A Rigorous Method for Synthesis of Offset Shaped ...oliker/papers/comp-lett-Dec-09.07.pdf · A Rigorous Method for Synthesis of Offset Shaped Reflector Antennas Vladimir I. Oliker∗

A Rigorous Method for Synthesis of OffsetShaped Reflector Antennas

Vladimir I. Oliker∗

Department of Mathematics and Computer Science,Emory University, Atlanta, Georgia

e-mail: [email protected]

Abstract

In this paper the problem of synthesis of offset shaped single reflector antenna isconsidered. This problem has to be solved when a reflector antenna system is requiredto control the field amplitude and/or phase on the far-field or on the output aperturein the near-field. Achieving high efficiency is a very important objective of the designand shaped reflector antennas are used for that purpose.

The equations of the problem are strongly nonlinear partial differential equationswhich can not be analyzed by standard techniques. Though the problem has been thesubject of study by many authors for over 40 years, up until recently, there were norigorous theoretical results resolving completely the questions of existence and unique-ness. With few exceptions, authors have attacked the problem with heuristic numericalprocedures, and, depending on the specific formulation, obtained different results notalways in agreement with each other.

In this paper a new method for solving the single reflector problem is presented.The new method allows a complete and mathematically rigorous investigation of thisproblem. Furthermore, the proposed method lends itself to a numerical implementationand we present here several examples.

Key words: reflector antenna, synthesis, nonlinear partial differential equations,numerics

Subject Classification: AMS(MOS): 78A50, 65N21; CR:G1.8.

∗Author’s note: This paper was submitted for publication to the Journal of Computational Methods inSciences and Engineering (JCMSE) on August 12, 2001 and accepted December 12, 2001. However, due toreasons related to publishing issues and completely beyond any control of the author the paper has neverbeen published. It was transmitted to Computed Letters on March 17, 2006.

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1 Introduction

Reflecting properties of quadric surfaces, such as ellipsoids, paraboloids and hyperboloids arewell known and have been widely used to build lenses, mirrors, antennas, and many other re-flecting and refracting devices. However, in many engineering problems of reflector/refractordesign and analysis the simple shapes of quadric surfaces are often not sufficient to satisfythe requirements of high efficiency and significantly more complex systems of surfaces haveto be designed and manufactured. A general problem that arises in the theory of synthesisof reflector antennas is that of shaping a reflecting surface so that it would redistribute agiven geometrical optics (GO) feed power pattern into a prespecified amplitude aperturedistribution [2], [3], [1], [4], [5] Restricting the design to axially symmetric surfaces or com-binations of ellipsoids, paraboloids and hyperboloids will produce in many cases a systemwith blockage and low efficiency.

Various versions of this problem arise also in the theory of synthesis of mirrors inoptics [9], [8], [10], [11], [12], heat transfer [6], [7], and other areas.

Because of important practical applications, this problem has been studied quite ex-tensively. In the axially symmetric case with the source on the axis the problem reducesto solving an ordinary differential equation which can be rigorously investigated and solvednumerically with any given precision [5]. Obviously, an axially symmetric design is too re-strictive. For example, blockage of the reflector by the feed can not be avoided in suchdesign. Thus, in many circumstances a non-axially symmetric and offset reflector antennasare desired. However, in this case the equations describing the problem are highly nonlin-ear partial differential equations which until recently could not be analyzed rigorously byavailable mathematical methods.

During the last four decades essentially three different approaches have been used forformulating the general problem analytically and solving it numerically. In the first approachthe problem is formulated as a system of first order partial differential equations and a methodresembling the method of characteristics is applied to solve the system numerically; see [2, 1],and other references there. In the second approach the same problem is formulated as aboundary value problem for a second order partial differential equation of Monge-Amperetype for a certain complex-valued function [4]. Then a linearization procedure is used toconstruct a solution close to some a priori selected solution. In both approaches a rigorousmathematical analysis of the resulting equations is lacking and consequently the validity ofthe numerics is never fully established. This led to a controversy regarding existence anduniqueness of solutions that until now has not been resolved [3, 2, 1]. A third approach basedon a rigorous mathematical analysis of a second order real-valued Monge-Ampere equationwas applied in [13] to prove existence and uniqueness in the special case when the solutionis required to be “close” to an axially-symmetric one.

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The purpose of this paper is to describe a new approach to the synthesis problemfor single reflector antennas. In contrast with the above mentioned first two approaches thenew approach is based on a rigorous mathematical theory and its validity does not dependon specific numerical examples. It also does not have the limitations present in the thirdapproach. The approach presented here was first developed in [14] for the far-field case, andthen, for the near-field case, in [15, 16]. A somewhat different but close “in spirit” approachto synthesis of dual reflector systems with a collimated source is described in [17].

