Lightwave Engineering

H	Derivation of the Orthogonality of the Eigenmode

Let us consider the electromagnetic fields E(^μ), H(^μ), E^(v), and H^(v) of two modes with different mode numbers. (Here, these fields involve only the part that is dependent on the spatial coordinates, excluding e^jωt.) First, let us consider the following vector operations:

$\nabla \cdot \left(E^{\left(\mu \right)}\times H^{\left(v\right)*}+E^{\left(v\right)*}\times H^{\left(\mu \right)}\right).$

(H.1)

When the vector formula (D.9) is used here, Equation (H.1) is rewritten by the following equation:

$\nabla \cdot \left(E^{\left(\mu \right)}\times H^{\left(v\right)*}+E^{\left(v\right)*}\times H^{\left(\mu \right)}\right)=H^{\left(v\right)*}\cdot \nabla \times E^{\left(\mu \right)}-E^{\left(\mu \right)}\cdot \nabla \times H^{\left(v\right)*}+H^{\left(\mu \right)}\cdot \nabla \times E^{\left(v\right)*}\cdot \nabla \times H^{\left(\mu \right)}.$

(H.2)

Moreover, when Equation (5.7) and (5.8) are used in ∇ × E and ∇ × H, and the refractive index is expressed as a real number (i.e., a medium with no gain and no absorption loss),

$\nabla \cdot (E^{\left(\mu \right)}\times H^{\left(v\right)*}+E^{\left(v\right)*}\times H^{(\mu }$