|
| 1 | +### A Pluto.jl notebook ### |
| 2 | +# v1.0.1 |
| 3 | + |
| 4 | +using Markdown |
| 5 | +using InteractiveUtils |
| 6 | + |
| 7 | +# ╔═╡ 7a1f0001-0000-4000-8000-000000000001 |
| 8 | +begin |
| 9 | + import Pkg |
| 10 | + Pkg.activate(@__DIR__) # the examples/ environment |
| 11 | + Pkg.develop(; path = dirname(@__DIR__)) # point TransitionMatrices at this repo |
| 12 | + Pkg.instantiate() |
| 13 | + using TransitionMatrices, Plots, LinearAlgebra |
| 14 | +end |
| 15 | + |
| 16 | +# ╔═╡ 7a1f0002-0000-4000-8000-000000000002 |
| 17 | +md""" |
| 18 | +# Near-field maps from a T-matrix |
| 19 | +
|
| 20 | +`TransitionMatrices.jl` reconstructs the electromagnetic field *around* a particle |
| 21 | +— not just the far-field cross sections — from any computed T-matrix. Given an |
| 22 | +incident plane wave, the scattered field is expanded in radiating vector spherical |
| 23 | +wave functions (VSWFs) with coefficients ``(p, q) = \mathbf{T}\,(a, b)``, where |
| 24 | +``(a, b)`` are the plane-wave expansion coefficients. The total external field is |
| 25 | +
|
| 26 | +```math |
| 27 | +\mathbf{E}_\text{tot}(\mathbf r) = \mathbf{E}_\text{inc}(\mathbf r) + \mathbf{E}_\text{sca}(\mathbf r), |
| 28 | +\qquad |
| 29 | +\mathbf{E}_\text{sca}(\mathbf r) = \sum_{n,m} p_{mn}\mathbf{M}_{mn}(k\mathbf r) + q_{mn}\mathbf{N}_{mn}(k\mathbf r). |
| 30 | +``` |
| 31 | +
|
| 32 | +This works for **any** T-matrix (Mie, EBCM, IITM, Sh-matrix). Here we map the field |
| 33 | +enhancement ``|\mathbf{E}_\text{tot}|/|\mathbf{E}_\text{inc}|`` around a dielectric |
| 34 | +sphere. |
| 35 | +
|
| 36 | +!!! note "Region of validity" |
| 37 | + The radiating expansion converges only **outside the sphere circumscribing the |
| 38 | + particle** (the Rayleigh hypothesis). For a sphere that boundary *is* the |
| 39 | + surface, so the map is valid right down to it; for a non-spherical particle, |
| 40 | + evaluate only outside the circumscribing sphere. |
| 41 | +""" |
| 42 | + |
| 43 | +# ╔═╡ 7a1f0003-0000-4000-8000-000000000003 |
| 44 | +md""" |
| 45 | +## 1. A scatterer and its incidence |
| 46 | +
|
| 47 | +A dielectric sphere of size parameter ``x = k a = 2.5`` and refractive index |
| 48 | +``m_r = 1.5``, illuminated by an ``x``-polarized plane wave propagating along |
| 49 | +``+z``. With ``\lambda = 2\pi`` the size parameter equals the radius, so ``a = 2.5``. |
| 50 | +
|
| 51 | +The incidence direction is ``(\vartheta_\text{inc}, \varphi_\text{inc}) = (0, 0)``; |
| 52 | +the polarization ``(E_\vartheta, E_\varphi) = (1, 0)`` is ``\hat{\mathbf x}`` because |
| 53 | +``\hat{\boldsymbol\vartheta}(0,0) = \hat{\mathbf x}``. |
| 54 | +""" |
| 55 | + |
| 56 | +# ╔═╡ 7a1f0004-0000-4000-8000-000000000004 |
| 57 | +begin |