Note. (Added in proof May 10, 2006). An alternative approach to the problemof designing optical and antenna systems consisting of one or two reflecting surfaces whichtransform the energy of the source into a prescribed energy distribution on a given targetset was given by T. Glimm and V. Oliker in [22, 23]. This approach is based on calculus ofvariations, in particular, on the Monge-Kantorovich transport theory.

In [14, 15, 16] the emphasis was on the mathematical aspects of the problems andthe results are not readily applicable to problems of practical interests. The present paperemphasizes the geometric side of the new approach and, in addition to providing a relativelyelementary exposition of it, contains also an important improvement and analysis leading topractical criteria permitting designs with no blockage.

The presented approach allows to not only resolve rigorously the questions of existenceand uniqueness of solutions but it also lends itself naturally to numerical procedures forconstructing numerical solutions with any user-specified accuracy. Such a procedure hasbeen implemented by us into a numerical code and we used it to develop offset shapedreflector antennas transforming a given GO feed power pattern into a prespecified amplitudedistribution across a given flat aperture.

In essence, the new approach is very geometric and quite elementary. While themathematics behind it is somewhat involved and uses relatively advanced concepts of weaksolutions to nonlinear partial differential equations, the main ideas are transparent and canbe explained by simple and familiar means.

The organization of this paper is as follows. In section 2 we give a detailed statementof the problem, in sections 3 and 4 we describe the new approach to its solution, in 5 weprovide conditions which guarantee that self-blockage and blockage by the feed are avoided.Finally, in section 6 we present numerical examples.

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2 Statement of the Problem

Let x, y, z be a Cartesian coordinate system with origin O. We denote also by O a non-isotropic point source of energy placed at the origin and let I(m) denote the feed powerpattern as a function of direction m. As usual, the geometrical optics approximation is usedto set up the synthesis problem [5, 11, 4, 1].

x

y

z

T

O

m

n

R r(m)

S

D

ut

ρ = |r(m)|

v = r + tu

Figure 1: Formulation of the problem

It is convenient to consider the direction m as a point on a unit sphere S centeredat O as shown on the Fig. 1. In this setting, m is the incidence direction passing throughthe input aperture D which is pictured as a region on the sphere S. We will assume that Dis a closed set on S in the sense of set theory. In order to derive the required relations weconsider a reflector surface R that intercepts the incident rays and reflects them according toSnell’s law. The surface R is represented by its polar radius ρ(m) with m varying in regionD. In vector form it is given by r(m) = ρ(m)m.

The ray of direction m is incident upon the surface R at the point r(m) and isreflected in direction u given according to Snell’s law as

u = m − 2(m · n)n, (1)

where n is the unit normal to surface R. It is assumed here that R is smooth and projectsradially onto D in a one-to-one fashion.

In the problem considered here we are also given in advance some region T in spacewhich the reflected rays must reach and produce there a certain power pattern. For simplicity,

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the energy delivered by rays reaching T directly from O is excluded from our analysis.Obviously, this energy can be easily accounted for in the final analysis.

In this paper we assume that T is a flat region on some plane. This assumptionis made only to simplify the presentation; it can be significantly relaxed [15]. It is alsoassumed that T is a bounded and closed set. We denote by L(v) a function defined onT which represents the required output power pattern. Finally, if we denote by t(m) thedistance from the surface R traveled by the reflected ray of direction u(m) to reach T , thenwe have a map Γ(m) : D → T from the input aperture D to the region T , that is, v = Γ(m).This map is given by

Γ(m) = r(m) + t(m)u(m). (2)

Next, we relate the input power pattern I(m) on D and the output power patternL(v) on T . Let J(Γ) be the Jacobian of the map Γ. The energy conservation law alonginfinitesimal ray tubes can be expressed as [18]

L(Γ(m))|J(Γ(m))| = I(m). (3)

Now, the reflector problem is formulated as the following inverse problem. Supposewe are given in advance the region D on S and a flat region T on some plane, a non-negativefunction I(m) defined on D and a non-negative function L(v) defined on T . The problemis to determine a reflecting surface R such that the map Γ, defined by (2), maps D onto Tand the condition (3) is satisfied for all m in the interior of D.

A necessary condition that must be satisfied by the functions I and L follows imme-diately from (3). Namely, integrating (3) and using the formula for changing the variableunder integral we obtain

∫

DI(m)dσ(m) =

∫

DL(γ(m))|J(γ(m))|dσ(m) =

∫

TL(v)dµ(v),

where dσ(m) is the area element on the sphere S and dµ(v) is the area element on T . Itfollows that if L(v) is a given output power pattern on the output aperture T then thefollowing “balance” equation is necessary for solvability of the reflector problem:

∫

DI(m)dσ(m) =

∫

TL(v)dµ(v). (4)

From the point of view of differential equations the equations (1), (2) and (3) are notconvenient for studying the reflector problem because the position vector of the unknownreflector R enters these equations in a very complicated way. Different ways for rewritingthese equations have been suggested [2, 3, 4, 5, 11, 19, 20]. Some of them are more convenientthan other but, except for the axially symmetric case, the resulting equations are alwayshighly nonlinear and complicated, and, consequently, difficult to investigate. Fortunately, inour approach we do not need these equations in explicit form.

