Generalization of the Ludwig-3 Definition for linear co-polarization and cross-polarization

This paper presents the generalization of the Ludwig-3 definition of linear co-polarization and cross-polarization. We say that our definition is a generalization because our definition agrees with the Ludwig-3 definition on boresight but departs further from the Ludwig-3 definition as the observation angle θ departs from zero as a result of the presence of a cos(θ) term in the generalized definition. The generalized definition assigns the linear co-polarized component at an arbitrary point on the observation sphere to be the electric field vector whose projection onto the aperture plane is oriented in the direction of a perfectly linearly polarized aperture field. The linear cross-polarized component at the same arbitrary point on the observation sphere is then the electric field vector orthogonal to the co-polarized component in the propagation plane. Since the Ludwig-3 definition does not take into account the additional rotation incurred by the presence of the cos(θ) term. a perfectly linearly polarized field in the aperture results in a zero cross-polarization at the observation point by the generalized definition, but not so in general by the Ludwig-3 definition. The discrepancy between the two definitions becomes significant for a general observation direction when θ > 15°, with the largest discrepancy being found for φ = 45°.