## Abstract

We analyse the electromagnetic (EM) characteristics of a self-adaptive material (SAM) surface whose support system is shielded from the SAM array by a perfectly conducting ground plane. This configuration enables subarrays consisting of EM sensors and actuators to be completely separated from the control unit, active components, beamformers, power hybrids and wiring harness. The radiation characteristics, spectral-domain Green dyadic function and array impedance are obtained in closed form, based on image theory. It is shown that the performance of a previously studied free-standing implementation can be achieved with the present implementation in a more practical manner. Numerical examples are given to illustrate the analytical results. An equivalent circuit representation is developed for a single sensor/actuator pair that takes all conductive and radiated EM interactions in this pair into account. The results find application in the design and operation of fully EM ‘intelligent’ surfaces, ‘smart’ antennas, and as novel active absorbing EM boundary conditions in computational electromagnetics.

## 1. Introduction

Artificial anisotropic and periodic structured surfaces with unusual properties for use in guided-wave structures, antennas and related electromagnetic (EM) applications have continued to evolve and attract interest for several decades (cf. Karbowiak 1958; Arnaud & Pelow 1975; Sievenpiper *et al*. 1999; Munk 2000; Lindell 2002). As a further development, self-adaptive materials (SAMs) were recently proposed (Arnaut 2002, 2003) as a class of fully EM ‘smart’ or ‘intelligent’ materials (cf. Takagi (1996) for a review) and some of their basic EM properties were analysed in that study. SAMs are synthetic adjustable hybrid active/passive materials that offer the prospect of overcoming a number of fundamental limitations inherent to conventional passive natural or man-made composite media. The constitutive parameters of passive dielectric or magnetic media suffer from inevitable limitations owing to manufacturing tolerances, sensitivity to process parameters, sample size, drift (including effects of ageing, contamination, physical or chemical instability, etc.) and limited range of practically available parameter values. Furthermore, in cases where these materials show exceptional and useful EM properties, these effects often derive from scattering, i.e. second-order phenomena—e.g. half-wavelength resonance mechanisms for inclusions exhibiting a specific geometry or constitution—which are often weak or narrow-band. Therefore, one seeks to improve on this situation by synthesizing a tailored medium whose useful properties are significantly pronounced and are of leading order.

Conventional structured materials and arrays (gratings, grids) operate most efficiently within their resonance band governed by the elements' sizes and interelement spacings. By contrast, SAMs were shown by Arnaut (2003) to become increasingly efficient for decreasing frequencies of operation, thus resulting in a low-pass wide-band frequency response where they exhibit their particular characteristics. For this reason, SAMs are most accurately conceived and modelled as effective media (continua), in the long-wavelength limit, rather than as distributed discretized antenna systems. They complement passive structured materials and surfaces that are based on scattering by discrete elements which typically need to be a sizeable fraction of the wavelength in order to be sufficiently efficient, and whose modelling as an effective medium becomes less adequate beyond the quasi-static limit. SAMs enable (self-)tunable materials or surfaces to be constructed through changes of their constitutive parameters in accordance with the dynamics of an EM environment or an on-demand specification. They can be used, for example, to construct ‘dial-a-permittivity’ reference materials for use in multi-purpose applications, in calibration and instrumentation, or in time-varying configurations or circumstances of operation. SAMs also enable one to realize and exploit interesting or unusual scattering properties, including the transfer of effects that normally manifest themselves predominantly or exclusively at relatively high frequencies to lower frequencies (e.g. microwave Raman scattering). A further application of the concept is to devise a realizable active absorbing boundary condition (ABC), as a modelling tool for truncating the computational domain in EM numerical simulation methods.

In Arnaut (2003), it was assumed that the SAM support system (SSS), containing the hardware pertaining to the control logic, active circuits, beamformers, power hybrids, interconnects, etc., forms an inherent part of the SAM (internal or embedded SSS). In such an implementation, the arrays of sensors and actuators form a more or less shielding ‘shell’ that encloses the SSS in its entirety (cf. figure 1*a*). For practical applications, however, it is advantageous to implement the SSS as an external subsystem, which is then connected to the sensor-plus-actuator (*s*/*a*) array (cf. figure 1*b*). This is particularly important with regard to the power supply of the active part of the SAM or when array thinning is of interest. However, such an implementation requires the SSS to be shielded from the *s*/*a* array in order to maintain system integrity in different circumstances. The result is a device that has sufficiently simple and stable radiation characteristics and is not being compromised by electromagnetic compatibility/electromagnetic interference (EMC/EMI) issues and by EM scattering associated with the presence of the SSS.

The presence of a ground plane modifies the procedure and optimization of the adaptive control and EM characteristics of SAMs. These modifications and their effects are considered in this paper. Only purely electric implementations (*viz*. electric dipole pairs used with perfect electric conductor (PEC) shields) are considered here, although the results can be extended to include magnetic arrays and/or perfect magnetic conductor (PMC) shields, on account of duality. The analysis is limited to the case of strictly collinear dipole sources, which were shown previously (Arnaut 2003) to yield the best SAM performance, resulting in an adaptive surface of arbitrarily small thickness in principle. Again, we focus on the primary function of the SAM, *viz*. the annihilation of radiated or naturally scattered power. In the present paper, however, we shall extend the results beyond those for the low-frequency limit.

