Mirrors in the air: mirages in nature and in the laboratory

Mirrors in the air: mirages in
nature and in the laboratory
M Vollmer
University of Applied Sciences Brandenburg, Germany
Abstract
Although mostly not perceived consciously, mirages are very familiar
phenomena of everyday life. They can generally occur if light is incident on
media with a gradient of refractive index. This article starts with the easiest
mirage effect, known as astronomical refraction, then presents examples for
inferior and superior multiple image mirages, and finally focuses on
demonstration experiments for the classroom.
If you love nature, you respond to her
phenomena as naturally as you breathe
.... Never think that the poetry of
nature’s moods in all their infinite variety
is lost on the scientific observer, for
the habit of observing refines our sense
of beauty and adds a brighter hue to
the richly coloured background against
which each particular fact is outlined.
These words of Marcel Minnaert in the
preface of his famous book Light and Colour in the
Outdoors of 1937 [1] expressed his feelings when
observing nature. Atmospheric optics offers a
huge variety of natural phenomena [1–4]. Many of
these phenomena can be observed nearly every day
and everywhere. Examples are rainbows, halos,
mirages, coronas (e.g. [5, 6]), glories, the colours
of the sky, sunsets and twilight phenomena, and
green flashes; and there are many more.
All these different phenomena are due to the
interaction of light with matter that is present in the
atmosphere. Mirages belong to the most simple
phenomena since they are due to pure air, i.e. air
consisting of gases only (N2, O2, Ar,CO2, etc)
but no water droplets, ice crystals, or aerosols.
They occur if light waves propagate in gases with
gradients in the index of refraction.
F EATURES
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Index of refraction of air
In general, light which is propagating in pure air
is partly scattered according to Rayleigh scattering
(from molecules and atoms within the air), the rest
propagating undisturbed in the forward direction.
The scattered light consists of sideways scattering
light and forward scattering light, the latter being
in phase with the incident light. As a consequence,
light in clear air is only slightly weakened
by sideways scattering, the forward propagating
light (being the superposition of incident light
and forward scattered light) dominating. The
weakening due to sideways scattering will not be
discussed here: it leads to the blue sky and related
phenomena.
In terms of geometrical optics, the forward
propagation of light in homogeneous media is
usually expressed using the refractive index n (for
tables of n, see[7]) and the law of refraction. n
depends on the temperature T (in K), pressure p
(in hPa), and humidity. Figure 1 depicts some
examples. Differences due to the typical humidity
of the air only amount to about 10 −6 compared to
dry air at the same atmospheric conditions, and are
neglected here.
Neglecting dispersion, equation (1) gives an
approximation for the real part of the index of
refraction n of dry air (its imaginary part is almost
zero in the visible region, since absorption features
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M Vollmer
index of refraction
1.00034
1.00032
1.00030
1.00028
1.00026
dry air
0°C
15°C
30°C
200 400 600 800
wavelength in nm
1000
Figure 1. Real part of the index of refraction for dry
air as a function of wavelength.
of air will only occur in the ultraviolet (N2, O2) or
infrared (CO2)).
n = 1 + 77.6p
T 10−6 . (1)
For example, T = 300 K and p =
1030 hPa yield n ≈ 1.000 266. Increasing the
air temperature by 40 K (e.g. air, directly above
the ground which is heated by the Sun) gives
n ≈ 1.000 235. Such small changes (here �n ≈
3 × 10−5 ) are responsible for all kinds of mirage.
Atmospheric refraction in astronomy
The most simple mirage effect is known from
astronomy: if an observer looks at an object in
the sky at a certain altitude, the actual position
vacuum
n = 1.0
refraction
air: n = 1.000266
refraction in arc seconds
2400
1800
1200
600
0
0
10 20 30 40 50 60 70 80 90
zenith distance in degrees
Figure 3. Astronomical refraction of standard
atmosphere as a function of zenith distance.
