Broadband focusing through a lens: A linear systems approach

Using linear transform techniques, the transient focal plane pattern is derived for a plane wave impulse normally incident on a converging lens. The resultant function of space and time generated in the focal plane can be regarded as the temporal impulse response of a linear, time‐invariant system consisting of the lens, its focal plane, and the intervening free space. The focal plane response to an incident plane wave disturbance with arbitrary time dependence is then obtained immediately by convolving the broadband waveform with the temporal impulse response of the lens. This impulse response is derived for two cases: a lens with a one‐dimensional (slit) pupil function, and a lens with a circularly symmetric pupil function. It is shown that while the monochromatic focal plane pattern for a one‐dimensional lens is proportional to the Fourier transform of the one‐dimensional pupil function, the impulse response of the one‐dimensional lens is proportional to the time derivative of the pupil function itself; also, while the monochromatic focal plane pattern for a circular lens is proportional to the Hankel transform of the pupil function, the impulse response of the circular lens is proportional to the time derivative of the Abel transform of the pupil function. These results also provide a valid description for the transient response of any broadband focusing system with linear, or circular, symmetry, including, for example, electronically focused linear, or circular, element arrays.