Characteristics of Surface Plasmon Polaritons with Effective Permittivity

Transcription

1 Journal of Optics Applications December 5, Volume 4, Issue, PP.7-4 Characteristics of Surface Plasmon Polaritons with ffective Permittivit Weiping Mao, eqing uang. School of Mechanical ngineering, Jiangsu Universit, Zhenjiang Jiangsu, China. Universit of Waterloo, Waterloo, NLG, China mail: Abstract ffective permittivit is a propagation constant of surface plasmon polaritons, which help visualise how surface plasmon ecitation travels in space. Compared with the traditional method with frequenc parameters, this method is more intuitive and practical. Kewords: Surface Plasmon Polaritons; Mawell's quations; Simulation INTRODUCTION A surface plasmon polaritons (SPPs) is an electromagnetic ecitation eisting between the surface of metal and dielectric. It is an intrinsicall two-dimensional ecitation whose electromagnetic field decas eponentiall with distance from the surface. In this paper, the principle of surface plasmon ecitation has been introduced. Under single flat interface and metal / dielectric multilaer structure sstem, transverse electric and magnetic wave modes have been derived from the Mawell equations. From the analses, with software Mathematica, propagation of surface plasmons has been simulated, analse effective permittivit varies with of a variet of metal /dielectric and wavelength in a single plane of the interface, and the changes of effective permittivit in the case with the thickness of the multilaer heterostructure and the wavelength changes. Simulation results have been compared with the theoretical value. A method as using the effective dielectric constant of the surface plasmons propagation has been proposed. BASIC CARACTRISTICS OF SPPS To analse the phsical properties of SPPs, a classical model of SPPs is illustrated in FIG.. FIG. CLASSICAL MOD OF SURFAC PLASMON POLARITONS Appl Mawell s equations to the flat interface between dielectric and metal. The electromagnetic equation is derived in the general form firstl. Use the curl equation for the electric field. B = () t - 7 -

2 According to the eternal charge ( ρ et D = µ t ) and the current densities ( J et ) are absent. D D = Jet + () t D = ε + P (4) P describes the electric dipole moment per unit volume inside the material, caused b the alignment of microscopic dipoles with the electric field. P is ero in this situation. Consider that there are no eternal stimuli and epand the formula. As the variation of the dielectric profile can be neglect, ε = ε() r. ε c t = D = (5) () ( µε = ) (6) c Consider the harmonic time dependence ( related to time and position) of the electric field. So elmholt equation, + ε = t k ω ( k = ) (7) c Define the propagation geometr in FIG.. Let ε depends onl on one spatial coordinate, so ε = ε( ), the waves propagate along the -direction and show no spatial variation in the perpendicular of a Cartesian coordinate sstem. FIG. DFIN T PROPAGATION GOMTRY IN CARTSIAN COORDINAT SYSTM Applied to electromagnetic surface, on the plane =, the traveling waves equation is i (,, ) = e ( ) β ( β = k) (8) β is the propagation constant of the propagating waves and relates to the constituent of the wave vector in the direction of propagation. The propagation direction is illustrated in FIG.. Rewrite the elmholt equation FIG. T PROPAGATION COMPONNTS OF SPPS ( ) + ( k ε β ) = t The equation for the magnetic field is similarl. The propagation direction is illustrated in FIG.4. (9) - 8 -

3 FIG. 4 T DYRCTION OF LCTRONMAGNTISM FOR SPPS From the equation (9), we can make the general analsis of guided electromagnetic modes in waveguides, even etend to its properties and applications. For determining the characteristics of spatial field profile and dispersion of propagating waves. The eplicit epressions for the different field components of and can be achieved b the curl equation (). As the propagation along the -direction, and is homogenies in the -direction. Simplifies the sstem of equation = iωµ (a) = = iωµ iωµ (b) (c) = iωε ε (d) = = iωε ε iωε ε (e) (f) There eist two sets of self-consistent presentation in this sstem with different polariation properties of the propagating waves. In transverse electric modes, the components of field,, and i ωε ε are nonero. (a) The wave equation for transverse electric mode become In the set of the transverse magnetic modes,, β (b) ωε ε + ( k ε β ) = and being nonero. i ωµ () (a) β = (b) ωµ - 9 -

4 The wave equation for transverse magneti mode become + ( k ε β ) = Consider with the most simple geometr sustaining SPPs as a single, flat interface between a dielectric, nonabsorbing half space ( > ) with positive real dielectric constant and an adjacent conducting half space ( < ) described via a dielectric function ε ( ω ). For transverse magnetic mode, in both half spaces ields separatel For > (5a) ( ) ia e e (4) k = (5b) ωεε ( ) A e e k (5c) ωεε k For < ( ) = Ae e (5d) i ( ) ia e e k (5e) ωεε β ( ) A e e k (5f) ωεε k is the component of the wave vector perpendicular to the interface in the two media k k ( i =, ).With evanescent deca in the perpendicular -direction, the reciprocal value, ˆ = / k. ẑ is the evanescent deca length of the fields perpendicular to the interface which quantifies the confinement of the wave. i i FIG. 5 T PROPGRATION OF ON-FLAT SPPS According to the continuit of and i ε at the interface. A k k = A (6a) ε (6b) ε The epression for further has to fulfil the wave equation. k k = β k ε (7a) = β k ε (7b) The central result represent the characteristics of SPPs at the interface between the two half spaces. ffective permittivit εε β = k = kn eff ε + ε represent characteristics of SPPs. (8) - -

