## Abstract

In this paper, we present an alternative approach to addressing the problem of designing a number of practical ‘microwave’ devices such as blankets serving as absorbers for radar targets, flat lenses and reflectarrays.

## 1. Introduction

In this paper, we present an alternative approach to addressing the problem of designing a number of practical ‘microwave’ devices, such as blankets serving as absorbers for radar targets, flat lenses and reflectarrays. Recently, these design problems have been dealt with by a number of researchers using the transformation optics (TO) algorithm, which is based upon transforming the geometry of an object from real space to virtual space, while keeping the Maxwell field solutions from real-to-virtual space intact. This type of transformation typically leads to designs that call for anisotropic *ε* and *μ* values in real space, in order to maintain the field variations unchanged as we navigate from the real space to virtual space and vice versa. Furthermore, depending on the geometry of the problem, the material values may be very unrealistic to realize in practice, even when artificially synthesized materials or metamaterials (MTMs) are employed for such realizations. The use of MTMs often leads to designs that are narrowband, lossy, dispersive and polarization-sensitive—attributes that are clearly undesirable for practical applications. In contrast to the TO algorithm, the proposed algorithm is based on ‘field transformation (FT)’, as opposed to geometry transformation. The FT algorithm has been designed to transform the electromagnetic field distribution in an input aperture, generated by a given source distribution, to a desired distribution in the exit aperture. We show how we can cast this design problem into the framework of the generalized scattering matrix (GSM) approach [1]. In this approach, the radar cross-section (RCS)-reduction problem is based on controlling only the *magnitude* of *S*_{11}, whereas for the lens or reflectarray problems, we specify only the desired *phase* of *S*_{12} without being concerned about its magnitude. In contrast to this, the TO imposes strict conditions on both the magnitude and phase characteristics of *S*_{11} and *S*_{12}, as we explain below, which in turn calls for anisotropic *ε* and *μ* MTMs. The scattering matrix/FT approach avoids these problems altogether and is able to work with *ε*-only materials for the lens and reflectarray problems, and is *realizable* with complex (*ε*, *μ*) materials that have wideband characteristics and do not suffer from the shortcomings of the MTMs.

A simple example of FT is the transformation of the spherical fields emanating from a point source into a planar *phase* front by using a flat lens whose behaviour mimics that of a convex lens, for instance. The same problem when tackled by the TO leads to a relatively complex design and calls for MTMs that are not only difficult to realize, but lead to performance characteristics that are not on par with the FT-based design, as we will demonstrate in this paper. This is true despite the fact that the latter design uses realistic materials, which can either be fabricated in the laboratory, or are commercially available off-the-shelf that are modified slightly to realize the desired material parameters. Such materials are wideband as well as relatively insensitive to the polarization and incident angle of the incoming wave.

Finally, we will point to a new interpretation of the TO algorithm that has recently been employed to design absorbers that help reduce the RCS of real-world targets. An example based on this type of design strategy will be included in the paper.

The concept of FT has been introduced in recent literature by Liu *et al.* [2] to manipulate wave propagation, and it has been explained in [2] how the FT they proposed complements the TO algorithm. (The reader is referred to the work cited above for further details.) It is useful to point out, however, that the FT approach introduced in this work differs from the one presented in [2], as well as from the TO, in two ways. First, the presented FT approach modifies the objectives of the TO, by redefining them and aiming only to reduce the scattering (magnitude only) from a radar target *solely* in the ‘backward’ hemisphere, rather than in *both* forward and backscattering hemispheres, as is the goal set in the TO algorithm, which aims to eliminate all field perturbations (magnitude and phase) due to scattering when the fields are mapped one-to-one from the ‘virtual’ to ‘real’ space, or vice versa. Furthermore, when dealing with transmission-type devices, e.g. lenses, the FT approach described in this paper only manipulates the transmission ‘phase’ of the impinging wavefront to design the lens, following the algorithm introduced by Luneburg [3] for instance. In contrast to this, the TO algorithm strives to strictly reproduce the field behaviour of a convex lens in its *entirety* by promoting a flat lens design whose transmission and reflection characteristics are identical to those of the convex lens—which the flat lens replaces—both in terms of magnitude *and* phase.