The outline of this paper is as follows. After deriving the optimum *s*/*a* controller transfer function in §2, we analyse arrayed sensors and actuators as receiving and transmitting antennae with independent reception of the signal on each element, respectively, in §3. It is clear that the introduction and design of an active controller for a specified transfer function involves considerable complexity in realizing EM self-adaptivity in practice, because of interactions between closely spaced elements, as well as the question of impedance matching and power flow between the sensor and actuator. These issues are addressed in §4. The main results are summarized in §5, where the limitations of the approach and further prospects are also discussed.

## 2. Electric SAM pair above planar PEC shield

### (a) Single direction of optimization

The radiation by a single electric dipole with moment (*α*=*s*, *a*) and located at is specified by the electric–electric and magnetic–electric Green dyadics and :(2.1)where (cf. Arnaut 1998)(2.2)(2.3)in which is the free-space wavenumber, represents a unit vector in the direction of and is the unit dyadic. A harmonic time dependence exp(j*ωt*) is assumed throughout and suppressed.

For a single *s*/*a* pair radiating in unbounded free space (two-dimensional configuration), it was shown previously (Arnaut 2003) that zero radiated power in the far field in an arbitrary single direction can be achieved, provided is driven by using an optimum controller with transfer function,(2.4)where the superscript ‘FS’ denotes a free-standing configuration. In (2.4), *θ*_{sr}, *θ*_{ar} and *θ*_{xr} represent oriented angles between the spatial direction of and the observation point, between the spatial direction of and the observation point, and between the PEC plane and the observation direction, respectively (cf. figure 1*a* in Arnaut 2003). Thus, the transfer function (2.4) defines the required impressed (secondary) current in function of an induced (primary) current .

When introducing a PEC ground plane, its effect on the radiation by can be taken into account by using image theory (figure 2). For an electric dipole at a height *h* above the plane and oriented at an elevation angle *θ*_{αx} with this plane, the effect on the dipole radiation in the half-space *z*>0 is found by replacing the plane with an image dipole with elevation angle , and centred at the mirror location with respect to the original plane, i.e. at *z*=−*h*. Choosing the origin (phase reference) as the geometrically equidistant centre for all sources *s*^{(}′^{)} and *a*^{(}′^{)}, as shown in figure 2*b*, then for radiation in the quadrant 0≤*θ*_{xr}≤*π*/2, the phase lead for radiation by *s* or the phase lag for radiation by *a* is(2.5)where the upper and lower signs correspond to *s* and *a*, respectively. This yields the phase difference between the actual sources as(2.6)which is independent of *h*. For radiation in the quadrant *π*/2≤*θ*_{xr}≤*π*, the same expressions apply except that the phase lead now becomes a phase lag, and vice versa. For the image sources, we have that , , whence the relative phase difference is preserved ().

Next, the relationship between *α* and *α*′ is required. Each source and its image are separated by a distance 2 *h* in the direction normal to the surface, whence(2.7)The phase difference between *α* and *α*′ reaches a maximum in the direction of *αα*′ (i.e. for the normal (broadside) direction *θ*_{xr}=*π*/2) and is zero in any direction perpendicular to it (i.e. for grazing (endfire) directions *θ*_{xr}=0 and *π*). Consequently, compared to a free-standing *s*/*a* pair, the phase lead or lag of the actual sources and their images can be taken into account by multiplying the source phasors by respective (space) phase factors,(2.8)where the upper and lower signs refer to *α* and *α*′, respectively. The are independent of *d* and the specific EM nature of the boundary (PEC or PMC).

Following the method in Arnaut (2003), the optimal transfer function for a dipole pair above a PEC plane can be expressed as(2.9)where(2.10)in which the corresponding coefficient for a free-standing implementation with internal SSS is(2.11)With (2.9), the radiated power in any target direction *θ*_{xr} vanishes, i.e. . For the special case where the orientations of and are parallel or perpendicular to the PEC plane, (2.9) reduces to(2.12)because in this case the image coefficients have the same value −1 or +1 for either *s* or *a*, respectively.

Figure 3 shows real and imaginary parts of *H*_{as,opt} as a function of *θ*_{xr} for controlled annihilation of radiated power in a single direction, at selected values of *kd*. For example, for *kd*=5*π*/4 the transfer function becomes purely real, *viz*. *H*_{as,opt}=+1, −1 and again +1 at *θ*_{xr}≈36.85, 90 and 143.11°, respectively. It becomes purely imaginary, *viz*. *H*_{as,opt}=−j and +j at *θ*_{xr}≈66.4 and 113.6°, respectively. For the endfire directions, . For increasing *kd*, the variability of *H*_{as,opt}(*θ*_{xr}) increases as well. The real and imaginary parts of *H*_{as,opt} show even and odd symmetry with respect to the normal direction, respectively. The associated residual radiated power (theoretically zero) in the direction of optimization, as shown in figure 4, is smallest in the normal direction. Even at 45 and 135°, where the residual power reaches a maximum, the active absorption appears to be sufficiently large for most practical purposes, in particular for application as an ABC in computational EM, which as a rule places high demands on the permissible level of residual reflection.

