Optics 051 · Interference, coherence, cavities, and metrology

Low-Coherence Interferometry & OCT

A 3D low-coherence fiber Michelson links a moving opaque reference mirror and weak layered phantom to four simulated phase-step detector exposures, quadrature A-scans and a fixed-scale B-scan.

Amplitude depth width / optical-path envelope width

\delta z_{\mathrm{amp}},W_{\Delta L}

$0.50$

Mirror displacement / physical gate depth

d,z_g

$50\%$

Recovered field / raw phase-I power

|\widehat C|,I_0/P_{\mathrm{in}}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Low-Coherence Interferometry & OCT

Vacuum centre wavelength

\lambda_0

850\,\mathrm{nm}

Equivalent wavelength bandwidth (power-spectrum FWHM)

\Delta\lambda_{\mathrm{eq}}

60\,\mathrm{nm}

Known nondispersive sample index

n

1.38

Reference mirror displacement (surface match is zero)

d

82.80\,\mu\mathrm m

Physical depth separation of first two layers

s

20\,\mu\mathrm m

Second effective field return / first return

\eta_2

0.80

Lateral scan coordinate (sample translation)

x_s

0\,\mu\mathrm m

Start with a single weak layer: move the gold reference mirror away from its match and the raw fringes fade. Widen the source bandwidth: the depth gate narrows. In the pair, shrink the physical layer separation and watch coherent peaks merge or cancel. In the B-scan, translate the sample to follow the curved layers; the selected A-scan and gold delay cursor use the same exposures. The full scan is pre-acquired; play only animates route markers. Drag to orbit, or use the four apparatus views.

Time-domain teaching model, not spectral-domain OCT or a clinical scanner. Exact Gaussian power spectrum in vacuum wavenumber; equivalent wavelength bandwidth is narrow-band shorthand (above 10% is flagged). Known constant phase/group index, prescribed weak effective returns and ideal achromatic phase stepping. Depth sampling 0.25 micrometres is not resolution. 61 columns span a 200 micrometre lateral window; no lateral PSF. No tissue calibration, speckle, noise, dispersion, absorption or multiple scattering. Diagrammatic arm lengths; sample depth and lateral shift use 1.1 world units per micrometre, reference motion 0.7. Display is amplitude dB, fixed to return 0.12 over 60 dB; raw power and linear amplitude remain separate.

Physics tutorial

Audit the physics of Low-Coherence Interferometry & OCT

BackgroundA broadband source still interferes with a delayed copy of itself, but only when the round-trip optical paths nearly match: $\Delta L_j=2(nz_j-d)$ . The reference mirror displacement is measured from the position matching the sample surface. The sample has a known constant phase and group index. Delayed weak returns from deeper layers are revealed by changing that displacement, not by painting a transparent specimen.

Why it mattersHow does a short coherence gate turn reflected light into a depth-resolved image?

Start with the essentials

Focus question: How does a short coherence gate turn reflected light into a depth-resolved image?
One-sentence intuition: The reconstruction uses four detector power measurements at each delay: $\widehat C=(I_0-I_\pi)+i(I_{\pi/2}-I_{3\pi/2})$ . Subtraction cancels the reference background and sample self-interference. Taking the complex magnitude then forms the A-scan. Repeating at lateral positions forms the B-scan. The known phantom is never used to insert reconstruction peaks.

Core mathematical model

One-way depth is not round-trip optical path

\begin{aligned}\Delta L_j&=2(nz_j-d)\\z_g&=d/n,\quad \Delta d=\lambda_0/2\end{aligned}

Depth increases along the sample beam; positive reference displacement lengthens the air reference arm. The two cancels when both arms make a round trip, so mirror match is index times physical depth. A mirror moves half a vacuum wavelength per fringe. Fixed background paths are assumed matched; scene arm lengths are diagrammatic, not the metric used to calculate phase.

Power spectrum becomes a coherence gate

\begin{aligned}S(k)&\propto e^{-(k-k_0)^2/(2\sigma_k^2)}\\\gamma(\Delta L)&=e^{-\sigma_k^2\Delta L^2/2}e^{ik_0\Delta L}\\\sigma_k&=\frac{2\pi\Delta\lambda_{\mathrm{eq}}}{\lambda_0^2\sqrt{8\ln2}}\end{aligned}

The mutual coherence is the normalized Fourier integral of source power, not a transform of spectral field amplitude. This Lab defines an exact Gaussian in vacuum wavenumber. The displayed equivalent wavelength FWHM is a narrow-band translation; it is not an exact Gaussian wavelength spectrum. Relative bandwidth above ten percent is flagged. The continuous SLD is not a train of ultrashort pulses.

