Laser physics, part 2

Up to this point, we have covered the following principles of how lasers work:

• A laser requires some sort of power source (usually an applied EM field) to bring the atoms (or ions or molecules, etc.) in some material to an upper (higher-energy) state (we call this laser pumping)
• Once the laser is in this state, any incident photons lead to the emission of two more photons
• The two photons start a chain reaction that leads to the exponential emission of monochromatic, coherent, strongly-directional light (or other form of EM radiation)

We now introduce some technical terminology. A laser is typically composed of an energy source as well as a (nearly) completely-sealed cavity, filled with some material, called the laser cavity (or optical cavity). The material is known as the gain medium or lasing medium^[9]; the atoms of the gain medium are excited by the energy source and emit light. Gain media (media is plural of medium) can be anything from a solid (e.g. crystalline solid), liquid solution (e.g. organic dyes), or gas (e.g. hydrogen/argon/carbon dioxide gas)^[6]. The gain media is then pumped with energy from the laser’s power source; typically, this is either a strong electric current, a very, very bright light (which is an EM field, since light is an EM wave), or another laser (which, again, is also an EM field).

To keep as many of the atoms as possible in the upper state, which is necessary for stimulated emission, the laser cavity has mirrors at its ends, which reflect the light back and forth throughout the material, continuously re-injecting energy back through the gain medium. When more than 50% of the atoms within the gain medium are in their upper state^[7], we say that a population inversion has occured. At this point, stimulated emission takes over, causing the chain reaction that leads to a rapid emission of more and more photons. One end of the laser cavity is a semi-transparent mirror^[10] that is designed to let a small fraction of the light through, while the rest of the light reflects off to continue the stimulated emission process inside the cavity. This mirror is often called an output coupler. The light that makes it through the output coupler forms the characteristic beam that emerges from one end of the laser.

The four main components of a laser - power source, gain medium, laser (optical) cavity, and output coupler^[11] - are each complex topics in and of themselves, and especially for high-performance lasers, each requires meticulous design and engineering. Lasers fine-tuned to specific tasks often have different requirements for the type of beam, wavelength/frequency, and power efficiency of the laser. Hence, we will discuss lasers in much greater depth in the following sections.

Lasing transitions and the gain medium¶

When designing a laser, we first want to consider the material composition of the gain medium. The gain medium could be composed of elemental atoms, ions, molecules (like the ammonia molecule used in the ammonia maser), crystals, semiconductors, or even a combination of different materials. For this reason, we will use the generic term “quantum system” to represent the atomic/molecular/ionic/etc. constituents of the laser’s gain medium, instead of specific terms like “atom” or “molecule”.

States and energy levels¶

First, we want to identify all the states of the quantum system. This is done by solving for the states of the quantum system with the time-independent part of the Hamiltonian. Just as we showed earlier in performing our calculations for the ammonia maser, if we let the total Hamiltonian of the system be $\hat H = \hat H_0 + \hat H_1(t)$, where $\hat H_0$ is the time-independent portion and $\hat H_1(t)$ is the time-dependent portion, then the states $|\psi_n\rangle$ of the system are found by solving the time-dependent Schrödinger equation:

Depending on the system’s complexity, an analytical solution may be possible to find (for instance, in the case of the hydrogen atom) or be impossible to find (for instance, in the case of all multi-electron atoms). This step therefore may often require using approximations or numerical methods to calculate the states and find the energy eigenvalues, from which we may obtain the emission spectrum (all the wavelengths the system can emit light).

Selection rules¶

Next, we want to find the possible transitions between different energy states. This involves the selection rules. A selection rule defines which transitions between states are possible and which transitions are impossible in a quantum multi-state system. A transition between two states $|\psi_m\rangle \to |\psi_n\rangle$ is said to be allowed if the quantity $\langle \psi_m |\mathbf{r} |\psi_n\rangle$, called the matrix element, satisfies:

However, if $\langle \psi_m |\mathbf{r} |\psi_n\rangle = 0$, a transition is said to be forbidden. For instance, the states of the hydrogen atom, which are parametrized by three quantum numbers ($n$, $m$, and $\ell$) and which we write as $|\psi_{m, n, \ell}\rangle$, satisfy the following selection rules:

In theory, we must compute the matrix element $\langle \psi_m |\mathbf{r} |\psi_n\rangle$ for all the possible combinations of states $|\psi_m\rangle$ and $|\psi_n\rangle$ to find all possible transitions; in practice, if the states of a quantum system already satisfy certain orthogonality relations, we can quickly tell which matrix elements are zero, and therefore which transitions are allowed or forbidden.

