Non-Linear Synthesis Problems for Plane Radiating Systems According to the Prescribed Power Directivity Pattern

A variational formulation of the synthesis problem for plane radiating systems according to the prescribed power directivity pattern (DP) is considered. The function representing the mean-square deviation of the prescribed and synthesized power DPs and containing the additional term with squared norm of the current or field in the antenna aperture is considered as the criterion of optimization. Freedom to choose the phase DP is used to improve the proximity of the prescribed and synthesized DPs. In such formulation, the classes of non-linear problems, for which the non-uniqueness of solutions, their branching and bifurcation are characteristic, arise. The properties of solutions depend on the electric size of radiating system and prescribed power DP. From a practical point of view, the existence of different solutions creating the same or similar DPs, gives the opportunity to choose the solution that has a simpler implementation. The synthesis problems for plane radiating systems and plane arrays are considered.

In many practical applications at the design stage of antennas and arrays the requirements are imposed only to the power DP [17,18].At that, the freedom of choice among the phase DPs is used for improvement of the approximation quality of the synthesized DP's squared module to the prescribed power DP.Appearing at that a necessity to approximate the real positive function (prescribed power DP) by square of module of the complex valued function generates the classes of the non-linear synthesis problems.Non-uniqueness and bifurcation of solutions are the characteristic feature of such problems.From the practical point of view, the existence of several types of solutions, which create the same or close power DPs, gives a possibility to choose that solution, which has simpler realization.
The problems of quantity and properties of the existing solutions using the methods of non-linear analysis [19] were investigated partially for the synthesis problems of linear antenna arrays in [10,11,15,18,20,21].It was shown that the non-uniqueness and branching of solutions are the characteristics for these problems in the both cases if the mutual coupling of separate radiators of array is not being taken into account, and by being taken into account of such coupling.The properties of existing solutions depend on the value of physical parameters of array and properties of the prescribed power DP.
Investigation of solutions of the nonlinear synthesis problems for plane radiating systems substantively differs from such investigation of the synthesis problems for linear antennas and arrays.In particular, unlike the existing the branching points for the synthesis problems of linear antenna and arrays, the lines of branching of the solutions appear in the synthesis problems for the plane antenna systems.The problem to search these lines is insufficiently investigated two-parameter non-linear spectral problem [22].
In the proposed work, the variational statement of the synthesis problems for the plane radiating systems according to the prescribed power DP is examined.These problems are reduced to the investigation and numerical solution of the non-linear integral equations of Hammerstein's type.The kernel of these equations contains two physical parameters, which describe the geometry of radiating aperture and region where the prescribed power DP is given.
The method for search of the bifurcation lines of the optimal solutions is developed; the justified algorithms to determine the optimal solutions of the synthesis problems and to search the existing solutions of various types for the synthesis equations are elaborated.It is shown that freedom in choosing the obtained optimal phase DP improves the quality of approximation of the synthesized DP to the prescribed one considerably, as compared with the synthesis in the class of the cophased DPs.In particular, this allows in some cases to decrease the dimensions of radiating system up to 30%, as compared with the synthesis in the class of cophased DPs, remaining the effectiveness of the synthesis the same.

