Cylindrical to rectangular coordinate transformation for planar phase front synthesis

Transcription

1 IET Microwaves, Antennas & Propagation Research Article Cylindrical to rectangular coordinate transforation for planar phase front synthesis ISSN Received on 6th July 207 Revised 9th October 207 Accepted on 28th Noveber 207 E-First on 2st February 208 doi: 0.049/iet-ap Seyed Hassan Sedighy, Caner Guclu 2, Mohaad Khalaj Airhosseini 3, Filippo Capolino 2 School of New Technologies, Iran University of Science and Technology, Tehran, Iran 2 Departent of Electrical Engineering and Coputer Science, University of California at Irvine, Irvine, CA 92697, USA 3 School of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran E-ail: sedighy@iust.ac.ir Abstract: Coordinate transforation technique is eployed to engineer a etaaterial slab that enables planar wave eission fro a planar or line localised source. First a rigorous cylindrical to rectangular coordinate transforation is applied resulting in with a sheet current generating a plane wave with ipedance atching at the boundary between the device and the surrounding ediu. Then, based on physical reasoning, the aterial constitutive paraeters are further reduced to having only one entry of the spatially varying pereability tensor instead of two entries as discussed in the references. This further reduction is equivalent to odifying the near field in the transfored doain. Finally, due to a specific u-near-zero condition near the origin, the sheet source is transfored to a localised one in the transfored doain and via full-wave siulations, the authors show that the field is still propagating as a plane wave. Moreover, the transission line () etaaterial ipleentation with discretised spatially varying pereability is designed. The capacitor- and inductor-loaded two-diensional grid is used to synthesise the required spatially varying relative pereability in the wavefront converter etaaterial slab. The etaaterial design is proven to successfully synthesise a planar wave front using a icrowave circuit siulator. Introduction The application of coordinate transforation in electroagnetics has received increasing attention in the last years. This technique, which relies on the for-invariance of Maxwell's equations through a spatial coordinate transforation, provides a ethodology to anipulate electroagnetic waves by engineering non-hoogeneous anisotropic constitutive paraeters, naely the perittivity and the pereability [ 4] based on a transforation schee. Novel applications of transforation electroagnetics also known as transforation optics include various devices such as the invisibility cloak [5], the electroagnetic concentrator [6, 7], illusion device [8], super absorber [9] and super scatter [0]. Here we present a directive eission device which can be integrated into antenna design as another application of the coordinate transforation. A coordinate transforation for converting cylindrical to plane waves was discussed in [], based on spatially dependent constitutive aterial paraeters, expressed as tensors in general. Moreover, as an application of the wave front conversion, a highly directive antenna was proposed in [2] by apping a half circle into a slab which needs infinite perittivity near the origin. A highly directive ulti-bea eission has been proposed in [3, 4] by proposing a wide triangle segent apping into narrow ones. A directive eission coposed of feeding source covered by an anisotropic coposite etaaterial has been proposed in [5, 6]. This structure is engineered by transforing a radiating cylindrical space into a rectangular one. A systeatic design procedure based on quasi-coordinate transforation technique has been proposed in [7] for the design of a lens antenna. In this procedure, the required three-diensional (3D) perittivity profile which is obtained by transforation of the tiny cells can liit the design and ipleentation. In this paper, a general coordinate transforation is applied to synthesise a etaaterial slab that enables planar TM-wave eission fro a planar or line localised source. The presented design also applies a reduction of constitutive paraeters which decreases the nuber of spatially dependent constitutive paraeters and at the sae tie preserves the advantages of the transforation schee such as eission with a planar wavefront IET Microw. Antennas Propag., 208, Vol. 2 Iss. 5, pp The Institution of Engineering and Technology 207 and wave atching to the surrounding