A digital video disc (DVD) records information in a spiral...

A digital video disc (DVD) records information in a spiral track approximately $1 \mu$ m wide. The track consists of a series of pits in the information layer (Fig. P35.47a) 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.

Transcript

00:01 Hi there! so for this problem we are told that a digital video disk records information in a spiral track approximately one micrometers wide.

00:15 Now the track consists of a series of bits in the information layer that is scattered alive from a laser beam sharply focused on them.

00:27 Now the laser shines in from below through a transparent plastic of thickness d, call this the thickness d, which is equal to 1 .20 millimeters.

00:45 And an indus of refraption that is also given.

00:49 So the index of refraption of plastic is equal to 1 .55.

00:58 So assuming that the width of the laser beam at the information ledger must be a, the ballast, of a, which is one micrometers, and from only one track and not from its neighbors.

01:18 So assume the width of the beam as it enters the transparent plastic width, w, which is equal to 0 .700 millimeters.

01:38 Now we are told that a length makes the beam converge into a con with an apis angle 2 times the angle teta 1.

01:49 Before it enters the dvd.

01:52 Now we need to find the incidence angle tita 1 of the light at the edge of the conical beam.

02:01 This design is relative immune to small dust particles degrading the video quality.

02:08 So the strategy to solve this problem is to use the right triangle geometry to determine the angle teta 2 in here, this angle and then applying a snail's law, we can obtain the angle teta 1.

02:29 Now the first thing that we can obtain from this figure that we are given is a relation between wb and a.

02:38 As you can see from this, w can be written as 2 times b plus a.

02:47 So if we can solve for b because b, we don't know b.

02:51 We are not given that value.

02:55 Then from this expression, we will obtain that b is equal to w minus a and this divided by 2.

03:06 And now we just simply substitute all of the values that we are given.

03:11 We know that w is 0 .700 millimeters.

03:16 We can write that into micrometers or tens to the minus six meters as 700 times 10 to the minus six meters...

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