PHYSICAL REVIEW B 93, 085429 (2016)

Transforming two-dimensional guided light using nonmagnetic metamaterial waveguides

Sophie Viaene,1,2 Vincent Ginis,1 Jan Danckaert,1 and Philippe Tassin2,1,*

1Applied Physics Research Group, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussel, Belgium2Department of Physics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden

(Received 30 September 2015; revised manuscript received 27 January 2016; published 19 February 2016)

Almost a decade ago, transformation optics established a geometrical perspective to describe the interactionof light with structured matter, enhancing our understanding and control of light. However, despite theirhuge technological relevance in applications such as optical circuitry, optical detection, and actuation,guided electromagnetic waves along dielectric waveguides have not yet benefited from the flexibility andconceptual simplicity of transformation optics. Indeed, transformation optics inherently imposes metamaterialsnot only inside the waveguide’s core but also in the surrounding substrate and cladding. Here we restorethe two-dimensional nature of guided electromagnetic waves by introducing a thickness variation on ananisotropic dielectric core according to alternative two-dimensional equivalence relations. Our waveguidesrequire metamaterials only inside the core with the additional advantage that the metamaterials need not bemagnetic and, hence, our purely dielectric waveguides are low loss. We verify the versatility of our theory withfull wave simulations of three crucial functionalities: beam bending, beam splitting, and lensing. Our methodopens up the toolbox of transformation optics to a plethora of waveguide-based devices.

DOI: 10.1103/PhysRevB.93.085429

Geometrical reasoning played a crucial role in the historicaldevelopment of optics as a scientific discipline, and itssuccesses are associated with the names of great scientistslike Snell, de Fermat, Huygens, Newton, and others. Tothis day, the design of many optical components, e.g.,microscopes, displays, and fibers, is based on the ray pictureof light, valid for electromagnetic waves inside media withslowly varying refractive index distributions [1]. Through theadvent of metamaterials and photonic crystals [2–8]—artificialmaterials whose electromagnetic properties are determinedby subwavelength unit cells—light may be manipulated byenhanced optical properties at the micro- and nanoscale. Asa result, there is a growing need for analytical tools to modeland design metamaterial devices that act upon the electric andmagnetic components of light [9].

With the design and experimental demonstration ofinvisibility cloaks [10–14], transformation optics proved tobe an adequate geometrical tool to explore the full potentialof metamaterials. Succinctly, transformation optics relies onthe form invariance of Maxwell’s equations to determineappropriate material properties, i.e., permittivity andpermeability distributions, that materialize unconventionallight flows based on the deformation of a coordinate system.Using this geometrical perspective of the interaction of lightwith structured matter, researchers have discovered andreconsidered many optical phenomena in three-dimensionalmetamaterials with regard to wave propagation [15–17],subwavelength sensing [18], Cherenkov radiation [19],effective gauges [20], and many more.

To enhance control on the propagation of surface wavesconfined to a single interface [21–23], several research groupssuccessfully applied the existing framework of transformationoptics to surface waves along graphene-dielectric [24,25] andmetal-dielectric interfaces [26–30]. Indeed, at frequencies faraway from the surface plasmon resonance of a metal-dielectric

*Corresponding author.

interface, the evanescent tails of the surface plasmons extendsubstantially into the dielectric material. As a result, surfaceplasmons can be made to follow coordinate-based trajectoriesif the dielectric is replaced by a metamaterial according to theconventional recipe of transformation optics. Unfortunately,the propagation range of surface plasmon polaritons is severelylimited by dissipative loss in the metallic substrate [31,32],especially at infrared and optical frequencies. Therefore,although a simple metal-dielectric interface is amenable totransformation optics, it is not an ideal platform for long-rangeprocessing of surface waves.

In this contribution, we introduce a radically differentformulation of transformation optics applicable to electromag-netic waves confined to a purely dielectric slab waveguide. Theresulting low-loss metamaterial waveguides can mold the flowof light in optical circuitry, optical detection, and actuationapplications [33–35]. In the first part of this contribution, weapply a two-dimensional conformal coordinate transformationto the symmetry plane of a slab waveguide [36]. These two-dimensional transformations are naturally compatible with aplanar waveguide structure and automatically lead to nonmag-netic metamaterial implementations according to our alterna-tive equivalence relations. In the second part of this contribu-tion, we discuss the effectiveness and versatility of our equiv-alence relations with three proof-of-concept waveguide com-ponents: a beam bender, a beam splitter, and a conformal lens.

I. TRANSFORMING TWO-DIMENSIONAL GUIDEDMODES

The main obstruction for the application of the conventionalframework of transformation optics is the following: in orderto implement a two-dimensional coordinate transformationof light flows along the symmetry plane of the waveguide[Figs. 1(a)–1(c)], the transformation optics recipe imposesa bulky, magnetic, and lossy three-dimensional materialof infinite extent. Indeed, because the transformation isinsensitive to the spatial coordinate perpendicular to the

2469-9950/2016/93(8)/085429(5) 085429-1 ©2016 American Physical Society

VIAENE, GINIS, DANCKAERT, AND TASSIN PHYSICAL REVIEW B 93, 085429 (2016)

(a)

(d)2a

β

2a

β~

2a

β~

(u,v) (x,y)

(b) (c)