Amplitude FWHM, intensity FWHM and sample pitch differ

\begin{aligned}\delta z_{\mathrm{amp}}&=\frac{2\ln2}{\pi}\frac{\lambda_0^2}{n\Delta\lambda_{\mathrm{eq}}}\\W_{\Delta L}&=2n\delta z_{\mathrm{amp}}\\\delta z_{\mathrm{power}}&=\delta z_{\mathrm{amp}}/\sqrt2\end{aligned}

The reported axial width is the field-amplitude envelope FWHM of one isolated reflector. A squared-amplitude peak has a narrower FWHM. Optical-path envelope width includes the round trip and index; neither is a new independent resolution. The 0.25 micrometre depth pitch samples this model but does not improve its resolution. Two coherent layers can merge or cancel; one width is not a universal two-layer resolving criterion.

Raw photodiode power includes background

\begin{aligned}C(d)&=\sum_j r_j\gamma\!\left(2(nz_j-d)\right)\\Q&=\sum_{j,l}r_jr_l\operatorname{Re}\gamma\!\left(2n(z_j-z_l)\right)\\I_\theta/P_{\mathrm{in}}&=\tfrac14\left[1+Q+2\operatorname{Re}(Ce^{-i\theta})\right]\end{aligned}

A balanced ideal coupler is passed twice. The prescribed effective real field returns include all previous transmission factors; they are not Fresnel coefficients inferred from bulk index. Reference power and sample self-interference remain in raw measurements. The complementary return port changes the interference sign; this Lab displays one photodiode, not a falsely labeled balanced pair.

Four exact phases recover complex coherence

\begin{aligned}\theta&\in\{0,\pi/2,\pi,3\pi/2\}\\\widehat C&=(I_0-I_\pi)\\&\quad+i(I_{\pi/2}-I_{3\pi/2})\\A(z_g)&=|\widehat C|\end{aligned}

Here the measurements are already normalized to incident source power. Ideal achromatic reference phases are a prescribed apparatus operation, not approximated by moving the broadband mirror a quarter wavelength. The 2021 four-phase OCM paper demonstrates why achromatic phase control matters, but this Lab does not reproduce its polarization optics or single-shot camera. Every B-scan pixel is reconstructed from its four stored simulated exposures.

A depth image is a processed measurement, not ground truth

D=\operatorname{clip}_{[-60,0]}\!\left(20\log_{10}\frac{A}{0.12}\right)

The B-scan uses a fixed amplitude-dB scale, not automatic contrast. It has 61 lateral columns over 200 micrometres and 801 depth samples from zero to 200 micrometres. The phantom surfaces in the instrument are a known reference; only the separate data row is the reconstruction. A gold outline is a virtual delay cursor, not glowing matter. No speckle, lateral PSF, noise, multiple scattering, material dispersion or clinical accuracy is solved.

Common difficulties

A sharp drawn layer is not a resolved OCT layer

Typical misconceptionAll visible 3D layers must appear as separated peaks, and a smaller sampling pitch guarantees sharper resolution.

Better mental modelThe apparatus is a known phantom reference. Only its measured interference supports reconstruction; overlapping coherent returns can cancel, and bandwidth fixes the single-reflector axial envelope width. Dense sampling cannot undo lost coherence information.

Run the experiment

01
Scene 1: Coherence envelope
Single layer: move the reference mirror around its match, then vary bandwidth and compare the raw zoom with the recovered envelope.
What to observe: Source and sample guides are false colour: infrared emission is not visible cyan. Moving markers illustrate route direction only, not photons, pulse propagation or acquisition speed.
02
Scene 2: Layered A-scan
Layer pair: reduce separation and second return amplitude. Distinguish the true phantom from overlapping measured peaks.
What to observe: Sample depth and lateral motion use 1.1 world units per micrometre; reference motion uses 0.7. Background arms are schematic with a matched baseline. Bulk index is known and nondispersive, so phase and group indices coincide.
03
Scene 3: Raster B-scan
B-scan: translate the sample laterally. The selected A-scan is acquired at the exact cursor coordinate; the B-scan has 61 fixed lateral positions, so only grid-aligned cursors share a stored column. Inspect the reference-mirror front.
What to observe: All exposures are noiseless expectations from a continuous Gaussian power spectrum. The full scan is pre-acquired at current settings; pause, orbit and reduced motion never change the measurements.