Surfaces with holes in them: new plasmonic metamaterials


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1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS J Opt A: Pure Appl Opt 7 5 S97 S oi:88/ /7//3 Surfaces with holes in them: new plasmonic metamaterials FJGarciaVial,LMartínMoreno an J B Penry 3 Departamento e Fisica Teorica e la Materia Conensaa, Universia Autonoma e Mari, E849 Mari, Spain Departamento e Fisica e la Materia Conensaa, ICMACSIC, Universia e Zaragoza, E59 Zaragoza, Spain 3 Imperial College Lonon, Department of Physics, The Blackett Laboratory, Lonon SW7 AZ, UK Receive June 4, accepte for publication 7 August 4 Publishe January 5 Online at stacksioporg/jopta/7/s97 Abstract In this paper weeplore the eistence of surface electromagnetic moes in corrugate surfaces of perfect conuctors We analyse two cases: oneimensional arrays of grooves an twoimensional arrays of holes In both cases we fin that these structures support surface boun states an that the ispersions of these moes have strong similarities with the ispersion of the surface plasmon polariton bans of real metals Importantly, the ispersion relation of these surface states is mainly ictate by the geometry of the grooves or holes an these results open the possibility of tailoring the properties of these moes by just tuning the geometrical parameters of the surface Keywors: surface plasmons, metamaterials, enhance transmission Some figures in this article are in colour only in the electronic version Introuction Since the appearance of the paper by Ebbesen et al [] reporting etraorinary optical transmission EOT in twoimensional D arrays of subwavelength holes in metallic films, the stuy of the optical properties of subwavelength apertures has become one of the most eciting areas in optics research In this seminal paper [], the relation between transmission resonances appearing in the spectra an the ecitation of the surface plasmon polaritons SPPs of the metallic surface was alreay pointe out The link between EOT an surface plasmons was corroborate theoretically three years after that [] Interestingly, in this last paper we also showe that similar anomalous transmission appears in arrays of subwavelength hole arrays perforate in a perfect conuctor It is well known that the surface of a perfect conuctor oes not support surface plasmons This seeme to suggest that the physical origins of EOT in real metals an in perfect conuctors were ifferent, leaing to iscussions asregars the true origin of the EOT phenomenon In this paper we solve this parao by showing that although a flat perfectly conucting surface supports no boun states, the presence of any perioic inentation of the flat surface for eample, D arrays of grooves or D hole arrays provokes the appearance of surface boun states that have strong similarities with the canonical SPPs of a flat metal surface [3, 4] Importantly, we also show that, as long as the size an spacing of the holes are much smaller than the wavelength, a perforate perfectly conucting surface behaves as an effective meium This meium is characterize by an effective ielectric function that has a plasmon form with a plasma frequency ictate by the geometry of the hole or the groove In other wors, the system behaves as a plasmonic metamaterial in which its electromagnetic response is governe by the surface moes that ecorate its surface It is worth commenting that this new class of metamaterials has some links with the metallic metamaterials invente in recent years in connection with the concept of negative refraction [5] D arrays of grooves First, we analyse the case of a D array of grooves rille in aperfect conuctor see figure a; a is the with of the grooves, h is the epth an the perio of the array We are intereste in looking at the surface EM moes supporte by /5/97+5$3 5 IOP Publishing Lt Printe in the UK S97
2 FJGarciaVial et al a b Figure a A oneimensional array of grooves of with a an epth h separate by a istance Weareintereste in ppolarize surface moes running in the irection with E lying in the z plane b In the effective meium approimation the structure isplaye in a behaves as an homogeneous but anisotropic layer of thickness h on top of a perfect conuctor this structure The proceure for calculating the ispersion relation of these surface moes, ωk,isthefollowing First, we calculate the reflectance of an incient ppolarize incient plane wave with parallel momentum k Asweareintereste in a truly surface moe, we will then analyse the epression for the reflectance for the particular case in which the incient plane wave is evanescent, k >ω/c a truly surface moe has tolive outsie the light cone The locations of the ivergences in the reflectance will give us the esire ispersion relation for the surface EM moes The electromagnetic EM fiels associate with the incient wave are E inc e ik e ikz z k /k z H inc e ik e ikz z k /k z where k is the wavenumber, ω/c,ank z k k The reflecte wave associate with the niffraction orer can be written as E ref,n e ikn H ref,n e ikn e ikn z z e ikn z z k n /k z n k /k z n where k n k +πn/ n,,,, an k z n k kn As we assume that the wavelength of light is much larger than the with of the grooves λ a, in the moal epansion of the EM fiels insie the grooves we only consier the funamental TE moe: E TE,± e ±ikz a H TE,± a e ±ik z Then, the EM fiels in region I vacuum can be epresse as