When a light ray impinges on an interface between different media, we saw that some of the light is reflected from that interface. However, in most cases some proportion of the light enters the second medium--the light is said to be transmitted. In this section we will concern ourselves only with transparent or dielectric, media. The main change upon entering different media is that for any dielectric medium, the permittivity, ε , and the permeability, μ , are different to those in free space. Thus the speed of light in various media is different: v = 1/ . We define the index of refraction of the medium as:
Of course, this reasoning implies that light can travel faster or slower than c in different media; this seems paradoxical since we have already said that photons can only exist at c . The resolution to this is that light does propagate through the void between the atoms in a medium with speed c , however when it encounters an atom it causes vibrations of the electrons which causes them to act like spherical wave sources. It is these spherical waves which cause the propagation through the medium, and they too travel at c between atoms. However, as you may know from studying driven oscillations, the resulting oscillation usually lags behind the driving oscillation by a factor between 90 o and 180 o , depending on whether the frequency is above or below resonance. Thus the electron vibrations, and hence the spherical waves, are out of phase with the incident light. As this spherical wave encounters another atom, there will again be a phase-lag for the light emitted from that atom, as so forth. Thus the light propagating through the medium is constantly being retarded (or advanced) in phase. Since we measure the speed of light effectively by watching the motion of a particular crest or trough, the continual phase shift amounts to a macroscopic 'slowing down' of the light.
We are now ready to examine what happens to the transmitted ray. To put it simply, light entering a medium at some non-zero angle with respect to the normal to the surface is bent. That is, upon entering a different medium, light changes direction. It is not difficult to show that the change of speed involved in moving from one medium to another implies this change in direction.
The two right triangles ABD and AED in share a common hypotenuse, AD , so we can write:
|n isinθ i = n tsinθ t|
|n i(×) = n t(×)|
We saw that dielectric media have an index of refraction defined by n = . For some media, however, this index is dependent on frequency. This effect causes different wavelengths of light to bend by different amounts upon entering a medium, causing a light beam made up of different colors to disperse. This phenomenon can be observed when white light enters a prism: the spectral pattern produced is due to the dependence of the index of refraction of the glass on wavelength (or color). To examine this effect we must once again consider the oscillations of the electron cloud caused by the oscillating electric and magnetic fields of the light wave (see the discussion of scattering). It is also possible that light waves can set up oscillations in ions or polar molecules of a substance, but their much greater mass means that rapidly oscillating fields have little effect. We can assume that electrons are bound in atoms in a fashion analogous to a mass on a spring, connecting the electron to the positive nucleus. Such a system has a natural or resonant frequency σ 0 = , where k will be determined by the strength of the attraction between the electron and the nucleus. The time-varying electric field of the light ray E(t) will cause a forced oscillation. The force due to this field will be F E = q e E(t) = q e E 0cos(σt) where σ is the angular frequency of the light. From Newton's Second Law we can write:
|q e E 0cosσt - kx = m e = q e E 0cosσt - m e σ 2 x|
|x(t) = E 0cosσt|
|n 2(σ) = 1 +|
In general, index of refraction increases with frequency, and hence blue light refracts more through glass than red light. This situation is called normal dispersion. There are some cases of anomalous dispersion, usually close to resonance, where index of refraction decreases for higher frequencies.
Imagine a beam of light in a dense medium incident on an interface with a vacuum (or any less dense medium). For small angles of incidence we know that some of the beam will be reflected and some transmitted into the vacuum. If we increase the angle of incidence, however, the angle the transmitted beam makes with the normal, θ t , increases at a greater rate. We know this because from Snell's Law we have:
|sinθ i = sinθ t|
|sinθ c = sin 90 o =|
Although we know that a wave incident on an interface will be partially reflected and partially transmitted along certain paths, we do not know what proportion of the energy of the ray will take each path. We can use Maxwell's Equations to help us solve this problem. Consider the diagram , in which the electric field is perpendicular to the plane of incidence.