On the design of a reflector antenna


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Introduction

In this paper we investigate the following problem which arises in geometric optics. In the
three-dimensional Euclidean space R3 fix a point O and suppose that a non-isotropic light
source is positioned there. Let S2 be a unit sphere with centre at O and  a domain on S2.
Denote by 0 a surface which projects radially in a one-to-one fashion onto . The surface
is supposed to have a perfect reflection property, that is, no loss of energy occurs when a
beam of light is reflected by it. Suppose a ray is originated from O in the direction x and
is reflected by 0, producing a reflected ray in the direction y. We identify a direction with
a point on S2. Then we get a mapping T of   S2 into S2.
The problem is to recover the reflecting surface 0 so that the reflected rays cover a
prescribed region D of a far field sphere and the density of the distribution of the reflected
rays is a function of the direction prescribed in advance. Below we refer to this as the light
reflection problem. For more details see [15].


Existence

Let  and D be two disjoint domains on the unit sphere S2 with Lipschitz boundaries
and f .x/ a positive function defined on D; suppose the rays originate from the origin
with density .x/; x 2 . We look for a surface 0 D 0 D fx
.x/I x 2 g whose
radial projection on S2 is , so that the directions of the reflected rays cover D and its
distribution density is equal to f . Here we identify a direction x with a point x on S2.
In section 1.1 we derive the analytic formulation for this problem, which is an equation of
Monge–Amp`ere type. In section 1.2 we introduce the concept of generalized solutions and
in section 1.3 we prove the existence of generalized solutions to the Dirichlet problem of the
equation. The basic idea in dealing with the Dirichlet problem is to approach the solution
by polyhedrons, which has been used by Alexandrov, Pogorelov and others in studying the
classical Monge–Amp`ere equation, see [12]. Using the existence result in section 1.3 we
prove in section 1.4 the existence of the reflecting surfaces.


Uniqueness and regularity

In this section we deal with the uniqueness and regularity for generalized solutions. The
uniqueness will follow from the definition of generalized solutions, see theorem 2.2 below.
The main task of this section is to prove the regularity for solutions of (1.2). There has
been a lot of work dealing with two-dimensional Monge–Amp`ere equations (see [12, 13]),
but no result has covered the equation (1.2) because of various conditions imposed in [12]
and [13]. The uniqueness of smooth solutions to the problem (1.2) and (1.3) has also been
discussed by Marder [6].