Four-level laser

A four-level laser uses four energy levels of its gain medium. Pump light raises the atoms or ions from the ground state (level 0) to a pump level (3), from which they relax rapidly and without radiation to the upper laser level (2). Laser action runs from level 2 to the lower laser level (1), and level 1 drains quickly back to the ground state. Because level 1 lies many $kT$ above the ground state, it is nearly empty in thermal equilibrium, and its fast decay keeps it empty during operation. Every ion placed in level 2 then contributes to population inversion, and gain appears as soon as pumping starts. Nd:YAG is the textbook case: its lower laser level lies about 2110 cm⁻¹ (0.26 eV) above the ground state, where the Boltzmann factor at 300 K is $4.0 \times 10^{-5}$.

Rate equations

With level 1 empty, the inversion is simply the upper-level population $N_2$. Below threshold it obeys

$$\frac{dN_2}{dt} = R_p - \frac{N_2}{\tau},$$

where $R_p$ is the pump rate per unit volume and $\tau$ the upper-state lifetime, so in steady state $N_2 = R_p\tau$. The gain coefficient $\sigma N_2$ grows linearly with pump rate from zero. Once the gain equals the cavity loss, at threshold, $N_2$ is clamped at its threshold value $N_\text{th}$ and every additional pump photon feeds stimulated emission into the laser mode, giving a linear output above threshold.

Two conditions make a system behave this way. The lower level must empty faster than it fills, $\tau_1 \ll \tau_{21}$; for Nd:YAG the ⁴I₁₁/₂ level relaxes by phonon emission in well under a microsecond, against 230 µs for the upper level. And the lower level must be far enough above the ground state that its thermal population is negligible, which requires an energy gap of many times $kT$ (208.5 cm⁻¹ at 300 K).

Comparison with three-level lasers

In a three-level laser such as ruby, the lower laser level is the ground state. With equal degeneracies, more than half the ions must be pumped out of it before any gain appears. The difference in pump requirement is large. The population inversion entry works out that a 1 cm Nd:YAG rod with a 95 % reflecting output coupler reaches threshold at $\Delta N = 9.2 \times 10^{16}$ cm⁻³, about 0.066 % of the ions in a 1 at.% crystal. Sustaining it with a 230 µs lifetime takes a pump rate of $4.0 \times 10^{20}$ cm⁻³ s⁻¹, an absorbed power of

$$\frac{\Delta N}{\tau}\,h\nu_p = 98\ \mathrm{W/cm^3}$$

at 808 nm, assuming unit quantum efficiency. Ruby with $1.58 \times 10^{19}$ Cr ions per cm³ must hold $7.9 \times 10^{18}$ cm⁻³ in the upper level merely to become transparent; with a 3 ms lifetime and pump light near 550 nm that is about 950 W/cm³, roughly ten times more, before any loss is overcome.

Quasi-three-level lasers

Many useful media fall between the two cases. In Yb:YAG the lower laser level for 1030 nm is a Stark component of the ground manifold only about 612 cm⁻¹ up; at room temperature it holds about 5 % of the ions. The crystal therefore absorbs at the laser wavelength unless pumped hard, threshold rises with temperature, and a minimum pump intensity is needed to bleach the reabsorption. The same applies to Er at 1.5 µm and Tm and Ho near 2 µm. The benefit is a small quantum defect; quasi-three-level media are run cold, at high pump intensity, or in geometries such as the thin-disk laser and fiber that supply both.

Other four-level systems

Vibronic lasers such as the Ti:sapphire laser and the dye laser act as four-level systems in which levels 1 and 3 are vibrationally excited states of the lower and upper electronic manifolds, emptied in picoseconds. Nd in glass and in YVO₄ also operates on the four-level scheme.

Pitfalls

The labels describe idealizations: real media have manifolds of levels, and a medium can be four-level at one wavelength and quasi-three-level at another, as Nd:YAG is at 946 nm, whose lower level lies in the ground manifold. Heating a four-level crystal populates its lower level and raises threshold. A slowly emptying lower level causes self-terminating behaviour, where lasing stops after a short burst even with the pump on.

Common questions

Why is a four-level laser more efficient than a three-level laser?

Its threshold does not require depleting the ground state, so far less pump power is spent before gain appears, and the lower level adds no absorption at the laser wavelength.

Is a helium-neon laser a four-level laser?

It is usually described as one: helium atoms transfer energy to an upper neon level, and the 632.8 nm transition ends on a level well above the neon ground state. See helium-neon laser.

References: A. E. Siegman, Lasers (University Science Books, 1986); O. Svelto, Principles of Lasers 5th ed. (Springer, 2010); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics 3rd ed. (Wiley, 2019); W. Koechner, Solid-State Laser Engineering 6th ed. (Springer, 2006).