Interferometric Microlensing Essentials

• Interferometric microlensing is a technique that uses angular observables—such as image separation, centroid shifts, and visibility phases—to directly probe the geometry of microlensing events.
• By measuring the angular Einstein scale and combining it with microlensing parallax, the method resolves degeneracies in lens mass, distance, and relative proper motion.
• Demonstrated in events like Gaia19bld and TCP J05074264+2447555, dual-field interferometry enables precise sub-milliarcsecond measurements essential for physical lens characterization.

Searching arXiv for recent and foundational papers on interferometric microlensing to ground the article. Interferometric microlensing is the use of interferometric, astrometric, or other directly angular observables to probe the image geometry of a microlensing event, rather than only its unresolved total magnification. In the classical single-lens problem, photometry constrains mainly the temporal parameters $t_0$, $u_0$, and $t_E$, whereas interferometric microlensing accesses the angular Einstein scale, the separation and orientation of the microimages, their flux ratio, and, in some regimes, visibility phase or closure phase. When these angular observables are combined with microlensing parallax, they break the standard mass–distance–proper-motion degeneracy and yield physical lens parameters such as mass, distance, and transverse motion (Lee, 2017, Cassan et al., 2016, Mroz et al., 2024).

1. Fundamental geometry and observables

In a point-lens event, the unresolved photometric magnification is

$$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$

The lens forms two images at

$$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$

so their instantaneous separation is

$\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$

These relations define the angular structure that interferometry attempts to measure directly, rather than infer only through the scalar light curve (Lee, 2017).

This distinction is central. In seeing-limited photometry the two microimages simply sum to produce $A(t)$, but in an interferometer they behave like a time-variable binary whose separation, orientation, and flux ratio are set by the microlensing geometry. For unresolved-image astrometry, the relevant observable is the centroid shift,

$$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$

which traces an ellipse on the sky and reaches a maximum of about $0.35\,\theta_E$ at $u=\sqrt{2}$ (Lee, 2017). This suggests that “interferometric microlensing” is best understood as a broader angular-measurement program whose observables include resolved image splitting, centroid motion, and visibility-based signatures.

A concrete single-lens example is Gaia19bld. Its photometric solution gave $u_0$0, so the two microimages were separated by roughly $u_0$1 mas near peak, a scale large enough to affect long-baseline near-infrared interferometric observables. The event was unusually favorable because it combined a very bright source, very high magnification, and sufficiently accurate peak prediction to permit VLTI scheduling (Rybicki et al., 2021).

2. Degeneracy breaking and physical inference

The basic microlensing scale is

$u_0$2

with $u_0$3. Photometry alone therefore mixes lens mass $u_0$4, relative parallax $u_0$5, and relative proper motion $u_0$6. The standard way to recover a physical solution is to measure both $u_0$7 and the microlensing parallax $u_0$8, which gives

$u_0$9

This is the core degeneracy-breaking relation to which interferometric microlensing contributes (Lee, 2017).

Several routes to $t_E$0 exist. Finite-source effects yield

$t_E$1

if the source angular radius $t_E$2 is known. High-resolution imaging at late times yields

$t_E$3

Astrometric trajectories give $t_E$4 from the scale of the centroid ellipse. Interferometry is distinctive because it can, in favorable cases, constrain not only the scalar Einstein radius but the vector Einstein radius,

$t_E$5

thereby encoding both angular scale and sky orientation (Cassan et al., 2016).

That directional information is often as important as the scale itself. In TCP J05074264+2447555, VLTI/GRAVITY supplied $t_E$6 directly and also constrained the direction of $t_E$7, hence the direction of $t_E$8. Combined with Spitzer plus ground-based parallax, this yielded $t_E$9 and $$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$0, in what was described as the first microlensing event whose lens mass was unambiguously measured by interferometry plus satellite parallax (Zang et al., 2019).

The same logic underlies more recent dual-field results. In OGLE-2023-BLG-0061 / KMT-2023-BLG-0496, GRAVITY Wide directly resolved the microlensed image pair and measured $$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$1. Combined with $$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$2 from the light curve, this gave $$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$3 and $$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$4 (Mroz et al., 2024).

3. Interferometric formalism and visibility modeling

The central observable of long-baseline interferometry is the normalized complex visibility,

$$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$5

For a point-source microlensing event with images at $$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$6 and magnifications $$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$7, the visibility becomes a phasor sum,

$$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$8

Image separation therefore controls fringe frequency, image orientation controls the dependence on baseline position angle, and image flux ratio controls fringe contrast (Cassan et al., 2016).

A particularly influential conceptual step is the “microlensing $$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}}, \qquad u(t)=\sqrt{\left(\frac{t-t_0}{t_E}\right)^2+u_0^2}.$$9 plane,” in which interferometric baselines are expressed in units conjugate to $$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$0. In this formulation the sampled Fourier coordinates depend not only on the projected baseline and wavelength but also on the vector Einstein radius. Interferometric fitting then constrains $$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$1 and $$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$2 directly, which is advantageous because component-wise combinations with $$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$3 can be more informative than scalar combinations alone (Cassan et al., 2016).

