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I want to determine the coupling length needed to achieve a certain power coupling coefficient of a directional coupler. Recently, I read a paper titled "Ultra-compact high order ring resonator filters using submicron silicon photonics wires for on-chip optical interconnects".

In the paper, the authors mentioned the "beating length" $L_b$, which is the length needed for the optical power to transfer completely from one waveguide to another in the directional coupler. $L_b$ is defined as

$$L_b = \frac{\lambda}{2(n_\text{even} - n_\text{odd})}$$

where $\lambda$ is the wavelength and $n_\text{even}$ and $n_\text{odd}$ are the effective indices of the fundamental (even) and first order (odd) modes of the two coupled parallel photonic waveguides operated at the specific lambda.

Here, my question is how to compute the $n_\text{even}$ and $n_\text{odd}$ for specific dimensions of the waveguide. Moreover, are $n_\text{even}$ and $n_\text{odd}$ always present in the waveguide, especially for a single-mode waveguide?

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If there are two weakly-coupled single-mode waveguides, then the whole system is almost guaranteed to have two modes (you can call them even & odd, bonding & antibonding, symmetric & antisymmetric, whatever notation you like).

The best (and usually only) way to calculate mode indices is electromagnetic simulation software.

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    $\begingroup$ +1 Especially for photonic wire like devices of the dimensions shown in the paper your last sentence is true! $\endgroup$ Commented Sep 2, 2015 at 12:43

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