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On the magnification factor of ring focus imaging
systems
Giuseppe Orlando
Thales Alenia Space, Antenna department, Rome, Italy, Giuseppe.orlando@thalesaleniaspace.com
Abstract— A closed formula that well approximates the
magnification factor of centered dual reflector ring focus
imaging systems is proposed. Numerical verifications carried
out on specific optics geometries provide a good correlation to
the analytical formula.
Index Terms— magnification factor, caustic, reflection, ray
tracing, UTD, GO, ring focus.
I.
I
NTRODUCTION
Ring focus dual reflector systems are widely used in
microwave antennas and are characterized by high aperture
efficiency, and avoiding back scattering toward the feed [1].
The most used configuration is the centered axially
displaced dual reflector [1] whose main reflector is a axially
displaced parabola, while subreflector is an axially
displaced hyperbola or ellipse. As outlined in [2] a ring
focus dual reflector system suffers of high order aberration
modes (particularly for scanned beams) and the bilinear
transformation derived in [3] cannot be applied to this
antenna family. Although the ring focus antenna is an odd
order aspherical system, recently it has been shown [4] that
surface shaping through odd aspherical modes can provide
better performances in terms of aberrations than evenorder
aspherical systems.
Imaging systems usually employ a phased array as a source
operating in a near field of subreflector [5].
In a canonical dual reflector imaging system it is important,
as a design parameter, to define preliminarily the
magnification factor M.
Dragone and Fitzgerald [6],[7] have firstly investigated such
property of confocal paraboloids. Their magnification factor
is substantially derived from the Abbe sine condition
introduced in [8].
A ring focus imaging system can therefore be modelled by
considering both its subreflector and main reflector as an
axially displaced parabola with confocal ring focus.
In this paper we investigate the caustic of such class of
reflectors first. Then, through the ray tracing technique, we
derive an analytical solution for the magnification factor,
finding a good correlation to the numerical results.
The caustic is defined as the envelope for either reflected or
refracted rays crossing an optical system or, equivalently,
the loci of singularities in the flux density [9],[10].
A clear and straightforward solution of a 3D single reflector
caustic is reported in [11]. In this work we follow that
approach to calculate the caustics of the reflector surface.
Section II introduces general definition of the ring focus
imaging system, while section III considers the
magnification factor of a canonical dual reflector system. In
section IV it is reported the calculation of the caustic for a
plane wave propagating parallel to the optical z axis and
impinging in a smooth reflector surface. Section V reports
the derivation of the aperture distribution through UTD for
the dual reflector system of interest. Section VI discusses the
ray tracing of the ring focus imaging system, while section
VII outlines the derivation of the relevant magnification
factor M. Finally section VIII reports correlations of the
obtained formula with numerical simulations through
GRASP10.
II. R
ING FOCUS IMAGING SYSTEM
In this section we extend the generalized classical axially
symmetric dual reflector antennas to an imaging optical
system. Following the published nomenclature [1], four
types of such imaging system can be defined. In this paper
we focus the attention on the Axially Displaced Cassegrain
Imaging (ADCI, also referred as reflaxicon in [12]) systems
but the reported considerations can be easily extended to
other three optical systems. To obtain an imaging system,
the subreflector (with a generating ellipse or hyperbola) is
replaced by a generating curve as a confocal displaced
parabola (see Fig. 1).
Using simple geometrical considerations, explicit formulas
can be defined for the ADCI antenna. For the subreflector
we have:
()
≤≤
+
+−=
2/0
4
2
S
S
SM
D
F
L
FFz
ρ
ρ
(1)
For the main reflector we have
()
≤≤+
+
=
2/'2/
4
2
MS
M
DLD
F
L
z
ρ
ρ
(2)
22
yx +=
ρ
F
M
and F
S
are main and subreflector focal lengths, D
S
and
D
M
are main and subreflector diameters, L’ is the clearance
and L is the shift of the parabola (see Fig. 1).
