462 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



the TEio mode. The sample extends a distance x into the cavity, x being 

 assumed to be small in comparison to the wavelength so that Hi does 

 not vary appreciably over the sample. The additional losses due to the 

 sample will then be proportional to 



A £ Ei sin' C^^\ dx ^ Cx' 



whereas the volume of the sample is proportional to x. The observed 

 voltage change AF is then given by 



AF ex Qo'F. oc /- — ^ \ {x) 



where Qo is the Q of the cavity without sample, and Qq the total Q of 

 both cavity and sample. Maximizing the voltage change AF we get 



^ = .-. C = 1 



dx 2Qo (30) 



.'• Qo = 3Q0 



Equation (30) tells us that we should load the cavity with the sample 

 until the Qo is reduced to § of its original value. It should be pointed out 

 that we optimized the signal and not the more important quantity, the 

 signal-to-noise ratio. Therefore the analysis is only valid as long as the 

 noise is not a function of Qo , (see Section YIB) and that we do not 

 saturate the sample (see Section VIII). If either condition does not 

 hold the amount of sample to be put in should exceed the above calcu- 

 lated value. 



b. Losses Proportional to Hi 



The losses do not vary along the sample so that we may write 



/ 1 



AFccQoFs°^ 



which clearly has no maximum for x. One should therefore put as big a 

 sample into the cavity as possible (compatible with the assumption that 

 if it be small in comparison to a wavelength), the same result as if one 

 had no losses at all. 



B. Noise Due to Frequency Instabilities 



Before considering the signal-to-noise ratio for specific sj^stems we 

 will investigate a noise source which is common to all of them. It arises 



