Distinguish between absorption and scattering as the two fundamental fates of a photon in tissue
Define the absorption coefficient ($\mu_a$) and reduced scattering coefficient ($ \mu_s’$) in plain language and explain what physical properties determine each
Write the relationship between $\mu_a$ and chromophore concentration, and identify the role of the extinction coefficient
Explain why tissue is overwhelmingly a scattering medium and what consequence this has for photon path length
Describe the concept of photon diffusion and explain why this is the physical regime relevant to fNIRS

In Unit 1, we learned that the Optical Window exists because tissue chromophores have low combined absorption between 650 and 950 nm, and the spectroscopic differences between HbO₂ and HbR are what fNIRS exploits to measure oxygenation. But knowing that photons can penetrate tissue is not the same as understanding what they do once inside.

This lesson takes you inside the fundamental interactions between light and tissue. It is the physical foundation on which everything in Units 2 through 5 is built. By the end, you will understand why extracting a clean hemoglobin signal from fNIRS data is not a trivial reading of an instrument; it requires careful modeling of a genuinely complicated physical process.

1. What Is Light?

Before we dive into what happens when light enters tissue, let’s just take a moment to understand what light is exactly. We can start by stating that light is a form of electromagnetic radiation: a self-propagating oscillation of electric and magnetic fields that travels through space at roughly 300,000 km/s. In Lesson 1.1, you encountered the electromagnetic spectrum and the concept of wavelength (the distance between successive peaks of the oscillating wave). Different wavelengths correspond to different colors in the visible range, and near-infrared light sits just beyond the red end of the visible spectrum at wavelengths between roughly 700 and 2500 nm.

This wave description of light is the more fundamental one physically. But in the early twentieth century, experiments showed that when light is emitted or absorbed by matter, it does so in discrete packets of energy rather than continuously. These packets are called photons. The energy of a single photon is determined by its wavelength: shorter wavelengths (higher frequency) carry more energy per photon, which is why ultraviolet light can damage DNA while infrared light cannot. In the NIR window we use for fNIRS, each photon carries enough energy to interact with biological molecules and drive electronic transitions, but not enough to ionize atoms or break chemical bonds, which is why NIR light is biologically safe.

Light therefore behaves as both a wave and a particle depending on how you interrogate it (this is one of the most counterintuitive features of quantum mechanics, and one that has no simple classical analogy). For most everyday phenomena involving light (color, reflection, refraction, interference), the wave description is the most useful. For phenomena involving the interaction of light with individual molecules (absorption, emission, photochemistry), the particle (photon) description is often more convenient.

📝 a relevant, but often ignored, conceptual reminder

In what follows, we treat light as if it were composed of discrete particles (photons) that travel through tissue and interact with molecules at specific points along their journey. This is the classical particle transport model, and it is the basis of the diffusion framework that underlies all fNIRS signal models.

It is important to be honest about what this model is and is not. In quantum mechanics, light propagates as a wave, and individual photons do not travel along definite trajectories. Strictly speaking, asking “how far did this photon travel?” or “what path did it take?” is a category error, as these are not well-defined quantities in quantum theory. The “path lengths” and “photon trajectories” we discuss in this lesson are model constructs that arise from the classical particle approximation, not literal descriptions of quantum reality.

This approximation works remarkably well for predicting macroscopic measurements of light transport in tissue, which is why it has been used successfully for decades. The quantities we derive, such as effective path length, should be understood as statistical properties of the light field, averages over many interactions, rather than as the journey of any individual quantum. With that caveat clearly stated, we proceed.

2. Two Fates for Light in Tissue

Working within the classical particle model clarified above, when a near-infrared photon enters biological tissue, two types of interaction with matter are possible. It is either absorbed or scattered. In the statistical sense of the model, there is no third option.

🔴 Absorption

The photon’s energy is transferred to a molecule (a chromophore), and the photon is destroyed. It does not reach the detector. The energy drives a chemical process through molecular transition or is dissipated as heat.

This is the process that carries physiological information. Different chromophores absorb at different wavelengths, which is the basis of spectroscopy and fNIRS.

