Photon diffusion approximates the spreading of radiation after many scattering events have largely randomized propagation directions. Individual photons travel along paths; their distribution can obey a diffusion equation on scales much larger than a transport mean free path. The longer arrival times arise from longer, repeatedly redirected paths.
For homogeneous, conservative isotropic scattering, the photon diffusion coefficient for isotropic scattering is , where is the transport speed and the scattering mean free path. Directional persistence replaces that scale by a transport mean free path in an appropriate transport model. Crossing a characteristic thickness then takes a time of order , with geometry-dependent factors, which can greatly exceed the direct travel time .
A diffuse photon time-of-flight experiment sends a short pulse into a scattering sample and measures the delayed transmitted or returning light. A pulsed source, a fast photodetector and time-correlated single-photon counting can build an arrival-time histogram from repeated pulses. The detector's timing response must be accounted for before attributing all measured broadening to the sample.[ref]
The 1989 study by M. S. Patterson, Britton Chance and B. C. Wilson used a slab diffusion model to relate picosecond-pulse reflectance and transmittance to absorption and scattering, comparing predictions with preliminary measurements and Monte Carlo simulations.[ref] The earliest nearly unscattered light, nearby boundaries and strongly directional transport can require a fuller radiative transfer description.
A diffuse photon time-of-flight experiment measures the distribution of delays experienced by light traversing a scattering sample. A short-pulse source illuminates the sample; a reference timing signal starts the clock, and a fast photodetector records returning or transmitted photons. Time-correlated single-photon counting accumulates their delays over repeated pulses into a histogram.[ref]
A useful laboratory comparison keeps the source-detector geometry fixed while changing sample thickness or scattering strength. The apparatus needs a defined optical path, a sample holder and the same detection/timing chain for a reference instrument response function. Longer, repeatedly scattered paths broaden the arrival distribution; absorption changes which paths survive to be detected.
Let denote the sample's optical impulse response in the chosen geometry and the normalized instrument response function. A simple model for the expected count-rate trace is
(1)
where denotes convolution, sets the signal scale and represents background counts. Fitting a transport model after including separates sample broadening from instrumental timing spread. A model for an infinite medium cannot simply be substituted for a finite slab's boundary-dependent detector response.
M. S. Patterson, Britton Chance and B. C. Wilson developed analytic slab responses and compared them with preliminary measurements and Monte Carlo simulations in 1989.[ref] The shape of the delay distribution provides information beyond total transmitted intensity.
The counting electronics also matter: if only the first arrival in each pulse period is recorded, multiple arrivals bias the sample toward earlier times. This photon pile-up must be represented or suppressed by the acquisition conditions, rather than mistaken for faster propagation.[ref]
For conservative photon transport at constant speed , suppose successive scattering directions are independent and isotropically distributed, and free-path lengths follow an exponential distribution of mean . In three spatial dimensions, the resulting long-time diffusion coefficient is
(1)
Here is the scattering mean free path. These assumptions describe homogeneous, memoryless interactions without absorption.
The velocity autocorrelation function gives a direct derivation. An initially isotropic ensemble has zero mean velocity. During a lag , the probability of no scattering is . No scattering preserves the direction; one or more independent isotropic scattering events give zero expected directional correlation. Consequently
(2)
The three-dimensional Green-Kubo formula for diffusion now yields
(3)
The factor converts the dot-product correlation into one Cartesian component under isotropy.
The same correlation gives the full mean squared displacement:
(4)
At short times this is ; at long times its leading term is . The model therefore connects direct flights to the diffusive many-scattering limit.
The free-path distribution matters. A fixed-step isotropic random flight with every length equal to has mean-square displacement after steps and elapsed time , giving . Replacing exponential flights by fixed-length flights thus changes the coefficient by a factor of two. For anisotropic scattering, directional persistence instead leads to a transport mean free path; strong absorption and boundaries require additional modeling.

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Photon diffusion refers to the process by which photons (particles of light) spread out as they scatter in a medium. This concept is particularly relevant in the context of light interacting with materials, especially those that are semi-transparent or opaque, like biological tissues, certain gases, and particulates in the atmosphere. In a medium where photons can scatter, the initial directional input of the light becomes randomized due to collisions with particles within the material.
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