FERRITES IN MICROWAVE APPLICATIONS 1349 



the original mode. When the above assumptions are valid the plane wave 

 theory can be extended easily to explain loaded waveguide behavior. In 

 the plane wave case it was shown that a negative effective permeability 

 results in attenuation of the positive circularly polarized wave. Quite a 

 different result is observed in waveguides containing very small cylinders 

 of ferrite. It appears that if the cylinder is small enough n(.i lo art as a 

 dielectric waveguide then the negative permeability inside the rod simply 

 is averaged with the permeability of the surrounding rof-ion so that \\w 

 rotation curve (which depends upon the difference between the s(|iiare 

 roots of the permeabilities seen by the two circular components of the 

 wave) follows the dispersion curve of the permeability of the pu.siiive 

 circularly polarized wave, even following the pattern of the permeability 

 when it is negative. 



In Fig. 8 are shown measured curves of the rotation of the wave and 

 the absorption of the positive circularly polarized component of the 

 wave as functions of applied dc field for comparison with Figs. 1 and 2. 

 To amplify our arguments we point out that the propagation constant 

 in a waveguide containing a very small pencil of ferrite is of the form: 



where /3e is the propagation constant of the empty guide 



ri is the radius of the ferrite cylinder 



ro is the radius of the waveguide 



€ is the dielectric constant of the ferrite 

 and juj. are the effective complex permeabilities (n db k) 



A is a constant = 3.2 

 We see that the expression within the brackets is finite for all values of 

 M± except iLi4. = — 1 + jO. Accordingly if the damping parameter (or 

 line width) is large enough to insure that fi+ can never take on this value, 

 there will always be a cylinder radius, r', for which the perturbation term 

 is small relative to /?/. However, the cylinder diameter does not have to 

 be very large before the above arguments fail and the rotation and loss 

 behavior become quite different. 



The cutoff wavelength of the TEn mode in round guide is 



Xe = 0.1708 d Vm^ 



where d is the diameter in centimeters of the waveguide and ^ and € are 

 the effective relative permeability and dielectric constant of the medium 

 contained therein. When a cylinder of ferrite is placed at the center of 



