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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, ...

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Niels Nielsen's answer is correct. Just to provide some calculations as to actual propagation speeds (not signal bandwidth): Typical optical fibers have $n\approxeq 1.5$ , so the propagation speed is $\frac{c}{1.5}$ , or roughly 2E8 m/s . This is the field, or photon, speed. Using the formula from Wikipedia, the speed of electromagnetic waves in a good ...

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Fiber cables have higher data transfer speed not because light travels faster in a fiber than electromagnetic waves in a copper wire, but because they can be modulated by signals having far higher frequencies than can be managed in the case of copper wires.

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