complex-coordinate non-hermitian transformation...

Complex-coordinate non-Hermitiantransformation optics

S Savoia, G Castaldi and V Galdi

Waves Group, Department of Engineering, University of Sannio, Benevento, I-82100, Italy

E-mail: [email protected]

Received 22 September 2015, revised 5 December 2015Accepted for publication 8 December 2015Published 1 April 2016

AbstractTransformation optics (TO) is conventionally based on real-valued coordinate transformationsand, therefore, cannot naturally handle metamaterials featuring gain and/or losses. Motivated bythe growing interest in non-Hermitian optical scenarios featuring spatial modulation of gain andloss, and building upon our previous studies, we explore here possible extensions of the TOframework relying on complex-valued coordinate transformations. We show that such extensionscan be naturally combined with well-established powerful tools and formalisms inelectromagnetics and optics, based on the ‘complexification’ of spatial and spectral quantities.This enables us to deal with rather general non-Hermitian optical scenarios, while retaining theattractive characteristics of conventional (real-valued) TO in terms of physically incisivemodeling and geometry-driven intuitive design. As representative examples, we illustrate themanipulation of beam-like wave-objects (modeled in terms of ‘complex source points’) as wellas radiating states (‘leaky waves’, modeled in terms of complex-valued propagation constants).Our analytical results, validated against full-wave numerical simulations, provide useful insightinto the wave propagation in non-Hermitian scenarios, and may indicate new directions in thesynthesis of active optical devices and antennas.

Keywords: metamaterials, transformation optics, non-Hermitian optics, parity-time symmetry

(Some figures may appear in colour only in the online journal)

1. Introduction

In the synthesis of metamaterials, the use of material con-stituents featuring gain is a well-established strategy toovercome the inevitable presence of losses, so as to attaineffectively lossless (or active) implementations [1]. Besidesthis more or less obvious scenario, recent studies have pro-vided theoretical, numerical and experimental evidence of awealth of anomalous effects and interactions associated withjudicious spatial modulation of loss and gain, includingdouble refraction [2], spectral singularities [3], spontaneoussymmetry breaking and power oscillations [4], unidirectionalinvisibility [5, 6], coherent perfect absorption [7, 8], negativerefraction and focusing [9], and bound states in the continuum[10], just to mention a few. This has generated considerableinterest in exploring a variety of related potential applicationsto optical and plasmonic scenarios including e.g. cavities[11, 12], lasers [13], couplers [14], switches [15], isolators[16], invisibility cloaks [17, 18], metamaterials [19, 20],

tunneling and waveguiding [21–23], composite dielectric/magnetic structures [24], and topological photonic struc-tures [25, 26].

Based on the formal analogy between classical optics andquantum mechanics, most of the above studies have beentriggered and inspired by the parity-time ( ) symmetryconcept introduced by Bender and co-workers [27–29] as apossible extension of quantum mechanics based on non-Hermitian, complex-valued Hamiltonians satisfying suitablesymmetry conditions. Accordingly, the general term ‘non-Hermitian optics’ is typically utilized when referring to theabove-mentioned optical counterparts. It is worth recallingthat certain aspects of the -symmetric quantum-mechanicsextension are still controversial (see e.g. [30–32]), but thisdoes not affect the physical soundness the non-Hermitianoptical counterparts.

Within this largely unexplored research area, a systematicand versatile design approach like transformation optics (TO)[33, 34], based on the form-invariance of Maxwellʼs and

Journal of Optics

J. Opt. 18 (2016) 044027 (13pp) doi:10.1088/2040-8978/18/4/044027

2040-8978/16/044027+13$33.00 © 2016 IOP Publishing Ltd Printed in the UK1

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Helmholtz equations with respect to coordinate transforma-tions, may provide valuable insights and may open up newapplication scenarios and perspectives. However, due to theinherently real-valued character of the underlying coordinatetransformations, conventional TO approaches cannot natu-rally handle ‘transformation media’ characterized by com-plex-valued constitutive parameters (typical of materialsfeaturing loss or gain), unless these are already present in theauxiliary space to be transformed. One possible TO extensionthat would allow one to overcome these limitations relies onthe analytic continuation of the coordinate transformations tothe complex plane.

Indeed, the ‘complexification’ of geometrical objects(spatial coordinates, curves, etc) is a well-known concept oflongstanding interest in electromagnetics (EM) and optics,with the prominent examples of ‘complex source point’ (CSP)[35, 36], ‘complex rays’ [37, 38], and complex-coordinate-stretching-based absorbing boundary conditions [39]. Withspecific reference to TO, recent complex-coordinate exten-sions have been proposed in order to enable field-amplitudecontrol [40], and to generate single-negative [41], and -symmetric [19] transformation media. Also worthy ofmention are very recent applications to reflectionlessmedia [42].

In particular, in [19], we proposed a complex-coordinateTO extension that could naturally handle non-Hermitianoptical scenarios. This allows, for instance, the metamaterial-based interpretation (and possible implementation) of com-plex-coordinates-inspired wave-objects and resonant states, aswell as the geometrical interpretation of the -symmetryconcept and the spontaneous symmetry breaking phenom-enon in terms of analytic features of the underlying coordinatetransformations.

In this paper, we proceed along the same lines, byexploring further examples and application scenarios. Inparticular, after a brief summary (section 2) of our previousresults from [19], we introduce (section 3) a general class ofnon-Hermitian metamaterial transformation slabs for themanipulation of beam-like wave-objects modeled in terms ofCSPs. Moreover, we also explore the manipulation of ‘leakywaves’ (LWs) associated with complex-valued solutions ofthe dispersion equation of open waveguides. Besides theanalytical developments (with some details relegated in fourappendices), we also present numerical validations based onfull-wave simulations. Finally, we provide some brief con-cluding remarks and hints for future research (section 4).

