482 BELL SYSTEM TECHNICAL JOURNAL 



SO that the total flow of light through the aperture at any position x is 



F,{x) = fm)E^(x + k)d^. (16) 



•^aperture * 



Since x is a constant with respect to the integration the exponential 

 term may be factored and the part involving x only may be brought 

 outside the integral sign. This gives 



+00 



Fi{x) = E Y{n)Anexpiir{nx/L), (17) 



n= — 00 



where 



Y{n) = fn^) exp *7r(w^/L)^^ (17') 



^ aperture 



For a symmetrical aperture (that is, about the origin of |) 



Y{n) = ^T{k) cos {irn^/L)d^ (17") 



•''aperture 



and Y{n) in this case is, therefore, a pure real quantity. 



The important conclusion to be drawn from equation (17) as to the 

 effect of a finite transmitting aperture is that it multiplies the complex 

 amplitude ^„ of each original image component by a quantity Y{n) 

 which is independent of the picture being scanned. This is entirely 

 similar to the effect of a linear electrical network in a circuit, and the 

 quantity Y{n) is quite analogous to the transfer admittance of that 

 network. 



The quantity Y{n) has been plotted for variously shaped apertures 

 in Fig. 12. For convenience in comparison, the ordinates of each curve 

 have been multiplied by a numerical factor to make F(0) = 1. The 

 curves show the characteristics that are by this time familiar, which 

 are that the effect of the finite size of the scanning aperture in the 

 transmitter is similar to that of introducing a low-pass filter in the 

 circuit, namely, cutting down the amplitudes of the signal components 

 for which n is numerically high, i.e., the high-frequency components. 



The curves are remarkable, however, in that in the useful frequency 

 band (i.e. from w = to something like half of the first root of Y{n) 

 = 0) all the distributions considered give practically the same transfer 

 admittance if the dimensions of the beam along the direction of scanning 

 are suitably chosen, as has been done in the figure. This results from 



* The integral is mathematically taken from — co to -|- oo but the regions outside 

 the aperture give no contribution since the integrand is there equal to zero. 



