/’
Sensor and Simulation Notes
Note 352
December
1992
Uniform Isotropic Dielectric Equal-Time
for Matching Combinations
Lenses
of Plane and
Spherical Waves
Carl E. Baum
and
tJoseph J. Sadler
Phillips
Laboratory
and
Alexander
P. Stone
PhiIIips Laboratory
and
University
of New lMexico
Abstract
This paper
considers
the general
design
of dielectric
lenses for transitioning
between
t’arious sphericaI waves. inciuding plane waves as special cases. such as one may wish to have
in antenna
and transmission
systems for fast-rising
sphericaI and plane waves through a single boundary
or one sheet of a hyperboloid
of two sheets.
is a fourth
equation.
order polynomial
surfaces these solutions
can be appIied
electromagnetic
pulses. For tran.sitioning
surface, this surface is a prolate spheroid
For the case of two spherical
For multiple
media with two (or more)
to each boundary
interest.
1
waves this surface
to generate
boundary
various lens designs of
1
.
1
Introduction
In transitioning
between various spherical
which the propagation
speeds (in general different)
trol the arrival times on the wavefronts
wavefronts,
including
application
in TEM transmission
plane wavefronts
nas such as IRAs (impulse
with very fast risetimes
general
in two or more media are used to con-
of interest.
as limiting
In uniform
radiating
antennas)
media spherical
Such lenses have
[1]. In these cases one requires
to cross-section
The problem
with a minimum
in two uniform
dielectric
cases, are of interest.
of distortion
dielectric
transit
times) be treated
is then one of mat thing
as waves (in
these waves from one
point of view, matching
media along their common boundary.
some reflection in general at the boundary,
that pulses
and reflection.
In this paper we treat the lens from an equal-time
wavefronts
involves a lens in
systems such as coaxial waveguides (cables) [2], and anten-
(compared
three dimensional).
region to another
waves, a general technique
from spherical
This introduces
and so such lenses are not ‘tperfect” transitioning
devices for TEM waves in the sense of [5]. However, in some cases the reflections
small and the wave passing into the second medium
dispersionless
TEM wave.
In a more detailed
consider the paths of the conductors
and the resulting
of the waves.
transmission-line
can usefully approximate
consideration,
the desired
such as in [2] one needs to
(for the desired TEM wave) through
impedances
can be
the lens medium
in the various media to optimize the matching
a
r
*
2
Diverging
Spherical
Wave
to Plane Wave
9
In [2] the general problem
from adiverging
spherical
ofgreater
permittivity)
involved a coaxial guiding system.
lens shape is more general,
considerations
a plane wave (in a medium
wave (inamedium
There the specific problem
impedance
of launching
As in Fig. 2.1 we have a spherical
structure.
wave propagating
requirement
outward
plane (z=
—
.
—.
+ /62 [2= – 26] =
=
z coordinate
from
= (0,%)
V6[%bo
has been evaluated
of the intersection
where 1 is the convenient
= rb Sh(db)
to some aperture
<%rb
where the constant
and not the
(2.1)
we have
Zb – Zlo = rb Cos(ob)} Wb
The equal-time
requirement,
Here, the result is extended.
(~,z)
On the lens boundary
h= been considered.
However, the result for the
as it relies only on the equal-time
for the transmission
of low permitiivity)
–
(2.2)
,za) is
constant
-%]+
by considering
of the lens boundary
~[%z–
Zbo
1
(2.3)
the ray along the z axis, ZbObeing the
with the z axis. Rearranging
scaling length as in [2].
=
~sin-l(dt,)
= [Zb j ‘bo +1] cos-l(db)
3
(2.5)
Yb
4?
APERTURE
PLANE
/
q
— = 2.26
Ez
LENS
BOUNDARY
f
RAY PATH
~29 Po
–1.261
–1:0
Obm: 48.3°
i’
A. Prolate spheroidal boundary
Y?b
t
/
/
/
LENS
BOUNDARY
●
/
4SYMPTOT[C
CONE
//
/
/
\
/
/
“
/
/
/
/
/
/
I
I APERTURE
I PLANE
I
I
RAY PATH [
#--
i
I
I
I
I
-1.0
4.3995
B. Hyperboloid-of-two-sheets
Fig 2.1:
boundary
Lens Boundary for Converting Expanding
Sphericat Wave into Plane Wave
4’
&
e
I
6
e
Converting
to cylindrical
coordinates
(2.2) gives
(;)2=
(2.6)
+’12
(?)2+[’’;”0
which with (2.5) gives
+
(:)2[
+,]2=
%;z’o
as the equation
A.
