Wavefront-Splitting and Amplitude-Splitting Interferometers

Every classical interferometer answers the same engineering question before it does anything else: how do you get two mutually coherent beams out of one source? A detector reads only time-averaged intensity, so the phase an interferometer is built to measure becomes visible only through the cross term of two overlapping coherent fields — and there are exactly two ways to manufacture that pair. You can take one wavefront and sample it in two different places (wavefront splitting: two slits, a split lens, a biprism, a grazing mirror), or you can take one ray and split its amplitude into a reflected and a transmitted copy (amplitude splitting: a beamsplitter, a thin film, a Fabry–Pérot cavity). Young's double slit, the Fresnel biprism, the Fresnel double mirror and Lloyd's mirror are the wavefront family; the Michelson, Mach–Zehnder, Sagnac, thin films and Newton's rings are the amplitude family. The classification earns its keep because it tells you, before any algebra, which kind of coherence controls the contrast. Wavefront splitting samples two separated points of the wavefront, so it succeeds only if those points lie within the source's spatial coherence area — which is why a biprism needs a small or distant source, exactly as the Van Cittert–Zernike theorem demands. Amplitude splitting duplicates a single wavefront point, so the two copies are mutually coherent no matter how big the source is; its enemy is instead temporal coherence — delay one copy past the coherence length and it no longer overlaps its own wave train. Both families end at the identical two-beam interference law; the split mechanism merely decides whether the spatial or the temporal coherence factor is the one near unity. Know which family an instrument belongs to and you know which knob — source size or bandwidth — will kill its fringes.

Key takeaways

Two ways to make two coherent beams from one source: wavefront splitting (sample two places on one wavefront) and amplitude splitting (partially reflect one ray into two copies).
Wavefront-splitting devices (Young's slits, Fresnel biprism, Fresnel double mirror, Lloyd's mirror) are limited by spatial coherence — they need a small or distant source (Van Cittert–Zernike).
Amplitude-splitting devices (Michelson, Mach–Zehnder, Fabry–Pérot, thin films, Newton's rings) tolerate extended sources but are limited by temporal coherence — path difference must stay below $\ell_c=\lambda^2/\Delta\lambda$ .
Both realize the identical two-beam law $I=I_1+I_2+2\sqrt{I_1I_2}\,|\gamma_{12}|\cos\delta$ ; only what sets $\gamma_{12}$ differs.
A Fresnel biprism makes two virtual sources separated by $d=2a(n-1)\alpha$ , giving Young-type fringes of spacing $\Delta y=\lambda(a+b)/d$ ; a Michelson advances one fringe per $\lambda/2$ of mirror travel.
Key idea: wavefront splitting divides one wavefront in space (spatial-coherence limited); amplitude splitting duplicates a ray by partial reflection (temporal-coherence limited). Same interference term, opposite coherence bottleneck — so the classification predicts which source property controls the fringes.

The core insight

Interferometers come in two families. Wavefront splitting samples two separated points of a single wavefront and is limited by spatial coherence (source size); amplitude splitting partially reflects one ray into two copies and is limited by temporal coherence (path difference vs coherence length). Both yield the same two-beam interference term — the classification's power is that it tells you, before any calculation, which coherence property sets the contrast.

Where this lives today

The amplitude-splitting Michelson is the most consequential measuring instrument of modern physics. Scaled to four-kilometre arms, its fringe shift reads out a mirror displacement of $10^{-19}$ m — a thousandth of a proton's width — which is how LIGO and Virgo detect gravitational waves. The same architecture, scanned rather than held fixed, is a Fourier-transform spectrometer: record intensity versus mirror position, and the interferogram's Fourier transform is the spectrum, the workhorse of infrared chemical analysis. Mach–Zehnder interferometers, amplitude-split on a chip, modulate the light in fibre-optic communications and encode qubits in photonic quantum computing. The wavefront family lives on wherever a source is naturally small or distant. Long-baseline optical interferometers — the VLTI, CHARA — are wavefront-splitting instruments that sample a stellar wavefront at two widely separated telescopes, turning the spatial-coherence limit into a measurement of stellar diameters and the orbits of close binaries. Electron and neutron biprisms extend the trick to matter waves: a charged wire replaces the glass prism, splitting an electron wavefront to produce interference fringes that confirm quantum superposition and enable electron holography. In every case the first design decision is the one this article names — divide the wavefront, or divide the amplitude — and it fixes whether the instrument's fortunes ride on the size of its source or the purity of its colour.

