Mirage


Contents

  1. Introduction
  2. Light Rays
  3. Terrestrial Parameters
  4. The Horizon
  5. Mirage Phenomena
  6. References

Introduction

Mirage is a French word meaning exactly the same thing as in English. It is a "thing seen," but an illusory thing. The word may refer to what are better known as hallucinations, produced by drugs or physical stress, but we are concerned here only with mirages produced by curved light rays. Mirages are, in fact, a kind of optical illusion where the natural and reasonable assumptions of the visual sense are led astray, in this case by the curvature of light rays, which are so often straight lines from the object seen to the eye, as the eye always assumes, but here are not. Mirages have been studied scientifically for a long while, and the literature is immense (although, unfortunately, inaccessible at the present time and place). The purpose of this article is to explain the nature and causes of the mirage, since this is only sketchily done, or not done well, in common reference materials. The reference by Minnaert surveys the observed phenomena in an excellent and interesting manner.

Light Rays

The concept of light rays will be quite familiar to most readers. Electromagnetic energy in a narrow wavelength range (400-700 nm) proceeds from an object point along a definite path to the observer's eye carrying energy that gives information on the object and its illumination. Of course, we can extend the concept to radiation that is not visible. The ray is, in general, a curved line normal to a family of wavefronts indentifying points of equal phase. If the wavefronts are not limited in the neighborhood of the ray, then the rays accurately describe the propagation of the light. If the wavefronts are limited, or there are caustic surfaces, then interference and diffraction effects may appear. In this article, we will assume that such effects are negligible, as they indeed are in most visual observation of the world.

The important parameter in ray propagation is the index of refraction n of the medium through which the light is passing. It may be a function of position, or even of the time. In the absence of matter, n = 1, and in material media n > 1. Diamond, with n = 2.4, has one of the highest indices of any transparent substance. Glass generally has n = 1.5, though variations from 1.4 to 1.7 are not unusual. In gases, n is close to 1, and (n - 1) is proportional to the density. The index of refraction may be defined operationally in terms of Snell's Law, n sin θ = n' sin θ', where primed and unprimed refer to media on either side of a smooth plane interface, and θ is the angle between the ray and the normal to the interface. The phase velocity of the radiation is determined by the index of refraction through v = c/n, where c is the speed of light in vacuum. If the index of refraction does not vary much with wavelength, the phase velocity is also the velocity of energy propagation. These things will not enter into our discussions, however.

The path of a ray is given by the variational principle δ∫ n ds = 0, called Fermat's Principle. The distance measured along the ray is s, so that n ds is c dt, the distance the light would travel in free space in the same interval of time, called the optical path length. By Fermat's Principle, the optical path length along the actual ray must be the same as along any closely neighboring path. This means, in practice, that the optical path length must be a minimum for the actual ray. This variational principle may be solved by standard methods of the variational calculus in terms of a function of position called the eikonal, E. The surfaces E = constant are the wavefronts, to which the rays are normal. E satisfies the differential equation (grad E)2 = n2 (which is not easy to solve in the general case), and a unit vector in the direction of the ray is given by s = (1/n) grad E. If r(s) is the parametric equation of a ray in terms of the distance s along it, then (d/ds)[n(dr/ds] = grad n is a differential equation for the ray, which is much easier to solve. Therefore, if we know the index of refraction as a function of position, we can determine the path of a ray connecting any two points in the medium.

The reader probably has shown as an exercise at one time or another that both Snell's Law and the Law of Reflection can easily be derived from Fermat's Principle, by the simple use of the calculus to determine the paths of minimum optical path length. If n = constant, the ray equation becomes d2r/ds2 = 0, which can be immediately integrated to find r = ss + a, where a and the unit vector s are arbitrary constant vectors chosen to satisfy the boundary conditions. This is the well-known result that the path of a ray in a homogeneous medium is a straight line, something our visual perception believes implicitly.

Terrestrial Parameters

Since we will be studying light rays at the surface of the earth, passing through the atmosphere, let's review the numerical magnitudes that we will need. In most cases, air is represented well enough by the ideal gas law, so that the density ρ = Mp/RT, where T is the temperature in K, p the pressure in pascal, M the molecular weight, and R = 8314.51 J/mol/K. Then, ρ will be in kg/m3. For air, M = 28.966 below approximately 90 km altitude. Pressure in meteorology is usually expressed in millibars (mb), where 1 mb = 1000 Pa. Standard atmospheric pressure at the surface, 1 atm, is 1013.25 mb, or 760 mmHg, or 29.921 inHg. You should be aware that the atmospheric pressure quoted in weather reports is reduced to sea level in a rather foggy way, and is not the actual atmospheric pressure, unless you happen to be at sea level. At Denver, 1609 m elevation, the standard pressure is 835.58 mb or 24.67 inHg. Above 90 km, the density of the atmosphere is so low that it will have no effect on electromagnetic propagation, and n can be taken as exactly unity. The density at sea level under standard conditions is 1.2250 kg/m3 (which is a lot larger than you might expect!).

