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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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.