(e) (f)~

FIG. 1. Our design uses a nonmagnetic uniaxial metamaterialwaveguide of varying thickness to impose two-dimensional flows oflight. (a) The symmetry plane of a slab waveguide is locally stretchedby a two-dimensional conformal coordinate transformation (b) sothat light bends over 90◦ as if it experiences a geometry γ (x,y) (c).(d) Accordingly, the vector space of the incident guided mode withpropagation constant β (green) is stretched in the symmetry plane(e). The total wave vectors lie respectively on elliptical (hyperbolic)isofrequency contours of the wave equation in blue (red) insidethe core (outer) region. According to the traditional recipe oftransformation optics, the exponential tails (k1) and thickness a arepreserved because metamaterials are implemented in the core andouter layers. (f) To preserve confinement and to impose a globallystretched propagation vector without metamaterials in the outerlayers, a thickness variation a ensures the continuity conditions atthe interfaces at the expense of changes in the exponential tails (k1).

waveguide, the material implementation of transformationoptics is also independent of this coordinate. Moreover,the conventional application of transformation opticsrequires impedance-matched magnetic metamaterials whoseimplementation is inherently lossy [37,38]. Here, wedemonstrate that in the case of two-dimensional conformaltransformations, the transformed guided light flows can bematerialized inside nonmagnetic metamaterial cores of varyingthickness without the need for material implementations inthe surrounding regions. To this aim, we introduce a set oftwo-dimensional equivalence relations.

Our analysis starts from the consideration that—instead ofthe full Maxwell equations—only those equations that governguided waves are required to construct a two-dimensionalframework. In particular, the transverse-magnetic guidedmodes of a slab waveguide with thickness 2a and dielectricprofile ε(z)—consisting of a high-index core layer εcore andlow-index outer layers εout—are determined by two scalarequations: the Helmholtz equation, which governs the in-planepropagation along the waveguide, and the dispersion relation,which imposes the continuity of the confined mode profile atthe material interfaces. This concept is illustrated in Fig. 1, bylooking at our design process in the spatial [Figs. 1(a)–1(c)]and reciprocal space [Figs. 1(d)–1(f)] of a guided mode.

For the initial isotropic, homogeneous waveguide[Fig. 1(d)], the reciprocal space is completely determined bythe propagation constant β along the waveguide symmetryplane and the angular frequency ω. Indeed, the guided

mode consists of a fixed confined transverse profile that ischaracterized by a transversal wave vector component k2 insidethe core region and an exponential decay with extinctioncoefficient k1 in the surrounding layers. These are defined interms of β, ω and the dielectric profile ε(z) by the Helmholtzwave equation of the in-plane magnetic field

[�xy ± k2

1,2 + εout,coreω2

c2

]H|| = 0, (1)

where �xy is the Laplacian associated with the waveguidesymmetry plane and the plus (minus) sign relates to k1 (k2).In other words, the Helmholtz equation [Eq. (1)] constrainsthe total wave vectors (β,k1,2) to a hyperbolic isofrequencycontour in the surrounding regions (red) and to an ellipsoidalisofrequency contour (blue) inside the core. Although bothcontours are compatible for a range of propagation constantsβ in the green band, only one specific propagation constantβ of the incident mode—indicated by the green line—allowsfor a continuous mode profile at the material interfaces z =±a. Mathematically, the selected propagation constant β of atransverse magnetic fundamental mode satisfies the followingdispersion relation:

tan[k2(ω,β)a] = εcore k1(ω,β)

εout k2(ω,β). (2)

To motivate our equivalence relations, we now consider theeffects of a two-dimensional conformal transformation on theincident mode [Fig. 1(e)]. In the Supplemental Material [39],we show that the Helmholtz wave equation, describing thepropagation along the waveguide with induced geometryγ (x,y), is materialized by a nonmagnetic, uniaxial materialε⊥ = γ (x,y)ε(z), ε|| = ε(z). To address the transformation inreciprocal space, we emphasize that each two-dimensionalconformal transformation locally reduces to a constant stretch-ing X of the symmetry plane. As a consequence, the transfor-mation locally stretches the in-plane propagation vector ofthe guided mode β = Xβ and preserves the transverse modeprofiles characterized by the variables k1 and k2. Accordingto the conventional three-dimensional equivalence relations oftransformation optics, this global deformation of the in-planewave vector needs to be imposed by nontrivial materials bothinside the core and the surrounding regions of the waveguide,leading to an inconvenient bulky design.

We now introduce the main idea of this paper: by modifica-tion of the core region’s thickness, we preserve the transformedin-plane solutions—identified by the in-plane propagationconstant β—without material implementations in the claddingand substrate regions [Fig. 1(f)]. In other words, we onlyimplement the nonmagnetic, uniaxial medium inside the coreregion, to preserve both the transverse wave vector k2 andthe desired propagation constant β, and we allow for changesin the extinction coefficient k2

1 = γ (x,y)β2 − εout(ω2/c2). Inaccordance with Eq. (2), the continuity of the fields atthe interfaces is restored by a variation of the thickness a

[Fig. 1(f)],

a = 1

k2arctan

⎛⎝εcore

√γ (x,y)β2 − εout

ω2

c2

εout k2

⎞⎠, (3)

