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TRANSCRIPT
2
Preliminary Schedule
No Date Subject Detailed Content
1 19.10. Introduction Introduction, optical measurements, shape measurements, errors,
definition of the meter, sampling theorem
2 26.10. Wave optics Basics, polarization, wave aberrations, PSF, OTF
3 02.11. Sensors Introduction, basic properties, CCDs, filtering, noise
4 09.11. Fringe projection Moire principle, illumination coding, fringe projection, deflectometry
5 16.11. Interferometry I Introduction, interference, types of interferometers, miscellaneous
6 23.11. Interferometry II Examples, interferogram interpretation, fringe evaluation methods
7 30.11. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods
8 07.12. Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight,
Scheimpflug setup
9 14.12. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology
10 21.12. Holography Introduction, holographic interferometry, applications, miscellaneous
11 11.01. Measurement of basic
system properties Bssic properties, knife edge, slit scan, MTF measurement
12 18.01. Phase retrieval Introduction, algorithms, practical aspects, accuracy
13 25.01. Metrology of aspheres
and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms
14 01.02. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous
15 08.02. Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method
Temporal coherence
Principle of optical coherence tomography
Light sources
Dispersion
Time domain OCT
Spectral domain OCT
Examples
OCT in biological tissue
3
Contents
Temporal Coherence
t
U(t)
c
duration of a
single train
Damping of light emission:
wave train of finite length
Starting times of wave trains: statistical
| ( ) |
c
Time-Related Coherence Function
( ) lim ( ) ( ) ( ) ( )* *
TT
T
TTE t E t dt E t E t
1
2
( ) ( ) ( )*0 E t E t IT
( )( )
( )
( ) ( )
( )
*
02
E t E t
E t
Time-related coherence function:
Auto correlation of the complex field E
at a fixed spatial coordinate
For purely statistical phase behaviour: = 0
Vanishing time interval: intensity
Normalized expression
Usually:
decreases with growing symmetrically
Width of the distribution: coherence time c
t /1
t
tA
sin)(
deAtE ti2)()(
Temporal Coherence
I()
Radiation of a single atom:
Finite time t, wave train of finite length,
No periodic function, representation as Fourier integral
with spectral amplitude A()
Example rectangular spectral distribution
Finite time of duration: spectral broadening ,
schematic drawing of spectral width
6
0
)( dSI
deS i2)()(
1c
dc
2)(
cc cl
Time-Related Coherence Function
Intensity of a multispectral field
Integration of the power spectral density S()
The temporal coherence function and the power
spectral density are Fourier-inverse:
Theorem of Wiener-Chintchin
The corresponding widths in time and spectrum are
related by an uncertainty relation
The Parceval theorem defines the coherence time
as average of the normalized coherence function
The axial coherence length is the space equaivalent of
the coherence time
7
8
Lateral and Axial Resolution
Intensity distributions
Aberration-free Airy pattern:
lateral resolution
axial resolution
lateral axial
Ref: U. Kubitschek
NADAiry
22.1
2NA
nRE
9
Lateral Resolution vs Depth of Focus
Gaussian beam as example:
Lateral resolution wo coupled to depth of focus zo
Increase of depth resolution : tight focussing
Measurement dilemma:
measurement of deep bore
holes with large divergence
impossible,
large depth of focus only for
bad lateral resolution
Imaging dilemma:
large spreaded light cone
gathers light from different
depth in volume imaging,
bad contrast conditions
2
0
0
0
0
2
wwz o
o
r
zwo
depth of focus
lateral resolution
weak
focussing
tight