(2.7)
for the lens boundary.
Case of Cl > Ez
This is the case considered
in [2]. It corresponds
(?)’++
This is the equation
of a prolate
1
It reaches maximum
cylindrical
a -
[4]. It interects
(2.7) as
[2’;2’012=0
(2.8)
[1-:1
spheroid
z— Zbo
to Fig, 2.1A, Rearranging
+
fjzb;zbo
o
+,]2
[$jzb;zbo
the z axis at
-1
=0,–21+
[
~
(1 Cl
(2.9)
radius W~~~=at
major radius
(2.10)
-
b ~ minor radius
Another
common parameter
of a prolate
spheroid
is the eccentricity
(2.11)
5
It has foci on the z axis at
–a[l + a]
,2— Zbo=
1
–a[l – a]
= –1
1–
= –
Q
[ 61
1+
(2.12)
?
;
{
the first of these being the Iaunch point for the spherical wave, and the second being a focus
for the reflection
As indicated
(ideally small) of the spherical wave from the lens boundary.
in Fig. 2.1A one need use only a certain portion
the lens. In particular
the lens boundary
for z – ZbO< –a where the lens radius is maximum
is not useful for focusing in the forward (+z) direction.
on the boundary
of the prolate spheroid for
In an angular sense 191is maximum
at
‘-4[:-’1’)
(2.13)
I$$m==
There are some limiting
cases. For el /cz near 1 we have
(2.14)
m
For large Cl/e2 we have
(2.15)
which is a sphere.
B.
Case of e2 > Cl
Now consider the converse case in which one is launching
permittivity
medium into a plane wave in a high permittivity
Fig. 2.lB. Rearrange
the spherical
medium.
(2.7) as
=
[:-’]{[’b;zb”
+[~+q-’
6
wave from a low
This corresponds
to
b
.
e
This is the equation
of two sheets [4]. Here we consider only the sheet in
of a hyperboloid
the positive z direction.
This body of revolution
intersection
about
being as in (2.9)).
the z axis at ,z~, (the other sheet
the z axis intersects
z this surface is asymptotic
For large positive
to a circular
+[fi+,]-’+~((zb;
“b — “b.
”bo)-’ ~
,
(2.17)
This cone has apex on the .z axis at ZCgiven by
II Q+ll
Lv~I”J
——
to (2.10)) with a half-cone angle
([
6C= arctan
●
(2.18)
4
1
(comparable
1-1
r-
~=— zb.
.-—
as indicated
1
5
(2.19)
Q – 1
Cl
1)
in Fig. 2.1 B.
There are some limiting
cases. For cz/el near 1 we have
Zc — .“bo
+–~,0c+Oas2
[
El
+
(2.20)
1+
For large e2/el we have
Zc — .“bo
1
which is a plane of constant
C.
+0,
‘ii’
Oc+—as
2
Q+cc
(2.21)
cl
z(= “bO).
Some Comments
As a practical
of permittivity
a constant
matter
one is often concerned
Co (free space).
(frequency-independent
For a transient
dispersionless
a plane wave into a medium
wave in medium
and lossless) Cl which is physically
larger than Coso that the frequency-independent
of light. Such a situation
with launching
propagation
implies Case A, the prolate spheroidal
7
1 we need
then required
to be
speed is less than c, the speed
lens. For special cases where
,
one wishes to launch a plane wave into a medium
water, etc.) then the hyperbo~oidal
The reader
can note that
plane wave is propagating
boundary
in medium
it k converted
of permittivity
C2 > co (such as earth,
lens may be of interest.
the wave-propagation
in medium
e
can be reversed.
2 in the –z direction.
Then in Fig.
