Transformation Electromagnetics and Optics › files › articles › 15328ab5bdc08e.pdfled to superscatterer [15] and superabsorber [16] designs. By manipulating the spatial loss

Forum for Electromagnetic Research Methods and ApplicationTechnologies (FERMAT)

Transformation Electromagnetics and OpticsDo-Hoon Kwon

Department of Electrical and Computer Engineering, University of Massachusetts Amherst, Amherst, MA 01003,USA

(Email: [email protected])

Abstract—An overview of the transformation electromagnet-ics/optics design technique is presented. Illustrative exampledesigns for invisibility cloaks, beam control devices, lenses as wellas applications to radiation problems are shown. Quasi-conformaltransformations in two dimensions and their applications tocloaks and virtual arrays are discussed.

Index Terms—Array antennas, coordinate transformations,invisibility cloaks, lenses, quasi-conformal transformations.

I. I NTRODUCTION

T RANSFORMATION Electromagnetics/TransformationOptics (TE/TO; abbreviated as TO hereafter) [1]–[4] isa systematic design methodology for novel microwave andoptical devices that feature unconventional wave-materialinteractions. The principle of TO was introduced in 2006 byPendryet al. [5] as a technique for controlling the behaviorof electromagnetic wave propagation. It was shown thatthis could be achieved generally by using anisotropic andinhomogeneous medium of customized permittivity andpermeability parameters, together with the introduction ofinvisibility cloaks as the first and prominent example. Thesame principle in the ray-optical limit was also demonstratedby Leonhardt [6]. Experimental demonstration of two-dimensional (2-D) invisibility cloaks in the microwavefrequency range [7] using the state-of-the-art metamaterialstechnology [8] sparked a great interest in the TO designtechnique and research on novel devices that can be designedbased on it.

The TO technique is formulated on the basis of ‘forminvariance’ of Maxwell’s equations under coordinate transfor-mations and its novel ‘materials interpretation’ [9]. Within thisbasic principle, the TO technique gave rise to several subfieldsdepending on the target application of a transformation orthe novelty of a transformation itself. Utilizing mappingsthattransform the bounding surface of a physical object, designrecipes for anisotropic coating materials for ‘reshaping’[10]or ‘masking’ [11] the scattering response have been presented.By incorporating an anti-object and/or an illusion object intothe coordinate transformation for invisibility cloaks, cloaking-at-a-distance [12] and illusion optics [13] were introduced.An anti-cloak [14] was designed to partially defeat the pur-pose of an invisibility cloak. Transformations that resultincomplementary media combined with scatterers and absorbersled to superscatterer [15] and superabsorber [16] designs.Bymanipulating the spatial loss profile of cloaking media, perfectwave absorption was shown to be possible and a practicalmetamaterial implementation was proposed [17]. In 2008,introduction of finite embedded coordinate transformation[18]

enabled designs that allowed modification of the incidentfields outside the device boundary. Designs based on the finiteembedded transformation include beam shifters/splitters[18],beam bends/expanders [19], flat-focusing lenses [20], reflec-tionless waveguide transitions and bends [21]–[23], and re-flectionless narrow slits [24]. Application to radiation prob-lems involves an antenna or array that is either inside or inclose proximity to a volume under transformation. Source-embedding transformation media that transform the shape ofan array without changing scanning characteristics have beenreported [25], [26]. Generation of single or multiple directivebeams from an electrically small source by transformationmedia realized using metamaterials has been proposed anddemonstrated [27], [28]. Designs for an array of virtual dimen-sions and an antenna at a virtual location have been reported[29], [30]. An application of the TO technique to the designof these devices typically calls for a medium that is not onlyrequired to be anisotropic and inhomogeneous, but magneticas well as electric at the same time. The magnetic property ofconstituent media poses challenges in fabrication and resultsin narrowband responses. To mitigate this problem, quasi-conformal (QC) coordinate transformation was introduced,together with the carpet cloak [31], [32]. Although limitedto the 2-D TE polarization, quasi-conformal TO (QCTO)devices can be realized using only dielectric constituent media,capable of broadband operation. Subsequently, a variety ofQCTO device designs and realizations have been reported,such as flattened Luneburg and fish-eye lenses [33], [34],three-dimensional (3-D) flattened Luneburg lens [35], andplanar focusing antennas [36], [37].

Plasmonics is the most recent area that is under active TOresearch. In [38], manipulation of surface plasmon polaritonpropagation was suggested via TO-enabled devices in theoptical regime. In a deep subwavelength scale in plasmonnanosystems, the TO technique is emerging as an elegantand versatile methodology for the analysis and design oflight-nanoparticle interactions [39]. Plasmonic particles andparticle systems for light harvesting and energy concentrationin a uniform background field were analyzed by applyingcoordinate transformations to analytical solutions of a dipoleradiation problem in planar geometries. Energy harvestingnanostructures such as nanowire pairs [40], crescent-shapedcylinders [41], and touching nanoparticle dimers [42] wereanalyzed. The TO technique provides valuable physical in-sight that is not provided by traditional numerical analysisapproaches.