🔵 Scattering

Light interacts with a small structure (e.g., a cell membrane, a mitochondrion, a collagen fiber) and changes direction without the energy being absorbed. In the particle model, we say the “photon is redirected” while remaining in the field.

This process carries information about tissue microstructure, but in standard fNIRS it is primarily a confound: it extends and randomises effective path lengths in ways that complicate the measurement.

Both processes reduce the amount of light reaching your detector: absorption by destroying photons, scattering by redirecting them away from the detector. A single intensity measurement cannot tell you how much of the reduction was due to each process. This is one of the central challenges of diffuse optical measurement, and we will return to it repeatedly in this unit.

3. Tissue Is Overwhelmingly a Scattering Medium

In the NIR Optical Window, scattering in biological tissue is approximately 100 times more probable than absorption per unit path length of tissue traversed. For every quantum of light that is absorbed, roughly 100 others are scattered. Tissue is not a slightly hazy medium with some absorption; it is an overwhelmingly scattering medium with a relatively small amount of absorption. This single fact is responsible for virtually every methodological complication in fNIRS.

Why does tissue scatter so much?

In the wave picture, scattering occurs when a propagating electromagnetic wave encounters a local change in the refractive index of the medium, such as a boundary between regions. Biological tissue is densely packed with such boundaries at many length scales:

Cell membranes: each cell is bounded by a lipid bilayer with different optical properties from the surrounding cytoplasm and extracellular fluid
Organelles: mitochondria, nuclei, and other structures have distinct refractive indices; mitochondria in particular are efficient scatterers due to their size (~0.5-1 µm) and dense internal membrane architecture
Collagen fibers: densely present in connective tissue, the extracellular matrix, and the meninges surrounding the brain
Lipid structures: myelin sheaths, fat droplets, and cell membrane lipid bilayers all contribute
Red blood cells: biconcave discs with high hemoglobin content that are both absorbers and effective scatterers

The structural heterogeneity of tissue, in other words, is the physical origin of its high scattering. This heterogeneity also encodes information about tissue composition, cell morphology, and health, which is why scattering-based measurements have clinical applications in tissue characterization and cancer detection. However, in a standard functional fNIRS experiment spanning seconds to minutes, this structural heterogeneity does not change appreciably. The assumption that scattering is stable over the duration of a functional paradigm is one of the key foundations of the Modified Beer-Lambert approach introduced in Lesson 2.2, and it is generally valid for healthy tissue under normal conditions. We will revisit this assumption in Lesson 2.4, when it can break down.

💭 Pause and Think

Scattering in tissue carries information about its microstructure – cell size, membrane density, organelle distribution. Can you think of a physiological or clinical scenario in which tissue microstructure would change significantly during a measurement, potentially violating the assumption that scattering is stable?

Hint: think about conditions that alter cell volume (edema, ischemia), membrane integrity (cell swelling or death), or tissue composition over longer time periods. This connects back to the limitations of CW fNIRS introduced in Lesson 1.3.

4. The Consequence of High Scattering: Light Does Not Travel in Straight Lines Through Tissue

In a clear, non-scattering medium (clean water, a glass cuvette, open air), light travels in straight lines. In the wave picture, the wavefront advances uniformly; in the particle model, photons travel direct rectilinear paths. Either way, you know exactly how far the light has traveled, because it is simply the geometric distance from source to detector.

The story is entirely different in tissue, however. Because scattering events happen thousands of times per centimeter, the effective propagation direction changes continuously. The statistical picture that emerges from a large ensemble of such interactions is one of light diffusing outward from the source rather than propagating as a beam. In the wave picture, the same phenomenon is described as the rapid randomization of the wavefront by the structural heterogeneity of the medium.

Diagram showing banana-shaped sensitivity region between fNIRS source and detector on the scalp

Figure 1. Light propagation in a non-scattering medium (clean water, left panel) and in a highly scattering medium (water + milk, right panel). In the first case, light travels in a straight line. In the latter, light is scattered several times, giving rise to a diffusive pattern through tissue.

