Optimization of some aperture antenna performance indices with and without pattern constraints

Optimization of some aperture antenna performance indices

with and without pattern constraints

Citation for published version (APA):

Worm, S. C. J. (1980). Optimization of some aperture antenna performance indices with and without pattern constraints. (EUT report. E, Fac. of Electrical Engineering; Vol. 80-E-112). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1980

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Optimization of some aperture antenna performance indices with and without pattern contraints

by

S.C.J.Worm

•

Department of Electrical Engineering

Eindhoven The Netherlands

OPTIMIZATION OF SOME APERTURE ANTENNA PERFORMANCE INDICES WITH AND WITHOUT

PATTERN CONSTRAINTS by S.C.J. Worm TH-Report 80-E-112 ISBN 90-6144-112-9 Eindhoven August 1980

Contents

Abstract.

Acknowledgements.

I. Introduction.

2. Fields and related parameters.

3. The optimization procedure.

4. Some examples of circular and annular apertures.

4. I. The circular aperture.

4.1.1. Aperture fields with f (I) z _{0 and f' (I)}

4. I. 2. Aperture fields with f (I)

i'

0 and f' (I) 4.1.3. Aperture fields with f(l) z 0 and f'(I)

4.2. The annular aperture.

4.2.1. Zero-edge aperture distributions.

References. Appendix. Figures.

i'

o.

z

o.

o.

0.2 0.3 1.1 2. I 3. I 4. I 4.2 4.2 4. IS 4.19 4.27 4.27 RI Al AS

ABSTRACT

For circular and annular apertures an investigation is carried out of the optimization of some antenna performance indices. These are the well-known efficiency

n,

the less well-known spread of radiated power

0

2 and a new index, namely the ratio

n/0

2. For the purposes of this investigation some new Bessel-function integrals have been evaluated. The performance indices are written as ratios of two Hermitian quadratic forms. The relative extremes of these ratios are the roots of characteristic equations. Optimizations have been carried out for several types of

aperture distribution. Analytical, numerical and graphical results are given for the optimization of unconstrained radiation patterns. Limit . values of the indi~es, corresponding modal excitation coefficients,

aperture fields and far fields are determined.

Furthermore, computations are made when pattern constraints are introduced. These constraints are introduced in such a way that the indices remain ratios of two Hermitian quadratic forms.

The constrained optimizations result in normal pencil beams, flat-top beams or patterns with constant 3dB beam width while, at the same time, the side lobe extremes are prescribed and a parameter is optimized. Results of these constrained optimizations are given in tables and graphs.

Worm, S.C.J.

OPTIMIZATION OF SOME APERTURE ANTENNA PERFORMANCE INDICES WITH AND WITHOUT PATTERN CONSTRAINTS.

Department of Electrical Engineering, Eindhoven University of Technology, 1980.

TH-Report 80-E-112

Address of the author:

ir. S.C . .). Worm,

Group Electromagnetism and Circuit Theory, Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513,

5600 MB EINDHOVEN, The Netherlands

ACKNOWLEDGEMENTS

The author wishes to express his thanks to the following persons: Dr. M. Jeuken and Dr. V. Vokurka of the Department of Electrical Engineering for stimulating discussions,

Prof.Dr. J. Boersma and drs. P. de Doelder of the Department of Mathematics for the evaluation of some Bessel-function integrals.

I. INTRODUCTION

In practical antenna applications there can be a need to tailor a radiation pattern to a specified shape and to improve a performance index of the antenna system at the same time. In this way the influence of unwanted signal sources can be reduced to an acceptable level or totally eliminated with the smallest possible index deterioration. This differs from synthesis techniques in which only the pattern shape is modified [12], [13], [20], [26], or from optimization techniques without pattern constraints [3], [18], [22].

Many performance indices of antennas can be expressed as ratios of two Hermitian quadratic forms. Commonly used indices are the specific directivity or efficiency and the ratio of the directivity to the antenna noise temperature. The relative extremes of such a ratio of quadratic forms are the roots of a characteristic equation [lOa].

The pattern constraints can be applied to sidelobe peak levels [23], [24], directions of pattern nulls or sidelobes [9] and to the main-lobe beam width [15].

Techniques for index optimization with pattern constraints are the method of Lagrange multipliers, which results in rather complicated numerical procedures, and a matrix method in which the dimensions of vectors and matrices reduce, depending on the number of constraints. This matrix method has been used for the maximization of the directive gain of an aerial array while specifying the directions of sidelobes and/or pattern nulls [9]. However, the method is applicable whenever a performance index can be written as a ratio of two Hermitian quadratic forms or a product of two independent ratios. For instance the constrained efficiency optimi-zation of a rectangular aperture results in the independent efficiency optimization in the two principal planes if the aperture distribution consists of the product of two independent distributions. Furthermore,

the method is not restricted to direction constraints but can be applied to level constraints as well.

In this report the above-mentioned matrix method is used for circular and annular apertures with several types of aperture distribution. The optimized performance indices are the efficiency

n,

the spread of radiated power

a

2

2 and a new performance index

n/a .

In section 2 the indices are written as ratios of two Hermitian quadratic forms. The variables are the modal excitation coefficients of the aperture field expansions. The elements of the Hermitian matrices depend on the index to be optimized and the type of aperture distribution.

In section 3 the optimization theorem for a ratio of two Hermitian

quadratic forms is outlined and pattern constraints are formally introduced. The theorem provides the optimum value of an index and the corresponding modal excitation coefficients.

In section 4 the aperture distributions are described and the elements of the Hermitian matrices are determined. Analytical and numerical optimizations are then carried out. Results are given in formulas, tables and graphs. At the end of the subsections summarizing conclusions are drawn.

In the appendix, results are given of the evaluation of some Bessel-function integrals. A full treatment can be found in [2], in which the private

2. FIELDS AND RELATED PARAMETERS

In this section we define fields and parameters relevant to our optimization problems.

Assume a circular or annular aperture with outer diameter D. The aperture points can be described with polar coordinates p and $, with

o ::

p :: D/2 and 0 :: $ < 2lT. Instead of p we will use the normalized variable r = 2p/D.

We restrict ourselves to $-independent aperture distributions fer) which can be written as series of N terms as follows,

N fer) =

_L

a e i f r l < r < 1 , n=1 n n (2. 1 ) = 0 i f r > I.

For circular apertures r

l = 0 and for annular apertures 0 < rl < I. The excitation coefficients a and the elementary functions e are taken to

n n

be real. In section 4 we will introduce functions en such that gn(u), eq. (2.2a), exists.

The far field g(u) and the partial far field g (u) are taken as

n

1

g(u) =

f

fer) J (ur)rdr o and g (u) n 1

f

e J (ur)rdr n 0 r l (2.2) (2.2a) In these expressions zeroth order and u

J is the Bessel function of the first kind and o

sin(8)lTD/A , (A is the wavelength and 8 is the angle o 0

with the main beam axis).

The far-field power pattern p(u) is

p(u) = g2(u).

The power Pr radiated by the aperture is 1 2 Pr

f

f (r)rdr. r l (2.3) (2.4)

The second moment m

2 of the far-field radiated power with respect to 2

the axis u = 0 is found by integrating u p(u). This results in 1TD

IA

2 m 2

=

J

_o u p(u)udu 1TD/A 3

J

u p(u)du. o (2.5)

In accordance with [16] the upper limit of the integral will be replaced by 00 if the aperture illumination is continuous and approaches zero at

the edges. This means that the reactive power is neglected and that the radiated power is set equal to the total power.