Statement of Problem
The statement of the synthesis problem according to the prescribed power DP is used in many practical applications [1,5,8].In spite of fact that the power DP N can be easy obtained from the amplitude DP f : ( ), and vice versa, the synthesis problems of the given amplitude DP F 0 and the given power DP are the different problems.For example, if will not be optimal solution of the analogous problem according to the prescribed power DP N 0 .
In general case the calculation of the DP f of antenna by the currents (fields) I in the antenna aperture is realized by action of some linear bounded operator A [11] The functions f and I can be either the vectors if the vector character of electromagnetic field is taking into account in the process of statement and solving the synthesis problem, or scalar functions if some simplifications for antenna are applicable.
The specific forms of operator A, functions f and I will be presented below in the process of solving the synthesis problems for the investigated radiating systems.We demonstrate here the peculiarities of the synthesis problem according to the prescribed power DP N 0 on example of a plane continuous aperture.
Introducing the specific generalized coordinate system [6] we pass from the vectorial statement of the synthesis problem to two scalar problems.Separation of the and -components of the DP and current's (field's) components in the antenna elements is assumed too.We consider below the scalar synthesis problems for plane radiating apertures and arrays.The geometry of synthesis problem for plane rectangular aperture is shown in Figure 1.The DP of plane aperture is determined by formula [3] , , where are the generalized dimensionless coordinates that allows to consider the synthesis problem in two relatively-perpendicular planes independently [1,6].The parameters is wave number.Use of generalized parameters c 1 and c 2 allows to consider the synthesis problem in wide region of frequencies, including the submm, mm, cm and longer lengths of the EM field's waves.
The variational formulation of the synthesis problem consists in minimization of functional The respective Euler's equation relative to optimal current distribution I is given by and respective equation for the DP f has form The kernel , , ; ,  K s s s s c c   in the general case is expressed as and depends essentially on the form of aperture.For the rectangular aperture (10) has form If the optimal solution * f to (9) is found, then optimal distribution of current is determined by formula Numerical solving Equation ( 9) is simpler than solving Equation ( 8), therefore we use below in the process of solving the specific synthesis problems solution to Equation (9) and determine the optimal current distribution * I by formula (12).

Determination of Bifurcation Lines: Case of the Plane Rectangular Aperture
The non-linear Equation ( 9) has only zero solution at small values of parameters c 1 and c 2 which contain in the kernel (11).The non-zero solutions appear, namely the bifurcation of solutions occurs, if c 1 and c 2 increase.The bifurcated solutions have the different properties depending on the form of the prescribed power DP N 0 , values c 1 and c 2 .For one's turn, these solutions correspond to the different optimal current distributions * I , calculated by (12).Of course, the different sets of functions * and * f I give the different values of functional   .In this connection, investigation of the branching process, search of different solutions, and determination of those provide the minimal value of   is of practical interest.Accordingly to the theory of branching the solutions [23], the bifurcation points of solutions to (9) are determined using the next linear equation On the other hand, this equation can be considered as the non-linear two-parameter spectral problem.We use the method of implicit function proposed in [22] for determination of values of the parameters c 1 and c 2 at which (13) has non-zero solutions.The non-zero solutions originate at the values c 1 and c 2 , for which except the evident eigen value   of Equation ( 13) appear.
Using the cubature formulas we obtain on the basis of (13) the homogeneous linear algebraic system (LAS) of P-th order for determination of eigenfunctions and corresponding eigenvalues Here the number P corresponds to quantity of discretization points.In Equation ( 14) , is vector of unknowns eigenfunctions, In order LAS (14) has different from zero solutions, it is necessary that parameters c 1 and c 2 be the solutions of equation , det , where P I is unit matrix.According to the method of implicit functions, search of solutions to (15) is reduced to the next Cauchy problem Initial conditions (17) are determined by solving the auxiliary non-linear one-parameter spectral problem on the beam 2 As result, we receive the next equation for finding the zeros of determinant which are determined by some known method, for example, by the bisection method.Consequently, solving the Cauchy problem, we obtain a set of bifurcation points of the solution to (9).As the matrix of ( 15) is analytical function of parameters c 1 and c 2 , the respective curves, which describe the spectrum's connected components of ( 14), are the continuous solutions of Cauchy problem ( 16)- (17).
Note that eigenfunctions of the problem (14), which correspond to bifurcation points, describe the properties of solutions of the initial Equation ( 9) in the first approximation.
We consider the results related to determination of the bifurcation lines for the synthesis problem of plane aperture at the prescribed power DP The eigenfunctions of LAS ( 14) are used in the process of construction of the bifurcated solutions of the nonlinear Equation (9), which branch off the zero solution.The properties of bifurcated solutions depend on the properties of symmetry (parity) of the prescribed DP N 0 and on the properties of eigenfunctions.In the process of construction of the bifurcated solutions one should take into account that the properties of the eigenfunctions differ for the various set of parameters c 1 and c 2 in the plane .

