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It is noted that the Niederer transformation can be used to find the explicitrelation between time-dependent linear oscillators, including the mostinteresting case when one of them is harmonic. A geometric interpretation ofthis correspondence is provided by certain subclasses of pp-waves; inparticular the ones strictly related to the proper conformal transformations.This observation allows us to show that the pulses of plane gravitational waveexhibiting the maximal conformal symmetry are analytically solvable.Particularly interesting is the circularly polarized family for which someaspects (such as the classical cross section, velocity memory effect andimpulsive limit) are discussed in more detail. The role of the additionalintegrals of motion, associated with the conformal generators, is clarified bymeans of Ermakov-Lewis invariants. Possible applications to the description ofinteraction of electromagnetic beams with matter are also indicated.

Niederer's transformation, time-dependent oscillators and polarized gravitational waves