Chapter 1
Geometrical Optics

1.1 Characteristics of Lasers

This book is about optical calculation methods and the principles for applying these methods to actual optical devices. Most of these devices use lasers, so we will begin by briefly examining the characteristics of lasers. The following exposition will be especially beneficial for readers whose understanding of lasers is rather limited.

The term LASER is an acronym for "Light Amplification by Stimulated Emission of Radiation." Lasers have special characteristics that distinguish them from most light-emitting devices: a narrow, low-divergence beam (sharp directivity), and a very narrow wavelength spectrum (monochromaticity). These features of lasers make them ideally suited for the generation of high intensity beams.

The history of lasers goes back about half a century to 1960, when the first working laser was demonstrated. Lasers are now widely used in a variety of fields, including optical storage (e.g., CD drives and DVD drives), fiber-optic communication, manufacturing (especially for cutting, bending, welding, and marking materials), scientific measurement, and medicine. This diversity of application is due to the following four properties of lasers, which give them a commercial and scientific edge over other light-emitting devices.

1. 1. Monochromaticity: Radiation that has a very narrow frequency band (or wavelength band) is said to possess the property of monochromaticity. Because the band is so narrow, the radiation can be regarded as having a single frequency (or alternatively, a single wavelength). Laser light typically has a very narrow frequency band (or wavelength band), which makes it an ideal example of the property of monochromaticity. Sunlight, by contrast, has a very broad band spectrum, with multiple frequencies and wavelengths.
2. 2. Beam directivity: The term directivity, when used in relation to lasers, refers to the directional properties of the electromagnetic radiation they emit. A laser beam exhibits a very sharp (narrow) directivity, allowing it to propagate in a straight line with almost no expansion. By contrast, normal light sources such as flashlights and car headlights have broader directivity than lasers, so their beams cannot travel as far.
3. 3. Coherence: If split laser beams which were emitted from the same source are superimposed, a fringe pattern will appear. This fringe pattern is never observed in an isolated beam. We refer to this phenomenon as interference caused by the wave characteristics of light. If these two beams traveling onto the same plane are superimposed while they are in phase (with their crests and troughs lined up), the resultant beam will appear brighter, but if these waves are superimposed with a 180° phase difference (i.e., if crests and troughs are superimposed), the beam will appear dark and the resultant amplitude will be zero. A laser can easily generate interference patterns because of its coherence (uniformity of phase). Sunlight cannot readily generate interference patterns, due to its incoherence: because its coherence time and coherence length are very short, and the phases of the superimposed waves will not come into phase very easily. Sunlight will only exhibit coherence over a very short interval both of time and space.
4. 4. High concentration of energy (high intensity): A sheet of paper can be burnt simply by focusing sunlight on it, using a convex lens. A laser is much more concentrated: it can even weld two pieces of steel together. This is not merely due to the high power of the laser, but also because of the extremely high intensity of the laser beam, where the light energy is narrowly focused. It is relatively easy to concentrate a laser beam on a small target. A high intensity beam can be generated very easily using a laser.

All of these characteristics of lasers can best be summed up by the word "coherent." A laser is both temporally coherent and spatially coherent. What this means is that a laser has a uniform phase over time at an arbitrary point in space, and it also has a uniform phase in space at an arbitrary point in time. Thus a laser has a uniform phase both in time and in space. A laser radiates a single-wavelength (more precisely, a very narrow wavelength spectrum) beam with a constant phase and it can propagate in a specific direction. An electric light, by contrast, radiates a multitude of different wavelengths at various phases and in all directions. Thus it is both spatially and temporally incoherent.

1.2 The Three Fundamental Characteristics of Light Which Form the Basis of Geometrical Optics

To understand geometrical optics, which is the most basic form of optics, we need to first study the fundamental characteristics of light - including light emitted by lasers. The following properties of light are confirmed by everyday experience:

1. 1. Light rays travel in straight lines in a uniform medium.
2. 2. Light rays are independent of one another.
3. 3. Light rays can be reflected and refracted: they change their direction at a boundary between different media, in accordance with the laws of reflection and refraction.

The whole science of geometrical optics can be derived from these three characteristics of light.

1.2.1 Light Rays Travel in Straight Lines

Many common optical phenomena attest to the fact that light rays travel in straight lines. For instance, when the sun is shining outdoors, a tree casts a shadow whose shape is identical with its own. Without using any lenses, we can construct a pinhole camera that can capture the image of an object, simply by making a pinhole in one of the walls of a black box as shown in Figure 1.1.

Figure 1.1 Pinhole camera

1.2.2 Light Rays Act Independently of One Another

Figure 1.2 illustrates the independent action of light rays using the example of three spotlights whose light is of different colors: red, blue, and green. When we irradiate the same area on a white sheet of paper with these three spotlights, we perceive the light as "white." However, if we replace the paper with a mirror, and then project the reflected light onto another sheet of paper, the three separate beams of light reappear in their original colors of red, blue, and green. The fact that these three beams reappear in their original colors demonstrates that light rays coming from different sources act independently of one another after being reflected by the mirror. Likewise, the fact that the light from the three spotlights appears white when they are all focused on the same area of paper can be explained in terms of the constituent light wavelengths (red, blue, and green) reaching our eyes and acting on our retinas independently. It is the superposition of waves which causes us to perceive them as white.

Figure 1.2 Independent action of light rays

1.2.3 Reflection of Light Rays

As shown in Figure 1.3a, when a light ray is reflected by a mirror at the point of incidence O, we can define the normal as an imaginary line through point O perpendicular to the mirror. The reflected ray will lie in the same plane as the incident ray and the normal to the mirror surface, and the angle of reflection will be the same as the angle of incidence. We are all familiar with this phenomenon from everyday experience. It can be described by the following equation:

Figure 1.3 (a) Reflection of a light ray and (b) refraction of a light ray

1.2.4 Refraction of Light Rays

An object lying in a tub of water appears to be at a shallower depth than it actually is. This phenomenon can be explained by the refraction of light at the interface between the water and the air. As shown in Figure 1.3b, when a ray is refracted at a boundary plane between different media, the relationship between the angle of incidence and angle of refraction can be described by the following equation (Snell's law):

where

1.3 Fermat's Principle

The law of rectilinear propagation and the law of reflection and refraction of light rays can both be derived from Fermat's principle, as explained below [1].

1. i. The velocity of light is inversely proportional to the refractive index of the medium in which light propagates. (In other words, light travels more slowly in a medium having a higher refractive index.)

   The velocity of light in a vacuum is a constant, c:

   1.3

   The velocity of light v in a medium having refractive index n is:

   1.4

   In optical calculations, the optical path length L is defined as follows:

   1.5

   where L0 = physical length traversed by light in the...