The performance indices which we want to optimize separately are the efficiency

n,

the normalized second moment

a

2 and the efficiency power-spread ratio

nla

Z defined respectively by

n

=

2 p(O)/Pr (Z.6) Z

m/

P

r

a

=

,

(Z.7) 2 2p(O)/m

_z·

nla

=

(2.8)

The formula for

n

can be deduced from the expression for the maximum gain G

M = nG o with G 0 the maximum gain of the uniformly illuminated aperture

[ Z5].

The normalized second moment of the far-field radiated power with respect to the beam axis is a measure of the spread of radiated power from the same axis.

We redefine the equations (Z.I) - (2.8) in terms of vectors and matrices in order to determine the optimum values of the performance indices and the related excitation coefficients. The optimum value of a performance index is an eigenvalue of an optimization problem and the excitation coeffi-cients are the elements of the corresponding eigenvector.

The aperture field can be written as an inner product

f(r) = <a,e> if r

l < r < I ,

(Z.9)

0 i f r > I ,

with e> aN element column vector having elements e _n' and with <a a N

element row vector having elements a

The far field is

g(u) = <a, T (e»

I (2.10)

with TI(e» = TI(e» a N element column vector with elements TI(e

n) when T I is defined by I T f =

f

fJ (ur)rdr. I 0 r l

The far-field power pattern is

<a,V a>,

with V a NxN matrix wit: elements

V ..

1J

(2.1 I)

(2.12)

(2.13)

whj C~l are functions of u. Expressions like <a, Va> are quadratic forms in the variables a •

n

The total power radiated by the aperture is

The The with p

=

<a,A a>. r NxN matrix A has I A .•

=

f

e.e.rdr. 1J 1 J r l second moment is m 2

=

<a,W a>, elements

W a NxN matrix with elements

00

J

3

w ..

u V •• du. 1J

0 1J

The efficiency expressed in vectors and matrices is n = 2<a,V(o)a> <a,Aa> (2.14) (2.15) (2.16) (2.17) (2.18)

The normalized second moment of the far-field radiated power with respect to the axis u = 0 is

<a,Wa>

<a,Aa> (2.19)

The combined performance index of eq. (2.8) is

2

n/o

2<a,V(o)a>

<a,Wa> (2.20)

In this way we have expressed each of the performance indices as a ratio of two quadratic forms. The real matrices

v,

A and Ware Hermitian because

v ..

=

V .. , A..

=

A.. and

1J J1 1J J1

equations (2.13), (2.15)

W ••

1J and

=

W .. , which 1S immediately evident from the

J1

(2. 17). If the denominator, <a, A a> or <a, Wa>, is positive definite, which means that the denominator is greater than 0 for any a

1

0, the optimization is easily performed by making use of a theorem on the properties of a function of a vector [lOa]. This theorem is given in the next section.

3. THE OPTIMIZATION PROCEDURE

Assume that we want to optimize the performance index PI given by

PI <8, H a>

<a, L a> ' (3. I)

in which <a is a real N element row vector, a> is a real N element column vector and Hand L are NxN real square matrices. The optimization theorem [lOa] states that with Hand L Hermitian and L positive definite, all the relative maxima and minima of the performance index are given by the eigenvalues determined from

H a> = A L a> , (3.2)

where A denotes the eigenvalues which must satisfy

det(H-AL) = O. (3.3)

Equation

(3.3)

is a characteristic equation with N roots AI ~ A2 ~ ..• ~ AN which are all real. The performance index is bounded by the smallest and the largest of these roots,

a> satisfies eq. (3.2) with

AI' A bound is achieved when the vector or A

=

AI' The restriction to real vectors and matrices is not necessary for the theorem.

If ln addition H is a one-term dyad which means that it can be expressed as "he product of a Nxl vector h> and a IxN vector <h, then there is only one nonzero eigenvalue. This is the maximum value of PI and is equal to [5]

-I

AI

=

<h, L h>. (3.4)

The corresponding eigenvector is

a> L-Ih> (3.5)

In the case of optimization of eq. (2.18) and (2.20) H is of the above-mentioned form,

Another special case occurs when both Hand L are diagonal matrices. Then the eigenvalues are derived from eq. (3.3) as

A. = H .. /L ..

1 11 11 i = 1,2, ... ,N, (3.6)

and the corresponding eigenvectors are (O ... Oa.O ... O) with a.

=

I. This

1 1

type of solution occurs when the normalized second moment is optimized (without constraints) for a circular aperture with a special type of aperture expansion function e as will be outlined in section 4.

n

First, however, we will show the technique which we have employed when there is a constrained optimization problem. The problem of optimizing a quantity subject to constraints can in general be approached by the method of Lagrange multipliers [17], [24]. This technique is characterized by the fact that as the number of applied constraints increases, so also does the number of manipulations required for the solution. The existence of

constraints, however, reduces the number of independent variables which can be adjusted for the optimization. Therefore a method which reduces the vector and matrix dimensions by the number of applied constraints is more appropriate for our optimization problems.

The constraints on the far field in M points V can be expressed as

m

m 1,2, .•. , M, (3.7)

if 1 is the prescribed relative value in V . In a shortened notation

m m

eq. (3.7) becomes

<a, q > = 0 ,

m m=I,2, ... ,M, (3.8)

with q > the independent constraint vectors. The N dimensional vector <a

m

is, according to eq. (3.8), required to be orthogonal to the M dimensional subspace of the constraint vectors. So the required vector <a lies in a

subspace of dimension N-M and has at least M elements equal to zero if an appropriate set of spanning vectors can be constructed. This is possible with the aid of the Gram-Schmidt procedure [6], [9], [II]. Form a NxN constraint matrix C having as the first M rows the vectors <~ and in the remaining N-M rows any arbitrary collection of N element independent vectors. From C we construct with the Gram-Schmidt procedure a matrix C orthogonal

by rows and columns, i.e. C

-,

,where T means transposition. The o

matrix C

o is the transformation matrix. Applied to a> satisfying eq. (3.8) it generates a N element vector a > which has zeros as the first M elements

c

because each of the first M rows of C is a linear combination of the o

constraint vectors <q . The vector a > is

m c

a > C a>.

c 0

The first element of a > is c The first row

~f

Co is <c,

=

«q,)/<q"q,>!

<q"a>/<q"q,>'. The second row of Co is

<c

=

{«q ) - «q ,c »«c )}/{<q ,q > - <q2,c,>2}! and the second element

2 2 2 ' , 2 2

of a > is the inner product of a> and c

matrix C o the first

are deduced in this way and M elements of a > are zero. _c

<c

2. Expressions if the vector a>

for the rows <c. of 1 satisfies eq. (3.8)

Rewriting the performance index of eq.

(3.')

we get a ratio of two quadratic forms in which the constraints are incorporated as follows

PI = <a,H a> <a,L a> <a, <a, <C a, C H CT C a> o 0 0 0 <C a, C L CT C a> o 0 0 0 <a ,H a > c c c <a ,L a > c c c where H c T = C H C and L o 0 c a

c are zero one can reduce C L

o the

(3.9) T

C . As the first M elements of the vector o

dimensions of a to N-M and of Hand L

c c c

to (N-M)x(N-M) (3.9) becomes

resulting in a vector ad and matrices Hd and L

d. Equation

<ad' Hd ad> PI =

-:--=--"7''---=:--<ad' Ld ad> (3.'0)

The problem of optimizing eq. (3.') with the constraints of eq. (3.8) is transformed into the problem of optimizing the unconstrained eq. (3.9).

