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Antenna Design Notes Part I Mario Orefice Dipartimento di Elettronica Politecnico di Torino c Copyright 2013 by the Author. All rights reserved.

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Aperture Antennas Lecture note orefice politecnico

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  • Antenna Design NotesPart I

    Mario OreceDipartimento di Elettronica

    Politecnico di Torino

    c Copyright 2013 by the Author. All rights reserved.

  • Contents

    1 INTRODUCTION 11.1 Summary on Maxwells equations . . . . . . . . . . . . . . . . . . . . 11.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.2.1 Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Antenna matching . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Ohmic eciency . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.5 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.6 Radiation pattern . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.7 Radiated eld . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.8 Eective area . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.9 Eective height . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.10 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.11 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2 APERTURE ANTENNAS 162.1 General expression of the radiation . . . . . . . . . . . . . . . . . . . 162.2 Equivalence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    2.2.1 Equivalence Theorem and planar apertures . . . . . . . . . . . 192.3 Radiation zones from an aperture . . . . . . . . . . . . . . . . . . . . 212.4 Gain ed eciency of an aperture . . . . . . . . . . . . . . . . . . . . . 232.5 Rectangular aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    2.5.1 Uniform amplitude and phase . . . . . . . . . . . . . . . . . . 242.5.2 Separable eld distributions . . . . . . . . . . . . . . . . . . . 252.5.3 Eects of the phase error . . . . . . . . . . . . . . . . . . . . . 28

    2.6 Circular aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6.1 Uniform amplitude and phase . . . . . . . . . . . . . . . . . . 312.6.2 Non uniform illumination . . . . . . . . . . . . . . . . . . . . 32

    2.7 Phase center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.8 Apertures with arbitrary eld distribution . . . . . . . . . . . . . . . 392.9 Horn antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

    2.9.1 Sectoral and pyramidal horns . . . . . . . . . . . . . . . . . . 412.9.2 Smooth circular horns . . . . . . . . . . . . . . . . . . . . . . 48

    1

  • M. Orece: Antenna design notes (2013) 2

    2.9.3 Beam symmetry and cross polarization . . . . . . . . . . . . . 502.9.4 Bimodal horn . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.9.5 Corrugated horns . . . . . . . . . . . . . . . . . . . . . . . . . 56

    2.10 Geometrical optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.10.2 Eikonal Equation . . . . . . . . . . . . . . . . . . . . . . . . . 672.10.3 Flux tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.10.4 Fermats principle . . . . . . . . . . . . . . . . . . . . . . . . . 722.10.5 Stationary phase method . . . . . . . . . . . . . . . . . . . . . 732.10.6 Reection from a surface . . . . . . . . . . . . . . . . . . . . . 76

    2.11 Surface Currents Method and Physical Optics . . . . . . . . . . . . . 802.12 Reector antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    2.12.1 Diraction from a revolution surface . . . . . . . . . . . . . . 852.12.2 Integration methods . . . . . . . . . . . . . . . . . . . . . . . 86

    2.13 Paraboloidal antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.13.1 Paraboloid design . . . . . . . . . . . . . . . . . . . . . . . . . 93

    2.14 Blocking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.14.1 Null eld hypothesis . . . . . . . . . . . . . . . . . . . . . . . 972.14.2 Central blocking . . . . . . . . . . . . . . . . . . . . . . . . . 982.14.3 Support structures blocking . . . . . . . . . . . . . . . . . . . 1002.14.4 Reaction on the feed . . . . . . . . . . . . . . . . . . . . . . . 103

    2.15 Oset paraboloid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.16 Defocusing errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

    2.16.1 Axial defocusing . . . . . . . . . . . . . . . . . . . . . . . . . 1072.16.2 Transverse defocusing . . . . . . . . . . . . . . . . . . . . . . . 108

    2.17 Eects of the surface tolerances . . . . . . . . . . . . . . . . . . . . . 1112.18 Total eciency of a reector antenna . . . . . . . . . . . . . . . . . . 1132.19 Reector antenna feeds . . . . . . . . . . . . . . . . . . . . . . . . . . 1142.20 Geometrical Theory of Diraction . . . . . . . . . . . . . . . . . . . . 117

    2.20.1 Diracted eld at shadow and reection boundaries . . . . . . 1242.20.2 Equivalent currents methods . . . . . . . . . . . . . . . . . . . 125

    2.21 Dual reector antennas . . . . . . . . . . . . . . . . . . . . . . . . . . 1262.21.1 Cassegrain antenna . . . . . . . . . . . . . . . . . . . . . . . . 1262.21.2 Simplied design of a Cassegrain antenna . . . . . . . . . . . . 1302.21.3 Twistreector . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312.21.4 Other types of reector antennas . . . . . . . . . . . . . . . . 134

    2.22 Shaped beam reectors . . . . . . . . . . . . . . . . . . . . . . . . . . 1362.22.1 Search radar pattern . . . . . . . . . . . . . . . . . . . . . . . 1372.22.2 Empirical design of shaped beam antennas . . . . . . . . . . . 1392.22.3 Cylindrical reector antenna . . . . . . . . . . . . . . . . . . . 1422.22.4 Reector synthesis with Geometrical Optics . . . . . . . . . . 145

    2.23 Lens antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1492.23.1 General concepts . . . . . . . . . . . . . . . . . . . . . . . . . 149

  • M. Orece: Antenna design notes (2013) 3

    2.23.2 Analysis of a lens antenna . . . . . . . . . . . . . . . . . . . . 1492.23.3 Lenses with refractive index < 1 . . . . . . . . . . . . . . . . . 1532.23.4 Aperture amplitude distribution . . . . . . . . . . . . . . . . . 153

    2.24 Travelling wave antennas . . . . . . . . . . . . . . . . . . . . . . . . . 1562.24.1 Surface wave antenna: radiation conditions . . . . . . . . . . . 1572.24.2 Leaky wave antennas: radiation conditions . . . . . . . . . . . 161

  • Chapter 1

    INTRODUCTION

    1.1 Summary on Maxwells equations

    Electromagnetic phenomena, of which the radiation from antennas is an importantexample, are ruled by the Maxwell equations, of which we will recall briey thefundamental aspects.

    In their most general expression, Maxwells equations are given by

    E = BtM

    H = Dt

    + J(1.1)

    with the divergence equations: B = m

    D = e(1.2)

    where the vectors E ,H,J ,M,D,B and the scalars e and m are quantities functionof time and space - e.g. E = E(P, t) - and represent, in the order, the followingquantities: electric eld, magnetic eld, electric current density, magnetic currentdensity, electric displacement, magnetic induction, electric charge density, magneticcharge density.

    Actually, some of these (magnetic current and charge density) are not real physi-cal quantities, but ctitious quantities which can be derived by others (in particularelectric eld, charge and current) but which may signicantly simplify the calcula-tions.

    Very often antennas are placed in a linear, homogeneous and isotropic medium,at least in their neighborhood, and with quasi-stationary signals. Actually, there arealso some important exceptions, like the antennas in plasmas, the EMP (Electro-Magnetic Pulse) simulators, the antennas made by non- homogeneous non-isotropicnon-linear material (e.g. antennas made by carbon bre), and others. However, in

    1

  • M. Orece: Antennas 2013, Introduction 2

    the most part of the cases, it is possible to use the assumptions of homogeneity,linearity and isotropy, and harmonic regime, and re-write eqs. (1.1-1.2) as:

    E = jH M

    H = jE + J

    H = m/

    E = e/

    (1.3)

    where vectors and scalars indicate the phasors, i.e. quantities depending on timewith ejt (this dependence will be omitted hereafter); and are respectively themagnetic permeability and the dielectric constant, or permittivity, of the medium.Note also that dierent fonts have been used for the quantities in eqs.(1.1-1.2) (ar-bitrary dependence on time) and those in eqs.(1.3), because the former representthe instantaneous values, the latter the phasors.

    Is is also known from the basic concepts of Electromagnetic Waves that, manip-ulating eqs.(1.1-1.2-1.3), we can derive the Wave Equations; in fact, from eqs.(1.3)we get:

    E k2E = jJ M

    H k2H = jM J(1.4)

    with k2 = 2, where k is the propagation constant of the medium. If this islossless, k is real and related to the wavelength by the equation:

    k = 2/ (1.5)

    Considering, as an example, the rst of eqs.(1.4), this can be rewritten as

    L(, j) E(P, ) = jJeq(P, ) (1.6)where Jeq is the equivalent current

    Jeq = J j

    M (1.7)

    and the operator L is given by:

    L(, j) = I 2I (1.8)Eq.(1.8) represents an important operator relationship between the electric eld andthe current density (see g.1.1).

    We recall also the boundary conditions necessary to solve the Wave Equationsin presence of dierent media:

    n (H2 H1) = Js(E2 E1) n = M s (1.9)

  • M. Orece: Antennas 2013, Introduction 3

    LJE

    Figure 1.1: Operator relationship between eld and current density.

    where Js and M s are the electric and magnetic surface currents at the separationsurface between two homogenous media 1 e 2, and H1,2, E1,2 are the magnetic andelectric eld on the separation surface, in each medium. This boundary condition isalso known as continuity of the tangential components of the electric and magneticelds.

    1.2 Denitions

    .We give in the following a number of denitions of the most important parameters

    for the electrical characterization of the antennas 1

    1.2.1 Antenna

    The antenna is that part of a radiofrequency (RF) electronic circuit which eectsthe conversion, in both directions, between transmitted or received electromagneticwaves, and the RF voltage or current (or power) guided along a transmission linkconnected to it.

    The antenna radiates (or receives) electromagnetic waves in a more or less freespace, and the presence of the near objects may modify its radiation characteristics.On the contrary, far objects may inuence anyway the propagation characteristicsand therefore the far radiated eld.

    For the description of the antenna characteristics, the preferred coordinate sys-tem is usually the spherical system (R, , ), because at large distance (which is thetypical antenna operating range) the eld radiated from an antenna can be expressedas spherical wave, with constant amplitude and phase on the sphere R=const.

