designing whispering gallery modes via transformation opticsappe/publication/2016/03... ·...

Designing whispering gallery modes viatransformation opticsYushin Kim1†, Soo-Young Lee2†, Jung-Wan Ryu2,3†, Inbo Kim2, Jae-Hyung Han2, Heung-Sik Tae2,Muhan Choi2* and Bumki Min1*

In dielectric cavities with a rotational symmetry, whisperinggallery modes (WGMs) with an extremely long lifetime (thatis, a very high Q factor) can be formed by total internal reflec-tion of light around the rim of the cavities. The ultrahigh Qfactor of WGMs has enabled a variety of impressive photonicsystems, such as ultralow threshold microlasers1–3, bio-sensors with unprecedented sensitivity4,5 and cavity optome-chanical devices6. However, the isotropic emission of WGMs,which is due to the rotational symmetry, is a serious drawbackin applications that require directional light sources. Considerableefforts have thus been devoted to achieving directional emis-sion by intentionally breaking the rotational symmetry7–9.However, all of the methods proposed so far have sufferedfrom substantial Q-spoiling. Here, we show how the mode prop-erties of dielectric whispering gallery cavities, such as the Qfactor and emission directionality, can be tailored at willusing transformation optics. The proposed scheme will opena new horizon of applications beyond the conventional WGMs.

It is well known from Einstein’s general theory of relativity thatlight propagates along a curved geodesic path. An analogy of thecurved light path due to gravity can be realized inside materialswith spatially varying electromagnetic properties as proposedin the seminal works by Pendry et al.10 and Leonhardt11. Thistheory, called transformation optics, has been receiving increasedinterest because it enables the achievement of exotic wavepropagation phenomena, such as invisibility cloak10,12,13, artificialblack hole14,15 and giant field enhancement in plasmonic nanostruc-tures for light-harvesting16. The theory is based on the fact that theform of Maxwell’s equations is preserved under general coordinatetransformations with renormalized electromagnetic fields and newconstitutive parameters (that is, the permittivity ε and permeability µ).This implies that light can be made to propagate along a chosencurved path if we implement a spatial distribution of the constitutiveparameters derived from the coordinate transformation that con-nects an original space with a target (physical) space. Transformationoptics has so far been applied mainly to controlling the propagationpath of electromagnetic waves in a desired manner. However, herewe show that it can also be exploited to manipulate resonance-mode properties of two-dimensional (2D) dielectric optical cavitiesfor various practical purposes.

The pursuit of an ultrahigh-Qmode in optical microcavities witha directional emission has been a long-standing challenge. Previousattempts to obtain directional emissions have been based essentiallyon breaking the rotational symmetry of the microcavities, forexample, by deforming the shape of a cavity7–9, by embedding a scat-terer near the cavity boundary17, or by making an annular cavity18.Among these methods, the most typical is to deform the cavity

shape, and for practical applications deformed cavity shapes thatsupport high-Q modes with unidirectional emissions have beenreported9. Even so, it is noteworthy that all of these schemes havebeen unable to take advantage of the beneficial properties of high-Q WGMs, because breaking the rotational symmetry is alwaysaccompanied by an appreciable Q-spoiling of the optical modes.In this work, based on the theory of transformation optics, wepropose a general scheme to obtain a highly directional light emis-sion solely from high-Q WGMs in deformed cavities, which doesnot suffer from the aforementioned problem and can be utilizedfor applications in advanced optoelectronic devices.

