## Abstract

We present an analytical study of the dynamic interplay among surface plasmon polarization charges, electromagnetic fields, and energy flow in the metal/dielectric interface and metal nanoslit structure. Particular attention is given to the regime where the energy flow in the metal side is significant compared to that in the dielectric side. The study reveals that a vortex-like circulation of energy is an intrinsic feature of surface plasmon propagation supported by a metal/dielectric interface, and, in general, a vortex can form when the permittivity and permeability values of the materials involved satisfy the following condition: {(*ε _{m}*/

*ε*) <-1 and (

_{d}*µ*/

_{m}*µ*) >-1} or {(

_{d}*ε*/

_{m}*ε*)>-1 and (

_{d}*µ*/

_{m}*µ*)<-1}.

_{d}©2006 Optical Society of America

## 1. Introduction

A surface plasmon (SP), a collective oscillation of electrons at the interface between a conductor and an insulator induced by optical excitation, is a well-known property of an electron gas [1]. The SP propagation on a metal/dielectric interface has been extensively studied involving various different geometries, such as planar metal surface, thin metal films and metal nanoslits. Much of the earlier work has been done on the study of dispersion relationships, by employing a relatively simple model of metal’s dielectric function, i.e., a lossless free electron gas model [1–10]. SP dispersion and dissipation on a planar interface is basically a macroscopic property of wave propagation, and the underlying mechanisms can be understood at a more fundamental level involving dynamic interactions between plasmonic fields and polarization charges across the metal/dielectric interface. The interactions between the fields and charges are mediated and governed by the dielectric functions of the materials at both sides of the interface. Metals show complex dielectric functions with dispersion characteristic distinctly different from those of dielectrics in the optical frequency range, and as such the plasmon dynamics in a given structure can be a strong function of operating wavelength [11–13]. The plasmon characteristics can also be changed by modifying the physical structure, such as by bringing two metal surfaces close for coupled interactions as in the case of a thin metal film or a slit with narrow gap. In this paper we elucidate the dynamic interplay among the surface plasmon polarization charges, electromagnetic fields, and energy flow in a metal/dielectric interface and a metal nanoslit structure. We pay a particular attention to the regime where the energy flow in the metal side is significant compared to that in the dielectric side. The study reveals that a vortex-like circulation of energy flow is an intrinsic feature of surface plasmon propagation and can be observed when the metal’s permittivity and permeability satisfy a certain requirement. Closer to the surface plasmon resonance point, the power circulation across the metal/dielectric interface becomes more significant, and in the case of silver, this backward power flow in the metal side can reach over 60 % of the flow in the dielectric region of a metal/dielectric/metal structure with a nanometer gap.

## 2. Plasmonic field distributions in a single infinite slit

The analysis of plasmon propagation in a relatively simple geometry can reveal a great deal about plasmon dynamics in general. A single infinite slit is one such structure that is relatively easy to analyze, while remaining practically relevant.

At a single, semi-infinite metallic interface, a surface plasmon–a polarization charge wave associated with an electron density fluctuation–will propagate along the surface with a velocity less than the speed of light in the dielectric side. The electromagnetic energy associated with the plasmon is bound to the interface, so an exponential drop off in the fields is to be expected in the direction perpendicular to the interface. The decay constants of the fields on each side of the interface and the propagation constant along the surface are expressed as follows [1].

In the regime where *ε _{D}* is positive real and

*ε*is largely negative real, it is obvious from the above equations that the decay constant in the metal side is always greater than that in the dielectric side. This indicates that the plasmonic fields extend farther into the dielectric side. The single-infinite slit structure is realized by bringing two such interfaces close together – resulting in coupling between the polarization charges at the two interfaces. The two sets of field distributions can add up or subtract each other, producing symmetric or anti-symmetric profiles, respectively (with the anti-symmetric profile having a nodal point in the middle of the gap region).

_{M}When the spacing between the two interfaces becomes smaller than the penetration depth of the plasmon fields in the dielectric side of a single isolated interface, the extent of the plasmon is effectively controlled by the gap size. In the case of the symmetric mode of SP, this has the effect of decreasing the penetration depth of the plasmon in the dielectric side (increasing the effective *γ _{D}*), and as a result, from Eq. (2),

*k*should increase. This suggests that squeezing the plasmons together in the transverse direction can result in an increase of the effective index of refraction that they feel.

