Scattering

Scattering is the redirection of light out of its original direction by anything that makes the refractive index of a medium non-uniform: molecules, dust, droplets, cells, density fluctuations in glass, or roughness on a waveguide wall. The incident field drives the electrons of the scatterer, which radiate a secondary wave in many directions. In most cases the scattered light keeps the incident wavelength (elastic scattering); a small fraction exchanges energy with molecular vibrations or acoustic waves and emerges shifted (inelastic scattering). Typical magnitudes span many orders: Rayleigh scattering accounts for most of the roughly 0.2 dB/km loss of standard silica fiber at 1550 nm, while fog with a 100 m visibility attenuates light by a factor of seven over 50 m.

Regimes and types

The behaviour depends mainly on the size parameter $x = 2\pi a/\lambda$, where $a$ is the particle radius, and on the relative index $m$.

• Rayleigh scattering ($x \ll 1$): scattered power scales as $\lambda^{-4}$ and the pattern is nearly symmetric between forward and backward. Molecules in air and frozen-in density fluctuations in glass are in this regime; see Rayleigh scattering.
• Mie scattering ($x$ of order 1 or larger): the full solution for a sphere, with weak wavelength dependence and strongly forward-peaked scattering. Cloud droplets, aerosols, cells and polymer beads fall here; see Mie scattering.
• Geometric scattering ($x \gg 1$): ray optics with reflection and refraction at each surface, plus diffraction around the edge.
• Inelastic scattering: Raman scattering from molecular vibrations (a 13 THz shift in silica) and Brillouin scattering from acoustic phonons (about 11 GHz in standard fiber at 1550 nm).

Fluorescence is sometimes confused with scattering. It differs in that the light is absorbed, the molecule resides in an excited state for nanoseconds, and emission follows at a longer wavelength; scattering has no such delay.

Scattering coefficient and attenuation

In a dilute medium with $N$ scatterers per unit volume, each with cross-section $\sigma_s$, the scattering coefficient is $\mu_s = N\sigma_s$ and the unscattered (ballistic) power falls exponentially, as in the Beer–Lambert law:

$$P(L) = P_0\, e^{-\mu_s L}.$$

When absorption is also present the total extinction coefficient is $\mu_s + \mu_a$. For a cloud of water droplets with radius 10 µm at 100 per cm³, the cross-section is close to twice the geometric area (see the Mie entry), giving $\mu_s = N \cdot 2\pi a^2 \approx 0.063$ m⁻¹, a mean free path of about 16 m.

Meteorologists express the same quantity as visibility, using the Koschmieder relation $V = 3.912/\mu_s$ for a 2% contrast threshold. A visibility of 100 m corresponds to $\mu_s = 0.039$ m⁻¹, and 50 m of that fog transmits $e^{-1.96} \approx 0.14$ of the direct beam.

In strongly scattering media such as tissue the photon is redirected many times, and the useful parameter is the reduced scattering coefficient $\mu_s' = \mu_s(1-g)$, where $g$ is the mean cosine of the scattering angle. Soft tissue in the red and near infrared has $\mu_s$ of order 10 mm⁻¹ and $g \approx 0.9$, so $\mu_s' \approx 1$ mm⁻¹: light becomes diffuse after about a millimetre.

Measurement

The loss due to scattering is measured with a transmission setup that rejects scattered light: a collimated beam, a long path and a small aperture in front of the detector, so that only ballistic light is counted. The angular distribution is measured with a goniometer that swings a detector around the sample. An integrating sphere collects all forward or backward scattered light and separates total transmittance from direct transmittance. In fiber, backscattered Rayleigh light is the signal used by optical time-domain reflectometry to map loss along a link.

Where it matters

Scattering sets the intrinsic loss floor of silica fiber, dominates propagation loss in many integrated waveguides through sidewall roughness, limits imaging depth in tissue, produces stray light and veiling glare in optical instruments, and makes coherent light scattered from rough surfaces form speckle. It is also a measurement tool: dynamic light scattering sizes nanoparticles, Raman spectroscopy identifies materials, and lidar reads aerosol profiles from atmospheric backscatter.

Pitfalls

A transmission measurement with a wide-aperture detector counts forward-scattered light as transmitted and underestimates the scattering loss, especially for large particles whose scattering is concentrated within a few degrees of the beam. Conversely, attenuation measured with a small aperture includes absorption, which must be separated by a second measurement. Multiple scattering invalidates the single-exponential law once the sample is more than a few mean free paths thick.

Common questions

Why is the sky blue?

Air molecules are Rayleigh scatterers, so sunlight at 450 nm is scattered $(700/450)^4 \approx 5.9$ times more strongly than at 700 nm. The eye's reduced sensitivity to violet and the solar spectrum shift the perceived colour to blue rather than violet.

Why are clouds white?

Cloud droplets are much larger than the wavelength, so their scattering efficiency is nearly the same for all visible wavelengths and the scattered light keeps the colour of sunlight.

What is the difference between scattering and absorption?

Absorption converts light energy into heat or excitation of the medium; scattering only changes the direction of the light. Both remove power from a beam, and both contribute to its extinction.

References: C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983); H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957); S. L. Jacques, Phys. Med. Biol. 58, R37 (2013).