Marcatili’s method is an approximate analytical method that describes how light propagates through rectangular dielectric optical waveguides . It was published by Enrique Marcatili in 1969.[1]
Optical dielectric waveguides guide electromagnetic waves in the optical spectrum (light). This type of waveguide consists of dielectric materials (e.g., glass, silicon, indium phosphide, etc). The core of the waveguide has a higher index of refraction than its surrounding and the light is guided due to total internal reflection. In a ray description, the light zig-zags between the walls.
The geometry of the waveguide dictates the light to propagate with specific velocities and specific distributions of the electric and magnetic fields, known as modes. For rectangular waveguides, these modes cannot be computed analytically. This can be done either using a numerical mode solver, or using an approximate method such as Marcatili’s method.
The method can also be understood from the behavior of two planar (slab) waveguides. In his approximate method, the light zig-zags simultaneously upwards-and-downwards as well as left-and-rightwards. In a wave description, the mode is formed by a standing wave in both horizontal x-direction and vertical y-direction with wave vector (
kx
ky
\beta
ky
kx
kx
ky
The propagation constant of the waveguide mode is then computed using:
\beta2=k2-
2 | |
k | |
x |
-
2 | |
k | |
y |
k2
kx
ky
Marcatili’s method used an Ansatz on the shape of the electromagnetic fields in the waveguide. In the core of the waveguide, the mode is a composed of a standing wave in the x- and y-directions. Outside the core, the field decays exponentially in horizontal and vertical directions. The outer quadrants of the rectangular waveguide are neglected.
For low-index-contrast waveguides
k ≈ \beta
kx ≈ ky\ll1
Marcatili’s method was extended to the regime of high-index-contrast, i.e., a large difference between the refractive index of the core of the waveguide and its surrounding. An example of such a waveguide is a silicon-on-insulator (SOI) waveguide.
Based on Marcatili’s propagation constant and the Ansatz on the shape of the electromagnetic field, Westerveld[2] and co-workers derived new relations for the distribution of the electric and magnetic fields. Maxwell’s equations require that the electromagnetic fields tangential to an interface between the core of the waveguide and its cladding are continuous. Several methods where proposed: The Extended Ey ≈ 0 method is based on continuity of the dominant electromagnetic field components. The amplitude optimization method in which the energy density associated with the discontinuity of the tangential field across the interfaces is minimized. These methods provided a more accurate description of the electromagnetic fields in subwavelength silicon waveguides.
RECTWG is an open-source implementation in Matlab of Marcatili's method as well as the extension to high-index-contrast. It allows computation of the effective index (propagation constant), the effective group index (dispersion), and the linear influence of external changes (e.g. temperature, cladding refractive index) and the electromagnetic fields of the modes in the waveguide. The method works for both TE-like and TM-like modes.