30
20
10
refraction in arc minutes
of the object is lower in the sky. This is due
to the refraction of light upon propagation in the
atmosphere with a gradient of n, giving rise to
curved light paths (figure 2).
For the standard atmosphere, refraction is
tabulated (e.g. [8]). In the zenith, the deviation
is zero; at 45 ◦ zenith distance it amounts to 1 ′
(1 minute of arc) increasing to more than 34 ′ at an
angle of 90 ◦ (tangentially incident light). Figure 3
depicts refraction as a function of zenith distance.
The maximum value of refraction has to
be compared with the angular sizes of the Sun
or Moon, which are about 30 ′ , i.e. half a
degree. Since the refraction is 34 ′ at a zenith
distance of 90 ◦ , but only about 28 ′ at a zenith
distance of 89.5 ◦ , the Sun or the Moon in an
otherwise undisturbed atmosphere (no inversion
apparent direction
refraction
actual direction
Figure 2. Two geometries for the atmosphere leading to astronomical refraction. Within an atmosphere of
constant density, i.e. constant index of refraction, any light ray incident from space is refracted. The
respective angular deflection is called refraction. It is calculated more realistically if the density and index of
refraction increase in a series of layers from top to bottom (due to either T or p changes). An incident light
ray will follow a polygon-like path. For infinitesimally small layer thickness, this leads to curved light paths.
Δx
n1
n2 > n1
n3 > n2
n4 > n3
n5 > n4
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horizon
Sun, geometrically
just below the horizon
observation at sea level:
large refraction, Sun
just above the horizon
Earth
Figure 4. The Sun is flattened close to the horizon
due to the change of astronomical refraction with
zenith angle (see figure 3).
layers) will thus be flattened (see figures 4
and 5).
Furthermore, the Sun or Moon can still be
seen above the horizon although geometrically
they are already below the horizon while setting.
The flattening effect can be nearly twice as large if
the Moon is seen through a twice as long optical
path through the atmosphere.
Inferior mirages
Mirages are images of objects which may be seen
in addition to real objects whenever there are
temperature differences in the atmosphere, which
give rise to unusual gradients in the index of
refraction (effects due to pressure variations are
much smaller), resulting in curved light paths. If
warm air is above the ground, n is smaller close to
the ground than above it. This leads to inferior
mirages like the well-known wet-looking streets
Mirrors in the air: mirages in nature and in the laboratory
1
3
2
Figure 6. Inferior mirages. Whenever the ground is
warmer than the air above it, the index of refraction
is lower close to the ground. Therefore light rays will
follow curved paths as shown (a) from ray tracing
results for a number of light rays scattered from a
palm tree. Some of them will hit the ground (not
shown), some will pass above or below the head of
the observer, and some of them (ray 4) may also
bend such that they reach the observer’s eye coming
from below. In this case, two light rays (rays 2 and
4) originate from the same spot of the object but
come from different directions. This is interpreted in
the brain as an object with a mirror image, as in (b).
Note that bundles of light rays are needed in order to
produce images. For the sake of simplicity, we will
always speak of light rays, meaning bundles of light
rays.
Figure 5. The Sun is flattened near the horizon close to sunset (Photo M Vollmer) and the Moon seen through
nearly twice the atmosphere from the ISS (Photo D Pettit, 28 April 2003, with kind permission of NASA).
March 2009 P HYSICS E DUCATION 167
(a)
(b)
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M Vollmer
Figure 7. Typical inferior mirage, observed on a hot summer street. Such mirages can be seen nearly every day
while driving a car, when the Sun shines for a long time on paved streets. (Photo E Tränkle, reproduced with kind
permission.)
Figure 8. Inferior mirage observed on the Wannsee in Berlin. The water had already warmed up in early spring
before cold arctic air was rushing in. As a result, a normal boat seems to look like a ‘sea daemon’. (Photo
ETränkle, reproduced with kind permission.)
on hot summer days. The formation of inferior
mirages is illustrated in figure 6.
