Rayleigh scattering angular pattern

{c}
{title2=$F(\theta)=(1+\cos^2\theta)/2$}

For incident <unpolarized light> and the isotropic <electric dipole> response of <Rayleigh scattering by a small sphere>, let $\theta$ be the angle between the incident beam's propagation direction and the outgoing direction. The <differential scattering cross-section> and its relative angular pattern are
$$
\frac{d\sigma}{d\Omega}=\frac{3\sigma}{16\pi}(1+\cos^2\theta),\qquad F(\theta)=\frac{1+\cos^2\theta}{2},
$$
where $d\Omega$ is an element of <solid angle>. Averaging over the two transverse directions of incident <polarization of an electromagnetic wave> gives this pattern. Forward scattering at $0^\circ$ and backward scattering at $180^\circ$ are equally strong: $F=1$. Sideways scattering at $90^\circ$ is weakest, but nonzero: $F=1/2$. https://farside.ph.utexas.edu/teaching/jk1/lectures/node107.html[Fitzpatrick, Thomson Scattering, equations 1283–1287: averaging the dipole angular pattern]{ref}.

\Image[/rayleigh-angular-scattering.png]
{title=<Rayleigh scattering angular pattern> for incident <unpolarized light>, measured from the incident beam}
{description=The plotted radial coordinate is $F(\theta)$, a relative <differential scattering cross-section>, rather than the distance travelled by <light>. The incident propagation axis points right: $0^\circ$ is forward, $180^\circ$ backward, and $90^\circ$ sideways. Rotating this plane about the incident axis gives the three-dimensional pattern.}

The normalization is consistent because
$$
\int(1+\cos^2\theta)\,d\Omega=2\pi\int_0^\pi(1+\cos^2\theta)\sin\theta\,d\theta=\frac{16\pi}{3}.
$$
Thus integrating $d\sigma/d\Omega$ gives the total <Rayleigh scattering cross-section> $\sigma$. The plotted $F$ has maximum one; it is not itself normalized to unit <solid angle> integral.

For a single <electric dipole> driven by incident <linear polarization>, the <angular distribution of electric dipole radiation> instead has a $\sin^2\alpha$ factor, where $\alpha$ is measured from the <electric dipole moment> direction, parallel to the incident <electric field>. That direction is perpendicular to the incident beam. Confusing $\alpha$ with the beam angle $\theta$ would incorrectly predict zero forward scattering. Averaging the dipole patterns over incident <unpolarized light> yields the displayed $F(\theta)$.


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