1) Grating coupling. One way to launch SPP waves is to etch grooves into the metal surface, then choose the incident angle so that the diffracted wave is along the surface. The goal is to match the phase velocity of the diffracted wave to that of the real part of the phase velocity of the SPP wave. The conventional form of the grating equation reads: n2 Sin[0 ] = m 4 + n] Sin[0j], where n is the refractive index of the incident medium (=1 here), m is the diffracted order, and 0 is the diffracted angle, measured from the opposite side of the surface normal from 0j. Multiplying by 2 It / do, this looks like a modification of Snell's Law, Kd = 2nm + ki the task is to make ko = +Re[k,], for ed = IT/2, which is along the surface. The sign determines which direction the plasmon goes. For a given input wavelength, there is some flexibility to choose a groove density and incident angle that couples the incident wave into the SPP mode in either direction. Do the calculations below for 1=1000nm and silver as the material. a) At first, pick 0; = 45 deg, and find a groove density that will couple to positive k, with the m = + 1 order. How many other diffracted orders will be present? This would be diffracted out into real angles into the outside medium (vacuum here), not along the surface. b) Second, the coupling efficiency will be best if the only diffracted order that exists (aside from m=0) is the one that couples to the SPP mode. Keep the angle of incidence the same and find a groove density that will make it so that the only diffracted order is along the surface. Hint: choose Re[k,]