= Chirp requirement for longitudinal bunch compression
{title2=$\sigma_{zf}^2=\sigma_z^2+2R_{56}\operatorname{Cov}(z,\delta)+R_{56}^2\sigma_\delta^2$}
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.
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