Flat Engineered Multi-Channel Reflectors

V. S. Asadchy1,2, A. Dıaz-Rubio1, S. N. Tcvetkova1,

D.-H. Kwon1,3, A. Elsakka1, M. Albooyeh1,4, and S. A. Tretyakov1

1Department of Electronics and NanoengineeringAalto University, P. O. Box 15500, FI-00076 Aalto, Finland

2Department of General Physics, Francisk Skorina Gomel State University, 246019 Gomel, Belarus3Department of Electrical and Computer Engineering,

University of Massachusetts Amherst, Amherst, MA 01002, USA4Department of Electrical Engineering and Computer Science

University of California, Irvine, CA 92617, USA

Recent advances in engineered gradient metasurfaces have enabled unprecedented opportunitiesfor light manipulation using optically thin sheets, such as anomalous refraction, reflection, or fo-cusing of an incident beam. Here we introduce a concept of multi-channel functional metasurfaces,which are able to control incoming and outgoing waves in a number of propagation directions si-multaneously. In particular, we reveal a possibility to engineer multi-channel reflectors. Under theassumption of reciprocity and energy conservation, we find that there exist three basic function-alities of such reflectors: Specular, anomalous, and retro reflections. Multi-channel response of ageneral flat reflector can be described by a combination of these functionalities. To demonstrate thepotential of the introduced concept, we design and experimentally test three different multi-channelreflectors: Three- and five-channel retro-reflectors and a three-channel power splitter. Furthermore,by extending the concept to reflectors supporting higher-order Floquet harmonics, we forecast theemergence of other multiple-channel flat devices, such as isolating mirrors, complex splitters, andmulti-functional gratings.

I. INTRODUCTION

Recently, it was shown that thin composite layers(called metasurfaces) can operate as effective tools forcontrolling and transforming electromagnetic waves, seereview papers [1–6]. A number of fascinating and uniquefunctionalities have been realized with the use of thin in-homogeneous layers, such as anomalous refraction [7], re-flection [8, 9], focusing [10], polarization transformation[11], perfect absorption [12], and more [1]. In particu-lar, significant progress has been achieved in controllingreflections of plane waves using structured surfaces.

Methods for engineering single-channel reflections(when a single plane wave is reflected into another sin-gle plane wave) are known. According to the reflectionlaw, light impinging on a flat and smooth (roughness isnegligible at the wavelength scale) mirror is reflected intothe specular direction. It is simple to show that if thereare induced electric or magnetic surface currents at thereflecting boundary, the conventional reflection law, ingeneral, does not hold when the surface properties (e.g.,surface impedance) smoothly vary within the wavelengthscale. Proper engineering of the induced surface currentgradients enables reflection of the incident light in a di-rection different from the specular one. This approachapplied to metasurfaces was exploited recently in [7] andformulated as the generalized law of reflection. It wasshown that the desired current gradient can be engi-neered using a metasurface realized as arrays of specifi-cally designed sub-wavelength antennas. A large varietyof metasurface designs for reflection control with differentpower efficiency levels were reported (e.g., [8, 9, 13–19]).More recently, it was shown that parasitic reflections in

undesired directions, inevitable in the designs based onthe generalized reflection law [7], can be removed [20, 21]via engineering spatial dispersion in metasurfaces [22].

While the recently developed anomalously reflectingmetasurfaces greatly extend the functionalities of con-ventional optical components such as blazed gratings [23],all these devices are designed to control and manipulateincident fields of only one specific configuration. For ex-ample, metasurface reflectors [8, 9] and blazed gratings[24–26] refract/reflect a plane wave incident from a spe-cific direction into another plane wave propagating in thedesired direction.

Recently, a possibility of multi-functional performanceusing several parallel metasurfaces each performing itsfunction at its operational frequency was considered in[27]. Using a single metasurface, it is in principle possibleto satisfy boundary conditions for more than one set ofincident/reflected/transmitted waves if one assumes thatthe surface is characterized by a general bianisotropic setof surface susceptibilities [28]. However, mathematicalsolutions for the required susceptibilities may lead to ac-tive, nonreciprocal or physically unrealizable parametervalues [28].

In this paper, we propose flat metasurfaces with en-gineered response to excitations by plane waves comingfrom several different directions. As a particular con-ceptual example of realizable flat multi-channel devices,here we study multi-channel reflectors: Lossless recipro-cal flat reflectors capable of simultaneous reflection con-trol from and into several directions in space. We explorethe possible functionalities of general surface-modulatedflat reflectors. To this end, we characterize an arbitraryflat periodically modulated reflector using Floquet har-

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monics as a multi-channel system and inspect all allowedreflection scenarios under the assumption of lossless andreciprocal response. We analyze the reflector responsein the framework of conventional scattering matrix no-tations. We find that in the case of periodic reflectorsdescribed by a 3 × 3 scattering matrix (three-channelreflectors), there are devices with three basic function-alities: General specular reflectors, anomalous mirrors,and three-channel retro-reflectors. Moreover, responseof an arbitrary multi-channel (N × N) reflector can bedescribed by a combination of these basic functionali-ties. Although this general classification reveals all pos-sible functionalities available with periodic reflectors, itdoes not provide a recipe for designing a metasurfacewith specific properties. To this end, exploiting the gen-eral surface impedance model [20], we rigorously designa three-channel retro-reflector and experimentally verifyits electromagnetic response. Furthermore, we synthesizetwo other multi-channel devices: A beam splitter that ismatched to the normally incident wave and a five-channelretro-reflector. We expect that the developed theory andrealizations of multi-channel flat reflectors lead to moresophisticated thin and flat N -channel metadevices, suchas power dividers, directional couplers, interferometersor multi-channel filters for a broad range of frequencies.

II. MULTI-CHANNEL PARADIGM OF FLATREFLECTORS

A. Definitions and notations

In general, reflection from planar surfaces can be con-trolled by surface structuring either on the wavelengthscale (conventional diffraction gratings) or on the sub-wavelength scale (metasurface-based gratings). Opera-tional principles of metasurface-based gratings differ fromthose of conventional blazed gratings [29]. The formerrely on proper phase gradient formed by sub-wavelengthscatterers, while the latter operate due to constructiveinterference of the rays reflected/refracted from differentgrooves. Planar topology of metasurface gratings is animportant advantage in fabrication and in some applica-tions. Here we explore the design flexibility offered bythe metasurface paradigm.

