Infrared and Optical Invisibility Cloak with Plasmonic Implants Based on Scattering Cancellation

Home , Infrared

University of Pennsylvania ScholarlyCommons

Departmental Papers (ESE) Department of Electrical & Systems Engineering

August 2008

Infrared and optical invisibility cloak with plasmonic implants based on scattering cancellation

Mário G. Silveirinha Universidade de Coimbra

Andrea Alù University of Pennsylvania, [email protected]

Nader Engheta University of Pennsylvania, [email protected]

Follow this and additional works at: https://repository.upenn.edu/ese_papers

Recommended Citation Mário G. Silveirinha, Andrea Alù, and Nader Engheta, "Infrared and optical invisibility cloak with plasmonic implants based on scattering cancellation", . August 2008.

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ese_papers/447 For more information, please contact [email protected]. Infrared and optical invisibility cloak with plasmonic implants based on scattering cancellation

Abstract In recent works, we have suggested that plasmonic covers may provide an interesting cloaking effect, dramatically reducing the overall visibility and scattering of a given object. While materials with the required properties may be directly available in nature at some specific infrared or optical frequencies, this is not necessarily the case for any given design frequency of interest. Here we discuss how such plasmonic covers may be specifically designed as metamaterials at terahertz, infrared, and optical frequencies using naturally available metals. Using full-wave simulations, we demonstrate that the response of a cover formed by metallic plasmonic implants may be tailored at will so that at a given frequency, it possesses the plasmonic-type properties required for cloaking applications.

Keywords light scattering, metamaterials, optical conductivity, optical materials, plasmons

This journal article is available at ScholarlyCommons: https://repository.upenn.edu/ese_papers/447 PHYSICAL REVIEW B 78, 075107 ͑2008͒

Infrared and optical invisibility cloak with plasmonic implants based on scattering cancellation

Mário G. Silveirinha,1,2 Andrea Alù,1 and Nader Engheta1,* 1Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 2Department of Electrical Engineering, Instituto de Telecomunicações, Universidade de Coimbra, 3030 Coimbra, Portugal ͑Received 18 March 2008; revised manuscript received 26 June 2008; published 11 August 2008͒ In recent works, we have suggested that plasmonic covers may provide an interesting cloaking effect, dramatically reducing the overall visibility and scattering of a given object. While materials with the required properties may be directly available in nature at some speciﬁc infrared or optical frequencies, this is not necessarily the case for any given design frequency of interest. Here we discuss how such plasmonic covers may be speciﬁcally designed as metamaterials at terahertz, infrared, and optical frequencies using naturally available metals. Using full-wave simulations, we demonstrate that the response of a cover formed by metallic plasmonic implants may be tailored at will so that at a given frequency, it possesses the plasmonic-type properties required for cloaking applications.

