In the advent of modern mobile satellite communications requiring rapid and adaptive multiple beams, this work studies the ability of reflector antennas fed by dense focal plane arrays (FPA) in achieving arbitrarily shaped and sized footprints to meet the demands. In this paper, the efficiencies of single off-axis as well as multiple beams of FPA-fed paraboloids are investigated. The offset FPA considered here comprises hard rectangular waveguides. The focal plane field, which the FPA samples, is synthesized by integration of the physical optics induced electric currents over the reflector surface caused by the off-axis incident plane wave arriving at that incidence angle of interest. Full mutual coupling analysis has been performed in the FPA sampling, thereby taking into account mutual coupling losses in the arrays. The fields over the tilted elliptical aperture of off-axis beams needed for calculation of the aperture efficiency are obtained by projecting the usual focal plane fields to this tilted aperture using geometrical optics. Results show that the total efficiency of the offset FPA-fed reflector decreases with increasing beam angle and increases with larger number of FPA elements. It is also found that the maximum directive gain of the reflector radiation patterns falls noticeably with beam angle when the FPA population is low, but the directivity can be maintained well when an adequate number of FPA elements are used. Multiple beams that are either closely overlapping or widely separated are also successfully investigated.
 Over the past two decades, there has been enormous growth in mobile satellite and radio communications. The constantly rising demand for increased capacity and enhanced performance of both commercial as well as military communications has led to continuous evolution of the technology. Satellite systems today provide services not only for aircrafts, ships, and land mobiles, but also for personal communications [Brisken et al., 1979; Zaghloul et al., 1990; Wu et al., 1994]. The applications include navigation, positioning [Sakagami et al., 1992], communications, tracking [Jacobs et al., 1991; Densmore and Jamnejad, 1993], range determination [Brisken et al., 1979], targeting, air traffic control, automated landing [Wu et al., 1994], and remote sensing, among many others. A schematic showing the services provided by a satellite to air, land, and sea mobile clients is given in Figure 1.
 In recent years, due to the rapid increase in the demand for high-speed wireless Internet and video conferencing, bandwidth is a highly priced commodity. Hence, there is a growing need to generate highly directive multiple beams with large signal bandwidth in order to satisfy such broadband multimedia applications [Maalouf and Lier, 2004]. Multibeam reflector antennas are attractive candidates for meeting this demand because they can be launched into orbit to provide spot beams (or footprints), each of which is allocated a certain frequency band of the available spectrum. In this way, frequencies are reused instead of assigning the entire spectrum to a single contoured beam [Rao and Tang, 2006]. Mobile users in two widely separated locations can then use the same frequency band with minimal interference [Brisken et al., 1979] (see Figure 1 again). By using a large number of spot beams supporting both uplink (ground-to-satellite) and downlink (satellite-to-ground) frequencies, such multiple beam antennas (MBAs) can provide service to specified geographical regions on the earth, either contiguously or noncontiguously. Associated advantages include enhanced antenna gain and thus higher effective isotropic radiated power (EIRP) on the downlink and higher gain-to-noise temperature (G/T) on the uplink, increased capacity, and smaller ground terminals [Rao and Tang, 2006]. Other than spatial isolation as described, frequency reuse can also be realized through polarization isolation [Zaghloul et al., 1990].
 Currently, MBAs are being used for direct-broadcast satellites (DBS), personal communication satellites (PCS), military communication satellites, and high-speed Internet applications [Maalouf and Lier, 2004]. In addition, MBAs have also been successfully implemented in geosynchronous satellites as well as low earth orbit (LEO) and medium earth orbit (MEO) satellites [Wu et al., 1994; Vatalaro et al., 1995; Godara, 1997]. Examples of commercial satellites employing frequency reuse MBAs include INTELSAT IV-A (1975), INTELSAT V (1981), INTELSAT VI (1989), NASA's ACTS satellite, and ITALSAT of Italy [Zaghloul et al., 1990]. As these satellites move at high velocities, they may be in view only for a short window of time. This gives rise to issues such as frequent handoffs between beams and rapid switching of beams. Therefore, it is evident that fast, dynamic, and highly adaptive beam steering of the antenna is required. Rapid beam forming and steering can be achieved by employing a beam forming network (BFN) to realize so-called adaptive antennas, whose radiation patterns are dynamically adjusted to suit the environment.
 Conventionally, multiple beams can be achieved by using a cluster of horns which is located in the focal plane of the reflector under a one-horn-per-beam configuration [Mrstik and Smith, 1981]. However, this practice is susceptible to aberrations and distortions associated with off-axis beams, thus possibly leading to a loss in aperture efficiency. In scenarios where a large contoured footprint on the earth's surface comprising several overlapping beams is required, the continuity of these beams is also important. It describes the ability to illuminate the service area uniformly and thus efficiently. The decrease in gain at the crossover point of adjacent beams (or crossover loss) is commonly used to characterize the quality of the overlapping beams. As mentioned by Ivashina et al. , cluster feed horns may sustain high crossover losses, due to their large sizes and thus coarse samplings of the focal plane fields. As such, a minimally required −3dB beam overlap for decent illumination may not be possible.
