Transverse mode output from a multimode fiber using TEM00 laser source?

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babreopti
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If a TEM00 output laser source (regarding both cases of single frequency vs multifrequency) is used as input into a multimode optical fiber, is it possible for the output of the optical fiber to be multimode?
 
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I think that it's inevitable- google "modal dispersion". Because a multimode fiber supports many transverse modes, energy will leak from the TEM00 to higher modes as the beam propagates due to mode coupling which varies as the square root of the length.
 
Actually that is one of the labs for telecom students: what happens if we mix single- and multi-mode fibres on a path between two singlemode telco devices running with 40 GHz clocks (using amplitude modulation, NRZ-OOK, so the frequency if modulation is equal to bit rate)?
Simple answer is: it works if the MM fiber is pretty short (tens of meters) - just signal is attenuated due to different apertures of core fibres; but if the MM segment is longer, modal dispersion obscures the signal to make it unreadable.
Full answer is: watch the output signal on the oscilloscope to see the dispersion.