| BMEG400C/591C
Optical Coherence Tomography (OCT)
| BMEG400C/591C
Optical Coherence Tomography (OCT)
Copyright By PowCoder代写 加微信 powcoder
| BMEG400C/591C
Gaussian Spectrum
Gaussian Power Spectrum Density
𝑆 𝜔 = 1 exp − 𝜔 − 𝜔” # 2𝜋𝜎! 2𝜎!#
𝜔!: Central angular frequency
𝜎”: Standard deviation of Gaussian distribution
FWHM!≜Δ𝜔=2 2ln2𝜎!
Similarly, for a spectral distribution expressed as a function of frequency:
𝑆𝜈= 2𝜋𝜎exp− 2𝜎$# FWHM$≜Δ𝜈=22ln2𝜎$,where𝜎!=2𝜋𝜎$
Δ𝜈 = 𝑐Δ𝜆 𝜆#”
: description in wavelength using 𝑐 = 𝜆𝜈
| BMEG400C/591C
Temporal Coherence & Spectrum
FWHM) = 𝜏’
Wiener-Khinchin Theorem
𝑆 𝜔 = 1 exp − 𝜔 − 𝜔” # 2𝜋𝜎! 2𝜎!#
Γ𝜏=F%&𝑆𝜔 𝜏=1exp𝑖𝜔”𝜏exp−𝜎!#𝜏# 2𝜋 2
𝛾 𝜏 = exp 𝑖𝜔”𝜏 exp − 2 = exp 𝑖𝜔”𝜏 exp −
𝜏’ ≜FWHM( ) =2 2ln2=4ln2𝜆#” 𝜎! 𝜋𝑐 Δ𝜆
= 2 2ln2 = 2 2ln2 =4ln2𝜆#” Δ𝜔⁄2 2ln2 2𝜋Δ𝜈⁄2 2ln2 𝜋𝑐 Δ𝜆
𝜎 !# 𝜏 # 1 #
| BMEG400C/591C
OCT – Equations
𝐸*+ 𝑡 = 𝛼𝐸” 𝑡
𝐸,+ 𝑡 = 𝛽𝑒-/⁄#𝐸” 𝑡
𝐸* 𝑡 = 𝛽𝑒-/⁄#𝑟*𝐸*+ 𝑡 = 𝛼𝛽𝑒-/⁄#𝑟*𝐸” 𝑡 𝐸, 𝑡+𝜏 = 𝛼𝑟,𝐸,+ 𝑡 = 𝛼𝛽𝑒-/⁄#𝑟,𝐸” 𝑡+𝜏
Schematic of a low-coherence interferometer measuring reflectivity, 𝑅#, of a single scatter at axial location 𝑧#.
OCT – Equations
𝐼012.𝜏= 𝐸*+𝐸,# =𝐸*∗𝐸* +𝐸,∗𝐸, +𝐸*∗𝐸, +𝐸*𝐸,∗ =𝛼𝛽𝑅* 𝐸” 𝑡 G𝐸”∗ 𝑡 +𝛼𝛽𝑅, 𝐸” 𝑡+𝜏 G𝐸”∗ 𝑡+𝜏
+𝛼𝛽 𝑅*𝑅, 𝐸”∗ 𝑡 G𝐸” 𝑡+𝜏 +𝛼𝛽 𝑅*𝑅, 𝐸” 𝑡 G𝐸”∗ 𝑡+𝜏 =𝛼𝛽 𝑅*𝐼” +𝑅,𝐼” +2 𝑅*𝑅, GRe Γ 𝜏
| BMEG400C/591C
=𝛼𝛽𝐼” 𝑅*+𝑅,+2 𝑅*𝑅,GRe𝛾𝜏
=𝛼𝛽𝐼” 𝑅* +𝑅, +2 𝑅*𝑅, Gcos 𝜔”𝜏 exp −4ln2 𝜏𝜏 # ‘
Re𝛾𝜏 =cos𝜔”𝜏exp−4ln2𝜏𝜏# ‘
2𝑣*𝑡−𝑧, 𝜔”𝜏=𝜔”2𝑣*𝑡−𝑧, =2𝜋𝜈2𝑣*𝑡−𝑧, 𝜏𝑡=𝑐𝑐
𝑐 =2𝜋𝜈#5!6%7″ =2𝑘𝑣*𝑡−𝑧, 8$
= 2𝑘𝑣*𝑡 − 2𝑘𝑧, = 2𝑘”𝑣*𝑡 − 𝜙
=𝛼𝛽𝐼” 𝑅* +𝑅, +2 𝑅*𝑅, Gcos 2𝑘”𝑣*𝑡−𝜙 exp −16ln2 𝑣*𝑡−𝑧, #
| BMEG400C/591C
OCT – Equations
𝐼012. 𝜏 =𝛼𝛽𝐼” 𝑅* +𝑅, +2 𝑅*𝑅, Gcos 2𝑘”𝑣*𝑡−𝜙 exp −16ln2 𝑣*𝑡−𝑧, # 𝑐𝜏’
• 𝑆 𝜔 = & exp − !%!$ % #/9# #9#%
• 𝜏’ = # #:;# 9#
• 𝜏 = # 7! %7″ ‘
• 𝑧* = 𝑣* A 𝑡
• 𝜏 𝑡 = # 5! 6%7″ ‘