DOI: 10.1103/PhysRevB.78.075107 PACS number͑s͒: 42.70.Ϫa, 78.66.Sq, 41.20.Jb

I. INTRODUCTION analogous to their homogeneous ideal models. The goal of the present study is to extend the concepts In our previous works,1–5 we have theoretically demon- introduced in Ref. 5 to the IR and visible domains, properly strated that isotropic plasmonic materials with ͑relative͒ per- taking into account the fact that at terahertz, IR, and optical mittivity below unity may be used to drastically reduce the frequencies, metals have finite conductivity, well modeled to specific scattering of moderately sized obstacles. It was a good approximation, as Drude plasmas. We derive some proven theoretically that such materials may behave as “an- simple design formulas for a class of novel metamaterial tiphase” scatterers, which may effectively cancel out the di- cloaks that operate at infrared and optical frequencies based polar radiation from the obstacle, inducing in this way elec- on plasmonic parallel-plate implants in a dielectric host, and tromagnetic invisibility. This phenomenon is possible we demonstrate with full-wave simulations how such because the polarization currents induced in a material with parallel-plate metallic covers may effectively reroute the in- permittivity less than unity are in opposite phase with respect coming light and induce a cloaking effect at the desired IR or to the local electric field. optical frequency. Recently, other groups have suggested alternative cloaking ideas6–11 based on coordinate transformation theory, anomalous localized resonances, and other related concepts. II. METAMATERIAL DESIGN Such configurations in general require anisotropic and/or in- homogeneous layers, and rely on the response of resonant The cloaking effect described in Ref. 1 exploits the nega- metamaterials and artificial magnetism. On the contrary, the tive polarizability provided by scatterers made of materials transparency phenomenon proposed by our group1 simply with ␧ negative ͑ENG͒ or ␧ near zero ͑ENZ͒͑␧ representing requires uniform plasmonic materials with an isotropic re- the material permittivity͒. For typical designs,1–5 the required ͑ ͒ ␧ sponse. Moreover, this transparency mechanism does not value of the real part of the permittivity c of the plasmonic ␧ Ͻ ͕␧ ͖Ͻ ␧ ␧ rely on a resonant effect, and so it is less affected by losses cover lies in the range −10 0 Re c 0.5 0, with 0 being and may have good tolerance with respect to changes in the the free-space permittivity. Since noble metals behave essen- geometrical or material parameters of the involved objects.2 tially as ENG materials at infrared and optical frequencies, While isotropic plasmonic materials with the required they may be directly used for cloaking purposes.1 A problem, electromagnetic properties may be readily available in nature however, may arise in that for frequencies one or two de- at certain specific IR or optical frequencies,12 in general one cades below the plasma frequency of the material of interest, may need to synthesize these materials as metamaterials to the ͑real part of͒ permittivity of such metals, even though operate with the required electromagnetic properties at a de- negative, may have an absolute value orders of magnitude sired frequency. Such ideas were explored in our previous larger than the permittivity of vacuum. For example, work,5 where we have effectively demonstrated at micro- following the experimental data tabulated in Ref. 13 waves how it may be possible to emulate the behavior of low at IR frequencies, silver may be well characterized by a ␧ / ␧ ␧ ␻2 / ␻͑␻ ⌫͒ ␧ or negative permittivity materials in cloaks by using parallel- Drude model Ag 0 = ϱ − p +i , with ϱ =5.0, ␻ ␲ϫ ͓ ͔ ⌫ ␲ϫ ͓ ͔ plate metallic implants embedded in a dielectric host. p =2 2175 THz , and =2 4.35 THz . This yields ␧ Ϸ ͑ ͒␧ Namely, we have shown how to effectively design metama- Ag 470 −1+0.04i 0 at 100 THz, which is 2 orders of terial cloaks using perfectly electric conducting ͑PEC͒ im- magnitude larger than the typical values required for cloak- plants. In this sense, we had to properly take into account ing applications. both interface effects and actual granularity of the structured To overcome this inconvenience, mimicking our micro- material, demonstrating numerically that these metamaterial wave setup5 we suggest to embed silver implants in a dielec- cloaks may indeed provide a drastic scattering reduction, tric region ͑with positive ␧͒, so that the resulting composite