The physical picture

Picture a wavefront as a moving sheet of synchronized phase. Wavefront splitting cuts two windows in that sheet and lets the light beyond each window overlap downstream — but the two windows must belong to the same coherent patch of the sheet, or their phases drift independently and the fringes never settle. That is the whole reason a Fresnel biprism or a double slit demands a pinhole or a far-off source: it is shrinking the source's angular size until a wide enough patch of wavefront marches in step. Amplitude splitting does something quite different: it makes a faint ghost copy of the entire sheet with a partial reflector, then delays the ghost. Because the ghost is a replica of the very same wavefront point-for-point, it interferes with the original however large the illuminated area — the only hazard is delaying the ghost so far that its wave train no longer overlaps itself. The mnemonic writes itself. Wavefront splitting asks "are these two places on the wavefront in step?" — a question about spatial coherence. Amplitude splitting asks "does this wave train still overlap its own delayed copy?" — a question about temporal coherence. Both then arrive at the same detector equation, so the physics of the source and geometry only decides which factor of the coherence function is the bottleneck. It is why you can watch Newton's rings form in the light of a broad table lamp (amplitude splitting shrugs at source size) but need a pinhole to see biprism fringes from the same lamp (wavefront splitting cannot).

Figure

The two families: wavefront splitting (Fresnel biprism → two virtual sources S′, S″ from one wavefront) versus amplitude splitting (a beamsplitter duplicates each ray into two arms)

The next figure turns the wavefront family's workhorse — the Fresnel biprism — into real numbers. A shallow glass prism, apex up, bends the top half of a beam down and the bottom half up, conjuring two overlapping virtual sources out of a single slit; their separation and the resulting fringe spacing follow from nothing but the prism angle and the geometry.

Figure

Fresnel-biprism fringe spacing: virtual-source separation d = 2a(n−1)α, fringe spacing Δy = λ(a+b)/d — for α = 0.5°, n = 1.5, a = 0.5 m, b = 1.0 m, λ = 633 nm this gives d = 4.36 mm, Δy = 0.218 mm

Notice what the biprism cannot escape: because it is a wavefront-splitter, those two virtual sources $S'$ and $S''$ must both lie within the coherence patch of the real slit $S$ . Widen the slit and the fringes fade — the spatial-coherence limit, made concrete. A Michelson in the same room would keep its fringes with a slit ten times wider, because its two beams are the same ray twice, not two places on a wavefront.

The core equations

Whatever the split mechanism, the detector sees one two-beam interference law. Adding two coherent scalar fields and time-averaging gives the two intensities plus a cross term carrying the coherence and the path-difference phase.

I = I_1 + I_2 + 2\sqrt{I_1 I_2}\,|\gamma_{12}(\tau)|\cos\big[\delta - \arg\gamma_{12}(\tau)\big], \qquad V=\frac{2\sqrt{I_1I_2}}{I_1+I_2}\,|\gamma_{12}| \qquad (1)

where: $I_1,I_2$ — beam intensities; $\gamma_{12}(\tau)$ — complex degree of coherence at delay $\tau$ ; $\delta=(2\pi/\lambda)\,\text{OPD}$ — phase from the optical path difference; $V$ — fringe visibility. The class fixes what $\gamma_{12}$ is: for wavefront splitting it is the transverse (spatial) coherence between two source-plane points; for amplitude splitting it is the temporal self-coherence at delay $\tau$ . For the wavefront family's biprism, the prism deviates each half-beam by $(n-1)\alpha$ , producing two virtual sources whose separation then feeds the Young fringe formula.

d = 2a\,(n-1)\alpha, \qquad \Delta y = \frac{\lambda\,(a+b)}{d} \qquad (2)

where: $a$ — slit-to-biprism distance; $\alpha$ — biprism wedge (base) angle; $n$ — prism index; $d$ — separation of the two virtual sources; $b$ — biprism-to-screen distance; $\Delta y$ — fringe spacing. The whole device is a compact way to realize Young's two-source geometry with far more light than two slits pass. For the amplitude family's Michelson, displacing one mirror by $\Delta L$ changes that arm's round-trip path by $2\Delta L$ , so fringes stream past at one per half-wavelength — the basis of interferometric length metrology.

