Exact Self-Imaging with Arbitrary Revival Spacings

Self-imaging represents a core hallmark of paraxial wave evolution; yet, across its many realizations and generalizations over the past two centuries, the uniformity of recurrence planes along the propagation axis has been considered fundamental. Here we reformulate the general phenomenon of self-imaging within the natural framework of canonical phase-space geometry, revealing a hidden canonical coordinate in which all exact self-imaging is indeed uniform, but which need not correspond to the physical propagation axis. This leads to a general law of self-imaging, in which the spacing of the physical recurrence planes can be prescribed through the choice of initial transverse phase structure. Using a single programmable spatial light modulator, we demonstrate the construction of Talbot carpets characterized by recurrence spacings that accelerate and decelerate along the propagation axis, as well as those that follow polynomial, exponential, and sinusoidal axial trajectories. These results reveal a hidden geometric freedom in paraxial wave propagation: exact self-imaging is rigid in canonical coordinates, but freely programmable in physical space, allowing for qualitatively new forms of optical recurrence.