space. This reduction procedure which is equivalent to odifying the near field in the transfored doain is justified analytically based on the Maxwell equations which have not been proved, previously. As a result, the final etaaterial design coprises constant constitutive paraeters with respect to position, except for only a single pereability tensor entry that varies only along a single principal direction (say x). Therefore, the resulting constitutive paraeters are siple and easy to ipleent in practice. Although our proposed transforation schee in general for is siilar to the ones proposed in [5, 6], the final proposed constructive paraeters are only spatially dependent in one direction copared with the ones introduced in these references which are spatially dependent in two directions. Moreover, the perfect ipedance atching at the outer boundary enhances our proposed slab perforance. In addition, as discussed in the next sections, due to a specific u-near-zero (MNZ) condition near the origin, the sheet source is transfored to a localised one in the transfored doain and therefore, the localised source can be planar or line eitter which enhances the design flexibility rather than the references works. The proposed schee of transforation is shown to successfully enable plane wave eission based on full-wave siulations. To validate the design concept with etaaterial ipleentation based on unit cells, the designed etaaterial slab with constitutive paraeters, (μ x, μ y, ε z ) is realised as a 2D transission line () etaaterial. For this purpose, the unit cells are loaded with sae proper luped reactive eleents to achieve the required effective pereability. Finally, the designed network is siulated at 00 MHz in a icrowave circuit siulator to validate the operation of the engineered etaaterial. We show that the transforation schee can be used for wavefront conversion even with discretisation of the spatial dependence function with respect to the etaaterial building block period for the ipleentation with a 2D etaaterial. The designed etaaterial requires the ipleentation of negative pereability which in principle cannot a wideband using foster circuit eleents. 84

2 The syste under investigation is invariant along the z-direction, thus it is essentially the 2D proble. Therefore assuing the presence of only transverse-to-z agnetic (TM) waves, the agnetic field has the x and y coponents and the electric field has only the z coponent. Effectively, only μ x, μ y and ε z in (2) govern the propagation of the TM waves (the subscripts x, y, and z denote the xx, yy, and zz entries of the tensors, respectively). As for the transforation schee, we adopt specific functions x = f ρ = wρ /R and y = g ϕ = Lϕ /π. Here the choice of g/ϕ and f /ρ as constants is otivated by having siple to ipleent functions. This leads to μ x = π L x, μ y = L π x, ε z = π L 2 R x (3) w Fig. Coordinate transforation schee fro virtual space to real space with cylindrical to rectangular coordinate transforation, where the colour coded lines illustrate the transforation apping Fig. 2 Required relative constitutive paraeters μ x, μ y and ε z for the ideal (dotted blue curves), odified (solid black curves) cases, and step-wise discretised μ x (dashed red curve) 2 Cylindrical to plane wave coordinate transforation The coordinate transforation concept is graphically shown in Fig.. For converting a cylindrical wave to a plane wave, the concentric circles in the virtual space x, y should be transfored to the parallel vertical lines in the real space x, y. There are various possible choices for this kind of transforation [2, 5, 6]. Here we start with a general schee which reads as x = f (ρ ) = f x 2 + y 2 (a) y = g(ϕ ) = g arctan y x (b) z = z (c) Assue that the virtual space (denoted by pried coordinates) is a vacuu, thus isotropic, (with relative perittivity and pereability as ε = I and μ = I, respectively, where I is the identity tensor). Based on the etric invariance of Maxwell's equations [, 3 5, 8, 9], the constitutive relative perittivity and pereability tensors of the ediu in the real space is ρ ( f /ρ ) ε = μ = g/ϕ ρ + z^z^. ( f /ρ )(g/ϕ ) x^x^ + g/ϕ ρ ( f /ρ ) y^ y^ (2) In the transfor schee, we expect the cylindrical wave in the virtual space to ap to plane wave propagation in the +x-direction in real space. The ipedance defined at x = w as μ y x = w ε z x = w η 0 = L πr η 0 (4) is required to be continuous between the slab and the free space at x = w. Indeed, when the slab