Optics 042 · Interference, coherence, cavities, and metrology

Young Double-Slit Bench

How do slit width, slit spacing, phase and coherence each change a double-slit pattern? The slit separation fixes the fringe spacing: the farther apart the slits, the closer the fringes. Each slit's width sets the broad diffraction envelope, and a fringe that falls on one of its zeros goes missing. A phase shift slides the fringes under that envelope, and partial coherence lowers their contrast.

Separate diffraction from interference on a 3D double-slit bench, then measure missing orders, the effect of partial coherence and the contrast when the two amplitudes differ.

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 $\text{valid model regime}$

Model checked against analytic results

Young Double-Slit Bench

Monochromatic wavelength

\lambda

532\,\mathrm{nm}

Actual width of each slit (sets the envelope)

a

20\,\mu\mathrm m

Slit centre separation (sets the 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 make one measured pattern. Widen the openings and the gold envelope shrinks; move their centres apart and the cyan fringes get closer together. In the missing-order preset, the third interference order lands exactly on a zero of the envelope. Shift-drag the white screen cursor to measure, or use its slider. Plain dragging orbits the apparatus, and Play only animates illustrative pulses along the paths.

The pattern is the scalar paraxial far field. The coherence between the slits is constant, each axis is magnified independently, and intensity is referenced to the current geometry.

Physics tutorial

Audit the physics of Young Double-Slit Bench

BackgroundA slit of finite width sends light out over a spread of angles, not as one narrow ray. In the scalar paraxial far field, the screen sees the Fourier transform of the aperture: $U(y)\propto\int t(s)e^{-i2\pi ys/(\lambda L)}\,ds$ . Light across each slit is coherent, and each slit is infinitely tall. Because the two slits have equal widths, they share one sinc amplitude. The two highlighted lines in the 3D view show phase paths to the probe; they are not a solved near-field intensity, and light does not travel only along them. The openings really change width and centre separation, and the screen really moves.

Why it mattersHow do slit width, slit spacing, phase and coherence each change a double-slit pattern?

Start with the essentials

Focus question: How do slit width, slit spacing, phase and coherence each change a double-slit pattern?
One-sentence intuition: Two independent length scales shape the pattern on the screen: $\Delta y=\frac{\lambda L}{d},\qquad y_1=\frac{\lambda L}{a}$ . The slit width sets the gold diffraction envelope, and the slit separation sets the cyan interference period. Because the pattern is their product, an order can go missing even where the two waves are in phase. Changing the phase moves the fringes but not 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 slit is centred at a negative position and the right slit at a positive one. The sign convention of the Fourier transform fixes which way a positive phase shifts the fringes. The sinc factor changes sign at each zero. Both slit amplitudes carry that same sign, so it has to stay in their phasors.

Mutual coherence weights only the cross term

\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

Lowering the coherence weakens only the cross term. The panel shows the coherent part and the incoherent part separately; partial coherence is not represented by one made-up field. The mutual coherence at the two slits is set to a constant real value. The SPEAR3 paper motivates the contrast measurement, 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}

An order where the two waves are in phase disappears if it lands on an envelope zero away from the centre. With zero phase and a separation three times the width, every third order is missing. The alignment ticks are not the exact local maxima of the full product. Visibility is measured by scanning the phase at a fixed point where the envelope is not zero; unequal amplitudes lower it even at full coherence. At an envelope zero the local contrast is undefined.

Reference scale and where the approximation holds

\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 infinitely tall slits, the one-dimensional Fresnel kernel gives this reference for the current geometry, relative to the incident intensity. It falls off as one over the distance, not as the inverse square that a finite two-dimensional aperture gives in the far field. Every plot runs from zero to one on this reference scale and is never rescaled to its current maximum. The far-field check uses the whole horizontal half-aperture, not just one slit width. Screen colours are illustrative RGB, not calibrated colorimetry. The model is an approximation; it is not a Maxwell solver, a Fresnel near-field calculation, a finite-height slit model or a 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 enter as different Fourier factors. Widening each opening narrows the envelope; moving the centres apart packs the fringes closer together. Changing the phase shifts the alignment ticks. Lowering the coherence fills in the dark gaps between fringes but leaves the envelope zeros in place. With the right amplitude at zero, a single-slit envelope remains, at one quarter of the equal-slit reference.

Run the experiment

01
Scene 1: Narrow-slit interference
In the narrow-slit preset, vary the centre separation, then the screen distance and the wavelength. Shift-drag the cursor, or use its slider, to put it on a dark fringe. Set the phase to 180 degrees and the centre turns dark.
What to observe: Changing the separation changes the fringe period but not the envelope; distance and wavelength stretch both. The apparatus, the amplitudes, the profile and the readout all use the same white probe.
02
Scene 2: Every third order missing
Choose the missing-order preset. Measure the third phase-alignment tick where it sits on the gold envelope zero, then change the slit width or the phase so the two no longer coincide.
What to observe: The third order is dark because the amplitude from each finite slit vanishes there; the two slits have not suddenly lost coherence. The initial phase does not move the envelope zero.
03
Scene 3: Partial coherence, not dimming
Choose partial coherence. Reduce the coherence to zero; then restore it and lower the right amplitude to zero instead. Compare the cross-term contribution with the baseline.
What to observe: With zero coherence, the screen shows just the sum of the two separate intensities. Blocking one slit removes the interference completely. Unequal amplitudes and partial coherence lower the contrast for different reasons.

Young Double-Slit Bench

Audit the physics of Young Double-Slit Bench

Core mathematical model

Two Fourier amplitudes

Mutual coherence weights only the cross term

Missing orders and phase-scan contrast

Reference scale and where the approximation holds

Common difficulties

Wider slits do not widen the envelope

Run the experiment

Scene 1: Narrow-slit interference

Scene 2: Every third order missing

Scene 3: Partial coherence, not dimming

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