Optics 055 · Interference, coherence, cavities, and metrology

Moiré & Fringe Metrology

How can broad fringe patterns reveal a tiny shift or the shape of a surface? Overlay two nearly identical gratings and broad moiré bands appear; they move much farther than the tiny shift that caused them. For surface shape, project fringes onto the part, step their phase and read height from the phase. A mirror needs another route: view a patterned screen in it, work out each point's slope from where on the screen its ray started, then integrate slope into height.

Use Moiré bands to magnify tiny grating shifts. Recover a diffuse surface from recorded phase-stepped fringes, see where unwrapping fails at a step, and integrate the profile of a mirror surface 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 check

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

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

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 the specimen shape)

h_{\max}

5\,\mathrm{mm}

Projection angle from vertical camera axis

\theta

30^{\circ}

Distance from the emitting screen to the flat reference

D

160\,\mathrm{mm}

Move the upper grating by a fraction of a millimetre and the broad Moiré bands travel much farther. Reverse the pitch mismatch or rotate the grating to change their direction. In Projection, light stripes bend over a diffuse cap; the camera records four fine and four coarse exposures, plus a flat reference for calibration. The data row compares the wrapped height, the spatially unwrapped height and the coarse–fine recovered height. Switch to Opaque step: spatial unwrapping picks the wrong fringe order, and the shadowed region stays missing. In Deflectometry, a screen is reflected in an opaque mirror; the measured phase gives the screen displacement, then the slope, which integrates to height. Use the specimen view to see the true shape, the camera view to inspect recorded intensity and the frame buttons to step through the phase shifts. Play only replays recorded frames and path markers.

The geometric model is ideal and calibrated, with point sampling and linear intensity records of a stationary object. The gratings are sinusoidal and in contact, projection is orthographic, shadows are masked, and the mirror cap is one-dimensional. Diffraction, noise, the optical PSF, gamma and phase errors from a moving object are not included. Coarse–fine reconstruction works from the recorded intensities alone; the known height is used only in the labeled accuracy check. Refractive-index measurement and general 2D deflectometry are not covered.

Start with the essentials

Focus question: How can broad fringe patterns reveal a tiny shift or the shape of a surface?
One-sentence intuition: With a calibrated parallel projector and light collected along the vertical, height gives $\Delta\phi=\frac{2\pi h\tan\theta}{p_f}$ . A specular mirror instead changes which point on the screen the camera sees. Recover that screen coordinate first, then infer the slope and integrate it to get 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}

In the contact limit, the two intensity transmissions multiply. The raw product contains the two carriers, their sum and their difference. A separate analytic beat panel shows the constant plus the difference term, and 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. The gap in the apparatus is exaggerated, and diffraction and parallax are left out.

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}

The upper grating moves along its own normal. The beat travel is reported along the normal to the difference vector, so the gain has a sign, and that direction itself flips when the pitch mismatch reverses. For parallel gratings, the beat period equals the product of the two pitches divided by the absolute value of their difference. Equal pitches with no rotation have no finite beat period and no defined normal; the model does not clamp this to an arbitrary giant number. The pattern repeats each time the upper grating moves one period. The displayed travel follows the continuous mathematical branch, so it is not an absolute displacement measured from a single image.

Four separate exposures recover the 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 keeps track of the quadrant. Subtracting opposite exposures removes the background. The object and a flat reference are each acquired at a fine and a coarse period, for sixteen images in total. Subtracting the reference removes the carrier. Pixels with zero modulation are marked as missing and get no phase value. The method assumes a 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, and height is positive upward. The incoming pattern travels to the right and downward, and the collected light goes straight up before a partial reflector folds it. The projector angle is measured from the vertical. Trace a ray backward to the pattern plane to get the shifted coordinate. Within the allowed control range, the smooth cap casts no shadow on itself. An opaque plateau casts a geometric shadow beyond its right edge, and that region stays missing in the reconstructed maps. The mesh, the rays and the recorded fringes all use the same height.

Unwrap from neighbouring pixels, or record a second 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 unwrapping starts from a calibrated flat left edge and adds up the principal-value differences between neighbours. A large phase jump hidden between samples breaks this assumption, and the result can look smooth and still be wrong. The coarse–fine method uses the acquired coarse phase to choose each fine fringe order on its own, even across a step. The coarse period is 512 mm, and the signed coordinate shift must stay within half of that period. The method cannot fill shadow gaps. A narrower fine pitch raises sensitivity but leaves less margin for spatial unwrapping. Noise and frequency-registration errors are not simulated.

Reflection measures slope; height also needs a boundary value

\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, extended unchanged across the transverse direction. A vertical camera ray, traced backward, reflects about the actual surface normal and lands on the emitting screen. A partial reflector sends the upward ray sideways to the camera; the opaque mirror never transmits it. The recovered screen shifts feed a Runge–Kutta integration that uses only the known screen distance and the left-edge height. The prescribed specimen height appears only in the separate gold comparison, never in the inversion. The small remaining error comes from integrating finite samples; it does not represent laboratory precision. General two-dimensional slopes, unknown absolute height and refractive-index fields are outside the model.

Common difficulties

A plausible surface is not proof of correct measurement

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

Better mental modelPhase repeats every cycle. Spatial continuity, extra acquired frequencies and calibrated geometry each supply a different piece of the missing information. Where no light arrived, the data stay missing. Comparing with a known shape is useful in a simulation, but it does not certify a real experiment.

Run the experiment

01
Scene 1: Moiré displacement magnifier
Move and rotate the upper grating. Compare the raw product with the analytic beat, which is labelled separately.
What to observe: Moiré magnifies motion, but the periodic ambiguity remains: shifts that differ by a whole period look the same.
02
Scene 2: Diffuse cap / fringe projection
Raise the diffuse cap and follow the four phase-stepped exposures through to the recovered height.
What to observe: Camera intensity is not a height map by itself; it takes calibrated phase and geometry.
03
Scene 3: Opaque step / unwrap failure
Switch to the opaque step. Compare the purple spatial recovery with the mint coarse–fine recovery, and look at the missing shadow region.
What to observe: The coarse acquisition restores the fringe order across the step, but it cannot restore data where no light fell.
04
Scene 4: Specular cap / deflectometry
In deflectometry, change the screen distance and the specimen’s peak height. Check the source, the specimen and the receiving face, then compare the recovered center profile with its separate comparison curve.
What to observe: Reflection measures direction. Integrating to height needs calibrated geometry and a boundary value.