The underlying principle of TO, i.e., the governing equa-tions of physical phenomena being form-invariant under co-

1


x

y

x’

y’

(a) (b)

Fig. 1. Coordinate transformation for an example 2-D TO device, a fieldconcentrator [49]. (a) Original(x, y, z) system. (b) Transformed(x′, y′, z′)system. The outer black circular contour defines the boundaryof this TOdevice. The inner contour bounds the region having concentrated fields.

ordinate transformations, is not limited to electromagnetics.Notably, the TO technique is being actively applied to novelacoustic devices, e.g., annular [43], [44] as well as ground-plane cloaks [45]. Recently, the coordinate transformationtechnique has been extended to thermodynamics and cloakingof heat flow [46], [47].

In this paper, an overview of the TO theory and devicedesigns are presented. Section II reviews the fundamental TOtheory. Invisibility cloaks are presented in Section III. Lensesas well as beam control devices are treated in Section IV.Applications to radiation problems by antennas and arraysare discussed in Section V. Section VI presents the QCTOtechnique and associated designs. Finally, Section VII providesa summary and conclusion of the TO technique.

II. T HEORY

The TO design approach is built on the basis of form invari-ance [48] of Maxwell’s equations under coordinate transfor-mations. In other words, for a given coordinate transformation,Maxwell’s equations retain the same form in the transformedcoordinates as that in the original coordinates. In the original(x, y, z) coordinates, consider time-harmonic electric fieldEand magnetic fieldH at an angular frequencyω = 2πf in amedium characterized by permittivity and permeability tensorsǫ andµ (an ejωt time dependence assumed and omitted). Atsource-free points, they satisfy Maxwell’s curl equations:

∇×E = −jωµH, (1a)∇×H = jωǫE. (1b)

Now, consider a transformation to a new(x′, y′, z′) systemdescribed by

x′ = x′(x, y, z), (2a)

y′ = y′(x, y, z), (2b)

z′ = z′(x, y, z). (2c)

Since Maxwell’s equations take the same form in any co-ordinate system, fields(E′,H′) in the transformed (primed)system satisfy the same set of curl equations as (1) at the same

frequency, i.e.,

∇′ ×E′ = −jωµ′H′, (3a)∇′ ×H′ = jωǫ′E′ (3b)

in terms of a new set of medium tensor parametersǫ′ andµ

′ in the (x′, y′, z′) system. Using (2) in (1), quantitiesin the transformed system can be found in terms of thosein the original system. Let the Jacobian matrixA of thetransformation be written as

A =∂(x′, y′, z′)

∂(x, y, z)=

∂x′/∂x ∂x′/∂y ∂x′/∂z∂y′/∂x ∂y′/∂y ∂y′/∂z∂z′/∂x ∂z′/∂y ∂z′/∂z

. (4)

Then,ǫ′ andµ′ are related toǫ andµ via [9], [48]

µ′ =

AµAT

|A| , (5a)

ǫ′ =

AǫAT

|A| . (5b)

Fields in the two coordinate systems are related via [48]

E′ = (A−1)TE, (6a)

H′ = (A−1)TH. (6b)

An example transformation is illustrated in Fig. 1, whichis associated with a field concentrator [49]. A square gridof constantx- and y-lines in Fig. 1(a) is mapped to thedistorted grid in Fig. 1(b). If a plane wave propagates in the+x̂ direction (to the right), the red lines represent individualrays in a tube of rays (indicated by arrows) and the blue linesrepresent equi-phase contours. The rays entering the deviceare bent toward its center leading to a higher field in the innerregion (higher ray density).

Both Figs. 1(a) and (b) represent a single electromagneticphenomenon, i.e., a plane wave propagating in free space, intwo different coordinate systems via (1) and (3), respectively.However, the latter permits an alternative interpretationdueto the form invariance. Although (3) is established in the(x′, y′, z′) system, it is a valid set of Maxwell’s curl equationsin any coordinate system. When (3) is interpreted in theoriginal (x, y, z) system, i.e., when primes are removed, (3)describes a new electromagnetic phenomenon, namely the fieldbehavior illustrated in Fig. 1(b) realized in the real(x, y, z)space. The medium parameters of the device enabling thisnew phenomenon are prescribed in (5). This is the ‘materialsinterpretation’ of a coordinate transformation [9].

Acceptable transformations used in the TO technique aregeneral. Any continuous or piecewise-continuous mappingsmay be used. Typically, transformation media will be inhomo-geneous (spatially varying), anisotropic (function of polariza-tion), and of magneto-dielectric nature (µ 6= µ0I andǫ 6= ǫ0I;µ0 = free-space permeability,ǫ0 = free-space permittivity,I = identity tensor). Strictly isotropic medium parametersresult if the transformation is conformal [50].

A TO device design boils down to finding a coordi-nate transformation that maps a known electromagnetic phe-nomenon into a novel, envisioned one. The medium parame-ters of the device are given by (5). Exact field distributionsare

2


x

y

b2a2

x’

y’

(a) (b)

(c) (d)

Fig. 2. 2-D circular annular cloak. (a) Original system. (b) Transformedsystem. Snapshots of the total magnetic field for (c) a bare PEC cylinder and(d) a PEC cylinder enclosed in a cloak. A TM-polarized incident plane wavepropagating along thex-axis illuminates a PEC cylinder of radiusa = 2λ(λ = free-space wavelength) in free space. The inner and outer radii of theannular cloak area = 2λ and b = 4λ. Numerical results in (c) and (d)are obtained using COMSOL Multiphysics, assuming a unit amplitude forthe incident magnetic field. Boundaries of the cloak are indicated by blackcircular contours.

found from (6). TO devices may be fabricated in a variety ofways. Among others, the metamaterials technology [8], [51],[52] is the prime candidate for realizing TO devices, whichhas been developing and maturing over the past decade.

