1) (ex) Laser light of wavelength / coming from the left passes through a single slit of width w, as shown. Diffraction causes the light to spread out beyond the slit. Many "wavelets" spread out from all points across the width of the slit; interference between these wavelets causes the light to make a Top view pattern on a screen a distance _ away, where L > >w. (The vertical and horizontal scales are very B, different on the drawing.) The pattern has a central maximum, or bright region, between the first dark -D spots or minima ( D,) to either side. The center of -B this bright region is B- Beyond this, there are alternating bright regions ("fringes") ( B, , B, , and X D so on) and dark spots ( D, , and so on) on either side L B of Bo: only the first of each are shown here. The angle 6, is the angle from the center of B, to the Plate with Screen one slit first minimum D, a) Now consider the two halves of the slit, (1) and (II), as shown in the figure. A wavelet leaves the top of (1) and travels to D, , shown by the top diagonal line; another wavelet leaves the top of (II) and travels to D, , shown by the lower diagonal line. Note that these two diagonal lines are in reality approximately parallel to each other, since L >> w. Therefore, each diagonal line makes approximately the same angle, 8, , with W/2I () the dashed line. These diagonal lines are W/21) different from each other in length by AL . To help produce a dark spot at D, , these two wavelets must be out of phase when they reach D, . Write an equation for AL , both in terms of (i) 1, and (ii) w and 6, . (Sketch a diagram to help.)h] Now, ooosider two rmre wavelets just sligmly below the wavelets in the previous parL When they arrive at D\" these two 1wavelets will also be out of phase just as we found for the wavelets in the previous part. In fact. every wavelet in {I} is out ofphase with a wavelet io {11} that is a distance will below it in the gure. Therefore, we have completely destructive interference at. [1'l _ Put togethertheeqjatjoos you found in the Previous pan to ndaoequation relating A, w, and HI _ c} Generalisie this equation for the at\" minimum {dark sprit}1 in terms of mi :1, W1 and '9. _