f [THz] 0 20 40 60 80 100 Ag T 0 0 y Host, h a silver -5 -1000

inc g

E A Artificial x inc z -10 material -2000

H and g A -15 -3000 Artificial -20 material FIG. 1. ͑Color online͒ Geometry of a truncated ͑semi-infinite͒ -4000 planar composite material formed by a periodic array of nanolayers FIG. 2. ͑Color online͒ Permittivity as a function of frequency: of silver embedded in a dielectric host. The truncated sample is Blue lines ͑associated with left-hand side scale͒: structured mate- illuminated by an incoming TEx plane wave. rial; Black lines ͑associated with right-hand side scale͒: silver. The artificial material is formed by silver slabs with thickness material has a tailored electromagnetic response suitable for T=0.0374a which are spaced by a=360 nm and embedded in a cloaking. host material with ␧h =6.5. The geometry of the proposed layered structure in its pla- ␧ ␧ ͓ nar version is depicted in Fig. 1. It consists of a periodic tric material with permittivity h =6.5 0 SiC has similar ␧ ͑ ͔͒ array of planar silver slabs with permittivity Ag inserted in a properties around 100 THz Ref. 15 . These values were ␧ ͕␧ ͖ ␧ dielectric host material with permittivity h. The thickness of chosen to obtain Re eff =−3.0 0 at 100 THz, and they will each silver layer is T and the lattice constant is a. The effec- be used in the following to design a plasmonic metamaterial tive response of such metamaterial for propagation in the x-y cloak. In the same graphic we have also plotted ͑using a plane with electric field along z may be readily characterized different scale͒ the permittivity of silver13 to show how dra- using the well-known dispersion characteristic of matically different the two dispersions are. Indeed, the per- E-polarized Floquet modes:14 mittivity of silver is 2 orders of magnitude larger than the ͑ ͒ ͑ ͒ ͓ ͑ ͔͒ one of the composite medium. cos kya = cos ky,2T cos ky,1 a − T The description of the composite material using the di- ͑ ͒ 1 ky,1 ky,2 electric function 2 relies on the assumption that the effect − ͩ + ͪsin͑k T͒sin͓k ͑a − T͔͒, ͑ 2 k k y,2 y,1 of the higher-order evanescent modes associated with the y,2 y,1 indices n=1,2,...͒ is negligible. In general such approxima- ͑ ͒ 1 tion is fairly accurate in the long-wavelength limit, especially ͑ ͒ if the thickness L of the considered metamaterial slab ͑mea- where k= kx ,ky ,0 is the wave vector of the Floquet mode, ␻ ͱ␧ ␮ ␻2 2 sured along x͒ is significantly larger than the lattice constant. is the angular frequency of operation, ky,1= h 0 −kx, ͱ␧ ␮ ␻2 2 However, when the metamaterial slab is relatively thin and L and ky,2= Ag 0 −kx. For a fixed frequency, the structured material supports an infinite countable number of electro- is comparable to a, interface effects become increasingly im- magnetic modes with propagation constants along x given by portant, and the effect of higher-order modes, even though ͑n͒ ͑n͒͑␻ ͒ small, is sufficient to detune the response of the material. kx =kx ,ky , n=0,1,2,..., which can be calculated from Eq. ͑1͒. It is simple to verify that for long wavelengths all the These effects have been carefully analyzed in our previous ͑ ͒ work5 for the case of PEC slabs at microwave frequencies, electromagnetic modes are attenuated ͑Im͕k n ͖0͒, even in x where it was demonstrated that the higher-order modes could the absence of losses. To homogenize the structured metama- modify the expected response of the composite material in terial, we will characterize the least attenuated electromag- several scenarios of interest. netic mode ͑associated with the index n=0͒. In this work, we Nevertheless, it is possible to take into account the granu- are particularly interested in propagation along the x direc- larity of the composite material and the existence of higher tion ͑k =0͒. In such situation, the composite material is de- y modes in a straightforward manner. The idea is to introduce scribed to a good approximation by an effective permittivity “virtual interfaces” that describe the effective boundaries of given by the formula: the composite material. These virtual interfaces are displaced ͑ ͒ ͉͑k 0 ͑␻͉͒ ͒2 at a distance ␦ with respect to the actual physical interface of ␧ x ky=0 ͑ ͒ eff = . 2 the layered material. This concept, first explored in Ref. 5,is ͑␻/c͒2 illustrated in the inset of Fig. 3. The proper choice of ␦ may Obviously, the effective permittivity depends not only on the indeed allow, also in this scenario, to describe a metamaterial permittivity of the components, but also on their volume slab with thickness L−2␦ adjoined by two dielectric layers ␧ ␦ fractions. Thus, by adjusting T and a it is possible to tune the with permittivity h and thickness effectively as a continu- response of the material according to our needs. This is il- ous medium with thickness L and dielectric function given ␧ ␧ ␧ ͑ ͒ ␦ lustrated in Fig. 2, where we plot Ј and Љ with eff by Eq. 2 . The thickness of these dielectric “gaps” de- ␧ ͑␧ ␧ ͒ = 0 Ј−i Љ , as a function of frequency for a structured ma- pends mostly on the lattice constant a and on the properties terial formed by silver nanolayers with thickness T of the metal. For example, for vanishingly thin PEC plates, =13.5 nm, spaced by a=360 nm, and embedded in a dielec- ␦Ϸ0.1a.5 In the more general present case for which the