On passing through
2.1 a
the lens
into a spherical wave converging (focusing) to z = ZIOon the z axis
1.
8
3
Plane Wave
to Diverging
Spherical
Wave
9
Now consider
the complementary
diverging spherical
the spherical
wave from anincoting
wave in medium
F’=
which is actually
boundary
problem
outward
(r,, O,) = (0,0,),
located in medium
of Section
plane wave asillustrated
2 is propagating
&
(to that
1. Matching
2), the launching
of a
in Fig. 3.1. In this case
from
(W,,z) = (0,2,.)
(3.1)
the plane and spherical
waves on the lens
we have
(3.2)
~b – .%, = rti cos(6’b), ~b = rb sin(~b)
The equal-time
requirement
is now from a source plane
(z = z.) to an aperture
sphere
(rz = r.) which gives
●
= ~[zbo
where the constant
z coordinate
is evaluated
of the intersection
– z.] +
by considering
-&[r.
- (Zbo -
the ray along the z axis, zb, again being the
of the lens boundary
firb
– fi[zb
(3.3)
22.)]
with the z axis. Rearranging
- .z~o]= @
t - zbO– ,z~O
where 1 is now redefined
the lens boundary
Comparing
subscripts
(compared
and the virtual
(3.4)
to Section 2) as the distance
focal point of the spherical
(3.4) to (2.4), note that
1 and 2. With this relationship
(along the z axis) between
wave.
they are the same except
then (2.7) is rewritten
for the interchange
for the present
of
case as
(3.5)
where the only difference is the inversion of the ratio of the permittivities.
9
9
Y?b
!
//
CONE
El
—
G
/-/
= 2.26
/
\
&29HtJ
/
I
i
0.6005
1
I
*Zb-
1.0
A. Hyperboloid-of-two-sheets boundary
Y?b
4?
\
\
\
\
E,
\
= 2.26
El
i1
\
\
“DAWTA
~o:~~8”30sl=L
I
Bc
i
I,/
I/
I
1.01.201
.
z
b-
b.
B. Prolate spheroidal boundary
Fig 3.1: Lens Boundary for Converting Plane Wave
into Expanding Spherical Wave
10
a
●
A.
Case of Cl > e2
This case is illustrated
in (3.5) as compared
(as in Fig. 2.lB).
in Fig. 3.1A. Note that, due to the interchange
to Section 2, the corresponding
The corresponding
equation
boundary
of the permittivities
is a hyperboloid
of two sheets
is
(32=[VI{[-+[6+T1]2-[E+
n-2}
““’)
where here we consider only the sheet in the positive z direction.
Again this sheet is asymp-
totic to a circular cone with apex at z. given by
ZC— Zbo
–1
:+1
=—
f
[[
with a half cone angle
/3.
= arctan
In a similar manner
a
all the formulae
1
(3.7)
~
(3.8)
~ – 1 2
([ C2
1)
of Section 2B can be simply converted
to the present
case.
B.
Case of e2 > Cl
This case is illustrated
the permittivities
in Fig.
3.2B. Again due to the interchange
in the lens-boundary
equation
this is a prolate
(from Section 2) in
spheroid
given by
(3.9)
It intersects
the z axis at
z— ~bo
–1
=,
~
–21+
1
The maximum
[fl
value of 132on the boundary
~
’52
is
=arctan([~-1]’)
@bm==
In a similar manner
all the formulae
of Section 2A can be simply converted
case.
●
(3.10)
11
(3.11)
to the present
C,
Some Comments
Note that in both cases illustrated
range of angles 62 centered on the z axis. In particular
for finite permittivities
the diverging rays do not fill a half space (with boundary
Again the wave-propagation
propagating
●
in Fig. 3.2 the diverging spherical wave has a restricted
direction
can be reversed.
as a plane of constant
A spherical
& < n/2, i.e.
z).
wave in medium
toward ,220on the .z axis is converted into a plane wave in medium 1 propagating
in the —z direction.
