Two-Photon Absorption
Two-photon absorption of semiconductors (GaAs, CdTe, ZnSe and others), silicon, fused silica, sapphire, BaF₂, BBO, LiNbO₃ and KTP: measured β scaled across the 2PA window, induced absorption βI and slab transmission 1/(1 + βI₀L).
GaAs: β = 23 cm/GW measured at 1064 nm, E_g = 1.42 eV, n = 3.43, E_p = 25.7 eV; Van Stryland et al. 1985, Table 1. 2PA window 873.1–1746 nm.
β at this wavelength
23 cm/GW
Eq. 1 predicts 20.1 cm/GW
Induced absorption α₂ = βI₀
23 cm⁻¹
Transmission T = 1/(1 + βI₀L)
30.3 %
plane wave, no linear loss
2PA loss
5.185 dB
β across the 2PA window
Transmission vs peak intensity (1 mm)
Model
dI/dz = −βI², T = 1/(1 + βI₀L), β ∝ F₂(2ħω/E_g) = (x − 1)^(3/2)/x⁵
Degenerate two-photon absorption between two parabolic bands, allowed for E_g/2 < ħω < E_g. A measured β moves to another wavelength by the ratio of F₂, which peaks at 2ħω = 1.43 E_g; Eq. 1 of Van Stryland et al. (1985) predicts β from n, E_g and E_p alone; it matches their measurements within the ±40 % error, except ZnTe, whose 1064 nm point sits at the edge of the window (4.5 against 0.71 cm/GW); scaling that point would carry the excess across the window (44 cm/GW at 800 nm), so the ZnTe preset uses Eq. 1.
T is for a plane wave of constant intensity. Gaussian beams and pulses transmit more at the same peak intensity, and free carriers made by 2PA add absorption for pulses longer than a few picoseconds.
Wide-gap solids (DeSalvo et al. 1996) use the effective direct gap E′_g the authors fitted to their 2PA data; they found the two-band dispersion diverges near the 2PA edge (BBO: F₂ scaling from 0.9 cm/GW at 266 nm gives 0.076 at 355 nm, measured 0.010). These rows are scaled only where 2ħω is at least 1.1 E′_g at both the measured and the chosen wavelength; five of the six were measured closer to the edge, so they give only the measured β. Indirect-gap silicon follows Bristow et al.'s fit, β = Σₙ 2C F₂⁽ⁿ⁾(ħω/E_gi) over allowed and forbidden phonon-assisted transitions (n = 0, 1, 2), C = 43 cm/GW, E_gi = 1.12 eV, from 850 nm to 2214 nm (2ħω = E_gi).
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Degenerate 2PA between two parabolic bands, valid for E_g/2 < ħω < E_g. A measured β is moved to another wavelength by the ratio of F₂(x) = (x − 1)^(3/2)/x⁵, with n held constant. Wide-gap solids use DeSalvo's fitted effective gap E′_g; away from their measured wavelength the scaling is order-of-magnitude (the two-band model diverges near the 2PA edge). Indirect-gap silicon uses Bristow's phonon-assisted fit β = Σₙ 2C F₂⁽ⁿ⁾(ħω/E_gi), C = 43 cm/GW, 850–2214 nm. Transmission is for a plane wave of constant intensity, without linear, free-carrier or reflection losses. Measured β carry about ±20–40 % error.
- Van Stryland E. W., Woodall M. A., Vanherzeele H., Soileau M. J. (1985). Energy band-gap dependence of two-photon absorption. Opt. Lett. 10, 490–492. Eq. 1–2 and Table 1. doi:10.1364/OL.10.000490
- Sheik-Bahae M., Hagan D. J., Van Stryland E. W. (1990). Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption. Phys. Rev. Lett. 65, 96–99. doi:10.1103/PhysRevLett.65.96
- DeSalvo R., Said A. A., Hagan D. J., Van Stryland E. W., Sheik-Bahae M. (1996). Infrared to ultraviolet measurements of two-photon absorption and n₂ in wide bandgap solids. IEEE J. Quantum Electron. 32, 1324–1333. Tables I–II: β, E′_g and n. doi:10.1109/3.511545
- Bristow A. D., Rotenberg N., van Driel H. M. (2007). Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm. Appl. Phys. Lett. 90, 191104. β = 2.1 cm/GW at 1220 nm; indirect-gap fit C = 43 cm/GW. doi:10.1063/1.2737359