Refraction
& Snell's Law
By David Knight
Refraction is the bending of a ray of light as it
passes from
one medium into another in a direction that 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 that 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 overall effect (in superposition) 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. Phase velocity can be greater or less
than the speed of light (but
information cannot
travel faster than the speed of light).
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.;
Sinθ_{1}
Sinθ_{2} |
= |
v_{p1}
v_{p2} |
= |
n_{2}
n_{1} |
where θ
_{1} and θ
_{2}
are the angles of deviation from perpendicular on going into and coming
out of the boundary;
v
_{p1} and v
_{p2} are the phase
(i.e, apparent) velocities in the respective media;
n
_{1} and n
_{2} are the
refractive indeces of the respective media;
and refractive index is defined as n = c / v
_{p}
where the velocity of light, c = 299 792 458 m/s (exactly, by
definition in the SI system of units).
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. The value for
water is often approximated as 4/3.
Snell's
Window
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
will see a 180° view of the world above water condensed into an
angle of 97°. This appears as a circular window, straight
above; outside of 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 θ
_{a} (in the air) has
become 90°, i.e., Sinθ
_{a}=1.
Snell's law then tells us that the critical angle θ
_{c}=Arcsin(1/n
_{w}).
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
_{w}=1.333 (fresh
water), then the
critical angle θ
_{c}=48.6°, so
Snell's window subtends an angle of 97.2° at the
camera. For sea water, n
_{w}=1.339, and
2θ
_{c}=96.6°.
Dispersion
Refractive index is a function of the wavelength (or
colour, or frequency) of the light. It is this variation of refractive
index with
wavelength that 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
image
radial
correction article for more information.
DWK.
© David W Knight.
2001, 2004, 2012, 2018
David Knight asserts the right to be recognised as the author of this
work.