2. Summary of previous work

For the sake of completeness, we compactly summarize themain results from [19]. As in conventional (real-valued) TO[34], we consider a time-harmonic EM field distribution E′,H′ (with suppressed texp i( )w- time-harmonic dependence)radiated by electric (J′) and magnetic (M′) sources in anauxiliary space with Cartesian coordinates r′≡(x′, y′, z′)filled by a lossless medium characterized by (real-valued)relative permittivity and permeability distributions ε′(r′) and

μ′(r′), respectively. By exploiting the form-invariance prop-erty properties of Maxwellʼs equations, the field/source-manipulation effects induced by a coordinate transformation

r F r 1( ) ( )¢ =

can be equivalently attributed to a new set of sources

J M r r r J M F r, det , , 21{ }( ) [ ( )] ( ) · { }[ ( )] ( )= L L ¢ ¢« «

-

radiating a new set of EM fields

E H r r E H F r, , , 3T{ }( ) ( ) · { }[ ( )] ( )= L ¢ ¢«

in a new physical space r≡(x, y, z) filled up by aninhomogeneous, anisotropic ‘transformation medium’ char-acterized by relative permittivity and permeability tensors

r F r r r r

r F r r r r

det ,

det . 4

T

T

1

1

( ) [ ( )] [ ( )] ( ) · ( )

( ) [ ( )] [ ( )] ( ) · ( ) ( )

e e

m m

= ¢ L L L

= ¢ L L L

« « «-«

-

« « «-«

-

In equations (2)–(4),

5

x

x

x

y

x

z

y

x

y

y

y

z

z

x

z

y

z

z

( )L º

¢ ¢ ¢

¢ ¢ ¢

¢ ¢ ¢

«

¶¶

¶¶

¶¶

¶¶

¶¶

¶¶

¶¶

¶¶

¶¶

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

indicates the Jacobian matrix of the transformation inequation (1), double-headed arrows are used to identifysecond-rank tensor quantities, det indicates the determinant,and the superscripts −1, T and −T denote the inverse, thetranspose, and the inverse transpose, respectively. In spite ofthe different notation utilized, it can be shown thatequations (3) and (4) can be reconciled with the resultsobtained via the covariant formulation [43].

It is readily observed from equations (4) and (5) that real-valued transformations inherently yield real-valued con-stitutive tensors, and hence lossless (and gainless) transfor-mation media, whereas complex-valued transformations mayinstead induce the presence of loss and/or gain.

In [19], we focused on particular classes of complex-coordinate transformations that generate media featuringbalanced loss and gain spatial distributions fulfilling the -symmetry condition [27–29]. More specifically, byrecasting the transformed Maxwellʼs equation in a pseudo-Hamiltonian form, we showed that the scalar (quantum-mechanics or paraxial-optics) -symmetry condition can begeneralized to our vector scenario as follows

r r r r, , 6( ) ( ) ( ) ( ) ( )* *e e m m- = - =« « « «

with * denoting complex conjugation. As for the scalar case[27–29], these represent necessary (but not sufficient)conditions for attaining a purely real eigenspectrum. More-over, we showed that coordinate transformations character-ized by

r r 7( ) ( ) ( )*L - = L« «

automatically satisfy such conditions. Similar to the scalarcase [27–29], the fulfillment of equation (7) guarantees that

2

J. Opt. 18 (2016) 044027 S Savoia et al

the eigenspectrum remains real as long as the ‘non-Hermiticity’ (gain/loss) level stays below a critical threshold.Above this threshold, an abrupt phase transition to a complexeigenspectrum (‘spontaneous -symmetry breaking’) maytypically occur. Remarkably, this phenomenon can be relatedto the (dis)continuity properties of the coordinate transforma-tion [19]. More specifically, continuous coordinate transfor-mations (1) (subject to equation (7)) from an auxiliary(lossless/gainless) space yield -symmetric transformationmedia that do not undergo spontaneous symmetry breaking,whereas discontinuous transformations may generally inducethis phenomenon.

As illustrative application examples, we consideredcontinuous transformations that provide a -symmetricmetamaterial interpretation (and implementation) of a CSP,and discontinuous transformations that induce symmetry-breaking phenomena associated with anisotropic transmis-sion resonances, such as ‘unidirectional invisibility’.

3. New results: Non-Hermitian transformation slabs

3.1. Problem geometry and generalities

Referring to figure 1 for illustration, we consider a two-dimensional (2D) scenario, with z-invariant geometry andfields, and transverse-magnetic (TM) polarization (i.e. z-directed magnetic field). More specifically, we focus ourattention on slab-type configurations, generated from anauxiliary space (x′, y′, z′) filled up with a nonmagnetic med-ium with generic relative-permittivity distribution ε′(x′)(figure 1(a)). To this auxiliary configuration, we apply ageneral class of coordinate transformations

x u x

y v x y w x

z z

,

,

, 8

( )( ) ( )

( )a

¢ =¢ = +¢ =

within the region x d∣ ∣ < , with u, v and w generally denotingcomplex-valued functions, and α a complex-valued constantparameter. Applying the TO formalism, the associated field-manipulation capabilities can be equivalently obtained via a

