Optics 051 · Interference, coherence, cavities, and metrology

Low-Coherence Interferometry & OCT

How can broadband light reveal the depth a reflection came from inside a sample? Broadband light interferes only when the sample and reference paths match within its short coherence length. That acts as a gate: set the reference delay, and only reflections from the matching depth make fringes. At each delay, four exposures with stepped reference phase give the fringe strength; stepping the delay through the sample builds the depth profile, the A-scan.

In a 3D low-coherence fiber Michelson interferometer, move the opaque reference mirror against a weakly reflecting layered phantom. Each position gives four simulated phase-stepped exposures, which build 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 $\text{valid model regime}$

Model checked against analytic results

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 (zero when matched to the surface)

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 the matched path length and the raw fringes fade. Widen the source bandwidth and the depth gate narrows. In the layer pair, shrink the separation between the layers and watch the coherent peaks merge or cancel. In the B-scan, translate the sample to follow the curved layers; the selected A-scan and the gold delay cursor use the same exposures. The full scan is recorded in advance, and Play only animates path markers. Drag to orbit, or use the four apparatus views.

This is a time-domain teaching model, not spectral-domain OCT or a clinical scanner. The power spectrum is exactly Gaussian in vacuum wavenumber; the equivalent wavelength bandwidth is a narrow-band shorthand, and values above 10% are flagged. The phase and group indices are known constants, the weak effective returns are prescribed, and phase stepping is ideal and achromatic. The 0.25 micrometre depth sampling is not the resolution. The 61 columns span a 200 micrometre lateral window, and no lateral PSF is computed. Tissue calibration, speckle, noise, dispersion, absorption and multiple scattering are not included. Arm lengths are diagrammatic; sample depth and lateral shift are drawn at 1.1 world units per micrometre, and reference motion at 0.7. The display shows amplitude in dB, referenced to a fixed return of 0.12 and spanning 60 dB; raw power and linear amplitude are kept separately.

Physics tutorial

Audit the physics of Low-Coherence Interferometry & OCT

BackgroundBroadband light still interferes with a delayed copy of itself, but only when the two round-trip optical paths nearly match: $\Delta L_j=2(nz_j-d)$ . The reference mirror displacement is measured from the position that matches the sample surface. The sample has a known, constant phase and group index. Weak, delayed echoes from deeper layers show up as you change that displacement.

Why it mattersHow can broadband light reveal the depth a reflection came from inside a sample?

Start with the essentials

Focus question: How can broadband light reveal the depth a reflection came from inside a sample?
One-sentence intuition: At each delay, the reconstruction uses four detector power measurements: $\widehat C=(I_0-I_\pi)+i(I_{\pi/2}-I_{3\pi/2})$ . The subtractions cancel the reference background and the sample self-interference. The magnitude of the complex result gives the A-scan, and repeating it at a row of lateral positions gives the B-scan. The reconstruction peaks come from these measurements alone; the known phantom is never used to place them.

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, and a positive reference displacement lengthens the reference arm in air. Both arms are round trips, so the factor of two cancels: the mirror position that matches a layer equals the index times its physical depth. Each fringe corresponds to a mirror move of half a vacuum wavelength. The fixed background paths are assumed to match; the arm lengths in the scene are only a diagram and are not used to compute phase.

The power spectrum sets the 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 the source power spectrum, not a transform of the spectral field amplitude. This Lab uses an exact Gaussian in vacuum wavenumber. The equivalent wavelength FWHM on display is a narrow-band conversion, so the spectrum is not exactly Gaussian in wavelength. Relative bandwidths above ten percent are flagged. The SLD emits continuously; it 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 FWHM of the field-amplitude envelope for one isolated reflector. The squared amplitude has a narrower FWHM. The envelope width in optical path includes the round trip and the index. None of these is a separate, new resolution. The depth samples are 0.25 micrometre apart; this spacing samples the model but does not improve its resolution. Two coherent layers can merge into one peak or cancel each other, so no single width works as a general rule for resolving two layers.

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}

Light passes an ideal balanced coupler twice. Each return is a prescribed, real effective field that already includes every transmission factor before it; it is not a Fresnel coefficient worked out from the bulk index. The raw measurements still contain the reference power and the sample self-interference. At the other return port the interference term has the opposite sign. This Lab shows a single photodiode, not a balanced detector pair.

Four exact phase steps recover the 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}

The measurements here are already divided by the incident source power. The reference phases are ideal and achromatic, applied directly by the apparatus; they are not approximated by moving the broadband mirror a quarter wavelength. The 2021 four-phase OCM paper shows why achromatic phase control matters, but this Lab does not reproduce its polarization optics or single-shot camera. Each B-scan pixel is rebuilt from its own 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 scale in dB, with no automatic contrast. It has 61 lateral columns across 200 micrometres and 801 depth samples from zero to 200 micrometres. The phantom surfaces drawn in the instrument are the known reference; only the separate data row shows the reconstruction. The gold outline is a virtual delay cursor, not glowing matter. The model does not solve speckle, a lateral PSF, noise, multiple scattering or material dispersion, and it makes no claim about clinical accuracy.

Common difficulties

A sharp drawn layer is not a resolved OCT layer

Typical misconceptionIf I can see the layers clearly in 3D, they must show up as separate peaks, and finer sampling must give sharper resolution.

Better mental modelThe apparatus shows the known phantom for reference. The reconstruction rests only on the measured interference. Overlapping coherent echoes can cancel, and the bandwidth fixes the axial envelope width for a single reflector. Denser sampling cannot bring back coherence information that was lost.

Run the experiment

01
Scene 1: Coherence envelope
With a single layer, move the reference mirror back and forth around the matched position. Then change the bandwidth and compare the raw zoom with the recovered envelope.
What to observe: The source and sample beams are drawn in false colour; the infrared light is not visible cyan. The moving markers only show the direction of the route, not photons, pulse travel or acquisition speed.
02
Scene 2: Layered A-scan
With two layers, reduce the separation and the amplitude of the second return. Tell the true phantom apart from the overlapping measured peaks.
What to observe: Sample depth and sideways motion are drawn at 1.1 world units per micrometre, and reference mirror motion at 0.7. The background arms are schematic, with matched baseline paths. The bulk index is known and has no dispersion, so the phase and group indices are equal.
03
Scene 3: Raster B-scan
In the B-scan, move the sample sideways. The selected A-scan is taken at the exact cursor position. The B-scan has 61 fixed lateral positions, so the cursor shares a stored column only when it sits on the grid. Look at the illuminated front of the reference mirror.
What to observe: Every exposure is the noiseless expected value for a continuous Gaussian power spectrum. The full scan is acquired in advance at the current settings. Pausing, orbiting the view and reduced motion never change the measurements.