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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 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 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.;

Sinθ1
Sinθ2
= vp1
vp2
= n2
n1

where θ1 and θ2 are the angles of deviation from perpendicular on going into and coming out of the boundary; 
vp1 and vp2 are the phase (i.e, apparent) velocities in the respective media;
n1 and n2 are the refractive indeces of the respective media;
and refractive index is defined as n = c / vp
where the velocity of light,  c = 299 792 458 m/s (exactly, by definition in the SI system of units).

snells-law

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
Photo: Dave Knight.
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 θa (in the air) has become 90°, i.e., Sinθa=1. Snell's law then tells us that the critical angle θc=Arcsin(1/nw). 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.

snellwin

If you take nw=1.333 (fresh water), then the critical angle θc=48.6°, so Snell's window subtends an angle of 97.2° at the camera. For sea water, nw=1.339, and 2θc=96.6°


Dispersion
The pot of gold is behind the tree on the bottom left.
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 computer lens correction article for more information.



© Cameras Underwater Ltd. 2001, 2004, 2012
David Knight asserts the right to be recognised as the author of this work.