Some examples from observations in everyday
life and nature are shown in figures 7 and 8.
The angular size of mirages is usually
about 0.5 ◦ –1 ◦ . For observations, binoculars and
cameras with telephoto lenses are used. The
mirages are often distorted and flickering due
to local density fluctuations in the air. This is
very similar to the flickering of stars at night,
seen through the atmosphere. Therefore, it
is not possible to get really ‘focused’ mirage
photographs.
Invisible objects: the vanishing line
A prominent feature of a mirage is the so-called
vanishing line of the object [2]. Light below
certain object points has no chance whatsoever of
reaching the eye of the observer, i.e. parts of the
object cannot be seen, irrespective of the direction
into which the light is scattered (see figure 9).
The concept of vanishing lines explains why
mirages often show incomplete images of objects,
which often makes it difficult to decide what is
actually being observed. Figure 10 depicts two
photographs taken close to the coast of the North
Sea in Germany.
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(a) (b)
Figure 9. Vanishing line in inferior mirages (after
[2]). For each light ray starting at the palm tree in
every possible direction, ray tracing is used to follow
the curved trajectories. All object points below a
critical line, here at the ape position, are invisible to
the observer, since there is no possibility that light
can take an allowed path to connect these points with
the eye of the observer.
Quantitative modelling of vanishing line
effects may be done via ray tracing with
computers. The various methods differ in their
choice of geometry and profile n(h) for the index
of refraction as a function of height above ground.
Mirrors in the air: mirages in nature and in the laboratory
Figure 11 depicts an example in which the object
is a car, seen from a height of 1 m, and at distances
from 600 m to 3 km for a chosen profile n(h).
The vanishing line effect shows up in
figure 11: for distances larger than 1.2 km, the
tyres are no longer visible, and above 2.2 km
the headlights are gone, too. A comparison
with observations gives qualitatively reasonable
agreement, as can be seen in figure 12 (typical
problems while observing are that one is proud of
getting a nice shot with a camera but one lacks
knowledge of the distance of an approaching car).
Superior mirages
Sometimes inversion layers are present in the
atmosphere: the air at a given height is warmer
than that below (and above), i.e., its index of
refraction is lower than that above or below this
height. This situation favours the observations of
superior mirages (figure 13). Figure 14 gives an
example of a fascinating superior mirage, observed
in Finland.
Figure 10. A church on the island of Pellworm, seen from the Hallig Hooge at high tide, and an inferior mirage of
it, seen from a distance of more than 11 km over mudflats at low tide. (Photos M Vollmer). The vanishing line is
just below the roof of the building beside the tower. Therefore some higher-lying parts of Pellworm seem to float in
the air and their base cannot be observed. The inferior mirage is a triple mirage image due to slightly more
complex atmospheric conditions close to the ground.
Figure 11. Theoretical inferior mirage of an object (car, left side). The observer’s eye is at 1 m and the
distance in km varies (from left to right): 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.6, 2.8 and 3.0.
(Simulation by E Tränkle, reproduced with kind permission.)
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M Vollmer
Figure 12. Some road inferior mirages, which resemble some of the theoretically expected features of figure 11.
The photographs were taken in Namibia close to Walvis Bay (Photos M Vollmer).
height
temperature
index of refraction
apparent
size of object
Figure 13. Superior mirages can occur if inversion layers are present in the atmosphere. The light rays from
the top leaf of a palm tree can follow many different curved paths to reach the eye of an observer. Some light
rays can enter the eye from quite large observation angles, i.e. an observer sees an image of an object which is
vertically stretched (called towering).
Magic of the air: three-fold mirage images
When studying the huge variety of vertical profiles
for the index of refraction, it becomes obvious that
a multitude of possible mirage effects is possible.
