Stokes parameters

Remarks

Every Stokes vector satisfies $S_0^2 \ge S_1^2 + S_2^2 + S_3^2$, by the Cauchy–Schwarz inequality for time averages. Equality holds exactly for fully polarized light. The ratio $p = \sqrt{S_1^2 + S_2^2 + S_3^2} / S_0$ is the degree of polarization, with $0 \le p \le 1$.

Example: light polarized at $45^\circ$ has constant amplitudes $E_x = E_y = 1/\sqrt{2}$. Then $S_0 = 1$, $S_1 = 0$, $E_x \overline{E_y} = 1/2$, so $S_2 = 1$ and $S_3 = 0$. The equality $S_0^2 = S_1^2 + S_2^2 + S_3^2$ holds, as it must for fully polarized light.

Non-example: the vector $(1, 1, 1, 0)$ is not the Stokes vector of any light. It fails the inequality, since $1^2 < 1^2 + 1^2$.

The sign of $S_3$, and thus which circular state counts as right-handed, depends on the convention for the time factor; texts differ on this point. Some texts write $I, Q, U, V$ for $S_0, S_1, S_2, S_3$.