Optics 053 · Interference, coherence, cavities, and metrology

Speckle Lab

Translate a deterministic phase diffuser on a 3D Fourier bench, acquire real simulated intensity maps, average the recorded frames and compare finite-sample contrast and displacement correlation with independent ensemble laws.

Recorded / ensemble contrast

K_{\mathrm{record}},K_{\mathrm{ensemble}}

$0.50$

Field-coherence grain scale / start displacement

s_0,d

$50\%$

Recorded / ensemble translation correlation and DFT closure

\rho,\epsilon_{\mathrm{DFT}}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Speckle Lab

Illuminated sites per side (50 micrometre pitch)

n

24

Monochromatic wavelength (programmed phase held)

\lambda

532\,\mathrm{nm}

Fourier lens focal length

f

150\,\mathrm{mm}

Starting screen translation in 50 micrometre steps

d/a

0

Frames actually recorded and averaged

M

8

Total travel of the exposure sequence, in sites

L/a

24

Fixed phase-screen sample

j_s

1

Start with one fixed phase screen. Enlarge the illuminated square to make smaller speckle grains; change focal length to physically move the screen and sensor. In Moving-diffuser averaging, all displayed frames are actually recorded and averaged as intensities. Set total travel to zero: more frames cannot smooth a stationary screen. In Displacement correlation, drag the illuminated screen centre in the diffuser view and compare the original and shifted records. Mint statistics come from the records; gold is a separate ensemble prediction. Play animates route markers only; pause never changes acquisition. The slider is also keyboard accessible.

Ideal finite point-site scalar model, not ground-glass Maxwell scattering or a calibrated flow camera. Fixed grayscale clips only the display; statistics retain all intensities. Discrete stopped-position frames, not continuous exposure.

Physics tutorial

Audit the physics of Speckle Lab

BackgroundA bright grain is not one scattering particle. It is constructive interference between many coherent contributions at one detector coordinate: $I(\boldsymbol q)=\left|\frac1{\sqrt N}\sum_j e^{i\phi_j}e^{-2\pi i\boldsymbol r_j\cdot\boldsymbol q/(\lambda f)}\right|^2$ . The coloured transmissive screen encodes programmed phase; the separate grayscale sensor records squared field amplitude. Their pictures need not resemble one another.

Why it mattersHow does random coherent phase create structured grains with predictable statistics?

Start with the essentials

Focus question: How does random coherent phase create structured grains with predictable statistics?
One-sentence intuition: A detector adds the intensities of sequential records, not their complex amplitudes. Correlation determines how much averaging helps: $K^2=\frac1{M^2}\sum_{j,k}\operatorname{Cov}(I_j,I_k)$ . Repeating a stationary frame leaves contrast unchanged. More closely spaced positions are not more independent observations.

Core mathematical model

Fourier geometry connects phase sites to the sensor

\begin{aligned}\boldsymbol q&=f\boldsymbol\alpha\\\boldsymbol h_\ell&=\boldsymbol r+\boldsymbol q\\\boldsymbol\alpha_{\mathrm{out}}&=\boldsymbol\alpha-\boldsymbol h_\ell/f=-\boldsymbol r/f\end{aligned}

The screen and sensor are in the front and back focal planes. A contribution starting at a phase site crosses the lens at its site coordinate plus the selected detector coordinate, then reaches that detector coordinate. Green and gold lines illustrate a small set of these paraxial contributions, not the full wave field. The detector face points toward the lens; its opaque housing is behind the absorbing plane.

The grain scale comes from illuminated width

\begin{aligned}D&=na,\quad a=50\,\mu\mathrm m\\s_0&=\lambda f/D\\\Delta q&=\lambda f/(128a)\end{aligned}

The square contains equal-amplitude phase sites on a regular lattice. The field coherence is a finite geometric sum, whose first zero sets the quoted characteristic grain scale. It is not the diameter of every visible grain or an imaging resolution. The full detector window rescales with wavelength and focal length; this keeps the same discrete angular samples. Phase values are held programmed when wavelength changes, not inferred from a fixed glass relief.