### (b) Integrated total radiated power

To obtain the total radiated power integrated across a hemisphere in the upper half-space (or, more generally, across a specified solid angle), we remove the PEC interface and calculate the power in this half-space as radiated by the combination of actual and image sources:(2.13)where the upper primes signify integrated power, triply underlined quantities symbolize tryadics (Arnaut 1998), and(2.14)For electric dipoles above a PEC plane, (2.14) vanishes for *αβ*=*αα*′ or *α*′*α*, as can be easily verified. Since the *F*_{α} now depends on the integration variable, the optimum controller that minimizes the integrated power cannot be obtained by mere multiplication of the for the free-standing configuration by these phase factors, unlike for the case considered in §2*a*. Instead, the integrations (2.14) must now be performed explicitly. As a result, the expression for the optimum controller now contains the *F*_{α} implicitly:(2.15)The dependencies of on *kd* and *kh* are shown in figure 5, for selected dipole separations *kh* and *kd*.

The associated normalized total integrated power, with reference to the power radiated by *s* and *s*′, is(2.16)which measures the residual radiation and, hence, quantifies SAM performance. Figure 6*a*,*b* show plots of (2.16) for collinear electric *s*/*a* pairs oriented parallel to a PEC plane, as a function of *kd* and *kh*, for selected dipole separations *kh* and *kd*, respectively. Qualitatively, the monotonic functional dependence on *kd* is not influenced by the value of *kh*. However, the presence of the PEC plane *reduces* the value of by up to approximately 1.5 dB, compared to the value for a free-standing SAM pair. The normalized power oscillates as a function of *kh*. The maximum of is reached at half-wavelength height (*kh*=*π*). While the value of for *kh*→0 serves as a near-optimum, its minimum minimorum value is marginally lower and is reached after one oscillation, at approximately *kh*≃4.5. Thus, an optimum height of an electric-dipole based SAM array above a PEC plane exists. When *kh* is increased further, the amplitude (envelope) of the oscillations steadily decreases and the normalized radiated power tends asymptotically toward its value for the free-standing pair when *kh*→+∞. The latter value has been calculated independently (Arnaut 2003) and is shown as dotted lines in figure 6*b*. When scanning across all heights for arbitrary *kd*, the dynamic range increases with decreasing *kd*, from 0 dB for *kh*→+∞ to values in excess of 6 dB for *kd*→0, as may be inferred from the figure. It is again noticed that the value of for *kh*→0 is smaller than the free-space value, which can be understood as follows. Physically, the sensor current and its PEC image are in complete anti-phase. This further reduces the residual radiation left following active control of the actual sensor by the actual actuator, as compared to a configuration without image sources. However, this anti-phase radiation between images is uncontrolled and, hence, acts optimally only when *kh*→0, in which case the improvement by the PEC image sources is therefore largest. As *kh* increases, the improvement diminishes because the annihilation of radiation from an actual source by its image in this way becomes less efficient, similar to the effect resulting when *kd* between actual sources is increased.

In summary, efficient low-pass (and hence, in principle, wide-band) non-resonant self-adaptive active absorbing surfaces can be achieved with a theoretically *arbitrarily small electrical thickness*, as measured by their distance above the PEC ground plane. At sufficiently low frequencies, they outperform conventional passive (magnetic and/or resonant) absorbers that have comparable or larger thickness, but at the expense of requiring additional circuitry and element loading as part of the SSS.

In practice, for quasi-static operation the *s*/*a* pairs are implemented as electrically small antennae, which show limited impedance matching to a free-space environment over all or part of the frequency band of intended operation. The problem of impedance mismatch is compounded by the presence of a conducting ground plane. However, in SAMs the problem may be at least partially alleviated on account of the following observations. First, does not necessarily need to merely compensate , but can be made to account for the scattered power from *s* by increasing the amplification factor . Thus, (2.4), (2.9) or (2.12) may have to be multiplied by this factor. Second, while the dipole lengths *l*_{a} and *l*_{s} are electrically small in quasi-static regime, they need not be equal. In fact, a design in which *l*_{a}/*l*_{s}≫1 allows for which is more practical. The issue of impedance matching is further discussed in §4. On the other hand, the bandwidth limitation is a more fundamental restriction, as discussed in §II-C of Arnaut (2003). Recent experimental and theoretical findings, however, indicate that in practice the bandwidth of ideal dipoles may be larger than the limiting values predicted by the Chu limit (Hujanen *et al*. 2005).

### (c) Radiated power in non-optimization directions. SAM radiation pattern

Even when optimization (annihilation) of the total radiation in a single chosen direction is the primary objective, the effect on the levels of radiated power in other (non-optimization) directions may be important. Following the approach in Arnaut (2003), the radiated power in a non-optimization direction is obtained by substituting by its expression in terms of and the optimum transfer function. At an arbitrary location of observation in the far field of a primary source which is being controlled for cancellation of radiation in a specified single direction , the received radiated power is given by (2.13), in which is replaced by . Upon substituting (2.12), we can express the total radiated power in the direction as a sum of self powers *W*_{αα} and mutual powers *W*_{αβ}, *viz*.(2.17)(2.18)with(2.19)(2.20)(2.21)(2.22)(2.23)(2.24)(2.25)(2.26)and for 0≤*θ*_{xr′}≤*π* in the two-dimensional configuration. The dependence of *W*_{r′} on is through the dependence of on *θ*_{xr}.