5 n eff = εε ε + ε As in transverse magnetic mode, the change of dielectric function ε ( ω) at metal interface, so we can use the Drude dielectric function. Set the dielectric material as air ( ε = ). ω p εω ( ) ω + iγω According to the metal dielectric constant and the wavelength data, we can use Mathematics to calculate the effective permittivit. The relationship of effective permittivit and wavelength is been plot in Fig 6. (9) () SPPS IN MULTILAYR SYSTMS FIG. 6 T FFCTIV PRMITTIVITY OF ON-FLAT SPPS Consider the SPPs in multilaer sstems, each single interface of conducting and dielectric thin films can sustain boundar situation. As the separation between the same interfaces is equal or smaller than the deca length ẑ of the interface mode, interactions between SPPs induce coupled modes. First one, a thin metallic laer sandwiched between two infinite thick dielectric claddings, which is an insulator/metal/insulator (IMI) heterostructure. And secondl a thin dielectric core laer (I) sandwiched between two metallic claddings (II, III), which is a metal/insulator/metal (MIM) heterostructure. FIG. 7 T MULTILAYRS SYSTM OF SPPS For the lowest-order bound modes, a general description of transverse magnetic modes has no oscillation in the Z- direction, the propagation direction perpendicular to the interfaces. Using the sstem of governing equations for transverse magnetic modes, we obtain the wave components of the field. k For > = Ae e (a) k = ia k e e (b) ωε ε - -

6 β A ke e k (c) ωεε k For < = Be e (d) k ib k e e (e) ωε ε β B ke e k (f) ωεε In order to simplif the discussion, the component of the wave vector perpendicular to the interfaces is sampled as ki ki. In the sandwiched region ( a< < a), the modes localied at the bottom and top interface couple. Since the requirement of continuit of and, we get the fellow relationship, a linear sstem of four coupled equations. The equation of further fulfill relationship between the three distinct regions. Using the sstem of linear equations to solve the dispersion relation between β and ω, the results in an implicit epression. e 4k k / ε + k / ε k / ε + k / ε k / ε k / ε k / ε k / ε Consider about a special condition that the dielectric response of the substructure (cladding II) and the superstructure (cladding III) are equall. In this special case, the dispersion relation is split into two different equations for odd and even modes. The modes of odd vector parit, as ( ) is odd, ( ) and ( ) are even functions tanh ka The modes of odd vector parit, as ( ) is even, ( ) and ( ) are odd functions tanh ka k ε k ε () (a) k ε k ε (b) The pair of equations can be applied to IMI and MIM heterostructure. We investigate the properties of the coupled SPP modes in IMI geometr - a thin metallic film of thickness a sandwiched between two insulating laers. As shown in Fig8, is the dielectric function of the metal, ε = ε ( ω), ε is the real dielectric constant of the dielectric cladding in substructure and the superstructure. FIG. 8 T SYMMTRICAL PROPATION MOD IN IMI MULTILAYR SYSTM OF SPPS FIG. 9 T ASYMMTRICAL PROPATION MOD IN IMI MULTILAYR SYSTM OF SPPS Newton s method is been used to solve equations. f ( ) n+ = n f '( ) (4) - -

7 We use Mathematica to simulation the relationship between n eff and thickness d. FIG. T FFCTIV CONSTANT OF COPP We can find that when the separation a increase, n eff will decrease in the same time. When the separation is relative small with the deca length, it almost have no effect on the effective permittivit. As the separation being ver large, the effective permittivit is jump seriousl. Then, we fied the separation a as 5 nm and change the dielectric constant of the metal and dielectric ε ( ) ω for different wavelength. and FIG. T FFCTIV CONSTANT OF GLASS AND MTAL The graphics are concave curve. When the wavelength increases, the permittivit of SPPs decreases. The maimum value is close to the dielectric constant. All of the plots describe the situation of odd mode. When the propagation mode is even, the curve upward bending. 4 CONCLUSIONS The parameter of effective permittivit is visualise how surface plasmon ecitation travels in space. With the rapid development of optical techniques, control and manipulation of light using SPPs on the nanometre scale ehibit significant advantages in nanophotonics devices with ver small elements, and SPPs open a promising wa in fields involving environment, energ, biolog, even medicine. Compared with the traditional method with frequenc parameters, effective permittivit provide a more intuitive observation of SPPs. In practice, controlled the sstem with adjust the effective permittivit can reduce the conversion process reach the anticipated result. ACKNOWLDGMNT Appreciate for the support b the International Conference on Photonics and Optoelectronics. RFRNCS [] omola, J. Present and future of surface plasmon resonance biosensors. Analtical and bioanaltical chemistr, [] Pitarke, J. M, Silkin,V. M, Chulkov,. V,& chenique, P. M. Theor of surface plasmons and surface-plasmon polaritons. Reports on progress in phsics, 7 [] STFAN A. MAIR. Plasmonics: Fundamentals and Applications. Centre for Photonics and Photonic Materials Department of Phsics, Universit of Bath, UK, 7 - -

8 [4] Zaats,A. V, Smolaninov, I. I, & Maradudin, A. A. Nano-optics of surface plasmon polaritons. Phsics reports, 5 AUTORS Weiping Mao (964), male, an, Postgraduate degree Associate Professor in School of Mechanical ngineering Jiangsu Universit, Research Interests: hdraulic transmission and control, laser processing technolog, Undergraduate degree (986), postgraduate degree (995) at School of Mechanical ngineering Jiangsu Universit, Zhenjiang, China. eqing uang (99), female, an, Undergraduate degree, Graduate student in Universit of Waterloo, Research Interests: optical detection, biophsics. Undergraduate degree (4) at Facult of Science in Jiangsu Universit, Zhenjiang, China. Undergraduate degree (4) in Wilfrid Laurier Universit, Waterloo, On, Canada