12
2
4
9
Diverging
Spherical
Wave
case, as indicated
in Fig.
to Diverging
Spherical
Wave
A. General Considerations
The more general
propagating
outward
4.1, has a spherical
wave in medium
1
from
F, = Zl,i,,
(~1, dl) = (O,%),
(Q, z) =
(UZIO)
/1= /?&– zlo
There is also a spherical
wave propagating
(4.1)
in medium 2 outward
from a position
in medium
las
F’2 = Z2J’2,
&
e
=
(T2,
26, ,—
02)
=
(0,
@2),
(~> z) = (0?
(4.2)
.220
Here, as before, z~o is the intersection
of the lens boundary
a subscript
to the lens boundary.
b on coordinates
applying
40)
with the z axis (symmetry
Introduce
a convenient
axis),
scaling
length as
L-j
= [&
which follows the usual focal-length
Relating
the coordinates
(4.3)
formula.
on the boundary
zb –
+ .f?;2
‘1
1
we have
210 = T’IbCOS(6j6) =
Zb
zbo + 41
(4.4)
These are manipulated
a
into the form
13
Y?b
LENS
BOUNDARY
I
//
-
4?
o
Fig 4.1: Two Diverging Spherical Waves
14
,
,
Zb — .?$o= rlb Cos(dlb) –/1
=
For the equal-time
condition
~b d(6~b)
= r2, ccIs(192,)
– l?~= ~b d(6jb)
(4.5)
– &
we have the time from ZIO~Z to the lens boundary
as
(4.6)
and from there to the spherical
wavefront of radius r2 (centered
t2=
–
@[7-2b
on 220~Z) as
(4.7)
r2]
Then we set
to
= -tI + t2 =
= -r,,+
●
= @ll
where
the last form
is found
by evaluating
constant
(for fixed
@[r2
- r.,]
+ @[r2
r2)
(4.8)
- ~2]
the time along the z axis as the ray path.
So we
have
(4.9)
as
the equal-time
subscripts
equation
condition.
Note
the symmetry
in this equation
with the interchange
of the
1 and 2. By clearing the square roots one finds that this is in general a quartic
(including
For convenience
permittivities
to ~
fourth powers of coordinates).
(4.9) can be put in parametric
form by normalizing
distances
to 10 and
as
n = 1, 2, X s
parameter
15
(4.10)
●
Varying x generat$s
coordinate
ll//.)
the lens surface, noting that for each x there are two equations
unknowns
(26 – z~O)/10 and IUb/lo. One still needs to specify cl/62 and 11/12 (or
to give a particular
lens surface. Note the particular
26 =26.,
To simplify the computation
~b=o,
‘z
~b
point common to all Iens surfaces
x=()
(4.11)
of the surface shape rewrite (4.10) as
1
Then eliminate
with
{[‘b–260
+/.]2
+ W:}*
(4.12)
as
‘$;‘t:
=
&{[zb
– Zbo +~2]2
– [Zb ‘Zbo
+fI]2}
(4,13)
where we also have
These combine to give
Note the special cases
(4.16)
Zb = Zbo
‘= :::[[:]’-[:]’]-l
{
16
~[:]’-2[;l’]
e
e
Then from (4. 12) we have
(4.17)
Since this is the same for both values of n we can average them (for greater
the expression)
symmetry
in
giving
Zb
2 &+/~zb-zbo
—Zbo
—[1
& – & /0
At this point with (4.15) we have explicit representations
for ~b and ,?!b– z~, in
Now we can see that in general ~b has terms ranging from x through
x and X2 terms.
Carrying
~b2
the substitution
a little further
(4.18)
of x.
kITTM
X4, while .Zb– zb, has
gives
.&)
~[21’-:[:l’]x2+-lo;:1:/,)
[[:l+-[:]*]X
/()
Cl
z ‘/’–/1
[1
~b
—Zbo
(4.19)
[1
2
—
.&
with the x term explicit.
Since W? must be positive then sign of x depends on the coefficient
of x in the above equation.
root exists
(in
This information
can be combined
which case the lens surface crosses the z – .ZbOplane).
one can carry the analysis further
to eliminate
and Zb. This is found in Appendix
A.