    Many types of antennas can be fount in the literature or in the market, but theycan be roughly grouped in a few categories:

    the aperture antennas, which are extended in two dimensions, usually largewith respect to the wavelength (up to thousands of wavelengths); for thisreasons, they are used especially at higher frequency (approximately, above

    1The antenna, being a structure often of large dimensions, and subject to various types ofenvironmental conditions, has also important mechanical characteristics (wind load, mechanicalresonant frequency, etc.) which, however, are in general out of the scope of this text.

  • M. Orece: Antennas 2013, Introduction 4

    around 1 GHz): typical examples of antennas of this type are the horn anten-nas, the parabolic antennas and others;

    the wire antennas, which are extended essentially in one dimension, usually ofsmall size, very often less than one wavelengths, so that they are used mainlyat low frequencies (approximately, below about a few GHz): typical examplesof antennas of this type are the dipoles on others derived from it;

    the array antennas, which are sets of radiating elements (of one of the previoustwo categories) arranged in the space to obtain particular radiation character-istics.

    1.2.2 Antenna matching

    From the circuit point of view, the transmitting antenna is a load, placed at the endof a transmission line (coaxial cable, microstrip, waveguide, etc.), which absorbs acertain amount of power. It is therefore characterized by an input impedance Z,whose knowledge is essential for a correct design of the transmission line and of thematching network, if any.

    The value of the input impedance is strictly dependent on the position of its inputterminals. These can be physically found in case of a coaxial or a two-conductorsfeeding; in case of a waveguide feeding, we shall refer to the equivalent line and toan input (or reference) section.

    Generally, however, rather than the input impedance, it is more interesting thematching to the transmission line. Consequently, very often the antenna impedancecharacteristics are given in terms of its reection coecient or of its modulus;alternatively they can be expressed by the standing wave ratio or the return loss.This latter is very used in practice, because it allows the immediate calculation ofthe amount of power lost because of the mismatch. These quantities are relatedeach other by the known relationships:

    S =1 ||1 + ||

    RL = 20 log10 ||(1.10)

    1.2.3 Ohmic eciency

    It is dened as the ratio between the power radiated by the antenna and the poweraccepted by the antenna at its input terminals:

    = Pr/Pa (1.11)

    From the circuit point of view, eq.(1.11) can be represented (neglecting possiblereactances) with a series circuit of the type shown in g.(1.2), where RL and Rr arethe resistances associated respectively to the losses and to the radiation. We obtain:

  • M. Orece: Antennas 2013, Introduction 5

    Pd Pr

    Rr

    Figure 1.2: Equivalent circuit for radiation resistance and loss resistance.

    =Rr

    RL + Rr(1.12)

    The ohmic eciency is usually less than 1 when the antenna size is much smallerthan the wavelength, the radiation resistance is low (a few ohms or even less) andcomparable to (or lower than) that associated to the losses 2. Antennas at microwavefrequency, and also at lowwer frequencies (above a few MHz) have generally 1,except when they contain dissipative materials (e.g. lossy dielectrics). For higherfrequencies (millimeter waves ad above) the losses in conductors can be signicant,even if with high conductivity (e.g. copper) because of the skin eect.

    1.2.4 Gain

    The gain is a function of the observation angles , around the antenna and it canbe dened for both receiving and transmitting antennas.

    For a transmitting antenna the gain is dened as the ratio between the powerdensity generated by the antenna in a point P at large distance, with angular coor-dinates , , and that generated in the same point by a lossless isotropic radiator,fed with the same power delivered to the considered antenna. This denition cantherefore be formulated as:

    G(, ) =S(, , r)

    Pd/4r2(1.13)

    where S is the real part of the Poynting vector in point P . At large distance, thepower density is inversely proportional to r2, and the gain can be rewritten, by usingthe angular power density dP/d, as:

    G(, ) = 4dP (, )/d

    Pd(1.14)

    2The resistance associated to the losses can be subdivided according to the various phenomenathat generate them, i.e., essentially, Joule eect in the conductors, dissipation in dielectrics, niteconductivity of the ground.

  • M. Orece: Antennas 2013, Introduction 6

    so that 4

    G(, )d = 4 (1.15)

    In reception, the gain is dened as the ratio between the available power at theoutput terminals of the antenna, when illuminated by a plane wave from a direction, , and that available at the output terminals of an isotropic radiator, illuminatedby the same plane wave.

    If the antenna, as often happens, is reciprocal, the gain in reception and trans-mission are the same.

    The term gain without other specications indicates usually the maximum valueof the function G(, ), and it is usually expressed in decibels3.

    This denition is clearly independent on the polarization and it refers to the totalpower (or power density). In practice, often there is a distinction between the powerassociated to the two orthogonal components of the electric eld. Depending on thepolarization characteristics and requirements for the antenna, it can be distinguishedbetween direct polarization and cross polarization, as it will be explained in one ofthe next paragraphs.

    1.2.5 Directivity

    The same denition holds as for the gain, except that, instead of the power deliveredto the antenna (Pe), the power radiated from the antenna (Pr) is considered:

    D(, ) = 4dP (, )/d

    Pr(1.16)

    From (1.11) it follows:G(, ) = D(, ) (1.17)

    1.2.6 Radiation pattern

    This term is very common in practice, and indicates the graphical (or numerical)representation of the gain or of the directivity functions (or of a part of them), typ-ically a particular cut for a xed value of one of the variables, in case normalizedto its maximum value; one of the most common cases is to represent the gain (di-rectivity) functions for =cost., for varying . For example, for an antenna with aspherical reference system as shown in g.(1.3), the most signicant diagrams arethose of G(, ) for = 0 and = /2 (see g.1.4). Such planes, in case of linear

    3The denition of gain compares essentially an antenna with the isotropic source. Anotherdenition of gain (a little obsolete but sometimes still used, especially for terrestrial broadcastingantennas) refers, for historical practical reasons, to the half-wave dipole, whose gain is about 2 dB.As a consequence, the gain of the same antenna may be expressed by two dierent numbers: theone referred to the isotropic radiator is greater than that referred to the dipole by 2 dB. To avoidconfusion, the notation dBi e dBd (not much appreciated by the metrologists...) is used,respectively. As a standard, however, the reference is the isotropic radiator.

  • M. Orece: Antennas 2013, Introduction 7

    polarization and with a suitable choice of the reference system, are coincident withthe principal planes of the radiation pattern, which are also called E and H planes,because they are dened respectively by the the electric (or magnetic) eld vectorand the propagation unit vector.

    d

    dS

    kE

    H

    P

    E plane

    k E

    H

    P

    H plane

    Figure 1.3: Reference system for an aperture antenna.

    The radiation pattern may also have a 3-D representation, as in g.(1.5), or atlevel curves as in g.(1.6). This latter is particularly used in the case of contouredbeam antennas.

    In order to have a rough characterization of a radiation pattern, just a fewcharacteristic parameters are enough, without the need of providing, or computing,the full pattern. The most signicant of such parameters are: the sidelobe level andthe half power beamwidth (HPBW), also called -3 dB angle since the half powercorresponds to a -3 dB level.

    The sidelobe level is the level of the highest of the side lobes (usually the oneadjacent to the main beam), and it is expressed in decibels with respect to the mainbeam. For more accurate characterizations, sometimes it is given as an envelopefunction: as an example, 29 25 log is the envelope of all the sidelobes of sometypes of antennas for earth stations in satellite communications: in this case thereference is the isotropic radiator.

    The half power beamwidth, often abbreviated with HPBW, is the full angle wherethe gain is maintained within -3 dB below the maximum. Sometimes other anglesand levels are used to indicate the width of a beam (e.g. -6 dB, or -10 dB, or therst minimum near the main beam) but in this case this must be clearly specied.

    There is also an approximate relationship between the HPBW (which may alsobe dierent in the principal planes) and the gain of a lossless antenna; this is given

  • M. Orece: Antennas 2013, Introduction 8

    50 40 30 20 10 0 10 20 30 40 5020

    18

    16

    14

    12

    10

    8

    6

    4

    2

    0

    (gradi)

    G

    (dB)

    Figure 1.4: Cuts in the principal planes of a radiation pattern.

    by

    G =K

    EH(1.18)

    where G is the maximum gain (expressed as a number, not in dB), E,H are theHPBWs in the principal planes (e.g. the E- and H-plane or anyway in the twosymmetry planes of the main lobe), and K is a suitable constant. If the angles arein degrees, the value of K is about 3 104. 4

    Eq.(1.18) is not valid for antennas directive in one plane only and omnidirectionalin the other plane (as a dipole or a linear array). It can be veried that for suchomnidirectional antennas the following relationship applies:

    G =Ko0

    (1.19)

    with Ko 100, where 0 is the HPBW.Exercise: Assuming that all the radiated power is all concentrated in the main beam, and this

    is symmetric about the axis and suciently narrow that in the region of interest it is sin ),demonstrate eq.(1.18) and compute the constant K in the three following cases of dependance on of the radiation pattern: a) quadratic in eld; b) quadratic in power; c) cosine to a power.

    4For low directivity antennas (e.g. up to 10-15 dB) the constant K is slightly higher, and itgoes to about 4 104.

  • M. Orece: Antennas 2013, Introduction 9

    105

    05

    10

    10

    5

    0

    5

    1040

    30

    20

    10

    0

    10

    Figure 1.5: Tri-dimensional representation of a radiation pattern (rectangular hornwithout phase error).

    The maximum value of the directivity, DMax (and as a consequence of the gain,if the ohmic eciency is known) can be computed more precisely than with eq.(1.18)if the behavior of the directivity function is known. In fact, assuming

    D = DMaxd(, ) (1.20)

    where obviously the maximum value of d(, ) is 1. Integrating both sides of eq.(1.16)on the whole solid angle we obtain

    DMax =4

    4d(, )d

    (1.21)

    Exercise: Compute the directivity of an antenna whose radiated power density is zero in thenegative z half space, and for positive z is expressed by the function d = cos , symmetric aboutthe z axis.

