718 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



of a multi-mode waveguide will not cause an appreciable exchange of 

 power between modes provided that the quantity ((81 — /32)/c is suffici- 

 ently large for the modes which are coupled by the boundary change. 

 Similarly, for disturbances in the coupled-wave system which takes place 

 OA'er a large number of wavelengths in the direction of propagation, the 

 coupled-wave theory indicates that all conversion will take place in the 

 forward direction and very little reflection in any mode will result. 



The tight coupling theory shows that for the case of identical complex 

 propagation constants, a periodic exchange of energy between waves 

 takes place along the coordinate of propagation. The only effect of the 

 existence of an attenuation constant for both waves (compared to the 

 dissipationless case) is to add the same exponential attenuation factor 

 (to the periodic energy exchange phenomenon) which would have existed 

 for a wave traveling on one of the lines in the uncoupled state. 



When the phase constants of the two coupled waves are not equal (and 

 the attenuation constants are either equal or negligibly small compared 

 to the coupling coefficient), the exchange of energy between waves is no 

 longer complete but remains periodic (Fig. 17). The quantity (0i — ^2)/c 

 determines the fraction of the total energy Avhich is exchanged, and also 

 modifies the period of the energy exchange phenomenon along the axis 

 of propagation. 



When the phase constants of the two lines are equal but the attenua- 

 tion constants are unequal, the energy transfer phenomenon differs only 

 slightly from that associated with equal propagation constants pro\'ided 

 that the quantity (ai — a2)/c is less than about —0.1. For (ai — a-2)/c 

 more negative than about —1, the periodicit}' of the energy transfer 

 phenomenon has largely disappeared (Fig. 23) and as (ai — a2)/c be- 

 comes on the order of — 10 or more, the principal effect of the coupling 

 for the low loss line is a minor alteration of the phase and attenuation 

 constants. The wave amplitude for unit input on the low-loss line be- 

 comes [from (33) for | (ai — 0:2) \/c » 1] 



7^ _ —lai—c-l(.ai—a2)+iic+0)]x (A^\ 



Through proper choice of the phase constants relative to the coupling 

 coefficient in two coupled transmission lines, it is possible to make di- 

 rectional couplers having an arbitrary transfer loss that is independent 

 of frequency despite ^'ariations in coupling strength with frequenc}^ 

 (equations 43-44). It was also shown that the coupled- wave approach 

 may be utilized to create highly freciuencj^-selective filters which may 

 operate between single-mode media or between selected individual modes 

 of a multi-mode system. 



