1152 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



modes while the odd values correspond to the odd modes. In this section 

 we shall exclude the case m = 0, which corresponds to the principal 

 mode discussed in the preceding section. From (389) and equation (330) 

 of Section IX, we get 



KO 



T 



(391) 



The fields of the ?nth mode are given by substituting these quantities 

 into (333) and (334) if m is even, or (335) and (336) if m is odd. 



From equation (388), making use of Clogston's condition, the propaga- 

 tion constant of the wth mode of this family is given by 



7" ;^ — co>oeo + m'-K /b' = — 47rVXo + nfT /b , (392) 



where Xo is the wavelength of a free wave in the main dielectric at the 

 operating frequency. To this approximation the values of y are the 

 same as the propagation constants of the family of modes (with H^ , 

 Ey , and E^ field components only) which are possible in a dielectric 

 slab of thickness b between perfectly conducting planes. The cutoff 

 wavelength of the mth mode is 



\c = 2b/7n, (393) 



the propagation constant being real, to the present approximation, 

 if Xo > X,. and pure imaginary if Xo < Xc . We see that the cutoff fre- 

 quency is inversely proportional to the width of the main dielectric, 

 so that this family of modes is not possible in a completel}^ filled Clog- 

 ston 2. 



It is worth noting that the effective skin depth of the stacks for the 

 mth mode is, from (390), 



A = 



Rer, 



ITT y g Xo K ^1^9 



If the mode is just above cutoff, then A is of the order of magnitude of 

 5i (= \/2/w/ii6ri), but as co increases indefinite!}^ A also increases in- 

 definitely, for the ideal stack of infinitesimally thin laminae. The physi- 

 cal explanation is simple: When the mode is near cutoff the phase 

 velocity is very high, but as the frequency is increased the phase velocity 

 approaches the velocity of a free wave in the main dielectric, for which 

 the effective skin depth of the stacks was designed by Clogston's condi- 