Figure 7*a* shows and , which is the ratio of the total power radiated in the direction to the power radiated by an uncontrolled source *s* in the presence of the PEC boundary (). This ratio is comparable with for an *s*/*a* pair in free space, in which case and *W*_{s}=*W*_{ss}. Compared to this free-space configuration, the figure demonstrates that as a result of the PEC plane, more radiated power is being redirected away from the direction of optimization (‘squeezed’ radiation). This is the penalty for improving performance in a single direction. Unlike , the ratio increases towards the endfire directions. Thus, the image quadrupole (*s*′/*a*′ pair) formed in the limit *kd*→0 modifies the radiation pattern of the actual quadrupole (*s*/*a* pair). Figure 7*b* indicates that for *kh*>*π*/2, one or more additional null(s) in this radiation pattern appear(s). The figure also shows that increasing *kh* allows for lower residual radiation in non-optimization directions, and is generally able to achieve better active absorption across a somewhat wider field-of-view.

Next, the radiated power can be integrated with respect to all directions to yield the integrated total power radiated across the upper hemisphere, as a single figure of merit, when the optimization criterion is the annihilation of the total radiated power in a *single* direction *θ*_{xr}. This must not be confused with the minimization of the integrated total radiated power itself, as considered in §2*b*. Figure 8 shows the normalized integrated power,(2.27)and demonstrates that, as the optimization direction is moved away from the normal, an increasing overall amount of power is radiated into other directions. This is reminiscent of the degradation of beam steering (scanning) performance toward endfire directions encountered in phased array antennae. The normalized integrated power (2.27) is independent of *kh* but, as expected, increases overall for increasing *kd*.

As pointed out in Arnaut (2003), a SAM dipole pair effectively creates a quadrupole element in the limit *kd*→0, offering a possibility for synthesizing ‘quadrupole media’. For conventional homogeneous or composite media, reflection and scattering characteristics are dominated by the re-radiation by induced dipole moments, whether individually or through magneto-electric coupling. Suppressing the dipole contribution to a level where radiation by second-order quadrupole moments becomes dominant offers scope for the realization of engineered materials with unusual EM reflection, transmission, absorption and diffusion properties, for one or several angles of incidence. By extension, higher-order ‘multipole media’ may be conceived and devised on the basis of yet more complex interconnection architectures.

## 3. SAM surface above planar PEC shield

Having established the radiation characteristics of a single SAM pair in the presence of a ground plane, we now turn to the characterization of a grounded effective SAM surface consisting of arrays of sensors and actuators. The surface will here be characterized in terms of its Green dyadic and its effective (array) and surface impedances. The corresponding free-standing configuration was analysed in Arnaut (2003).

### (a) Spectral-domain dyadic Green function

In this section, we make the notational convention that a vector quantity carrying 2*m*+1 primes corresponds to the PEC mirror image of the corresponding quantity with 2*m* primes. For example, if and denote locations of an observation and a source point, then and represent their respective images. For an electric dipole of length *l* centred at , oriented along the *x*-direction and located above a PEC plane with normal , for TM waves (cf. Lindell 1992),(3.1)The associated current is specified by(3.2)

Consider a SAM surface consisting of two interleaved planar periodic arrays of *s*/*a* dipoles, for example as shown in figure 9. The electric field radiated from an *s*/*a* pair above a PEC plane is again the coherent phasor sum of the fields radiated by the actual sources (*s* and *a*) and the image sources (*s*′ and *a*′). Therefore, the field radiated by the sensor array and its image, evaluated at a location situated above the PEC plane, then follows as(3.3)in which and refer to unit vectors characterized by the direction cosines for the forward or specular direction of propagation for the relevant plane-wave component, respectively. The subscripts ‘+’ and ‘−’ refer to observation points located above the SAM surface (*z*>*h*; scattering in specular direction back into the upper half-space) and in between the SAM array and PEC surfaces (0<*z*<*h*; scattering in forward direction below the *s*/*a*-plane), respectively. An explicit definition of , including its dependence on *q* and *v*, was given in §III-B.3 of Arnaut (2003), where it was denoted as . The component *ϱ*_{n}≡*ϱ*_{z} is the projection of onto the normal direction. For electrically short dipoles,(3.4)with(3.5)(3.6)(3.7)The total field, as obtained for identical combined *s* and *a* elements, remains formally the same as for the free-space configuration, *viz*.(3.8)with(3.9)(3.10)Here, the superscript ‘‡’ denotes a quantity associated with the actuator *a* (as opposed to the sensor *s*, for which associated quantities carry no superscript), but now with (2.12) or (2.15) for . Equation (3.8) then serves as a definition for the electric–electric Green dyadic via , *viz*.(3.11)

### (b) SAM array impedance

Following Munk (2000), the voltage induced by in an external rectilinear test wire element having wire radius *w*, which is oriented along and located at , is(3.12)Letting and , we obtain the array self-impedance as(3.13)where . Since this relates to the interaction with an incident wave whose electric field component is polarized in the direction , (3.13) defines an anisotropic wave impedance . By adding supplementary orthogonal elements in the transverse plane, this impedance can be made transversally isotropic, i.e. . If the *x*- and *y*-directed array elements are interconnected, then, on account of current continuity, cross-polarization impedance terms arise which, in general, involve additional contributions and effects, also on the wave polarization (Arnaut & Davis 1995).

