Amplitude Interferometry: Methods & Applications

• Amplitude interferometry is a technique that measures complex field amplitudes and coherence to reconstruct detailed brightness distributions.
• It employs methods like aperture synthesis, homodyne, and heterodyne detection to capture both amplitude and phase information for high resolution imaging.
• The approach is pivotal in applications ranging from astronomical imaging and exoplanet detection to metrology and electron interferometry.

Amplitude interferometry denotes a family of interferometric methods in which the measured quantity is the complex field amplitude, the complex degree of coherence, or a directly related quadrature amplitude, rather than intensity alone. In astronomical aperture synthesis it is also called complex-visibility interferometry and measures the complex coherence of the electric field received at separated antennas in order to reconstruct the sky brightness distribution; in synchrotron radiation interferometry it refers to extracting fringe visibility amplitudes from interferograms; in homodyne and heterodyne settings it refers to direct measurement and processing of field amplitudes or beat-note amplitudes and phases; and in phase-space treatments of resonators it means analyzing the full complex transmitted and reflected amplitudes rather than only power coefficients (Marti-Vidal, 2017, Thyagarajan et al., 2 Apr 2025, Zhang et al., 6 Feb 2025, Hechenblaikner, 2013, Monzon et al., 2015). A common contrast throughout these literatures is with intensity interferometry, which measures second-order intensity correlations such as $g^{(2)}$ and typically accesses $|\gamma|^2$ rather than a per-baseline complex phase (Bojer et al., 2021).

1. Conceptual foundations and terminological scope

In astronomical usage, amplitude interferometry is phase-sensitive first-order coherence measurement. It estimates the complex degree of coherence or complex visibility and uses the Van Cittert–Zernike theorem to connect those observables to a source brightness distribution (Marti-Vidal, 2017, Bojer et al., 2021). In the notation of aperture synthesis,

$$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$

where $I(l,m)$ is sky brightness as a function of direction cosines $l$ and $m$, and $V(u,v)$ is the complex visibility at spatial frequency coordinates $(u,v)$ set by the projected baseline in units of wavelength (Marti-Vidal, 2017). In the broader first-order coherence notation,

$$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$

and, for thermal light,

$$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$

This formal distinction underlies the usual contrast between amplitude interferometry and intensity interferometry (Bojer et al., 2021).

The contrast is not merely semantic. Amplitude interferometry correlates voltages or field amplitudes from coherent receivers, returning a complex visibility whose amplitude and phase map directly to spatial Fourier components of the source. Intensity interferometry correlates detected powers or intensity fluctuations, is robust to phase errors, but returns only second-order information and no per-baseline phase (Marti-Vidal, 2017, Thyagarajan et al., 2 Apr 2025). In optical astronomy, where direct phase measurements are not phase-stable across very long baselines, amplitude interferometry therefore requires fringe tracking, phase referencing, and coherent beam combination, whereas intensity interferometry can tolerate much larger physical separations because only arrival-time correlations are required (Bojer et al., 2021).

Outside astronomy, the term is used more broadly. In homodyne-based optical work, amplitude interferometry means revealing interference by measuring and adding optical field amplitudes—specifically homodyne-measured quadratures—rather than by first superposing fields and then measuring intensity (Zhang et al., 6 Feb 2025). In heterodyne metrology, it denotes how temporal fluctuations of beam amplitudes and phases propagate through beat-note detection and how reference-phase subtraction suppresses common-mode disturbances (Hechenblaikner, 2013). This suggests that the most stable cross-domain characterization is not a single hardware architecture but an observable: amplitude interferometry treats complex amplitudes, or observables linearly related to them, as the primary interferometric quantities.

2. Aperture synthesis, Fourier sampling, and image formation

In radio and millimeter astronomy, amplitude interferometry underpins aperture synthesis. For a projected baseline vector $|\gamma|^2$0, the interferometric coordinates are

$|\gamma|^2$1

and in the narrow-field approximation the $|\gamma|^2$2-term can be neglected so that visibilities live on the 2D $|\gamma|^2$3-plane (Marti-Vidal, 2017). The angular resolution is set by the longest projected baseline,

$|\gamma|^2$4

so shorter wavelengths and longer baselines tighten the synthesized beam, whereas longer wavelengths and shorter baselines improve sensitivity to extended structure (Marti-Vidal, 2017).

Earth rotation synthesis converts fixed physical baselines into time-dependent $|\gamma|^2$5-tracks whose shapes depend on array latitude, source declination, and hour-angle coverage. The sampling function $|\gamma|^2$6 determines imaging fidelity and sidelobe structure (Marti-Vidal, 2017). The corresponding dirty image is

$|\gamma|^2$7

where $|\gamma|^2$8 is the Fourier transform of $|\gamma|^2$9, or equivalently

$$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$0

Changing weighting toward natural or more uniform weighting reweights $$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$1 and therefore changes sidelobes, sensitivity, and effective resolution (Marti-Vidal, 2017).

