arXiv:1404.5868v1 [physics.optics] 22 Apr 2014 osbet pl hs tutrsntol swvgie bu waveguides as only metamaterials. it for not blocks makes structures building what these as also parameters, apply electromagnetic to cylind possible extreme anisotropic an with as behaves an Fin rod waveguide modes the degenerated. that guided highly shown TE becomes is foun only relation is supports dispersion waveguide It their the constant. corrugation limit, of periodicity this number their in the keeping which in but limit infinite, con the the as materia in defined analyzed dielectric meta is limit, waveguide a corrugated this with of longitudinally relation filled dispersion a corrugations wavegui with on cylindrical plasmon-l cylinder open consists this an structure work, design The to this mim employed In to is plasmons. ability behaviour their surface to of due behaviour attention strong received cently locniee.I hsae pca eto eevsthe [9]–[12 electromagnet deserves cylinders of anisotropic mention highly realization using special the metamaterials area concerning been this work have new author’s In realization found, of metamaterials considered. been variety of also have wide methods applications several a and and material, phenomena effective exciting has metamateria and resulting structures naming the By designed attention. to artificially strong of received recently means by properties nteps o h ento n elzto fsf n hard and soft of realization and studi [8]. definition been boundaries the perme- already electromagnetic for and had past permitivity behavior the effective its in anisotropic This an of as [7]. values behave ability surfaces extreme closed with such showi material that waves, as plasmon-like surfa well these spherical as of or localization microwave cylindrical the the closed b allows has in texturing it plasmons that Furthermore, [2]–[6]. shown mimic bands to frequency terahertz used or be can surfaces employe the materials metallic to design. or limited their dielectric are the otherwise of which properties structure, characteri waveguide propagation a the Periodici of over [1]. control properties additional relevant pinges their to due structures P httewvgiealw nytepoaaino Emodes; TE of propagation the demonstr only be allows will waveguide It the Fig.1. wi in that filled shown as are material, grooves dielectric the a longitudin where a cylinder on metallic op consists corrugated lossless surface a in The placed environment. plasmons circular of localization the on based ORA FL OF JOURNAL ne Terms Index Abstract nti otx,tecnrlo h fetv electromagnet effective the of control the context, this In corrugated periodically that shown been has it Recently, nti etrw nlz yidia aeud structure waveguide cylindrical a analyze we letter this In aebe ieysuidfrteruea waveguide as use their for studied widely surfaces metallic been and/or have dielectric corrugated ERIODIC nlsso ea-ilcrcWvgie with Waveguides Metal-Dielectric of Analysis Tesuyo ealccruae ufcshsre- has surfaces corrugated metallic of study —The A T E LS IE,VL ,N.1 AUR 2007 JANUARY 1, NO. 6, VOL. FILES, CLASS X IEta,junl L journal, —IEEEtran, .I I. ailTret eioGmn,VcneE oi n o´ S´ Jos´e and Boria E. Vicente Gimeno, Benito Torrent, Daniel NTRODUCTION A T E ,ppr template. paper, X, iclrSectors Circular tinuous that, d yim- ty ly it ally, .The l. cthe ic ]. stics rical ated in d ally een ces is s llic de. ike ng ed en th ic ic ls d t h aeud sdvddinto divided is waveguide The ϕ n le ihtepoaainmtra aiga angle an having material propagation the with filled one ealccr fradius of core metallic n h te n le ihtepretcnutrmaterial conductor perfect the with angle filled an one having other the and h aeud sdfie uhta h rpgto material propagation the that such defined is waveguide the meability simre nfe pc,caatrzdby characterized