1238 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



ficulty from (50), (51), and (98). The first few coefficients are: 



dai) = -0.18454/3a/6, 



0.18638(/3a)Vb 



\/h[oi]ah[n]a ' 



(99) 

 _ _0.31150(^a)V6 



d[u] — 



\/h[oi]ah[i2]a 



0.02751 (^a)Vb 

 \/h[oi]ah[i3]a 



Some numerical values are recorded in Table I. 



The phase constant of any mode in the compensated bend is equal 

 to the phase constant of the same mode in a guide filled with material 

 of constant permittivity eo . We may therefore set /3[oi] — ^[im] equal 

 to /i[oi] — h[im] . Strictly speaking, X is then the wavelength of a free wave 

 in a medium of permittivity eo ; but eo differs little from the permittivity 

 of vacuum if the compensator is made from a foam dielectric. The ratio 

 of total coupling coefficient to difference in phase constants, namely 

 2(c + d)/{ho — hi), which determines the maximum power transfer by 

 (97), is given in Table I for the |-inch and 2-inch guides at X = 5.4 mm. 



For large I3a the leading terms in C[i„,] and rf[i„,] , which are proportional 

 to /Sa, cancel each other, and C[im] + d^m] decreases like 1/jSa. The differ- 

 ence in phase constants, /i[oi] — /i[im] , is also proportional to l//3a for 

 large ^a. Hence the ratio of coupling coefficient to difference in phase 

 constants approaches a finite limit as l3a approaches infinity; to three 

 decimal places the limiting values are the same as those given in Table I 

 for ^a = 29.554. 



If we choose a sufficiently large value of a/h, the maximum power 

 transferred to a given mode may be made to approach any preassigned 

 small value as X/a approaches zero. This is a special property of the 

 geometrical optics compensator. For other compensator designs c -\- d will 

 be of the order of jSa while ho — hi will be of the order of l/|8a, so that 

 2(c + d)/{ho - hi) will tend to infinity like ((3a)^. Hence in the short- 

 wavelength hmit the bend output will be a jumble of modes all at com- 

 parable power levels. Practically, one must expect the same end result 

 with a geometrical optics compensator, because of the impossibility of 

 meeting exact mathematical specifications; but a carefully-designed 

 geometrical optics compensator should work satisfactorily up to a con- 

 siderably higher frequency than any other type. 



