Refraction and Snell’s Law
Recall that refraction is the bending of a light ray as it passes across the boundary between two materials of different optical properties, e.g., as when a light ray moves between water and air. The ray moves in a straight line before and after the refraction at the boundary.
In 1621, the Dutch mathematician and physicist Snell created the self-named law describing mathematically the relationship between the angles of incidence and the angle of refraction. It is based on the ratios of the Index of Refractions of the two materials through which the ray is travelling. A material’s index of refraction (IR) value compares the speed of light (3 * 10 8 m/s) in a vacuum with its speed in the material. Expressed in a mathematical way, this definition of the IR of a material becomes an equation: IR = speed of light in vacuum Still with equations, we can describe the relationship between
speed of light in material the optical velocities and IR’s of a couple of transmitting materials in the following equation. See that the relationship is an inverse one, i.e., the optical speed of a
v1 = IR2 material is the inverse of its IR. In other words, an optically “slow” medium has a large
v2 IR1 index of refraction.
Now, another way of thinking about the index of refraction is “bending power.” There are two situations of refraction – when the ray moves into a material with a lower IR, the refracted ray bends away from the reference line (the “normal”). The other situation is when the ray enters a material with a larger IR; now, the ray bends toward the normal, toward the reference line.
Look at the following equation. It is the form of Snell’s Law that allows us to describe mathematically the path change of the light ray as it moves from one material into the other. See that the equation relates IR into = sin ∡ i the IR’s of the materials through the light ray passes and the ray’s angles of
IR from sin ∡ r incidence and of refraction as it crosses the boundary where the two materials meet. The material in which the ray originates is the “from” material; the material toward which it moves is the so-called “into” material.
Refraction Effects Sometimes the effects of refraction are easy to see. Just lower a straw into a glass of water and see it “break.” Refraction makes it look like the two sections of the straw have separated. Another common example is the rainbow sparkle from a suncatcher. Taken to a huge scale, this suncatcher example becomes the creation of rainbows when light rays refract from raindrops. In this situation, the sun must be behind us and the raindrops in front, everything at the proper angle and height. Sunlight enters the raindrops, reflects from their back walls and refract out the front, toward us. The geometry of the situation lines up the colors from violet (on the inside of the continuous spectrum arc) to red (on the rainbow arc’s outside). Another effect of refraction is the sparkle and fire of diamonds when their faces have been cut at angles resulting in maximum refraction and internal reflection.
If we investigate the rainbow (spectral) colors, the optical properties of each one differs in a predictable, progressive fashion from those of its neighbours. For example, red has the longest wavelength at 750 * 10 – 9 m or 750 nanometers, nm, it shows the least refraction and it has the fastest speed in glass. In contrast, violet has the shortest wavelength (450 nm), the greatest diffraction and the slowest speed in glass. A prism splits white light into its components because, since each rainbow color refracts to a different degree, they spill out of the prism all in different paths. As noted above, this is what we are seeing when we look at the pretty swirl of colors coming out of a window sparkler twisting in the sunlight.
Critical Angle In the case of a ray moving into a material with a lower IR, recall that the refracted ray bends away from the reference line, creating a large angle of refraction. If the angle of incidence happens to exceed the critical angle, the ray is unable to escape across the boundary and instead reflects back into the original material at an angle equal to the angle of incidence. Total internal reflection occurs. You can predict the critical angle by the following equation. In it, the “i c” stands for the “angle of incidence called the critical angle”. If the IR of the material into which the ray tries to move is very small, the
sin ∡ i c = IR into critical angle becomes large and almost any angle of incidence will be too large
IR from to allow the refracted ray to escape and instead, total internal reflection will be the result.
Some optical devices require total internal reflection. One example are binoculars. To work
well in low light, the objective lenses are made
large to gather as much light as possible and so
must be located much farther apart than are our
eyes. After the light rays pass through the objective
lenses, they must be directed toward the close set
ocular (eyepiece) lenses and this is done through
reflection. High quality glass prisms are used because
they absorb less light than would plane mirrors.
A second use of total internal reflection is with fiber optic cables. These cables have come into their own as efficient ways of moving the almost unimaginable amounts of information – voice, images, and text – flying among telephones and computers around the globe. And, they are still a mainstay in minimally invasive operations and in investigative procedures in hospitals. Also, a variety of high tech repairs rely on them by allowing the technicians a look inside cramped devices without the nuisance of disassembly. In each case, light rays are introduced to the glass fiber interior and, as they move along the length of the cable, they must not leak out because that is lost information, lost revenue. Total internal reflection is responsible for keeping the light information inside the cable where it belongs. As a ray approaches the surface of the glass fiber core, it encounters a coating with a lower IR, the angle of incidence will exceed the critical angle and so total internal reflection occurs. The light ray is unable to escape from the glass fiber core; anytime it approaches the surface of the cable, total internal reflection heads it back into the cable.