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The Nature of Light

Page 4:  More Reflections

Other examples of total internal reflections include mirages. A common mirage is the appearance of water-like reflections in the linear distance on a hot day.

Figure 4.1:  Mirage in desert sand.

Figure 4.1 shows what appears to be water around a building in the desert. However, there is no water there or anywhere else in that scene. The building is on sandy ground, not on water.

Figure 4.2:  Closeup of Figure 4.1.

The building and its surroundings are being reflected into a mirror image below the building as if there was a reflecting pool of water there. In fact, there is only sand there.

What is occuring is that the sand is very hot, causing the air near the ground to heat up. On a hot day, that layer of air may be 20 degrees Celsius hotter than the air above, causing it to have a lower refractive index than the air above.

In hotter air, the molecules of the air vibrate more, pushing themselves further apart, causing the air to become less dense; with molecules pushing apart, there becomes less molecules per cubic millimeter of the air near the ground.

The air close to the ground becomes a rare optical medium, with the air above more dense (Fig. 3.2 upside-down). When viewed from far away, light from above approaches the rarer medium with approach angles (alpha-2 in right side of Fig. 3.2) that are too shallow to refract into the rarer medium (hot air below) and thus are reflected back into the dense medium (cooler air above) by total internal reflection toward the viewer.

On flat ground, the reflections are like shown in Fig. 4.1 and 4.2 above, with a mirror image reflected below, in this case the building, surrounding objects and background mountainside all reflected upside down in a mirror image below all of those objects. Where there are no objects or land to be reflected, the sky is reflected.

If the ground is not flat, the interface between higher cooler air and lower hot air may be nonuniform and make some of the reflections rightside up instead of upside down, because of almost unnoticable concavities causing light ray caustics (see van der Werf, “Noninverted images in inferior mirages”, Applied Optics, Oct. 2011).

Frustrated Total Internal Reflection

There is a fascinating exception to total internal reflection. It turns out that if another dense medium is close to the propagating dense medium, with rare optical medium separating the two dense media, photons will appear in the second dense medium, without photons appearing in the intervening rare medium.

This is called frustrated total internal reflection. It is also called evanescence or optical tunneling, and is due to evanescent waves of electromagnetic radiation.

Figure 4.2:  Tunneling across rare optical medium d. [Lucas]

Evanescence can be used to tap and spoof fiber optics cables.

External Reflection

So far we have been considering light reflecting in dense optical media, such as water, glass, etc. We now consider light reflecting in a rare medium, such as air.

When light transmitting in air approaches a dense medium, if the angle of approach (alpha-1 in left side of Fig. 3.2) is shallow enough, all of the light will be reflected off the interface between the rare and dense media back into the rare medium. This always happens before the angle of approach vanishes, and is the reason why you see reflections on the surface of a lake at a distance.

As the angle of approach becomes smaller, before all of the light is reflected, some of the light will be reflected (into the rare medium) and some will be refracted (into the dense media).

In the case of light approaching glass, not much light is reflected when the angle of approach is head-on (perpendicular to the glass surface) or nearly head-on. As the angle of approach becomes smaller, for many angles most of the light is still refracted into the glass. But as the angle of approach gets smaller than about 30 degrees, a sharp drop-off occurs in the amount of light refracted, with the extra light (that is not refracted) reflected back into the air. This is illustrated in the graphs of Fig. 5.1.3 in Duffie and Beckman, Solar Engineering of Thermal Processes 4th ed., and Fig. 9.18e in Lechner, Heating, Cooling, Lighting 4th ed.

Reflecting all incident light in a rarer medium is called mirroring or total external reflection. All materials do total external reflection (reflect all light) if the approach angle of the light is shallow enough.

“When an electromagnetic wave hits the mirror surface, its electric field sets into vibration the electrons which then become emitting sources: a reflected wave is thus generated.”
Chartier, Introduction to Optics, p. 81

Total external reflection allows xray telescopes to be built. Xrays are more energetic than visible light, and therefore need a shallower approach angle to achieve total external reflection. Even with xrays, total external reflection occurs before the approach angle vanishes.

Figure 4.3:  Xrays are so energetic, they would not be reflected by “mirrors” other than at shallow angles of approach. [NASA]

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