075107-2 INFRARED AND OPTICAL INVISIBILITY CLOAK WITH… PHYSICAL REVIEW B 78, 075107 ͑2008͒

L 0.14 L-2 RRc 1.4 Only nm Q 1 9 dielectric max 10 0.12 ENG cover h h h (arb. units) 5 obj 3.0 2 k 0 0.1 x eff 2 Rnm 382 a / c SiC a 0.08 Object in free- Tnm13.5 1 nm 2 16 space 0.06 0.5 0.04 Cloak without Only metallic 0.02 0.1 gaps implants 0.05 0.25 0.5 0.75 1 1.25 1.5 Cloaked object s 50 100 150 200 T f [THz] FIG. 3. ͑Color online͒ Virtual interface position ␦ as a function FIG. 4. ͑Color online͒ Peak in the scattering width Q ͑normal- of the skin depth of the metal ⌬s at 100THz. The parameters T, a, ized to arbitrary units͒ as a function of frequency, for different and ␧h are as in Fig. 2. The inset illustrates the equivalence between a composite material slab with two adjoining dielectric layers with scenarios of interest. The inset represents a combined object-cloak thickness ␦, and a continuous material described by the dielectric system ͑the different parts of the cloak are not drawn to scale͒. function ͑2͒. However, it is noted that in the configuration shown in the thickness and the conductivity of the metallic plates are fi- inset of Fig. 4 the spacing between the plates is not uniform, nite, ␦ may be calculated using formula ͑A14͒ in the Appen- and strictly speaking it depends on the radial coordinate: ͑ ͒ ␲ / dix. To illustrate the dependence of ␦ with the properties of a=a r =2 r N. This implies that in general the effective 5 the metal, in Fig. 3 we plot ␦ as a function of the skin-depth permittivity of cloak is also a weak function of r. In prin- ⌬ ciple, it is possible to design a cloak with uniform permittiv- s of the metal at 100 THz. The composite material is char- ␧ ity by varying the thickness of the implants T as a function of acterized by the same parameters a, T, and h as in Fig. 2, ␧ r so that the effective permittivity would be independent of and the permittivity of the metallic layers m is assumed to be the free parameter ͑the skin depth is given by the radius. However, this solution may be challenging from a ⌬ /␻ͱ ␧ ␮ ⌬ technological point of view. It is much simpler to assume s =1 − m 0; for silver s =1.64T at the considered fre- quency͒. It is seen that ␦ depends appreciably on ⌬ and that the thickness T is constant and replace a by its s ␲͑ ͒/ ⌬ average value amed= R+Rc N in the design formulas for decreases when s increases, or equivalently as the conducting properties of the metal deteriorate. For example, when the planar geometry. Proceeding in this way, we obtain ␻ / ͑ ͒ the metal is made of silver we get ␦=0.035a. amed c=0.75. Feeding this value to Eq. 2 and imposing ͕␧ ͖ ␧ that Re c =−3 0, it is found that the required thickness for the metallic plates is T=0.0374 amed=13.5 nm. The corre- ␧ ␧ ͑ ͒ III. CLOAK DESIGN sponding effective permittivity is c = 0 −3.0+0.2i at 100 THz. Having introduced the necessary homogenization con- As depicted in the inset of Fig. 4, we introduce two small ␦ ␦ cepts, we are now ready to design a plasmonic metamaterial SiC cylindrical shells with thicknesses 1 and 2 at the with the desired permittivity at IR or optical frequencies and interfaces of the structured material with the object and with apply it to a cloak. Consider a dielectric cylindrical object the air region, respectively. As described before, these ␧ ␧ with permittivity obj=3.0 0 at 100 THz and diameter shells are necessary to fully take into account the effects of ␮ ␭ ␭ ͑␻ / ͒ 2R=0.76 m=0.25 0 =0.44 diel 2R c=1.6 at 100 THz . higher-order modes and of the granularity of the artificial To reduce its visibility and scattering, we may design a suit- material. Following the results in the Appendix, it may be ␦ ͑ ͒ able plasmonic cloak. Assuming that the cloak permittivity is verified that the cloak design requires 1 =0.030a R and ␧ ␧ 1–5 ␦ ͑ ͒ ␦ ␦ c =−3 0, our theory shows that the scattering width of the 2 =0.039a Rc , which yields 1 =9 nm and 2 =16 nm. cloaked system is drastically reduced when the cloak radius Using the full-wave electromagnetic simulator CST Mi- 17 is equal to Rc =1.40R. crowave Studio™, we have calculated the variation of the Inspired by the planar metamaterial configuration of Fig. peak in scattering width Qmax of the combined object-cloak 1 and by our work at microwaves,5 we design a cylindrical system vs frequency, as reported in Fig. 4 for different con- cloak as shown in the inset of Fig. 4. The cloak is formed by figurations of interest. The system is always illuminated by a N=8 silver implants with thickness T, oriented along the plane wave that propagates along the x direction and has radial direction, and uniformly spaced along the azimuthal electric field parallel with the axis of the cylinder ͑z direc- direction.16 The metallic implants are embedded in SiC, tion͒. It is seen that when the object stands alone in free ␧ Ϸ ␧ ͑ ͒ which is characterized by h 6.5 0 at 100 THz Ref. 15 space its scattering width ͑black line in Fig. 4͒ increases ͑for simplicity, in this work we will neglect the frequency monotonically with its electrical size, consistent with what dependence of the permittivity of SiC͒. The effective permit- one may intuitively expect. Quite distinctly, however, when tivity of the proposed cloak may be estimated as in Eq. ͑2͒. the object is covered with the metamaterial cloak ͑dark blue

075107-3 SILVEIRINHA, ALÙ, AND ENGHETA PHYSICAL REVIEW B 78, 075107 ͑2008͒

(a) (b)