L’ and L parameters are respectively constrained by
S
M
S
SM
F
F
D
LDD =
−−
2/
'2/2/
(3)
Fig. 1. Ray tracing of the ADCI antenna and optical parameters.
S
M
S
M
F
F
LD
LD =
+
+
2/
2/
For parallel rays propagating along the optical z axis the
following relation between axial ray coordinates holds true
),(
2/
'2/2/
'2/
),(
0SSSS
S
SM
S
SSM
yxk
D
LDD
LD
yx
ρρρ
ρ
+=
−−
++=
=
(4)
where the subscript “M” is for main reflector and “S” stands
for subreflector while
22
SSS
yx +=
ρ
.
III. M
AGNIFICATION FACTOR OF A CANONICAL IMAGING
DUAL REFLECTOR SYSTEM
In a canonical imaging system made by two parabolic
confocal reflectors the magnification factor is equal to [13]
S
M
S
M
F
F
D
D
M==
(5)
In Tx mode, such systems convert a plane wave generated
by the feedarray into a spherical wave (after first
reflection); due to the second reflection, the spherical wave
is then converted again into a plane wave.
For the ADCI antenna it can be easily verified that (5) is not
valid, in fact, assuming absence of the central blockage, (3)
holds between focal lengths and diameters. This means, for
instance, in Tx mode parallel rays are transformed into a
cylindrical locus of output rays and again into parallel rays.
IV.
CAUSTIC GENERATED BY A FRONT PHASE DIRECTED
AS

Z AXIS
The procedure to calculate the two caustics (sagittal and
tangential) of a smooth reflector is well detailed in [11].
Following this procedure, in the appendix A we calculate an
explicit expression of the two caustics generated by a plane
wave directed as zaxis:
++−
+−±+
−
=
=+=
22
22
*
2,1
*
2,1
1
2
2
4)(
1
ˆ
1
yx
y
x
xyyyxxyyxx
ff
f
f
fffff
f
y
x
b
k
r
ξ
(6)
where f=f(x,y), f
x
=f
x
(x,y) is the first derivative respect x,
f
xx
=f
xx
(x,y) is the second derivative respect x, etc. and
[]
T
yxfyxr ),(=
;
[]
T
a100
ˆ=
;
nnaab ˆ
)
ˆˆ
(2
ˆ
ˆ⋅−=
;
f(x,y) represents the reflector smooth surface,
n
ˆ
is the unit
Fig. 2: Sagittal and tangential caustic of ADCI main reflector for a rotated
front phase around Y axis (α = 1°)
normal vector,
a
ˆ
and
b
ˆ
are the optical wave vectors of the
incident and reflected rays, respectively.
For the main reflector (2), the sagittal caustic is a segment
laying on the z axis, and can be derived from (6):
T
M
MM
M
L
L
F
F
L
F
L
F
+
−++=
ρ
ρ
ξ
44
00
2
*
1
The tangential caustic is a ring of radius L laying on the
z=F
M
plane, from (6) we have
T
M
F
y
L
x
L
−−=
ρρ
ξ
*
2
The procedure of appendix A can be extended to tilted
wavefronts around Y axis (for example) and, for the ADCI,
the two caustics are derived and depicted in Fig. 2 for a
rotated front phase of α = 1°.
We observe that the eigenvalue λ
1,2
(see (A11)) inside the
equation (6) is only a function of second derivatives of the
reflector surface. Therefore to generate an offset version of
the main reflector we can simply add to the f(x,y) function
another first order function that doesn’t change the
eigenvalue λ
1,2
.
The offset version of main reflector (2) will be
()
ax
F
L
z
M
+
+
=4
2
ρ
with a real number. The resulting tangential caustic will be
a rotated ellipse (around Y axis).
V. A
PERTURE
D
ISTRIBUTION
The aperture distribution of ADCI can be calculated
following [1] and [14] (we omit the phase contribution that
is not relevant for this discussion) as
()()
21
21
)( rRrR
rr
A
SMSM
S
++
=
ρ
(7)
where R
SM
is the distance between the sub and main
reflector surfaces along the reflected ray, r
1
and r
2
are the
subreflector reflected wavefront principal radii of curvature
at the reflection point (associated with the sagittal and
tangential caustics, respectively). We can see that for a
canonical dual reflector imaging system with confocal
paraboloids, for normal incidence, the asymptotic aperture
distribution (7) is constant and is equal to the inverse of (5).