focussing
zo=wo2/
Resolution in OCT
1. Axial resolution limited by spectral bandwidth
2. Lateral resolution: diffraction limited, improvement by confocal setup
3. Usually low NA
22
4413.02ln2
cohzLow NA High NA
2zdiff
2x 2x
2zcoh
2zdiff
10
Lateral resolution:
Airy profile
Penetration depth:
axial resolution
u
fx
sin
24
22ln2resz
Resolution of OCT
Log x
100 m
10 m
1 m
100 m 1 mm 1 cm 10 cm depth
lateral
resolution
confocal
microscopy OCT
ultra
sound
Log z
11
OCT Setup
Basic principle of OCT
Michelson interferometer
Time domain signal
receiverfirst mirrorfrom
source
signal
beam
reference
beam
beam
splitter
second
mirror
moving
overlap
lc
z z
relative
moving
I(z)
wave trains
with finite
length
-4 -2 0 2 4
0
0,2
0,4
0,6
0,8
1,0
-4 -2 0 2 4
0
0,2
0,4
0,6
0,8
1,0
primary
signal
filtered
signal
t
12
z
I
measured
signal
filtered
signal
measured
position
axial length of coherence
m mn
mnmnm IIII cos2
2
42
zzk
0,,2)()()( 2121 rrrIrIrI
)()(
),,(
21
21
minmax
minmax
rIrI
rr
II
IIK
Interference Contrast
Superposition of plane wave with initial phase
Intensity:
Radiation field with coherence function :
Reduced contrast for partial coherence
Measurement of coherence in Michelson
interferometer:
phase difference due to path length
difference in the two arms
(Fourier spectroscopy)
13
Basic method of optical coherence tomography:
- short pulse light source creates a coherent broadband wave
- white light interferometry allows for interference inside the axial coherence length
Measured signal:
- low pass filtering
- maximum of envelope gives
the relative length difference
between test and reference arm
For Gaussian beam:
axial coherence length
High frequency oscillation
depends on z
Principle of OCT
z
I(z)
signal
measured
filtered
signal
measured position
coherence length
14
24ln 2 o
cl
2
ol
2
42
zzk
Example:
sample with two reflecting surfaces
1. Spatial domain
2. Complete signal
3. Filtered signal
high-frequency content removed
15
Optical Coherence Tomography
Ref: M. Kaschke
Signature of OCT signale for thin layer measurements at the resolution limit
16
Optical Coherence Tomography
Ref: R. Leach
1. Decreas of contrast as a result of the coherence gating
17
Optical Coherence Microscopoy
Ref: R. Leach
Signal
spectral profile
Basic setup:
Michelson interferometer
OCT - White Light Interference
z-scan by moving
mirror
white light
source
spectrumI
signal
surface
under
test
z
I
18
Fiber Based OCT Interferometer
Basic setup
LED
source
source
spectrum
I
fiber coupler
reference arm
measuring
arm
surface
under test
z-scan
detector
fiber fiber
fiberfiberI signal
z
19
Typical light sources used for OCT
Light Sources of OCT
No Type of source wavelength
[nm]
axial resolution
remark
1 Superluminescent diode 800-830 10 m
2 Swept laser source 1050-1070 2.8 kHz swept rate
3 Supercontinuum fiber laser 450-1700
4 Photonics crystal fiber (PCF) illuminated by a fs-Ti:Sapphire laser
550-950 < 1 m [3]
6 PCF source 1300 2 m
5 Ti-Sapphire laser 675-975 1 m [4]
22
Left column:
optical spectrum
Right column:
signal in the spatial
domain
23
OCT Sources and PCI Signal
Multimode
laser diode
Super
Luminescent
Diode
Ti-Saphir-
Laser
Ref: M. Kaschke
Pulse transmission through dispersive medium
1. input pulse
2. after propagation with dispersion
24
Dispersion
Ref: M. Brezinsky
Dispersive material in OCT:
- wavelength-dependent phase delay
- group velocity dispersion
- degradation of the axial resolution
- the dispersion causes a distortion of the pulse
shape during propagation
Dispersion:
k not linear changing with /
Group velocity
1st derivative
Group velocity dispersion
2nd derivative
GVD Dispersion
0( , )
i t kzE z t E e
t k z
3 2(2)
2 2''
2
z d nk z
c d
3 2
2 22j
j
d nt D z z
c d
