Resolving power of a diffraction grating ($\mathcal{R}=\lambda /\delta \lambda =|m|N$)

For $N$ coherently illuminated grating lines used in diffraction order $m$, the Rayleigh criterion yields the displayed ideal resolving power of a diffraction grating. The angular maximum-to-first-zero separation is $\lambda /(ND\cos {\theta }_m)$, while nearby wavelengths are separated by $|m|\delta \lambda /(D\cos {\theta }_m)$. The order must exist, carry appreciable light intensity and remain free from dominant instrumental broadening. Line count alone therefore does not determine whether a particular instrument resolves a specified doublet.

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.

At normal incidence, an ideal grating of $N$ lines and period $d$ resolves a small wavelength separation $\delta \lambda$ in order $m$ by the Rayleigh condition $|m|N\gtrsim \lambda /\delta \lambda$. Combining this with $|m|\lambda =d|\sin \theta |$ yields $W=Nd\gtrsim {\lambda }^2/(\delta \lambda |\sin \theta |)\ge {\lambda }^2/\delta \lambda$. The final expression is a limiting lower bound as diffraction approaches grazing. Finite usable diffraction angle, unequal line strengths, aberrations and intrinsic linewidth can all demand a larger instrument or a more detailed criterion.