The practical consequence is the same in either description: the light that reaches your detector has, on average, traveled a much longer and more tortuous path through the tissue than the straight-line source-detector distance would suggest. The characteristic shape of the sensitivity region between source and detector is a broad, curved volume often referred to as a “banana” because of its curved profile. Photons that sampled deeper tissue have, on average, interacted more with the medium and are less likely to have reached the detector, which is why the central, shallower portion of the banana dominates the measurement.

Diagram showing banana-shaped sensitivity region between fNIRS source and detector on the scalp

Figure 2. The characteristic banana-shaped sensitivity region between a source and detector placed on a highly scattering medium. Because light diffuses through the scattering medium rather than propagating as a beam, the detected signal is most sensitive to a broad curved volume beneath the scalp. Increasing source-detector separation shifts this region deeper.

💡 NOTE: The Unknown Effective Path Length

Because light diffuses through strongly scattering tissue, the effective optical path length, i.e., the statistical average distance light travels between source and detector in the particle model, is much longer than the straight-line separation between source and detector. The true optical path length can be three to six times longer in the head.

The Beer-Lambert Law, which relates light attenuation to absorber concentration, requires path length as an input. In tissue, the effective path length depends on the scattering properties of the medium, which a CW intensity measurement cannot determine independently. This is the root cause of every methodological complication addressed in Lessons 2.2 through 2.4.

5. Quantifying the Two Processes: the optical coefficients

To reason precisely about light transport in tissue, we need two quantitative descriptors, one for absorption and one for scattering. These are defined within the classical particle transport model and appear throughout the fNIRS literature and in every model of light propagation in tissue.

The absorption coefficient (μₐ)

The absorption coefficient $\mu_a$ (pronounced /mu-a/) represents the probability per unit length of tissue that a photon is absorbed. Its units are inverse length (typically mm⁻¹ or cm⁻¹). A tissue with a high $\mu_a$ absorbs photons quickly over short distances, while a tissue with low $\mu_a$ allows photons to travel further before being absorbed.

$\mu_a$ is determined by the chromophores present and their concentrations. Because absorption is additive, the total absorption coefficient is the sum of each chromophore’s individual contribution. For N different chromophores in the medium,

💡 Absorption Coefficient

$$ \mu_a(\lambda) = \sum_{i=1}^N \epsilon_i(\lambda) \cdot c_i $$

where the sum runs over all chromophores i present in the tissue:

$\epsilon_i(\lambda)$ is the molar extinction coefficient of chromophore i at wavelength λ. The extinction coefficient is a fixed molecular property measurable in a clean solution and tabulated, and it quantifies how strongly a given molecule interacts with light at a given wavelength.
cᵢ is the molar concentration of chromophore i in the tissue. In fNIRS, this is exactly the quantity we want to determine for HbO₂ and HbR.
The parenthesis reflects the dependences of these variables; μₐ depends on wavelength λ because extinction coefficients are wavelength-dependent; this is the spectral variation that gives each chromophore its distinctive absorption fingerprint, as described in Lesson 1.2.

This relationship was proposed by August Beer in 1852 and is the direct connection between the physics of light absorption and the biology we are interested in. The extinction coefficients ε for HbO₂ and HbR have been carefully measured and tabulated, and these tables underlie the calculations in virtually every fNIRS analysis pipeline. The unknown in this expression is cᵢ: the concentration we want to recover. If we can determine μₐ at two wavelengths, and we know ε at those wavelengths for both chromophores, we can in principle solve for both c(HbO₂) and c(HbR).

🎍 MUST-READ Paper

Cheong, W.F., Prahl, S.A. & Welch, A.J. (1990). “A Review of the Optical Properties of Biological Tissues.” IEEE Journal of Quantum Electronics 26(12), 2166–2185.

This reference provides a comprehensive compilation of the theoretical foundations and experimental approaches to measure the optical properties of the main tissue chromophores.

→ doi:10.1109/3.64354

The Oregon Medical Laser Center maintains an updated list of tabulated molar extinction coefficients for several chromophores, including hemoglobin in water, that you can use to implement the equation below: https://omlc.org/spectra.