space, free in immersed is radius 1 of Fig. cylinder in long infinitely depicted An as waveguide, cylindrical inhomogeneous the aeo lentn etr fapretcnutrcombined permittivity conductor of perfect material a dielectric of a with sectors alternating of made oeta plctoso hs yidia tutrsfo structures metamaterials. the cylindrical electromagnetic providing of these thus realization of given, applications are waveguide potential propertie proposed electromagnetic identical novel Final the being anisotropic work. similar, this extraordinary in quite explained the conditions be multipo special to some different under shown the of are relation components dispersion the moreover, waveguide. layered angularly the of Geometry 1. Fig. p h emtyo h rpsduiomwvgiei an is waveguide uniform proposed the of geometry The 2 = z xsi osdrd h rs eto fteclne is cylinder the of section cross The considered. is axis π/N µ d en ahsco iie notosubsectors, two into divided sector each being , aldte“rpgto aeil;testructure the material”; “propagation the called , ϕ m = a anchez-Dehesa I T II. spae ttecne ftecylinder. the of center the at placed is ϕ p − HEORY ϕ N d h nua emtyof geometry angular The . dnia etr fangle of sectors identical b ihisai aallto parallel axis its with ε d n antcper- magnetic and µ 0 and ε the r 0 of s A . ϕ lar ly, d 1 , . JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 2

(n) sectors start at ϕd = (n 1)ϕp, while the conductor sectors for the Hz field. The objective of this work is to study this (n) − start at ϕm = (n 1)ϕp + ϕd, with n =1, 2,...,N. type of waveguide in its simplest form, thus it will be assumed Considering a time-harmonic− dependence with frequency that a = 0, since finite but small values of a only contribute f = ω/(2π) and invariance along the z axis, it is known that to the dispersion relation. the transverse components of the electromagnetic guided fields Once the solution for the fields at both sides of the boundary can be obtained from the z components as follows [13] r = b is established, standard mode matching method is j directly applied. Boundary conditions at r = b implies the Et = − [β∇tEz + ωµzˆ ∇tHz] (1a) E H µεω2 β2 × continuity of the z and ϕ components of the and fields j− at the boundaries of the propagating material sector, as well as Ht = − [β∇tHz ωεzˆ ∇tEz] (1b) the cancellation of E and E at the corresponding boundaries µεω2 β2 − × z ϕ − of the perfect metallic sectors. In order to apply the mode- where β is the modal propagation constant, and µ and ε matching technique, the equations for Ez and Eϕ must be represent the magnetic permeability and the dielectric permit- multiplied by the modal solutions outside the waveguide, while tivity, respectively; it should be emphasized that both µ and the equations for the Hz and Hϕ must be multiplied by the ε are functions on the transverse vector position. Boundary solution inside the waveguide. After some algebraic manip- equations, together with the guided wave condition, will be ulations, equations are integrated within the corresponding forced in order to obtain the electromagnetic fields for rb, while at r = b mode matching method will grant AE, AH ,BE and BH defined in equations (2) and (3). For the satisfaction of the corresponding boundary conditions. For q q ns ns instance, the continuity condition of the Ez component is r > b only evanescent solutions are allowed, thus the z imposed after multiplying by the exponential factor e−jqϕ and components of the fields will be expressed as integrated over ϕ [0, 2π], ∞ ∈ E jqϕ ∞ Ez(r, ϕ)= A Kq(Γr)e , (2a) N q 1 ∗ −∞ E E (n) q=X Aq Kq(Γb)= BnsJsγ(ktdb)Nsq , (6a) ∞ 2π nX=1 Xs=1 H jqϕ Hz(r, ϕ)= Aq Kq(Γr)e , (2b) q=X−∞ where means complex conjugate operation, and N is the total number∗ of angular sectors. Next, the axial magnetic field where Kq( ) are modified Bessel functions of second kind, (n) · component is multiplied by cos sγ(ϕ ϕd ), and integrated which are evanescent-like solutions which cancels as r , (n−1) (n) − 2 2 2 → ∞ over ϕ [ϕ , ϕ ], then we obtain for the n-th angular and Γ = β µ0ε0ω , Γ being the propagation constant for ∈ d d r>b. − sector the following relationship Inside the waveguide, that is, for r