| 58 | + λ = 2π |
| 59 | + k = 2π / λ |
| 60 | + a = 2.5 |
| 61 | + mᵣ = 1.5 + 0.0im |
| 62 | + x = k * a |
| 63 | + N = ceil(Int, x + 4 * cbrt(x) + 2) |
| 64 | + 𝐓 = TransitionMatrices.MieTransitionMatrix{ComplexF64, N}(x, mᵣ) |
| 65 | + |
| 66 | + # Incident plane wave: +z propagation, x-polarized. |
| 67 | + ϑ_inc, φ_inc = 0.0, 0.0 |
| 68 | + Eθ, Eφ = 1.0 + 0im, 0.0im |
| 69 | + |
| 70 | + # The scattered coefficients (p, q) = 𝐓 (a, b) depend only on the incidence, |
| 71 | + # so compute them once and reuse them at every field point. |
| 72 | + p, q = scattering_coefficients(𝐓, ϑ_inc, φ_inc, Eθ, Eφ) |
| 73 | + nothing |
| 74 | +end |
| 75 | + |
| 76 | +# ╔═╡ 7a1f0005-0000-4000-8000-000000000005 |
| 77 | +md""" |
| 78 | +## 2. Sample the total field on a grid |
| 79 | +
|
| 80 | +We evaluate ``|\mathbf{E}_\text{tot}|`` in the ``x``–``z`` plane (the plane of |
| 81 | +incidence and polarization). Points inside the sphere are masked — the external |
| 82 | +expansion is not meant to represent the interior field (that is the *internal* |
| 83 | +field, a separate reconstruction). |
| 84 | +""" |
| 85 | + |
| 86 | +# ╔═╡ 7a1f0006-0000-4000-8000-000000000006 |
| 87 | +begin |
| 88 | + xs = range(-3a, 3a; length = 161) |
| 89 | + zs = range(-3a, 3a; length = 161) |
| 90 | + enhancement = fill(NaN, length(zs), length(xs)) |
| 91 | + for (i, zz) in enumerate(zs), (j, xx) in enumerate(xs) |
| 92 | + |
| 93 | + r = hypot(xx, zz) |
| 94 | + r ≤ a && continue # inside the sphere: masked |
| 95 | + pos = [xx, 0.0, zz] |
| 96 | + E_inc = incident_field(λ, ϑ_inc, φ_inc, Eθ, Eφ, pos) |
| 97 | + E = E_inc + scattered_field(p, q, λ, pos) |
| 98 | + enhancement[i, j] = norm(E) / norm(E_inc) |
| 99 | + end |
| 100 | + nothing |
| 101 | +end |
| 102 | + |
| 103 | +# ╔═╡ 7a1f0007-0000-4000-8000-000000000007 |
| 104 | +md""" |
| 105 | +## 3. The field-enhancement map |
| 106 | +
|
| 107 | +The incident wave travels upward (``+z``). The map shows the standing-wave |
| 108 | +interference of the incident and scattered fields, the forward-scattering |
| 109 | +concentration on the far (``+z``) side, and the shadow on the near side. |
| 110 | +""" |
| 111 | + |
| 112 | +# ╔═╡ 7a1f0008-0000-4000-8000-000000000008 |
| 113 | +let |
| 114 | + hm = heatmap(xs, zs, enhancement; |
| 115 | + aspect_ratio = 1, c = :inferno, clims = (0, maximum(filter(isfinite, enhancement))), |
| 116 | + xlabel = "x", ylabel = "z", colorbar_title = "|E| / |E₀|", |
| 117 | + title = "dielectric sphere, x = $(round(x; digits = 2)), mᵣ = 1.5", |
| 118 | + size = (640, 560)) |
| 119 | + # outline the sphere |
| 120 | + θ = range(0, 2π; length = 200) |
| 121 | + plot!(hm, a .* cos.(θ), a .* sin.(θ); label = "", lw = 1.5, lc = :cyan) |
| 122 | +end |
| 123 | + |