III. I NVISIBILITY CLOAKS

Invisibility cloaks [5], [6] are the most-prominent designbased on the TO technique. They are based on the principleof mapping a zero-scattering entity in the original space into afinite volume in the transformed space. A point (zero volume)in three dimensions and a line (zero cross section) in twodimensions are candidates in the original space. Coordinatesystems for a 2-D circular annular cloak are illustrated inFigs. 2(a)–(b). Using the circular cylindrical(ρ, φ, z) systemdefined in terms of Cartesian coordinates by

ρ =√

x2 + y2, (7a)

φ = tan−1y

x, (7b)

z = z, (7c)

a cloak transformation from(ρ, φ, z) to (ρ′, φ′, z′) is obtainedusing the following mapping for the radial coordinate:

ρ′ =

{

b−ab

ρ+ a, 0 ≤ ρ ≤ bρ, ρ > b

. (8)

The two remaining coordinates are unchanged. Figs. 2(a)–(b)show constant-x and -y lines and their transformed contoursfor the case ofb = 2a. In thexy-plane, a single pointρ = 0is mapped to a circleρ′ < a; the circleρ < b is mapped to acircular annulus ofa ≤ ρ′ ≤ b.

From (5), the medium parameters (normalized to free-spacequantities) are found to be

ǫ′ρρ = µ′ρρ =

ρ′ − aρ′

, (9a)

ǫ′φφ = µ′φφ =

ρ′

ρ′ − a, (9b)

ǫ′zz = µ′zz =

(

b

b− a

)2ρ′ − aρ′

, (9c)

with the remaining elements of the tensors being equal tozero. It is noted that the circumferential components divergeto infinity and those in other directions converge to zero on theinner boundary . These pose challenges in accurate realizationof such a cloak design.

The performance of a cloak design may be tested andvalidated using numerical simulation.1 As an example, TM-mode plane wave scattering by a perfect electric conductor(PEC) cylinder is analyzed in Figs. 2(c)–(d). A bare PECcylinder scatters the incoming plane wave in all directions[Fig. 2(c)]. When it is enclosed in a cloak, the incident fieldis guided around the cylinder and emerges on the other sideas if it had passed through an empty space [Fig. 2(d)].

Numerous variations of the basic cloak design of Fig. 2have been reported from the geometrical viewpoint of thecloak transformation. As the first design that has edges in thecross section, a square cloak [49] was shown; 2-D polygonalcloaks [53], [54] were presented. Elliptical annular cloaks [55]and annular cloaks having uniform thickness [56] were de-signed. Cloaks having arbitrary curved cross sections [57], [58]were designed and numerically validated.

Various aspects of invisibility cloaks are of research interest.The cloak medium is inherently dispersive [5] leading tonarrow operating bandwidth, having a bandwidth upper bounddue to causality [59]. On the realization side, complexity of thecloak medium in both permeability and permittivity combinedwith unavailability of strong magnetic material and magneticresonances of dielectric material at higher frequency towardthe optical regime has raised interest in approximate sets ofmedium parameters. Such ‘reduced’ cloak parameters [60]require only one set of parameters out of permeability and per-mittivity tensor elements to be synthesized. An all-dielectric2-D cloak design in the near-IR regime has been reported in[61]. In [62], a unidirectional 2-D cloak realizing full-tensormedium parameters was presented.

1All simulation results in this work have been obtained using the commer-cial package COMSOL Multiphysics.

3


x’

y’

x’

y’

(a) (b)

b

(c) (d)

Fig. 3. Design and performance of 2-D beam control devices under a TE-polarized Gaussian beam illumination. (a) Transformed coordinates and (b)the electric field snapshot of a beam shifter. (c) Transformedcoordinates and(b) the electric field snapshot of a beam expander. The incident Gaussian beampropagates in the+x̂ direction and the minimum waist is4λ at x′ = 0. Forboth designs, device parameters area = 1 and the thicknessb = 8λ. Thedevice boundaries are indicated by black lines.

IV. B EAM CONTROL DEVICES AND FLAT LENSES

The cloaking transformation (8) belongs to a class oftransformations that are continuous across the outer boundaryof a device. Transformations for designing field concentrators[49] (Fig. 1) and field rotators [63] belong to this category.By definition, these devices cannot affect the field distributionoutside the device from that of the original configuration.Transformation devices that change the external field canbe obtained by introducing a discontinuity in the mappingacross the outer boundary. To this end, the ‘finite embedded’transformation [18] was introduced.