085429-2

TRANSFORMING TWO-DIMENSIONAL GUIDED LIGHT . . . PHYSICAL REVIEW B 93, 085429 (2016)

R

)d()a(

(e)

(f)

(g)

xyz

xy

z

x

yz

RR-w

Ani

sotro

py1.98

1

Ani

sotro

py

2.53

0.91

Ani

sotro

py

4

1

1

-1

H||

1

-1

H||

(c)( )

1

-1

Hρ

1

-1

Hρ

(b)

FIG. 2. Demonstration and numerical verification of the versatility of the two-dimensional equivalence relations for a beam bender (a)–(c),beam splitter (d)–(e), and Mobius lens (f)–(g). In all these examples, both the anisotropy (visualized by surface coloring on the symmetryplane of the waveguide) and the thickness variations of the core medium (visualized by the height of the floating surface representing theupper interface z = a) manipulate the in-plane magnetic fields in a desired way and correspond to technologically feasible parameters. Withoutthickness variations (c), the in-plane magnetic fields cannot complete the bend.

which preserves the initial dispersion (β,ω) despite the modi-fied extinction coefficient. The transformed wave equation andthe continuity of the tangential fields are thus imposed by theanisotropic permittivity and the thickness variation of the core,respectively.

II. NONMAGNETIC METAMATERIAL WAVEGUIDES

In the second part of this paper, we illustrate the versatilityand effectiveness of our nonmagnetic equivalence relationsusing three conformal transformations: an exponential mapimplementing a beam bender [Figs. 2(a)–2(c)], a curve-factor Schwarz-Christoffel transformation implementing abeam splitter [Figs. 2(d)–2(e)], and a Mobius transformationimplementing a lens [Figs. 2(f)–2(g)]. For each of theseexamples we have visualized the upper half of our symmetricwaveguide, representing the anisotropy of the metamaterialcores with a surface coloring on the waveguide symmetryplane and the thickness variation of the upper interface atz = a with a floating surface. Furthermore, we note thatconformal transformations have a rich history in physics andengineering [40,41]; e.g., they also contributed to severalthree-dimensional transformation-optical devices [16], so thata variety of two-dimensional conformal transformations areavailable from literature.

As a first example we use a logarithmic map [16] todesign a beam bender [Figs. 2(a)–2(c)]. This two-dimensionalimplementation requires modest thickness variations andanisotropies, illustrated for a beamwidth w = 24a and outerradius R = 82.7a in terms of the initial thickness 2a =0.4 μm. Qualitatively, comparisons of in-plane magnetic fields(Fig. 2(b) and Figs. S4–S7 of the Supplemental Material [39])to the conventional transformation-optics implementation con-firm that the combined variation of the anisotropy (ε⊥/ε||) =γ (x,y) in the core and the thickness a(x,y) lead to efficient

beam bends, although eventually the size of such a beambend will be limited by the Miller limit [42]. Moreover, as isshown by Fig. 2(c), anisotropic beam bends without thicknessvariations cannot preserve a global propagation constant due toincompatible continuity conditions. Therefore, guided wavescannot propagate without thickness variations.

To test the equivalence in a quantitative way, we determinethe throughput while reducing the outer radius R of thebends at a fixed initial width w (Fig. 3). In this way, theanisotropy of the core (ε⊥/ε||) increases with the benchmark(w/R). Impressively, these throughputs range from 84% to93% for microscale inner radii between 3.26 μm and 25.9 μm,comparable to designs by three-dimensional transformationoptics (86% to 95%). However, when using an isotropicmedium εcore = γ (x,y)εcore together with the required thick-ness variations, the throughputs fall considerably at radii closeto the effective incident wavelength λ = 1.5 μm, leading topoor beam benders (Fig. 3 and Fig. S6). Indeed, isotropicmedia do not preserve the incident polarization of the guidedmode so that the thickness variation only partially applies,i.e., only to the transverse-magnetic part of the light. Thus,we have established that anisotropy and thickness variationsare both indispensable to our equivalence on a qualitative andquantitative level.

As a second example, we design a beam splitter inFigs. 2(d)–2(e) that relies on a curve-factor Schwarz-Christoffel transformation [41]. The coordinate lines and lightflows follow the outline of a polygon with parametrizedcorners and curved boundaries specified by the curve factor[Figs. 2(d)]. Our simulations confirm that the in-plane mag-netic field splits successfully when it reaches the first vertex[Fig. 2(e)]. To estimate the performance of the beam split-ter, we calculate the splitting efficiency η = 1 − (Pin/Ptotal),which compares the transmitted power inside the excludedregion (Pin) to the total power at the end facet of the splitter

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VIAENE, GINIS, DANCKAERT, AND TASSIN PHYSICAL REVIEW B 93, 085429 (2016)

0

0.10.2

0.3

0.4

0.50.6

0.70.8

0.91.0

6 9 12 15 20 253

147 79 54 41 33 24 19

Isotropic

Two-dimensional

Three-dimensional

Throughput after bend

Benchmark (w/R)

Inner radius (μm)

FIG. 3. The throughput of our beam bender is evaluated forseven outer radii R at fixed beam width w, where increases inanisotropy are represented by increases in the benchmark w

R, as a

way to demonstrate the effectiveness of our equivalence relations. Inparticular, we compare three implementations: the conventional de-sign of transformation optics (red dots), our two-dimensional design(purple squares), and an isotropic implementation with appropriatethickness variation (blue triangles). The throughputs of our two-dimensional metamaterial cores are impressive, lying close to those ofthe three-dimensional implementation while isotropic metamaterialcores cannot maintain their performance as inner radii approachthe free space wavelength λ = 1.5 μm. Mesh convergence studiesresulted in negligible error bars, although we suspect a systematicerror since the three-dimensional implementations represent the idealimpedance-matched implementation corresponding to a theoreticalthroughput of unity.

