Optical sensor: At optical wavelengths ( = 633 nm), the real part of the refractive index for gold is ng = 0.197, and for water, the refractive index is nw = 1.33. Now answer the following (15 pts) 

a) Calculate the effective refractive index (Neff) for an SPR wave traveling across a gold/water interface. 

b) We now grow a 20 nm-thick thin film made of proteins on the gold surface and find out that the protein thin film has a real refractive index of np = 1.40. What is the new effective refractive index? Assume the fraction of the evanescent wave in the protein layer is f=0.33 Hint: Assuming the fraction of the evanescent wave in the protein layer is f=0.33, use the weight factor of 0.33 for the protein layer and a weight factor of (1-f) = (1-0.33) = 0.67 for water. Then find the total refractive index of the dielectric which is now a combination of water and protein. 

c) What is the % change in the speed of light for the propagating surface plasmon wave induced by adding a protein layer.