Crucially, μₐ changes when brain activity changes. As we will see in more detail, when a cortical region activates, local hemoglobin concentrations change: HbO₂ rises and HbR falls, thereby altering μₐ at wavelengths where their extinction coefficients differ. This changes the amount of light reaching the detector, producing the fNIRS signal.

The reduced scattering coefficient ($\mu_s’$)

The reduced scattering coefficient $\mu_s’$ (pronounced /mu-s prime/) describes how strongly tissue scatters light, corrected for the directionality of the scattering (in other words, it quantifies how rapidly light loses all memory of its original propagation direction due to repeated scattering interactions). The “reduced” qualifier accounts for the fact that scattering in tissue is predominantly forward-directed: each interaction deflects the propagation direction by a relatively small angle rather than randomly in all directions. The reduced coefficient folds this directionality into a single parameter that describes effective diffusion. Its units are also inverse length.

Unlike $\mu_a$, which is determined by chemistry (which molecules are present and how much), $\mu_s’$ is determined by tissue structure: the size, density, and refractive index contrast of scattering elements (mitochondria, cell membranes, collagen, etc.). It varies with wavelength according to an approximate power law that reflects how waves of different frequencies interact with structures of different sizes:

💡 Reduced Scattering Coefficient

$$ \mu_s'(\lambda) \propto A\cdot \lambda^{-b}$$

where:

A is a scaling factor related to the density and refractive index contrast of scattering structures in the tissue. More scatterers, or greater optical contrast between scatterers and their surroundings, produces a larger A and stronger overall scattering.
b is an exponent related to the size distribution of scatterers. Larger scatterers such as whole cells tend to give a smaller b; smaller scatterers such as mitochondria and collagen fibrils give a larger b. Typical values in brain tissue fall in the range b ≈ 0.5-2.
The λ^−b dependence means scattering decreases with increasing wavelength; longer wavelengths diffuse through tissue more readily, contributing to why the NIR window allows deeper penetration than visible light.

In a short functional experiment, it is assumed that $\mu_s’$ does not change. Neural activation alters hemoglobin concentrations and therefore μₐ, but it does not meaningfully alter cell membranes, mitochondria, or collagen fibers over the timescale of seconds to minutes. The scattering properties of healthy brain tissue are stable during a typical fNIRS paradigm. This stability is what allows the Modified Beer-Lambert approach to work: it treats $\mu_s’$ as a fixed quantity and focuses on detecting changes in $\mu_a$ alone.

However, $\mu_s’$ does vary between individuals, between tissue types, and between age groups. Infant brain tissue scatters differently from adult brain tissue. It also changes with pathological conditions that alter tissue microstructure. CW fNIRS cannot measure $\mu_s’$ directly, so it must be assumed or estimated from the literature. Time-domain and frequency-domain systems can separate $\mu_a$ from $\mu_s’$ independently, which is precisely what makes them more powerful measurement tools.

6. From Individual Interactions to Predictable Bulk Behavior

Knowing that individual scattering interactions are random might seem to make fNIRS modeling intractable. If you cannot predict the effective path of any individual photon, how can you extract a meaningful signal? The answer comes from a fundamental principle of statistical physics: while individual events are unpredictable, the collective behavior of a large number of particles is highly predictable and follows smooth deterministic laws.

A useful analogy is the diffusion of a drop of ink in a glass of still water. You cannot predict where any individual ink molecule will be after one second. But you can predict with confidence that the ink cloud as a whole will spread outward in a smooth, symmetric pattern described by a diffusion equation. The randomness of individual paths averages out into smooth, deterministic bulk behavior.

The same principle applies to light in tissue. The strongly scattering medium rapidly randomizes the light wavefront, and the energy propagates diffusively from the source. In the particle transport model, photons undergo many random redirections, and the resulting spatial distribution of light energy follows the photon diffusion equation, a well-established partial differential equation that describes how the local light intensity evolves through space and time in a strongly scattering, weakly absorbing medium. This is called the diffuse regime or diffusion approximation, and it is the physical regime that all practical fNIRS operates in.

