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Electromagnetic Radiation, Attenuation, and. Scattering Using Self-Adaptive Material Systems. Luk R. Arnaut, Senior Member, IEEE. Abstract—Adaptive control ...
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 7, JULY 2003

Adaptive Control and Optimization of Electromagnetic Radiation, Attenuation, and Scattering Using Self-Adaptive Material Systems Luk R. Arnaut, Senior Member, IEEE

Abstract—Adaptive control of radiation and scattering by integrated antenna arrays in the long-wavelength limit is investigated. A hybrid active/passive system with embedded control logic is considered and analyzed as a synthetic composite medium with adaptive constitutive parameters. Optimal control techniques yield the transfer function of the controller for a given internal or external stimulus and cost function. The method is applied to the minimization of radiated or scattered power at large distance. A limiting condition for omnidirectional power reduction is derived. A chequered planar array of interleaved subarrays for wave conditioning in reflection and transmission is analyzed in detail. Its dyadic Green function, impedance, equivalent permittivity, and reflection, transmission and radiation coefficients for general transfer functions are calculated. For random configurations, the effective permittivity and frequency response of the reflection and transmission by a free-standing slab of the self-adaptive medium are analyzed. The results may find application in the design and operation of electromagnetic smart materials. Index Terms—Adaptive control, effective medium, homogenization, intelligent materials, phased arrays, self-adaptive material systems, spectral-domain array Green function.

I. INTRODUCTION

I

N THE quest for more versatility and improved performance of electromagnetic (EM) materials, important developments in passive micro-structured composite materials and surfaces continue to arise [1]–[7]. With advances in adaptive antennas, micro-electronics, control systems, signal processing, nanotechnology and computer architectures, the prospect of an active synthetic engineered material on a single substrate becomes feasible. Sensors, T/R modules, power hybrids, beamformers, micro-actuators, etc., may eventually be integrated with control logic into a single system, presenting itself as a self-adaptive material system (SAM) to incident waves. In this way, the EM response of the material is no longer restricted by its physico-chemical properties. The integration of controllers, sensors and actuators yields a single system with a manually or automatically adjustable response for the scattering, absorption or transformation of incident EM waves, and which can be treated as an effective medium. It holds the prospect of realizing the ultimate tailored or multifunctional material, Manuscript received June 6, 2000; revised May 17, 2002. This work was supported by the NPL 1999–2001 Strategic Research Programme under Contract 9SRPE040. The author is with the National Physical Laboratory, Center for Electromagnetic and Time Metrology, Middlesex TW11 0LW, U.K. (e-mail: luk.arnaut@ npl.co.uk). Digital Object Identifier 10.1109/TAP.2003.813617

perhaps even with different properties to different simultaneous observers. The specified transfer function of the system thus becomes an integral part of the constitutive properties. The term “material system” emphasizes the embedded control logic and the (self-)adaptive character in a design where all characteristic dimensions are small relative to the wavelength. The aim is of course to implement response functions that are expensive, difficult or impossible to achieve otherwise. Conventional passive materials are characterized by a negawithin tive net energy density of the material (lossless or disany arbitrarily small volume sipating). They may be regarded as homogenized collections of microscopic receiving and reradiating (scattering) antennas which are electrically small and densely packed. For active main an infinitesimal volume , i.e., part or all terials, of the medium generates EM energy. By combining active and passive properties into a single entity, SAMs offer the prospect of making their constitutive properties dependent on the incident wave itself. Thus, a fundamentally nonlinear material is obtained. These concepts have also been referred to as “smart” or even “intelligent” materials [8]–[10]. The latter term refers to a SAM which adapts itself on the basis of the EM characteristics of the incident wave to achieve a user-specified response (the so-called object function). The detailed manner how the objective is met is in this case left to the internal system management itself. As will be shown, a SAM can be realized by interleaving an array of receiving (Rx) passive antennas (sensors) with one of transmitting (Tx) active antennas (actuators), using control logic to interconnect and intraconnect both subarrays. Although the techniques are akin to those used with frequency-selective surfaces, phased arrays and smart antennas [11], [12], a SAM distinguishes itself in a number of ways. a) The control of the Tx by the Rx is the main focus and is fundamental to the operation of the SAM, unlike adaptive arrays which are usually either Rx or Tx and externally controlled by the operator. Hence, the cost functions and optimization criteria are usually different from those in user-adaptive arrays. b) The array elements are, by definition, electrically small and operate out-of-band. c) Both the Tx and Rx are adaptive, because the SAM is to be perceived as “smart” for arbitrary directions of incidence and radiation for energy and polarization. By contrast, Rx adaptive arrays discriminate between desired, interfering and noise signals.

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d) The transformation of waves, rather than their simple detection or generation, is the primary goal. e) Parasitic scattering off both arrays is to be taken into account. f) Analysis of near-field and mutual coupling effects is of key importance. g) The effect of the host medium (antenna support) on the Rx and Tx signal is inherent to efficient SAM operation. The “smartness” of a system has been defined as its ability to reduce its local entropy, i.e., the information contents of the system [8, Sec. 1.3]. Interpreted in an EM context, this means that information about EM characteristics of a simple source or scatterer can be obtained, in principle, by analyzing its radiation or scattering at a distance for a known excitation field. By changing these characteristics and appearance through adaptive control, for a given excitation, in a manner that is unknown to an observer, some or all information about the source or scatterer is lost. In case of perfect absorption or total destructive interference, the presence of the source may even be totally obscured, in which case the entropy of the system becomes minimal. A further characteristic of any smart material is that it cannot be a single, homogeneous material, for the latter would respond to its environment without reducing its own information contents [8]. Hence smart materials are necessarily hybrid systems (mixtures) of electromagnetically different components. This paper focuses on one particular but fundamental aspect of control of radiation, viz., the minimization of the radiated power based on destructive interference between radiating and scattering sources. This may be conceived as the control of artificial active shadows for arbitrary directions of observation. Since conventional passive shadows occur owing to (partial or complete) destructive interference between the incident and forward scattered waves, such shadows can only exist in the forward half-space, as defined by the propagation direction of the incident wave. Active shadows, however, are formed by nulls in antenna radiation patterns for the radiated though not for the parasitically scattered field. The direction and width of active shadows can be controlled via destructive interference using a controlled radiating source. Similarly, active stimulated emission can be realized using constructive interference and is not limited to backscattering directions. The analysis, design and synthesis of SAMs is a multidisciplinary and synergetic task, embracing such diverse fields as electromagnetics, microwave engineering, network topology, physics, material science, antenna theory and engineering, control system theory, digital electronics, signal processing, etc. Only the electromagnetics and some of the antenna and control aspects are discussed here. These define the goals for the other enabling technologies, although the state-of-the-art of the latter ultimately defines the limitations on achievable performance of the implemented system. II. ADAPTIVE CONTROL OF RADIATION COUPLED SOURCES