For the radar scattering case, our FT approach aims to design a blanket which reduces the level of the scattered field in the backscattering hemisphere, in contrast to a TO- or FT-based cloak design which aims to totally suppress all scattered fields to faithfully reproduce the incoming wavefront once the electromagnetic fields incident on the object have traversed past it and have exited in the forward direction. The cloak is obviously the preferred choice over the blanket from the performance point of view, especially if it can make the radar target disappear totally without a trace, as an ideal cloak can indeed do in principle; unfortunately, the TO-based cloak design is unrealistic to fabricate because it calls for materials that are difficult if not impossible to realize in practice over the frequency range of interest. The FT algorithm proposed herein offers an alternative approach to addressing the RCS-reduction problem, as shown below in this work.

Finally, the FT approach presented in this work totally differs from the one described in [4] in another aspect. Specifically, it presents the algorithm in the framework of scattering matrix, which neither the TO nor the FT approaches described in [2] do. The TO and the previously introduced FT approach in [2] both lead to unrealistic designs that often call for MTMs, whereas the proposed FT approach systematically modifies the objectives of the design so as to lead to realistic devices that can be fabricated with available materials. These materials are often commercial off-the-shelf (COTS) type, or slight modifications thereof, realized by using the dial-a-dielectric (DaD) approach as mentioned below in this paper in the context of flat lens design. For the RCS-reduction problem, the materials can be fabricated in the laboratory, as has been adequately demonstrated in [5], for instance. It is important to recognize that, unlike MTMs, the materials called for in the present FT design have many desirable features, namely broad bandwidth; desirable loss and dispersion characteristics; and desirable levels of insensitivity to angle of incidence and polarization, as shown below in this paper.

## 2. Field transformation in the context of generalized scattering matrix approach

When designing an electromagnetic device, such as a lens antenna, we typically begin with a given source, such as a feed horn, and specify that the field radiated by the horn be transformed into a planar phase front, for instance. As is well known, this task is easily accomplished by placing the feed horn at the focal point of a conventional convex or plano-convex lens, and the procedure for designing such a lens based on the ray optics (RO) approach is well established in the literature; in fact, it is a just a textbook case.

The situation is different when we stipulate that the shape of the lens be flat instead of convex. To address this problem, the TO algorithm offers an elegant recipe based on transforming the geometry of the original convex lens into a planar one. The TO algorithm then shows us how to derive the requisite material parameters of the planar lens by using well-established relationships [6] between the *ε* and *μ* values of the original convex lens and its surrounding medium, and those of its planar counterpart. These relationships involve the Jacobians of the geometry transformation and are relatively straightforward to find, even for arbitrary geometries being transformed from the real space to virtual space. The caveat, though, is that the *ε* and *μ* values are anisotropic in general, and may be difficult if not virtually impossible to realize in practice. It is not uncommon, therefore, to set the *μ* values equal to *μ*_{0}, i.e. that of free space, to ignore the *ε* values less than unity, and to only work with isotropic dielectrics, albeit at the risk of compromising the performance of the lens in comparison to that of the original TO design prior to introducing the modifications. What is equally important to realize is that there is no clear roadmap provided by the TO algorithm that tells us how we can improve the performance of the modified design, should we need to do so.

Given this background, we pose the following question for ourselves: can we modify the problem statement that forms the basis of the TO algorithm to circumvent the problems alluded to above, without compromising the performance relative to that of the convex lens, in a way such that we can still use realizable materials found in nature, without having to resort to MTMs? We will now present an approach based on the GSM method [1], which indeed offers a way to address the problem at hand, as we have just enunciated above.

To introduce the GSM approach [1] in the context of the FT method [7], we refer the reader to figure 1, where we have defined the input and output ports to correspond to interfaces that bound an electromagnetic device. The field distribution in the input port, which is illuminated by the source located at the left of the port, can be expressed in terms of a set of coefficients (vector) associated with the basis functions used to represent this ‘incident’ field in the absence of the device when there are no reflections. Next, we insert the electromagnetic device, whose scattering matrix we desire to describe, inside the region bracketed by the input and output ports. We define a set of coefficients , again associated with the same basis functions as we used to define , to represent the outgoing fields scattered by the electromagnetic device, i.e. the ‘reflected’ fields that originate from the device and propagate back towards the source. We can similarly define a set of coefficients , associated with the field distribution in the output port, through which these fields propagate in the free-space region to the right of this port, and are termed the ‘transmitted’ fields. Our next step is to place the illuminating source to the right of the output port, which we have previously defined when the source was at the left, and reverse the roles of the input and output ports to correspond to the new source location. The incident, reflected and transmitted fields are now characterized by a new set of coefficients , and , where fields now propagate to the left of the device, whereas the fields do the opposite, i.e. propagate to the right.

