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Refraction, Refractive Index, Snell's
Law, Dispersion, Snell's Window, Critical Angle, Total Internal
Reflection. |
Refraction
Refraction is the bending of a ray of light as it passes from
one medium into another in a direction which is not perpendicular
to the boundary between the two media. This property of boundaries
between transparent media makes it possible to design lenses,
but it is also responsible for unexpected behaviour of optical
systems when they are taken underwater.
The standard high-school explanation for refraction is that light
slows down on going from a less-dense to a more-dense medium.
The same text-books which tell you that light is slowed down
as it passes through solids, liquids, and gases however, will
also tell you, that even in a solid, the gaps between the constituent
atoms or molecules are large compared to the sizes of the atoms
or molecules themselves; so how fast does the light travel as
it passes between the atoms? The reality is that light doesn't
slow down at all. The speed of light c is a constant, 299 792
458m/s, in all media; solid, liquid, gas, and vacuum, but each
object (atom, or molecule) in its path scatters it, i.e., absorbs
it and re-radiates it. A scattering object looks like a new light
source, and so the resultant beam of light is the combination
of the light which appears to radiate from all of the tiny scattering
objects in its path. You can add the effects of all of these
tiny parasitic light sources by using a mathematical technique
called 'superposition', and the net result is that a refractive
medium progressively alters the phase of the light, i.e., it
squashes or stretches the wave. If a light beam hits a block
of glass squarely, nothing appears to happen; but if it hits
at an angle, one side of the beam starts to get phase-shifted
before the other, and the net effect is a change in direction
(For the proper explanation see "The Feynman Lectures on
Physics" ISBN 0-201-02010-6-H). We can however, account
for the refracting power of a medium by referring to an apparent
velocity for light travelling in it. This is known as the 'phase
velocity', and is a mathematical convenience rather than a physical
reality. |
Refractive Index & Snell's Law
The refractive index of a medium tells you how much a beam will
slew for a given angle of incidence, according to an equation
known as `Snell's law'; i.e.;
( SinA / SinB ) = n,
where A and B are the angles of deviation from perpendicular
on going into and coming out of the boundary, and n is the relative
refractive index. It is conventional to write the equation so
that n normally comes out greater than 1 (in which case A corresponds
to the less refractive of the two media), and that completes
the definition. It is also convenient to record the refractive
index of a material relative to a vacuum, and this is known as
the `absolute refractive index' (usually omitting the word `absolute').
The relative index between two media is then simply the ratio
n1/n2.
The refractive index relative to vacuum is defined as:
n = speed of light / apparent (phase) velocity of light in the
medium.
For visible light, in the range 10-20 degrees Celsius, you may
assume approximately: n=1.00028 for air at 1 bar, n=1.333 for
fresh water, and n=1.339 for sea water. The value for air is
so close to 1 that it is normally good enough to assume it to
be 1. |
Refractive index is a function of the wavelength (or colour,
or frequency) of the light. It is this variation of refractive
index with wavelength which causes a glass prism or a raindrop
to split (i.e., disperse) white light into its component colours.
Dispersion is a problem in camera lenses, because it causes chromatic
aberration, i.e., colour fringing, which worsens as you move
away from the centre of the image. Correction for chromatic aberration
is usually achieved by designing a lens with multiple elements
(a compound lens) in such a way that colours dispersed by one
element are brought back together by another. Unfortunately,
complete correction across the whole visible spectrum is impossible
to achieve, and so a lens is usually designed to be corrected
exactly at two or three wavelengths, with a reasonable compromise
at all others. A lens corrected at two wavelengths is called
an 'achromat'. A lens corrected at three wavelengths is called
an 'apochromat'. It is also possible to minimise chromatic aberration
by using various types of special low-dispersion glass, but such
glasses are made from exotic rare-earth elements and are only
used in very expensive lenses.
Should chromatic aberration prove to be a problem with a particular
optical system, and your pictures are destined to be processed
digitally, you can use software to apply a radial correction.
See the software lens correction article for more information. |
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One of the consequences of refraction, which may be of artistic
interest, is the phenomenon known as `Snell's window' (or the
'optical manhole'). If you lie on the bottom of an absolutely
still pool, you see a 180° view of the world above water
condensed into an angle of 97°. This appears as a circular
window, straight above; outside which you can't see through the
water at all, you can only see a reflection of the bottom. You
can only capture the whole width of Snell's window on film if
your lens has a coverage greater than 97° (as does the Sea
& Sea 12mm fisheye used above). To work out what's happening,
consider a ray travelling from the camera to the surface (the
geometry works just the same regardless of which way the light
is going). As you stray away from the perpendicular, you eventually
reach an angle (the critical angle), at which the
ray can no longer escape from the surface, because it runs along
the surface. In this case, angle A (in the air) has become 90°,
i.e., SinA=1. Snell's law then tells us that the critical angle
C=Arcsin(1/N). Beyond the critical angle, total internal
reflection occurs, i.e., our notional ray from the camera
bounces off the surface and goes back down, and so the surface
outside Snell's window becomes a mirror.
If you take n=1.333 (fresh water), then the
critical angle C=48.6°, so Snell's window subtends an angle
of 97.2° at the camera. |
Text and illustrations © D. W. Knight 2001,
2004 |
