waveguide as a commixic atm )\ mkoium 12(>1 



Appendix 



tiiix)i;ktu'al analysis for coxTixuors modk coxversion 



Whereas the travelling-pulse type ol' thcorclicnl analysis utilized in 

 the body of this paper can be extended to a realistic spacial distribution 

 of conversion points and to a series of modes instead of only one, J. I^ 

 Pierce suggested that the assumption of iniiform motie conversion along 

 the axis of propagation would lead to a solution in closed form and would 

 probably show the general properties being sought. 'J'his suggestion was 

 adopted and Pi is designated as the signal jjower, 1\ as the power in the 

 unused mode, and P,, as the power which has transferred from mode-a; 

 l)ack to the signal mode, mode 1. We assume (luadrature addition of 

 conversion components, and write a series of dilt'crential e(juations ex- 

 ])ressing the power flow between the modes along the axis of propagation, 

 including the heat loss effects: 



~ = -auPi - ai.Pi (9) 



dz 



dP 



V = -a^hPx - a.iPx + OixPi + auPn (10) 



dz 



dPn 



-J^ = -aihPn — CluPn + dxlPx (11) 



dz 



in which the symbols have the following definitions: 



ttih = the heat loss coefficient - mode 1 



flix = the mode conversion coefficient from mode 1 to mode x 



Qxh — the heat loss coefficient — mode x 



0x1 = the mode conversion coefficient from mode x to mode 1 

 z = distance along the axis of propagation 

 Note that the above heat loss coefficients are those associated with power 

 rather than attenuation coefficients associated with amplitudes, (2ai = 

 ciih , 2ax = ttjch)- The above equations also imply mode con\-ersion in the 

 forward direction only. 



In a phenomenological way, these eciuations represent the decay of 

 power in the signal mode and the build up of power in both the unused 

 mode X and reconverted energy P„ in mode 1. The general plan is to 

 solve these equations for P. and P„ in terms of the input wave power. 

 P„ is maintained separate mathematically from Pi , even though both 

 of them are in the same mode, so that we can clearly identify tlie energy 

 which has been at one time in the unused mode x. 