The standard temperature at sea level is 288.16K (15°C) and at 1609 m, 277.70K (4.6°C). The standard lapse rate of 6.5°C per 1000 m prevails up to the tropopause, at about 11 km altitude (though this varies considerably with latitude), above which the temperature is roughly constant at 216.66K (-56.5°C), showing what the temperature of the earth would be in the absence of the "greenhouse" effect. The rapid decrease in pressure with altitude has the major effect on the index of refraction, the effect of temperature being much less important. This contrasts sharply with the propagation of sound, where temperature has the major effect on velocity, which is independent of density to a very good approximation. Moist air has a smaller density than dry air, and so a reduced index of refraction. To allow for this, in the correction for pressure and temperature, use P - 0.156Pm, where the atmospheric pressure P and the water vapor pressure Pm are in mb.

The index of refraction of dry air is given pretty well by the formula (n - 1) x 10-7 = 2875.66 + 13.412/λ2 + 0.3777/λ4, where λ is the wavelength in Å divided by 10-4. This formula applies to 0°C and 1 atm. For any other temperature or pressure, multiply by 0.26958(p/T), where p is in mb and T in K. For 550 nm, the formula gives n = 1.0002924. The index is larger in the blue, and smaller in the red, so the dispersion of air is normal. However, it is very small, and dispersive effects are usually negligible. Refraction in ice or raindrops is much more dispersive, and produces well-known color effects. The spectral character of light in the atmosphere is much more strongly affected by scattering and absorption. Rayleigh scattering is proportional to λ-4, so blue light is more strongly scattered by density fluctuations than red light. Water vapor has some rotational absorption in the red that can turn light blue or green under some conditions.

An interesting example of the effect of meteorological conditions on the index of refraction is presented by the HP 3805A distance meter. This is an old infrared instrument, but I happened to have the instruction manual on hand. There are tables for the correction to be applied for temperatures from -20°C to +60°C, and pressures from 450 mmHg to 800 mmHg, but curiously, no correction for humidity. The instrument is advertised to have an accuracy of ±(7 mm + 10 mm per km). Let's see what effect the absence of a humidity correction has. At 20°C, 100% humidity corresponds to a partial pressure of water vapor of 17.535 mmHg. This means that when the barometer reads 1013 mb, the air pressure is 989 mb, while the water vapor pressure is 23 mb. The presence of the water vapor makes the medium less dense, and so the index of refraction is decreased. The net effect can be expressed as an effective pressure po - 0.156pwo, where po is the total pressure, and pwo is the water vapor pressure. For our example, the effective pressure is 1009 mb. This is a difference of 4 mb, or about 4 parts in 1000, and will be reflected in the reported distance. This is an error of 4000 mm per km, much larger than the advertised accuracy, even at 10% relative humidity! It is clear that humidity corrections are necessary, but HP ignored them. Incidentally, the manual gives no hint of how the instrument works, and the correction is an arbitrary scale number, so that even if you knew the corrections, you would not be able to use them. HP apparently did not anticipate that their instrument would fall into the hands of intelligent life.

The equatorial radius of the earth is 6378.14 km, while the effective radius at 45° latitude is 6357 km. The polar radius is 6356.75 km. The flattening, f = (a - b)/a = 1/298.57. In atmospheric optics, the earth can be considered to be a sphere of any radius in the range given; use 6378 km if this figure has stuck in your mind as the radius of the earth.

The Horizon

Horizon is from the Greek for "division," specifically the line of division between the sea and the sky, which is "horizontal." It is used as a reference point for the measurement of altitudes of stars, sun and moon by mariners using a sextant. If the earth were flat, its direction would be perpendicular to the vertical at any point. Since the earth is round, it is a little below that, by the amount of the dip. If the observer's eye were on the surface, the horizon would be immediate, and the dip would be zero. As the eye is raised, the horizon recedes and the dip increases. More and more of the surroundings can be seen. From an aircraft, a wide circle of circle is seen, bounded by the distant horizon on all sides, and the dip is noticeable even to the naked eye. Although ancient navigation instruments were not precise enough to detect the dip, the sinking of ships and the coast below the horizon was evident to all, and realized to be an effect of the sphericity of the earth, which was commonly accepted in the Hellenistic world before 300 BCE. Only more recent, more ignorant people have assumed a flat earth, and not thought about this well-known experience of all who have gone to sea.