In transformation optics, the new ε and µ in the transformedspace are given in tensor forms under a general coordinate trans-formation, so the corresponding medium becomes anisotropicand inhomogeneous in general. Owing to this complexity, practicalrealizations of that kind of medium are a challenging task, so only afew examples with relatively simple geometries for a given polariz-ation have been reported12,13,19,20. Fortunately, in 2D systems(when the transformation is a conformal mapping), only linearisotropic materials with scalar ε and µ for a specific polarizationof electromagnetic waves need to be considered in designing trans-formation optical devices21,22. For this reason, our scheme takesadvantage of conformal mapping and isotropic dielectric materials.The idea is as follows: deform a 2D dielectric disk via optical con-formal mapping11. If both the inside and outside regions of thedisk are transformed, the WGMs will be deformed with their intrin-sic characteristics intact because the optical conformal mappingtransforms the full wave equation in 2D. However, if only theinterior of the disk is transformed, the output coupling can bedrastically altered without sacrificing the WGMs. In the lattercase, the transformed WGMs therefore retain their ultrahigh Qfactors, even in considerably deformed cavities, and also exhibitanisotropic emission.

As a simple example of our approach, we start with a deformedboundary: the so-called Pascal’s limaçon23, which reads in (r, θ)polar coordinates, r(θ) = 1 + 2αcosθ, where α is a deformation par-ameter. The corresponding conformal transformation, whichmaps the unit circle to the limaçon, is given by:

z = β(w + αw2) (1)

wherew and z are complex variables that denote positions in the twocomplex planes respectively, and β is a positive scaling factor thatwill be given later as a function of α. Figure 1a,b shows that auniform grid in the complex w plane (w = u + iv) can be trans-formed into a curved grid in the complex z plane (z = x + iy) accord-ing to equation (1). It is worth pointing out that although a straight

1Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea. 2School of ElectronicsEngineering, Kyungpook National University, Daegu 41566, Republic of Korea. 3Center for Theoretical Physics of Complex Systems, Institute for BasicScience, Daejeon 34051, Republic of Korea. †These authors contributed equally to this work. *e-mail: [email protected]; [email protected]

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ray trajectory inside the circle on the w plane is mapped to a curvedtrajectory in the z plane, the incident angle χ of a ray in the w planeis preserved in the z plane, which is an intrinsic property of the con-formal mapping. Therefore, the incident angle at every reflection inthe limaçon cavity is kept constant, and it provides us with a classi-cal ray picture for restoring WGMs in the conformally deformedmicrocavity. With this conformal mapping, an inhomogeneouscavity can be defined by imposing spatially varying refractiveindices derived from transformation optics theory.

First let us consider a homogeneous dielectric disk cavity of unitradius, as shown in Fig. 1a. The resonance modes in the disk cavityon the w plane can be obtained easily by solving the Helmholtzequation with an outgoing boundary condition, and aWGM solutionwith transverse magnetic polarization is depicted in Fig. 1c, where therefractive index of the disk cavity is taken as n0 = 1.8. Next, we want toobtain the corresponding mode in the inhomogeneous limaçon cavityon the z plane, described by the conformal mapping equation (1) withα = 0.2 and β = 0.714, as shown in Fig. 1b. The resonance modes ofour concern are solutions of the Helmholtz equation

[∇2 + n2(x, y)k2]E(x, y) = 0 (2)

where ∇2 is the 2D Laplacian and E(x, y) is the normal component ofthe electric field with respect to the cavity plane (z plane), k is the freespace wavenumber and the refractive index n(x, y) is given by

n(x, y) =n0

dzdw

∣∣∣∣∣∣∣∣−1

=n0

β��1 + 4αz/β

√∣∣∣∣∣∣, inside the cavity

1, outside the cavity.

⎧⎪⎨⎪⎩

(3)

The only difference from the case of a homogeneous disk cavity in thew plane is the introduction of the gradually varying refractive index

profile n(x, y) inside the cavity on the z plane, given by the ratiobetween the local length scales in both planes. In our cavity model,the conformal mapping is applied only to the inside of the cavity,and a uniform refractive index of nout = 1 is assigned to the outsideregion of the cavity on the z plane (by considering the cavity infree space). Related to this uniform setting of nout = 1, a conditionis needed to ensure total internal reflections in the conformallydeformed cavity, that is, |dz/dw|−1 ≥ 1 and, in our case, the conditionis given by β ≤ βmax = 1/

��1 + 4α(1 + α)

