Optical Coherence Theory

Coherence time and length of a Gaussian source, OCT axial resolution 0.4413 λ²/(nΔλ), and the shot-noise-limited Fourier-domain SNR and sensitivity.

Model: Textbook approximation

Centre wavelength λ₀ (nm)Bandwidth Δλ, FWHM (nm)Group index of the sample nPower on the sample (mW)Sample reflectivity R_sInterferometer efficiency κ (1 = lossless)Detector responsivity ρ (A/W)A-scan rate (kHz)

Results

Axial resolution in air7.573 µm

Axial resolution in the sample5.694 µm

Coherence length l_c (path difference)15.15 µm

Coherence time τ_c50.52 fs

Spectral width Δν17.47 THz

Signal power κ·R_s·P6.000e+4 nW

Shot-noise SNR (Fourier domain)98.7 dB

Sensitivity (1/R_s at SNR = 1)118.7 dB

SNR = ρ P_s Δt/(2e) with Δt = 1/(A-scan rate) and the reference arm strong enough for shot-noise-limited detection. Spectral width Δλ/λ₀ = 0.076; the Gaussian formulas assume it is small.

Coherence gate in the sample

Source spectrum

Axial resolution against bandwidth

Model

|γ(τ)| = exp(−(πΔν τ)²/(4 ln 2)), Δν = cΔλ/λ₀², τ_c = (4 ln 2/π)/Δν, l_c = cτ_c

Δz = l_c/(2n) = (2 ln 2/π) λ₀²/(n Δλ) = 0.4413 λ₀²/(n Δλ)

P_s = P · κ · R_s, SNR = ρ P_s Δt/(2e) (Choma 2003, eq. 10), sensitivity = ρ P κ Δt/(2e); κ = 1 is a lossless 50/50 interferometer, whose detector receives P_s/2

A Gaussian source spectrum with Δλ ≪ λ₀ and matched dispersion in the two arms. The coherence length is the FWHM of |γ| against the optical path difference; the path difference is twice the depth, so the axial resolution is half of it, divided by the group index (Fercher et al. 2003). The SNR is the shot-noise limit of a spectrometer or swept-source system (Choma et al. 2003, eq. 10) as a power ratio; excess intensity noise, read noise, the sensitivity roll-off with depth and losses in the spectrometer are not included.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Gaussian source spectrum with Δλ ≪ λ₀ and matched dispersion; n is the group index. The SNR is the shot-noise limit of a spectrometer or swept-source system with a strong reference arm and the A-scan time as exposure; read noise, excess noise and the depth roll-off are not included.

Fercher A. F., Drexler W., Hitzenberger C. K., Lasser T. (2003). Optical coherence tomography — principles and applications. Rep. Prog. Phys. 66, 239–303. Axial resolution (2 ln 2/π) λ₀²/Δλ, the round-trip coherence length. doi:10.1088/0034-4885/66/2/204
Choma M. A., Sarunic M. V., Yang C., Izatt J. A. (2003). Sensitivity advantage of swept source and Fourier domain optical coherence tomography. Opt. Express 11, 2183–2189. Fourier-domain SNR ρP_sΔt/(2e), eq. 10. doi:10.1364/OE.11.002183

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