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Wave optics

When the wavelength is comparable to the gravitational radius of the lens, geometric optics fails. Diffraction and interference then appear as frequency-dependent modulations of the wave.

Diffraction patterns carry information about the objects they probe. Here simulated monochromatic light is shot through a single opening whose size and shape change (top right). The pattern it casts (in full at left, and as a cut through the middle at bottom right) follows the size of that opening: widen it and the pattern tightens, stretch it one way and the pattern stretches the other.

Beyond magnification: chromatic distortions

In most lensing situations, propagation can be understood in terms of geometric optics: each path between source and observer behaves as a ray, a well-defined trajectory that is deflected, magnified and delayed by the lenses along the way. A lens can split a source into multiple images. Resolving those images is another matter: most astronomical sources are extended and fairly static, so only configurations with large angular separations or long time delays can be told apart, and what is left in most cases is an overall magnification.

Coherent transients change that. Gravitational waves and fast radio bursts come from extremely compact sources, so they are far less limited by their own finite size, and they vary fast enough that interference between images separated by very small time delays becomes observable.

Wave optics is the right description whenever the lensing time delays are not long compared with the period of the signal, and its effects are most pronounced when the two are comparable, equivalently when the wavelength approaches the gravitational radius of the lens, λ ∼ GM/c². Diffraction there cannot be read as a finite number of well-defined images at all. The source is smeared across the lens plane instead.

Nothing switches off below that. Diffraction continues to every longer wavelength with its imprint progressively diluted, until at λ ≫ GM/c² the amplification factor tends to unity and the lens leaves no measurable trace. The one real boundary sits at the other end, where the delays grow long compared with the period and the description collapses back to a finite set of images interfering in the ordinary way. The figure below sweeps the whole range, together with the amplification factor in the frequency domain, its time-domain counterpart, and the waveform of a typical gravitational wave.[1]

What the detector records is then the unlensed signal multiplied by an amplification factor: a complex, frequency-dependent function that encodes the lens. Its oscillations are the interference between paths. Their spacing measures time delays, and their envelope measures how the mass is distributed. This quantity fully specifies wave-optics lensing.

The same lens across the whole frequency range, in both domains at once. Nothing about the lens changes across the sweep: what changes is the ratio between its time-delay scale and the period of the wave. Wave optics describes all of it but the last step. While the period is long the imprint is too weak to measure, and as the ratio grows the amplification factor develops the oscillations that carry the lens. Once the delays are long compared with the period, the images separate and interfere in the ordinary geometric-optics way.

Which lenses can a wave feel

Diffraction sets its own scale, and for an isolated lens that scale is the ratio of the signal's wavelength to the lens's gravitational radius. This fixes a characteristic redshifted lens mass for each band: the mass whose time delay matches one period of the wave. It is of order 107–108 M☉ for LISA at millihertz frequencies, tens to hundreds of M☉ for LIGO–Virgo–KAGRA, and around 10−5 M☉ (a few tens of Earth masses) for fast radio bursts at gigahertz frequencies. An isolated lens far below its band's mass is invisible. One far above it is back in the geometric-optics regime.

These effects grow as the alignment between source and lens tightens, and near caustics, where the geometric-optics magnification diverges. Wave optics is what keeps a prediction finite and continuous across a caustic. The animation below crosses one.

Two consequences follow, and both matter for coherent sources. A single signal probes a range of scales at once, because it spans a range of frequencies. And the scales it reaches are set by the wavelength rather than by spatial resolution, which is how a gravitational wave can be sensitive to objects far too small and too dark to image: the subject of dark matter.

That mass scale assumes a single, isolated lens. Where matter is spread along the line of sight the relevant quantity is a length rather than a mass: the Fresnel scale, the transverse distance over which the paths reaching the observer stay in phase. It grows as the square root of the wavelength, so a lower-frequency wave averages over a wider region, and structure smaller than the Fresnel scale is smoothed over rather than resolved. It is what sets the reach of a wave when there is no single lens to point at.

Lensing also cannot always be reduced to an isolated object. Collective effects matter: a stellar field of thousands of stars leaves a far stronger diffraction signature than any single star or compact object would.[2] That is the subject of microlensing.

The impact parameter falls from 2.0 to 0.1 Einstein radii and the source crosses a caustic mid-sweep. As the alignment tightens, the amplification factor acquires the oscillations that are the signature of diffraction.

Wave-optics lens features

In the wave-optics regime, all paths connecting the source and the observer contribute, but some do more than others: at high frequencies, the signal propagates through a discrete set of trajectories, which are associated with the images in the geometric-optics approximation. This is because of Fermat's principle, stating that all but the stationary points of the time-delay surface cause destructive interference. This picture becomes incomplete at lower frequencies, when diffraction becomes important.

