Chebyshev self-imaging from inverse-sampled angular spectra

For nearly two centuries, conventional self-imaging has obeyed a single rule: a periodic wave reconstructs itself at a fixed distance, independent of its bandwidth. Here we show this law is programmable. Sampling the angular spectrum in inverse powers of the mode index makes each mode rephase with its own integer period, deferring exact revival to the least common multiple of all periods: $z_{\mathrm{r}\mathrm{e}\mathrm{v}}=z_{\mathrm{T}}{e}^{r\psi (M)}$, with $\psi$ the second Chebyshev function. We observe these revivals optically as a prime-power staircase in propagation distance, turning free-space diffraction into a readout of the multiplicative structure of the integers.