How is the momentum change of a photon is related to the grating periodicity and diffraction order? I came across with this question while studying surface plasmon polaritons (SPPs) and how to excite them. According to a book(Plasmonics by Maier) on this subject we have the following equation:
$$\beta = k \sin\theta \pm nG$$
Where $\beta$ is the in-plane(as far as I understood the surface of the grating metal interface) wave-vector of the SPP, $\theta$ is the angle of incidence and n is the diffraction order. I know that the laws of momentum and energy conservation should apply in diffraction but too confused about the process as I think when the light gets an increase in its in-plane momentum the other components of the wave-vector might become imaginary to keep its magnitude fixed. I would appreciate if someone can point out at the details.
EDIT: I will try to explain my concern. I have read that question before posting and my questions actually asks for something a bit different. The in-plane wave vector for the SPP is larger than the free space propagation wave vector magnitude wise. Therefore in order to excite an SPP we need to have an in-plane wave vector whose magnitude is bigger than the free space wave vector. Therefore the component of the wave vector perpendicular to grating and metal-air interface should be imaginary to fulfill the energy conservation law. Is my reasoning correct? Because even at grazing incidence the wave vectors in-plane component is not enough to excite the SPPs without the aid of a prism or grating.