Lightwave Engineering

L	Derivation of the Coupling Mode Equation

Here, let us derive the equation that describes the coupling between two waveguides, which are in close proximity by a distance of about a wavelength, as shown inFigure L.1c.

First, if the electromagnetic field distributions of the mode propagating in waveguide I are designated as E⁽¹⁾ and H⁽¹⁾ in the case of Figure L.1a where waveguide I exists alone, and the electromagnetic field distributions are designated as E⁽²⁾ and H⁽²⁾ in the case of Figure L.1b where waveguide II exists alone, then they will satisfy the following Maxwell’s equations:

$\nabla \times E^{\left(\sigma \right)}=-j\omega {\mu }_0{H}^{\left(\sigma \right)},$

(L.1)

$\nabla \times H^{\left(\sigma \right)}=-j\omega {\epsilon }_0{n}^{{\left(\sigma \right)}^2}\left(x,y\right)E^{\left(\sigma \right)}.$

(L.2)

Here, σ = 1 or 2, and n⁽¹⁾(x, y) and n⁽²⁾(x, y) represent the refractive index distributions, ...