloids

2. one outgoing, one incoming wave

rotational ellipsoids Ref: W. Osten

jk j k

1 2 1 2 122 cosI I I I I

2jk N

(2 1)jk N

0 122 1 cosI I

1 2k r k r

1 2k r r

1 2k r r

9

Intensity

CCD is not able to detect phase due to time averaging

Measuring of intensity with simple detector

Measured intensity is time average

Interferometry and holography:

coding of phase information into measurable intensity variation

Conrast / visibility:

normalized difference of two different intensities

(typically maximum / minimum values)

Value between 0...1

General case of two-wave interference

Ref: W. Osten

22

0

1

2r ot

I P E A

max min

max min

I IC

I I

1 2 121 cosI I I C

10

Interference of Two Plane Waves

Two plane waves with normals ek

angles against x-axis

Equations of interference

Location of maxima: straight lines

Distance of maxima:

along x / z / angle

fringe distance

Ref: W. Osten

fringe

maxima

plane

wave

normals

z

1 2 1 1 2 2

1 2 1 2

2 2cos sin cos sin

2cos cos sin sin

k r k r x z x z

x z

1 2 1 2cos cos sin sinx z N

1 2

1 2 1 2

sin sin

cos cos cos cos

Nx z

1 2 1 2

,cos cos sin sin

x z

1 1

cos

2sin2

h z

1 2tan tan2

z

x

Scalar:

Helmholtz equation

Vectorial:

Maxwell equations

Scalar / vectorial Optics

0)(2 rEnko

k

E

H

k

0

Bk

iDk

BEk

jiDHk

EJ

Jk

MHB

PED

r

r

0

0

Description of electromagnetic fields:

- Maxwell equations

- vectorial nature of field strength

Decomposition of the field into components

Propagation plane wave:

- field vector rotates

- projection components are oscillating sinusoidal

yyxx etAetAE )cos(cos

z

x

y

Basic Notations of Polarization

1. Linear components in phase

2. circular phase difference of 90° between components

3. elliptical arbitrary but constant phase difference

x

y

z

E

E

x

y

z

EE

x

y

z

E

E

Basic Forms of Polarisation

Elimination of the time dependence:

Ellipse of the vector E

Different states of polarization:

- sense of rotation

- shape of ellipse

0° 45° 90° 135° 180°

225° 270° 315° 360°

2

2

2

2

2

sincos2

yx

yx

y

y

x

x

AA

EE

A

E

A

E

Polarization Ellipse

Descriptions of Polarization

E

Parameter Properties

1

Polarization ellipse

Ellipticity ,

orientation only complete polarization

2

Complex parameter

Parameter

only complete polarization

3

Jones vectors

Components of E

only complete polarization

4

Stokes vectors

Stokes parameter So ... S4

complete or partial

polarization

5

Poincare sphere

Points on or inside the

Poincare sphere only graphical representation

6

Coherence matrix

2x2 - matrix C

complete or partial

polarization

Polarizer with attenuation cs/p

Rotated polarizer

Polarizer in y-direction

p

s

LIN c

cJ

10

01

z

y

x

TA

2

2

sincossin

cossincos)(PJ

10

00)0(PJ

Polarizer

Polarizer and analyzer with rotation

angle

Law of Malus:

Energy transmission

TA

z

y

x

TA

linear

polarizer y

linear

polarizer

E

E cos

2cos)( oII

I

0 90° 180° 270° 360°

Pair Polarizer-Analyzer

parallel

polarizer

analyzer

perpendicular

Phase difference between field

components

Retarder plate with rotation angle

Special value:

/ 4 - plate generates circular polarized light

1. fast axis y

2. fast axis 45°

2

2

0

0

i

i

RET

e

eJ

z

y

x

SA

LA

ii

ii

Vee

eeJ

22

22

cossin1cossin

1cossinsincos),(

iJ V

0

01)2/,0(

1

1

2

1)2/,4/(

i

iiJ V

Retarder

Rotate the of plane of polarization

Realization with magnetic field:

Farady effect

Verdet constant V

cossin

sincosROTJ

z

y

x

VLB

Rotator

Law of Malus-Dupin:

- equivalence of rays and wavefronts

- both are orthonormal

- identical information

Condition:

No caustic of rays

Mathematical:

Rotation of Eikonal

vanish

Optical system:

Rays and spherical

waves orthonormal

wave fronts

rays

Law of Malus-Dupin

object

plane

image

plane

z0 z1

y1

y0

phase

L = const

L = const

srays

rot n s

0

AP

OE

OPL rdnl

)0,0(),(),( OPLOPLOPD lyxlyx

R

y

WR

y

y

W

p

''

p

pp

pp y

yxW

y

R

u

yy

y

Rs

),(

'sin

'''

