Arthur Ludwig defines cross polarization in a paper published in the IEEE Transactions on Antennas and Propagation in 1973. The paper discusses three different definitions of cross polarization used in the literature and proposes one as the best choice. The definition that corresponds to a standard measurement practice is suggested as the most suitable. The paper also presents a simplified formula for calculating the electric field $E_z$ for a frill, which is more efficient and accurate than previous methods. This formula is derived by substituting equations into a general expression and simplifying the integrand. The resulting formula is exact and applicable for all observation points not on the frill. The paper also compares the calculated $E_z$ using this formula with the numerical differentiation method and finds that the new formula is much more efficient, requiring approximately one-tenth the computer time. The paper also discusses the importance of cross polarization in antenna systems, particularly in the context of polarization purity and interference. It highlights the need for accurate definitions of cross polarization in order to ensure proper performance of antenna systems. The paper also discusses the use of cross polarization in the calculation of antenna feed or aperture illumination efficiency, where cross polarization must be included as a gain loss factor. The paper concludes that the proposed definition of cross polarization is the best choice for describing antenna patterns. The paper also acknowledges the contributions of a reviewer and a professor who helped in the development of the formula. The paper is supported by several references to other works in the field of antenna theory and design.Arthur Ludwig defines cross polarization in a paper published in the IEEE Transactions on Antennas and Propagation in 1973. The paper discusses three different definitions of cross polarization used in the literature and proposes one as the best choice. The definition that corresponds to a standard measurement practice is suggested as the most suitable. The paper also presents a simplified formula for calculating the electric field $E_z$ for a frill, which is more efficient and accurate than previous methods. This formula is derived by substituting equations into a general expression and simplifying the integrand. The resulting formula is exact and applicable for all observation points not on the frill. The paper also compares the calculated $E_z$ using this formula with the numerical differentiation method and finds that the new formula is much more efficient, requiring approximately one-tenth the computer time. The paper also discusses the importance of cross polarization in antenna systems, particularly in the context of polarization purity and interference. It highlights the need for accurate definitions of cross polarization in order to ensure proper performance of antenna systems. The paper also discusses the use of cross polarization in the calculation of antenna feed or aperture illumination efficiency, where cross polarization must be included as a gain loss factor. The paper concludes that the proposed definition of cross polarization is the best choice for describing antenna patterns. The paper also acknowledges the contributions of a reviewer and a professor who helped in the development of the formula. The paper is supported by several references to other works in the field of antenna theory and design.