With three or more telescopes one also measures closure phase,

$$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$4

which is sensitive to asymmetry in the image configuration and is robust against telescope-based phase errors. In the dual-field GRAVITY Wide analysis of OGLE-2023-BLG-0061 / KMT-2023-BLG-0496, the two microlensed images were modeled as an unresolved binary with

$$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$5

where $$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$6 is the minor-to-major image flux ratio. The Einstein scale then followed from

$$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$7

so a single interferometric epoch can, in principle, determine $$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$8 directly from the resolved image geometry (Mroz et al., 2024).

Finite source structure complicates this picture. For a uniformly bright source, the exact visibility is the Fourier transform over the full lensed image domain, and for single lenses a robust exact integration scheme can be written in polar coordinates. A computationally important approximation is the “thin-arcs approximation,” developed for medium- to high-magnification single-lens events near peak. It treats the images as narrow arcs whose thickness is unresolved but whose opening angle is retained. The approximation was reported to run six to ten times faster than the exact calculation and to provide accurate results for a large fraction of potential observational targets (Cassan, 2021).

4. Optical and infrared demonstrations

The modern empirical development of interferometric microlensing has been driven by a small number of exceptionally favorable events. These events established, in sequence, the feasibility of measuring $$\theta_\pm=\frac{1}{2}\left(u\pm\sqrt{u^2+4}\right)\theta_E,$$9, constraining the direction of motion, breaking the $\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$0 degeneracy, and combining interferometry with satellite or annual parallax to obtain lens masses.

Event	Interferometric role	Physical inference
TCP J05074264+2447555	VLTI/GRAVITY measured $\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$1 and motion direction	$\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$2, $\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$3
Gaia19bld	VLTI/PIONIER measured image geometry during peak	$\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$4 agrees with photometric value
OGLE-2023-BLG-0061 / KMT-2023-BLG-0496	GRAVITY Wide resolved the two images directly	$\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$5, $\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$6

TCP J05074264+2447555 showed the full interferometry-plus-satellite-parallax architecture. The preferred luminous-lens solution adopted $\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$7 and $\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$8, implying $\Delta\theta_{+-}=\theta_E\sqrt{u^2+4}.$9 and a heliocentric relative proper motion $A(t)$0. The blend was found to be consistent with lens light, and the large proper motion implied that source and lens could later be resolved with AO imaging (Zang et al., 2019).

Gaia19bld demonstrated a different mode of synergy. It was primarily a photometric and spectroscopic mass measurement, with a preferred solution

$A(t)$1

$A(t)$2

leading to

$A(t)$3

Here interferometry was not the sole route to mass. Instead, VLTI/PIONIER provided an independent measurement of the angular Einstein radius, a directional constraint on the relative proper motion, and a resolution of the $A(t)$4 sign degeneracy in favor of $A(t)$5. The paper explicitly presented this as a demonstration that interferometry can become a mass-measurement channel for objects that would otherwise remain undetectable, including stellar-mass black holes (Rybicki et al., 2021).

OGLE-2023-BLG-0061 / KMT-2023-BLG-0496 marked the dual-field transition. The source had baseline $A(t)$6, too faint for conventional on-axis interferometric microlensing, but a nearby star at separation $A(t)$7, with $A(t)$8 and $A(t)$9, served as fringe-tracking and AO guide star. GRAVITY Wide measured image separations around $$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$0 mas in two epochs and obtained a subpercent Einstein-radius determination,

$$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$1

with fractional precision $$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$2. The paper described this as the first successful microlensing observation with GRAVITY Wide and argued that dual-field interferometry increases the pool of interferometrically accessible microlensing events by two orders of magnitude (Mroz et al., 2024).

5. Wave-optical, time-delay, and structured-source variants

Although long-baseline optical/IR interferometry is the most direct realization of the subject, the same microlensing image geometry also appears in other interferometric regimes. In diffractive microlensing, when the observing wavelength approaches the Schwarzschild scale of the lens, the relevant observable becomes the phase gradient of the unresolved wavefield. For a point lens the wave-optics amplification is written as

$$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$3

and the centroid is

$$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$4

A simple bound found numerically is

$$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$5

This framework predicts oscillatory centroid displacements and chromatic astrometric signatures, with explicit discussion of radio facilities such as the proposed SKA (Heyl, 2010).

A different coherence-based route is the use of second-order optical coherence. For a point-mass microlens, the two unresolved images have a relative arrival-time delay

$$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$6

or equivalently a form determined by the observed total magnification $$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$7. The proposal in “Microlensing masses via photon bunching” is that narrow-band unresolved lensed light should show Hanbury Brown–Twiss side peaks at $$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$8, with

$$\delta\boldsymbol{\theta}_C=\frac{\boldsymbol{u}}{u^2+2}\,\theta_E,$$9

The paper argued that a single delay measurement would provide the lens mass without degeneracies, but also concluded that the required photon statistics appear infeasible at present except possibly for very bright, compact early-type stars observed with 30 m-class telescopes (Saha, 2019).