BY

PAIR

OF

A. Introduction Fundamental to this study is the response of a hybrid pair of electrically short, coupled dipoles. Either an incident wave ex-

(a)

(b) Fig. 1. (a) Adaptive control of radiation from primary (master) source s by secondary (slave) source a, with relative source spacing d and observation point at distance r and (b) adaptive control of scattering using sensor/actuator pair.

cites a Rx antenna (sensor ) and scatters part of the incident field, or a primary (master) source generates radiated power. The impressed or induced current or voltage in is then used to drive a Tx antenna (slave source, actuator ) by impressing a current or voltage to yield the desired overall (combined) reand , sponse. Consider two sources of respective volumes apart [Fig. (1a)]. For general electric spaced a distance and , the radiated fields and magnetic source currents follow from the Green six-dyadics for both sources d (1) The coupling between and will be explicitly taken into account through a feedback mechanism, instead of via computation on the basis of the relative distance and orientation of both sources [13] which would require a dual sensing/actuating func-

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tion for . Mutual coupling is of paramount importance in such hybrid systems. At sufficiently large distances of observation, where the cost function is evaluated, both sources can be represented by their multipole moments. As will be demonstrated, adaptive source control is most efficient (i.e., the radiation by is maximally interfered with by ) at relatively low frequencies, in which case dipole modeling of the source currents usually yields a sufficiently accurate equivalent representation. For radiation by infinitesimal dipole elements

where the superscripts and denote transposition and complex conjugation, respectively, is difficult to express in the form of a single integral. However, for dipole radiation (6)

with for , symmetric tryadic

is defined by

,

and

with all other elements being zero and [14], [15]. The use of enables a cross product to be converted into a bilinear form of single-dot products, e.g., , which simplifies the minimization procedure , it is easier to (cf. infra). Instead of directly minimizing minimize , i.e.,

(3)

(7)

(2) for electric and magnetic dipole moments

or . Here, the Levi-Cività anti-

, where [14]

(4) and . We shall further denote . The , present analysis is limited to simple electric sources ( ). The extension to higher-order multipole moments requires specification of multipole Green polyadics [14] which belongs to the null space of some become important if . The control and optimization procedure can be formulated in a very general way, by defining a cost function for an EM quantity of interest, which is then evaluated at a specified location and distance of observation or across a spatial region. In one important case the cost function is formulated for the radiated power at a given field point or through a spherical surface at a pair. Other optimization criteria may large distance from the be applied (e.g., field uniformity, transformation of polarization, etc.), which can often also be expressed in terms of an energy functional. In fact, the initial stage in the synthesis of some smart function is the annihilation of the natural (passive) response of the material, so that the minimization of power serves as a fundamental first step in designing a SAM.

With (2)–(4), the explicit expressions for the

and

are

with

B. Single Observation Point 1) General Case: The minimization of the radiated power may be required at a single location at arbitrary distance from the sources. In general, the complex Poynting vector for the instantaneous power density, viz.:

(8)

(9) where

and

, in and are Cartesian unit vectors. For arbitrary which and given , (7) is a quadrilinear form in the orientation of . Local and absolute excontrol variable are reached for the roots of . tremes of 2) Special Case: Far-field Approximation: Consider a location of observation in the far field of two sources with coplanar and at respective distances and . dipole moments , , With the far-field approximations , , we obtain

,

d d

(5)

(10)

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(11)

(12)

(13) and . The oriented angles with are measured from to [Fig. 1(a)]. The reference direction is along the line connecting the centres of both dipoles and is oriented from to . With (5), the real Poynting vector reads

H

(14)

(15) , (15) is a bilinear form in For arbitrary is specified as minimum provided that

. Hence,

is

Fig. 2. Magnitude of transfer function direction of observation, as a function of 

(in dB) optimized for single and  , with kd = 0:0189.

in the null direction of the radiation pattern of then cannot influence the radiation by in that direction, if is to remain finite. The other extreme situation occurs when or the observation is in the null direction of , i.e., . In this case, so that any source control for the radiated power density is then obsolete. Although the optimum and , it is control is achievable for arbitrary orientations convenient to choose them as parallel

(16)

(18)

which is obtained by calculating the roots of the first-order with respect to and individually. This derivatives of and can be varied independently. Thus, presumes that to the transfer function of the optimum controller linking follows as:

Equation (18) defines two anti-parallel complex vectors, corresponding to two identical polarization ellipses whose phasors [16, Ch. 1]. Fig. 2 shows are separated in phase by as a function of and for . The is near 180 deg for all angles of orientation. As phase of for and for . expected, , as obThe minimum cumulative radiated power served in the direction of observation , is obtained by substituting (16) into (14). This power is zero regardless of the , , and , although the required to values of achieve this minimum does depend on each one of these quantities. Thus, perfect active field annihilation (zero total radiated power) is possible in the far field in an arbitrary single direction. Compared with passive absorbers, the key difference is that now a variable and arbitrarily large suppression of the power radiated arbitrarily by the primary source can be achieved by taking small. From (16), it follows that if the direction of observation toward broadside then the phase of moves away from relative to decreases from to . The amplitude is greater or smaller than depending on whether is smaller or greater than . This is again due to the and are small. inefficient radiation by the dipoles if

(17) This can be implemented using a two-stage serial or parallel , PID controller with or for a low-pass or high-pass filter where . implementation, respectively, and can be conceived as a bilinear form By the same token, for arbitrary and is minimized for in the variable . With (16) this yields an identity. Thus, both optimizations lead because depends to the same specification for and depends on . It is emphasized that the on , optimization in consecutive steps was possible because and can be varied independently. From (16) it follows that no stable optimum controller can be realized for or if nor , for the bilinear form is then no longer positive definite. Physically, if the observation point is situated

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For parallel dipoles, both sources combine to a quadrupole in , i.e. . the limit C. Total Integrated Radiated Power The reduction of the combined radiated power in one arbitrary direction does not necessarily imply a reduction of the total integrated power radiated through a spherical surface (or part thereof) at large distance, but can be achieved by choosing the appropriate control function. Consider the 2-D problem of a circular boundary located in the plane of the two dipoles and with radius [Fig. 1(a)]. The centred around total power flux density crossing this boundary is obtained by integrating (14)

where is the density of the power radiated . For , by the uncontrolled source , i.e., for (22) can be expanded using as

(24) Likewise, (24) reduces for

to

(19) with , i.e.: (20) sinc (25) For a pair of collinear parallel dipoles ( (21) and . Using the 0th- and 1st-order spherical Bessel functions of the , i.e., sinc , first kind , the optimum controller which min, imizes the total integrated power, viz. has the transfer function where we have used

or ) (26) (27)

whereas for strictly parallel dipoles, perpendicular to their connecting line

(28)

(22) (29) Hence, for parallel sources, the optimum control source radiates a spherical dipole mode and a spherical quadrupole mode, . The associated whereby the latter varies as is

(23)

, these correspond to collinear and doublet In the limit quadrupoles, respectively. Thus, to leading order, a quadratic deon is found in both cases. For pendence of arbitrary , the collinear configuration yields smaller levels of radiated power than the doublet (although the opposite situation holds for the directivity of arrays having collinear and strictly parallel interconnections; cf. Section VI). This is because the doublet, unlike the collinear pair, exhibits an additional magnetic dipole moment of the same order of magnitude as the electric quadrupole moment. Practically, this implies that when using arrays of triple dipole sensors and actuators, a lower overall radiated power is obtained if each sensor dipole is made to control the collinear actuator dipole in the adjacent triple dipole sensor which is located in the direction of these dipoles

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H

Fig. 4. Optimum transfer function for total integrated radiated power as a function of  and  with kd = 0:0189.