(n) H Finally, the equation for Hϕ is multiplied by sin sγ(ϕ ϕd ), Solving for Bn0 from (8b), and inserting it into (8c), we (n−1) (n) − find and integrated over ϕ [ϕd , ϕd ], resulting in ∈ ∞ ∞ µ0 ′ H µd H jqβ H ωε0 ′ E (n) Kq(Γb)Aq = χqsAs Ks(Γb), (9) Kq(Γb)Aq + Kq(Γb)Aq Nsq = − Γ k Γ2a Γ  td s=X−∞ q=X−∞ where the χ matrix elements are defined as π sγβ H ωεd ′ E qs 2 Jsγ (ktdb)Bns + Jsγ (ktdb)Bns . (6d) −2γ k a ktd  ′ N td J (k b) γ ∗ 0 td (n) (n) (10) χqs = 2 M0q M0s , Equations (6) define a homogeneous system of equations for J0(ktdb) 2π E H E H nX=1 the unknown coefficients Aq , Aq ,Bns and Bns. The fre- quency values that make the determinant of the corresponding which still keeps finite values of both N and γ. matrix equation vanishes for positive values of Γ2 define the Although the above system of equations could be used as guided modes. The discussion on the several solutions of this an approximated solution to (6), for large values of N and γ, system is out of the scope of the present work. Here we are the goal of this work is to analyze the continuous limit. As mainly interested in the analysis of a limiting case where a consequence, the infinite limit of both N and γ has to be equations simplify, which corresponds to a type of waveguides taken, leading to whose properties can be easily understood. ′ N ′ n J0(ktdb) ξ i(s−q)ϕ( ) J0(ktdb) lim χqs = lim e d = ξ δqs. N→∞ J0(ktdb) N→∞ N J0(ktdb) III. CONTINUOUS LIMIT nX=1 (11) Let us define the filling fraction ξ as the ratio of the area This result demonstrates that in the continuous limit all occupied by the propagation material divided by the sector the multipolar q components are fully decoupled, and the area of the waveguide, dispersion relation simply becomes as follows ′ ϕd N µ K (Γb) µ ξ J (k b) ξ = , (7) 0 q = d 1 td . (12) ≡ ϕp 2γ Γ Kq(Γb) ktd J0(ktdb) where (4) has been employed. This equation defines a transcendental equation to determine The continuous limit is defined as the limit when the number the dispersion relation β = β(k0) (where k0 = ω√ε0µ0 is of cells goes to infinity, N , while the filling fraction ξ the wavenumber in free space), for the continuous limit of the remains constant, so that we→ also ∞ have that γ , which is sectorial waveguide previously analyzed in its general form. (n) (n) → ∞ obvious as in this limit ϕd ϕm . In this limit, the structure The solutions for β = β(k0) cannot be obtained analytically will behave as a continuous→ and homogeneous material, as it from (12), and a numerical root finding algorithm must be will be demonstrated next in this section. employed. In the continuous limit, it is easy to see that the matrix Note that for the case q = 0, µd = µ0 and ξ = 1, (n) elements Nsq cancels, therefore from (6a) we easily obtain the eigenvalue equation for the T E modes is recovered in (12); however, for the case q = 0 the eigenvalue equation E E A Kq(Γb)=0 A =0. (8a) 6 q ⇒ q is completely different even when ξ = 1. In fact, even though the case indicates that the full waveguide is This result shows that this structure only allows the propa- ξ = 1 made of dielectric material, the boundary condition due to gation of T E modes. In the same limit, the Bessel functions the perfect metallic walls is still present in the equations and, J ( ) are identically zero for all value of s but s = 0, thus sγ as a consequence, the result turns in a completely different again· it is easy to see that