Optics 050 · Interference, coherence, cavities, and metrology

Division-of-Wavefront Interferometers

Three connected 3D wavefront-division benches derive virtual source images, finite-surface ray access and one simulated screen record from Lloyd reflection, a thin Fresnel biprism and two exact plane mirrors.

Source-image separation / central spacing

d_v,\Delta q_0

$0.50$

Probe intensity / relative phase

I/I_{\max},\Delta\phi

$50\%$

Overlap in the screen window / branch masks

W_{\cap},M_A,M_B

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Division-of-Wavefront Interferometers

Monochromatic vacuum wavelength

\lambda

532\,\mathrm{nm}

Axial screen distance (Lloyd/prism: source; mirrors: hinge)

D_{\mathrm{ax}}

1.20\,\mathrm m

Lloyd real-source height above mirror

h

1\,\mathrm{mm}

Single wedge angle / relative mirror tilt

\alpha

0.12^{\circ}

Prescribed constant biprism index

n

1.50

Biprism half-aperture (geometry follows)

w

4\,\mathrm{mm}

Length of each Fresnel mirror

\ell

80\,\mathrm{mm}

Lloyd mirror length / source-screen distance

f

0.75

Screen probe coordinate (not mirror dragging)

q

0.50\,\mathrm{mm}

Start with Lloyd: move the source upward and fringes tighten; shorten the actual mirror and the overlap shrinks. Switch to the biprism: both halves refract, and the central fringe is bright. Switch to Fresnel mirrors: both branches reflect once, so their equal reflection phases cancel. Scan the mint screen probe through overlap, single-beam and unlit regions. Only relevant controls are enabled. Drag to orbit; optional purple dashed virtual-source lines are not transmitted light.

Ideal coherent monochromatic scalar unit-amplitude branches. Lloyd uses a prescribed phase-reversing reflection; Fresnel mirrors share that phase. The biprism is a paraxial thin-phase-screen model, not thick-prism Snell tracing. Hard geometric edges omit diffraction, real coating phase, losses, dispersion, source width and noise. Transverse scale is 33.3 times axial; prism thickness is enlarged 1500 times axial plus a minimum visible skin. Screen brightness uses display gamma; data intensities are linear. Overlap is sampled at 5 micrometres within a fixed 12 mm window. Play only moves slowed route markers.

Physics tutorial

Audit the physics of Division-of-Wavefront Interferometers

BackgroundA wavefront can be divided without two physical source lamps. Reflection or refraction redirects different parts of one coherent field, and the detector compares their optical paths: $\Delta\phi=\frac{2\pi}{\lambda}(L_B-L_A)+\phi_B-\phi_A$ . Virtual sources are geometric back-projections, not emitters behind opaque mirrors. This Lab uses one real source in every apparatus.

Why it mattersWhy is Lloyd’s contact fringe dark, while a symmetric biprism or double mirror has a bright centre only where both branches arrive?

Start with the essentials

Focus question: Why is Lloyd’s contact fringe dark, while a symmetric biprism or double mirror has a bright centre only where both branches arrive?
One-sentence intuition: A pair of virtual images does not guarantee light everywhere. Each finite optic supplies a geometric branch mask: $\frac I{I_{\max}}=\frac{M_A+M_B+2M_AM_B\cos\Delta\phi}{4}$ . Two arriving branches interfere; one has quarter-scale intensity without fringes; neither is unlit. The hard boundaries neglect edge diffraction.

Core mathematical model

Lloyd: one real source and its mirror image

\begin{aligned}d_v&=2h\\\Delta L&=\sqrt{D^2+(q+h)^2}\\&\quad-\sqrt{D^2+(q-h)^2}\\\phi_B-\phi_A&=\pi\end{aligned}

The real source is above an opaque horizontal mirror; its virtual image is equally far below. The reflected path uses the actual mirror intersection. A prescribed ideal phase-reversing reflection makes the contact fringe dark. A real coating has wavelength, angle and polarization dependent phase; that material model is not implemented.

Lloyd: a finite mirror limits overlap

\begin{aligned}x_{\mathrm{hit}}&=\frac{Dh}{q+h}\\D(1-f)&\le x_{\mathrm{hit}}\le D\\0&\le q\le\frac{hf}{1-f}\end{aligned}

This apparatus deliberately ends the mirror at the screen, so the ideal contact limit is observable. The length fraction moves the other end. Above the reflected footprint only the direct beam remains; below the mirror plane both modeled branches are absent. The reported overlap is clipped to the fixed screen window and sampled at five micrometres.

