Radio Science

On the receiving cross section of an antenna in infinite linear and planar arrays

Authors


Abstract

[1] A rigorous derivation is given for the receiving cross section of an antenna located in infinite linear and planar arrays. For linear arrays the result involves the active input impedance and the azimuthal directivity of the array. For planar arrays it depends on the active input impedance and on geometrical parameters of the array. The expression of the receiving cross section of an element can be used to derive the generally accepted expression for the active element pattern without, however, making use of intuitive arguments. Results are first presented for conditions without grating lobes; they are then generalized to conditions where grating lobes enter the visible space.

1. Introduction

[2] One way of characterizing antenna arrays consists of describing the active element pattern of each antenna, i.e., the pattern of the element of interest, considering the presence of all other array elements, which are terminated in matched loads [Pozar, 1994; Hansen, 1998]. In general, due to mutual coupling between the antennae, the active element pattern is quite different from the pattern of the same element taken in isolation. For large regular arrays consisting of identical elements, the element patterns tend to be rather homogeneous in the array, except for antennae near the edges, which are in a quite different environment. For very large arrays, the active element pattern is a very useful concept, because the pattern of the fully excited array can be expressed as the product of the active element pattern and the array factor, while still accounting for mutual coupling effects.

[3] It is generally accepted that, for an infinite planar array, the active element pattern in a given direction has a simple relation with the active reflection coefficient of the same array when it is fully excited and scanned at the same angle [Mailloux, 1994, p. 338]. However, we think that the derivations given in the literature are not entirely satisfactory, because they are either partly based on intuitive arguments, or they are limited to a specific class of antennae, as explained by Hansen [1998, p. 222]. Most publications on the active element pattern refer to a paper written by Hannan [1964]. In that paper, the ratio between the realized and directive gains is expressed as 1 − ∣Γa (θ, ϕ)∣2, where Γa is the active reflection coefficient for the array scanned in the direction of interest, defined by the angles θ and ϕ. Then, for a situation without grating lobes, the peak directive gain of the array is related to the projected aperture of the array, by using the standard formula for “apertures large compared with a wavelength.” Finally, the directive gain is scaled down to a single element of the array, which yields:

equation image

where A is the area allotted to each antenna in the array plane, θ is the scan angle from broadside, and λ is the wavelength.

[4] While this result is correct, the derivation presented in the work of Hannan [1964] involves large arrays of finite size (with apertures “large compared with a wavelength”), and it does not explain why situations with grating lobes must (indeed) be excluded to comply with result (1). Another derivation is given in the work of Pozar [1994], but it is limited to elements which can be characterized by a single radiation mode, as short dipoles and thin slots.

[5] These derivations are based on the transmit-only situation. We think that a rigorous and more general derivation should be obtained by explicitly relating the transmit and receive situations, with the help of the Lorentz reciprocity theorem, as is done for isolated antennae in the work of Collin and Zucker [1969] and De Hoop and De Jong [1974]. The approach presented here consists of first deriving an expression for the receiving cross section of the antenna in its terminated array environment. The receiving cross section [Collin and Zucker, 1969] with respect to a given direction is defined as the ratio between the power received in the terminating load (which can be the input impedance of an amplifier) and the power density of the incoming plane wave. It differs from the effective area through the fact that the latter quantity does not account for the potential impedance mismatch between the antenna and the load in which it is terminated. Once the receiving cross section is obtained, reciprocity applied to the port of the antenna immediately yields the active element pattern (also called “scan element pattern” in the work of Hansen [1998]) of the antenna in its array environment.

[6] The methodology followed here to obtain the receiving cross section can be seen as an extension of the work done by De Hoop and De Jong [1974] for an isolated element. This will provide an explanation of why reciprocity applies to the active input impedance of infinite arrays, as it is stated without explicit proof in the work of Hannan [1964, section 3]. The proofs will be given here for nonlossy antennae. For the sake of clarity, this will be done first for situations without a grating lobe, then the results will be extended to conditions allowing grating lobes.

