Mie scattering calculations for perfect spheres based on miepython. Extensive documentation is at (https://miepython.readthedocs.io).

MieScattering.jl is a Julia package to calculate light scattering of a plane wave by non-np.absorbing, partially-np.absorbing, or perfectly conducting spheres.

The extinction efficiency, scattering efficiency, backscattering, and scattering asymmetry for a sphere with complex index of refraction m, diameter d, and wavelength lambda can be found by:

    qext, qsca, qback, g = ez_mie(m, d, λ0)

The normalized scattering values for angles µ=cos(θ) are:

    Ipar, Iper = ez_intensities(m, d, λ0, µ)

If the size parameter is known, then use:

    mie(m, x)

Mie scattering amplitudes S1 and S2 (complex numbers):

    mie_S1_S2(m, x, μ)

Normalized Mie scattering intensities for angles µ=cos(θ):

    i_per(m, x, µ)
    i_par(m, x, µ)
    i_unpolarized(m, x, µ)

D_calc(m, x, N)

Compute the logarithmic derivative using best method.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• N: order of Ricatti-Bessel function

Output

The values of the Ricatti-Bessel function for orders from 0 to N.

D_downwards!(z, N, D)

Compute the logarithmic derivative by downwards recurrence.

Parameters

• z: function argument
• N: order of Ricatti-Bessel function
• D: gets filled with the Ricatti-Bessel function values for orders from 0 to N for an argument z using the downwards recurrence relations.

D_upwards!(z, N, D)

Compute the logarithmic derivative by upwards recurrence.

Parameters

• z: function argument
• N: order of Ricatti-Bessel function
• D: gets filled with the Ricatti-Bessel function values for orders from 0 to N for an argument z using the upwards recurrence relations.

Lentz_Dn(z, N)

Compute the logarithmic derivative of the Ricatti-Bessel function.

Parameters

• z: function argument
• N: order of Ricatti-Bessel function

Output

This returns the Ricatti-Bessel function of order N with argument z using the continued fraction technique of Lentz, Appl. Opt., 15, 668-671, (1976).

ez_intensities(m, d, λ0, μ, n_env = 1.0, norm = :albedo)

Return the scattered intensities from a sphere. These are the scattered intensities in a plane that is parallel (ipar) and perpendicular (iper) to the field of the incident plane wave. The scattered intensity is normalized such that the integral of the unpolarized intensity over 4𝜋 steradians is equal to the single scattering albedo. The scattered intensity has units of inverse steradians [1/sr]. The unpolarized scattering is the average of the two scattered intensities.

Parameters

• m: the complex index of refraction of the sphere [-]
• d: the diameter of the sphere [same units as lambda0]
• λ0: wavelength in a vacuum [same units as d]
• µ: the cos(θ) of each direction desired [-]
• n_env: real index of medium around sphere, optional.

Output

ipar, iper: scattered intensity in parallel and perpendicular planes [1/sr]

ez_mie(m, d, λ0, n_env = 1.0)

Calculate the efficiencies of a sphere.

Parameters

• m: the complex index of refraction of the sphere [-]
• d: the diameter of the sphere [same units as lambda0]
• λ0: wavelength in a vacuum [same units as d]
• n_env: real index of medium around sphere, optional.
• use_threads (optional): Flag whether to use threads (default: true)

Output

• qext: the total extinction efficiency [-]
• qsca: the scattering efficiency [-]
• qback: the backscatter efficiency [-]
• g: the average cosine of the scattering phase function [-]

generate_mie_costheta(μ_cdf)

Generate a new scattering angle using a cdf. A uniformly spaced cumulative distribution function (CDF) is needed. New random angles are generated by selecting a random interval μ[i] to μ[i+1] and choosing an angle uniformly distributed over the interval.

Parameters:

• μ_cdf: a cumulative distribution function

Output:

• an array of random scattering angle cosines based on the CDF supplied.

i_par(m, x, μ; norm = :albedo)

Return the scattered intensity in a plane parallel to the incident light. This is the scattered intensity in a plane that is perpendicular to the field of the incident plane wave. The intensity is normalized such that the integral of the unpolarized intensity over 4π steradians is equal to the single scattering albedo.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• µ: the angles, cos(theta), to calculate intensities

Output

The intensity at each angle in the array µ. Units [1/sr]

i_per(m, x, μ; norm = :albedo)

Return the scattered intensity in a plane normal to the incident light. This is the scattered intensity in a plane that is perpendicular to the field of the incident plane wave. The intensity is normalized such that the integral of the unpolarized intensity over 4π steradians is equal to the single scattering albedo.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• µ: the angles, cos(theta), to calculate intensities

Output

The intensity at each angle in the array µ. Units [1/sr]

i_unpolarized(m, x, μ; norm = :albedo)

Return the unpolarized scattered intensity at specified angles. This is the average value for randomly polarized incident light. The intensity is normalized such that the integral of the unpolarized intensity over 4π steradians is equal to the single scattering albedo.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• µ: the angles, cos(theta), to calculate intensities