Assuming
with (4. 16) to see if the X2
x and obtain
With (4.15) and (4.19)
a quartic
equation
relating
~b
that the coefficients of x in (4,15) and (4.19) are non zero, then we have
~bibo
[%]2=~[:l’-$[:l’]-’%$
[[:l[:l’]’]
as x + O
+O(x)x
+O(x)
as
x -+ O
17
(4.20)
If this expression
is positive
thelens
to the left (near the z axis). Fitting
surface is concave to the right and if negative
(4.21)
/, - ‘E[$wi]-’[[:r-[ :l’]
be a parameter,
*
this to a sphere, the center is at z.r, with
2= — Zbo
To illustrate
concave
this solution consider a special permittivity
ratio ez/el = 2.26 and let &/ll
giving the family of lens surfaces in Fig, 4.2.
B. Special Spherical Lens Surface
As a special case choose
fit,
which can be described
respective
as equal electrical distances
medium permittivities.
[
This is conveniently
(4.22)
= &l?,
to the two wave centers in terms of the
gives
Then (4.9)
‘b
;Izb”
+112 + [212=
[’b;,””
+’12+
[32
a quadratic
equation
which can be rearranged
[[zb
– Z~o] + &]2
+ ~;=&
(4.23)
as
(4.24)
which is a sphere of radius to centered on –lor= as in Fig. 4.3.
Noting that
(4.25)
then only one parameter
ratio needs to be specified to constrain
the relative permeabilities
and relative wave centers (foci). Note that
(4.26)
18
‘lo
4’2
\—
‘9
3.0
=
11
8
LENS
BOUNDARY
‘7
6
0
FLAT
/
\
4
1.25-—
/
/
/
/’
//
3
.
CONVERGING >\\
LENS
\
\
/
\’
2
1.0//“C-Y
/
/-/
DIVERGING
LENS
/
x
/
ml
i
f
I
!
1
-6
-5
i
I
i
1
1
–4
–3
–2
–1
)
Fig 4.2: Lens Surface for Fixed Permittivity Ratio
19
Y?
f!o
_b
‘2
LENS ~
BOUNDARY
t,
—=
/’
~
/
q
/
/
/
/
–3
.5
–1
–2
Fig 4.3: Special Spherical Lens Surface
20’
‘
(4.27)
c.Maximally
Another
Flat Lens Surface
special
representation
case is found
e
of x in (4.15) to zero in the
to X2 near the z axis, a case which can be referred
With this constraint
flat.
the relative permittivities
Now the
the coefficient
of ~b — ZbOas
This makes ~b – z~Oproportional
maximally
by setting
Only kHJ.2 k
only one parameter
ratio needs to be specified to specify
and relative wave centers (foci).
Zb – zbO is
Zb — Zbo
lo
proportional
=;42:11
tO x’ as
[[:]*
-[:]*]X2
1
=—-2X2
which is negative
to as
for real x indicating
crossing the zb = zbOplane.
that
(4.29)
the lens surface is concave to the left, never
Then (4.19) becomes
(4.30)
and x is positive,
Near the z axis (small X) we have
(4.31)
The curvature
of the lens surface at the z axis is zero (infinite radius of curvature).
21
.
Figuie4.4shows
thelens
(4.30) the symmetrical
surface with 12/11 (and hence cz/cl)as
roles of 41 and /z, so that a particular
a parameter.
Note from
value of 12/41 corresponds
to
e
the same value of n/42.
D. Nearly Matched
Consider
Permittivities
now the case that
&2
—=l+v,
pl <<l
(4.32)
61
where v can be positive or negative
can occur when one medium
depending
on which permittivity
is free space (air)
and the other
is larger,
a low density
Such a case
dielectric
(such
as foam polyethylene).
For small x the leading term in (4.15) gives
Zb
“bO=[l+
f?rJ
0(v)]X+O(X2)a
sv~OandX~
O
(4.33)
Note that we are taking v + O before x ~ O. Similarly from (4.19) we have
~b2
[V+ O(V2)]X+O(X2)asv -+0 and
[1 =/O(~!ll)
z-
which means x takes the sign of v/(12 – /1). Combining
~ ~ o
(4.34)
these gives
(4.35)
This is the equation
positive
v/(lj
of a paraboloid
which (for positive/l
and /2) is concave to the right for
–/l ), and to the left if negative,
As a limiting
case let /2 = ca so that the second wave (in medium
2) is a plane wave,
giving
asv~Oand
22
X+O
(4.36)
●
Y
_b
f!0
10
9
f?