The diffusion approximation is valid precisely when scattering strongly dominates over absorption (i.e., when $\mu_s’ \gg \mu_a$), which, as we have established, is the case in biological tissue in the NIR window. Under this condition, the spatial distribution of light intensity in tissue is smooth and predictable. Analytical solutions to the diffusion equation for simple geometries give closed-form expressions for the light intensity as a function of source-detector separation.

The Modified Beer-Lambert Law, which we will study next, is a linearized approximation derived from one such solution (specifically, from the semi-infinite homogeneous medium solution of the diffusion equation). Understanding this lineage matters because the MBLL is not an empirical rule of thumb, but a physically grounded approximation with known assumptions and known failure modes. Knowing those assumptions is what allows you to judge when the MBLL can be trusted and when it cannot, which is the subject of Lessons 2.2 through 2.4.

💭 Pause and Think

You now know that $\mu_a$ changes with neural activity but $\mu_s’$ does not, over short time windows. This means that any change in detected light intensity between a baseline and a task period reflects a change in absorption alone. Does this mean CW fNIRS can cleanly recover the change in $\mu_a$?

Think carefully: even if $\mu_s’$ is stable and only $\mu_a$ changes, does a single intensity measurement tell you the magnitude of that change? What else do you need to convert a change in detected light intensity into a change in chromophore concentration? This is the question we will answer next.

📌 Key Takeaways

Light can be described as both a wave and as discrete quanta of energy (photons). In tissue optics, a classical particle transport model is used in which photons travel through tissue and interact with matter at specific points. This is a physically motivated approximation; quantities such as “path length” and “photon trajectories” are statistical constructs of this model, not quantum-mechanically exact descriptions.
Within this model, light in tissue has two fates: absorption (photon removed from the field, physiological information encoded) or scattering (propagation direction changed, effective path length extended).
In the NIR window, scattering is ~100× more probable than absorption. This dominance arises from the structural heterogeneity of tissue: cell membranes, mitochondria, collagen, and lipid structures all create refractive index boundaries that scatter light.
The absorption coefficient is determined by chromophore concentrations and changes with neural activity. The scattering coefficient is determined by tissue microstructure, is stable over short functional time windows, but varies between individuals and tissue types. CW fNIRS cannot measure $\mu_s’$ independently.
Because scattering strongly dominates, light in tissue propagates diffusively and follows the photon diffusion equation. The Modified Beer-Lambert Law is a linearised approximation derived from a specific solution of this equation, with known assumptions and known failure modes.

📚 Further Reading & Landmark References

Cheong, W.F., Prahl, S.A. & Welch, A.J. (1990). “A review of the optical properties of biological tissues.” IEEE Journal of Quantum Electronics, 26(12), 2166–2185. [doi] — The foundational reference for tissue optical properties; comprehensive tables of $\mu_a$ and $\mu_s’$ across wavelengths and tissue types. Cited in virtually every tissue optics paper.
Patterson, M.S., Chance, B. & Wilson, B.C. (1989). “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties.” Applied Optics, 28(12), 2331–2336. [doi] — Landmark paper applying the diffusion approximation to extract $\mu_a$ and $\mu_s’$ from time-resolved tissue measurements. The theoretical framework underpinning TD fNIRS.
Chance, B. et al. (1988). “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain.” Proceedings of the National Academy of Sciences, 85(14), 4971–4975. [doi] — Early experimental demonstration that time-resolved measurements can separate absorption from scattering in tissue; the physical basis for TD fNIRS being more informative than CW.
Ishimaru, A. (1978). Wave Propagation and Scattering in Random Media. Academic Press, New York. — The foundational text on electromagnetic wave scattering and the radiative transfer equation, from which the diffusion approximation is derived. Advanced reading for those wanting the complete mathematical framework.
Prahl, S.A. (1999). Tabulated molar extinction coefficients for haemoglobin in water. Oregon Medical Laser Centre. [omlc.org/spectra/hemoglobin] — The definitive tabulation of ε(HbO₂) and ε(HbR) used by virtually all fNIRS analysis software to implement the μₐ formula.

Society for Functional Near-Infrared Spectroscopy. Content is for educational purposes.

Absorption and Scattering: What Happens when Light Enters Tissue?