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The angular multipole of order two in a local radiation intensity distribution. A monopole is isotropic and a dipole has one directional axis; a quadrupole has a second-order angular pattern. Thomson scattering of an incident radiation quadrupole can generate linear polarization, which is central to Cosmic microwave background polarization.

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 arewhere $d\Omega$ 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: $F=1/2$. ^[ref].

The normalization is consistent becauseThus 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 $F(\theta )$.

The rotatory power is the rotation of transmitted linear polarization per unit path length, in $\mathrm{rad}{\mathrm{m}}^{-1}$. With equally transmitted circular eigenmodes $({\mathbf{e}}_1\mp i{\mathbf{e}}_2)/\sqrt{2}$ and time convention $e^{-i\omega t}$, their accumulated phase difference gives $\theta =({\varphi }_{+}-{\varphi }_{-})/2$. For uniform circular birefringence, $\rho =\pi (n_{+}-n_{-})/{\lambda }_0$. Generic linear birefringence need not admit one rotation angle common to all incident polarization directions. In a weak longitudinal Faraday effect, $\rho =V{B}_{\parallel }$, with signed Verdet constant $V$.

Split one stationary scalar optical field into equal copies travelling at right angles with common transverse linear polarization. For wavevectors $k(\hat{\mathbf{x}}\pm \hat{\mathbf{y}})/\sqrt{2}$, the relative geometric delay is $\sqrt{2}y/c$ in vacuum, while the delay on the $x$ bisector is zero. The electric-field intensity convention gives the displayed pattern and $I(x)=2I_0$. The mean Poynting vector points along $+x$, with magnitude $I(y)/\sqrt{2}$ in that convention. A transverse detector can record the $y$ profile if it resolves the period $\lambda /\sqrt{2}$; a longitudinal $x$ profile is not a transverse camera image.

For ideal polarizers with successive transverse transmission directions ${\theta }_j$, the Jones matrix is the rank-one orthogonal projection $P_j={\mathbf{u}}_j{\mathbf{u}}_j^{\mathsf{T}}$. The identity ${\mathbf{u}}_j^{\mathsf{T}}{\mathbf{u}}_{j-1}=\cos ({\theta }_j-{\theta }_{j-1})$ reduces the transmitted probability amplitude analogue, here the optical complex amplitude, to a product of projections. For unpolarized light, the first element transmits half the incident light intensity, giving the displayed product. For a specified incident linear polarization, replace the first half-factor with its actual projection.

A waveplate uses birefringence to change polarization of an electromagnetic wave through phase retardation. Its fast axis has the smaller refractive index, hence the larger phase velocity; its slow axis has the larger index. Consider normal incidence with the optic axis in the plate face, so both components follow the same straight path. A linear polarization at $45^{\circ }$ to these axes excites equal wave amplitudes.

A complementary experiment uses two sheet polarizers with a cellophane sheet between them. First cross the polarizers and remove the sheet: the field is dark. Insert and rotate the sheet about the beam: light returns at intermediate orientations, but extinction recurs at two perpendicular orientations. The sheet has changed the polarization of an electromagnetic wave, without requiring a resolved double image. With white light, different wavelengths can produce different transmitted colors.^[ref]. Brightness alone does not establish circular polarization: an ideal rotating linear polarizer also gives orientation-independent light intensity for unpolarized light.