WAVEGUIDE AS A COMMUNICATIOX MKDIUM 1235 



^x = - («x + MZ 



k = amplitude of the converted wave for a single conversion ]ioint 

 and for unit wave incident on the conversion ])<)int. 

 In deriving the series represented by (2), (3) and (4), it is assumed that 

 all of the converted power travels in the forward direction, and that re- 

 flection effects are negligible. These conditions are tyi)ical of imperfec- 

 tions in practical multi-mode waveguides. 



When the input pulse for the idealized line is sufficiently short, the 

 various terms of (3) and (4) (representing successive conversions and 

 reconversions) are non-overlapping pulses. It is instructive to write 

 down the ratio of the signal-wave amplitude to the reconverted wave 

 ;uni)litu(l(' which is separated from the amplitude of the signal wave at 

 the same point by the time difference z{l/vx — 1/vi), in which v^ and Vi 

 are group velocities. This ratio is [ratio of (2) to the first term of (4)] 



(1 - 1^ ,_. 



(o) 



(n - l)/cV^"'~"^^^ 



It is clear from this ratio that the signal wave may be smaller than the 

 reconverted wave if n is sufficiently large. Physically, what happens is 

 that the reconverted amplitude created at each successive conversion 

 point adds in phase with the reconverted wave amplitude present at that 

 point due to previous conversions and reconversions. This liappcMis of 

 course because the line contains identicall}^ spaced conversion points. 

 ^\'ith random location of conversion points, a less severe build-up of 

 reconverted wave energy would certainly occur. 



The first and second reconverted pulses [i.e., the first and second terms 

 in (4)] are separated by the time difference 2(1 /t'x — l/vi) and the ratio 

 of the amphtude of the second to the first reconverted pulse is 



(n - 1) 



The largest amplitude existing in the unused mode at a point immedi- 

 ately following the n'^ conversion is the component converted at the n"' 

 conversion point and the ratio of this component to the amplitude of 

 the signal pulse after the n^'' conversion is: 



^ (7) 



(1 - k-y- 



Xote that this ratio is independent of n. The unu.sed-mode anijjlitude 

 converted at the first conversion point is, after n trips and relative to 