Let us consider a periodic metasurface in free spaceilluminated by a plane wave at an angle θi as shown inFig. 1(a). Reflection from a periodic structure, in general,can be represented as interference of the infinite numberof propagating and evanescent plane waves (Floquet har-monics). Here, fully reflecting surfaces (no transmissionthrough the metasurface) are studied. The tangentialwavenumber krx of a reflected harmonic of order n isrelated to the incident wave wavenumber ki and to theperiod of the structure D as krx = ki sin θi +2πn/D. Thecorresponding normal wavenumber of the n-th harmonickrz =

√k2i − k2rx indicates whether it is a propagating or

evanescent wave. Figure 2(a) shows the normal compo-

(a)

< < 2

= 0

= 1= 1port 3

port 2

port 1

(b)

FIG. 1: (a) Illustration of a periodic metasurfaceilluminated by a plane wave impinging from an angle θi.(b) A periodic metasurface illuminated at θi = 0◦. The

three propagating harmonics of the metasurface areanalogous to a three-port network.

nent of the reflected wave vector of all possible propagat-ing modes from a flat reflector (for simplicity the inci-dent angle θi = 0◦ is assumed) with respect to the periodD. The reflection angle of the corresponding harmonicsθr = arcsin(krx/ki) is shown in Fig. 2(b). The operation

0 0.5 1 1.5 2 2.5 3 3.5D/λ

0

0.2

0.4

0.6

0.8

1|ℜ

(krz)|/k0

n = -3n = -2n = -1n = 0n = 1n = 2n = 3

(a)

0 0.5 1 1.5 2 2.5 3 3.5D/λ

-90

-60

-30

0

30

60

90

θr(deg)

n = -3n = -2n = -1n = 0n = 1n = 2n = 3

(b)

FIG. 2: (a) The real part of the normal wavenumber fordifferent Floquet harmonics versus the periodicity of thereflector. (b) The reflection angle of the corresponding

Floquet harmonics. The incidence angle is θi = 0◦.

of the reflector strongly depends on the periodicity. Tohighlight it, we divide the plots in Fig. 2 into three char-acteristic regions using vertical dashed lines. The first re-gion corresponds to reflectors with the periodicity smallerthan the wavelength λ. Such reflectors (e.g., most natu-ral materials, uniform antenna arrays and metasurfaces)illuminated normally exhibit only the usual mirror reflec-tion (harmonic n = 0). The second region corresponds toflat reflectors with the period λ < D < 2λ. Under normalillumination, they may provide anomalous reflection intothe +θr or −θr directions (harmonics n = 1 and n = −1,respectively). Hereafter, we adopt the usual conventionof counting θi counter-clockwise and θr clockwise fromthe z-axis (as shown in Fig. 1). Most of the recentlyproposed gradient metasurface reflectors [7–9, 13–22, 30]operate in this periodicity region. Reflectors with the pe-riodicity D > 2λ (the third region) illuminated normallyhave more than three “open” channels for propagatingreflected waves. Reflectors operating in this region werenot widely studied mainly due to the more complicated

3

design procedures (more channels where uncontrolled re-flections occur become “open”). It should be noted thatin the case of oblique incidence the number of propagat-ing channels is generally different from that found for thenormal incidence. For example, as it will be shown be-low, for a sub-wavelength periodicity 0.5λ < D < λ morethan one mode become propagating.

B. Fundamental classes of periodical reflectors

Here we explore to what extent and using what meta-surface topologies it is possible to engineer reflectionsthrough all open channels, which are defined by the sur-face modulation period. As an example, let us first con-centrate on “three-channel” reflectors because of theirsimple and concise analysis (reflectors with the highernumber of channels are discussed in Section V). Thethree-channel regime can be realized in metasurfaces witha period D < 2λ if one of the channels corresponds tothe normal incidence [see an illustration in Fig. 1(b)]. Itis convenient to represent the three propagation chan-nels of this system as a three-port network using ananalogy from the circuit theory. The channels are num-bered according to Fig. 1(b). One can associate a scat-tering matrix with this system Sij (i, j = 1, 2, 3) whichmeasures the ratio of the tangential electric field com-ponents (normalized by the square root of the corre-sponding impedance Zi,j) of the wave reflected into thei-th channel when the reflector is illuminated from thej-th channel [31], i.e., Sij = (Et,i/

√Zi)/(Et,j/

√Zj) =√

Et,iHt,i/√Et,jHt,j . Assuming that the reflector is re-

ciprocal and lossless, we have two constraints on the scat-tering matrix: It must be equal to its transpose Sij = Sjiand unitary S∗

ki Skj = δij , where δij is the Kroneckerdelta, and the index notations are used.

1. General specular reflectors

Obviously, for a usual mirror made of perfect electricconductor (PEC), the scattering matrix [with ports de-fined as in Fig. 1(b)] is anti-diagonal with all non-zeroelements equal to the reflection coefficient of PEC, i.e.,S13 = S22 = S31 = −1. Assuming possibly arbitrary re-flection phases, in the most general form this matrix canbe written as

S =

0 0 ejφor2

0 ejφor1 0ejφor2 0 0

. (1)

Here φor1 and φor2 represent phases of ordinary reflec-tions when the reflector is illuminated normally andobliquely at θr, respectively. The matrix satisfies bothpreviously defined constraints. In contrast to usual metalmirrors, the reflection phase in gradient reflectors de-scribed by (1) can be engineered arbitrarily and inde-pendently for normal and oblique illuminations. In other

words, there are no physical limitations that would forbidus from creating a reflector which, for example, operatesas a metal mirror when illuminated normally and as a“magnetic” mirror [32] when illuminated obliquely.

2. Anomalous reflectors

In the case of flat reflectors exhibiting ideal anomalousreflection [22], the scattering matrix can be fully deter-mined using the symmetry constraints. Indeed, assumingthat a normally illuminated reflector sends all the inci-dent power to the n = 1 channel and, reciprocally, powerfrom n = 1 channel to the normal direction, we immedi-ately fix its response for illumination from n = −1 chan-nel. The scattering matrix in this scenario reads

S =

0 ejφan1 0ejφan1 0 0

0 0 ejφis1

, (2)

where φan1 and φis1 are the phases of anomalous reflec-tion and reflection for illumination from the n = −1 chan-nel, respectively.

