Optical Coherence Tomography: Andrew Gomez Daniel Kim Jiwon Lee Kenny Tao

Optical Coherence Tomography Andrew Gomez Daniel Kim Jiwon Lee Kenny Tao

Theory of OCT Reference Reflector

ER =

Ei i 2 kz R e 2

Es =

zR

[

Ei rs ( z s ) ⊗ ei 2 kzs 2

Ei = s ( k , ω )ei ( kz −ωt ) Light Source

Z=0

k = 2π / λ

ω = 2π ν

Sample

zS1 zS2

Beamsplitter (50/50)

]

rS ( z s )

rS ( z s ) = rS 1δ ( z S − z S 1 ) + rS 2δ ( z S − z S 2 ) + ...

Ei 2

zS1

{

Sample Reflections

zS2

Es =

iD = ρ ER + ES

1 2 3

[

Ei rs ( z s ) ⊗ e i 2 kz s 2

]

2

Detector

Schematic of a Michelson interferometer used in OCT.

Exemplary model for a sample comprising a series of discrete reflectors.

Izatt, Joseph A. Theory of Optical Tomography, 2006

Discrete Reflectors ER =

I D (k , ω ) = ρ E R + E S

2

Ei rR e i 2 kz R 2

Es =

= ρ ( E R + E S )( E R + E S )

Ei 2

(r

S1

)

e i 2 kz S 1 + rS 2 e i 2 kz S 2 + ...

∗

i ( kz −ωt ) For z=0 at beamsplitter andEi = s (k , ω )e

s (k , ω ) i ( 2 kz R −ωt ) s(k , ω ) I D (k , ω ) = ρ rR e + rS 1e i ( 2 kzS 1 −ωt ) + rS 2 e i ( 2 kzS 2 −ωt ) + ... 2 2

(

)

2

S (k )  S (k ) ( RS1 + RS 2 + ...)  " DC Terms" I D (k ) = ρ  RR + 2  2   S (k )  + ρ rR rS 1 e i 2 k ( z R − z S 1 ) + e −i 2 k ( z R − z S 1 ) + rR rS 2 e i 2 k ( z R − z S 2 ) + e −i 2 k ( z R − z S 2 ) + ...   2 

[ [

(

)

) ]

(

 S (k )  + ρ rS 1rS 2 e i 2 k ( z S 1 − z S 2 ) + e −i 2 k ( z S 1 − z S 2 ) + ...   2  RS 1 = rS1

2

) ]

(

" Auto - correlation Terms" S (k ) = s(k , ω

2

" Cross - correlation Terms"

Fourier Domain OCT 1.0 0.5 0

γ(z)

S(k)

1 ∆k π F

lc

∆k 0

0 2

2

F γ ( z ) = e − z ∆k ←→ S (k ) =

1 ∆k π

e

 ( k − k0 )  −   ∆k 

lc =

2 ln(2) 2 ln(2) λ0 2 = ∆k π ∆λ

k0

2

1 F [δ ( z + z 0 ) + δ ( z − z 0 )] ←→ cos kz 0 2 F x( z ) ⊗ y ( z ) ←→ X (k )Y (k )

iD ( z ) =

ρ  γ ( z) [ RR + RS1 + RS 2 + ...]  " DC Terms"  2 2  ρ + [ γ ( z ) ⊗ [ rR rS 1 ( δ ( z ± 2( z R − z S 1 )) ) + rR rS 2 ( δ ( z ± 2( z R − z S 2 )) ) + ...] ] " Cross - correlationTerms" 2 ρ γ (z)  +  ⊗ [ rS 1rS 2 ( δ ( z ± 2( z S 1 − z S 2 )) ) + ...]  " Auto - correlationTerms" 2 2 

Results iD ( z ) =

ρ [γ ( z )[ RR + RS1 + RS 2 + ...] ] 4 ρ + [ rR rS 1 ( γ [2( z R − z S 1 )] + γ [−2( z R − z S 1 )]) + rR rS 2 ( γ [2( z R − z S 1 )] + γ [−2( z R − z S 1 )]) + ...] 2 ρ + [ rS 1rS 2 ( γ [2( z S 1 − z S 2 )] + γ [−2( z S 1 − z S 2 )]) ] 4 rS ( z s ) Example field reflectivity function Delta function reflectors

0

zR

zS1

zS

zS2

iD (z ) “A-Scan” DC term

AutoCorrelation terms

Cross-correlation terms

2(zR-zS2) 2(zR-zS1)

0

Mirror image artifacts

-2(zR-zS1) -2(zR-zS2)

z

Experimental Setup 

First Experiment: Low-Coherence Interferometry



Second Experiment: Optical Coherence Tomography

Light Source

Fiber Coupler (50/50 Beamsplitter)

Reference Reflector & Detector Array (1-D CCD Camera)

Microscope

-Dichroic Mirror -Sample Stage

Methods 

Experiment 1: Low-Coherence Interferometry 

Purpose to obtain spectral interferogram data to measure center wavenumber ko, standard deviation ∆k and the power reflectivity of the slide surface

Low Coherence Interferometry 

Procedure 









Adjust reference arm micrometer such that there are no interference patterns across the spectrum. Turn micrometer known distance till a fringe pattern similar to the one shown in the theory writeup is observed (Fig 1). Calibrate spectrogram plot to be able to calculate power reflectivity of slide surface. Obtain spectrogram (Fig 2) of reference arm only by blocking light from reaching the microscope. Use to measure ko and ∆k. The value of ko is where the spectrum is at maximum and ∆k is the difference in wavenumber between maximum and 1/e of maximum. Turn on Fourier processing to observe A-scan plot.

I D (k ) Single Reflector

rS1

I D (k ) Multiple Reflectors

π z R − z S1

[1 + RS 1 ] 2 0

k0

k

k0

k

Figure 1 1.0 0.5 0

γ(z)

S(k)

1 ∆k π

lc

0

F

∆k 0

Figure 2

k0

Methods 

Experiment II: Optical Coherence Tomography 

Purpose to take two and three dimensional images of internal biological tissue microstructure.

Optical Coherence Tomography 

Procedure 







Take B-scan of IR card using “DC removal” and dual-axis scanning mirror. Repeat for fingertip. Obtain 3D image by setting “scan pattern” to “rectangular volume”. This allows 100 sequential B-scans to be taken. Stop scan and select “volume image” to obtain 3D rendering of data. Try using with a coin. Experiment using the 3D rendering program “3DView” on acquired data.

LCI Results – Single Reflector

z R − z S1

z R − z S 1 = 110 µm average wavenumber period = 33.327 pixels resolution = 8.57 ⋅10-4 k/pixel

LCI Results – Reference Arm k 0 = 4.799rad/μm Δk = 0.179rad/μm Rs1 = 9.481 ⋅10 −3

k 0 = 4.799rad/μm Δk = 0.179rad/μm Rs1 = 9.481 ⋅10

−3

λ0 = 1.309 μm Δλ = 0.08128 μm lc = 9.302 μm

LCI Results – A-scan

FWHM = 0.6 pixels resolution = 13.1μm/pixel lc = 7.86μm

OCT Results – B-scan

0.2mm

OCT Results – B-scan Complex conjugate artifacts Sweat glands 0.2mm

OCT Results – 3D Scan