2 2
2 2 2
1 2
gr
d c d k d nD
d v d c d
0
2 ( ) ( )( )
n nk
c
25
oc
nk
)(2)(
d
dk
vgr
2
11
2 3 2
2 2 2
1 1
2 gr
d k d d nD
d dv v c d
Dispersion relation
Expansion
Rearrangement with as variable
introduction of group velocity and GVD
GVD Dispersion
2
00000
2
02
2
00
)(''2
1)(')(
)(
2
1)()()(
00
kkk
kkkkS
00 0
0
2 ( )( )
nk k
26
gvd
dnn
c
n
d
d
cd
dk
d
d
d
dk 11)(2
vDd
nd
cd
dk
d
d
d
kd
2
1)(
2
'2
2
2
3
2
2
Numerical stable calculation with Sellmeier formula
Refractive index
Derivatives
GVD Dispersion
3
12
2
1)(j j
j
L
Kn
2
2
1 j j
jj
K Ldn
d n L
222
2 32 3 2 2
31 1 j j jj j
j jj j
K L LK Ld n
d n nL L
27
Example:
n, n' and n'' for three types of glasses
GVD - Group Velocity Dispersion
BK7 SF6 FK54
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
n( )
1.51
1.52
1.53
dn/d
-150
-100
-50
0
0
2
4
6
8
1.75
1.8
1.85
1.9
-600
-400
-200
01.430
1.435
1.440
1.445
-80
-60
-40
-20
0
-2
0
2
4
6
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
x 105
d2n/d 2
0
20
40
60
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
28
Reference arm:
- allows for z-scan in the depth
- z-discrimination by axial length of coherence, defined by the bandwidth
(coherence gating)
- Large spectral width of illumination source:
good time/spatial z-resolution
Measured signals:
- reflected light and scattered light
- SNR above 10-10 can be resolved
Problems:
- refractive interfaces are dispersive
- group refractive index is important
Typical:
- fast axial scan by moving mirror or rotating cube (A-scan)
- slow lateral x-y-scans (B-scans)
Properties of OCT
29
OCT Enables 3D Imaging in Ophthalmology
OCT combines unique
resolution with high
sensitivity
• axial resolution is independent
of the numerical aperture (pupil)
• by heterodyne detection a
photon-noise limited sensitivity is
achieved (R 10-10)
optical resolution
z = lc 0.44 /
5 µm
z 2/NA²
1.7 µm (1.33 1/24)-2
555 µm
within Rayleigh range of imaging optics
Ref: M. Kempe
Dimensions of OCT imaging:
a) only depth (A-scan), one-dimensional
b) depth and one lateral coordinate (B-scan), two-dimensional
c) all three coordinates, volume imaging
32
Optical Coherence Tomography
Ref: M. Kaschke
OCT for Fundus Imaging
Ref: M. Kempe
OCT imaging in Ophthalmology
- between 800 nm and 1100 nm reasonable contrast
- here no dazzling of the eye
- lateral and depth resolution appropriate for single cells
Basic assumption: Michelson interferometer
Coherent field of the reference arm
factor 2: double pass in the arm
Coherent field of the sample/test arm
The spectral distributions E() correspond to a low coherence light source
(ultra short pulses)
Simplification: only a single mirror is assumed as sample
Interference signal
integrating over all spectral components
of the modulated signal part
Phase difference determines the fast
oscillating signal
depends on z-difference and on dispersion
36
Theory of Time Domain OCT
tizik
rorrreEE
)(2
tizik
sossseEE
)(2
*22Re2)( srsr EEEEI
deEE
dEEI
i
soro
srTD
)(*
*
Re2
Re2
rrss zkzk )()(2)(
With expansion of the dispersion functions
Definition of the spectral cross correlation
S() between refernce and sample arm
assumed to be symmetrical
Rs: reflectivity in sample arm
Phase difference with phase and group
velocity
Final result of TD-OCT signal
Interpretation:
- cos-prefactor: high oscillating term, proportional to time difference
- integral term: envelope of signal, depends on dispersion
is determined by the light source spectrum bandwidth
37
Theory of Time Domain OCT
gopo
g
o
p
o ttcc
z
)(2
od
dkkk os
)()()( 0
*)( soros EESR
deStRzI go ti
posTD )(cos)(
Spectral Domain OCT
Spectral Domain-OCT:
- broad band source
- reference mirror fixed in position, no A-scan necessary
- signal splitted by spectrometer