 These above challenges may be well managed by using a dense focal plane array (FPA) feed comprising many electrically small elements (with diameters less than half wavelength). Recent studies on the use of dense FPAs as feeds for parabolic reflector antennas are found in the works of Ivashina et al. , Ng Mou Kehn et al. , Ng Mou Kehn and Kildal , and Ivashina et al. . The multiple beams, closely overlapping or widely separated, are achieved by grouping the FPA elements into overlapping subarrays that are excited independently and simultaneously. One of the main features of such FPA feeds is their ability to adapt to the deformed and skewed focal plane field rings of off-axis beams, and thus correct for aberrational errors. High-performance scanning reflector antennas may thus be achieved. In addition to that, closely separated overlapping beams may be achieved, thus increasing the size of the continuous footprint on the target area. Aside from this, many tightly bonded beams are able to create any arbitrarily shaped contour beam as a collective whole. As such, instead of having fixed beam sizes and shapes, such adaptive beam forming capabilities of FPA-fed reflectors allow dynamic shaping and sizing of the footprints. The usefulness of this can be seen from Figure 1, in which the land or sea mobile crafts may be distributed over a region of any geographical shape and area. Multiple beams that are widely separated can also be effectively generated, providing service to regions that are not within close proximity to each other. Furthermore, because FPAs are realized using high-speed digital beamformers, very rapid beam forming, steering, and adaptation can be achieved. This makes FPA-fed reflectors well suited for mobile satellite applications requiring highly agile beams to keep up with the fast-moving users.
 A schematic of such FPA-fed reflector antennas is given in Figure 2, illustrating two overlapping beams for dual-beam applications. Two sets of overlapping focal plane fields can be seen, corresponding to closely separated beams. Both can be sampled to fine resolution by the dense FPA. Any FPA element located within overlapping field regions on the focal plane contributes to all those overlapping beams concerned. Therefore, by properly sampling the focal plane fields of each closely separated beam, the field of continuous coverage is enlarged.
 Observing Figure 2, the off-axis beam angle measured from the axis is denoted by θb. The ray directions indicated are those of incident plane waves on receive mode, which give rise to the likewise overlapping concentric field rings on the focal plane. This field distribution is to be discretely sampled by the FPA on transmit mode, and the resultant far-field overlapping beams are also shown. The corresponding feed displacement angle is represented by θd. The beam deviation factor is thus θb/θd [Lo, 1960].
 Previous investigations on the use of array feeds to scan reflector antennas include those of Mrstik and Smith , Assaly and Ricardi , and Davis et al. . It was shown by Assaly and Ricardi  that an adequate number of array elements are needed to attain a decent beam quality of the considered parabolic cylindrical reflector. In the work of Mrstik and Smith , a likewise parabolic cylindrical reflector fed by axially polarized line source feeds was treated, and the factors affecting the degradation of the main beam and sidelobe levels were characterized. However, mutual coupling effects within the array feeds had been neglected there, and the performance in terms of the usual reflector subefficiencies was not conveyed.
 A more rigorous study of the aforementioned strengths offered by FPA feeds is conducted in this paper, this time for the common parabolic dish reflector. At least three major factors are of concern when dealing with a more thorough investigation: (1) the decoupling efficiency of the FPA feed when it samples the focal plane fields corresponding to off-axis or overlapping beams (thus taking full mutual coupling effects into account), (2) the aperture efficiency of those beams, comprising the usual well known reflector subefficiencies (spillover, illumination, polarization, phase, and blockage), and (3) the far-field radiation pattern of the reflector. It is thereby the objective of this work to look into these aspects.
 In the groundwork of Ng Mou Kehn et al. , only axial beams were considered, and the FPA-sampled focal plane fields associated with on-axis plane wave incidence were approximated by the theoretical Airy pattern in closed form. However, there is no likewise simple analytical function for the focal plane field distribution corresponding to off-axis incidence. Rigorous analysis would be required instead. In this work, the focal plane fields which the FPA samples are synthesized by integration of the physical optics (PO) electric currents flowing over the metal surface of the parabolic reflector. These currents arise from one or more incident plane waves arriving at any incidence angle, which is the desired beam angle on transmit. The elements of the FPA are then excited according to this field distribution, in a discretized and truncated manner, in the same way as had been done in the work of Ng Mou Kehn et al. . Also as in the latter work, full mutual coupling analysis is performed in the FPA sampling in this work, thereby taking into account all mutual coupling losses in the array. The same type of array element engaged in the work of Ng Mou Kehn et al.  shall be reused here: the sidewall dielectric-loaded hard rectangular waveguide (HRW) element [Ng Mou Kehn and Kildal, 2005].
 The well-known approach of obtaining the aperture efficiency of a parabolic reflector radiating an axial beam is by using geometrical optics (GO) to project the fields radiated by the feed and observed at the reflector surface up to the focal plane aperture, thereby acquiring the focal plane fields expressed in terms of the feed radiation pattern, which can then be used to calculate the various subefficiencies [Kildal, 2000]. However, the present state of the literature on this technique so far only considers axial beams, thus corresponding to the focal plane aperture on which the feed-radiated fields are to be GO-projected. However, when off-axis beams are concerned, the aperture onto which the fields of the feed are to be projected is no longer the circular focal plane, but rather, a tilted elliptical aperture plane. This is due to the projection of the circular focal plane aperture onto a plane that is perpendicular to the direction of the off-axis beam, thus resulting in a tilted and reduced effective aperture, as required and well known. Therefore, the formerly accustomed GO-projected feed fields on the focal plane must be further multiplied by an appropriate phase shift term, to account for the additional raypath traveled from the focal plane to the new elliptical aperture. The mathematical details for this shall be presented later.