√. Hereafter, the cavity

defined in this way will be called the transformation cavity. Theuniform refractive index setting implies that the high-Q resonancemode in the transformation cavity (see Fig. 1d) is not identical tothe transformed counterpart EWGM(z) of the WGM in the homo-geneous disk cavity; EWGM(z) = EWGM(w(z)), where EWGM(w) is theWGM solution of homogeneous disk cavity in the entire w plane.However, the internal field pattern of a high-Q resonance solution inthe transformation cavity would be nearly the same as that of thecounterpart because the condition for total internal reflection is still ful-filled in the transformation cavity. Meanwhile, the emission directional-ity of the transformation cavity can deviate substantially from that of thetransformed counterpart due to the spatial variation of the output coup-ling at the dielectric–air interface (that is, the cavity boundary), whicharises from the uniform setting of nout = 1 outside the cavity.

The intensity pattern of a high-Q mode can be calculatedthrough a mode matching method based on a virtual spaceGreen’s function by introducing an auxiliary space that is derivedfrom the transformation cavity via conformal mapping (seeSupplementary Section I). The calculated intensity pattern isshown in Fig. 1d, where the following parameters are used: n(x, y)with n0 = 1.8, α = 0.2 and β = βmax. Henceforth, we will call thiskind of mode a conformal WGM (cWGM) because the incidentangle of the light ray is invariant inside the cavity. The field intensityof the cWGM is localized near the boundary, as in the conventional

w z

χχ

4.2

1.8

3.0

2.4

3.6

1.0

0.0

0.5

a b

c d e1.0

0.0

0.5 n

|E/Emax | 2

|E/Emax | 2

Figure 1 | A homogeneous disk cavity versus a limaçon-shaped transformation cavity. a, A ray trajectory in the (u, v) plane (that is, a homogeneous diskcavity in the original space) is shown. b, Space distortion in the (x, y) plane and the limaçon-shaped cavity with a curved ray trajectory, generated byconformal mapping (equation (1)) with α =0.2 c, A WGM with transverse magnetic polarization in the homogeneous disk cavity; its azimuthal mode numberm is 16. Emax, maximum value of E field inside the cavity. d,e, Restored WGM in the limaçon-shaped cavity (d) with the spatially varying refractive index (e).

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WGMdepicted in Fig. 1c. The length scale inside the cavity varies inaccordance with the refractive index n(x, y) (Fig. 1b,e) and the dis-tance between the adjacent nodes in the intensity pattern of thecWGM therefore scales down in the high-refractive-index regionof the cavity. This is the characteristic feature of cWGMs that is dis-tinct from the conventional WGMs.

Next, we examine the Q factor variation as a function of thedeformation parameter α; the results are shown in Fig. 2a. It is inter-esting to note that the Q factor of the cWGM in the transformationcavity is comparable to that of the WGM in the homogeneous diskcavity even in the severely deformed case of α = 0.25 (the square dotlabelled (iii) on the red line in Fig. 2a). This unique feature of thecWGM is starkly contrasted with the exponential Q-degradationof conventional deformed cavities with a homogeneous refractiveindex (see the blue line for α≳ 0.12). The intensity patterns ofcWGMs are localized strictly along the cavity boundary whereasthe mode intensity patterns in the homogeneous cavity start todeviate from their boundary shape, forming a polygonal patternas the deformation increases (for example, Fig. 2a(vi)), for whichthe corresponding ray dynamics in the homogeneous cavity isfully chaotic. In quantum chaos, this intensity localization isknown as the ‘scarring’ phenomenon by the polygonal classicalperiodic orbit24.