Output

The intensity at each angle in the array µ. Units [1/sr]

mie(m, x)

Calculate the efficiencies for a sphere where m or x may be vectors.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere

Output

• qext: the total extinction efficiency
• qsca: the scattering efficiency
• qback: the backscatter efficiency
• g: the average cosine of the scattering phase function

mie_An_Bn(m, x)

Compute arrays of Mie coefficients A and B for a sphere. This estimates the size of the arrays based on Wiscombe's formula. The length of the arrays is chosen so that the error when the series are summed is around 1e-6.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere

Output

• An, Bn: arrays of Mie coefficents

Calculate the scattering amplitude functions for spheres. The amplitude functions have been normalized so that when integrated over all 4*π solid angles, the integral will be qext*pi*x^2. The units are weird, $sr^{-0.5}$.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• µ: the angles, cos($θ$), to calculate scattering amplitudes
• norm (optional): The normalization. Must be one of :albedo (default), :one, :four_pi, :qext,

:qsca, :bohren or :wiscombe"

• use_threads (optional): Flag whether to use threads (default: true)

Output

S1, S2: the scattering amplitudes at each angle µ [$sr^{-0.5}$]

mie_cdf(m, x, num; norm = :albedo)

Create a CDF for unpolarized scattering uniformly spaced in cos(θ). The CDF covers scattered (exit) angles ranging from 180 to 0 degrees. (The cosines are uniformly distributed over -1 to 1.) Because the angles are uniformly distributed in cos(theta), the scattering function is not sampled uniformly and therefore huge array sizes are needed to adequately sample highly anisotropic phase functions. Since this is a cumulative distribution function, the maximum value should be 1.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• num: length of desired CDF array

Output

• µ: array of cosines of angles
• cdf: array of cumulative distribution function values

mie_cdf(m, x, num; norm = :albedo)

Create a CDF for unpolarized scattering for uniform CDF. The CDF covers scattered (exit) angles ranging from 180 to 0 degrees. (The cosines are uniformly distributed over -1 to 1.) These angles mu correspond to uniform spacing of the cumulative distribution function for unpolarized Mie scattering where cdf[i] = i/(num-1). This is a brute force implementation that solves the problem by calculating the CDF at many points and then scanning to find the specific angles that correspond to uniform interval of the CDF. Since this is a cumulative distribution function, the maximum value should be 1.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• num: length of desired CDF array

Output

• µ: array of cosines of angles (irregularly spaced)
• cdf: array of cumulative distribution function values

mie_phase_matrix(m, x, μ; norm=:albedo)

Calculate the phase scattering matrix.

The units are $sr^{-1}$. The phase scattering matrix is computed from the scattering amplitude functions, according to equations 5.2.105-6 in K. N. Liou (2002) - An Introduction to Atmospheric Radiation, Second Edition.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• μ: the angles, cos(theta), at which to calculate the phase scattering matrix

Output

• p: The phase scattering matrix [$sr^{-1}$]

normalization_factor(a, b, x; norm)

Figure out scattering function normalization.

Parameters

• m: complex index of refraction of sphere
• x: dimensionless sphere size
• norm: symbol describing type of normalization

Output

scaling factor needed for scattering function

small_conducting_mie(m, x)

Calculate the efficiencies for a small conducting spheres. Typically used for small conducting spheres where x < 0.1 and real(m) == 0.

Parameters

• x: the size parameter of the sphere

Output

• qext: the total extinction efficiency
• qsca: the scattering efficiency
• qback: the backscatter efficiency
• g: the average cosine of the scattering phase function

small_mie(m, x)

Calculate the efficiencies for a small sphere. Typically used for small spheres where x<0.1

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere

Output

• qext: the total extinction efficiency
• qsca: the scattering efficiency
• qback: the backscatter efficiency
• g: the average cosine of the scattering phase function

Calculate the scattering amplitude functions for small spheres (x<0.1). The amplitude functions have been normalized so that when integrated over all 4*π solid angles, the integral will be qext*pi*x^2. The units are weird, $sr^{-0.5}$

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• µ: the angles, cos($θ$), to calculate scattering amplitudes

Output

S1, S2: the scattering amplitudes at each angle µ [$sr^{-0.5}$]

small_mie_conducting_S1_S2(m, x, μ)

Calculate the scattering amplitudes for small conducting spheres. The spheres are small perfectly conducting (reflecting) spheres (x<0.1). The amplitude functions have been normalized so that when integrated over all 4𝜋 solid angles, the integral will be qext(𝜋x²). The units are weird, $sr^{-0.5}$.

Parameters

• m: the complex index of refraction of the sphere
• x: the size parameter of the sphere
• µ: the angles, cos($θ$), to calculate scattering amplitudes

Output

S1, S2: the scattering amplitudes at each angle µ [$sr^{-0.5}$]