In the figure, O represents an observer at an altitude h, and OQ a straight light ray tangent to the earth at P, the horizon point. If d = OP is the distance to the horizon, then Pythagoras says that d2 + R2 = (R + h)2. Then, d2 = 2hR + h2. Since h will be on the order of metres, and R is 6378 metres, the second term will be much smaller than the first, and can be neglected to a good approximation. We find that d = √(2hR) = 3571 √h, where h is in metres. Also, if D is the dip angle, cos D = R/(R + h) = (1 + h/R)-1 = 1 - (h/R) + ... = 1 - D2/2 + ..., so approximately D2 = 2h/R, and D = 0.000560 √h radians, or D = 1.925' √h. Since eye level is approximately 2 m, this gives d = 5050 m and D = 2.7' of arc. The flat plains of the Llano Estacado in Texas are flat enough to detect the curvature of the earth, with grain elevators sinking beneath the horizon that is about 3 miles off.

The path of a horizontal light ray is, however, not a straight line. Consider a wavefront 1000 m high. At the surface the index is 1.0002924, but at 1000 m it is about 1.0002666. This means that the top of the wavefront is moving 7740 m/s faster than the bottom. In a microsecond, the light moves about 300 m ahead, but the top moves 7.7 cm further than the bottom. This tilts the wavefront, deflecting the ray downward slightly. The radius of curvature of the ray can be estimated at about 38,000 km. If the earth was of this radius, you would see the back of your head if you looked horizontally. This is, however, about 6 times the radius of the earth, so not only is the ray pretty straight over a short distance, but it eventually gets away from the earth. Such a ray is shown as OQ' in the diagram, and the horizon has moved to P', further away than P. The dip D has also been reduced. The calculation has been carried out to some accuracy by Woodard and Clemence, who find that D' = 1.75' √h, about a 10% decrease. Using 38,000 km, I have found d' = 5534 km, about a 10% increase. This is a good estimate of the effect of ray curvature at the surface.

If the ray OQ' is continued on, it soon leaves the atmosphere and proceeds in a straight line in space. The angle between the final direction and the horizontal at P' is 33' 51" according to the Pulkovo tables of refraction (given in Tricker). The angular diameter of the sun at mean distance is 32', and that of the moon, 31', so when the sun's lower limb is resting on the horizon, the sun is actually already completely below the horizon at the point of observation. The full moon can be seen in the east when the sun is still fully visible in the west. Refraction gives us four minutes more of sunlight than we would otherwise enjoy. To this must be added the dip D, of course, if we are observing from an elevation. At 35,000 ft, the dip is nearly 3°, so we get an extra 12 minutes of sun.

As we raise our eyes to the heavens, the curvature becomes less until it vanishes in the zenith, when the ray goes straight up. The difference between the actual elevation and the apparent elevation is called the refraction. Astronomical observations must be corrected for refraction unless they are of a very low order of precision. At most the refraction is 34', at the horizon. It would be much simpler to calculate the refraction if the earth were flat, but it is not. At small zenith distances z (90° - altitude) the refraction is closely approximated by ξ = K tan z, where K = 16.271" (P - 0.156Pm)/T, where P is the atmospheric pressure in mb, and Pm is the water vapor pressure, also in mb. Under standard conditions at sea level, K = 57", or about 1', the refraction at z = 45°. At a z = 80°, this formula gives 5° 23', while a more accurate value is 5° 13'. It should be remembered that refraction varies with pressure, temperature and humidity.

Under normal conditions, the curvature of light rays has little effect, except on accurate astronomical observations. In surveying, with sights less than a mile in length, light rays are usually considered straight lines. When electromagnetic radiation is used for distance measurement, whether radio, infrared or visible (laser), the index of refraction must be known to assure accurate results, though the assumption of rectilinear propagation is normally adopted. This is as true, of course, with the Global Positioning System as with any other method. GPS and similar methods render unnecessary the corrections for ray curvature that once had to be made in long triangulation sights. For casual observations, then, the assumption of rectilinear propagation made by our visual sense is valid.

