# Geometric Optics

### Refraction

#### Dielectric Media

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:

 nâÉá =

Generally substances substances that are transparent are non-magnetic, so μ = mu 0 , and in this case n = .

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.

#### Refraction

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.

Figure %: Refraction of a wavefront.
The wave-crest AB is just hitting the interface at A . As soon as the wavefront enters the new medium it must begin to travel with the new speed v t . The part of the wavefront at B , however, must continue traveling in a straight line to D , with the speed v i ; say that the time taken for the part of the wavefront at B to move to D is Δt . Thus the distance BD is given by v i Δt . In this same time the part of the wavefront at A must continue in some direction a distance v t Δt . The crucial point is that all the wavefronts must remain continuous; no wavefront (corresponding to a peak or trough in the light wave) can get out of phase with itself. Indeed, this would contradict the notion of a wavefront, which is just a line joining points of constant phase. If the transmitting medium is denser that the incident medium, v i < v t , and hence the distance v i Δt < v t Δt . Thus different parts of the wavefront have traveled different distances. It is not hard to see that it would be impossible to maintain the continuity of the wavefront without bending it at some point. Since we know all parts of the wavefront travel at a constant speed in one medium, the only place the bend could be is at the interface. Clearly, the bending of the wavefront implies the bending of the light ray.

The two right triangles ABD and AED in share a common hypotenuse, AD , so we can write:

 =

But BD = v i Δt and AE = v t Δt , so we can write the law of refraction, or Snell's Law as:

 n isinθ i = n tsinθ t

where n i = c/v i and n t = c/v t . Moreover, the incident, reflected and refracted rays all lie in the same plane. From Snell's Law it is not hard to deduce that a light ray entering a denser (higher index of refraction) medium bends towards the normal, while a ray entering a rarer (lower index of refraction) medium bends away from the normal. In vector notation, for the normal to the surface , and incident and transmitted waves and respectively, the law of refraction can be written:

 n i(×) = n t(×)

#### Dispersion

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

You can check that the solution to this differential equation is x = x 0cosσt . We can substitute this back into the equation to solve for x 0 and we find:

 x(t) = E 0cosσt

Now, when a dielectric medium is subjected to an applied electric field, , the internal charge distribution is altered, creating a dipole moment. In other words, the external field separates positive and negative charges, creating an additional electric field called the electric polarization, . The two fields are related by (ε - ε 0) . But | is also given by the product of the magnitude of the charge, the displacement, and the total number of displaced charge per unit volume, N: | , for displaced electrons. Substituting the (time-averaged) value we found for x , above, we have:

 |

We can then write the relationship between and as ε = ε 0 + = ε 0 + = . Finally, n 2 = ε/ε 0 so we can express the index of refraction as a function of the angular frequency of the light as:

 n 2(σ) = 1 +

This is known as the dispersion equation. For frequencies above resonance (σ 0 2 - σ 2) < 0 , the oscillators are about 180 o with the driving electric field (see the discussion of phase-lag) , and n < 1 . When the frequency is below resonance, (σ 0 2 - σ 2) > 0 , the oscillation is in phase with the applied electric field and n > 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.

#### Total Internal Reflection

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

Since the transmitting medium is less dense, < 1 , so θ t > θ i . Eventually it reaches the situation where θ t = 90 o and the transmitted beam just grazes the surface. If θ i is increased still further, the transmitted beam disappears. This is illustrated in :
Figure %: Total Internal Reflection
The incident angle at which this occurs is called the critical angle, θ c , and is given by:

 sinθ c = sin 90 o =

When θ I > θ c all the light is reflected inside the dense medium. This is known as total internal reflection (TIR). TIR is employed to great benefit in technologies such as optic fibers in which a laser beam is shined down a narrow transparent fiber; total internal reflection maintains the laser signal within the fiber, irrespective of the shape into which it is bent.

#### The Fresnel Equations

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.

Figure %: Directions of fields of reflected and transmitted rays.
Maxwell's Equations imply certain boundary conditions. The first is that components of the electric field perpendicular to the interface must be continuous across the interface, hence: . The other relevant boundary condition in this case is that the perpendicular component of the magnetic field divided by the permeability, . Thus:

 + =

where the positive direction is that of increasing x . Recall from that | . We can use these three equations to find the two ratios and . The first of these is known as r âä¥ , the amplitude reflection coefficient, and the second as t âä¥ , the amplitude transmission coefficient. These equations tell us the relative amplitudes of the reflected and transmitted rays for light incident on an interface at some angle θ i . When the permeabilities of the two media are taken to be both equal to μ 0 , which is usually true to a good approximation, we find:

 r âä¥ âÉá = t âä¥ âÉá =

When the electric field vector is in the plane of the incident ray, the coefficients are:

 r || = t || =

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