Resolvance of Grating

Resolvance or "chromatic resolving power" for a device used to separate the wavelengths of light is defined as

The limit of resolution is determined by the Rayleigh criterion as applied to the diffraction maxima, i.e., two wavelengths are just resolved when the maximum of one lies at the first minimum of the other.

Since the space between maxima for N slits is broken up into N-2 subsidiary maxima, the distance to the first mimimum is essentially 1/N times the separation of the main maxima. This leads to a resolvance for a grating of Show where N is the total number of slits illuminated and m is the order of the diffraction.

Example

Examples of Resolvance

A standard benchmark for the resolvance of a grating or other spectroscopic instrument is the resolution of the sodium doublet. The two sodium "D-lines" are at 589.00 nm and 589.59 nm. Resolving them corresponds to resolvance

Another standard example is the resolution of the hydrogen and deuterium lines, often done with a Fabry-Perot Interferometer. The red lines of hydrogen and deuterium are at 656.3 nm and 656.1 nm, respectively. This requires a resolvance of

Resolvance of Grating

An approximate development of the resolvance expression can be done by using the small angle approximation to the condition for maxima.

This gives the basic ideas, but the assumptions are shaky, so you might want a real derivation.

Resolvance of Grating

This approach to the resolvance of a grating has made use of the fact that the phase is a continuous variable which can be represented analytically, and that the differential of this variable is also well-defined. Since the Rayleigh criterion places the peak of one order at the first minimum of the adjacent order, the phase associated with being "just resolved" is determined to be 2π/N. Taking the differential of that phase gives an expression which contains the differential of wavelength dλ which allows the quantity λ /dλ to be evaluated. In practice, the resolvance is stated in the form R=λ /Δλ for applications like the observation of the sodium doublet. We know the wavelength difference to be Δλ = .59 nm, so the resolvance can help us to anticipate whether a particular diffraction grating could resolve that difference.