_{sp}It is well-established that the TE field does not propagate in this sort of structure, so we will analyze only the TM case. The magnetic field should exhibit (1) propagation along the interface (define this as the *y*-direction), (2) exponential decay normal to the interface (define this as the *x*-direction) and (3) if the slit structure is sufficiently long, the fields should be roughly constant along the *z*-direction (making the problem effectively two dimensional). Of course, the magnetic field can be rigorously expressed as a product of complex exponentials, but the final solution will end up having the above three features under the conditions of interest.

After applying Maxwell’s equations and the relevant boundary conditions (continuity of the tangential electric and magnetic fields) to the structure shown in Fig. 1, the results are as follows in region 1 (*x*<0), region 2 (0<*x*<*a*), and region 3 (*x*>*a*).

Here the (+) sign indicates the symmetric mode, and the (-) sign the anti-symmetric mode. In describing the symmetry we refer to the functional form of the magnetic field distribution along the transverse direction. The symmetric mode defined this way corresponds to the case where polarization charges of opposite signs align directly across the gap, as described in the sections below. From Eq. (5), it is obvious that the anti-symmetric mode has a nodal point in the middle of the gap, whereas the symmetric mode maintains high intensity in the gap region. This suggests that the symmetric mode of a metal slit has an intrinsic tendency to confine the SP energy to the dielectric side, whereas the anti-symmetric mode has the opposite characteristic, i.e., confining the energy to the metal side. The *x* and *y* components of the electric field have a similar form and follow directly from Maxwell’s equations. In this case, Eqs. (2) and (3) are still valid, but the propagation constant changes. The eigenvalue equation for the propagation constant follows directly from the boundary conditions.

Here, the (+) and (-) signs correspond to the symmetric and anti-symmetric modes, respectively. In order to determine the propagation constant, Eq. (7) must be solved numerically in terms of *k _{sp}*. In a real, finite structure, one would have to be concerned with coupling light into the slits. Considering the symmetry of field distributions, a TM-polarized light normally incident to a slit is expected to couple more efficiently to the symmetric mode rather than to the anti-symmetric mode. Details of the mode excitation depend on the coupling configuration used, and once excited the mode will propagate along the slit with propagation constant determined by the dispersion relationship, Eq. (7). The anti-symmetric mode in a metal nanoslit is expected to show higher loss in propagation, and this can be understood in view of the mode profile showing the SP energy tends to propagate in the metal side. In this work, we will focus on SP propagation via the symmetric mode, and the results with the anti-symmetric mode will be presented in another paper.

## 3. Polarization charge distribution and power flow

While there is no free charge in the system, the surface plasmon wave is characterized by a propagating polarization charge disturbance at the interface between the metal and dielectric. From the Gauss’ law for a medium without free charge, ∇·**D**=0, the polarization can be expressed as follows [14, 15].

This polarization charge causes the discontinuity in the normal component of the electric field at the interface. From the solution of the fields in the slit in the previous section, the form of this polarization charge density at *x*=0, *σ- _{P}(y)* can be calculated as follows for the case of the symmetric mode.

The polarization charge density on the interface at *x=a* can be derived similarly.

We can see that along the propagation direction, the polarization charge density will fluctuate with the same period as the fields. The amplitude of the charge density wave has four major components which are multiplied together: a normalization constant in front to give the proper units $(\frac{{H}_{0}}{c})$, an expression related to the dielectric contrast between the two media $(\frac{1}{{\epsilon}_{M}}-\frac{1}{{\epsilon}_{D}})$, the effective index $({N}_{\mathit{eff}}\equiv \frac{{k}_{\mathit{sp}}}{{k}_{0}})$, and an additional coupling factor [1+ exp(-*γ _{D}a*)]. The dielectric contrast is entirely dependent upon the material properties of the system, while the effective index and coupling factor are dependent upon both the material and the geometry (slit width).

The evolution of the surface charge distribution is basically connected to the energy flow across the interface. One way of looking at this is by deriving the Poynting vector, which should indicate the rate and direction of energy flow.

