tmmcore

Transfer-matrix method for multilayer thin-film optics, with exact analytic derivatives.

Documentation · Python · Source

Takes a stack of layers and returns reflectance, transmittance and absorptance for absorbing and dispersive materials, at any angle of incidence, in s and p polarization. Alongside them it returns the exact thickness Jacobian and Hessian, and phase, group delay, GDD and third-order dispersion with their thickness derivatives, all computed analytically rather than by finite differences.

This is the Python package of tmmcore. The wheel carries the same C kernel as the npm package, compiled for your platform and called through ctypes. It needs numpy and nothing else.

Install

pip install tmmcore

Wheels are built for Windows (x86-64), macOS (Intel and Apple silicon) and Linux (x86-64 and ARM64, glibc), for Python 3.9 and later.

Use

import tmmcore

# A quarter-wave MgF2 layer on glass, at 550 nm, normal incidence.
stack = {"incident": 1.0, "substrate": 1.52,
         "layers": [{"n": 1.38, "d": 550 / (4 * 1.38)}]}   # d in nm

print(tmmcore.rta(stack, 550)["R"])   # 0.012600790214630274

Layers run from the incident medium toward the substrate. In rta_spectrum any n or k can be one value per wavelength. A quarter-half-quarter antireflection coating on BK7-like glass (Macleod, Thin-Film Optical Filters 5th ed., ch. 4), with Cauchy indices:

import numpy as np
import tmmcore

def cauchy(a, b):
    return lambda lam: a + b / lam**2

low, high, mid, glass = cauchy(1.377, 2000), cauchy(2.170, 8000), cauchy(1.680, 5000), cauchy(1.504, 4200)
lam = np.arange(400.0, 701.0, 2.0)   # nm

# Quarter, half and quarter wave at 510 nm, the layer facing air first.
layers = [{"n": m(lam), "d": waves * 510 / m(510)} for m, waves in ((low, 0.25), (high, 0.5), (mid, 0.25))]
coated = tmmcore.rta_spectrum({"incident": 1.0, "substrate": {"n": glass(lam)}, "layers": layers}, lam)

print(f"{coated['R'].mean():.2%}")   # 0.25%, against 4.25% for bare glass

Calls

Each call takes the stack, then one wavelength or, in a _spectrum call, deposition_spectra and growing_stack, an array of them. angle and pol are keyword arguments. angle is the angle of incidence, 0 when left out, and over an array of wavelengths may be one angle per wavelength. pol is "s", "p" or "avg" (the default, the mean of the two) in the R/T, derivative and needle calls, and "s" or "p" in the phase calls, which may leave it out at normal incidence, where the two are the same. In place of angle the phase calls take sine, the sine of the angle of incidence as a jet, one per wavelength in a _spectrum call, for an angle fixed outside a dispersive incident medium. The needle calls take candidates, the materials they try inserting, each an index given as a medium is, and fractions, where inside each layer they also try them. The monitoring calls take angle and no pol, and return s and p together with the reflectance seen from the substrate side; monitor_curve also takes top, the growing layer's index, and thicknesses, the thicknesses of it to sample. Results are dicts. At one wavelength the values are numbers, each first derivative an array with one value per layer and each second derivative an (N, N) array; a needle call's gaps and intra are dicts of dR, dT and dA, arrays of shape (N + 1, nCand) and (N, nFrac, nCand). Over a grid each gains a leading wavelength axis. The phase calls return one dict for r and one for t, and at one wavelength None for a coefficient that is exactly zero. monitor_curve returns one value per thickness, deposition_spectra arrays of shape (N, nLam), row k after k + 1 layers, and each sample of growing_stack one value per wavelength.

Conventions

Agreement with the npm package

The package's tests run every call on the cases of the npm package's equivalence suite and compare each value with its WebAssembly build: R, T and A to within 1e-9, everything else to within 1e-7 of its size or 1e-12, whichever is larger. Both run the same C, so they differ only in the last digits, where the platform's math library rounds differently from the one compiled into the WebAssembly.

They also run the comparison with Essential Macleod on its committed results, to the same tolerances as the npm package.

Licence

MIT © Andrey Achapovsky