equation (6b) becomes scenario. ∞ π It is interesting to note that all the modes resulting from (12) AH K (Γb)M (n) = BH δ J (k b), (8b) q q sq γ n0 0s 0 td have the same cut-off frequency, corresponding to the zeros of q=X−∞ Bessel function J0(ktdb). Effectively, the cut-off frequency is where we still keep the γ factor in the right hand side because obtained in the limit β k0 which corresponds to Γ 0. In → → it will be cancelled by the γ factor in the denominator of this limit (12) can only be hold if J0(ktdb)=0, which leads (n) Msq . to Similarly, the reduced form of equation (6c), reminding that k b = µ ε ω2 β2b = ωb√µ ε µ ε = κ , (13) AE =0, can be cast as follows td d d d d 0 0 i q p − − where κ are the zeros of the zero order Bessel function, i.e., N i ωµ0 ′ H 1 (n)∗ ωµd H ′ J0(κi)=0. Thus, the set of cut-off frequencies are given by Kq(Γb)Aq = M0q Bn0J0(ktdb). (8c) − Γ 2π ktd nX=1 κi ωib = . (14) √µdεd µ ε Finally, from Eq. (6d) we arrive to a trivial solution, because − 0 0 the left hand side is zero due to the cancellation of the matrix It should be noted that these frequencies are independent (n) elements Nsq , while the right hand side is zero due to the of q and ξ, being therefore only dependent of the material terms involving Jsγ . employed to construct the waveguide. JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 4

q = 0 q = 5 1.5 q = 10 1.5 q = 15 = 0 9 q = 20 1.4 ξ . 1.4 q = 25 q = 0 0 1.3 q = 1

0 q = 2 1.3 q = 3 β/k 1.2 q = 4 ( 1 ) β/k β ∞ ( 2 ) 1.1 1.2 β ∞

1 1.1 0.5 1 1.5 2

q = 0 1 q = 5 0 0.5 1 1.5 2 1.5 q = 10 = 15 q k 0a /π 1.4 ξ = 0.1 q = 20 q = 25 0 1.3 Fig. 3. The dispersion relationship of a dielectric rod with εd = 2.56 ε0 and

β/k = 1.2 µd µ0 (dots), is compared with the limiting dispersion relation β∞/k0 (continuous lines) of the angularly layered waveguide filled with the same 1.1 dielectric material. 1 0.5 1 1.5 2 (εd = 2.56 ε and µd = µ ). Note that the modal dispersion k a /π 0 0 0 is larger for the dielectric rod, where the single-mode regime Fig. 2. Dispersion relationship for the angularly layered waveguide. is much smaller than for the angularly layered waveguide, as it is reported in [1].

Another interesting result is found when we take the limit V. MODAL SOLUTION AND EFFECTIVE q , then we have that ELECTROMAGNETIC TENSOR → ∞ ′ µ K (Γb) It has been already shown that the polarization of the EM lim 0 q = (15) →∞ fields traveling along the waveguide is TE, since Ez =0. Let q Γ Kq(Γb) ∞ us deduce the expressions for the other EM-field components and the dispersion relation is found from the condition β(k0) in the continuous limit defined in the previous section. The Hz J0(ktdb)=0 or ktdb = κi, easily obtaining H component is given by (3b), while the Bns is obtained from (i) 2 2 (8b), showing that this coefficient cancels for all s but s =0, β∞ = µdεdω (κi/b) . (16) − thus it can be shown that the Hz field inside the cylinder, for p (n) (n) Next section solves (12) for a particular case of practical ϕ [ϕ , ϕm ], is given by ∈ d interest, and it will be shown that such equation can be ∞ K (Γb) γ properly used to characterize this novel type of waveguide. H (r, ϕ)= AH q J (k r) M (n) (17) z q J (k b) 0 td π 0q q=X−∞ 0 td IV. NUMERICAL RESULTS The factor γM (n) is, in the limit γ , proportional to Let us consider the case where εd =2.56 ε0 and µd = µ0, 0q jqϕd jqϕ → ∞ e e , thus the Hz component inside the cylinder has which corresponds to the dielectric material used in [1] to ≈ analyze the dielectric rod waveguide. the following form Fig. 2 shows the dispersion relationship obtained from (12) ∞ H jqϕ for two filling fraction values of ξ = 0.9 (upper side) and Hz(r, ϕ)= J0(ktdr) Bq e (18) ξ = 0.1 (lower side). The several depicted dots are the ones q=X−∞ corresponding to the multipolar components q =0 to q = 25 which shows