As is seen from (2), the n = −1 channel (port 3) iscompletely isolated from the other open channels of thismetasurface. Incident light from the −θr direction is al-ways fully reflected back at the same angle. Moreover,no light from other directions can be reflected into thisdirection. This behaviour is imposed by the reciprocityand energy conservation. As an example, we consider theanomalous reflector proposed in [22] which under normalillumination reflects 100% of power at θr = 70◦. Fig-ure 3(a) depicts the reflection angles for different propa-gating harmonics in this system versus the illuminationangle θi. When the reflector is illuminated from port 1or port 2, it creates an additional tangential momentumadded to the incident wavevector kx = 2πn/D, wheren = +1. The grey dashed line denotes the specular reflec-tion angles for an equivalent mirror tilted at θi/2 = 35◦

(for such a mirror θr = θi + 70◦). As is expected, thereflector imitates the behaviour of a titled mirror at twoangles, when θi equals to 0◦ and −70◦ (marked by cir-cles in the plot). Figure 3(b) illustrates this result byeyes of an external observer. However, when θi = +70◦

(from port 3), the reflector emulates a mirror titled at70◦ (for such a mirror θr = θi − 140◦) which correspondsto the solid grey line in Fig. 3(a). In this case the n = +1mode is evanescent (krx > ki), while the n = −2 modebecomes propagating and the reflector creates a nega-tive tangential momentum kx = −4π/D responsible forretroreflection. Thus, an external observer looking atthe reflector at θi = 70◦ would see his/her own image[see Fig. 3(b)]. Experimental verification of this isolationproperty of the reflector proposed in [22] is demonstratedin supplementary materials [33]. At all other angles, thereflection from the structure constitutes a combination ofseveral harmonics whose amplitudes depend on the spe-cific design of the reflector. Interestingly, the isolation

4

-90 -60 -30 0 30 60 90θi (deg)

-90

-60

-30

0

30

60

90

θr(deg)

port 3 →

← port 2

← port 1

n = -2n = -1n = 0n = 1n = 2mirrorat 35 degmirrorat 70 deg

(a)

!

" = #/sin70°

35°

70°

70°$70°

35°

(b)

FIG. 3: (a) Operation of the anomalous reflectorproposed in [22]. Reflection angle versus incidence anglefor different Floquet harmonics. All the harmonics areconsidered. The circles show at what angle θr all the

incident energy is reflected from the consideredanomalous reflector when it is excited from the

corresponding port. (b) The flat anomalous reflectorappears to an external observer at different angles asdifferently tilted mirrors. At θi = 0◦ and θi = −70◦

angles, it appears as a tilted at 35◦ mirror. However, atθi = 70◦ angle, the observer would see himself as in a

mirror tilted at 70◦.

effect of anomalous reflectors was not widely discussedin literature. A recent study [8] erroneously reports thatwhen such a reflector is illuminated from port 3, it formsa surface wave bound to the reflector.

Likewise, the normally incident power can be redi-rected to the n = −1 channel, yielding the scatteringmatrix

S =

ejφis2 0 00 0 ejφan2

0 ejφan2 0

, (3)

where φan2 and φis2 are the phases of anomalous reflec-tion and reflection for illumination from the n = 1 chan-nel, respectively. It should be noted that scattering ma-trices (2) and (3) represent equivalent physical proper-ties. Matrix (3) corresponds to the metasurface modelledby (2) but rotated by 180◦ around the z-axis.

3. Three-channel retro-reflectors

By analysing the structure of scattering matrices (1),(2) and (3), we can observe that one more matrix formwith three non-zero components is possible, which satis-fies the symmetry constraints of the reciprocity and en-ergy conservation:

S =

ejφis1 0 00 ejφis2 00 0 ejφis3

, (4)

where ejφis1,2,3 denote independent reflection phases fordifferent illumination angles. It is simple to check that allother tensor structures with three non-zero components

are forbidden. Remarkably, the three classes of reflectorswith scattering matrices defined by (1), (2) and (4) [ma-trix (3) is equivalent to (2)] constitute a peculiar basisof most general three-channel reflectors. From the phys-ical point of view, the three elemental functionalities ofthese reflectors form a basis of all possible reflection func-tionalities achievable with such periodic flat structures.Moreover, it can be shown that the response of an arbi-trary multi-channel (N×N) reflector can be described bya combination of the three basic functionalities: Generalspecular, anomalous, and retro-reflection.

III. THREE-CHANNEL RETRO-REFLECTOR

A closer look at scattering matrix (4) of a three-channel retro-reflector reveals that it corresponds to aso-called “isolating” mirror: In contrast to the previ-ously considered structures, here all three channels areisolated from one another. This retro-reflector (or iso-lating mirror) under the assumption of lossless responserepresents a sub-wavelength blazed grating in the Lit-trow mount [24–26]. Conventional realizations of Lit-trow gratings employ triangular or binary groove profileswith the total thickness of several wavelengths (metallicor dielectric gratings) [26, 34, 35]. While this thicknessis practically acceptable for optical gratings, it becomesvery critical for their microwave counterparts resulting inseveral-centimeters-thick structures. Therefore, for mi-crowave and radio frequency applications there is a needin sub-wavelength thin designs. A good candidate forlow-frequency multi-channel mirrors are flat reflectarrayantennas of slim profile [36]. However, to the best ofour knowledge, there is only one work devoted to thedesign of flat reflectarrays comprising a dense array ofsub-wavelength resonators redirecting energy in the di-rection of incidence [37], and the isolating nature of thesystem was not explored in this work. In comparison,traditional retro-directive array antennas typically use ahalf-wavelength element spacing. In addition, they em-ploy a collection of delay lines in passive designs or anactive phase-conjugating circuitry in active designs be-hind the array plane [38]. Therefore, we next rigorouslydesign a flat three-channel retro-reflector (isolating mir-ror) and experimentally investigate its response from allthree open-channel directions.

Here, the general approach [20] based on the surfaceimpedance concept is used. The design methodologystarts with the definition of the total fields at the meta-surface plane that ensure the desired functionality ofport 3 (alternatively, it could be port 1). We requirethat no evanescent waves are excited for illuminationfrom port 3 and the reflection constitutes a single planewave. Considering as an example the TE polarization(electric field polarized along the y-axis), one can relatethe tangential total electric and magnetic fields through

5

the surface impedance Zs:

Eie−jki sin θix + Ere

jki sin θix

= Zscos θiη

(Eie

−jki sin θix − Erejki sin θix

),

(5)where Ei and Er are the amplitudes of the incident andreflected waves at port 3, and η is the intrinsic impedancein the background medium. Ensuring that all the inputenergy is reflected back into the same direction Ei = Er,the surface impedance reads Zs = j η

cos θicot(ki sin θix).