It can be shown [9] that if H is Hermitian Hc and Hd are also Hermitian. The same applies to L, Lc and L

d. The quadratic form <a,L a> is positive definite as it represents the radiated power or the spread of the radiated power. The quadratic forms associated with Lc and Ld represent the

radiated power or its spread in the constrained case, so that they must also be positive definite. The optimization of eq. (3.10) is similar to that of eq. (3.1) except for the reduced dimensions. The relative maxima and minima of eq. (3.10) are given by the eigenvalues determined from

(3.11 )

with AC the constrained eigenvalues satisfying

(3.12)

c c c

Eq. (3.12) has F-M roots

Al :

A2 : ... :

A

N_M which are all real. Ordered according to numerical size as with the unconstrained eigenvalues it can be shown that A~ lies between

A.

and

A.

M [11, p.174]. This implies that

1. 1. 1. +

the constrained minimum is always equal to or greater than the unconstrained minimum and that the constrained maximum is always equal to or less than the unconstrained maximum.

If H is a term dyad it can be shown [9] that Hc and Hd are also one-term dyads, namely

and with H c h > <h , c c h > <h d d h >

=

C h> c 0 (3.13) (3.14) (3.15)

and with hd> formed from h

c> by deleting the first M entries. In this case the constrained eigenvalue and the constrained eigenvector are respectively

A

C <h d, -1 hd> (3.16) 1 Ld

,

and ad> = _Ld -1 _hd> (3.17)

After ad> has been determined from eq. (3.11) or (3.17) one can form the vector a

c> by adding M zeros to ad>' The vector a> can then be computed with

a> a >.

c

The same result is obtained if formed by deleting the first M

0.18)

ad> is multiplied by the matrix which is T

columns of C . o

In this section we have given the formulas for the (constrained) optimization, for instance of the normalized second moment of the far-field radiated power, the efficiency of an antenna or the efficiency power-spread ratio. In the next section we will be more specific in the treatment of some examples.

4. SOME EXAMPLES OF CIRCULAR AND ANNULAR APERTURES

I n sect~on . 4 we Wl . 11 optltnlze . . t h e per ormance 1U lees f . d'

n,o

2 an d

n

/ 2

a

defined respectively by the equations (2.18), (2.19) and (2.20). We

will start with the circular aperture having successively three different types of functions e . We will then treat the annular aperture with one

n

type of functions e .

n

It is found [16] that the aperture illumination function F(x,y) satisfying the minimum condition for 02 is the lowest mode solution of the Helmholtz equation for the given planar aperture,

a

2F - - +

ax

2

a

2F 2 - + k F 0, (4.1)

ai

2

where k

=

0 2 and F(x,y) is continuous and approaches zero at the closed boundary. This illumination is called the optimum illumination for the given aperture with regard to 02

F of the circular aperture, in polar coordinates defined by 0 < r < opt

and 0 < ~ < 2n, 1S the lowest mode of 00 00

F(r,~) =

L

[C sin n~ + D cos n~]J (k r),

urn nm nnm

n=o m=o

with C and D constants and k

nrn nrn urn solution of J n (k)

annular aperture, in polar coordinates defined by 0 <

o

< ~ < 2n, is the lowest mode of

00 00 (4.2) = 0. F of the opt r 1 ::: r < 1 and F(r,~) =

I

L

[C

sin n~ +

D

cos n~][J

(k

r ) -nrn nm n nrn n=o m=o

with k the solutions of nm

J (k)

n O. (4.4)

(4.3)

For our optimization (constrained or unconstrained) we use, for instance, the elements of the ~-independent form'of eq. (4.2) in the circular aperture case and of eq. (4.3) in the annular aperture case.

4. I. The circular aperture

As the first aperture illumination we take fer) consisting of the above-mentioned elements of eq. (4.2). This can be represented by a series in

the following form

N fer) =

L

n=1

o

a J (u r) n o n i f 0< r < I, if r > I,

with un the solutions of Jo(u) = 0, which means that u

l = 2.4048, u

2 = 5.5201, etc. In Fig. I the lowest mode is shown.

o

Fig. I. Lowest mode of eq. (4.5).

The far field g(u), eq. (2.2) with eqs. (AI) and (AZ), is

N

2 2

g(u) = J (u)

_L

au JI(u )/(u -u ) if u

l'

u ,

0 n=1 n n n n n 2 if !aJI(u) u u . n n n

The vector <e, eq. (2.9), can be written as

The vector <TI(e), eq. (2.10), is

2 2 2 2

<TI(e) = Jo(u) {u1J1(ul)/(ul-u ) ..• ~Jl(~)/(~-u )} .

(4.5)

(4.6)

(4.7)

The vector <TI{e(o)} needed for the optimization of nand n/a2 is

(4.9)

The elements of the matrix A, eq. (2.15), are computed with eqs. (AI) and (A2) as A .. = 0 1J i f i

f

j (4.10) A ..

jJ~(ui)

1J i f i j. 2.4048, u

2

=

5.520 etc., matrix A is a positive definite diagonal matrix. The aperture radiated power is

N

~

L

i= I 2 2 . a. J I (u.). 1 1 (4.11)

The elements of the matrix

w,

eq. (2.17), are 00 u3J 2(u)

w

..

= u.u·JI(u·)JI(u.)

f

0 duo (4.12)

1J 1 J 1 J 2 2 2 2

0 (u.-u )(u.-u )

1 J

Evaluation of the integrals results in [I] , [2]

w ..

0 if i

f

j,

1J

(4.13)

w ..

!u~J~(u.)

i f i j.

1J 1 1

(See appendix A2) .

Matrix W is also a positive definite diagonal matrix. The second moment of the far-field power pattern is

N \' 222 m2 =

!

L a.u.J I (u.) i= 1 1. 1. 1. (4.14)

We are now able to derive some simple results for the unconstrained performance indices. The unconstrained maximum value of

n

and the corresponding eigenvector are determined with eqs. (3.4) and (3.5). With N modes in the aperture field this results in

-2 -2 -2

4(u

l +u2 + ••. +un ), (4.15)

If N is allowed to become infinite, the value of ~,N becomes I, [Z7,p.50Z], which corresponds to a uniformly illuminated aperture. The elements of <a are, apart from a constant, equal to the coefficients of expansion of a constant function C on 0 < r < I in a series of the form

[10,p.319] 00 C =

L

n=1 s J (u r) n o n

with u the solutions of J (u)

=

O.

n 0

The coefficients s are [27, p. 580]

n l

s

=

ZJI-Z(u )

J

rCJ (u r)dr

=

2C/{u JI(u )}.

n n o n n n

o

The aperture illumination for optimum

n

is with N modes

N

fer)

=

I

J (u r)/{u JI(u )}

n=1 0 n n n

=

0

The corresponding far field is

PI

(u )/u

n n

The second moment is then

m

=

jN 2

and the normalized second moment is

N

=

NO

n=1

This means that

and 2 n_{MAX N •}

_,

a

-Z -I

u } n 4N if if i f i f 0< r < I, r> I. u

#-u = u • n (4.17) (4.18) (4.19) (4.Z0) (4.21) (4.22) (4.23) (4.24)

As the number of modes grows, the aperture illumination tends to uniform illumination with

n

=

I having 02

=

00 if the upper limit integration in eq. (4.12) is taken to be infinite. If, however,

the of 2 •

o

1S

calculated for the uniform illumination with the upper limit of integration t

=

ITD/A it follows that

t 2

2

J

uJ

1 (u)du (4.25)

o

With D/A 100 this is approximately equal to 200.