The characteristics of the cWGM can be seen more clearly in thephase-space representation; that is, the so-called Husimi function25

H(s, sinχ), where s is the boundary coordinate. The Husimi functionat the dielectric interface is obtained by the overlap of a Gaussianwave packet with a resonant mode of the system in phase space.The Husimi function for the cWGM is plotted in Fig. 2b, where sis normalized with the cavity boundary length L. Along theordinate, the maxima of H(s, sinχ) are located near the lines ofsinχ = ±0.8108, where the χ value is nearly the same as theincident angle of the disk cavity given by the semiclassical relation,sinχ =m/nkR = 0.8125 where m is the azimuthal mode number, n isthe index of refraction inside the cavity, k is the free space wavenum-ber and R is the radius of the disk. The sign of sinχ denotes the senseof circulation (+ for anticlockwise (ACW), − for clockwise (CW)) ofthe wave components inside the cavity. The maximum line nearsinχ = 0.8108 is located much higher than the critical line denotedby the solid yellow line, which implies that the waves are well con-fined by total internal reflection. This is the reason why the cWGMhas such a high-Q value. The corresponding ray trajectory, launchedwith the incident angle χ = arcsin (0.8108), is depicted in Fig. 2c.The ray trajectory produces a caustic inside the cavity and isrestricted in the area between the caustic and the boundary, wherethe intensity pattern of the corresponding cWGM is formed inaccordance with the ray trajectory, as can be seen in Fig. 2a. Onthe other hand, in the Husimi function for the scar mode in thehomogeneous cavity with α = 0.25 (shown in Fig. 2d), four localizedmaxima exist for each circulating wave component, and the corre-sponding incident angles deviate from χ = ±arcsin (0.8125), whichimplies that the mode is not a WGM. It is well known that the dis-tribution tail of the Husimi function near the critical line is affectedby the classical unstable manifold structure and is responsible fordirectionality of the refractive emission26,27. When the emission ofa mode is refractive, the mode Q factor cannot be so high,because light is no longer confined by total internal reflection.The chaotic ray trajectory (shown in Fig. 2e and generated withthe same initial condition as for Fig. 2c) illustrates that the incidentangle of the ray is frequently smaller than the critical angle, resultingin refractive escape from the homogeneous cavity.

On the other hand, the emission property of the cWGM in thelimaçon-shaped transformation cavity is distinguished from thatof the corresponding mode of the homogeneous limaçon cavity justmentioned above. To illustrate this more clearly, let us consider thelimaçon-shaped transformation cavity with α = 0.15 (see Fig. 2a(ii)),

the refractive index profile of which is shown in Fig. 3a. The fieldintensity of the cWGM is displayed inside a circle of radius 100R,and the far-field distribution is drawn on the cylindrical screen(Fig. 3b). Directional emissions along the y axis are not refractivebecause internal waves are confined by total internal reflection. Itturns out that the emission beam of the cWGM does not directly

(i) (ii)(iii)

(iv)

(v)

(vi)

Q fa

ctor

103

104

Transformation cavityHomogeneous cavity

1.0

0.0

1.0

0.0

(iv) (v) (vi)

(i) (ii) (iii)

0.5

0.5

1.0

0.0

0.5

s/L0 1

s/L0 1

sin χ

−1.0

−0.5

0.0

0.5

1.0

sin χ

−1.0

−0.5

0.0

0.5

1.0

H/H

max

H/H

max

s

χ

a

b c

d e

0.00 0.05 0.10 0.15 0.20 0.25

|E/Emax | 2

α

Figure 2 | Comparison of a transformation cavity and a homogeneouscavity with limaçon shapes. a, Q factor variation of high-Q modes as afunction of the deformation parameter α (the resonance mode at α = 0 is aWGM with m = 16). The Q value of the cWGM (red line) is retained evenwhen the resonator is strongly deformed, whereas the Q factor of the high-Qmode in the homogenous cavity starts to exhibit an exponential drop fromα≈0.12. b, Husimi function of the cWGM at α=0.25: the solid yellowcurves indicate the varying critical angle for total internal reflection. Thedotted white lines are maximum lines of H(s, sinχ). Hmax, maximum value of H.c, Well-ordered ray trajectory in a limaçon-shaped transformation cavitywith sinχ =0.81. d, Husimi function of the high-Q mode of the homogeneouslimaçon-shaped cavity with α =0.25. The solid yellow lines indicate theconstant critical angle of the total internal reflection. e, Chaotic ray trajectoryin the limaçon-shaped homogeneous cavity: the ray is launched with thesame initial condition as in c.