Two devices that are based on the finite embedded transfor-mations are shown in Fig. 3—a beam shifter [18] and a beamexpander [19]. They are designed for controlling propagationbehavior of a beam entering its planar interface on the leftat x′ = −b/2. Both devices modify they coordinate withx, z unchanged. In the beam shifter design [Fig. 3(a)], theycoordinate is linearly shifted in the+ŷ direction with respectto x within the device(−b/2 ≤ x′ ≤ b/2). The originalcoordinates are defined by constant-x and -y grid lines [e.g.,Fig. 2(a)]. An appropriate transformation fory is

y′ =

y, y < −b/2y + a

(

x+ b2)

, −b/2 ≤ x < b/2y, y ≥ b/2

, (10)

wherea is a constant defining the slope of the transformedconstant-y line segments. The mapping is continuous atx =−b/2, but discontinuous atx = b/2. The Jacobian matrix isof the form

A =

1 0 0a21 a22 00 0 1

(11)

and the resulting medium parameters are given by

µ′ = ǫ′ =

1

a22

1 a21 0a21 a

221 + a

222 0

0 0 1

. (12)

Fig. 3(b) shows a Gaussian beam entering from the left shiftedup as it propagates through the device. Upon exit, the beampropagates in the original direction of propagation. It is notedthat this beam shifter has a reflectionless exit boundary.

The transformed coordinates for a 2-D beam expander isshown in Fig. 3(c), where they coordinate is expanded linearlyfrom x′ = −b/2 to x′ = b/2. The transformation fory is givenby

y′ =

y, x < −b/2[

a(

x+ b2)

+ 1]

y, −b/2 ≤ y < b/2y, x ≥ b/2

, (13)

which is discontinuous atx′ = b/2. The constanta definesthe rate of expansion. The medium tensors are found from(12) using appropriate Jacobian matrix elements associatedwith (13). Fig. 3(d) shows the beam expansion function fora 2-D Gaussian beam. Unlike the beam shifter, 2-D beamexpanders suffer from reflections from the discontinuous exitboundary of the device atx′ = b/2. In a 2-D expansion, onlyone coordinate in the transverse plane to beam propagation,e.g.,y in (13), is expanded. This leads to modification to onefield quantity from the value in the original coordinates [Hin the TE polarization of Fig. 3(d)], resulting in impedancemismatch with the ambient medium (free space). This short-coming can be removed in a 3-D expander design if bothcoordinates transverse to the beam propagation direction arestretched by the same rate. Such an impedance-matched beamexpander/compressor design has been reported in [64].

In introducing discontinuities in a coordinate transformationto modify the field behavior outside a TO device, the originaland transformed volumes do not have to coincide. A 2-D beam bend can be designed by mapping a rectangulararea in the original coordinates into a circular sector. Thetransformed coordinates for a right-angle bend design areillustrated in Fig. 4(a); a square given by0 < x < a,0 < y < a is transformed into a right-angle sector0 ≤ ρ′ < a,0 ≤ φ′ < π/2. Specifically,

ρ′ = y, (14a)

φ′ =π

2

(

1− xa

)

, (14b)

z′ = z. (14c)

Continuity of the mapping atx′ = 0 results in a reflectionlessinterface. Although the interface aty′ = 0 is discontinuous,there is no stretch or compression of coordinates along the

4


x’

y’

a

(a) (b)

Fig. 4. 2-D right-angle beam bend. (a) Transformed coordinates. (b) Snapshotof the electric field due to a TE-polarized Gaussian beam propagating in the+x̂ direction. The dimension of bend is given bya = 16λ. The minimumwaist of the beam is4λ. The boundary of the device is indicated by a blackcontour.

boundary. Hence, they′ = 0 interface is also reflectionless.In contrast, the curved boundaryρ′ = a will be reflective.Written in the cylindrical coordinates, the medium parametersare found to be

µ′ρρ = ǫ′ρρ =

2a

πρ′, (15a)

µ′φφ = ǫ′φφ =

πρ′

2a, (15b)

µ′zz = ǫ′zz =

2a

πρ′. (15c)

The simulated performance of the right-angle bend undera TE-polarized Gaussian beam illumination is shown inFig. 4(b). A device of arbitrary bend angle can be easilyobtained by modifying (14b). The electrical dimension of thebend, i.e., the number of wavelengths between the entry andexit interfaces, can be changed by modifying thex-dimensiona of the original rectangular area with they-dimension fixed.

An important class of devices that can benefit from anapplication of the TO technique are lenses, for which tra-ditional design process amounts to curved interface shapingusing homogeneous dielectric media. The TO technique makesboth the lens shape as well as the lens medium availablefor design and optimization. A planar lens is amenable toeasier integration in an imaging system than is a curved lens.In free space, a far-field focusing lens converts the curvedwavefronts of a point source into planar wavefronts. Hence,a TO lens can be designed by mapping a curved equi-phasesurface into a planar surface. In 2D, a circular arc centeredat a source can be mapped into a straight line segment. Anexample transformation is illustrated in Figs. 5(a)–(b), where acircular segment is mapped to a rectangle such that the chordcoincides with the bottom side. If we choose to modify theycoordinate only, an appropriate mapping is given by [20]

y′ =

{

l√a2−x2−g

(y − g) + g, g ≤ y ≤√a2 − x2

y, otherwise. (16)

x

y

w2

x’

y’

(a) (b)

(c) (d)

g

Focus

al

Fig. 5. 2-D far-zone focusing lens. (a) Original coordinates. (b) Transformedcoordinates. (c) Snapshot and (d) magnitude of the electric field due to a TE-polarized cylindrical wave illuminating the bottom face of the lens originatingfrom the focal point. The width and thickness of the lens are2w = 16λ andl = 1.33λ. The distance between the focal point and the entry face of thelens isg = 5.33λ. The cylindrical wave having a Gaussian amplitude profilewith respect to angleφ′ that illuminates the lens is obtained using a customelectric current distribution inz < 0 [65] (not shown).

