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Checks that need no other code: closed forms, limits and conservation laws, the same series summed in extended precision, and independent routes through PyStratify's own physics. Each case plots PyStratify against its reference on the same axes.
decay rates against Mie's series in 40 digits
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20 40 60 80 100 10⁻² 10⁻¹ 1 10 10² 10³ 10⁴ distance from the surface [nm] Γ / Γ₀
Source Source type Electric point dipole
λ [nm] 633
r min [nm] 51
r max [nm] 150
points 41
R 50 nm
p r 60 nm
⊥ radial ⊥ ∥ tangential ∥
R 50 nm
p r 60 nm
⊥ radial ⊥ ∥ tangential ∥ Sphere · Electric dipole Label Material n k Thickness [nm] C Silver const 0.093 4 50 H Air const 1 0 ∞
decay rates inside and outside against transfer matrices in 60 digits
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20 40 60 80 100 120 140 10⁻¹ 1 10 10² 10³ 10⁴ distance from the centre [nm] Γ / Γ₀
Source Source type Electric point dipole
λ [nm] 614
r min [nm] 5
r max [nm] 150
points 44
R 55 nm
p r 70 nm
⊥ radial ⊥ ∥ tangential ∥
R 55 nm
p r 70 nm
⊥ radial ⊥ ∥ tangential ∥ Sphere · Electric dipole Label Material n k Thickness [nm] C Silica const 1.45 0 50 1 Au const 0.27 2.93 5 H Water const 1.33 0 ∞
a magnetic dipole's decay rates against 60-digit transfer matrices
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50 100 150 200 250 300 10⁻¹ 1 10 10² distance from the surface [nm] Γ / Γ₀
Source Source type Magnetic point dipole
λ [nm] 800
r min [nm] 102
r max [nm] 400
points 41
R 100 nm
m r 120 nm
⊥ radial ⊥ ∥ tangential ∥
R 100 nm
m r 120 nm
⊥ radial ⊥ ∥ tangential ∥ Sphere · Magnetic dipole Label Material n k Thickness [nm] C Silicon const 3.7 0.005 100 H Air const 1 0 ∞
total and radiative rates against the series in 40 digits
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50 100 150 200 250 300 10⁻² 10⁻¹ 1 10 10² 10³ 10⁴ distance from the surface [nm] Γ / Γ₀
Source Source type Electric point dipole
λ [nm] 600
r min [nm] 26
r max [nm] 325
points 36
R 25 nm
p r 35 nm
⊥ radial ⊥ ∥ tangential ∥
R 25 nm
p r 35 nm
⊥ radial ⊥ ∥ tangential ∥ Sphere · Electric dipole Label Material n k Thickness [nm] C Metal const 0.05 3.5 25 H Air const 1 0 ∞
the radiative rate against the power reaching the far field
50 100 150 200 250 300 350 400 10⁻¹ 1 10 10² distance from the centre [nm] Γ_rad / Γ₀
Source Source type Electric and magnetic dipoles
λ [nm] 614
r min [nm] 5
r max [nm] 400
points 38
R 90 nm
p r 120 nm
⊥ radial ⊥ ∥ tangential ∥
R 90 nm
p r 120 nm
⊥ radial ⊥ ∥ tangential ∥ Sphere · Electric dipole Label Material n k μ Thickness [nm] C Core const 1.45 0 1.3 50 1 Au const 0.27 2.93 1 20 2 Shell const 1.6 0 1.7 20 H Water const 1.33 0 1.1 ∞
The calculator has no permeability: this setup opens there only without μ.
the Green's dyadic between two points, order by order, against 80 digits
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20 40 60 80 100 120 10⁻¹⁴ 10⁻¹² 10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 1 order ℓ |g_ℓ|
Source Source type Electric point dipole
Offset from origin · r [nm] 56
observed at r [nm] 60
R 55 nm
p r 56 nm
⊥ radial ⊥ ∥ tangential ∥
R 55 nm
p r 56 nm
⊥ radial ⊥ ∥ tangential ∥ Sphere · Electric dipole Label Material n k Thickness [nm] C Silica const 1.45 0 50 1 Au const 0.27 2.93 5 H Water const 1.33 0 ∞
the quadrupole's coupling to each order against finite differences
1 2 3 4 5 6 7 8 10⁻¹¹ 10⁻⁸ 10⁻⁵ 10⁻² 10 10⁴ 10⁷ 10¹⁰ order ℓ |coupling|
A quadrupole couples to the vector spherical waves W = M + N \mathbf{W} = \mathbf{M} + \mathbf{N} W = M + N
1 6 ∑ i j Q i j ∂ j W i , \tfrac{1}{6}\sum_{ij} Q_{ij}\,\partial_j W_i, 6 1 ij ∑ Q ij ∂ j W i , which PyStratify evaluates in closed form; the reference takes central differences of the same waves in 40-digit arithmetic. The axial (m = 0) family is drawn, 47 nm from the centre with k = 0.013 nm⁻¹: the regular wave fades with order, the outgoing one grows. The test covers all five families.
the Casimir–Polder potential against its non-retarded limit −C₃/d³
10 20 30 40 50 10⁻²⁷ 10⁻²⁶ 10⁻²⁵ 10⁻²⁴ 10⁻²³ 10⁻²² 10⁻²¹ 10⁻²⁰ 10⁻¹⁹ distance from the surface [nm] |U| [J]
Close to the surface an atom sees a flat, non-retarded half-space:
U ( d ) → − C 3 d 3 , C 3 = ℏ 4 π ∫ 0 ∞ α ( i ξ ) ε ( i ξ ) − 1 ε ( i ξ ) + 1 d ξ U(d) \to -\frac{C_3}{d^3}, \qquad C_3 = \frac{\hbar}{4\pi}\int_0^\infty \alpha(i\xi)\,\frac{\varepsilon(i\xi) - 1}{\varepsilon(i\xi) + 1}\,d\xi U ( d ) → − d 3 C 3 , C 3 = 4 π ℏ ∫ 0 ∞ α ( i ξ ) ε ( i ξ ) + 1 ε ( i ξ ) − 1 d ξ PyStratify sums the sphere's full retarded response. The two meet within a nanometre (0.3 % at 0.5 nm) and part further out by design, as retardation and the sphere's curvature weaken the potential (37 % at 50 nm).