(Ptotal). The efficiency is adversely affected by the singularitiesof a Schwarz-Christoffel transformation, associated to verticesthat impose vanishing rescalings X = 0. Indeed, rescalingsbelow a specific threshold (Fig. S2) lead to optically dilutecores which cannot confine light. This is expressed mathemat-ically by an imaginary extinction coefficient k1. Fortunately,subcritical rescalings—crucial to some applications such as theinvisibility cloak [10–12]—can be eliminated by combiningembedded transformations and truncations with appropriateglobal rescalings (Fig. S8). For example, our beam splittereasily attains a splitting efficiency of 81%.

As a final example, we implement a two-dimensional lensbased on the Mobius transformation [16]. Our design inFigs. 2(f)–2(g) requires realistic anisotropies and thicknessvariations and connects continuously to the untransformedwaveguide behind the lens thanks to a suitable embedding(Fig. S10). Figure 2(g) confirms that the embedding does notaffect the performance: the lens focuses in-plane magneticfields extremely well, and more importantly, behaves in the

FIG. 4. Instead of manipulating individual light modes withnumerous distinct fibers or waveguides, two-dimensional transfor-mation optics manipulates an incident plane wave holistically bycombining beam splitters, beam benders, and lenses in an integratedsetup. Individual rays of an incident plane wave (yellow) are splitand bent by the geometry-induced anisotropic material (visualizedby surface coloring on the symmetry plane of the waveguide) andthe thickness variation (visualized by floating surface representingthe upper interface z = a). The deformed coordinate grid is projectedboth onto the waveguide’s symmetry plane and the core-claddinginterface.

same way as a three-dimensional transformation-optical lens(Fig. S11). Indeed, the spot size (determined by 1

eof the

maximal amplitude) is as small as 1.6λ0 with free spacewavelength λ0.

While this report is focused on a transformation opticsframework applicable to two-dimensional waveguides, wewant to briefly discuss the feasibility of the designs discussedabove. Each of the three devices requires only dielectricresponse and no magnetic response. The ε|| component ofthe permittivity tensor is constant over the entire structureand can be chosen arbitrarily, and the anisotropies are fairlysmall. Such structures can be fabricated using 3D printing(direct laser writing) by using small subwavelength ellipsoidsas constituent elements [7,38,43], similar to how opticalground-plane cloaks have been realized [43].

The beam bender, beam splitter, and conformal lens consti-tute three independent examples that illustrate how our two-dimensional equivalence relations allow for the manipulationof guided waves in coordinate-designed waveguides withrealistic material parameters. Therefore, we envision that ourresults open up the geometrical toolbox of transformationoptics for the manipulation of two-dimensional guided waveson multifunctional optical chips (Fig. 4).

ACKNOWLEDGMENTS

S.V. and V.G. acknowledge fellowships from the ResearchFoundation Flanders (FWO-Vlaanderen). Work at VUB waspartially supported by the Research Council of the VUBand by the Interuniversity Attraction Poles program of theBelgian Science Policy Office, under Grant No. IAP P7-35“photonics@be”.

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085429-5

Supplemental Material:

Transforming Two-Dimensional Guided Light

Using Nonmagnetic Metamaterial Waveguides

Sophie Viaene,1, 2 Vincent Ginis,1 Jan Danckaert,1 and Philippe Tassin1, 2

1Applied Physics Research Group, Vrije Universiteit Brussel, Pleinlaan 2, B-1050, Brussel, Belgium2Department of Applied Physics, Chalmers University of Technology, SE-412 96, Goteborg, Sweden

(Dated: January 16, 2016)

In this supplemental material, we discuss the foun-dation of our two-dimensional equivalence relations andcompare the performance of three key applications—a beam bender, a beam splitter and a conformalMobius lens—to traditional three-dimensional designswith transformation optics. In particular, we show(1) how wave equations with respect to inhomogeneoustransformations are mapped by uniaxial electromagneticmedia, (2) some technical information about the perfor-mance of our conformal devices and (3) a recipe to applyour equivalence relations in a numerical way.

CONFORMAL TRANSFORMATION OPTICS

In transformation optics, electromagnetic light flowsextremize their path length with respect to coordinate-induced geometries [Figs. S1(a)-(c)]. In particular, ten-sor transformation rules specify the coordinate-inducedgeometry g = JTJ in terms of the Jacobian

J =

∂xX(x, y) ∂yX(x, y) 0∂xY (x, y) ∂yY (x, y) 0

0 0 1

, (S1)

due to a two-dimensional coordinate transformation from(u, v) to (x, y) coordinates

u = X(x, y),v = Y (x, y),z = Z.

