Simplified method of calculating legendre coefficients for computing optical properties of colorants

The Kubelka‐Munk (K‐M) theory is widely used for instrumental colour measurement and colorant formulation in modern colour technology. The K‐M theory is derived with certain assumptions and hence has several limitations. A more rigorous theory called the many‐flux (M‐F) technique was recently proposed by Mudgett and Richards to overcome the limitations of the K‐M theory. Application of the M‐F technique requires lengthy computations and hence is not suitable for industrial organizations for day‐to‐day work. It has been shown that the limitations of the K‐M theory and the complexity of the M‐F technique can be significantly reduced if simple equations are developed to calculate Legendre coefficients. Empirical equations to calculate Legendre coefficients (a n /a o , n = 1,2) are developed. The validity and utility of these equations have been confirmed by computing K‐M scattering coefficients and phase functions, and comparing them with those computed using Mie‐theory values. The use of these equations to determine the values of refractive index of colorants is also shown. The work reported in this article simplifies to some extent the computation of reflectance using the M‐F technique and enable the estimation of K‐M scattering coefficients with change in size of pigments. Utility of the proposed equations is established by computing reflectance by the M‐F technique and comparing it with values obtained by using rigorous Mie equations for Legendre coefficients.