Such surface impedance can be realized as a single-layerstructure. According to the theory of high-impedancesurfaces [39], an arbitrary reactive surface impedance canbe achieved using a capacitive metal pattern separatedfrom a metal plane by a thin dielectric layer. Since thecotangent function repeats with a period π, the periodic-ity of this impedance profile is D = λ/(2 sin θi). This an-alytical solution gives periodicity smaller than the wave-length if port 3 is at an angle θi > 30◦ from the normal.It should be noted that the derived impedance expres-sion is somewhat similar to that for conventional reflec-tarrays obtained in [22]. This similarity appears becauseboth formulas were derived under the assumption thatthe metasurface possesses local electromagnetic response.

For the actual implementation, we design a three-portretro-reflector whose ports 1, 2, and 3 are directed atangles −70◦, 0◦, and +70◦, respectively [see Fig. 1(b)].Due to the sub-wavelength periodicity D = λ/(2 sin 70◦)imposed on the system, port 2 is isolated from the othertwo ports. The functionality of port 1, in the absence ofdissipation loss, is automatically satisfied because of thereciprocity of the system. The same scenario occurs inlossless blazed gratings in the Littrow mount [24–26]. In-terestingly, if the metasurface is lossy, this conclusion isnot generally correct and reflections in port 1 and port 2in principle could be tuned differently and independently.Note that there are other possible solutions for losslessisolating mirrors with periodicity λ < D < 2λ, however,they imply excitation of evanescent waves when the il-lumination is from either of the three channels, whichcomplicates the theoretical analysis.

Figure 4 shows numerical simulations [40] of a retro-reflector modelled as an ideal inhomogeneous sheet withthe surface impedance calculated from Eq. 5. Fig-ures 4(a–c) depict the real part of the scattered electricfield (the instantaneous value), while Figs. 4(d–f) showthe amplitude of this field. When the metasurface is illu-minated from port 3 [see Figs. 4(a) and 4(d)], a perfectreflected plane wave (no evanescent waves) is generatedin the same direction, fulfilling the design condition. Fig-ures 4(b) and 4(c) show the scattered electric fields whenthe metasurface is illuminated at θi = 0◦ and θi = −70◦,respectively. Likewise, in these scenarios the incidentwave is fully reflected back at the same angle. How-ever, as it is seen from Figs. 4(e) and 4(f), evanescentfields naturally appear in order to satisfy the boundaryconditions at the metasurface.

0 0.5 1x/D

0

0.2

0.4

0.6

0.8

1

z/λ0

-1

-0.5

0

0.5

1ℜ

(

Er/Ei

)

(a)

0 0.5 1x/D

0

0.2

0.4

0.6

0.8

1

z/λ0

-1

-0.5

0

0.5

1ℜ

(

Er/Ei

)

(b)

0 0.5 1x/D

0

0.2

0.4

0.6

0.8

1

z/λ0

-1

-0.5

0

0.5

1ℜ

(

Er/Ei

)

(c)

0 0.5 1x/D

0

0.2

0.4

z/λ0

0

1

2|Er/Ei|

(d)

0 0.5 1x/D

0

0.2

0.4

z/λ0

0

1

2|Er/Ei|

(e)

0 0.5 1x/D

0

0.2

0.4

z/λ0

0

1

2|Er/Ei|

(f)

FIG. 4: Numerical simulations of the three-channelmirror modelled as an inhomogeneous sheet with

surface impedance calculated from Eq. (5). Real part(instantaneous value) and magnitude of the scattered

normalized electric field when the reflector isilluminated from port 3 [(a) and (d)], port 2 [(b) and

(e)] and port 1 [(c) and (f)].

The purely imaginary surface impedance facilitatessimple implementation of our design using conventionaltechniques. For demonstration purpose, we implementa three-port isolating mirror in the microwave regime(8 GHz) using rectangular conducting patches over ametallic plane separated by a dielectric substrate [seeFig. 5(a)]. Each unit cell consists of five patches with

zy

x

(a)

x

y

(b)

FIG. 5: (a) Schematic representation of one unit cell ofthe designed three-channel isolating mirror

(retro-reflector). (b) A photograph of the fabricatedprototype.

the same width of 3.5 mm and different lengths alignedalong the y = 0 line. The dimensions of the unit cellare D = λ/(2 sin 70◦) = 19.95 mm and λ/2 = 18.75 mmalong the x-axis and y-axis, respectively. For designingthe array, each patch was placed in a homogeneous ar-ray of equivalent patches and the length was calculated

6

-90 -60 -30 0 30 60 90

-90

-60

-30

0

30

60

90

0

0.2

0.4

0.6

0.8

1

(a)

!

" = #/(2sin70°)

70°70°

70°$70°

(b)

FIG. 6: (a) Simulated results of the three-channelretro-reflector made of a patch array. Distribution of

the normalized reflected power across Floquetharmonics propagating at different angles θr versus

incidence angle θi. (b) Appearance of the flat isolatingmirror for an external observer. At θi = 0◦ and

θi = ±70◦ angles, the observer would see himself as in amirror normally oriented with respect to him.

to ensure the reflection phase dictated by the surfaceimpedance of the system. Although the efficiency of themetasurface synthesized using this approach was high(about 90% of retro-reflection in each channel), it wassubsequently improved by numerical post-optimizationof the patch dimensions. The numerically calculated effi-ciency of retro-reflection of the final metasurface is 92.8%,95.4%, and 94.3% when excited from ports 1, 2, and 3,respectively. The rest of the energy is absorbed in themetasurface. The final lengths of the patches are 11,11.8, 18.1, 8.4, and 9.8 mm (listed along the x-axis). Thesubstrate material between the patches and the metal-lic plane is Rogers 5880 (εd = 2.2, tan δ = 0.0009)with thickness 1.575 mm (λ/24 at 8 GHz). Figure 6(a)presents the results of numerical simulations of the fi-nal design. The colour map represents the normalizedreflected power into the θr direction when the system isilluminated from the θi direction. We can clearly iden-tify the directions of the three open channels as the lightcoloured regions when θi = −θr. From the colour mapone can see that the metasurface reflects energy back inthe diffracted (non-specular) direction when illuminatednot only at ±70◦, but in the range of angles from around±62◦ to ±90◦ (the amount of this energy decreases whenθi deviates from ±70◦). For example, when θi = 75◦,about 88.8% is diffracted back at the angle θr = −66◦.Such diffraction regime resembles reflection from a metalmirror tilted at 70◦ (to be precise, this mirror would havesome curvature since θi 6= −θr). Extending the previ-ously shown explanations, Figure 6(b) schematically rep-resents the appearance of the flat three-channel isolatingmirror for an external observer. For simplicity, the cur-vature of the equavalent mirrors tilted at ±70◦ is omittedin the illustration. Looking along any of the three direc-tions of open channels, the observer always will “see” amirror orthogonal to the observation direction. Noticethat in contrast to conventional blazed gratings whose

antenna

(a)

