The trajectory of the polarization state of a monochromatic beam passing through a fixed linear polarizer and a rotating elliptical retarder on the Poincaré sphere is found to be a three-dimensional 8-shaped contour, which is determined as the line of intersection of a right-circular cylinder with the Poincaré sphere. The cylinder is parallel to the S3 axis, and the projection of the contour on the S1S2 plane is a circle whose center and radius are determined. A method of projecting the three-dimensional geometric relationships to the two-dimensional S1S2 plane to locate the position of the polarization state of the emerging beam on the Poincaré sphere for a given azimuth of the elliptical retarder is presented, and applied to solve a problem of polarization optics. The proposed graphic method substantially simplifies the polarization state analysis involving elliptical retarders.
2. Goldstein, D. H., Polarized Light, CRC Press, 2017.
3. Gil, J. J. and R. Ossikovski, Polarized Light and the Mueller Matrix Approach, CRC Press, 2017.
4. El-Hosseiny, F., "Methods for determining the optical parameters of elliptic retarders," J. Opt. Soc. Am., Vol. 65, 1279-1282, 1975.
5. Pancharatnam, S., "Achromatic combinations of birefringent plates," Proceedings of the Indian Academy of Sciences --- Section A, 137-144, Springer, 1 1955.
6. Gottlieb, D. and O. Arteaga, "Optimal elliptical retarder in rotating compensator imaging polarimetry," Opt. Lett., Vol. 46, 3139-3142, 2021.
7. Lin, P.-L., C.-Y. Han, and Y.-F. Chao, "Three-intensity measurement technique and its measurement in elliptical retarder," Opt. Commun., Vol. 281, 3403-3406, 2008.
8. Eugui, P., D. J. Harper, A. Lichtenegger, M. Augustin, C. W. Merkle, A. Woehrer, C. K. Hitzenberger, and B. Baumann, "Polarization-sensitive imaging with simultaneous bright-and dark-field optical coherence tomography," Opt. Lett., Vol. 44, 4040-4043, 2019.
9. Chung, D., H. S. Park, F. Rotermund, and B. Y. Kim, "Measurement of bending-induced birefringence in a hollow-core photonic crystal fiber," Opt. Lett., 5872-5875, 2019.
10. Vella, A. and M. A. Alonso, "Poincaré sphere representation for spatially varying birefringence," Opt. Lett., Vol. 43, 379-382, 2018.
11. Yu, C.-J., "Fully variable elliptical phase retarder composed of two linear phase retarders," Rev. Sci. Instrum., Vol. 87, 035106, 2016.
12. Rajagopalan, S. and S. Ramaseshan, "Rotating elliptic analysers for the automatic analysis of polarised light --- Part I," Proceedings of the Indian Academy of Sciences --- Section A, 297-312, Springer, 1964.
13. Azzam, R., T. Bundy, and N. Bashara, "The fixed-polarizer nulling scheme in generalized ellipsometry," Opt. Commun., Vol. 7, 110-115, 1973.
14. Sabatke, D., M. Descour, E. Dereniak, W. Sweatt, S. Kemme, and G. Phipps, "Optimization of retardance for a complete Stokes polarimeter," Opt. Lett., Vol. 25, 802-804, 2000.
15. Azzam, R., "Poincaré sphere representation of the fixed-polarizer rotating-retarder optical system," J. Opt. Soc. Am. A, Vol. 17, 2105-2107, 2000.
16. Salazar-Ariza, K. and R. Torres, "Trajectories on the Poincaré sphere of polarization states of a beam passing through a rotating linear retarder," J. Opt. Soc. Am. A, Vol. 35, 65-72, 2018.