    1.2.7 Radiated eld

    The electromagnetic eld radiated from an antenna is computed, in general, byusing very complex formulas, obtained by inverting the operator (1.8). However, atdistances suciently large from the antenna, it can be written in formally simplemanner as:

    E = V0F (, )ejkr

    rp(, ) (1.22)

  • M. Orece: Antennas 2013, Introduction 10

    0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.20.2

    0.15

    0.1

    0.05

    0

    0.05

    0.1

    0.15

    0.2

    u

    v

    103

    20

    20

    20

    20

    20

    3030

    30

    30

    30

    30

    30

    30

    3040

    4040

    40

    4040

    40

    40

    40

    40

    Figure 1.6: Representation of the radiation pattern of g.(1.5) through level curves.

    where:

    V0 is a normalization constant, having the dimensions of a voltage; F is a complex scalar function called radiation force, which depends only on

    the angular coordinates. By choosing

    V0 =

    Z0Pe4

    (1.23)

    we have|F |2 = FF = G (1.24)

    p(, ) is the polarization unit vector of the antenna, and in general it dependson the observation angles.

    1.2.8 Eective area

    This parameter, as well as the following one (the eective height) has a particularpractical importance because it allows to transform the electromagnetic quantities(elds, power density) in electrical circuits quantities (voltage, power).

  • M. Orece: Antennas 2013, Introduction 11

    In reception, the eective area Ae is the quantity which, multiplied by the in-cident power density onto the antenna, gives the available power at the outputterminals of the antenna. In formula:

    Pav = AeS (1.25)

    under the assumption of matching between the polarization unit vectors of theincident wave and of the receiving antenna. If this is not, the polarization mismatchis taken into account by the dot product between such unit vectors, (pinc pant)2.

    Moreover, it is known the relationship between the eective area and the gain,which may be derived through the Reciprocity Theorem:

    G =4

    2Aeq (1.26)

    The eective area has the physical dimensions of square meters, which explainsthe name. For aperture antennas it exists also a relationship between the eectivearea and the geometric area of the aperture A, given by:

    Aeq = A (1.27)

    where is the aperture eciency5, and it has, in general, values ranging between0.5 and 0.85, depending on the types of antennas. In some particular cases (superdi-rective antennas) the value of 1 may be exceeded, but in a narrow bandwidth.

    1.2.9 Eective height

    In reception, it is the (vector) quantity which, multiplied (with a dot product) by theelectric eld incident on the antenna, gives the open-circuit voltage at the antennaoutput terminals. In formula:

    V = he Ei (1.28)The eective height has the physical dimensions of meters, which explains the

    name. For simple wire antennas, there is a simple relationship between the eectiveheight and the length of the antenna (see Tab.1.1).

    These values can be easily derived from the denition of the eective height intransmission, which relates it to the generalized electric moment P e through theequation

    P e = Ihe (1.29)

    where I is the antenna input current, and P e is related to the radiated eld by

    E =jZ02r

    ejkrP e (1.30)

    5Actually, for many aperture antennas, the aperture eciency is only one of the gain reductionfactors, together with others (spillover, blockage, etc.); in practice, it is more common to dene as the antenna eciency, including all factors.

  • M. Orece: Antennas 2013, Introduction 12

    Antenna type he(lenght=)hertzian dipole short dipole /2resonant dipole /2 2/

    Table 1.1: Eective height for a few simple types of wire radiators.

    By comparing the available power from this equivalent source with eq.(1.25) wemay derive the relationship between the eective area and the eective height, incase of polarization matching:

    Aeq =h2eZ04Rirr

    (1.31)

    1.2.10 Polarization

    In the most general case, the polarization of an electromagnetic wave radiated by anantenna (usually intended as the polarization of the electric eld) has an ellipticalpolarization. In practice, however, all antennas have a nominal polarization, whichis the ideal polarization the antenna should have, depending on the design and onthe service for which it is used: usual nominal polarizations are linear (vertical,horizontal, 45) or circular (right- or left-hand circular, often abbreviated withRHC, LHC, also called respectively clockwise or counterclockwise (CW or CCW)).However, since the actual polarization is not exactly as the nominal one, there isalso a spurious polarization component which may be non negligible, especially outof the intended coverage region of the antenna.

    We may therefore dene two eld components, the direct polarization and thecross polarization (or also wanted and unwanted), which are the projections on thetwo polarization unit vectors, indicated respectively with p and q, which correspondrespectively to the desired (or nominal) polarization, and to the spurious polar-ization, orthogonal to the previous one. As it will be seen in the following, theirdenition can be given in dierent ways, and it is therefore important to make a fewdistinctions6.

    Possible denitions of polarization

    The direct and cross polarization unit vectors may be dened in dierent ways,so that the various couples of unit vectors, although coincident in the direction ofmaximum radiation, dier each other signicantly for dierent directions.

    6This subject has been widely discussed by A.C.Ludwig in The denition of cross polarization,in IEEE Transactions on Antennas and Propagation, Jan.1973, pp.116-119, and to this paper thereader is addressed for a thorough examination of this subject.

  • M. Orece: Antennas 2013, Introduction 13

    Figure 1.7: First and second denition of cross polarization.

    Lets consider a directive antenna (as an example, an aperture with arbitraryshape in the x, y plane) with the maximum radiation direction along z. If the eldon the aperture is mainly directed in one direction, e.g. along y, the radiated eldalong the axis will have essentially the same direction, as shown in g.(1.7). As aconsequence it may seem reasonable to assume:7

    p = y ; q = x (1.32)

    which in spherical coordinates become:

    p = sin sin r + cos sin + cos

    q = sin cos r + cos cos sin (1.33)

    If this denition is correct on the z axis, for other directions it is no longer accept-able because the polarization unit vectors would have radial components that theactual radiated eld has not. As a consequence this rst denition is not applicable,if we want a denition which can be used out of the z axis.

    Considering that the radiated far eld is always transverse (perpendicular tor) one could think of using as polarization vectors , : but this is not possible,because on the z axis (the polar axis, = 0 or ) the variable is undetermined,and so are the unit vectors.

    A second possible denition is to associate to the aperture a spherical referencesystem with axis directed along y. In this case we may assume8

    p = ; q = (1.34)7S.Silver:Microwave antennas theory and design, New York: Mc.Graw-Hill, 1949, pp.424 and

    557-564.8V.P.Narbut, N.S.Khmelnitskaya: Polarization structure of radiation from axisymmetric re-

    ector antennas, Radio Eng. Electron. Phys., vol.15, pp.1786-1796, 1970.

  • M. Orece: Antennas 2013, Introduction 14

    where primes indicate a spherical reference system with polar axis y (see g.1.7).Such unit vectors, transformed in the main reference system (polar axis z), areexpressed as:

    p =sin cos + cos

    1 sin2 sin2 q =

    cos sin cos 1 sin2 sin2

    (1.35)

    Again, this denition is insucient in the xy plane for = /2, where the unitvectors are not dened.

    A third denition considers as direct polarization of the antenna that dened bythe usual method of measurement of the antenna radiation pattern9.

    y

    x

    z

    p

    pq

    =

    0

    0

    0=

    q

    0

    Figure 1.8: Cuts for the radiation pattern measurement and third denition of crosspolarization.

    As it is known, this latter is obtained by rotating a probe, inclined according thedirect (or cross) polarization, about the antenna under test (AUT) in various planesat =const. (v.g.1.8); this technique is perfectly equivalent to the most usual one ofkeeping xed the probe (or the probes if both polarizations are taken simultaneously)and rotating the AUT about y for each cut, by successively increasing by (around

    9See J.D.Kraus:Antennas, New York: Mc.Graw-Hill, 1950, Cap.15, or in Silver, ibid., pp.557-564.

  • M. Orece: Antennas 2013, Introduction 15

    z) both the AUT and the probe. It follows that the denition of the unit polarizationvectors is:

    p = cos + sin

    q = sin + cos (1.36)

    A source of this type is coincident with the Huygens source.The eld lines for the direct and cross polarizations associated to the unit vectors

    of the three denitions are shown in g.(1.9).

    Y

    Z X X

    Y

    Z

    XZ

    Y

    XZ

    Y

    X

    Y

    ZZ

    Y

    X

    Figure 1.9: Field lines for the three denitions of direct and cross polarization:(above, the reference polarization, below, the cross polarization).

    Exercise: Show that the eld lines of the 3rd denition are obtained by intersecting a spherewith a bundle of planes with axis parallel to y with origin in the point 0,0,-1.

    Also in the third denition it exists an indetermination point, corresponding to = . In fact, while the singularity for = 0 can be eliminated by continuity(lim0

    p = y for any ), for = we have that p tends to y for = /2, and to+y for = 0 o = .

    1.2.11 Bandwidth

    It is the frequency range where a certain parameter (e.g. gain, or impedance, orsidelobe level, etc.) is maintained within the prescribed limits: consequently, thereare as many bandwidths as the parameters of interest. Particularly important is theoperating bandwidth, which is the intersection of all the bandwidths of interest fora given antenna.

  • Chapter 2

    APERTURE ANTENNAS

    2.1 General expression of the radiation

    As already mentioned in the introduction, the operator expression bounding theelectric eld and the current density, expressed by

    L(, j) E(P, ) = jJeq(P, ) (2.1)where Jeq is the equivalent current

    Jeq = J j

    M (2.2)

    and the operator L is given by:

    L(, j) = I k2I (2.3)may be inverted in order to derive the electric eld from the currents. In eq.(2.3)k2 = 2; k is the propagation constant in the medium (which may be also complex,with negative imaginary part)1. The inverse relationship is clearly given by:

    E(P, ) = jL1(, j) Jeq (2.4)It is known that eq.(2.4) may be expressed as:

    E(P ) = jVG(P P ) Jeq(P )dV (2.5)

    where P is the observation point, V is any volume which encloses all the sources,P is a generic point in V , and G is the Greens dyadic function given by:

    G =[I +

    k2

    ](P ) (2.6)

    1In practice, since the most common case is when antennas radiate in air, we will always assume(except where otherwise indicated) k = k0 =

    00

    16

  • M. Orece: Antennas 2013, Aperture antennas 17

    where

    (P ) =1

    4rejkr (2.7)

    is the scalar Greens function and I is the identity dyadic.If P is at distance such that the condition |P P | >> is satised for any P ,

    eq.(2.6) can be simplied to:

    G(P ) It,r(P ) (2.8)

    where It,r

    is the transverse identity dyadic with respect to r, given by:

    It,r

    = + (2.9)

    and eq.(2.5) becomes:

    E(P ) = jVIt,r Jeq(P )

    ejkr

    4rdV (2.10)

    Then, if P is at distance such that all the segments from P to any P can beconsidered as parallel, I

    t,rcan be extracted from the integral, and, moreover, can

    be assumed as practically constant in amplitude. We can therefore apply the fareld approximation

    ejkr

    r= e

    jk(RR)

    R(2.11)

    which may be considered valid if |P P | RF , where RF = 2D2/ is the Fraun-hofer distance, and D is the maximum dimension of the volume in a direction trans-verse to to r. Eq.(2.5) becomes

    E(P ) = jejkR

    4RIt,rVJeq(P

    )ejkRrdV (2.12)

    where R is the (xed)distance of the observation point P from an origin O placedin the region of the sources (although arbitrary), and r = OP .

