Ray model and Reflection

We have seen that light is an electromagnetic wave.

If a EM wave is moving in the $x$ direction there is no reason for it to change it's amplitude or direction if it is traveling in free space.

This is the core principle that leads to the rules governing geometric optics.

The first rule we have is for specular reflection.

We define an angle of incidence $\theta_{i}$ relative to the surface normal and find that the angle of reflection $\theta_{r}$, also defined relative to the surface normal is given by

$\theta_{r}=\theta_{i}$

Parabolic mirror

A parabolic reflector has the special property that any ray coming in to the mirror parallel to the principal axis of the parabola will be reflected through the focus of the parabola.

For a parabola $y=ax^{2}$ the focus is at $y=\frac{1}{4a}$

Get a feel for how the focal point changes with the parameter of the parabola here.

Spherical mirror

Because spherical surfaces are easier to make than parabolic ones, most curved mirrors are actually spherical reflectors. In the case where the width of the mirror is small compared to it's radius of curvature the incoming rays are focused with a focal length

$f\approx\frac{r}{2}$

for small $\theta$

$2x\approx r$

$x\approx \frac{r}{2}\to f\approx\frac{r}{2}$

Mirror equation

Two sets of right triangles formed by the object and the image are similar triangles which give us

$\frac{h_{0}}{h_{i}}=\frac{d_{o}}{d_{i}}$ (green triangles)

and

$\frac{h_{0}}{h_{i}}=\frac{d_{o}-f}{f}$ (purple triangles)

hence

$\frac{d_{o}}{d_{i}}=\frac{d_{o}-f}{f}$

which can be rearranged to give

$\frac{1}{d_{o}}+\frac{1}{d_{i}}=\frac{1}{f}$

The magnification of a mirror $m$

$m=\frac{h_{i}}{h_{o}}=-\frac{d_{i}}{d_{o}}$

Sign Conventions:

• The image height $h_i$ is positive if the image is upright, and negative if inverted.
• $d_i$ or $d_o$ is positive if image or object is in front of the mirror (real) or object is behind the mirror (virtual).
• Magnification is positive for an upright image and negative for an inverted image
• $r$ and $f$ are also negative when behind the mirror

Curved mirror examples

Wikipedia has a nice set of diagrams for the various imaging conditions of curved mirrors and you can find some java applets for mirrors from the University of Florida here.

Speed of light in a medium

We derived earlier that light in a vacuum, as it is an electromagnetic wave, should travel with a constant speed

$c=\sqrt{\frac{1}{\mu_{0}\varepsilon_{0}}}=3.00\times10^{8}\,\mathrm{m/s}$

What about when it travels in a medium?

Many materials absorb light, others reflect it, and in transparent materials, light is absorbed and remitted as it passes through the material.

This has the effect of slowing it down.

The extent to which this happens depends on the wavelength of the light and we will see later that this leads to dispersion.

See here for a cartoon animation of the process.

Refractive index

A useful way to describe a material in terms of it's refractive index. The velocity of light in a medium is related to it's velocity in free space by the refractive index $n$ through the equation

$v=\frac{c}{n}$

Refractive indices of materials are typically somewhere between 1 (vacuum) and ~2.5 (diamond, strontium titanate).

Glass will typically have a refractive index of about 1.5 though the exact value depends on the type of glass.

We can recall that the velocity of a wave is given $v=f\lambda$, when light is traveling in a medium, the frequency $f$ does not change, the wavelength $\lambda$ changes according to

$\lambda=\frac{\lambda_{0}}{n}$

Huygen's Principle

When light enters a medium with a different refractive index as well as changing it's speed it also changes it's angle.

This can be understood based upon the wave nature of light by applying Huygen's principle.

Huygen's principle states that every point on a wavefront can be considered to be the centre of a new secondary spherical wave, and the sum of these secondary waves determines the form of the wavefront at any subsequent time.

Click here for a diagram showing how refraction follows from Huygen's principle.

Here is a good demo to get better intuition for Huygen's principle.

Now see what happens when light travels from vacuum into a medium with a refractive index greater than 1 here.