Answer Happy

[VW33d] Diffraction Grating Screen IN-1 N-1 di 72 ro le L Consider a diffraction grating consisting of N thin slits at a distance d apart. A plane wave with wavelength 1 = 27/k is incident from the left. The screen is very far away from the grating, i.e. L>(N − 1)d, and so we will work in the far-field approximation. All the lines from the slits to the point P are therefore at essentially the same angle 0 with respect to the horizontal. Take note of the results derived in VW31 of the last tutorial. (a) Within the far-field approximation, how are rn and ro related? Use known results from class. (b) Provide an explanation of why you expect the brightest spots (points of high inten- sity) to occur at angles which satisfy d sin(0) = ml with me Z. =
= (c) Take the wave at P coming from the n’th slit to be A(0) cos(wt – krn). Show that the relative total intensity at point P is given by 40) IO) A RO) sin(Na/2) I(0) A(0) N sin(a/2) with a = kd sin(0). Here [A(0)/A(0)]? = cos(6) as usual. (d) Typically a diffraction grating will have a very large number of slits. We are there- fore interested in how the interference pattern described by R(0) will change as we increase N. Here we need to consider two cases: (i) At one of the angles identified in (b), what is the value of R(O)? How does it behave as we increase N? Show your steps clearly. (ii) Away from one of the angles identified in (b), how does R(O) behave as we increase N? Explain your reasoning. (e) Describe, using simple but neat figures and full sentences, how the interference pat- tern on the screen will change with increasing N, and what you expect to see when N > 1. It is recommended that you make some plots in Mathematica, say for kd = 20, to check your predictions.