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Metrology and Sensing
Lecture 14: OCT
2018-02-01
Herbert Gross
Winter term 2017
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
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
9
1. Decreas of contrast as a result of the coherence gating
10
Optical Coherence Microscopoy
Ref: R. Leach
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)
11
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
12
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
13
Optical Coherence Tomography
Ref: M. Kaschke
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
14
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
15
Achronyms in literature
16
Optical Coherence Tomography
Ref: R. Leach
Left column:
optical spectrum
Right column:
signal in the spatial
domain
17
OCT Sources and PCI Signal
Multimode
laser diode
Super
Luminescent
Diode
Ti-Saphir-
Laser
Ref: M. Kaschke
Examples:
SLD = super luminescent diodes
18
OCT Light Sources
Ref: M. Kaschke
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]
19
Pulse transmission through dispersive medium
1. input pulse
2. after propagation with dispersion
20
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
21
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
22
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
23
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
24
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
25
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
26
27
OCT Measurement of Distances
Technical application of white light interferometry
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
28
Signature of OCT signale for thin layer measurements at the resolution limit
29
Optical Coherence Tomography
Ref: R. Leach
Example of OCT Imaging
Example: Fundus of the human eye
30
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
31
Optical Coherence Tomography
Ref: M. Kaschke
32
White Light Interferometry
Examples
Ref: R. Kowarschik
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
33
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
34
Theory of Time Domain OCT
gopo
g
o
p
o ttcc
z
)(2
od
dkkk os
)()()( 0
*)( soros EESR
deStRzI go ti
posTD )(cos)(
Notations:
Illumination field: E0(t)
Impulse response from the sample: H(t)
Field at detector for time delay
between test and reference arm
Measured time averaged signal
Autocorrelation between two delayed
signals:
1. general, coherence function
2. here, with modulated sample field
The highly oscillating signals without
averaging contain the information:
OCT signal
Wiener Chintchine theorem:relation
of sample response with spectrum
Fourier transfrom of time signal:
spectrum of sample
35
Transfer Function Theory of TD-OCT
E t E t E t H tDet o o( , ) ( ) ( ) ( )
I t E t dtDet
T
( , ) ( , ) 2
0
( ) ( ) ( ) ( ) ( )* *
E t E t dt E t E to o o o
*
2 2
0
( , ) ( ) ( ) ( ) ( )
( ) ( ) ( )
T
o o
I t H t t H t t
E t E t H t dt
I H t t H t tOCT ( ) ( ) ( ) ( ) ( )*
H E e do
i( ) ( )
2 2
I H Eo( ) ( ) ( ) 2
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
36
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
37
Fourier Domain OCT
Fourier Domain-OCT:
setup
Signals:
a) intensity spectrum
b) spatial intensity distribution
38
Ref: M. Kaschke
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
39
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)
40
Fourier Domain OCT Signal
Rkzi
RR erkE
E 20
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
41
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)
42
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
43
FD OCT in 3D Gaussian Beams with EDF
Illumination profile
Coherence gating
OCT signal
Integration over the sample volume
xs, ys : B-scan
zs : A-scan
2
2
2
2
22
2
2
2
1),(
Sn
w
r
zpz
non
w
r
z
Sdill
w
eee
w
eePRzrI
Ssbsnons
refs
refs
refs
zzzikz
zzz
zzzkkik
k
s
eez
dkeezzH
0
2
0
2
22ln4
)2(2ln4
1
),(
dzydxzzHzyyxxIzyxrCzyxS sssillsssOCT ),(),,(),,(),,(
44
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
45
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
46
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
47
FD OCT in 3D Gaussian Beams Example
5 discrete scatter planes with finite extend
Without EDF-caustic
48
FD OCT in 3D Gaussian Beams Example
5 discrete scatter planes with finite extend
With EDF-caustic factor 4
49
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
50
Overview on OCT Setups
51
Ref: M. Kaschke
52
Scattering in Turbid Media
Different strengths of interaction
Ref: M. Gu
a) ballistic photon
b) snake photons
c) multiple
scattered photons
53
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
54
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
55
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
56
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
57
Propagation in Scattering Tissue
Transverse coherence length
decreases with z
Correlation of statistical phase
normal distribution
Intensity inside scattering medium
srms
cz
nzL
3)(
2
)(1),(
zL
r
zz
PTcss eeezr
0 0.5 1 1.5 210
-4
10-3
10-2
10-1
100
101
Lc [mm]
z [mm]
212
*
1212
*
010 ,,,,,),,( DDDinDinDDPTDDill rdrdrErEzrrzrrGzrrGzyxI
58
Propagation in Tissue with Gaussian Beams
Intensity inside the scattering
medium
First term: unscattered part
coherent
Second term:
scattering underground
Broadening by coherence decrease
Lateral resolution in focus plane unchanged
With g increasing incoherent scattered light decreases contrast
2
2
2
2
22
2
2
2
1),(
Sn
w
r
zpz
non
w
r
z
Sdill
w
eee
w
eePRzrI
Ssbsnons
22
22 1
L
Lnonw
z
f
zww
222
22 21
cL
LSL
z
w
z
f
zww
59
Propagation in Tissue with Gaussian Beams
log Iill
(r,z)z
r
g = 0 g = 0.9 g = 0.99
0 1 2 3 410
-4
10-3
10-2
10-1
100
log Iill
(z)
z
g = 0
g = 0.5
g = 0.9
g = 0.99
log Iill
(r,zF)
r-0.4 -0.2 0 0.2 0.4
10-4
10-3
10-2
10-1
100
g = 0
g = 0.5
g = 0.9
g = 0.99
60
OCT Heterodyne Signal
tissue
surface
n
signal
layerlens f
illumination
fiber back
scattering
d
z
lens f
z
x
signal
B-scan
n1
Setup:
4f confocal
(B-scan omitted)
Detector current
Coherence function reference and
sample arm
Signal power
rdrErEzCzi SR
)()(Re)()( *
),(),(),,(
)()(),(
2
*
121
2
*
121
zrEzrEzrr
rErErr
SSS
RRR
212121
222 ),(),,(Re)(2)( rdrdrrzrrzzI RS
61
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
62