Dispersion

The angular spread Δβ of a spectrum of order m between the wavelength λ and λ + Δλ can be obtained by differentiating the grating equation, assuming the incidence angle α to be constant. The change D in diffraction angle per unit wavelength is therefore

D = dβ/dλ = m / dcosβ = (m/d)secβ = Gm secβ   (2-13)

where β is given by Eq. (2-4). The quantity D is called the angular dispersion. As the groove frequency G = 1/d increases, the angular dispersion increases (meaning that the angular separation between wavelengths increases for a given order m).

In Eq. (2-13), it is important to realize that the quantity m/d is not a ratio which may be chosen independently of other parameters; substitution of the grating equation into Eq. (2-13) yields the following general equation for the angular dispersion:

D = dβ/dλ = (sinα + sinβ) / λcosβ   (2-14)

For a given wavelength, this shows that the angular dispersion may be considered to be solely a function of the angles of incidence and diffraction. This becomes even more clear when we consider the Littrow configuration (α = β), in which case Eq. (2-14) reduces to

D = dβ/dλ = (2/λ) tanβ, in Littrow.   (2-15)

When |β| increases from 10° to 63° in Littrow use, the angular dispersion can be seen from Eq. (2-15) to increase by a factor of ten, regardless of the spectral order or wavelength under consideration. Once the diffraction angle β has been determined, the choice must be made whether a finepitch grating (small d) should be used in a low diffraction order, or a coarse-pitch grating (large d) such as an echelle grating (see Section 12.5) should be used in a high order. [The fine-pitched grating, though, will provide a larger free spectral range; see Section 2.7 below.]

For a given diffracted wavelength λ in order m (which corresponds to an angle of diffraction β), the linear dispersion of a grating system is the product of the angular dispersion D and the effective focal length r'(β) of the system:

r'D = r'(dβ/dλ) = mr'/dcosβ = (mr'/d)secβ = Gmr'secβ (2-16)

The quantity r'Δβ = Δl is the change in position along the spectrum (a real distance, rather than a wavelength). We have written r'(β) for the focal length to show explicitly that it may depend on the diffraction angle β (which, in turn, depends on λ).

The reciprocal linear dispersion, formerly called the plate factor P, is more often considered; it is simply the reciprocal of r' D,

P = dcosβ / mr'   (2-17)

usually measured in nm/mm (where d is expressed in nm and r' is expressed in mm). The quantity P is a measure of the change in wavelength (in nm) corresponding to a change in location along the spectrum (in mm). [It should be noted that the terminology plate factor is used by some authors to represent the quantity 1/sinΦ, where Φ is the angle the spectrum makes with the line perpendicular to the diffracted rays (see Figure 2-6); in order to avoid confusion, we call the quantity 1/sinΦ the obliquity factor.] When the image plane for a particular wavelength is not perpendicular to the diffracted rays (i.e., when Φ ≠ 90°), P must be multiplied by the obliquity factor to obtain the correct reciprocal linear dispersion in the image plane.