Usage

Define a shape

Spheroid

The following defines a spheroid with semi-major axis a=1.0, semi-minor axis c=0.5, and a refractive index of m=1.311.

spheroid = Spheroid{Float64, ComplexF64}(1.0, 0.5, 1.311);

The type specification {Float64, ComplexF64} can be omitted, and in that case, the real and complex types used in the calculations will be inferred from the input parameters.

Note

In the example above, if {Float64, ComplexF64} is ommited, do remember to use complex(1.311) instead of 1.311 to denote that the refractive index is a complex value. Otherwise errors would occur in later calculations.

Cylinder

The following defines a cylinder with radius r=1.0, height h=2.0, and a refractive index of m=1.5+0.01im. Note that the type specification {Float128, ComplexF128} is used to specify the real and complex types used in the calculations.

cylinder = Cylinder{Float128, ComplexF128}(1.0, 2.0, 1.5+0.01im);

Chebyshev particle

The following defines a Chebyshev particle with radius r₀=1.0, deformation coefficient ε=0.1, order n=4, and a refractive index of m=1.5+0.01im. Note that the type specification {Arb, Acb} is used to specify the real and complex types used in the calculations.

chebyshev = Chebyshev{Arb, Acb}(1.0, 0.1, 4, 1.5+0.01im);

Calculate the T-Matrix

After defining a shape, the T-Matrix can be calculated using the transition_matrix function. It has an alias calc_T for convenience.

Take the spheroid defined above as an example, and suppose the wavelength is λ=2π. The T-Matrix can be calculated iteratively as follows:

𝐓 = calc_T(spheroid, 2π)

To see the process, you can turn on debugging by setting ENV["JULIA_DEBUG"] = "TransitionMatrices", and then run the above code.

There are many optional keyword arguments, and you can refer to the detailed documentation of the transition_matrix.

You can also calculate the T-Matrix directly by specifying the truncation order and number of Gauss-Legendre quadrature points:

𝐓 = transition_matrix(spheroid, 2π, 10, 100)

Post-processing

After getting the T-Matrix, you can calculate the far-field scattering properties using the following functions:

amplitude_matrix, a.k.a. calc_S
phase_matrix, a.k.a. calc_Z
extinction_cross_section, a.k.a. calc_Cext
scattering_cross_section, a.k.a. calc_Csca
absorption_cross_section, a.k.a. calc_Cabs
albedo, a.k.a. calc_ω
asymmetry_parameter, a.k.a. calc_g

And the orientation-averaged scattering matrix:

scattering_matrix, a.k.a. calc_F

Automatic differentiation

TransitionMatrices.jl supports automatic differentiation using ForwardDiff.jl.

Here is an example of calculating the gradient of the scattering cross section with respect to a, c, mᵣ, mᵢ and λ:

using ForwardDiff, TransitionMatrices

function f(x)
    s = Spheroid(x[1], x[2], complex(x[3], x[4]))
    T₀ = TransitionMatrices.transition_matrix(s, x[5], 10, 100)
    Csca = calc_Csca(T₀)
end

gradient = ForwardDiff.gradient(f, [2.0, 3.0, 1.311, 0.02, 2π])

# output

5-element Vector{Float64}:
  19.872009666582716
   5.740718904091896
  89.1305069099317
 -45.84568759716399
  -9.066448506675068