Fig. 5. Normalized density of total radiated power associated with H Fig. 4.

(b) Fig. 3. (a) Collinear versus and (b) strictly parallel element interconnections.

(i.e., the respective dipoles of a given triple dipole sensor controls an actuator dipole in three different triple dipole actuators), whereas controlling all dipoles of a triple dipole sensor by one single triple dipole actuator only results in suboptimal control of total radiated power (Fig. 3). Consequently, a 3-D configuration offers the possibility of better performance over a planar array. , the power generated by is increasingly conAs verted from radiated into reactive power stored in the near field or, depending on the input impedance and (mis)match, absorbed acts by the SAM support system (cf. Section VI), whence as a sink of energy for . Proper consideration needs to be given therefore to the power handling and heat dissipation, together with the possible use of matching networks. given by (22) for Fig. 4 shows . It is seen that for parallel or anti-parallel dipoles, the actuator should be steered or in-phase , in anti-phase respectively. On a linear scale, small deviations from (anti-)parallelism are seen to be not critical to the optimal tuning , making the control in this case robust. For of

of

orthogonal dipoles , the controller be. The tuning of comes a pure phase shifter is now maximally sensitive to deviations from dipole is a weak function of orthogonality. Generally, . For , its value is relatively small. Fig. 5 as defined by (24). The superishows ority of parallel and anti-parallel dipoles, and the worst case of perpendicular dipoles are noticed. Collinear dipoles are marginally superior over a strictly parallel pair. The (near-) are stable with respect to misalignment minima at for a few relative orientations. errors. Fig. 6 shows , most although not all cases require the controller For increases, the controller to steer in anti-phase. When gradually switches off in the mean because, for the chosen criterion, active control becomes increasingly ineffective. Moreover, the strong oscillations of for but relatively flat characteristics outside this range indicate that the tuning accuracy is most critical at intermediate frequencies. Fig. 7 shows the associated . For , collinear parallel dipoles allow for a threefold power reduction ( 4.77 dB) compared to strictly parallel dipoles. The steady reduction with decreasing demonstrates the advantage of continued miniaturization, in spite of its increasing demands put on power handling. Rotating both dipoles from a collinear to a strictly planar configuration , making yields increasingly larger oscillations for collinear interconnections a fortiori the preferred choice. Other

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(a)

orientations yield almost no reduction of the total radiated power, as expected. In summary, adaptive source control is most efficient and its efficiency decreases with increasing when . If both dipoles are electrically short ( , ), then the general transfer function for controlling the acfollows from tual currents , where is the current ammay cause instability, making plification factor. A large distributed control and amplification schemes recommended. In practice, miniaturization is limited by the fundamental radiation inefficiency of small antennas: for an antenna contained , the minimum within a sphere of radius where quality factor is [17]. For a short thin dipole of length and wire radius , , but is lower-limited by the power handling capacity of the individual element or array. D. Total Radiated Power in Nonoptimization Directions A reduction of radiated power in a chosen direction should preferably not increase the power radiated in other directions. To derive a criterion that guarantees for such an increase not to occur for any direction different from the optimization direction, at an arbitrary farwe calculate for a primary source whose radiation field location is being controlled for the single direction

(30) Upon substituting (16) we find (b)

H

(31)

Fig. 6. Optimum transfer function for total integrated radiated power as a function of kd for some relative source orientations specified by ( ,  ).

(32) (33) not to exceed , where For the latter represents the power density at in the direction solely due to with no adaptive control applied, we must have that

(34)

Fig. 7. Normalized density of total radiated power associated with H of Fig. 6.

which defines a condition for omnidirectional power reduction. and , (35) poses an upper limit on the spacing: For given . For

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Fig. 9 W =W as a function of  and  for parallel dipoles (  ) for kd = 5=4 (top) and kd = =8 (bottom).

Fig. 8. Maximum values of kd for achieving omnidirectional reduction of radiated power.

parallel dipoles, this limit is independent of the relative direction of : in this case

=

III. MULTIPLE SOURCES A. Arbitrary Configuration primary sources and control Consider now , with each type grouped into source vectors . sources For the far-field radiated power we form the vector bilinear form

(35) , then the maximum possible , i.e., , is smaller than . This power decreases for and is increasingly being drained via to the T/R increasing , the maxmodule, or converted to reactive power. For increases. imum ratio increases from 1 to 4 as for Fig. 8 shows the maximum permissible values given dipole orientations. Note that although an orthogonal enables the maximum spacing to be sensor dipole increased beyond the value for a parallel pair, it was found in Fig. 7 that in this case adaptive control becomes increasingly . Fig. 9 shows inefficient when for and , for parallel dipoles . , correspond to power The darker regions, where is clearly not omnidirectional. reduction which for , the maximum power always occurs for For orthogonal to . For decreasing values of , the main radiation lobe (central dark region near in the Figure) steadily broadens and eventually only remains. A corresponding analysis for near-field optimization is in general considerably more complex. Repeating the previous optimization method, mutatis mutandis, it can be shown that the minimization of the near-field power in a single direction, for the special case of isotropic sensor and actuator systems, is achieved for . For triple dipole sources, the replaced by previous analysis is to be repeated with , where and are the 90-degree rotation dyadics in azimuth and elevation, respectively. If

(36) with .. .

.. .

(37)

only differ For periodic arrangements, the elements of the by a phase factor, governed by the inter-element spacing and configuration. For power radiated in a single direction , the are given by (10)–(12) with , . Optimization is performed by computing the firstsources. For the order derivatives with respect to the vector of optimum control sources (38) In the computation of

, a critical factor is the phase

between and as difference observed in the direction . The previous results for a source , , and pair can be used upon replacing by , ,

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2) Adaptive Control for SAM Arrays: Denoting the singleelement fields or by , the total field for the actuator array is

Fig. 10. Planar chequered array of interleaved primary (radiating) sources and secondary (control) sources. Only one quarter of the array is shown.

