Basic Laser Physics
Introduction
Lasers can be found all around us in the modern age from computer mice to laser guided missiles. Since their invention in the 60's they have evolved many different forms and uses. In this article we shall look at the underlying concepts and physics of laser theory, I have also written a seperate article describing other laser types and some of their uses. Here are a few characteristics of laser beams due to their stimulated emission process.
- Coherence
- Narrow Spectral Bandwidth
- High Intensity
Also we get a high level of directionality from the resonator configuration. As you can imagine some of these properties might be very useful, lasers are often used for industrial applications where cutting/drilling is required, as well as in the medical world and tele-communications.
Energy Levels
Energy levels are key to the operation of all lasers, an energy level is a measure of the excitation of an atom we definer a population density for these levels($N_1$ and $N_2$ units are $m^{-3}$). Basically we can excite atoms by giving them energy(absorbing a incident photon with energy $h\nu$), when we give the electrons in an atom more energy they will move to a higher energy levels this is called absorption.
Probability for absorption is given as below where $B$ is the Einstein B coefficient and $\rho$ is the energy density.
The electrons will stay in this excited state for a short time and then drop back down to the 1st level. The natural relaxation of these electrons gives rise to a photon been emitted this is called spontaneous emission. The interesting thing is we can stimulate a similar effect by introducing another photon, this has a certain probability of stimulating an emission depending on the population density in the upper energy level.
As you can probably see from the formulae we need to have a population inversion in order to generate optical gain, where $N_2$ > $N_1$ in order to have a good probability of stimulated emission. This can never occur in a two level material such as above as the population can only be shared 50:50 that is to say there is equal probability of absorption or emission. At this point the material becomes transparent to any incoming radiation so as any incident radiation. Note: we tend to ignore spontaneous emission as a contributor to the laser output
III and IV energy level materials
For optical gain to exist in a material we need to look carefully at the spread of population across the layers. We will now look at a material with 3 energy levels(like a ruby laser). We have seen that the problem with a two level material is that the population can never be sustained above a 50:50 ratio.
The Diagram to the left shows a 3 level material, the second level $E_2$ is a temporary stop as far as the electrons are concerned. The electrons naturally relax to the third energy level in ~$10^{-12}$s this transfer is a radiation less transfer. The third level is what we call metastable where the electrons can reside for a relatively long 3ms before relaxing. As we can see this third level allows a population inversion to build up easily and thus giving us optical gain.
We can build upon the idea of a 3 level material by adding a fourth level just above $E_1$ this level functions similar to that of the second level we looked at previously and naturally relaxes to the 1st level very fast. This means that the 4th level is basically unpopulated thus making a population inversion between levels 3 and 4 almost instantaneously. A 4 level material therefore makes a very efficient gain medium.
Optical Gain
Optical gain is required to make any laser a reality, the basic idea is that for any photon entering the gain material more photons will come out the other end given that the material is sufficiently excited(or pumped). Here we will consider a 3 level gain material(eg. a ruby laser) though optical gain can be achieved wherever a population inversion is present.
Now if we take a cut of the gain material of area A and length dx then the change in intensity over that dx can be shown as.
Where $\sigma$ is respectively the cross section for stimulated emission and the cross section for absorption, in our case simply replaced with A. This can then be rearranged by taking out the common factor I. We can now see that if we have $N_3$ > $N_1$ we will obtain a positive change in intensity giving us optical gain. We then simply integrate this across the length of the gain medium to work out $I_{out}$ given $I_{in}$.
We can now define a gain coefficient as $\alpha = \sigma(N_3-N_1)$ with units $m^{-1}$.
The Laser Oscillator
Right so far we have a material that can amplify and input to get a more intense beam we could either make the gain material longer as is the case in fibre lasers, but for now we are going to concentrate on the more conventional laser the laser oscillator or cavity. This method utilises 2 mirrors to reflect the beam back through the gain material so as to stack the amplification. Ah you say how does the light escape? Well one of the mirrors is not 100% reflective, this allows a portion of the light to escape every time it passes the mirror giving us our nice laser beam.