NATURE PHOTONICS DOI: 10.1038/NPHOTON.2016.184 LETTERS





emerge from the cavity boundary. This fact cannot be explained bySnell’s law; indeed, it is a tunnelling emission due to the wave prop-erty without any corresponding classical ray trajectory. The tunnel-ling emission can be identified by the output along the tangentialdirection emerging from a free-space point off the cavity bound-ary28,29. To verify this, we inspected the near-field intensity distri-bution on a projection plane near the cavity29. The plane was setperpendicular to the direction of the far-field maximum, that is,the y direction in this case, and the normal incident componentof the near-field could be projected onto that plane. Figure 3cshows the result of such projections, which clearly indicates afeature of the tunnelling emission: light coming out from a free-space point off the boundary. The tunnelling emissions emergefrom the right end of the cavity, where the refractive index islowest. This observation is consistent with the fact that the distancebetween the critical line and the maximum value line of H(s, sinχ)near sinχ = 0.8108 is minimized at both ends of the Husimi distri-bution in Fig. 2b (see also Supplementary Section II). Generally,in conventional cases, the leakage is maximized where the boundarycurvature is highest because the largest bending loss occurs there. Inour case, however, the ratio of the inside refractive index proximal tothe cavity boundary to that outside of the transformation cavities,which varies along the boundary position, plays a decisive role inthe emission mechanism at the dielectric boundary.

As shown above, in the limaçon-shaped transformation cavity,the cWGM exhibits bidirectional far-field emissions. More interest-ingly, we can also form a unidirectional output beam from cWGMsby choosing an appropriate boundary shape along with an internalrefractive index profile. Unidirectional emission can be attained ifthe two emission beam directions are oriented to be nearly parallel

to the symmetric axis (that is, the x axis in Fig. 3). In addition, astrong asymmetry is required between the CWand ACWwave com-ponents at each emission point. The presence of two parallel emis-sion beams, as well as a considerable difference between theintensities of the oppositely rotating wave components at each emis-sion point, are the conditions that are considered sufficient and arewidely used for obtaining the unidirectional emission in deformedcavities. This task can be accomplished by a triangular deformationgiven by a conformal mapping z(w) that transforms a unit circle intoa rounded triangular boundary:

z(w) = z3 ○ z2 ○ z1(w)

where

z1(w) = αw + δ

1 + wδ, z2(w) = i

1 + w1 − w

,

z3(w) = ∫w

0eiπ/6(h + 1)−2/3(h − 1)−2/3dh

(see Supplementary Section III for more details). In this mapping, wehave two control parameters, α (0 < α ≤ 1) and δ (a complex valueinside the cavity boundary); α changes the cavity boundary shapefrom triangular (α = 1) to circular (α ≪ 1) and δ provides anadditional variation of the refractive index profile without changingthe boundary shape. When δ is real, this conformal mapping has amirror symmetry about the horizontal axis, that is, the x axis. Theboundary shape and the refractive index profile of the triangulartransformation cavity for (α, δ) = (0.68, 0.2) are depicted in Fig. 3d.The far-field distribution of the cWGM in Fig. 3e shows a

