Wave-Optics and Interference Tests

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When a wave’s period is comparable to the lens-induced delay, separate geometric images cease to be the right description. Diffraction and interference produce a complex, frequency-dependent amplification. That structure can reveal compact or structured lenses and can test whether the assumed propagation equation is adequate, but waveform error and detector noise can mimic a fringe over a finite band. The status assessment is current through 2026-08-19.

Required background. Geodesics and Jacobi Fields supplies geometric-optics rays and stationary paths; Probability and Statistical Inference supplies model comparison, selection, and injection-calibrated significance.

Helpful background. Weak-Field Gravity supplies the lensing time-delay potential; Relativistic Cosmology supplies redshifted lens masses and propagation distances.

Diffraction integral and its limits

With $h(t)=\int_0^\infty\tilde h(f)e^{-2\pi i f t}df+\mathrm{c.c.}$ , propagation through a thin lens gives

\tilde h_L(f)=F(f,\boldsymbol y)\tilde h_0(f), \qquad F(w,\boldsymbol y)= \frac{w}{2\pi i} \int d^2\boldsymbol x\, \exp\!\left[iwT(\boldsymbol x,\boldsymbol y)\right].

The two-dimensional angles $\boldsymbol x$ and $\boldsymbol y$ are image and source positions scaled by the Einstein angle. For a point mass, a fully specified dimensionless delay is

T(\boldsymbol x,\boldsymbol y) =\frac12\lvert\boldsymbol x-\boldsymbol y\rvert^2 -\ln\lvert\boldsymbol x\rvert-T_{\min}(\boldsymbol y),

where $T_{\min}$ is chosen so the earliest stationary ray has $T=0$ . Adding a constant to $T$ only shifts the common arrival time, a fitted source parameter; relative delays and interference remain unchanged. For a point mass,

w=8\pi G M_L(1+z_L)f.

Here $f$ is detector-frame frequency, $M_L$ is physical lens mass, and $z_L$ is lens redshift; SI units restore a factor $c^{-3}$ . The scale is $w\simeq1.24(M_L(1+z_L)/100M_\odot)(f/100\,\mathrm{Hz})$ , illustrating why lens mass and observing band must be stated together. The oscillatory integral is defined with a vanishing positive convergence regulator. Without a lens, integrating $\exp[iw\lvert\boldsymbol x-\boldsymbol y\rvert^2/2]$ gives $2\pi i/w$ , which checks the prefactor and $F=1$ normalization.

For $w\ll1$ at fixed source position, diffraction makes $F\to1$ . For $w\gg1$ away from a caustic, stationary phase yields the geometric-optics sum

F(f)\simeq \sum_j\lvert\mu_j\rvert^{1/2} \exp\!\left(2\pi i f\Delta t_j-i\pi n_j\right).

Here $\mu_j$ is signed image magnification, $\Delta t_j$ is its detector-frame delay, and $n_j=0,\tfrac12,1$ for a minimum, saddle, or maximum of the arrival-time surface. For two images with a relative delay $\Delta t$ and Morse-index difference $\Delta n$ ,

\lvert F\rvert^2 =\lvert\mu_1\rvert+\lvert\mu_2\rvert +2\sqrt{\lvert\mu_1\mu_2\rvert} \cos(2\pi f\Delta t-\pi\Delta n).

Thus the fringe spacing is $\Delta f=1/\lvert\Delta t\rvert$ , while its contrast depends on both image amplitudes. Near a caustic, stationary points merge and the image approximation can fail even at large $w$ ; the diffraction calculation remains finite. Takahashi and Nakamura give the amplification and its geometric limit in Takahashi and Nakamura 2003, §II, Eqs. (11)–(18).

Wave optics is coherent: averaging over a frequency bin wider than a fringe, a source larger than the lens coherence scale, or an unresolved distribution of delays suppresses interference. A macromodel plus stars requires a magnification map or diffraction calculation, not multiplication of isolated point-lens factors without checking coupling. The calculation must declare the scalar or tensor propagation equation, polarization, lens plane, boundary condition, branch choices, and whether frequency is measured in the detector or source frame.

From amplification to a measured residual

The data prediction passes $F(f)\tilde h_0(f;\lambda)$ through each detector response and calibration model. A likelihood over strain segments must include power-spectral-density uncertainty, waveform parameters $\lambda$ , calibration, glitches, and lens variables. Selection depends on both unlensed source properties and lens magnification: a distorted signal may enter or leave the catalog, and a search bank built from unlensed templates can have lens-dependent efficiency.