is chosen with the diensions L/π = R = w =.5, the ipedance at x = w will be atched to the surrounding free space wave ipedance. For the chosen size, the perittivity and pereability (called direct paraeters fro now on) versus x are plotted in Fig. 2 with blue dashed curves. In order to validate the design forula, full-wave siulations based on the frequency-doain finite eleent ethod (ipleented via COMSOL Multi-Physics) are carried out for the design with the above direct paraeters. In the 2D siulation doain, the fields are invariant in the z-direction. The slab is excited by a z-directed surface current density (here referred to as sheet source) at x = 0. The sheet source corresponds to the line current in the virtual space noting that the origin of the virtual space aps to the edge of the etaaterial at the left (x = 0) side. The etaaterial doain is surrounded by vacuu on all sides of Fig. 2. Perfectly atched layers are ipleented at the boundary of the full-wave siulation doain to suppress reflections and iicking unbounded space. In Figs. 3a and b, we report E z and H y, respectively, in isotropic x y free space at a noinal frequency of 00 MHz, where the cylindrically eanating waves are observed fro a z-directed current line source. Then in Figs. 3c and d, we report the case after coordinate transforation, leading to the etaaterial slab excited by the sheet current. In practical cases, we consider a etaaterial slab only on one side of the source. In this case, ost of the power is eitted towards the free space side (the x direction) due to the high ipedance is seen fro the source sheet looking towards the etaaterial side (+x direction). This can be attributed to the growing μ y and decrease ε z close to x = 0. In order to eliinate the inefficient eission towards the etaaterial and to further siplify the spatial dependence of the constitutive paraeters, we ake another step of reduction in the constitutive paraeters under certain assuptions introduced hereafter (refer to Appendix for ore details). However, the reduction approach has been introduced in [20], here we focus on analytical prove of this approach. Let us look at the TM wave equation (derived in Appendix) in ters of the constitutive paraeters relevant to TM polarisation that vary only as a function of x ε z μ y 2 x 2E z + ε z μ x 2 y 2E z + ω 2 ε 0 μ 0 E z + ε z x μ y x E z = 0. Substituting the constitutive paraeters in (3) into (5) yields (5) IET Microw. Antennas Propag., 208, Vol. 2 Iss. 5, pp The Institution of Engineering and Technology

3 In (8), the constitutive paraeters only appear in two places as the products ε z μ x and ε z μ y which are, in general, the function of x. The wave equation in (8) guides us to ake another step in the reduction of the constitutive paraeters as μ x = μ x /a, μ y = μ y /a, and ε z = aε z (by using a spatially varying paraeter a) where the products ε z μ x and ε z μ y are sae as the products ε z μ x and ε z μ y, respectively. With the odified constitutive paraeters μ x, μ y, and ε z we can rewrite the wave equation as 2 ε z μ y x 2E z + 2 ε z μ x y 2E z + ω 2 μ 0 ε 0 E z + ε z x μ y x E z = 0 (9) Fig. 3 Noralised instantaneous z-polarised electric field (and y- polarised agnetic field in the second row) distribution in the x y plane for (a), (b)isotropic ediu, (c), (d) Converter slab using direct paraeters excited by a surface electric current sheet source at x = 0 along the edge of the etaaterial slab R/w 2 2 x 2E z + + R/w 2 x (π/l)(r/w)x 2 2 y 2E z + ω 2 μ 0 ε 0 E z x E z = 0. It can be observed that if we consider a plane wave solution (i.e. a wave not varying in y and z) of (6) this is not a plane wave with constant aplitude ainly because of the last ter, and indeed the field aplitude would vary in x. However, for large x, the solution tends to be that of a plane wave with a constant aplitude that propagates along x (guaranteeing that the second ter regarding the derivative with respect to y also vanishes). For large x, such a wave is a solution of the wave equation as (6) R/w 2 2 x 2E z + ω 2 μ 0 ε 0 E z = 0 (7) where the last ter in (6) has been neglected. This plane wave is consistent with the transforation in (2) because in the original x y doain [Fig. a], the corresponding cylindrical field solution is represented by Hankel function that near the origin does not behave siply as e jkρ / ρ due to near field represented by a logarithic function. Only far fro the origin, the cylindrical wave asyptotically behaves as e jkρ / ρ which corresponds to a pure plane wave in the transfored real space. Therefore, if in the transfored doain, we neglect the last ter of (6), the equation becoes the one that adits a constant-aplitude plane wave as a solution. We ai to ap ainly the propagative far-field fro the source in the virtual x y space onto a plane wave in the transfored space, whose far-field solution (i.e. for large x) is directly obtained by (5) after dropping its last ter. We will exploit this fact in siplifying the constitutive paraeters with the next step of odification. Rewriting (5) by neglecting the last ter, the TM wave equation in the etaaterial slab reduces to ε z μ y 2 x 2E z + ε z μ x 2 y 2E z + ω 2 ε 0 μ 0 E z = 0. (8) Note that the first line in (9) and the left-hand side ters in (5) are the sae since ε z μ x = ε z μ x and ε z μ y = ε z μ y. Moreover, by choosing μ y = μ y /a as a constant (thus a μ y ) we can guarantee that the last ter in (9) is zero. Accordingly, (8) and (9) becoe identical, and the etaaterial with the odified paraeters is expected to guide the waves in the sae anner as the original etaaterial under the approxiations that these equations are derived. Therefore the solution of (9) is asyptotically (for large x) equal to the one of (6). Eventually, we have defined a new step of odifying the constitutive paraeters without altering the wave equation of the propagation waves for large x, whose validity is also verified by full-wave siulations. Consider the three constitutive paraeters μ x, μ y, ε z that the TM waves propagation depends on, then divide μ x, μ y by a = μ y / χ and ultiply ε z by a = μ y / χ, where χ is a proper constant used for atching purposes. This further paraeter odification in the transfored doain keeps ε z μ x and ε z μ y functions intact. The odified relative constitutive paraeters are thus expressed as μ x = π L 2 χx 2, μ y = χ, ε z = R 2 w χ (0) Note that now only μ x depends on x whereas μ y and ε z are invariant with respect to position. A plane wave propagating along the x-direction at the aterial boundaries (in the real space) should be atched to the isotropic region to avoid reflection. This iplies that the ipedance TM wave propagating along the x-direction should be atched to the surrounding ediu's wave ipedance. Here the surrounding ediu is taken as a vacuu with the wave ipedance η 0, and the wave ipedance η TM = η 0 μ y /ε z should be atched to η 0 at x = w. Accordingly, the paraeter χ is set to be equal to R/w. With this condition, the final relative constitutive paraeters are siplified as μ x = Rπ2 wl 2 x2, μ y = R w, ε z = R w () In Fig. 2, the odified constitutive paraeters with L = 2R = 2w = 3, are reported with solid black curves where μ y = ε z. The only varying ter is μ x whose range is fro 0 to 2.5. Moreover, Fig. 2 shows a possible stepwise approxiation for the only varying paraeter μ x, which will be used in Section 3. The converter slab ade of etaaterial with the odified constitutive paraeters reported in Fig. 2 with solid black curves is also siulated in the full-wave environent. In Figs. 4a and b, we report a tie-shot of the z-polarised electric and agnetic field distribution at 00 MHz, respectively, using the etaaterial slab with odified paraeters excited by an electric current sheet. Note that this tie, the plane wave eanates in both + and x directions with equal aplitude. However, it is interesting to show that not only the electric surface current but also a line current source leads to plane wave eission when using the etaaterials slab. To this ai, we report the electric and agnetic field in Figs. 4c and d 86 IET Microw. Antennas Propag., 208, Vol. 2 Iss. 5, pp The Institution of Engineering and Technology 207

4 excited by a line source, all noralised by their respective axia at the plane wave regions. Strikingly, the slab is able to facilitate eission of a plane wave also by a line source. Note that for the transforation schee in Fig., every point in the cylindrical virtual space, except the origin, aps to a single point in the rectangular slab in the real space. The origin in the cylindrical virtual space, on the other hand, aps to the edge on y line in the rectangular real space. In the ideal transfored case, the z-directed line current at the origin in virtual space corresponds to a z-directed sheet current along the x = 0 side of the slab. However, we have shown that the etaaterial slab can lead to planar wave front radiation even for the line current at the origin. This