Light through the head
In the near-infrared 'optical window' (~700–900 nm) tissue is relatively transparent and the dominant absorbers are oxy- and deoxy-haemoglobin. Functional near-infrared spectroscopy shines NIR light from a scalp source, and a detector a few centimetres away catches the diffusely back-scattered photons that took a banana-shaped path through cortex. The source–detector separation sets the depth sampled — roughly half the separation.
\Delta A = \log_{10}\!\frac{I_{0}}{I} = \varepsilon\,\Delta c\,d\,\cdot\mathrm{DPF}Modified Beer–Lambert law: measured attenuation change is proportional to chromophore concentration change Δc, source–detector distance d, and a differential pathlength factor (DPF) that accounts for the extra distance scattering adds.
Shine light into the head and less comes out when blood absorbs more of it. This law makes that quantitative: the drop in light (attenuation) is proportional to how much the absorbing molecule's concentration changed, times the distance travelled, times a fudge factor (DPF) for the extra winding path that scattering forces the light to take. It is the heart of fNIRS.
- \Delta A
- The change in attenuation — how much dimmer the light got.
- \varepsilon
- The extinction coefficient — how strongly the molecule absorbs at this wavelength.
- \Delta c
- The change in chromophore (e.g. haemoglobin) concentration — what we want.
- d\cdot\mathrm{DPF}
- Source-detector distance times a factor for scattering's extra path.
If two source-detector pairs sit at different distances d, the same blood change produces a proportionally larger \Delta A at the wider spacing — which is how depth is teased apart.
One wavelength cannot separate oxy- from deoxy-haemoglobin, so fNIRS measures at (at least) two. With two wavelengths and two unknowns you invert a small extinction-coefficient matrix per channel to recover ΔHbO and ΔHbR.
\begin{bmatrix}\Delta A_{\lambda_1}\\[2pt] \Delta A_{\lambda_2}\end{bmatrix} = d\,\mathrm{DPF}\begin{bmatrix}\varepsilon^{\lambda_1}_{\mathrm{HbO}} & \varepsilon^{\lambda_1}_{\mathrm{HbR}}\\[2pt] \varepsilon^{\lambda_2}_{\mathrm{HbO}} & \varepsilon^{\lambda_2}_{\mathrm{HbR}}\end{bmatrix}\begin{bmatrix}\Delta c_{\mathrm{HbO}}\\[2pt] \Delta c_{\mathrm{HbR}}\end{bmatrix}Two wavelengths, two unknowns: inverting the 2×2 extinction matrix recovers oxy- and deoxy-haemoglobin changes — the core fNIRS computation.
You care about two things — oxygenated and deoxygenated haemoglobin — so you take two measurements, at two light colors that each "see" the two molecules differently. That gives two equations in two unknowns; inverting the small 2\times2 table of how much each color is absorbed by each molecule cleanly separates the two blood signals. This is the core fNIRS computation.
- \Delta A_{\lambda_1},\,\Delta A_{\lambda_2}
- The measured attenuation changes at the two wavelengths.
- \varepsilon_{\mathrm{HbO}},\,\varepsilon_{\mathrm{HbR}}
- How strongly oxy- and deoxy-haemoglobin absorb at each wavelength.
- \Delta c_{\mathrm{HbO}},\,\Delta c_{\mathrm{HbR}}
- The two unknowns — the blood changes you solve for.
Choosing two wavelengths where the absorptions differ a lot makes the matrix well-conditioned and the separation reliable; pick them too similar and the inversion amplifies noise.
The measurement is hemodynamic, not neural
This is the honest ceiling of the whole optical family. fNIRS does not see neurons firing; it sees the neurovascular response they trigger — a haemodynamic wave that lags activity by ~4–6 seconds and is intrinsically low-pass. No electronics can make a slow biological signal fast. That caps information rate far below electrical BCIs, regardless of how good the optics get.
There is also a confound problem: much of the light path is scalp and skull, so systemic physiology — scalp blood flow, heart rate, blood pressure Mayer waves — contaminates the signal. The standard defence is short-separation channels (a source–detector pair too close to reach cortex) that measure the superficial signal and regress it out.
Up the resolution ladder
Channel-based fNIRS is the entry rung. Above it, high-density diffuse optical tomography overlaps many overlapping source–detector measurements and solves an inverse problem to reconstruct volumetric images, approaching fMRI-like maps at the scalp. See HD-DOT.
Two refinements improve quantification. Time-domain fNIRS sends picosecond pulses and times individual photons' flight — late photons travelled deeper, so time-of-flight separates cortical from scalp signal and yields absolute concentrations. Diffuse correlation spectroscopy instead reads the speckle-decorrelation rate of coherent light to measure blood flow directly rather than oxygenation.
The most tantalising optical signal is the fast optical signal (EROS) — a scattering change tied to neuronal activity itself, and therefore fast (milliseconds) rather than hemodynamic. It sidesteps the speed ceiling in principle, but its SNR is minuscule, which is why it remains a research curiosity rather than a BCI workhorse.
Functional ultrasound: hemodynamics at depth
Functional ultrasound (fUS) uses ultrafast plane-wave imaging and power-Doppler to map cerebral blood volume with remarkable spatio-temporal resolution and real depth penetration — far beyond fNIRS. The catch is the skull: ultrasound is strongly reflected and attenuated by bone, so in adult humans fUS needs an acoustic window (a thinned skull, a craniotomy, or an implanted transparent skull), placing it in the minimally-invasive column, not the fully non-invasive one. In neonates it can image through the fontanelle.