These relations are operational rather than purely formal. Missing short spacings suppress extended emission and bias deconvolution, while minimum-redundancy geometries suppress point-spread-function sidelobes relative to highly redundant layouts (Marti-Vidal, 2017). Primary-beam attenuation multiplies the sky brightness in image space and convolves visibilities in $$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$2-space; heterogeneous arrays alter sensitivity to different spatial scales according to antenna diameters and relative subarray weights (Marti-Vidal, 2017). A practical implication is that array geometry, wavelength, weighting, and observing span are all part of the interferometric measurement itself, not merely post-processing choices.

Sensitivity follows the usual single-visibility noise estimate

$$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$3

so increasing bandwidth, integration time, and baseline count improves signal-to-noise ratio and $$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$4-fidelity (Marti-Vidal, 2017). In this sense amplitude interferometry is a joint problem in coherence measurement and sampling design: the complex visibility is the datum, but imaging performance is governed by where and how often that datum is acquired.

3. Measurement equations, closure invariants, and reconstruction

A central formalism of amplitude interferometry is the measurement equation. In aperture synthesis,

$$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$5

with $$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$6 representing antenna-based amplitude and phase gains and $$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$7 thermal noise (Marti-Vidal, 2017). In synchrotron radiation interferometry, the same structure appears as

$$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$8

with $$V(u,v) = \iint I(l,m)\, e^{-2\pi i (u\,l + v\,m)} \; dl \; dm,$$9 given by the normalized Fourier transform of the source brightness (Thyagarajan et al., 2 Apr 2025). Amplitude errors bias fluxes and effective $I(l,m)$0-weights, producing flux-density errors, sidelobe asymmetries, and limited dynamic range; phase errors smear structure and can create antisymmetric artifacts (Marti-Vidal, 2017).

Closure quantities remove part of this calibration burden. Closure phase on antennas $I(l,m)$1 is

$I(l,m)$2

and for antenna-based phase errors the phases cancel, leaving a source-structure observable (Marti-Vidal, 2017). A common closure amplitude on antennas $I(l,m)$3 is

$I(l,m)$4

which is robust to antenna-based amplitude gains to first order (Marti-Vidal, 2017). In optical synchrotron radiation interferometry, closure amplitudes are written

$I(l,m)$5

so the per-hole gains cancel exactly even when they are time-varying (Thyagarajan et al., 2 Apr 2025).

For a 2D Gaussian synchrotron beam, the degree of coherence is

$I(l,m)$6

and the logarithm of the closure amplitude becomes linear in the parameters $I(l,m)$7 (Thyagarajan et al., 2 Apr 2025). The resulting system,

$I(l,m)$8

admits the inverse-covariance-weighted least-squares solution

$I(l,m)$9

Using a non-redundant 5-hole mask at ALBA, closure-amplitude-based recovery from a single interferogram yielded

$l$0

in agreement within a few percent with non-linear amplitude self-calibration, a rotating two-hole mask method, and LOCO-based optics-inferred profiles (Thyagarajan et al., 2 Apr 2025).

Image reconstruction in astronomical amplitude interferometry typically proceeds through deconvolution. CLEAN iteratively models compact components and restores them with a Gaussian CLEAN beam, reducing sidelobes within the limits imposed by sampling and SNR (Marti-Vidal, 2017). Best practice, as reinforced by interactive aperture-synthesis exercises, is to ensure adequate $l$1-coverage, match baselines and wavelength to the target angular scales, perform phase calibration before amplitude calibration, monitor closure quantities, and compare measured $l$2 versus baseline length to the expected source model in order to identify systematic amplitude biases that limit dynamic range (Marti-Vidal, 2017).

4. Precision limits, superresolution, and the amplitude–intensity comparison

A major contemporary theme is whether amplitude interferometry can evade the classical Rayleigh deterioration in separation estimation. In a quantitative comparison based on two distant thermal point sources, amplitude interferometry was treated as a phase-sensitive, first-order coherence method whose signal and precision scale linearly with the degeneracy parameter $l$3, whereas intensity interferometry scales quadratically with $l$4; both precisions scale quadratically in effective numerical aperture (Bojer et al., 2021). The same study gave representative thermal values $l$5 at $l$6 nm and $l$7 K, and $l$8 at $l$9 nm (Bojer et al., 2021). It also emphasized that at optical wavelengths the effective numerical aperture of amplitude interferometry is practically limited by phase stability to baselines typically less than a few hundred meters, whereas intensity interferometry can use multi-km baselines (Bojer et al., 2021).