FIG. 5. ͑Color online͒ Amplitude of the electric field for an object with ͑a͒ no cloak, ͑b͒ metamaterial cloak, ͑c͒ cloak with only metallic implants, and ͑d͒ dielectric cloak. line in Fig. 4͒, the scattering width of the combined system is To further support our theory, the amplitude of the z com- greatly reduced around the design frequency of 100 THz, ponent of the electric field is reported in Fig. 5 for the dif- despite the fact that the physical size of the system has in- ferent configurations discussed above. When the object is creased when the cloak is added. Such property stems from cloaked with the metamaterial ͓panel ͑b͒ of Fig. 5͔, the field the plasmonic-like properties of the metamaterial cloak, in the air region is nearly uniform, showing that even in the which ensure that the polarization currents induced in the near-field region the scattered field is very weak. Quite dif- cloak are out of phase with the polarization currents induced ferently, for the other configurations in which the object is in the object, effectively eliminating the dipolar-type compo- either uncloaked ͓panel ͑a͔͒, or cloaked with only metallic nent of the scattered field.1–5 This property is further sup- implants or only SiC ͓panels ͑c͒ and ͑d͒, respectively͔, the ported by the simulations obtained for the cases in which the field in the air region is highly nonuniform due to strong object is just covered with SiC ͑red line in Fig. 4͒, or sur- dipolar scattering from the system. Since the scattered field rounded by silver implants ͑pink line in Fig. 4͒. In fact, in of the cloaked system is very weak in the near field at the these two scenarios the scattering width is greatly enhanced design frequency, the coupling between an arbitrary number as compared to the uncloaked case. Finally, we have also of such cloaked objects would be negligible and their com- plotted the scattering width of the system when virtual inter- bined scattering width would stay very small, even if the ␦ ␦ ͑ faces are ignored and 1 and 2 are set equal to zero light objects are closely spaced, consistent with our findings for blue line in Fig. 4͒. It is seen that in this last scenario the the case of continuous material cloaks.3 This situation is re- overall response of the system is detuned, even though Qmax ported in Fig. 6, where we show the transmission coefficient may still become significantly lower than the corresponding ␶ as a function of frequency, when a plane wave propagating value for an uncloaked object around the design frequency. along the x direction illuminates a periodic array of cloaked We have also verified that the response of the metamaterial objects arranged along the y axis, as depicted in the figure cloak ͑with the virtual interfaces͒ mimics very closely the inset. Independently from the distance d between the cloaked response of an equivalent continuous material cloak with objects, the periodic array is effectively transparent to radia- ͑ ͒͑ ϫ permittivity given by Eq. 2 not shown here for sake of tion around 100 THz, even when d is as small as 1.1 2Rc. brevity͒. Moreover, following the results in Ref. 4 we may envision

