WAVEGUIDE MODES IN GYROMAGNETIC MEDIA 423 



nizing that pi = m~'P2 , we have from (30) 



^ J/(pi) + n -^-^ = * (31) 



K Pi 



where pi = ikzyT^R. 

 Equation (21) may be modified by a recurrence relationship to become 



tj^^^P^JpM (32) 



For /x > the quantity pi is a pure imaginary for large real values of 

 A-j . Since the n order Bessel function is monotonic in imaginary argu- 

 ments and possesses the multiplier (i)", the right-hand side is negative 

 for n positive. For n > 0, propagation occurs for 



\k\ > \n\ 

 sgn K = — sgn fji 



Inspection of (31) reveals that a reversal of the sign of n is equivalent 

 to reversing the sign of k. This conforms to the physical situation in 

 which reversal of the sense of circular polarization is equivalent to the 

 reversal of magnetic field. Thus for n < and n > 0, 



\ k\ > \ n\ 



Sgn K = sgn /Lt 



We find, from the above arguments, that just one sense of circular 

 polarization propagates in an undersized circular guide for n > and for 

 a given direction of the magnetic field. It will be demonstrated shortly 

 that propagation occurs f or ju < 0, but with an entirely different struc- 

 ture of modes. The right-hand side of (32) is monotonic as a function of 

 p for ^t > 0, leading to only one solution for each value of n. This will 

 not be the case for n < 0. 



It is of interest first, however, to observe the limiting approach to 

 M = in the region of ^ > 0. The right-hand side of (32) is finite for 

 finite imaginary values of pi , so that the only solution as /x approaches 

 zero is that for which the magnitude of pi becomes infinitely great. The 

 Bessel function is asymptotically expansible as a cosine divided by a 

 square root of its argument. Thus 



Jn(pi) = - a/- (f'('" + f^"+^>^'^^''» + ^-i(Pl+12n+l](./4))) (33) 



2 V Pi 



* Equation (31) maj' likewise be obtained from the small radius limit in (34) 

 of Reference 2. 