\delta = \frac{2\pi}{\lambda}\,(2\,\Delta L), \qquad N = \frac{2\,\Delta L}{\lambda} \quad\Longrightarrow\quad \Delta L_{\text{per fringe}} = \frac{\lambda}{2} \qquad (3)

where: $\Delta L$ — mirror displacement; the factor $2$ is the double pass; $N$ — number of fringes crossing the field. One fringe per $\lambda/2$ : for a HeNe laser ( $\lambda=632.8$ nm) that is a $316.4$ nm step, the working resolution of laser interferometry.

Derivation AWhy amplitude splitting tolerates a broad source and wavefront splitting does not

The aim is to prove the classification's central claim — that amplitude-split fringes survive an arbitrarily extended source while wavefront-split fringes wash out — by tracking what a transverse displacement of one source point does to the fringe phase in each geometry. We displace a source point by $s$ and follow the induced phase shift $\Delta\delta$ through both mechanisms.

Step 1. Amplitude splitting: for a plane-parallel film, the two interfering rays are the front- and back-surface reflections of one incident ray, and their path difference $2n_f t\cos\theta_t$ depends only on the inclination angle $\theta_t$ , not on which source point launched the ray. Why: both beams are the same ray split in amplitude, so they share every accident of the source — the source's identity cancels out of their difference.

Step 2. A source point displaced by $s$ merely changes which screen point a given inclination reaches; every source point still contributes a fringe of the same order to that inclination. Hence $\Delta\delta=0$ and the fringes (of equal inclination) stay sharp for a source of any size. Why: with the fringe order tied to inclination alone, all source points paint the same pattern, so summing over an extended source only adds brightness, not blur.

Step 3. Wavefront splitting: in Young/biprism geometry the two beams sample the wavefront at a fixed separation $d$ . A source point displaced by $s$ at distance $R$ tilts the incident wavefront, adding a slit-to-slit phase $\Delta\delta=(2\pi/\lambda)(d\,s/R)$ . Why: the two beams come from two different places on the wavefront, so tilting the wavefront changes their relative phase — the source's position now matters.

Step 4. Different source points therefore produce mutually shifted patterns; integrating over a source of half-width $a_s$ smears them, and contrast collapses once $d\,a_s/R\sim\lambda$ , i.e. once $d\gtrsim\lambda R/a_s=d_c$ , the transverse coherence width. Why: this is the owned result — the source-size sensitivity is zero for amplitude splitting and linear for wavefront splitting, which is exactly the Van Cittert–Zernike spatial-coherence criterion. *Verified: the biprism virtual-source separation $d=2a(n-1)\alpha=4.36$ mm and fringe spacing $\Delta y=\lambda(a+b)/d=0.218$ mm, plus the zero-vs-linear source-shift responses, in verify_wavefront-amplitude-splitting.py.*

Go deeper · Derivation B · 1 minMichelson fringe count as a length standard

Derivation BMichelson fringe count as a length standard

The goal is to turn the amplitude-splitting Michelson into a ruler: relate the number of fringes that cross the detector to the distance one mirror travels, the relation that underlies interferometric metrology and once helped define the metre. We convert the mirror displacement into a round-trip path change and then into fringes.

Step 1. Displacing one mirror by $\Delta L$ lengthens that arm's round trip by $2\Delta L$ — the light goes out and back. Why: the interfering beams traverse each arm twice, so a mirror move counts double in the optical path.

Step 2. Convert to phase, $\delta=(2\pi/\lambda)(2\Delta L)$ , and note that each increase of $\delta$ by $2\pi$ slides the pattern by one whole fringe, so $N=\delta/2\pi=2\Delta L/\lambda$ . Why: the fringe pattern is periodic in $\delta$ with period $2\pi$ , so counting fringes counts multiples of the wavelength in the path.