D
M
D
S
L’
F
M
F
S
L
Fig. 3: UTD aperture distribution overlaid to simulated at 5, 25, 50 GHz
(solid, dash, dot, dashdot respectively) for first optical system (OS1)
In this case sagittal and tangential caustic degenerates into
the focal point and the aperture distribution is simply the
inverse of the magnification factor.
For the ADCI system the asymptotic aperture distribution
(7) simply becomes:
)(
)(
L
L
A
MM
SS
+
+
=
ρρ
ρρ
(8)
where ρ
M
and ρ
S
are defined in (4). We verified this aperture
distribution finding a good agreement with this result (see
Fig. 3, details of the optics geometry are explained in
section VIII).
VI.
RAY TRAC ING OF A RING FOCU S IMAGIN G SYSTEM
For a ray tracing we need to calculate normal vectors of the
two reflecting surfaces, and the two bundle of reflected rays.
We also need to solve for the intersection of a generic ray
with the reflecting surface (solving a fourth order
polynomial). Sub and main reflector surfaces are defined in
(1) and (2). In the following the subscript “S” is used for the
subreflector and “M” for the main reflector.
Optical wave vector of incident rays is indicated as
[]
T
a
αα
cos0sin
ˆ=
Due to the symmetry of the system, only a rotation around
Y axis is considered.
The relation between a ray reflected by the subreflector
impinging the main reflector is
+=
+=
+=
)3(
)2(
)1(
SSM
SSM
SSM
lbzz
lbyy
lbxx
(9)
with
[]
T
SSSSSS
bbbnnaab
)3()2()1(
ˆ
)
ˆˆ
(2
ˆ
ˆ=⋅−=
where b
S(1)
, b
S(2)
, b
S(3)
, indicate the x, y and z components
respectively of optical wave vector of reflected rays
S
b
ˆ
, l
represents the distance between reflectors,
S
n
ˆ
is the unit
normal vector of subreflector.
Starting from the relation of the main reflector:
2.0E05
1.5E05
1.0E05
5.0E06
0.0E+00
5.0E06
1.0E05
1.5E05
2.0E05
0.0E+00 5.0E06 1.0E05 1.5E05 2.0E05 2.5E05 3.0E05 3.5E05 4.0E05
v
u
Spot diagram (reflected ray directions) for
α=0.01°
Fig. 4: Spot diagram (reflected ray directions) in the uv plane associated to
uniform cylindrical gridbased distribution (top right) for a ray bundle
rotated of
α =
0.01° around Y axis. Optical parameters are: F
M
/F
S
= 5.0,
D
M
/D
S
=6.3, L/F
S
=0.43.
;
4
)(
)3(
2
SS
M
M
lbz
F
L+=
+
ρ
(10)
;;
2222
MMMSSS
yxyx +=+=
ρρ
and substituting each parameter inside, we find a fourth
order polynomial in respect the distance l between sub and
main reflector that can be numerically solved (see appendix
B for more details). Once we have calculated the l
parameter, we can finally obtain the optical wave vector of
reflected rays at main reflector as
[]
T
MMMMMSSM
bbbnnbbb
)3()2()1(
ˆ
)
ˆ
ˆ
(2
ˆˆ =⋅−=
and the ray tracing is concluded.
The Optical Path Length (OPL) can be written as
)cos1(
)cos(),()cos(),(
)3(
α
α
α
S
MMMSSS
bl
yxzlyxzOPL
−=
=−+=
If α = 0, OPL becomes equal to 2l.
