3 A.  Snell's laws of refraction

 Greek physicist Claudius Ptolemy in 140 AD measured and tabulated the angles of incidence and refraction through water. However, the reason why refraction occurs was known at that time.   Ibn Saul, an Arabian mathematician in 984 AD wrote a treatise "On Burning Mirrors and Lenses". His work which explained how light is bent by curved lenses and mirrors, is credited with the discovery of refraction phenomenon of light and the laws of refraction. The Dutch mathematician Willebrod Snell in 1621 independently formulated the laws of refraction that explained the measurements of incident and refracted angles. Snell's laws of refraction are the result of a careful experimental study of the behavior of the incident, reflected and the refracted rays. However, it is just a mathematical expression; no proof was known at that time. The proof had to wait till early 1860s when James Clerk Maxwell showed through his equations that fluctuations of electric and magnetic fields travel in vacuum with a constant speed c and that light is an electromagnetic wave.  The electromagnetic waves cover a wide range of spectrum from long radio waves to short gamma rays. The light rays fall in a small window of this spectrum.

 

  1. THE FIRST LAW: The incident, reflected, refracted rays and the normal to the surface of the second medium at the point of contact lie in the same plane.  In the figure on the left, consider S is the source of light. Consider a ray from this source striking the surface of another medium at the point O. The incident ray SO, the reflected ray OA, the refracted ray OT and the normal ON (extended as OM in the second medium) are in the same plane.   

An inquisitive student may point out that this figure is conveniently drawn on the paper or screen and is perpendicular to our line of sight in this case.  What if the source S is in front of or behind the plane of the paper?  Or the line of sight is not perpendicular to this plane? Let us call the incident, reflected, and refracted rays and the normal to the surface the optical plane. The three figures below show these three cases.  The line of sight is such that we are looking at the surface of the second medium from above.  The part of the plane containing the rays in the second medium is shaded in different hue.  Figure 5B is the same as we discussed above but we are looking at the surface of separation from above at an acute angle.  The line I1 - I2 is the line of intersection of the optical plane with the surface of the second medium. The portion of the plane below the suface is shown shaded. The source is in front of the point O in figure 5A and it is behind in the figure in Figure 5C. The optical plane will be intact in any scenario but will be tilted as shown in these figures.  As an exercise, the student can check that the perpendicular drawn from S on the surface of medium 2 will lie on the line of intersection of the optical plane and the medium 2 at point C.

THE SECOND LAW:   For a given pair of media, the ratio

sin i / sinr  =  1μ2                                       (Eq.2)

is constant for light of a given wavelength.

If the first medium is vacuum,  then 

sin i / sin r  = μ ,                               (Eq. 3)

where μ is called the refractive index of the medium. The refractive index of the medium depends on the wavelength or color of light.  If the wavelength of the measured value of μ is not specified it is taken to be the yellow radiation of Sodium with average wavelength 589.3 nano meter ( 1 nano meter = 10 -9 meter). Equations 1 and 2 represent definition of the refractive index which is dependent on the value of the angle of refraction and the angle of incidence. In other words, this refers to the amount of bending of the refracted ray towards the normal. The refractive indices of some of the materials are given below.  A comprehensive list can be found here.

 

Refractive indices of some materials       

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Material                  Refractive index

Vacuum                  1  (by definition)

Air at STP                 1.0003

water                        1.333

normal glass          1.52 

Polycarbonate        1.60  (used in   

                                   spectacles)

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As the refractive index of air at STP is close to that of vacuum,  refractive index of any medium with respect to air is  appoximated to the value with respect to vacuum. 

 

 

 

 

The animated figure on the right shows the variation of the angle of refraction with the angle of incidence for a block of common glass with a refractive index of 1.52.  Note that at normal incidence, the ray goes straight through the glass.

 

 

Snell's laws of refraction are the result of a careful experimental study of the behavior of the incident, reflected and the refracted rays. The value of the ratio derived by Eq. 3 is same as the ratio of the speed of electromagnetic wave in vacuum to its speed in the given medium which is the modern accepted definition of   μ  (Eq.  1).   It may be puzzling to note that two ratios one derived from the angles and another from speeds are indeed equivalent. For a curious student, the following sub section gives a  geometrical proof.