The 3D on-axis space-time scalar impulse response field of the parabolic reflector in the focal plane

For the parabolic reflector system, the scalar theory of diffraction and the spatial integration of reflected Dirac-type impulse wavelets are employed to derive a novel easily-visualized closed-form expression for the 3D space-time impulse response in the focal plane. It is shown that this impulse response has a 3D region of support in space-time that closely approximates the exterior of a cone and a 3D pulse-amplitude that closely approximates the time-derivative, in the electromagnetic case, of a `scaled-semi-circular pulse' having a temporal duration that is proportional to the distance in the focal plane from the focal point and an intensity that is inversely proportional to this distance. This result enables general non-monochromatic focal plane space-time fields to be derived by direct 3D convolution in space-time and is applicable in such fields as radio astronomy, radar, mobile wireless communications, ultrasound and acoustical imaging.