VII. E
STIMATION OF THE MAGNIFICATION FACTOR
From the ray tracing, we can calculate the spot diagram for
a rotation α of direction of incident ray bundle. An example
is depicted in Fig. 4 for the case of small α angle. The
toroidal properties of this system can be easily noticed and
also high order aberrations and coma are evident from the
spot diagram. In Fig. 4 are reported 5 main points that can
be calculated in an approximated form. We indicate points
as P
i
≈ (u
i
; v
i
) in the uv plane for small α. With some
geometrical consideration we can write
()
()
=
≈
=
≈
0;0;
0;0;
22
11
s
D
D
P
s
F
F
P
M
S
M
S
α
α
−
−
+
≈
−+
≈
2
;
2
2
;
2
2121
4
2121
3
ssss
P
ssss
P
Point P
1
represents the direction of rays with the classical
P
2
P
1
P
4
P
3
P
5
magnification ratio of the two focal lengths and is associated
to all rays reflected in the y
s
=0 plane of the subreflector.
The point P
2
is associated to rays impinging the sub
reflector at Q
2
=(x
s
;y
s
)=(0; ±Ds/2). P
3
and P
4
are associated
to rays impinging the subreflector at Q
3
= (±Ds/(2√2);
±Ds/(2√2)) and Q
4
= (
Ds/(2√2); ±Ds/(2√2)) respectively.
Point P
5
is calculated by simulation and is associated to the
effective magnification of the optical system. It is also
associated to the direction of maximum radiation of the far
field beam after second reflection and is a consequence of
the aperture radiation integral. Despite the availability of the
complete solution of the ray tracing, it is not straightforward
in this case to apply the procedure of [2] to calculate the
magnification factor, due to the complexity of such solution
(see appendix B).
A good approximation of point P
5
is found to be
≈0;
1
*
5
M
P
α
with the effective magnification factor equal to
M
S
M
S
M
S
M
S
M
S
F
F
D
D
D
D
F
F
D
D
M3
1
3
4
3
11
*
−=
−−=
(11)
Using this equivalence, we obtain a relative error between
simulated mean value M
SIM
(mean value of magnifications
obtained from each simulation) and calculated M
*
value
from (11) less than 0.6%. The relative error is defined as
SIMSIMREL
MMM /)(
*
−=
ε
Details of simulations are reported in the next paragraph.
VIII. N
UMERICAL RESULTS
For the verification of the aperture distribution (8) we
considered the following first Optical System (OS1 in the
following): freq. equal to 5, 25, 50 GHz, F
M
/F
S
= 5.0,
D
M
/D
S
=6.3, L/F
S
=0.43, D
M
=10 m.
The simulation is carried out through a plane wave source
rotated of a selected angle, reflector surface currents are
calculated via Physical Optics (PO) and Physical Theory of
Diffraction (PTD) through Grasp10. In Fig. 3 it is reported
the comparison between UTD model and simulated near
field at main reflector aperture level.
For the estimation of the magnification factor we considered
firstly OS1 with freq. = 5 GHz, and α from 0 to 4°, step
0.5°: simulated mean value M
SIM
is 6.94 vs 6.9 from (11).
The second optical system (OS2) considered has following
parameters: F
M
/F
S
= 4.4, D
M
/D
S
=5.6, L/F
S
=0.406, D
M
=6 m,
freq.= 8 GHz: simulated mean value M
SIM
is 6.17 vs 6.16
from (11). The OS3 has following parameters: F
M
/F
S
= 5.3,
D
M
/D
S
=6.8, L/F
S
=0.397, D
M
=8 m, freq.= 6.5 GHz:
simulated mean value M
SIM
is 7.55 vs 7.51 from (11).
Finally we considered OS4 with : F
M
/F
S
= 3.4, D
M
/D
S
=4.5,
L/F
S
=0.505, D
M
=7 m, freq.= 7.5 GHz: simulated mean
value M
SIM
is 5.05 vs 5.04 from (11).
Fig. 5 shows the estimation of the magnification factor as a
function of the steering angle of the plane wave source for
the four optical systems considered.