Mirage Phenomena

We now turn to the unusual phenomena in which illusions, that is, perceptions at variance with reality, are exhibited. In most cases, we will be satisfied by qualitative explanations, since the actual conditions vary widely, and every experience is unique. These happen when the light rays are significantly deviated by large, unusual changes in density. These changes often occur in layers, with the vertical variations much more abrupt than the horizontal ones. Also, the light rays involved are almost horizontal, and the effects are upward or downward curvatures where there is a strong vertical gradient of density. These gradients will be much larger than the average gradients than cause a radius of curvature on the order of 38,000 km, which we have considered above. Of course, these variations will naturally involve the perception of the distance and dip to the horizon. We assume that we are observing objects at a distance of a mile or more, but many phenomena also can be observed on a smaller scale.

The most commonly observed mirage is the inferior mirage, which is usually seen as a shiny patch on a hot surface that is the reflection of the sky. Summer sunlight may raise the temperature of an exposed surface as much as 80°C or more than the air not far above it. A strong temperature gradient is established just above the surface, and with it a relatively thin layer of reduced density. Shimmering, caused by the turbulence induced by the strong temperature gradient, is an evidence of this. Rays incident at small angles with the surface (angles of incidence near 90°) are reflected. By Snell's Law, n' = n sin θ, where n is the index of refraction of the air outside the layer of strong heating, n' the minimum index in the heated layer, and θ is the limiting angle of incidence for total reflection. For glancing incidence, total reflection is possible for temperatures commonly occurring. This reflection must be distinguished from the increased reflectivity of a rough body at small angles of incidence, which can give the same effect. This reflectivity can be eliminated by a thin layer of sand when performing an experiment. In most cases, however, it is quite clear that we are dealing with the inferior mirage, and not with normal reflection.

The usual form of the inferior mirage is shown in the figure. The air is normal except for a thin heated layer, so the rays are mostly straight lines, but strongly curved upwards in the heated layer. An object AB appears to the observer at O reflected at A'B', inverted and below the horizon, in front of the shining pool that is the reflection of the sky. Except for inversion, the object is not distorted. Note that the rays to the top and bottom of the object are crossed. This crossing is necessary to produce the reflection. The object AB is also visible by direct rays in the normal way. It is not very unusual to see such inverted images, but in many cases there is nothing to be imaged, so just the shining pool of the sky is seen. The inferior mirage with this geometry is frequently seen while driving in the summer, and cars ahead may be seen reflected in it. It can also be seen on walls exposed to the sun by placing yur eye close to the wall. Dark rocks on a sandy plain can appear to be vegetation around open water, the classic desert mirage of fable.

A less common variant of the inferior mirage is shown at the left, which is known as stooping. The observer at O sees a reduced image A'B' of an object AB against the sky. The curvature of the rays is greatly enhanced in this diagram to show what is happening. Note that there are no direct rays in this case, and no reflection, since there are no crossed rays. The erect image A'B' is all that is observed. The point P is observed on the horizon, and objects further away are invisible, as below the horizon. The curvature of the rays for this phenomenon is induced by an abnormal density aloft, as by a cold layer on top of a warm surface layer. Since the change in density is not large, this is a subtle effect, but one that should exist rather commonly, but in amounts too small to be perceived without careful observation and comparison of far and near objects. A distant object will seem to be smaller with respect to a nearby object than it normally is, and the horizon will draw in, making the earth seem smaller.

A superior mirage is created when the hot layer is above the observer. In meteorology, this is called an inversion, since temperatures are usually lower aloft, not higher. A very strong inversion is necessary before the superior mirage is noticeable. There is no analogue to the heated surface and sharp temperature gradient above it in this case, so the superior mirage with crossing of rays and reflection is seldom observed. The superior mirage is also called the polar mirage, because it is most frequently observed in that region, over a very cold surface. Waves of cold water breaking on a very hot beach may produce a superior mirage if the eye is low enough, perhaps 30 cm above the sand.

The diagram shows ray conditions in a superior mirage, in the phenomenon known as looming. Ray curvature is again greatly enhanced to make clear what is observed. The horizon is seen apparently above the horizontal, so the earth seems to be saucer-shaped, and the horizon is distant. An object AB appears as an enlarged erect image A'B'. As in stooping, there is no direct view with straight rays. This figure assumes that the density gradient is uniform from the surface up, giving approximately circular rays. A very strong general inversion, as above snow-covered ground in cloudy weather, may produce looming, which lends a very mysterious appearance to the landscape. It may be possible to see objects that are normally hidden behind obstacles appear above them and looking closer.