The high-frequency content of the signal is analyzed
The frequency is proportional to the depth z,
measured is the overlay beat-signal of all scatterers
Broadband
source
spectrometer
reference mirror
Sample
in z fixed
interferogramm
frequency domainFourier transform
spatial domain
data
processingz
z
38
Fourier Domain OCT
Fourier Domain-OCT:
setup
Signals:
a) intensity spectrum
b) spatial intensity distribution
39
Ref: M. Kaschke
Principle of OCT
Comparison of
1. time domain
2. spectral domain
source
k
z
z, t
z beam
splitter
intensity
Probe
reference mirror
Interference signal
TD:
sequentiell detection
with photodetector
SD:
parallel detection with
spectrometer
)cos(2~ zkIII rsIF
Intensity spectrum
Fourier-transform
and
Envelope
detection
envelope
detection
A-scan
A-scan
z
z k
Ref: M. Totzeck
40
Frequency Domain OCT has dramatic signal-to-
noise advantage
Time Domain OCT Frequency Domain OCT
Shot-
noise-
limited
detection
Spectral Domain OCT Swept Source OCT
Detection in M wavelength channels
TD-OCT
FD-OCT
R. Leitgeb et al., Opt.Expr. 11(2003), 889
𝑆𝑁𝑅𝑇𝐷 =𝜂𝑅𝑠𝑃0𝐸𝑝𝐵
𝑆𝑁𝑅𝑆𝐷 =𝜂𝑅𝑠𝑃0𝑀𝜏𝑖𝑛𝑡
2𝐸𝑝 𝑆𝑁𝑅𝑆𝑆 =
𝜂𝑅𝑠𝑃0𝑀𝜏𝑠𝑤𝑒𝑒𝑝2𝐸𝑝
• coherent addition of M
waves
• signals from all depths
contribute
1
𝐵= 𝜏𝑖𝑛𝑡 = 𝜏𝑠𝑤𝑒𝑒𝑝 for
𝑆𝑁𝑅𝑆𝐷 = 𝑆𝑁𝑅𝑆𝑆 =𝑀
2𝑆𝑁𝑅𝑇𝐷
Ref: M. Kempe
Properties of Fourier Domain OCT
the modulation frequency depends on the path length difference
simultaneous measurement of all backscattering contributions: larger sensitivity
- Fourier transform adds signals coherent
- noise is added incoherent
faster image processing due to missing A scan
signal drop-off with increasing depth
spectrometer resolution changes over depth
positive and negative z cannot be distinguished
42
Field in the reference arm
Field in the sample arm
Interference field
Discrete scattering model:
OCT signal
with reflectivities rj
Final signal evaluation
1st term: DC signal (underground)
2nd term cross correlation with
reference, interesting term
3rd term autocorrelation between
scatterers (small)
43
Fourier Domain OCT Signal
Rkzi
RR erkE
E20
2
),(
Skzi
SS ezrkE
E20 )(
2
),(
2
SRFD EEI
2
22
2
0
2
),(
2),(
j
tkzi
Sj
tkzi
RFDSjR erer
kEkI
j
SjSSjSS zzrzr )(
22( , ) ( )
4
( ) cos 2 ( )2
( ) cos 2 ( )4
FD R Sj
j
R Sj R Sj
j
Sj Sm Sj Sm
j m
I k S k r r
S k r r k z z
S k r r k z z
Fourier Domain OCT Signal
Typical Fourier Domain
OCT signal
only z-part
First term: DC
Second term: interference, cross-correlation, contains information
Third term: autocorrelation between scatterers
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20
0.2
0.4
0.6
0.8
1
1.2
1.4
interference
signal
DC signal
contrast
mj
SmSjSmSj
j
SjRSjR
j
SjRFD
zzkrrkS
zzkrrkSrrkSkI
)(2cos)(4
)(2cos)(2
)(4
),(22
44
Fourier Domain OCT Signal
Signal complexity depends on
scatter-distribution
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20
0.2
0.4
0.6
0.8
1
1.2
1.4
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20
0.2
0.4
0.6
0.8
1
1.2
1.4
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20
0.2
0.4
0.6
0.8
1
1.2
1.4
1 scatterer
2 scatterer
3 scatterer
rS(z
S)
zSz
S2z
S1z
R
reflectivity
samplereference
ID(z)
z-2(z
R-z
S1)0
A-scan
DC term
auto
correlation
cross correlation mirror image artifacts
-2(zR-z
S2)2(z
R-z
S2) 2(z
R-z
S1)
45
Fourier Domain OCT Signal
)(16
1
2)(
2)(ˆ)(ˆ)(
2
211 zrA
nn
rrz
n
rkSFkIFzI SC
SS
SR
S
S
)(
)()( 1
kS
kIF
r
nzr
R
SS
)(
114
)(
0
2
24
2
22
22
22
2
22
zI
eew
weee
ww
we
w
PPzI
zzp
S
nonzzzp
Snon
nonz
non
bSR
cohgate
ssbsssbs
Signal evaluation: inverse Fourier transform
DC-term subtracted by difference measurement
Auto-correlation: mostly negligible
Heterodyne efficiency:
Decrease in signal strength due to scattering underground for gaussian beams
46
Fourier Domain OCT Heterodyne Efficiency