01

2

−10

−2

2

−1

1xyACW

CW

CW

ACW

x

y

x

y

|Efar-field|2

a 3.3

1.8

2.55

b c

3.4

2.0

2.7

d e f

|Efar-field|2

CW

ACW

CW

ACWxy

−2−1

01

2−2

−10

12

nn

Figure 3 | Bidirectional and unidirectional far-field emission of the cWGMs of a limaçon-shaped transformation cavity and a triangular transformationcavity. a, Limaçon-shaped transformation cavity and its internal refractive index profile. b, Far-field distribution of the cWGM (m = 16) at α = 0.15 is plottedon the cylindrical screen located on the circle of radius 100R. Efar-field, far-field amplitude on the screen. The intensity distribution of the emission is plottedon the bottom. The x and y axes are the same as in c. c, Intensity plot of the near-field components normally incident to the projection planes: the projectionplanes are located perpendicular to the directions of the far-field maxima in b. Both planes are normal to the y axis. The tunnelling emissions of the CW/ACWwave components are shown on the solid white line leading to the peaks of the near-field intensity plotted on the projection planes. The dotted whiteline is a tangent at the cavity boundary point closest to the solid line. d, A triangular-shaped transformation cavity and its internal refractive index profile.e, The unidirectional far-field distribution is plotted on the cylindrical screen located on the circle of radius 100R. The intensity distribution of the emission isplotted on the bottom. The x and y axes are the same as in f. f, Intensity plots of the near-field components of the cWGM (m = 22) normally incident tothe projection planes oriented perpendicular to the maximum-intensity directions of the tunnelling emissions from the CW and ACW wave components,respectively. The tunnelling emissions of the CW/ACW wave components are depicted on the solid white lines leading to the peaks of the near-fieldintensity plotted on the projection planes. Each of the dotted white lines is a tangent at the cavity boundary point closest to the nearby solid line.






unidirectional emission. As in the case of the limaçon transformationcavity (Fig. 3c), a tunnelling emission is again implied by the near-field intensity distribution projected onto the planes set perpendicular

to the directions of the far-field maxima (Fig. 3f). Therefore, the far-field emission is the unidirectional tunnelling emission, the firstexample for a 2D dielectric cavity, while all of unidirectional emis-sions reported so far have been refractive emissions. Furthermore,the high-Q cWGMs can be configured to exhibit a multiple direction-ality, for example, three or four directions, by imposing a certaingeometric symmetry on the conformal mapping.

In practice, a spatially varying refractive index profile can beimplemented effectively by drilling subwavelength-scale air holes ina dielectric slab or by arranging dielectric posts with high refractiveindices19,20. First, we numerically examined the validity of thiswell-known approach on a circular dielectric cavity with auniform hole or post distribution. Figure 4b shows the convergenceof the Q factor of a WGM depicted in Fig. 4a to the ideal radiative Qvalue of the corresponding WGM in a homogeneous disk cavity asthe density of holes or posts is increased (while keeping the sameeffective refractive index neff = 2.5). The deviation of the Q factoris reduced to within 10% of the ideal Q value when the number ofscatterers (holes or posts) per wavelength (free-space wavelength/neff)is greater than 20. The convergence of the Q factors on the idealvalue is attributed to the reduction of the scattering loss in morehomogenized cavities. Examples of limaçon-shaped transformationcavities with such hole and post distributions (α = 0.08 for the cavitywith holes and α = 0.15 for the cavity with posts) are also shown inFig. 4c,e. For a clear visualization, much lower densities of holes andposts are chosen for these plots. For limaçon-shaped transformationcavities with higher densities of holes or posts, the intensity patternsof cWGMs were calculated using the finite element method (FEM)and shown in Fig. 4d,f, respectively. It can be seen in these intensityplots that the distance between adjacent nodes is scaled down inregions with a higher effective refractive index. This characteristic is inaccordancewith the previous results for limaçon-shaped transformationcavities with gradually varying refractive index profiles.

To verify our design principle, a triangular transformation cavityis implemented and its associated cWGM is experimentally charac-terized at microwave frequencies30. Here, aluminium oxide posts(n = 3.1) of a diameter of 6 mm are pinned at theoretically predeter-mined positions on a thin polyvinyl chloride (PVC) foam sheet. Thesubstrate permittivity could be assumed to be that of air becausehigh-index alumina posts were pinned on a very thin PVC foamsheet. It should also be noted that the operating frequency ischosen carefully for a clear visualization of the cWGM, but the pro-posed scheme would be applicable in various frequency regimes.A photographic image of the implemented triangular transform-ation cavity, of which the parameters are α = 0.58 and δ = 0.2 withn0 = 1.8, is shown in Fig. 5a. The base index value n0 is chosen tomatch the range of refractive indices that is realizable by the distri-bution of the alumina posts. The intensity pattern at the cWGMresonance (2.648 GHz) is plotted in Fig. 5b. The experimentally

