Attenuation: scattering and absorption

In 3-photon microscopy, as in other forms of microscopy, attenuation by biological tissue reduces the intensity of ballistic photons at the focal volume. Scattering and absorption each contribute to attenuation, with scattering playing a greater role than absorption.

The strength of scattering and of absorption by biological tissue are often expressed as scattering and absorption coefficients (proportional to the number of photons absorbed or scattered per unit distance) or as length constants, the distances over which intensity is reduced e-fold. Coefficients and length constants are reciprocals.

Attenuation is the sum of scattering and absorption. Hence,

μe = μs + μa      and      1/le = 1/ls + 1/la

where,

μe is the attenuation coefficient, μs is the scattering coefficient, and μa is the absorption coefficient.

le is the attenuation length constant, ls is the scattering length constant, and la is the absorption length constant.
 

At visible wavelengths and near-IR wavelengths commonly used for 2-photon excitation, scattering dominates. Scattering length constants are ~100 um at 500 nm and ~200 um at 910 nm (Theer & Denk, 2006). Length constants for absorption by water are ≥3 orders of magnitude greater, at >100 mm at 500-900 nm.

Scattering and absorption each change with wavelength. Scattering decreases (the scattering length constant increases) with increasing wavelength. The wavelength-dependence of absorption depends on the absorbing species, major absorbing species in the brain being water and haemoglobin. Broadly, absorption increases with increasing wavelength. Scattering length constants are ~2- and 4-fold greater at 1300 and 1700 nm than at 900 nm (Jacques, 2013). Length constants for water absorption at 1300 nm and 1700 nm are ~10 mm and 1 mm. Although absorption increases 2 orders of magnitude from 900 to 1700 nm, scattering still dominates.

Attenuation by biological tissues declines with increasing wavelength across the spectrum used for multi-photon excitation. Attenuation lengths have been measured in rodent neocortical grey matter (Kleinfeld & Denk, 2000; Kobat et al., 2009; Wang et al., 2018a; Yildirim et al., 2019b):

le ≈ 130 μm at 775 nm

le ≈ 200 μm at 800-850 nm

le ≈ 250-300 μm at 1300 nm

le ≈ 400 um at 1700 nm

Tissue attenuation necessitates an increase in illumination intensity with imaging depth, to maintain intensity at the focus. The increase per unit depth is less for 3- than for 2-photon excitation. 3-photon excitation at ~1 mm is often possible with surface intensities of only ~25-50 mW.