    Eq.(2.12) is theRadiation Integral in far eld.

    2.2 Equivalence Theorem

    This theorem allows to simplify the Radiation Integral, transforming a volume in-tegral into a surface integral.

    Lets consider a distribution of sources, all enclosed in a surface S, which denestwo volumes V1 e V2 (inner and outer to S) respectively with and without sources,as in g. 2.1. Here the normal to S is directed outwards. The currents radiate aeld distribution E1, H1 in V1 and E2, H2 in V2.

    If we want to replace the eld E1, H1 in V1 with an arbitrary distribution (e.g.E,H), it will be necessary, to the external eects in region 2, the introduction of

  • M. Orece: Antennas 2013, Aperture antennas 18

    Distribuzionedi campo

    E , H_ _2 2

    Distribuzionedi campo

    E , H_ _2 2

    V2

    V2

    n^

    1

    V1 Ms

    E_ n^= x

    H_s^J = n x

    E , H_ 1 1_

    (J , H)__Sorgenti

    di campoe distribuzione

    S

    V

    Camponullo

    Figure 2.1: Equivalence Theorem

    surface currents on S to make the the distribution E,H compatible with E2, H2.The continuity conditions of the tangent component on S imply the presence on thesurface of a distribution of electric and magnetic surface currents given by:

    Js = n (H2 H)SM s = n (E2 E)S (2.13)

    where the values of the electric and magnetic eld are those on S. Such currentsmay be interpreted as those which, together with the sources internal to V1, radiatea eld E,H in V1 and E2, H2 in V2.

    The eld distribution E,H can be chosen arbitrarily. If we choose, for example,E = 0, H = 0 inside S, the currents on S will be:

    Js = n (H2)SM s = n (E2)S (2.14)

    This theorem may be formulated in other two dierent forms:

    1. Since the eld in V1 is zero, we can put in V1, and immediately below S, aperfect electrically conducting (PEC) surface.

    The electric currents will ten be short-circuited and will not radiate; con-versely, the magnetic current components will be doubled. By consequence,the external led can be computed by using only the magnetic current, as(omitting the sux S for the eld):

    M s = 2n E2 (2.15)

    2. Dually, inserting a perfect magnetic conductor (PMC), we can obtain the eldfrom the electric currents:

    Js = 2nH2 (2.16)

  • M. Orece: Antennas 2013, Aperture antennas 19

    The equivalence theorem has, for the radiation, a similar signicance as theThevenin theorem in Electric Circuits, because it allows, given an arbitrarily com-plex structure (network), to determine electric and magnetic currents (voltage andinternal resistance for Thevenin) equivalent to it, only relatively to the external prob-lem.

    2.2.1 Equivalence Theorem and planar apertures

    Despite of the advantage of replacing a volume integral with one surface integral (ortwo, if the formulation with both magnetic and electric currents is chosen) a fewproblems are connected with the use of the Equivalence Theorem. For example, thechoice of the arbitrary surface enclosing the sources must be done in appropriatemanner, in order to simplify the computation of the elds and of the equivalentsurface currents on it, and of the integral.

    To this purpose, one of the simplest surfaces to be chosen is that consisting of aplane and of the half sphere at innity enclosing all sources.

    Thanks to the radiation conditions at the innity, the eld on the half sphereat innity vanishes, so that the sources on one side of the plane are equivalent, forthe purpose of computing the eld on the other side, to the surface currents on theplane, computed by means of eqs.(2.14), or equivalent forms.

    Lets call Ea, Ha the transverse eld components on the plane. If we dene thedouble Fourier Transforms of such vectors 2 as:

    ft(kx, ky) = fxx + fyy =

    SEa(x, y)e

    j(kxx+kyy)dx dy

    gt(kx, ky) = gxx + gyy =

    SHa(x, y)e

    j(kxx+kyy)dx dy

    kx = k sin cosky = k sin sin

    (2.17)

    where the scalar functions fx,y, gx,y are the Fourier Transforms of the transverseeld components of the aperture Ea,x,y, Ha,x,y.

    Carrying out the computations, from (2.12) we get for the three formulations,respectively:

    JS, MS:

    E =jkejkR

    4R[fx cos + fy sin + Z0 cos (gy cos gx sin)]

    E =jkejkR

    4R[cos (fy cos fx sin) Z0(gy sin + gx cos)]

    (2.18)

    2See R.E. Collin, F.J. Zucker: Antenna Theory, Part I; New York: Mc. Graw-Hill, 1969, pp.69-74.

  • M. Orece: Antennas 2013, Aperture antennas 20

    2JS:E =

    jkZ0 cos ejkR2R

    (gy cos gx sin)

    E = jkZ0ejkR2R (gy sin + gx cos)(2.19)

    2MS:E =

    jkejkR2R

    (fx cos + fy sin)

    E =jk cos ejkR

    2R(fy cos fx sin)

    (2.20)

    The radiated eld from a volume of sources is therefore expressed through thedouble Fourier Transforms of the tangential eld components on a plane which hasall the sources on the same side. It his however apparent that this eld distributionis non-zero only in a limited region of the plane, in order to be, sooner or later, infar eld conditions and therefore use eq.(2.12).

    Lets see now the relationship between the eld components on the aperture,directed along x, y, and those of direct and cross polarization, along p, q, of theradiated eld. The polarization unit vectors, according to the 3rd Ludwigs de-nition, are given, for p y on the normal to the aperture, by eq.(1.36), and thecomponents, in matrix form, are:

    Ep(, )

    Eq(, )

    =

    sin cos

    cos sin

    E

    E

    (2.21)

    Replacing eq.(2.20) in eq.(2.21) we obtain:

    Ep(, )

    Eq(, )

    = K cos2 /2

    1 t

    2 cos 2 t2 sin 2

    t2 sin 2 1 + t2 cos 2

    fy(, )

    fx(, )

    (2.22)

    where K = jejkR/R, and t = tan /2.Eq.(2.22) is an exact expression for the far eld radiated from an aperture in its

    front region, and it allows to note that, for t small, the direct and cross polarizationcomponents are directly proportional to the Fourier transforms of the scalar compo-nents along y and x of the eld. Since the cross polarization is of particular interestmainly in correspondence of the main beam, this means that for a large aperture(whose main beam is narrow) there will be a low cross polarization if the eld onthe aperture is linearly polarized (for example, a rectangular horn). On the con-trary, if the aperture is small and the main beam is wide (as it is, for example, for areector antenna feed) the relationship providing the polarization is more complex,and consequently to minimize the cross polarization the eld on the aperture shallnot be linearly polarized.

  • M. Orece: Antennas 2013, Aperture antennas 21

    2.3 Radiation zones from an aperture

    From the approximation of eq.(2.6) to eq.(2.8) it appears that, for distances largerthan a few wavelengths from the source volume (about 10 for a good accuracy, oreven less if a lower accuracy is accepted) the radiation integral is strongly simplied:we may therefore dene,for lower distances, a near eld zone, where the expressionof the Greens dyadic must be complete, given by eq.(2.6), and another for the eldfar from the sources.

    P

    ^r r

    RR^x

    y

    z

    P

    Figure 2.2: Coordinates for the radiation from an aperture.

    To analyze the possible approximations in this latter region, lets consider aplanar aperture on the xy plane as in g.(2.2). By applying the equivalence theorem,eq.(2.5) reduces to the surface integral

    E(P ) = jSIt,r Js,eq(P )

    ejkr

    4rdS (2.23)

    where S is the surface of the whole plane, and Js,eq(P) is the equivalent electric

    surface current, which is obtained from eq.(2.2) using the electric and/or magneticcurrents deriving from the Equivalence Theorem. We will assume that the equivalentcurrent is non-zero within a nite area A of the plane.

    Let x, y be the coordinates of a point on the aperture, and x, y, z (or the corre-sponding r, , ) those of the observation point P .

    Eq.(2.23) becomes:

    E(P ) =Z02

    A( + ) (J(x, y) + J(x, y))e

    jkr

    rdS (2.24)

    The zones in which the eld can be subdivided are mathematically determined,although without precisely dened sharp boundaries, by the approximations thatcan be made in the integral (2.24) and by the structure of the eld.

    In the intermediate eld zone, or Fresnel zone, we may consider negligible, inaddition to 1/r with respect to k, the change of direction of the unit vectors of thetransverse dyadic on the aperture. This corresponds to assume almost parallel, ing.(2.2), the unit vectors r and R.

  • M. Orece: Antennas 2013, Aperture antennas 22

    Moreover we neglect in eq.(2.24) the amplitude variation of 1/r, which is takenequal to the constant value (for each observation point) 1/R.

    Conversely, we must deal more carefully with the variation of r on the aperture,as far as the phase term ejkr is concerned. The distance of the observation pointP from the generic source point P is, in general:

    r =[(x x)2 + (y y)2 + z2

    ]1/2(2.25)

    In the paraxial region, i.e. for small and z |x x|, |y y|, eq.(2.25) maybe developed as:

    r = z

    [1 +

    (x x)2z2

    +(y y)2

    z2

    ]1/2= = z + (x x

    )2

    2z+

    (y y)22z

    + ... (2.26)

    The approximation Fresnel zone consists in neglecting the terms of order higherthan the second in the development (2.26).