can be attributed to the fact that a line current in the virtual space at an infinitesial distance fro the origin, eits cylindrical waves alost identical to the source at the origin; at the sae tie, it aps to a point at an infinitesial distance to the origin at the rectangular real space. Fro constitutive paraeters aspect, we observe that μx vanishes at the x = 0 edge of the slab. This corresponds to a resonance condition with MNZ where the field profile of the line source at y = 0 is transferred along the y-direction alost without losing intensity and in phase. The transforation schee introduced here would also hold for the dual case, where TE waves could be altered by siply ipleenting a single spatially-varying perittivity tensor entry. Fig. 4 Noralised instantaneous z-polarised electric field (and ypolarised agnetic field in the second row) distribution in the x y plane for (a), (b)converter slab using odified paraeters excited by a surface electric current sheet source at x = 0 along the edge of the etaaterial slab and (c), (d) Converter slab using odified paraeters excited by a line source at location (x, y) = (0, 0) 3 etaaterial synthesis For a etaaterial deonstration by synthesising the designed constitutive paraeters, a approach is selected. In the approach, for a 2D periodic esh grid loaded by series luped ipedance as shown in Fig. 5, the effective diagonal pereability and perittivity can be described by [2] μx, y μ0 = Ldis + Fig. 5 Unit cell of the etaaterial with luped series eleents Z y, x, jωd (a) Sketch ap of the converter etaaterial surrounded by ISO ediu, synthesised by grids ade of (b) capacitor loaded cell for μx < μxiso, (c) Isotropic cell with μx = μxiso, (d) Inductor loaded cell for μx > μxiso IET Microw. Antennas Propag., 208, Vol. 2 Iss. 5, pp The Institution of Engineering and Technology 207 (2) where Ldis (H/) and Cdis (F/) are the distributed (per-length) inductance and capacitance of the, respectively, d is the icrostrip grid period, and Z x, y are the series luped ipedances along the x and y directions, respectively. The surrounding isotropic ediu (ISO ediu) is ipleented as the isotropic grid (without series ipedance loading) shown in Fig. 6c. The unloaded icrostrip is the background ediu, so the constitutive paraeters in the slab are noralised by the ISO ediu's perittivity and pereability which are calculated by inserting Z x, y = 0 in (2) leading to μxiso μ0 = μyiso μ0 = Ldis and εzisoε0 = 2Cdis. In the designed slab, we need spatially varying μx, whereas the required μy and εz (relative to the isotropic background) are constant and equal to the isotropic surrounding ediu. Therefore, it is sufficient to load the icrostrip grid only along the y-direction, and then constitutive paraeters of the etaaterial slab are μx Zy =+, ISO jωdl μx dis Fig. 6 The sketch ap of the proposed converter εzε0 = 2Cdis μy = μyiso εz =. εziso (3) To synthesise the constitutive paraeters expressed in () with these relations, we chose L = 2R = 2w = 3 sae as the siulated case above. Note that the only spatially varying paraeter is μx = (π 2 /4) x/w 2 μxiso. To ipleent μx < μxiso, the should be loaded with capacitor (Fig. 6b) whereas for μx > μxiso, the should be loaded with inductor (Fig. 6d). The grid ade of.5 wide icrostrip line is designed on an FR4 substrate with relative perittivity εfr4 = 4.5, dielectric thickness h = Thus, the distributed inductance and capacitance of the icrostrip are Ldis = 306 nh/ and ISO Cdis = 28.5 pf leading to μxiso = 28.96ε0,, y = 0.243μ0 and εz respectively. The eshsize is chosen as d = 5 c which is less than λg /8 at the design frequency, 00 MHz, where λg =.3 is the 87

5 Table Design paraeters for the converter slab μ x /μ x ISO Region Loading type Coponent < x/w < 0. capacitor pf < x/w < 0.2 capacitor pf < x/w < 0.3 capacitor pf < x/w < 0.4 capacitor pf < x/w < 0.5 capacitor 3.2 pf < x/w < 0.6 capacitor pf 0.6 < x/w < 0.7 without loading none < x/w < 0.8 inductor nh < x/w < 0.9 inductor nh < x/w < inductor 5.66 nh Fig. 7 2D noralised voltage distribution in the network (a) tiso ediu, (b) Converter slab surrounded by ISO ediu equivalent guided wavelength in the isotropic grid. Also, the whole structure is shown in Fig. 6a consists of unit cells (3.98λ g 3.98λ g ) that the converter slab is ade by 0 20 (2.65λ g.32λ g ) cells. The continuous change of μ x with respect to x, cannot be applied to the 2D grid etaaterial since the structure is ade up of unit cells with finite diensions. The