Finite-grating incoherent doublet profile ($I(\theta )\propto {\sum }_j[\sin (N{u}_j)/(N\sin u_j){]}^2$)

For narrow slits and equally bright mutually incoherent spectral lines, $u_j=\pi p\sin \theta /{\lambda }_j$ and the intensities add as displayed. The Rayleigh criterion uses a main-peak-to-first-zero separation $\delta \lambda \simeq \lambda /(|m|N)$. A slightly sub-Rayleigh doublet can still have two local maxima: the criterion is not a theorem of statistical non-identifiability. photon counts, background, unequal strengths and uncertainty in the instrumental profile govern actual inference.