# 6.3: Lasers

- Page ID
- 17212

## Properties of Laser Light

Lasers are so ubiquitous in daily life, that it's easy to overlook what a technological miracle they are. They have countless applications, ranging from devices as crude as pointers, to tweezers for manipulating microscopic objects. The wide range of applications for this device stems from just a few properties of the light that is not shared by "normal" light. The primary operative word for describing laser light is that it is *coherent*. Coherence is the condition where all the photons involved are in phase with each other. In addition, laser light is also close to being monochromatic (the photons are generated at the same frequency, but for various reasons there is some small spread in what comes out), and is usually collimated.

Besides the coherence, a key feature is the way that the power output of a laser is focused from a rather disorderly beginning to a highly-ordered final product. Indeed, the word "laser" is in fact an acronym, with the first two letters standing for "light amplification." We will complete this acronym shortly.

Finally, the design of certain gas lasers is such that the emitted laser light is polarized as well. Besides the usefulness of this feature for certain applications, it comes from an integral part of the design of the device – it is not simply an "add-on."

## Stimulated Emission

For now, let's make our discussion as simple as possible, by assuming we are talking about the interaction of light with an atom that has only two energy levels. There is a collection of these atoms present, and they are in an assortment of states, some in the ground state, and some in "the" excited state. Our discussion of photon emission to this point has generally centered around spontaneous emission, the triggering of which we attributed to a perturbation in the hamiltonian that comes from (if nothing else) vacuum fluctuations.

Suppose some of these atoms spontaneously emit photons, and now the collection of atoms is not sitting in a total vacuum, but is in the presence of many photons as well. We now consider what effects these present photons can have on the atoms. The answer depends upon which state the atom is in when it encounters the photon. If the atom is in the ground state, then it can absorb the photon and go to the excited state. If the atom is in its excited state, it can't absorb the photon, but the time-varying EM fields of the photon present a nice perturbation to the electron's hamiltonian (at just the right frequency!) to induce the emission of a photon. This is called *stimulated emission*. The photon that induces the emission is unaffected by the interaction, and continues on its merry way. The emitted photon naturally has the same frequency as the passing photon (the energy gap assured this), but it also emerges moving in the same direction as, *and** in phase with* the stimulating photon. This completes the acronym - * l*ight

*mplification by the*

**a****timulated**

*s**mission of*

**e***adiation.*

**r**It turns out that if a photon happens by two of these atoms, the first in the ground state and the second in the excited state, the probability that it will be absorbed by the first is equal to the probability that it will stimulate emission in the other – neither of these results is preferred over the other. Now consider a collection of these atoms in a state of some fixed total energy. This total energy comes from the number of atoms in the ground state multiplied by the energy of that state, plus the number of atoms in the excited state multiplied by the number of atoms in that state. As we will discuss in a future chapter, the populations of these two states subject to the constraint of the total energy can be shown to be a function that depends upon the energy difference of the two states and the temperature of the collection of atoms. Specifically, if \(N_1\) is the number of atoms in the ground state, \(N_2\) is the number of atoms in the excited state, and the energy difference is \(\Delta E\), then the following relation holds:

\[ N_1 = N_2 \;e^{\dfrac{\Delta E}{k_B T}}\;, \]

where \(T\) is the temperature and \(k_B\) is Boltzmann's constant. Clearly \(N_1\) is greater than \(N_2\), but how much greater? We can do a very quick calculation of this using an approximation that all physicists have at their fingertips (**take note!**): At room temperature, the constant \(k_B T\) is about \(\frac{1}{40} eV\). The energy difference between two energy states of an atom is on the order of a few electron volts (think of hydrogen!). Let's say the energy difference is \(1eV\). Then the number of ground state atoms exceeds the number of excited state atoms by a factor of \(e^{40}=2.4\times 10^{17}\)!

Okay, so a photon passing through this collection has an equal probability of being absorbed or stimulating emission if it happens by an atom prepared for one of those actions. But there are so many more atoms available to absorb the photon, that this has a much greater chance of happening. While waiting for a photon to stimulate emission, an atom in an excited state will impatiently emit a photon spontaneously, which will then be much more likely to be absorbed than stimulate emission, and things continue like this, with some atoms spontaneously sending out photons while others absorb them... no laser.

## Population Inversion

The problem is clear – there are just far too few atoms in the excited state. If we can somehow pump some energy into the system to get a much higher proportion of atoms in the excited state, then the probability of a photon stimulating emission would increase. Clearly one way to do this is to fire more photons in (called *optical pumping*) – they will be absorbed, and the number of excited states goes up. Another way is to raise the temperature of the collection of atoms in any way you can (usually via *electrical discharge pumping*), since this increases the denominator of the exponent in Equation 6.3.1.