We are now ready to define the scattering matrix [**S**], which we will use to characterize the device, as follows:
2.1or explicitly,
2.2where ** b**={

*b*^{1},

*b*^{2}} and

**={**

*a*

*a*^{1},

*a*^{2}} represent the weights of the outgoing and incoming field representations at the input and output ports, respectively. Equation (2.2) provides us a convenient way to characterize an electromagnetic device in terms of its response to a plane wave, regardless of whether the illuminating source is incident from the left or the right of the device.

We will now explain how we can specify the desired characteristics of a device in terms of its **S**-parameter description. Towards this end we will consider three types of electromagnetic devices for our examples: (i) a lens, be it convex or flat, (ii) a blanket for RCS reduction, and (iii) a reflectarray or a metasurface.

### (a) Specification of lens performance based on its S-parameter representation

Consider the lens problem depicted in figure 2. We begin by using our knowledge of the incident field emanating from the feed located at the focal point of the lens to derive *a*^{1}. Next, we specify that the phase distribution in the output plane (port) be uniform, for instance, so that the lens realizes a maximum gain while pointing at boresight. We could, of course, specify instead that the phase distribution have a linear taper, and cause the beam to tilt in a desired direction; however, for now, we consider the former uniform distribution case, simply for the sake of convenience. We could also stipulate the behaviour of the amplitude distribution in the output port, or we could simply choose to let it float. The ramification of specifying both the amplitude and phase distributions in the output plane is significant, and we defer the discussion of this case until later.

We assume for the sake of illustration that the incident field is spherical in nature and, hence, it produces a quadratic phase distribution at the input port. Then a straightforward and classical solution to the lens design problem that meets these specifications is to use a convex (or plano-convex) lens comprising a uniform dielectric material, assuming that we are free to choose the shape of the lens. However, if we next add the stipulation that the lens shape be flat on both sides, then it becomes intuitively clear that we must use a lens design in which the refractive index must at least vary in the transverse direction, and the result would be a GRIN (graded index) lens [8], which is also a classical design. What is more important, however, is that this lens design can also be realized—just like in the case of the convex lens—by using dielectric-only materials that are often readily available commercially, or can be synthesized conveniently as we show below.

Next, let us change the design specification for the flat lens as follows: we now stipulate that the flat lens perform precisely like the convex lens in its entirety, implying that the fields at the input and output ports be identical for the convex and flat lens designs. Obviously, such a performance of the flat lens would be highly desirable because it would deliver the proven performance of the classical convex lens design. In fact, this is exactly the goal of the TO-based design, which has been attempted by a number of authors [9–14]. What is not so obvious, however, is that such a seemingly minor change in the specifications, in which we insist that the flat lens design should not only manipulate the phase distribution at the output port but also the amplitude distribution, has a highly significant impact on the realizability of this type of flat lens, which we refer to herein as the TO-based design. Some of the difficulties arising in the realization of the TO design are: (i) it calls for both dielectric and magnetic materials, (ii) these materials must be anisotropic in general, and (iii) typically some of the values called for can only be realized by using MTMs, which can be lossy, dispersive and narrowband, forcing us to use substitutes that are likely to compromise the performance of the lens. It is equally important to realize that there is no simple way out of this conundrum without paying a high price in terms of cost and complexity, and/or sacrificing the performance of the lens. In summary, the FT approach to flat lens design in which we require that the device transforms the electromagnetic field distribution as it traverses through the device and makes its way from the input to the output port, by manipulating only the phase distribution and not its amplitude, offers a realistic and viable solution to the flat lens problem, which the TO algorithm fails to do because it insists on controlling both the amplitude and phase distributions of the field distributions at both the input and output ports.

### (b) Reduction of scattering from radar targets

We now turn to the problem of reducing the level of scattering from radar targets—a problem that has been extensively researched in the context of the TO. Once again, we will cast the problem in the language of scattering matrices, to help us understand why the TO approach leads us to untenable situations and/or to solutions which call for MTMs that have many drawbacks as we have pointed out earlier. We will also show how the FT approach mitigates the problems alluded to above by restating the design objectives and modifying them slightly from those associated with the TO-based designs.