In Table 1 we give nMAX,N' the 3dB beam width u_3dB' 02 and the first 5 side lobe levels (denoted by sll) in dB for some values of N

N 2 3 4 5 6 7 00 2 s11 I s11 2 s11 3 s11 4 s11 5 nMAX,N u3dB 0 0.6917 4.15 5.7831 27.5 36.4 42.6 47.3 51.2 0.8229 3.62 9.7212 16. I 30.5 37.3 42.4 46.4 0.8764 3.47 13.6930 17.0 21.5 32.8 38.6 42.9 0.9051 3.41 17.6771 17.3 22.9 25.2 34.8 39.8 0.9231 3.37 21. 6668 17.4 23.3 26.7 27.9 36.5 0.9353 3.34 25.6597 17.4 23.5 27.2 29.6 30. I 0.9442 3.33 29.6544 17.5 23.6 27.4 30. I 31.8 3.20 17.6 23.8 28.0 31.1 33.6

Table I. Unconstrained optimum

n,

corresponding 3dB beam width,

02 and side lobe levels with N modes in the aperture field of eq. (4.5).

Data in the first and last rows of Table I are reference data representing the fields of the lowest mode Jo(ulr) and of the uniform illumination. With only one mode there is no optimization possible, all the parameters are completely determined.

In Fig. 2 we show the normalized aperture illuminations which maximize n

for N is 3,5 and 7 while the illumination for N

=

I is added for reference. The normalized numerical values of the first 7 components of the optimum excitation vector are

Fig. 2. Normalized aperture illuminations for unconstra'ined optimum 11 with N

modes in the aperture field of eq. (4.5).

2

The unconstrained minimum value of 0 and the corresponding eigenvector <a can be computed with eq. (3.6) because both Wand A are diagonal matrices. The relative extremes A. of the unconstrained 02 follow from

det (W-AA) =

o.

This yields

A

1

1

The minimum value of 02 is therefore

02

MIN,N

u~=

5.7831

The corresponding eigenvector <a is

<a=(IO ••• O).

(4.26)

(4.27)

(4.28)

The aperture illumination needed for minimum 02 is Jo(utr). This is in accordance with the results of [16]. The far field having

O~IN

is

Some numbers associated

-2 11 4u_I = 0.6917, 2 2 4 110 _{110M1N =}

,

11/02 11/o_M1N 2 4u-4 I = if i f

with this far

0.1196. field are (4.30) (4.31) (4.32) (4.33) 2

Using eqs. (3.4) and (3.5) the unconstrained optimization of 11/0 results, with N terms 1n the aperture field of eq. (4.5), in

2

(11/0 )MAX N

_,

4 (u -4 I + u-4 ₂ +... + uN ), -4 (4.34) which has a limi t value of 1/8 if N + 00 [27,p.502].

The eigenvector <a with eq. (4.34) is

(4.35)

Tite normalized numerical values of the excitation coefficients are

{I; -0.12615; 0.04104; -0.01894; 0.01050; -0.00652; 0.00437; ••. }.

The efficiency is now N

-4 2 N -6 11 = 40: u. } /

I

u.

i=1 1 i=1 1

which for N + 00 becomes 3/4.

And the normalized second moment is

2 N o =

U

i=1 N

I

i=1 -6 u. 1

which for N + 00 becomes 6.

(4.36)

The product of n and 0 2 is now 2 N -4 3 N -6 2 no = 40: u. }

_/

{I

u. }

,

i=1 1 i=1 1 (4.38) which is 41 2 for N -+ 00.

The aperture field for optimum n/02 with N terms is

N 3

f(r) =

L

J (u r) / {u JI(u )}.

n=i 0 n n n

(4.39)

The far field corresponding to this f(r) is

N -2 2 2-1 g(u) J (u)

I

u (u -u ) i f u '" u n' 0 n=1 n n _(4.40) -3 if = PI (u )u u u n n n 2 In Table 2 we show the results of the unconstrained optimization of n/o for Some values of N, while N = is added for reference. Convergence towards the end values J/8, 3/4 and 6 is fast.

N 2 3 4 5 6 7 2 (n/o )MAX,N 0.1196 0.1239

o.

1246 O. 1248 0.1249 0.1250 0.1250 n 0.6917 0.7374 0.7455 0.7480 0.7489 0.7494 0.7496 2 (J 5.7831 5.9507 5.9822 5.9917 5.9955 5.9972 5.9982 sl1 sll 2 sll 3 sll 4 sll 5 4.15 27.5 36.4 42.6 47.3 51.2 4.01 3.99 3.99 3.99 3.98 3.98 24.0 34.7 24.5 32.9 24.6 33.4 24.6 33.5 24.6 33.5 24.6 33.6 41.1 40.3 39. I 39.5 39.6 39.7 46.0 45.3 44.8 43.8 44.3 44.4 49.9 49.3 48.9 48.5 47.8 48. I

Table 2. Results of unconstrained optimization of n/02 with N terms in the aperture field of eq. (4.5).

In Fig. AI we show for N = 7 the aperture fields and the far fields for optimum n/02 and n. Also, the lowest mode with optimum 02 is added. Patterns with optimum n/02 have envelopes which lie between the envelopes

f . h . . 2 d .

Bl'CZIlLSt' or tilL' v<lLue of I}, the sidelobe extremes and the 3dll beam width InvcBtigatc' till" relationship of the

. 1 1 . 1 . . 2 W1t 1 t 1e well-known 1 lunllnatlon l-r we wprc led to 2 optimum q/o illumination for [25,p.195]. Expand 2

i-r for 0 < r < 1 in a series [27,p.580] as follows

00 2

I-r

l:

s J ( u r ) ,

n o n (4.41)

n=1

with II the n-th positive zero of J (u). Then the coefficients s are

n o n given by .) s n r( I-r-).J (u r)dr o n (4.42) whicll results in s n

Apart from the constant 8, tIle a of eq. (4.35). The aperture

coefficient s of eq. (4.43) is

n

field of eq. (4.39) for optimum

(4.43) equal 2 r,/a , to ~ . . 2

Wltl! N ->- co, 1S equal to j-r apart from a constant multiplier. Stated

I . 2 . / 2

ot 1eYWl se, )-r has maXllTIUm fl a .

The constrained optimization of 11, 02 or

n/a

2 can easily be performed using the technique outlined in section 3. The constrained optimization is done while prescribing for instance a level (relative to the level In u = 0) at a point of the main beam and/or the extremal values of a numilcr of sidelohe peaks. The prescribed level in a point of the main beam can be used, for instance, to get a certain 3dB beam width or a socallcd flat-top beam. As the positions of the side lobe peaks are not foreknown we must choose stnrting values for them. Suitable starting

points ;Irl' midwny iwtw(,l'n tilL' zerOR of J (u). If the computed levels of o

tilL' t'xlrl'lIu.'s nrc then found not tu be in accordance with the prescribed values we repcilt the computation with new locations midway between the old starting points and the computed positions of the extremes. We can thus, in a number of iterations, obtain the prescribed sidelobe levels to any desired degree of accuracy. In our computations the iterative

prOCCRS is stopped when the differences between the computed and prescribed

If no parameter 1S to be optimized we need N Q P + 1 source terms if

the levels at P points are prescribed. If a parameter is optimized while at the same time the levels at P points are prescribed, we need N > P+2 source terms.

