Optics Lessons: Part 3 - Geometrical Optics

Let's face it. Optics is pretty complicated. Light travels in waves, and waves are not easy to analyze. That's why in this lesson we are going to try to explain as many phenomena of reflection and refraction as possible with only straight lines and angles instead of waves to represent light.

Before we begin, an important concept to note is the speed of light, or .

Now that you know the speed of light in meters per second, you will be able to apply it to many calculations that you may come across in geometrical optics and beyond. The two major phenomena in geometrical optics are reflection and refraction, both of which we will not go into extreme detail in this lesson.

Wave Fronts and Rays

Waves, whether it be electomagnetic or other, are conveniently described in terms of wave fronts. A wave front is the line or surface defined by adjacent protions of a wave that are in phase.--If an arc is drawn along one of the crests of a circular water wave moving out from a point source, all the particles on the line will be in phase.

The curvature of a short segment of a spherical or circular wave front is small. The segment may be approximated as a linear wave front or a plane wave front, just as we assume the surface of the Earth to be locally flat.

In a uniform medium wave fronts propagate outward from the source at a wave speed characteristic of the medium. For example, the speed of light travels fastest in a vacuum:

The geometrical description of a wave in terms of wave fronts tends to neglect the fact that the wave is actually sinusoidal. The concept of a ray simplifies the wave description even further. A ray is a line drawn perpendicular to to a series of wave fronts and pointing in the direction of propagation. A beam of light can be simplified and represented by a group of parallel rays or just a sinlge ray.

Reflection and Refraction

We said that we are only going to talk about light in terms of straight lines and rays in this lesson; we lied. For the most part, we are still going to talk about light in terms of straight lines and rays in this lesson, but to explain the concept of reflection, we will need to talk a little bit about the electromagnetic waves that compose light.

The two most important methods of light propagation are reflection and refraction. Reflection involves the absorbtion and re-emission of light by means of complex electromagnetic vibrations in the atoms of the reflecting medium. Refraction refers to the change in direction of a wave at a boundary where the wave passes fro m one medium into another.

A light ray incident on a surface is desribed by an angle of incidence. This angle is measured relative to a line perpendicular, or normal, to the reflecting surface. Likewise, the reflected ray is described by an angle or reflection. This relationship between these angles is given by the law of reflection:

The angle of incidence is equal to the angle of reflection.

Typically when a ray strikes a smooth surface separates two different transparent mediums (i.e. air and water) part of the ray is reflected and part of the ray is refracted. or transmitted into the second medium. There are two forms of reflection: specular reflection, which entails the reflection at a definite angle from a very smooth surface, and scattered reflection, which entails the reflection at various different angles. Both forms of reflection may occur in either opaque or transparent materials. Refraction is the light propagation where the transmitted light passes through two transparent mediums and consequently changes its direction. An index of refraction or refractive index of an optical material, denoted by n, is the ratio of the speed of light, c, in vacuum to the speed v in the material.

(index of refraction)

The index of refraction of an optical material (also called the refractive index) denoted by n above, plays a central role in geometric optics. It is the ratio of the speed of light c in vacuum to the speed v in the material.

Wave speed v is inversely proportion to the index of refraction n. THe greater the index of refraction in material, the slower the wave speed in that material. Failure to remember this point can lead to hopeless confusion!

The Laws of Reflection and Refraction

1. The incident, reflected and refracted rays and the normal to the surface all lie in the same plane.
2. The angle of reflection is equal to the ang;e of incidence for all wavelengths and for any pair of materials.
3. The ratio of the sines of the angles, a and b, where both angles are measured from the normal to the surface, is equal to the inverse ratio of the two indexes of refraction:

(law of refraction)

This experimental result, together with the observation that the incident and refrated rays and the normal all lie in the same plane, is called the law of refraction or Snell's law, after the Dutch scientist Willebrord Snell.

Total Internal Reflection:

According to Snell's law, the larger the angle of incidence, the larger the angle of refraction, or the greater distance the refracted ray diverges from the norm.

There is a restraint to this divergence, however. With an particular angle of incidence called the critical angle, the resulting angle of refraction is 90 degrees. This angle carries the refracted light ray along the boundary between the two mediums involved in the refraction. If the angle of incidence is greater than the critical angle, or in other words, if the angle of refraction is greater than 90 degrees, then the light is internally reflected. This state is known as total internal reflection.

The critical angle necessary for total internal reflection to occur can be found by plugging in 90 degrees° for the angle of refraction and simplifying to the equation on the right.

Fiber Optics works on the principle of total internal reflection. Light travels along a fiber, reflecting off the walls at an angle greater than the critical angle due to the narrow, linear shape of the fiber. This causes the light to be reflected into the fiber again and again. To learn more about fiber optics, feel free to refer to the Fiber Optics lesson on this page.

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