2. Subefficiencies of the FPA-Fed Reflector
 The foremost discussion in this paper would be to define the subefficiencies that are used as figures of merit to characterize the FPA-fed reflector. Even though these are already well known, they are still briefly described here for completeness. The subefficiencies computed here are attributed to two factors, (1) the reflector geometry and feed radiation pattern, and (2) losses in the FPA feed itself. The former dictates the usual spillover, illumination, phase, and polarization subefficiencies [Ng Mou Kehn et al., 2009]. The total radiation efficiency of the latter quantifies how much power is lost within the FPA due to mutual coupling between the elements as well as through dissipation losses arising from nonideal lossy materials involved in the feed structure. This total radiation efficiency may then be decomposed further into two subefficiencies: (1) the conventionally well known radiation efficiency accounting for dissipation losses, and (2) the decoupling efficiency taking care of mutual coupling losses. This definition has been presented by Ng Mou Kehn et al. . The decoupling efficiency accounts for both mutual and self coupling losses, in which the latter may also be referred to as reflection or mismatch losses. This decoupling efficiency will always be existent, even if ideal lossless materials are involved. For prime focus paraboloids, upon taking into account central blockage losses via a blockage efficiency, the product of all these subefficiencies together yields the total efficiency of the reflector system, i.e.,
in which the subscripts are representative of what they suggest (those subefficiencies described earlier). These subefficiencies are computed and presented later in section 8. The decoupling efficiency ɛcoup will be discussed more in the next section.
3. Mutual Coupling Analysis
 In this section, the mutual coupling analysis is described for an embedded array element by using an infinite array approach. This embedded element is the only one in the infinite array that is excited, while all the rest are passively match terminated. This formulation paves the way for the definition and computation of the decoupling efficiency ɛcoup. Full modal analysis is performed here, for which a multiplicity of modes is present over the open-ended waveguide apertures, according to the theory of Ng Mou Kehn and Kildal .
3.1. Embedded Element Efficiency Using Infinite Array Approach
 Let us consider an infinite planar array of open-ended waveguide elements in the xy plane, with periods dx and dy along the x and y directions. The (u, v)th element center is located at (x = udx, y = vdy), where u and v are integers. Any waveguide is fed by a positive z traveling incident mode. When only a single element (say the (0, 0)th one for the present illustration) is excited by, say its fundamental pith mode, with all the rest in the infinite array being passively match-terminated, the coupling to the pth mode in the (u, v)th element is defined as
where Auv,p− = modal amplitude of the −z traveling pth mode into the (u, v)th element A00,pi+ = modal amplitude of the +z traveling incident pith mode exciting the (0, 0)th element.
 These modal amplitudes may be generally complex. This coupling coefficient of (2) is under the situation whereby only the (0, 0)th element is excited in fundamental mode, with the presence of all other match-terminated elements taken into consideration. This pertains to the “embedded” scenario, in which the radiation field pattern from this solely excited element is the so-called embedded element pattern. Linear superposition of separate such embedded situations collectively yields the entire “all excited” array scenario. This fact constitutes the fundamental concept in obtaining the coupling coefficients from the “all excited” array situation.
 Inversely, swapping the excited and received elements, this (2) leads to
noting the superscript minus sign introduced to the coupling coefficient to signify the inverse situation from (2).
 Let us now consider the case of all elements being used as in a conventional uniformly excited and linearly phase-steered infinite array. For a uniform amplitude of Ai and with the (0,0)th element at zero phase, this (3) becomes
where kx0 = k0 sin θ00 cosϕ00 and ky0 = k0 sinθ00 sinϕ00 are the phase progressions per unit distance along x and y, to achieve a certain steered main-beam direction (θ00, ϕ00) of the fundamental (0, 0)th Floquet modal plane wave. Subsequently, by linear superposition, (4) allows us to write
The summation runs through all (u, v)th elements in the infinite planar array. The bracketed superscript “all” on the left-side quantity is to symbolize that it is under phase-steered array conditions in which all elements are excited.
 Subsequently, by reversing the phase to get back the desired form of the coupling coefficient as expressed by (2), this (5) leads to
With this (6) recognized as a transform from (u, v) to (kx0dx, ky0dy) domain, the relevant inverse transform may be written as
where Φx = kx0dx and Φy = ky0dy. Owing to the periodicity, the limits of integration range only within the Brillouin zone [−π, π]. This (7) constitutes the ultimate expression for the coupling coefficient Cuv,p that is treated computationally. The integrations can be handled easily in the existing computer program code used by Ng Mou Kehn and Kildal  for treating a phase-steered infinite array of dielectric loaded hard rectangular waveguides, by repeatedly solving for numerous cases of phasings (phase-steer angles). The trapezoidal method is used in the present work to numerically integrate the two-dimensional integrand.
 Upon computing Cuv,p, the modal amplitude Auv,p− of the −z traveling pth mode into the (u,v)th element may be obtained via (1), with a known incident modal amplitude A00,p+ in the solely excited (0,0)th element. Subsequently, this Auv,p− enables us to determine the total −z traveling coupled (real) powers into all elements [−∞ < u, v < ∞], including the excited (0, 0)th element itself (reflected power). This total coupled plus reflected power constitutes the coupled power loss: Pc in (8) below, since it is not radiated outward from the array aperture, i.e., lost power carried by modal fields traveling backward into all elements. The embedded element decoupling efficiency is then defined as
where Pc is the coupled power loss as mentioned above, Prad is the total radiated power, and Pinc is the total power injected into the array. For an actual array solution, Pinc is known and Pc is calculable.
3.2. Finite Array Decoupling Efficiency: Finite Number of Arbitrarily Excited Elements
 When the coupling coefficient between any two elements under the embedded element scenario is determined as just discussed, the treatment of a planar focal plane array of open-ended rectangular waveguides (a finite number of them) illuminated according to any function may be done by linear superposition using an element-by-element approach. This means that we determine the modal amplitude coefficients of a particular receiving element resulting from the superposed coupling contributions from all other excited elements (including itself), performing each constituent excitation case separately, i.e., one at a time. Each excitation case consists of a solely excited embedded element. This may then be repeated for all modes considered in that receiving element. Doing the same for all other receiving elements yield the solution of the entire configuration. The same formula of (8) for the decoupling efficiency reapplies for arbitrarily excited finite arrays, but just that the Pinc comprises the power injected into all elements (not only just one, as for the embedded element case), and Pc is the total coupled power losses in the entire array caused by mutual coupling among all elements. It is this decoupling efficiency which has been computed and presented later on, for FPAs in our present case.