4.00
4.50
5.00
5.50
6.00
6.50
7.00
7.50
8.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
Magnification
alpha [deg]
Magnification estimation
OS1, Freq.=5 [GHz]
OS2, Freq.=8 [GHz]
OS3, Freq.=6.5 [GHz]
OS4, Freq.=7.5 [GHz]
Fig. 5: Magnification estimation of first ,second ,third and fourth optical
system (OS1, OS2, OS3, OS4) vs direction of propagating plane wave
source
IX. C
ONCLUSIONS
In this article we investigate optical properties of a ring
focus imaging system through its ray tracing. We calculate
explicit expressions of tangential and sagittal caustic
considering a plane wavefront propagating parallel to the
optical z axis. We calculate the aperture distribution with
the UTD. We obtain a closed formula of the magnification
factor of ADCI system. Numerical simulations to check the
validity of the approach to estimate dual reflector optical
system parameters have been presented confirming the
validity of results.
A
PPENDIX
A
Following the notation of [11] we calculate the two caustics
of a smooth surface S in case of a plane wave propagating
along z axis. The surface of the reflector is
[]
T
yxfyxr ),(=
(A1)
and its derivatives are
[]
T
x
fr 01
1
=∂
;
[]
T
y
fr 10
2
=∂
.
The unit vector of the impinging plane wave is
[]
T
a100
ˆ=
The unit normal vector we take as
−
++
=
1
1
1
ˆ
22 y
x
yx
f
f
ff
n
In the following we indicate the scalar product between two
vectors u and v as (u, v). The first and second fundamental
form of the surface S respectively can be written as
+
+
=∂∂=
2
2
1
1
),(
yyx
yxx
ji
fff
fff
rrg
(A2)
−−
−−
++
=∂=
yyxy
xyxx
yx
ji
ff
ff
ff
nrB
22
,
1
1
)
ˆ
,(
(A3)
The first and second fundamental form of the reflected
wavefront are
)
ˆ
,)(
ˆ
,(
*
arargg
ijijij
∂∂−=
(A4)
22
*
1
1
cos;cos2
yx
ff
withBB
++
−=−=
θθ
(A5)
More explicitly we can write
;),
ˆ
(
1x
fra =∂
;),
ˆ
(
2y
fra =∂
(A6)
and hence substituting (A2), (A3) and (A6) into (A4) and
(A5) we obtain
=
−
+
+
=10
01
1
1
2
2
2
2
*
yyx
yxx
yyx
yxx
fff
fff
fff
fff
g
(A7)
and
−−
−−
++
=
yyxy
xyxx
yx
ff
ff
ff
B
22
*
1
2
(A8)
The Weingarten matrix becomes
−−
−−
++
==
−
yyxy
xyxx
yx
ff
ff
ff
BgA
22
*1**
1
2
)(
(A9)
To find the two caustics we need to calculate the two
eigenvalues of the matrix A
*
, and hence
0det =
−−−
−−−
λ
λ
yyxy
xyxx
ff
ff
(A10)
The characteristic polynomial is
2
4)()(
;0)(
22
2,1
22
xyyyxxyyxx
xyyyxxyyxx
fffff
fffff
+−±+
−=
=−+++
λ
λλ
(A11)
and finally
)1(
4)(
22
22
*
2,1
yx
xyyyxxyyxx
ff
fffff
k++
+−±+
−=
(A12)
To complete the calculation we need to obtain the reflection
vector, and hence
++−
++
=−=
22
22
1
2
2
)1(
1
ˆ
)
ˆ
,
ˆ
(2
ˆ
ˆ
yx
y
x
yx
ff
f
f
ff
nnaab (A13)
The caustic is
b
k
rˆ
1
*
2,1
*
2,1
+=
ξ
(A14)
Substituting (A1), (A12) (A13) into (A14) we obtain the
final expression as
++−
+−±+
−
=
22
22
*
2,1
1
2
2
4)(
1
yx
y
x
xyyyxxyyxx
ff
f
f
fffff
f
y
x
ξ
(A15)
Following the same procedure we have calculated the
caustic of a rotated front phase (see Fig. 2) but it is a lengthy
algebra and we don’t report the solution.