Step 3. Invert: one fringe corresponds to $\Delta L=\lambda/2$ . For HeNe light ( $\lambda=632.8$ nm), one fringe is a $316.4$ nm displacement; counting $N$ fringes measures a length $N\lambda/2$ to sub-wavelength precision. Why: this is the owned result — a displacement readout in units of half a wavelength, the principle of gauge-block calibration and gravitational-wave mirror sensing. *Verified: $\Delta L=\lambda/2=316.4$ nm for $\lambda=632.8$ nm, and $N=2\Delta L/\lambda$ , in verify_wavefront-amplitude-splitting.py.*

Limiting cases

Case	Result	Meaning
ideal point, monochromatic source	$\\|\gamma\\|\to1$	both families give full-contrast fringes; the classification becomes academic (a good laser)
extended source, wavefront-splitting	$\\|\gamma_{12}\\|=\\|2J_1(\pi a_s\theta/\lambda)/(\cdots)\\|$	visibility set by Van Cittert–Zernike; fringes vanish once $d\gtrsim\lambda/\theta$
extended source, amplitude-splitting	$\\|\gamma\\|\approx1$ (source size irrelevant)	equal-inclination fringes tolerate any source; only temporal coherence limits contrast
large path imbalance	$\Delta L\gg\ell_c=\lambda^2/\Delta\lambda$	amplitude-split fringes die when the delay exceeds the coherence length

Common trap“You need a laser to see interference”

Amplitude-splitting instruments work beautifully with broad thermal light — Newton's rings and soap-film colours appear in plain daylight — because every point of the source contributes fringes of the same order, so source size simply does not enter. What such devices do demand is temporal coherence: keep the two paths within the coherence length. It is the wavefront-splitting devices (double slit, biprism, Lloyd's mirror) that need a small or distant source, because they sample two separated points of the wavefront and so require spatial coherence. The reverse trap is just as common: a source can be spatially coherent (a laser) yet still ruin amplitude-splitting fringes if it is broadband and the path imbalance exceeds $\ell_c$ . Match the coherence you have — spatial or temporal — to the family of instrument you are using.

Worked examples

1 · Fringe spacing from a Fresnel biprism (confidence builder)

A Fresnel biprism of index $n=1.5$ and base angle $\alpha=0.5^\circ$ is placed $a=0.5$ m from an illuminated slit, with the screen $b=1.0$ m beyond the biprism. For $\lambda=633$ nm, find the separation of the two virtual sources and the fringe spacing.

Step 1. Each half of the biprism deviates its beam by $(n-1)\alpha=(0.5)(0.5^\circ)=(0.5)(8.727\times10^{-3}\ \text{rad})=4.36\times10^{-3}$ rad. The two deviated beams appear to come from two virtual images of the slit.

Step 2. Virtual-source separation: $d=2a(n-1)\alpha=2(0.5)(4.36\times10^{-3})=4.36\times10^{-3}$ m $=\mathbf{4.36\ mm}$ . The biprism has split one slit into two coherent sources $4.36$ mm apart.

Step 3. Fringe spacing on the screen (Young geometry with total source-to-screen distance $a+b$ ): $\Delta y=\dfrac{\lambda(a+b)}{d}=\dfrac{633\times10^{-9}\times1.5}{4.36\times10^{-3}}=2.18\times10^{-4}$ m $=\mathbf{0.218\ mm}$ . Fringes about a fifth of a millimetre apart — comfortably resolved with a low-power eyepiece, and far brighter than a true double slit because the biprism wastes no light on opaque bars. d = 4.36 mm, Δy = 0.218 mm.

2 · Lloyd's mirror and its telltale dark centre (exam level)

In Lloyd's mirror a slit and its grazing-incidence reflection in a flat mirror act as two coherent sources. A student expects a bright central fringe as in Young's experiment but observes a dark one. Explain and quantify.

Step 1. The geometry is wavefront-splitting: the direct beam and the reflected beam are two portions of one wavefront, separated by $d=2h$ where $h$ is the slit's height above the mirror plane. The fringe spacing follows the same law, $\Delta y=\lambda(a+b)/d$ .

