Rare-earth ions (as laser and amplifier dopants)

Rare-earth ions are the lanthanide elements, doped into a crystal or glass host in their trivalent form, where they replace a host cation at concentrations from a fraction of a percent to several percent. They supply the gain in most solid-state and fiber lasers and in optical amplifiers: neodymium at 1064 nm in the Nd:YAG laser, ytterbium at 1030 nm pumped at 976 nm, erbium at 1530–1560 nm in the erbium-doped fiber amplifier, thulium near 2 µm and holmium near 2.1 µm. The shared feature is that their optical transitions take place inside the partly filled 4f shell, which gives narrow lines, small cross sections and upper-state lifetimes of hundreds of microseconds to about 10 ms.

Shielded 4f transitions

A trivalent lanthanide ion has the configuration [Xe]4fⁿ, with n running from 1 for Ce³⁺ to 13 for Yb³⁺ (Nd³⁺ is 4f³, Er³⁺ is 4f¹¹, Tm³⁺ is 4f¹²). The 4f orbitals lie inside the filled 5s and 5p shells, which screen them from the electric field of the neighboring host atoms. The energy levels are therefore close to those of the free ion and change little from one host to another: erbium amplifies near 1.53 µm in silica, fluoride glass and many crystals alike. Transition-metal ions such as Ti³⁺ in sapphire differ: their outer d electrons couple strongly to the lattice and give broad, host-dependent bands.

Electric-dipole transitions within one configuration are forbidden by parity in the free ion. At a site without inversion symmetry, the odd-parity part of the crystal field mixes in a small amount of other configurations and makes them weakly allowed, so oscillator strengths are small. Peak cross sections are therefore low, roughly 5–7 × 10⁻²¹ cm² for erbium in silica and 2.8 × 10⁻¹⁹ cm² for Nd:YAG at 1064 nm, as listed under absorption cross section. Radiative lifetimes are long, and the upper-state lifetime is about 230 µs in Nd:YAG, about 1 ms for ytterbium and about 10 ms for erbium in silica.

Stark splitting and linewidth

The weak crystal field still lifts the degeneracy of each free-ion level (a $J$ manifold) into several Stark components, spaced by tens to a few hundred cm⁻¹. Thermal population among these components decides whether a transition behaves as three-level or four-level. In Yb:YAG the lower laser level is the ground-manifold component about 612 cm⁻¹ up; with $k_BT$ = 208.5 cm⁻¹ at 300 K, the Boltzmann factor

$$e^{-612/208.5} = 0.053$$

puts about 5 % of the ions there, so the crystal reabsorbs its own emission unless pumped hard.

Each transition between Stark components has a homogeneous width set mainly by phonon scattering at room temperature, far wider than the 0.7 kHz implied by the 230 µs lifetime: about 0.45 nm (119 GHz) for Nd:YAG at 1064 nm. In glass, ions occupy many slightly different sites, and the inhomogeneous spread joins the Stark lines into a smooth band; the 1530–1560 nm erbium window spans 3.8 THz. Line broadening treats both mechanisms.

Saturation intensity and quantum defect

The small cross section and long lifetime enter the saturation intensity

$$I_\text{sat} = \frac{h\nu}{\sigma\,\tau}.$$

For Nd:YAG at 1064 nm, with $\sigma$ = 2.8 × 10⁻¹⁹ cm² and $\tau$ = 230 µs, this is 2.9 kW/cm². Long storage suits these ions to Q-switched and amplified pulses: inversion builds up over the lifetime and is released in one pulse.

The quantum defect, $1 - \lambda_p/\lambda_s$, sets the minimum heat load. It is 5.2 % for ytterbium pumped at 976 nm and lasing at 1030 nm and 24 % for neodymium pumped at 808 nm and lasing at 1064 nm. The small value for Yb is one reason high-power fiber lasers use ytterbium-doped fiber.

Ion-ion effects and pitfalls

At high doping, neighboring ions exchange energy: cross-relaxation and cooperative upconversion empty the upper level, and clustering of erbium in silica produces pairs that saturate quickly and limit amplifier efficiency. A measured lifetime well below the low-concentration value is a sign of this quenching. Ytterbium-doped fiber at high inversion can suffer photodarkening, a slow growth of background loss. In thulium, cross-relaxation is useful: one ion excited at 793 nm can leave two ions in the upper laser level of the thulium fiber laser.

Common questions

Why are rare-earth laser lines narrower than those of Ti:sapphire or dyes?

The 4f electrons are shielded from the lattice, so vibrations of the host broaden the transitions only weakly. In Ti:sapphire and dyes the active electrons couple strongly to vibrations, and emission spreads over hundreds of nanometers.

Why is erbium used at 1550 nm?

The ⁴I₁₃/₂ to ⁴I₁₅/₂ transition of Er³⁺ falls at 1530–1560 nm, where silica fiber has its lowest loss, and the 10 ms lifetime keeps the gain from following the bit pattern of a data signal.

Does the host change the wavelength?

Only slightly in center wavelength. The Stark structure, linewidth, cross sections and the non-radiative decay rate, which depends on the host's highest phonon energies, can change considerably.

References: G. H. Dieke, Spectra and Energy Levels of Rare Earth Ions in Crystals (Interscience, 1968); E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994); W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006); A. E. Siegman, Lasers (University Science Books, 1986).