  Spacetime perturbations due to scalar, vector, and tensor fields on a fixed
background geometry can be described in the framework of Teukolsky's equation.
In this work, wave scattering is treated analytically, using the Green's
function method and solutions to the separated radial and angular differential
equations in combination with a partial wave technique for a scalar and
monochromatic perturbation. The results are applied to analytically describe
wave-optical imaging via Kirchhoff-Fresnel diffraction, leading to, e.g., the
formation of observable black hole shadows. A comparison to the ray-optical
description is given, providing new insights into wave-optical effects and
properties. On a Kerr-de Sitter spacetime, the cosmological constant changes
the singularity structure of the Teukolsky equation and allows for an
analytical, exact solution via a transformation into the Heun's differential
equation, which is the most general, second-order differential equation with
four regular singularities. The scattering of waves originating from a point
source involves a solution in terms of the so-called Heun's function $Hf$. It
is used to find angular solutions, which form a complete set of orthonormal
functions similar to the spherical harmonics. Our approach allows to solve the
scattering problem while taking into account the complex interplay of Heun's
functions around local singularities.