A
PPENDIX
B
To calculate the l parameter we start substituting (9) into
(10) and taking into account that
()
S
S
SMS
F
L
FFz 4
2
+
+−=
ρ
(B1)
we square (10) to eliminate the square root of ρ
M
and after
some algebra we get
0)2(
)2()2(
13
2
343032132
2
2021131
2
2
3
1021
4
2
1
=+++++++
++++++++
pqqplqppqpqq
lqppqpqqqlqpqqlq
(B2)
with
+=
+=
+=
22
3
)2()1(
2
2
)2(
2
)1(
1
22
SS
SSSS
SS
yxq
bybxq
bbq
++−=
+−=
=
−−=
−=
22
42
4
)3(2)3(
2
3
2
)3(2
2
2
1
)3(
0
168
328
16
82
8
SMSM
SSMSM
SM
SM
SM
zFLzFLp
bzFbFLp
bFp
zFLp
bFp
(B3)
Finally to stabilize the polynomial is better to pose the
substitution
hFl
M
=
and solve numerically respect h (for
example with the tangent method).
A
CKNOWLEDGMENT
The author is grateful to Roberto Mizzoni and Alexander
Yampolsky for valuable assistance and comments.
R
EFERENCES
[1] Fernando J. S. Moreira, Aluizio Prata, “Generalized Classical Axially
Symmetric DualReflector Antennas”, IEEE Trans. on Antennas and
Propagation, vol.. 49, no. 4, Apr. 2001
[2] Fernando J. S. Moreira, “First order aberrations on generalized
classical axiallysymmetric dual reflector antennas”, IEEE Trans. on
Antennas and Propagation, 2000
[3] C. Dragone, “A FirstOrder Treatment of Aberrations in
Cassegrainian and Gregorian Antennas," IEEE Trans. on Antennas
and Propagat., AP30, No. 3, pp. 331339, May 1982.
[4] Takao Tanabe, Masato Shibuya, “Aberration properties of oddorder
surfaces in optical design,” Opt. Eng. 55(12), 125107 (2016).
[5] Fitzgerald, W. D., “Limited electronic scanning with an offsetfeed
nearfield Gregorian system," ESDTR71272, Technical Report
486, Lincoln Laboratory, 1971.
[6] Dragone, C. and M. J. Gans, “Imaging reflector arrangements to form
a scanning beam using a small array," Bell Syst. Tech. J., Vol. 58,
No. 2, 501515, 1979.
[7] B. Houshmand, S. Lee, Y. RahmatSamii, P. T. Lam “Analysis of
NearField Cassegrain Reflector: Plane Wave Versus Elementby
Element Approach”, IEEE Trans. on Antennas and Propagation, vol..
38, no. 7, July 1990
[8] E. Abbe "On the Estimation of Aperture in the Microscope". Journal
of the Royal Microscopical Society. 1 (3): 388–423.
[9] Stavroudis O. N.,” The Mathematics of Geometrical and Physical
Optics, The Kfunction and its Ramifications” WileyVCH Verlag
GmbH & Co. KGaA 179186 (2006)
[10] D. G. Burkhard and D. L. Shealy. “Flux Density for Ray
Propoagation in Gemoetrical Optics”. Journal of the Optical Society
of America, 63(3):299304, March 1973.
[11] A. Yampolsky and O. Fursenko, “Caustics of wave fronts reflected by
a surface”, Journal of Mathematical Sciences and Modelling, 1 (2)
(2018) 131137
[12] Walter LaVaughn Hales, Dietrich Korsch, “Design and analysis of
afocal, twomirror system for arbitrary intensity transformations”,
TECHNICAL REPORT RH812, US Army Missile Command,
October 1980.
[13] S. K. Sharma, S. Rao, L. Shafail, “Handbook of reflector antennas
and feed systems.” pag 221
[14] P. H. Pathak, G. Carluccio, and M. Albani, “The Uniform
Geometrical Theory of Diffraction and Some of Its Applications”,
IEEE Antennas and Propagation Magazine, Vol. 55, No. 4, August
2013