On the choice of the phase constant of the Fresnel transmission coefficient of a slab

The Fresnel reflection and transmission coefficients for a homogeneous isotropic layer of finite thickness can be derived by using the Airy summation method or by applying the boundary conditions for the electric and magnetic fields at the boundaries of the slab. The two methods result in different expressions for the transmission and reflection coefficients. The reflection coefficients can be reduced to the simplest form. However, the transmission Fresnel coefficients cannot be transformed to the simplest form. Instead, they can be reduced to two different forms which have different phase constants. The transmission coefficient derived based on the Airy summation method does not depend on the index of refraction of the medium behind the slab while the transmission coefficient derived based on the electrodynamics boundary condition method does depend on the index of refraction of the medium behind the slab. The difference between the two expressions for the transmission coefficients stems from the choice of the phase of the transmitted wave as assumed in the two methods. The two different forms for the transmission coefficient of a slab lead to quantitatively and qualitatively different dependences of the real and imaginary parts of the transmission coefficient as a function of the index of refraction of the medium behind the slab. The striking difference in the numerical results is the oscillating form of the real and imaginary parts of the transmission coefficients and the phase constant as a function of the index of refraction of the medium behind the slab. We propose a generalized form of the Fresnel transmission coefficient of a slab.