Poincaré sphere and Stokes parameters of orbital angular momentum wave based on Euler’s formula

The orbital angular momentum (OAM) Poincaré sphere has been developed using the mathematical relationship between Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes. However, it did not exhibit correspondence to spin angular momentum (SAM), the other angular momentum (AM) form. Here, based on Euler's formula, a definition method for general OAM waves is presented, in which the OAM wave is superposed by two orthogonal trigonometric-function waves. As SAM waves exhibit an ellipse in a direction-space, OAM waves exhibit an ellipse in a function-space. Based on this finding, the OAM Poincaré sphere is constructed through polar-spherical coordinate transformation. Finally, a physical monitoring method to monitor OAM Stokes parameters is presented and verified by Rayleigh-Sommerfeld's (RS) diffraction theory computation. This study can be used in AM theoretical research, OAM generation and manipulation, and OAM multiplexing and information encoding.