| 124 | +# ╔═╡ 7a1f0009-0000-4000-8000-000000000009 |
| 125 | +md""" |
| 126 | +## 4. A line cut along the optical axis |
| 127 | +
|
| 128 | +Along ``x = 0`` the field shows the shadow/illumination asymmetry and the |
| 129 | +forward enhancement directly. |
| 130 | +""" |
| 131 | + |
| 132 | +# ╔═╡ 7a1f000a-0000-4000-8000-00000000000a |
| 133 | +let |
| 134 | + zline = range(-3a, 3a; length = 400) |
| 135 | + amp = map(zline) do zz |
| 136 | + abs(zz) ≤ a && return NaN |
| 137 | + pos = [0.0, 0.0, zz] |
| 138 | + E_inc = incident_field(λ, ϑ_inc, φ_inc, Eθ, Eφ, pos) |
| 139 | + E = E_inc + scattered_field(p, q, λ, pos) |
| 140 | + norm(E) / norm(E_inc) |
| 141 | + end |
| 142 | + plot(zline, amp; lw = 2, xlabel = "z (optical axis)", ylabel = "|E| / |E₀|", |
| 143 | + label = "", title = "axial cut (x = y = 0)", size = (720, 320)) |
| 144 | + vspan!([-a, a]; alpha = 0.15, c = :gray, label = "sphere") |
| 145 | +end |
| 146 | + |
| 147 | +# ╔═╡ 7a1f000b-0000-4000-8000-00000000000b |
| 148 | +md""" |
| 149 | +## Takeaway |
| 150 | +
|
| 151 | +`scattering_coefficients` once, then `scattered_field` / `total_field` at any point |
| 152 | +gives the full external field for any T-matrix — sphere, spheroid, cylinder, |
| 153 | +Chebyshev, or a general particle. Reusing the precomputed `(p, q)` keeps a dense |
| 154 | +field grid cheap. |
| 155 | +
|
| 156 | +For the field **inside** the particle, use `internal_field`, which reconstructs |
| 157 | +the regular internal solution from `internal_coefficients`. Those coefficients |
| 158 | +are the VSWF expansion amplitudes for the field inside the particle; compute them |
| 159 | +once and reuse them just like `(p, q)`: |
| 160 | +
|
| 161 | +```julia |
| 162 | +c, d = internal_coefficients(x, mᵣ, ϑ_inc, φ_inc, Eθ, Eφ; nmax = N) |
| 163 | +E_inside = internal_field(c, d, mᵣ, λ, [0.2a, 0.0, 0.0]) |
| 164 | +``` |
| 165 | +
|
| 166 | +The Tier-2 implementations live in `src/common/near_field.jl` for homogeneous |
| 167 | +spheres and `src/EBCM/near_field.jl` for axisymmetric particles. |
| 168 | +""" |
| 169 | + |
| 170 | +# ╔═╡ Cell order: |
| 171 | +# ╟─7a1f0002-0000-4000-8000-000000000002 |
| 172 | +# ╠═7a1f0001-0000-4000-8000-000000000001 |
| 173 | +# ╟─7a1f0003-0000-4000-8000-000000000003 |
| 174 | +# ╠═7a1f0004-0000-4000-8000-000000000004 |
| 175 | +# ╟─7a1f0005-0000-4000-8000-000000000005 |
| 176 | +# ╠═7a1f0006-0000-4000-8000-000000000006 |
| 177 | +# ╟─7a1f0007-0000-4000-8000-000000000007 |
| 178 | +# ╠═7a1f0008-0000-4000-8000-000000000008 |
| 179 | +# ╟─7a1f0009-0000-4000-8000-000000000009 |
| 180 | +# ╠═7a1f000a-0000-4000-8000-00000000000a |
| 181 | +# ╟─7a1f000b-0000-4000-8000-00000000000b |
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