Fast transients provide yet another limit. For FRBs, the source–lens geometry is effectively frozen during the burst, so the observable is a superposition of pulses from individual microimages rather than a standard months-long microlensing light curve. The paper “FRBs Lensed by Point Masses II” found that relative time delays between pulses can reach $0.35\,\theta_E$0–$0.35\,\theta_E$1 ms for stellar-mass lenses, making temporally resolved multi-peaked bursts possible with current facilities (Chen et al., 2021). In gravitational waves, microlenses embedded in a macromodel near a critical curve can generate delays of order milliseconds and thereby produce interference distortions in LIGO/Virgo-band strains, especially for negative-parity macroimages (Diego et al., 2019).

The concept also extends to resolved, structured sources. In “Microlensing Black Hole Shadows,” each point on a circular true shadow boundary is differentially lensed, so the observable is a change in the center, mean radius, and asymmetry of the shadow. For Sgr A$0.35\,\theta_E$2, microlensing can create an asymmetry of up to approximately $0.35\,\theta_E$3 and enhance the size by $0.35\,\theta_E$4 of the true shadow. The paper concluded that terrestrial EHT baselines lack the required resolution, but future Moon or L$0.35\,\theta_E$5 baselines could potentially detect such events (Verma et al., 2023).

6. Selection effects, limitations, and observational outlook

Interferometric microlensing is intrinsically selection-limited. The most favorable events are bright, high-magnification, and predictable near peak; they must also have image separations of order $0.35\,\theta_E$6 large enough to perturb the interferometric transfer function. Simulations for VLTI/PIONIER suggested that two interferometric observations are sufficient to obtain good constraints on the vector Einstein radius, while three are better for robustness, with the recommended strategy being epochs close to peak and spread over about $0.35\,\theta_E$7 hours (Cassan et al., 2016).

Event-rate estimates have therefore been central to the field. Using OGLE-IV EWS alerts from 2011–2014, about $0.35\,\theta_E$8 events, one study found that the first potential target appears at about $0.35\,\theta_E$9, that $u=\sqrt{2}$0 events in the sample had $u=\sqrt{2}$1 at peak, and that this corresponds to about $u=\sqrt{2}$2–$u=\sqrt{2}$3 events per year. The same analysis argued that increasing the limiting magnitude by one magnitude, to $u=\sqrt{2}$4, would increase the number of accessible targets by about an order of magnitude (Cassan et al., 2016).

Dual-field interferometry substantially changes these constraints. In GRAVITY Wide, a nearby bright reference star within roughly $u=\sqrt{2}$5 performs fringe tracking while the fainter microlensed target is observed simultaneously. In dense bulge fields the probability of finding a suitable fringe-tracking star with $u=\sqrt{2}$6 within $u=\sqrt{2}$7 was stated to be about $u=\sqrt{2}$8–$u=\sqrt{2}$9. Practical target selection prioritized events near or past maximum brightness, image contrast ratio smaller than $u_0$00, corresponding to $u_0$01 or minimum event amplitude $u_0$02, and long timescales $u_0$03 because these are more compatible with massive lenses. For UTs, the adopted practical limits were about $u_0$04 and $u_0$05, although in practice the selection was often conservative, $u_0$06 (Mroz et al., 2024).

The principal conceptual limitation is that interferometry alone does not usually determine the lens mass. It can measure or constrain $u_0$07, image geometry, and the direction of relative proper motion, but without $u_0$08 the absolute mass scale remains incomplete. Gaia19bld makes this distinction explicit: the mass measurement fundamentally required photometric parallax and source-size information, while interferometry provided an independent geometric check and resolved the sign of $u_0$09 (Rybicki et al., 2021). The same pattern appeared in TCP J05074264+2447555, where the uncertainty budget was dominated by the parallax amplitude rather than the interferometric $u_0$10 measurement (Zang et al., 2019).

Instrument-specific systematics remain important. In the GRAVITY Wide demonstration, correlated/systematic closure-phase errors were reported to reach $u_0$11, motivating an extra noise term and bootstrap resampling over interferograms (Mroz et al., 2024). More generally, blending, finite-source structure, guide-star availability, weather, sparse $u_0$12 coverage, and real-time scheduling all affect performance (Cassan et al., 2016, Cassan, 2021).

The broader outlook is therefore bifurcated. In optical/IR stellar microlensing, dual-field interferometry moves the subject from a handful of exceptional demonstrations toward a realistic follow-up program for long-timescale events and candidate isolated remnants (Mroz et al., 2024). In wave-optical and structured-source variants, the relevant observables change—phase-gradient astrometry, delayed intensity correlations, frequency-domain fringes, or shadow-shape distortions—but the unifying principle remains the same: microlensing creates multiple image paths whose angular, temporal, or coherence structure can be measured interferometrically, and these angular observables are the route from a light curve to a physical lens (Lee, 2017).