Let us consider an arbitrarily shaped radar target placed in the region between the input and output ports, as shown in figure 3. Next, let us suppose that our objective is to reduce the level of scattering from the target, both in the forward and backward directions to the extent that the target becomes totally invisible to the incident field, say from an interrogating radar. We could cast this objective in the language of the scattering matrices, by specifying that *S*_{11} be identically zero at the input port and *S*_{12} be such that the field distribution at the output port is identical to the incident field, as though the scatterer was totally transparent or invisible. This is precisely the premise upon which the TO algorithm is based, and the TO shows how to achieve this goal by filling the centre region, which contains the target, with materials that cloak the target to render it invisible, just as though it were absent. While such a goal is highly attractive, it is also very unrealistic, insofar as the physical realization of the cloak is concerned, for several reasons: (i) the materials called for by the TO algorithm are highly anisotropic, even for simple targets such as cylinders, (ii) the requisite material values span a wide range, as shown in figure 4 for a cylindrical target, (iii) these materials can only be realized by artificially synthesizing them, i.e. by using MTMs that are inherently lossy, dispersive and narrowband (bandwidth is typically on the order of a few per cent of the centre frequency), (iv) the design is dependent on the incident angle as well as polarization of the incident wave—which makes it very impractical for real-world applications (note that, in principle, the ideal three-dimensional cloak would work for all incident angles and polarizations, but it calls for unrealistic material parameters that have never been realized in practice), and (v) the thickness of the TO-based cloak is much too large for real-world applications involving microwaves. Although the mantle cloak [15] is much thinner than the TO cloak, its design principles are different; furthermore, the mantle cloaks are considerably more complex in design than the one we propose herein, and these mantle cloaks are inherently narrowband.

To circumvent these problems that we encounter when attempting to use the TO to meet the ideal but unrealistic goal of making the target altogether invisible, we turn to the FT approach and modify our stated objectives, again in the context of **S**-parameters. In contrast to the TO, this time we ask that *S*_{11} be small—in terms of magnitude only—but not 0, as we demanded in the case of TO. Furthermore, we do not impose any restrictions on *S*_{12}, as we did in the case of the TO when we stipulated that the scattered fields at the output port be identically zero, so that the total field there be just the incident field. While we concede that the performance of the FT-based cloak or the blanket will not be as ideal as the performance of the TO-based design would have been if we could realize it, we certainly stand to gain considerably when we follow this strategy, as we can now obtain realizable solutions for the cloak that are very wideband and can cover the entire frequency range of 2–18 GHz, for instance, if we so desire. Furthermore, the cloak (or blanket) can now be very thin (only a few millimetres) and it would work for arbitrary incident angles and also for polarizations (an FT-treated radar target is shown in figure 5). In short, we can say that this strategy of following the FT-based algorithm to design the cloak enables us to circumvent all the problems we encounter when employing the TO-based design strategy instead. Of course, we compromise the performance of the cloak in the forward scattering direction when we use the FT-based strategy, though that is not a problem for either monostatic or bistatic radars that are only concerned with the backscattered fields.

Some examples of such realistic blanket designs based on the FT strategy can be found in [5,7,16–18] and are omitted here for the sake of brevity. The above work also shows how to design the blankets for arbitrary targets by borrowing some ideas from the TO algorithm to improve their design.

Before closing this discussion, we would like to mention that the strategy for designing carpet cloaks [19] is very similar. We can use the image theory to recast the problem of designing a carpet cloak so that it is totally equivalent to the cloak design problem for radar targets that we have discussed above in this section.

## 3. Metasurface design

The subject of metasurfaces, or artificially synthesized surfaces, to manipulate the field behaviours is a topic of great interest today, as evidenced by a surfeit of publications on this topic in recent years. Let us consider a simple but practical example of designing a flat surface that mimics a parabolic reflector, which may be either symmetric or offset. We assume that the reflector is illuminated in the usual manner, by using a feed horn which emanates a spherical wave from its phase centre (see figure 6). The reflector serves to convert this wavefront to a planar one to generate a directive beam in a specified direction, which depends on the choice of the feed location.