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3 The New Approach

3.1 An Observation and the Main Strategy

In some sense the approach described below resembles the classical approach in which onewould like to use the known reflecting properties of quadrics to build reflectors. Considerfirst the special case when the output aperture T consists of only one point v. The requiredoutput power is interpreted in this case as the Dirac δ− function with a positive “mass” L

v

concentrated at the point v. Since the right hand side in the balance equation (4) is thetotal energy required at v, this equation in this case assumes the form

∫

DI(m)dσ(m) = L

v. (5)

In the following our constructions will involve ellipsoids. To avoid confusion, let usnote that throughout the paper by an ellipsoid we always mean the surface of the solid.

Let E(v) be an ellipsoid of revolution with the axis of revolution passing through thepoints O and v, with one of the foci at O and the other at v. Its polar radius is given by

ρ(m) =d

1 − ǫm · v , (6)

where d is the focal parameter of E(v), ǫ the eccentricity, and

v =v

|v| .

Here and throughout the paper we identify the point v with the vector originating at O andterminating at v. Since the distance |Ov| (= |v|) between the foci is specified, fixing apositive d fixes one such ellipsoid. The eccentricity of E(v) can then be determined fromthe known formula

|v| =2ǫd

1 − ǫ2, (7)

which can be resolved as

ǫ =

√

√

√

√1 +d2

|v|2 − d

|v| . (8)

The well known geometric properties of such an ellipsoid imply that any ray of direc-tion m originating at O is reflected by E(v) and reaches v which is a caustic point of thisGO flow. The total power of the source

∫

DI(m)dσ(m)

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is carried to the focus v by the GO rays passing through the aperture D and transformedby E(v). By (5)

∫

DI(m)dσ(m) = L

v.

Thus, in this special case the piece of the ellipsoid E(v) that projects radially onto D is asolution of the reflector problem.

Uniqueness. It is clear that any ellipsoid of revolution with the same foci O and vis also a solution in this case. However, as it follows from (6) and (8), the relation betweenpolar radii of any two such solutions is not linear. In particular, the two solutions are notrelated by a homothety relative to O. Thus, we see that in this case the reflector problemhas infinitely many geometrically distinct solutions.

3.2 Reflectors Defined by a Finite Number of Ellipsoids

Consider now a more general case when the set T consists of a finite number of points and theoutput power pattern required on T is a collection of masses concentrated at these points.In this case, our plan is to construct the reflector surface from pieces of confocal ellipsoids ofrevolution with the common focus at O and axes with directions defined by the points in T .

More specifically, let T be given as a finite set of points, T = {v1, ...,vN}, and therequired output power pattern on T is given as a sum

LT =N

∑

i=1

Liδ(vi),

where Li are positive numbers equal to the output powers required at points vi. The functionLT should be understood here in the distributional sense, that is, when applied to a unitprobe at the point vi, it produces the power Li. Similarly, the right hand side of the balanceequation (4) should be understood in this case as the total power Q =

∑Ni=1 Li and we assume

that the numbers Li are such that the following balance equation is satisfied:∫

DI(m)dσ(m) = Q. (9)

Ultimately, the reflector surface R in this case will be built of pieces of confocalellipsoids of revolution E(vi) with the common focus at O and axes of revolution alongOvi, i = 1, ..., N . However, the construction of such a reflector in this case is a bit moreinvolved than before and we need to introduce precise definitions.

To simplify the notation we put Ei ≡ E(vi). The polar radius of Ei is given by

ρi(m) =di

1 − ǫi(m · vi), (10)

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where di > 0 and ǫi, 0 < ǫi < 1, are, respectively, the focal parameter and eccentricity of Ei,while m is an arbitrary point on the sphere S. The solid body bounded by Ei is denoted byBi.

Next, we consider the intersection of the bodies Bi and put

B =N⋂

i=1

Bi.

Since each of Bi is a convex body, the set B is also convex.