Say we have our gain material and we want to find out what reflectivity we should have for our mirror. To find out we should look at the path of the beam at various points throughout one cycle of the cavity. Say a photon starting at A and travelling to B passes through the gain material this amplifies it according to $ I_{out} = I_{in}e^{\alphaL} $ this is then reflected off the $R_1$ before it passes through the gain material again. Then a portion is reflected off $R_2$ back to A and the rest continues to escape the cavity.
We can then say to sustain a constant intensity in the beam we can equate $I_{A'}$ and $I_{A}$ thus we require $ R_1R_2e^{2\alphaL}=1 $. If the threshold condition is met then the laser will produce a beam of very constant intensity.
Free Running Laser
This is probably the most simple type of laser, to excite the laser(3 level ruby) we will use a flash lamp powered by a a charged capacitor. I think the easiest way to see how the laser operates is to look at how the intensity and population inversion vary with time.
The top graph shows the intensity of the flash lamp over time this is what pumps the gain material. The second graph shows the population inversion between the 3rd and 1st level you can see that this starts as a negative value but increases rapidly towards the threshold value. Once this threshold is reached the laser can start lasing. The third graph shows the intensity of the beam. We can see that once the threshold is reached the intensity peaks for a very short pulse. This is due to the population inversion been depleated and needs to re-establish itself before it can continue to function. As energy is still been supplied from the flashlamp the populating reaches threshold again and we see a series of pulses. These are called relaxation oscillations and occur on timescales of $\mus$. After a few pulses the flashlamp intensity decreases in such a way that a threshold inversion cannot be sustained and the laser stops.
I will admit right now that the graphs to the left are not accurate at all are not based on experimental data but are just to illustrate the ideas involved. I hope you will bear this in mind as they were only drawn in photoshop :)
Perhaps it is wise to mention something of efficiency in laser systems. The free running laser is deffinately not efficient for a lamp energy of $1kJ$ the Energy in the laser pulse is ~$1J$ giving a rather measly efficiency of 0.1%. I think that the highest efficiency for a laser runs at about 2% though this might be improved with future designs.
Continuous Wave Lasers
What if instead of using a flashlamp to pump our laser we use an arclamp a continuous power source? Well this will provide us with a continuous beam of a sustainable intensity. It is worth noting that the relaxation oscillations are still visible in the beam intensity but the laser will eventually settle down to a constant intensity when the threshold population inversion is secure.
Q-Switching
The Q or Quality of a laser cavity is defined by a Q value. A low loss system is assigned a high Q value and a lossy system a high one. We can create a low Q system by blocking the light from the gain material reflecting off one of the mirrors.
If we switch from a high quality system to a low one then we can build up a population inversion $(N_3-N_1)_{max}$. If we then switch back to a high quality system at the peak of this inversion then we can create a pulse. The length of this pulse depends on the size of the cavity and the gain medium but the pulse will usually depleat the population inversion within a few trips. $\deltaT_p \sim 3xT_{RT} $ where $T_{RT}$ is the time for one round trip and $\deltaT_p$ is the pulse FWHM(Full width half maximum).
If for example our laser cavity has a length of 1m and a average refractive index of 1.3 then we get:
If we say that the pulse contains 1J of energy the we can work out power $P = \frac{\deltaE}{\deltaT_p} = \frac{1}{26.01x10^{-9}} = 38.4MW$
We should now look at how we can implement Q-switching in a laser cavity. There are two ways we can do this, influencing the cavity externally called an active system or using a passive system where the switching is achieved with no input external to the cavity.First we shall look at active switching. An easy way to achieve this is to have a rotating prism within the laser cavity controlled by a motor. This motor can rotate the prism such that the prism will only reflect the beam back to the gain material at a set period timed to coinside with the time required for peak population.
A passive system can be implemented in a simmilar way but instead of a motorised prism all we need to do is stick a piece of 2 state material between the gain medium and the mirror. As we said earlier the two state material becomes transparent to radiation at a certain intensity as the probability of absorption is the same as emission. We can utilise this to block light until a required intensity is reached, when this is reached it will become transparent the population in level 2 then relaxes and becomes opaque blocking the mirror once more.