4. Focal Plane Field Synthesis by Physical Optics
 Consider an incident plane wave on a paraboloidal reflector as shown in Figure 3. For a fixed incidence angle θinc″ measured from the z axis (being the axis of rotational symmetry of the paraboloid), the magnetic field vector of the plane wave observed at any point (x″, y″, z″) on the reflector surface is expressed as
where k0 is the usual free space wave number, H0inc is the amplitude of the magnetic field, and
are the unit vectors along x and z whose spherical coordinate components are dependent on the angular coordinates (θ″, ϕ″) defining the positional vector of the point of interest on the paraboloidal surface, being at (x″, y″, z″). These x″, y″, and z″, will later on be the source coordinates of the equivalent PO electric currents on the PEC reflector surface, and they are written as
where F is the focal length of the paraboloid, well known as
in which D is the diameter of the reflector and θ0 is the half-subtended angle. All these are shown in Figure 3. Therefore, for any angular coordinates (θ″, ϕ″) directed toward the reflector surface, (11) and (12) define a certain point (x″, y″, z″) on the surface in Cartesian coordinates, at which the magnetic field vector is defined by (9), having x and z components defined by (10). Note from (9) that the phase reference point is at the global origin, or the focal point of the paraboloid, for the present case.
 Hence, assuming H0inc = 1, the impinging magnetic field on any point on the reflector surface of (9) may be reexpressed as
 Then with the well-known unit normal of the paraboloidal surface stated as
the PO electric current on the reflector surface is
where the unit normal vector and the incident magnetic field vector are obtained from (15) and (14), respectively. The Cartesian coordinate components, Jx, Jy, and Jz of this current are then easily acquired from these spherical coordinate components.
 Subsequently, the electric field radiated by the PO currents defined by (16) and observed at any general point o = (xo,yo,zo), or particularly on the focal plane for our present interest, is expressed by the following well known field integral
where the integration spans over the surface of the parabolic reflector: [0 ≤ θ″ ≤ θ0], [0 ≤ θ″ ≤ 2π], represents the unit dyadic, and the elemental area ds″ on the paraboloidal surface takes on the known form:
The in (17) is the unit vector along the direction from any source point on the reflector surface to any field observation point, i.e.,
 Therefore, by evaluating the integral of (17), the fields arising on the focal plane due to an impingent plane wave of generally oblique angle of incidence may be obtained.
5. Off-Axis Beam Treatment by Geometrical Optics
 When treating off-axis beams with oblique angle α measured from the axis of the parabolic reflector, the original circular aperture plane needs to be tilted from the usual circular focal plane for on-axis beams to an elliptical tilted aperture plane (the red line). Consider an off-axis beam radiated along a direction in the xz plane, (corresponding to ϕinc = 0 on receive of an incidence plane wave), as shown in Figure 4. Hence, for all points on the original focal plane along a certain constant x line, (parallel to y), their fields on this focal plane along that line are multiplied by the same phase exponential factor:
 Consider now, an ellipse with major and minor axis lengths a and b, respectively, as shown in Figure 5. The length ρ0 of the radial line circumscribing the ellipse, expressed in terms of a, b, and ϕ is written as
 In our present context, the major and minor axis lengths a and b are D/2 and (D/2)cosα, respectively, thus leading to
 The area of the ellipse Aell, is simply π multiplied by the product of the major and minor axis lengths, i.e.,
 Let us now define the radial coordinates with respect to the circular and elliptical apertures of the original focal plane and the tilted one as ρcir and ρell, respectively, as indicated in Figure 4b. Hence, we have the following equations.
From the already well known
In addition, upon combining (23a) and (24a), we have
With this (26) and using (22b), the angle corresponding to the rim of the tilted elliptical aperture (ρell = ρ0) is
noting the prime attached to θ0 to distinguish it from the half-subtended angle of the parabolic reflector. When α = 0 (axial beams), then θ0′ = θ0.
 At present, the focal plane fields obtained by GO projection of the fields radiated by the feed from the reflector surface to the focal plane are of the form [Kildal, 2000]
in which Gf is the far field functions of the feed radiation, and F is the focal length. With (24a) used in x = ρcir cos ϕf, which in turn is placed into (21), the correction phase factor from the original circular focal plane to the tilted elliptical plane is obtained as
noting that ϕf = π − ϕ and thus cos ϕ = −cos ϕf have been invoked. This phase factor shall then be multiplied by that original focal plane aperture field of (28) to obtain the field on the tilted elliptical aperture of the off-axis beam. Hence, we have
Subsequently, the copolar and cross polar field components on the elliptical aperture are expressed as [Kildal, 2000]
 Therefore, upon suppressing the subscript f of the ϕ coordinate for simplicity, the various reflector subefficiencies are obtained in the follow equations. Before that, several factors in these expressions are first noted. The θ(ρell) is explicitly stated as a function of ρell as seen from (26). The ρfeedrim and θfeedrim are the elliptical radial coordinate and theta coordinate, respectively, corresponding to the rim of the feed. The latter is illustrated in Figure 4a. These occur as the lower limits of the integrations in the numerators of the efficiency expressions, to account for blockage. It is noted that doing so only approximates the blockage effects, but it is highly accurate when α is small. The θ0′(ϕ) and Aell are from (27), and (22c), respectively. Only the illumination, polarization, and phase subefficiencies are affected by this new study of off-axis beams. The previous well-known formula for the spillover efficiency remains unchanged. By using (25) for the elemental area of the elliptical surface with the subefficiency formulas of Kildal , we obtain the following subefficiencies.
