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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


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


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


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


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


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


)(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


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


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

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