‘transformation slab’ occupying the region x d∣ ∣ < in theactual physical space (x, y, z) (figure 1(b)), with constitutivetensors (cf equation (4))

x y

xx y

,,

0

0

0 0

, 9

v

u

vy w

u

vy w

u

u vy w

uvuv

2 2( )( )

( ) ( )˙

( ˙ ˙ )˙

( ˙ ˙ )˙

[ ˙ ( ˙ ˙ ) ]˙

˙

ee

m¢ ¢

= =

-

-

a a

a a

a

««

+

+ + +

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

where the overdot denotes differentiation with respect to theargument, and the explicit x-dependence has been omitted fornotational compactness. It is worth pointing out that theanalytic mapping (3) is strictly valid in the presence ofcontinuous transformations. In our case, assuming the exteriorregion x d∣ ∣ > filled up with vacuum (cf figure 1(b)), this isensured by the matching conditions

v d v d 1, 10( ) ( ) ( )a- = = =

which imply that the transformations in equation (8) reduce tothe identity at the interfaces x=±d, apart from irrelevant(possibly complex-valued) shifts along the x- and y-directions. Loosely speaking, this means that the reflection/transmission properties of the auxiliary space are preserved,apart from (possibly complex-valued) phase factors.

Moreover, we observe from equation (9) that, for theassumed TM polarization, the resulting transformation med-ium is effectively nonmagnetic (i.e. μzz=1) provided that

v xu x

1. 11( )

˙ ( )( )

a=

Finally, we point out that the coordinate transformationsin equation (8) are formally similar to (though slightly moregeneral than) those previously considered by us in [44].However, while the study [44] is restricted to real-valuedtransformations, here we allow the functions u, v, w and theparameter α to be generally complex-valued, therebyexploring general classes of non-Hermitian transformationslabs.

One of the pivotal characteristics of conventional TO isits intuitive geometrical interpretation in terms of local dist-ortion of the geodesic path of light rays, which essentiallydrives the conceptual design [33, 34]. One may argue thatsuch an attractive feature is lost in our complex-coordinateextension. Nevertheless, in spite of the general complications,there are in fact instances of modeling tools and concepts inEM and optics that inherently rely on spatial and/or spectralquantities residing in a complex-coordinate space, and admitremarkably simple geometrical and physical interpretations.In what follows, we explore two notable cases.

3.2. Complex-source-point-based beam manipulation

As the first application example, we consider the manipula-tion of CSP-based beams. The CSP formalism is a well-known and powerful tool in EM and optics for the modelingof beam-like excitations in configurations for which thesolution pertaining to point-source illumination (Greenʼs

Figure 1. Problem schematic. (a) Auxiliary space, filled up with anonmagnetic medium with generic relative permittivity distribution

x( )e¢ ¢ . (b) Physical space containing the transformation slab, withconstitutive tensors given by equation (9). Different color shadingsare used to indicate regions that are generally inhomogeneous.

3


function) is known analytically in exact or approximate forms[35, 36]. In our example, we start considering a unit-ampl-itude (V/m2), z-directed magnetic line-source,

M x y x x y y, , 12z s s( ) ( ˜ ) ( ˜ ) ( )d d= - -

radiating in vacuum. In equation (12), δ denotes the Dirac-delta function, and

x x b y y bi cos , i sin , 13s s s s s s s s˜ ˜ ( )q q= + = +

represents a complex-valued position, with xs, ys, bs and θsassumed as real-valued, and bs>0. Here and henceforth,complex-valued spatial quantities are identified by a tilde ∼. Itcan be shown [35, 36] (see also appendix A) that thecorresponding radiated field (calculated via analytic continua-tion of a 2D Greenʼs function) corresponds to a beam-likewave-object which, in the paraxial limit, is well approximatedby a Gaussian beam [45] with waist located at (xs, ys),propagation axis forming an angle θs with the x-axis, andRayleigh parameter equal to bs. This is exemplified infigure 2, which shows two (magnitude) field maps forrepresentative values of the CSP parameters, from which thebeam-like behaviour is evident. Also shown are themagnitude and phase distributions along a linear cut,compared with those pertaining to the associated Gaussian

beam, from which the approximate equivalence can beobserved in the paraxial region. Here, and henceforth,λ0=2πc/ω=2π/k0 denotes the vacuum wavelength, withc and k0 being the corresponding wavespeed and wavenum-ber, respectively.

We now consider a CSP beam in the presence of ourmetamaterial transformation slab. In all examples below, weassume that the auxiliary space is entirely filled up by vacuum(i.e. ε′(x′)=1 in figure 1(a)), the reflectionless conditions(10) hold, and the CSP beam waist is located outside the slab,say xs−d. Under these conditions, it can be shown (seeappendix B for details) that the response at the other side ofthe slab (x d ) can be effectively interpreted as generated byan ‘image’ CSP

x x b y y bi cos , i sin , 14i i i i i i i i˜ ˜ ( )q q= + = +

which is related to the input CSP and transformationparameters via

x x d u d u dy y w d w d

2 ,. 15

i s

i s

˜ ˜ ( ) ( )˜ ˜ ( ) ( ) ( )= + - + -= - + -

Equation (15) describes in a remarkably simple andphysically-incisive fashion the image-formation propertiesof our transformation slab, via direct CSP-mapping betweenthe input and output beams. Within this framework, a fewconsiderations and remarks are in order. First, we observe thatthe results in equation (15) depend only on the boundaryvalues of the mapping functions, and not on their actualbehaviour in the slab region; this potentially leaves usefuldegrees of freedom in the design. Moreover, real-valuedtransformations can only affect the CSP real part, which, inphysical terms, translates into the manipulation of the beamwaist location. This is consistent with the typical manipula-tions (e.g. beam shifting, perfect radome, etc) explored in [44]without resorting to the CSP formalism. Conversely, thecomplex-coordinate TO extension proposed here can manip-ulate the imaginary part too, thereby enabling for full controlof the CSP beam parameters, including the propagationdirection and Rayleigh parameter. In what follows, weillustrate some representative examples.