[7] Besides providing a more rigorous derivation, the approach adopted here can also be used for estimating the receiving cross section of antennae located in infinite linear arrays. The result depends on the azimuthal directivity of the infinite array, and probably cannot be obtained intuitively. To the authors best knowledge, the relation between element pattern and azimuthal directivity of an antenna located in an infinite linear array has never been given in the literature.

[8] This paper is organized as follows. The expressions for the receiving cross section and power pattern are developed in section 2 for the linear array case. In section 3, a similar derivation is applied to the planar array case. Parts of the argumentation refer to intermediate results obtained in section 2. The extension to grating lobe conditions is presented in section 4. The use of reciprocity, instead of the establishment of a power budget, automatically involves the polarization aspects, so that it is not necessary to introduce the polarization mismatch a posteriori. The results are commented and conclusions are drawn in section 5.

2. Infinite Linear Array

2.1. Equivalent Circuit of the Antenna on Receive

[9] The infinite linear array of interest has an element spacing a, and, in the transmitting case, it is fed by current sources of amplitude Ia, with a phase shift ψ between successive elements (Figure 1). This phase shift corresponds to scanning in a direction defined by the unit vector equation image, at an angle θ from broadside.

Figure 1.

The infinite linear array of antennae in receiving conditions.

[10] In this section, we will establish the equivalent circuit of the receiving antenna located in the infinite linear array. A plane wave is incident on the array, from a direction defined by the unit vector equation image, with an amplitude Ebo for the electric field, and with a polarization defined by the unit vector equation imageb. The derivation of the receiving cross section will be carried out in a way which can be seen as an extension of the approach adopted by De Hoop and De Jong [1974] for a single antenna. The keys of the proof are the surface equivalence principle [Harrington, 1961] and the Green's function associated with an infinite array of infinitesimal dipoles.

[11] A unit cell of the array is isolated by introducing two planes perpendicular to the array line, S2 and S3, separated by a distance a. Besides this, a surface S1 is wrapped around the shielded antenna contained in the unit cell. A surface S0 also contains the antenna and is itself located in the unit cell.

[12] We denote by (equation imagea, equation imagea) the electric and magnetic fields generated by the array when it is active, and by (equation imageb, equation imageb) the total (incident and scattered) fields due to the incoming plane wave. The volume contained between surfaces S0 and S1 contains no sources, and the Lorentz reciprocity theorem [Chen-To, 1992], applied to it can be written as:

equation image

where equation image is the following field product: equation image = equation imagea × equation imagebequation imageb × equation imagea. The unit normal vectors equation image0 and equation image1 are pointed outside the volume comprised between surfaces S0 and S1.

[13] Following the same procedure as in the work of Collin and Zucker [1969] for the integration over S1, we can write:

equation image

where (Va, Ia) and (Vb, Ib) are the antenna port voltages and currents, which are generated by the active antenna array and the incoming wave, respectively. It results from the restriction of the integral over S1 to a cross section of the transmission line which feeds the antenna [Collin and Zucker, 1969, pp. 94–95]. The next steps consist of finding a new expression for the integral over S0 in equation (2).

[14] First, we prove that the contribution of the scattered fields (equation imagebs, equation imagebs) in the integral over S0 is zero. To this end, we apply the Lorentz reciprocity theorem to the volume V delimited by the surface S0 and surfaces S2 and S3. In this case, the function equation image = equation images is constructed with the fields (equation imagea, equation imagea) radiated by the array, and the fields (equation imagebs, equation imagebs) scattered by it when exposed to a plane wave incident from the direction of interest: equation image = equation imagea × equation imagebsequation imagebs × equation imagea. Hence we have:

equation image

where the unit normal vectors equation image0, equation image2 and equation image3 are pointed inside the volume V.

[15] If we denote by superscripts (2) and (3) the fields on surfaces S2 and S3, respectively, the periodicity of the electric fields in the array can be expressed as:

equation image
equation image

and similar relations can be written for the magnetic fields. From there, taking into account the fact that the unit normals are opposite on planes S2 and S3, the integrals over these surfaces in (4) annihilate each other.