7 7.5 8 8.5 9Frequency (GHz)

-3

-2

-1

0

1

2

ξ r(dB)

(b)

7 7.5 8 8.5 9Frequency (GHz)

-20

-15

-10

-5

0

5

ξ r(dB)

θi = −66 degθi = −70 degθi = −74 deg

(c)

7 7.5 8 8.5 9Frequency (GHz)

-15

-10

-5

0

5

ξ r(dB)

θi = 66 degθi = 70 degθi = 74 deg

(d)

FIG. 7: (a) Illustration of the experimental set-up (topview). Measured reflection efficiency of the metasurface

when illuminated (b) normally, (c) at angles nearθi = −70◦, and (d) at angles near θi = +70◦.

design procedure based on the coupled-mode theory isapproximate and requires heavy post-optimization, oursynthesis method provides a straightforward and rigor-ous solution (an exception is work [35] which appearedduring the review process of the current paper).

The metasurface sample was manufactured using theconventional printed circuit board technology and com-prised 14 × 14 unit cells in the xy-plane [see Fig. 5(b)]with the total dimensions of 7.5λ = 282 mm and 7λ =262.5 mm along the x and y axes, respectively. The ex-periment was performed in an anechoic chamber emulat-ing the free-space environment. The orientation of thesample, defined by the angle θi, was controlled by a ro-tating platform as shown in Fig. 7(a). The sample wasilluminated by a quad-ridged horn antenna (11 dBi gainat 8 GHz) which was located at a distance of 5.5 m (about147λ) from the sample and also played the role of a re-ceiving antenna. To filter out parasitic reflections fromthe chamber walls and antenna cables, the conventionaltime gating post-processing technique was exploited. Inorder to find the reflection efficiency ξr of the sample,defined as the ratio of the incident and reflected powerdensities, we measured the received signal from the sam-ple and then normalized it by the corresponding receivedsignal from a reference uniform aluminium plate of thesame cross section size.

The measured reflection efficiency of the sample whenit is illuminated normally (from port 2; θi = 0◦) is shownin Fig. 7(b). As it was discussed above, under the normalillumination the sub-wavelength periodicity of the meta-surface allows only the n = 0 propagating channel (spec-ular reflection). Therefore, at the resonance frequency

7

8 GHz and below, the structure reflects back nearly allthe incident power. Interestingly, at 8 GHz the reflectionefficiency reaches 1.288 dB (135% of the power). Thisnon-physical result is a consequence of the normalizationerror which wrongly implies that the currents induced atthe metasurface and the reference metal plate should beequally uniform. More details on the proper normaliza-tion procedure in this case can be found in [33]. Fig-ures 7(c) and 7(d) depict the reflection efficiency whenthe sample was illuminated at angles near the directionsof the other two open channels ±70◦. As is seen, thereare strong peaks of reflection at 8 GHz: ξr = 0.3 dB andξr = 1.17 dB for incidence at −70◦ and +70◦, respec-tively. These results, with admissible measurement errorlevel of 1 dB for such type of measurements, well confirmthe high reflection efficiency of all channels predicted bythe full-wave simulations.

IV. THREE-CHANNEL POWER SPLITTER

To further illustrate the versatility of the proposedmulti-channel paradigm, we synthesize a three-channelpower splitter. The response of the splitter can beconsidered as a combination of two basic functionalities(anomalous reflections at two angles) dictated by (2) and(3). Beam splitters have been studied intensively in theform of diffraction gratings (see e.g., [41, 42]), however,to the best of our knowledge, the possibility to tailorthe proportion of power divided into different directionswas not reported. Usually, the power is split equally be-tween all channels. In contrast, our synthesis techniquecan be used to design a high-efficiency beam splitter witharbitrary energy distribution between the ports and ar-bitrary transmission phases. In the following example,we target a metasurface that under normal incidencesplits incident energy towards −70◦, 0◦, and +70◦ an-gles with the proportion 50:0:50. A metasurface with thesame functionality and designed using the same designapproach was recently proposed in [43] (only as a theo-retical impedance sheet), but no physical implementationwas demonstrated.

In our design, normally incident plane waves (fromport 2) are split between ports 1 and 3 [see Fig. 1(b)]without any reflection into port 2, i.e., port 2 is matchedto the free-space wave impedance. Considering TE-polarized modes, we write the total tangential electricEt and magnetic Ht fields on the reflecting surface atz = 0 as

Et = Ei + Er1e−jkr sin θrx+jφ1 + Er3e

jkr sin θrx+jφ3 ,

Ht =Ei

η− cos θr

η

(Er1e

−jkr sin θrx+jφ1 + Er3ejkr sin θrx+jφ3

),

(6)where Er1, Er3 and φ1, φ3 refer to the amplitudes andphases of the reflected waves at ports 1 and 3, respec-tively, and kr = ki is the wavenumber of the reflectedwaves. Equal power division occurs when Er1 = Er3 =Ei/√

2 cos θr. Interestingly, if phases φ1 = φ3 = 0◦

-0.4 -0.2 0 0.2 0.4-3

-2

-1

0

1

2

3

(a)

-0.5 0 0.50

0.5

1

1.5

-2

-1

0

1

2

(b)

FIG. 8: A 50:0:50 power splitter. (a) Surface impedancedictated by (7) shown with solid lines and optimizedsurface impedance shown with circle marks. (b) A

snapshot of the scattered normalized electric field (realvalue) in the proximity of the splitter located at z = 0.The data corresponds to the splitter implementation

shown in Fig. 9(a).

are chosen, the surface impedance Zs = Et/Ht becomespurely real, meaning that the the Poynting vector hasonly the normal component (negative or positive in dif-ferent locations) at the impedance boundary. Designinga metasurface to implement this scenario is not straight-forward with our approach which implies optimizing unit-cell elements to follow the imaginary part of the requiredsurface impedance. Therefore, in our example, we choosereflection phases φ1 = 180◦ and φ3 = 0◦ to simplify themetasurface design. In this case, the corresponding sur-face impedance is expressed as

Zs =η√

cos θr

√cos θr + j

√2 sin(kr sin θrx)

1− j√

2 cos θr sin(kr sin θrx), (7)

which is a periodic function with the period D =λ/ sin θr. Notice that this periodicity is the double ofthat in the previous example of an isolating mirror. Thesurface impedance is plotted in Fig. 8(a). The surface re-sistance Re{Zs} oscillates between positive and negativevalues, associated with locally lossy and active proper-ties, respectively. Averaged over the period, the surfaceis in overall lossless.