A first interesting case is given by the condition

u d u d w d w d, , 16( ) ( ) ( ) ( ) ( )= - - =

which, from equation (15), yields

x x d y y2 , . 17i s i s˜ ˜ ˜ ˜ ( )= + =

In other words, the illuminating field generated by theinput CSP at the interface x=−d is exactly reproduced at theoutput interface x=d, as if the slab were not present. Thetransformation slab therefore acts as an EM ‘nullity’. Work-ing with real-valued coordinate transformations, such a con-dition for the function u can only be attained by enforcing aconstant value (which would yield extreme-parameter trans-formation media) or a non-monotonic behaviour (whichwould correspond to a combination of complementarymaterials with positive and negative parameter values, similarto those studied in [46]). Within this framework, complex-

Figure 2. (a) Magnetic-field magnitude (Hz∣ ∣) map pertaining to aCSP with xs=ys=0, bs=λ0/2 and θs=0 (cf equation (13)). (b),(c) Corresponding magnitude and phase cuts, respectively, atx=5λ0 (blue-solid curves) compared with those pertaining to aconventional Gaussian beam with waist located at (xs, ys),propagation axis forming an angle θs with the x-axis, and Rayleighparameter equal to bs (red-dashed curves). (d)–(f) Same as panels(a)–(c), but for CSP parameters xs=ys=0, bs=λ0 and θs=π/4,and with linear cuts at x=10λ0. Field cuts are normalized withrespect to the peak values.

4


coordinate transformations provide more flexibility andadditional degrees of freedom. For instance, considering thefollowing transformation

u xx

dv x w xi 1 , 1, 0, 18( ) ∣ ∣ ( ) ( ) ( )c= - = =⎜ ⎟

⎛⎝

⎞⎠

with χ denoting a real positive constant, we obtain apiecewise-homogeneous bi-layer with uniaxial media char-acterized by a balanced spatial distribution of gain and loss inthe relevant constitutive-tensor components,

x xd

x x xd

sgni

, sgni

,

19

xx yy zz( ) ( ) ( ) ( ) ( )

( )

ec

e mc

= = = -

which fulfills the -symmetry condition (6) (with sgndenoting the signum function). Such transformation mediumwas also considered in [19] within a different context, and apossible multilayered implementation was explored. Similaruniaxial media, simultaneously featuring gain and loss alongdifferent directions, have also been recently studied in [47].

Figure 3 shows the response of such an EM ‘nullity’metamaterial transformation slab (with d=0.5λ0, transverseaperture of 30λ0, and χ=0.3λ0) due to a CSP beam with waistlocated at (xs=−λ0, ys=0) and diffraction length bs=0.1λ0,

impinging along the positive x-direction (θs=0). More speci-fically, figure 3(a) shows the finite-element-computed (seeappendix C for details) magnetic-field (real-part) map, fromwhich it is rather evident that the impinging wavefront at theinput slab interface x=−d is reproduced at its output (x=d).For a more quantitative assessment, figures 3(b) and (c) com-pare the magnitude and phase field distributions, respectively, atthe slab output with the theoretical prediction in equation (17)for an infinite slab. The agreement is very good, with the smalloscillations towards the ends of the slab aperture attributable totruncation effects.

It is worth stressing that in the above example (as well asthe following ones) the parameter configuration was chosenfor a simple and direct illustration of the phenomenon, whilemaintaining a moderately-sized computational domain. Atthis stage, no particular attempt was made to optimize thegeometry and material parameters in order to ensure thetechnological feasibility. For instance, the magnetic characterin equation (19) may be traded off (via equation (11)) with amore complex (inhomogeneous, and with spatially-variantoptical axis) spatial distribution.

In the above example, the manipulation was restricted tothe CSP waist location, and therefore it could have been

Figure 3. (a) Finite-element-computed magnetic-field (Hz) real-part map pertaining to the -symmetric transformation bi-layer inequation (19), with d=0.5λ0, χ=0.3λ0 and finite aperture of 30λ0, excited by a CSP with xs=−λ0, ys=0, bs=0.1λ0, and θs=0. Thecorresponding material parameters (cf equation (19)) are εxx=mi5/3 and εyy=μzz=±i3/5 for −d

achieved as well via real-valued coordinate transformations.We now consider a different class of transformations that canenable more general manipulations. In particular, we focus onthe possibly simplest form

u x u x v x w x w x, 1, , 1, 200 0( ) ( ) ( ) ( )a= = = =

with u0 and w0 denoting complex-valued constant parametersrepresented in polar form as

u u w wexp i , exp i . 210 0 0 0∣ ∣ ( ) ∣ ∣ ( ) ( )f y= =

Such transformations satisfy the matching conditions inequation (10) and, as it can be observed from equation (15),they affect both the real (via u0) and imaginary (via w0) partsof the CSP. The resulting transformation medium (cf (9)),with μzz=u0 and

u

w

u

w

uu

w

u

u

10

1 0

0 0

, 220

0

0

0

00

02

02

0

( )e =

-

- +«

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤

⎦

⎥⎥⎥⎥⎥⎥

is homogeneous and generally anisotropic. However, condi-tions can be derived on transformations (20) so that theanisotropy reduces to a simpler uniaxial form (with tiltedoptical axis). As detailed in appendix D, such a condition canbe expressed in a simple analytical form,

uw1 sin

sin 2, 230

02

∣ ∣ ( ∣ ∣ )( )

( )yy f

=+

-

which admits feasible solutions provided that

sin

sin 20. 24

( )( )y

y f->

We observe, for instance, that for real-valued w0 (i.e.n n, y p= Î ), the only feasible (u 00 ¹ ) solution of

equation (23) implies n n1 2 ,( ) f p= + Î ), i.e. eitherreal or purely imaginary u0.