Identifying transitions¶

Next, we need to sort through all the allowed transitions to identify the ideal transitions. Atoms usually have many different possible radiative transitions (also called decay modes, “radiative” means that the transition leads to a photon being emitted), so we want to find the best decay modes. A decay mode is ideal when all the below conditions are satified^[12]:

1. The transition wavelength of the decay mode corresponds with the desired wavelength of the laser light. For instance, you would look for transition wavelengths between 780nm-2500nm if your target wavelength for the laser is in the near-infrared range.
2. It should be easy to create a population inversion. In practical terms, it means that the transition rate (probability of a transition per unit time) of stimulated emission should be much higher than the transition rate of spontaneous emission in the system.
3. The upper state in the decay mode should have a generally long lifetime $\tau$, so that the quantum system maintains a higher population in the upper state compared to the lower state (which again is what leads to a stable population inversion)
4. The lower state in the decay mode should be quickly depopulated by some mechanism^[13], so that the lower state does not simply re-absorb the same radiation it emitted. This depopulation mechanism should be non-radiative (meaning it doesn’t involve releasing a photon), instead causing the lower state to decay by some other means, like the emission of a phonon (a type of structural vibration) for solid gain media or by molecular collisions that transfer away energy in gases^[15].
5. The decay mode can lead to another decay, meaning that the system in the lower state can decay to an even lower state. This requires that the lifetime of the lower state is not too long, and that it has allowed transitions (by the selection rules) with a high transition probability to decay to another state. Multiple decays allows building multi-level lasers, which have many advantages over two-level lasers (lasers that use a transition from only one upper state to the ground/lower state)

Nearly all of these conditions can be checked against with (tedious) calculations. The transition wavelength comes from calculating the energy eigenvalues associated with each of the states of the system. Using the energy eigenvalue expression $E_j = \langle \psi_j |\hat H|\psi_j\rangle$, which we learned from the section on the matrix representation of operators, we can write the transition wavelength for a transition from upper state $|\psi_m\rangle$ to lower state $|\psi_n\rangle$ as follows:

From calculating all possible transition wavelengths for the allowed transitions, we can build up the spectrum of the system, giving all the wavelengths of light that the system can (theoretically) emit. We need to then filter the transitions by our other conditions. The transition rates $\Gamma_{fi}$ for stimulated and spontaneous emission can be calculated using Fermi’s golden rule the same method we outlined for the ammonia maser, from which we may easily obtain the lifetime $\tau$ of the upper state with $\tau = 1/\Gamma_{fi}$, as we mentioned earlier. Multi-level decays can be checked with the selection rules’ matrix elements $\langle \psi_m |\mathbf{r} |\psi_n\rangle$ as well as further computation of transition rates.

In the case of optically-pumped lasers (and masers), we also need the gain medium to absorb light strongly around the pump wavelength, which is the wavelength of the light used to pump the laser/maser. The wavelength of the laser’s light does not always match the pump wavelength, and the key reason for this is that not all lasers are two-level systems; three-level and four-level lasers (which we’ll get to soon) are excited to their highest-energy state by the pump source, but decay in several steps back to the lowest-energy state, where stimulated emission takes place in (usually) one of these transition. Such lasers typically use a mixture of different atomic species or molecules (species means the same as “type”), and especially for solid-state lasers that use a combination of solids (usually crystals) with ions of a different element packed into their crystal lattice. In such materials, one of the species is typically the one that absorbs light and quickly decays to a lower energy state, triggering additional decays in the next species. For solid-state lasers in particular, the species responsible for light absorption is typically the solid, which (again) is typically a crystal such as $\mathrm{Y3Al5O12}$ (yttrium aluminum garnet) or (yttrium lithium fluoride).

