Dispersion and attenuation of surface plasmon polariton at metal–dielectric interface

In the present work the behavior of a monochromatic electromagnetic wave propagating along the interface between dielectric and metallic media was rigorously studied. The exact analytical expressions of the electric and magnetic fields were derived. On the basis of the experimental data of the frequency dependent optical parameters of the media the dispersion and the attenuation of the surface plasmon polariton were determined by means of the numerical calculations


Introduction
The fundamental and applied studies on the effects of the plasmon resonance in the electron gas of metal as well as of the large variety of related physical phenomena have led to the emergence of a new scientific area in photonics: plasmonics. The plasmon resonance propagating along the interface between a metallic and dielectric media is called the surface plasmon polariton (SPP). The highly successful theoretical and experimental researches on SPP have led to the rapid development of photonic technologies resulting in the creation of surface plasmon subwavelength optics [1] with the fabrication and exploitation of miniaturized photonic circuits and new types of photonic devices for data storage, optical elements for light generation, focusing, refraction and total internal reflection [2], surface plasmon waveguides [3][4][5][6] etc.
The theory of SPP in the interface between a metal and a dielectric was briefly presented in [7]. In particular, the spatial and temporal dependence of the monochromatic oscillating electromagnetic field at this interface was discussed in the simple case of the electron gas in the metal described by the Drude model. The dispersion curves in the case of the silver/ air and silver/silica interface with the values of the frequency dependent complex dielectric constant determined from the experimental data on the index of refraction n and absorption coefficient k of Johnson and Christy [8] were also plotted in [7] without presenting the calculation method. However, for the theoretical study of plasmonic phenomena and processes related to the SPP it is necessary to use the explicit analytical formulae expressing the dispersion of the electromagnetic field at the interface. The derivation of the expressions analytically determining the dispersion of the electromagnetic field of the SPP is the purpose of the present work. On the basis of the analytical formulae determining the electric and magnetic fields, the propagation of the electromagnetic waves along the metal-dielectric interface and their attenuation will be investigated in detail.

Formulation of the problem and basic equation
The subject of the present work is the study of the propagation of a monochromatic wave at the plane interface between two non-magnetic isotropic homogeneous media, one being a dielectric medium with the (frequency independent) positive dielectric constant ε > 0, another being a medium with For a monochromatic electromagnetic wave the electric field E(r,t), magnetic field H(r,t) and electric induction vector D(r,t) can be presented in the form For the definiteness we chose the axis Oz to be perpendicular to the interface between two media such that the upper half-space is the dielectric medium and the lower half-space is the metallic one (figure 1). Then between electric field E(r,t) and electric induction vector D(r,t) there exists the following relation The vector fields E(r), H(r) and D(r) must satisfy the system of Maxwell equations where c is the light velocity in vacuum. From the system of equations (6)-(9) it follows the d'Alembert equation for the vector fields E(r) and H(r), for example In this work we consider the electromagnetic fields of a special form: the electric and the magnetic fields E(x, y, z) and H(x, y, z) are independent of y and describe the electromagnetic wave propagating along axis Ox with the wave vector k: ikx ikx and the system of Maxwell equations becomes x z Using the Ostrogradski-Gauss and Stoke theorems, from the Maxwell equations (13a)-(14c) we derive following boundary condition at the interface: x y x y , , x y x y , , They mean that the tangential components of the electric and magnetic fields as well as the normal components of the electric induction vector must be continuous across the interface. Maxwell equations in the form similar to equations (13a)-(14c) and the boundary conditions (15)-(17) have been used previously in [7]. We must use these formulae in our subsequent calculations for deriving exact expressions of the solution of the Maxwell equations and therefore here we rewrite them for convenience in the presentation of our reasoning.

