# Diffraction Efficiencies and Evanescent Waves

For a binary phase grating (duty cycle = 0.5, phase contrast = $$\pi$$), the diffraction efficiency $$\eta_q$$ for an odd diffraction order $$q$$ is given by $$\eta_q = \left( \frac{2}{\pi q} \right)^2$$ (see [I, page 9], [II, page 3], [III, page 8], [IV, page 2]. All these efficiencies sum up to one, i.e., $$\sum_{q \text{ odd}} \eta_q = \sum_{q \text{ odd}} \left( \frac{2}{\pi q} \right)^2 = 2\sum_{k=1}^\infty \left( \frac{2}{\pi \, (2k-1)} \right)^2 = 1.$$ Especially in [II, page 3] this is taken as a sign that total amount of diffracted light is 100%, since phase gratings do not absorb any light.

This argument assumes the existence of an infinite number of diffraction orders, i.e, an infinite grating period. For realistic cases, however, the grating period $$\Lambda$$ is finite and due to the grating equation, only orders $$q$$ for which $$-\frac{\Lambda}{\lambda} \leq q \leq \frac{\Lambda}{\lambda}$$ are non-evanescent and carry light into the far field.

My question is now, how one could reconcile these two requirements? A possible solution that I see is that one could renormalize the diffraction efficiencies over the non-evanescent orders $$q$$ (with $$q_\text{min} \leq q \leq q_\text{max}$$) by dividing the diffraction efficiencies by the sum $$\sum_{q_\text{min} \leq q \leq q_\text{max}}^{q \text{ odd}} \left( \frac{2}{\pi q} \right)^2$$ to ensure that all light is transmitted into the non-evanescent diffraction orders.

Is this a sensible approach? Or this there a more correct way to do this?

Note that in this description, normal incidence is assumed. Furthermore, transmission and reflection effects due to the material of the phase grating -- which require the use of the Fresnel equations -- are omitted.