The Reciprocity Theorem

A useful property of grating efficiency is that embodied in the reciprocity theorem,100 which states that (under certain conditions) reversing the direction of the beam diffracted by a grating will leave its diffraction efficiency unchanged. Stated another way, the reciprocity theorem says that the grating efficiency for wavelength λ in order m is unchanged under the transformation α ↔ β. This equivalence follows from the periodic nature of the grating and is strictly true for perfectlyconducting gratings and lossless dielectric gratings illuminated by an incident plane wave.

Three consequences of the reciprocity theorem should be noted:

  • The zeroth-order efficiency E(λ,0) is a symmetric function (of angle α) about α = 0.
  • Rotation of the grating groove profile through 180° (while keeping α constant) does not affect E(λ,0); moreover, if only two diffraction orders are propagating (say, m = 0 and m = 1), the efficiency E(λ,1) will be unchanged as well.
  • The efficiency E(λ,m) for a given diffraction order m is a symmetric function of vs. sinα about the Littrow condition (α = β).
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