Using (11)–(12), the medium parameters are found to be

µ′xx = ǫ′xx =

√a2 − x′2 − g

l, (17a)

µ′xy = ǫ′xy =

x′(y′ − g)l√a2 − x′2

, (17b)

µ′yy = ǫ′yy =

1√a2 − x′2 − g

[

x′2(y′ − g)2l(a2 − x′2) + l

]

, (17c)

µ′zz = ǫ′zz =

√a2 − x′2 − g

l. (17d)

Using an angle-limited cylindrical wave originating from thefocal point (coordinate origin), Fig. 5(c) demonstrates itsconversion to planar wavefronts.

Stretch or compression of they-coordinate in transformingthe circular segment into the rectangle results in modificationof the wave impedance at the exit interfaceρ′ = a. Reflectionfrom the exit interface is recognizable from the ripples in|E| as shown in Fig. 5(d). An improved lens design havinga trapezoidal cross section has been reported in [65], wherereflection is significantly reduced by mapping the circular arcinto a straight line having the same length that result in a betterimpedance match.

V. A PPLICATION TO ANTENNAS AND ARRAYS

All the preceding TO-based device designs involve interac-tion between waves and materials in source-free regions. Whena space under transformation contains sources, they undergothe same transformation as the medium. Transformed sourcescan be different from the originals in location, size, shape, and

5


x

y

R222R1

x’

y’

(a) (b)

(c) (d)

Fig. 6. Conformal array using a TO shell. (a) Original coordinates. (b)Transformed coordinates. (c) 10-element linear array scanning in 30° offbroadside. (d) Conformal array on the surface of a perfect magnetic conductorcylinder with a TO-material shell. Green dots indicate the 2-D electric linesource locations in both systems. The length of the array isd = 2λ with anelement spacing of2λ/9. In (c)–(d), the inner and outer radii of the shell areR1 = 0.5λ, R2 = 1.1λ.

distribution. In particular, it is noted that even the dimensionof a source can be changed. For example, a 3-D transformationthat is capable of mapping a line current of finite length intoa surface current over a sphere has been reported [66].

A source transformation can be utilized to manipulate thegeometry of current distribution without changing its radiationcharacteristics. Fig. 6 shows a conformal array on a curvedmounting surface, which is enclosed by a TO medium shell.The objective is to make the conformal array generate thesame fields as a linear array in free space. This is achievedby transforming a free-space volume containing a linear array[a uniform linear array of electric line currents in Fig. 6(a)]into a curved array enclosed by a TO medium shell [a circulararray in Fig. 6(b)]. An appropriate transformation is givenby[25]

x = a(ρ′) cosφ′, (18a)

y = b(ρ′) sinφ′, (18b)

where

a(ρ′) =R2 − (d/2 + δd)

R2 −R1(ρ′ −R1) +

d

2+ δd, (19a)

b(ρ′) =R2

R2 −R1(ρ′ −R1). (19b)

ρ

z

ρ’

z’ρf( )

ρp( ’)ρg( ’)

(a) (b)

(d)(c)

ca

d

h

t

Fig. 7. Virtual monopole antenna created using a folded geometry. (a)Original coordinate with a monopole on a PEC ground plane. (b)Transformedcoordinates with a monopole at the bottom of a ground recess, embeddedin TO media. Snapshots of the circumferential magnetic fieldH′

φfor (c)

a monopole in an empty recess and (d) a monopole at the bottom of arecess filled with TO media. The geometries are rotationally symmetric withrespect to thez-axis. Green line segments indicate the quarter-wave monopoleantenna. The region in gray in (b) indicate a TO medium with negative-indexparameters. The geometry of the example case are given bya = λ, c = 2λ,h = t = λ/2, andd = λ.

Designed for transforming a linear array of lengthd into acurved array on a circle of radiusR1, (18) maps a line segment−d/2 − δd < x < d/2 + δd, y = 0 into a circleρ′ = R1.The transformation is continuous atρ = ρ′ = R2. Sincethe original coordinates (x, y) are expressed in terms of thetransformed coordinates (ρ′, φ′), it is easier to find the Jacobianmatrix A using the relationship

A =∂(x′, y′, z′)

∂(x, y, z)=

[

∂(x, y, z)

∂(x′, y′, z′)

]−1

. (20)

The medium tensor parameters are found via (5). For the given2-D sources and the mapping (18), the transformed sourcesare the same line sources at new locations carrying the samecurrents [67].

For a 2-D, 10-element linear array along thex-axis with aconstant element spacing, original and transformed coordinatesand source locations are illustrated in Figs. 6(a)–(b). Theelement spacing of the conformal array is not uniform. Fora scan at 30° from broadside, snapshots of the electric fieldsin Figs. 6(c)–(d) show identical distributions outside theTOshell. Without the shell, the same cylindrical array cannothavethe same scanning characteristics.