Hence, with the feed radiation patterns already obtainable previously, as in the work of Ng Mou Kehn et al. , these subefficiencies for off-axis beams may be computed.
6. Treatment of Overlapping Beams by Superposing Individual Off-Axis Beams
 When overlapping beams are concerned, the synthesis of the focal plane fields to be sampled by the FPA can be performed by simply superposing the separately synthesized focal plane fields of the various beam angles involved. In other words, the focal plane fields arising from the plane wave arriving from any one beam direction (on receive) are first synthesized individually. Repeating for the other beam directions and then adding up the respective focal plane fields yields the final focal plane field distribution, to be subsequently sampled by the FPA in the usual discretized and truncated manner.
7. Description of the FPA Element and Conjugate Field Matching Using a Digital Beamforming Network
 This section describes the concept of the use of a digital beamformer for implementing the FPA. As mentioned earlier, the FPA elements are divided into subarrays. In this work, the subarrays are excited with unique sets of complex coefficients according to the so-called conjugate field matching (CFM) method [Minett et al., 1968; Rahmat-Samii, 1990]. In other words, the excitations of the FPA elements are conjugate-matched to the focal plane field distribution that is synthesized according to the PO integration approach described in section 4. This is illustrated in Figure 6 for an example of a single axial beam. The upper diagram is an array of hard rectangular waveguides (HRW), each (square) element being loaded by dielectric slabs on the two side E plane walls. The slab thickness d is related to the wavelength at the TEM frequency as shown. In our investigations, the aperture-to-unit cell ratio is fixed at 0.981, and the thickness d of each dielectric slab is also constant at 1/40 of the waveguide width. The relative permittivity ɛr of the slab is such that the TEM frequency is maintained, i.e., always operated at the hard condition [Ng Mou Kehn and Kildal, 2005]. The unit cell size considered here is 15mm × 15mm. Figure 6 (bottom left) shows a typical set of synthesized concentric focal plane field rings (in continuous form), with a reference grid frame overlay dictating the resolution of the discretized sampling. Each FPA element is excited by generally different amplitude and phase according to its location in the field distribution: the CFM as just described. The color (or shading) correspondence between the top and bottom of Figure 6 is noted, whereby each color represents a certain amplitude. In this way, we portray the idea how the continuous focal field distribution of Figure 6 (bottom) can be discretely reconstructed by 5 × 5 sample points of a rectangular grid FPA depicted by Figure 6 (top). It is to be mentioned, however, that this Figure 6 only demonstrates the sampling of the amplitude but not the phase of the focal plane fields. In reality, both amplitude and phase will have to be sampled. Hence, the illustration here serves only to convey the concept of the CFM. Figure 6 (bottom right) is a generic block diagram of the beamformer, which ensures that the FPA elements are excited with complex coefficients according to the CFM method. For any one beam, each FPA element is excited according to the complex conjugate of the focal plane field value at the point coinciding with the center of that element. This is repeated for the subarray elements pertaining to all other desired beams. Hence, each achievable beam direction from the reflector pertains to a certain subarray with its own unique set of amplitude and phase excitations.
 The same concept of this Figure 6 can be translated to multiple overlapping beams, for which the focal plane fields are likewise discretely sampled by the FPA, albeit just a difference in the field pattern – this time consisting of the respective sets of overlapping concentric field rings (see Figure 19 later for a dual-beam configuration example). More details on the complexity of combining subarrays to achieve multiple beams will be discussed in a later paragraph of this section, with particular emphasis on the beamforming concepts.
 In practice, CFM can be performed only over a portion of the focal plane field distribution with infinite extent, although the bulk of the field intensity is concentrated around the center of the diffraction pattern. In general, the more sidelobes (rings) of the focal plane field pattern are excited by the subarrays, the better will be the illumination of the reflector and thus the higher will be the aperture efficiency (when aperture blockage is neglected) [Rahmat-Samii, 1990; Kildal, 1984]. This focal plane field sampling is also discretized due to the finite number and finite size of the FPA elements. The consequence of this truncation and discretization is a reduced aperture efficiency of the reflector.
Figure 7 presents the logical concept of combining several subarrays for generating a multiplicity of reflector beams. An example of four beams is demonstrated, labeled as A, B, C, and D. The FPA schematics of Figures 7a–7d convey the individual subarray excitations for beams A–D, respectively. A total of 24 FPA elements, as depicted by boxes in the diagrams, are selected for the illustration. The encircled number within each box denotes the FPA element index. For any one subarray, the excitation coefficient of each element is represented by a certain numeric appending its associated beam-labeling alphabet, e.g., B7 of Figure 7b denotes the coefficient of the seventh (central) element of subarray B. By mere superposition of the four separate subarrays of Figures 7a–7d, the combined subarray excitation configuration is given by Figure 7e. As observed, the central elements of the combined FPA are each tagged with multiple excitation coefficient labels. This means that each of these elements contributes to various beams. Particularly, each of the central four elements (indices 1–4) contributes to generating all the four beams. Every one of the eight elements (indices 5–12) surrounding those four central ones helps to create three beams, whereas each of the remaining 12 peripheral elements (indices 13–24) assists in forming only one beam.
