Planar layers and S-matrices¶

An SMatrix is the scattering matrix of a planar two-port network: it maps the waves arriving at both sides to the waves leaving both sides. The sides are named after the z axis of the ports' basis:

the negative side, below (z < 0), with medium negative_medium;
the positive side, above, with medium positive_medium.

SMatrix in treams

treams calls this whole network SMatrices, and calls one of its four blocks an SMatrix. In treams-rs one block is a ScatteringBlock. See Coming from treams.

Building a network¶

Function	Builds
`interface(basis=, k0=, negative_medium=, positive_medium=)`	one flat interface
`slab(basis=, k0=, thickness=, material=)`	one homogeneous layer
`multilayer_slab(basis=, k0=, thicknesses=, materials=)`	layers from the negative to the positive side
`propagation(distance=, basis=, k0=)`	a gap of homogeneous medium
`PeriodicResponse.to_smatrix(ports)`	a periodic array; see Periodic arrays
`stack([first, second, ...])`	the networks in order from the negative to the positive side

The basis is a PlaneWavePorts: PlaneWavePorts.default([0, 0]) holds the two pol values at normal incidence; PlaneWavePorts.diffr_orders holds the diffraction orders of a lattice.

import numpy as np
import treams_rs as tr

ports = tr.PlaneWavePorts.default([0, 0])
glass = tr.interface(basis=ports, k0=1.0, negative_medium=1, positive_medium=2.25)
incident = tr.plane_wave(direction=[0, 0, 1], pol="positive_helicity", k0=1.0)
power = glass.power(incident)
# Fresnel reflectance at normal incidence from n = 1 to n = 1.5.
np.testing.assert_allclose(power.reflection, ((1 - 1.5) / (1 + 1.5)) ** 2)
two = tr.multilayer_slab(
    basis=ports, k0=1.0, thicknesses=[0.3, 0.4], materials=[2.25, 4]
)
layers = [
    tr.slab(basis=ports, k0=1.0, thickness=d, material=e)
    for d, e in ((0.3, 2.25), (0.4, 4))
]
np.testing.assert_allclose(two.array, tr.stack(layers).array, atol=1e-15)

Results¶

Method	Returns
`power(incident)`	`PowerBalance(transmission, reflection)` as fractions of the incident power
`scatter(negative=..., positive=...)`	`ScatteredPorts(negative, positive)`: the waves leaving each side
`block("positive", "negative")`	one `ScatteringBlock`: the waves leaving the first side for waves arriving from the second; this one is the transmission upwards
`circular_dichroism(incident)`	the contrasts of transmission and total outgoing power between the two helicities
`bands(period=...)`	`BandModes(wavenumbers, eigenvectors)` of the network repeated along z
`transfer_matrix()`	the transfer matrix of one period

power and scatter read the incident side from the direction of the wave: a wave with a positive z component arrives from the negative side. A wave from above uses direction=[0, 0, -1] and the positive medium; the layer order stays the same.

import numpy as np
import treams_rs as tr

ports = tr.PlaneWavePorts.default([0, 0])
glass = tr.interface(basis=ports, k0=1.0, negative_medium=1, positive_medium=2.25)
transmission = np.asarray(glass.block("positive", "negative"))
# The field transmission coefficient 2 n1 / (n1 + n2), the same for both pols.
np.testing.assert_allclose(transmission, 0.8 * np.eye(2), atol=1e-15)
from_above = tr.plane_wave(
    direction=[0, 0, -1], pol="positive_helicity", k0=1.0, medium=2.25
)
np.testing.assert_allclose(glass.power(from_above).reflection, 0.04)
layer = tr.slab(basis=ports, k0=1.0, thickness=0.5, material=2.25)
bands = layer.bands(period=0.5)
# A period filled with one material carries waves with kz = +-n k0 = +-1.5.
np.testing.assert_allclose(np.sort(bands.wavenumbers.real), [-1.5, -1.5, 1.5, 1.5])

cascade joins two networks like stack does for a list. The treams method names add, illuminate, tr, cd, periodic and bands_kz call these methods; the name map lists them.