(S2)

To be compatible with the waveguide geometry, we willrestrict ourselves to two-dimensional conformal transfor-mations satisfying the Cauchy-Riemann equations

{

∂xX(x, y) = ∂yY (x, y)∂yX(x, y) = −∂xY (x, y)

, (S3)

so that the geometry

g =

γ(x, y) 0 00 γ(x, y) 00 0 1

, (S4)

is characterized by a single scalar field γ(x, y) =

(∂xX(x, y))2+(∂xY (x, y))

2representing the induced ge-

ometry on the waveguide’s symmetry plane.

COMPLEX REPRESENTATION

The complex analytical function w = f(z) with coor-dinates w = u + iv and z = x + iy provides an intuitiverepresentation of a two-dimensional conformal transfor-mation

Re[f(z)] = X(x, y),Im[f(z)] = Y (x, y).

(S5)

Because differentials of w are proportional to differentialsof z

dw = f ′(z)dz, (S6)

the modulus of the transformation |f ′(z)| represents a lo-cal coordinate stretching while the argument representsa rotation angle α = arg(f). It is the modulus of the ana-lytical function that elegantly defines the induced scalargeometry γ(x, y) = |f ′(z)|2 (Eq. S4). Therefore, the com-plex representation is of great value both for understand-ing the local behavior of a transformation and for obtain-ing the geometry.A careful reader may object that the main paper

only discusses the microscopic limit of conformally trans-formed regions in terms of local rescalings. Local angularrotations are not included. These rotations are, however,safely neglected because an analytical function preservesangles between transformed coordinate lines from (x, y)to (u, v). Only the magnitude of the rescaling contributesto the induced geometry so that the microscopic validityof our equivalence relations is established by real, homo-geneous rescalings X.

GUIDED TRANSFORMATION OPTICS

The three-dimensional framework of transformationoptics makes use of uniaxial electromagnetic media in-side the core and outer layers

ǫ =

{

ǫ|| = ǫ(z),ǫ⊥ = ǫ(z)γ(x, y).

µ =

{

µ|| = 1,µ⊥ = γ(x, y),

(S7)to impose a conformal geometry onto light. The actualmaterial parameters are distinguished from those of theinitial waveguide by a tilde. Indeed, Fig. S1 shows how

2

(a)

(d)

εout

εout

εcore

2a

β

εout, || ε

out, ||ε

core, ||

2a

~ ~ ~

β~

εout

εout

εcore, ||

2a

~

β~

(u,v) (x,y)

(b) (c)

(e) (f)~

εout,

εout,

εcore,

εcore,

~ ~ ~ ~

2a 2aεcore

εcore, ||

~

2a~

┴ ┴┴ ┴

εcore,

~ε

core,~

εcore, ||

~

┴ ┴

εout

εout

εout

εout

εout, ||

~

out, ┴ε~

εout, ||

~

┴ε out,

β β β~ ~

(g) (h) (i)

FIG. S1. Our design uses a nonmagnetic uniaxial metamaterial waveguide of varying thickness to impose two-dimensionalflows of light. The symmetry plane of a slab waveguide (a) is locally stretched by a two-dimensional conformal coordinatetransformation (b) so that light bends over 90 degrees as if it experiences a geometry γ(x, y) (c). (d) Accordingly, the vectorspace of the incident guided mode with propagation constant β (green) is stretched in the symmetry plane (e). The totalwave vectors lie respectively on elliptical (hyperbolic) isofrequency contours of the wave equation in blue (red) inside the core(outer) region. According to the traditional recipe of transformation optics, the exponential tails (k1) and thickness a arepreserved because metamaterials are implemented in the core and outer layers. (f) To preserve confinement and to impose aglobally stretched propagation vector without metamaterials in the outer layers, a thickness variation a ensures the continuityconditions at the interfaces at the expense of changes in the exponential tails (k1). (g) The initial transverse-magnetic profile(with intensity in color online) is either stretched in a global way by nontrivial materials in both the core and the outer regions(h) or in a partial way by one nontrivial material in the core region complemented with a thickness variation (i).

an incident transverse profile [Fig. S1(g)] is stretched in aglobal way by the traditional three-dimensional metama-terial [Fig. S1(h)] if both the core and outer layers consistof a uniaxial metamaterial. Here we derive equivalence

relations for inhomogeneous coordinate transformationsto mold the flow of guided waves without metamaterialimplementations in the outer layers, as shown in Fig.S1(i).

A two-dimensional coordinate transformation splits the Helmholtz wave equation into in-plane xy and transverse z

contributions

i = xy 1γ(x,y)

[

∂2x + ∂2

y

]

Hi + ∂2zHi +

ǫ(z)ω2

c2 Hi +∂jγ(x,y)γ(x,y)2 [∂iHj − ∂jHi] +

∂iγ(x,y)γ(x,y) ∂zHz = 0,

i = z 1γ(x,y)

[

∂Zx + ∂2

y

]

Hz +ǫ(z)ω2

c2 Hz = 0,(S8)

due to distinct in-plane and transverse transformations of the Laplacian ∆xy. In the spirit of transformation optics—which compares the Maxwell equations of a geometry and to those of an electromagnetic material—we compare thisset of wave equations to the wave equations of an electromagnetic uniaxial medium to emulate the effect of the

3

transformation-induced geometry

i = xy[

∂2x + ∂2

y

]

Hi +ǫ⊥ǫ||

∂2zHi +

ǫ⊥µ||ω2

c2 Hi +∂j ǫ⊥ǫ⊥

[∂iHj − ∂jHi] +∂iγ(x,y)γ(x,y) ∂zHz = 0,

i = z[

∂2x + ∂2

y

]

Hz +ǫ||µ⊥ω2

c2 Hz +∂xǫ||ǫ||

[∂zHx − ∂xHz] +∂y ǫ||ǫ||

[∂zHy − ∂yHz] = 0.(S9)

Notice that these equations contain first-order deriva-tives, in contrast to those of constant rescalings, as ex-pected for wave equations beyond the geometrical limit.