The focal parameters of the ellipsoids Ei can be selected so that the points vi liestrictly inside B for all i. In order to establish this we first note that by (10) we have for allm on S

ρi(m) > di/2. (11)

The inequality in (11) is strict because 0 < ǫi < 1. The inequality (11) means that a ballof radius di/2 with the center at O fits strictly inside Ei. Because of the symmetry of Ei aball of the same radius but with the center at the second focus vi is also contained insideEi. Let ω(T ) denote the diameter of the set T , that is, the maximal distance between anytwo points in T . If we choose

di ≥ 2ω (12)

for all i = 1, ..., N than each of the ellipsoids defined by di will contain the set T strictlyinside. Therefore, the solid B and, of course, the surface bounding it will also contains Tstrictly inside. Because the required reflector will be constructed as a part of the boundaryof the solid B, the above property will be important for establishing that no part of thereflector constitutes a blockage to reflected rays (prior to reaching the points vi of the outputaperture). Now we can state the precise definition of a reflector.

Definition 1 The convex surface bounding the solid B is called a reflector defined by the

ellipsoids E1, · · · , EN .

Obviously, any family of ellipsoids Ei, i = 1, ..., N , defines a reflector redirecting therays from the source O to their respective foci vi. However, the amount of energy deliveredat each of the vi depends on how much of the total energy from O is “intercepted” by thecorresponding ellipsoid prior to other ellipsoids in the family. Thus, the main problem is toselect the ellipsoids Ei so that for each i = 1, ..., N the energy arriving at vi is equal to Li.As we will see in the following sections, this is accomplished by solving a system of equationsfor the required focal parameters.

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E 1

E2

O v1

v2

A

C

B1

B2

Figure 2: The geometry of the reflector in 2D

3.2.1 A Special 2D Case

Let us explain the main idea for constructing the reflector in the simple planar case shownon Fig. 2. In this figure E1 and E2 are two ellipses with a common focus O and T consistsof two points v1 and v2. The latter are also the second foci of E1 and E2, respectively. Itfollows from definition 1 that the reflector in this case is the closed arc AB1CB2A. Denotethis reflector by R. Note that every ray emanating from O and reflecting off R passes eitherthrough v1 or v2.

The input aperture is the angle B1OB2. It is convenient to assume that the inputaperture is extended to the entire unit circle centered at O. In order to account for suchaperture extension we assume that outside of the actual input aperture B1OB2 the feedpower density is zero. Then, only the rays in the angle B1OB2 are of interest. If the sumof powers required at v1 and v2 is equal to the total input power (that is, the equation (9)applied to this case is satisfied), then the contribution of each of the ellipses to the power atthe corresponding focus depends on how much of the total input power is intercepted andredirected by the corresponding ellipse. This, on the other hand, can be controlled by theappropriate choice of the focal parameters.

For example, if the focal parameter of E2 is so large that E1 is strictly inside the setbounded by E2 then all the rays from O are intercepted by E1. In this case, there is nopower contribution to v2 and all power goes to v1. Similarly, if the focal parameter of E2 issufficiently small then E2 will intercept all the rays from O and all power will be deliveredto v2. In the former case, keeping the focal parameter of E1 fixed and decreasing the focalparameter of E2 we can achieve any desired distribution of the total power between thepoints v1 and v2. Once the desired distribution is achieved, the part of the reflector outsidethe arc B1AB2 is deleted. Such deletion does not affect the attained energy distribution,since the input power density for directions outside the angle B1OB2 is zero. Then, the

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remaining part of the reflector contained in the angle B1OB2 gives the required reflector.

3.2.2 Constructing Reflectors With Ellipsoids

We now use the same idea to build a reflector in 3D in case when the number of ellipsoidsdefining the reflector may be arbitrary (but finite). We will see later that this is the importantcase in practice. In order to describe the required construction we need to introduce tworelated notions.

First of all, it is convenient to assume that the feed power pattern I(m) is definedover the entire sphere S by setting I(m) ≡ 0 outside D. Then we can work with the entiresphere S as the input aperture. The resulting reflector will be a closed surface. Once thisreflector is constructed we will delete the unnecessary part and obtain the required reflector.

Let T = {v1, ...,vN} and let Ei, i = 1, ..., N , be a family of ellipsoids defining thereflector R as in definition 1. Let Ej be one of the ellipsoids of the family. By definition,the solid B(vj) must contain the reflector R. We need to distinguish the cases when thereflector R is lying strictly inside B(vj) and when such inclusion is not strict, that is, whenEj and R have common points.

Definition 2 We say that Ej is supporting to R if the set Cj = Ej

⋂

R is not empty. The

set Cj is called the “contact” set of Ej with R.

This definition is illustrated in the 2D case on Fig. 3. In this figure, all three ellipses E3

are supporting. However, if the focal radius d3 of E3 is increased the ellipse E3 ceases to besupporting and C3 becomes empty. We need to allow for that because when we construct areflector we do not know a priori if a particular ellipsoid from a given family constitutes apart of the reflector or not.

Now we define a set that determines the contribution of each ellipsoid to the outputpower pattern.

Definition 3 Let R be a reflector as above and E(v) an ellipsoid from the family defining

R. Denote by V (v) the radial projection of C(v) on S by rays from O. This set is called the

visibility set of v.

