Terahertz (THz) Spectroscopy
THz time-domain transmission of a slab: n and α of liquid water (double Debye), a phonon mode (Lorentz) or a constant medium, Fresnel losses, time delay, the extraction n = 1 + cφ/(ωd) and the dynamic-range limit on α.
The water model is fitted below 1 THz; above about 2 THz the intermolecular modes (5.3 THz) take over, so values there are extrapolated.
n + iκ at 1.00 THz
2.070 + 0.533i
ε = ñ²; time delay (n − 1)d/c = 0.3569 ps
Absorption Coefficient α
223.4 cm⁻¹
α_max = 1356 cm⁻¹ at 60 dB
Power Transmission |T|²
8.305 %
Fresnel loss 22.4 % at two faces, e^(−αd) = 0.1071
Extracted from T (real Fresnel)
n 2.0261, α 224.4 cm⁻¹
error -0.044 in n, 0.44 % in α
Photon at Read-out
33.36 cm⁻¹
4.136 meV, λ = 299.8 µm
Thermal Energy kT
6.108 THz
203.7 cm⁻¹ at 293.15 K
Unmeasurable Points
0 of 400
where α exceeds the dynamic-range limit
T(ω) = [4ñ/(ñ+1)²] e^(i(ñ−1)ωd/c), n = 1 + cφ/(ωd), α = −(2/d) ln[|T|(n+1)²/(4n)], α_max = (2/d) ln[DR·4n/(n+1)²]
The sample is a free-standing slab at normal incidence, measured against the empty path with the Fabry–Pérot echoes cut out of the time window. The extraction treats the Fresnel factor as real, which is accurate when κ ≪ n; for strong absorbers such as water its error is shown above. α_max is the Jepsen–Fischer limit for a field dynamic range DR (taken flat here; a real spectrometer's falls with frequency).
Water: Liebe, Hufford & Manabe (1991) double Debye model, τ₁ = 1/(2πγ₁) ≈ 8.2 ps at 25 °C. A liquid sits in a cell, whose windows add their own reflections. Lorentz: ε = ε∞ + Δε f₀²/(f₀² − f² − iγf), a user-defined mode.
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
Free-standing slab at normal incidence with the Fabry–Pérot echoes windowed out; extraction with a real Fresnel factor. The water model is the Liebe–Hufford–Manabe double Debye fit, fitted up to 1 THz to data from −4 to 40 °C (the page allows −5 to 40 °C); the dynamic range is taken as flat in frequency.
- Liebe H. J., Hufford G. A., Manabe T. (1991). A model for the complex permittivity of water at frequencies below 1 THz. Int. J. Infrared Millim. Waves 12, 659–675. doi:10.1007/BF01008897
- Duvillaret L., Garet F., Coutaz J.-L. (1996). A reliable method for extraction of material parameters in terahertz time-domain spectroscopy. IEEE J. Sel. Top. Quantum Electron. 2, 739–746. doi:10.1109/2944.571775
- Jepsen P. U., Fischer B. M. (2005). Dynamic range in terahertz time-domain transmission and reflection spectroscopy. Opt. Lett. 30, 29–31. doi:10.1364/OL.30.000029
- Withayachumnankul W., Naftaly M. (2014). Fundamentals of measurement in terahertz time-domain spectroscopy. J. Infrared Millim. Terahertz Waves 35, 610–637. doi:10.1007/s10762-013-0042-z