Calculating the Propagation of Light Based on Modes

Using the calculated set of mode, one can calculate the resulting field profile for an arbitrary input profile after any distance of propagation in the fiber:

• First, calculate the excitation amplitudes of all guided modes via complex overlap integrals of the input amplitude profile with the complex conjugates of all mode amplitude profile.

• Then change the complex amplitudes of all modes according to their β values.

• Construct the final beam profile by adding up all contributions from the modes.

Note that this procedure is not computationally difficult, unless the fiber has a huge number of modes and/or complicated modes in case of an index profile which is not radially symmetric. The amount of computations does not depend on the propagation distance. (For numerical beam propagation, longer distances would generally require more time.)

There are cases where mode coupling calculations are helpful. For example, one may calculate the modes of an “undisturbed” fiber and then calculate mode coupling caused by some additional effect. For example, a periodic index modulation in a fiber Bragg grating can couple counter propagating or copropagating modes.

Limitations of the Mode Approach

The concept of modes is often very convenient for doing calculations, as shown above. However, it also has its limitations:

• In some cases – for example, for arbitrary index profiles – the modes are very difficult to compute.
• In cases with a huge number of modes, it may also not be convenient to do calculations based on them. Note: if cladding modes need to be taken into account, one would end up with a huge number of modes in the calculation, even if the core has only few guided modes.
• Additional disturbances such as bending already make the calculation of modes much more difficult.
• The concept is also at least more difficult to apply if the mode properties change along the length of a fiber – for example, for tapered fibers.

For such reasons, direct numerical beam propagation is often required instead of the mode concept.

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