---
title: "Multilayer slab, transfer matrices"
module: mode::multilayer
summary: "The bound modes and leaky waves of any planar stack, and its reflection and transmission of a plane wave, exactly, from 2 × 2 field-transfer matrices."
order: 4
papers:
- cite: "J. Chilwell, I. Hodgkinson, J. Opt. Soc. Am. A 1, 742 (1984)"
doi: 10.1364/JOSAA.1.000742
validation:
- mode/multilayer-bound-chilwell
- mode/multilayer-leaky-chilwell
- mode/multilayer-power-chilwell
- mode/multilayer-fresnel
- mode/multilayer-bragg
examples:
- multilayer_chilwell
---
Films 1 … J lie between a semi-infinite cover and substrate. In each medium of index n,
$\alpha = \sqrt{n^2 - \beta^2}$ (β the effective index), and $\gamma = \alpha$ for TE or $\alpha/n^2$ for TM.
## The method
Each film's field-transfer matrix (Chilwell and Hodgkinson's Eq. 10), with phase thickness
$\Phi_j = k \alpha_j d_j$:
$$
M_j = \begin{pmatrix} \cos\Phi_j & -\frac{i}{\gamma_j}\sin\Phi_j \cr -i\gamma_j \sin\Phi_j & \cos\Phi_j \end{pmatrix},
\qquad M = \prod_{j=1}^{J} M_j .
$$
A mode is a root of the modal-dispersion function (Eq. 26):
$$
\chi(\beta) = \gamma_c m_{11} + \gamma_c \gamma_s m_{12} + m_{21} + \gamma_s m_{22} = 0 .
$$
In the cover and substrate, α takes the root that decays away from the stack
($\operatorname{Im}\alpha \gt 0$), except for a **leaky wave**: where $\operatorname{Re}\beta$ is below that medium's
index, the outgoing root ($\operatorname{Re}\alpha \gt 0$), which grows away from the stack (Section 3.C).
The paper's convention, $e^{i(k\beta y - \omega t)}$ with $\operatorname{Im}\beta \gt 0$, is photonoxide's.
- **Bound modes** of a lossless stack: real roots between the largest bounding index and the
largest film index, bracketed and bisected.
- **Leaky waves** and lossy stacks: complex roots, by secant iteration from a guess, or all of
them in a region of the complex plane, from the minima of $|\chi|$ on a grid.
- **Fields** (Eq. 34) and each layer's **share of the power** (Eq. 42).
## Reflection and transmission
The same matrix answers how much of a plane wave the stack reflects. Incident from the cover at
angle θ (so $\beta = n_c \sin\theta$), with χ the left side of Eq. 26 (Eqs. 13–16):
$$
r = \frac{\gamma_c m_{11} + \gamma_c \gamma_s m_{12} - m_{21} - \gamma_s m_{22}}{\chi}, \qquad
t = \frac{2\gamma_c}{\chi}, \qquad
r and t are ratios of U: the tangential E for TE, the tangential H for TM. Beyond total internal
reflection $\gamma_s$ is imaginary and T is zero. `Multilayer::reflection` returns all four; the
cover must be lossless, so that the incident wave is defined. This is the exact reference for
the plane-wave checks of the FDFD solver.
## Validation
The paper's four-layer guide: cover 1.0; films 1.66, 1.53, 1.60, 1.66 (500 nm each); substrate
1.50; 632.8 nm.
- **Table 3**, the 8 bound modes: all within 5e-7 (printed to 6 decimals).
- **Table 3**, all 48 shares of power per layer: within 0.05 % (printed to 0.1 %).
- **Table 2**, the 5 TE leaky waves: nine of the ten printed numbers are ours rounded. m = 5's
real part is printed 1.38250 against our 1.3824892, one unit in the last place. With the
bound modes agreeing to 7 digits, that is most likely the 1984 table's rounding.
- One film reproduces the [three-layer slab](slab.md) to 1e-12.
- **Reflection:** at one interface Eq. 13 is Fresnel's equations, to 1e-12 at 0–70° for TE and
TM, with no TM reflection at Brewster's angle. Eight quarter-wave pairs reflect as the closed
form, to 1e-12. A lossless stack's R + T is 1 to 1e-11 at every angle, and beyond the critical
angle R is 1 and T is 0.
The multilayer slab is also the exact reference for the [PML](pml.md): an SOI slab's leakage
through its buried oxide into the substrate.