Three and four-level systems¶

The ammonia maser, which we’ve been analyzing, is a two-level laser. Unfortunately, it is not a very practical laser, and emits miniscule amounts of power in its microwave beam. This is all to do with the fact that it operates as a two-state system, where we have only one upper and one lower state.

In a two-state system, any atom (or ion or molecule) that decays from the upper state must necessarily increase the population of the lower state. But a laser operates by achieving a population inversion, where the upper state has a greater population than the lower state, so we need to constantly boost the atoms back into the upper state using our pump source. This can theoretically be done, but uses a lot of energy. A good analogy, which we borrow here from the textbook Physical Chemistry from Libretexts^[14], is reversing the flow of water in a waterfall; while possible, it is incredibly energy-consuming and inefficient.

Any form of pumping can only establish a thermal equilbrium between the two states, where the population $N_2$ of the upper state and the population $N_1$ of the lower state are equal, that is, $N_1 = N_2$. Thus, $N_2/N_1 = 1$, which cannot create a population inversion, since a population inversion is created (by definition) when the population of a upper state $N_j$ and the population of a lower state $N_i$ satisfies $N_j/N_i > 1$. The ammonia maser gets around this issue by not using a pump source at all; it uses an electric field to separate ammonia molecules that happen to naturally be in the higher-energy state from those of the lower-energy state, using the fact that they have different angular momenta. Unfortunately, it also means that it outputs very little power. We can also get around this issue by running the laser in pulses, so that there is not enough time for thermal equilibrium to develop, and we can immediately pump more energy after each pulse to raise the population of the upper level. But it is impossible to make a pumped two-level laser (or maser) operating continuously - you either need to give up continuous operation and make the laser pulsed, or give up pumping and end up with a very weak laser/maser beam.

So it should come as no surprise that almost all lasers are either three-level lasers or four-level lasers. The most efficient lasers, particularly for continuous operation, are four-level lasers.

Case studies of lasers¶

We have seen the general mechanism of all lasers, but let’s now examine specific types of lasers to gain a deeper understanding of how real lasers work. The five laser/maser types we will examine are all very well-known and represent a variety of different laser designs:

The ammonia maser¶

The first of the two microwave laser (maser) types we will cover is the ammonia maser. The ammonia maser is one of the simplest types of lasers/masers; this is because it is a two-level system, composed of just one ground state and one excited state, which we have already analyzed in-depth previously.

The lasing action of the ammonia maser results from the ammonia molecule transitioning between the two states. This physically occurs as a result of the nitrogen atom in the ammonia molecule “flipping” to the opposite side of the molecule, which is often called a nitrogen inversion or umbrella inversion transition^[1]. As these two states have slightly-different energies, there exists an energy difference between the two states of about $\mathrm{97.8 \mu eV}$. As you can see, this is an extremely small energy difference, hence why it leads to the emission of photons in the microwave range (which carry much less energy than visible or UV light).

Note: Technically-speaking, the ammonia molecule has a lot more possible states, but only two of them are relevant in the ammonia maser, so we can just consider those two states. Additionally, the lower-energy state that we have been calling the “ground state” is not technically the ground state. We have only chosen to use this terminology for its much greater familiarity.

The umbrella inversion transition happens spontaneously as a result of quantum tunneling “through” the molecule. The configuration of the ammonia molecule can be modelled as a harmonic potential with two stable equilibria (representing the nitrogen atom on the left and on the right of the ammonia molecule), along with a Gaussian potential barrier in between. That is, we have:

Where $k, c, b$ are some constants to fit to empirical data. Solving the time-independent Schrödinger equation $\hat H \psi = E \psi$ allows us find the allowed energy levels, from which we calculate the frequency of the umbrella inversion transition to be 24 GHz, or in terms of wavelength, 1.26 cm.