Elliptic Aperture
The synthesis problem for an elliptic aperture is reduced to minimization of functional (7), and the respective Euler's equation for determination of optimal current in the aperture is determined by formula similar to (8).By this, integration is carried out over the elliptic domain of the antenna aperture.
The aperture DP is determined by formula similar to (2).The kernel (10) for elliptic aperture is given by , , , ; , ; , 2 cos where a and b are semiaxes of ellipse.The optimal current is determined by the DP * f found from (9) as follows The numerical results of synthesis for specific prescribed power DPs N 0 will be given in Subsection 5.2.

Plane Equidistant Rectangular Array, Bifurcation of Solutions
In the case of scalar consideration of the synthesis problem one can restrict by the following representation for DP (array multiplier) , where .In this connection, one can consider the synthesis problem for such array within the limits of one period.
The minimizing functional has form similar to (7), the Euler's equation for the optimal currents I and DP f are determined by formulas similar to ( 8) and ( 9), respectively.
The kernel K is degenerated and it is presented as the product of two independent functions   ,    c ,c ) , , , , , lines obtained by solving the Cauchy problem ( 16) with the initial values ( 17) are presented in Figure 5. Similarly to the case of rectangular plane aperture, the bifurcation lines are symmetrical relatively the axes 1 , 2 , and intersection of the curves corresponding to the second and third eigenvalues is observed.

Plane Hexagonal Array, Bifurcation of Solutions
The DP of hexagonal array has the form similar to (21): , The investigation of solution to respective Euler's equations is carried out in the manner similar to the case of plane aperture.If the optimal DP * f by solving equation type of ( 9) is found, the optimal current distribution * I is given by The number of linear subarrays along the axes (coordinate 2 Oy s ) is odd, and number of radiators in the central linear subarray along the axes (coordinate 1 Ox s ) is odd too.For such array's geometry, the kernel of integral Equation ( 9) can be written in form (27).
In the case when the above assumption regarding the number of radiators in the subarrays does not satisfied, (27) has more complicate form.

(25)
The numerical results for rectangular array are presented for the problems of determination of the bifurcation lines for the array consisting of N 2 × M 2 = 11 × 11 radiators for the prescribed power DP Search of bifurcation lines for the solutions to nonlinear Equation ( 9) is carried out for array consisting of 61 radiators for the prescribed power DP the respective beams are presented in Figure 6.The points ( ), for which eigen values ) by solving the transcendental equation type of ( 18) is carried out.The points ( ) considered as initial approximations in this procedure.The bijection method is used for this goal.In the third finish stage after the specifications of the values ( ), the bifurcation lines in the plane ( 1 2 ) are determined by solving Chauchy problem ( 16)- (17).In Figure 7, the curves corresponding to the first four eigenvalues , , shown.Similarly to previous investigations, the cross section of curves corresponding to the second and third eigenvalues is observed.
Note that the problem of specification of the roots of Equation ( 18), as well as correct solving the Cauchy problem ( 16)-( 17) are quite complicate computational problems, because they require for their solving a series of computational experiments with the various initial approximations.

Computational Results of Synthesis
The results of numerical calculations are presented for the synthesis problems of several prescribed power DPs.We use the iterative process [24]    for numerical solution of ( 9).Parameter is used for convergence acceleration of (28).
The numerical results demonstrate that for the certain ranges of the physical parameters of antennas there exist different types of solutions to (9).These solutions provide different degree of approximation of the synthesized power DP to the prescribed one.The investigations which show the dependence of the mean square deviation of the DP on the type of parity of the prescribed initial approximations are carried out for the various types of the considered antennas.
In the process of numerical calculations one substantiated that for the prescribed power DP with two axes of symmetry, there exist the solutions of different types of parity relative to the coordinates 1 s and odd relatively 2 s ; 3) odd relatively 1 s and even relatively 2 s ; 4) odd relatively 1 s and 2 s .In order to obtain these solutions, one should to give initial approximation of the respective type for (28).