Exact expressions of the transverse solution
The solution of Maxwell equations (13a)-(14c) with the boundary conditions (15)-(17) with the y-independent electric and magnetic fields E(r) and H(r) of the form (11) can be considered as a vector field in the coordinate plane xOz. It is called the transverse magnetic solution if the magnetic field ( ) z e H 0,0, ikx is perpendicular to the plane xOz and therefore has only transverse component parallel to the axis Oy: x z y Below we shall see that there exists only transverse magnetic solution satisfying condition (18).
Consider now this transverse magnetic solution. From condition (18) and equation (14b) it follows immediately that We want to find the non-vanishing component ( ) x pz p z Similarly, from equation (14c) it follows that According to the d'Alembert equation (10) for both electric and magnetic fields E(r) and H(r) three parameters p, ′ p and k must satisfy following two equations: From these equations we obtain the formula which was derived in [7]. We set Since ε is real but ε ω ′( ) is complex ξ(ω) must be a complex function of ω: It can be shown that Since the electromagnetic wave propagating along the Ox axis can attenuate but cannot augment in the propagation process, the real part ξ 1 (ω) and imaginary part ξ 2 (ω) must satisfy condition 1 2 Formulaes (30a), (30b) and condition (33) determine the phase velocity of the electromagnetic wave propagating along the Ox axis and the special size of the wave packet, called also the propagation length [1].
The behavior of the electric and magnetic fields in the direction perpendicular to the interface, i.e. their z-dependence, is determined by two complex parameters p (in the half-space z > 0) and ′ p (in the half-space z < 0). Introduce notations Solving equations (25) and (26), we obtain following result:   Because the electromagnetic field must vanish at → ± ∞ z , real part p 1 and ′ p 1 of p and ′ p must satisfy conditions 1 1 Similarly, the surface wave propagating along the Ox axis must attenuate, real part k 1 and imaginary part k 2 of k must satisfy conditions > k k 0.
(43) 1 2 Moreover, from equations (25) and (26) it follows that    of the penetration of SPP from the interface into the metal (gold) on the frequency as well as the dielectric constant ε of the dielectric is more complicated, as can be seen in figure 4(a). For the comparison let us consider the special case, when the dielectric function ε ω ′( ) of gold is presented by the Drude formula with their values determined from the experimental data in [8]. Unfortunately, the fit for ε ω ( ) 2 can be done only in the frequency range ω < 2.5 × 10 15 Hz, as this can be seen in figures 6(a) and (b). The obtained values of the parameter are ε ∞ = 9.84, ω p = 13.8 × 10 15 Hz, γ = 1.017 × 10 14 Hz.
Using Drude formula for the dielectric function ε ω ′( ) we have calculated the physical characteristic parameters of SPP.       index and the absorption coefficient of the metal as well as the given value of the dielectric constant of the dielectric, we have calculated numerically the frequency dependent values of the physical characteristic parameters of SPP. The obtained results show that there is the significant deviation of the dispersion of the real part k 1 of the wave vector of SPP from that of the wave vector of the light in the dielectric medium if the dielectric constant ε of this medium is much larger than 1. This is the case of the titania/gold interface.
For comparison we have calculated the values of the physical characteristic parameters of SPP using Drude formula of the complex dielectric function ε ω ′( ) of metal and always obtain the deviation from the dispersion lines of the light in dielectric in the frequency region where there exists the discrepancy between the values of ε 2 (ω) determined by the Drude formula and those derived from the experimental data in [8].
The fast and large fluctuating variations of the values of dynamical characteristic parameters, in particular of the real part k 1 of the wave vector of the SPP propagation along the interface, can take place according to two mechanisms: -very small values of the imaginary part ε 2 (ω) of the complex dielectric function ε ω ′( ) of the metal; -cancellation of the positive value of the dielectric constant ε of the dielectric and the negative values of the real part ε 1 (ω) of the complex dielectric function ε ω ′( ) of the metal.
Since in the Drude free electron theory the imaginary part ε 2 (ω) is nearly vanishing at the frequency region w > 2.5 × 10 15 Hz, the variation of the parameters (in this region) in the case of the Drude model is more significant in comparison with the case when large values of ε 2 (ω) are derived from the experimental data of [8]. Besides, because the large positive value of the dielectric constant ε of titania is comparable with the absolute values of the negative real part ε 1 (ω) of the complex dielectric function ε ω ′( ) and nearly cancels ε 1 (ω), the variation of many parameters of SPP at the titania/gold interface is larger than those of other interfaces. Similarly, the variation of many parameters of SPP at the air/ gold interface is the smallest one in comparison with other interfaces because the dielectric constant ε of air has the smallest value.
The new results presented above show that the exact expression of the dynamical characteristic parameter of SPP are very useful.