075107-4 INFRARED AND OPTICAL INVISIBILITY CLOAK WITH… PHYSICAL REVIEW B 78, 075107 ͑2008͒

0 dR3.5 2 c ͑ ͒ ͑ ͒ Ͼ ͑ ͒ E r = ͚ cnEn r;kn , x 0, A1 -10 n

where cn are the unknown coefficients of the expansion and -20 dR2.0 2 ͑n͒ c kn =kx uˆ x +kyuˆ y is the wave vector associated with the mode ͑ ͒ En r;kn . Since in a scattering problem the component of the dB -30 d wave vector parallel to the interface is preserved, the component ky is completely determined by the angle of incidence -40 dR1.1 2 c ␪ ͑␻/ ͒ ␪ Einc i of the incoming plane wave: ky = c sin i. On the other ͑n͒ inc hand, kx depends on the considered mode and is both a -50 H ␻ function of frequency and of ky. Since the periodic mate- 25 50 75 100 125 150 175 rial is invariant to translations along the x and z directions f [THz] and the incoming wave propagates in the x-y plane, it is clear ͑ ͒ that En r;kn may be assumed of the form, FIG. 6. ͑Color online͒ Amplitude of the transmission coefficient ͑ ͒ ͑n͒ ͑in dBs͒ for an array of cloaked objects illuminated by a plane wave ͑ ͒ n ͑ ͒ +ikx x ͑ ͒ En r;kn = Ez y e uˆ z, A2 ͑normal incidence͒. The distance between adjacent objects is d. The ͑n͒ ͑ Ն ͒ geometry is depicted in the inset. where the propagation constants kx n 0 may be calcu- ͑ ͒ lated numerically by solving Eq. 1 with respect to kx for ␻ given and ky. For example, for vanishingly thin PEC slabs, the possibility of using different metamaterial covers to op- ͑n͒ ͑n͒ erate the cloak simultaneously at different frequencies. the propagation constants kx are given by kx =iͱ͓͑n+1͒␲/a͔2 −͑␻/c͒2. For a fixed frequency and for n ͑n͒ sufficiently large kx is complex imaginary, which is consis- IV. CONCLUSIONS tent with the fact that the number of propagating Floquet We have demonstrated the possibility of designing realis- modes supported by the unbounded crystal is finite. tic metamaterials that achieve an effective plasmonic-type Using the results of Refs. 18 and 19, formula ͑A2͒, and response at selected infrared and optical frequencies for taking into account that the problem is two dimensional, it which natural materials with the same characteristics may may be proven that the unknown coefficients cn satisfy the not be readily available. The proposed design fully takes into following infinite linear system of equations: account the finite conductivity of metals and the granularity ϱ 1 of the artificial materials. Full-wave simulations have dem- Einc␦ = ͚ A , l =0,Ϯ 1, Ϯ 2, ... , ͑A3͒ z l,0 n,l ␥ ͑n͒ onstrated that such metamaterial cloaks may drastically re- n=0 l + ikx duce the scattering width of a given object or array of objects ␥ ͱ͑ ␲ / ͒2 ͑␻/ ͒2 at the mid-IR band. Similar results may be obtained at other where l = ky +2 l a − c and infrared or optical frequencies. This suggests exciting poten- c ␻ 2 2␲ A = n ͩ ͪ q ͩk + lͪ, tials for metamaterials with a plasmonic-type response. n,l ␥ n y 2 0 c a

ACKNOWLEDGMENTS with

1 ͑n͒ Ј This work is supported in part by Fundação para Ciência ͑ Ј͒ ͵ ͑␧ ͒ ͑ ͒ −ikyy ͑ ͒ qn k = r −1 E y e dy. A4 e a Tecnologia under Project No. PDTC/EEA-TEL/71819/ y a z 2006. diel. ␧ In the above, r represents the relative permittivity of the ͑ Ј͒ APPENDIX plasmonic slabs and the integral in the definition of qn ky is calculated over the plasmonic slab in the unit cell. Similarly, ␦ In this Appendix, we calculate the displacement of the following Refs. 18 and 19 it may be proven that the ampli- virtual interfaces with respect to the physical interfaces. To ref tude of the reflected electric field at the interface Ez verifies: this end, first we will obtain the reflection coefficient for a ϱ plane wave that illuminates a semi-infinite structure formed 1 ref ͚ ͑ ͒ by a periodic array of stacked plasmonic slabs. The geometry Ez =− An,0 ͑n͒ . A5 n − ␥ + ik of the problem is shown in Fig. 1. It is assumed that the =0 0 x incident electric-field vector is along the z direction, and has Hence, it is possible to compute the reflected field by: ͑i͒ inc ͑ ͒ amplitude Ez at the interface. For simplicity, the host mate- solving the infinite linear system A3 with respect to the ͓ rial is assumed to be air. unknowns cn notice that the entries of the linear system are ͑n͒ ͑n͒͑ ͒ ͑ The main idea for solving this electromagnetic problem written in terms of kx and Ez y , which may be numeri- has been proposed in Refs. 18 and 19 and it uses the property cally͒ determined by computing the eigenmodes of the asso- that the transmitted field ͑in the region xϾ0͒ may be ciated unbounded periodic stratified material͔; ͑ii͒ substitut- ͑ ͒ ͑ ͒ ref written in terms of the electromagnetic modes En r;kn ing the calculated cn into Eq. A5 to obtain Ez . This ͑n=0,1,2,...͒ supported by the associated unbounded peri- procedure yields the exact reflection coefficient, but requires, odic material, manifestly, significant computational efforts. It is however