Plane Rectangular Aperture
The computational results related to solving the synthesis problem of plane rectangular aperture are shown at the prescribed power DP 0 1  N  (Figure 8).The problem is reduced to solving the non-linear Equation ( 11) by the successive approximation method (28).In order to obtain OJAPr the different types of solutions, the initial approximations of type 1) -4) were chosen for the iterative process (28).The synthesized power DP axis.This optimal current distribution to solution of Equation ( 9) with initial approximation of type 2).The mean-square deviation of the prescribed DP N 0 and synthesized DP Ox Oy 2 f (the first summand in functional ( 7)) for this type of so-   lution is approximately three times smaller than the mean-square deviation for symmetrical solution of type 1).
In that way, the optimal symmetrical power DP can be created by the non-symmetric current's distribution in the antenna aperture.
As the numerical calculations show, the effectiveness of synthesis essentially depends on the form of the prescribed power DP N 0 , on the parameters 1 and 2 describing size of aperture, and on the type of obtained optimal solution.c c

Plane Circular Aperture
The numerical results for circular aperture (particular case of elliptic aperture considered in Subsection 4.1) are shown for the synthesis of the prescribed power DP in the form of the figure of revolution with zero value in the centre (Figure 12), which is even on 1 s and 2 s  

s s s s s s N s s s s
The synthesized DP 2 f (Figure 13) and optimal amplitude current distributions I (Figures 14-16) at  The cross sections of the synthesized power DPs in the plane 1 demonstrate effectiveness of the synthesis for the obtained optimal current distributions (see Figure 17).Curve 1 corresponds to prescribed power DP N 0 , curves 2, 3, and 4 correspond to amplitude distributions of current presented in Figure 16, Figures 14 and 15, respectively.One can see that the current distribution presented in Figure 16 is more effective, because the synthesized power DP created by it approximate the prescribed DP in the best way; this properties the most evidently is observed in the neighborhood of central zero point.

Plane Rectangular Equidistant Array
The results of synthesis for the prescribed power DP 0 are shown for the solutions with properties 1) (Figure 18) and properties 2) (Figure 19); the equidistant plane array consisting of radiators is examined.In spite of the fact that phase DP    The effectiveness of synthesis also depends on the

Plane Hexagonal Array
The synthesis results are presented for prescribed power DP 0 1 2 ( , ) 1 N s s  and DP (29).The array consisting of 127 radiators is considered, parameter .The synthesis problem is reduced to solving Equation ( 11) by the method of successive approximations (28).The main lobe of synthesized power DP The quality of approximation to the prescribed power DP depends also on the parameter in functional (7).The dependence of mean-square deviation (msd) of the prescribed and synthesized power DPs on the parameter at the various 1 on the beam 2 1 is shown in Figures 27 and 28.Such ratio of 1 and 2 provide the regularity of the array geometry, and, as the   29) is equal to 1.43685.As is easy to see that in the general the value of msd diminishes as decreases, but the value of the second term in functional (7) (containning terms with the current's norm) grows considerably that yields in total increase of .
  

Conclusions
The generalized technique for the synthesis of plane radiating systems according to the prescribed power DP is proposed on the basic of the variational approach.The non-linear synthesis equation is received, the investigation of their solutions is carried out, and it is shown that bifurcation of solutions is a characteristic feature of ob-tained solutions.The numerical results of synthesis, demonstrated for several prescribed power DPs, testify about existence of the various types of solutions which differ by the parity of phase DP.It is shown that the same value of msd in functional   is attained at the various values of 1 and 2 depending on the type of initial approximation.This gives the possibility to choose the solution corresponding to smaller values of 1 and , and as result, to reduce the antenna size on 10% -20%.The use of the generalized parameters 1 and 2 describing the electrical size of antenna and region of the prescribed power DP N 0 allows examining the directivity properties of the investigated radiation systems in the wide range of frequencies and geometrical parameters of antennas.

f
is optimal solution for some variational synthesis problem according to the amplitude DP F then 2 * f

1  , 2 
of antenna; a, b are the geometrical parameters of aperture; are the angles characterizing the domain where DP N 0 is prescribed (N 0

Figure 1 .
Figure 1.Geometry of synthesis problem for plane aperture.