step-wise approxiation μ x is applied in such a way that each cell bears the value of μ x corresponding to the x-position of the cell's centre, as presented in Fig. 2. The odified paraeters sapled at the centre of each cell, are tabulated in Table, with the corresponding luped reactive series coponent required to realise the. The required values of μ x range fro 0 to 2.5, and this range of values are realised by three different types of unit cells: capacitor loaded grids shown in Fig. 6b for μ x < μ x ISO, isotropic grids shown in Fig. 6c for μ x = μ x ISO, and inductor loaded grids shown in Fig. 6d for μ x > μ x ISO. Also, based on () and (3), the required capacitors and inductors are calculated and tabulated in Table. The grids should be terinated with a proper ipedance to iniise the reflection fro the edges, iicking an unbounded environent. This proper terination ipedance can be coputed via Bloch theory [9] and it depends on the propagation direction of a Bloch wave. Based on the siulation results presented in Figs. 3 and 4, we expect to observe a wave with alost planar wavefront propagating in the x-direction inside the ISO ediu with the wavenuber k x = ω μ x ISO μ 0 ε z ISO ε 0. Therefore, the edges of the grid on x and +x sides are terinated by the ISO ediu noral-incidence Bloch ipedance of 29 Ω which atches the ain coponent of the spatial spectru of the waves propagating along the x-direction. Note that since the power is expected to propagate ainly along x, the power propagating along y is negligible. Therefore, we leave the y-direction edges of the ISO ediu open, which supports the plane wave propagating along the x-axis. On the other hand, when siulating the ISO ediu without the anisotropic slab, the choice of terination ipedances is strongly dependent on the terination point for which the incidence angle of the ain Bloch wave coponent varies as a function of position since the waves eanate cylindrically fro the source. The Bloch ipedance varies for propagation directions defined by (k x, k y ) pairs varying between 29 and 42 Ω on the edges along the x and y directions. We used the 34 Ω Bloch terination value as a weighted average of the required atching ipedances over the terination nodes at the edges of the ISO ediu. The axiu reflection coefficients turn out to be 0. and indeed as shown in the following results, the little isatch led to no observable reflection at the grid edges. We use a coercial icrowave circuit siulator (Agilent's Advanced Design Syste, ADS) to siulate the structure. An AC voltage source is used to excite the network, which is located half of the cell away fro the left slab boundary. First, we siulate the network without the converter slab, i.e. the source in the ISO ediu. The noralised voltage distribution of the grid nodes is plotted in Fig. 7a at 00 MHz. The cylindrical wavefront can be seen in this figure clearly. In Fig. 7b, the network with a converter slab surrounded by ISO ediu is siulated. The noralised voltage distribution at the grid nodes, i.e. the centre of the cells as shown in Fig. 5, is plotted in Fig. 7b. The designed slab converts the cylindrical to the planar wavefront. Coparing the noralised electric field distribution in Figs. 3 and 4 and noralised voltage distribution in Fig. 7 verifies the ability of the designed converter slab to convert the cylindrical wavefront to planar one. 4 Conclusion The coordinate transforation was applied for cylindrical to plane wavefront conversion. In addition to conventional transforation ethod, the constitutive paraeters are engineered in an additional step to further siplify the spatial dependence. The final constitutive paraeters are constant except only a single spatially varying paraeter. Therefore, the proposed constitutive paraeters, perittivity, and pereability are siple and easy to ipleent in practice. First using full-wave siulations of a bulk etaaterial slab, we proved that the designed etaaterial enables localised sources to eit planar waves. The etaaterial slab with single pereability tensor entry is shown to convert TM wave front via full-wave siulations. The dual case for converting TE waves by requiring only a single spatially-varying perittivity tensor entry could be also utilised since controlling perittivity could be ore convenient, depending on the frequency of operation. Finally, a etaaterial ade of a planar icrostrip line grid loaded with luped capacitors and inductors were used to synthesise the proposed etaaterial converter slab. Both the fullwave siulation of a bulk etaaterial slab and the circuit siulation of the grid verified the concept of circular to a planar wave converter that is also ipedance atched to the free space region. 88 IET Microw. Antennas Propag., 208, Vol. 2 Iss. 5, pp The Institution of Engineering and Technology 207

