Show how a single block of glass can be used to turn a p-polarized beam of light through \(180^{\circ}\), with the light suffering (in principle) zero reflective loss. The light is incident from air, and the returning beam (also in air) may be displaced sideways from the incident beam. Specify all pertinent angles and use \(n=1.45\) for glass. More than one design is possible here.

Understanding Brewster's angle is crucial when working with polarized light beams. It is the specific angle of incidence at which p-polarized light is perfectly transmitted through a boundary between two media, eliminating any reflection of that polarized component. This phenomenon is named after Sir David Brewster, a Scottish physicist who discovered this property in the 19th century.

At Brewster's angle, the reflected and refracted rays are perpendicular to each other. Mathematically, it is given by the formula \( \tan( \theta_B ) = \frac{n_2}{n_1} \), where \( \theta_B \) is the Brewster's angle, \( n_1 \) is the refractive index of the first medium (from which the light is coming), and \( n_2 \) is the refractive index of the second medium (into which the light is going). When light hits a surface at this precise angle, the p-polarized component does not reflect, thus minimizing losses due to reflection.

Applying this knowledge, a beam of p-polarized light can enter a glass block at Brewster's angle without reflective loss. This principle is essential when designing optical devices, such as the scenario in our exercise where we use a single block of glass to turn a p-polarized beam through 180 degrees without any reflective loss.

Snell's law, also known as the law of refraction, is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through the boundary between two different isotropic media. This fundamental law in optics is expressed as \( n_1 \sin(\theta_i) = n_2 \sin(\theta_r) \), where \( n_1 \) and \( n_2 \) are the refractive indices of the first and second medium, respectively, while \( \theta_i \) and \( \theta_r \) are the angles of incidence and refraction.

In the context of the given problem, Snell's law helps to determine how the light beam will bend when entering the glass block from air. Once the light enters at Brewster's angle, we can use Snell's law to calculate the angle at which the light will travel inside the glass. This calculation is pivotal for designing the path that the light beam will follow through the glass, enabling the desired 180-degree turn. Understanding and applying Snell's law is fundamental for students involved in designing complex optical systems and exploring the behavior of light at interfaces.

The refractive index is a dimensionless number that describes how light propagates through a medium. It's represented by the symbol \( n \) and defined as the ratio of the speed of light in a vacuum to the speed of light in the medium: \( n = \frac{c}{v} \), where \( c \) is the speed of light in vacuum, and \( v \) is the speed of light in the material. Each material has its unique refractive index, and it can have a considerable impact on how light behaves when transitioning from one medium to another, including how much it bends, or refracts.

In our exercise, the refractive index of glass is given as 1.45, indicating that light travels through glass at a speed slower than it does in air (which has a refractive index of 1). The refractive index is central to calculations using Snell's law; it determines the angles of incidence and refraction, critical to achieving the required 180-degree rotation of the p-polarized light beam within the glass block. Students should appreciate the importance of this physical property in designing optical instruments and systems, where precision in light manipulation is essential.