Figure 4 .
Figure 4.The bifurcation points in the plane for rectangular array.
solving the series of auxiliary one-parameter problems, are shown.The bifurcation plot data in this stage.In the second stage, the specification of the values (

Figure 5 .Figure 6 .
Figure 5.The bifurcation lines in the plane for rectangular array. 1 2 (c ,c ) Figure 7.The bifurcation lines in the plane for hexagonal array. 1 2 (c ,c )

2 s
. It is ascertained that for this case there exist various types of solutions to(9) with the next properties of the phase of synthesized DP, namely function

2 f 1 2s 2  2 Figure 11 )
and phase DP arg f are shown in Figures 9 and 10 at , .One can see that the phase DP arg f presented in Figure 10 is even relative to 1 and odd relative to 2s .This indicates that phase DP arg f has the same parity properties that the eigenfunction which corresponds to the eigenvalue   in Figure3(curve corresponds to this value of   creating the synthesized power DP is symmetrical relative to axis and non-symmetrical relative to

Figure 8 Figure 9 . 2 f
Figure 8.The prescribed power DP for rectangular aperture.0 N 1 

Figure 10 .
Figure 10.The phase DP arg f for the rectangular aperture.

Figure 11 .
Figure 11.The optimal non-symmetrical distribution of current.

Figure 12 .
Figure 12.The prescribed power DP (29) in form of the figure of revolution.

1 2 ares and 2 s 2 f
presented.The optimal current distributions correspond to various types of bifurcated solutionscorresponds to amplitude distribution I of current presented in Figure14.As a result this distribution is symmetric relatively two coordinate axes.Even relative to 1 s and odd respect to 2 correspond to non-symmetrical relative to axis amplitude distribution of current in aperture, presented in Figure15.In Figure16, the amplitude distribution I of current, which correspond to odd relative to 1 , is shown.The synthesized power DP presented in Figure13is created by this current distribution.This current distribution essentially differs on those presented in Figures14 and 15 .

Figure 13 . 2 f
Figure 13.The synthesized power DP

Figure 14 .
Figure 14.The symmetric optimal current distribution.

Figure 15 .
Figure 15.The optimal current distribution non-symmetric relative to Oy axis.

Figure 16 .Figure 17 .
Figure 16.The optimal current distribution non-symmetric relative to both axes.

2 f is even relative to 1 s and 2 s
arg f for the latter solution is odd respect to the coordinate 2 s the synthesized power DP , but the value of mean-square deviation of the prescribed and synthesized power DPs for the latter solution is equal to 0.0562 against 0.0897 for the solution of type 1).In order to demonstrate the effectiveness of synthesis for the different type of solutions, the value of functional are shown for the solutions of type 1) -4) at 2 1 , 1 , (see Figure20).The numbers of lines corresponds to types of solutions.One can see that the solution of type 4) provides the worst approximation of the prescribed power DP.The even relative to s solution is more optimal for such

Figure 18 .
Figure 18.The synthesized power DP

Figure 19
Figure 19.The synthesized power DP

Figure 20 .
Figure 20.The value of functional for different types of solution to Equation (11), . 

and 2 s 1 cFigure 21
Figure 21.The synthesized power DP

Figure 22 .
Figure 22.The value of functional  for different types of solution to Equation (11) for prescribed power DP (30).

2 f
for the prescribed N 0 is narrower than the main lobe of prescribed DP.The maximal deviation of the both DPs does not exceed −10 dB in the main lobe; the level of the first side lobe does not exceed −45 dB.The mean-square deviations (values of the first summand in (9)) are equal to 0.3774 and 0.2218 for the first and second prescribed DPs, respectively.The level of the first side lobe for the prescribed DP (29) does not exceed −35 dB.The plots of synthesized power DPs 2 f are shown in Figures 23 and 24.In Figures 25 and 26, the cross-sections of both the prescribed and synthesized DPs in the planes 1 s 0  and 2 s 0  are shown.

Figure 23 Figure 24 . 2 f s  2 f
Figure 23.The synthesized power DP

Figure 27 .
Figure 27.The msd value versus parameter  for the prescribed DP 0 1 N  .

Figure 28 .
Figure 28.The msd value versus parameter  for the prescribed DP (29).numerical calculations showed, gives possibility to obtain close DPs in the planes and . 1 2The maximal value of msd for the prescribed DP 0