Rayleigh scattering

Rayleigh scattering is the elastic scattering of light by particles, or fluctuations in density, much smaller than the wavelength: air molecules in the atmosphere, frozen-in density fluctuations in glass. Its strength rises as $1\mathrm{/}{\lambda }^4$, so blue light at 450 nm scatters 4.4 times more strongly than red light at 650 nm. That is why the clear sky is blue and the low sun red, and why silica fiber loses about 0.16 dB/km to scattering at 1550 nm but 1.8 dB/km at 850 nm. The scattered light keeps the wavelength of the incident light, unlike Raman and Brillouin scattering, and light scattered at 90° to the incident beam is linearly polarized. It is not the same thing as the Rayleigh range of a laser beam or the Rayleigh criterion for resolution, which carry the same physicist's name.

The 1/λ⁴ law

A particle much smaller than the wavelength sits in a nearly uniform oscillating field, which induces an oscillating dipole moment in it. An oscillating dipole radiates a power proportional to the fourth power of its frequency, so the scattered power rises as ${\omega }^4$, or $1\mathrm{/}{\lambda }^4$. For a small sphere of diameter $d$ and refractive index $n$ in air, the scattering cross-section is

$${\sigma }_s=\frac{2{\pi }^5}{3}\frac{d^6}{{\lambda }^4}{\left(\frac{n^2-1}{n^2+2}\right)}^2\mathrm{.}$$

The sixth power of the diameter means that halving a particle's size cuts its scattering 64 times. A medium containing $N$ such scatterers per unit volume attenuates a beam by ${\alpha }_s=N{\sigma }_s$, the same exponential law as absorption (Beer–Lambert), although the light is redirected rather than absorbed.

The formula holds for particles up to roughly a tenth of the wavelength. Larger particles scatter according to Mie theory: the scattering depends only weakly on wavelength and is concentrated in the forward direction, which is why fog and clouds, made of droplets several micrometers across, look white.

Angular pattern and polarization

For unpolarized incident light, the scattered intensity varies with the scattering angle $\theta$ as

$$I(\theta )\propto 1+{\cos }^2\theta ,$$

equal in the forward and backward directions and half as strong at 90°. The light scattered at 90° is linearly polarized, perpendicular to the plane containing the incident and scattered rays, because the induced dipoles radiate nothing along their own axis. Skylight is therefore most strongly polarized in the part of the sky 90° from the sun, though multiple scattering keeps it from being fully polarized; a polarizing filter on a camera darkens that part of the sky the most.

Blue sky and red sunsets

Air molecules are about a thousand times smaller than visible wavelengths, so they scatter in the Rayleigh regime, and the blue part of sunlight is scattered out of the direct beam across the sky. Violet scatters even more strongly, but sunlight contains less violet than blue and the eye is much less sensitive to it, so the scattered mixture looks blue. Near sunset the direct light crosses many times more air, most of its blue is scattered away before it arrives, and the sun and the clouds it lights look orange and red.

Rayleigh scattering in optical fiber

Glass is a liquid that solidified without crystallizing, and it keeps the density and composition fluctuations of the melt, frozen in on scales of a few nanometers. They scatter light in the Rayleigh regime, with a loss ${\alpha }_R=A\mathrm{/}{\lambda }^4$. The coefficient $A$ is roughly 0.8 to 0.9 dB·µm⁴/km for silica fibers, higher with more germania in the core, which gives 0.14 to 0.16 dB/km at 1550 nm, where ${\lambda }^4=5.77\mu {\text{m}}^4$. For $A$ = 0.9 dB·µm⁴/km, the upper end of that range:

Scattering is the largest loss in silica fiber at telecom wavelengths. Beyond about 1.6 µm the infrared absorption of the glass rises steeply, so the total fiber attenuation has its minimum near 1550 nm: 0.18 to 0.20 dB/km for standard single-mode fiber, and about 0.14 dB/km for the best pure-silica-core fibers. The steep wavelength dependence is a large part of why optical communication moved from 850 nm to 1310 nm and then to 1550 nm.

A small part of the scattered light is recaptured by the fiber in the backward direction. An OTDR sends pulses into the fiber and records this backscatter against time, so the slope of the trace gives the local loss and steps and peaks locate splices, connectors and breaks. The backscatter coefficient of standard single-mode fiber at 1550 nm is about −82 dB for a 1 ns pulse: the light scattered back from the 0.1 m of fiber that the pulse resolves is 82 dB below the launched peak power, about −72 dB per meter of fiber. Rayleigh backscatter also returns light toward transmitters, contributing to a link's optical return loss, and its phase is the signal in distributed acoustic sensing and OFDR. Hollow-core fibers guide light in air, where scattering is far weaker, which is how they get below the silica floor; Hollow-core fiber: below the Rayleigh floor describes how.

In integrated waveguides the equivalent loss comes from sidewall roughness, which also backscatters light: in a high-Q ring resonator the backscatter couples the clockwise and counterclockwise modes and splits resonances, and it supplies the feedback in self-injection locking.

Rayleigh, Mie, Raman and Brillouin scattering

Common questions

Why is the sky blue and not violet?

Violet light scatters even more strongly than blue, but sunlight contains less violet than blue, and the eye is much less sensitive to it. The mixture of scattered wavelengths that reaches the eye is seen as blue.

What is the difference between Rayleigh and Mie scattering?

Rayleigh scattering applies to particles much smaller than the wavelength and rises steeply as the wavelength falls, as $1\mathrm{/}{\lambda }^4$, with equal forward and backward lobes. Mie scattering applies to particles comparable to or larger than the wavelength, depends only weakly on wavelength and is concentrated forward.

Is Rayleigh scattering elastic?

Yes. The scattered light has the same frequency as the incident light; only its direction changes. Raman and Brillouin scattering are the inelastic processes, which shift the frequency.

How much fiber loss is due to Rayleigh scattering?

At 1550 nm, about 0.14 to 0.16 dB/km of the 0.18 to 0.20 dB/km total loss of standard single-mode fiber. The rest comes mostly from infrared absorption of the glass and from waveguide imperfections.

References: Lord Rayleigh (J. W. Strutt), "On the light from the sky, its polarization and colour," Philosophical Magazine 41, 107 and 274 (1871). C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010).