In Tables 3 and 4 we show some results of the optimization of the efficiency when side lobe values are specified. In Table 3 we have two side lobes of -25dB and in Table 4 four sidelobes of -30dB. With

increasing N the efficiency and the 3dB beam width come close to a limit which depends on the requirements. Adding more modes in the aperture field will have less and less effect upon nand u_3dB• The efficiencies shown in Tables 3 and 4 are less than the unconstrained efficiencies in Table 1 for the for N 3 4 5 6 7 N 5 6 7 8 9

same values of N. In Fig. A2 far fields and aperture fields are shown two cases of _{Table 4"}

nOPT 02 u 3dB sl1 3 sl1 4 sl1 5 0.8040 7.9210 3.81 36.0 41.6 45.9 0.8495 14.6343 3.64 24.6 35.4 40.6 0.8705 19.0847 3.58 26.2 27.4 36.7 0.8838 23.2823 3.55 26.7 29.2 29.7 0.8934 27.1914 3.53 27.0 29.8 31.5

Table 3. Results of constrained efficiency optimization with N

modes in the aperture field of eq. side lobes prescribed at -25dB.

nOPT 02 u 3dB sl1 5 0.8033 11.3297 3.80 39.4 0.8236 19. 1341 3.72 29.2 0.8345 23.7835 3.69 31.1 0.8423 28.0795 3.67 31.7 0.8480 32.4256 3.65 32.0

Table 4. Results of constrained efficiency optimization with N modes in the aperture field of eq. (4.5). First 4 side lobes prescribed at -30dB.

In 1'.1111('s 5-8 we givC' results of the optimization of 02 with N modes in

eq. (4.5) when sidelobe values arc prescribed. These are -25, -30, -40

2

and -SOdB. With increasing N, a and u

3dB diminish to limit values while tlte change in Tl towards a limit value depends on the prescribed sidelobe level. In Table 5 n decreases and in Tables 6-8 '1 increases with increasing

N. The values of

O~PT

_,

N are greater than the minimum possible value 5.7831. In Fig. A3 two examples of Table 6 are shown.

N 2 3 4 N 2 3 4 5 6 N 3 4 5 6 7 5.8623 5.8493 5.8478 0.7240 4.05 0.7124 4.09 0.7110 4.09 sl1 2 35.2 37.0 36.6 sl1 3 41.6 42.4 43.1 sl1 4 46.4 47. I 47.3 sl1 5 50.3 50.9 51.1 2

Table,S. Results of constrained 0 optimization with N modes in the aperture field of eq. (4.5). First sidelobe prescribed at -25dB. 5.8311 5.8192 5.8185 5.8182 5.8182 0.6640 4.23 0.6746 4.20 0.6756 4.19 0.6759 4.19 0.6761 4.19 sl1 2 37.5 35.9 36.2 36.2 36.2 sl1 3 43.5 42.7 42.2 42.4 42.4 sl1 4 48.2 47.5 47.3 47.0 47.1 sl1 5 52.0 51.4 51.2 51.1 50.6

Table 6. Results of constrained 02 optimization with N

modes in the aperture field of eq. (4.5). First side lobe prescribed at -30dB.

6.2894 6.1961 6.1852 6.1822 6.1813 0.5968 4.44 0.6227 4.35 0.6255 4.34 0.6267 4.33 0.6268 4.33 sl1 3 45.9 40.0 40.9 41.1 41.2 sl1 4 50.4 47.2 45.3 46.0 46.2 sl1 5 54.1 51.5 50.7 49.4 50.0

Table 7. Results of constrained 02 optimization with N modes in the aperture field of

eq. (4.5). First 2 sidelobes prescribed at -40dB.

N _°OPT 2

_n

u sl1 7 3dB

7 6.7919 0.5621 4.55 56.3

8 6.7819 0.5662 4.54 53.4

9 6.7774 0.5677 4.53 54.5

Table 8. Results of constrained 02 optimization with N modes in the aperture field of eq. (4.5). First 6 sidelobes prescribed at -50dB.

In Tables 9 and 10 we show optimization results for 02 when there is also a prescription for the main beam of the radiation pattern. This yields flat-top beams with prescribed sidelobes and optimum 02 or

beams with a prescribed 3dB beam width,prescribed sidelob~s and optimum 2

° . These kinds of pattern can even be synthesized while another

parameter is optimized. The patterns in Figs. A4 and A5 are flat-top in type and in Fig. A6 a pattern with prescribed 3dB beam width is shown.

N 2

°OPT

n

u3dB sl1 4 sl1 5

a)6 20.0990 0.0760 12.28 32.6 39.2

b)6 15.6062 0.1765 9.08 37.8 44.8

Table 9. Constrained

o~PT

with N modes of eq. (4.5).

Requirements: a) Three side lobes of -30dB, and for u I the relative power is 0.5dB.

b) Three side lobes of -35dB, and for u I the relative power is OdB.

N _°OPT 2

n

u sl1 2 s11 3 s11 4 s11 5

3dB

4 6.1083 0.6129 4.40 42.4 52.9 53.4 56.4

5 6.1024 0.6130 4.40 41.7 49.0 57. I 57.6

6 6.0996 0.6131 4.40 41.5 48.5 54.0 60.5

Table 10. Constrained

o~PT

with N modes of eq. (4.5).

Requirements: First side lobe prescribed at -30dB, and for u

=

2.20 the relative power is -3dB.

In Tables II-I) results of the optimization of are given when side lobe values are specified.

n/er2 with N modes of eq. 2

The values of 'I/O are,

with constraints, less than the maximum unconstrained value 1/8. The

(4.5)

2

values of n and a , however, can be less or greater than the unconstrained

limit values 3/4 and 6. This depends on the constraints chosen. In Fig. A7 two patterns and corresponding aperture fields of Table 13 are shown.

N 2 0.1226 3 0.1229 4 0.1230 5 0.1230 6 0.1231 7 0.12'31 n O.75H3 6.1773 0.7647 6.2219 0.7668 6.2346 0.7676 6.2395 0.7683 6.2411 0.7685 6.2423 3.95 3.93 3.93 3.93 3.92 3.92 511 2 34.0 33. I 33.4 33.5 33.5 33.5 sl1 3 40.5 40.0 39.3 39.6 39.6 39.7 sl1 4 45.4 45.0 44.7 44. I 44.3 44.4 Table II. Constrained (n/er2)OPT with N modes of eq. (4.5). First

sidelobe prescribed at -22.5dB. N 2 3 4 5 6 7 0.1139 0.1188 0.1194 0.1196 O. I 197 0.1197 TJ 0.6640 0.7064 0.7109 0.7130 0.7139 0.7144 5.8311 5.9455 5.9550 5.9623 5.9656 5.9673 4.23 4.10 4.09 4.08 4.08 4.08 sl1 2 37.5 31.8 32.7 32.8 32.9 32.9 sl1 3 43.5 40.5 38.4 39. I 39.2 39.3 sl1 4 48.2 45.6 44.7 43.2 43.9 44.0 Table 12. Constrained (n/er2)OPT with N modes of eq. (4.5). First

sidelobe prescribed at -30dB. N 4 5 6 7 8 2 (n/er ) OPT 0.1005 O. 1027 D.1032 0.1034 0.1035 0.1036 n 0.6227 0.6418 0.6461 0.6476 0.6486 0.6492 6.1952 6.2478 6.2587 6.2628 6.2658 6.2678 4.35 4.28 4.27 4.26 4.26 4.26 sl1 4 47.2 41.3 42.5 42.9 43.0 43.0 sl1 5 51.5 48.4 45.6 46.6 46.9 47.0 Table 13. Constrained (n/er2)OPT with N modes of eq. (4.5).