In some cases (see e.g. [2]) the paths of light from
objects—as governed by the law of refraction—
can even give rise to multiple images on top of the
object. Figure 15 depicts possible light paths from
an object (here the letter F) in a situation which
resembles inversion layers in the atmosphere.
The top of the letter F may, for example, be
seen using light bundles 2, 3, or 5, all entering
the eye of the observer at different angles, thus
leading to a three-fold image. The first wellknown
natural mirages of this type were reported
in 1798 by Reverend Vince. Looking at the North
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Mirrors in the air: mirages in nature and in the laboratory
Figure 14. Superior mirage, observed in Kustavi in a lake area in the southwest of Finland. Portions of distant land
are vertically stretched and appear as wall-like objects. In the middle, part of the distant land is hidden (similar to
the vanishing line effect of figure 9) giving rise to a ‘bridge’-like structure. The maximum angular height of the
mirage is governed by the distance of the object and the height of the inversion layer. (Photo Pekka Parviainen.)
Many more spectacular photographs of mirages and other optical phenomena can be found at [9].
5
4
2
1
Figure 15. Light paths leading to three-image
superior mirages.
3
Sea he observed very strange mirages of sailboats
for hours. Some of his sketches are shown in
figure 16.
Demonstration experiments
Several experiments can be used to illustrate the
concepts of mirages while teaching. These include
demonstrations of curved light paths as well as
inferior and superior mirages [4, 11–14]. Inferior
mirages were already simulated nearly 100 years
ago. In nature, they are observed over distances
of hundreds of metres to kilometres. Typical
laboratory dimensions are smaller by about a
factor of 1000; hence, the temperature differences
must be higher and the objects smaller. Usually,
a plate of metal (1–2 m long) is heated (e.g. by
gas flames, which by itself is a spectacular
demonstration). Covering the plate with a layer
of sand ensures that no actual reflections from the
metal are possible. Figure 17 depicts a mirage seen
5
4
3
2
1
above such a hot plate. In this case, the object was
a toy car (for other examples, see [4, 11]).
The curvature of light paths as well as
superior mirages can be demonstrated very easily
in liquids with a gradient of the index of
refraction [4, 11, 12]. The equipment needed
consists only of a water tank (typical dimensions
10 cm × 30 cm × 20 cm height), a light beam
(for example from a He–Ne laser or, more cheaply,
a laser pointer) as well as layered sugar or salt
solutions.
The solutions may be prepared in a variety
of ways. In the following, only the procedure
for the salt solution is described. Either fresh
water is placed on top of a saturated salt solution
(for example, by placing a thin foil on top of the
salt solution which is later cautiously removed)
or a saturated salt solution is placed below the
fresh water layer (for example, by means of a
regulated flow through a thin rubber tube, ending
at the bottom of the tank). In both cases it is
possible to generate the layered systems without
much mixing, by carefully avoiding turbulence
and vibrations when filling the tank.
After preparation, the two liquids slowly mix
via diffusion processes, resulting in a gradient
of the index of refraction between the saturated
salt solution (n ≈ 1.364) and pure water (n ≈
1.332) at λ = 632.8 nm. Therefore, this setup
qualitatively resembles an inversion layer in
the atmosphere, where warm air (smaller n) is
above colder air (larger n). The water tank
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M Vollmer
Figure 16. Two-fold and three-fold superior mirage images, observed on 1 August 1798 by Reverend S Vince in
Ramsgate (after [10]).
Figure 17. Inferior mirage in the laboratory using a
1.8 m long metal plate, covered by sand and heated
using gas flames.
should not be exposed to vibrations, since rapid
diffusion decreases the period for observations.
Without disturbing the solution, experiments for
demonstration of curved light beams may be
performed for a couple of hours, those of three
part superior mirages (see below) for even longer
times, and some mirage distortions like towering
or stooping may be seen for one or two days.
