A digital video disc (DVD) records information in a spiral track approximately 1$\mu \mathrm{m}$ wide. The track consists of a series of pits in the information layer (Fig. P35. 47 $\mathrm{a}$ ) that scatter light from a laser beam sharply focused on them. The laser shines in from below through transparent plastic of thickness $t=1.20 \mathrm{mm}$ and index of refraction 1.55 (Fig.P35.47b). Assume the width of the laser beam at the information layer must be $a=1.00 \mu \mathrm{m}$ to read from only one track and not from its neighbors. Assume the width of the beam as it enters the transparent plastic is $w=0.700 \mathrm{mm}$ . A lens makes the beam converge into a cone with an apex angle 2$\theta_{1}$ before it enters the DVD. Find the incidence angle $\theta_{1}$ of the light at the edge of the conical beam. This design is relatively immune to small dust particles degrading the video quality.