5 10 15 20 25Number of holes (posts) per

wavelength

103

104

Q fa

ctor

HolePost

b

d

f

c

e

a

1.0

0.0

0.5

1.0

0.0

0.5

1.0

0.0

0.5

|E/Emax | 2

|E/Emax | 2

|E/Emax | 2

Figure 4 | Numerical verification of WGMs and cWGMs in the cavitiesimplemented by holes and posts. a, The uniform distribution of holes (n = 1)or posts (n = 3.4) on a dielectric disk substrate (n = 3.4 for holes and n = 1.4for posts), giving an effective refractive index neff = 2.5. The intensity patternof the WGM examined is overlain as a colour plot. b, Convergence of theQ factor of the WGM on the ideal Q value (dashed line) of the correspondingWGM in homogeneous disk cavity as the number of scatterers (holes orposts) per wavelength increases. c, Limaçon-shaped transformation cavitywith α =0.08 implemented by a hole distribution on a dielectric substrate(n = 3.4). d, Intensity pattern of a cWGM (m = 8) formed in thetransformation cavity of c. e, Limaçon-shaped transformation cavity withα =0.15 implemented by a post (n = 5) distribution on a dielectric substrate(n = 1.4). f, Intensity pattern of a cWGM (m = 9) formed in thetransformation cavity of e (for a clear visualization, much lower densities ofholes (or posts) are depicted in c,e than the actual densities used in thenumerical calculation of cWGMs).

a b c 1.0

0.0

0.5

1.0

0.0

0.5

|E/Emax | 2

|E/Emax | 2

Figure 5 | Experimental implementation of a triangular transformation cavity. a, Realized triangular transformation cavity with alumina posts (n = 3.1)pinned on a thin PVC foam sheet (with a thickness of 2 mm). b, Measured spatial mode profile of the cWGM constructed in a triangular transformationcavity at 2.648 GHz with a mode number of 10. c, Numerically calculated mode pattern in a triangular transformation cavity with the same mode numberas the measured data.






obtained intensity pattern of the cWGM matches well with thenumerically calculated intensity pattern (Fig. 5c) from the FEMmodelling of the transformation cavity implemented with aluminaposts. Despite the highly deformed cavity shape, it should benoted that the intensity pattern of the cWGM is localized alongthe cavity boundary and the distance between adjacent nodes (orantinodes) varies in accordance with the refractive index distri-bution, which are the characteristic features of cWGMs in thetransformation cavities.

The unique properties of the cWGM, that is, the high Q factorand the emission directionality, will be essential to attain directionalcoherent light from ultralow threshold microlasers and to achieveextreme sensitivities in photonic bio-sensing devices with animproved free-space optical coupling. The tailored modes achievedby the proposed scheme will be able to improve the performance ofcavity-based optoelectronic devices. Although we focused here onthe modes of electromagnetic waves, the basic idea of our schemecould be extended to the resonance modes of various kinds ofwaves, such as acoustic and elastic waves.

MethodMethods and any associated references are available in the onlineversion of the paper.

Received 20 October 2015; accepted 17 August 2016;published online 26 September 2016

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AcknowledgementsThis research at KNU (S.-Y.L, J.-W.R., I.K, J.-H.H andM.C) was supported by Basic ScienceResearch Program through the National Research Foundation of Korea (NRF) funded bythe Ministry of Education (No. 2013R1A1A2065357). J.-W.R was supported by ProjectCode (IBS-R024-D1). H.-S.T was supported by the NRF funded by the Ministry ofEducation (2013R1A1A4A03008577). KAIST (Y.K and B.M) was supported byNano·Material Technology Development Program (2015036205), theWorld Class InstituteProgram (No. WCI 2011-001) and the Pioneer Research Center Program(2014M3C1A3052537) through the NRF as funded by the Ministry of Science, ICT andFuture Planning (No. 2012R1A2A1A03670391 and No. 2015001948). This work wassupported by the Center for Advanced Meta-Materials funded by the Ministry of Science,ICT and Future Planning as Global Frontier Project (CAMM-2014M3A6B3063709).