Looking off into the distance over a hot desert, a silvery surface appears that looks like water but is the reflection of the sky. This is the apparent horizon of the convex-appearing earth, and we see the sky beyond. Objects may project up through it, like ships beyond the usual horizon, and they seem to be floating on the water. This is the usual, and very common, case. On rare occasions, there may be a second horizon in the sky, over which the tops of mountains may project. These mountain tops may be reflected in the second horizon as inverted images. This is the three-image mirage. Under some conditions, the second horizon may be diffuse and appear like a featureless fog bank, called the Fata Brumosa.

Mirages have often been seen over distances of 40-50 miles, as across the English Channel, when the opposite coast seems to be brought within 10 miles, and things usually invisible below the horizon become evident. Mirages have been reported at much larger distances. A ship sailing between Cape Farewell, Greenland and Iceland, at 63° 42' N and 33° 32' W, reported a mirage of southwest Greenland, with the recognizable mountain Snaefells Jökull seeming only 30 miles away, when the actual distance was 335-350 miles. A less credible report claims that the skyline of Bristol, England was seen over the glacier of Mt. Fairweather in southeastern Alaska. Mt. Fairweather, 15,318', 59° N 138° W is 125 miles west of Juneau, and is the highest mountain in British Columbia (it is on the border). The distance to Bristol is 2500 miles, and it would seem that without lateral magnification (which does not occur in mirages) it would be too tiny to see. The explanation is probably a chance resemblance of part of the vertically-magnified mirage scene to the spire of St. Mary Redcliffe, and the masts of ships. That there is a mirage is not disputed, and it is regularly seen, but the masts are probably still there, while they have disappeared from Bristol.

The Fata Morgana is named after the legendary Morgan le Fay, the enchantress sister of King Arthur. It is a simultaneous occurrence of superior and inferior mirages, famously seen across the Straits of Messina from Reggio towards the Sicilian coast. It is not observed in the opposite direction. The observer sees a fairy city in the water, and also in the sky, with fantastic buildings and gardens. Although the name does not appear in print before 1617, it was probably given by the Normans who drove out the Saracens in the 11th century. Mirages were well-known to the ancients, and this was the haunt of Scylla and Charybdis, malevolent haunters of the Straits who caused dangerous tidal currents there. The illusions of the Fata Morgana were reputed to lure sailors seeking port to destruction on the rocky beaches. The Fata Morgana also occurs on the Great Lakes, but it is always an extrordiary occurrence, and depends on placing the eye at a critical level. There is usually the appearance of two horizons, with the fairy city between, and grotesque reflections above and below. The Fata Morgana requires hot layers above and below, with the eye in an intermediate cool layer. These are the conditions for the "three-image" mirage mentioned above.

Fraser and Mach have explained how the astigmatism of the atmospheric imaging, with its redistribution of intensity, and periodic gravity waves when the temperature profile is not laterally horizontal, can give rise to the pilasters, towers, windows and arches of the Fata Morgana, so that it does not have to be the distorted image of a real object.

I have observed the inferior mirage frequently, and looming occasionally. I have not observed the superior mirage, stooping or the Fata Morgana, but plan to take pains to keep an eye out in future for them. These phenomena are very easily overlooked unless specifically sought. Mirages are most easily seen over water and desert plains, and in bright sunlight.

References

M. Minnaert, Light and Colour in the Open Air (New York: Dover, 1954). pp. 39-63. An excellent introduction to the subject.

R. A. R. Tricker, Introduction to Meteorolgical Optics ( New York: American Elsevier, 1970). pp. 11-23.

M. Born and E. Wolf, Principles of Optics (London: Pergamon, 1959). pp. 108-131.

Born and Wolf do not mention the mirage, and there is no entry in the McGraw-Hill Encyclopedia of Physics for it. Optics texts either ignore the mirage, or show one or two misleading diagrams, as in R. W. Wood, Physical Optics (New York: Dover, 1967; reprint of third edition, 1934), p. 87f, or E. Hecht and A. Zajac, Optics (Reading, MA: Addison-Wesley, 1974), p. 69. A concave-upwards ray is shown, and an indication of an inverted image, but little else, in these two references. An encyclopedia article usually has an unsatisfactory explanation, but mentions the two kinds of ordinary mirage as well as the Fata Morgana, though there are no diagrams.

A. B. Fraser and W. H. Mach, Mirages, Scientific American, January 1976.

The "green ray" is a phenomenon closely related to the mirage, since it occurs near the horizon. There is an excellent website, The Green Flash, that deals with it. A link to this site is also given in the article Green Flash on this website.

W. R. Corliss, Handbook of Unusual Natural Phenomena (Glen Arm, MD: The Sourcebook Project, 1977). pp. 164-169 and 240-249.


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Composed by J. B. Calvert
Created 28 January 2003
Last revised 30 July 2003