The TO approach to handling the problem of designing the flat reflector is relatively straightforward: transform the parabolic reflector geometry into a planar one and embed it in a medium that surrounds it so that the fields reflected off the surface of the planar reflector covered by the TO-designed coating mimic those associated with the original parabolic surface. We can set up equivalent apertures for each of these reflectors (parabolic and flat) and stipulate that these two distributions be identically the same, in accordance with the TO strategy. The Jacobians of the transformation provide a relatively easy way to determine the parameters of the medium in which we must embed the flat reflector so that it delivers a performance that is identical to that of the parabolic reflector. But there are several problems we must address before we can use this design in practice. Not unexpectedly, these problems are similar to the ones we have encountered earlier with the TO designs, namely that it is plagued by difficulties in realizing the desired low-loss, low-dispersive and wideband materials—especially, when we must use MTMs—with which to cover the surface of the flat reflector. In addition, we need to ensure that the thickness of the coating is small, so that the resultant flat reflector (with coating) has a low profile, which was our principal motivation of the flat reflector project to begin with. To the best of our knowledge, a systematic procedure for designing such thin coatings has not yet been developed. In fact, such surfaces that are also known as reflectarrays, are almost always designed to match only the phase of the reflected field, rather than to control the amplitude distribution of this field in its aperture. Incidentally, reflectarrays are inherently narrowband in comparison to the parabolic reflector, at least when they are fabricated by using metasurfaces, as they almost always are.

We now show that this problem of designing a surface with specific reflection characteristics can also be cast in terms of the scattering matrix formulation. Because the metasurface is backed by a perfectly electrically conducting plane, the output port is short-circuited; hence, we only use a one-port description of this surface and deal with only the *S*_{11}. To design the metasurface in the context of TO, we stipulate that the *S*_{11} for the metasurface be identical to that of the original parabolic reflector, in terms of both magnitude and phase. However, the FT approach only requires that the field incident from a spherical source which impinges upon the metasurface be transformed into a planar *phase* front, with no constraint imposed on the magnitude. An important consequence of this is that the FT-based metasurface can be fabricated by using dielectric-only materials, and can use COTS materials with minor modifications [4,20], whereas the TO-based approach would encounter the same range of difficulties that it did when we considered the lens design problem.

Examples of an FT-based lens and metasurface designs are presented in figure 7, which shows how the concept known as DaD [4,20] can be used to tweak the properties of available COTS materials (dielectric only) and achieve the desired phase behaviour (constant) over a wide frequency band, and thereby focus the beam of the horn antenna used to illuminate the surface, fulfilling the design goal of the lens.

To illustrate some practical applications of the DaD approach, we compare the results we obtain for the performance characteristics of DaD-based lenses and reflectarrays, with those of similar devices designed by using legacy approaches. We show that we can typically achieve a better performance by using the DaD approach than by using MTMs that are typically called for in TO-based designs. The DaD approach uses COTS materials, as a starter, and then tailors their material properties slightly in order to achieve the desired refraction or reflection characteristics. While lens designs based on RO do not suffer from the drawback of TO-based designs, as they do not call for MTMs, they still require dielectric materials that are typically unavailable off-the-shelf. Furthermore, there appears to be no systematic approach available in the literature for addressing this important problem and artificially synthesizing the desired material properties. Significantly, the losses and dispersive effects of the DaD-based materials are relatively small compared with those of MTMs used for the same purpose. We illustrate the application of the DaD technique with a number of practical examples involving flat lenses and reflectarrays and present some representative results in figures 8–11. A performance comparison of TO- and DaD-based lenses is shown in figure 10, whereas figure 11 demonstrates the efficacy of the reflectarray (metasurface) design.

We mention here that, to the best of our knowledge, a TO-based design that delivers a comparable performance has yet to be developed, and the question of its realizability remains unanswered as yet.

## 4. Conclusion

We have discussed the problem of designing electromagnetic devices by using the FT method, as an alternative to the TO approach. Both the FT and TO approaches have been viewed using the framework of the GSM method in order to explain why the TO approach leads us to unrealistic and unrealizable designs, while the FT approach does not suffer from these shortcomings, albeit at the cost of slightly compromising the performance of the electromagnetic device under consideration, in a manner that is acceptable for most real-world applications.

## Competing interests

We declare we have no competing interests.

## Funding

We received no funding for this study.

## Acknowledgements

The authors are grateful to our colleagues Ravi Kumar Arya and Shaileshachandra Pandey for providing helpful information on the dial-a-dielectric approach mentioned in §3.

## Footnotes

One contribution of 14 to a Theo Murphy meeting issue ‘Spatial transformations: from fundamentals to applications’.

- Accepted June 12, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.