(40) and or , respectively. The latter two expressions are related by , hence specifying either or is sufficient. Thus

(39) refers to source where a planar array.

at the reference position (0,0) in

The expression for by replacing by tively, where1

follows by and

in (41), respecand are the direction cosines with respect to the target directions (axes of the main beams of and ). The linear phase tapers and control the direction of the beam axes. They are specified by elevations and azimuths , as defined by [20]. The distances and are the interelement spacings in -and -directions, respectively. The previous far-field optimization for a source pair can be repeated here in a similar manner. For example, the far-field now reads radiated power density in a direction defined by

B. Regular Rectangular Chequered Array of Interleaved Sources elements in 1) Description: By configuring the and a spatially symmetric way, equal flexibility is achieved in scanning and beamforming for both incident and radiated waves, allowing for a reciprocal SAM. To this end, the layout can be chosen as a 2-D “chessboard” pattern (Fig. 10), in which a hexagonal “white” subarray of active sources is interleaved with a similar “black” subarray of passive sensors. Compared with rectangular lattices, such hexagonal lattices have their grating lobes maximally shifted toward higher frequencies, thus increasing the operational bandwidth. Here we consider -subarray as part of an idealized infinite planar a array, neglecting edge effects. The issue of interconnectivity between elements (“wiring harness”) and its repercussions are not addressed here. Because of the symmetric configuration, the mutual coupling [18], [19] within and between the two subarrays is , then the details of the elements’ curuniform. If then rent distribution affect this coupling. If is not significantly influenced and the SAM operates as an effective medium (quasistatic). Because of the latter inequality, and in view of the above results for a source pair, this ensures efficient control but inefficient radiation, unless element loading is applied.

(41) and is minimized for a controller defined by (42)

For small deviations off the main beam axis and the approximate expressions for sinc sinc control of the radiated power is then defined by

sinc sinc

sinc sinc

, are similar: . Optimum

(43)

order not to overload the notation, we have abbreviated  as  and as  . No confusion arises because only angles between the orientation of the dipoles p and the associated target direction r , i.e., of the same kind , are considered.



1In

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where depends on the chosen cost function, e.g. (16). times the value of The associated minimum power is for a source pair. With (16), for ,

sinc

sinc (44)

(48) . where This characterizes the SAM electric Green dyadic via d . Thus, the array becontrol is accounted for by the field coupling factor tween the Rx and Tx subarrays. For

2-, i.e., minimum planar cell , we obtain and . If we can furthermore approximate by for control of the far field, then

(49)

(45)

(50)

For a 2

The result for optimum control of a single source of radiation by a single control source is retrieved when retaining only the first term in the expressions for and (minimum linear cell). 3) Spectral-Domain Dyadic Green Functions: If the spatial in a reference distribution of the current is known, then the radiated field sensor element at can be found from the vector potential of the individual and subarrays [21], [22]. For the chequered configuration, the field scattered by the subarray is

d

d

which now defines the SAM magneto-electric Green function via d. Special cases include the control of radiation in retro-direction ), forward direction ( ) and directions of ( ) or inverted (i.e., double-negaspecular reflection ( ). For a general discussion on the tive) refraction ( and , cf., accuracy of the thus obtained e.g., [22, Secs. 4.1 and 4.13.2]. 4) SAM Impedance and Equivalent Permittivity: The (real) of the array for self-impedance the main Rx and Tx beams, assuming no grating lobes and quasistatic excitation of thin-wire and elements, is

(46) (51)

where and , in which , and are the cosines for the propagation direction of the incident wave. The lower (resp. upper) sign refers to reflected, i.e., back-scattered (resp. transmitted, i.e., forward scattered) fields in the half-space (resp. ). The field radiated by the subarray of sources differs from (47) by a phase factor , a scale (because the reference current is now ), diffactor and for the main beam, and ferent direction cosines , for the grating lobes different modal propagation directions . In quasistatic operation ( ) no with indexes surgrating lobes occur, whence only the term for . Thus, the total vives, and is field

d

(47)

and is the where array impedance of the subarray. The impedance coupling factor between both subarrays hence depends on their “look” direcand relative to . The current shape factors tions d depend on the elements’ current distribution in Rx (Tx) mode, e.g., for a long thin dipole, for a Hertz dipole or for an Abraham dipole of length and amplitude . The latter can be used as a rooftop basis function when evaluor numerically [23], for different angles of ating incidence. For nonmagnetic homogenizable SAM arrays, (52) can be for , yielding the equivformally equated to alent refractive index

(52)

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Consider two special cases, viz., retro-radiation and specular radiation, in more detail. In either case, for -directed elements, , whence

(53) For

collinear

and

,

from

(16), .

Thus, for retro-radiation, , whereas for specular . If, in addition, incidence radiation, , , , ), then we is normal ( . obtain for Hertz dipoles 5) SAM Radiation and Reflection Coefficients: In Appendix A, the reflection coefficient for passive arrays [22] is generalized to radiation and scattering by hybrid active/passive arrays, for arbitrary directions of observation in the principal planes. By expressing the excitation of the subarray in terms instead of the induced , the SAM of the incident field can be calculated based radiation2 coefficient vector , i.e., no adaptive control, then the on (49). If radiated fields are the usual reflected (i.e., scattered) fields. For , i.e., , ), we retro-radiation ( obtain from (80)

(54) In this case, the radiation matrix for the apparent reflection is

(55) For TE incidence in the principal plane ( ), at an angle of incidence

, , we obtain . For forin (56)

), the coefficients ward radiation ( . are to be replaced by A more general control strategy can be performed, using a more complex wiring harness, by subdividing the array into subarrays with full intra- and interconnectivity. Iteratively repeating this process yields a hierarchy of element interconnections. This is required, for example, for the control of multiple or inhomogeneous waves by a single SAM array. The direction of arrival and polarization per subarray can be estimated using adaptive array techniques [24] and may perhaps be facilitated by using commercially available EM six-vector sensors [25], [26]. Specific results on local array optimization are given in [27]. 2Since space waves can be radiated by SAMs in any direction, not just in specular direction, the radiation coefficient is a generalization of the notions of reflection and transmission coefficients.

H

H

Fig. 11. Optimum transfer functions and function of kd , kd with M = N = 10,  =  = =2,  ' = ' = 0, f = 10 MHz and L = L = 1 m.

as a

= 0,

C. Numerical Results as in (43) , thereby choosing (16) to dethen , whence . The condition for the avoid, must be ance of grating lobes, viz., observed. Fig. 12 shows dependencies of TE/TM reflection coefficients of an infinite chequered SAM array on the / spacing , angle of incidence , and element load impedance , all for the case of specular radiation. Strictly parallel interconnections are confirmed to give rise, on average, to slightly larger magnitudes of the apparent reflection compared to collinear arrangements. Element loading is seen to profoundly affect the obtainable minimum power. Further numerical results, also for the complex field reflection coefficients, are given in [27], [28].

