Stokes Phenomena in Lensing

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As lensing of coherent astrophysical sources, e.g., pulsars, fast radio bursts, and gravitational waves, becomes observationally relevant, the mathematical framework of Picard–Lefschetz theory has recently been introduced to fully account for wave optics effects. Accordingly, the concept of lensing images has been generalized to include complex solutions of the lens equation referred to as “imaginary images,” and more radically, to the Lefschetz thimbles, which are a sum of the steepest descent contours connecting the real and imaginary images in the complex domain. In this wave-optics-based theoretical framework of lensing, we study the “Stokes phenomena” as the change of the topology of the Lefschetz thimbles. Similar to the well-known caustics at which the number of geometric images changes abruptly, the corresponding Stokes lines are the boundaries in the parameter space where the number of effective imaginary images changes. We map the Stokes lines for a few lens models. The resulting Stokes line-caustics network represents a unique feature of the lens models. The observable signature of the Stokes phenomena is the change of interference behavior, in particular the onset of frequency oscillation for some Stokes lines. We also demonstrate high-order Stokes phenomena where the system has a continuous number of effective images but with an abrupt change in the way they are connected to each other by the Lefschetz thimbles. Their full characterization calls for an analogy of the catastrophe theory for caustics.