For equal-brightness sources, the same comparison reported a constant optimal separation QFI,

$m$0

showing that the Rayleigh limit is a property of the measurement rather than of the quantum state (Bojer et al., 2021). For unequal brightness, it reported a small-separation precision drop $m$1 in the specified benchmark model (Bojer et al., 2021). A later quantum-estimation treatment of two unequal-brightness thermal sources in a two-telescope interferometer reached a different conclusion for the far-field CDC parameterization: the fundamental precision

$m$2

remains constant, establishing superresolution in the sub-Rayleigh regime (Hu et al., 16 Sep 2025). That analysis further decomposed the separation information into amplitude and phase contributions of the CDC and found that the amplitude information is the robust contributor when the geometric position parameter is unknown (Hu et al., 16 Sep 2025). This suggests that superresolution claims are sensitive to the exact source model, nuisance-parameter treatment, and measurement architecture.

The 2025 analysis also compared first-order and second-order schemes directly. With nulling, defined by $m$3 or $m$4 and alignment to $m$5, amplitude interferometry suppresses the brighter source and, for small $m$6, the Fisher information under the nulling strategy matches the separation QFI exactly; in that regime nulling is quantum optimal across all $m$7 (Hu et al., 16 Sep 2025). Intensity interferometry, by contrast, yielded a separation Fisher information that vanishes as $m$8 in the lossless case, so it fails to achieve superresolution in that setting (Hu et al., 16 Sep 2025). However, the same work showed that optical loss reduces amplitude-interferometric performance by $m$9, and that under strong loss and insufficient photon flux, intensity interferometry can become competitive or even superior, especially when ultra-long baselines and many synthesized baselines are available (Hu et al., 16 Sep 2025).

Taken together, these results define a nuanced comparison rather than a universal ranking. Amplitude interferometry offers phase-sensitive access to the CDC, linear scaling in photon number, and, under appropriate conditions, quantum-optimal nulling-based superresolution (Hu et al., 16 Sep 2025). Intensity interferometry forfeits per-baseline phase and pays a $V(u,v)$0 or $V(u,v)$1 penalty, but can compensate through extreme baselines, relaxed phase-stability requirements, and electronic baseline synthesis (Bojer et al., 2021, Hu et al., 16 Sep 2025).

5. High-precision astronomical applications

Beyond conventional imaging, amplitude interferometry has been proposed and demonstrated in several precision astronomical settings. In two-photon amplitude interferometry for relative astrometry, two geographically separated stations each mix light from two sky sources on local beam splitters, exchange only classical time tags, and form cross-station coincidences whose phase depends on $V(u,v)$2 (Stankus et al., 2020). The resulting difference-of-sums observable is

$V(u,v)$3

so the relative opening angle is encoded without a phase-stable optical link between stations (Stankus et al., 2020). For two bright stars of visual magnitude $V(u,v)$4 at $V(u,v)$5, baseline $V(u,v)$6 m, $V(u,v)$7 s, and spectral multiplexing across $V(u,v)$8 disjoint 1 GHz sub-bands, the projected statistical precision is $V(u,v)$9as in a single night (Stankus et al., 2020).

A different proposal uses stellar amplitude interferometry for gravitational-wave detection. Here two space-separated receivers sample coherent portions of the wavefront from a distant star and a host satellite combines the fields to form fringes whose phase is modulated by a passing gravitational wave (Park et al., 2019). The spatial coherence baseline must remain below the source coherence length,

$(u,v)$0

and the interferometer response is

$(u,v)$1

For visible-light spatial coherence interferometry using the Crab pulsar at $(u,v)$2 nm and $(u,v)$3 km, the quoted 5$(u,v)$4 characteristic-strain sensitivity reaches $(u,v)$5 at $(u,v)$6 Hz (Park et al., 2019). The same framework also motivates primordial-black-hole lensing parallax and neutron-star size estimation through intensity-mode spatial correlations (Park et al., 2019).

High-contrast stellar interferometry provides a third application class. For two unequal-brightness thermal sources, nulling amplitude interferometry suppresses the brighter source through destructive interference and is explicitly connected to exoplanet detection (Hu et al., 16 Sep 2025). The quantitative comparison in that work included the Keck Interferometer Nuller, LBTI, VERITAS, and a CTA concept with baselines up to 2 km (Hu et al., 16 Sep 2025). A practical implication is that amplitude interferometry remains the preferred high-contrast strategy when visibility is high, optical loss is tolerable, and phase control is sufficient to maintain the null.

