Terrestrial Mirages

Mirages are not illusions, any more so than are reflections in a pond. Reflections of plants growing at its edge are not interpreted as plants growing into the water. If the water is ruffled by wind, the reflected images may be so distorted that they are no longer recognizable as those of plants. Yet we would not call such distorted images illusions. And so it is with mirages. They are images noticeably different from what they would be in the absence of atmospheric refraction, creations of the atmosphere, not of the mind. An example of a true illusion is the moon illusion, a moon seen to be larger on the horizon than overhead. This seemingly enlarged moon is a creation of the mind, not the atmosphere. And yet the moon illusion is still often attributed to atmospheric refraction even though this has been known not to be true for at least 1000, possibly 2000 years, and can be verified by simple measurements of the angular size of the moon at different elevations.

Wavelength (nm)

Figure 8.17: Refractive index of dry air at a pressure of one atmosphere and for the two temperatures noted. From the compilation by Penndorf (1957).

Wavelength (nm)

Figure 8.17: Refractive index of dry air at a pressure of one atmosphere and for the two temperatures noted. From the compilation by Penndorf (1957).

Mirages are vastly more common than is realized. Look and you shall see them. Contrary to popular opinion, they are not unique to deserts. Mirages can be seen frequently even over ice-covered landscapes and highways flanked by deep snowbanks. Temperature per se is not what produces mirages but rather temperature gradients.

Because air is a mixture of gases, the polarizability for air in Eq. (8.37) is an average over all its molecular constituents, although their individual polarizabilities are about the same at visible and near-visible wavelengths. The vertical refractive index gradient can be written so as to show its dependence on pressure p and absolute temperature T by way of the ideal gas law p = NkBT and Eq. (8.37):

Pressure decreases approximately exponentially with height [i.e., exp(-z/H)], where the scale height H is about 8 km. The first term on the right side of Eq. (8.39) is therefore about 0.1 km-1. Temperature usually decreases with height in the atmosphere. An average lapse rate of temperature (i.e., its decrease with height) is about 6 °C km-1. A characteristic temperature in the troposphere, within about 15 km of the surface, is 280 K. Thus the magnitude of the second term in Eq. (8.39) is about 0.02 km-1. On average, therefore, the refractive index gradient is dominated by the vertical pressure gradient. But within a few meters of the surface, conditions are far from average. On a sun-baked highway your feet may be touching asphalt at 50 °C while your nose is breathing air at 35 °C, which corresponds to a lapse rate thousands of times the average. Moreover, temperature near the surface can increase with height. In shallow surface layers, in which pressure is nearly constant, the temperature gradient dominates the refractive index gradient. In such shallow layers mirages, which are caused by refractive index gradients, are seen.

Figure 8.18: Deviation of incident light because of refraction by a uniform slab with refractive index n.

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Figure 8.18: Deviation of incident light because of refraction by a uniform slab with refractive index n.

Cartoonists with fertile imaginations unfettered by science and careless textbook writers have engendered the notion that atmospheric refraction can work wonders, lifting images of ships, for example, from the sea high into the sky. A back-of-the-envelope calculation dispels such notions. The refractive index of air at sea level is about 1.0003 (Fig. 8.17). Light from free space incident on a uniform slab (Fig. 8.18) with this refractive index is displaced from where it would have been in the absence of refraction by an angle S given by Snel's law sin = n sin = n sin($; — S) = n(sin cos S — sin S cos ), which at glancing incidence = 90 °) yields c 1

Because n « 1, and hence S C 1, we can approximate Eq. (8.41) as

For n — 1 = 0.0003, Eq. (8.42) gives an angular displacement of about 1.4°, which is a rough upper limit.

Trajectories of light rays in nonuniform media can be expressed in different ways. According to Fermat's principle of least time, which ought to be extreme time, the actual path taken by a ray between two points is such that the path integral n ds

is an extremum; strictly, this integral is stationary, which includes the possibility of a point of inflection. That is, of all possible paths between 1 and 2, that taken by a light ray is such that Eq. (8.43) is either a minimum (least time) or a maximum (greatest time). Why time?

Figure 8.19: Ray trajectory from object point O to image point I in air with a temperature decreasing at a rate more than 100 times the average rate in Earth's atmosphere. To an observer at I it is as if the light from O comes from an object displaced downward from the line of sight OI by an angle 5. Note that the horizontal and vertical scales differ by a factor of about 600, which creates the impression that 5 can be much larger than it is in reality (~1°).

Figure 8.19: Ray trajectory from object point O to image point I in air with a temperature decreasing at a rate more than 100 times the average rate in Earth's atmosphere. To an observer at I it is as if the light from O comes from an object displaced downward from the line of sight OI by an angle 5. Note that the horizontal and vertical scales differ by a factor of about 600, which creates the impression that 5 can be much larger than it is in reality (~1°).

Because n is the ratio c/v, where c is a universal constant (the free-space speed of light) and v is the phase speed, and hence except for the constant factor c, Eq. (8.43) has the dimensions of time. But this time is not the time it would take a signal to propagate from 1 to 2 except in a non-dispersive medium (see Sec. 3.5). The principle of least time has inspired piffle about the alleged efficiency of nature, which directs light over routes that minimize travel time, presumably giving light more time to attend to important business at its destination.

The scale of terrestrial mirages is such that in analyzing them we may pretend that Earth is flat. On such a planet, with an atmosphere in which the refractive index varies only in the vertical, Fermat's principle yields a generalization of Snel's law:

where $ is the angle between the ray and the vertical direction. We could have bypassed Fermat's principle to obtain this result.

A ray in such a medium is a curve z = f (y), where the yz-plane is the plane of incidence and z is the vertical coordinate. The slope of this curve is dz 1

Square Eqs. (8.44) and (8.45) and combine them to obtain

Take the derivative with respect to y of both sides:

Here n « 1, and if we restrict ourselves to nearly horizontal rays (i.e., ■ « n/22), we can set both n and C equal to 1 in Eq. (8.47) to obtain the approximate differential equation satisfied by nearly horizontal rays:

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