 The beamforming network of combined subarrays for generating multiple beams is schematized in Figure 8a, for the same four-beam configuration of Figure 7e as an example. The amplitude and phase of each FPA element are independently controlled by an integral network comprising a 90° hybrid, amplifiers, and a combiner, as represented by small boxes along the leads in Figure 8a and schematized by Figure 8b. By adjusting the two amplifiers, this network is able to generate output signals with any arbitrary complex coefficients. By observing the output levels labeled within the boxes, it can be seen how the excitation of each antenna element in Figure 8a is appropriate with the configuration of Figure 7e. Every beamformer (A–D) has a predefined set of complex excitation coefficients assigned to all FPA elements, according to the synthesized focal plane fields (as described in sections 4 and 6), which are then realized by the network of Figure 8b.
8. Numerical Results and Discussion
 The investigation of single off-axis beams is first performed. Later on, simultaneous multiple beams will be studied – both closely overlapping and widely separated. For a parabolic reflector with 5-m diameter and 60 deg half-subtended angle and illuminated by a y-polarized incident plane wave at 10 GHz, the amplitude of the copolar and cross polar components (y and x components) of the focal plane E field synthesized by PO integration for various incidence angles (0.5 deg, 1 deg, 1.5 deg, and 2 deg) are shown in Figure 9. Here, the computed discrete points in the pattern are separated by 10 mm along both orthogonal coordinates in the focal plane, and there are 31 × 31 grid points centered at the focal point of the paraboloid. Hence, the fields over a 20.67λ by 20.67λ portion of the focal plane are computed. As can be seen, as the incidence angle increases, the copolar field pattern (comprising a central main lobe surrounded by concentric rings) gets more skewed from the focal point of the paraboloid, being the center point of the plots, as expected. As also observed, the cross-polar field levels are null at the centers of the field patterns (the foci of the incident plane waves), but are maximum along the diagonal 45 deg azimuthal lines with respect to those foci (different from the center of each plot, being the focal point of the paraboloid). Table 1 gives the displacement (in wavelengths) of the center of the focal plane field diffraction pattern skewed from the focal point of the paraboloid, for various beam angles θb (see Figure 2). The angular displacement θd of the feed (Figure 2 again) and the beam deviation factor defined earlier as θb/θd [see Lo, 1960] are also tabulated.
Table 1. Skew Radial Distances of Diffraction Patterns From Focal Point for Various Beam Angles and the Respective Beam Deviation Factorsa
Beam Angle, θb (deg)
Skew Distance, λ
Feed Angular Displacement, θd (deg)
Beam Deviation Factor, θb/θd
Diameter of 60 deg paraboloid equal to 5 m, with focal length 2.165 m.
 When this same parabolic reflector is fed on transmit mode at the same 10 GHz frequency by an offset HRW FPA of a certain array population, this finite number of FPA elements are excited according to the synthesized focal plane field distribution, thus sampling the (semi)continuous form of the focal plane fields shown in Figure 9 in a discretized manner. The distributions of the copolar and cross polar E field excitation (amplitude) coefficients of the FPAs for various subarray populations ranging from 3 × 3 to 17 × 17 are depicted in Figure 10, for an example of 0.5 deg beam angle. A total of 21 × 21 FPA elements, each of size 15mm × 15mm, are present, of which various fractions (subarray populations) are excited according to the synthesized focal fields. This same total of 21 × 21 FPA elements reapplies to other investigated beam angles. The central (11, 11)th element always coincides with the centers of the synthesized focal field patterns resulting from the various incident plane waves (on receive). In this way, the FPA feed is offset from the focal point. For any subarray population, the unexcited passive elements are indicated by their zero levels in Figure 10. Unlike Figure 9, the center of each subplot here is now the center of the diffraction pattern on the focal plane due to a 0.5 deg incident plane wave. The graphs of the subefficiencies versus the subarray population for 0.5 deg, 1 deg, 1.5 deg, and 2 deg beam angles are shown in Figure 11, for population ranges of 3 × 3 to 17 × 17. These subefficiencies are those defined earlier in section 2. The element size of 15mm × 15mm is half the wavelength at the investigated 10 GHz frequency, which has been found to be optimal [Ng Mou Kehn et al., 2008]. Clearly, for all beam angles, the total efficiency rises as the number of elements increases, due to the increase in the aperture efficiency. The decoupling efficiency however is not seen to be significantly affected by the subarray population, so long as it is higher than 5 × 5. Hence this suggests that the decoupling efficiency is negligibly affected by both the array excitation function as well as the extent of this function. It is primarily the electrical element separation which influences the decoupling efficiency [Ng Mou Kehn et al., 2008]. Another observation of Figure 11 is that, for any one subarray population, as the beam angle increases, the total efficiency falls. This is as expected. It is also evidential that the total efficiency starts to rise rapidly from low subarray population, but then tapers off and saturates at around −1.2 dB from about 13 × 13 upward, with the exception of the on-axis beam. This saturation effect applies to all beam angles. Hence the improvement in performance diminishes as the subarray population increases. The benefit is thus the greatest when more elements are initially added to a lowly populated FPA; but as more elements are added, the benefit dwindles. This saturation phenomenon will be reportrayed later on by the radiation patterns of Figure 12. Therefore, a subarray population of around 13 × 13 provides the best tradeoff between performance and the number of elements (which is proportional to cost and complexity), for all investigated beam angles. This consistent optimal-tradeoff subarray population is highly advantageous, since the same number of FPA elements can be reapplied to different beam angles.