Fig. 11 compares with . If fine

IV. ADAPTIVE CONTROL OF SCATTERING When the primary source is one of scattering of an externally incident wave (sensor antenna) [Fig. 1(b)], this wave is usually incident onto as well, causing additional parasitic scattering and mutual coupling which must be accounted for by . In general, consists of several contributions, viz., impressed by the controller, induced by the inciinduced by the nearby sensors into (i.e., dent wave, and / mutual coupling): . Likewise, consists of three contributions, but an impressed moment is usually absent in adaptive control of scattering, whence . For and , the additional contributions of higher-order multiple scattering between and are often comparatively small. They can be accounted for by specifying a mutual impedance [18] or by defining modified dipole moments [13], but in general only the first-order scatin SAMs, tering needs to be considered. Since is usually negligible, but can be the contribution of important.

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(a)

(b)

(c)

Fig. 12. TE/TM reflection coefficients for specular radiation off chequered sensor/actuator array, for collinear and parallel interconnections: (a) as a function of = R ), (b) as a function of incidence angle  for 0  < =2, kd = 0:5, Z = R =2, and (c) spacing kd for 0:1 kd 100, Z = 0,  = 0 (R as a function of element load impedance Z for 10 Z 10 , kd = 0:5,  = 0 (R = R ).











The analysis is again confined to the 2-D case. For a moment, . A plane wave characterized we ignore the contribution of with respect to . The by ( , , ) is incident at an angle and induced differ by a phase factor3 and a radiating relative orientation (polarization mismatch) factor. Observation , i.e., pointing toward in the direction the source of the incident plane wave, results in a phase factor . The polarization mismatch follows from projecting , the unit vector along the direction of , specified by as defined by ( onto the direction or ). Thus, and , yielding

In the far field, the second and fourth terms in (60) vanish when and are blind to one another by proper choice of their mutual orientations (nonoverlapping radiation patterns; ), which we shall further assume. Then, (60) simplifies to

(60) Thus, the actuator in effect controls two “sources” of scattering, viz., the scattering by and the scattering by relative to . Alternatively, (61) can be written as

(56) and by replacing with and , Now consider respectively, insofar as the dipole approximation remains valid for calculating the mutual coupling. The propagation in the diis now governed by the radiation from the nearby rection of or . For dipoles of length (57) (58) is the dyadic impedance of . where , the optimum control which acUsing (16) with is counts for scattering and defined by

(61) signifying an actively controlled perturbed scatterer , whose nonlocal perturbation is caused by the presence of the nearby . The associated minimum radiated power in the direction is again zero. Two special cases are (a) zero retro-reflection ) (

(62) which for parallel dipoles further reduces to , i.e., a purely real ; (b) zero ), for which specular reflection ( (59) 3In general, the wavenumbers of the incident and conditioned wave may be different (k = k , e.g., for Raman radiation), but this renders the notion of electrical distance between a and s ambiguous and in effect a does then not interfere with the incident radiation.

6

(63) If now, in addition, the moments are parallel then .

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So far, we have assumed and to be elements at different spatial locations, but this is not strictly necessary. A dual element could be devised by operating a single antenna alternatingly between Rx and Tx mode (Fig. 14), each operating during one half cycle. In this case the phase delay is a due to an elapsed . time, rather than because of a spatial offset for zero cumulaIn such an implementation, the direction tive radiated power cannot be steered. However, the reduction in integrated total radiated power is spatially uniform along the radiation pattern of , due to its complete overlap with that of . Thus, a dual implementation has the advantage of maximum , but requires efficiency of adaptive control because of higher speeds defined by the rate of change of the signal at (field envelope) to maintain control at all times. V. RANDOM SAMs

H

Fig. 13 Optimum control transfer function for adaptive control of scattering as a function of  and  for parallel dipoles ( = ) and kd = 5=4.

The

can also be specified for the source fields. For , where is the local field for which consists of the incident field and a depolarization field, and , where is the impressed control field, we obtain from (61)

A. Microscopic Constitutive Modeling of Source Pair pair We first calculate the dipolarizability of an [29]. The induced electric dipole moment is defined by d . With (75), and

as defined in Section III-B-4

(65)

(64)

defined by (61) Fig. 13 shows the optimum ) with . Both its for collinear dipoles ( in-phase and quadrature components vary maximally, i.e., between 2 and 2, thus covering double the range of for a single radiating source (cf. Fig. 4), because the controller must now account for scattering by both and . At broad, the scattered power is maximum side ( . The adaptive control is then maximally efand , the controller is closest to ficient. For ): in this case, either no scatswitching itself off ( or ; ) tered power is generated by ( or no scattered field reaches the detector in the blind direction ; or ). The case of zero specular re( and are flection need not be re-examined, because for true sources. Even in the general case proportional to and , there are no extra terms for interaction between . Upon replacing by , all rewhence sults for the adaptive control of radiating sources thus remain applicable. Equation (61) further shows that information on is required to implement the controller, which requires a direction finding or direction synthesizing capability for or , respectively, using a single vector sensor or multiple scalar sensors (cf. Section III-B-5).

For example, for randomly oriented pairs of short iden, and tical dipoles, i.e., , this yields an average dipolariz, where ability is the equivalent capacitance of the dipole element. B. Effective Permittivity Since a SAM was found to operate most efficiently in the qua), a description in the framework sistatic regime ( of effective medium theory (EMT) is possible. EMT applies to random configurations in general, and to regular dipole arrays (“artificial crystals”) in particular, provided there is a sufficient number of particles in every direction. pair as a single SAM particle When considering each may of specific orientation and dipolarizability (66), be calculated directly from the Lorenz—Lorentz formula. However, this formulation is known to be sufficiently accurate only for relatively low particle densities. On the other hand, the two-phase asymmetric Bruggeman formula [29], [30] appears to be well suited for high-density mixtures of conducting particles [31]—which are typical for efficient implementations of SAMs—but requires the individual particle permittivities to be known. We here extend the latter formulation to multiphase mixtures. Consider a SAM of volume as a composite of individual passive and active particles with random orientations, with negligible scattering losses in the quasistatic regime. The effective properties can then be calculated using a three-phase mixture

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(a)

(b) Fig. 14. Adaptive control of (a) source radiation and (b) scattering using separated channels vs. single channel. The controller, interconnections and support system present a load impedance Z to the elements s and a.

formula. A mixture of active and passive inclusions inside a dican then be characterized reelectric matrix of permittivity cursively, by adding phase after phase, via [30]

polarization factors, and is the apparent permit, tivity [29, Sec. 3.4]. With and (quasicrystalline approximation)

(68) (69)

(66)

, 2, where , , and . for In order to apply (67), the relationship between the particle’s dipolarizability and permittivity is needed. Consider therefore spherically symmetric and of diameter as particles with ). The principal comisotropic dipolarizabilities ( ponents of their relative permittivity follow from as (67) valid for