Instead of an inhomogeneous surface with locally ac-tive and lossy regions, the reflecting surface may be im-plemented as a strongly non-local lossless metasurfacebased on energy channelling along the surface [21, 22].Indeed, the surface impedance (7) has been derived as-suming that there are only three plane waves propagat-ing above the metasurface: A single incident and tworeflected ones. This assumption is very limiting and ingeneral can be relaxed by allowing evanescent waves gen-eration at the interface. While they would not contributeto the far-field radiation, decaying quickly near the in-terface, their contribution to the total fields (6) at z = 0

8

would ensure purely imaginary surface impedance Zs ateach point of the metasurface. Although analytical de-termination of the required set of evanescent waves is acomplicated task, the solution can be obtained exploit-ing the concept of leaky-wave antennas [22]. In this im-plementation, the energy received over the lossy region(where Re{Zs} > 0) is carried by a surface wave propa-gating along the x-axis before being radiated into spacein the active region of the surface (where Re{Zs} < 0).In what follows, we discard the real part of impedance (7)and numerically optimize [40] the imaginary part of theimpedance to ensure the desired amplitudes of the re-flected plane waves. The optimized impedance profile(discretized into 15 elements with a uniform impedance)is shown by circle markers in Fig. 8(a). The numericallyoptimized impedance profile (the imaginary part) turnsout to closely follow that of the active-lossy impedancedictated by (7).

Fig. 8(b) depicts numerical results for the real part (theinstantaneous value) of the normalized electric field re-flected by the metasurface with the optimized impedanceprofile. An interference pattern between two plane wavespropagating in the xz-plane towards the −70◦ and +70◦

directions with a 180◦ phase difference is observed. Asis expected, at the interface a set of evanescent fields isgenerated. The incident wave is split equally betweenports 1 and 3 with 49.8% of incident power channelledinto each direction.

Following the same implementation approach as forthe three-channel isolating mirror, we use rectangularpatches over a metallic ground plane. Ten equidistantpatches with the same width of 2.69 mm and differentlengths were placed in each unit cell. As an initial es-timation, the length of each patch was chosen in orderto ensure the reflection phase described by the imagi-nary part of the optimized impedance when the patch isplaced in a homogeneous array. After that, we built theunit cell consisting of ten patches and accurately tunedthe lengths by using post-optimization for obtaining thebest performance. The final lengths of the patches listedalong the +x-direction are 4.9, 7.4, 9.2, 10.6, 11, 18,11.6, 11.5, 10.6, and 11.2 mm [see Fig. 9(a)]. The di-

z y

x

(a)

z y

x

(b)

FIG. 9: Schematic representation of single unit cell ofdesigned (a) three-channel power splitter and (b)

five-channel retro-reflector.

mensions of the unit cell are D = λ/ sin 70◦ = 39.91 mmand λ/2 = 18.75 mm along the x-axis and y-axis, respec-tively. The substrate material and patches material arethe same as in the previous example. The numerical re-

sults indicate that normally incident power is split by themetasurface towards−70◦ (port 3), 0◦ (port 2), and +70◦

(port 1) with the proportion 49.71% : 1.06% : 45.74%.About 3.49% of incident power is absorbed in the meta-surface due to material losses. When the splitter is illu-minated from port 1 or 3, it is not matched to free space,and energy is split in all three ports with the propor-tion of 23.26% : 45.74% : 28.51% (when illuminated fromport 1) or 25.19% : 49.71% : 23.26% (when illuminatedfrom port 3). This result is not surprising and in agree-ment with the basic multi-port network theory whichstates that lossless passive three-port splitters matchedat all ports cannot exist (since it would contradict thecondition of unitary scattering matrix S∗

ki Skj = δij [44]).

Next, we experimentally verify operation of the meta-surface in an anechoic chamber. The metasurface sam-ple consists of 11 × 14 unit cells along the x and yaxes, respectively. The overall dimensions of the sam-ple along the x and y axes are 11.7λ = 439 mm and7λ = 262.5 mm.

First, the sample was illuminated from the normaldirection (θi = 0◦) by a horn antenna as is shown inFig. 7(a), and the reflected signal was measured by thesame antenna. Similarly to the experiment in the previ-ous section, the signal was normalized by that of a refer-ence uniform aluminium plate of the same size. The mea-sured level of retro-reflection ξr along direction θi = 0◦

is depicted in Fig. 10(a). At the resonance frequency

7 7.5 8 8.5 9Frequency (GHz)

-30

-25

-20

-15

-10

-5

0

5

ξ r(dB)

(a)

transmitting

antenna

receiving

antenna

(b)

-90 -60 -30 0 30 60 90θi (deg)

-90

-80

-70

-60

-50

-40

-30

|S21,m|(dB)

θa = −70 degθa = +70 deg

(c)

-90 -60 -30 0 30 60 90θi (deg)

-70

-60

-50

-40

-30

-20

|S21,p|(dB)

(d)

FIG. 10: (a) Efficiency of retro-reflection from thepower splitter under normal illumination. The splitter

is matched for normal incidence at 8.2 GHz. (b)Illustration of the experimental set-up for the power

splitter (top view). (c) Signals measured by thereceiving antenna versus the orientation angle of the

metasurface θi for two positions of the receivingantenna. (d) Signal from a reference metal plate

(θa = +70◦).

9

of 8.2 GHz, the measured curve has a deep drop, cor-responding to reflected power of 0.3%. This result isin excellent agreement with the simulated data and con-firms that the power splitter is well matched from port 2.A slight shift of the resonance frequency in the experi-ment can be explained by manufacturing errors and thetolerance of the substrate permittivity value.

