Instrumentation & Measurement Magazine 25-1 - 73

Fig. 5. A focusing MS was designed, resulting the susceptibilities shown in: (a) in (b)  and (c) The transmission coefficient. The MS was then mm ;
xx
yy ;
ee
illuminated with three different incident fields: (d) Ideal illumination; (e) Horn illumination; and (f) Superposition, N = 8.
transmission. We desire the transmitted field to be a cylindrical
wave, given at the point ,xzr
by:
EE Hk rr
1

t,y
r 
HE0
t,x
r 
HE0
t,z
r  
where H are Hankel functions of the first kind, represent
1
0,1
ing
inward propagating cylindrical waves, and the fields have
been scaled such that 
EE . Meanwhile, the incident
t,y
0,0
February 2022
field is a normally incident plane wave normalized such that
0f

Hk r
1
jz z H k rr
1
 
1
 Hk

rr
jx x H k 
1
1
0f
0f

f1 f



 Hk

rr
0f

f1 f

rr

(5a)
Ei,y = E0 andHE  
i,x
0 / , and furthermore, we desire there to
be no reflection, i.e., a matched lens. For this field transformation,
suitable susceptibilities are [16]:
,0
,0
 
x
ee
mm x
(5b)
yy xx
yy
t ,
i,
t ,
i,
 
xx yy
xx
t ,


j E x Ex,0
f H x Hx,0
t ,
,0
,0
i,
i,
  
  
 
 
j H x Hx,0
 f E x Ex,0

 
 
 
 
 
(6a)
(6b)
(5c)  mm x
quired MS is passive.
yy x
ee


≈ 1 and ∠Ei
Fig. 5a and Fig. 5b show these synthesized susceptibilities
at f = 30 GHz, with  xz  0 cm,10 cm. Note that
ff,
0 and  0, which indicates that the rexx
These
susceptibilities correspond to the transmission coefficient
plotted in Fig. 5c. Next, the MS is illuminated using
a field having Ei
signed for. (More precisely, a Gaussian beam with w0
and beam waist at z = 0 is used, with negligible error
IEEE Instrumentation & Measurement Magazine
= 0°; i.e., what the MS was de=
15 cm
73

Instrumentation & Measurement Magazine 25-1

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