: Gaussian spectral power density
: Coherence time of the Gaussian source
: Time delay due to a path mismatch between reference and sample arms
: Position of the reference mirror moving at constant velocity, 𝑣*. : Time delay (function of time)
| BMEG400C/591C
OCT – Summary
𝐼012. 𝜏 =𝛼𝛽𝐼” 𝑅* +𝑅, +2 𝑅*𝑅, Gcos 2𝑘”𝑣*𝑡−𝜙 exp −16ln2 𝑣*𝑡−𝑧, # 𝑐𝜏’
cos 2𝑘”𝑣*𝑡−𝜙 =cos 2𝜋2𝑣*𝑡−𝜙 =cos 2𝜋2𝑣* 𝑡−𝜙 =cos 2𝜋𝑓𝑡−𝜙
𝜆” 𝑖1;<1:=>1 𝑡 ∝ exp −16ln2
FWHM1;<1:=>1 = 2 2ln2
𝑐𝜏 𝑐𝜏 2ln2𝜆#
1= 𝜆” 𝑓 2𝑣*
𝛼𝛽𝐼”𝑅* 𝑧,⁄𝑣* 𝑡
𝛿𝑧= ‘×𝑣*= ‘= ” 2𝑣* 2𝜋Δ𝜆
| BMEG400C/591C
OCT – Equations (Multiple Scatters)
Time-domain OCT
: 𝑟? is the field reflectivity of scatter 𝑛 at axial location 𝑧,?. #
= K 𝑟,?𝛿 𝑧, − 𝑧,?
𝐼012.=𝛼𝛽 𝑟*𝐸”𝑡+V𝑟,?𝐸”𝑡+𝜏?
| BMEG400C/591C
OCT – Equations (Multiple Scatters)
𝐼012.=𝛼𝛽𝐼” 𝑅*+V𝑅,?+ V 𝑟,?𝑟,CRe𝛾𝜏?−𝜏C +2𝑟*V𝑟,?Re𝛾𝜏?
𝐼012. 𝑡 ≈𝐾 𝑅*+2𝑟*V𝑟,?Re𝛾𝜏?
• 𝐾 : constant
• 𝑅*:MaincomponentofDC
• Neglecting the autocorrelation term
𝑣*𝑡−𝑧,? # 𝐼012. 𝑡 ≈𝐾 𝑅*+2𝑟*V𝑟,?cos2𝑘”𝑣*𝑡−𝜙? exp −16ln2 𝑐𝜏’
| BMEG400C/591C
OCT – Equations (Multiple Scatters)
𝐼012. 𝑡 ≈𝐾 𝑅*+2𝑟*V𝑟,?cos2𝑘”𝑣*𝑡−𝜙?
Filtering envelope by first rectifying the signal and then applying a low-pass filter
𝑣*𝑡−𝑧,? # exp −16ln2 𝑐𝜏’
A-lines from three scatters
| BMEG400C/591C
OCT Imaging
§ Lateral resolution o Gaussian Optics
§ Lateral field of view
o FOV = 2 x f x tan(scan angle)
§ Axial resolution
o Decoupled from the optics
o Dependent on the light source
Sample arm beam (maximum scan angle 𝜃&'()
𝐹𝑂𝑉 = %*-%$
sin, sin./ 𝑁𝐴 2
Axial field of view
Objective lens (focal length 𝑓, numerical aperture 𝑁𝐴)
Lateral resolution
𝛿𝑥 = 0.37 𝜆! 𝑁𝐴
Axial resolution
𝛿𝑧 = 𝑙+ = 2ln2 𝜆,! 𝜋 Δ𝜆
Lateral field of view
𝐹𝑂𝑉$%&'(%$ = 2 * 𝑓 * 𝜃)%*
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com