This notion is illustrated on Fig. 4. Put

G(R;v) =∫

V (v)I(m)dσ(m).

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E 1

E2

O v1

v2

A

C

B1

B2

E 3

v3

Figure 3: Supporting ellipses

E1

E2

v 1

v 2

V(v )2

V(v )1

S

Figure 4: Visibility sets

Evidently, the visibility set V (v) consists of directions m of all rays emitted from O which arereflected by R and reach v. Therefore, the quantity G(R;v) gives the total power deliveredby reflector R to the point v. Note, that because of the way we redefined I(m) over S, onlythe contribution to this integral from the set D

⋂

V (v) plays a role. Also, note that if C(v)is empty then V (v) is empty, and G(R;v) = 0.

Now, the reflector problem with the output aperture consisting of a finite number ofpoints can be formulated as follows. Let T = {v1, ...,vN} and L1, ..., LN are positive numberssuch that the balance condition (9) is satisfied. The problem is to determine a reflector Rdefined by ellipsoids Ei with foci at O and vi, i = 1, 2, ..., N, such that

G(R;vi) = Li for each i = 1, 2, ..., N. (13)

Since each of the ellipsoids Ei is uniquely defined by its focal parameter di, the determinationof the reflector R means determining N focal parameters d1, ..., dN so that the correspondingreflector satisfies (13).

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The system (13) is a nonlinear system with respect to the variables d1, ..., dN and inthe next section we describe a procedure for solving it.

3.2.3 The Algorithm for Solving System (13)

The system (13) is solved by an iterative procedure starting with an initial reflector R0. Thereflector R0 is constructed from the data. We first describe this construction.

Let R be some reflector defined by ellipsoids Ei, i = 1, ..., N , with one of the foci atO and the other at v1, ...,vN , respectively. We want to show that the focal parameters ofEi can be selected so that for all m in D and all i > 1 we have

ρ1(m) ≤ ρi(m). (14)

This inequality implies that all the energy from the source O is redirected by the reflectorto the point v1.

We let, as before, M = maxi |vi| and ω = diameter of T . Naturally, it is alwaysassumed that the set T is at a positive distance from O, and thus mini |vi| > 0. Let alsoγi = maxD m · vi and γ = maxi γi. Also, it is assumed that the rays from the source passingthrough the input aperture are separated from the rays from the source directed towardspoints on the output aperture, that is,

γ < 1. (15)

Putd1 = αM, (16)

where α is a positive number which is a design parameter whose specific choice will bediscussed in section 3.2.4. Then, using (7), we obtain

2ǫ1

1 − ǫ21

=|v1|d1

≤ 1

α

andǫ1 ≤

√1 + α2 − α.

It follows from (10) that for the polar radius of the ellipsoid E1 over the input aperture Dwe have the estimate

ρ1(m) =d1

1 − ǫ1m · v1≤ αM

1 − (√

1 + α2 − α)γ1

, (17)

that is, the inequality (14) is satisfied.

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Put β = 2α

1−(√

1+α2−α)γ1

and di = βM. Then for points m in domain D we obtain,

using (10) and (17),

ρi(m) =di

1 − ǫim · vi

≥ di

2=

βM

2=

αM

1 − (√

1 + α2 − α)γ1

≥ ρ1(m). (18)

Let R0 denote the reflector constructed with ellipsoids E1, ..., EN . Because of (18)the rays from O through D are all intercepted first by E1. Using the notation introducedearlier, and the assumption (9), we can express this fact as

G(R0;v1) =∫

SI(m)dσ(m) = Q,

G(R0;vi) = 0, for i > 1.

The construction of the initial reflector is now complete.

We now begin modifying the reflector R0. In this process only the focal parameters ofellipsoids Ei, i > 1, will be changing, while E1 will remain fixed. We begin with E2. Decreasecontinuously the focal parameter d2 and consider the reflectors defined by the same ellipsoidsas before except for E2 which is replaced by the ellipsoid with reduced focal parameter d2.The “intermediate” reflectors are denoted by R0

int. The parameter d2 is decreased untilthe equality G(R0

int;v2) = L2 is achieved. This must happen because G(R0int;v2) changes

continuously with d2 on the segment [0, Q]. The continuity of G(R0int;v2) follows from

the fact that it depends continuously on the visibility set V (v2) and the latter increasescontinuously as d2 decreases. More precisely, if in the initial position the ellipsoid E2 issupporting then V (v2) will increase when d2 is decreased. If E2 is not supporting at theinitial stage then V (v2) is empty until E2 becomes supporting (when d2 becomes sufficientlysmall). In addition, if d2 is near zero then the ellipsoid E2 is intercepting all the rays fromO through D and the corresponding visibility set V (v2) = S. Then

G(R0int; V (v2)) =

∫

SI(m)dσ(m) = Q.

