Low-Coherence Interferometry (2D): OCT A-Scan From Numerical Spectral Synthesis

Source Spectrum

Center wavelength λ₀ 1050 nm

Spectral bandwidth Δλ 60 nm

Sample

Layer separation Δd 15 μm

Reference Mirror

Mirror depth z_ref 20.0 μm

Coherence Gating (measured, not formula)

Coherence length L꜀ (numeric)

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Axial resolution Δz (numeric)

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Detector signal |Γ(z_ref)|

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Spectral terms summed K

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Two layers resolved? —

How it works

This is the 2D counterpart of the OCT interferometer simulator. The 3D scene evaluates a ready-made Gaussian formula for the coherence envelope. This version never uses that formula — it builds the interferogram bottom-up by discretizing the source's power spectrum into K monochromatic components and coherently summing their individual interference terms, then extracting the envelope by quadrature (I/Q) demodulation, exactly as a real spectral-domain OCT processing pipeline does.

Source spectrum, K discrete bins over λ₀±4Δλ:
  wᵢ ∝ exp[-4ln2·(λᵢ-λ₀)²/Δλ²],  Σwᵢ = 1

Per reflector j at depth dⱼ, reflectivity Rⱼ, round-trip mismatch OPD = 2(z-dⱼ):
  I(z) = Σⱼ Rⱼ Σᵢ wᵢ·cos(2π·OPD/λᵢ)     (in-phase)
  Q(z) = Σⱼ Rⱼ Σᵢ wᵢ·sin(2π·OPD/λᵢ)     (quadrature)
  envelope(z) = √(I(z)² + Q(z)²)          — the demodulated A-scan

Axial resolution Δz is then MEASURED: the full-width-half-max
of envelope(z) around a single reflector — never assumed.

Raw fringe panel — zoomed near the current mirror depth, this plots I(z) directly: the actual oscillating optical carrier at the source's center wavelength, riding under the coherence-gated envelope. This is what a real photodetector signal looks like before demodulation.
A-scan panel — the demodulated envelope across the full 0–70 μm range, built from the same K-term sum, no shortcut formula.
λ₀, Δλ — a wider Δλ makes the summed terms de-phase faster with depth, numerically shrinking the measured coherence length — the same resolution/bandwidth trade-off, arrived at from first principles here.
Layer separation Δd — when the numerically measured Δz exceeds Δd, the two reflectors' envelopes overlap into one peak in the A-scan panel.

Cross-check: for Δλ ≪ λ₀ this bottom-up sum converges to the textbook closed-form L_c ≈ 0.44λ₀²/Δλ used by the 3D sim; at larger Δλ/λ₀ ratios (e.g. Δλ=160 nm at λ₀=800 nm) the two diverge — a real limitation of that closed-form approximation that only shows up once you actually integrate the spectrum, which is exactly what this engine does.