, i.e., is the ambient wavelength,

, where are the de-

. which can then be used with (67) to yield and as a function of the Fig. 15 shows , for a mixture of volume loading fraction particles embedded within a Kapton maspherical isotropic m , trix. In this example, m , and . Both the Maxwell Garnett (MG) and asymmetric Bruggeman (AB) formulations are seen to give nearly identical results. For reference, the effective permittivity for the same SAM in its ) is also shown. The effective permitpassive state ( or 1 at to 1 or tivity changes from at . Fig. 16 shows the resulting reflection coefficient for a single free-standing slab of effective SAM, for normal incidence, compared to the corresponding reflection for the passive SAM and host medium only. This confirms the result of Fig. 12 spacings, are required that high particle densities, i.e., small to obtain a substantial reduction in reflection. For example, a reduction of reflection by 20 dB compared to the value at requires . The reflection approaches the theoretical because of the restriction to a dipole model of null if each particle. In reality, the quadrupole and higher-order multipole moments result in some nonzero nonspecular reflection for nonzero , which requires further analysis [32].

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(a)

!

Fig. 16. Reflection coefficient as a function of loading fraction of sensor/ actuator pairs for kd 0, for passive and active SAM, based on Maxwell Garnett (MG) and asymmetric Bruggeman (AB) mixing formulas.

The forward scattering is obtained from the projection of the vector scattering amplitude, i.e., for parallel (TM) orientation and for perpendicular (TE) polarization. For a linear dipole, and . Refraction by array is then governed by the relative refractive the , where is the index elements. For actuators, volume density of . Thus, for the pairs

(b)

(71)

!

Fig. 15. Real and imaginary part of complex effective permittivity as a 0 function of volume loading fraction v of sensor/actuator pairs for kd relative to host medium or free space, for passive and active SAM, based on Maxwell Garnett (MG) and asymmetric Bruggeman (AB) mixing formulas.

C. Scattering Matrix, Reflection and Transmission Coefficients for Homogenized SAM Slab A scattering-matrix formalism can be used to calculate reflection and transmission coefficients and their frequency dependence for a free-standing effective SAM slab. At long wavepair is negligible comlengths, the scattering by an efficient pared to its absorption in the overall attenuation (extinction) budget. The formalism is valid because we consider complex scattering coefficients, provided that each particle can be assumed to be excited by the external incident wave only. The TE/TM scattering by a single passive element is characterized by its scattering matrix [33, Sec. 3.4]

(70)

transmitted across a composite slab of The field containing SAM pairs follows from the thickness , summation of all scattered fields. For sufficiently large this can be replaced in EMT by an integration, leading . The reto , evaluated at the air/SAM interface, is flected field . Thus, the field transmission coefficient and field reflection coefficient follow as:

(72)

(73)

ARNAUT: ADAPTIVE CONTROL AND OPTIMIZATION

The power absorption rate is then and the energy extinction cross section

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is

.

VI. IMPLEMENTATION ASPECTS OF SAMs Cost functions involving a countable number of discrete spatial directions for optimization of the total radiation, as in Section II, require a high directivity of the arrays. Since , the SAM array the 3-dB beamwidth is proportional to should therefore have relatively large characteristic lengths , typically as a minimum. By contrast, adaptive control across solid spherical angles (including hemispheres) requires only modest directivity and can be achieved by SAMs that are relatively compact compared to the wavelength. , result in wide-band Small interelement spacings SAMs. Capacitive loading of the array elements can be used to reduce their size and spacing. Whereas efficient and robust operation and stability of resonance when varying the scan , , the angle and polarization direction require array bandwidth is only limited by the onset of grating lobes, , . Since the latter condition is less stringent, viz., the economy afforded by array thinning affects the efficiency of self-adaptivity and the field-of-view more strongly than its bandwidth. The sensitivity of SAMs to incident TE/TM polarization may be reduced and the radiation efficiency improved by using inductive loading of array elements or different element geometries, e.g., loops [22, Ch. 2]. However, the resulting magnetic or multipole moments make adaptive control and performance of a SAM array more complex. Alternatively, dielectric sub- or superstrates can reduce polarization sensitivity and stabilize the resonance frequency with respect to scan angle [36]. Since normal operation conditions of SAMs are quasistatic, i.e., out-of-band, one should account for large impedance mismatches in the SAM support system. The latter comprises controllers, T/R modules, beamformers, feed and combiner networks, phase shifters, mixers, compensation networks [38], etc. This requires special attention to the power handling capabilities of the individual subsystems. Quarter-wave transformer slabs may be used to match the SAM support system to the array impedance. Semiconductor substrates can provide nonlinear or active element loads, e.g., grids of PIN diodes or FETs (for control of Tx amplitude) or varactors (for additional control of Rx phase) to control the impedance of the active subarray [39], [40]. The SAM support system, signal processors and hybrids cause additional parasitic scattering when left unshielded. This can be remedied by placing a ground plane in between the array (Fig. 14). This, in effect, control hardware and the loads the array inductively. The above results still apply in to account for the image general, but now with a different sources. High densities of array elements necessitate miniaturization. Due to fundamental limitations on the radiation efficiency of

electrically small antennas [17] and their maximum available or dissipative power, this restricts the distance from the array at which wave control can be achieved. Expressions for the gain of a linear array of strictly parallel or collinear short dipoles (e.g., [34, (2.56) and (2.58)]) show that parallel interconnections yield a marginally higher directivity. Generally, increasing the directivity of active arrays requires a larger number of source elements, because large source spacings would compromise the effectiveness of the interleaved arrays. This creates superdirective arrays, allowing for active wave control in a single receiving and transmitting direction but at the expense of narrowing the array bandwidth. Making the elements more resistive increases the bandwidth, but decreases the radiation resistance and, hence, the efficiency. For each one of the interleaved planar subarrays , the gain is d d . In a simple design, the sensor and actuator functions are assigned a priori to each individual element. In a reconfigurable array, the controller allocates the Rx or Tx function to each array , based on the situation at . In this way, element at time the array organizes itself so that a minimum cost function can be maintained as time progresses. Such a feature is useful, for example, if the amplitude of the primary source fluctuates and the output power of the elements is restricted. In such a case the number of control sources needs to be changed accordingly in order to maintain the overall power level. Reconfigurable arrays are also useful for self-diagnosis and self-healing in the case of failure of one or more array elements. The signal processing techniques for SAMs can be borrowed from those for adaptive antenna systems [12]. In adaptive arrays, the antenna pattern may be steered using a superresolution beam-forming or direction-finding algorithm, e.g., based on an eigenspace decomposition (MUSIC [35], two-dimensional unitary ESPRIT [37], etc.) for fast response. The algorithms can be made robust with respect to element failure and steering errors. The difficulty of computational complexity in tracking multiple signals in real time may be reduced by using six-element electric-magnetic vector sensors which estimate locally and instantaneously [25]. Here, coupling between the electric and magnetic subsensors requires additional calibration and signal processing in dense SAM arrays. This difficulty makes individual electric or magnetic triple dipole sensors, combined with local optimization schemes [27], more attractive [24]. VII. CONCLUSION In this paper, we studied the mechanism and implementation of a self-adaptive material system, by considering adaptive control of a primary (master) radiating source or scattering sensor by a secondary juxtaposed (slave) source. The system was found to have its highest control efficiency, but lowest radiation efficiency when operating at relatively low frequencies. A source pair optimized for zero radiated power in a single direction defines an optimal control transfer function (17). For optimization of radiation across a spherical surface, the optimum controller