Measure contrast from unclipped records

\begin{aligned}\overline I_p&=\frac1M\sum_m I_{m,p}\\K_{\mathrm{record}}&=\frac{\sqrt{\langle(\overline I_p-\langle\overline I\rangle)^2\rangle_p}}{\langle\overline I\rangle_p}\\\langle I\rangle_{\mathrm{DFT}}&=1\end{aligned}

Each frame is calculated by Fourier propagation, then squared, stored and averaged with equal weights. Parseval fixes the mean over the full discrete angular tile without per-frame auto exposure. The grayscale code reaches white at normalized intensity four; brighter values remain in the statistics. A finite realization can have contrast above one. The closure check is discrete normalization, not calibrated optical power.

Translation preserves the overlapping phase sites

\begin{aligned}N&=n^2\\O(\delta)&=n\max(0,n-|\delta|/a)\\\Gamma_I(\delta)&=\frac{O(\delta)[O(\delta)-1]}{N^2}\\\rho_{\mathrm{ensemble}}(\delta)&=\frac{\Gamma_I(\delta)}{1-1/N}\end{aligned}

This finite equal-phasor covariance is an analytic limit derived for integer lattice translations. Overlapping sites keep exactly the same seeded phase. Disjoint sets have zero ensemble covariance, but a pair of finite recorded maps need not have zero Pearson correlation. The mint curve is computed from recorded maps; the gold dashed curve is the ensemble reference, not a fit and not a manufactured change to the detector image.

Correlated averages do not obey the independent-frame shortcut

\begin{aligned}K_{\mathrm{ensemble}}^2&=\frac1{M^2}\sum_{j,k}\Gamma_I(d_j-d_k)\\K_{\mathrm{independent}}&=\sqrt{\frac{1-1/N}{M}}\\K_{\mathrm{stationary}}&=\sqrt{1-1/N}\end{aligned}

For four frames separated by a full illuminated width, their phase-site sets are disjoint and the ensemble contrast halves. Packing sixteen frames into the same travel range does not generally quarter contrast. With zero travel, all frames are identical and no amount of repetition helps. These are equal-duration stopped positions, not a quadrature of a continuous-motion exposure. The familiar time-exposure covariance integral explains the same weighting principle but is not claimed as this camera simulation.

The exponential curve is a static large-site limit

p(I)=\frac1{\langle I\rangle}e^{-I/\langle I\rangle},\quad N\to\infty

Many independent uniform phase contributions approach a circular complex Gaussian field, giving an exponential single-record intensity distribution. The histogram uses all unclipped averaged records and separately reports its tail. The dashed exponential is a static reference; it is not the distribution of a correlated average. This model omits vector scattering, finite-pixel integration, a site form factor, finite laser coherence, polarization diversity, photon noise and material calibration. It does not measure blood flow.

Common difficulties

More frames do not automatically mean independent speckle

Typical misconceptionSixteen frames always divide speckle contrast by four.

Better mental modelThat limit requires uncorrelated intensity records. Repeated or overlapping phase-site sets retain covariance, and coherent amplitude addition is a different operation altogether. Compare the actual frame sequence before applying a square-root rule.

Run the experiment

01
Scene 1: Fully developed speckle
In the static preset, enlarge illumination and change focal length. Check the physical apparatus moves while the field-coherence grain scale changes.
What to observe: Grain size changes with optical scale, while the detector is not a direct image of the phase screen.
02
Scene 2: Moving-diffuser averaging
In averaging, choose eight sites per side, four frames and total travel 24 sites. Then set travel to zero, keeping four frames.
What to observe: Mean discrete intensity remains fixed; correlated averages suppress fluctuations less than independent ones.
03
Scene 3: Displacement correlation
In correlation, use the diffuser detail view and drag the illuminated centre. Compare recorded and ensemble correlation across screen samples.
What to observe: Finite-screen deviations are visible rather than forced onto an ideal reference curve.