Another TO application to antennas of interest is virtual

6


antennas and arrays, where the dimension or location of aphysical system is significantly modified from that of theoriginal system without changing its radiation characteristics.Fig. 7 illustrates a virtual monopole design realized usingafolded geometry [68]. With reference to the original (virtual)radiation configuration of a monopole antenna on a PECground plane atz = 0 shown in Fig. 7(a), it is desired that aphysical antenna located inside a ground recess and enclosedby proper TO media be designed as shown in Fig. 7(b) suchthat it has the same far-field radiation characteristics. Thisis of particular interest for a pattern having the maximumdirectivity right on the ground plane because the transformedconfiguration will not have any physical structure above thexy-plane.

The volume bounded byz = 0 andz = f(ρ), indicated bythe black dash-dot contour in Fig. 7(a), is folded below thexy-plane into a recess. It is mapped to the volume betweeng(ρ′) andp(ρ′) in Fig. 7(b). This is achieved by the followingmapping inz [29]:

z =

z′, z ≥ 0f(ρ′)p(ρ′) , p(ρ

′) ≤ z′ < 0f(ρ′)

p(ρ′)−g(ρ′) [z′ − g(ρ′)], g(ρ′) ≤ z′ ≤ p(ρ′)

. (21)

Coordinatesρ andφ remain unchanged. Conducting boundaryis inherited from the original problem for the slanted and flatwalls of the recess. A negative slope betweenz and z′ inp(ρ′) ≤ z′ < 0 results in a negative-index material (NIM)in the region, as indicated by the gray shade in Fig. 7(b). Ifh = t = d/2 is chosen, the volume containing the translatedmonopole(ρ′ < a,−d < z′ < −t) becomes free space; theNIM has both permittivity and permeability equal to −1, whichare the parameters of the Veselago-Pendry superlens [69]. Thevolume with a triangular cross sectionρ′ > a, g(ρ′) < z′ <p(ρ′) is an anisotropic medium that guides waves up along theslated recess wall without reflection.

Fig. 7(c) shows the magnetic field due to a monopole atthe bottom of an empty recess, exhibiting severe distortionfrom what is expected of a monopole on a flat ground plane.Once the recess is filled with the required TO media, the fieldassociated with a monopole on a ground plane is reproducedby the physical system inz′ > 0 as shown in Fig. 7(d). Withoutany physical presence above thexy-plane, the embeddedmonopole will transmit and receive just like a monopole thatprotrudes above a flat PEC ground.

Other TO applications to antennas and arrays have beenreported. Using a sequence of rotation and compressiontransformations, an embedded linear array capable of endfirescanning was reported [70]. A NIM shell was employed tocreate electrically large virtual sources from smaller sources togenerate highly directive radiation patterns [30], [71]. In [72],[73], a line source and surrounding cylinder of free space weretransformed into a strip current and a TO lens for achievinghigh directivity. A directive emission was later demonstratedusing a metamaterial lens in the microwave regime [27]. Otherapplications of the TO technique to radiation problems donot modify the sources themselves, but transform the spacearound them to change the radiation patterns. A broadband TO

lens for highly directive multibeam radiation from a monopoleantenna was designed and experimentally demonstrated [28].Using a rotation transformation similar to (14) for a rotatorand its metamaterial realization, broadside directive radiationfrom a patch antenna was converted into azimuthal directiveradiation [74].

VI. QUASI-CONFORMAL TO AND DEVICES

Typical TO-medium parameters are inhomogeneous,anisotropic, and require both electric and magnetic responses.Although the metamaterials technology provides buildingblocks for synthesizing a wide range of permittivity (e.g.,wiremedia [75] and electric-LC resonators [76]) and permeability(e.g., split-ring resonators [77]), such resonant subwavelengthinclusions result in narrowband responses. Furthermore,magnetic resonances become increasingly weak at higherfrequencies toward the optical regime and begin to accompanysignificant absorption loss. Since a TO medium specificationis determined by the original medium and the coordinatetransformation via (5), it is of interest to identify a classoftransformations that result in material parameters that areamenable to synthesis using available techniques, such asthose of all-dielectric media.

The QCTO technique [31] based on quasi-conformal co-ordinate transformation addresses this question. For 2-D TEpolarization, a QC grid can be numerically generated to mini-mize the anisotropy introduced by the mapping. The resultingmagnetic permeability can be approximated by that of theambient medium. The resulting device may be accuratelyrealized using isotropic dielectrics, i.e., traditional graded-index material. To illustrate, Fig. 8 shows the design andperformance of a carpet cloak or a ground plane cloak [31].The original system has a flat PEC ground in thexy-plane.It is desired that a cloak be designed to hide a smooth bumpin the ground so that the presence of the raised conductingsurface is concealed.