a sum of the incient plane wave an the reflecte ones: E I E inc + H I H inc + n n ρ n E ref,n 3 ρ n H ref,n 4 where ρ n is the reflection coefficient associate with the iffraction orer n InregionIIinsiethe grooves, the EM fiels can be written as a linear combination of the forwar an backwar propagating TE moes: E II C + E TE,+ + C E TE, H II C + H TE,+ + C H TE, By applying the stanar matching bounary conitions at z continuity of E at every point of the unit cell an continuity of H y only at the groove s location an at z h; E must be zero, we can easily etract the reflection coefficients, ρ n : i tank hs S n k /k z ρ n δ n itank h n S n k /k z n 6 where S n is the overlap integral between the nthorer plane wave an the TE moe: S n a a/ e ikn a/ a sink n k n a/ 5 a/ 7 In principle, we coul calculate the surface bans of our system by just analysing the zeros of the enominator of equation 6 [6] The calculation is much simpler if we assume λ Then, all the iffraction orers can be safely neglecte ecept the specular one an ρ takes the form ρ +is tank hk /k z is tank 8 hk /k z For the case k > k k z i k k,wecan calculate the ispersion relation of the surface boun state by calculating the location of the ivergences of ρ : k k S k tank h 9 This is the ispersion relation of the surface EM moes supporte by a D array of grooves in the limit λ an λ a It is interesting to note here that the same ispersion relation coul be obtaine if we replace the array of grooves S98
3 Surfaces with holes in them: new plasmonic metamaterials y z Figure 3 Atwoimensional square array ofsquare holes sie aperforate on a semiinfinite perfect conuctor Figure The ispersion relation ωk ofthe surface boun states supporte by a D array of grooves with geometrical parameters a/ anh/ asobtaine with equation 4 by a single homogeneous but anisotropic layer of thickness h on top of the surface of a perfect conuctor see the schematic rawing in figure b The homogeneous layer woul have the following parameters: ɛ /a ɛ y ɛ z As light propagates in the grooves in the y or z irections with the velocity of light, ɛ µ y ɛ µ z an, hence, µ y µ z ɛ µ After some straightforwar algebra the specular reflection coefficient, R, for a ppolarize plane wave impinging at the surface of a homogeneous layer of thickness h with ɛ an µ given by equations an can be written as R ɛ k z k + k + ɛ k z e ikh 3 ɛ k z + k k ɛ k z eikh Again, by etening this formula to the case k > k an looking at the zeros of the enominator of R we can calculate the ispersion relation of the surface moes: k k a k tank h 4 Note that this epression coincies with equation 9 in the limit k a In figure we plot the ispersion relation equation 4 for the particular case a/ anh/ We have checke that this epression equation 4 gives accurate results for the range of wavelengths analyse in this case λ > 4h bycomparing them with the ispersion relation obtaine by calculating the zeros of the enominator of equation 6 in which the approimation λ is not applie It is worth commenting on the similarities between this ispersion an the one associate with the bans of SPPs supporte by the surfaces of real metals In a SPP ban, at large k, ω approaches ω p /, whereas in this case, ω approaches πc /h that is, the frequency location of a cavity waveguie moe insie the groove in the limit a/, the locations of the ifferent cavity waveguie moes correspon to the conition cos k h 3 D hole array Now we consier the caseofsquare holes of sie a arrange on a lattice perforate on a perfect conuctor semiinfinite structure see figure 3 [7] We assume that the holes are fille with a material whose ielectric constant is ɛ h Asinthe case of the array of grooves, we are intereste in looking at the possible surface states supporte by this system by looking at ivergences of the reflection coefficient of a ppolarize plane wave impinging at the perforate surface As we are intereste in the long wavelength limit λ, now we only take into account the specular reflecte wave The normalize EM fiels associate with the incient an specular reflecte waves are E inc eik e ikz z k /k z H inc eik e ikz z k /k z E ref e eik ikz z k /k z H ref e eik ikz z k /k z 5 6 Insie the holes, as we are intereste in the limit λ a, we assume that the funamental eigenmoe will ominate because it is the least strongly ecaying The EM fiels are zero insie the perfect metal but insie the holes they take the form E TE a eiqzz sin πy a where q z H TE ɛ h k π /a a eiqzz sin πy q z /k a iπ/ak 7 S99
4 FJGarciaVial et al Again, the EM fiels in region I can be epresse as a sum of the incient plane wave an the reflecte one: E I E inc + ρ E ref H I H inc + ρ H ref 8 where ρ is the specular reflection coefficient, an in region II insie the holes, as we are ealing with a semiinfinite structure, we only have to consier the ecaying moe: E II τ E TE H II τ H TE 9 where τ is the transmission coefficient In the matching proceure at z, E must be continuous over the entire unit cell an y ranging from an whereas H y has to be continuous only at the hole This woul yiel ρ k S q zk z k S + q zk z where S is the overlap integral of the incient plane wave an the funamental moe insie the hole: a a S e ik y sin πy a a y a sink a/ π k a/ By analysing the zeros of the enominator of ρ an etening the epression to k > k, we can etract the ispersion relation of the surface states supporte by the D hole