Splines¶

Knotted depth laws. Product Spline is cubic/natural. The same Spline object can drive Mix volume fraction (phi) or free optical diagonals (delta_o / delta_e / beta_o / beta_e).

Builders: case_volfrac_spline, case_free_optical_spline

Shared preamble:

from refloxide.model import MaterialSLD, FreeUniTensor, Mix, ReflectModel
from refloxide.profiles import (
    Diffusion,
    SecondOrderTransition,
    DepthProfile,
    Spline,
    Polynomial,
    CallableField,
    CallableDepthProfile,
)

vac = MaterialSLD("", 0, name="vacuum")(0, 0)
si = MaterialSLD("Si", 2.33, name="si")(0, 3.0)
mat = FreeUniTensor(energies=[284.4], name="film")
mat_a = FreeUniTensor(energies=[284.4], name="a")
mat_b = FreeUniTensor(energies=[284.4], name="b")
mat_c = FreeUniTensor(energies=[284.4], name="c")

Changing spline order ($\phi$)¶

Same six control points; linestyles mark $k=1$ (piecewise linear), $k=2$ (quadratic), and $k=3$ (cubic). Linear mix optics; residuals vs cubic.

phi = Spline(
    knots=[0.0, 30.0, 70.0, 110.0, 150.0, 200.0],
    values=[0.20, 0.95, 0.25, 0.80, 0.15, 0.55],
)
layer = DepthProfile(
    material=Mix([mat_a, mat_b], rule="linear"),
    thickness=200.0,
    phi=phi,
)
stack = vac | layer | si

$volfrac spline depth$

$volfrac spline reflectivity$

Changing control-point count (cubic $\phi$)¶

Fix $k=3$ and sample a high-frequency target with $N=4$, $6$, and $10$ equally spaced knots. Sparse polygons flatten the mid-film swings; denser knots recover them. Reflectivity residuals vs $N=10$.

import numpy as np

t = 200.0


def target(z: np.ndarray) -> np.ndarray:
    x = z / t
    return np.clip(
        0.50
        + 0.45 * np.sin(3 * np.pi * x)
        + 0.30 * np.sin(7 * np.pi * x)
        + 0.18 * np.cos(5 * np.pi * x),
        0.02,
        0.98,
    )


stacks = {}
for n in (4, 6, 10):
    knots = np.linspace(0.0, t, n)
    phi = Spline(knots=knots.tolist(), values=target(knots).tolist())
    layer = DepthProfile(
        material=Mix([mat_a, mat_b], rule="linear"),
        thickness=t,
        phi=phi,
    )
    stacks[f"N={n}"] = vac | layer | si

$volfrac spline knots depth$

$volfrac spline knots reflectivity$

Free optical channels¶

Builder: case_free_optical_spline

Extreme seven-knot cubic Splines on $\delta_o$, $\delta_e$, $\beta_o$, and $\beta_e$ (large anti-correlated swings; aux = dichroism). No Mix.

knots = [0.0, 35.0, 70.0, 110.0, 160.0, 210.0, 280.0]
delta_o = Spline(
    knots=knots,
    values=[0.40e-3, 2.20e-3, 0.35e-3, 2.00e-3, 0.50e-3, 1.80e-3, 0.70e-3],
)
delta_e = Spline(
    knots=knots,
    values=[2.30e-3, 0.45e-3, 2.40e-3, 0.55e-3, 2.10e-3, 0.60e-3, 1.90e-3],
)
beta_o = Spline(
    knots=knots,
    values=[1.0e-4, 6.0e-4, 1.5e-4, 7.0e-4, 1.2e-4, 5.5e-4, 2.0e-4],
)
beta_e = Spline(
    knots=knots,
    values=[7.0e-4, 1.5e-4, 8.0e-4, 1.0e-4, 7.5e-4, 1.8e-4, 6.5e-4],
)
layer = DepthProfile(
    material=mat,
    thickness=280.0,
    delta_o=delta_o,
    delta_e=delta_e,
    beta_o=beta_o,
    beta_e=beta_e,
)
stack = vac | layer | si

free optical spline depth

free optical spline reflectivity

Multi-form stacks that include free-optical Splines are on Stacking examples.

Splines

Splines¶

Changing spline order (\(\phi\))¶

Changing control-point count (cubic \(\phi\))¶

Free optical channels¶