Diffraction gratings and resolving power

Consider $N$ identical slits separated by pitch $d$, illuminated coherently at normal incidence. In far-field direction $\theta$, adjacent slit contributions have phase difference of magnitude $\beta=2\pi d\sin\theta/\lambda$. Using the negative sign appropriate to the aperture-transform convention, the array amplitude is a finite sum of $e^{-ij\beta}$. Summing the geometric progression produces a sine ratio times an overall phase, which disappears from intensity.

Principal maxima occur when adjacent phases agree modulo $2\pi$. Their amplitudes are $N$ times the contribution of one slit and their intensities $N^2$ times its intensity in that direction. This does not violate energy conservation: coherent addition narrows and redistributes the pattern. The finite width and transmission of each slit impose a separate envelope. A mathematically permitted grating order can be weak or entirely missing when that envelope has a zero there.

Near a principal order, the first array zero occurs when the adjacent phase changes by $2\pi/N$. Converting that phase difference to angle gives a one-sided peak width approximately $\lambda/(Nd\cos\theta_m)$. Differentiating the grating equation at fixed order gives angular dispersion $m/(d\cos\theta_m)$. Both quantities grow near grazing propagation, so the angular magnification by itself does not give a free improvement in spectral resolving power.

For two narrow mutually incoherent spectral lines with comparable brightness, the Rayleigh convention places one peak at the first zero of the other. Equating displacement and one-sided width yields resolving power $|m|N$. It is an ideal instrument criterion, not a universal theorem about detectability. A finite entrance slit, aberrations, pixel sampling, background, and line-shape fitting can change practical performance. The zero order has no wavelength dispersion and cannot use this spectral-resolution formula as a positive resolving power.

At fixed illuminated physical width, more grooves per millimetre increase the number of illuminated grooves and change the accessible orders. At fixed total groove count instead, changing pitch changes dispersion but leaves the ideal resolving power of a chosen order unchanged. A specification stating only groove density therefore does not determine resolution. The beam footprint, incidence geometry, and usable order must also be known before counting effective contributions.

Different order-wavelength pairs can satisfy the same grating equation. For example, first-order light at one wavelength overlaps second-order light at half that wavelength. This ambiguity requires spectral filtering or another dispersive stage; sharper peaks alone cannot remove it. A blazed groove profile redistributes efficiency among orders but does not replace the finite-aperture resolution argument. Always distinguish order location, order brightness, and order width when interpreting a grating spectrum.