Question

A \(45^{\circ}-45^{\circ}-90^{\circ}\) triangular prism can be used to reverse a light beam: The light enters perpendicular to the hypotenuse of the prism, reflects off each leg, and emerges perpendicular to the hypotenuse again. The surfaces of the prism are not silvered. If the prism is made of glass with in dex of refraction \(n_{\text {glass }}=1.520\) and the prism is surrounded by air, the light beam will be reflected with a minimum loss of intensity (there are reflection losses as the light enters and leaves the prism). a) Will this work if the prism is under water, which has index of refraction \(n_{\mathrm{H}_{2} \mathrm{O}}=1.333 ?\) b) Such prisms are used, in preference to mirrors, to bend the optical path in quality binoculars. Why?

Short answer

Answer: No, the prism will not work underwater, as total internal reflection will not occur, and the light will not be fully reflected. b) Why are these prisms preferred over mirrors to bend the optical path in quality binoculars? Answer: Prisms are preferred due to their compact design, superior image quality, durability, and better performance in various optical properties, including chromatic aberration correction.

Step-by-step solution

Question a: Determine if total internal reflection occurs underwater

First, let's recall Snell's Law: $$ n_1 \cdot \sin{\theta_1} = n_2 \cdot \sin{\theta_2} $$ Where \(n_1\) and \(n_2\) are the indices of refraction of the first and second medium, respectively, and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction, respectively. Since the light is entering perpendicular to the hypotenuse, we have \(\theta_1 = 90^{\circ}\). The angle of refraction will be related to the critical angle, which is the angle for which total internal reflection occurs. Let's calculate \(\theta_2\) when light enters from water to glass: $$ \theta_2 = \arcsin{\left(\frac{n_{H_2O}}{n_{glass}}\right)} $$ Now, let's compute the critical angle, \(\theta_c\), for total internal reflection: $$ \theta_c = \arcsin{\left(\frac{n_{H_2O}}{n_{glass}}\right)} $$ We will compare \(\theta_2\) to \(\theta_c\) to determine if total internal reflection occurs when the prism is underwater.

Calculating angle of refraction and critical angle

Now, we will compute the angle of refraction, \(\theta_2\), and the critical angle, \(\theta_c\): $$ \theta_2 = \arcsin{\left(\frac{1.333}{1.520}\right)} \approx 61.2^{\circ} $$ $$ \theta_c = \arcsin{\left(\frac{1.333}{1.520}\right)} \approx 61.2^{\circ} $$ Since \(\theta_2\) is equal to \(\theta_c\), total internal reflection will not occur when the prism is underwater, and the light will not be fully reflected. Thus, it will not work if the prism is underwater.

Question b: Benefits of prisms over mirrors

There are several reasons for using a prism instead of a mirror to bend the optical path in quality binoculars: 1. Prisms provide a more compact design, as they can fold the light path, making binoculars smaller and more portable. 2. Mirrors can alter the image quality due to imperfections in the reflective coating, while prisms do not require a coating and maintain image quality. 3. Mirrors are subject to wear and tear due to oxidation and other environmental factors, while prisms are made of solid glass, thus providing a more prolonged and reliable performance. 4. Prisms can offer chromatic aberration correction, which leads to better image quality in binoculars. In conclusion, the preference for prisms over mirrors in quality binoculars is due to their compact design, superior image quality, durability, and better performance in various optical properties.

Key concepts

Snell's Law

Snell's Law is a fundamental principle in optics that dictates how light waves change direction, or refract, when they pass from one medium into another with a different index of refraction. The law is mathematically expressed as \begin{align*} n_1 \cdot \sin{\theta_1} &= n_2 \cdot \sin{\theta_2}, \end{align*}where \(n_1\) and \(n_2\) represent the indices of refraction, and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction, respectively.

Understanding Snell's Law is critical when designing optical instruments like binoculars, where precision in light handling can significantly impact the clarity and brightness of the image seen. As light enters a new medium at an angle, it bends accordingly; the relationship given by Snell's Law allows us to predict precisely how it will bend, which is essential in managing the optical path within complex systems.

Critical Angle

The critical angle is a key concept in understanding total internal reflection. It's defined as the angle of incidence beyond which light cannot pass through the interface of two substances and instead reflects entirely back into the original substance. The critical angle can be determined when light travels from a denser medium to a less dense medium by using the formula: \begin{align*} \theta_c &= \arcsin{\left(\frac{n_2}{n_1}\right)}, \end{align*}where \(n_1\) is the index of refraction of the denser medium and \(n_2\) is that of the less dense medium.

For total internal reflection to occur, the angle of incidence must be greater than the critical angle. This is pivotal in applications like fibre optics and certain optical instruments, where it is important to keep light confined within a material without significant loss.

Optical Path in Binoculars

The optical path in binoculars is carefully engineered to provide an amplified, upright image in a compact space. Prisms are a critical component in this design, specifically due to their ability to fold the light path. This capability comes from the principles of reflection and refraction, particularly the process of total internal reflection, which occurs without the need for a reflective coating. Unlike mirrors, which could distort the image due to imperfections or degrade over time, prisms made of high-quality glass maintain the integrity of the light path, ensuring durability and exceptional image quality.

Furthermore, prisms can correct for chromatic aberration, a type of distortion where colours separate because different wavelengths of light refract through a medium at slightly different angles. This correction is crucial for clear and color-accurate viewing. The robustness and intricate optical manipulation possible with prisms explain their preference over mirrors in high-quality binocular construction.