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 package 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 (factor)
t_s and t_p	Transmission coefficient	Change in amplitude of E-field upon transmission (factor)

Quickstart

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

julia> using FresnelEquations

julia> # Defining toy-input for functions

julia> n1 = 1; n2=2; angle_incident=deg2rad(45);

julia> # Looping over each of the 8 exported functions

julia> for func in (R_s, R_p, T_s, T_p, r_s, r_p, t_s, t_p)
           println(func, " = ", func(n1, n2, angle_incident))
       end
R_s = 0.20377661238703051
R_p = 0.04152490775593412
T_s = 0.7962233876129692
T_p = 0.9584750922440658
r_s = -0.4514162296451364
r_p = 0.20377661238703063
t_s = 0.5485837703548635
t_p = 0.6018883061935153

The functions have detailed docstrings, which are concidered the documentation of this package. Type ? to enter the REPL help mode, and enter the name of the function to see the docstring. For example:

help?> R_s
search: R_s r_s

  R_s(n₁, n₂, θᵢ)
  R_s(n₁, n₂, θᵢ, θₜ)


  Calculate the reflectance for s-polarized light, i.e. the fraction of incident light energy that is reflected.

  The transmitted angle θₜ defaults to asin(n₁ / n₂ * sin(θᵢ)), which Snell's law states.

  Arguments:
  ≡≡≡≡≡≡≡≡≡≡

    •  n₁: The refractive index of the original medium

    •  n₂: The refractive index of the medium transmitted into

    •  θᵢ: The incident angle in radians, meansured from the surface normal

    •  θₜ: The transmitted angle in radians, measured from the surface normal

This package uses unicode in docstrings and internals, but takes care to never force the user to use unicode for anything.

Correctness conciderations

Definition of transmittance

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:

1. The interface between the media is flat
2. The media are homogeneous and isotropic
3. 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: