Optics 044 · Interference, coherence, cavities, and metrology

Interference Surface Metrology

An air-gap cutaway links exact spherical sag, a real tilted flat and a smooth test-surface pit to monochromatic fringes and a four-frame phase inverse.

Shared centre-line probe / gap / normalized intensity

x_p,t_p,I_p

$0.50$

Recovered curvature, wedge or finite-window departure

\widehat R,\widehat\alpha,\mathrm{PV}_{\mathrm{res}}

$50\%$

Sampling grid / maximum adjacent phase jump

N,\max|\Delta\phi|/\pi

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Interference Surface Metrology

Monochromatic measurement wavelength

\lambda

600\,\mathrm{nm}

Spherical radius of curvature (not specimen radius)

R

4\,\mathrm m

Reference-flat tilt (zero means parallel)

\alpha

0.25\,\mathrm{mrad}

Base gap (not an automatic phase scan)

t_0

0\,\mathrm{nm}

Test-surface pit depth

d_{\mathrm{pit}}

200\,\mathrm{nm}

Shared centre-line probe position

x_p

1.50\,\mathrm{mm}

Contact is dark. Increase the radius of curvature: the cyan reference face really flattens and the rings spread, while specimen width stays fixed. Try the wedge and then a pit: its depth moves the gold test face and bends the fringes. Four captured phase steps recover the profile using a known corner and calibrated plane. Drag to orbit; the slider moves the shared probe. Play does not fake surface motion.

Balanced coherent two-return, normal-incidence scalar approximation; fixed normalized intensity, not absolute reflectance. Independently magnified vertical scale, schematic support, ideal phase steps and calibrated anchor. No noise, arbitrary defect recovery or instrument-accuracy claim.

Physics tutorial

Audit the physics of Interference Surface Metrology

BackgroundTwo coherent returns sample the opposite faces of a thin air gap. Its round-trip phase is $\phi=4\pi t/\lambda$ . One glass–air return and one air–glass return have opposite reflection signs. Contact is therefore dark, not bright. The 3D faces, camera patterns and inverse profile all share the same gap function.

Why it mattersHow can fringe geometry reveal thickness, curvature, and tiny surface defects?

Start with the essentials

Focus question: How can fringe geometry reveal thickness, curvature, and tiny surface defects?
One-sentence intuition: Adjacent equal-brightness orders differ in height by $\Delta t_{\mathrm{fringe}}=\lambda/2$ . A single image cannot distinguish an integer number of those steps, or a bump from a pit. Four ideal phase steps recover the wrapped phase; smooth sampling and one independently calibrated corner fix a continuous gap. They do not create an absolute-height measurement from nothing.

Core mathematical model

Why optical contact is dark

\phi=\frac{4\pi t}{\lambda},\qquad I_k=\frac{1-\cos(\phi+k\pi/2)}{2}

Intensity is normalized to the fixed constructive maximum of two balanced coherent returns. The minus sign accounts for the reflection reversal. The four frames represent independently imposed reference phases, not a fictitious time-dependent surface. Multiple reflections and the exact Fresnel imbalance are neglected; this is not an absolute reflectance calculation.

Curvature radius is not specimen radius

\begin{aligned}t(r)&=t_0+\frac{r^2}{R+\sqrt{R^2-r^2}}\\r_m^2&=2Rs_m-s_m^2,\quad s_m=m\lambda/2-t_0\end{aligned}

The specimen stays 8 mm square while changing the metre-scale radius of curvature changes its actual spherical face. Larger curvature radius means flatter glass and more widely spaced rings. Dark-ring radii use exact sag, not just the small-sag approximation. The listed first noncentral dark circle can be clipped by the square window.

A tilted flat becomes a thickness ruler

t(x)=t_0+(x+a)\tan\alpha,\qquad\Delta x=\frac{\lambda}{2\tan\alpha}

The half-window is 4 mm. Tilt really moves the reference surface; wavelength changes the pattern without moving either face. At zero tilt spacing is infinite and the image becomes uniform. Wedge angle is recovered from the two measured edge gaps, not copied from the slider.

Four images, one wrapped phase

\phi_w=\operatorname{atan2}(I_1-I_3,I_2-I_0),\qquad\widehat t=\frac{\lambda}{4\pi}\phi_u

Spatial nearest-neighbour unwrapping receives only the four intensity arrays, wavelength and calibrated corner height. The UI reports the maximum adjacent phase jump; it must remain below half a cycle. All allowed control extremes obey this condition on the 257-square grid. Without the anchor, a global integer order remains unknown. Noise, discontinuities and phase-step errors are not simulated.

A pit bends fringes; a calibrated plane separates it

\begin{aligned}t(x,z)&=t_0+(x+a)\tan\alpha+h(x,z)\\h(x,z)&=d\exp\!\left[-\frac{(x-x_c)^2+z^2}{2\sigma^2}\right]\end{aligned}

The test face has a smooth pit centred 0.8 mm right of the centre with width 0.85 mm. Positive pit depth increases the air gap and lowers the actual test face. Subtracting the independently calibrated piston and tilt reveals the recovered departure; the colour scale is fixed, not auto-normalized. Peak-to-valley is evaluated over the finite sampled window, not equated to the input depth.

Common difficulties

More rings do not mean more height information

Typical misconceptionA single fringe photograph uniquely reveals absolute height and nanometre accuracy.

Better mental modelIntensity is periodic and even in phase. Known phase steps remove the sign ambiguity but not the integer order without calibration. This noise-free forward/inverse agreement checks the implementation, not instrument accuracy. Reference-face errors, vibration, finite coherence, Fresnel imbalance and imaging resolution need separate models.

Run the experiment

01
Scene 1: Newton rings
Start at contact. Increase curvature radius: the cyan face flattens, the square stays the same size, and dark rings move outward.
What to observe: Changing wavelength never changes measured-face geometry.
02
Scene 2: Air wedge
Choose the air wedge, then lower tilt to zero. Change the base gap by half a wavelength: geometry moves but the intensity pattern repeats.
What to observe: Identical images can correspond to gaps differing by half a wavelength.
03
Scene 3: Fizeau optical-flat test
Choose the optical-flat pit. Raise its depth and watch fringes bend; set tilt to zero to expose why one image loses the sign. The four-step inverse still recovers the calibrated profile.
What to observe: Curvature survives piston removal; a flat wedge disappears after calibrated-plane subtraction.