Absorption and emission cross section (σ)

The cross section $\sigma$ of an optical transition is the area that a single atom, ion or molecule effectively removes from (absorption) or adds to (emission) a beam passing through it. It converts a density of emitters into a coefficient per unit length: $N$ ions per cm³, each with absorption cross section $\sigma_a$ in cm², give an absorption coefficient $\alpha = \sigma_a N$ in cm⁻¹. Typical peak values span several orders of magnitude: about 2.8 × 10⁻¹⁹ cm² for the 1064 nm line of Nd:YAG, about 3 × 10⁻¹⁹ cm² for Ti:sapphire near 800 nm, roughly 5–7 × 10⁻²¹ cm² for erbium in silica near 1530 nm, and of order 10⁻¹⁶ cm² for laser dyes in solution, whose allowed transitions are far stronger than the parity-forbidden 4f transitions of rare-earth ions.

Absorption, gain and the two cross sections

A level pair with $N_1$ ions in the lower state and $N_2$ in the upper state has a net optical gain coefficient

$$g(\lambda) = \sigma_e(\lambda) N_2 - \sigma_a(\lambda) N_1 ,$$

where $\sigma_e$ is the stimulated-emission cross section. When every ion is in the ground state, this reduces to $-\sigma_a N$, the small-signal absorption. For a four-level medium such as Nd:YAG the lower laser level is nearly empty, so $g \approx \sigma_e \Delta N$ with $\Delta N$ the population inversion. A gain of 0.6 cm⁻¹ in Nd:YAG then needs $\Delta N$ = 0.6/(2.8 × 10⁻¹⁹ cm²) = 2.1 × 10¹⁸ cm⁻³, the figure used in the material gain entry.

As an illustration for absorption, a silica core with $N$ = 10¹⁹ erbium ions per cm³ and $\sigma_a$ = 5 × 10⁻²¹ cm² absorbs $\alpha$ = 0.05 cm⁻¹, or 22 dB/m in the doped glass; the absorption a fiber datasheet quotes is lower by the fraction of the mode that overlaps the doped region.

Measurement

Absorption cross sections come from a transmission spectrum of a sample with known dopant concentration: the Beer-Lambert law gives $\alpha(\lambda)$, and dividing by $N$ gives $\sigma_a(\lambda)$. Emission cross sections are derived from the fluorescence spectrum and the radiative lifetime through the Füchtbauer–Ladenburg relation, described under stimulated emission. Quoted values for the same material differ by a factor of two or more, depending on how the Stark structure is treated and on polarization in anisotropic crystals.

McCumber relation

For rare-earth ions whose upper and lower manifolds are each in internal thermal equilibrium, McCumber theory links the two cross sections at each frequency:

$$\sigma_e(\nu) = \sigma_a(\nu)\, e^{(\varepsilon - h\nu)/kT} ,$$

where $\varepsilon$ is the energy at which the two are equal. On the long-wavelength side of that crossover, emission exceeds absorption. With $kT$ = 25.85 meV at 300 K and a crossover at 1530 nm, the ratio $\sigma_e/\sigma_a$ is 1.5 at 1550 nm and 1.8 at 1560 nm. This asymmetry is why an erbium-doped fiber amplifier can show gain at longer wavelengths with only partial inversion, and why a single $\sigma$ does not describe a quasi-three-level medium.

Saturation and the στ product

The cross section and the upper-state lifetime $\tau$ together set the saturation intensity of a four-level medium:

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

For Nd:YAG, $h\nu$ = 1.165 eV = 1.87 × 10⁻¹⁹ J at 1064 nm and $\tau$ = 230 µs, so $I_\text{sat}$ = 2.9 kW/cm² and the saturation fluence $h\nu/\sigma$ is 0.67 J/cm². Ti:sapphire, with about the same $\sigma$ and $\tau$ = 3.2 µs, saturates near 260 kW/cm². The product $\sigma\tau$ is a figure of merit for continuous-wave operation, since threshold pump power in a four-level laser scales inversely with it: Nd:YAG gives 6.4 × 10⁻²³ cm²·s and erbium in silica (6 × 10⁻²¹ cm², 10 ms) gives 6 × 10⁻²³ cm²·s, almost the same, because erbium's longer lifetime offsets its smaller cross section. Saturation power is this intensity multiplied by the mode area.

Scattering cross sections

A particle with scattering cross section $\sigma_s$ removes $\sigma_s I$ watts from a beam of intensity $I$, and $N$ particles per unit volume give an extinction coefficient $N\sigma_s$. For particles much smaller than the wavelength, $\sigma_s$ scales as $d^6/\lambda^4$, the Rayleigh scattering law.

Pitfalls

Cross sections are spectral quantities: a single number refers to a peak, and the value at a neighboring wavelength can be several times smaller, following the transition's lineshape. Peak values also fall as the line broadens with temperature, since the integrated strength is fixed (see homogeneous and inhomogeneous broadening). Some authors fold degeneracy factors into $\sigma$, which matters when comparing tables.

Common questions

How is the cross section related to the absorption coefficient?

$\alpha = \sigma_a N$ for a uniform density $N$ of absorbers all in the lower state. When some are excited, the absorption falls and can turn into gain, following $g = \sigma_e N_2 - \sigma_a N_1$.

Why is the emission cross section different from the absorption cross section?

At a given wavelength the two differ when the upper and lower levels are manifolds of thermally populated sublevels, as in erbium and ytterbium. For a transition between two single levels of equal degeneracy the two are equal.

References: A. E. Siegman, Lasers (University Science Books, 1986); W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006); E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (Wiley, 1994); D. E. McCumber, "Einstein relations connecting broadband emission and absorption spectra," Physical Review 136, A954 (1964); C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).