FresnelEquations

From the wikipedia article on the Fresnel Equations: "The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media."

This function defines 8 functions that implement the equations found in the wikipedia article. An overview of the functions is given below.

Function	Description	Physical meaning
`R_s` and `R_p`	Reflectance	Fraction of energy reflected
`T_s` and `T_p`	Transmittance	Fraction of energy transmitted
`r_s` and `r_p`	Reflection coefficient	Change in amplitude of E-field upon reflection
`t_s` and `t_p`	Transmission coefficient	Change in amplitude of E-field upon transmission

Examples

Usage of this package is quite simple. Below is a demonstration of the usage of all 8 functions defined.

julia> using FresnelEquations

julia> let
           n1 = 1
           n2 = 2
           θ_i = deg2rad(10)
           [f(n1, n2, θ_i) for f in (R_s, R_p, T_s, T_p, r_s, r_p, t_s, t_p)]
       end
8-element Vector{Float64}:
  0.11454557997889622
  0.10771599997760896
  0.8854544200211036
  0.892284000022391
 -0.3384458301987132
  0.32820115779443704
  0.6615541698012867
  0.6641005788972185

Note that the transmittance T could be defined as 1 - R, guaranteeing perfect energy conservation. This implementation can however suffer catastrophic cancellation as R approaches 1. Instead, the transmission coefficient t is used directly, meaning that conservation of energy is only accurate to numerical precision.

julia> let
           n1 = 1
           n2 = 2
           θ_i = deg2rad(10)
           @show R_s(n1, n2, θ_i) + T_s(n1, n2, θ_i)
           @show R_p(n1, n2, θ_i) + T_p(n1, n2, θ_i)
           nothing
       end
R_s(n1, n2, θ_i) + T_s(n1, n2, θ_i) = 0.9999999999999999
R_p(n1, n2, θ_i) + T_p(n1, n2, θ_i) = 1.0

Assumptions

Note that some assumptions are made in deriving these equations:

The interface between the media is flat
The media are homogeneous and isotropic
The media are non-magnetic, i.e. with a permeability equal to that of vacuum.

You can read more about the assumptions and the sources referenced in the wikipedia article on the Fresnel Equations: