instrumental width

When monochromatic light passes through an optical instrument, diffraction creates fringes, but those fringes are not infinitely sharp. Instrumental width is the physical breadth of the transition zone at the edge of a bright fringe, measured from the point where intensity drops to half its peak value on one side to the half-intensity point on the other. This quantity determines how fine a detail the instrument can actually resolve, regardless of what theory might promise.

The width arises from the finite aperture of the optical system. A circular aperture produces the Airy disk pattern; a rectangular slit produces sinc-function fringes. In both cases, diffraction causes the sharp edges predicted by geometric optics to spread into gradual intensity gradients. A spectrometer with a narrow entrance slit yields narrower fringes than one with a wide slit, because the source itself becomes smaller and better approximates a point. The instrumental width thus depends on slit geometry, lens focal length, and wavelength.

This width has immediate consequences for spectroscopy and astronomy. Two emission lines separated by less than the instrumental width of the spectrograph will appear as a single blurred feature; they cannot be resolved. Similarly, in stellar imaging, two stars closer than the instrumental width (usually quoted as the full width at half maximum, or FWHM) will blur together into a single point spread function. A spectrometer with instrumental width of 0.1 nm cannot distinguish spectral features only 0.05 nm apart, no matter how stable the instrument or how long the exposure.

Relationship to instrumental resolution

Instrumental width and instrumental resolution are closely related but not identical. Resolution is often stated as a single number, such as 50,000 for a high-resolution spectrograph, meaning the instrument can distinguish wavelengths differing by lambda divided by 50,000. Instrumental width is the physical manifestation of that resolution in real space or on a detector, typically reported in nanometers, nanometers per pixel, or as an angular measure in arcseconds.

Factors that reduce instrumental width include narrowing the entrance slit, increasing focal length, and using shorter wavelengths. However, narrowing the slit also reduces light throughput, introducing a classic trade-off between light-gathering power and resolving power. In practical instruments, instrumental width is often the limiting factor in measurement precision, particularly when analyzing weak sources or detecting faint absorption features.