The latter is impossible, since by construction d2 was decreasing only while G(R0int;v2) ≤ L2.

Let us show that the equality G(R0int;v2) = L2 is attained while

d2 > d1(1 − γ)/2. (19)

This estimate is required in order to show that the ellipsoid E2 during the above modification(and the subsequent ones to be described below) does not degenerate into a straight linesegment joining O and v2.

To establish the required property assume that for some intermediate position ofE2 we have d2 = d1(1 − γ)/2. (Since in the initial position d2 ≥ 2d1 and d2 is changing

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continuously, it is clear that if (19) is not true then d2 = d1(1 − γ)/2 must hold for someconfiguration of ellipsoids forming an intermediate reflector.) Then for points m in D wehave

ρ2(m) =d1(1 − γ)

2(1 − ǫ2m · v2)≤ d1(1 − γ)

2(1 − ǫ2γ2)≤ d1(1 − γ)

2(1 − γ2)≤ d1

2.

On the other hand, it follows from (10) that

ρ1(m) ≥ d1

2.

Therefore, for points m in Dρ2(m) ≤ ρ1(m).

Then we must haveG(R0

int;v2) =∫

SI(m)dσ(m) = Q > L2,

which is in contradiction with the way E2 was modified. The estimate (19) is established.

We fix the value d2 for which G(R0int;v2) = L2 and the corresponding ellipsoid E2.

If N = 2 then we are done; otherwise, the same procedure is repeated for each of theremaining ellipsoids for i > 2 (it is assumed that N ≥ 2; the case when N = 1 was describedin section 3.1). The resulting reflector we denote by R1. It is clear that if the ellipsoid Ei issupporting and the focal parameter di is decreasing then the energy G(R0

int;vi) is increasingwhile G(R0

int;vj) are non-increasing for all j 6= i. That is, the intermediate reflectors R0int

and R1 satisfy the inequalitiesG(R1;v1) ≥ L1,

G(R1;vi) ≤ Li, for i > 1,

andN

∑

i=1

G(R1;vi) =N

∑

i=1

Li =∫

SI(m)dσ(m).

If G(R1;vi) = Li for each i = 1, 2, ..., N, the process terminates. Otherwise, itcontinues by resetting R0 = R1 and repeating the above steps. As a result, a sequence ofreflectors Rk, k = 0, 1, ..., with focal parameters dk

1, ..., dkN is constructed. For each i > 1

dki ≥ dk+1

i . (20)

In the terminology of partial differential equations the reflectors Rk are supersolutions ofthe system (13). The estimate (19) holds for any di defining ellipsoid Ek

i for i > 1 and anyreflector Rk. The monotonicity (20) of the focal parameters implies that the sequence ofreflectors Rk converges to some reflector R and because of the estimate (19), applied to eachof the elements of the sequences dk

i , none of the ellipsoids Eki degenerates into a straight line

segment. If we denote by d1, ..., dN the focal parameters of the ellipsoids forming R then byconstruction we have

di ≤ dki , for all i ≥ 1 and all k = 0, 1, 2, ... (21)

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Let us show that the focal parameters d1, ..., dN of the reflector R satisfy the system(13). Suppose that for some j > 1 we have

G(R;vj) < Lj .

Then we can decrease dj by a sufficiently small amount and construct a new reflector R′ forwhich

G(R′;v1) ≥ L1,

G(R′;vi) ≤ Li, for i > 1.

For this reflector we have d′j < dj which is impossible because of (21).

It can be shown (see [16]) that for a given tolerance the above process terminates ina finite number of steps with the determined reflector R satisfying (13) within the specifiedtolerance.

The reflector determined by the above algorithm is a closed convex surface. However,its portion outside of the part that projects radially onto D transmits no energy (sinceI(m) ≡ 0 when m ∈ S \ D). Deleting the part of the reflector surface over S \ D we obtainthe final solution (within a user specified tolerance). By letting the tolerance tend to 0 weobtain a sequence of reflectors converging to a reflector satisfying the system (13) exactly.

3.2.4 Uniqueness of Solution to System (13)

As the note at the end of section 3.1 indicates we can not expect that the determined solutionis unique. The same observation applies to reflectors constructed by solving the system (13).Indeed, in the construction of the reflector described in preceding section, the first ellipsoidwas fixed only up to the choice of the parameter α and therefore another reflector satisfying(13) can be constructed by simply choosing differently this parameter.

However, the uniqueness still holds in the following sense. Let R and R′ be twosolutions of the system (13). Let R be determined by ellipsoids with focal parametersd1, d2, ..., dN and R′ by d′

1, d′2, ..., d

′N . Suppose that the two reflectors have a common ellipsoid.