To preserve the in-plane polarization of the transversemagnetic field, both the transformed wave equations (Eq.S8) and the wave equations of the medium (Eq. S9) im-pose a vanishing transverse component Hz. Therefore,the in-plane dielectric components ǫ|| are homogeneous.Moreover, to preserve the in-plane components of themagnetic field Hx,y in equation (Eq. S8), the dielec-tric properties of the incident waveguide are preservedin the plane ǫ|| = ǫ(z) and stretched proportionally tothe geometry outside of the plane ǫ⊥ = ǫ(z)γ(x, y). Inthis way, we exactly map the Helmholtz equations witha nonmagnetic medium inside the core of the waveguide.

To compensate for the lack of metamaterials in theouter layers of the waveguide, we impose an analyticalthickness variation based upon the transformed incidentmagnetic field. For simplicity, we continue with an ex-pression of the magnetic field in the plane wave approxi-mation

Hi ∝√

γ(x, y) eiβ√

γ(x,y)1β ·r, (S10)

0 1 2 3 4 5 6

1

√ε

out

εcore

√ εcore

εout-

"/2

Single mode

waveguide cutoff

√εout

c

ωc

β

Frequencycω

a

Cutoff rescaling Xc

FIG. S2. The cutoff rescaling X2

c = ǫoutω2

β2c2expresses a min-

imal stretch that is performed on a guided mode before los-ing confinement. It decreases monotonically with frequency

from 1 to√

ǫoutǫcore

. To stay within the single mode regime of

the waveguide, frequencies are necessarily smaller than thefundamental cutoff frequency ωc of the incident waveguide(green). For example, an incident waveguide with refractiveindex ncore = 1.5 in the core and vacuum nout = 1 outside ofthe core is limited to rescalings above 0.79.

emphasizing that our formalism is valid beyond theregime of geometrical optics. When the Helmholtz equa-tion is applied to this magnetic field, the extinction coef-ficient associated to the guided mode changes accordingto

k2out = γ(x, y)β2 − ǫoutω2

c2, (S11)

so that a thickness variation is required to preserve thedispersion relation and the associated continuity condi-tions

a =1

kcorearctan

ǫcore

ǫout

√

γ(x, y)β2 − ǫoutω2

c2

kcore

. (S12)

When dealing with applications, we notice that both thethickness variation and the anisotropy of the mediumare crucial to the performance of our devices. To obtainequivalence relations beyond the plane wave approxima-tion, the transformed in-plane field (Eq. S10) should con-tain the Jacobian of the transformation and a nonlinearphase β (X(x, y)1x + Y (x, y)1y).

Because the thickness (Eq. S12) is ill-defined for rescal-ings below a frequency-dependent cutoff Xc (Fig. S2)

X2c =

1

β2

ǫoutω2

c2, (S13)

some coordinate transformations which require largephase velocities, such as invisibility cloaks and singulartransformations, are not directly amenable to our tech-nique. We show later that a combination of truncationsand additional global rescalings ensure that the thresholdrescaling Xc is avoided without modifying the propaga-tion of the guided waves.

APPLICATIONS

Beam bender

As explained in the main paper, the logarithmic mapof a beam bender

{

u = R2 Log

(

x2 + y2)

,

v = arctan(

yx

)

,(S14)

provides a reliable way to probe the performance of thetwo-dimensional equivalence relations as compared to

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

6 9 12 15 20 253

147 79 54 41 33 24 19

Isotropic

Two-dimensional

Three-dimensional

Troughput after bend

Benchmark (w/R)

Inner radius (μm)

6

4.5

3

1.5

0

2

1.5

1

0.5

06 9 12 15 20 253

147 79 54 41 33 24 19

Max. Anisotropy

Max. Relative Thickness Variation

Benchmark (w/R)

Inner radius (μm)

(a) (b)

FIG. S3. Realistic material parameters lead to effective beam bends as demonstrated for seven outer radii R at fixed beamwidth w, where increases in anisotropy are represented by the benchmark w

R. (a) The maximal anisotropy ǫ⊥

ǫ||(red dots)

and relative thickness aa(purple squares) are perfectly realizable with existing fabrication methods [38]. (b) The throughput

provides a way to demonstrate the effectiveness of our equivalence relations. In particular, we compare three implementations:the conventional design of transformation optics (red dots), our two-dimensional design (purple squares) and an isotropicimplementation with appropriate thickness variation (blue triangles). The throughputs of our two-dimensional metamaterialcores are impressive, lying close to those of the three-dimensional implementation while isotropic metamaterial cores cannotmaintain their performance as inner radii approach the free space wavelength λ = 1.5µm. Mesh convergence studies resultedin negligible error bars, although we suspect a systematic error since the three-dimensional implementations represent the idealimpedance-matched implementation corresponding to a theoretical throughput of unity.