2

Relationships

Concrete calculation of wave aberration:

addition of discrete optical path lengths

(OPL)

Reference on chief ray and reference

sphere (optical path difference)

Relation to transverse aberrations

Conversion between longitudinal

transverse and wave aberrations

Scaling of the phase / wave aberration:

1. Phase angle in radiant

2. Light path (OPL) in mm

3. Light path scaled in

)(2

)(

)(

)()(

)()(

)()(

xWi

xki

xi

exAxE

exAxE

exAxE

OPD

21

R

y

WR

y

y

W

p

''

Relationship to Transverse Aberration

Relation between wave and transverse aberration

Approximation for small aberrations and small aperture angles u

Ideal wavefront, reference sphere: Wideal

Real wavefront: Wreal

Finite difference

Angle difference

Transverse aberration

Limiting representation

22

yp

z

real ray

wave front W(yp)

R, ideal ray

C

reference

plane

y'

reference sphere

u

idealreal WWWW

py

W

tan

Ry'

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

23

Diffraction at the System Aperture

Self luminous points: emission of spherical waves

Optical system: only a limited solid angle is propagated, the truncaton of the spherical wave

results in a finite angle light cone

In the image space: uncomplete constructive interference of partial waves, the image point

is spreaded

The optical systems works as a low pass filter

object

point

spherical

wave

truncated

spherical

wave

image

plane

x = 1.22 / NA

point

spread

function

object plane

Fraunhofer Point Spread Function

Rayleigh-Sommerfeld diffraction integral,

Mathematical formulation of the Huygens-principle

Fraunhofer approximation in the far field

for large Fresnel number

Optical systems: numerical aperture NA in image space

Pupil amplitude/transmission/illumination T(xp,yp)

Wave aberration W(xp,yp)

complex pupil function A(xp,yp)

Transition from exit pupil to

image plane

Point spread function (PSF): Fourier transform of the complex pupil

function

1

2

z

rN

p

F

),(2),(),( pp yxWi

pppp eyxTyxA

pp

yyxxR

i

yxiW

pp

AP

dydxeeyxTyxEpp

APpp

''2

,2,)','(

''cos'

)'()('

dydxrr

erE

irE d

rrki

I

PSF by Huygens Principle

Huygens wavelets correspond to vectorial field components

The phase is represented by the direction

The amplitude is represented by the length

Zeros in the diffraction pattern: destructive interference

Aberrations from spherical wave: reduced conctructive superposition

pupil

stop

wave

front

ideal

reference

sphere

point

spread

function

zero

intensity

side lobe

peak

central peak maximum

constructive interference

reduced constructive

interference due to phase

aberration

0

2

12,0 I

v

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: x < z

Ref: M. Kempe

Scaled coordinates according to Wolf :

axial : u = 2 z n / NA2

transversal : v = 2 x / NA

Perfect Point Spread Function

NADAiry

22.1

2NA

nRE

Abbe Resolution and Assumptions

Assumption Resolution enhancement

1 Circular pupil ring pupil, dipol, quadrupole

2 Perfect correction complex pupil masks

3 homogeneous illumination dipol, quadrupole

4 Illumination incoherent partial coherent illumination

5 no polarization special radiale polarization

6 Scalar approximation

7 stationary in time scanning, moving gratings

8 quasi monochromatic

9 circular symmetry oblique illumination

10 far field conditions near field conditions

11 linear emission/excitation non linear methods

Abbe resolution with scaling to /NA:

Assumptions for this estimation and possible changes

A resolution beyond the Abbe limit is only possible with violating of certain

assumptions

log I(r)

r0 5 10 15 20 25 30

10

10

10

10

10

10

10

-6

-5

-4

-3

-2

-1

0

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%

Defocussed Perfect Psf

Perfect point spread function with defocus

Representation with constant energy: extreme large dynamic changes

z = -2RE z = +2REz = -1RE z = +1RE

normalized

intensity

constant

energy

focus

Imax = 5.1% Imax = 42%Imax = 9.8%

0,0

0,0)(

)(

ideal

PSF

real

PSFS

I

ID

2

2),(2

),(

),(

dydxyxA

dydxeyxAD

yxWi

S

Important citerion for diffraction limited systems:

Strehl ratio (Strehl definition)

Ratio of real peak intensity (with aberrations) referenced on ideal peak intensity