As a first example, figure 4 shows the response of a slabdesigned so as to ‘transform’ a conventional line-source(xs=−λ0, ys=0, bs=0) into an image CSP representing abeam propagating along the negative y-direction(xi=−1.118λ0, yi=0, bi=0.5λ0, θi=−π/2). Parameters

Figure 4. (a) Finite-element-computed magnetic-field (Hz) real-part map pertaining to the transformation slab in equation (20), withu0=1.118, w 0.5,0∣ ∣ = ψ=π/2, d=0.5λ0 and finite aperture of 30λ0, excited by a real (i.e. bs=0) line-source located at xs=−λ0,ys=0. (b) Magnitude and (c) phase distributions (red dots) computed at the interface x=d+λ0/100, compared with the theoreticalpredictions (blue-solid curves) in terms of an image CSP with xi=−1.118λ0, yi=0, bi=0.5λ0, θi=−π/2 (cf equation (15)). Results arenormalized with respect to the theoretical solution evaluated at x=d+λ0/100 and y=−3.875λ0. The dashed contour delimits the slabregion. Also shown in panel (a) is the principal reference system (ξ, υ), rotated by an angle γ=π/4, in which the transformation mediumassumes a uniaxial form, with relevant components εξξ=0.894−i0.447, ευυ=0.894+i0.447, μzz=1.118.

6


are chosen so as to fulfill the conditions in equations (23) and(24), and hence the arising transformation medium reduces toa homogeneous, uniaxially-anisotropic material with a 45°-tilted optical axis, and components exhibiting both gain andloss along different directions. As it can be observed from thefield-map (figure 4(a)), as well as the relevant magnitude(figure 4(b)) and phase (figure 4(c)) distributions, the isotropicradiation produced by the real line-source is converted by theslab in a markedly directional beam, in excellent agreementwith the theoretical prediction.

Figure 5 shows a further example, where an obliquely-incident CSP beam with xs=−3λ0, ys=0, bs=λ0,θs=−π/6 is converted into a beam represented by an imageCSP with xi=xs, yi=0.25λ0, bi=bs, θi=−0.379π,thereby producing a shift (along the y-direction) of the waistlocation and an angular steering of nearly 40°. Once again,the resulting transformation medium assumes a uniaxially-anisotropic form (with a ∼25°-tilted optical axis, and com-ponents exhibiting both gain and loss), and the agreementbetween numerical simulations and theoretical predictions isvery good.

To sum up, the proposed classes of transformationsallow, in principle, rather general manipulations of CSP-based beam-like wave objects. Clearly, suitable restrictionsshould be enforced in order to achieve structural simplifica-tions such as uniaxial anisotropy, positive real-parts of theconstitutive parameters, and feasible levels of gain. Wehighlight that, besides the possible applications to the designof active optical devices, such as lasers, lenses and imagingsystems, the above framework also provides a simple and yetpowerful geometrical interpretation of the complex-coordi-nate TO extension, with the CSP-based beam propagatorsplaying the role of light rays in conventional TO. Within thisframework, it is also worth mentioning that CSP-based beampropagators can be utilized as efficient basis functions tomodel general illuminating field distributions [48].

3.3. Leaky-wave manipulation

As a second illustrative application, we consider the manip-ulation of LWs, associated with complex-valued roots of thedispersion equations of open (e.g. dielectric) waveguides [49].To this aim, we start considering an auxiliary space

Figure 5. (a) Finite-element-computed magnetic-field (Hz) real-part map pertaining to the transformation slab in equation (20), withu 1.0850∣ ∣ = , f=−0.150π, w 0.5,0∣ ∣ = ψ=π/3, d=0.5λ0 and finite aperture of 30λ0, excited by a CSP with xs=−3λ0, ys=0, bs=λ0,θs=−π/6. (b) Magnitude and (c) phase distributions (red dots) computed at the interface x=d+λ0/100, compared with the theoreticalpredictions (blue-solid curves) in terms of an image CSP with xi=xs, yi=0.25λ0, bi=bs, θi=−0.379π (cf equation (15)). Results arenormalized with respect to the theoretical solution evaluated at x=d+λ0/100 and y=−5.866λ0. The dashed contour delimits the slabregion. Also shown in panel (a) is the principal reference system (ξ, υ), rotated by an angle γ=0.137π, in which the transformation mediumassumes a uniaxial form, with relevant components εξξ=0.829−i0.628, ευυ=0.766+i0.581, μzz=0.970+i0.495.