However we do it, to get over the hump and get some "light amplification" going, with equal probabilities of absorption and stimulated emission, we need the number of excited states to exceed the number of ground states, a condition known as a *population inversion*. While pumping can greatly increase the population of the excited state, it cannot actually invert the population. If the population is balanced, there's no way to force the next several injected photons to be absorbed (causing the inversion), since they are equally-likely to stimulate emission, reducing the excited state population. Looking at Equation 6.3.1, we see that raising the temperature to infinity only serves to create equality between \(N_1\) and \(N_2\), so that is not the solution either.

## Metastable States

The solution is to discard this two-energy-level atom, in favor of one that has (at least) three energy levels, with the middle level being quite special. We discussed selection rules, we saw that some transitions are significantly faster than others. Consider a three-energy-level atom where the transition from the middle energy level to the ground state is slowed from spontaneous emission by a selection-rule-like mechanism, but is still subject to stimulated emission. Such an energy state is called *metastable*.

If these atoms start in (mostly) ground states, and we begin pumping in energy, the top and middle states will begin populating. Once again, the pumping alone will not invert the populations between any two states, but there is a difference here. The atoms that go to their top state will spontaneously emit photons quickly (because that state is not metastable), with some of these atoms returning to their ground state, and others dropping to the middle state. When the middle state reaches the population of the ground state, direct population of that state from the ground state is no longer possible, but the atom can still take the ground-to-top, top-to-middle route to populate its metastable state. And because this two-step process happens so much faster than the spontaneous decay of the metastable state, the metastable state population inverts relative to the ground state. Then stimulated emission that exceeds absorption can occur. So long as we keep pumping the ground state into the top state, the production of coherent photons from stimulated emission can continue – a working laser!

The efficiency of the laser is improved the more that we can increase the population inversion. That is, the greater the ratio of \(N_2\) to \(N_1\), the greater the rate of simulated emission, because there are fewer atoms competing with states primed to absorb photons. A way to improve the inversion ratio is to use atoms with *four* (or more) energy levels. In this case, the metastable energy level is the next-to-highest, so there is an intermediate energy state between the metastable state and the ground state. Once again the pumping populates the top state, which in turn overpopulates the metastable state. But now we create photons that stimulate emission with transitions between the metastable state and the next lowest state below it. This intermediate state is not metastable, so it spontaneously emits photons quickly (going back to the ground state). The effect of this is that the population of this intermediate state remains very low (much lower than the ground state), improving the ratio. With very little absorption competition for the incoming photons, each one is far more likely to stimulate emission, improving efficiency.

## Design

When designing a laser, the medium is obviously quite important – it needs to have the desired structure of energy states, with a well-placed metastable state. The two main types of media are solid-state and gas. It also requires some sort of pumping mechanism. Typically solid-state lasers are optically-pumped, while gas lasers are generally pumped by electrical discharge. Besides their sizes (solid state lasers can be significantly more compact), one generally can tell these apart from the voltages they require – it requires a reasonably high voltage to induce the necessary electrical discharge, while low power solid state lasers can run on AAA batteries.

Another design element that does not involve the medium or pumping is called the Fabry-Perot cavity. This cavity encloses the region where the medium is housed, and is constructed so that photons causing stimulated emission can be "recycled." This is achieved by placing two reflecting surfaces on opposite ends of the medium, so that stimulated photons are reflected back into the medium in order to stimulate more emissions. The spacing of the reflecting surfaces needs to be such that a standing wave is set up between them, meaning that the length of the cavity is an integer number of half-wavelengths of the laser light. Of course, if the reflecting surfaces don't let any of the light out, it isn't of much use, so one of them is only partially-silvered, allowing some fraction to escape.

There is another important reason to include the Fabry-Perot cavity in the design. It was stated earlier that stimulated photons emerge not only with the same frequency and in phase with their parent photons, but they also emerge moving in the *same direction*. When the first photons emerge due to the initial pumping, they can be going in any number of directions. The reflecting surfaces select photons moving parallel to the axis joining the mirrors to return and stimulate more photon emissions, which emerge also moving parallel to the axis. Those that miss the reflecting surfaces don't return, and don't stimulate more photons to move along their directions.

Often these reflectors are curved in a concave shape to focus stray photons back toward the axis of the cavity, to provide more leeway for the standing wave – it would be a shame to let any perfectly good photons escape unnecessarily.