Orthogonality between optical fibre modes

[thomas430]

Hi there,

I've just read the following:

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.
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Thanks for any help,

Thomas.

 

[mighty2000]

Hello Thomas,

Thank you for your question. The mathematical expression given does indeed indicate orthogonality between the j'th and k'th bound modes in a fiber. To understand why, we need to break down the components of the expression and look at them individually.

First, let's look at the cross product between the electric field of the j'th mode (e_j) and the complex conjugate of the magnetic field of the k'th mode (h_k*). The cross product between two vectors results in a third vector that is perpendicular to both of the original vectors. In this case, the resulting vector will be perpendicular to both the electric and magnetic fields of the two modes. This is because the electric and magnetic fields are perpendicular to each other in electromagnetic waves.

Next, let's look at the dot product between the resulting vector and the unit vector in the direction of propagation (z^). The dot product between two vectors results in a scalar (a single number) that represents the magnitude of the projection of one vector onto the other. In this case, the dot product is taken over the infinite cross-section of the fiber, which means that it represents the projection of the resulting vector onto the direction of propagation at every point along the cross-section. Since the direction of propagation is perpendicular to the cross-section, the projection will be 0 at every point along the cross-section. This means that the integral of the dot product over the entire cross-section will also be 0.

Therefore, we can conclude that if the integral of the dot product is 0, then the resulting vector must be orthogonal to the direction of propagation at every point along the cross-section. And since the resulting vector is perpendicular to both the electric and magnetic fields, this means that the two fields must also be orthogonal to each other at every point along the cross-section. This is why the mathematical expression given indicates orthogonality between the j'th and k'th bound modes in a fiber.

I hope this helps to clarify the concept for you. If you have any further questions, please don't hesitate to ask.