Step 2. But reflection at the mirror occurs going from a rarer to a denser medium at grazing incidence, which imprints a $\pi$ phase shift — an extra half-wave — on the reflected beam. The interference condition is therefore inverted: where a direct-source Young pattern would be bright (zero path difference at the mirror plane), Lloyd's mirror is dark.

Step 3. The signature is unmistakable at the point where the two paths are geometrically equal — the mirror surface itself. A path-difference-only argument predicts a bright fringe there; the observed dark fringe is direct evidence of the reflection phase shift, the same $\pi$ that governs thin-film and Newton's-ring interference. Historically this was one of the cleanest early demonstrations that reflection can flip a wave's phase. The central fringe is dark, shifted by half a fringe (λ/2) from the naive Young prediction, because grazing reflection adds a π phase shift.

Physics of the portrait

The interferometer is not a person but an instrument idea — the machine that makes phase visible — and its two-family architecture is the organizing principle of a century and a half of precision optics.

Origins · 1 mininterferometer and how this was discovered

interferometer

the machine that measures phase · Young 1801 → Michelson 1881 · from a biprism to gravitational-wave detectors

The interferometer is the device that converts an optical path difference into a countable intensity pattern, and its history is a ledger of the two splitting strategies. The wavefront branch opened with Thomas Young's double slit (1801–1804) and matured through Augustin Fresnel, whose double mirror (1816) and biprism (1819) produced clean two-source fringes without diffracting edges and helped settle the wave theory of light; Humphrey Lloyd's single-mirror arrangement (1834) added the striking proof that reflection can flip a wave's phase. The amplitude branch is dominated by Albert Michelson, who from 1881 built the beamsplitter interferometer that bears his name — first to chase the luminiferous ether (the null result of the 1887 Michelson–Morley experiment), then to redefine the metre in wavelengths of cadmium light, then, mounted as a stellar interferometer, to measure the diameter of a star. The same amplitude-splitting architecture, scaled to four-kilometre arms and stabilized to a fraction of a proton's width, is how LIGO detected gravitational waves in 2015. Every one of these instruments obeys the single two-beam law; what distinguishes them is only how they made their two coherent beams — by dividing a wavefront in space, or by dividing a ray in amplitude.

History

1801

— Young demonstrates two-source interference by wavefront division at a double slit

1816

— Fresnel produces fringes with a double mirror (wavefront splitting)

1819

— Fresnel introduces the biprism, giving two virtual coherent sources

1834

— Lloyd's single-mirror experiment reveals the π phase change on reflection

1881

— Michelson builds the amplitude-splitting interferometer

1887

— the Michelson–Morley experiment returns its famous null result

2015

— LIGO's kilometre-scale Michelson interferometers detect gravitational waves

Going further

Texts. Hecht, Optics, 5th ed., §9.3 (Wavefront-splitting interferometers) and §9.4 (Amplitude-splitting interferometers) · Born & Wolf, Principles of Optics, 7th ed., Ch. 7 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., Ch. 2 and Ch. 11 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 3.

Provenance. The wavefront/amplitude classification and the localization argument follow Hecht §9.3–9.4 and Born & Wolf Ch. 7. A verifier recomputes: the biprism virtual-source separation $d=2a(n-1)\alpha=4.36$ mm and fringe spacing $\Delta y=\lambda(a+b)/d=0.218$ mm; the Michelson per-fringe displacement $\lambda/2=316.4$ nm for HeNe $632.8$ nm; and the zero-vs-linear source-shift responses that distinguish the two classes. The coherence length $\ell_c=\lambda^2/\Delta\lambda$ (temporal limit) and the transverse coherence width $d_c\sim\lambda/\theta_s$ (spatial limit) are the standard Van Cittert–Zernike and Wiener–Khinchin scales. (Base note: the Step-1 draft mislabeled the thin-film OPD equation's left side as an index; the correct statement is $\text{OPD}=2n_f t\cos\theta_t+\lambda/2$ , treated in the thin-film article.) All numbers are reproduced in verify_wavefront-amplitude-splitting.py.

