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 (${\displaystyle k_{x}}$, ${\displaystyle k_{y}}$, ${\displaystyle \beta }$). The up-and-downwards zig-zag path of the light (${\displaystyle k_{y}}$) is given by a horizontal planar waveguide that is formed by the rectangular waveguide but with infinite width. The left-and-right zig-zag path (${\displaystyle k_{x}}$) is given by a vertical planar waveguide that is formed by the rectangular waveguide but with infinite height. Knowing the zig-zag paths (${\displaystyle k_{x}}$, ${\displaystyle k_{y}}$) allows to compute the propagation velocity of the light in the waveguide (or equivalently, the effective refractive index).

The propagation constant of the waveguide mode is then computed using: ${\displaystyle \beta ^{2}=k^{2}-k_{x}^{2}-k_{y}^{2}}$, where ${\displaystyle k^{2}}$ is the wavenumber corresponding to the core material of the waveguide, ${\displaystyle k_{x}}$ and ${\displaystyle k_{y}}$ are the wave-numbers corresponding to the standing waves in the x- and y-direction, and beta is the wavenumber in the propagation direction of the waveguide, also known as the propagation constant.

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 ${\displaystyle k\approx \beta }$ because modes are not guided otherwise, so ${\displaystyle k_{x}\ll \beta }$ and ${\displaystyle k_{y}\ll \beta }$. Marcatili's method neglects these terms in the second order, and computes the electromagnetic fields in the waveguide based on this assumption and the Ansatz of the shape of the fields.