The most probable orientational distribution functions of rod-like polar molecules contained in a noncentrosymmetric uniaxial system are established using the first-rank and third-rank Legendre polynomials, ‹<i>P</i> <sub>1</sub>(cos θ)› and ‹<i>P</i> <sub>3</sub>(cos θ)› order parameters, and the maximum entropy method. Emphasis is put on the different domains of existence in the (‹<i>P</i> <sub>1</sub>›, ‹<i>P</i> <sub>3</sub>›) plane for the various shapes of the distributions: it is thus shown that, for any positive ‹<i>P</i> <sub>1</sub>(cos θ)› value and for decreasing ‹<i>P</i> <sub>3</sub>(cos θ)› values, the distribution function may exhibit either a distorted oblate form with an intense maximum at 0°, or a three-leaved rose curve with maxima at 60°, 180°, and 300°, and finally another markedly oblate shape with a strong maximum at 180°. As an illustrative example, we have considered the azobenzene molecular orientations in an electrically poled p(DR1M) homopolymer thin film after a thermal process and several relaxation periods. We have made use not only of the ‹<i>P</i> <sub>1</sub>› and ‹<i>P</i> <sub>3</sub>› parameters determined from polarized second-harmonic generation (SHG) measurements, but also of the ‹<i>P</i> <sub>2</sub>› values extracted from UV-visible spectra and of the ‹<i>P</i> <sub>4</sub>› values adjusted according to the information entropy theory. In such a thin film with very large nonlinear properties (d<sub>33</sub> coefficients were varying from 437.0 to 117.0 pm/V at 1064 nm) it is evidenced that a strong polar order is maintained even after a long relaxation period of 42 days. So, the distribution functions demonstrate that the poling treatment was quite efficient and they emphasize the importance in the determination of both couples of odd and even order parameters in such uniaxially oriented optical elements.
V. Rodriguez, F. Lagugné-Labarthet, and C. Sourisseau, "Orientation Distribution Functions based upon both ‹P 1›, ‹P 3› Order Parameters and upon the Four ‹P 1› up to ‹P 4› Values: Application to an Electrically Poled Nonlinear Optical Azopolymer Film," Appl. Spectrosc. 59, 322-328 (2005)