Interference patterns for simple lens models in wave-optics regime

TL;DR

This work analyzes how wave-optics effects shape interference patterns in gravitational-wave lensing by simple lens models (point mass, Chang-Refsdal, binary) in the LVK band around $100$ Hz. It develops and applies a Fourier-contour (UG) method to compute the frequency-dependent amplification factor $F(f)$ and compares full wave-optics results to geometric optics across lens types, with special attention to caustic regions. The key finding is that near caustics the wave-optics oscillations in $|F|$ differ significantly from geometric-optics predictions, while away from caustics the two approaches converge; in particular, CR and binary lenses with total masses near $100-200\,M_\odot$ can induce notable de-amplification around $f\sim 100$ Hz, offering a potential diagnostic to distinguish lens models in LVK data. These results enhance interpretation of lensed GW signals and help break degeneracies between lens configurations using wave-optics signatures.

Abstract

This work studies interference patterns created by simple lens models (point mass, Chang-Refsdal, and binary lens) in the wave optics regime, primarily in the context of lensing of gravitational waves (GWs) in the LIGO band at frequencies around 100 Hz. We study how the interference patterns behave close to the caustic curves which mark the high magnification regions in conventional geometric optics. In addition, we also look at the formation of highly de-amplified regions in the amplification maps close to caustics and how they differ under wave and geometric optics. We see that for a source close to caustics, the oscillations in the amplification factor (their amplitude and location of crests and troughs) can differ significantly in wave optics compared to geometric optics. As we move away from caustics, the wave optics amplification factor starts to converge towards geometric optics one, especially the frequencies at which crests and through occur in the amplification factor, although the amplitude of these oscillations can still be considerably different. For Chang-Refsdal and binary lens with ${\sim}100\:{\rm M_\odot}-200\:{\rm M_\odot}$ can introduce significant de-amplification at frequencies ${\sim}100$ Hz when the source is close to caustics, which may help us distinguish such lenses from the point mass lens.

Figures (7)

• Figure 1: Amplification factor, $F(f)$, for point mass lens. Top-row: Source plane maps of the absolute value of amplification factor ($|F|$) at three different frequencies. Bottom-left: $\Tilde{F}(t)$ curves normalized by $2\pi$ for three different values of source positions marked by the black solid stars in the top row. Bottom-middle: Absolute value of amplification factor ($|F|$). The black dashed curves represent $|F|$ calculated using the analytical formula in Equation \ref{['eq:pm_analytic']}, and coloured solid curves are calculated using the UG method. The dashed-dotted curves show the $|F|$ under geometric optics approximation. Bottom-right: Phase value of amplification factor ($\theta_F$). Like the middle panel, the black dashed curves are estimated using the analytic formula. Coloured solid curves are calculated using the UG method, and dashed-dotted curves are the results under the geometric optics approximation.
• Figure 2: Amplification factor, $|F|$, maps for Chang-Refsdal lens with positive values of external shear (i.e., $\gamma > 0$). The three rows represent the source plane maps corresponding to $\gamma = 0.5/\sqrt{2}, 0.8/\sqrt{2}, 0.9/\sqrt{2}$ (i.e., $\mu_{\rm m} = 1.33, 2.78, 5.26$), respectively. In each row, left, middle and right panels correspond to $f=50, 100, 500$ Hz. In each panel, black solid curves represent the caustic. The black and green stars in the top and middle rows represent the source position for which amplification factor curves are shown in Figure \ref{['fig:cr_Ff_curves']}.
• Figure 3: Amplification factor, $|F|$, maps under geometric optics approximation for the Chang-Refsdal lens with $M_z=200~{\rm M_\odot}$ at $f=100$ Hz. The left, middle, and right panels correspond to three different $\gamma$ values, which are shown in the top-right part of each panel. The corresponding interference maps of $|F|$ in the wave optics are shown in the middle column of Figure \ref{['fig:cr_interf_comb']}. The amplification factor curves for black and green stars in the left and middle panels are shown in Figure \ref{['fig:cr_Ff_curves']}.
• Figure 4: Example amplification factor, $F(f)$, curves for the Chang-Refsdal lens with $\gamma = 0.5/\sqrt{2}$ in the top panel and $\gamma = 0.8/\sqrt{2}$ in the middle and bottom panels, respectively. The source positions in the top panel are shown in the top panels of Figure \ref{['fig:cr_interf_comb']} and are marked by solid black stars. Source positions for the middle and bottom panels are shown by black and green stars in the middle panels of Figure \ref{['fig:cr_interf_comb']}. The left, middle, and right panels show the $\Tilde{F}(t)$, $|F|$, and $\theta_F$ curves, respectively. The solid curves are obtained using the UG method and the corresponding geometric optics approximation results are shown by dashed curves.
• Figure 5: Amplification factor, $|F|$, maps for binary lens with $(q_1, q_2) = (0.5, 0.5)$ and $M_T=200~{\rm M_\odot}$. In each row, left, middle, and right panels are corresponding to $f=50, 100, 500$ Hz. From top-to-bottom, we decrease the distance (2$L$) between the two-point masses. In each panel, black solid curves represent the caustic structure. The black solid stars are source positions for which we show the amplification factor in Figure \ref{['fig:bml_Ff']}.