The time-averaged Poynting vector is written in a complex notation.

Like the fields, the Poynting vector is described in three regions – the dielectric region inside the slit and the metal regions outside the slit.

$$\times \mathrm{exp}\left[2\mathrm{Re}\left({\gamma}_{M}\right)x-2\mathrm{Im}\left({k}_{\mathit{sp}}\right)y\right]$$

$$\times \left\{\mathrm{cosh}\left[\mathrm{Re}\left({\gamma}_{D}\right)\left(a-2x\right)\right]+\mathrm{cos}\left[\mathrm{Im}\left({\gamma}_{D}\right)\left(a-2x\right)\right]\right\}$$

$$\times \mathrm{exp}\left[2\mathrm{Re}\left({\gamma}_{M}\right)\left(a-x\right)-2\mathrm{Im}\left({k}_{\mathit{sp}}\right)y\right]$$

$$\times \mathrm{exp}\left[2\mathrm{Re}\left({\gamma}_{M}\right)x-2\mathrm{Im}\left({k}_{\mathit{sp}}\right)y\right]$$

$$\times \left\{\mathrm{Re}\left(\frac{{\gamma}_{D}}{{\epsilon}_{D}{k}_{0}}\right)s\mathrm{in}\left[\mathrm{Im}\left({\gamma}_{D}\right)\left(2x-a\right)\right]-\mathrm{Im}\left(\frac{{\gamma}_{D}}{{\epsilon}_{D}{k}_{0}}\right)s\mathrm{inh}\left[\mathrm{Re}\left({\gamma}_{D}\right)\left(a-2x\right)\right]\right\}$$

$$\times \mathrm{exp}\left[2\mathrm{Re}\left({\gamma}_{M}\right)\left(a-x\right)-2\mathrm{Im}\left({k}_{\mathit{sp}}\right)y\right]$$

Generally [see the discussion following Eqs. (28)–(30)], if a lossless dielectric function is assumed for metal, i.e., Im(*ε _{M}*)=0, then Im(

*k*)=Im(

_{sp}*γ*)=Im(

_{D}*γ*)=0 from Eqs. (1)–(3). It is then obvious from Eqs. (17)–(19) that the transverse component (

_{M}*x*-component) of the time-averaged Poynting vectors would also be zero everywhere for such a lossless system, and there would be no power transfer across the interface when averaged over time. It is also confirmed from Eqs. (14)–(16) that there would be no energy loss along the propagation direction (

*y*-direction). Metals, however, usually show a complex dielectric constant, and all three propagation/decay constants (

*k*,

_{sp}*γ*, and

_{D}*γ*) have an imaginary part. The Poynting vectors would then have a non-zero transverse component [Eqs. (17)–(19)], aligned along the negative

_{M}*x*-direction in both the metal and dielectric sides. As it propagates it would lose energy [Eqs. (14)–(16)]. It is important to note that the energy loss occurs only in the metal side, and not in the dielectric side. This can be proven by calculating ∇·

*S*in each side, i.e., a calculation shows that ∇·

*S*=0 in the dielectric side and ∇·

*S*>0 in the metal side. The non-zero transverse component of the Poynting vector in the dielectric side then accounts for the energy flow that goes into the metal side, supplying for the loss occurring in that region, as discussed below.

For the case of a single interface (limit as *a*→∞), it is interesting to note that the orientation of the time-averaged Poynting vector, tan^{-1}(*S _{x}/S_{y}*), is fixed, independent of position in each region, and can be expressed as follows (the angles are defined in the fashion illustrated on Fig. 8; with respect to -y on the metal side and with respect to +y on the air side).

The above expressions for the Poynting vector orientation can be understood in view of the fact that the imaginary part of *γ _{D}* and

*γ*corresponds to the propagation component of a damped oscillation in each region. The expression given in Eq (20) remains valid even when the effects of coupling (slit width) are included, while Eq (21) does not – in the slit structure, the orientation of the Poynting vector changes significantly in the slit region.