Biprism: thin phase screen, not a thick Snell solver

\begin{aligned}\delta&=(n-1)\alpha,\quad a=0.25\,\mathrm m\\d_v&=2a\delta,\quad D=a+b\\u_{\pm}&=\frac aD(q\pm b\delta)\\\Delta L&=\frac{2a\delta q}{D}\end{aligned}

The upper and lower prism halves give opposite angular kicks toward the centre. Each ray must cross its own half within the selected aperture. The phase uses the stationary paraxial optical path through that same thin phase screen, not an unrelated exact spherical-distance formula. Symmetry gives a bright centre. Index and wedge angle are constant; dispersion and interface loss are omitted.

Fresnel mirrors: exact images, one reflection each

\begin{aligned}\mathbf S_j'&=2\mathbf t_j(\mathbf t_j\cdot\mathbf S)-\mathbf S\\L_j&=|\mathbf P-\mathbf S_j'|\\d_v&=2r\sin\alpha,\quad r=0.25\,\mathrm m\\\phi_B-\phi_A&=\pi-\pi=0\end{aligned}

Two finite mirror segments share a hinge and differ by the selected small angle. Both source and screen are on the reflective side. The source has a fixed two-degree grazing angle relative to the mean mirror plane. Coordinates are rotated to place the mean outgoing axis horizontally. Every accepted ray obeys reflection, remains inside its own segment and is checked against interception by the other mirror. The displayed stop blocks direct source light.

Central spacing is a local phase gradient

\begin{aligned}\Delta q_0&=\frac{\lambda}{|\partial_q\Delta L|_{q=0}}\\\Delta q_{0,\mathrm L}&=\frac{\lambda\sqrt{D^2+h^2}}{2h}\\\Delta q_{0,\mathrm B}&=\frac{\lambda D}{2a\delta}\\\Delta q_{0,\mathrm F}&=\frac{\lambda\sqrt{(R+r\cos\alpha)^2+(r\sin\alpha)^2}}{2r\sin\alpha}\end{aligned}

Lloyd and double-mirror records use exact geometric path lengths, so the local spacing is not assumed uniform over arbitrarily large angles. The biprism spacing is uniform within its paraxial model. With no image separation there is no finite fringe period; a bright central extrapolation is not a finite illuminated overlap region.

A simulated screen is not a reconstruction

\begin{aligned}E_A&=M_Ae^{i\varphi_A}\\E_B&=M_Be^{i\varphi_B}\\I/I_{\max}&=|E_A+E_B|^2/4\end{aligned}

The physical screen texture, intensity profile and probe readout use these same fields and masks. The separate diagnostic fixes the first field phase to zero and shows the relative phase of the second; an absent branch has no vector. Unit amplitudes neglect spreading and loss; normalization is to the equal-field constructive intensity, not total energy flux. Screen display gamma and pixel averaging only affect presentation.

Common difficulties

A mirror image is not transmitted light

Typical misconceptionDashed rays from virtual sources are real light passing through the mirror, and a two-source formula applies over the whole screen.

Better mental modelOnly the solid physical branches reach the screen. Dashed purple lines, when enabled, are backwards constructions. Finite mirror and prism extents can leave one or neither branch available, even when two virtual images still exist.

Run the experiment

01
Scene 1: Lloyd mirror
Start with Lloyd; raise the source and shorten the actual mirror. Scan the probe across the footprint boundary.
What to observe: Vertical coordinates are enlarged 33.3 times relative to axial distances. Mirror vertices and ray endpoints use one transform, but apparent display angles are not physical incidence angles.
02
Scene 2: Fresnel biprism
Switch to the biprism; vary angle, index and aperture. Compare a bright centre with no-overlap settings.
What to observe: The prism is an enlarged shape around an ideal thin interaction plane. Axial thickness is scaled 1500 times plus a visible minimum; it is not an exact thick-glass path.
03
Scene 3: Fresnel mirrors
Switch to Fresnel mirrors; change the angle and both mirror lengths. Inspect side and screen-front views.
What to observe: Play animates slowed route markers only. Pause, orbit or reduced motion never changes the acquired data. Hard edges, a line-invariant field and no detector noise are idealizations.