METALLIC DELAY LENSES 81 



Resonance Effects 



It was stated earlier that the size of the elements should be small relative 

 to a half wavelength in order for the refractive index to be independent of 

 frequency. A qualitative idea of this criterion can be obtained by an 

 elementary analysis of forced oscillations of dipoles. It is known that a 

 dielectric medium which is composed of elements that resonate under the 

 action of an alternating electric field, such as atoms having bound electrons, 

 will exhibit a dielectric constant which varies with frequency :'*^ 



e. = 1 + 1 T2, (26) 



Jo — J 



where /o is the frequency of resonance of the element,/ the frequency of the 

 incident radiation and k a proportionality constant. Thus when/ is small 

 relative to/o, er is practically independent of/. 



As a means of estimating the change in refractive index with frequency of 

 metal delay lenses, we consider a specific example : Let n = 1 .50 when the 

 elements are X/4 in length, i.e. when /- = J/o, then the last term of (26) 

 equals 1.25. Decreasing / by 20% reduces n from 1.50 to 1.46. Thus the 

 change in n from midband to the edges of a =b 10% band is about .02. 

 From this, at 7 cm, the phase front, even for a lens 30'' thick, should remain 

 plane to within ±1^ wavelength over this 20% band of wavelengths. If 

 the members had been made | wavelength long, the variation in n from the 

 design frequency all the way down to D.C. would have amounted to only 



1.2%. 



Summary 



A metallic dielectric is constructed by arraying conducting elements in a 

 three-dimensional lattice structure. For electromagnetic waves whose 

 wavelength is long compared to the size and spacing of the elements, this 

 structure displays an effective dielectric constant and index of refraction 

 which is sensibly constant over wide frequency bands. Lenses can be 

 designed according to these principles which will focus microwaves and 

 longer radio waves as a glass lens focusses light waves. Such lenses have 

 the advantage of broad-band performance over the earlier waveguide type 

 metal lenses and they retain the advantages of light weight over dielectric 

 lenses. As microwave antennas, they are superior to parabolic dish reflec- 

 tors from the standpoint of warping and twisting tolerance, profile tolerance, 

 directional properties and impedance match. By eliminating the steps in 

 the lens, the directive patterns are made cleaner and an increase in absolute 



" See for example, Joos, Theoretical Physics, Blackie & Son, Book 4. Chapte- 4. 