traditional transformation-optical designs. The geome-

try γ(x, y) = R2

x2+y2 leads to realistic material param-eters, deviating the strongest from the incident waveg-uide at inner radii of the bend (Fig. S3). Dependingon the initial beam width w and the outer beam radiusR, anisotropies and inhomogeneities are introduced toput our equivalence relations to the test. Excitingly, ifthe incident guided mode is sufficiently narrow and/orconfined, the inner radius may become smaller than thefree-space wavelength. In this regime, beam bends be-come subwavelength (Fig. S4).

To obtain reliable estimates about the throughput,we performed a mesh convergence analysis of our finite-element numerical simulations. For increasingly densemeshes, the difference in throughput provides an esti-mate of the meshing error. The error bars are muchsmaller than the point sizes on the graph and are, there-fore, negligible.

Additionally, the three-dimensional beam bend simu-lations allow estimating the overall numerical error. Be-cause three-dimensional equivalence relations exactly re-produce the Maxwell equations of a curved geometry, weexpect that the throughput approaches unity, modulosome reflections when coupling into and out of the de-vice. To validate our equivalence relations, differences inthroughput of two- and three-dimensional transformationoptics should be small.

The in-plane magnetic fields behave as expected foranisotropic and isotropic implementations of the beam

bend (Figs. S5 and S6). Over a wide span of outer radii,the anisotropic implementations bend guided light in asatisfactory way while isotropic implementations havedifficulties to impose the turn at inner radii. This isrelated to the polarization of the incident wave, which isonly preserved by anisotropic implementations.In Fig. S7, the in-plane magnetic fields and the mag-

netic norm in the symmetry plane of the waveguide arecompared for two- and three-dimensional designs. Over-all, both simulations agree very well. The magnetic normonly deviates slightly at the outer radius, which explainswhy throughputs differ for both implementations.

Beam splitter

Schwarz-Christoffel transformations are useful to manyapplications in physics, such as in hydrodynamics, acous-tics and electrostatics. In this contribution, we constructa beam splitter from a traditional Schwarz-Christoffeltransformation with exterior angles qπ and (n − q)π atcorners u1,4 = ∓b and w2,3 = ∓a

dz =R Cn(w)

(w2 − b2)q(w2 − a2)n−qdw, (S15)

containing a double curve-factor

C(w) = w2 − 1

2

(

a2 + b2)

+ i([

w2 − a2] [

w2 − b2])1/2

.

(S16)

5

1

-1

H||

1

-1

H||

6 μm

1.2 μm

(a) (b)

w w’

FIG. S4. We compare two beam benders with equal bench-mark factors w

R= 0.4, i.e. (a) R = 50a and w = 20a with

a throughput of 91.2% and (b) R = 10a and w′ = 4a withthroughput 84.2%. For our two-dimensional beam bends, themaximal rescaling at the inner radius of the beam benderonly depends on the ratio of the incident width w over R.Therefore, the effective free-space wavelength 1.3µm of thefundamental incident mode may be larger than the inner ra-dius of the bend. Still, we notice that the performance de-creases for smaller bends due to diffraction and depends onthe initial confinement of the guided mode. The inner radiiare indicated in micrometer.

1

-1

H||

1

-1

H||

1

-1

H||

1

-1

H||

3.26 µm 11.74 µm

25.88 µm 37.20 µm

(a) (b)

(c) (d)

FIG. S5. To test our two-dimensional beam benders, wegradually decrease the outer radii at fixed incident widthsto turn on inhomogeneities and anisotropies. The in-planemagnetic fields bend very well as indicated by modestly de-creasing throughputs (Fig. S3), i.e. (a) 87.7%, (b) 91.2% (c)91.2%, (d) 93.4%. The inner radii are indicated in micrometerand increase from (a)-(d).

1

-1

H||

1

-1

H||

1

-1

H||

1

-1

H||

3.26 µm 11.74 µm

25.88 µm 37.20 µm

(a) (b)

(c) (d)

FIG. S6. For isotropic two-dimensional implementations withisotropic media and appropriate thickness variations, the in-plane magnetic fields do not bend efficiently for small outerradii due to mode coupling with transverse-electric modes.The inner radii are indicated in micrometer and increase from(a)-(d), with increasing throughputs (a) 4%, (b) 31.0%, (c)87.2%, (d) 75.2%.

1

0

|H|

1

0

|H|

1

-1

H||

1

-1

H||

(a) (b)

(c) (d)

11.74 µm11.74 µm

11.74 µm11.74 µm

FIG. S7. For a beam bend of R = 16.54µm and incidentwidth w = 4.6µm, the norm and the in-plane magnetic fieldfor the two-dimensional (a), (c) and three-dimensional (b), (d)implementations agree very well with throughputs of 91.2%(c) and 93.2% (d).

