1224 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



where the symbols with double subscripts are defined by (28), (31), and 

 (32). 



1.3 Coupling Coefficients Involving the TEoi Mode 



I 



In Part II we shall consider a well-compensated bend in which the 

 total power in all spurious modes is everywhere low compared to the 

 power level of TEoi . (This is somewhat more restrictive than merely 

 assuming that the power in any one spurious mode is everywhere small 

 compared to the power in TEoi •) To first order, therefore, we may com- 

 pute the power abstracted from TEoi by mode conversion by assuming 

 that the TEm mode crosstalks into each spurious mode independently. 

 For this calculation we need the values of the forward coupling coeffi- 

 cients between TEoi and all other modes. The crosstalk into backward | 

 modes will be negligible in all practical cases. 



We shall use the customary double-subscript notation for the various i 

 modes in a round guide, but shall continue to denote TM waves with ; 

 parentheses and TE waves with brackets. We assume that the distribution j 

 of dielectric is symmetric with respect to the plane of the bend, so that TEqi j 

 is coupled to a definite polarization of each spurious mode. The normal- ; 

 ized T-f unctions are then: j 



J, ^ /en Jn{X(nm)p) SlTl nxp 



V TT n'(nm)J n-l{rC{nm)) 



Jn{x[nm]p) COS Hif 



(42) 



where 



and 



[nm]" n~)^Jn\K[nm]) 



k{nm) — X(nm)tt, JnV^\nm)) — 0, 



K[nm] = Xlnm](l, J n \k[nm]) — 0, 



1, n = 0, 



(43); 



(44) 



2, n ^ 0. 



1.3.1 Coupling Coefficients due to Curvature 



We know that to first order the curvature of the guide can couple the 

 TEoi mode only to modes of angular index unity. Let us calculate the 



