A digital videodisc records information in a spiral track approximately 1 m wide

Question

A digital videodisc records information in a spiral track approximately 1 m wide. The track consists of a series of pits in the information layer (shown below) that scatter light from a laser beam sharply focused on them. The laser shines in through transparent plastic of thickness t = 1.4 mm and index of refraction 1.55. Assume the width of the laser beam at the information layer must be a = 1.00 m to read from only one track and not from its neighbors. Assume the width of the beam as it enters the transparent plastic from below is w = 0.640 mm. A lens makes the beam converge into a cone with an apex angle 21 before it enters the videodisc. Find the incidence angle 1 of the light at the edge of the conical beam. This design is relatively immune to small dust particles degrading the video quality. Particles on the plastic surface would have to be as large as 0.7 mm to obscure the beam.

Explanation / Answer

We need 2 first. Notice that b = (w-a)/2 = 3.2 * 10-4 m and b/t = tan(2), so 2 = arctan(b/t)

= arctan(3.2 * 10-4 m/1.4 * 10-3 m) = 12.8555 degrees

From Snell's law, sin 1 = n sin 2 = (1.55)(.222494) = 0.344866

so 1 = arcsin (0.344866) = 20.17 degrees

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