Convergence and efficiency of different methods to compute the diffraction integral for gravitational lensing of gravitational waves

Gravitational Faraday rotation of gravitational waves by a Kerr black hole

“…The lensed and unlensed waveform is denoted by subscript 'len' and 'unlen'. The complex transition factor F (f ) is calculated from Kirchhoff integral [37,38,41,45,46]. The amplitude and phase factor of F (f ), |F | and arg(F ), describe the amplitude enhance and phase shift of lensing GW with relative to that of the unlensed one.…”

“…With the discovery of GW events generated by the mergers of compact binaries [28][29][30][31][32], the gravitationallylensing effects of GW signals attract more and more attention in both observation [33][34][35][36] and theory [22,27,37,38]. In addition to the lensing effects such as multiple images, magnification, shift of the GW phase, interference and diffraction [37][38][39][40][41][42][43][44][45], the polarization plane of GW can be dragged by the angular momentum of the lens, i.e. the GFR of GWs [10,27].…”

Wave-optical Effects in the Microlensing of Continuous Gravitational Waves by Star Clusters

“…Standard numerical methods that involve finite cutoffs, for example, fail to return an adequate evaluation because, depending on whether one truncates at a trough or a crest of the time delay t d , the integral will either be under-or overestimated, respectively. To make progress, we make use of the ideas behind PL theory, as described by Feldbrugge et al (2019; see also Diego et al 2019 andGuo &Lu 2020 for alternative approaches).…”

“…2. Various regularization techniques are possible, including (i) approximating ψ near the singularities by a different function that is regular there (see Christian et al 2018;Guo & Lu 2020), and (ii) cutting out regions of some size surrounding the singularities. These approaches can be difficult to tune, however, because if too much of the contour is cut, or if ψ is poorly approximated, significant errors can be introduced.…”

“…Generally speaking, the main challenge in performing legitimately wave-optical calculations is the oscillatory nature of the relevant Fresnel-Kirchhoff diffraction integral (Peters 1974); given a phase profile j, the physical optics calculation involves integrating the exponential e i j over the aperture, which is infinitely oscillatory owing to Euler's formula. Feldbrugge et al (2019), following a mathematical program outlined by Witten (2010), have recently developed a method based on the application of Picard-Lefschetz (PL) theory that is useful in this regard (see also Guo & Lu 2020, for a catalog of other methods).…”