that the radial dependence of the field is com- in steps of 5. Note also that even for the case of a low pletely decoupled from the angular dependence, as it might be metallic filling fraction (ξ = 0.9) the modal dispersion is deduced from (12). small, being the case of ξ = 0.1 of practically zero modal Additionally, from Eqs. (1) and (3), together with the fact H H dispersion. Moreover, for the two cases shown in Fig. 2, the that Bns = Bn0δ0s, we can conclude that the fields component modal dispersion relationship for large values of q are quite Er and Hϕ, both implying terms proportional to the angular H similar. Thus, the dispersion relation β∞ can be used as a derivative of Hz, therefore to sBns, will vanish inside the reference for characterizing this type of waveguides, since cylinder, then Er(r, ϕ)=0 and Hϕ(r, ϕ)=0. from Figure 2 it can be concluded that all the modes present The latter of these equations will be also a boundary a similar dispersion behavior. condition, since the tangential component of the fields must (i) Fig. 3 shows the dispersion response β∞ for i = 1, 2 be continuous at an arbitrary interface. Equations (1) show (continuous lines) together with the dispersion curves for a also that both Eϕ and Hr will be different than zero and dielectric rod (dotted line) made of the same dielectric material proportional to ∂rHz. JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 5

In summary, the only nonzero components of the field be a solution of the two dimensional wave equation in an within this structure, in the continuous limit, will be Hz,Hr anisotropic medium, given by [14] and Eϕ. In this limit, the expression for the Hz field is given 2 1 ∂ ∂Hz εϕ 1 ∂ Hz 2 by (18), but the expressions for Hr and Eϕ cannot be obtained (24) r 2 2 = ω µzεϕHz by a direct application of (1), since these equations are not − r ∂r  ∂r  − εr r ∂ϕ valid in the continuous limit. whose solutions are linear combinations of cylindrical harmon- In this limit, the cylindrical structure analyzed in the present ics and Bessel functions of real order, i.e. work behaves as a homogeneous and anisotropic cylinder, thus ∞ the relationship between the electromagnetic field components jqϕ Hz(r, ϕ)= AqJqP (kr)e (25) must satisfy the tensorial generalization of Eqs. (1), obtained q=X−∞ from Maxwell equations in tensor form as indicated in [14] where P = εϕ/εr and k = ω√εϕµz. It is straightforward E + jωµ H =0 (19a) to show thatp for the above equation be consistent with (18) ∇× · H jωε E =0 (19b) we need that µz = µr = ξµd. Note that since εr = , P =0 ∇× − · and the radial dependence of the field will be proportional∞ to where the tensors µ and ε are diagonal matrices of the form J0(ω√εdµdr). The only remaining tensor components to be determined are εr 0 0 εt 0 εz and µϕ. Since Ez =0, from (22d) we must conclude that ε =  0 εϕ 0  = (20)  0 εz  εz = , but none of the equations require any assumption to 0 0 εz ∞   determine the value of µϕ. However, it can be obtained if a and two dimensional scattering process under Ez illumination is µr 0 0 assumed. µt 0 µ =  0 µϕ 0  = (21) Thus, under these conditions, it is known that the equations  0 µ  0 0 µz for Hz and Ez are decoupled, and, in this case, boundary   conditions would only imply continuity of Ez and Hϕ. The ˆ After replacing ∇ by t jβz in (19), these equations can previous analysis has shown that E must cancell at the ∇ − z be cast as boundary of the cylinder, which is enough to define the scattering problem. However, it has been also shown that jβEt + jωzˆ (µ Ht) = tEz (22a) − × t · ∇ Hϕ = 0, and, since these fields are related in the two- jβHt jωzˆ (εt Et) = tHz (22b) dimensional case by − − × · ∇ zˆ ( t Et) = jωµzHz (22c) · ∇ × − ∂Ez ˆ H jωµϕHϕ = (26) z ( t t) = jωεzEz (22d) ∂r · ∇ × if µϕ is allowed to have a finite value, the