After preparation of the layered liquids
in the tank, the curvature of a laser beam
in the inhomogeneous solution can be easily
demonstrated by illuminating from below and
varying the angle of incidence. Figure 18 depicts
a photograph of a curved laser beam directed
through the tank and viewed from the side.
Even a horizontally incident beam will be
curved downward due to the vertical gradient of
the index of refraction. This is due to the finite
dimension at the beam waist: the upper and lower
portions see different values of refractive index
and will be bent differently. Consequently, the
whole ray will be curved.
Once the solution is prepared and curved light
paths are observed, one should also be able to
observe superior mirages when looking through
the water tank. The size of the objects depends
on the height of the salt solution and fresh water
layers. Dimensions of several centimetres work
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Mirrors in the air: mirages in nature and in the laboratory
Figure 18. A laser beam is illuminating a layered solution from below. It illustrates a curved light ray.
well. Parameters for the observation are: relative
height of the object and the boundary (diffusion)
layer, the distance of the object behind the water
tank, and the distance of the observer as well as
the angle of incidence of light rays with respect to
the eye of the observer.
Once the solution is prepared, optimum
observation conditions are usually found within
a few minutes. For better results, one may
illuminate the objects and observe through the
smaller dimension (width) of typically 10 cm
rather than the length of the tank. Depending
on the observation angle, one can readily observe
three-fold images (see figure 19).
As mentioned above, in nature, typical
dimensions are of the order of hundreds of metres,
and changes of the index of refraction are of
the order of several 10 −5 . In the water tank
experiment, the distances are only of the order of
10 cm, but this is compensated by the change of
the index of refraction of the order of 10 −2 .
Among the most fascinating properties of
mirages in nature are transient effects due to
air fluctuations. They can be demonstrated by
introducing water turbulence: for example, by
slowly stirring the system. Wavelike disturbances
in the artificial inversion layer lead to similar
wavelike patterns in the mirages.
More experiments
More water tank experiments are possible using
different liquids. Besides layered salt solutions,
sugar solution also works very well [13], although
it is more messy to clean up. However, sugar
allows the preparation of multilayer solutions
which can help to understand more complex
mirages in nature. Quite a few more complicated
chemical recipes have also been reported in the
literature; however, most are with controlled
substances which are not suited for teaching
purposes. Using the salt water tank set-up with the
three-image mirage, it is also possible to observe
colourful effects from black and white objects
which are due to dispersion effects within the
artificial inversion layer [14].
In nature, one may also observe lateral
mirages, for example, as mirages from heated
walls. Experiments along these lines were realized
in the laboratory, by using a U-shaped, flameheated
channel [4, 14]. This allowed observation
of three lateral images of the object.
Finally, it is possible to observe mirages with
hot plates as small as 14 cm diameter, although
the image effects are small. However, hot plates
are also ideally suited to demonstrate what we
might call the deflection of light beams by human
breath [14]. A laser is set up to illuminate a hot
plate at grazing incidence such that some of its
light is scattered by the surface. In addition, one
also observes the position of the light beam on a
distant wall. Blowing onto the hot plate results
in an easily observable upward deflection of the
beam spot on the wall while, simultaneously, the
scattered light on the surface of the plate is no
longer visible.
Acknowledgments
Special thanks go to Pekka Parviainen for letting
me use his photograph (figure 14) and in particular
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M Vollmer
Figure 19. A layered solution of fresh water above salt water resembles an artificial inversion layer. It allows the
observation of three-fold mirages (bottom) of objects (top).
to my former friend E Tränkle, who died much too
early.
Received 1 October 2008, in final form 20 November 2008
doi:10.1088/0031-9120/44/2/008
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Michael Vollmer is professor of physics
at the University of Applied Sciences in
Brandenburg, Germany, working in the
fields of infrared thermal imaging,
spectroscopy, atmospheric optics and
didactics of physics. Besides teaching
and research he is also involved in
in-service-teacher training.
174 P HYSICS E DUCATION March 2009