6

(a)

1

-1

H||

1

-1

H||

(b)

1

0

|H|

1

0

|H|

(d)

(c)

FIG. S8. Comparison of the two-dimensional design (a)-(c) and the three-dimensional design (b)-(d): The two-dimensional design imposes all features of traditional imple-mentations without using metamaterials inside the outer lay-ers. Both the in-plane magnetic field and the magnetic norm,related to the power flow, agree in a qualitative way.

To obtain an interferometer of splitting angle 2πq with-out corners at w = ±a, we equate n and q. In addition,we insert a normalization factor R to deal with singu-larities at the other vertices w = ±b. After numericalintegration, we obtain the geometry

γ(u, v) =1

| dzdw (u, v)|2, (S17)

which is easily expressed with respect to z-coordinates.Then, our equivalence relations readily define the appro-priate uniaxial core and thickness variation.

Figure S8 compares two-dimensional and three-dimensional implementations of the beam splitter. Thein-plane magnetic fields and norm agree very well, con-firming that the two-dimensional equivalence relationsmimic the propagation of guided waves in the coordinate-designed space.

Because Schwarz-Christoffel transformations often in-duce singularities, e.g., vanishing rescalings at the ver-tices, we truncate the coordinate transformation in asmall neighborhood of the vertices before applying ourtwo-dimensional equivalence relations. Indeed, whenrescalings are smaller than the cutoff rescaling in Fig.S2, the core is too dilute to apply dispersion engineering.The extent of the truncated neighborhoods is reduced byan additional global rescaling, i.e. introducing the nor-malization parameter R. Figure S9 confirms that designsincluding low threshold rescalings X = 0.66 outperformthose with high threshold rescalings X = 0.8, both qual-itatively and in terms of their splitting efficiencies, i.e.81% and 60%.

(a)

1

-1

H||

1

-1

H||

(b)

1

0

|H|

1

0

|H|

(d)

(c)

FIG. S9. Different global rescaling factors R allow includinginitial rescalings above X = 0.66 and X = 0.8, respectivelyshown in (a), (c) and (b), (d), with splitting efficiencies of 81%and 60%. The in-plane magnetic norm (a)-(b) and magneticfield (c)-(d) clearly reflect this trend.

Mobius lens

The Mobius transformation [16]

z = i

(

i+ w

i− w

)

, (S18)

converts a point source on the rim of a lens with radius Rto a plane wave at the opposite edge. In real coordinates,the transformation (Eq. S18) is equal to

{

x = R 2uu2+(1−v)2 ,

y = R 1−u2−v2

u2+(1−v)2 .(S19)

Figure S10 illustrates how the transformed Cartesiancoordinates result in a lens with moderate thickness vari-ations and anisotropies. The associated geometry is equalto

γ(x, y) =4R4

(

[x− x0]2+ [y − y0]

2)2 , (S20)

with central coordinates (x0, y0). In particular, we embedthe transformation starting from ystart = y0+R to ystop =y0 +2R and take x0 in the middle of the incident face ofthe lens with radius R. In this implementation, there is aregion beyond y = ystart+(

√2−1)R for which rescalings

are smaller than one. We truncate the transformation atthis point, embedding it into the surrounding flat space,both to avoid a dilute core and to connect continuously tothe untransformed waveguide. We expect that, similar tothe beam splitter, optimized global rescalings may reducethe side tails and finite spot widths in Fig. S11.

7

R

(b)

x

yz

y

An

iso

tro

py

4

1

1

-1

H||

(a)

R(√2 - 1) R

4

3

2

1

1/4

Embedded geometry

Unbounded geometry

(d)

-R R0

(c)

FIG. S10. Our two-dimensional Mobius lens with radius R focuses an incident plane wave—propagating along the y-direction—with modest thickness variations and anisotropies. (a) The anisotropy (color online on symmetry plane of dielectric waveguide)is maximal at the incident face and approaches unity at the end, leading to a gradual symmetric thickness variation (visualizedby the floating surface representing the z = a interface), shown for the upper half of the waveguide. (b) Our implementationsuccessfully focuses the in-plane magnetic field, coming from the left, with minor developments of side tails that also occur inthree-dimensional implementations (Fig. S11). (c) Deformation of Euclidean coordinate lines due to the Mobius transformation.Only the geometry below the red line is implemented in our embedding, to avoid the asymptotic value of 1/4 which is belowcutoff and does not connect continuously to the untransformed waveguide after the lens. (d) Illustration of the embedded(purple) and unbound (black) geometry along the y-axis of the lens, representing the anisotropy of the core.

1

-1

H||

1

-1

H||

1

0

|H|

1

0

|H|

(a) (b)

(c) (d)

FIG. S11. We compare a two-dimensional implementationof the Mobius lens with the conventional bulky implementa-tion. The in-plane magnetic field (a) and norm (c) of our two-dimensional design confirm that guided waves focus similarlyto the transformation-optical design of a bulk metamaterialwith in-plane magnetic field (b) and norm (d).

CONCLUSION

The implementation of two-dimensional and three-dimensional equivalence relations leads to strikinglysimilar in-plane magnetic norms and fields (Figs. S7, S10and S13). In this way, we established three independentdemonstrations of the validity of our two-dimensionalequivalence relations, a uniaxial dielectric metamaterialcore of varying thickness. The anisotropies of the coreand the thickness variations are realistic and techno-logically feasible. Relying on our equivalence relations,guided wave optics can now also benefit from the toolsof transformation optics.