—=
i92 \
10,0.1
8
\
‘7
6
3.0.33
5
4
3
2
1
-6
–5
–4
-3
-2
–1
Fig 4.4: Maximally Flat Lens Surfaces
23
)
z~-zb
4?
0
0
.
Assuming
v positive the lens surface is concave to the right.
For a fixed lens surface
(4.37)
so that small v implies large 10 (focal length),
For comparison,
if this surface were a reflector
[3] we would have
~j
~b — zbO
F s focal length
24
= 4F
(4.38)
5
Diverging
Spherical
Wave to Less Divergent
ical Wave
to Plane Wave
Spher-
@
Now let us combine
As illustrated
in Figs.
the results of previous
sections
to have three waves in three media.
5.1 and 5.2 there are three regions,
the lens and the first and last (third)
the second (middle)
having the same electrical
This defines two lens surfaces designated
1 and 2 (superscripts).
properties
one being
in the examples.
These may intersect
at
(5.1)
depending
on the choices of the various parameters.
tions of medium parameters,
Tracing
medium
our primary
3. In medium
this corresponds
2 this corresponds
by a quartic
equation.
permittivities,
to a prolate
wave) in
of two sheets,
for lens surface 2. For e2 < e3
In medium
1 the ray is on
in Section 4 lens surface 1 is in general
in Figs. 5.1 and 5.2 are determined
described
(z~~) – Z$)) and the
Zjl)
ez/el (= 2.26) with
(2)
2[:)– ,z~o
and 42 gives the lens thickness
zb(’) – &
then determines
by first specifying
220 from zbO as
1=
Choosing 42/n
spheroid
For a specified set of /l, /2, the lens thickness
Then choosing 1 determines
Then choosing
to the z axis (plane
the two lens surfaces are determined.
The examples
El,
parallel
to a ray on a radial from F2 = 22012.
+ As discussed
to one sheet of a hyperboloid
a radial from T’l = ZIO~Z. As discussed
e3 =
has medium 2 as a lens with C2> el = C3.
the ray back we have a ray specified
in Section 2 for e2 > e3 this corresponds
●
interest
While there are many possible combina-
on axis
= / –/2
/l, 40 and lens surface 1.
25
(5.2)
(5.3)
.
/?
—
1
1.5
=
1?o
1?
—
/?2
f?
2
—=3
4)
= 1.5
t 2
PROLATE SPHERE
(LENS SURFACE 2)
—=2
! 1
——_
-\_
LENS SURFACE 1
q
//
//
MM
//
//
//
PO
/“
Z*O
Zi
Z,.
L
L~
4
t0
4
1 2
z(l)
b.
Fig 5.1: Converging Lens Separating Two
Identical Media
26
E’2
—
= 2.26
El
4?
—
z’2
=
1.5
t
1
— =
to
12
—
=
1.67
2.5
Y
4?
o
4?
2
=
PROLATE SPHERE
(LENS SURFACE 2)
1.5
Y
T 1
LENS SURFACE 1
x
(MAXIMALLY FLAT)
i
)
,
I
t
Z*0
‘ilo
L/?
z,
4?
0
*
9
1
Fig 5.2: Converging Lens with Maximally Flat
First Lens Surface
27
,
6
Transmission
In passing through
of Waves
the lens boundaries,
of the wave is transmitted,
Associated
there is in general a partial
reflection
the previous
{
corresponds
lens of permittivity
to the transmission
1 there is a transmission
parameters
As discussed
Cz separating
two media
~forn=l,2
en
for the curvature
and reflection
in
p. the media have wave impedances
(6.1)
in Fig. 6.1, consider the simple problem
Except
is met, not all
of the medium
of the wave at each lens boundary.