Then the two reflectors coincide. This is proved in [15]. Thus, in particular, fixing theparameter α fixes uniquely a solution.

4 Distributed Output Power Patterns

Let now T be a region on some plane and L(v),v ∈ T, a function defining the requiredoutput power pattern on T . Assume that the balance equation (4) is satisfied. The reflector

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approximating the required reflector within any user specified tolerance is constructed in thiscase via a procedure which reduces this problem to the problem of constructing a reflectordefined by a finite number of ellipsoids. This procedure is as follows. Partition the region Tinto small subregions Ti, i = 1, 2, ..., N so that T =

⋃Ni=1 Ti. Put

Li =∫

Ti

L(v)dµ(v).

Pick a point vi in each Ti and solve the system (13) with the points v1, ...,vN and powersL1, ..., LN at the corresponding points. Denote the resulting reflector by RN . It can beshown that as the diameters of the sets Ti shrink to zero the corresponding sequence ofreflectors {RN} converges to a reflector R solving the reflector problem with the powerpattern L specified on T . Furthermore, for a given tolerance one can construct a reflectorapproximating R within this tolerance in a finite number of steps. The mathematics here issomewhat involved and we refer the reader for further details to the papers [15, 16].

5 Avoiding Blockage

5.1 Avoiding Self-blockage

Let us show now that by appropriate choice of the parameter α in (16) the reflector can bedesigned so that self-blockage can be completely avoided. By self-blockage we mean herea situation when a part of the output aperture T is blocked by some parts of the reflectorsurface from the rays reflected off other parts of the reflector.

Again, from the point of view of numerics and applications, the case when the reflectoris constructed with a finite number of ellipsoids is the one that needs to be considered thoughthe described result is also valid in general. It has been explained at the end of section 3.2that with our construction of a reflector the self-blockage will not happen if for all ellipsoidsforming the reflector the focal parameters satisfy the inequality (12). Therefore, it followsfrom (19) that if α is chosen so that

α ≥ 4ω

M(1 − γ)(22)

then, setting d1 = αM , we are guaranteed that all ellipsoids in the constructed reflector willcontain T inside and no self-blockage will occur.

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5.2 Avoiding Blockage by the Feed

A common problem in axially symmetric reflectors is that if the radiation source is positionedon the axis then it becomes a secondary scatterer and that leads to energy losses.

From the construction described in section 3.2.3 it follows that the following conditionguarantees that no blockage of reflected rays will occur. Let CD denote a cone generated byrays from O through the points in the input aperture D. Let l1, ..., lN be the rays from Opassing through v1, ...,vN , respectively. Since our reflectors are constructed of ellipsoids, itis clear that for a reflected ray to pass through O and vi the input aperture D should includea direction m = −vi. Such possibility is excluded if we require that no two rays, one fromCD and one from the set l1, ..., lN , form together a complete straight line.

When the target is a specified region not limited to a finite number of points thiscondition should be formulated as follows. Let CT be a cone generated by rays from Othrough the points in the output aperture T . In order to avoid blockage of reflected rays bythe source O, the positions of the input aperture D and output aperture T should be suchthat no two rays, one from CD and one from CT , form together a complete straight line. Inpractice, of course, one should also take into account the size of the feed.

6 Examples

In this section, we present four examples calculated numerically with the algorithm de-scribed in section 3.2.3. The code used for computing solutions presented in this paper is asignificantly improved version of the original code used for calculating the results in [16].

EXAMPLE 1. In the first example, shown schematically on Fig. 5, there were used25 nodes uniformly distributed over the output aperture T . The solution was constructedwith 25 ellipsoids with one focus at the source O and second foci at the nodes on T . Inthis example, the diameter of T ω(T ) =

√2m, distance from the source to the central

node on T is h = 200m, the maximum among all distances from the source to nodes onT is ≈ 200.005m, and the parameter γ ≈ −.4938639. Using the inequality (22), it canbe determined that self-blockage will be avoided if the focal parameter of the first ellipsoidd1 ≥ 4ω

1−γ≈ 3.786727. The actual computations were carried out with d1 = 3.8. This gives

for the parameter α ≈ .019. Since 3.79971 ≥ 2ω = 2√

2, the condition (12) is satisfied. Infact, this inequality shows that in this case we could have taken d1 just slightly bigger than2√

2 (see example 2 below).

