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