Chirp requirement for longitudinal bunch compression (${\sigma }_{zf}^2={\sigma }_z^2+2R_{56}\mathrm{Cov}(z,\delta )+R_{56}^2{\sigma }_{\delta }^2$)

A magnetic chicane's momentum-dependent path length does not specify a compression factor from initial length and marginal energy spread alone. A suitable longitudinal position-energy correlation is required. With zero correlation the variance above is minimized at $R_{56}=0$, so such a chicane cannot shorten that ideal uncorrelated bunch. An imposed chirp and its uncorrelated residual determine the achievable minimum.