    An alternative for of expansion of eq.(2.25) is obtained by using spherical coor-dinates fore the observation point P (see g.2.2):

    x = R sin cos = Ruy = R sin sin = Rv

    z = R cos (2.27)

    By introducing eqs.(2.27) in (2.25) we obtain, after having developed in Taylorseries up to the rst three terms, and neglecting the powers of 1/R of order higherthan 1 (which is equivalent to assume ux/R, vy/R 1):

    r = R (ux + vy) + x2 + y2 (ux + vy)2

    2R= R + rb (2.28)

    Eq.(2.24) becomes therefore:

    E(P ) = jZ02

    ejkR

    R

    AJt(x

    , y)ejkrbdxdy (2.29)

    In the Fraunhofer region further approximations are made on the phase termejkr. We neglect in (2.28) the terms of order higher than 1 in the coordinates y

    and x on the aperture, obtaining:

    E(P ) = j2

    ejkR

    R

    AJt(x

    , y)ejk sin (x cos+y sin)dxdy (2.30)

    Consequently, in the Fraunhofer zone the eld distribution is similar (i.e. scaledby a factor) on all spheres centered in the origin, where the aperture is placed: thismay therefore be considered as a point source.

    If we write:kx = k sin cosky = k sin sin

    (2.31)

  • M. Orece: Antennas 2013, Aperture antennas 23

    eq.(2.30) becomes:

    E(P ) =j

    2RejkRF (kx, ky) (2.32)

    withF (kx, ky) =

    AJt(x

    , y)ej(kxx+kyy)dxdy (2.33)

    Lets consider the plane z = 0. The sources (related to the eld) on the aperturemay be seen as a function Jt(x, y) dened on all the plane z = 0:

    Jt(x, y) = Jt(x, y) inside A

    Jt(x, y) = 0 outside A(2.34)

    F (kx, ky) is clearly the double Fourier transform of Jt(x, y):

    F (kx, ky) = F [Jt(x, y)] =

    Jt(x, y)ej(kxx+kyy)dx dy (2.35)

    and, vice versa:

    Jt(x, y) = F1[F (kx, ky)] = 142

    F (kx, ky)ej(kxx+kyy)dkxdky (2.36)

    As it will be seen later,in the part of the Fresnel zone nearer to the sources, theeld is essentially given by the geometrical optics contribution, with additional eldoscillations due to diraction eects. In general, it is assumed as maximum limit ofthis region the distance D2/2 (i.e. 1/4 of the Fraunhofer distance) called Rayleighdistance; the region within this limit is called Rayleigh zone. Here, the phase surfacesradiated by a plane aperture illuminated in phase remain approximately planar andparallel each other. A typical example is the paraboloid, whose phase surfaces on theaperture are parallel planes, so that in the near region we can consider a propagationby parallel rays.

    The Raleigh zone can be considered as the near eld region for the aperture,and therefore with dimension depending on the aperture itself, while the near eldregion for the sources depends only on the wavelength.

    2.4 Gain ed eciency of an aperture

    Eq.(1.21) requires the integration of the angular power density within the solid angle,which in many case, in particular when dealing with apertures, is usually expressedby complicated functions, often not analytically integrable.

    For an aperture it exists a dierent formulation, usually simpler, to express thepower at the denominator of eq.(1.14) or (1.16) as the integral of the power densityon the aperture. Assuming that the eld on the aperture is TEM (which it is truefor apertures larger than one wavelength) the power density is S = |E|2/Z0, and the

  • M. Orece: Antennas 2013, Aperture antennas 24

    directivity on the z axis (which is usually the direction of the maximum) is givenby:

    G(0) D(0) = 42

    AE(P )dA

    2A|E(P )|2dA

    (2.37)

    where P is a generic point on the aperture, and A is the aperture area.If E(P ) is constant on the aperture, the maximum gain is clearly

    G(0) =4

    2A (2.38)

    and the eective area is coincident with the geometric area. We may therefore denethe aperture eciency as the ratio between the directivity of the aperture and thatof the same aperture uniformly illuminated: the eciency is given by

    =1

    A

    AE(P )dA

    2A|E(P )|2dA

    (2.39)

    2.5 Rectangular aperture

    Lets consider a rectangular aperture on the (x, y) plane, with dimensions a and b(see g. 2.3) and area A. Eq.(2.33) becomes:

    F (, ) = a/2a/2

    b/2b/2

    Jt(x, y)ejk sin (x

    cos+y sin)dxdy (2.40)

    We will now consider a few particular cases of practical interest.

    2.5.1 Uniform amplitude and phase

    For a uniformly illuminated aperture it is U(x, y) = 1 (normalized value); theintegral (2.40) can be easily computed and we nd:

    F (, ) = A

    sin

    (asin cos

    )asin cos

    sin

    (bsin sin

    )bsin sin

    (2.41)

    The diagrams in the principal planes (x, z) and (y, z) are particularly interesting.In the former ( = 0) eq.(2.41) becomes:

    F () = Asin

    (asin

    )asin

    (2.42)

    In the latter ( = /2) the eld is still given by eq.(2.42) with b instead of a: there-fore in both principal planes the radiation patterns have, except for the horizontal

  • M. Orece: Antennas 2013, Aperture antennas 25

    -15 0 150

    0.5

    1

    xy

    z^

    ^b

    a

    Figure 2.3: Normalized diagram of the eld intensity radiated from a rectangularaperture uniformly illuminated (linear scale).

    scale, the same behavior. The normalized radiation pattern is shown in g.(2.3) asa function of the normalized variable

    w =

    (k02

    ab

    )sin (2.43)

    The scale in the ordinates is linear in eld; in this case the minima are nulls, andare located in the points w0 = n, with n = 1, 2, .... In the u, v plane, the nullsare on a rectangular grid; moreover the highest secondary lobes are in the principalplanes, while in the other planes = const., e.g. the diagonal ones, their level is muchlower. A 3-D representation of the radiation pattern for a rectangular aperture, indB scale, is shown in g.(2.4), for the case of uniform illumination: we may note thesymmetry with respect to the principal planes, typical of the rectangular apertureswith separable variables.

    2.5.2 Separable eld distributions

    A common type of aperture illumination is that where the eld distribution is sep-arable in the product of two functions:

    Jt(x, y) = u1(x)u2(y) (2.44)

    In this case the integral (2.40) is easy to be solved. Eq.(2.40) becomes:

    F (, ) = a/2a/2

    u1(x)ejkx

    sin cosdx b/2b/2

    u2(y)ejky

    sin sindy (2.45)

    The patterns in the principal planes (x, z) and (y, z) are therefore respectively:

  • M. Orece: Antennas 2013, Aperture antennas 26

    21

    01

    2

    2

    1

    0

    1

    240

    30

    20

    10

    0

    10

    Figure 2.4: Three-dimensional radiation pattern of a square aperture with uniformillumination.

    F (, 0) =

    [ b/2b/2

    u2(y)dy

    ] a/2a/2

    u1(x)ejkx

    sin dx

    F (, /2) =

    [ a/2a/2

    u1(x)dx

    ] b/2b/2

    u2(y)ejky

    sin dy

    (2.46)

    The eects of the non uniform illumination and the errors in the phase patternson the principal planes can therefore be studied as one-dimensional problems, pro-vided that the aperture eld be separable both in amplitude and phase in the formgiven by eq. (2.44).

    With these approximations, the expressions of the radiated eld can be reducedto Fourier transforms of a single variable functions, many of which are found inanalytical form on the Fourier transforms tables. A few examples of such functions,with the analysis of the main radiation characteristics, are shown in Table 2.1.

    The behavior of the radiated eld from an aperture has, in general, some commoncharacteristics, which are connected to the tapering 3 of the aperture illuminationfunctions.

    In fact, from Table 2.1we may observe that, for constant size of the aperture,with the increase of the tapering the eciency decreases, the main beam widens

    3Tapering is the ratio (usually in dB but sometimes in number) between the eld intensity atthe center of the aperture and that at its edges. Usually this ratio is greater than 1, and thereforein dB it is positive.

  • M. Orece: Antennas 2013, Aperture antennas 27

    Type of HPBW Position First side Aperturedistribution of rst null lobe level eciency

    (rad) (rad) (dB)Uniform 0.88/L /L -13.2 1Parabolic

    1 (1)x2 = 1 0.88/L /L -13.2 1 = 0.8 0.92/L 1.06/L -15.8 0.994 = 0.5 0.97/L 1.14/L -17.1 0.97 = 0 1.15/L 1.43/L -20.6 0.833

    cosn x2

    |x| < 1n = 0 0.88/L /L -13.2 1n = 1 1.2/L 1.5/L -23 0.81n = 2 1.45/L 2/L -32 0.667n = 3 1.66/L 2.5/L -40 0.575n = 4 1.93/L 3/L -48 0.515

    Triangular1 |x| |x| < 1 1.28/L 2/L -26.8 0.75

    Table 2.1: Characteristics of the radiation patterns for various types of aperturedistributions.

    (and consequently also the angle of the rst null), and the sidelobes decrease. Wemay also note that also the derivatives of the eld at the edges of the aperture haveinuence on the radiation pattern, in the sense that a low value of such derivativeincreases the eect of the tapering.

    The choice of one aperture distribution or another thus will depend on the designspecications (for example, whether a high gain or low sidelobes are preferred). Inpractice, however, the actual aperture distributions can be represented only roughlyby those reported in the Table, and it will be therefore often necessary perform nu-merical integrations or Fourier transforms, or to represent the aperture distributionwith a sum of functions which may be transformed analytically.

    By introducing a suitable normalization, the previous equations, for exampleeq.(2.45), can be rewritten in a more universal form.

    Considering a separable distribution where the aperture eld is expressed by agiven function along y; for example, if it is uniform it is u2(y

    ) = 1. The radiationpattern in the plane x = 0 ( = /2), except for a multiplicative constant, is, aspreviously seen, of the type:

    F () =sin

    (bsin

    )bsin

    (2.47)

  • M. Orece: Antennas 2013, Aperture antennas 28

    while in the plane y = 0 ( = 0):

    F () = a/2a/2

    u1(x)ejkx

    sin dx (2.48)

    If we now introduce new variables:

    x =2x

    au =

    a

    sin (2.49)

    then u1(x) f(x) and F () F (u); so that eq.(2.48) becomes:

    F (u) =a

    2

    11

    f(x)ejuxdx (2.50)

    It is clear that, if we have the same type of distribution on two apertures withdierent size, these will produce the same radiation pattern as a function of thevariable u dened in the second of eqs.(2.49).