First 3 side lobes prescribed at -40dB.

sl1 5 49.3 49.0 48.7 48.5 47.9 48.2 sl1 5 52.0 49.7 49.0 48.3 47.2 47.8

Conclusions

In section 4.1. I we have analytically and numerically optimized three performance indices for aperture illuminations consisting of a series of Bessel functions J (u r) with u positive solutions

o n n

Results of unconstrained and constrained optimizations

of J (u) = O. o

are given. Unconstrained optimization of the efficiency results in excitation coefficients which are equal to the coefficients of expansion of a uniform illumination into a Fourier-Bessel series. The unconstrained optimum efficiency is a function of the zeros of a Bessel function and has I as its limit value. The unconstrained optimum of the spread of radiated power occurs

'II ' , h 2

1 UID1nat10n as a

=

with the aperture illumination J (2.4048r). This

2 0

5.7831,

n

=

0.6917 and

n/o =

0.1196. The uncon-strained optimization of the efficiency power-spread ratio results in the aperture illumination I - r2. This distribution has limit values

2 2

n/o=

1/8,

n

=

3/4 and 0

=

6.

The applied constraints on the values of the far fields are various. This finds expression in

- flat-top patterns with prescribed sidelobes and an optimized index, - patterns with prescribed 3dB beam width, prescribed sidelobes and

an optimized index,

- patterns with prescribed sidelobes and an optimized index.

The values of the performance indices depend on the constraints and the nL@ber of modes. The influence on an index is greatest for the lowest modes.

2

The advantages of

n/o

optimizations are that, with smooth aperture distributions, efficiencies can be attained which are higher than those of 02 optimizations and that the non-prescribed sidelobes decay faster

Other aperture functions e which can be used for optimization purposes n

are Bessel functions .f (u r) where

o n u n is a positive solution of JI (u) = 0 The aperture distribution consists

N f(r) ~

I

n=1

o

a J (u r) n o n of a truncated series i f

o

< r < 1, (4.44) if r > I.

The values of un are U

I ~ 0, U2 ~ 3.8317, u3 the modes with u

1 and u2 are shown.

7.0156 etc. In Fig. 3

1 k :

-•

•

Fig. 3. J (u.r) for u l o 1

1

r

-The far field g(u) is, wi til eg. (AI) and (A2),

g(tI) N u.i I (tI)

l:

n= I 2

!

a .1 (u ) n o n

The vector <e 18 now

2 2 a J (u ) / (u -u ) n o n n i f if J(ur)}. o N u ;& u , n u

=

u • n 3.8317. (4.45) (4.46)

The vector <TI(e) is

(4.47)

In u

=

0 this becomes

<T I {e (O)} =

l

{I 0 ... O} • _(4.48)

The elements of the matrix A can be computed with the eqs. (AI) and (A2) which results in A .. 0 i f i

I<

J • 1J (4.49) A .. p2(u. ) if i j. 1J. o 1

Matrix A is a positive definite diagonal matrix. The'power radiated by the aperture is N p

=

1

L

r i=1 2 2 a.J (u.). 1 I 1

The elements of the matrix Ware

W •• 1J J (u.)J (u.) o 1 0 J

J

5 2 u J I (u) ( u 2

_'--u-co~c-)-(-u72-_u-2=-.

)- d u • 1 J (4.50) (4.51)

Because of the discontinuous aperture distributions the reactive power cannot be neglected. The integration limits in eq. (4.51) must therefore be equal to 0 and nD/A.

After partial fractioning in eq. (4.51) there is one integral which

diverges. the others are convergent for upper limit 00. (see appendix A3).

If 02 is involved in an optimization the elements of W with an upper limit of nD/A must be evaluated numerically.

The unconstrained optimi~ation of

n

with the aperture field of eq. (4.44) results in

n

MAX•N <a = (1.0 ... 0).

=

n

MAX

=

I. The corresponding eigenvector is and the aperture field

well-known uniform illumination with far

is fer) = I. This field J I (u) /u.

The unconstrained optimizations of

0

2 or

n/0

2 do not yield simple results due to the complexity of the elements of matrix W.

The constrained optimization of the efficiency is interesting as we can compare results with those in the literature [14), [19). To be able to make these comparisons we have made computations with N modes when N-I sidelobes are prescribed (which in fact is not an optimization) and when less than N-I sidelobes are prescribed. Our results are

summarized in Tables 14-17. In Table 17 results for the combinations (N,N-I)

=

(4,3), (6,5) etc. are given for -40dB sidelobe& Computations for the combinations (N,N-2) showed that N should be ~ 14 to get the first not-prescribed sidelobe below the -40dB level. For instance with

(N,N-2) (N,N-2)

(13,11) the twelfth sidelobe was -39.8dB and with

(14,12) the thirteenth sidelobe was -4I.IdB. However, the first not-prescribed sidelobe will decrease somewhat if N is increased while the number of prescribed side lobes is unchanged.

Comparing our efficiencies with [19) shows that they are greater than those for the Taylor distributions. This is because the prescribed side-lobes do not decay as in the Taylor distributions, and because we are not restricted to N modes and N-I prescribed sidelobes.

If we compare our results with those in [14) it appears that the

efficiencies are almost the same. In some cases we have computed higher efficiencies. The optimum distributions in [14), however, are determined in another way. In Fig. A8 the aperture illumination of two cases are shown.

Furthermore it should be noted that the lowest 3dB beam width is not achieved with the highest efficiency.

n_OPT u 3dB

~

3 4 5 6 3 4 5 6 2 0.9434 0.9482 0.9487 0.9489 3.463 3.420 3.415 3.413 3 - 0.9455 0.9456 0.9456 - 3.390 3.394 3.394 4 - - 0.9263 0.9301 - - 3.336 3.361 5 - _{- - 0.8919 - - - 3.296}

Table 14. Results of efficiency optimization with N modes in eq. (4.44) and with K sidelobes prescribed at -25dB.

TlOPT u3dB

~

4 5 6 7 10 4 5 6 7 3 0.8780 0.8882 0.8894 0.8901 0.8906 3.631 3.569 3.559 3.557 4

-

0.8872 0.8899 0.8902 0.8906

-

3.585 3.561 3.557 5

-

- 0.8886 0.8888 0.8890

-

- 3.548 3.546 6 - - - 0.8832 0.8839 -

-

- 3.520 9 -

-

-

-

0.8383 -

-

-

-Table 15. Results of efficiency optimization with N modes in eq. (4.44) and with K sidelobes prescribed at -30dB.

Additional results for -25 and -30dB sidelobe levels are given in Table 16 for N

=

15 modes.

Number of equal side lobe level (dB) _TlOPT u 3dB side lobes

2 -25 0.9491 3.411

2 -30 0.8907 3.553

Table 16. Results of efficiency optimization with 15 modes in eq. (4.44).

N 4 6 8 10 12 14

0.7200 0.7347 0.7475 0.7554 0.7594 0.7604 Tl

4.037 3.993 3.951 3.920 3.898 3.881 u

3dB

Table 17. Efficiencies and corresponding 3dB beam widths when,with N modes in eq. (4.44), N-I side lobes are prescribed to have the level of -40dB.

Conclusions

---In section 4.1.2 we have investigated aperture illuminations consisting of a series of Bessel functions J (u r) with u positive solutions of

o n n

JI(u)

=

O. We have restricted the optimization to the efficiency index. Comparison of our results with those in the literature shows that the computational method which we have employed provides efficiencies which are higher than those of the Taylor distributions [19] and which are much the same as those of the optimum distributions derived in [14].