The general result (3.13) can be specialized to the following particular cases of practical relevance:

For radiation in the specular outward direction—i.e. in the direction with a component of the propagation direction in the positive normal direction—and assuming uniform current distributions (Hertzian dipoles), so that , the surface impedance becomes(3.14)For its numerical calculation using the periodic method of moments (Blackburn & Arnaut 2005), the accuracy can be improved by using triangular currents with shape factor .

At sufficiently low frequencies (quasi-static regime), only the term for

*q*=*v*=0 in (3.13) remains, whence and , where*s*_{x},*s*_{y}and*s*_{z}are the cosines of the forward propagation direction (). In this case, with ,(3.15)For normal incidence (

*s*_{x}=*s*_{y}=0 and ) and quasi-static operation (*q*=*v*=0), (3.13) becomes(3.16)where(3.17)showing the reactive effect of the ground plane when*h*≠0.

### (c) SAM surface impedance

Expression (3.13) defines the self-impedance of a periodic SAM array based on an equivalent lumped circuit model for the entire array, i.e. relating the voltage impressed by a spectrum of incident plane waves to the induced current in a reference element *s*_{00}. Since the effect of the active array is entirely contained within the control dyadic , the same results and analyses as in the case of a purely passive array apply formally. In particular, *Z*_{A} can be shown to be related to the complex reflection coefficient *ρ* of the array (Munk & Burrell 1979)—which for SAMs is to be interpreted as a radiation coefficient for a rectilinear dipole array, for incidence in one of the principal planes (transverse electric polarization (TE) or transverse magnetic polarization (TM) incidence)—via , hereby allowing for element loading by a lumped impedance *Z*_{L}. On the other hand, for normally incident plane waves and in the absence of grating lobes, *ρ* defines an equivalent surface impedance *Z*_{S} of the planar SAM array via(3.18)because of the abovementioned reason. Hence, for the principal planes of incidence,(3.19)This surface impedance can be explicitly related to with the aid of (3.13). Furthermore, (3.19) applies to obliquely incident waves as well, because in the absence of grating lobes the SAM array does not refract the incident wave into directions other than the direction of incidence. Therefore, the angles of incidence and refraction are identical, causing their cosines to cancel out in the expression of *ρ* for TE and TM coefficients.

### (d) Numerical results

As an example, we consider a chequered planar array of interleaved rectilinear sensors and actuators above a PEC ground plane. The elements are assumed to tessellate the plane of the array entirely (*l*/*d*_{x}→1, 2*w*/*d*_{y}→1). The sensors and actuators are interconnected pairwise in collinear fashion, and are assumed to have uniform current distributions. Figure 10 shows *Z*_{A}/(*Z*_{0}*d*_{x}*d*_{y}) as a function of the non-optimization direction when the surface is optimized for radiation in the normal direction (*θ*_{xr}=90°), at selected values of *kd* and with *kh*=*π*/4. In the limit *kd*→0, the array impedance reaches its asymptotic value that is independent of *θ*_{xr}. Figure 11 shows the magnitude of the associated radiation coefficient *ρ* for TM incident waves. As one moves away from the optimization direction *θ*_{xr}, |*ρ*| increases and may even exceed unity, because the SAM surface is EM active. The array reflection is found to exhibit only a weak dependence on *kh* (not shown), confirming similar findings for radiation by the *s*/*a* dipole pair shown in figure 8.

For minimized total integrated power, figure 12 shows as a function of an observation direction , at selected values of *kd* and with *kh*=*π*/4. The associated |*ρ*| is shown in figure 13. Unlike for the case of optimization for a single direction in figure 11, the reflection in the normal direction is now non-zero, because the total radiated power now involves integration across a finite-sized solid angle. As *θ*_{xr} is moved away from broadside, |*ρ*| again increases monotonically. The dependence on *kh* (not shown) is generally weak, and even so only for observation near broadside and sufficiently large *kd*.

## 4. Equivalent circuit

In any practical implementation of a SAM surface, mutual interactions between *s* and *a* affect its overall characteristics and performance. This coupling can be categorized into two types, *viz*. conducted transfer (via the controller and its wired interconnections) and radiated transfer (via radiated fields between *s* and *a* caused by their currents *I*_{s} and *I*_{a}). Both types will be identified by second subscripts *c* and *r*, respectively. Their effect on the specification of the adaptive controller is now investigated. In this section, a voltage or current carrying *m*+1 primes refers to an impressed output caused by an input voltage or current denoted by *m* primes.

