Optics 055 · Interference, coherence, cavities, and metrology

Moiré & Fringe Metrology

Amplify tiny grating shifts with Moiré bands, recover a diffuse surface from acquired phase-stepped fringes, expose step-unwrapping failures and integrate a specular profile from reflected screen coordinates.

Beat scale / signed gain, or acquired fringe periods

P_M,M_n;p_f,p_c

$0.50$

Grating travel / beat travel, or records / illuminated fraction

d,\Delta s_n;N,\eta

$50\%$

Intensity product, or known-specimen height audit

T_{12};\epsilon_{\mathrm{sp}},\epsilon_{\mathrm{cf}}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Moiré & Fringe Metrology

Physical period of the first grating

p_1

1.50\,\mathrm{mm}

Second grating relative period change (signed)

\epsilon_p

4\%

In-plane rotation of the second grating

\alpha

1.50^{\circ}

Translation along second grating normal

d

0.25\,\mathrm{mm}

Fine fringe period (reacquires)

p_f

6\,\mathrm{mm}

Specimen peak height (changes actual shape)

h_{\max}

5\,\mathrm{mm}

Projection angle from vertical camera axis

\theta

30^{\circ}

Emitting screen to flat reference distance

D

160\,\mathrm{mm}

Move the upper grating a fraction of a millimetre: the broad Moiré bands travel much farther. Reverse pitch mismatch or rotate the grating to change their direction. In Projection, light stripes bend over an actual diffuse cap; the camera acquires four fine and four coarse exposures, plus flat-reference calibration. The data row compares wrapped, spatially unwrapped and coarse–fine recovered height. Switch to Opaque step: spatial unwrap picks the wrong fringe order and the shadow stays missing. In Deflectometry, a screen reflects in an opaque mirror; measured phase gives screen displacement, then slope and integrated height. Use the specimen view to see actual shape, camera view to inspect acquired intensity, and frame buttons to follow phase stepping. Play only replays acquired frames and route markers.

Ideal calibrated geometric model with point sampling and stationary, linear intensity records. Contact sinusoidal gratings, orthographic projection, shadow masking and a one-dimensional specular cap. No diffraction, noise, optics PSF, gamma or moving-object phase errors. Coarse–fine reconstruction uses acquired intensities, never prescribed height. Known height is used only for the explicitly labelled audit. No refractive-index or general 2D deflectometry claim.

Start with the essentials

Focus question: How can coarse fringe motion amplify and recover tiny displacement or surface shape?
One-sentence intuition: For the calibrated parallel projector and vertical collection axis, height gives $\Delta\phi=\frac{2\pi h\tan\theta}{p_f}$ . A specular mirror instead redirects which screen coordinate is seen. Recover that coordinate before inferring slope and integrating height.

Core mathematical model

Geometric Moiré, not wave interference

\begin{aligned}T_j&=\tfrac12(1+\cos\varphi_j)\\T_{12}&=T_1T_2\\\mathbf f_M&=\mathbf f_1-\mathbf f_2\\P_M&=1/|\mathbf f_M|\end{aligned}

Multiply intensity transmissions in the contact limit. The raw product contains the two carriers, their sum and their difference. A separate analytic beat panel shows the constant plus difference term; it is not labelled as filtered acquired data. Rotation changes the second spatial-frequency vector. The finite 80 by 60 mm overlap is cropped from a larger upper grating. Gap in the apparatus is exaggerated; diffraction and parallax are omitted.

A tiny translation gives signed, amplified travel

\begin{aligned}p_2&=p_1(1+\epsilon_p)\\M_n&=-\frac{1}{p_2|\mathbf f_M|}\\\Delta s_n&=M_nd\end{aligned}

Translation is along the upper grating normal. Beat travel is reported along the difference-vector normal, so gain is signed; the vector direction itself changes when mismatch reverses. For parallel gratings the beat period equals the product of pitches divided by their absolute difference. Equal unrotated pitches have no finite beat period or defined normal, not an arbitrarily clamped giant number. Motion repeats after one upper-grating period; the displayed travel is the continuous mathematical branch, not an absolute displacement measurement from one image.