10 3.553 3.553 3.543 3.531 3.463

If the side lobe extremes of a far-field pattern must lie below a certain level it is favourable from the efficiency point of view to keep the number of prescribed side lobe extremes as low as possible.

In this section we will investigate aperture distributions which have low and rapidly decaying sidelobes. The decay rate of the sidelobes is dependent on the behaviour of the aperture distribution at the edge of the aperture [26].

The general expression for the aperture illumination f(r) is

N

f(r)

I

a {J (u r) - J (u )} i f

o

< r < I , n=1 n o n o n

(4.52)

0 i f r > I.

The values of un are u

l = 3.8317, u2 = 7.0156 etc., that is the positive zeros of JI(u)/u. This type of series has been used for synthesis

purposes before in [28]. The lowest mode is shown in Fig. 4. Z1 o ~ .... ::s

...

~

i

...

'"

::s

!<

_~

-""

..

..

,.

~ ~ .. - r j 'I:--~-~-~-~-"'-: o 1

Fig. 4. Lowest mode of eq. (4.52), normalized to I.

The far. field is, with eqs. (AI) and (A2),

J I (u) N g(u) = - ' - -

I

u n=1 2 ja J (u ) n o n 2 2 2 a u J (u )/(u -u ) n n o n n i f i f u

=

u • n (4.53)

For the lowest mode the far field is given in Fig. A9 while the numerical values of the side lobe levels are summarized in the first row of Table 18.

The vector <e is

<e

=

{J (u r) - J (u ) o I 0 I The vector <T I (e) = u In u

=

0 we have (4.54) (4.55) (4.56)

The elements of the matrix A can be computed with eqs. (AI) and (A2), which yields A .. lJ A •. lJ

lJ

(u.)J (u.) o 1 0 J 2 J (u.) o 1 if i

:f

j, (4.57) if i j.

As <a,Aa> represents the aperture-radiated power, matrix A is positive definite. The power radiated by the aperture is

N P =

HI

r i=1 2 2 a.J (u.) 1 0 1 N +

I

i=1 The elements of matrix Ware

N

I

a.a.J (u.)J (u.)}.

. I 1 J 0 1 0 J

J=

W •• lJ

2 2 ooJ

UJ~(U)

= u. u. J (u.)J (u.) -""2-;;'2-""2"----;2;- duo

1 J 0 1 0 J 0 (u -u.)(u -u.)

1 J

Evaluation of these integral yields, see appendix A4,

W .. = 0 lJ if i

"

j , 2 2 j . W .. = ju.J (u.) i f i lJ 1 0 1 (4.58) (4.59) (4.60)

The second moment is N

=

!

I

n=1 {auJ(u)}2. n n o n (4.61)

The unconstrained optimum efficiency with N modes can be derived from the equation 2V(O) a> = AAa>. By inspection it is found for nMAX,N that, due to the special nature of the matrices involved,

n_MAX N = N/(N+I).

,

(4.62)

The corresponding eigenvector is

< a = _(4.63)

If N is allowed to become infinite, the value of

n

MAX

,

N becomes I, which corresponds to the efficiency of a uniformly illuminated aperture.

The aperture illumination for optimum efficiency with N modes is

N fer) =

I

n=1 {J (u r) - J (u )}/J (u ) o n o n 0 n if = 0, i f

The corresponding far field is

g(u) J 1 (u) N

L

u n=1 2( 2 2)-1 u u-u n n i f = jJ (u ) o n if

With optimum nthe normalized second moment is 2

a u 2

n

as the radiated power power p = jN(N+I).

0< r < I, r > I. u

=

u • n (4.64) (4.65) (4.66) 2 r 2

Formulas for nMAX,N a and nMAX,N/a are easy to derive with eq. (4.62).

In Table 18 we give some results of the unconstrained optimization of the efficiency with N modes of eq. (4.52).

N _{I1MAX ,N} 0 2 u sl1 I sl1 2 sl1 3 sl1 4 sl1 5 3dB 0.5000 7.3410 4.85 35. I 46. I 53.7 60.0 65. I 2 0.6667 10.6501 4.02 16.2 36.9 45.9 52.4 57.7 3 0.7500 13.9501 3.76 17. I 21.6 38.4 46.2 51.9 4 0.8000 17.2461 3.62 17.3 22.9 25. I 39.8 46.7 5 0.8333 20.5401 3.55 17.4 23.3 26.7 27.8 41.0 6 0.8571 23.8330 3.49 17.5 23.5 27.2 29.5 29.9 7 0.8750 27.1251 3.46 17.5 23.5 27.4 30. I 31.7

Table 18. Unconstrained optimum n, corresponding 3dB beam width, 02 and side lobe levels with N modes of eq. (4.52).

The far fields and aperture fields with N = I and N = 7 are shown in Fig. A9 • .

The normalized numerical values of the first 7 components of the optimum excitation vector are:

{I; -1.3420; 1.6129; -1.8445; 2.0500; -2.2368; 2.4091}.

The unconstrained optimum of 02 gives no simple results with the aperture functions of eq. (4.52). Using eq. (3.4) the unconstrained optimization of 11/02 with

N

modes yields

2 (11/0 )MAX,N N

L

n=1 (4.67)

which for N ~ 00 becomes 1/8 [27,p.502].

The maximum value of eq. (4.67) is reached if the excitation vector equals

<a

where use has been made of eq. (3.5).

The efficiency is now N 11 =

q:

n=1 -4 N un +

n

n=1 which is 3/4 if N ~ 00 [27,502]. (4.68) (4.69)

The normalized second moment is N

0:

n=1 N u-4 +

n

n n=1

which for N ~ 00 equals 6 [27,p.502].

The aperture field for optimum n/02 with N modes is

N

fer)

=

I

{J (u r) - J (u )}/{u2 J (u )}.

o n o n n o n

n=1

The far field is then

J I (u) N (2 2)-1 g(u) _u

I

u -u i f u

f

n=1 n -2 if jJ (u )u u = o n n u n u n (4.70) (4.71) (4.72) In Table 2

19 we show the results of the unconstrained optimization of

11/0 for some values of N, while N is added for reference.

Convergence to the limit values 1/8, 3/4 and 6 is not as fast as in Table 2. N 10 20 30 40 50 0.0681 0.5000 0.1156 0.7169 0.1201 0.7348 0.1217 0.7399 0.1225 0.7424 0.1230 0.7439 7.3410 6.2259 6.1171 6.0791 6.0597 6.0479 4.85 4.06 4.02 4.01 4.00 4.00 s11 I 35. I 24.6 24.6 24.6 24.6 24.6 s11 2 46. I 33.4 33.5 33.6 33.6 33.6 s11 3 53.7 39.5 39.7 39.7 39.7 39.7 s11 4 60.0 44.0 44.4 44.4 44.5 44.5 s11 5 65. I 47.7 48.2 48.3 48.3 48.3

Table 19. Results of the unconstrained optimization of

n/a

2 with N modes in eq. (4.52).

Fig. AID gives the aperture fields for N = I and N = 50. The values of

the parameters

n,

0 2 and n/o 2 which result from the optimization of the

efficiency power-spread ratio have led us to the assumption that for N

~

00 the aperture field equals I - r2, apart from a constant multiplier.

This means that, with c a constant, 00

I

n=1 {J (u r) - J (u )}/{u2J (u )} o n o n n o n 2 c (I-r ).

Integrating both sides, for the far field u = 0, yields

00

-!

I

n=1 -2 u n c/4. (4.73 (4.74)

with [27,p.502] it follows that c = -1/4. Substituting c and r = 0 in eq. (4.73) gives

00

I

-1/8.

n=1

It should be noted here that eq. (4.73) can be proved exactly by expanding l_r2 in the Dini series with H = V =:0 [27,§18.·3].

