Polarization time and length for random optical beams

We investigate the dynamics of the instantaneous polarization state of stationary, partially polarized random electromagnetic beamlike fields. An intensity-normalized correlation function of the instantaneous Poincaré vector is introduced for the characterization of the time evolution of the polarization state. This polarization correlation function enables us to define a polarization time and a polarization length over which the polarization state remains substantially unchanged. In the case of Gaussian statistics, the polarization correlation function is shown to assume a simple form in terms of the parameters employed to characterize partial coherence and partial polarization of electromagnetic fields. The formalism is demonstrated for a partially polarized, temporally Gaussian-correlated beam, and black-body radiation. The results are expected to find a range of applications in investigations of phenomena where polarization fluctuations of light play an important role.

Figure 1

Description of the instantaneous polarization state by a point on the Poincaré sphere of unit radius. The vectors $\widehat{\mathbf{s}}\left(t\right)$ and $\widehat{\mathbf{s}}(t+\tau )$ represent the polarization states at two instants of time with time difference $\tau$, and $\cos \alpha$ equals the scalar product $\widehat{\mathbf{s}}\left(t\right)\cdot \widehat{\mathbf{s}}(t+\tau )$.

Figure 2

Illustration of polarization time as discussed in the text. The topmost solid curve corresponds to a fully polarized field for which the polarization time is infinite. The second solid curve from the top represents a partially polarized field for which the polarization time is infinite as well. The third curve corresponds to a partially polarized field for which one can relate a polarization time ${\tau }_{p3}$. The lowest curve corresponds to a fully unpolarized field for which the polarization time is ${\tau }_{p4}$. The quantities $P_i$, with $i=1–4$, are the degrees of polarization related to the fields.

Figure 3

Behavior of ${\gamma }_P\left(\tau \right)$ for a uniformly partially polarized, Gaussian correlated beam. The solid curves correspond to the degrees of polarization $P=0$, $P=0.5$, and $P=0.95$. The vertical dotted lines mark the polarization times, which are ${\tau }_p=0.96\sigma$ and ${\tau }_p=1.20\sigma$ for $P=0$ and $P=0.5$, respectively.

Figure 4

Behavior of ${\gamma }_P\left(\tau \right)$ for black-body radiation at temperatures $T=10\mathrm{K}$, $T=30\mathrm{K}$, and $T=300\mathrm{K}$. The temperatures correspond to the polarization times ${\tau }_p=3.8\times {10}^{-13}\mathrm{s}$, ${\tau }_p=1.3\times {10}^{-13}\mathrm{s}$, and ${\tau }_p=1.3\times {10}^{-14}\mathrm{s}$, respectively. The polarization times are marked with vertical dotted lines.