The three-dimensional inverse-scattering and inverse-source problems with a planar aperture

Area of Science:

• Wave scattering and inverse problems
• Electromagnetics and acoustics
• Computational imaging

Background:

• Planar aperture measurements are crucial for characterizing 3D scattering potentials and incoherent sources.
• Existing methods often rely on approximations or numerical solutions for complex imaging scenarios.
• A unified theoretical framework is needed for the weak scattering limit.

Purpose of the Study:

• To derive a fundamental integral equation governing imaging from planar apertures.
• To analytically solve this equation for both 3D scattering potentials and 3D spatially-incoherent sources.
• To establish a robust theoretical basis for inverse scattering and source imaging problems.

Main Methods:

• Formulation of an integral equation involving the product of two Green's functions.
• Application of the weak scattering limit (Born approximation).
• Analytical derivation and solution of the resulting imaging equation.

Main Results:

• A novel integral equation is derived for planar aperture measurements.
• The equation is shown to be applicable to both scattering potentials and incoherent sources.
• An analytical solution to this fundamental imaging equation is obtained.

Conclusions:

• The derived analytical solution provides a powerful tool for inverse problems in wave scattering and source localization.
• This work unifies the treatment of scatterers and sources under the Born approximation.
• The findings have implications for various imaging modalities utilizing planar sensor arrays.