Author contributionsY.K., S.-Y.L., J.-W.R., I.K., M.C. and B.M. conceived the original idea. S.-Y.L., I.K. and M.C.performed theoretical calculations. Y.K., S.-Y.L. and J.-W.R. performed numericalsimulations. Y.K. prepared the experimental set-up and performed the experiments.Y.K., S.-Y.L., J.-W.R., I.K., M.C. and B.M. analysed the data. Y.K., S.-Y.L., J.-W.R., I.K.,J.-H.H., H.-S.T., M.C. and B.M. discussed the results. Y.K., S.-Y.L., J.-W.R., I.K., M.C. andB.M. wrote the manuscript, and all authors provided feedback.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints. Correspondence andrequests for materials should be addressed to M.C. and B.M.

Competing financial interestsThe authors declare no competing financial interests.







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MethodsResonance mode calculation. One of the popular methods used to find theresonance modes of cavities is the FEM. In spite of the versatility of the FEM, it ishard to avoid the disturbance from backscattering completely at the boundary of thecalculation domain because the absorption of electromagnetic waves at theboundary is not perfect. The powerful alternative method in the field of deformeddielectric cavities is the boundary element method (BEM), which is based on Green’stheorem, with advantages including the efficiency in computational cost and thefaithful generation of far-field patterns31. However, the BEM cannot be applied toinhomogeneous dielectric cavities, including our transformation cavity, becauseGreen’s function does not exist in a closed form in those systems. In this work, toovercome the above problem in using the BEM for a transformation cavity, we utilizea virtual space Green’s function by introducing auxiliary virtual space that is derivedfrom the transformation cavity via conformal mapping and can thereby find theresonance modes in the inhomogeneous transformation cavity successfully (seeSupplementary Fig. 2 in Supplementary Section I).

Our goal is to calculate the resonance modes of the transformation cavity inphysical (x, y) space, that is, to find the solutions of the Helmholtz equation with theindex profile n(x, y). In auxiliary virtual (u, v) space derived from (x, y) space viaconformal mapping (w =w(z), defined as reverse conformal mapping), whichtransforms the physical cavity back to the original virtual disk cavity, the refractiveindex inside the cavity is constant (see Supplementary Fig. 2). Thus we know theinside Green’s function (in (u, v) space) and it can be used to construct the insideboundary integral equation (BIE). On the other hand, the outside Green’s function

known in (x, y) space can be used to construct the outside BIE, which can betransformed into (u, v) space through the conformal mapping. With both BIEs, wecan find the resonance solutions of the Helmholtz equation by matching them at thecavity boundary in the (x, y) space.

Microwave measurements. The field intensity distribution of the resonantmode was obtained by a 2D scanning method with two monopole antennas assource and probe, both of which are connected to a network analyser (AgilentPNA-L n5230c)30. The antenna directions are set to excite and probe cWGMs withtransverse magnetic polarization. The source antenna is placed in the proximity ofan antinode of the resonant mode, whereas the other antenna is attached to amoving arm and scanned over the cavity surface and its periphery. To reduce thenoise and the distortion of the resonant mode patterns by the source antenna,averaging was performed with two independently measured field intensity profiles,each of which was obtained by positioning the source antenna at a differentantinode. The transmission power spectrum was also recorded and analysed toquantify the Q factors, and the Q factor analysis is included in SupplementarySection IV.

References31. Wiersig, J. Boundary element method for resonances in dielectric microcavities.

J. Opt. A 5, 53–60 (2003).


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