Consider the diagram for general mutual interaction within a single *s*/*a* pair as schematized in figure 14*a*. A first contribution to the mutual coupling between *s* and *a* is the conducted forward transfer, which is intentional and defined by the transfer function *H* of the controller itself, i.e. . Second, the radiated forward transfer contributes and can be characterized by a mutual impedance between *s* and *a* for this oriented direction, i.e. from *s* to *a*, *viz*. . Hence,(4.1)where *Z*_{aa} is the self-impedance of the actuator antenna element, e.g.(4.2)in the case of a short lossless thin-wire transmitting dipole of diameter *δ*_{a}, and where *Z*_{a} is the source output (line) impedance of the transmitter. Therefore, for the total forward transfer,(4.3)With regard to the backtransfer from *a* to *s*, its conducted part is determined by the transfer admittance of the controller for the reverse direction, , whence(4.4)For an ideal controller, the reverse isolation is perfect (*Y*_{T}, *G*=0). Finally, radiated backtransfer is governed by the associated mutual impedance *Z*_{sa} between *a* and *s* for this reverse direction, as defined by , through(4.5)where *Z*_{s} is the load input impedance of the receiver and *Z*_{ss} is the self-impedance of the sensor, defined in a similar manner as *Z*_{aa} above. In a reciprocal system, *Z*_{sa}=*Z*_{as}. In practice, since must at least compensate *I*_{s}, the radiative coupling of *a* with *s* is usually considerable, irrespective of whether the sensors and actuators are designed using identical or significantly different (in type or size) antenna elements. The total actuator current, however, also contains a contribution by the impinging incident wave to which *a* is equally exposed, whence(4.6)The dual effect, i.e. radiation and scattering by *s*, is taken into account by the radiation resistance contained within *Z*_{ss}. Hence, for the overall round-trip of current, from (4.3)–(4.6),(4.7)This expression shows the effect of *s*/*a* interactions, through their dependence on the perturbed sensor current, on (i) the intrinsic antenna characteristics of *s* and *a* (*Z*_{ss} and *Z*_{aa}); (ii) the characteristics of the receiving and transmitting circuits (*Z*_{s} and *Z*_{a}); (iii) the conducted interaction (*H* and *Y*_{T} or, equivalently, *G*); (iv) the radiated interaction (*Z*_{sa} and *Z*_{as}); and (v) the configuration of the pair (via *ϕ*_{a}−*ϕ*_{s}). It also contains, implicitly, the effect of any impedance mismatch between the *s* and *a* antenna elements and their respective receiver and transmitter circuits ( and as functions of *Z*_{ss} and *Z*_{aa}; see below).

The conducted and radiated couplings affect the Thévenin equivalent source voltage and input impedance for both *s* and *a*. For an *s*/*a* pair in free space with given *I*_{s}, the perturbed source voltage is, using (4.3),(4.8)whence the corresponding active terminal (i.e. driving-point) impedance of *s* is(4.9)The dependence of on *H* via the product *HZ*_{sa} expresses the fact that the forward conduction is relayed back to the sensor via radiation, whereas the term represents the round-trip radiative coupling for *s*. Similarly for *a*, from for given , and using (4.4) and (4.5),(4.10)The and represent the source and load input impedances of the open-port equivalent circuits for *s* and *a*, respectively (cf. figure 14*b*). The sums (4.9) and (4.10) show that mutual coupling within each *s*/*a* pair can be incorporated into decoupled element impedances via calculable additional impedances that are placed in series with the self-impedances *Z*_{αα} for the isolated elements *α*. Conduction loss in *s* or *a* can be represented by a supplementary series resistance in the usual manner. The remaining additional mutual coupling between *s*/*a* pairs in a SAM array, *viz*. between elements of the same type (function) *α*, is of the same sort as that encountered in conventional phased arrays and frequency selective surfaces (diffraction gratings), and is therefore not discussed here in further detail but can be calculated numerically from the impedance matrix of the array (e.g. Blackburn & Arnaut (2005)). Indeed, it suffices to consider either each *s*/*a* pair as a single element of a single SAM array, or to calculate the interaction between like elements in the *s* and *a* subarrays separately.

To obtain the Thévenin voltage sources for the equivalent circuit, we use the known result that the e.m.f. of the equivalent source for an opened-circuit single receiver (Rx) placed in the vicinity of a single transmitter (Tx) is (e.g. Kraus (1988)). In a connected *s*/*a* pair, *s* is excited by both the externally incident field and the field radiated by *a*, with respective associated induced open-circuit voltages *V*_{0} and , as well as through *I*_{a} inducing a voltage in the receiver circuit if , via reverse conducted transfer. Thus, for the overall voltage of the sensor source,(4.11)Note that the effect of *G* can always be incorporated by redefining *H* as *H*/(1−*GH*). Therefore, without loss of generality, we shall further assume *G*=0. Similarly, *a* is excited by the sensor via *H*, the mutual coupling of *a* with *s*, and the incident external phase-shifted field to which *a* is also exposed (parasitic sensing by actuator). Hence,(4.12)Solving (4.11) and (4.12) yields the source voltages of the respective open-port circuits for an *s*/*a* pair in free space, with reference to *V*_{0}, as(4.13)(4.14)With(4.15)and , we finally arrive at(4.16)This is the general expression for *H*_{as}^{FS} when interaction and mismatch effects between a single SAM pair and the associated receiving and transmitting circuits are taken into account. The transfer function *H* that is required for the design of a controller implementing a specified *H*_{as}^{FS}, such as e.g. (2.4), follows upon inversion of (4.16).