Four independent records recover phase

\begin{aligned}I_\delta&=a+b\cos(\varphi+\delta)\\S&=I_{3\pi/2}-I_{\pi/2}\\C&=I_0-I_\pi\\\varphi&=\operatorname{atan2}(S,C)\\\Delta\varphi_w&=\mathrm{wrap}(\varphi_o-\varphi_r)\end{aligned}

The two-argument arctangent retains the quadrant. Opposite exposure differences remove background. Object and flat reference are independently acquired at fine and coarse periods: sixteen records in total. Reference subtraction removes the carrier. Zero modulation is masked as missing, never converted into a made-up phase. This assumes stationary geometry, exact quarter-turn steps and linear intensities.

Projection geometry measures height

\begin{aligned}q&=x+h\tan\theta\\\Delta\varphi&=\frac{2\pi h\tan\theta}{p_f}\\\widehat h&=\frac{p_f\Delta\varphi}{2\pi\tan\theta}\end{aligned}

Coordinates and lengths are in millimetres. Positive height points upward; the incoming pattern travels rightward and downward, while collection is vertical before a partial fold. The projector angle is measured from vertical. Trace the ray backward to its pattern plane to derive the shifted coordinate. The smooth cap has no self-shadow in the allowed controls. An opaque plateau casts a geometric shadow after its right edge; that region remains absent from reconstructed maps. Mesh shape, rays and recorded fringes use the same height.

Unwrap by neighbours, or acquire another scale

\begin{aligned}|\Delta\varphi_{\mathrm{adj}}|&<\pi\\k&=\mathrm{round}\!\left(\frac{(p_c/p_f)\varphi_c-\varphi_f}{2\pi}\right)\\\varphi_{\mathrm{cf}}&=\varphi_f+2\pi k\end{aligned}

Spatial unwrap starts at a calibrated flat left edge, adding principal neighbour differences. A large unseen phase jump violates that assumption; a visually smooth wrong answer is possible. Coarse–fine uses acquired coarse phase to choose each fine fringe order independently, including across a step. The coarse period is 512 mm and its signed coordinate shift must stay inside half that period. It cannot fix shadow gaps. Narrow fine pitch improves sensitivity but reduces the spatial-unwrapping margin. Noise and frequency-registration errors are not simulated.

Reflection encodes slope, then height needs a boundary

\begin{aligned}\Delta x_s&=-\frac{2s(D-h)}{1-s^2}\\s&=-\tan\!\left[\tfrac12\arctan\!\frac{\Delta x_s}{D-h}\right]\\h'&=s,\quad h(-40)=0\end{aligned}

The mirror is a smooth one-dimensional cap extruded across the transverse direction. Reflect a vertical backward camera ray about the actual surface normal and intersect the emitting screen. A partial reflector relays the upward received ray sideways; the opaque mirror never transmits it. Recovered screen shifts enter a Runge–Kutta integration using only the known screen distance and left-edge height. Prescribed specimen height enters the separate gold audit, not inversion. The remaining small error is finite-sampling integration error, not laboratory precision. General two-dimensional slopes, unknown absolute height and refractive-index fields are outside scope.

Common difficulties

A plausible surface is not proof of correct measurement

Typical misconceptionAny wrapped fringe phase can be turned into the original surface.

Better mental modelPhase is periodic. Spatial continuity, additional acquired frequencies and calibrated geometry supply different missing information. Missing light remains missing. A known-shape audit is useful in simulation but is not an external experimental certificate.

Run the experiment

01
Scene 1: Moiré displacement magnifier
Translate and rotate the upper grating; compare raw product with the separately labelled analytic beat.
What to observe: Moiré amplifies motion but retains periodic ambiguity.
02
Scene 2: Diffuse cap / fringe projection
Raise the actual diffuse cap; follow four phase-stepped exposures into the recovered height.
What to observe: Camera intensity is not itself a height map; calibrated phase and geometry are required.
03
Scene 3: Opaque step / unwrap failure
Switch to the opaque step. Compare purple spatial recovery and mint coarse–fine recovery; inspect the missing shadow.
What to observe: Coarse acquisition restores step fringe order, not unilluminated data.
04
Scene 4: Specular cap / deflectometry
In deflectometry, change screen distance and actual peak height. Inspect source, specimen and receiving face, then compare the recovered centre profile with its separate audit.
What to observe: Reflection measures direction; integrated height needs calibrated geometry and a boundary.