Mie Scattering Calculator

Particle radius

Sphere radius a.

Wavelength (vacuum)

λ in free space; medium sets λ/ n.

Medium refractive index

n_m (real, non-absorbing).

Particle index (real)

n of m = n + i k.

Particle index (imag)

k controls absorption (k ≥ 0).

Series order (nmax)

Auto uses nmax ≈ x + 4x^(1/3) + 2.

Angle range (deg)

min max

0° forward, 180° backward.

Number of angles

2 to 721 points (higher costs time).

Intensity normalization

Applied to I values only.

Reset

Tip: If results look unstable, increase nmax or reduce angles.

Example data table

Case	Radius (µm)	Wavelength (µm)	n + i k	Medium n	Notes
Latex bead	0.25	0.532	1.59 + i0.00	1.00	Visible scattering with clear angular structure.
Soot-like	0.10	0.550	1.75 + i0.44	1.00	Strong absorption, lower single-scattering albedo.
Water droplet	1.00	0.650	1.33 + i0.00	1.00	Forward-peaked intensity; g tends to increase.

Formula used

The size parameter is computed in the surrounding medium:

x = 2π a n_m / λ

The relative refractive index is:

m = (n + i k) / n_m

Mie coefficients are built from Riccati–Bessel functions ψ_n and ξ_n:

a_n = (m ψ_n(mx) ψ′_n(x) − ψ_n(x) ψ′_n(mx)) / (m ψ_n(mx) ξ′_n(x) − ξ_n(x) ψ′_n(mx))
b_n = (ψ_n(mx) ψ′_n(x) − m ψ_n(x) ψ′_n(mx)) / (ψ_n(mx) ξ′_n(x) − m ξ_n(x) ψ′_n(mx))

Efficiencies follow standard series expressions:

Q_sca = (2/x²) Σ(2n+1)(|a_n|²+|b_n|²)
Q_ext = (2/x²) Σ(2n+1)Re(a_n+b_n)
Q_abs = Q_ext − Q_sca

Angular amplitudes use S₁, S₂ with π_n, τ_n recurrences:

S₁(θ)= Σ (2n+1)/(n(n+1)) (a_nπ_n + b_nτ_n)
S₂(θ)= Σ (2n+1)/(n(n+1)) (a_nτ_n + b_nπ_n)

How to use this calculator

Enter particle radius and vacuum wavelength with units.
Provide particle optical constants as n and k.
Set the surrounding medium refractive index n_m.
Choose auto nmax, or enter a manual value.
Select angle range and number of angular samples.
Press Compute to view results above the form.
Use PDF or CSV buttons for exportable outputs.

Article: Advanced interpretation of Mie scattering outputs

1) Physical model solved here

This calculator computes electromagnetic scattering by a homogeneous sphere using the Mie-series solution of Maxwell equations. The particle uses n + i k, embedded in a lossless medium with refractive index n_m.

2) Inputs and practical ranges

Radius a and vacuum wavelength lambda set geometry, while n, k set optical contrast and absorption. Common values include a from 0.05 to 5 micrometers, lambda from 0.3 to 1.6 micrometers, n from 1.3 to 2.5, and k from 0 to 0.5.

3) Size parameter as the regime map

The size parameter x = 2*pi*a*n_m/lambda controls behavior. For x < 0.3, scattering is Rayleigh-like and g is near 0. For x around 1 to 30, resonances and strong angular oscillations appear. For x > 50, scattering becomes sharply forward-peaked.

4) Series cutoff and convergence

Mie predictions require truncating partial waves at nmax. The automatic rule nmax about x + 4*x^(1/3) + 2 is a standard accuracy-speed balance. As a reference, x = 10 often needs about 21 terms. Increase nmax if Q values drift or Q_abs becomes slightly negative.

5) Reading Q values as cross sections

Efficiencies convert to cross sections through sigma = Q*pi*a^2. Q_ext may exceed 2 near resonances because extinction includes forward interference. Absorption is computed by Q_abs = Q_ext - Q_sca. A helpful summary is omega0 = Q_sca/Q_ext, which drops as k increases.

6) Angular intensity and anisotropy g

The unpolarized intensity is I(theta) = (|S₁|^2 + |S₂|^2)/2 on your chosen angle grid. Larger x creates narrow forward lobes, so coarse angular sampling can miss peak structure. The asymmetry parameter g = <cos(theta)> typically rises from about 0 (small x) toward 0.7 to 0.95 (large x).

7) Backscatter relevance

Q_back quantifies 180-degree return relative to pi*a^2, useful for lidar-style backscatter comparisons. In the Mie regime, Q_back can oscillate with x; higher k usually damps these oscillations by suppressing internal resonances.

8) Quality checks and performance tips

Check Q_ext >= Q_sca and Q_abs >= 0 within small numerical error. Runtime scales roughly with (angles) times nmax, so reduce angles first for fast scans. For higher fidelity, increase angles and confirm convergence by raising nmax by 10 to 20 percent.

FAQs

1) Why can Qext be greater than 2?

Extinction includes scattering plus absorption and also forward-direction interference. Near resonances, the effective interaction area can exceed the geometric area, so Q_ext above 2 is physically allowed.

2) What does k represent in n + i k?

k is the extinction coefficient that models internal loss. Larger k increases absorption, lowers the single-scattering albedo, and typically smooths sharp angular oscillations by damping internal resonances.

3) When should I switch from auto nmax to manual?

Use manual nmax when you need strict convergence control, such as matching benchmark data. Increase nmax until Q values and the intensity curve change negligibly, typically less than 0.1%.

4) Why does the intensity plot show ripples?

Ripples come from interference between partial waves in the Mie series and internal reflections within the sphere. They become more pronounced as x increases and when absorption is low.

5) What does g tell me in one number?

g is the mean cosine of the scattering angle. g near 0 indicates nearly symmetric scattering, while g near 1 indicates strongly forward-peaked scattering, common for larger particles.

6) How many angles should I use?

For quick checks, 181 angles is usually fine. For narrow forward lobes at large x, use 361 to 721 angles, or reduce the maximum angle step size to resolve sharp features.

7) Why do my results look unstable for large particles?

Large x requires more terms and can amplify roundoff. Increase nmax, avoid extremely fine angular grids at the same time, and confirm that Q_abs stays nonnegative within small tolerance.