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Take the below image for example: enter image description here Let's imagine the cladding is really just perfect mirrors, where all light energy is reflected back towards the core. I understand that modes arise by imposing the condition that round-trip (two bounces) light must accumulate 2π radians of shift.

However, clearly with perfect mirrors, the light will continue to reflect forever. In this case, what does the light look like? Do we tend to represent it as linear combinations of the modes?

Also, if the answer is "the light interferes with itself destructively", then where does the power go?

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One of your assumptions is incorrect: there is no need to have any "round-trip" phase match. All the photons (or wavefronts) just keep reflecting off the mirror surface. The reflections will always contain a vector component in the longitudinal direction (parallel to the axis of the fiber), so all the light will end up at the far end of the fiber, ignoring reflection losses, absorption in the core, etc.

So, other than phase flips of $\pi$ upon every reflection, any "mode" that successfully enters the fiber will exit unchanged.

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