075107-5 SILVEIRINHA, ALÙ, AND ENGHETA PHYSICAL REVIEW B 78, 075107 ͑2008͒

ref ͑ ͒ possible to considerably simplify the problem by making m=0,1,2,...,N, and Ez =−f −z0 , respectively, and so the some reasonable assumptions. In fact, let us suppose that reflection coefficient is given by: Ӷ T a, i.e., the thickness of the plasmonic slab is much N smaller than the lattice constant. In such conditions it is valid Eref f͑− ␥ ͒ p − ␥ p − ␥ z + ␥ ␳ ϵ z =− 0 =− 0 0 ͟ n 0 n 0 . to make the approximation q ͑k +2␲ / al͒Ϸq ͑k ͒ for the inc ͑␥ ͒ ␥ ␥ ␥ n y n y Ez f 0 p0 + 0 n=1 pn + 0 zn − 0 range of integers l for which the term e−i2␲l/ay is approxi- ͑ ͒ mately constant over the plasmonic material, i.e., for A11 ͑ ␲ / ͒ Ӷ␲ 2 a lT . It is clear that the thinner the plasmonic slab, In order to obtain the solution of the infinite linear system the better is this approximation. Also the approximate iden- ͑A6͒, we may now let N go to infinity. It may be verified tity is more accurate for smaller values of l, which corre- that, for the infinite product in Eq. ͑A9͒ to converge for ͑ spond to the lowest-order Fourier harmonics which are the N→ϱ, it is sufficient that the sequences of poles and zeros most important, since the higher-order evanescent harmonics grow to infinity, ͉z ͉→ϱ and ͉p ͉→ϱ, and in addition that ͒ n n are in principle weakly excited . Using this approximation, it ͚ ͉1/z −1/p ͉Ͻϱ.18 The latter condition is verified when, Ϸ n n n follows that An,l An,0, and thus the infinite linear system for n sufficiently large, the poles and the zeros alternate in ͑ ͒ A3 can be rewritten as: the real line.18 It may be verified that if the poles and zeros of ϱ the problem are ordered in such a way that Re͕p ͖ and Re͕z ͖ 1 n n inc␦ ͑ ͒ are monotonically increasing sequences, the convergence of Ez m,0 = ͚ An,0 , m = 0,1,2, ... , A6 n=0 zm − pn the infinite product is ensured. →ϱ ͑ ͒ ͑n͒ ͑ ͒ By letting N , Eq. A11 can be rewritten as where we defined pn =−ikx n=0,1,2,... and z ͑m=0,1,2,...͒ such that z =␥ and ͑0͒ ␥ m 0 0 ␥ ␦ − ikx − 0 ␳ = ␳ e2 0 , ␳ =− , ͑A12͒ ␥ ␥ ͑ ͒ ͑ ͒ e e ͑0͒ ␥ z2l = −l; z2l−1 = l l =1,2,... . A7 − ikx + 0 ␳ In this scenario, following the ideas of Refs. 18–21 the so- where e is the reflection coefficient that would be obtained ͑ ͒ ͑ ͒ lution An,0 n=0,1,2,... of the system A6 may be calcu- under the hypothesis that the effect of all higher-order modes lated in closed analytical form. To simplify the discussion we is negligible and the scattering problem can be described ␥ ␦ will truncate the infinite series in Eq. ͑A6͒ so that it reduces using only the fundamental mode. The factor e2 0 is a cor- to the sum of a finite number ͑N͒ of terms. This is clearly rection term ͑which takes into account the effect of higher- possible on physical grounds since, as mentioned above, the order modes͒ given by: higher-order evanescent electromagnetic modes are weakly ϱ ␥ ␥ excited. Thus, Eq. ͑A6͒ becomes: ␥ ␦ pn − 0 zn + 0 e2 0 = ͟ . ͑A13͒ ␥ ␥ N n=1 pn + 0 zn − 0 1 inc␦ ͑ ͒ Ez m,0 = ͚ An,0 , m = 0,1,2, ... ,N. A8 ␥ ␥ ͱ͑␻/ ͒2 2 n=0 zm − pn Notice that 0 is complex imaginary: 0 =−i c −ky.Itis simple to verify that in the long-wavelength limit and in case ͑ ͑ Ն ͒ Consider now the auxiliary complex function f w is the of negligible losses, the zeros zn and the poles pn n 1 are complex variable͒ given by: all real valued. Thus, it is evident that in such conditions ␥ ␦ ͉e2 0 ͉=1, and thus the parameter ␦ is real valued. As ex- N / C 1−w zn plained in our previous work,5 the parameter ␦ determines f͑w͒ = ͟ , ͑A9͒ / / 1−w p0 n=1 1−w pn the displacement of “the virtual interface” with respect to the physical interface. From the point of view of an incoming ͑ ͒ inc where the constant C is calculated so that f z0 =Ez .Itis wave, the stratified semi-infinite material behaves effectively ͑ ͒ ͑ ͒ clear that f w has zeros at the points w=zm m=1,2,... and as a continuous material, characterized by the effective per- ͑ ͒ ͑ ͒ poles at the points w=pn n=0,1,2,... . In addition f w mittivity obtained from the slope of the dispersion character- converges to zero at the same rate as 1/w,asw goes to istic of the fundamental mode, being the effective interface infinity. In particular, applying the residues theorem it is with air positioned at the plane x=−␦ instead of at the physi- simple to verify that cal interface ͑x=0͒. From Eq. ͑A13͒, it is clear that ␦ is N ͉ ͑ ͉͒ given by: Res f w=p ͑ ͒ n ͑ ͒ ϱ f w = ͚ , A10 ͉␥ ͉ ͉␥ ͉ w − p 1 0 0 n=0 n ␦ = ͚ arctanͩ ͪ − arctanͩ ͪ. ͑A14͒ ͉␥ ͉ z p where ͉Res͑f͉͒ is the residue of f͑w͒ at the pole p . How- 0 n=1 n n w=pn n ͑ ͒ inc ͑ ͒ ͑ ͒ ␦ ␻ ever, since f z0 =Ez and f zm =0 m=1,2,... ,itis In general, is a function of frequency , wave vector ky, evident—comparing Eq. ͑A8͒ to Eq. ͑A10͒—that the solu- and of course of the thickness and permittivity of the dielec- tion of the truncated infinite linear system is given by tric slabs. For normal incidence, zn is such that z2l =z2l−1 A =͉Res͑f͉͒ , which may be calculated explicitly using ͱ͑ ␲ / ͒2 ͑␻/ ͒2 ͑ ͒ ͑n͒ ͑ Ն ͒ n,0 w=pn = 2 l a − c l=1,2,... , and pn =−ikx n 1 are Eq. ͑A9͒. This yields the exact solution of the truncated the attenuation constants in the stratified unbounded material ͑ linear system for arbitrary N. In particular, we note that for ky =0 excluding the lowest attenuation constant which is ͑ ͒ ͑ ͒ inc␦ ͑ ͒ ͒ Eqs. A5 and A8 are equivalent to Ez m,0= f zm , associated with the fundamental mode n=0 .

