Information in a compact disc is stored in "pits" whose depth is essentially one-fourth the wavelength of the laser light used to "read" the information. That wavelength is \(780 \mathrm{nm}\) in air, but the wavelength on which the pit depth is based is measured in the \(n=1.55\) plastic that makes up most of the disc. Find the pit depth.

When light passes from one medium into another, its speed changes which, in turn, affects its wavelength. This change is quantified by the refractive index, a dimensionless number that describes how light propagates through that medium. The refractive index of a material () is defined as the ratio of the speed of light in a vacuum (c_v) to the speed of light in that material (c_m):

c_v/c_m. This relationship implies that materials with a higher refractive index will slow down light more than materials with a lower refractive index.

In the case of our exercise, the compact disc information pits are dependent on the refractive index of the plastic, which is given as 1.55. This number means that light travels 1.55 times slower in the plastic than it does in a vacuum. Understanding this concept is crucial because it directly affects the wavelength of the light within the plastic and subsequently determines the pit depth used in the storage of information on a compact disc.

The wavelength of light is the distance over which the wave's shape repeats. It determines the color of light that we perceive and is a critical parameter in various optical technologies, such as the reading of compact discs. In different media, the wavelength changes due to the alteration of light's speed, as dictated by the refractive index.

In the exercise, we deal with the wavelength of laser light used in reading CD information. As given, this wavelength is 780 nm in air. However, for pit depth calculations, we are interested in the wavelength of this light within the plastic of the disc, which has different optical properties than air. To find the wavelength in the plastic (λ_p), we must adjust the air's wavelength (λ_a) by dividing it by the refractive index of the plastic. A lower wavelength in the plastic means that the pits can be shallower for the laser to read the information correctly.

The speed of light in any medium is slower than its speed in a vacuum, which is approximately 299,792 kilometers per second. Media with higher refractive indices significantly reduce the speed of light. This reduction is crucial for various applications where controlling the phase and velocity of light is necessary, such as in optical data storage devices like compact discs.

In essence, the speed of light in a medium can be calculated by dividing the speed of light in a vacuum by the medium's refractive index. For example, the speed of light in the plastic of a CD (c_m) can be calculated as:c_v/n, where c_v is the speed of light in a vacuum, and n is the refractory index of the plastic. The knowledge of how light's speed changes in different media allows engineers and physicists to design systems that precisely manipulate light for various technological applications, including determining the optimal pit depth for information storage in a compact disc.