Optics 042 · Interference, coherence, cavities, and metrology

Young Double-Slit Bench

Separate diffraction from interference on a real-opening 3D bench, then measure missing orders, partial coherence and unequal-amplitude contrast.

Shared screen probe

I(y_p),y_p

$0.50$

Fringe spacing / first envelope zero

\Delta y,y_1

$50\%$

Phase-scan visibility / width ratio / far-field check

V_{\mathrm{phase}},d/a,F_A

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Young Double-Slit Bench

Monochromatic wavelength

\lambda

532\,\mathrm{nm}

Actual width of each slit (envelope)

a

20\,\mu\mathrm m

Slit centre separation (fringe spacing)

d

180\,\mu\mathrm m

Mask to screen distance (screen moves)

L

1.50\,\mathrm m

Right-slit relative initial phase

\phi

0^{\circ}

Mutual coherence between slits

\mu

1

Right amplitude ratio (zero for one slit)

r

1

Shared screen probe

y_p

0\,\mathrm{mm}

Two slits, one measured pattern. Widen the openings to shrink the gold envelope; separate their centres to tighten the cyan fringes. In the missing-order preset, the third phase-alignment order lands exactly on an envelope zero. Shift-drag the white screen cursor to measure, or use its slider. Ordinary dragging orbits the apparatus; play animates illustrative path pulses only.

Scalar paraxial far field, constant slit coherence, independently magnified axes and a current-geometry intensity reference.

CONTINUE EXPLORING

Aperture, propagation, measurement

Change one physical assumption at a time: slit count, coherence, aperture shape or propagation distance.

Physics tutorial

Audit the physics of Young Double-Slit Bench

BackgroundA finite slit sends an angular spread, not one narrow ray. In the scalar paraxial far field, the aperture is Fourier transformed: $U(y)\propto\int t(s)e^{-i2\pi ys/(\lambda L)}\,ds$ . Each slit is internally coherent and vertically infinite. Their equal widths give one common sinc amplitude. The two selected 3D lines illustrate phase paths to the probe, not a solved near-field intensity or the only places light travels. The openings really change width and centre separation, and the screen really moves.

Why it mattersHow do slit width, separation, phase, and coherence jointly shape one fringe pattern?

Start with the essentials

Focus question: How do slit width, separation, phase, and coherence jointly shape one fringe pattern?
One-sentence intuition: Two independent scales control the screen: $\Delta y=\frac{\lambda L}{d},\qquad y_1=\frac{\lambda L}{a}$ . Width sets the gold diffraction envelope; separation sets the cyan interference period. Their product can lose an order even when its two waves would align in phase. Phase shifts move the fringes without shifting the envelope.

Core mathematical model

Two Fourier amplitudes

\begin{aligned}u_-&=\operatorname{sinc}(\beta)e^{i\alpha},\quad u_+=r\operatorname{sinc}(\beta)e^{i(\phi-\alpha)}\\\beta&=\frac{\pi ay}{\lambda L},\quad\alpha=\frac{\pi dy}{\lambda L},\quad\operatorname{sinc}(x)=\frac{\sin x}{x},\quad\operatorname{sinc}(0)=1\end{aligned}

The left centre is negative, the right centre positive. The Fourier sign convention fixes the positive-phase fringe shift. Sinc changes sign beyond each zero; both slit amplitudes share that sign, so it must not be silently discarded from their phasors.

Mutual coherence, not fake attenuation

\frac{I}{I_{\mathrm{ref}}}=\frac{|u_-|^2+|u_+|^2+2\mu\operatorname{Re}(u_-^*u_+)}4=\mu\frac{|u_-+u_+|^2}4+(1-\mu)\frac{|u_-|^2+|u_+|^2}4

Coherence suppresses only the cross term. The panel shows a coherent contribution and an incoherent contribution, never a single fictitious field for partial coherence. A constant real mutual coherence is prescribed at the two slits; the SPEAR3 paper motivates contrast measurements but its source-size inversion and optical train are not simulated here.

Missing orders and phase-scan contrast

y_m=\left(m+\frac{\phi}{2\pi}\right)\frac{\lambda L}{d},\quad y_n=n\frac{\lambda L}{a},\quad V_{\mathrm{phase}}=\frac{2\mu r}{1+r^2}

Phase-alignment orders disappear where they meet a noncentral envelope zero. With zero phase and separation three times width, every third order is missing. Alignment ticks are not the exact local maxima of the full product. Visibility is measured by scanning phase at a fixed nonzero envelope; imbalance lowers it even at full coherence. At an envelope zero the local contrast is undefined.

Reference scale and approximation boundary

\frac{I_{\mathrm{ref}}}{I_{\mathrm{inc}}}=\frac{4a^2}{\lambda L},\quad F_A=\frac{[(d+a)/2]^2}{\lambda L}\le0.081,\quad\frac{|y|}{L}\le0.020

For infinite vertical slits the one-dimensional Fresnel kernel gives this current-geometry reference relative to incident intensity: inverse distance, not the inverse-square law of a finite two-dimensional far-field aperture. Every plot remains zero to one on that stated reference, not auto-gained to its current maximum. The full horizontal half-aperture, not just one slit width, is used for the far-field check. Screen hues are illustrative RGB, not calibrated colorimetry. This is an approximation, not Maxwell, Fresnel near-field, finite-height or photon-counting simulation.

Common difficulties

Wider slits do not widen the envelope

Typical misconceptionChanging slit width and separation should have the same effect because both make the aperture bigger.

Better mental modelThey are different Fourier factors. Wider individual openings narrow the envelope; larger centre separation tightens fringe spacing. Phase changes shift the alignment ticks, while reducing coherence fills dark interference gaps without moving envelope zeros. Zero right amplitude leaves a single-slit envelope at one quarter of the equal-slit reference.

Run the experiment

01
Scene 1: Narrow-slit interference
In the narrow-slit preset, vary centre separation, then screen distance and wavelength. Shift-drag the cursor or move its slider to a dark fringe. Set phase to 180 degrees to darken the centre.
What to observe: Separation changes the period without changing the envelope; distance and wavelength scale both. The same white probe is used by hardware, amplitudes, profile and readout.
02
Scene 2: Every third order missing
Choose the missing-order preset. Measure the third phase-alignment tick at the gold envelope zero, then change slit width or phase to break their coincidence.
What to observe: The third order is dark because each finite slit amplitude vanishes there, not because the two slits suddenly lose coherence. Initial phase does not move the envelope zero.
03
Scene 3: Partial coherence, not dimming
Choose partial coherence. Reduce coherence to zero, then restore it and lower the right amplitude to zero. Compare the cross-term and baseline contributions.
What to observe: No coherence leaves the sum of separate intensities; one blocked slit removes interference entirely. Amplitude imbalance and partial coherence are different causes of low contrast.