7


containing a homogeneous, lossless slab of thickness 2d, i.e.

xx d

x d

, ,

1, ,25b( ) ∣ ∣

∣ ∣( )e e¢ ¢ =

¢ <¢ >

⎧⎨⎩with 1be ¹ , in figure 1(a). As is well known, this simplewaveguiding structure can support modal (source-free)solutions that satisfy the dispersion equation [49]

k dk

k

k

kcot 2

i

2

i

2, 26xb

b x

xb

xb

b x

( ) ( )ee

¢ =¢¢

+¢¢

where ky¢ and k k kx y02 2¢ = - ¢ denote the spectral-domain

wavenumbers associated to the y′ and x′ variables, respec-

tively, and k k kxb b y02 2e¢ = - ¢ . The solutions of equation (26)

are generally to be found in the complex ky¢ plane. In view ofthe underlying symmetry, without loss of generality, we focuson the case Re(ky¢)>0 (i.e. propagation along the positive y′direction). In particular, real-valued solutions,

k k k kIm 0, , Im 0, 27y y x0( ) ( ) ( )¢ = ¢ > ¢correspond to the well-known guided (or bound) modes thatpropagate without attenuation along the y′-direction, anddecay exponentially along the x′-direction [49]. On the otherhand, we can also identify a class of complex-valued solutions

k k kRe , Im 0, 28y x0( ) ( ) ( )¢ < ¢

that are typically referred to as LW modes [49]. We stress thatthe complex-valued character of these roots is generally notrelated to the presence of loss/gain; in fact, these solutionsmay also be found in the lossless (and gainless) scenarioconsidered here. These LW solutions belong to the so-called‘improper’ spectrum, as they grow exponentially along thex′-direction, thereby violating the radiation condition [49].Though not directly observable in the far field, LW modes canbe utilized to represent the field distribution nearby the slabaperture. Such LW-based modeling can effectively capturethe radiation propertties of the slab, and its use is widespreadin the EM and antenna communities [50]. In essence,assuming that a LW with complex propagation constantkys¢ =±(β+iη) is the dominant mode excited (via properillumination), the field radiated by the slab can beapproximated as generated by an equivalent aperture fielddistribution [50]

H y i y yexp exp , 29zeq ( ) ( ∣ ∣) ( ∣ ∣) ( )( ) b h¢ ¢ ~ ¢ - ¢

which, in the far region, yields the radiation pattern [50]

H

kcos

sin 2, 30

zeq 2

22 2

02 2 2 2 2 2

∣ ( )∣

[ ( )] ( )( )

( ) q

qb h

q b h bh

¢ ¢

~ ¢+

¢ - - +

⎧⎨⎩⎫⎬⎭

with θ′ denoting the radiation angle measured from the x′-axis. Therefore, also in this case, the real and imaginary partsof the LW propagation constant admit a simple and insightfulphysical interpretation that is tied with the main radiationcharacteristics (direction and beamwidth). It appears thereforesuggestive to explore to what extent our complex-coordinateTO extension can manipulate these quantities so that, given

an LW solution supported by the dielectric slab in theauxiliary domain, one may design a metamaterial transforma-tion slab that supports (among others) an LW with a suitablymodified propagation constant. To this aim, we consider thepossibly simplest class of transformations

u x u x v x v w x u v, , 0, , 310 0 0 0( ) ( ) ( ) ( )a= = = =

with u0 and v0 denoting complex-valued constant parameters.Such simple transformations essentially constitute a complexcoordinate scaling. However, unlike the previous examples,they do not satisfy (apart from the trivial case u0=v0=1)the matching conditions in equation (10), while they do fulfillthe condition in equation (11) for an effectively nonmagneticcharacter. Although the dispersion equation in the trans-formed domain cannot be obtained via direct mapping ofequation (26), it can still be readily derived by following thestandard procedure [49] and taking into account thecoordinate scaling (31). We obtain

k u du k

k

k

u kcot 2

i

2

i

2, 32xb

b x

xb

xb

b x0

0

0( ) ( )e

e= +

with ky and k k kx y02 2= - denoting the spectral-domain

wavenumbers associated to the y- and x- variables, respec-

tively, and k k k vxb b y02 2

02e= - . Assuming that the slab in

the auxiliary domain supports a LW with propagationconstant kys¢ , a sufficient condition for the transformed slabto support a LW with modified propagation constant kys canbe obtained by equating the left- and right-hand sides of thedispersion equations (26) and (32) evaluated for ky¢=kys¢ andky=kys, respectively. Accordingly, by equating the right-hand sides, we obtain a bi-quadratic equation which directlyyields

v

,

,33

k

k k k

k k

k k k k k

0

ys

b b xs xs

ys xs

b xs xs xs ys

02

02 2

( )( )

( )

=

¢

¢

¢ ¢

e e

e

-

- +

⎧⎨⎪⎪

⎩⎪⎪

with k k kxs ys02 2= - and k k kxs ys0

2 2¢ = - ¢ . Then, byequating the cotangent arguments (thereby disregardingmultiplicities), we get a simple algebraic equation that canbe solved with respect to u0, yielding

uk k

k. 34

b ys

bk

v

002 2

02 ys

2

02

( )e

e=

- ¢

-

Equations (33) and (34) above enable for the desiredmapping between the LW propagation constants kys¢ and kys inthe two domains. It is worth pointing out that the abovetransformation is not restricted to the LW solution kys¢ ofspecific interest, but it rather affects the entire eigenspectrumof the slab in the auxiliary domain. Therefore, care should betaken to make sure that the transformed LW is actually thedominant mode.