6. Metrology, electron interferometry, and amplitude-space representations

In optical homodyne implementations, amplitude interferometry can be realized without first forcing the interfering fields into a common optical mode. For a field $(u,v)$7 and local oscillator $(u,v)$8, balanced homodyne detection yields

$(u,v)$9

and with two orthogonal modes and two matched local oscillators,

$$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$0

The average current power then becomes

$$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$1

with

$$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$2

On this basis, interference was recovered for orthogonal polarization modes, temporally non-overlapping pulses, and path differences far beyond the source coherence length, with the recovery controlled by detector response overlap or by delayed photocurrent addition rather than by pre-detection mode projection (Zhang et al., 6 Feb 2025).

Single-electron amplitude interferometry extends the concept to mesoscopic electronic systems. In an electronic Fabry–Perot interferometer at integer quantum Hall filling $$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$3, a time-dependent gate voltage on one arm imprints a phase on a single-electron wavepacket, and fitting the interferometer transmission versus a dc gate offset yields both the instantaneous phase shift $$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$4 and the contrast $$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$5 (Bartolomei et al., 2024). The voltage is reconstructed through

$$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$6

while contrast suppression tracks phase fluctuations (Bartolomei et al., 2024). The demonstrated performance was a time resolution of a few tens of picoseconds and a voltage sensitivity of approximately $$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$7, corresponding to approximately $$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$8 photons at $$\gamma(\mathbf{r}_1,\mathbf{r}_2) = \frac{\langle E^*(\mathbf{r}_1)\, E(\mathbf{r}_2)\rangle}{\sqrt{\langle |E(\mathbf{r}_1)|^2\rangle \langle |E(\mathbf{r}_2)|^2\rangle}},$$9 GHz (Bartolomei et al., 2024).

Free-electron Ramsey-type interferometry uses two sequential coherent light–electron interactions, a known reference and an unknown sample nearfield, to form an effective coupling

$$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$0

so that the energy-sideband populations become phase-scanned interferometric observables (Bucher et al., 2023). In the weak-sample limit, the gain-side signal is linear in $$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$1,

$$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$2

rather than quadratic as in conventional weak-field PINEM (Bucher et al., 2023). The optimal reference strength satisfies

$$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$3

giving $$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$4, and the reported sensitivity gain is almost two orders of magnitude in minimal detectable interaction strength relative to conventional PINEM (Bucher et al., 2023).

In precision heterodyne metrology, amplitude interferometry appears in the relation between beat-note amplitude, demodulated phase noise, and common-mode rejection. For the differential phase obtained by subtracting a reference interferometer, the residual coupling of common-mode amplitude noise is proportional to $$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$5, while the residual coupling of common-mode phase noise is proportional to $$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$6 (Hechenblaikner, 2013). Complete suppression of high-frequency common-mode amplitude noise occurs at $$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$7, whereas complete suppression of high-frequency common-mode phase noise occurs at $$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$8 (Hechenblaikner, 2013). In LISA-representative two-beam heterodyne interferometers, beam tilt reduces the heterodyne efficiency approximately as

$$g^{(2)}(\mathbf{r}_1,\mathbf{r}_2) = 1 + |\gamma(\mathbf{r}_1,\mathbf{r}_2)|^2.$$9

with $|\gamma|^2$00 the curvature-mismatch parameter, and the resulting phase-noise terms scale approximately as $|\gamma|^2$01 (Pizzella et al., 20 Feb 2026). The same analysis identified curvature mismatch as the key parameter driving amplitude asymmetry across quadrant segments and excess phase-noise coupling under tilt (Pizzella et al., 20 Feb 2026).

A geometrically distinct but conceptually related formulation appears in the Fabry–Perot interferometer. For a symmetric, lossless cavity,

$|\gamma|^2$02

and the complex reflected and transmitted amplitudes trace specific loci in the complex plane as the optical path is varied (Monzon et al., 2015). The reflected amplitude satisfies

$|\gamma|^2$03

so it lies on a circle, while the transmitted amplitude lies on a hippopede,

$|\gamma|^2$04

(Monzon et al., 2015). This phase-space treatment makes explicit that amplitude interferometry retains information—mirror parameters, phase response, losses, and alignment—that is invisible in intensity-only descriptions.

Across these implementations, the unifying theme is consistent. Amplitude interferometry measures observables that preserve phase-sensitive or complex-amplitude information: complex visibilities in aperture synthesis, gain-invariant visibility ratios in closure analysis, quadratures in homodyne detection, beat amplitudes in heterodyne metrology, electron-sideband couplings in free-electron interferometry, and full complex reflection or transmission coefficients in resonant cavities. The precise experimental architecture varies widely, but the governing principle is the same: interference is analyzed at the level of amplitudes rather than reduced at the outset to intensities alone.