_{M}The total power carried in each region (the metal, dielectric gap, and metal sides) can be calculated by integrating Eqs. (14)–(16) over the *x-z* the plane (for *x*<0, 0<*x*<*a*, and *x*>*a*, respectively, and over an arbitrary distance along the *z*-direction).

$$\times \left\{\frac{\mathrm{sinh}\left[\mathrm{Re}\left({\gamma}_{D}\right)a\right]}{\mathrm{Re}\left({\gamma}_{D}\right)}+\frac{\mathrm{sin}\left[\mathrm{Im}\left({\gamma}_{D}\right)a\right]}{\mathrm{Im}\left({\gamma}_{D}\right)}\right\}$$

Some interesting features associated with the Poynting vector field can be deduced from Eq. (12) qualitatively. The magnetic field (*H _{z}*) and the tangential component of the electric field (

*E*) are continuous at the metal-dielectric interface, so

_{y}*S*should also be continuous. The normal component of the electric field, however, is discontinuous by a factor of

_{x}*ε*. In the regime where Re(

_{D}/ε_{?}*ε*) is negative,

_{M}*E*will be pointed in the direction opposite to that of the field in the dielectric side. This leads to a conclusion that the longitudinal component of the energy flow (

_{x}*S*) will also be pointed in the opposite direction in the metal side, i.e., along the negative y-direction as opposed to the positive y-direction in the dielectric side. The backward energy flow has been predicted in a metal/dielectric system, assuming a lossless dielectric function of metal (i.e., Im(

_{y}*ε*)=0) [3,4,9], and more recently in negative index materials [8,10].

_{M}## 4. Single interface case

A single silver-air interface can be treated as a special case of the derivations above by considering the limit where the slit width a goes to infinity.

In this case, the only degree of freedom of the system is the dielectric constants of two regions. The effect of changing the dielectric constant alone can be simply examined by removing the effect of geometry and changing the incident wavelength. In silver, the dielectric constant is largely negative with a relatively small imaginary part around 650 nm wavelength. The real part gradually decreases in its magnitude and crosses -1 at 337 nm wavelength and zero at about 324 nm [11, 12].

In Fig. 2 (upper panels), ‘snapshots’ of the time-dependent Poynting vector field associated with a propagating surface plasmon wave are shown at a single silver interface (silver fills the region *x*<0 and air fills the region *x*>0), for three different free-space wavelengths: (a) 650 nm (*ε _{Ag}*=-17+

*i*1.2), (b) 400 nm (

*ε*=-3.8+

_{Ag}*i*0.68), and (c) 350 nm (

*ε*=-1.8+

_{Ag}*i*0.60) [12]. The lower panels (d, e, f) of Fig. 2 show the time-averaged Poynting vectors calculated at each corresponding wavelength. Several important features of SP propagation can be extracted from the images. First, an attenuation of the power is clearly observed along the propagation direction (the +

*y*direction) at each wavelength.

As the free-space wavelength approaches the SP resonance point (around 350 nm), attenuation along the propagation direction increases significantly. This is because the SP field (and thus power) distribution shifts towards the metal side from the dielectric as the real part of the metal’s dielectric constant decreases in its magnitude. The imaginary part of the metal’s dielectric constant still remains significant around the SP resonance wavelength, causing the energy flow to be a lossy process. Also, the SP wavelength decreases substantially. Of course, this is partially due to the decrease in the free-space wavelength, but it is also caused by the steady increase in the effective index of refraction seen by the surface plasmon. For example, the effective index calculated as the ratio of the SP wavelength to the free-space wavelength increases from 1.03 at 650 nm to 1.15 at 400 nm and to 1.36 at 350 nm. If the system is assumed to be lossless like in bulk of earlier work reported in this field, as *ε _{M}* approaches (-)

*ε*(i.e., -1 for air), the denominator approaches zero in Eq. (1). The effective index given by

_{D}*k*would then blow up. In a real system, however, there is an upper bound for the effective index values that can be taken by SPs [13]. In the case of air/silver interface, this occurs at 344 nm wavelength with a maximum effective index of 1.4.