Prerequisites. Superposition Principle of Electromagnetic Waves · Two-Wave Interference and the Interference Term · Conditions for Observable Interference · Optical Path Difference and Phase Retardation. Leads to. Young's Double-Slit Experiment · Thin-Film Interference · Fringes of Equal Inclination and Equal Thickness · Spatial Coherence and the Van Cittert–Zernike Theorem · Newton's Rings.

What is the practical difference between splitting a wavefront and splitting an amplitude?

Splitting a wavefront means sending two different pieces of the same wavefront on to overlap — the top and bottom of a beam through a biprism, or the light through two slits. Splitting an amplitude means taking one piece of wavefront and making a partial-reflection copy of it — the transmitted and reflected halves at a beamsplitter or a film surface. The practical consequence is which coherence you must supply. Two different pieces of a wavefront are only in step if the source is small enough (spatial coherence), so wavefront-splitters need pinholes or distant sources. Two copies of one piece are automatically in step regardless of source size, so amplitude-splitters accept broad sources but demand you keep their path difference short (temporal coherence).

Why does a Michelson interferometer work with an extended source but a double slit does not?

Because the Michelson is amplitude-splitting. Its two beams are the same ray reflected off two mirrors, so every point of an extended source produces fringes at the same order (fringes of equal inclination) and the patterns from all source points coincide — a broad source just adds brightness. The double slit is wavefront-splitting: its two beams come from two separated points of the wavefront, so a wider source tilts the wavefront by a range of angles and each source point lays down a shifted fringe pattern. Once those shifts span a full fringe, the sum washes out. This is exactly the spatial-coherence criterion $d\lesssim\lambda R/a_s$ — the double slit is spending the source's angular size, the Michelson is not.

Is a Fabry–Pérot etalon wavefront or amplitude splitting?

Amplitude splitting — but multiple-beam amplitude splitting. A single ray entering the cavity is partially reflected back and forth between the two mirrors, and at each pass a fraction of its amplitude is transmitted; those many transmitted copies, all derived from one ray, interfere. Summing the geometric series of amplitudes gives the sharp Airy transmission function whose peak width narrows as the mirror reflectance rises. Because it is amplitude-splitting, the etalon tolerates extended sources and produces its ring pattern (fringes of equal inclination) in broad light; its resolving power is limited instead by temporal coherence and by how many round trips the light survives, captured by the finesse.

Why does Lloyd's mirror give a dark central fringe when Young's experiment gives a bright one?

Because one of Lloyd's two beams is reflected, and reflection at grazing incidence off a denser medium adds a $\pi$ phase shift — an extra half-wave. At the point where the direct and reflected paths are geometrically equal (on the mirror plane) the naive path-difference argument predicts constructive interference and a bright fringe, but the extra $\pi$ makes it destructive instead, so the centre is dark and the whole pattern is shifted by half a fringe relative to Young. It is a clean, visible proof that reflection can flip a wave's phase — the same phase shift that governs the half-wave bookkeeping in thin films and Newton's rings.

Both families end at the same interference equation — so why bother classifying them?

Because the classification tells you, before you compute anything, which experimental knob controls your fringes. If an instrument is wavefront-splitting and your fringes are faint, the fix is a smaller or more distant source (improve spatial coherence); widening the source will only make it worse. If it is amplitude-splitting and the fringes fade as you unbalance the arms, the fix is a narrower spectral filter or a shorter path difference (improve temporal coherence); source size is irrelevant. The two-beam law is the same, but the coherence factor $\gamma_{12}$ in it means different physics in each case — transverse correlation versus longitudinal correlation — and the family label is a one-word diagnosis of which one you are fighting.

Can matter waves be split the same two ways?

Yes, and both families have matter-wave versions. The electron biprism — a fine charged wire that deflects an electron wavefront to either side — is a wavefront-splitting device, and it produces the electron interference fringes that demonstrate quantum superposition and enable electron holography. Amplitude splitting is realized for atoms and neutrons with gratings and crystal interfaces that coherently divide a matter wave into transmitted and diffracted copies, as in neutron and atom interferometers. The underlying two-beam law is identical to the optical case, with the de Broglie wavelength in place of $\lambda$ — which is why interferometry is one of the clearest bridges between classical wave optics and quantum mechanics.