In the xy-plane, the original system is a rectangle ofdimensionsw × h shown in Fig. 8(a), which is in free space.The transformed system has the same area less the raisedportion of the ground, as illustrated in Fig. 8(b). In the TEpolarization,E = ẑEz, H = x̂Hx + ŷHy and E′ = ẑE′z,H′ = x̂H ′x + ŷH

′y. For the permittivity, anisotropic relative

parameter

ǫ′ =1

|A| =1

√

|g|(22)

satisfies (5b) for the given E-field polarization. Here,g is thecovariant metric tensor given by

g = ATA. (23)

Now, the anisotropy of the TO medium is determined by the2×2 permeability tensor in thexy-plane. If we denote theprincipal values of this permeability tensor byµL and µT ,they satisfy [31]

√

µLµT

+

√

µTµL

=Tr(g)√

|g|, (24a)

µLµT = 1. (24b)

7


x’

y’ ε’zz

0.8

1

1.2

1.4

1.6

1.8

x

y

x’

y’

(d)

(b)

(c) (e)

(a)

w

h

Fig. 8. 2-D carpet cloak for TE polarization based on the QCTOtechnique. (a) Original (virtual) system. (b) Transformed (physical) system. (c) The permittivityprofile of the cloak. (d) Snapshot of the electric field for a bump on a PEC ground with a carpet cloak under a 2-D Gaussian beam illumination at an obliqueincidence. (e) Snapshot of the electric field for the same bump without the cloak. The profile of the modified ground plane is given byy′ = 0.4λ cos2(πx′/λ),−2λ < x′ < 2λ. The axis of the incident 2-D Gaussian beam makes a −45° with the x-axis and the minimum waist is1.0λ. The dimension of the cloak isgiven by8λ× 3λ and its boundary is indicated by a black contour.

If a transformation can be found such that the anisotropy,i.e., the difference betweenµL and µT , is minimized, theanisotropy may be ignored and let the isotropic relativepermeability be set equal to the geometric mean ofµL andµT , or

µ′ = 1. (25)

A mapping that minimizes the anisotropy is numericallyobtained by minimizing a properly defined functional [31],[78]. The result is a QC map. The medium described by(22) and (25) is a graded-index (inhomogeneous), isotropicdielectric medium.

Fig. 8(b) shows a QC map obtained by solving the inverseLaplace’s equation [78]. To obtain the transformed contoursof constant-x grid lines, the two boundaries atx′ = ±w/2were fixed and the remaining two boundaries were allowedto slip. The contours associated with constant-y grid lineswere obtained similarly. The resulting permittivity map forthis carpet cloak is plotted in Fig. 8(c). Most of the volumehas ǫ′(x′, y′) > 1, but there are small areas whereǫ′ < 1.If a host dielectric is a background medium rather than freespace, all of the cloak volume can have the index of refractionlarger than unity, as was done in the microwave experiment ofa metamaterial realization [32]. In the presence of the carpetcloak, an incident Gaussian beam on the bump in the groundplane reflects and leaves the cloak volume as if the incidentbeam underwent reflection from a flat ground plane, as shownin Fig. 8(d). If the curved ground is exposed to the incomingwave, Fig. 8(e) shows that the beam is noticeably disturbed.Since a QCTO device may be realized by using graded-index

all-dielectric media, a broadband response can be expectedcompared with a typical TO design that needs to be realizedusing magneto-dielectric constituent material.

A QC transformation can also be applied to volumes con-taining sources. An interesting QCTO application to arrayantennas is QCTO lenses for conformal arrays [79]. For a 2-D physical linear array mounted on a flat ground plane (e.g.,an array of slots), it is desired that a lens be designed usingthe QCTO technique such that the physical array with theQCTO lens behaves equivalently to a virtual cylindrical phasedarray mounted on a semi-circular PEC platform. The original(virtual) system for a semi-circular conducting platform ofradius a is illustrated in Fig. 9(a). The transformed systemis a rectangle ofw × t having constantx′- and y′-grid lines[cf. Fig. 8(a)].

Except for the different shapes of the deformed ground, thetransformation is recognized to be the inverse of the carpet-cloak mapping in Figs. 8(a)–(b). Hence, the Jacobian matrixisgiven by the inverse matrix of that associated with the carpet-cloak design. Essentially, the permittivity of the QCTO lensǫ′zz is the reciprocal of Fig. 8(c) once the different shape ofthe mounting platform is accounted for. Fig. 9(b) shows theresulting permittivity map. Fig. 9(c) shows the field distribu-tion for the original conformal cylindrical array scanningat45° from broadside. In Fig. 9(d), a linear array in the flatground plane scans in the same direction with the QCTO lensapplied on top. Outside the lens, the original field distributionis reproduced. The QCTO lens enables a physically lineararray to have the characteristics of a cylindrical array, e.g.,a constant beamwidth independent of scan angle.

8


x’

y’ ε’zz

0

1

2

3

4

x

y

2a

(c)

(d)(b)

(a)

w

t

Fig. 9. QCTO lens for converting a physical linear array intoa virtual cylindrical array. (a) Original (virtual) system.(b) Permittivity map of the QCTO lens.(c) Snapshot ofEz for a nine-element cylindrical array mounted on a cylindricalconducting platform of radiusa, for a scan angle of 45° off broadside. (d)Snapshot ofEz for a physical linear array on the flat ground plane with the QCTO lens. The geometry of the lens is given byw = 2t = 8a = 16λ andits boundary is indicated by a black contour. In the virtual array, the element spacing isλ/2. Each source is modeled by a circumferentially directed, shortmagnetic current. The array has a uniform amplitude distribution.

