On the Dynamic Response of the Ideal Paraboloidal Reflector

The ideal paraboloidal reflector (IPR) is analyzed as a 4D spatio-temporal linear system having a dynamic focal plane response that is characterized by an idealized scalar Dirac plane-wave (PW) signal on the aperture. Using the same path-difference equations that are used for classical steady-state quasi-monochromatic (QMC) analysis, simple algebraic expressions are derived for the spatio-temporal focal plane response
$h_{fp}\left ({\boldsymbol {x},ct }\right)$
to the Dirac-PW. These expressions for
$h_{fp}\left ({\boldsymbol {x},ct }\right)$
are used to directly determine the focal plane response to far-field on-axis short-time transient signals, thereby avoiding the complexities of QMC-based methods for the analysis of such signals. The derived first-order approximation of
$h_{fp}\left ({\boldsymbol {x},ct }\right)$
describes its spatio-temporal region of support (ROS) and amplitude whereas the second-order approximation includes a further spatio-temporal distortion that is the dynamic equivalent of the Petzval aberration. Examples of the focal plane response to highly transient far-field pulses are described.