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(22) results in a minimum total radiated power (24), which increases quadratically with small increasing values of . The upper limit (36) for the source spacing guarantees omnidirectional reduction of radiated power when optimizing for a single direction of radiation. A planar chequered configuration of interleaved sensing and actuating arrays yields (43), with (44) and (45) as the optimum controller for radiated power in directions with small deviations off the main beam axis and the associated density of radiated power, respectively. Expressions for the electric and magneto-electric array dyadic Green functions were given based on (48) and (50). The array impedance and equivalent permittivity were expressed as a function of the optimum controller function in (52) and (54), respectively. By considering the master source to be one of scattering, the sensor/actuator coupling was accounted for. Optimum control is then governed by (60). In a constitutive model, the dipolarizability (66) of a SAM pair was expressed in terms of the controller and used with two different mixing formulas to calculate effective permittivity and reflectivity of a free-standing SAM slab. With regard to the practical implementation, the major challenges were identified as the response time of the controller to enable wave control in real time, and the required level of power radiated by the actuator array. In this paper, only one but fundamental aspect, viz., the minimization of radiated power according to a user-specified criterion, was treated in detail. This primary annihilation action is needed in order to suppress the “natural” (passive) EM reflection and transmission for a conventional material or scatterer. A chosen secondary function may then be further added, via superposition or modulation of the actuator excitation using adaptive array techniques, to create the desired response. SAMs may then find application as an enhancement over conventional materials (e.g., tunable, wide-band or adaptive RF absorbing or shielding media), by adding new capabilities to HF, RF, microwave or millimeter-wave materials (e.g., Raman scattering surfaces, nonspecular reflection, retrodirective glory, etc.) or by extending their capabilities to handle different signals, polarizations and frequencies simultaneously or in distinct manners (e.g., dual polarization or frequency, adaptive anisotropy, spectral or angular filtering, etc.). While the operation of a SAM shows some similarities with adaptive array techniques, the present paper has emphasized how Rx and Tx antennas are to be combined and controlled in order to minimize a global cost function for their overall EM effect response, including scattering. A compact constitutive description was obtained that succinctly characterizes the adaptive active effective material system as a synthetic effective medium. A distinctive feature of the SAM is the user-specified transformation of incident to radiated EM waves, rather than either detection or synthesis of waves with conventional adaptive arrays.

trol and configurational characteristics of the SAM array. We decompose in (49) in components parallel and perpendicular to the plane of incidence (cf. [22, Ch. 4]) as

(74)

where we have omitted the subscripts “ ” for simplicity. , and Here, are unit vectors parallel and perpendicular to the plane of incidence for the sensors and to the plane of secondary radiation for the actuators, respectively, is the unit vector inward normal to the SAM and is given by (52). Furthermore, array. The array impedance per array element. we allowed for a load impedance From the antenna reciprocity theorem, the antenna terminal voltage is

(75)

(76)

denotes the current shape function of the elements where when operating in Rx (Tx) mode. Defining

(77)

(78)

APPENDIX A SAM RADIATION COEFFICIENTS Based on the electric field integral and the reciprocity theorem, the field radiation can be expressed in terms of the con-

(79)

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this yields for the total radiated field generated by a plane incident wave

(80)

or correspond to TE components where the terms in with respect to the plane of incidence or the plane of secondary (actuator) radiation, respectively, whereas terms in correspond to TM components with respect to these planes. and account for possible The terms containing cross-polarization effects between incident and (re-)radiated fields. An important subclass of propagation problems consists of SAM radiation and scattering in the same plane as the plane of incidence, defined by the Snell law, i.e., and . Then, for the components of the total field

(81)

Important special cases such as specular, forward and retro-radiation are discussed in Section III-B. REFERENCES [1] D. L. Jaggard, A. R. Mickelson, and C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., vol. 18, pp. 211–216, 1979. [2] M. M. I. Saadoun and N. Engheta, “A reciprocal phase shifter using novel pseudochiral or medium,” Microw. Opt. Technol. Lett., vol. 5, pp. 184–188, 1992. [3] L. R. Arnaut, “Chirality in multidimensional space with application to characterization of multidimensional chiral and semi-chiral media,” J. Electromagn. Waves Appl., vol. 11, no. 11, pp. 1459–1482, 1997. [4] E. O. Kamenetskii, “Magnetostatically controlled bianisotropic media: a novel class of artificial magnetoelectric materials,” in Advances in Complex Electromagnetic Materials. ser. NATO ASI Series 3, A. Priou, A. Sihvola, S. A. Tretyakov, and A. Vinogradov, Eds. Dordrecht, The Netherlands: Kluwer, 1997, vol. 28. [5] J. Romeu and Y. Rahmat-Samii, “Dual band FSS with fractal elements,” Electron. Lett., vol. 35, no. 9, pp. 702–703, Apr. 1999. [6] F. Auzanneau and R. W. Ziolkowski, “Artificial composite materials consisting of nonlinearly loaded electrically small antennas: operationalamplifier-based circuits with applications to smart skins,” IEEE Trans. Antennas Propagat., vol. 47, pp. 1330–1339, Aug. 1999. [7] E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Amer. B, vol. 10, pp. 283–294, 1993.