For quasi-static operation (*ϕ*_{a}−*ϕ*_{s}→0) in a reciprocal medium (*Z*_{as}=*Z*_{sa}) with, in addition, identical symmetric *s* and *a* circuits that are terminated1 according to , and assuming identical *s* and *a* antenna elements (*l*_{a}=*l*_{s}),(4.17)(4.18)where we have introduced the reduced mutual impedance . Radiative transfer is seen to cause nonlinearity in the transformation between *H* and *H*_{as}^{FS}.

For negligible radiative coupling (|*z*_{as}|≪1), *viz*. when *s* and *a* are spaced more than half a wavelength apart, . This shows that, as expected, the total actuator current then consists of the same current that is externally induced in the sensor plus half of the conductively transferred sensor current *HI*_{s}^{FS}, owing to *Z*_{aa}=*Z*_{a}. More significantly, if *kd*→0 and assuming *z*_{as} is real for the sake of discussion, then |*z*_{as}|→1 and . In the absence of conductive transfer (*H*=0), implying no adaptive control, the actuator current consists entirely of the current induced by the externally incident wave and is identical to the current carried by *s*, whence *H*_{as}^{FS}=1 as again expected.

It is emphasized that *Z*_{ss}=*Z*_{s} and *Z*_{aa}=*Z*_{a} do *not* correspond to matched sensor and actuator circuits, even when the reactive parts of *Z*_{ss} and *Z*_{aa} are tuned out separately: because of the conducted and radiative coupling, such matching would require *Z*_{ss}^{(a)}=*Z*_{s} and *Z*_{aa}^{(a)}=*Z*_{a}. Although the impedance mismatch of the sensor to the receiver in a free-space environment gives rise to increased scattering from *s*, this is partially compensated for by the correspondingly increased radiation from *a*, via the increased amplification in *H*_{as}^{FS} that results from conductive and radiative coupling, at the expense of reception and radiation becoming less efficient. This mismatch depends on the amplitude and phase of both *H* and *Z*_{as}, as is apparent from (4.9).

With regard to practical SAM design, it is of interest to relate ranges for *H*_{as}^{FS} to corresponding intervals for *H*. These ranges depend, in general, on the value of *z*_{as} for the design in question, i.e. on *l*_{a}, *l*_{s}, *d*. We discuss this briefly for ranges of *z*_{as}, *H* and *H*_{as}^{FS} limited to real values only, although it should be remembered that at least *H*_{as,opt}^{FS} has, in general, a (small) imaginary part to compensate for the spatial phase due to non-zero separations. If *z*_{as}=0, then a passive controller (i.e. 0≤*H*≤1) results in 1≤*H*_{as}^{FS}≤3/2. Conversely, −1≤*H*_{as}^{FS}≤0 requires −4≤*H*≤−2: the connection of *s* and *a* to receiving and transmitting networks gives rise to a higher required amplification |*H*| than the specified |*H*_{as}^{FS}|, as expected. By contrast, for asymptotically strong radiative coupling (*z*_{as}→1), a passive controller *H* now yields 1≤*H*_{as}^{FS}≤5/2, whereas −1≤*H*_{as}^{FS}≤0 now imposes −3/2≤*H*≤−1/2. Clearly, these ranges are dependent on the ratios *Z*_{ss}^{(a)}/*Z*_{s} and *Z*_{aa}^{(a)}/*Z*_{a}, so that the specification of the active control and the design of the actual controller cannot be separated from the EM characteristics of the receiving and transmitting circuits. In particular, the impedance (mis)match in the receiver depends on *H*, and may require separate tuning elements. Figure 15 shows conversion charts between implemented and realized *s*/*a* transfer functions for voltage control (4.17) or current control (4.18) at selected real values of *z*_{as}.

In implementations based on microelectronic components, negative real parts of *H* as prescribed by (2.4) can be realized using an inverter circuit based on a transconductance operational amplifier, for which the value of *H* can be achieved with an appropriate choice of the loop impedance relative to the impedance at its input port. Expression (4.17) can be used in an alternative implementation of *H* based on a voltage amplifier. Furthermore, a transistor circuit with common emitter can be used if a fast response (real-time tracking) or high-power generating capability is required. Further aspects of the practical implementation have been detailed and discussed in §VI of Arnaut (2003).

When *s* and *a* are placed above a PEC ground plane,(4.19)(4.20)in which and in case of and being parallel to the plane, whence(4.21)(4.22)The source voltages now follow as(4.23)(4.24)where(4.25)(4.26)Solving (4.23) and (4.24), upon substituting (4.25) and (4.26), and performing similar manipulations as above for the *s*/*a* pair in free space leads to a general, but now much more cumbersome expression for *H*_{as}^{PEC} instead of (4.16).