A representative example is illustrated in figure 6. Westart from a slab in the auxiliary domain with d=0.125λ0and εb=0.05, which supports a LW mode with propagation

8


constant kys¢ =±(0.257+i0.120) k0. Figure 6(a) shows thefinite-element-computed (see appendix C for details) field-map pertaining to a line-source excitation located atx y 0= = , from which we can observe the radiated fieldaccompanied by an exponentially-decaying (along the y-direction) behaviour of the field inside and nearby the slab, inaccord with the LW-based modeling in equation (29). This isquantitatively more evident in figure 6(b), which comparesthe numerically computed far-field radiation pattern with theLW-based theoretical prediction in (30). The fairly goodagreement is indicative of the dominant character of the LWmode. We now apply the coordinate transformations inequation (31) (with (33) and (34)), with the aim of designing ametamaterial transformation slab that supports an LW with amodified propagation constant kys=(0.707+i0.05)k0, so asto steer the beam direction to θ=π/4 and achieve a sig-nificantly smaller beamwidth (i.e. higher directivity).Accordingly, from our design equations (33) and (34), weobtain u0=1.170+i0.0212 and v0=2.582−i0.756.Figures 6(c) and (d) show the field map and radiation pattern,respectively, pertaining to the corresponding transformationslab (with parameters in the caption) excited by the same linesource. As it can be observed, the desired radiation char-acteristics are obtained, in fairly good agreement with thetheoretical predictions.

By comparison with conventional TO-based approachesfor tailoring the waveguiding [51, 52] and radiation [53–55]

properties, which rely on real-valued coordinate transforma-tion to suitably shape the wavefronts or effectively scale thestructure (so that, e.g., a small-sized antenna can exhibitradiation characteristics comparable with a larger one), ourapproach here manipulates directly the dispersion equation.The above results may pave the way to the development ofnew strategies for the design of active antennas, where thebeam steering may be achieved (and actively controlled) viasuitable modulation of loss and gain, rather than conventionalphase shifters.

4. Concluding remarks and perspectives

In summary, we have explored some general classes of non-Hermitian metamaterial slabs inspired by complex-coordinatetransformations. Analytical and numerical results from ourprototype studies indicate that complex-coordinate extensionsof TO can be naturally and fruitfully utilized in conjunctionwith powerful EM analytic tools and formalisms based on the‘complexification’ of spatial and spectral quantities (the mostprominent examples being CSP-based beams and LWs,respectively), as well as pervasive paradigms such as -symmetry. In particular, we focused on the interestingmanipulation effects enabled by transformation slabs which,in their simplest configurations, are made of piecewisehomogeneous, uniaxal (possibly with tilted optical axis)

Figure 6. (a) Finite-element-computed magnetic-field (Hz) real-part map pertaining to a slab in the auxiliary domain with εb=0.05,d=0.125λ0 and finite aperture of 30λ0, excited by a magnetic line source located at x′=y′=0. (b) Corresponding radiation pattern (red-dashed curve) compared with theoretical LW prediction (blue-solid curve) in (30), with kys¢ =±(0.257+i0.120)k0. (c), (d) Same as panels(a) and (b), respectively, but pertaining to a homogeneous, uniaxial transformation slab (cf (31), with u0=1.170+i0.0212 and v0=2.582−i0.756) with εxx=0.305−i0.195, εyy=0.068+i0.003, μzz=1, which supports a LW with kys=±(0.707+i0.05)k0.

9


media with constitutive parameters featuring both loss andgain along different directions.

We expect that these results may provide useful insightinto the EM wave interactions with non-Hermitian scenarios,and may indicate new avenues in the design of active opticaldevices and radiating systems, where the use of gain is notlimited to attain loss compensation. Within this framework,current and future research is aimed at extending the class ofpossible transformations and manipulations (also to 3D sce-narios as well and cylindrical and spherical configurations).Also of interest is the optimization of the parameter config-urations in order to address their practical implementationwith feasible material constituents.

Appendix A. Essentials on CSP modeling

Without loss of generality, we consider in equation (13) thecase xs=ys=0 and θs=0, i.e.

x ib y, 0, A.1s s s˜ ˜ ( )= =

to which any arbitrary case can be reduced via suitablecoordinate shifts and rotations. The exact solution ofMaxwellʼs equations pertaining to the line-source (12)radiating in vacuum can be derived in closed form viaanalytical continuation of the well-known 2D Greenʼsfunction [56], viz.,

H x y k s,4

H , A.2zs 0

01

0˜ ( ) ( ˜) ( )( ) ( )we= -

where ε0 denotes the vacuum permittivity, H01 is the zeroth-

order Hankel function of first kind [57], and

s x b yi A.3s 2 2˜ ( ) ( )= - +

represents a complex distance. Two main choices can bemade for the branch-cut in (A.3): either sRe 0( ˜) (generallyreferred to as the source-type solution) or sIm 0( ˜) (generally referred to as the beam-type solution) [36]. Thesource-type solution satisfies the radiation condition, andyields strong beam-like radiation in the halfspace x>0, withonly negligible radiation in the halfspace x0, and yields strong radiation both for x>0 and x0 and within the paraxial range, the exact CSP-based solution in equation (A.2) approximately matches astandard Gaussian beam with Rayleigh distance

xk W

b0

2. A.8R s

02 ( ) ( )º =

On the other hand, it is also evident from equation (A.5)that the radiated field in the halfspace xd), the beam-type solution would be theproper choice in order to correctly model the field within theregion d

with the second equality derived from the spectral–integralrepresentation of the Hankel function [56], andk k k k, Im 0x y x0

2 2 ( ) = - . Via straightforward plane-wave algebra, the field transmitted in the region x>d inthe presence of the transformation slab can be written as

H x y H k k T k k

k x k y k

,1

2, ,

exp i d , B.2

zi

zs

x y x y

x y y

( ) ˆ ( ) ( )

[ ( )] ( )