[16] Besides this, the contribution of the fields on the surface closing the volume V at infinity can also be omitted: this surface increases proportionally with the distance from the array, while ∣equation images∣ decreases slightly faster than the inverse distance, as soon as infinitesimal losses are introduced. This completes the proof that, in (2), the contribution of the scattered fields (equation imagebs, equation imagebs) in the integral over S0 is zero, such that the incident field can be considered alone.

[17] The incident field on a point equation image of S0 can be written as:

equation image
equation image

where equation image is a unit vector in the direction of observation, k = 2π/λ is the wave number, and η is the free-space impedance.

[18] Introducing expressions (7), (8) and (3) into (2), we find:

equation image

where equation image is the normal to S0, pointing outside.

[19] The remaining part of the proof consists of showing that the integral over S0 in (9) can be expressed as a function of the far field radiated in the transmit situation. The latter can be obtained with the help of the scalar Green's function associated with an infinite linear array of dipoles [Mailloux, 1982] with orientation equation imaged, amplitude Id and spacing a. The vector potential can be written as:

equation image

with

equation image

where R is the distance from the array axis and k = 2π/λ is the wave number. Expression (10) can be regarded as the decomposition of the vector potential into an infinite number of cylindrical waves centered on the array axis. A cylindrical wave is propagating if kn2 > 0. Its direction of propagation, given by the angle θn from the array axis, is defined by cos θn = kn/k.

[20] The far field radiated by the active array can be obtained by considering equivalent electric and magnetic currents on surfaces identical to S0, surrounding each antenna of the infinite array (see Figure 1). If we denote by S the ensemble of these surfaces, the field radiated far from the array can be written as:

equation image

where equation image is a point on S, r = ∣equation imageequation image∣ and equation images = (equation imageequation image)/r.

[21] Exploiting the periodicity of the fields in the transmitting array, we can write the far field (12) as an integral over S0 only, provided that the periodic Green's function (10) is used. In the absence of grating lobes, the series (10) can be restricted to the n = 0 term. Hence the far field reads:

equation image

with the normalized field:

equation image

[22] By taking the scalar product of both members of (14) with equation imageb, we have:

equation image

[23] Taking into account the fact that equation image and equation imageb are perpendicular to each other, it can be shown that the integrals over S0 in equations (9) and (15) are identical. Consequently, (9) becomes:

equation image

where ω is the radian frequency, and μ is the magnetic permeability. After division by Ia, we obtain:

equation image

The right side of equation (17) is the open-circuit voltage of the antenna (all antennae are open-circuited), Za = Va/Ia is the active input impedance of the infinite array for the scan angle θ, and cos ϕp = equation imagea · equation imageb is a complex number describing the polarization mismatch. This situation corresponds to the equivalent circuit shown in Figure 2, where we see that the active input impedance on transmit also is the source impedance on receive.

Figure 2.

Equivalent circuit for a receiving antenna in an infinite linear array.

2.2. Receiving Cross Section of the Antenna

[24] Given that the array is infinite, in the absence of grating lobes, all the far radiated fields propagate with an angle π/2 − θ from the array axis (see Figure 3). Within this cone of directions, a pattern can be defined as a function of the azimuth angle. Then the azimuthal directivity Dθ(ϕ) of the transmitting array is defined as the ratio Pr/Pz, where Pz is the power radiated by an antenna, and Pr is the power that would be radiated per unit cell if, for all azimuth angles about the array axis, the field intensity were the same as in the direction of observation (θ, ϕ). Hence:

equation image

where R is the radius of the cylinder over which the power density is integrated, and ∣equation imagea∣ = ∣equation imagea(θ, ϕ)∣ is the magnitude of the field radiated in the azimuth angle of interest. Referring to equation (13), and using the approximation of the Hankel function for large arguments, the latter can be written as a function of the normalized field, i.e., ∣equation imagea∣ = ∣Eaoequation image. Then the directivity (18) can be written as:

equation image
Figure 3.

