Near-Infrared (NIR) Spectroscopy
NIR spectroscopy: measured water absorption bands and transmittance from 780 to 2500 nm, Morse overtone positions, and haemoglobin changes from the two-wavelength modified Beer–Lambert law with an age-dependent DPF.
Water absorption, 780–2500 nm
Pure water (Segelstein 1981 compilation) scaled by the volume fraction, T = exp(−μ_a L). The peaks are the O–H combination and overtone bands of liquid water: about 970 nm (2ν₁ + ν₃), 1200 nm (ν₁ + ν₂ + ν₃), 1450 nm (ν₁ + ν₃) and 1940 nm (ν₂ + ν₃), found here as maxima of the data. Scattering is not included: in tissue it lengthens the path, which the modified Beer–Lambert law below accounts for.
Overtone positions, Morse oscillator
| Band | ν̃ (cm⁻¹) | λ (nm) | ν̃ / ν̃(0→1) |
|---|---|---|---|
| 0 → 1 (fundamental) | 2885.31 | 3465.8 | 1.000 |
| 0 → 2 (1st overtone) | 5664.98 | 1765.2 | 1.963 |
| 0 → 3 (2nd overtone) | 8339.01 | 1199.2 | 2.890 |
| 0 → 4 (3rd overtone) | 10907.4 | 916.81 | 3.780 |
| 0 → 5 (4th overtone) | 13370.2 | 747.93 | 4.634 |
ν̃(0→v) = v ω_e − ω_e x_e v(v + 1). Highest bound level v_max = 27.0, Morse well depth D_e = ω_e²/(4ω_e x_e) = 42342 cm⁻¹. Defaults: H³⁵Cl (ω_e 2990.946, ω_e x_e 52.8186 cm⁻¹); observed bands 2886, 5668, 8347, 10 923 cm⁻¹. Overtones fall short of v times the fundamental, which is why the C–H, O–H and N–H overtones land in the NIR. Overtone intensities need the dipole function and are not computed.
Haemoglobin changes, modified Beer–Lambert law
Δ[HbO₂]
1.003 µM
oxyhaemoglobin
Δ[HbR]
-0.3010 µM
deoxyhaemoglobin
Δ[HbT]
0.7022 µM
total, Δ[HbO₂] + Δ[HbR]
DPF λ₁ / λ₂
6.30 / 5.68
mean path d·DPF = 18.9 / 17.0 cm
ΔOD(λ) = [ε_HbO₂(λ) Δ[HbO₂] + ε_HbR(λ) Δ[HbR]] · d · DPF(λ), solved at the two wavelengths. ε are decadic molar extinction coefficients (Prahl's compilation); DPF(λ, A) = 223.3 + 0.05624 A^0.8493 − 5.723·10⁻⁷ λ³ + 0.001245 λ² − 0.9025 λ. The result averages over the sampled tissue and assumes the change is homogeneous; wavelengths on either side of the 800 nm isosbestic point give the best-conditioned system.
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
Water spectrum is measured data (no scattering); overtones use the Morse oscillator, which drifts at high v; the modified Beer–Lambert law assumes homogeneous changes and a fixed DPF fitted for the adult frontal head (690–832 nm, 0–70 years).
- Delpy D. T. et al. (1988). Estimation of optical pathlength through tissue from direct time of flight measurement. Phys. Med. Biol. 33, 1433–1442. doi:10.1088/0031-9155/33/12/008
- Scholkmann F., Wolf M. (2013). General equation for the differential pathlength factor of the frontal human head depending on wavelength and age. J. Biomed. Opt. 18, 105004. doi:10.1117/1.JBO.18.10.105004
- Prahl S. Tabulated molar extinction coefficient for hemoglobin in water (Gratzer and Kollias data). omlc.org
- Segelstein D. J. (1981). The complex refractive index of water. M.S. thesis, University of Missouri–Kansas City. omlc.org
- Curcio J. A., Petty C. C. (1951). The near infrared absorption spectrum of liquid water. J. Opt. Soc. Am. 41, 302–304. doi:10.1364/JOSA.41.000302
- Banwell C. N., McCash E. M. (1994). Fundamentals of Molecular Spectroscopy, 4th ed., ch. 3. McGraw-Hill.