Several degeneracies are diagnostic. A smooth phase correction can resemble a shift in masses or spins. A short noise transient can resemble one oscillation of a beat pattern. Precession and higher harmonics produce physical modulation tied to detector orientation and binary geometry. A real lens transfer function is common to all detectors after antenna response and predicts correlated amplitude and phase across the full usable band. Alternate waveform families, off-source noise injections, and detector-by-detector residuals are therefore indispensable.

The full-O3 LVK search tested point-mass microlensing and Morse-phase distortions using calibrated event strain and found no compelling evidence in its declared sample LIGO–Virgo–KAGRA Collaboration 2024, §§II, IV–V and Appendix B. Its products are preserved in the LVK 2023 official data release. At the cutoff, the submitted O4a analysis also searches isolated point-mass distortions but does not turn one investigated outlier into an established lens LIGO–Virgo–KAGRA Collaboration 2025, official DCC release P2500419-v10. That draft discloses a likelihood-normalization correction affecting parts of the underlying parameter estimation, with analysis-specific mitigation and further updates anticipated LIGO–Virgo–KAGRA Collaboration 2025, §3. A current GWTC-5.0 event list is only an input to a new search, not a new wave-optics conclusion.

Results and limitations through 2026-08-19

Evidence or position	Result supported	Limitation or scope
Diffraction integral and point-lens solutions	Predict the transition from $F\to1$ to geometric images and frequency fringes	Thin-lens, coherence, propagation-equation, and lens-profile assumptions apply
Peer-reviewed full-O3 injection-calibrated search	No compelling point-lens wave-optics evidence in its tested events	Sensitivity, source waveforms, glitch distribution, and point-lens prior bound the null
Submitted O4a distortion search	Extends the empirical search to newer strain and retains an outlier for study	Submitted status, disclosed likelihood correction, and a single lens family prevent a settled interpretation

The conservative position treats present searches as null constraints within particular lens masses and waveforms. A competing position may attribute a localized strain modulation to diffraction, but noise or waveform alternatives count as evidence against that claim unless the coupled amplitude–phase pattern repeats across detectors and analysis choices. The absence of fringes does not exclude lenses whose delays lie outside the coherent band.

The adversarial check injects point, extended, and macro-plus-micro lenses into real off-source noise while recovering them with several waveform families and unlensed-glitch alternatives. Coverage, false-alarm probability, and detection efficiency must remain calibrated across $w$ and impact parameter. The physical boundary remains unresolved and model-dependent if only one oscillation is visible or the inferred lens scale follows a waveform-model change.

What could change the conclusion. The interpretation could change after a superseding calibrated strain or dedicated lensing release, a validated non-point lens search, or an injection study showing that source-waveform and glitch alternatives materially alter wave-optics detection efficiency.

Exercise

A point lens has $w=0.2$ at $50\,\mathrm{Hz}$ . What are $w$ at $25\,\mathrm{Hz}$ and $500\,\mathrm{Hz}$ , and which qualitative regimes do they probe?

Solution

Because $w$ is proportional to frequency, the values are $0.1$ and $2$ . At $25\,\mathrm{Hz}$ the wave is strongly diffractive and the amplification tends toward unity, so the lens is difficult to recognize. At $500\,\mathrm{Hz}$ the system enters the interference transition, where coupled amplitude and phase structure can appear. It is not yet safe to replace the diffraction integral by a geometric-image sum without checking convergence at the actual impact parameter.

References

LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2023), “Search for Gravitational-Lensing Signatures in the Full Third Observing Run of the LIGO–Virgo Network,” data release, version 1.0.0, 17 April 2023. Zenodo. DOI.
LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2024), “Search for Gravitational-Lensing Signatures in the Full Third Observing Run of the LIGO–Virgo Network,” Astrophysical Journal 970, 191. DOI.
LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2025), “GWTC-4.0: Searches for Gravitational-Wave Lensing Signatures,” submitted manuscript P2500419-v10. Official DCC record.
Takahashi, R., and Nakamura, T. (2003), “Wave Effects in the Gravitational Lensing of Gravitational Waves from Chirping Binaries,” Astrophysical Journal 595, 1039–1051. DOI.