    2.5.3 Eects of the phase error

    We may have a phase error on a radiating aperture for various reasons: for example,in the case of a parabolic reector antenna, when the primary feed is displaced withrespect to the focus; or because of the presence of a phase error in the primary feed;or, for a horn antenna, it may be due to the sphericity of the eld wavefront incidenton the aperture.

    Moreover, we will assume that the eld on the aperture is given by a separablevariables function, so that we will consider only one variable at a time. Assuming theaperture distribution along x of the type f(x)ej(x), where f(x) is the amplitude(real) and (x) is the phase function, the expression for the eld radiated from theaperture becomes proportional to:

    F (u) =a

    2

    11

    f(x)ej[ux(x)]dx (2.51)

    We will limit the discussion to the following particular forms of (x):

    linear error: (x) = x quadratic error: (x) = x2

    (a) Linear error - By inserting the expression (x) = x in eq.(2.51) we obtain:

    F (u) =a

    2

    11

    f(x)ej(u)xdx (2.52)

    It is clear that eq.(2.52) has the same form as eq.(2.50) with u replaced by (u).Consequently the distribution of the radiated eld is equal to that obtained with

  • M. Orece: Antennas 2013, Aperture antennas 29

    constant phase distribution, but shifted along u by a quantity . The maximumwill be for u = , i.e. in the direction:

    0 = sin1

    a(2.53)

    The radiation pattern is therefore the same of an aperture with constant phasedistribution, but tilted by an angle 0 in opposite direction to the increasing senseof the phase. Note that, since u is proportional to sin , in the variable there isnot only a rotation but also a deformation of the radiation pattern in abscissas.

    (b) Quadratic error - In this case the radiated eld is proportional to:

    F (u) =a

    2

    11

    f(x)ej(uxx2)dx (2.54)

    where, with this normalization, is the maximum phase error (obviously at theedges of the aperture).

    The computation of this integral is in general very dicult and usually it is donenumerically or with approximate methods.

    The eect of a quadratic phase error is illustrated in g.(2.5) for two types ofillumination: uniform (f(x) = 1), and tapered (f(x) = cos2 x

    2). In both cases

    = /2. The maximum is always for = 0 and, since the phase error is symmetricwith respect to the center of the aperture, also the radiation pattern will be sym-metric with respect to the direction = 0. However we nd that, when becomessuciently large, in some cases the main beam bifurcates, and we have the largestmaxima on both sides of the direction = 0.

    -10 -5 0 5 1 00

    0 .1

    0 .2

    0 .3

    0 .4

    0 .5

    0 .6

    0 .7

    0 .8

    0 .9

    1

    -10 -5 0 5 1 00

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    Figure 2.5: Eect of the quadratic phase error, for (a) uniform illumination; (b)illumination of the type cos(x/2), for dierent values of the maximum phase error(from the bottom: 0, /4, /2, ).

    The general eect of the quadratic phase error is of reducing the maximum gain,widening the main beam, increasing the sidelobes level and lling the nulls. For a

  • M. Orece: Antennas 2013, Aperture antennas 30

    tapered illumination, since the sidelobes of the starting pattern are low, the llingof the nulls makes often impossible to distinguish the rst sidelobe from the mainbeam (see g.2.5, b). The gain loss due to this type of phase error is shown, inpercentage, in g.(2.6) for a uniformly illuminated aperture.

    0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.540

    50

    60

    70

    80

    90

    100

    beta/2pi

    %G

    Figure 2.6: Gain loss due to the quadratic phase error, as a function of the maximumerror (in wavelengths): uniform illumination.

    2.6 Circular aperture

    The considerations and the results for a rectangular aperture, concerning to therelationship existing between the eld on the aperture and the radiation pattern,can also be applied to a circular aperture. However, in this case it is convenient touse cylindrical coordinates (see g.2.7), which are related to y and x by:

    x = cos y = sin (2.55)

    Indicating with U(, ) the eld distribution on the aperture, the expression forthe radiated eld (2.33) becomes:

    F (, ) = 20

    a0

    U(, )ejk sin cos()dd (2.56)

    where a = D/2 is the aperture radius. By introducing the normalized variables

    r =

    au =

    2a

    sin =

    D

    sin (2.57)

  • M. Orece: Antennas 2013, Aperture antennas 31

    2a

    y

    x

    z

    Figure 2.7: Circular aperture and coordinate systems.

    U(, ) and F (, ) are transformed into the functions f(r, ) and F (u, ), respec-tively; moreover we will assume normalized to 1 the maximum of f(r, ). Theradiated eld then becomes:

    F (u, ) = a2 20

    10

    f(r, )ejur cos()rdrd (2.58)

    Note that eq.(2.58) is simply the expression of the Fourier transform in cylindricalcoordinates, and it is called Fourier-Bessel transform of f(r, ).

    2.6.1 Uniform amplitude and phase

    Assuming f(r, ) = 1 in eq.(2.58) and carrying out the integration , we obtain4:

    F (u) = 2a2 10

    J0(ur)rdr (2.59)

    where J0(ur) is the Bessel function of order 0. Then, by integrating with respect tor we obtain:

    F (u) = 2a2J1(u)

    u(2.60)

    Note that F (0) = a2, i.e. the aperture area.The behavior of F (u) is easily predictable if we consider that J1(u) has the

    oscillating behavior shown in g.2.8.

    4Integrate from 0 to 2 in the relationships (see, e.g., M.Abramowitz, I.Stegun: Handbook ofmathematical functions, New York: Dover Publ., 1965, pp.361)

    cos(z sin) = J0(z) + 2

    k=1

    cos(2k)J2k(z)

    sin(z sin) = 2

    k=1

    sin[(2k + 1)]J2k+1(z)

    recalling that only the integrals with k = 0 are non-zero.

  • M. Orece: Antennas 2013, Aperture antennas 32

    -0.6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    0 2 4 6 8 10 12 14 16

    J

    x

    Figure 2.8: Behavior of the Bessel functions of order 0,1,2,3,4.

    The normalized eld pattern F (u) is shown in g.(2.9) as a function of u. TheHPBW (in radians) is given by:

    = 2 sin1(0.51/D) = 1.02/D (2.61)

    and the rst sidelobe is 17.5 dB below the main beam.Such values are comparable with the corresponding values ( = 0.88/a for the

    HPBW and -13.5 dB for the rst sidelobe) derived for the rectangular aperture. Thedierent levels of sidelobes should not however suggest that the circular aperturehas less power in the sidelobes than the rectangular aperture: in fact, in this lattercase, the rst SLL is constant in all the azimuth planes, while in the rectangularaperture there are high sidelobes only in the principal planes (see g. 2.4).

    2.6.2 Non uniform illumination

    The eects of the tapering are similar to those seen for the rectangular aperture:gain reduction, main beam widening and increase of the sidelobes. These eects canbe illustrated considering the series of aperture eld distributions

    f(r) = (1 r2)p (2.62)

    with p positive integer. The aperture distribution is therefore symmetric with re-spect to . The normalized radiated eld is given by:

    Fp(u) = 2a2 10(1 r2)pJ0(ur)rdr (2.63)

    from where

    Fp(u) = a22

    p+1p!Jp+1(u)

    up+1(2.64)

  • M. Orece: Antennas 2013, Aperture antennas 33

    0 5 10 1560

    50

    40

    30

    20

    10

    0

    u

    G d

    B

    Figure 2.9: Radiation pattern of an uniformly illuminated circular aperture.

    The main characteristics of the radiated eld for aperture distributions of thistype with p = 0, 1, 2, 3, 4, are summarized in table (2.2), where is the apertureeciency, is the HPBW and 0 is the angular position of the rst null, and shownin g.(2.9) and (2.10).

    In general, an arbitrary aperture distribution (provided that it is regular andindependent on ) can be approximated with the sum

    f(r) = a0 + a1(1 r2) + a2(1 r2)2 + a3(1 r2)3 + ... =

    an(1 r2)n (2.65)

    where each term give rise to an integral which may be solved analytically (eq. 2.64).Moreover, by using the change if variable x = 1r2 the coecients ai of the RHS ofeq.(2.65) are given directly by those of a polynomial expansion of f(x): the radiationfrom the aperture will therefore be expressed by:

    F (u) = a2N0

    ap2pp!Jp+1(u)

    up+1(2.66)

    Another interesting type of distribution is the following:

    f(r, ) = 1 r2 cos2 (2.67)

  • M. Orece: Antennas 2013, Aperture antennas 34

    p 0 SLL0 1.00 1.02/D 1.22/D 17.61 0.75 1.27/D 1.63/D 24.62 0.56 1.47/D 2.03/D 30.63 0.44 1.65/D 2.42/D 364 0.36 1.81/D 2.79/D 40.5

    Table 2.2: Characteristics of the radiation from circular apertures withillumination of the type (1 r2)p.

    0 5 10 1560

    50

    40

    30

    20

    10

    0

    u

    G d

    B

    Figure 2.10: Radiation patterns from circular apertures with illumination of thetype (1 r2)p.

  • M. Orece: Antennas 2013, Aperture antennas 35

    which is uniform in the plane = /2 and has some tapering (cosine square) in = 0. By replacing eq.(2.67) in eq.(2.58) we get:

    F (u, ) = 2a2[J1(u)

    u 1

    2

    20

    10

    r2 cos2 ejur cos()r dr d

    ](2.68)

    The integral in eq.(2.68) is solved by using the expansion:

    ejur cos() =

    n=

    Jn(ur)ejn() (2.69)

    and using some integral expression of the Bessel functions that allow to reduce it tocombinations of eqs.(2.64).

    The radiation patterns in the planes = 0 and = /2 are shown g. 2.11.Note that the HPBW is larger in the = /2 plane than in = 0 , clearly becausethe aperture distribution is tapered in that plane, while it is uniform in the other.However, the sidelobe level is not the same that it would be with the same taperingin both planes: in other words, in a circular aperture the tapering in a principalplane inuences the sidelobe level also in the other plane, unlike the rectangularaperture.

    0 5 10 1540

    35

    30

    25

    20

    15

    10

    5

    0

    u

    G(u)

    Figure 2.11: Radiation pattern of a circular aperture with distribution of the typef(r, ) = 1 r2 cos2 , in the planes = 0 (-) and = /2 (- - -).

