# 5. Optics

## 5.1 Basic Optics

### 5.1.1 What is Light?

What is light? You know the answer, of course. To refresh your memory, here is the definition:
 Light is the common name for electromagnetic waves with wavelengths just below a micrometer (400 nm - 800 nm). Light is the common name for photons with energies just above 1 eV (1,8 eV - 3,2 eV).
So we still have the good old dichotomy between the wave picture, championed by Huygens and the particle picture first championed by Newton. As you know, Newton lost the fight but was redeemed to some extent by Einstein in 1905. More to that in the link.
It is of course quantum theory that reconciles the otherwise incompatible viewpoints. Either you have some ideas how this works or you don't. In the latter case you need to do some work on your own. I cannot go into this kind of "details" here.
In case of doubt think about what you learned about "electron waves". For example the y = y0 · exp(ikr) wavefunction for a free electron that turned a particle into a wave, and the |y|2 that turns a wave back into a particle. It's just as easy for photons.
If you don't get it - tough luck! We're not doing quantum mechanics here. Just accept and follow the simple rule:
 For the propagation of light: use the wave model For the generation and disappearance (= absorption) of light: use the photon model
Now we need to consider a few very basic numbers and relations.
The key properties and parameters that should come to mind when considering light propagating in vacuum or in some transparent material with dieelectric constant er, magnetic permeability mr (always » 1 for optical frequencies) and index of refraction n are:
Relations concerning light
Propagation in vacuum Propagation in material with index of refraction n
Wavelength l0 l0/n
Frequency n n
Energy hn = w hn = w
Propagation speed
c0  =  natural
constant
=  n · l

=   (e0 · m0)–½
c(n)  =   c0
n
=   1
(e0er · m0mr)½

n  =   (er · mr)½   » er½
Wave vector
|k|  =   2p/l0
=   w/c0
|k|  =   2p/l
=   2pn/l0  =   w/c
Momentum p = k = w/c0 p = k = w/c
Snellius law n = sina/sinb with a, b the angle of incidence or propagation, resp.
We can actually derive all the materials stuff and Snellius law as given in the last entry very easily by considering energy and momentum conservation. We will do that in a litle exercise.
 Exercise 5.1.1 Derivation of Snellius' law
The following table gives a few basic numbers for these quantities that you must know .

Numbers concerning light
Rough order
of ten value
Better value
Wavelength  » 1 µm 500 nm (390 to 750 nm)
Frequency  » 1015 Hz 5 · 1015 Hz
Energy  » 1 eV 2.5 eV
c0 (vacuum)   300 000 km/s = 3 · 108 m/s
Momentum ratio
p(2,5 eV electron)
p(2,5 eV photon)
»   103
The momentum entry serves to remind you that photons have very little momentum relative to electrons (and phonons) of the same energy.
At a slightly higher level of sophistication we remember that light is an electromagnetic wave consisting of an interwoven electric and magnetic field; E and H.
In complex notation (with the understanding that we only use the real part; in contrast to quantum theory) a standard plane wave with amplitude E or H, wavelength l = 2p/|k| and circle frequency w that propagates in k direction writes and looks as shown below.
E(r,t); H(r,t)   = E0; H0 · exp{i(krwt)}
The electric or magnetic field are vector quantities, always perpendicular to each other. Light thus always has a polarization vector associated with it, defined as the direction of the electric field vector (always perpendicular to the propagation direction).
Where does light come from? The sun, of course, is a major producer of light and so is any other hot body. Max Planck, as you know, first described the spectrum of light emitted by a hot "black body" in his famous work that was the beginning of quantum theory. The link gives a short and simple derivation
What Planck calculated and what the sun actually does is shown in the following pictures.
The sun comes pretty close to a black-body spectrum and the same is true for a light bulb or any other light source relying on high temperatures.
It is clear to a Materials Scientist or Engineer that the sun is hot because nuclear fusion going on in its interior delivers the necessary energy, and that the radiation energy flooding the earth is the one and only energy on which life depends.
Right now (2011) we enter the age of massive solar energy harvesting via solar cells and wind or water power. The necessary materials science and engineering for doing this on a large scale will provide work and jobs for many years to come - but that will not concern us here.
Besides hot bodies we also have "cold" light sources like light emitting diodes (LEDs) and Lasers. We will come to that later in more detail.
Most light sources and all hot bodies produce incoherent light (travelling in all directions with random phases) and multi-chromatic light (having all kinds of frequencies), which is a far cry from the E = E0 · exp(ikr) fully coherent and mono-chromatic plane wave that we like to use as mathematical representation. Sun light or artificial light sources used for illumination thus generate extremely "messy" light from a purists viewpoint. The messy light is nevertheless quite important in a general sense (imagine it missing!) but not of much technical interest - besides generating it.
Laser light, by contrast, is a good approximation to the plane wave model but not of much use for illuminating your kitchen.
What we are interested in here is working with light. That means we have to consider manipulating it by running it through or off materials. What comes to mind in this context are optical products and components. The table below gives an incomplete list of a few catchwords that you should know in this context
Products and components around light technology
Components Products Field
Lenses (and apertures) Microscope, Glasses, Camera, Film projector Geometric optics
Mirrors Reflector telescope; steppers, optical MEMS
Prisms Binocular; Reflectors
Filters Color photography etc.
Diffraction gratings Spectrometer ß
Anti reflection coatings Solar cells, glasses, lenses
Linear polarizers Cameras, sun glasses, optical measurements,
Circular polarizers 3-D cinema, advaned measurements "Tensor" Optics

Interference Optics
Interference filters; Interferometers Optical precision measurements
Phase shifters, High resolution optics Lithography for Microelectronics
"Digital" optics Beamers, Displays, Cameras, optical MEMS