Z.=
wave incidence.
requirement
with the” discontinuity
Cl. With common permeabilities
As illustrated
Lens Boundaries
while the equal-time
section one can have a dielectric
of permittivity
normal
Through
of two planar lens surfaces with
of the lens surfaces
in Section
5, this
of the ray along the z axis. At lens surface
coefficient (for the electric field) using the usual one-dimensional
formula.
T1 =
2z~
ZI+ Z2
(6.2)
‘2’W+W
At lens surface 2 we similarly have
(6.3)
This gives an overall transmission
coefficient of
(6.4)
This applies to the first portion
tions at the lens boundaries
of a transient
wave reaching medium
3. There are reflec-
which can appear in medium 3 at a later time due to multiple
reflections.
Since media 1 and 3 have the same constitutive
the relative
power
transmission
(at early times)
from medium
parameters,
T2 < 1 represents
1 to medium
3.
Ideally one would like T to be one, such as for the ideal lenses discussed in [5], this being
accomplished
by controlling
the permeability
and/or
28
medium inhomogeneity
and anisotropy
LENS
SURFACE
1
RAY
PATH
LENS
SURFACE
2
.
T= TIT2
T2
MEDIUM 1
MEDIUM 2
*
MEDIUM 3
Fig 6.1: Wave Normally Incident on Two Lens Surfaces
29
.
as well as special angles (e.g. Brewster
sacrifices ideal performance
angle) at the lens boundary.
for simplicity
of the lens medium.
In the present case one
If the two permittivities
are
e
only slightly different so that as in (4.32) we write
(6.5)
For small u we have
T=~osh-2~n((~)’))
= 1 – #n2(l
=cosh-2(~ln(l+~))
+ v) +
0(4n4(l
+v))= 1 –
So, for small v, not much is lost in transmission
through
removed from the initial wave by the lens-boundary
1–
T2= ~ln2(l
as v -0
as v ~ O
(6.6)
The fractional
power
+O(V3)
the lens.
reflections
+ v) + O(ln’(1
= ~Z/2 + 0(v3)
1
~v2
is
+ v))
(6.7)
o
30
.
a
7
Plane
Wave
to Diverging
Spherical
Wave
to More
—
Divergent
Spherical
The considerations
wave through
Wave
in Section 5 can readily be extended
a lens in which the wave is spherically
the rate of spherical
expansion
a plane wave in medium
to lenses for going from a plane
expanded
is even greater as illustrated
to a third medium
in Figs. 7.1 and 7.2. Begin with
1 where a ray is parallel to the z-axis.
In medium
2 the ray is on
a radial from ?2 = Z20r2. As discussed in Section 3 for C2> c1 this corresponds
spheroid for lens surface 1. For 62 < Cl this corresponds
sheets.
In medium
As discussed
3 the spherical
wave is described
to a prolate
to one sheet of a hyperboloid
by the ray on a radial from 7“ =
in Section 4 lens surface 2 is in general described
specified set of 12, ~3 the lens thickness
in which
by a quartic
(2)
equation.
(Z60 – Z60
“)) and the three permittivities,
of two
Z30rz
For a
the two lens
surfaces are determined.
Note now that
With our primary
interest
in e2 > Cl = C3, the two lens surfaces have closest approach
near
the axis where one can choose Z60
‘1) – z~o
‘2). III some cases the z axis may be in a region where
there is no wave (e.g. inside a conductor
such as the center conductor
(cable)).
In such cases we can have .z~~)> z~~) and the intersections
described
as in (5.2).
.
31
of a coaxial structure
of the two surfaces
E,.—
— –
El
1?
4?
2
—=1
2.26
!3
—
=
1.5
1?o
t
2
—=3
to
1?
2
—=2
13
LENS SURFACE 2
PROLATE SPHERE
(LENS SURFACE 1)
Fig 7.1: Diverging Lens Separating Two
Identical Media
Y
t
3
—
=
1.67
z
2.5
40
4?
t2
—
—=1
1’0
t 2
1 2
—
/
1.5
=
LENS SURFACE 2
\
3
PROLATE SPHERE
(LENS SURFACE 1)
Y?