    Exercise: Show that the aperture eciency for a distribution with tapering t of the typet + (1 t)(1 r2) (called parabola-on-a-pedestal), is given by:

  • M. Orece: Antennas 2013, Aperture antennas 36

    =34

    (1 + t)2

    1 + t + t2

    and that for a distribution of the type t+(1t)(1r2)2 (called square-parabola-on-a-pedestal),is given by:

    =53

    (1 + 2t)2

    3 + 4t + 8t2

    2.7 Phase center

    The phase center of an aperture and, in general, of an antenna, is the center ofcurvature of the radiated phase surface and therefore, in the assumption of far eldconditions, is the center of the spherical wave radiated by the antenna, i.e. ofthe constant phase (spherical) surfaces. From this denition, this point should beindependent on the angle and the distance of observation.

    But in practice this is not true. In fact, computing the eld radiated by anaperture with the various formulas previously shown (e.g. with the Fourier transformof the aperture eld), we nd that on a spherical observation surface, any point betaken as center, the phase is not constant, except that for some particular case, forexample when the eld distribution on the aperture plane is symmetric and withconstant phase. And this is valid only for the main beam: in fact, the Fouriertransform of real symmetric function is also real and symmetric, but usually with achange of sign for any successive lobe: consequently, on a sphere at large distancea previously dened, the phase will switch between 0 and 180 from one lobe toanother, passing through an amplitude null.

    In general, the phase of the radiated eld on a sphere al large distance will alwaysvary with the observation angle, and it doesnt exist a point which may be takenas the center of a sphere where the phase of the radiated eld is exactly constantfor all angles. An example of the behavior of the phase is shown in g.(2.12).In conclusion, the constant phase surface, even at large distance, is not exactly asphere, and therefore the phase center varies its position with the observation angle.However, it is possible to nd a xed point for which the phase error is small in areduced angle (for example, that corresponding to the main beam of the antenna)which can be considered, with these limitations, the phase center.

    The determination of the position of the phase center is particularly importantfor the sources used as reector antenna feeds: as an example, in a paraboloid thewave radiated from the feed must be spherical with center in the paraboloid focus,and any error with respect to this requirement produces a degradation of the antennapattern. In this case, we must aim to minimize the phase error on the paraboloidaperture by nding a spherical surface where the phase variation of the eld radiatedfrom the feed is minimum5, and making coincide its center with the focus of theparaboloid.

    5The minimization criterion can be variable, as it will be seen in the following.

  • M. Orece: Antennas 2013, Aperture antennas 37

    -10 0 100

    0.5

    1

    -10 0 10-2.5

    -1

    0.5

    Figure 2.12: Example of a radiation pattern (in amplitude, left, and phase, right)from an aperture with phase error.

    We must also note that the phase pattern of the eld radiated from an aperture isusually dierent in the principal planes, so that a certain point may be the optimumphase center for one plane but not for the other: this is the case of apertures withunequal eld distribution in the principal planes (for example, the rectangular andcircular horns, with fundamental mode only).

    0

    d

    r

    0

    -a / 2

    a / 2

    Figure 2.13: Geometry for the determination of the phase center.

    The approximate position of the phase center can be found observing that, bymoving the origin of the reference system from O to O with a displacement d(positive or negative) along the z axis (see g.2.13), the vector r describing thespace of the sources becomes

    = xx + yy dz (2.70)so that the radiated eld E on the sphere with center in O is related to the eld E0on the sphere with center in O from the relationship

    E = E0ejkd cos (2.71)

    Thus the displacement of the origin produces only a phase variation.If we consider, in order to nd the phase center, the relative phase shift ()

    with respect to the direction = 0 (the z axis), i.e. xing (0) = 0, we will obtain

    () = 0() kd(1 cos ) (2.72)

  • M. Orece: Antennas 2013, Aperture antennas 38

    where the suxes have the same meaning as in the previous equation.With a suitable choice of d we can therefore obtain that for a given angle of

    observation 0 it is (0) = (0), i.e. that the constant phase surface surface istangent to the sphere and moreover it intersects it in 0. If then 0 0, we willobtain the osculating circle to the curve at constant phase in its intersection withthe z axis: on this circle, the phase variation will be innitesimal of higher order,and its center will be coincident with the phase center on the z axis, and in thisregion it will be () = (0).

    By expanding, in eq. (2.72), the cosine in Taylor series and expressing bothphase and observation angles in degrees, we obtain that the distance of the phasecenter from the reference point (respect to which the phase was computed) isgiven by:

    d/ = 18.24(0(0) 0(0))20

    (2.73)

    The sign indicates that, if the computed (or measured) phase variation vs. , on asphere with a given center, is positive, the phase center is on the opposite directionwith respect to the radiation direction.

    The previous considerations have been carried out for the z axis. If we compute(or measure) the phase with respect to the point determined in this way, that willnot be exactly constant: in fact, it has been attened in the neighborhood of theaxis but not for wide angles. For using an antenna as a reector feed, covering widerangles, it may be more convenient to choose as phase center (i.e. where the focus ofthe paraboloid must be) a point minimizing the phase error according some dierent(and in some way arbitrary) criteria; some examples of these are shown in g.(2.14):as an example, the same phase on the axis and on the maximum illumination angle(A), or on the axis and for a given angle or a given tapering level, or minimizationof the maximum absolute value of the error (B), weighting the dierent illuminationin the center and at the edges, etc.

    0

    A

    B

    Figure 2.14: Behavior of the phase of the radiated eld from an aperture, for dierentpositions of the phase reference point.

  • M. Orece: Antennas 2013, Aperture antennas 39

    2.8 Apertures with arbitrary eld distribution

    The eld radiated by an aperture can be expressed analytically only in particularcases, when the shape of the aperture and the eld distribution function are verysimple. More generally, it is necessary to resort to numerical integrations which, inmany cases, can be simple two-dimensional FFTs.

    Alternatively, if the aperture shape is relatively simple, the eld can be ex-pressed as a sum of analytically integrable functions: an example (dened in anunidimensional domain) is the expansion of eq.(2.65). Techniques of this type wereparticularly useful when fast computers were not available, but they are anywayvalid when optimizations requiring repeated integrations are to be carried out, toreduce the total computational load.

    One of the rst examples of expansion in two-dimensional domain is that of anelliptical aperture6, which can be suciently easily analyzed by using an illuminationfunction expressed in Fourier series, and then using the Fourier-Bessel transforms ofsinusoidal functions.

    The radiation patterns computed through this assumption are in good agreementwith those determined experimentally, especially for what concerns the gain, theHPBW, the rst sidelobe and the position of the other sidelobes. Also in this casethe sidelobes level in one of the principal planes depends, although less signicantly,on the illumination in the other principal plane. Fig.(2.15) shows the sidelobe level

    2 4 6 8 10 12 14 16 1838

    36

    34

    32

    30

    28

    26

    24

    22

    20

    18

    Tapering piano H (dB)

    Liv.

    lobo

    sec

    onda

    rio (d

    B)

    Figure 2.15: Sidelobe level in the H (-) and E (- - -) planes, as a functions of theedge illumination in the H plane (from Adams and Kelleher).

    computed for apertures obtained with reector antennas, by using measured feed

    6See R.J. Adams e K.S. Kelleher: Pattern calculation for antennas of elliptical aperture,Proc. IRE, vol.38, Sept. 1950. The paper is in fact a short summary, where it is also reported thediagram of g.(2.15), which was later reported on Jasiks Antenna engineering handbook and oftenused for the preliminary design of reector antennas.

  • M. Orece: Antennas 2013, Aperture antennas 40

    radiation patterns, keeping constant (-13 dB) the taper in the E plane and varyingit in the H plane: we may observe that in the plane where the edge taper varies,the sidelobe level decreases for increasing taper, while in the other there is a slightincrease, still however remaining almost constant within a couple of decibels.

  • M. Orece: Antennas 2013, Aperture antennas 41

    2.9 Horn antennas

    2.9.1 Sectoral and pyramidal horns

    The sectoral horns are obtained from a rectangular waveguide having dimensionsallowing the propagation of the fundamental mode TE10 alone, by widening to adihedron two of the four walls, as shown in g.(2.16): the cross section widensgradually in one direction, while it remains constant in the perpendicular one. It istherefore possible to have sectoral horns with directive pattern only in the H or Eplane.

    lh le

    a

    E

    b

    b

    E

    a

    Figure 2.16: Sectoral horns in the H (left) or E (right) planes.

    The higher order modes arising in correspondence of the discontinuity are belowcut and therefore are exponentially attenuated, and in the horn cross sections thatwould allow their propagation their amplitude is almost negligible: therefore thereis always only the fundamental mode TE10. On the aperture the amplitude ofthe transverse electric and magnetic eld will vary following the fundamental modecharacteristics, with half-sinusoid in one of the principal directions and constant inthe other.

    Between the two dihedral walls the wave is approximately cylindrical, with axisin the intersection of the two planes. Consequently the eld phase behavior isapproximately quadratic, with a maximum deviation (assuming a are angle ofthe horn not too large) equal (in radians) to

    =a2

    4(2.74)

    where a is the aperture width and is the length of the oblique side of the horn,still measured from the vertex of the dihedron.

    It can be easily derived that the exact expression of the phase error as a functionof the dimension and the aperture angle is

    =a

    tan

    2(2.75)

    The radiation patterns are shown in g.(2.17) for the two principal planes. Thecorresponding phase patterns are also shown in g.(2.18): both are plotted vs. the

  • M. Orece: Antennas 2013, Aperture antennas 42

    dimensionless quantities (b/) sin and (a/) sin , having as parameter the maxi-mum phase error in wavelength, s or t in both planes, given respectively by:

    s =b2

    8Et =

    a2

    8H(2.76)

    It can be seen that, when the maximum deviation is high, the main beam in theE plane (and also, although less noticeably, in the H plane) splits in two. In general,the sidelobes are higher in the E plane than in the H plane.

    The patterns of g.(2.17) are suciently precise for horns of a few wavelengths;for smaller apertures, up to about 2, the values obtained shall be multiplied by anobliquity factor which depends on the reection coecient and on the wavelength onthe horn aperture, With sucient approximation this factor is given7 by cos(/2).

    The gain of a sectoral horn can be computed from the curves of g.(2.19), pro-vided that the smallest dimension of the aperture is at least one wavelength. Mea-surements carried out on several horns show that the dierences between theoreticaland experimental values are within a few tenths of dB.