_{sp}/k_{o}Another point of interest is that over a given spatial period, the power flow exhibits a vortex-like pattern. This becomes clearer as the magnitude of the dielectric constant on the metal side becomes comparable to that on the air side. It is important to note that the vortices observed here are not symmetric in their shape. The power flow in the dielectric side is significantly larger than that in the metal side, although the asymmetry lessens for shorter wavelengths. Along the positive y-direction, the power flow also significantly attenuates as discussed earlier. These two features are clearly confirmed in the time-averaged Poynting vectors plotted in the lower panels of Fig. 2. The orientation of the time-averaged Poynting vectors is mostly parallel to the propagation direction with low tilt angle (0.5° in the air side and 8.4° in the metal side) at 650 nm. The tilt angle increases to 3.5°/13° (air/metal) at 400 nm and to 13°/26° at 350 nm. As discussed above, the tilt angle corresponds to the ratio of the two propagation constants of a wave at a given spatial point, one along the longitudinal direction (*y*-direction) and the other along the transverse direction (*x*-direction). The increase of tilt angle observed at shorter wavelengths then indicates more deviation from the ideal situation of SP confinement, i.e., from the evanescent field distribution that shows an exponential decay profile along the transverse direction to the damped oscillation profile with increasing aspect of propagation (oscillation) component. The observed tendency is well understood in view of the wavelength dependence of silver’s dielectric constant in this spectral range, i.e., that the real part of the dielectric constant decreases in its magnitude and the imaginary part becomes significant over the real part [11,12]. It should be noted that ∇·*S* is nonzero (positive) only in the metal side, and therefore power loss occurs only in the metal. The transverse component of the time-averaged power flow is along the negative x-direction at both sides of the interface, and this accounts for the supply of the power being dissipated in the metal region.

## 5. Vorticity of power flow

Mathematically, the vortex circulation is characterized by three different properties: (1) the *y*-component of the Poynting vector oscillates with twice the frequency of the fields and takes only positive values on the air side (and negative values on the metal side), (2) the *x*-component of the Poynting vector also oscillates with twice the field frequency, but oscillates between positive and negative values, and (3) the *y*-component changes sign across the interface while the *x*-component is continuous. The first property results from the *H _{z}* and

*E*fields oscillating in phase (if

_{x}*k*is real), the second results from

_{sp}*H*and

_{z}*E*oscillating about 90° out of phase (satisfied if

_{y}*γ*and

_{D}*γ*are real). In other words, the vorticity is a direct result of the fact that electromagnetic wave is bound to the surface [3, 4]. But, in general, in what kind of systems should this type of power flow be observed ?

_{M}Consider, in general, an infinitely long planar interface with a dielectric on one side (*ε*
_{1}>0 and *µ*
_{1}>0 for *x*>0) and some other material, which may have positive or negative permittivity and permeability on the other side (*ε*
_{2} and *µ*
_{2} for *x*<0). Tsakmakidis et al. examined a similar, more general system in [10], but this special case is important because our interest lies mostly with metal-dielectric systems. In order to have the type of bound, propagating surface wave described previously, the fields must be oscillatory along the interface (*y*-direction, with effective index of refraction *N _{eff}*) and evanescent normal to the interface (+

*x*and -

*x*directions, with decay constants

*γ*

_{1}and

*γ*

_{2}). The TE and TM modes must each satisfy three relationships.

The quantities *γ*
_{1} and *γ*
_{2} are defined so that values greater than zero indicate that the fields decay exponentially along the +*x* and -*x* directions, respectively, while values less than zero indicate that the fields grow exponentially. This means that one requirement placed on the material parameters by Eq. (26) is as follows:

For the TE case, Eqs. (24), (25) and (26) can be solved simultaneously for *N _{eff}*,

*γ*

_{1}and

*γ*

_{2}.

Eqs. (28), (29) and (30) must all be greater than zero for bound TE surface wave propagation to occur. It is tempting to argue that Eq. (28)’s being positive is sufficient for the existence of a propagating surface wave. In actuality, lateral boundedness is an important distinction – for example, the vortex-like evolution of the time dependent Poynting vector is a direct result of the field dying off normal to the surface in both directions. If the wave propagates on both sides, the system is analogous to simple refraction. If it propagates along one direction and decays along another, the propagation along the surface would be analogous to the Goos-Hänchen effect [14]. There are two different ways for Eqs. (28)–(30) to be greater than zero.

or

The TM conditions can be derived by simply switching *ε*
_{2} and *µ*
_{2}, and *ε*
_{1} and *µ*
_{1}.

or

The regions where surface plasmon propagation is expected are shown on Fig. 3. Keep in mind that loss is ignored in this formulation (i.e. all quantities were assumed to be real).

