Optics 050 · Interference, coherence, cavities, and metrology

Division-of-Wavefront Interferometers

Why is the fringe dark where Lloyd’s mirror meets the screen, but bright at a biprism’s center? Reflection off Lloyd's mirror flips the wave by half a cycle. Where the mirror meets the screen the two paths are equal in length, so the flip alone decides, and the direct and reflected waves cancel. A biprism or double mirror adds no flip, so its center is bright, but only where the finite optics let both beams reach the screen.

Three linked 3D benches split one wavefront in different ways: Lloyd's mirror, a thin Fresnel biprism and two exact plane mirrors. Each derives its virtual sources, finds which rays its finite surfaces can pass and produces one simulated screen record.

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 $\text{valid model regime}$

Model checked against analytic results

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

Height of the real source above the Lloyd 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

Probe position on the screen (does not move a mirror)

q

0.50\,\mathrm{mm}

Start with Lloyd's mirror: raise the source and the fringes get closer together; shorten the mirror and the overlap region shrinks. Switch to the biprism: both halves refract, and the central fringe is bright. Switch to Fresnel's mirrors: each branch reflects once, so the equal reflection phases cancel. Scan the mint screen probe through the overlap, single-beam and unlit regions. Only the controls that apply to the current device are enabled. Drag to orbit. The optional purple dashed lines to the virtual sources are construction lines, not light passing through the mirror.

Both branches are ideal, coherent, monochromatic and scalar, with unit amplitude. Lloyd's mirror uses a prescribed phase-reversing reflection, and Fresnel's mirrors share that phase. The biprism is a paraxial thin phase screen; it is not traced through a thick prism with Snell's law. Hard geometric edges leave out diffraction, real coating phases, losses, dispersion, source width and noise. The transverse scale is 33.3 times the axial scale; prism thickness is enlarged 1500 times relative to the axial scale, with a minimum thickness so it stays visible. Screen brightness uses display gamma, while the data intensities stay linear. The overlap is sampled every 5 micrometres within a fixed 12 mm window. Play only moves slowed path markers.

Physics tutorial

Audit the physics of Division-of-Wavefront Interferometers

BackgroundYou do not need two lamps to divide a wavefront. Reflection or refraction sends different parts of one coherent field along different routes, and the screen compares their optical paths: $\Delta\phi=\frac{2\pi}{\lambda}(L_B-L_A)+\phi_B-\phi_A$ . A virtual source is the point the redirected rays seem to come from when traced backward; nothing emits light behind the opaque mirror. Every apparatus in this Lab has only one real source.

Why it mattersWhy is the fringe dark where Lloyd’s mirror meets the screen, but bright at a biprism’s center?

Start with the essentials

Focus question: Why is the fringe dark where Lloyd’s mirror meets the screen, but bright at a biprism’s center?
One-sentence intuition: Having two virtual images does not mean light reaches everywhere. Each finite optic sets a geometric mask for its branch: $\frac I{I_{\max}}=\frac{M_A+M_B+2M_AM_B\cos\Delta\phi}{4}$ . Where both branches arrive, they interfere. Where only one arrives, the intensity is a quarter of the normalized maximum and shows no fringes. Where neither arrives, the screen is dark. The sharp boundaries ignore diffraction at the edges.

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 sits above an opaque horizontal mirror, and its virtual image is the same distance below. The reflected path goes through the point where the ray actually meets the mirror. The reflection is set to reverse the phase ideally, which makes the fringe at the contact line dark. A real coating gives a phase that depends on wavelength, angle and polarization; that material model is not included.

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}

Here the mirror deliberately ends at the screen, so you can see the ideal contact limit. The length fraction moves the other end of the mirror. Above the area lit by the reflection only the direct beam remains, and below the mirror plane neither modeled branch arrives. The reported overlap is clipped to the fixed screen window and sampled every five micrometres.

Biprism as a thin phase screen, not a thick prism traced with Snell's law

\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 halves of the prism bend the light toward the centre in opposite directions. Each ray has to cross its own half inside the chosen aperture. The phase comes from the stationary paraxial optical path through that same thin phase screen, not from a separate exact spherical-distance formula. By symmetry the centre is bright. The index and the wedge angle are constants, and dispersion and interface losses are left out.

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 meet at a hinge and are tilted against each other by the chosen small angle. The source and the screen are both on the reflecting side. The source sits at a fixed two-degree grazing angle to the mean mirror plane. The coordinates are rotated so that the mean outgoing axis is horizontal. Every accepted ray obeys the law of reflection, lands inside its own segment and is checked for being blocked by the other mirror. The stop in the scene blocks light going straight from the source to the screen.

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}

The Lloyd and double-mirror calculations use exact geometric path lengths, so the local fringe spacing is not assumed to stay the same at arbitrarily large angles. In the paraxial biprism model the spacing is uniform. When the two images do not separate, there is no finite fringe period; extending the bright centre to that limit does not give a finite lit 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 screen texture, the intensity profile and the probe reading all come from these same fields and masks. The separate diagnostic sets the phase of the first field to zero and shows the phase of the second relative to it; a branch that does not arrive has no vector. Unit amplitudes ignore spreading and loss. Intensity is divided by the constructive peak of two equal fields, not by the total energy flux. Display gamma and pixel averaging affect only how the screen looks.

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 branches carry real light to the screen. The dashed purple lines, when shown, are backward constructions. Because the mirrors and prisms are finite, a point on the screen may receive one branch or none, even though both virtual images still exist.

Run the experiment

01
Scene 1: Lloyd mirror
Start with Lloyd. Raise the source and shorten the mirror, then scan the probe across the edge of the reflected footprint.
What to observe: Vertical distances are drawn 33.3 times larger than axial distances. Mirror corners and ray endpoints share one transform, but the angles on screen are not the physical angles of incidence.
02
Scene 2: Fresnel biprism
Switch to the biprism and vary the angle, index and aperture. Compare a setting with a bright centre against one where the beams do not overlap.
What to observe: The prism is drawn as an enlarged shape around an ideal thin plane where the light is deflected. Its axial thickness is scaled up 1500 times, with a minimum so it stays visible; it does not show the exact path through thick glass.
03
Scene 3: Fresnel mirrors
Switch to Fresnel mirrors and change the angle and both mirror lengths. Look at the side view and at the front of the screen.
What to observe: Play only animates slowed markers along the routes. Pausing, orbiting or reducing motion never changes the recorded data. Hard edges, a field that does not vary along the screen width and a noise-free detector are all idealizations.