Next, it is important to verify that the splitter in factreflects normally incident power equally towards +70◦

and −70◦ directions. For this purpose, we add a se-cond (receiving) antenna to the experimental setup [seeFig. 10(b)]. The position of the second antenna is de-termined by the angle θa. The distances from the meta-surface center to the receiving and transmitting antennaswere 2.1 m and 5.4 m, respectively. Figure 10(c) depictsthe signal measured by the receiving antenna |S21,m| fortwo different positions of the antenna (when θa = +70◦

and θa = −70◦) versus the incidence angle θi. The redcurve (θa = +70◦) has three distinct peaks. The centralpeak at θi = 0◦ corresponds to the case of normal inci-dence (from port 2) and strong reflection towards port 1(θr = +70◦). The peak at θi = +70◦ corresponds toexcitation of the metasurface from port 3 and reflectiontowards port 2 (θr = 0◦). The peak at θi = +35◦ appearsdue to specular reflection from the metasurface when itis illuminated not from its main ports. Three peaks ofthe blue curve for the case of θa = −70◦ can be explainedlikewise. Interestingly, the central peaks on the red andblue curves do not occur exactly at θi = 0◦ and thereis a small shift between them. This result is expectedand means that when the power splitter is illuminated ata small oblique angle, the reflection towards one direc-tion increases at the expense of the decrease of reflectionin the other direction. The total efficiency (sum of thenormalized reflected power towards +70◦ and −70◦ di-rections) is maximum exactly when θi = 0◦.

In order to calculate the power reflected by the meta-surface towards θr = +70◦ and θr = −70◦, we per-form an additional measurement with a reference alu-minium plate (position of the receiving antenna was atθa = +70◦). As is seen from Fig. 10(d), a specular re-flection peak is detected by the antenna when the metalplate is oriented at θi = +35◦. Following the procedurereported in [22], we calculate the reflection efficiency ξr ofthe power splitter towards θr = +70◦ (for the red curve)and θr = −70◦ (for the blue curve) as

ξr =1

ξ0

|S21,m(θi = 0◦)||S21,p(θi = +35◦)|

. (8)

Here ξ0 is a correction factor which gives the ratio be-tween the theoretically calculated signal amplitudes froman ideal power splitter (of the same size and made of loss-less materials) and a perfect conductor plate [22]. Usingthis formula, the calculated reflection efficiencies towardsθr = +70◦ and θr = −70◦ directions were found to benearly equal to ξ+70◦

r ≈ ξ−70◦

r ≈ −3.17 dB. In the linearscale and expressed in terms of power, the reflection ef-ficiency towards each direction (ports 1 and 3) is about

48%. This result is in good agreement with the simulateddata (45.74% and 49.71%).

V. FIVE-CHANNEL RETRO-REFLECTOR

In this section, we demonstrate that the concept ofmulti-channel flat reflectors can be extended to deviceswith more than three ports. In what follows, we syn-thesize a mirror which fully reflects incident waves backfrom five different directions. Since such functionality re-quires fast variations of the surface impedance over thesub-wavelength scale (due to excitation of required addi-tional evanescent fields for satisfying the boundary condi-tion), it is unattainable with conventional blazed gratingswhose grooves are of the wavelength size. Designing con-ventional gratings with sub-wavelength-sized grooves isimpractical with today’s nano-fabrication technologies.

In the case of the three-channel retro-reflector de-scribed above, engineering back reflections in one portautomatically ensured proper response of the reflector inthe other two ports (due to reciprocity and negligible dis-sipation). However, when the number of ports increasesto five, one should design a metasurface with prescribedresponse for several different illuminations.

The directions of the five ports cannot be chosen arbi-trary and can be determined using Floquet-Bloch anal-ysis: θrn = arcsin(sin θi + nλ/D), where n = 0,±1,±2is the number of the harmonic and θrn is the reflectionangle corresponding to the n-th harmonic. There aretwo scenarios of retro-reflection in five ports. In the firstscenario, the periodicity of the reflector D is chosen tosatisfy inequality 2λ < D < 3λ. In this case, all the fiveports are “open” when the reflector is illuminated fromany of them. The directions of the five ports are givenby θrn = arcsin(nλ/D), where n = 0,±1,±2.

In the second scenario, the reflector periodicity D ischosen so that λ < D < 2λ. Here, at normal illu-mination θi = 0◦, only three ports are open: At an-gles θr0 = 0◦ and θr±1 = ± arcsin(λ/D) [see Fig. 2(b)],where subscript indices correspond to the harmonic num-ber n. The same ports remain open also in the case whenθi = ± arcsin(λ/D) [see Fig. 11(a) where D ≈ 1.064λis chosen as an example]. However, illuminated fromθi = ± arcsin(0.5λ/D), the metasurface has only twoopen ports corresponding to θr−1,0 = ± arcsin(0.5λ/D),as is shown in Fig. 11(b). Thus, although the retro-reflector has five channels, three of its channels are iso-lated from two others.

Both scenarios of the retro-reflector provide similar re-sponse for the illuminations of the five channels. Forour design we use the second scenario since it requiresto ensure retro-reflection only in three ports (response inothers will be automatically satisfied due to reciprocity).We choose periodicity D ≈ 1.064λ so that the five portsare directed at 0◦, ±28◦, and ±70◦ from the normal.

It can be shown that an ideal five-channel retro-reflector illuminated from either channel necessarily gen-

10

0 0.5 1 1.5 2 2.5 3 3.5D/λ

-90

-60

-30

0

30

60

90θr(deg)

n = -3n = -2n = -1n = 0n = 1n = 2n = 3

(a)

0 0.5 1 1.5 2 2.5 3 3.5D/λ

-90

-60

-30

0

30

60

90

θr(deg)

n = -3n = -2n = -1n = 0n = 1n = 2n = 3

(b)

FIG. 11: The reflection angles of different Floquetharmonics for a five-channel retro-reflector. Here, the

reflector periodicity D ≈ 1.064λ is chosen. Illuminationis at (a) θi = 70◦ and (b) θi = 28◦. The grey dashed line

indicates the periodicity of the reflector. For clarity,only seven harmonics are shown.

erates evanescent waves. Analytical determination ofthese required evanescent waves is a complex problemand could be a subject for a separate study. Next, wefirstly design a metasurface for retro-reflection when il-luminated from θi = ±28◦ directions and then exploitnumerical post-optimization to gain proper response inthe other channels.

