An algebraic method to reconstruct the relative phases and polarization of a complex vector in N dimensions based on 3(N-1) amplitude measurements

A real vector X in n dimensions whose components are sinusoidally varying with time can be represented by an n-dimensional complex factor Ze^jωt whose real part is X. In different real coordinate systems, the components of X transform linearly, but the amplitudes of the components of Z transform nonlinearly. The author describes a method to reconstruct the relative phases of the (complex) components of Z (and, therefore, of X) based on the amplitude measurements of its n components in an orthogonal coordinate system and at least 2n-3 additional amplitude measurements in different directions. The author shows the necessary and sufficient condition on these additional directions to ensure uniqueness of the phase and polarization reconstruction for any arbitrary vector X, and presents an algebraic method for the reconstruction which offers substantial reduction in computing time over the method of reconstruction by nonlinear optimization. The result of this phase of reconstruction is the complete characterization of the polarization of X except for chirality