A coherent lensed signal can be described as the convolution of the unlensed signal with a kernel, or response function, h(t) = ∫ dτ G(τ) h0(t − τ) with G = (2π)−1 dI/dτ.[3] This kernel can be evaluated by sampling the lens plane: curves of constant arrival time, the isochrones of the Fermat potential, expand outward from the first image, and what they enclose as they grow is the response itself. Images correspond to the points where families of contours are created or destroyed. This translates into discontinuities or singularities in the response function. In addition to these features, there are additional peaks in the response function that have purely diffractive origin and can only be observed in a certain frequency range.

These peaks are wave-optics lens features (WOLFs): the smooth counterparts of geometric images. Each gravitational lens felt by the signal typically produces a WOLF,[4] whose shape contains information about the lens total mass and its spatial profile: the stochastic signal from many lenses along the signal's path provides means to constrain dark matter,[5] as forecast for LISA by Choi, Urrutia & Zumalacárregui. WOLFs become particularly strong in configurations that are close to forming additional images, where the geometric magnification becomes very large:[6] they may therefore be crucial to discover distant and highly magnified sources.

Building the arrival-time response of the best-fitting lens for GW231123, a point mass in an external potential. Left: isochrones of the Fermat potential sweeping the lens plane, colored by the local contribution to the response function. Right: the total response I(τ) (top), the wave-optics part after images are removed (bottom). The movie stops when the two images and the WOLF form.

Not only gravity: plasma and dispersion

A gravitational lens is not the only thing that can cause frequency-dependent distortions. Ionized gas does it too: a radio pulse crossing a cloud of free electrons is refracted and dispersed, and where the gas is structured on small enough scales the same wave-optics treatment applies. This is plasma lensing, and for fast radio bursts it is not a correction but a probe in its own right[7] : a way of weighing ionized matter that is otherwise close to invisible.

The physics differs in one instructive way. Gravitational lensing is achromatic in the geometric limit, so all its frequency dependence comes from diffraction. Plasma lensing is chromatic to begin with, because the refractive index of a plasma depends on frequency, much as it does in the water droplets that make a rainbow. The two therefore leave distinguishable signatures, and a lens that contains both can be taken apart.

Radio sources also force the treatment onto more than one lens plane. A fast radio burst is not only lensed somewhere along the way: it can be plasma-lensed in its host galaxy and again in the Milky Way, and those planes have to be handled together.

Dispersion of a different kind appears again when gravity itself is modified: a frequency- and polarization-dependent propagation speed for gravitational waves would be a smoking gun for departures from Einstein's theory on cosmological scales, with implications for dark-energy dynamics.

Algorithms for realistic lenses

Computing wave-optics lensing is numerically hard. The amplification factor is an integral of a highly oscillatory function whose convergence is poorly behaved, and only a handful of idealized lenses (a point mass among them) give a closed-form expression for it. Realistic configurations, such as fields of stars and remnants, need better methods, and those methods have to be fast enough to be used in data analysis.[8]

Making wave-optics lensing tractable for general matter distributions is a major goal of the project, and the foundational work has crystallized into the GLoW code. Extending it to populations of lenses, and to the multi-plane geometry that plasma lensing needs, is where the methodological effort goes now.

References

  1. C. Leung et al., Wave mechanics, interference, and decoherence in strong gravitational lensing (2023)
  2. X. Shan, G. Li, X. Chen et al., Wave effect of gravitational waves intersected with a microlens field II: An adaptive hierarchical tree algorithm and population study Sci. China Phys. Mech. Astron. 68, 219512 (2025)
  3. S. Savastano, G. Tambalo, H. Villarrubia-Rojo, M. Zumalacárregui, Weakly lensed gravitational waves: probing cosmic structures with wave-optics features Phys. Rev. D 108, 103532 (2023)
  4. M. Zumalacárregui, Lens Stochastic Diffraction: A Signature of Compact Objects in Gravitational-Wave Data arXiv preprint (2024)
  5. G. Tambalo, M. Zumalacárregui, L. Dai, M. H.-Y. Cheung, Gravitational wave lensing as a probe of halo properties and dark matter Phys. Rev. D 108, 103529 (2023)
  6. J. M. Ezquiaga, R. K. L. Lo, L. Vujeva, Diffraction around caustics in gravitational wave lensing Phys. Rev. D 112, 043544 (2025)
  7. J. M. Cordes et al., Lensing of fast radio bursts by plasma structures in host galaxies Astrophys. J. 842, 35 (2017)
  8. H. Villarrubia-Rojo, S. Savastano, M. Zumalacárregui et al., GLoW: novel methods for wave-optics phenomena in gravitational lensing Phys. Rev. D 111, 103539 (2025)

Where this is done

Work packages
Microlensing diffraction · Plasma lensing
Papers
Effective description of lensed gravitational waves diffracted by stellar fields
Microlensing → All topics