## 5. Conclusion

Self-adaptive surfaces constitute extensions of bianisotropic surfaces and ideal EM boundaries beyond passivity. In this paper, we proposed and analysed an alternative implementation of such surfaces suspended above a planar PEC boundary. The effect of such a ground plane on the adaptive control and radiated power was analysed for various optimization criteria. For optimization of radiation in a single direction by a single SAM pair, the effect of a ground plane on the optimum transfer function can simply be taken into account by pre- and post-multiplication by the phase factor (2.8). For minimization of the hemispherically integrated power, however, these factors need to be incorporated into the kernel as in (2.14). In this case, the radiated power (2.18) involves cross-product terms involving contributions by all combinations of *s*, *a* and their images. The effect of the ground plane on the total radiated power is a ‘squeezing’ of energy away from the surface normal. Increasing the electrical distance of the SAM pair above this plane (layer thickness) enables lower residual radiation in non-optimization directions to be achieved, combined with a wider field-of-view. The minimum (non-zero) total integrated radiated power is obtained for surfaces with electrical thickness *kh*≃4.5, although arbitrarily thin surfaces (*kh*≤1) provide a near-optimum solution. In either case, the integrated power including the PEC plane is lower than in the free-standing configuration.

For an effective SAM array with PEC plane, expressions for the dyadic Green function (3.11), array impedance (3.13) and radiation coefficient (3.18) were obtained. Depending on the geometry and configuration of the individual sensor and actuator elements and their subarrays, a transversally isotropic or anisotropic array impedance is obtained that can be related to an equivalent surface impedance (3.19) for reflection of plane waves incident in the principal planes. For an appropriate array configuration and tessellation of the surface, the performance of this new design was shown to be comparable to that of a free-standing implementation. The magnitude of the radiation coefficient is found to increase steadily with increasing angle of the non-optimization direction (for single-direction optimization) or the optimization direction (for total integrated radiated power) away from broadside, and the value of this coefficient may start to exceed unity.

An equivalent circuit model of a SAM pair was developed, valid for general implementations and inclusive of scattering, coupling and impedance mismatch effects, as well as the characteristics of the receiving and transmitting circuits through their load input impedance *Z*_{s} and line impedance *Z*_{a}, respectively. The model distinguishes between individual contributions by the conducted and radiated transfer in the mutual coupling within an *s*/*a* pair. For a SAM pair in free space, the contributions of the various sources of interaction were identified for the net-induced current (4.7) and for the overall transfer function of the current (4.16). The voltage and current transfer functions expressed in terms of the reduced mutual impedance *z*_{as} were obtain as (4.17) and (4.18), respectively, showing that current transfer—unlike voltage transfer—gives rise to a nonlinear mapping between implemented and realized transfer functions. Implicit expressions (4.23) and (4.24) for the equivalent voltages for the SAM pair in the presence of a PEC plane were also given.

The analysis in this paper was based on two idealizations. First, it made use of image theory, which presumes that the ground plane is perfectly conducting. Effects of finite conductivity are to be investigated using an explicit calculation of a series of reflections from the plane. Second, the model assumed ideal dipoles, i.e. equivalent sources, which makes abstraction of the physical carrier of the current (thin wire) of the *s*/*a* elements. In such a model, the incidence of a plane wave onto the ground plane is not perturbed by diffraction (shadowing) caused by the physical elements, as though they were not present, irrespective of their planar density. In reality, these conducting elements will more or less shield the incident radiation from the ground plane. The optimum controller that takes the shadowing on the ground plane into account is expected to evolve from (2.9) to (2.4), as the SAM surface density increases. This requires further investigation.

As in Arnaut (2002, 2003), the focus in this paper was on active absorption of the natural scattering response, as a fundamental first step towards generating an artificial secondary response that can be realized using more conventional array techniques. Of course, if one is solely concerned with annihilation of the primary scattering and reflection, then an implementation based on ideal passive impedance matching could achieve the same result. An active implementation, however, enables one to overcompensate the received power, thus allowing for overcoming residual scattering caused by imperfect impedance matching (cf. §IV of Arnaut (2003)) and, more generally, permits in addition adaptivity. Second, if the objective is the realization of a synthetic secondary response using stimulated radiation, rather than the mere conversion of the incident energy at the wavelength of operation, then incorporation of active radio-frequency (RF) elements to realize a SAM surface is inevitable. An exception would be a scenario in which the secondary response can be generated through ‘energy harvesting’ from the environment in which the SAM surface is placed, e.g. using a photoelectric, bioelectric, thermoelectric or other mechanism for converting other sources of energy into EM energy, crucially, at the intended wavelength of operation. Such mechanisms for generating RF output remain largely speculative at present, but offer the attractive prospect of not requiring a dedicated energy source.

## Acknowledgments

This work was supported by the 2003–2006 National Measurement System Electrical Programme of the UK Department of Trade and Industry.

## Footnotes

↵Typically, for

*kl*_{a},*kl*_{s}≪1, the self-impedances*Z*_{aa},*Z*_{ss}are strongly capacitive, unless*s*and*a*are themselves loaded dipoles. Therefore, matched termination then requires the reactive parts to be tuned out separately.- Received October 4, 2004.
- Accepted October 18, 2005.

- © 2006 Crown Copyright