Array geometry and power radiated at an angle π/2 − θ = π/6 from the array axis, as a function of the azimuth angle ϕ. Pθ is the power radiated with polarization along equation imageθ, and Pϕ is the power radiated with polarization along equation imageϕ. Dθ(ϕ) is the directivity, displayed as a function of ϕ. The wavelength is four array spacings.

[25] On receive, the real power PL dissipated in the load ZL can be written as:

equation image

which, after a few algebraic transformations, becomes:

equation image

where the last ratio corresponds to the active reflection coefficient, Γa, defined with respect to the load impedance ZL.

[26] The receiving cross section can now be written as the ratio between the power PL and the incident power density:

equation image

[27] Using expression (17) for the open-circuit voltage, and expressing ∣Eao2 as a function of the directivity with the help of (19), the receiving cross section can be rewritten as:

equation image

[28] Figure 3 shows an example obtained with an infinite linear array of broad plate dipoles, fed by a delta-gap source. The progression of phases along the array is such that radiation occurs inside a cone of directions defined by the angle π/2 − θ = π/6 from the array axis. The plot on the right side shows, as a function of the azimuth angle ϕ along this cone, the two polarization components of the radiated power, as well as the azimuthal directivity Dθ.

[29] Applying reciprocity to one element of the array, taking into account the presence of the other (terminated) elements, we also have the standard result:

equation image

where G is the directive gain of an element in the terminated array, and Γ is the reflection coefficient for the element in the passively terminated array, defined with respect to the load impedance ZL. Comparing this expression with (23), we obtain two different expressions for the realized gain of an element in an infinite linear array:

equation image

[30] This expression also defines the element pattern of the antenna in its array environment. It is important to realize that the reflection coefficient appearing in the left member of (25) refers to a passively terminated array, while the one appearing in the right member is an active reflection coefficient. Because of mutual coupling, the latter depends on the scan angle θ, while the directivity Dθ introduces the azimuthal dependence in the new expression of the realized gain.

3. Infinite Planar Array

[31] Let us now consider an infinite planar array of antennae with spacings a and b and phase shifts ψx and ψy, along the X and Y axes, respectively, as sketched in Figure 4. The receiving cross section of an antenna in such an array can be obtained using a technique similar to what is done in section 2 for the linear array. In this case, the unit cell is defined by four surfaces, whose contributions to the Lorentz integral annihilate each other as a result of periodicity.

Figure 4.

The infinite linear array of antennae in receiving conditions.

[32] If, as in section 2.1, a surface S0 is defined around the antenna of interest, following the same procedure and notations as for the linear array, we obtain:

equation image

[33] The far field radiated by the active array is now computed with the help of the Green's function related to an infinite planar array of dipoles [Mailloux, 1982], which we recall here for the vector potential:

equation image

with

equation image

[34] In (27), equation imaged is the orientation of the dipoles, and Id is the amplitude of their currents. For the propagating modes (kmn2 > 0), the angle of propagation from broadside is defined by cos θmn = kmn/k.

[35] Now, the far fields can be expressed as radiated by the equivalent currents on the replicas of the surface S0, surrounding each antenna of the array. The use of the infinite planar array Green's functions allows us to write this field as an integral over just one of these surfaces. If there are no grating lobes, the m = n = 0 term only should be kept, and the far fields read:

equation image

with

equation image

[36] In (29), k0 = k cos θ, and equation imageaEao can be regarded as a normalized field. Taking the scalar product of (30) with equation imageb, and recognizing that the integrals in (26) and (30) are identical, we obtain:

equation image

where cos ϕp = equation imagea . equation imageb and Vo is the open-circuit voltage.

[37] The receiving cross section of the antenna is defined as the ratio between the power absorbed by the load, and the incident power density (cf. section 2):

equation image

[38] Using the expression (31) for Vo, and rearranging the factors, we obtain:

equation image

where the numerator and the denominator of the fraction each correspond to the power generated per antenna in the active array situation (The array is assumed to radiate into the z > 0 half-space only). Hence they annihilate each other, which leads to the following result:

equation image

[39] In this case, it should be noted that all elements of the array are assumed to be terminated by the impedance ZL. Comparing expressions (24) and (34) for the receiving cross section of an antenna in the array, we obtain the realized gain of an element:

equation image

where Γ is the reflection coefficient at one port of the array when the other ports are terminated by identical passive loads. From the left to the right member of (35), the dependence on the angular coordinates is transferred from the gain G to the active reflection coefficient ∣Γa∣.

