Resolving Power in Diffraction Gratings

The comments above regarding resolving power and resolution pertain to planar classical gratings used in collimated light (plane waves). The situation is complicated for gratings on concave substrates or with groove patterns consisting of unequally spaced lines, which restrict the usefulness of the previously defined simple formulas, though they may still yield useful approximations. Even in these cases, though, the concept of maximum retardation is still a useful measure of the resolving power, and the convolution of the image and the exit slit is still a useful measure of resolution.

The resolving power R of a grating is a measure of its ability to separate adjacent spectral lines of average wavelength λ. It is usually expressed as the dimensionless quantity

R = λ / Δλ (2-18)

Here Δλ is the limit of resolution, the difference in wavelength between two lines of equal intensity that can be distinguished (that is, the peaks of two wavelengths λ1 and λ2 for which the separation |λ1 – λ2| < Δλ will be ambiguous). Often the Rayleigh criterion is used to determine Δλ – that is, the intensity maxima of two neighboring wavelengths are resolvable (i.e., identifiable as distinct spectral lines) if the intensity maximum of one wavelength coincides with the intensity minimum of the other wavelength.

The theoretical resolving power of a planar diffraction grating is given in elementary optics textbooks as

R = mN (2-19)

where m is the diffraction order and N is the total number of grooves illuminated on the surface of the grating. For negative orders (m < 0), the absolute value of R is considered.

A more meaningful expression for R is derived below. The grating equation can be used to replace m in Eq. (2-19):

R = Nd(sinα + sinβ) / λ (2-20)

If the groove spacing d is uniform over the surface of the grating, and if the grating substrate is planar, the quantity Nd is simply the ruled width W of the grating, so

R = W(sinα + sinβ) / λ (2-21)

As expressed by Eq. (2-21), R is not dependent explicitly on the spectral order or the number of grooves; these parameters are contained within the ruled width and the angles of incidence and diffraction. Since

|sinα + sinβ| < 2 (2-22)

the maximum attainable resolving power is

R_MAX = 2W / λ (2-23)

regardless of the order m or number of grooves N under illumination. This maximum condition corresponds to the grazing Littrow configuration, i.e., |α| ≈ 90° (grazing incidence) and α ≈ β (Littrow).

It is useful to consider the resolving power as being determined by the maximum phase retardation of the extreme rays diffracted from the grating. Measuring the difference in optical path lengths between the rays diffracted from opposite sides of the grating provides the maximum phase retardation; dividing this quantity by the wavelength λ of the diffracted light gives the resolving power R.

The degree to which the theoretical resolving power is attained depends not only on the angles α and β, but also on the optical quality of the grating surface, the uniformity of the groove spacing, the quality of the associated optics in the system, and the width of the slits (or detector elements). Any departure of the diffracted wavefront greater than λ/10 from a plane (for a plane grating) or from a sphere (for a spherical grating) will result in a loss of resolving power due to aberrations at the image plane. The grating groove spacing must be kept constant to within about one percent of the wavelength at which theoretical performance is desired. Experimental details, such as slit width, air currents, and vibra-tions can seriously interfere with obtaining optimal results.

The practical resolving power of a diffraction grating is limited by the spectral width of the spectral lines emitted by the source. For this reason, systems with revolving powers greater than R = 500,000 are not usually required except for the study of spectral line shapes, Zeeman effects, and line shifts, and are not needed for separating individual spectral lines.

A convenient test of resolving power is to examine the isotopic structure of the mercury emission line at λ = 546.1 nm. Another test for resolving power is to examine the line profile generated in a spectrograph or scanning spectrometer when a single mode laser is used as the light source. The full width at half maximum intensity (FWHM) can be used as the criterion for Δλ. Unfortunately, resolving power measurements are the convoluted result of all optical elements in the system, including the locations and dimensions of the entrance and exit slits and the auxiliary lenses and mirrors, as well as the quality of these elements. Their effects on resolving power measurements are necessarily superimposed on those of the grating.

While resolving power can be considered a characteristic of the grating and the angles at which it is used, the ability to resolve two wavelengths λ1 and λ2 = λ1 + Δλ generally depends not only on the grating but on the dimensions and locations of the entrance and exit slits (or detector elements), the aberrations in the images, and the magnifi-cation of the images. The minimum wavelength difference Δλ (also called the limit of resolution, spectral resolution or simply resolution) between two wavelengths that can be resolved unambiguously can be determined by convoluting the image of the entrance aperture (at the image plane) with the exit aperture (or detector element). This measure of the ability of a grating system to resolve nearby wavelengths is arguably more relevant than is resolving power, since it considers the imaging effects of the system. While resolving power is a dimensionless quantity, resolution has spectral units (usually nanometers).

The (spectral) bandpass B of a spectroscopic system is the range of wavelengths of the light that passes through the exit slit (or falls onto a detector element). It is often defined as the difference in wavelengths between the points of half-maximum intensity on either side of an intensity maximum. Bandpass is a property of the spectroscopic system, not of the diffraction grating itself.

For a system in which the width of the image of the entrance slit is roughly equal to the width of the exit slit, an estimate for bandpass is the product of the exit slit width w' and the reciprocal linear dispersion P:

B ≈ w'P (2-24)

An instrument with smaller bandpass can resolve wavelengths that are closer together than an instrument with a larger bandpass. The spectral bandpass of an instrument can be reduced by decreasing the width of the exit slit, but usually at the expense of decreasing light intensity as well.