The inverse function of the QCTO lens can also beconsidered—a QCTO lens that transforms a physical cylin-drical array into a virtual linear array [79]. This mappingis none other than the cloaking transformation itself. Forexample, if a QCTO lens for the physical cylindrical arrayin Fig. 9(c) is wanted to make it operate as a linear array in aflat ground plane, the permittivity of the lens will be given bythe reciprocal ofǫ′zz in Fig. 9(b), but plotted in the physicaldomain [same as the map area in Fig. 9(a)].

QCTO-based cloaks and devices are under active researchin design, fabrication, and experimental validation. Opticalground-plane cloaks were fabricated and tested [80], [81].Byrotating the ground-cloak permittivity map, a 3-D broadbandcloak was designed and measured in the microwaves [82].A 3-D optical carpet-cloak was demonstrated for broadbandoperation at near-IR wavelengths [83]. The QCTO techniquealso finds various applications in lenses. Flattened 2-D Luneb-urg and Maxwell’s fisheye lenses were obtained [33], [34],whereby integration with flat imaging sensors and circuits isfacilitated. A 3-D variant of the flattened Luneburg lens hasbeen investigated [35]. A flat lens based on a discrete trans-formation and quasi-orthogonal grids has been designed [37].Using a similar technique, a planar zone-plate lens was re-ported [84]. Various planar photonic device designs such asalight source collimator, waveguide adapters and crossingshavebeen presented [85]. For antenna applications, the parabolic re-flector surface of a reflector antenna has been transformed intoa flat conductor loaded with QCTO lenses/superstrates [36],[37] and experimentally verified [86]. The QCTO techniquewas applied to a free-space volume centered at a source toobtain directive radiation with multiple narrow beams for 2-

D [87] and 3-D [88] applications.

VII. C ONCLUSION

The fundamental theory of transformation electromagneticsand optics has been reviewed together with device designsaimed toward practical applications. Based on the materialsinterpretation of Maxwell’s equations under coordinate trans-formations, the TO technique reduces device design to thatof finding a coordinate transformation from a known electro-magnetic phenomenon to a desired wave-material interaction.The resulting device medium specifications depend on thespecific transformation employed. Hence, mappings that yieldpermittivity and permeability tensors that are amenable toconvenient practical fabrication are desirable.

Using a variety of available material synthesis techniquesincluding the advancing metamaterials technology, the focusof the TO research is shifting to accurate fabrication of noveldevices and faithful realization of envisioned features that arenot available via traditional designs and approaches. Currently,the main obstacles to adoption of TO devices in practiceremain relatively narrow bandwidths and losses from materialas well as conductor constituents.

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[88] Q. Wu, Z. H. Jiang, O. Quevedo-Teruel, J. P. Turpin, W. Tang, Y. Hao,and D. H. Werner, “Transformation optics inspired multibeam lensantennas for broadband directive radiation,”IEEE Trans. AntennasPropag., vol. 61, no. 12, pp. 5910–5922, Dec. 2013.

Do-Hoon Kwon received the B.S. degree fromKorea Advanced Institute of Science and Technology(KAIST), Korea in 1994, and the M.S. and Ph.D.degrees from the Ohio State University, Columbus,OH in 1995 and 2000, respectively, all in electricalengineering. He was a senior engineer at the Cen-tral RD Center and Samsung Advanced Institute ofTechnology of Samsung Electronics, Co. in Koreafrom 2000 to 2006. During 20062008, he was apost-doctoral researcher with the Material ResearchScience and Engineering Center and the Department

of Electrical Engineering of the Pennsylvania State University. In August2008, he joined the Department of Electrical and Computer Engineering,University of Massachusetts Amherst, as an associate professor. He is affiliatedwith the Antennas and Propagation Laboratory and the Centerfor AdvancedSensor and Communication Antennas of the ECE department. In 2011, hereceived the inaugural IEEE Antennas and Propagation Society Edward E.Altshuler Prize Paper Award for the article Transformation electromagnetics:an overview of the theory and applications, published in theIEEE Antennasand Propagation Magazine (co-authored with Prof. Douglas H. Werner). Heserved as the Student Program Co-Chair at the 2010 IEEE InternationalSymposium on Phased Array Systems Technology. He spent the summerof 2011 as a Summer Faculty Fellow with the Sensors Directorateof the AirForce Research Laboratory.

He is the inventor or co-inventor of 18 U.S. patents in antenna and wirelesscommunication areas. His main research interests include antenna scatteringtheory, small/wideband antennas, frequency selective surfaces, metamaterials,cloaking, and transformation electromagnetic/optical device designs. He is asenior member of the IEEE.

EDITORIAL COMMENT

Perhaps there is no other topic that has created so muchrecent stir in the EM and Optics community as has the topicof Transformation Optics (TO) aka Transformation Electro-magnetics (T-EM), and for good reason. TO offers a totallynew way of designing flat lenses, invisibility and carpet cloaks,scannable Fabry-Perot antennas, and other exotic devices thatwere not on the radar screen even just a few years ago. Kwonexplains the principles of the TO algorithm in this contributionin a lucid manner that is easy to follow even by those whoare relatively new and uninitiated to the field. In this articleKwon does a very comprehensive job of explaining the basicprinciples of TO, mentions a plethora of applications of thesame, and points out a few of its caveats as well.

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Documents

Transformation Electromagnetics and Optics › files › articles › 15328ab5bdc08e.pdfled to superscatterer [15] and superabsorber [16] designs. By manipulating the spatial loss