The 3^{rd} quadrant of Fig. 3 is representative of left-handed materials (where both *ε* and *µ* are negative). It can be seen that certain left-handed systems support the same kind of waves that appear in metallic systems. This illustrates that the Poynting vector circulation observed in left-handed materials [8, 10] is fundamentally the same as SP propagation in metal systems.

A further understanding of the vortex can be gained by comparing the Poynting vector field to the electromagnetic fields and surface polarization charge distributions. Figure 4 shows the plots calculated for a single semi-infinite interface at 400-nm free-space wavelength, using the analytical expressions described above. Note that the Poynting vector is tangential near the extrema of the polarization charge wave and normal to the surface around the point where the polarization charge changes sign. Around the point where the Poynting vector orients parallel to the interface, the electric and magnetic fields show maxima. It is interesting to note that the vortex of Poynting vector is skewed towards the metal side in terms of the magnitude and orientation of the vector field. In other words, the energy flow entering into the metal side at one end of the vortex’s major axis is greater than that of the flow coming out of the metal at the other end. Overall then there will be a net power flow from the air side to the metal. This accounts for the supply of the power being dissipated in the metal side. Also it is important to note that the electric and magnetic fields are slightly asynchronized to each other. This phase difference can explain the observed skewedness of Poynting vector fields. The more out of phase the two fields, the more skewed the vortex is. As we understand the relationship between the vortex and the electromagnetic fields, a next question is what is the core of a vortex ? Or what drives the vortex formation ? To answer this question, we took the curl (which characterizes the vorticity of a vector field) of the Poynting vector on each side. In the relationship, ∇×(**E**×**H**)=**E**∇·**H-H**∇·**E**+(**H**∇·)**E**-(**E**∇·)**H**, the third term on the right-hand side only can be nonzero on the interface and all other terms are zero everywhere of the given metal/dielectric system. On the interface, ∇·**E**=(-1/*ε*
_{0})∇·**P**=*ρ _{s}*/

*ε*, and the curl of the Poynting vector can be expressed as follows [14].

_{0}Overall the curl of Poynting vector fields is zero in both the metal and dielectric sides and is nonzero only on the interface as expressed above. It is important to note that the polarization charge density *ρ _{s}* and the magnetic field

*H*take the same polarity, therefore ∇×

_{z}**S**always takes the same direction (along the positive

*z*-direction). This explains why each vortex shows the same orientation. The magnitude of ∇×

**S**takes a maximum value when the product of the polarization charge density and the magnetic field has peak values. Since the polarization charge and magnetic fields are mostly in phase, the curl of the Poynting vector will be with maximum magnitude and orients tangential to the interface around the peak of the two fields.

In the regime where the metal-side dielectric constant has much larger magnitude than that of the dielectric side, the tangential component of Poynting vector on the metal side is proportionally small and the overall power flow is dominated by the one in the dielectric side. As the dielectric magnitude contrast becomes less significant (as *ε _{M}/ε_{D}* approaches -1), the backward power flow becomes more significant. Intuitively, this should result in a slower propagation of energy, because most of the energy flow forwarded in the dielectric side will circulate back through the metal side [16].

From the previous discussion of the effective index and its tendency to blow up as *ε _{M}/ε_{D}* approaches -1, it is tempting to say that the forward and reverse power flow could be perfectly balanced at this point. Even though we assume a lossless dielectric function, i.e., Im(

*ε*)=0, this cannot be true. This is because of the intrinsic asymmetry of spatial extents of the fields on each side. Examination of Eqs. (2) and (3) reveals that the decay length of the fields on the air side is still larger than that on the metal side, meaning that the overall power flow is still positive.