Remember that for stimulated emission to occur, we must first bring the gain medium (in this case, the ammonia molecules) to an excited state. We may accomplish this by utilizing the fact that at any one time, the ammonia molecule may be in either one of its two aforementioned states. So, on average, in a certain quantity of ammonia gas, there will be some ammonia molecules in the ground state, as well as some ammonia molecules in the excited state. Other than having different energies, the two states are also distinct in another respect: they possess a different electric dipole moment. Thus, if we apply an electric field, we can separate the ammonia molecules in the excited state from those in the ground state, without the need for any pump source!

Thus, the ammonia maser is designed as follows. Ammonia gas, stored in a container, is slowly discharged and flows through a collimator that keeps it focused in a tight beam. An electric field is then applied to separate the ammonia molecules in the excited state from those in the ground state. This means that only the ammonia molecules that are in the excited state are fed into the optical cavity (technically, microwave cavity) of the maser. The cavity is a resonant cavity, meaning that is constructed to restrict the electromagnetic waves within the cavity to those that match the resonant frequency of the transition - that is, 24 GHz. Once in the resonant cavity, the excited ammonia molecules decay to their ground state by the process of stimulated emission, emitting photons in the microwave range as a result (in more familiar terminology, microwaves). Those emitted photons, bolstered by the resonant characteristics of the cavity, continue to reflect within the chamber and re-excite ammonia molecules to their excited state, ensuring that stimulated emission continues.^[2]

Note: we say “photon” and “electromagnetic wave” interchangeably here, because electromagnetic waves are (the classical description of) photons.

We have therefore achieved a population inversion, where most of the ammonia molecules in the resonant cavity are in their excited state, which allows stimulated emission to take place continually. The output coupler allows some of the microwaves to pass through, giving the ammonia maser its characteristic microwave beam. This beam, however, is extremely weak, in some cases on the order of nanowatts^[3], so the ammonia maser is mostly relegated to scientific research and has very few other applications.

The hydrogen maser¶

The hydrogen maser is very similar to the ammonia maser: it uses a gas as its gain medium, emits microwaves (though of a longer wavelength), it requires no pump source, and it produces a very weak beam. However, its gain medium uses atomic hydrogen rather than ammonia, and it operates on a different type of transition, known as the hyperfine transition.^[4]

The hyperfine transition in the hydrogen atom results from a tiny energy level just above the ground state, caused by the spin of its electron. Even when it is in its ground state, the hydrogen atom possesses a slightly different energy depending on whether its electron is spin-up or spin-down. The hyperfine transition occurs when the electron “flips” its spin, meaning it changes from spin-up to spin-down (or spin-down to spin-up). The energy difference between the two states is about $\mathrm{5.87 \mu eV}$, so the transition wavelength is about 21 cm (corresponding to a frequency of 1.43 GHz). This is also known as the hydrogen line and is very important in astronomy, but we will focus only on its relevance to the hydrogen maser here.

In a hydrogen maser, hydrogen gas is slowly discharged from a container, where, just like in the ammonia maser, it is collimated. Then, an electrical discharge is passed through the gas, ionizing the molecular hydrogen ($\mathrm{H2}$) into atomic hydrogen (individual atoms of hydrogen). Within the hydrogen gas, there will be some hydrogen atoms in the spin-up state, and some in the spin-down state. Crucially, hydrogen atoms in the spin-up and spin-down electron states are deflected differently in the presence of a magnetic field; thus, a magnetic field is used to separate hydrogen atoms in the excited state from those in the lower-energy state. Only the hydrogen atoms in the excited state are fed into a resonant cavity, which, similar to the ammonia maser’s resonant cavity, is tuned to 1.43 GHz. Once again, stimulated emission takes place; the output coupler lets out a fraction of the microwaves, producing a very stable microwave beam - so stable, that hydrogen masers are frequently used as high-precision clocks.^[5] The microwave beam produced by hydrogen masers is even weaker than that of ammonia masers, and is often around only 1 picowatt (or even lower!).^[8] Thus, they are useful for only a few applications, such as (again) scientific research and timekeeping.