Composition law for polarizers

The polarization process when polarizers act on an optical field is studied. We give examples for two kinds of polarizers. The first kind presents an anisotropic absorption—as in a Polaroid film—and the second one is based on total reflection at the interface with a birefringent medium. Using the Stokes vector representation, we determine explicitly the trajectories of the wave light polarization during the polarization process. We find that such trajectories are not always geodesics of the Poincaré sphere as is usually thought. Using the analogy between light polarization and special relativity, we find that the action of successive polarizers on the light wave polarization is equivalent to the action of a single resulting polarizer followed by a rotation achieved, for example, by a device with optical activity. We find a composition law for polarizers similar to the composition law for noncollinear velocities in special relativity. We define an angle equivalent to the relativistic Wigner angle which can be used to quantify the quality of two composed polarizers.

Figure 1

Representation of the Stokes vector $\vec{s}$ and the Poincaré sphere ${\mathcal{S}}^2$.

Figure 2

In the case of a nonabsorbing polarizing device, the Stokes vector $\vec{s}$ is rotated around the polarization vector $\vec{p}$.

Figure 3

(Color online) Transformation (31) of the spatial components $\vec{s}\left(z\right)$ of the Stokes four-vector $\mathbf{s}\left(z\right)$. The polarization vector is $\vec{p}={\vec{e}}^1$. The incoming light wave has an initial polarization $\vec{s}\left(0\right)=\mathbf{(}1∕\sqrt{3}\mathbf{)}({\vec{e}}^1+{\vec{e}}^2+{\vec{e}}^3)$ represented by the thick red (light gray) dashed vector. The Stokes vector for $\gamma \to \infty$ is aligned along the polarization vector $\vec{p}={\vec{e}}^1$ and is represented by the thick blue (dark gray) vector. The other vectors are drawn to show the intermediate positions of the Stokes vector $\vec{s}$ as the parameter $\gamma$ (or equivalently the parameter $z$) increases from 0 to ∞.

Figure 4

(Color online) Representation of typical evolutions of the Stokes vector $\vec{s}\left(z\right)$ and of the normalized Stokes vector $\underset{̱}{\vec{s}}\left(z\right)$ according to the transformations (29, 32), respectively.

Figure 5

(Color online) Typical representation of the Stokes vector $\vec{s}\left(z\right)$ evolution according to the general transformation (40) or (26). The two dashed lines give the direction of the initial Stokes vector $\vec{s}\left(0\right)$ and the direction of the polarization vector $\vec{p}$, respectively. The initial Stokes vector $\vec{s}\left(0\right)$ is represented by the thick black dashed vector. The Stokes vector for $\gamma \to \infty$ is represented by the thick black vector oriented along the direction of the polarization vector $\vec{p}$. The other vectors are the successive Stokes vectors $\vec{s}$ as the parameter $\gamma$ (or equivalently the parameter $z$) increases from 0 to ∞.

Figure 6

(Color online) (a) Generalized Malus law: representation of the ratio $I\left(z\right)∕I\left(0\right)$ as a function of $\alpha \left(0\right)=2\theta \left(0\right)$ [angle between the polarizer vector $\vec{p}$ and the incoming wave polarization $\vec{s}\left(0\right)$] for different values of the absorption term $\gamma \left(z\right)$. (b) Angle $\alpha \left(z\right)$ between the polarizer vector $\vec{p}$ and the wave polarization $\vec{s}\left(z\right)$ as a function of $\alpha \left(0\right)$ for different values of the absorption term $\gamma \left(z\right)$.

Figure 7

Two successive boosts with respective directions ${\vec{p}}_1$ and ${\vec{p}}_2$ can be decomposed in a boost of direction $\vec{p}$ and a rotation around the direction $\vec{q}$.

Figure 8

The hyperbolic triangle $ABC$ corresponding to the polarizers $P_1$, $P_2$ and to the resulting polarizer $P$ in the Poincaré disk.

Figure 9

Representation of the Wigner angle $w$, (94), as a function of $\alpha$ for different values of the product $\sqrt{{\Gamma }_1{\Gamma }_2}$ from 0.6 (dotted line), 0.8 (double-dot-dashed line), 0.95 (dot-dashed line), 0.99 (dashed line), and 1 (solid line).