308 



BELL SYSTEM TECHNICAL JOURNAL 



In our work we shall consider only those modes corresponding to a fixed 

 value of m (or of n) and the order is almost automatically fixed. 



The factors which determine the propagation of the typical terms in the 

 summations (1.1-3) and (1.1-4) for A and B are 



Ctmniz) = gin ^^" "*", ^mn{z) = fi« 6 '^ '"" (1.1^9) 



The column matrices obtained by arranging these quantities in the same 

 order as in (1.1-8) will be denoted by a{z) and 18(2). We may write 



^(3) = e~'^' d+ 



a{z) = .-^"« g+, 



(1.1-10) 



whereexp(— zFa) andexp(— zr^) are square matrices defined by power series 

 each term of which is a square matrix: 



,-zT 



7-51 + ^-^ 

 1! ^ 2! 



3! 



+ 



(1.1-11) 



/ is the unit matrix and F^ is the diagonal matrix" 



r« 



Tio 





















 



(1.1-12) 



in which the order of the diagonal elements is the same as the order of the 

 elements in the column matrix g^. Similarly F^ is a diagonal matrix whose 

 elements are Foi, F02, Fu, F03, • • • , the order being fixed by ^+. When F is 

 replaced by Fa in (1.1-11) it is easy to obtain Fa, Fa, etc. and sum the 

 resulting series to obtain 



e-"' = 



■zTu 



(1.1-13) 



A similar expression exists for exp(— sF^). The expression (1.1-10) for 

 a{z) is seen to be true when the square matrix (1.1-13) is multiplied, by 

 matrix multiplication, into the colurnn g+. 



It turns out that the field in a circular bend (in a rectangular guide) may 

 be represented by a generalization of the foregoing expressions. In this 

 generalization, which will be studied in the following sections, the square 

 matrices Ya and IV no longer have the simple form of diagonal matrices. 



* That is, a square matrix in which all of the elements other than those in the principal 

 diagonal are zero. 