Because the output energy pattern was required to be uniform and the variation of the

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TOUTPUTAPERTURE T

INPUT APERTURE D,CIRCULAR, WITH ANGULAR RADIUS OF 15

ENERGY FEED I( θ,φ) = 10exp(−3 θ), θ − ANGULAR DISTANCEFROM CENTRAL RAY

FEED

REFLECTOR

OUTPUT ENERGY PATTERN L(v) IS REQUIRED TO BE U NIFORM

L(v)

h

h − DISTANCE IN MET ERS FROM SOURCE TO THE P LANE OF OUTPUT APERTURE

A

OB

AOB = 135 (= 90 IN EXAMP LE 2)

o

o

o

Figure 5: Geometry of the single reflector in example 1.

feed energy density is not very significant all of the resulting ellipsoids had focal parametersclose to 3.8 with the maximum = 3.80618 and minimum = 3.79972. It took 348 iterationsto reach a solution for which

max | computed output density distribution − required uniform distribution| ≤ 0.01.

The wall clock run-time was 19 minutes on a 300MHZ Silicon Graphics computer with oneprocessor. The designed reflector has the following dimensions: the distance from the sourceto the reflector along the central ray is 2.25m and the diameter of the reflector is 1.27m.

It is important to note that in the above inequality we are comparing output densitydistribution with the density distribution for the true solution and the bound on the righthand side is user-specified. If this bound is decreased the number of iterations will increase,in some cases, quite significantly.

A snapshot of the computer generated picture of the found reflector is shown in Fig.6. Note that the apex of the reflector is displaced slightly in the downward direction. Thisis due to the fact that the output aperture is at an angle with the input aperture ( = 135o

between central rays from the source to the input aperture and to the target; see Fig. 5). Asthe angle AOB becomes smaller this displacement becomes more significant; see example 2below.

EXAMPLE 2. In the second example the setting was similar to the example 1,but the following parameters were changed: 6 AOB = 90o and the focal parameter of thefirst ellipsoid was taken = 1.7. The diameter of this reflector is 1.3m. The distance from

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Figure 6: A view (from the back) of the reflector in example 1.

the source to the reflector along the central ray is 1.7m. The run-time in this case was 13minutes. A snapshot of the computer generated picture of the reflector is shown in Fig. 7.It clearly shows further displacement of the surface apex in downward direction.

EXAMPLE 3. In the third example all data were the same as in example 1 exceptfor the distance h to the target aperture which was taken = 200, 000m. This resulted onlya small change in the choice of the focal parameter of the first ellipsoid which was taken= 3.772. The diameter of the determined reflector is 1.26m and the distance from the sourceto reflector along the central ray is 2.23m. A picture of this reflector is shown on Fig. 8.This example shows that the proposed techniques can be also used for designing reflectorswith pre-specified energy patterns across far-field regions.

EXAMPLE 4. The purpose of this example is to illustrate the behavior of thealgorithm when the number of ellipsoids in the problem increases. In this example, all ofthe data remains the same as in example 1 except for the number of ellipsoids which wastaken to be 36. In this case, the algorithm made 613 iterations; the run-time was 1 hour 20minutes. Its diameter is 1.27m and the distance from source to reflector along the centralray is 2.24m.

The example 4 and other experiments that we performed show that the computationaltime grows with the number of ellipsoids used in the solution. While testing the code onnumerous examples we observed that the algorithm finds quickly a relatively good approxi-mation and then it slows down as the iterates approach the true solution. Such behavior is

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Figure 7: A view of the reflector in example 2.

very typical for numerical schemes dealing with equations containing strong nonlinearities.In [21] we presented a strategy for improving significantly the convergence properties in asimilar case of a nonlinear problem. Currently, we are investigating the applicability of thisstrategy to the problem considered here.

It should be noted, however, that for practical design applications the current versionof the code will most likely be sufficient. The fact that the algorithm does not require theknowledge of an initial approximation makes it particularly attractive. Of course, the factthat it is supported by a rigorous mathematical justification is of critical importance, sincethis makes the numerical results reliable.

7 Conclusion

In this paper a new method for synthesis of offset, shaped, single reflector antennas is pre-sented. The synthesis problem is considered in geometrical optics approximation. In con-trast with previously known GO synthesis methods the validity of the method presentedhere is established by mathematically rigorous arguments. Our results completely resolvethe questions of existence and uniqueness of solutions to the above synthesis problem. Is-sues regarding self-blockage and blockage by the feed have been investigated and conditionswhich guarantee avoidance of such blockages are presented. This leads to designs of reflectorantennas with high efficiency. The new method has been implemented in a computer code

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Figure 8: A view of the reflector in example 3.

and several design cases are presented here. The corresponding numerical procedure is aniterative one which starts with an initial solution. A important advantage of our numericalprocedure is that the initial solution can be explicitly and easily determined from the initialdata.

The proposed method and its numerical implementation should have applicationsin various important problems of design of contour beam shaping spacecraft and groundsystems in which high gain is required and offset geometry must be utilized to achieve highefficiency.

Acknowledgement. The research of Vladimir Oliker was partially sponsored by theAFOSR under contract F49620-97-C-0007 and by the Emory University Research Com-mittee.

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