Geometric Optics
The Law of Reflection
A mathematical theory of the bending, or refraction, of light as it passes from one material to another is beyond the scope of this course, but a qualitative description is possible. According to classical theory in which light behaves as a wave, when an electromagnetic wave enters a transparent, uniformly dense material its electric field interacts with the electrons in the atoms that compose the material causing them to vibrate as if they were bound by tiny springs. The "vibrating" electrons then re-radiate, or scatter, the electromagnetic wave in such a way that the scattered waves is out of phase with the original wave. The scattered waves are out of phase by an amount that depends upon the frequency of the original wave and the vibrational frequency of the electrons in the atoms.
When the scattered waves from all the electrons in a thin layer of the material are combined to make a secondary we find that the addition of the secondary wave and primary wave produces a resultant wave that lags behind the primary wave and appears to move slower. This reduction in wave speed through a material produces the phenomenon of refraction. The index of refraction is simply the speed of light in vacuum c divided by the speed of light through the material v:
n = c/v .
As stated above, the phase relation between the electric field of the primary wave and the vibrating electrons depends on the primary wave frequency and the natural frequencies of the electrons in the atoms making up the material. It should not be surprising, then, that the change in the speed of light through a particular material depends on the primary wave frequency. Since visible (white) light is composed of electromagnetic waves of many different frequencies, each color will be refracted by a different amount and pass through a material a a different speed. The phenomenon of dispersion is based on the observation that shorter wavelengths (i.e., higher frequencies such as found at the violet end of the visible spectrum) travel slower through a medium and are refracted at a larger angle than longer (redder) wavelength with lower frequencies (see Figure). Dispersion is responsible for the spectrum seen when sunlight passes through a prism or into water droplets to produce a rainbow.
Snel's law describes the relationship between the primary wave—in this instance called the incident wave—and the resulting wave, called the transmitted wave, as it moves through a material:
n1sinθ1 = n2sinθ2 .
The equation is based on original work of the French mathematician-philosopher Rene Descartes (1596–1650). The figure below shows a light wave traveling through a material medium of refractive index n1 striking the surface of another material medium of refractive index n2 at an incident angle of Θ1 (measured from the perpendicular to the surface). The resulting wave is refracted by an angle Θ2 as it travels through the medium of refractive index n2. If n2>n1, the resulting wave is bent towards the "normal" while n2 is bent away from the normal when n2<n1. Based on this simple principle, the refraction of light as it passes through a medium with a curved surface (e.g., a lens) results in waves striking different portions of the curved surface to be refracted by different angles to that all the waves eventually converge at a point called the focal point of the lens.
The Law of Reflection
In the case of a reflective surface, the figure below shows the relationship between the incident wave's angle of incidence Θ1 and the angle at which the reflected wave is transmitted Θ2. The two angles are the same, as seen in the figure. Since the angles are measured from the normal to the reflective surface, a curved reflector will cause the waves striking it surface to be reflected as various angles such that they eventually converge at a focal point.
Images
Two types of images may be produced by a system of lenses or mirrors. If the resulting waves actually pass through the position where the image is formed, the image is said to be real. On the other hand, if the resulting rays do not pass through the location where the images if formed, the image is said to be virtual.
When the object being observed is located beyond the focal point of a lens, the image will be inverted (upside down) while in the case of a mirror, the image will be reversed left to right.