\
/
0
0
w%
/
//
Z*O
/.~’,0’
0-
/
/
/
/“’
0
/
z 3*
Fig 7,2:
/
0
0
I
0
I
il) = ~$)
%o
Diverging Lens with Maximally Flat
SecondLens Surface
33
8
Concluding
Remarks
The case of a single lens surface separating
two media of different
rather simple results, allowing only for prolate spheres and hyperboloids
provided
permittivities
(one of two sheets),
one of the two waves is a plane wave. With two surfaces separating
the situation
is in general
more
complicated.
described
by a fourth order-polynomial
examples
of this, in particular
and the wave in either medium
The
equation.
surface
matching
1 or 3 (but not both)
three media
two spherical
Here we have considered
where the first and third media
results
another
different
permittivities
and/or
in Section 4 and Appendix
three spherical
have the same permittivities,
However, the results of
is a plane wave.
waves as well.
A one can go from a spherical
spherical wave in an adjacent
medium,
34
waves is
some interesting
this paper are more general in that they can be applied to a wider set of situations
three
gives
Utilizing
involving
the general
wave in one medium
and then on to another,
etc,
to
.?
Appendix
A: Non-Parametric
Form of Lens Surface
@
Section 4 considers the general case of a lens surface matching
the general solution
zb — zbo
&
is cast in terms of a parameter
two spherical waves. There
x. From (4.15) and (4.19) this is
‘~~2~~1{[[:1’-[:1’1x2
+2
E[:l’-$[:l’lx}
[?l’+[zbizbol’=l,!,, [:: [ 1‘ -H:]*]X2
21~1,
‘10(12
Rewrite
[[:]*
_[y]x
(Al)
– 1*)
this in the convenient
form
Zb — Zbo
– CIX2 +
C,x
[#i2+i’bizb012=
@2+@
with the four constants
(A.’)
as above.
Solving the first of (A.2) for #
and substituting
for X2 in the second gives
[%12+
[Zbizbol’=:zbizbo
+c1c4ic2c3x (A.3)
Solving for x we have
(A.4)
C1 {[~]2+[zb;ozq2}-c,c4:c2c3zb;ozbo
x = C~C4 – C’C3
which can be placed back in the first of (A.2) for both X2 and x to give
c:
Zb — Zbo
&
= [CIC~ – C,C3]2
+
{c1{[:]2+[yq2}-c3zb;fl}2
{C,{[;]2+[’7J’]
2}-C3Z7’}
– CZC3
ClC4 C2
35
(A.5)
+
With the four constants
given, this is a fourth order equation
The special combination
UI C4 – C2C3
of constants
relating ~b/.& and (zb – %bO)/~o.
reduces to
[[:]
’-[:1’]
[[$-w]
=(1,
-e,)’
‘: ~[~]i-:[:l’]k[:l’-%
[:l’l
–(1?’
–l?,)’
& C2
{!A[Q]4-;[%I*}
(A.6)
‘/2–/1
& C2
‘“2
lo
With Cl, C2, and C3 from (Al)
A special case treated
corresponds
flat
this gives the constants
in (A.5).
in Section 4 is the special spherical
to the combination
in (A .6) being zero. Another
lens surface in (4.28). This corresponds
to C2 = O.
36
lens given in (4.22).
This
special case is the maximally
●
References
8
[1] C. E. Baum,
Radiation
321, November
[2] C. E. Baum,
Dielectric
of Impulse-Like
Transient
Fields, Sensor and Simulation
Note
1989.
J. J. Sadler,
Lens Feedinga
and A. P, Stone,
Circular
A Prolate
Spheroidal
Coax, Sensor and Simulation
Uniform
Isotropic
Note 335, December
1991.
[3] E. G. Farr and C. E. Baum,
Radiating
Antenna,
Prepulse
Associated
Sensor and Simulation
[4] G. A. Kern and T. M. Kern,
Mathematical
with the TEM Feed of an Impulse
Note 337, March 1992.
Handbook for
Scientists and
Engineers,
2nd
Ed., McGraw Hill, 1968.
[5] C. E. Baum and A. P. Stone,
tromagnetic
Transient Lens Synthesis:
Theory, Hemisphere
Publishing
Corp., 1991.
37
..
Differential
Geometry
in Elec-