4. Grating Lobe Conditions

4.1. Linear Array

[40] In this section, we extend the solutions obtained in section 2 to situations where grating lobes can appear. In this case, in the Green's function (10), all the propagating terms (denoted here under by n = MN) must be considered. Hence the far field reads:

equation image

with

equation image

where equation imagen is a unit vector in the direction of the n-th grating lobe.

[41] Applying the same procedure as in section 2.1 for a plane wave coming from direction equation imagen with amplitude Ebn and polarization equation imagebn, we obtain for the open-circuit voltage:

equation image

where Ibn and Vbn are the port current and voltage for a plane wave indicent from grating lobe n, and cos ϕpn = equation imagean . equation imagebn stands for the polarization mismatch.

[42] The azimuthal directivity, defined in section 2.2, in the direction of the n-th grating lobe is now:

equation image

[43] Finally, the receiving cross section with respect to a plane wave coming from the n-th grating lobe direction is:

equation image

4.2. Planar Array

[44] The far field radiated by the active array is:

equation image

with

equation image

where the different values taken by the indices m and n correspond to the main and grating lobes, and equation imagemn are unit vectors in the corresponding directions of radiation.

[45] Following the same procedure as in section 3, we obtain for the open-circuit voltage with respect to a plane wave coming from the grating lobe direction equation imagemn with amplitude Ebmn and polarization equation imagebmn:

equation image

where cos ϕpmn = equation imageamn. equation imagebmn, Vmn is the open-circuit voltage, and θmn is the angle the grating lobe makes with the broadside direction.

[46] The corresponding receiving cross section Armn is:

equation image

[47] Contrary to the case without grating lobes, the fraction is not equal to one, because the numerator corresponds to the energy radiated by the active array in the direction of the grating lobe of interest only. Hence the expression of the receiving cross section with respect to a given direction cannot be simplified. However, adding the powers radiated in different directions (numerator in (44)), we obtain an expression which links the receiving cross sections in the different directions in which the array is sensitive:

equation image

[48] Except for the polarization factor in the denominator, this is equivalent to Hannan [1964, expression (8)].

5. Conclusion

[49] We gave a rigorous derivation for the receiving cross section of an antenna located in an infinite nonlossy receiving array, without and with grating lobes. This derivation makes explicit use of the Lorentz reciprocity theorem. To the authors knowledge, the result obtained for the linear array is not given in the literature. It depends on the directivity of the azimuthal pattern of the array. Table 1 shows a parallel between the receiving cross sections of the isolated antenna, and of the antennae in linear and planar infinite arrays, when there are no grating lobes. In terms of the dependence on wavelength and array spacing, the linear array clearly appears as an intermediate situation between the isolated antenna and the planar infinite array. The result obtained for the planar array is generally accepted, but, in the derivation presented here, no assumption regarding the type of elements has been considered, and intuitive reasoning has been avoided.

Table 1. Isolated Antennae, Infinite Linear, and Infinite Planar Arrays, Factors Appearing in the Expression of the Receiving Cross Section of an Element
 WavelengthSpacingDirectivityConstantMatchingPolarization
Isolated antennaλ21G(θ, ϕ)1/(4π)(1 − ∣Γ∣2)∣cos ϕp2
Infinite linear arrayλaDθ(ϕ)1/(2π)(1 − ∣Γa(θ)∣2)∣cos ϕp2
Infinite planar array1abcos θ1(1 − ∣Γa(θ, ϕ)∣2)∣cos ϕp2

Acknowledgments

[50] The authors would like to thank D. H. Schaubert, from the University of Massachusetts, Amherst, and A. T. De Hoop, from the Technical University of Delft, for their helpful comments on an early version of the manuscript.

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