[8] B. Culshaw, Smart Structures and Materials. Boston, MA: Artech, 1996. [9] T. Takagi, “Recent research on intelligent materials,” J. Intell. Mater. Syst. Struct., vol. 7, pp. 346–352, 1996. [10] K. L. Ford and B. Chambers, “Smart microwave absorber,” Electron. Lett., vol. 36, no. 1, pp. 50–52, Jan. 2000. [11] R. T. Compton Jr., Adaptive Antennas: Concepts and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1988. [12] L. C. Godara, “Application of antenna arrays to mobile communications—II. Beam-forming and direction-of-arrival considerations,” Proc. IEEE, vol. 85, pp. 1195–1245, Aug. 1997. [13] L. R. Arnaut and L. E. Davis, “Mutual coupling between bianisotropic particles: a theoretical study,” in Progress in Electromagnetics Research, J. A. Kong, Ed. Cambridge, MA: EMW, 1997, vol. 16, PIER 16, pp. 35–66. [14] L. R. Arnaut, “Recursive de-embedding procedure for computation of mutual coupling between bianisotropic or complex sources and scatterers,” Int. J. Electron. Commun., vol. 52, no. 1, pp. 1–8, 1998. [15] T. B. Drew, Handbook of Vector and Polyadic Analysis. New York: Reinhold, 1961. [16] I. V. Lindell, Methods for Electromagnetic Field Analysis. Oxford, U.K.: Clarendon, 1992. [17] S. Zouhdi, A. Sihvola, and M. Arsalane, Eds., Advances in Electromagnetics of Complex Media and Metamaterials. Dordrecht, The Netherlands: Kluwer, 2002, vol. 89. [18] H. E. King, “Mutual impedance of unequal length antennas in echelon,” IRE Trans. Antennas Propagat., vol. 5, no. 3, pp. 306–313, 1957. [19] H. Steyskal and J. Herd, “Mutual coupling compensation in small arrays,” IEEE Trans. Antennas Propagat., vol. 38, pp. 1971–1975, Dec. 1990. [20] R. S. Elliot, Antenna Theory and Design. Englewood Cliffs, NJ: Prentice-Hall, 1981. [21] B. A. Munk and G. A. Burrell, “Plane wave expansion for arrays of arbitrarily oriented piecewise linear elements and its application in determining the impedance of a single linear antenna in a lossy half-space,” IEEE Trans. Antennas Propagat., vol. 34, pp. 331–343, Mar. 1979. [22] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000, ch. 4. [23] A. W. Glisson and D. R. Wilton, “Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces,” IEEE Trans. Antennas Propagat., vol. 28, pp. 593–603, May 1980. [24] J. Li and R. T. Compton Jr., “Angle and polarization estimation using ESPRIT with a polarization sensitive array,” IEEE Trans. Antennas Propagat., vol. 39, pp. 1376–1383, Sept. 1991. [25] A. Nehorai and E. Paldi, “Vector-sensor array processing for electromagnetic source localization,” IEEE Trans. Signal Process., vol. 42, pp. 376–398, Feb. 1994. [26] J. Li, “Direction and polarization estimation using arrays with small loops and short dipoles,” IEEE Trans. Antennas Propagat., vol. 41, pp. 379–387, Mar. 1993. [27] L. R. Arnaut, “Adaptive control and optimization of electromagnetic radiation, attenuation and scattering using self-adaptive material systems,” NPL, Tech. Rep. CETM/RFMW/DGSS/000 505, May 2000. , “Self-adaptive material systems,” in Advances in Electromag[28] netics of Complex Media and Metamaterials. ser. NATO ASI, S. Zouhdi, A. Sihvola, and N. Arsalane, Eds. Dordrecht, NL: Kluwer, 2002, vol. 89, pp. 421–438. [29] A. H. Sihvola and I. V. Lindell, “Polarizability modeling of heterogeneous media,” in Progress in Electromagnetics Research, A. Priou, Ed. New York: Elsevier, 1992, vol. 6, PIER 6, pp. 101–151. [30] L. R. Arnaut, “Statistical characterization of complex media in random fields,” Int. J. Electron. Commun. (Archiv Elektr. Übertr.), vol. 55, no. 4, pp. 211–223, 2001. [31] W. M. Merrill, R. E. Diaz, M. M. LoRe, M. C. Squires, and N. G. Alexopoulos, “Effective medium theories of artificial materials composed of multiple sizes of spherical inclusions in a host continuum,” IEEE Trans. Antennas Propagat., vol. 47, pp. 142–148, Jan. 1999. [32] L. R. Arnaut, “Numerical multipole modeling of bianisotropic and complex composite materials,” in Proc. 13th Appl. Computat. Electromagn. Symp.. Monterey, CA, March 17–21, 1997, pp. 789–795.

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[33] C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983. [34] R. C. Hansen, Phased Array Antennas. New York: Wiley Interscience, 1998. [35] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propagat., vol. 34, pp. 276–280, Mar. 1986. [36] S.-J. Yu and J.-H. Lee, “Efficient eigenspace-based array signal processing using multiple shift-invariant subarrays,” IEEE Trans. Antennas Propagat., vol. 47, pp. 186–194, Jan. 1999. [37] M. D. Zoltowski, M. Haardt, and C. P. Matthews, “Closed-form 2-D angle estimation with rectangular arrays in element space or beamspace via unitary ESPRIT,” IEEE Trans. Signal Process., vol. 44, pp. 316–328, Feb. 1996. [38] S. I. Maslovski, A. A. Sochava, V. V. Yatsenko, S. A. Tretyakov, and L. R. Arnaut, “Wide-band active RF absorber,” NPL, Tech. Rep. CETM S74, May 2000. [39] W. W. Lam, H. Z. Chen, K. S. Stolt, C. F. Jou, N. C. Luhmann Jr., and D. B. Rutledge, “Millimeter-wave diode-grid phase shifters,” IEEE Trans. Microwave Theory Tech., vol. 36, pp. 902–907, May 1988. [40] L. B. Sjogren, H.-X. Liantz Liu, X.-H. Qin, W. Wu, E. Chung, C. W. Domier, and N. C. Luhmann Jr., “A monolithic millimeter-wave diode array beam transmittance controller,” IEEE Trans. Microwave Theory Tech., vol. 41, pp. 1782–1790, Oct. 1993.

Luk R. Arnaut (S’94–M’95–SM’01) received the B.S. degree in applied physics and electrical engineering from the University of Gent, Gent, Belgium, in 1989 and the M.Sc. and Ph.D. degrees in communication engineering and digital electronics from the University of Manchester Institute of Science and Technology (UMIST), Manchester, U.K., in 1991 and 1994, respectively. At UMIST, he was an Associate Professor of electrical engineering. During Summer 1994, he was a Post-doctoral Research Scientist at the Defence Research Agency, Farnborough, U.K., working on the fabrication and measurement of synthetic chiral composites. In 1995, he was a Visiting Scientist at the Naval Research Laboratory, Washington, DC. From 1995 to 1996, he was a Consultant to British Aerospace (Operations), Bristol, U.K., responsible for the technical management of the RUSSTECH International Programme on hydraulic ram, acoustic lasers (sasers), wave catastrophes, and complex composite materials. Since 1996, he has been a Senior Research Scientist at the National Physical Laboratory, Teddington, U.K., involved in work on dielectric resonators, structured materials, EMC, antennas, and arrays. He is principal author of over 40 refereed publications and co-author of IEC 61 000-4-21 (Joint Task Force CISPR(A)/TC77B). His current research interests include electromagnetic interaction effects, reverberation chambers, statistical electromagnetics, complex media, and image processing techniques. Dr. Arnaut received the Rayleigh Prize in 2002. He is listed in Who’s Who in the World.