The figure below shows a schematic of a surface plasmon resonance sensor for detecting the binding of biomolecules to antibody ligands at the surface of a gold film. The gold film is 50 nm thick (permittivity of -15.6 + i 1.2647 at the free- space wavelength of 671 nm), the refractive index of the water in the channel is 1.33 and the refractive index of the substrate (glass under the gold) is 1.52. (a) Calculate an expression for the reflection amplitude as a function of angle above the critical angle. (b) Find the angle at which the reflection is minimized? (c) Make a small change to the refractive index of the water in the channel and calculate the sensitivity in refractive index units/radians (RIU/rad). Sensor chip { Polarized incident light Light source Flow channel Prism Biomolecule Ligand -Gold film Reflected light Decreased intensity at the resonance angle Detector Schematic of SPR biosensor. A semiconductor laser has real part of the refractive index 3.3 at a wavelength of 980 nm. The laser is 0.7 mm long. Using the transfer matrix approach, calculate the imaginary part of the refractive index to obtain self-sustained oscillation (lasing - defined as the limit when there is output for vanishing input).