As the first approximation, we synthesize the meta-surface to fully reflect incident waves from θi = +28◦ atangle θr = −28◦. This case is similar to that consideredabove in Section III and, therefore, the impedance profilecan be calculated using Eq. (5). The obtained metasur-face rigorously works as a retro-reflector from ports 2and 4 [see ports notations in Fig. 11], while it producesparasitic energy coupling to ports 1 and 5 when excitedfrom the normal direction (port 3). Next, we numeri-cally optimize [40] the obtained impedance profile to en-sure maximized back reflection (retro-reflection) in allthe ports. The optimized unit cell of the metasurface isshown in Fig. 9(b) and contains ten patches with the fol-lowing lengths listed along the +x-direction: 12.3, 11.2,10.4, 6.8, 4.1, 16.9, 11.1, 11.1, 11.0, and 12.3 mm. Theunit cell dimensions are the same as those of the three-channel power splitter considered above. The width ofthe patches is 3.49 mm. The numerically obtained results

can be represented by the following matrix (at 8 GHz)

(|S|2)ij =

0.796 0 0.136 0 0

0 0.815 0 0.156 00.136 0 0.784 0 0.064

0 0.156 0 0.807 00 0 0.064 0 0.873

, (9)

where (|S|2)ij characterizes the ratio of power reflectedinto the i-th port when the metasurface is illuminatedfrom the j-th port [see definitions of the ports in Fig. 11].As is seen, for illumination from either port, the metasur-face effectively reflects energy back in the incident direc-tion with the average efficiency of 80%, i.e. (|S|2)ii ≈ 0.8for i = 1, 2 . . . 5. The non-ideal operation can be im-proved further by increasing discretization of the unitcell.

Next, we fabricated a metasurface sample with thesame dimensions as those presented in Section IV. Theexperimental setup was identical to that shown in Sec-tion III for a three-channel retro-reflector. Figure 12shows the measured efficiency of back reflection (retro-reflection) when the metasurface is illuminated at θi =0◦, ±28◦, and ±70◦. At the frequency of 8.03 GHz, retro-reflection in all channels is high and nearly equal. Tocompare the measured results at 8.03 GHz with the sim-ulated ones, we write them in the matrix form similarto (9):

(|S|2)ij =

0.776 · · · ·· 0.772 · · ·· · 0.802 · ·· · · 0.801 ·· · · · 0.800

. (10)

Here the dotes denote unknown values that were not mea-sured in the experiment. The values at the main diagonalrepresent reflection efficiencies ξr (into the i-th port) ex-pressed in the linear scale for illumination from the samei-th port. As can be seen, the simulated and measuredresults in (9) and (10) are in good agreement.

VI. DISCUSSIONS AND CONCLUSIONS

To summarize, combining the microwave circuit anddiffraction gratings theories, we proposed a concept offlat engineered multi-channel reflectors. We revealed theexistence of three basic functionalities available with gen-eral flat reflectors: General specular, anomalous, andretro- reflections. Next, we designed three differentmulti-channel devices whose response can be describedby a combination of these basic functionalities.

As was demonstrated, proper engineering the period-icity and surface impedance of a metasurface enablesstraightforward design of various multi-channel reflec-tors. Indeed, with the increase of the period D the num-ber of “open” channels grows (see Fig. 2(b)), leading to

11

7 7.5 8 8.5Frequency (GHz)

-4

-3

-2

-1

0ξ r

(dB)

θi = 0 deg

(a)

7 7.5 8 8.5Frequency (GHz)

-80

-60

-40

-20

0

ξ r(dB)

θi = 24 degθi = 28 degθi = 32 deg

(b)

7 7.5 8 8.5Frequency (GHz)

-40

-30

-20

-10

0

ξ r(dB)

θi = −24 degθi = −28 degθi = −32 deg

(c)

7 7.5 8 8.5Frequency (GHz)

-50

-40

-30

-20

-10

0

ξ r(dB)

θi = 66 degθi = 70 degθi = 74 deg

(d)

7 7.5 8 8.5Frequency (GHz)

-60

-40

-20

0

ξ r(dB)

θi = −66 degθi = −70 degθi = −74 deg

(e)

FIG. 12: Measured reflection efficiency of the five-channel isolating mirror when illuminated from (a) port 3, (b)port 4, (c) port 2, (d) port 5, and (e) port 1. The dashed line shows the operating frequency.

3 < < 4

(a)

2 < < 3

port 5

port 3

port 1

port 2port 4

(b)

FIG. 13: (a) N -port isolating mirror. Wave impingingfrom different angles (grey arrows) is reflected back to

the source (coloured arrows). (b) Multi-functionalreflector. Normally incident light is split between

ports 2 and 4. Ports 1 and 5 are isolated.

great design freedom. One example of possible multi-channel reflector device is a N -port isolating mirror (flatretro-reflector) depicted in Fig. 13(a). All the channels(in principle, the number of channels can be very big)of this mirror are isolated from one another and, there-fore, the mirror illuminated at almost any angle wouldfully reflect energy back to the source. Such a mirrorwould possess unprecedented physical properties: Ob-servers standing around the mirror would see only im-ages of themselves but not other observers. In fact, thequestion of what exactly an observer would see in thismirror is not straightforward and strongly depends onthe number of the “open” channels and their isolationefficiency.

Alternatively, large periodicity of the reflector couldbe used to combine different functionalities in one re-flector or even to overcome the design limitations ofthree-channel structures. Figure 13(b) demonstrates abeam splitter which at the same time acts as a mirrorat other angles. Modifying the response from ports 1and 5, one can greatly extend the properties which thereflector exhibits when excited from ports 2–4. For thesake of simplicity, in this paper we confined our designof multi-channel reflectors to the case when one of thechannels corresponds to the normal plane-wave incidenceeven though the proposed metasurfaces operate also atoblique angles. The scenario with the normal incidencechannel being closed further extends design possibilities,but we do not consider this case here.

Other exciting functionalities become possible by ex-tending the multi-channel paradigm in three other di-rections: Partially transparent films (additional chan-nels appear in transmission), multi-channel polarizationtransformers, and non-periodical structures (e.g., to em-ulate a convex or spherical mirror response using a singleflat surface).

ACKNOWLEDGMENTS

This work was supported in part by the Nokia Founda-tion and the Academy of Finland (project 287894). Theauthors would like to thank Muhammad Ali and AbbasManavi for technical help with the experimental equip-ment.

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