| BMEG400C/591C
Fourier Domain OCT
| BMEG400C/591C
Fourier Domain Detection
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§ Previous introduction to OCT was “Time Domain” OCT
o Not commonly used, but easier to understand the concepts
§ Next discuss Fourier Domain (FD) OCT
o Different from Time Domain in that the detector is spectrally resolved
• Swept source / optical frequency domain imaging (OFDI) • Spectral domain
| BMEG400C/591C
Spectrally Resolved Detection
§ Wavenumber (wavelength) encoded in time
𝐼 ! ” # 𝑘 = 𝜅 𝑆 𝑘 𝐸 $% + 𝐸 &% + 2 𝐸 $ 𝐸 & c o s 2 Δ 𝑧 ‘ 𝑘
o 𝑆 𝑘 : Spectrum / 𝜅: Constant
o Δ𝑧! is fixed
o 𝑘 (wavenumber) is the independent variable
Fringe frequency ←→ Distance
| BMEG400C/591C
Spectrally Resolved Detection
o “Swept source”
o Source is a rapidly tunable laser o Detector is simple photodiode
o High speed digitizer
o Acquire 𝜆 vs time
o “Spectral Domain”
o Low coherence source (SLD, laser) o High resolution spectrometer
o Line scan camera
o Acquire 𝜆 vs pixel
| BMEG400C/591C
Wavelength Swept Source
§ Broad band gain medium
§ A mechanical scanner swept rapidly in time § Output is a rapidly changing wavelength
High-speed wavelength-scanning filter Spectrum and intensity output of a wavelength-swept source
| BMEG400C/591C
Spectral Domain OCT
Spectrometer
• All wavelengths incident on sample at the same time • Detect interference signal using a spectrometer
• High phase stability
| BMEG400C/591C
FD OCT – Equations
Reference arm
𝐸$ 𝑘,𝜔 = 𝛼𝛽𝐸, 𝑘,𝜔𝑒-/⁄0𝑟$exp𝑖2𝑘𝑧$−𝜔𝑡
Sample arm
𝐸) 𝑘,𝜔 = 𝛼𝛽𝐸, 𝑘,𝜔 𝑒-/⁄0 0𝑟)% exp 𝑖 2𝑘𝑧)% −𝜔𝑡 %&’
Time-averaged Interference
𝐼 ! ” # 𝑘 = 𝐸 $ + 𝐸 ) 0 1 = 𝐸 $ 0 1 + 𝐸 ) 0 1 + 𝐸 $∗ 𝐸 ) 1 + 𝐸 $ 𝐸 )∗ 1
𝐼!”# 𝑘 =𝛼𝛽𝑆𝑘 𝑅$+0𝑅)%+ 0 𝑟)%𝑟)+cos2𝑘𝑧)%−𝑧)+ +20𝑟$𝑟)%cos2𝑘𝑧$−𝑧)% %&’ %*+&’ %&’
SD OCT in which the reference arm mirror is fixed and the photodiode is replaced by a grating dispersing the wavelengths onto an array of detector
| BMEG400C/591C
Fourier Transform – Review
§ Fourier transform is a common tool for extracting the frequencies of signals § Fourier transform pair of 𝑧 is 𝑘 (not 𝜆)
oArgumentofcosineis𝑘𝑧= 2𝜋⁄𝜆𝑧butacquisitionisapproximatelylinearin𝜆 o Inverting the axis to 𝑘 introduces non-uniform sampling resulting in a ‘chirp’
o Need to resample the signal to be linear in 𝑘 prior to take Fourier transform
→ Resampling or Rescaling process
| BMEG400C/591C
FD OCT – Equations
𝐼!”# 𝑘 ∝𝑆𝑘 B 𝑅$+0𝑅)%+ 0 𝑟)%𝑟)+cos2𝑘𝑧)%−𝑧)+ +20𝑟$𝑟)%cos2𝑘𝑧$−𝑧)%
𝑖!”# 𝑧 ∝𝛾 𝑧 ⊗E 𝑅$ +0𝑅)% %&’
+ 0 𝑟)% 𝑟)+ %*+&’
𝛿 𝑧 ± 2 𝑧)% − 𝑧)+
“Auto-correlation Terms”
“Cross-correlation Terms”
+ 2 0 𝑟$ 𝑟)% 𝛿 𝑧 ± 2 𝑧$ − 𝑧)% H %&’
| BMEG400C/591C
FD OCT – Equations
= F c o s 2 𝑘 ∆ 𝑧
4 𝑒-05∆7 + 𝑒3-05∆7 = +3 4 2 𝑒
3-057 1 4 𝑑 𝑧 = 2 +3 4 𝑒
-05∆7 3-057 4 3-05∆7 3-057
𝑒 𝑑 𝑧 + +3 4 𝑒 𝑒 𝑑 𝑧
1 4 -05(∆737) =2 +34𝑒
4 3-05(∆7:7) 1
𝑑𝑧++34𝑒 𝑑𝑧 =2 𝛿(𝑧−∆𝑧)+𝛿(𝑧+∆𝑧)
𝐹𝑘 𝑓𝑧 cos 2𝑘𝑧!
| BMEG400C/591C
FD OCT – Equations
𝑖!”# 𝑧 ∝𝛾 𝑧 𝑅$ +0𝑅)% “DCTerms” %&’
+ 0 𝑟)% 𝑟)+ 𝛾 2 𝑧)% − 𝑧)+ + 𝛾 −2 𝑧)% − 𝑧)+ “Auto-correlation Terms” %*+&’
+2 0 𝑟$ 𝑟)% 𝛾 2 𝑧$ − 𝑧)% + 𝛾 −2 𝑧$ − 𝑧)% “Cross-correlation Terms” %&’
| BMEG400C/591C
FD OCT – Characteristics
§ FD OCT has HIGH sensitivity
§ but also has new sources of artifact o DC subtraction
o Auto-correlation
o Sensitivity roll-off
o Complex conjugate symmetry o Aliasing
| BMEG400C/591C
DC Subtraction
§ Source spectrum needs to be subtracted from interferogram
o Remove DC & Fixed Pattern Noise (FPN)
o Acquire DC spectrum with sample blocked; OR
o Compute a representative DC spectrum by taking
mean the interferograms in a B-scan
• Assumesufficientsignalvariationforthefringestocancelout
| BMEG400C/591C
Auto-correlation Artifact
§ Strong reflections in the sample will interfere with each other o Appears as low frequency noise (Ghost image)
) 𝑟:4𝑟:6 𝛾 2 𝑧:4 −𝑧:6 +𝛾 −2 𝑧:4 −𝑧:6 45678
| BMEG400C/591C
Sensitivity Roll-off
§ SNR in FD OCT is related to distance
Dynamic range for signal detection at a high frequency
is decreased !
o High frequencies cannot be sampled by the limited spectral resolution o Convolution of optical PSF and pixel width
o 𝑧;<= = =
| BMEG400C/591C
Complex Conjugate Artifact
§ Interferometric signal is sinusoidal
§ Fourier transform has positive and negative frequencies
§ Signal is a real valued function, so positive and negative are indistinguishable
o Need to ensure that the entire sample is to one side of the reference position
§ Sample depth is limited
o Can’t stay close to DC (Sign ambiguity)
o Can’t go to far from DC (Sensitivity Roll-off)
| BMEG400C/591C
§ High frequencies of OCT signal get aliased
FD OCT image of a finger pad (A), folded (B) as a result of depth degeneration.
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