Diffraction Grating Efficiency

A complete treatment of grating efficiency requires the vector formulation of electromagnetic theory (i.e., Maxwell's equations) applied to corrugated surfaces, which has been studied in detail over the past few decades. While the theory does not yield conclusions easily, certain rules of thumb can be useful in making approximate predictions.

The simplest and most widely used rule of thumb regarding grating efficiency (for reflection gratings) is the blaze condition

mλ= 2dsinθ_B    (2-30)

where θ_B (often called the blaze angle of the grating) is the angle between the face of the groove and the plane of the grating (see Figure 2-9). When the blaze condition is satisfied, the incident and diffracted rays follow the law of reflection when viewed from the facet; that is, we have

α – θ_B = β – θ_B    (2-31)

Because of this relationship, it is often said that when a grating is used at the blaze condition, the facets act as tiny mirrors. This is not strictly true; since the dimensions of the facet are often on the order of the wavelength itself, ray optics does not provide an adequate physical model. Nonetheless, this is a useful way to remember the conditions under which a grating can be used to enhance efficiency.

Eq. (2-30) generally leads to the highest efficiency when the following condition is also satisfied:

2K =  α – β = 0    (2-32)

where 2K was defined above as the angle between the incident and diffracted beams (see Eq. (2-6)). Eqs. (2-30) and (2-32) collectively define the Littrow blaze condition. When Eq. (2-32) is not satisfied (i.e., α≠β and therefore the grating is not used in the Littrow configuration), efficiency is generally seen to decrease as one moves further off Littrow (i.e., as |2K| increases).

For a given blaze angle θ_B, the Littrow blaze condition provides the blaze wavelength λ_B, the wavelength for which the efficiency is maximal when the grating is used in the Littrow configuration:

λ_B = 2d/m sinθ_B, in Littrow    (2-33)

Many grating catalogs specify the first-order Littrow blaze wavelength for each grating:

λ_B = 2d sinθ_B, in Littrow (m = 1)    (2-34)

Unless a diffraction order is specified, quoted values of λ_B are generally assumed to be for the first diffraction order, in Littrow.

The blaze wavelength λ_B in order m will decrease as the off-Littrow angle α–θ_B ncreases from zero, according to the relation

λ_B =2d/m sinθ_Bcos(α–θ_B)   (2-35)

Computer programs are commercially available that accurately predict grating efficiency for a wide variety of groove profiles over wide spectral ranges.