    The pyramidal horn - of which a few samples, operating in various frequencybands, are shown in g.(2.20) - is obtained by widening in pyramidal shape all the4 planes which bound the rectangular waveguide. In general, the two couples ofplanes do not intersect each other at the same distance from the aperture, so thatin fact it is not a perfect pyramid.

    The pyramidal horn, thanks to its good reproducibility and easy manufacturing,and to the excellent agreement between theoretical and experimental results, isfrequently used as a Standard Gain Horn (SGH) in antenna gain measurements asa reference standard. Its gain can be computed precisely through the relationship(in decibel):

    G = 10(log 0.81 4ab2

    ) Le Lh (2.77)The rst term of eq.(2.77) is the gain of a rectangular aperture illuminated witha TE10 mode and constant phase, and it can be computed analytically througheq.(2.37). The other two terms represent the gain loss due to the phase error,respectively in the E and H plane, and can be derived from the curves of g.(2.21).For not too high phase errors (e.g. 0.8 in the E plane and 0.6 in the H plane),this gain reduction can be approximated by (dashed curves g.(2.21))

    Le = 0.405 2

    Lh = 0.177 2 (2.78)

    The gain can also be derived from the product of the two values that can be readin the ordinates in g.(2.19), with a further multiplication by /32.

    7See R.C.Johnson, H.Jasik: Antenna engineering handbook - 2nd ed., New York: Mc. Graw-Hill,1984, p.15-5.

  • M. Orece: Antennas 2013, Aperture antennas 43

    0 0.5 1 1.5 2 2.5 340

    35

    30

    25

    20

    15

    10

    5

    0

    s=1

    3/4

    1/23/81/41/8

    0

    u

    E/Em

    ax

    0 0.5 1 1.5 2 2.5 340

    35

    30

    25

    20

    15

    10

    5

    0

    t=1

    3/4

    1/23/81/4

    1/8

    0

    u

    E/Em

    ax

    Figure 2.17: Radiation patterns of sectoral horns with phase error, in both principalplanes. In abscissas u = (b/) sin and u = (a/) sin , respectively.

  • M. Orece: Antennas 2013, Aperture antennas 44

    0 0.5 1 1.5 2 2.5 3

    -150

    -100

    -50

    0

    50

    100

    150

    t=1

    3/4

    1/23/81/4

    1/80

    b/ s in

    Pha

    se

    (E

    pl

    .)

    0 0.5 1 1.5 2 2.5 3

    -150

    -100

    -50

    0

    50

    100

    150

    t=13/41/2

    3/81/41/8

    0

    a/ s in

    Pha

    se

    (H

    pl

    .)

    Figure 2.18: Phase patterns of sectoral horns with phase error, in both principalplanes. In abscissas u = (b/) sin and u = (a/) sin , respectively.

  • M. Orece: Antennas 2013, Aperture antennas 45

    5 10 15 20 250

    50

    100

    150

    l=100

    75

    50

    30

    2015

    1086

    b/lambda

    G

    5 10 15 20 250

    50

    100

    150

    l=100

    50

    20

    10

    6

    75

    30

    15

    8

    a/lambda

    G

    Figure 2.19: Gain of sectoral horns in the E and H planes (respectively for a/ = 1and b/ = 1).

  • M. Orece: Antennas 2013, Aperture antennas 46

    Figure 2.20: Examples of pyramidal horns.

    From the behavior on g.(2.17) we may observe that the radiation patterns of asquare horn are not symmetric about the axis, but they dier in the E and H planes.However, with a suitable choice of the b/a ratio, it is possible to obtain a relativelysymmetric beam, e.g. by imposing that the HPBWs be equal (or the angles at10 dB, etc.). If the criterion chosen is that of the HPBWs, from g.(2.17) we derivethat they are respectively (in radians) 0.88/b for the E plane (uniform distribution,constant phase) and 1.2/a for the H plane (cosine distribution, constant phase).For the symmetry of the beam at this level, it must be a/b = 1.2/0.88 = 1.36. As apractical rule, usually the ratio 4/3 is used.

    In general, both to reduce the phase error on the aperture and to avoid the per-sistence of the higher order modes generated by the discontinuity, it is recommendedthat the are angle be small, and consequently the horn should be long; problemsmay therefore arise in order to the excessive axial length.

    To avoid an excessive length, it is worth noting that, for constant , with theincrease of the aperture angle also the aperture area becomes larger (and so thegain) but there is also a larger phase error (which reduces the gain): so the eectson the gain are contrasting. The maximum gain is obtained when the maximumphase error on the aperture is respectively of 3/8 for the H plane and of /4 for

  • M. Orece: Antennas 2013, Aperture antennas 47

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

    2

    4

    6

    8

    10

    12

    s

    Le

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

    1

    2

    3

    4

    5

    6

    7

    t

    Lh

    Figure 2.21: Gain loss due to the phase error in the E and H planes.

    the E plane. This condition is called optimum horn. This corresponds to:

    a =3h

    b =2e

    (2.79)

    where a and b are the dimensions of the aperture of the optimum horn in the H andE planes, respectively. The gain of an optimum horn, in decibels, is:

    G = 10(0.808 + logab

    2) (2.80)

    with a loss of 2 dB with respect to the case without phase error.The optimum horn, since it maximizes the gain for a given horn length, cor-

    responds to the locus of the maxima on the curves of g.(2.19). However, with

  • M. Orece: Antennas 2013, Aperture antennas 48

    constant geometry, the gain is not stationary with respect to the frequency, since inthis case the ratio / changes, and with it the parameter of the curve in g.(2.19).

    Example:Design a rectangular horn in the Ku band (12-18 GHz) with a gain of 20 dB at 15

    GHz, with symmetric main beam.Since the gain is high, we should apply the criterion of the optimum horn, and we get

    from (2.80) that the area must be 15.562; since a/b = 4/3, it results: a = 4.555(9.11 cm);b = 3.416(6.83 cm). The lengths in the two planes, from (2.79), result to be respectively6.92 (13.84 cm) in the H plane and 5.83 (11.66 cm)in the H plane.

    Since the standard waveguide in that band is the WR62 (15.8 7.9 mm), with asimple geometrical construction we may derive that the axial length of the longitudinalhorn section should be respectively of 10.80 cm in the H plane and 9.86 cm in the E plane(half are angles of 19.22 and 17.03). A waveguide/horn connection at dierent lengthsis complicated, so that it is preferred to assume the same value of distance aperture -waveguide junction for both planes: as an example, the average value of 10.33 cm. Keepingthe same size of the horn aperture, the new values of become respectively 13.30 cm and12.17 cm, with phase errors of 0.240 and 0.390: the gain remains the same within 0.1 dB.At lower and upper frequencies, the gain is respectively 18.6 dB and 20.7 dB.

    2.9.2 Smooth circular horns

    Smooth circular horns are, together with the rectangular horns, among the oldesttypes of horn antennas. The earliest works published on this subject are of thethirties of the 20th century.8 They are derived from the circular waveguide, andare obtained by connecting to it a truncated cone. For this reason the antenna isalso called conical horn. The prole, instead of being rectilinear, can be shaped(to reduce the phase error on the aperture) or tapered (to reduce the reectioncoecient in the junction waveguide - cone). To this family belongs also the conicalhorn with zero are angle, i.e. the truncated cylindrical waveguide.

    The eld topography on the aperture of a conical horn can be considered asderived from that of a circular waveguide, whose fundamental mode is the TE11.The modes in a conical waveguide are dierent from those in a cylindrical waveguide(associated Legendre and spherical Bessel functions instead of Bessel and exponentialfunctions), but for small angles it can be shown their substantial coincidence.

    Actually, as for the rectangular horns, this eld topography is not exactly asthat of the waveguide, because the truncation of the cone also generates higherorder modes. In particular, there may be the generation of currents, and thereforeradiation, on the outer surface of the horn and of the guide, especially for smallaperture diameters.

    8Se e.g. G.C.Southworth and A.P.King: Metal horns as directive receivers of ultra-shortwaves, Proc.IRE, Vol.27, pp.95-102, Feb.1939. This paper, together with many others concerningthe horn antennas, was collected by A.W.Love in Electromagnetic horn antennas, IEEE Press,1976.

  • M. Orece: Antennas 2013, Aperture antennas 49

    L

    dm

    1

    L

    d

    BOCCA DELLA TROMBA

    f

    Figure 2.22: Geometry of a conical horn.

    Assuming a eld topography on the aperture given by the circular TE11, with nophase error, and integrating to obtain the radiated eld9 we obtain an illuminationeciency of about 0.83 (0.8 dB). In presence of phase error, the gain decreases,and it is given by the formula:

    G = 20 logC

    L (2.81)

    where C is the circumference of the circular aperture; L is the term taking intoaccount the gain loss for non uniform illumination and for phase error, and it isshown in g.(2.23). The curve may be approximated with the quadratic expressionL = 17 s2, where s is the phase error measured in wavelength.

    0 0.1 0.2 0.3 0.4 0.5 0.60.5

    1

    1.5

    2

    2.5

    3

    3.5

    4

    4.5

    5

    5.5

    s

    L, d

    B

    Figure 2.23: Correction factor for the gain of a conical horn (solid curve) and itsapproximate expression (dashed line).

    Also in this case, keeping constant , by increasing the are angle the aperturearea increases as well as the phase error, which have opposite eects on the gain.

    9Details on the computation ar reported in S.Silver,op.cit., pp.336-341.

  • M. Orece: Antennas 2013, Aperture antennas 50

    The optimum value (optimum horn condition) is for s = 3/8 = 0.375, corresponding,with the small are angles approximation, to

    d =3 (2.82)

    For wide are angles, eq.(2.82) is no longer valid, and it must be expressed exactlyusing trigonometric functions. The gain of an optimum conical horn, in dB, is givenby:

    Gopt = 20 logC

    2.82 (2.83)

    Consequently, and similarly to the rectangular case, the optimum horn has again 2 dB less than a horn of the same size and constant phase on the aperture.In g.(2.24) the diameter and the corresponding axial length are reported for anoptimum horn for a required gain, while in g.(2.25) is shown the gain of a conicalhorn vs. diameter, with the axial length