_{M}## 6. Isolated metal nanoslit: effects of slit width

While many of the general features of plasmon propagation in a slit can be observed at a single interface, it is important to understand what happens when two metal-dielectric interfaces couple with each other. Figure 4 illustrated the interaction between the charge and power flow at a single interface calculated at 400 nm wavelength – the single interface case is equivalent to the limit as slit width goes to infinity. Charge density plots are given in units of *H _{0}/c*. Figures 5 and 6 introduce a second silver surface 100 nm and 25 nm away from the first one respectively (all at 400 nm wavelength). The result of coupling to the second surface is similar to approaching resonance – the effective wavelength of propagation in the slit decreases and the propagation length decreases. For an isolated silver interface the air-side part of the plasmon extends about 200 nm, so the coupling between the two surfaces should start to become significant at slit widths on the order of 100 nm. As the slit width decreases the amplitude of the charge wave also increases – as predicted by Eq. (11). The charge polarization waves on the two surfaces are 180 degrees out of phase, meaning that inside the slit they produce electric fields in the same direction, which reinforces the idea that the symmetric mode is characterized by constructive interference between the two plasmons.

Returning to the discussion of time-averaged power flow culminating in Eqs. (21) and (22), the ratio of the power flowing in the metal to the power flowing in the slit region can be determined analytically. This result is shown in Fig. 7 for the case of silver/air/silver system. It can be seen that over a large range of slit widths, the power flow is uniformly positive (the forward propagating air-side power is larger than the backward propagating metal-side power), though they become more balanced as the slit width becomes very small and as the wavelength approaches resonance.

Figure 8 shows the orientation of the time-averaged Poynting vector in the metal and slit regions calculated at 650 nm and 400 nm wavelength. Here the orientation is expressed as the tilt angle with respect to the +y direction on the air side and the -y direction on the metal side. Hence, the x-component of the Poynting vector has the same sign as the angle on the air side and the opposite sign as the angle on the metal side. The lateral position is normalized by slit width, and the Poynting vector angle in the slit region is illustrated in the right panels. First of all, the tilt angle in the metal side is found to be constant, independent of position for a given slit width and wavelength, whereas the angle varies in the slit region. At 650 nm wavelength the tilt angle monotonically decreases from 8 degrees to 6 degrees as slit width is reduced from 200 nm to 25 nm. The angle in the slit region proportionally decreases from 0.46 degree to 0.35 degree at the interface. The decrease of tilt angle with reduced slit width is explained by the fact that squeezing a surface plasmon in the lateral direction has the effects of increasing the effective index (propagation constant) along the longitudinal direction and also of reducing the propagation component along the transverse direction [See Eqs. (2) and (3)]. Since the tilt angle is basically determined by the ratio of two propagation constants (along the lateral and the longitudinal directions) as shown in Eqs. (20) and (21), the change in the lateral confinement (therefore the wave vectors) results in the change of the Poynting vector orientation. As the wavelength approaches resonance the tilt angle becomes significantly larger than that at 650 nm. At 400 nm, the metal-side Poynting vector angle starts at around 13° at 200 nm slit width, drops slightly at 100 nm slit width to 12.9°, then goes back up to about 14° at 25 nm. At the interface, the angle on the dielectric side stays at about 3.5° for all slit widths. It is interesting to note that at 650 nm wavelength the tilt-angle-versus-slit-width shows a monotonic relationship with the largest slit width corresponding to the largest angle, while at 400 nm the dependence is not as strong. In part (c), the 200 nm, 100 nm and 50 nm curves are separated by less than a degree – making it difficult to distinguish between them. This might be due to the fact that the imaginary part of metal’s dielectric function becomes significant near resonance.

## 7. Conclusion

The dynamic interplay among the surface plasmon polarization charges, electromagnetic fields, and energy flow in the metal/dielectric interface and metal nanoslit structure is elucidated. A particular attention is paid to the regime where the energy flow in the metal side is significant compared to that in the dielectric side. The study reveals that a vortex-like circulation of energy flow is an intrinsic feature of surface plasmon propagation and can be observed when the metal’s permittivity and permeability satisfy a certain requirement. Closer to the surface plasmon resonance point, the power circulation across the metal/dielectric interface becomes more significant. In the case of silver, this backward power flow in the metal side can reach over 60 % of the flow in the dielectric region of a metal/dielectric/metal structure with a nanometer gap. The imaginary part of metal’s dielectric function is found to play an important role in surface plasmon dynamics, limiting the propagation distance and effective index, and shaping the energy flow.

## Acknowledgment

This work has been supported by the US National Science Foundation.

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