 The far field radiation patterns of the 60 deg parabolic reflector at 10 GHz for various beam angles have also been computed. They are calculated by Fourier integration of the focal plane aperture fields of (28), which are related to the computed feed radiation patterns of any offset FPA configuration. Figure 12 shows the copolar directive gain plots for various subarray populations and beam angles, with each plot pertaining to a common subarray population, thus comparing between the beam angles. As observed, the main lobes of the patterns are achieved perfectly at their respective beam angles. The maximum gains toward the various beam directions are also seen to be fairly equal at around 52 to 53 dBi and the first sidelobe levels are about 17 dB below the main beam level for array populations larger than 7 × 7. This comparable main-beam performance among all beam angles is chiefly attributed to the virtually consistent effective area of the parabolic reflector for all the small beam angles considered. This aforementioned axial gain level is close to the maximum directivity of 54.4 dBi of the reflector aperture with 5-m diameter, being around 167λ at the 10 GHz frequency (the 4πA/λ2 formula, where A is the aperture area). Note that since the beam angles are small, the elliptical effective aperture is still very close to a circular one. It is also observed that the −3dB beam width is about 0.36 deg for all the investigated beam angles. This value is very close to the theoretically predicted one of λ/D = 0.344 deg for large D and small beam angles, where D is the dish diameter. It also retrieves the same maximum directivity value of 54.4 dBi when used in the formula Dmax = 36,000/(θEθH), where θE and θH are the half-power beam widths along the E and H plane, respectively, both in degrees (for our symmetric case here, θE = θH). Therefore, at least up to the maximum 3 deg beam angle investigated here, these offset FPA-fed reflectors are able to scan up to at least 9 beam widths and still maintain good levels of gain performance. In addition, it can be seen that when the subarray population is low, specifically 3 × 3 and 5 × 5, the drop in the maximum gain level from a small to large beam angle is the most pronounced. In fact, although not so apparent from the plots, we have zoomed in to scrutinize more closely the difference in the maximum gain levels in our source plots. It is found that at 3 × 3 subarray population, the level of the 1 deg beam is higher than that of the 2 deg beam by around 0.7 dB (52.1 dBi versus 51.4 dBi). For the 5 × 5 subarray, the difference drops to about 0.3 dB (52.7 dBi versus 52.4 dBi). When the subarray population is 7 × 7 and larger, the difference in maximum gain levels between the 1 deg and 2 deg beams becomes comparatively insignificant, albeit the slight drop. This goes to show that with such offset FPA feeds sampling the focal plane fields to proper resolution levels, the usual loss in maximum gain levels as the beam is steered from boresight suffered by offset one-element-per-beam type of feeds can be alleviated, so long as the FPA population is not too low – around 7 × 7 and above, from the example just illustrated, and the beam angle is not too large (up to 2 deg). This demonstrates an advantage of FPA feeds over single-element feeds. Further observation of Figure 12 indicates that the gain patterns for all investigated beam angles get stabilized with increasing subarray population, the rate of change being the highest when the population is initially increased from a low value. This coheres with a former statement about the saturation phenomenon. In particular, for our illustrated cases here, the threshold population for pattern stability is 13 × 13 elements. Coincidentally, this concurs with the tradeoff subarray population mentioned earlier. All these thus make good sense of the computed results. Therefore, not only does this subarray population provide a good tradeoff between system complexity and performance, it is also the threshold condition for stable radiation patterns.
 The variations of the subefficiencies with beam angle for various subarray populations, for the 60 deg paraboloid at 10 GHz with 15mm × 15mm elements, are presented in Figure 13. It is clear that as the beam angle increases, the aperture efficiency falls. This is due to the degradation of the phase, spillover, and illumination subefficiencies with increased obliquity of the beam. For array populations larger than 7 × 7, the rate of degradation in performance with increasing beam angle is also seen to rise, in view of the increasingly negative slopes of the curves. This accelerated deterioration with increasing beam angle is an intuitively satisfying phenomenon. The decoupling efficiency of the FPA, unlike the aperture subefficiencies, does not seem to be affected by the changes in the beam angle, or correspondingly, variations in the array excitation function. This is again in concurrence with earlier findings [Ng Mou Kehn et al., 2008] which asserted that the decoupling efficiency is only affected by the electrical element separation, but is independent of the array excitation function. It is thus found that as the beam angle increases, the total efficiency of the FPA-fed paraboloid degrades. Another interesting aspect to note from here is that, as the array population increases, the slopes of the curves get gentler. Therefore, the larger the array population, the less susceptible the performance is to degradation with increase in beam angle. This indicates another desire for adequately high subarray populations.
 In addition to the radiation plots of Figure 12, the contour pattern of the copolar directive gain of the 17 × 17 FPA-fed 60 deg paraboloid (5 m diameter) at the same 10 GHz for a 1 deg single off-axis beam angle is presented in Figure 14. The same 15 mm × 15 mm element size is maintained. The main lobe corresponds to the beamed direction, and it is surrounded by concentric sidelobes. The ability of FPA-fed reflectors in steering spot beams is thus clearly demonstrated.
 The variations of the maximum cross-polar field levels relative to the respective maximum copolar levels with subarray population for various beam angles are given in Figure 15. Evidently, the maximum cross-polar level for the axial beam drops with increasing number of FPA elements, although the improvement gets less rapid as the subarray population grows. On the contrary, the maximum cross-polar level increases with rising subarray population for off-axis beams, but only mildly; besides, this degradation is also seen to taper away as the FPA population gets large. In addition, with the exception of the 2 deg beam, the cross-polar performance degrades with increasing beam angle.