array: k k k S k π /a ɛ h k As in the case of D arrays of grooves, we woul like to test whether the semiinfinite perfect conuctor perforate with holes coul be replace by a semiinfinite homogeneous system, characterize by an effective ielectric constant an an effective magnetic permeability Due to the symmetry of the structure, ɛ eff ɛ yeff ɛ eff an µ eff µ yeff µ eff Asthe ispersion of the waveguie moe insie the hole is unaffecte by parallel momentum, ɛ zeff µ zeff In a homogeneous structure, the reflection coefficient for a normally incient plane wave can be epresse as a function of the impeance of the meium, Z µ ɛ : R Z Z + 3 Then, the effective impeance of a D hole array perforate on a perfect conuctor can be easily calculate by analysing equation in the particular case of a normally incient plane wave k, k z k : µeff Z eff S k 4 ɛ eff q z where S S k a/π The other equation linking ɛ eff an µ eff can be obtaine from π q z k ɛeff µ eff i a ɛ hk 5 Figure 4 The ispersion relation ωk oftheppolarize surface boun states supporte by a D array of holes with a/ 6an fille with a ielectric material with ine of refraction n h 3, as obtaine with equation 9 Combining equations 4 an 5 we can write own the effective magnetic permeability an effective ielectric permittivity of our system: µ eff µ yeff S 6 ɛ eff ɛ yeff ɛ h π S a ɛ h k ɛ h π c 7 S a ɛ h ω which isthe canonical plasmon form with a plasma frequency, ω pl πc / ɛ h athisfrequency is just the cutoff frequency of a square waveguie of sie a fille with a material characterize by a ielectric constant ɛ h The net step is to calculate the ispersion relation of the surface moes supporte by this effective meium an compare it with equation For an interface between vacuum an a semiinfinite structure characterize by ɛ eff,thesurface moes have to fulfil the equation k z + q z ɛ eff 8 where k z k i k is the inverse of the ecaying length of the surface moe insie the vacuum, e k z z,anq z is the analogue magnitue in the effective meium By using ɛ eff from equation 7 we obtain k k k S k π /a ɛ h k 9 which coincies with equation in the long wavelength limit when the effective meium approimation makes sense, k a In figure 4 we plot the ispersion relation of these surface moes for the particular case a/ 6 anɛ h 9 3 D arrays of holes of finite epth imples It is quite interesting to analyse also the case of a D square array ofsquare holes sie a offiniteepth, h The proceure for calculating the ispersion relation of the surface S
5 Surfaces with holes in them: new plasmonic metamaterials moes supporte by this type of structure is quite similar to the one presente for the previous case The only ifference is that in equation 9 we have to consier not only the ecaying moe e qz z but also the growing one, e + qz z : E II C + E TE,+ + C E TE, H II C + H TE,+ + C H TE, 3 Apart from the continuity equations at z, we have to a the conition E atthebottom of the hole, z h By oing straightforwar algebra, we en up with a ispersion relation of the surface moes: k k k S k π /a ɛ h k e qz h 3 +e qz h Note that in the limit h, k k light line an for h we recover equation 9, as we shoul 4 Conclusions We have emonstrate that a semiinfinite perfect conuctor perforate with a oneimensional array of grooves or a twoimensional array of holes can be optically escribe in the long wavelength limit as an effective meium characterize by a ielectric function of plasmon form in which the plasma frequency only epens on the geometry of the inentation groove or hole The surface moes supporte by this system have close resemblances with the surface plasmon polaritons of a real metal In these new plasmonic metamaterials, their electromagnetic response coul be engineere by tuning the geometrical parameters efining the corrugate surface Then, these tailore surface plasmons coul be moifie at will at almost any frequency because metals are nearly perfect conuctors from zero frequency up to the threshol of the THz regime Surface electromagnetic moes ecite at a metal surface can also be analyse as propagating waves in two imensions [8, 9] Our results coul be use as an alternative way to control the flow of light in the surface of a metal by just playing with the geometry size an separation of the inentations ispose at the surface Acknowlegments Financial support by the Spanish MCyT uner grant BES an contracts MAT534 an MAT 39 an by EC project FP6NMP4CT Surface Plasmon Photonics is gratefully acknowlege References [] Ebbesen T W, Lezec H J, Ghaemi H F, Thio T an Wolff P A 998 Nature [] MartínMoreno L, GarciaVial F J, Lezec H J, Pellerin K M, Thio T, Penry J B an Ebbesen T W Phys RevLett 86 4 [3] Ritchie R H 957 Phys Rev [4] A very recent review of works relate to surface plasmons can be foun in: Barnes W L, Dereu A an Ebbesen T W 3 Nature [5] Penry J B, Holen A J, Robbins D J an Stewart W J 998 J Phys: Conens Matter 4785 Penry J B, Holen A J, Robbins D J an Stewart W J 999 IEEE Trans Microw Theory Tech [6] GarciaVial F J an MartínMoreno L Phys Rev B [7] A previous theoretical analysis of this case although using a slightly ifferent approach can be foun in: Penry J B, MartínMoreno L an GarciaVial F J 4 Science [8] Bozhevolnyi S I, Erlan J, Leosson K, Skovgaar P M an Hvam J M Phys RevLett [9] Ditlbacher H, Krenn J R, Schier G, Leitner A an Aussenegg F R Appl Phys Lett S
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