Typical z-dependence
Dependence on scattering coefficient
for different focus locations
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
z
(z)
single
scattering
multiple
scattering
0 2 4 6 8 10 12 14 16 18 2010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
(s)
s
z = 0.5
z = 1.0
z = 1.5
z = 2.0
47
Fourier Domain OCT Heterodyne Efficiency
Dependence on numerical aperture
for different scattering coefficients
Dependence of the mean signal
strength on z
(NA)
s = 2
0 0.05 0.1 0.15 0.2 0.2510
-7
10-6
10-5
10-4
10-3
10-2
10-1
NA
s = 10
s = 10
s = 40
<P>
NA0 0.05 0.1 0.15 0.2 0.25
102
103
104
105
106
48
OCT Heterodyne Signal Strength
Decrease with growing z
Peak in focus
z0 0.5 1 1.5 2 2.5
10-9
10-8
10-7
10-6
10-5
10-4
10-3
I(z)
focus
49
Fourier Domain OCT Example Calculation
k /ko
inte
nsit
y
0 50 100 150 200 250 3000
0.2
0.4
0.6
0.8
1FD Signal / IFT of deconvolved dI(k)
Depth z in my
refl
ecti
vit
y
0.9 0.95 1 1.05 1.10
0.2
0.4
0.6
0.8
1Interferogram
Only z-dependence
2 discrete scatterers
50
Fourier Domain OCT with Swept Source
Special setup for Fourier domain OCT:
- SS-OCT: swept source instead of broad spectral source
- source has tunable wavelength
- advantage: no spectrometer necessary
- usually faster than classical Fourier domain OCT
- tunable laser expensive
Narrow source
frequency sweep
detector
reference mirror
Sample
in z fixed
primary signal
frequency domain spatial domain
data
processingz
z
frequency
time
samplereference
51
Overview on OCT Setups
52
Different realization options for OCT:
Properties Time domain Spectral domain /
Fourier Spectral swept source
Source broad band broad band tunable laser
z-Scan, moving reference mirror yes no no
speed slow fast very fast
data gathering sequential parallel, spectrometer sequential
detector photodetector line camera photodetector
signal processing filtering, envelope Fourier transform Fourier transform
data encoding time frequency frequency
signal to noise behavior bad good good
54
Scattering in Turbid Media
Different strengths of interaction
Ref: M. Gu
a) ballistic photon
b) snake photons
c) multiple
scattered photons
55
Scattering in Turbid Media
Change of light properties
Ref: M. Gu
a) spectral shift
frequency
frequency
b) spatial broadening
x
w
x
c) temporal broadening
time t t time t
snake
d) polarization
snake
56
Scattering in Turbid Media
Imaging of a circular disc through
a turbid medium with
growing scattering strength
a)...d)
Ref: M. Gu
Scattering in Tissue
Description of the light propagation in tissue:
1. Coefficient of absorption a
Loss of energy on the path.
2. Coefficient of scattering s
Probability of directional change per unit length of the path
3. Phase function p(q)
Mean angle distribution of the scattering process.
Frequently used model: Henyey-Greenstein
The sum of both coefficients is called the total extinction coefficient
sat
57
Henyey-Greenstein Scattering Model
Henyey-Greenstein model for human tissue
Phase function
Asymmetry parameter g:
Relates forward / backward scattering
g = 0 : isotropic
g = 1 : only forward
g = -1: only backward
Rms value of angle spreading
Typical for human tissue:
g = 0.7 ... 0.9
)1(2 grms
2/32
2
cos21
1
4
1),(
gg
ggpHG
forward
30
210
60
240
90
270
120
300
150
330
180 0
g = 0.5
g = 0.3
g = 0.7
g = 0.95
g = -0.5
g = -0.8
g = 0 , isotrop
z
58
Light Propagation in Tissue
Extended Huygens-Fresnel principle
Greens function with statistical
phase term
Coherent primary beam without scattering
Incoherent scattered intensity contributions
tissue
surface
n n1
signal
layerlens f
illuminating
beam
back
scattering
d
z
a
)',(
''22
22
2)',( rri
DrrrArB
ik
eeB
ikrrG
Pill
(z)
z
coherent
incoherent
straylight
')',()'()( 0 rdrrGrErE
59