 We now arrive at the results of multiple beams. Figure 16 shows the directive gain patterns of four combination pairs of closely overlapping beam angles: (1) 0.5 deg with 1 deg, and (2) 0.5 deg with 1.5 deg, (3) 1 deg with 1.5 deg, and (4) 1 deg with 2 deg. The maximum gain levels at the two beamed directions of all four pairs are achieved. However, the gain levels of the overlapped beams (for all pairs) are reduced from those of the solitary beams by about 3 dB (fall from around 53 dBi to about 50 dBi). This drop in gain level associated with beam broadening is as expected. The copolar contour gain pattern for the overlapped 1 deg and 1.5 deg beams is given in Figure 17. As seen, the main lobe is now elongated, as required by the two beams. The crossover loss with respect to the two individual main-beam levels is thus minimal. In this way, the coverage area can be enlarged without compromising the quality of service near the boundary regions of the two overlapping footprints. When many more such tightly merged beams are generated, arbitrarily shaped and sized footprints can be realized. This is highly attractive for modern day mobile satellite communications, in which the users are so mobile that the demanded service area changes all the time and never stays constant. As mentioned earlier, these beams are electronically created and controlled by high-speed digital devices. Hence, the switching and steering of beams are rapid, agile and dynamic, making FPA-fed reflectors well suited for meeting the demands of highly mobile users. Last, the contour gain plot for a pair of widely separated beams is given in Figure 18, for 0.5 deg and 2 deg beams. Two distinctively strong spot beams toward the desired directions can be clearly seen. Satellite links can thus be established for regions that are geographically far away from each other, thus serving as a very essential feature of satellite applications. There is a good example in the case of continental USA and the islands of Hawaii. If a satellite is due south in continental U.S., the link with Hawaii will entail beams that are widely separated from the continental ones. The U.S. satellites must serve both areas, and they usually encounter problems in covering Hawaii with conventional satellites. The use of FPA-fed reflectors could potentially be an excellent solution to this challenge.
 The distributions of the copolar and cross polar E field excitation (magnitude) coefficients of the FPAs for a couple of overlapping beam pairs (0.5 deg with 2 deg, and 1 deg with 2 deg) are given in Figure 19. Note that we have intentionally selected beam pairs which are widely separated, so as to illustrate the two sets of overlapping focal plane field patterns more distinctively. As indeed observed, the superposed focal plane field patterns feature two peaks in each copolar distribution.
 When the FPA is synthesized for generating simultaneously overlapping beams, each resultant individual beam has its own aperture efficiency calculated with respect to its tilted elliptical aperture. Table 2 presents such individual total efficiencies of beams for various combinations of overlapping pairs. Table 2 is to be read in the following way. Excluding the topmost row and leftmost column containing beam angle identifiers, the numerical value situated at the pth row and qth column is the total efficiency (in dB) of the beam radiated toward the angle labeled by the pth angle down the leftmost column, in the presence of an overlapped beam radiated toward the angle indicated by the qth angle across the topmost row. The values located along the diagonal are the total efficiencies of the respective individual beams existing in solitude, i.e., without any accompanying beams. As can be observed, with respect to the solitary beam, a drop in the total efficiency by approximately 3dB is generally sustained when an overlapping beam is added. This is as expected of the usual loss in gain with beam broadening. There are, however, peculiar cases whereby the efficiency of one of the two beams is notably higher than the other, thus indicating that a larger proportion of the total radiated power is contained within this stronger beam, instead of being shared out in a more balanced manner for the other majority of instances (with approximate half-power loss for both beams). Two such unusual cases identified from Table 2 would be the stronger beam of 1.5 deg in the presence of a 0.5 deg beam, and the 2 deg beam accompanied by a weaker 1 deg beam.
Table 2. Total Efficiency for Various Pairs of Overlapping Beamsa
Beam Angle (deg)
Units are in dB.
 The demands of modern mobile satellite communications call for high-speed, adaptive, and reconfigurable multiple beams for achieving rapidly varying contoured footprints that can adapt quickly and dynamically to the radio environment. This work has investigated the potentials of FPA-fed reflectors in meeting these requirements. Toward this end, the theoretical approach for treating off-axis beams of parabolic reflectors fed by offset FPA feeds has been formulated. This constitutes the core solution for treating multiple beams, by linear superposition of individual beams. Physical optics (PO) integration of the electric currents over the reflector surface is used to synthesize the focal plane fields arising from any off-axis incident plane wave. For beams radiated along such off-axis directions on transmit mode, the feed-radiated fields reradiated by the reflector surface are integrated over a tilted elliptical aperture for calculations of the subefficiencies. These fields are obtained by projecting the usual focal plane fields onto the tilted aperture using geometrical optics (GO). Results have shown that the total efficiency of off-axis beams radiated by offset FPA-fed parabolic reflectors falls with increasing beam angle, but rises with increasing number of FPA elements. The benefit of adding FPA elements diminishes though, and a good tradeoff subarray population appears to be around 13 × 13. The loss in directivity with increasing beam angle that is typically suffered by single-element feeds can be averted with offset FPA feeds, so long as a fairly decent number of FPA elements are used, apparently 7 × 7 and higher as suggested by our investigations. Results for closely overlapping as well as widely separated beam pairs have also been presented. It is shown that the two or any other larger number of simultaneously radiating beams can indeed be achieved using FPA feeds. The usual loss in efficiency of individual beams with beam broadening of overlapped beams is demonstrated.
 Therefore, the studies conducted in this paper have successfully presented the prospects of FPA-fed reflectors in achieving adaptive multiple beams. The important characterizing factor of the FPA decoupling efficiency has been conveyed. The illustration of the tilted aperture by GO ray projection and the emphasis of its relevance to off-axis beams are also clearly presented. In addition, the performance of simultaneous beams has been investigated and the loss in efficiency is quantified. We have presented comprehensive sets of results, showing how the various subefficiencies and radiation patterns vary with beam angles and subarray populations, thus providing good insights into the effects of these parameters. In this way, the potentials of FPAs as reflector feeds for achieving arbitrarily shaped and sized beams have been thoroughly characterized.
 The research for this work was supported by the Natural Sciences and Engineering Research Council of Canada and the Department of Electrical and Computer Engineering of the University of Manitoba.