[Paper Review] Wave asymptotics and their application to astrophysical plasma lensing

Gianfranco Grillo, J. M. Cordes|arXiv (Cornell University)|Oct 22, 2018

Pulsars and Gravitational Waves Research10 citations

TL;DR

This paper develops a wave asymptotic formalism extending geometric optics to model two-dimensional astrophysical plasma lenses of arbitrary shape, accurately capturing amplification near fold caustics and in shadow regions. It enables efficient simulation of multifrequency time-of-arrival residuals, revealing significant deviations from $ u^{-2}$ dispersion scaling—especially at low frequencies—critical for pulsar timing and fast radio burst studies.

ABSTRACT

Plasma lensing events can have significant observational consequences, including flux density modulations and perturbations in pulse arrival times. In this paper we develop and apply a formalism that extends geometrical optics to describe the effects of two dimensional plasma lenses of arbitrary shape. We apply insights from catastrophe theory and the study of uniform asymptotic expansions of integrals to describe the lensing amplification close to fold caustics and in shadow regions, and explore the effects of image appearance and disappearance at caustics in the time of arrival (TOA) perturbations due to lensing. The enhanced geometric optics approach successfully reproduces the predictions from wave optics and can be efficiently used to simulate multifrequency TOA residuals during lensing events. Lensing will introduce perturbations both in the way the residuals change as a function of frequency and also in the magnitude and sign of the residuals averaged over a frequency band. The deviations from the expected dispersive $ν^{-2}$ scaling will be most significant when including observations at low frequencies. We examine the consequences of lensing in the context of precision pulsar timing and touch on its potential relevance to the study of FRBs.

Motivation & Objective

• To extend geometric optics to describe plasma lensing effects for two-dimensional lenses of arbitrary shape, overcoming limitations near caustics.
• To apply catastrophe theory and uniform asymptotic expansions of integrals to model amplification in fold caustic regions and shadow zones.
• To simulate multifrequency time-of-arrival (TOA) residuals during lensing events with high efficiency and accuracy.
• To quantify deviations from the standard $ u^{-2}$ dispersion scaling in TOA residuals, particularly at low frequencies.
• To assess the implications of lensing for precision pulsar timing and fast radio burst (FRB) phenomenology.

Proposed method

• Uses wave asymptotic methods derived from the geometrical theory of diffraction and catastrophe optics to model lensing near caustics.
• Applies uniform asymptotic expansions to describe amplification in regions where geometric optics fails due to singularities.
• Employs a root-finding algorithm with contour plotting to locate real solutions of the lens equation near caustics.
• Extends the method to complex conjugate solutions by using the smallest imaginary part solutions near the caustic shadow side.
• Validates results against full numerical solutions of the Kirchhoff diffraction integral (KDI) using FFT-based methods.
• Models diverse lens shapes (Gaussian, super-Gaussian, Lorentzian, ring-like, double lenses) with varying phase and scale parameters.

Experimental results

Research questions

• RQ1 How can wave asymptotic methods accurately describe amplification near fold caustics in two-dimensional plasma lenses?
• RQ2 To what extent do TOA residuals deviate from the $ u^{-2}$ dispersion scaling when lensing effects are included, especially at low frequencies?
• RQ3 How do image formation and disappearance at caustics affect the observed time-of-arrival perturbations in pulsar signals?
• RQ4 Can the extended geometric optics formalism efficiently and accurately simulate multifrequency TOA residuals without solving the full KDI?
• RQ5 What are the observational signatures of plasma lensing in dynamic spectra and pulse timing, particularly in the context of FRBs and extreme scattering events?

Key findings

• The wave asymptotic formalism successfully reproduces full KDI results for lensing amplification, especially near caustics and in shadow regions.
• The method enables efficient simulation of multifrequency TOA residuals, significantly outperforming direct KDI computation.
• Significant deviations from $ u^{-2}$ dispersion scaling are predicted, particularly when observations include low-frequency bands.
• Image coalescence and disappearance at caustics lead to sharp, observable perturbations in TOA residuals, detectable in high-precision pulsar timing.
• The formalism is robust across diverse lens shapes, including Gaussian, super-Gaussian, ring-like, and double lenses, with accurate prediction of up to 17 simultaneous images.
• Complex ray solutions near caustics are reliably found using a modified root-finding approach with minimal imaginary part constraints.