DS takes values between 0...1

DS = 1 is perfect

Critical in use: the complete

information is reduced to only one

number

The criterion is useful for 'good'

systems with values Ds > 0.5

Strehl Ratio

r

1

peak reduced

Strehl ratio

distribution

broadened

ideal , without

aberrations

real with

aberrations

I ( x )

31

Approximation of

Marechal:

( useful for Ds > 0.5 )

but negative values possible

Bi-quadratic approximation

Exponential approach

Computation of the Marechal

approximation with the

coefficients of Zernike

2

241

rms

s

WD

N

n

n

m

nmN

n

ns

n

c

n

cD

1 0

2

1

2

0

2

12

1

1

21

Approximations for the Strehl Ratio

22

221

rms

s

WD

2

24

rmsW

s eD

defocusDS

c20

exac t

Marechal

exponential

biquadratic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

32

Psf with Aberrations

Psf for some low oder Zernike coefficients

The coefficients are changed between cj = 0...0.7

The peak intensities are renormalized

spherical

defocus

coma

astigmatism

trefoil

spherical

5. order

astigmatism

5. order

coma

5. order

c = 0.0

c = 0.1c = 0.2

c = 0.3c = 0.4

c = 0.5c = 0.7

33

Resolution of Fourier Components

Ref: D.Aronstein / J. Bentley

object

pointlow spatial

frequencies

high spatial

frequencies

high spatial

frequencies

numerical aperture

resolved

frequencies

object

object detail

decomposition

of Fourier

components

(sin waves)

image for

low NA

image for

high NA

object

sum

pppp

pp

vyvxi

pp

yxOTF

dydxyxg

dydxeyxg

vvH

ypxp

2

22

),(

),(

),(

),(ˆ),( yxIFvvH PSFyxOTF

pppp

pp

y

px

p

y

px

p

yxOTF

dydxyxP

dydxvf

yvf

xPvf

yvf

xP

vvH

2

*

),(

)2

,2

()2

,2

(

),(

Optical Transfer Function: Definition

Normalized optical transfer function

(OTF) in frequency space

Fourier transform of the Psf-

intensity

OTF: Autocorrelation of shifted pupil function, Duffieux-integral

Absolute value of OTF: modulation transfer function (MTF)

MTF is numerically identical to contrast of the image of a sine grating at the

corresponding spatial frequency

I Imax V

0.010 0.990 0.980

0.020 0.980 0.961

0.050 0.950 0.905

0.100 0.900 0.818

0.111 0.889 0.800

0.150 0.850 0.739

0.200 0.800 0.667

0.300 0.700 0.538

Contrast / Visibility

The MTF-value corresponds to the intensity contrast of an imaged sin grating

Visibility

The maximum value of the intensity

is not identical to the contrast value

since the minimal value is finite too

Concrete values:

minmax

minmax

II

IIV

I(x)

-2 -1.5 -1 -0.5 0 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Imax

Imin

object

image

peak

decreased

slope

decreased

minima

increased

Number of Supported Orders

A structure of the object is resolved, if the first diffraction order is propagated

through the optical imaging system

The fidelity of the image increases with the number of propagated diffracted orders

0. / +1. / -1. order

0. / +1. / -1.

+2. / -2.

order

0. / +1. -1. / +2. /

-2. / +3. / -3.

order

Polychromatic MTF

38

Polychromatical MTF:

Cut off frequency depends on

Spectral incoherent weighted

superposition of monochromatic MTF’s

0

)( ),()()( dvgSvg OTF

poly

OTF

0.8

1.0

0.6

0.4

0.2

0

0 max

/2

27.5 Lp/mm

max

55 Lp/mm

gMTF

= 350 nm

= 400 nm

= 450 nm

= 500 nm

= 550 nm

= 600 nm

= 650 nm

= 700 nm

polychromatic

ideal 350 nm

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

),(),(),( yxPTF vvHi

yxMTFyxOTF evvHvvH

/ max

00

1

0.5 1

0.5

gMTF

x p

y p

area of

integration

shifted pupil

areas

f x

y f

p

q

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

Interpretation of the Duffieux Iintegral

Interpretation of the Duffieux integral:

overlap area of 0th and 1st diffraction order,

interference between the two orders

The area of the overlap corresponds to the

information transfer of the structural details

Frequency limit of resolution:

areas completely separated

Contrast and Resolution

High frequent

structures :

contrast reduced

Low frequent structures:

resolution reduced

contrast

resolution

brillant

sharpblurred

milky

41

Contrast vs contrast as a function of spatial frequency

Typical: contrast reduced for

increasing frequency

Compromise between

resolution and visibilty

is not trivial and depends

on application

Contrast and Resolution

V

/c

1

010

HMTF

Contrast

sensitivity

HCSF

42