(4.75)

In Table 20 results are given of the optimization of the efficiency when sidelobe values are prescribed. In Fig. All two cases are shown.

N _110PT

a

2 u_3dB sll 3 sll 4 sll 5

3 0.6880 8.2977 4.12 41.5 49.2 54.9 5 0.7856 18.2239 3.77 26.2 27.4 41.3 10 0.8614 34.9761 3.59 27.3 30.3 32.6 15 0.8891 51.8375 3.53 27.4 30.5 33.0

Table 20. Results of constrained efficiency optimization with N modes in eq. (4.52). First 2 sidelobes prescribed at -25dB.

Tables 21 and 22 summarize the results of the constrained optimization of

a

2 with N modes of eq. (4.52) and with a number of prescribed side-lobes. In Figs. AI2 and AI3 some cases are shown graphically.

N 2 °OPT 11 u3dB 811 2 811 3 sl1 4 sl1 5 2 6.8954 0.5376 4.69 43.9 52. I 58.3 63.4 3 6.5172 0.5905 4.48 33.7 47. I 54.2 59.6 4 6.3537 0.6114 4.41 35. I 39.5 50.7 56.9 5 6.2534 0.6242 4.36 35.5 41.0 43.9 53.8 6 6.1853 0.6327 4.33 35.7 41.5 45.5 47.5 10 6.0440 0.6501 4.28 36.0 42.1 46.7 50.3 15 5.9702 0.6594 4.25 36.0 42.2 46.9 50.7

Table 21. Results of constrained 0 2 optimization with N modes in eq. (4.52). First sidelobe prescribed

3 7.2476 4 6.8414 5 6.6943 6 6.6032 7 6.5408 10 6.4312 15 6.3462 at -30dB. 0.5095 0.5570 0.5731 0.5831 0.5898 0.6015 0.6104 4.80 4.59 4.53 4.50 4.47 4.42 4.39 sl1 3 51.4 38.3 39.8 40.3 40.6 40.9 41.1 sl1 4 58. I 51.2 43. I 44.7 45.3 45.9 46.2 sl1 5 63.3 57.6 53.9 46.7 48.5 49.6 50.0 2

Table 22. Results of constrained 0 optimization with N modes in eq. (4.52). First 2

side lobes prescribed at -40dB.

In Tables 23 and 24 results of the optimization of 11/02 with N modes of eq. (4.52) are given when a number of sidelobe values is prescribed. In Figs. A14 and A15 some of these results are shown in graphical form.

2 2 s11 2 s11 3 s11 4 N

_(n/a

_)OPT

_n

a u 3dB s11 5 2 0.0780 0.5376 6.8954 4.69 43.9 52.1 58.3 63.4 3 0.0923 0.6126 6.6372 4.40 30.6 45.3 52.7 58.2 4 0.0985 0.6398 6.4952 4.31 31.9 36.5 48.6 54.9 5 0.1023 0.6549 6.3999 4.26 32.3 38.0 41.0 51.5 6 O. 1050 0.6652 6.3353 4.23 32.5 38.5 42.6 44.5 10 O. 1106 0.6857 6. 1993 4.16 32.8 39.1 43.7 47.4 15 O. 1136 0.6956 6.1261 4.13 32.9 39.2 44.0 47.7

Table 23. Constrained optimization of

n/a

2 with N modes of eq. (4.52). First side lobe prescribed at -30dB.

N

_(n/a

2 _)OPT

_en

a 2 u sl1 4 sl1 5 3dB 4 0.0801 0.5494 6.8548 4.63 52.2 58.4 5 0.0864 0.5829 6.7490 4.49 40.0 52.2 6 0.0895 0.5969 6.6720 4.44 41.6 43.7 7 0.0916 0.6061 6.6174 4.41 42.1 45.4 8 0.0931 0.6123 6.5739 4.38 42.4 46. 1 9 0.0943 0.6173 6.5438 4.37 42.6 46.3 10 0.0953 0.6206 6.5131 4.35 42.8 46.6 15 0.0981 0.6311 6.4331 4.32 43. I 47.0

Table 24. Constrained optimization of

n/a

2 with N modes of eq. (4.52). First 3 sidelobes prescribed at -40dB.

Conclusions

---In section 4.1.3 we have optimized the performance indices

n,

a 2

2

n/a .

The aperture fields comprise series of functions with both field and its derivative equal to zero at the aperture edge r

=

This ensures lower and more rapidly decaying sidelobes than with functions having only one of them equal to zero.

and the 1.

Unconstrained optimization of the efficiency with N modes results in

n

=

N/(N+1) with I as limit value. The unconstrained optimum of

n/a

2

is a function of the zeros of a Bessel function. The limit value is 1/8 and occurs with the aperture illumination 1_r2 having

n

=

3/4 and

a

2

=

6. The convergence towards the limit values is slower than with the functions of section 4.1.1.

The values of the constrained performance indices depend on the constraints and the number of modes. The influence on an index is greatest for the lowest modes. Several more modes, compared with section 4.1.1., are nevertheless needed in order to achieve certain values for the optimized indices. However, it is not known whether or not the limit values of an index with constraints is the same for both types of aperture functions.

4.2. The annular aperture

In section 4. I we have treated optimization for an unblocked aperture. Now we will introduce blocking of the central part of the aperture. The ¢-independent form of eq. (4.3) will be used to generate aperture

distributions. The aperture illumination can be written as

N J (u r l) f(r)

_L

a {J (u r) - o n Y (u r)} i f

o

< r l < I , Y (u r l ) r < n=1 n 0 n o n o n 0 i f 0 < r < r . r > I '

Y is the Bessel function of the second kind and zeroth order. With o u positive solutions of n (4.76) I. J (u) -o Y o (u) = 0, (4.77)

we have illuminations which are zero at the aperture edges r = r

l and r = I. In Fig. 5 we show the lowest mode for r

l = O. I having ul = 3.31394. z

gl

5 '" ~

to

~

....

..

'"

~QS'

'"

~ ~

...

~

~

~ ~o

--•

o.S f

The far field 's, with eqs. (A') and (A2), g(u) and g(u)

=

N

I

n='

a. a u {J (u)a n n o n 2 2 + J (ur,)8

}/(u

-u ) o n n 2' {J,(ui)ai + r,J,(uir,)Si} + if N

+

I

au {J (u.)a + J (u.r,)8

}/(u~_u2)

n n 0 ~ n 0 1 n 1 n

n='

nfoi

u fo

u , n (4.78) if u = u.. (4.79)

,

In these equations an and Sn are introduced for convenience. They are respectively a ='-J (u ) n ' n .~ (4.80) (4.8' )

Here we have used the Wronskian relation ['0,p.252]

J ,(x)Y (x) - J (x)Y

+,

(x) = 2/(rrx).

n+ n n n (4.82)

From eqs. (4.78) and (4.79) it is evident that the partial far-field patterns are not orthogonal. Each mode of the aperture field contributes

t·) every far-field point u .. The far field of the lowest mode of eq. (4.76)

,

with r,

=

0.' is given in Fig. A'6. It clearly shows the irregular behaviour.

The vector <e is

<e

=

{J _a (u r) -_, Jo(u,r,) Yo(u,r,)

The vector <T,(e) is

u,

[2 2 {Jo(u)a, + J_o(ur,)8,} ..• u -u, and <T,{e(O)} is Jo(uNr,) Yo(uNr,) (4.85) (4.83) (4.84)