075107-6 INFRARED AND OPTICAL INVISIBILITY CLOAK WITH… PHYSICAL REVIEW B 78, 075107 ͑2008͒

*Author to whom correspondence should be addressed. engheta M. R. Querry, Appl. Opt. 24, 4493 ͑1985͒. @ee.upenn.edu 14 L. Brillouin, Wave Propagation in Periodic Structures, 2nd ed. 1 A. Alù and N. Engheta, Phys. Rev. E 72, 016623 ͑2005͒. ͑Dover, New York, 1953͒. 2 ͑ ͒ A. Alù and N. Engheta, Opt. Express 15, 3318 2007 . 15 W. G. Spitzer, D. Kleinman, and D. Walsh, Phys. Rev. 113, 127 3 ͑ ͒ A. Alù and N. Engheta, Opt. Express 15, 7578 2007 . ͑1959͒. 4 A. Alù and N. Engheta Phys. Rev. Lett. 100, 113901 ͑2008͒. 16 A geometry that may somehow resemble the one suggested here 5 M. G. Silveirinha, A. Alù, and N. Engheta, Phys. Rev. E 75, has been recently proposed to achieve cloaking at optical fre- 036603 ͑2007͒. quencies ͑Ref. 11͒. However, despite some apparent geometrical 6 J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 ͑2006͒. similarities, the cloaking technique, the material parameters, and 7 U. Leonhardt, Science 312, 1777 ͑2006͒. the physical behavior of the cloak presented here are completely 8 D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, different than those presented in Ref. 11. Moreover, the two A. F. Starr, and D. R. Smith, Science 314, 977 ͑2006͒. cloaks work for two different ͑orthogonal͒ polarizations of the 9 G. W. Milton and N. A. Nicorovici, Proc. R. Soc. London, Ser. A electromagnetic wave. 462, 3027 ͑2006͒. 17 CST Microwave Studio™ 5.0, CST of America, Inc., 10 W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Nat. www.cst.com Photonics 1, 224 ͑2007͒. 18 M. G. Silveirinha, IEEE Trans. Antennas Propag. 54, 1766 11 W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, Opt. ͑2006͒. Express 16, 5444 ͑2008͒. 19 P. A. Belov and C. R. Simovski, Phys. Rev. B 73, 045102 12 C. F. Bohren and D. R. Huffman, Absorption and Scattering of ͑2006͒. Light by Small Particles ͑Wiley, New York, 1983͒. 20 G. D. Mahan and G. Obermair, Phys. Rev. 183, 834 ͑1969͒. 13 M. A. Ordal, Robert J. Bell, R. W. Alexander, Jr., L. L. Long, and 21 C. A. Mead, Phys. Rev. B 17, 4644 ͑1978͒.

075107-7