above problem from which Et and Ht can be easily obtained in terms of Ez would be not well defined, since the scattering problem would and Hz. It has been shown that Ez is equal to zero for the case considered in this paper, therefore the tensorial form of require that both Ez and ∂rEz be zero at the surface, which is not possible. Thus, we conclude that µϕ = , which means (1) are ∞ that ∂rEz is different than zero and the problem of scattering jβ ∂Hz must be solved by imposing that . (23a) Ez =0 Hr = 2 − 2 ω εϕµr β ∂r In summary, all the tensor components of the sector cylinder − jβ 1 ∂H have been determined in the continuous limit, being z (23b) Hϕ = 2 − 2 ω εrµϕ β r ∂ϕ − 0 0 jωµϕ 1 ∂Hz ∞ (23c) ε =  0 εd/ξ 0  (27) Er = 2 2 −ω µϕεr β r ∂ϕ 0 0 − jωµ ∂H  ∞  r z (23d) Eϕ = 2 2 and ω µrεϕ β ∂r − ξµd 0 0 µ =  0 0  (28) With these relationships and the solution for the dispersion ∞ 0 0 ξµd relation given in (12), it is possible to deduce the effective   tensor describing the waveguide in the continuous limit. Once the tensor components have been obtained, the non-zero As mentioned before, both Er and Hϕ must be zero, which, electric and magnetic field components inside the cylinder can from (23b) and (23c) allows to conclude that εr = . ∞ be obtained from (23a) and (23d), together with (18), Additionally, the dispersion relation (12) is obtained from the ∞ continuity of the Hz and Eϕ fields, therefore if this solution jβ H jqϕ Hr(r, ϕ) = J1(ktdr) Bq e (29a) must be consistent with (23d), it is required that µr = ξµd k r td q=X−∞ and µrεϕ = µdεd, which is equivalent to εϕ = εd/ξ. ∞ jωξµd H jqϕ The solution for the Hz field given by (18) will also hold Eϕ(r, ϕ) = J (ktdr) B e (29b) − k 1 q in the limit β 0, with ktd = ωεdµd. In this case, Hz must td q=X−∞ → JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 6

Q=0 Q=0 In the continuous limit the waveguide becomes azimuthally symmetric, therefore the different multipolar components q −1 −1 are uncoupled each other. In principle, a different dispersion relation for each multipolar mode is found; however, it has 0 0 been shown that such modal dispersion is low, becoming negli- y/b gible when the metal filling fraction is increased. The obtained 1 1 dispersion behavior has been compared with a dielectric rod −1 0 1 −1 0 1 waveguide, and it has been found that these dispersion curves are similar each other, but with the peculiarity that the sectorial Q=1 Q=1 waveguide presents negligible modal dispersion. −1 −1 It has been also demonstrated that in the continuous limit the waveguide behaves as an anisotropic cylinder, and the effective 0 0 dielectric and magnetic corresponding tensors describing it y/b have also be found. 1 1 Finally, let us conclude that the main application of these waveguides would be as polarizers, because only the TE po- −1 0 1 −1 0 1 larization is allowed, and the dispersion behavior is similar to Q=2 Q=2 propagation in free space. Also, the extraordinary anisotropic properties of the rod in the continuous limit suggest its use as −1 −1 building blocks for new electromagnetic metamaterials. 0 0 y/b ACKNOWLEDGMENT 1 1 This work was partially supported by the Spanish Ministerio −1 0 1 −1 0 1 de Ciencia e Innovacion (MICINN) under Contracts No. TEC2010-19751 and No. CSD2008-66 (the CONSOLIDER Q=3 Q=3 program) and by the US Office of Naval Research. −1 −1 REFERENCES 0 0 y/b [1] R. E. Collin, Field Theory of Guided Waves. Ieee Press, 1991. [Online]. Available: 1 1 http://books.google.com/books?id=7DZ9KwAACAAJ&pgis=1 [2] J. B. Pendry, L. Mart´ın-Moreno, and F. J. Garcia-Vidal, “Mimicking −1 0 1 −1 0 1 surface plasmons with structured surfaces.” Science (New York, N.Y.), x/b vol. 305, no. 5685, pp. 847–8, Aug. 2004. [Online]. Available: x/b http://www.ncbi.nlm.nih.gov/pubmed/15247438 [3] A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental Fig. 4. Modal solution of the field components Hz(left column) and Eϕ verification of designer surface plasmons.” Science (New York, N.Y.), (right column) for the multipolar solutions q = 0, 1, 2 and 3. vol. 308, no. 5722, pp. 670–2, Apr. 2005. [Online]. Available: http://www.ncbi.nlm.nih.gov/pubmed/15860621 [4] F. Garc´ıa de Abajo and J. S´aenz, “Electromagnetic Surface Modes in Structured Perfect-Conductor Surfaces,” Physical Review Letters, Fig. 4 shows the modal solution of the fields Hz (left column) vol. 95, no. 23, p. 233901, Nov. 2005. [Online]. Available: and Eϕ (right column) corresponding to q =0, 1, 2 and 3 for http://link.aps.org/doi/10.1103/PhysRevLett.95.233901 the same waveguide parameters as those considered in the [5] F. J. Garc´ıa de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Reviews of Modern Physics, previous section with ξ = 0.1. The free space wavenumber vol. 79, no. 4, pp. 1267–1290, Oct. 2007. [Online]. Available: is k0a = π. Notice that all these modes have the same radial http://link.aps.org/doi/10.1103/RevModPhys.79.1267 dependence, proportional to for , and to for , and [6] M. a. Kats, D. Woolf, R. Blanchard, N. Yu, and F. Capasso, J0 Hz J1 Eϕ “Spoof plasmon analogue of metal-insulator-metal waveguides.” Optics that they are also highly degenerated, with βa 4.35, being express, vol. 19, no. 16, pp. 14 860–70, Aug. 2011. [Online]. Available: the only difference among them the angular dependence≈ of the http://www.ncbi.nlm.nih.gov/pubmed/21934847 EM fields. [7] A. Pors, E. Moreno, L. Martin-Moreno, J. Pendry, and F. Garcia-Vidal, “Localized Spoof Plasmons Arise while Texturing Closed Surfaces,” Physical Review Letters, vol. VI. CONCLUSIONS 108, no. 22, p. 223905, May 2012. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.108.223905 In this work, the dispersion relation of a waveguide with [8] P. S. Kildal, “Artificially Soft and Hard Surfaces in Electromagnetics,” alternating metallic and homogeneous material sectors has IEEE Transactions on Antennas and Propagation, vol. 38, no. October, been analyzed in the continuous limit, that is, when the number 1990. [9] J. Carbonell, D. Torrent, A. D´ıaz-Rubio, and J. S´anchez- of sectors goes to infinity but the relative filling fraction Dehesa, “Multidisciplinary approach to cylindrical anisotropic remains constant. It has been found that this waveguide metamaterials,” New Journal of Physics, vol. 13, allows propagation of only T E modes, since the waveguide no. 10, p. 103034, Oct. 2011. [Online]. Available: http://stacks.iop.org/1367-2630/13/i=10/a=103034?key=crossref.f345fc30d8d63f2d62fb74366ad51ddb effectively behaves as a homogeneous metallic cylinder for the [10] D. Torrent and J. S´anchez-Dehesa, “Multiple Scattering Formulation of TM polarization. Electromagnetic and Acoustic Metamaterials,” New Journal of Phsyics. JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 7

[11] J. Carbonell, A. D´ıaz-Rubio, D. Torrent, F. Cervera, M. A. Kirleis, a. Piqu´e, and J. S´anchez-Dehesa, “Radial Photonic Crystal for detection of frequency and position of radiation sources.” Scientific reports, vol. 2, p. 558, Jan. 2012. [Online]. Available: http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=3412275&tool=pmcentrez&rendertype=abstract [12] J. Carbonell, D. Torrent, and J. S´anchez-Dehesa, “Radial Photonic Crys- tal Shells and Their Application as Resonant and Radiating Elements,” vol. 61, no. 2, pp. 755–767, 2013. [13] J. D. Jackson, Classical Electrodynamics Third Edition. Wiley, 1998. [Online]. Available: http://www.amazon.com/Classical-Electrodynamics-Third-Edition-Jackson/dp/047130932X [14] W. C. Chew, Waves and fields in inhomoge- neous media. IEEE Press, 1999. [Online]. Available: http://books.google.com/books?id=sIgpAQAAMAAJ&pgis=1