## Orthogonality between optical fibre modes

Hi there,

 Each bound mode of a fibre is orthogonal to all other bound modes. Physically this means that if a particular mode is propagating along a fibre, it cannot excite any other mode. Mathematically, orthogonality between the j'th and k'th bound modes is expressible in terms of the vanishing of the integral off a triple scalar product of electric field of the j'th mode and the magnetic field of the k'th mode over the infinite cross-section of the fibre.
The expression that is given is:
$$\int_{A \infty} e_j \times h_k* \cdot \widehat{z} dA = 0$$

where * denotes the complex conjugate, and z^ is the unit vector in the direction of propagation (along the axis of the fibre).

Can anyone explain why this mathematical expression indicates orthogonality? I am trying to interpret the mathematics rather than just accept it.
What I've done:
I tried to explain it by saying that the dot product between two perpendicular vectors will be 0. But in this sense, I only see the indication that a component perpendicular to both e and h (cross product) is perpendicular to the direction of propagation.
Thanks for any help,

Thomas.
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 Tags modes, optical fibers, orthogonality