Premed · Premed · Physics 2
Lecture 15: Geometric Optics: Reflection and Refraction
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- State and apply the law of reflection
- State and apply Snell's law of refraction
- Explain total internal reflection and calculate the critical angle
- Describe dispersion and its role in phenomena such as rainbows
- Apply Fermat's principle to derive the laws of reflection and refraction
Lecture Content
I. The Ray Model of Light
Geometric optics treats light as rays that travel in straight lines through uniform media. This approximation is valid when the objects and apertures involved are much larger than the wavelength of light; it breaks down when diffraction effects become significant, requiring the wave optics treatment covered in later lectures.
A ray indicates the direction of propagation of a wavefront and is always perpendicular to the wavefront. Light travels at speed c = 3 x 10^8 m/s in vacuum. In a medium with index of refraction n, the speed is v = c/n, where n >= 1 for all materials (n = 1 for vacuum). Common values include air (1.000), water (1.333), glass (1.5-1.9), and diamond (2.42).
II. Reflection
When light hits an interface between two media, some light is reflected and some is transmitted. The law of reflection states that the angle of incidence equals the angle of reflection: theta_i = theta_r. Both angles are measured from the normal to the surface, not from the surface itself, and the incident ray, reflected ray, and normal all lie in the same plane.
Specular reflection occurs from smooth surfaces and produces clear images, as in mirrors. Diffuse reflection occurs from rough surfaces and scatters light in many directions. The law of reflection still holds locally at each point on a rough surface; it is the variation in the surface normals that produces the scattering. Diffuse reflection is the mechanism by which we see most objects in our everyday environment.
III. Refraction and Snell's Law
Refraction is the bending of light as it passes from one medium to another, caused by the change in the speed of light between the two media. Snell's law (the law of refraction) relates the angles: n_1 sin(theta_1) = n_2 sin(theta_2), where theta_1 is the angle of incidence and theta_2 is the angle of refraction, both measured from the normal.
When light enters a denser medium (n_2 > n_1), it slows down and bends toward the normal (theta_2 < theta_1). When light enters a less dense medium (n_2 < n_1), it speeds up and bends away from the normal (theta_2 > theta_1). The frequency of light does not change upon refraction; only the wavelength changes, according to lambda_n = lambda_0 / n.
<image>A diagram showing a light ray hitting a flat interface between two media (n_1 above, n_2 below, with n_2 > n_1). The incident ray comes from the upper left at angle theta_1 to the normal (dashed vertical line). The reflected ray goes to the upper right at angle theta_r = theta_1 to the normal. The refracted (transmitted) ray bends toward the normal and continues into the lower medium at angle theta_2 < theta_1. All three rays and the normal are in the same plane. Snell's law n_1 sin(theta_1) = n_2 sin(theta_2) is written below. Wavefronts (perpendicular to each ray) are shown, with shorter wavelength in the denser medium.</image>
IV. Total Internal Reflection
Total internal reflection occurs when light travels from a denser medium to a less dense medium (n_1 > n_2). As the angle of incidence increases, the refracted angle increases more rapidly. At the critical angle theta_c, the refracted ray travels along the interface (theta_2 = 90 degrees), giving sin(theta_c) = n_2 / n_1. For angles of incidence greater than theta_c, total internal reflection occurs: no light is transmitted, and all of it is reflected back into the denser medium.
Total internal reflection requires that n_1 > n_2, meaning light must travel from a denser medium into a less dense one. Its applications are numerous and important. Optical fibers guide light by bouncing it inside a glass core (n approximately 1.5) surrounded by cladding with a lower index of refraction. This technology is used in telecommunications, medical endoscopes, and the internet backbone. Prisms in binoculars and periscopes use total internal reflection instead of mirrors. The exceptional brilliance of diamond arises because its high index of refraction (2.42) produces a small critical angle of about 24.4 degrees, trapping light inside the gem and creating intense sparkle.
<image>Three panels showing total internal reflection. Panel A: Light in glass (n_1 = 1.5) hitting the glass-air interface at a small angle — most light refracts into air with only partial reflection. Panel B: At the critical angle theta_c, the refracted ray travels along the interface (theta_2 = 90 degrees). The equation sin(theta_c) = n_2/n_1 is shown. Panel C: At an angle greater than theta_c, total internal reflection occurs — no refracted ray exists, and all light is reflected back into the glass. A fourth small panel shows an optical fiber with light zigzagging inside the core via repeated total internal reflection.</image>
V. Dispersion
The index of refraction depends on wavelength: n = n(lambda). For most transparent materials, n is larger for shorter wavelengths (violet and blue) and smaller for longer wavelengths (red). This is called normal dispersion.
Dispersion is the separation of white light into its component colors resulting from this wavelength dependence. When white light enters a prism, violet light is refracted more strongly (because it has a higher n and bends more toward the normal) while red light is refracted less. The result is a spectrum of colors spread across space.
Rainbows are caused by the combined effects of dispersion, refraction, and internal reflection inside water droplets. The primary rainbow involves one internal reflection, with red on the outside (at about 42 degrees) and violet on the inside (at about 40 degrees). The secondary rainbow involves two internal reflections, producing reversed colors and a fainter image. Chromatic aberration in lenses occurs because dispersion causes different colors to focus at different points, and it is corrected by using achromatic doublets that combine lenses made of different types of glass.
VI. Fermat's Principle and Applications
Fermat's principle states that light follows the path that takes the least time between two points. More precisely, the optical path length, defined as the integral of n dL, is stationary (a minimum, maximum, or saddle point). From Fermat's principle, both the law of reflection and Snell's law can be derived.
Several practical consequences follow from refraction. Apparent depth makes an object in a denser medium appear closer to the surface than it actually is: the apparent depth equals the actual depth multiplied by n_2/n_1 for near-normal viewing. For example, a swimming pool appears shallower than it is because n_water = 1.33. Lateral displacement occurs when a ray passes through a parallel-sided slab such as a glass window: the ray emerges parallel to its original direction but shifted laterally by an amount that depends on the slab's thickness, the angle of incidence, and the index of refraction. Atmospheric refraction, caused by the gradient of n in Earth's atmosphere (which decreases with altitude), makes stars and the sun appear slightly higher than their true positions. Mirages are produced by refraction through layers of air at different temperatures.