( ) ( )òp=´ +

-¥

¥

where Hzsˆ ( ) is the spatial Fourier transform of the incident field

in equation (B.1), viz.,

H k k H x y k x k y y

kk x k y

, , exp i d

2exp i ,

B.3

zs

x y zs

x y

xx s y s

0

ˆ ( ) ( ) [ ( )]

[ ( )]( )

( ) ( )òwe

= - +

=- - +

-¥

¥

whereas T denotes the plane-wave transmission coefficient ofthe transformation slab. Assuming, for now, real-valuedcoordinate transformations, in view of the matching condi-tions in equation (10), the transmission coefficient can bederived by solving the straightforward problem in theauxiliary space (plane-wave propagation in vacuum) andsubsequently mapping it in the actual physical space.Accordingly, we obtain a unit-magnitude transmissioncoefficient which essentially accounts for the phase accumu-lation through the (reflectionless) slab, viz.,

T k k k u d u d

k w d w d

, exp i

i . B.4x y x

y

( ) { [ ( ) ( )][ ( ) ( )]} ( )

= - -+ - -

By substituting equations (B.3) and (B.4) in the spectralintegral (B.2), we finally obtain

H x yk

k x x

k y y k

k x x y y

,4

1exp i

d

4H ,

B.5

zi

xx i

y i y

i i

0

001

02 2

( ) { [ ∣ ∣

( )]}

[ ( ) ( ) ]( )

( )

( )

òwep

we

=- -

+ -

=- - + -

-¥

¥

where the absolute value stems from the branch-cut choice forkx, and we have defined the real-valued image coordinates

x x u d u d dy y w d w d

2 ,. B.6

i s

i s

( ) ( )( ) ( ) ( )

º + - - +º + - -

The CSP mapping in equation (15) can be finally obtained asthe analytic continuation of equations (B.5) and (B.6), by

considering complex-valued source coordinates (x y,s s˜ ˜ ) aswell as complex-valued mapping functions u and w.

Appendix C. Details on numerical simulations

For our full-wave numerical validations (figures 3–6), weutilized the finite-element-based commercial software pack-age COMSOL Multiphysics (RF module) [58]. For theexamples in figures 3 and 5, the excitation was defined (at theleft boundary of the computational domain) as a magnetic-field distribution given by the CSP model in equation (A.2);the remaining three boundaries of the computational domainwere terminated by a wavelength-thick perfectly-matchedlayer (PML). For the examples in figures 4 and 6, a magneticline-source was instead utilized, with PML terminations at allfour boundaries of the computational domain. The far-fieldradiation patterns in figures 6(b) and (d) were obtained usinga dedicated post-processing tool available in the RF module.In all examples, we utilized the MUMPS frequency-domaindirect solver, and a triangular mesh with adaptive element size(ranging from λ0/60 to λ0/20), resulting into a number ofdegrees of freedom ranging from three to five million.

Appendix D. Details on the uniaxiality condition (23)

In order to derive the condition in equation (23), we observethat the tensor in equation (22) can be diagonalized in theprincipal reference (ξ, υ, z) constituted by its orthogonaleigenvectors. However, in view of the generally complex-valued nature of its components, the eigenvectors are alsogenerally complex-valued, and hence the diagonal form doesnot necessarily admit a physical interpretation in terms of auniaxial medium with a tilted optical axis. In order to derive theconditions under which such physical interpretation is valid, westart expressing in Cartesian (x, y, z) form the tensor pertainingto a medium that is uniaxial in a rotated (by an angle γ in the x–y plane) orthogonal reference system (ξ, υ, z),

with εξξ and ευυ denoting the relevant components (eigenva-lues) along the principal axes ξ and υ, respectively.The sought condition can be derived by equating comp-onent-wise the above tensor with equation (22), solvingwith respect to εξξ, ευυ and γ, and enforcing real values of γ.By equating the diagonal terms, we obtain two linear

1

2cos 2 sin cos 0

sin cos cos 2 0

0 0 1

, D.112

[ ( ) ( )] ( )

( ) [ ( ) ( )]( )

e e e e g e e g g

e e g g e e e e g

+ + - -

- + + -

xx uu xx uu xx uu

xx uu xx uu uu xx

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

11


equations in εξξ and ευυ, which yield

uu w u w

1

21 sec 2 1 ,

D.20

02

02

02

02[ ( )( )]

( )

e g= + + - + -xx

uu w u w

1

21 sec 2 1 .

D.30

02

02

02

02[ ( )( )]

( )

e g= + + + + -uu

Finally, by equating the off-diagonal terms, and sub-stituting the above solutions, we obtain

w

u wtan 2

2

1. D.40

02

02

( ) ( )g =+ -

By assuming w 00 ¹ (otherwise the uniaxial condition istrivially satisfied), and multiplying by w0* both the numeratorand denominator of the right-hand side of equation (D.4), itreadily follows that the rotation angle γ is real if

w u wIm 1 0, D.50 02

02[ ( )] ( )* + - =

from which the condition in equation (23) is straightforwardlyobtained by substituting the polar representations inequation (21).

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http://dx.doi.org/10.1063/1.3264085http://dx.doi.org/10.1063/1.3131843

1. Introduction2. Summary of previous work3. New results: Non-Hermitian transformation slabs3.1. Problem geometry and generalities3.2. Complex-source-point-based beam manipulation3.3. Leaky-wave manipulation

4. Concluding remarks and perspectivesAppendix A.Essentials on CSP modelingAppendix B.Details on the CSP mapping (15)Appendix C.Details on numerical simulationsAppendix D.Details on the uniaxiality condition (23)References

complex-coordinate non-hermitian transformation...

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