6 5 References [] Sedighy, S.H., Khalaj-Airhosseini, M.: Effect of optical transforation order on design of arbitrary polygonal invisible cloak, JOSA A, 202, 29, (2), pp [2] Kwon, D.-H., Werner, D.H.: Transforation electroagnetics: an overview of the theory and applications, IEEE Antennas Propag. Mag., 200, 52, pp [3] Leonhardt, U.: Optical conforal apping, Science, 2006, 32, pp [4] Leonhardt, U., Philbin, T.G.: Transforation optics and the geoetry of light, Prog. Opt., 2009, 53, pp [5] Pendry, J.B., Schurig, D., Sith, D.R.: Controlling electroagnetic fields, Science, 2006, 32, pp [6] Hasan Sedighy, S., Khalaj-Airhosseini, M.: Arbitrary n-sided irregular polygonal electroagnetic transfored edia, J. Opt. Soc. A. A, 202, 29, pp [7] Li, T., Huang, M., Yang, J., et al.: Three diensional electroagnetic concentrators with hoogeneous aterial paraeters, Prog. Electroagn. Res. M, 20, 8, pp [8] Lai, Y., Ng, J., Chen, H., et al.: Illusion optics: the optical transforation of an object into another object, Phys. Rev. Lett., 2009, 02, p [9] Ng, J., Chen, H., Chan, C.: Metaaterial frequency-selective superabsorber, Opt. Lett., 2009, 34, pp [0] Yang, T., Chen, H., Luo, X., et al.: Superscatterer: enhanceent of scattering with copleentary edia, Opt. Express, 2008, 6, (22), pp [] Jiang, W.X., Cui, T.J., Ma, H.F., et al.: Cylindrical-to-plane-wave conversion via ebedded optical transforation, Appl. Phys. Lett., 2008, 92, pp [2] Zhang, J., Luo, Y., Chen, H., et al.: Directive eission obtained by coordinate transforation, Prog. Electroagn. Res. (PIER), 2008, 8, pp [3] Jiang, Z.H., Gregory, M.D., Werner, D.H.: Experiental deonstration of a broadband transforation optics lens for highly directive ultibea eission, Phys. Rev. B, 20, 84, p. 65 [4] Wu, Q., Jiang, Z.H., Quevedo-Teruel, O., et al.: Transforation optics inspired ultibea lens antennas for broadband directive radiation, IEEE Trans. Antennas Propag., 203, 6, (2), pp [5] Tichit, P.H., Burokur, S.N., Gerain, D., et al.: Design and experiental deonstration of a high-directive eission with transforation optics, Phys. Rev. B, 20, 83, (5), p [6] Tichit, P.-H., Burokur, S., Gerain, D., et al.: Coordinate-transforationbased ultra-directive eission, Electron. Lett., 20, 47, pp [7] Ebrahipouri, M., Quevedo-Teruel, O.: Bespoke lenses based on quasiconforal transforation optics technique, IEEE Trans. Antennas Propag., 207, 65, (5), pp [8] Tang, C.A.W., Kallos, E., Song, W., et al.: Discrete coordinate transforation for designing all-dielectric flat antennas, IEEE Trans. Antenna Propag., 200, 58, pp [9] Cai, W., Chettiar, U.K., Kildishev, A.V., et al.: Optical cloaking with etaaterials, Nature Photonics, 2007,, pp [20] Schurig, D., Mock, J.J., Justice, B.J., et al.: Metaaterial electroagnetic cloak at icrowave frequencies, Science, 2006, 34, (580), pp [2] Wong, J.K., Balain, K.G., Eleftheriades, G.V.: Fields in planar anisotropic transission-line etaaterials, IEEE Trans. Antennas Propag., 2006, 54, pp Appendix The TM wave equation in ters of z-directed electric field is derived next. Assuing invariance along the z-axis together with diagonalisable constitutive paraeter tensors, Maxwell's equations can be separated for TM-to-z and TE-to-z waves. TM-to-z waves possess only H x, H y, and E z that satisfy the equations (derived fro the curl expression in Maxwell's equations) y E z = jωμ x H x, x E z = jωμ y H y, x H y (4) y H x = jωε z E z By anipulating the first two equations above as y jωμ 0 μ x y E z x jωμ y x E z = y H x = x H y (5) We can construct the suation ters on the left hand side of the last line in (4). Then the agnetic field coponents where the su of the left hand side of the equations in (5) is equal to jωε z E z. As such, the wave equation for the electric field is obtained as x μ y x E z + y μ x y E z + ω 2 μ 0 ε 0 ε z E z = 0 (6) Keeping in ind that pereabilities above are functions of x only in our transforation schee, the derivatives can be expanded to reach the wave equation given as ε z μ y 2 x 2E z + ε z μ x 2 y 2E z + ω 2 μ 0 ε 0 E z + ε z x μ y x E z = 0. (7) IET Microw. Antennas Propag., 208, Vol. 2 Iss. 5, pp The Institution of Engineering and Technology